Finite Group Databasebeta

← #34019094

Subgroups — #34019094

Order 18688.

748 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626673C211yesmaximal, characteristic, complemented
2D4×#11924037C211yesmaximal, characteristic, complemented
3#22625564C211yesmaximal, characteristic, complemented
4#22625562C211yesmaximal, characteristic, complemented
5#22625533C211yesmaximal, characteristic, complemented
6#22625532C211yesmaximal, characteristic, complemented
7#22625532C211yesmaximal, characteristic, complemented
8#22625530C211yesmaximal, characteristic, complemented
9#22625511C211yesmaximal, characteristic, complemented
10D8×#10596833C211yesmaximal, characteristic, complemented
11#22625500C211yesmaximal, characteristic, complemented
12#22625276C211yesmaximal, characteristic, complemented
13C2×D4⋊#10596833C211yesmaximal, characteristic, complemented
14#22625132C211yesmaximal, characteristic, complemented
15#22619926C211yesmaximal, characteristic
16C2×#14556678—21no—
17C22×#11924037—21no—
18#17989622C2211yescharacteristic
19#17989622C2211yescharacteristic
20#17989622C2211yescharacteristic, complemented
21#17989621C2211yescharacteristic, complemented
22#17989619—21no—
23#17989618—21no—
24#17989618—21no—
25#17989616—21no—
26#17989615C2211yescharacteristic, complemented
27D4×#10596833C2211yescharacteristic
28D4×#10596833C2211yescharacteristic
29D4×#10596833C2211yescharacteristic, complemented
30#17989590C2211yescharacteristic
31C2×C4⋊#10596833C2211yescharacteristic
32#17989560—21no—
33#17989557—21no—
34C4×#11924037C2211yescharacteristic
35C2×C4×#10596833C2211yescharacteristic
36#17989451C2211yescharacteristic, complemented
37#17989451C2211yescharacteristic, complemented
38#17989450C2211yescharacteristic, complemented
39#17989450C2211yescharacteristic, complemented
40#17989435C2211yescharacteristic, complemented
41#17989435C2211yescharacteristic, complemented
42D4⋊#10596833C2211yescharacteristic
43D4⋊#10596833C2211yescharacteristic
44D4⋊#10596833C2211yescharacteristic
45D4⋊#10596833C2211yescharacteristic
46D4⋊#10596833C2211yescharacteristic, complemented
47D4⋊#10596833C2211yescharacteristic, complemented
48#17989429C2211yescharacteristic, complemented
49#17989429C2211yescharacteristic, complemented
50C8×#10596833C2211yescharacteristic, complemented
51C8×#10596833C2211yescharacteristic, complemented
52#17989132C2211yescharacteristic
53C2×D4×Dic73C2211yescharacteristic
54#17988901C411yescharacteristic
55#17988898C411yescharacteristic
56#17988865C411yescharacteristic
57#17988865C411yescharacteristic
58#17988780C411yescharacteristic
59#17988780C411yescharacteristic
60#17988721C411yescharacteristic
61D8×Dic73C2211yescharacteristic
62D8×Dic73C2211yescharacteristic
63#17988667C2211yescharacteristic
64#17988667C2211yescharacteristic
65#17988637C2211yescharacteristic
66#447×C73C411yescharacteristic
67#14556678—44no—
68C2×#11924038—21no—
69C2×#11924037—21no—
70C2×#11924037—21no—
71C2×#11924037—21no—
72C2×#11924037—21no—
73C2×#11924037—44no—
74C2×#11924037D411yescharacteristic
75C2×#11924037D411yescharacteristic
76C2×#11924036—21no—
77C22×#10596833—21no—
78C22×#10596833—21no—
79C22×#10596833—21no—
80C22×#10596833—21no—
81C22×#10596833—21no—
82#14556658—21no—
83C4⋊#10596833—21no—
84C4⋊#10596833C2311yescharacteristic
85C4⋊#10596833C2311yescharacteristic
86C4⋊#10596833C2311yescharacteristic
87C4⋊#10596833C2311yescharacteristic
88#14556648—21no—
89#14556648—21no—
90C22⋊#10596833—21no—
91C22⋊#10596833—21no—
92C22⋊#10596833—21no—
93C22⋊#10596833—21no—
94C22⋊#10596833—21no—
95C22⋊#10596833—21no—
96#14556646D411yescharacteristic
97#14556646D411yescharacteristic
98C4×#10596833—21no—
99C4×#10596833C2311yescharacteristic
100C4×#10596833C2311yescharacteristic
101C4×#10596833C2311yescharacteristic
102C4×#10596833C2311yescharacteristic
103C2×#11924027—21no—
104C23×Dic73—21no—
105C2×#11924013C2×C411yescharacteristic
106C2×#11924013C2×C411yescharacteristic
107#14556590C2×C411yescharacteristic
108#14556587C2×C411yescharacteristic
109#14556577—21no—
110#14556565C2×C411yescharacteristic
111#14556563—21no—
112D4×Dic73—21no—
113D4×Dic73—21no—
114D4×Dic73C2×C411yescharacteristic
115D4×Dic73C2×C411yescharacteristic
116D4×Dic73C2311yescharacteristic
117D4×Dic73C2311yescharacteristic
118D4×Dic73C2311yescharacteristic
119#14556553C2311yescharacteristic
120#14556551—21no—
121C2×C4⋊Dic73C2×C411yescharacteristic
122C2×C22⋊Dic73—21no—
123C2×C4×Dic73C2311yescharacteristic
124#14556525C2×C411yescharacteristic
125#14556525C2×C411yescharacteristic
126#14556524C2×C411yescharacteristic
127#14556524C2×C411yescharacteristic
128#14556520C2×C411yescharacteristic
129#14556520C2×C411yescharacteristic
130#14556513C2×C411yescharacteristic
131#14556513C2×C411yescharacteristic
132#14556512C2×C411yescharacteristic
133#14556512C2×C411yescharacteristic
134#14556512C2×C411yescharacteristic
135#14556512C2×C411yescharacteristic
136C22⋊#10596831C2×C411yescharacteristic
137C8×Dic73C2311yescharacteristic
138C8×Dic73C2311yescharacteristic
139C22×D4×C73C2×C411yescharacteristic
140C2×D8×C73C2×C411yescharacteristic
141C2×D8×C73C2×C411yescharacteristic
142#121×C73C2×C411yescharacteristic
143#102×C73C2×C411yescharacteristic
144#102×C73C2×C411yescharacteristic
145C22⋊C8×C73C2×C411yescharacteristic
146#11924038—42no—
147#11924037—22no—
148#11924037—42no—
149#11924037—42no—
150#11924037—42no—
151#11924037—42no—
152#11924037D814yescomplemented
153#11924036—42no—
154C2×#10596833—21no—
155C2×#10596833—21no—
156C2×#10596833—21no—
157C2×#10596833—21no—
158C2×#10596833—21no—
159C2×#10596833—21no—
160C2×#10596833—42no—
161C2×#10596833—42no—
162C2×#10596833—42no—
163C2×#10596833—42no—
164C2×#10596833—42no—
165C2×#10596833C2×D411yescharacteristic
166C2×#10596833C2×D411yescharacteristic
167C2×#10596833C2×D411yescharacteristic
168C2×#10596833C2×D411yescharacteristic
169#11924027—42no—
170#11924027—42no—
171C2×Dic146—21no—
172C2×D292C22×C411yescharacteristic
173C2×C22⋊D73—21no—
174C2×C22⋊D73—21no—
175C2×C22⋊D73—21no—
176C22×Dic73—21no—
177C22×Dic73—21no—
178C22×Dic73—21no—
179C22×Dic73—21no—
180C22×Dic73—21no—
181C22×Dic73—21no—
182C22×Dic73—42no—
183C22×Dic73—42no—
184C22×Dic73C2×D411yescharacteristic
185C22×Dic73#4111yescharacteristic
186C2×C4×D73—21no—
187#11924013—22no—
188#11924013—41no—
189C2×#10596831C22×C411yescharacteristic
190C2×#10596831C22×C411yescharacteristic
191C4⋊Dic73—21no—
192C4⋊Dic73C22×C411yescharacteristic
193C4⋊Dic73C22×C411yescharacteristic
194C4⋊Dic73C22×C411yescharacteristic
195#11924007—21no—
196C22⋊Dic73—21no—
197C22⋊Dic73—21no—
198C22⋊Dic73—21no—
199C22⋊Dic73—21no—
200C22⋊Dic73—21no—
201#11924005—21no—
202#11924005—21no—
203C4×Dic73C22×C411yescharacteristic
204C4×Dic73C2×D411yescharacteristic
205C4×Dic73#4111yescharacteristic
206C4×Dic73C2411yescharacteristic
207C24×C73—21no—
208C2×D4×C73—22no—
209C2×D4×C73—21no—
210C2×D4×C73—21no—
211C2×D4×C73—41no—
212C2×D4×C73C22×C411yescharacteristic
213C2×D4×C73C22×C411yescharacteristic
214C2×D4×C73C22×C411yescharacteristic
215C22×C4×C73C22×C411yescharacteristic
216D8×C73—22no—
217D8×C73—41no—
218C2×C8×C73C22×C411yescharacteristic
219C2×C8×C73C22×C411yescharacteristic
220C4⋊C4×C73C22×C411yescharacteristic
221C22⋊C4×C73—21no—
222C23×C73—21no—
223C23×C73—21no—
224C23×C73—21no—
225C23×C73—42no—
226C23×C73—42no—
227C23×C73—81no—
228C23×C73C4×D411yescharacteristic
229C22×D73—21no—
230D4×C73—21no—
231D4×C73—21no—
232D4×C73—21no—
233D4×C73—21no—
234D4×C73—21no—
235D4×C73—42no—
236D4×C73—42no—
237D4×C73#13112yescomplemented
238C2×C4×C73—21no—
239C2×C4×C73—21no—
240C2×C4×C73—21no—
241C2×C4×C73C4×D411yescharacteristic
242C2×C4×C73C23×C411yescharacteristic
243C22⋊D73—42no—
244C22⋊D73—42no—
245C22⋊D73—81no—
246C2×Dic73—21no—
247C2×Dic73—21no—
248C2×Dic73—21no—
249C2×Dic73—41no—
250C2×Dic73—42no—
251C2×Dic73—41no—
252C2×Dic73—41no—
253C2×Dic73—41no—
254C2×Dic73—41no—
255C2×Dic73—41no—
256C2×Dic73—41no—
257C2×Dic73—42no—
258C2×Dic73—42no—
259C2×Dic73C4×D411yescharacteristic
260C2×Dic73C4×D411yescharacteristic
261C2×Dic73C2×D812yes—
262C2×Dic73#13512yes—
263C2×Dic73C22×D411yescharacteristic
264C2×Dic73C2×#4111yescharacteristic
265D292—21no—
266D292—21no—
267C4×D73—41no—
268C4×D73—41no—
269Dic146—42no—
270#10596833—22no—
271#10596833—21no—
272#10596833—21no—
273#10596833—41no—
274#10596833—41no—
275#10596833—41no—
276#10596833—41no—
277#10596833—41no—
278#10596833C2×D812yes—
279#10596833C2×D812yes—
280C8×C73—21no—
281C8×C73—21no—
282#10596831—21no—
283#10596831—21no—
284C22×C73—21no—
285C22×C73—21no—
286C22×C73—22no—
287C22×C73—41no—
288C22×C73—41no—
289C22×C73—41no—
290C22×C73—41no—
291C22×C73—41no—
292C22×C73—41no—
293C22×C73—41no—
294C22×C73—41no—
295C22×C73—84no—
296C22×C73C4×D812yes—
297C22×C73C2×C4×D411yescharacteristic
298C2×D73—41no—
299C2×D73—41no—
300C2×D73—41no—
301C4×C73—21no—
302C4×C73—41no—
303C4×C73C2×#13111yescharacteristic
304C4×C73#56811yescharacteristic
305Dic73—21no—
306Dic73—21no—
307Dic73—41no—
308Dic73—41no—
309Dic73—41no—
310Dic73C4×D812yes—
311Dic73C22×D811yescharacteristic
312Dic73C2×#13511yescharacteristic
313#17994—731nomaximal, Sylow
314C2×C73—22no—
315C2×C73—42no—
316C2×C73—81no—
317C2×C73—81no—
318C2×C73#293311yescharacteristic
319C2×C73C2×C4×D811yescharacteristic
320C2×C73#295611yescharacteristic
321D73—81no—
322#3100—731no—
323#2935—731no—
324D4×M16—731no—
325#1589—731no—
326#1589—731no—
327#1581—731no—
328#1581—731no—
329#1581—731no—
330#1581—731no—
331C8×D8—731no—
332C8×D8—731no—
333#1571—731no—
334#1492—731no—
335C2×D4⋊C8—731no—
336C8×M16—731no—
337C73#1799411yescharacteristic, complemented, Sylow
338C2×#131—1461no—
339C22×M16—1461no—
340#532—731no—
341C2×C4×D4—731no—
342#480—731no—
343#480—731no—
344#474—731no—
345#474—731no—
346#459—731no—
347#459—731no—
348#447—731no—
349#447#6440811yescharacteristic
350C4×D8—731no—
351C4×D8—731no—
352C4×D8—731no—
353C4×D8—731no—
354#436—731no—
355#436—731no—
356#436—731no—
357#436—731no—
358#435—1461no—
359#435—1461no—
360#435—1461no—
361#435—1461no—
362C8×D4—731no—
363C8×D4—731no—
364C8×D4—731no—
365C8×D4—731no—
366#431—731no—
367#423—731no—
368C2×C4⋊C8—731no—
369#408—1461no—
370#407—1461no—
371C4×M16—731no—
372C2×C4×C8—731no—
373#335—731no—
374#335—731no—
375#335—731no—
376#335—731no—
377D4⋊C8—731no—
378D4⋊C8—731no—
379D4⋊C8—731no—
380D4⋊C8—731no—
381D4⋊C8—731no—
382D4⋊C8—731no—
383D4⋊C8—731no—
384D4⋊C8—731no—
385#322—731no—
386#322—731no—
387C82—731no—
388C82—731no—
389C2×#41—1461no—
390C22×D4C2×#6440811yescharacteristic
391C23×C4—1461no—
392C2×D8—731no—
393C2×D8—731no—
394C2×D8C2×#6440811yescharacteristic
395C2×D8C2×#6440811yescharacteristic
396#131—2924no—
397C2×M16—2924no—
398C2×M16—731no—
399C2×M16—731no—
400C2×M16—1461no—
401C2×M16—1461no—
402C2×M16—1461no—
403C2×M16—1461no—
404C2×M16—1461no—
405C22×C8—1461no—
406C22×C8—1461no—
407C22×C8—1461no—
408C22×C8—1461no—
409C22×C8—1461no—
410C22×C8—1461no—
411C4⋊Q8—731no—
412#127—731no—
413#124—1461no—
414#121—731no—
415#121—1461no—
416#121C2×#6440811yescharacteristic
417C4×D4—731no—
418C4×D4—731no—
419C4×D4—731no—
420C4×D4—731no—
421C4×D4—731no—
422C4×D4—731no—
423C4×D4—1461no—
424C4×D4—1461no—
425C4×D4—1461no—
426C2×C4⋊C4—731no—
427C2×C22⋊C4—1461no—
428C2×C42—731no—
429#107—731no—
430#107—731no—
431#107—731no—
432#107—731no—
433C4⋊C8—731no—
434C4⋊C8—731no—
435C4⋊C8—731no—
436C4⋊C8—731no—
437C4⋊C8—731no—
438C4⋊C8—731no—
439C4⋊C8—1461no—
440C4⋊C8—1461no—
441#102—731no—
442#102—731no—
443#102—731no—
444#102—731no—
445#102—731no—
446#102—731no—
447#102C2×#6440811yescharacteristic
448#102C2×#6440811yescharacteristic
449C22⋊C8—731no—
450C22⋊C8—1461no—
451C22⋊C8—1461no—
452C22⋊C8—1461no—
453C22⋊C8—1461no—
454C22⋊C8—1461no—
455C22⋊C8—1461no—
456C22⋊C8—1461no—
457C22⋊C8—1461no—
458C22⋊C8C2×#6440811yescharacteristic
459#97—731no—
460#97—731no—
461C4×C8—731no—
462C4×C8—731no—
463C4×C8—731no—
464C4×C8—731no—
465C4×C8—731no—
466C4×C8—731no—
467C4×C8—1461no—
468C24—21no—
469#41—2922no—
470#41—2922no—
471C2×Q8—1461no—
472C2×D4—22no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—41no—
476C2×D4—731no—
477C2×D4—1461no—
478C2×D4—1461no—
479C2×D4—1461no—
480C2×D4C22×#6440811yescharacteristic
481C2×D4C22×#6440811yescharacteristic
482C2×D4C22×#6440811yescharacteristic
483C22×C4—2922no—
484C22×C4—2922no—
485C22×C4—731no—
486C22×C4—731no—
487C22×C4—1461no—
488C22×C4—1461no—
489C22×C4—1461no—
490C22×C4—1461no—
491C22×C4—1461no—
492C22×C4—1461no—
493C22×C4—1461no—
494C22×C4C22×#6440811yescharacteristic
495D8—22no—
496D8—41no—
497D8—2921no—
498D8—1462no—
499M16—2922no—
500M16—2922no—
501M16—2922no—
502M16—2922no—
503M16—2922no—
504M16—734no—
505M16—1462no—
506C2×C8—2922no—
507C2×C8—2922no—
508C2×C8—2922no—
509C2×C8—2922no—
510C2×C8—2922no—
511C2×C8—2922no—
512C2×C8—731no—
513C2×C8—731no—
514C2×C8—731no—
515C2×C8—731no—
516C2×C8—731no—
517C2×C8—731no—
518C2×C8—1461no—
519C2×C8—1461no—
520C2×C8—1461no—
521C2×C8—1461no—
522C2×C8—1461no—
523C2×C8—1461no—
524C2×C8C22×#6440811yescharacteristic
525C2×C8C22×#6440811yescharacteristic
526C4⋊C4—731no—
527C4⋊C4—731no—
528C4⋊C4—731no—
529C4⋊C4—1461no—
530C4⋊C4—1461no—
531C4⋊C4C22×#6440811yescharacteristic
532C22⋊C4—21no—
533C22⋊C4—1461no—
534C22⋊C4—1461no—
535C22⋊C4—1461no—
536C22⋊C4—1461no—
537C22⋊C4—1461no—
538C22⋊C4—1461no—
539C22⋊C4—1461no—
540C42—731no—
541C42—731no—
542C42—731no—
543C42—731no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—42no—
548C23—42no—
549C23—81no—
550C23—1461no—
551C23D4×#6440811yescharacteristic
552Q8—2922no—
553D4—21no—
554D4—21no—
555D4—21no—
556D4—21no—
557D4—21no—
558D4—42no—
559D4—42no—
560D4—2922no—
561D4—2922no—
562D4—5841no—
563D4—1461no—
564D4—1461no—
565D4#1455667812yescomplemented
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—2921no—
570C2×C4—2921no—
571C2×C4—2922no—
572C2×C4—2921no—
573C2×C4—2921no—
574C2×C4—2921no—
575C2×C4—2921no—
576C2×C4—2921no—
577C2×C4—2922no—
578C2×C4—2922no—
579C2×C4—2921no—
580C2×C4—2921no—
581C2×C4—731no—
582C2×C4—731no—
583C2×C4—732no—
584C2×C4—732no—
585C2×C4—731no—
586C2×C4—731no—
587C2×C4—1461no—
588C2×C4—1461no—
589C2×C4—1461no—
590C2×C4D4×#6440811yescharacteristic
591C2×C4C23×#6440811yescharacteristic
592C8—21no—
593C8—21no—
594C8—2921no—
595C8—2921no—
596C8—2921no—
597C8—2921no—
598C8—2921no—
599C8—732no—
600C8—732no—
601C8—1461no—
602C8—1461no—
603C8—1462no—
604C8—1461no—
605C8—1461no—
606C22—21no—
607C22—21no—
608C22—22no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—41no—
617C22—84no—
618C22—2921no—
619C22—2921no—
620C22—2921no—
621C22D8×#6440812yes—
622C22C2×D4×#6440811yescharacteristic
623C4—21no—
624C4—41no—
625C4—2921no—
626C4—2921no—
627C4—2921no—
628C4—731no—
629C4—731no—
630C4—732no—
631C4—1461no—
632C4—1461no—
633C4C2×#1455667811yescharacteristic
634C4#1798966511yescharacteristic
635C2—22no—
636C2—42no—
637C2—81no—
638C2—81no—
639C2—5841no—
640C2#2262666311yescharacteristic
641C2C2×D8×#6440811yescharacteristic
642C2#2262670511yescharacteristic
subgroupG/H
centerC22C2×D4×#64408
commutatorC2×C4×C73C23×C4
FrattiniC2×C4C23×#64408
Fitting#447×C73C4
socleC22×C73C2×C4×D4
prime radical#22619926C2
cosocleC4×Dic73C24
hypercenter#447#64408
hypocenterC73#17994
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #22626673C2semidirect
2. D4×#11924037C2semidirect
3. #22625564C2semidirect
4. #22625562C2semidirect
5. #22625533C2semidirect
6. #22625532C2semidirect
7. #22625532C2semidirect
8. #22625530C2semidirect
9. #22625511C2semidirect
10. D8×#10596833C2semidirect
11. #22625500C2semidirect
12. #22625276C2semidirect
13. C2×D4⋊#10596833C2semidirect
14. #22625132C2semidirect
20. #17989622C22semidirect
21. #17989621C22semidirect
26. #17989615C22semidirect
29. D4×#10596833C22semidirect
36. #17989451C22semidirect
37. #17989451C22semidirect
38. #17989450C22semidirect
39. #17989450C22semidirect
40. #17989435C22semidirect
41. #17989435C22semidirect
46. D4⋊#10596833C22semidirect
47. D4⋊#10596833C22semidirect
48. #17989429C22semidirect
49. #17989429C22semidirect
50. C8×#10596833C22semidirect
51. C8×#10596833C22semidirect
152. #11924037D8semidirect
237. D4×C73#131semidirect
337. C73#17994semidirect
565. D4#14556678semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#226266732
D4×#119240372
#226255642
#226255622
#226255332
#226255322
#226255302
#226255112
D8×#105968332
#226255002
#226252762
C2×D4⋊#105968332
#226251322
#226199262
#1799473