Finite Group Databasebeta

← #34018689

Subgroups — #34018689

Order 18688.

678 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626821C211yesmaximal, characteristic, complemented
2#22626778C211yesmaximal, characteristic, complemented
3#22625552C211yesmaximal, characteristic, complemented
4#22625550C211yesmaximal, characteristic, complemented
5#22625532C211yesmaximal, characteristic, complemented
6#22625530C211yesmaximal, characteristic, complemented
7#22625528C211yesmaximal, characteristic, complemented
8#22625527C211yesmaximal, characteristic, complemented
9#22625511C211yesmaximal, characteristic, complemented
10D8×#10596833C211yesmaximal, characteristic, complemented
11#22625502C211yesmaximal, characteristic, complemented
12#22625270C211yesmaximal, characteristic, complemented
13#22625250C211yesmaximal, characteristic, complemented
14#22625148C211yesmaximal, characteristic, complemented
15#22619926C211yesmaximal, characteristic
16#17989622C2211yescharacteristic, complemented
17#17989621C2211yescharacteristic, complemented
18#17989619—21no—
19#17989616—21no—
20#17989615C2211yescharacteristic, complemented
21D4×#10596833—21no—
22D4×#10596833—21no—
23D4×#10596833C2211yescharacteristic
24D4×#10596833C2211yescharacteristic
25D4×#10596833C2211yescharacteristic
26D4×#10596833C2211yescharacteristic
27D4×#10596833C2211yescharacteristic, complemented
28#17989608C2211yescharacteristic
29#17989603C2211yescharacteristic
30#17989603C2211yescharacteristic
31#17989561—21no—
32#17989560—21no—
33C2×C22⋊#10596833—21no—
34C2×C22⋊#10596833—21no—
35C4×#11924037C2211yescharacteristic
36#17989451C2211yescharacteristic, complemented
37#17989450C2211yescharacteristic, complemented
38#17989449C2211yescharacteristic, complemented
39#17989448C2211yescharacteristic, 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
50#17989428C2211yescharacteristic, 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
67C2×#11924038—21no—
68C2×#11924037—21no—
69C2×#11924037—21no—
70C2×#11924036—21no—
71C22×#10596833—21no—
72C22×#10596833—21no—
73C22×#10596833—21no—
74C22×#10596833—21no—
75C22×#10596833—21no—
76C4⋊#10596833D411yescharacteristic
77C4⋊#10596833D411yescharacteristic
78C4⋊#10596833C2311yescharacteristic
79C4⋊#10596833C2311yescharacteristic
80C4⋊#10596833C2311yescharacteristic
81C4⋊#10596833C2311yescharacteristic
82#14556648—21no—
83#14556648—21no—
84C22⋊#10596833—21no—
85C22⋊#10596833—21no—
86C22⋊#10596833—21no—
87C22⋊#10596833—21no—
88C22⋊#10596833—21no—
89C22⋊#10596833—21no—
90C22⋊#10596833—21no—
91C22⋊#10596833—21no—
92C22⋊#10596833D411yescharacteristic
93C22⋊#10596833D411yescharacteristic
94#14556646—21no—
95#14556646—21no—
96C4×#10596833—21no—
97C4×#10596833C2311yescharacteristic
98C4×#10596833C2311yescharacteristic
99C4×#10596833C2311yescharacteristic
100C4×#10596833C2311yescharacteristic
101C2×#11924027—21no—
102C23×Dic73—21no—
103C2×#11924013C2×C411yescharacteristic
104C2×#11924013C2×C411yescharacteristic
105#14556590C2×C411yescharacteristic
106#14556587C2×C411yescharacteristic
107#14556577—21no—
108#14556565C2×C411yescharacteristic
109#14556563—21no—
110D4×Dic73—21no—
111D4×Dic73—21no—
112D4×Dic73C2×C411yescharacteristic
113D4×Dic73C2×C411yescharacteristic
114D4×Dic73C2311yescharacteristic
115D4×Dic73C2311yescharacteristic
116D4×Dic73C2311yescharacteristic
117#14556553C2311yescharacteristic
118#14556551—21no—
119C2×C4⋊Dic73C2×C411yescharacteristic
120C2×C22⋊Dic73—21no—
121C2×C4×Dic73C2311yescharacteristic
122#14556525C2×C411yescharacteristic
123#14556525C2×C411yescharacteristic
124#14556524C2×C411yescharacteristic
125#14556524C2×C411yescharacteristic
126#14556520C2×C411yescharacteristic
127#14556520C2×C411yescharacteristic
128#14556513C2×C411yescharacteristic
129#14556513C2×C411yescharacteristic
130#14556512C2×C411yescharacteristic
131#14556512C2×C411yescharacteristic
132#14556512C2×C411yescharacteristic
133#14556512C2×C411yescharacteristic
134C22⋊#10596831C2×C411yescharacteristic
135C8×Dic73C2311yescharacteristic
136C8×Dic73C2311yescharacteristic
137C22×D4×C73C2×C411yescharacteristic
138C2×D8×C73C2×C411yescharacteristic
139C2×D8×C73C2×C411yescharacteristic
140#121×C73C2×C411yescharacteristic
141#102×C73C2×C411yescharacteristic
142#102×C73C2×C411yescharacteristic
143C22⋊C8×C73C2×C411yescharacteristic
144#11924038—81no—
145#11924037—42no—
146#11924037—81no—
147#11924036—81no—
148C2×#10596833—21no—
149C2×#10596833—21no—
150C2×#10596833—21no—
151C2×#10596833—21no—
152C2×#10596833—21no—
153C2×#10596833—21no—
154C2×#10596833—81no—
155C2×#10596833—81no—
156C2×#10596833—81no—
157C2×#10596833—81no—
158C2×#10596833—81no—
159C2×#10596833C2×D411yescharacteristic
160C2×#10596833C2×D411yescharacteristic
161C2×#10596833C2×D411yescharacteristic
162C2×#10596833C2×D411yescharacteristic
163#11924027—81no—
164#11924027—81no—
165C2×Dic146—21no—
166C2×D292C22×C411yescharacteristic
167C2×C22⋊D73—21no—
168C2×C22⋊D73—21no—
169C2×C22⋊D73—21no—
170C22×Dic73—21no—
171C22×Dic73—21no—
172C22×Dic73—21no—
173C22×Dic73—21no—
174C22×Dic73—21no—
175C22×Dic73—21no—
176C22×Dic73—81no—
177C22×Dic73—81no—
178C22×Dic73C2×D411yescharacteristic
179C22×Dic73#4111yescharacteristic
180C2×C4×D73—21no—
181#11924013—22no—
182#11924013—22no—
183C2×#10596831C22×C411yescharacteristic
184C2×#10596831C22×C411yescharacteristic
185C4⋊Dic73—21no—
186C4⋊Dic73C22×C411yescharacteristic
187C4⋊Dic73C22×C411yescharacteristic
188C4⋊Dic73C22×C411yescharacteristic
189#11924007—21no—
190C22⋊Dic73—21no—
191C22⋊Dic73—21no—
192C22⋊Dic73—21no—
193C22⋊Dic73—21no—
194C22⋊Dic73—21no—
195#11924005—21no—
196#11924005—21no—
197C4×Dic73C22×C411yescharacteristic
198C4×Dic73C2×D411yescharacteristic
199C4×Dic73#4111yescharacteristic
200C4×Dic73C2411yescharacteristic
201C24×C73—21no—
202C2×D4×C73—22no—
203C2×D4×C73—22no—
204C2×D4×C73—21no—
205C2×D4×C73—21no—
206C2×D4×C73C22×C411yescharacteristic
207C2×D4×C73C22×C411yescharacteristic
208C2×D4×C73C22×C411yescharacteristic
209C22×C4×C73C22×C411yescharacteristic
210D8×C73—22no—
211D8×C73—22no—
212C2×C8×C73C22×C411yescharacteristic
213C2×C8×C73C22×C411yescharacteristic
214C4⋊C4×C73C22×C411yescharacteristic
215C22⋊C4×C73—21no—
216C23×C73—24no—
217C23×C73—21no—
218C23×C73—21no—
219C23×C73—21no—
220C23×C73—81no—
221C23×C73—81no—
222C23×C73C4×D411yescharacteristic
223C22×D73—21no—
224D4×C73—24no—
225D4×C73—21no—
226D4×C73—21no—
227D4×C73—21no—
228D4×C73—21no—
229D4×C73—81no—
230D4×C73#13112yes—
231D4×C73#13112yes—
232C2×C4×C73—21no—
233C2×C4×C73—21no—
234C2×C4×C73—21no—
235C2×C4×C73C4×D411yescharacteristic
236C2×C4×C73C23×C411yescharacteristic
237C22⋊D73—81no—
238C22⋊D73—81no—
239C22⋊D73—81no—
240C2×Dic73—21no—
241C2×Dic73—21no—
242C2×Dic73—21no—
243C2×Dic73—21no—
244C2×Dic73—21no—
245C2×Dic73—41no—
246C2×Dic73—41no—
247C2×Dic73—41no—
248C2×Dic73—41no—
249C2×Dic73—41no—
250C2×Dic73—41no—
251C2×Dic73—41no—
252C2×Dic73—81no—
253C2×Dic73—81no—
254C2×Dic73—81no—
255C2×Dic73C4×D411yescharacteristic
256C2×Dic73C4×D411yescharacteristic
257C2×Dic73C22×D411yescharacteristic
258C2×Dic73C2×#4111yescharacteristic
259D292—21no—
260D292—21no—
261C4×D73—41no—
262C4×D73—41no—
263Dic146—81no—
264#10596833—22no—
265#10596833—21no—
266#10596833—21no—
267#10596833—21no—
268#10596833—21no—
269#10596833—41no—
270#10596833—41no—
271#10596833—41no—
272#10596833—41no—
273#10596833—41no—
274C8×C73—21no—
275C8×C73—21no—
276#10596831—21no—
277#10596831—21no—
278C22×C73—21no—
279C22×C73—22no—
280C22×C73—22no—
281C22×C73—21no—
282C22×C73—21no—
283C22×C73—41no—
284C22×C73—41no—
285C22×C73—41no—
286C22×C73—41no—
287C22×C73—41no—
288C22×C73—41no—
289C22×C73—41no—
290C22×C73—84no—
291C22×C73C2×C4×D411yescharacteristic
292C2×D73—41no—
293C2×D73—41no—
294C2×D73—41no—
295C4×C73—21no—
296C4×C73—41no—
297C4×C73C2×#13111yescharacteristic
298C4×C73C2×#13111yescharacteristic
299Dic73—21no—
300Dic73—21no—
301Dic73—21no—
302Dic73—41no—
303Dic73—41no—
304Dic73—41no—
305Dic73#57611yescharacteristic
306Dic73#57611yescharacteristic
307#17922—731nomaximal, Sylow
308C2×C73—41no—
309C2×C73—42no—
310C2×C73—42no—
311C2×C73—81no—
312C2×C73#292911yescharacteristic
313C2×C73#294811yescharacteristic
314C2×C73#295611yescharacteristic
315D73—81no—
316#3100—731no—
317#2981—731no—
318#2972—731no—
319#1586—731no—
320#1586—731no—
321#1581—731no—
322#1581—731no—
323#1580—731no—
324#1580—731no—
325C8×D8—731no—
326C8×D8—731no—
327#1572—731no—
328#1490—731no—
329#1484—731no—
330#1453—731no—
331C73#1792211yescharacteristic, complemented, Sylow
332#532—731no—
333C2×C4×D4—731no—
334#480—731no—
335#480—731no—
336#474—731no—
337#474—731no—
338#459—731no—
339#459—731no—
340#447—731no—
341#447#6440811yescharacteristic
342C4×D8—731no—
343C4×D8—731no—
344C4×D8—731no—
345C4×D8—731no—
346#436—731no—
347#436—731no—
348#435—1461no—
349#435—1461no—
350C8×D4—731no—
351C8×D4—731no—
352C8×D4—731no—
353C8×D4—731no—
354C8×D4—731no—
355C8×D4—731no—
356C8×D4—1461no—
357C8×D4—1461no—
358#432—731no—
359#431—731no—
360#431—731no—
361#431—731no—
362#408—1461no—
363#408—1461no—
364C2×C22⋊C8—1461no—
365C2×C22⋊C8—1461no—
366C4×M16—731no—
367#335—731no—
368#335—731no—
369#334—731no—
370#334—731no—
371D4⋊C8—731no—
372D4⋊C8—731no—
373D4⋊C8—731no—
374D4⋊C8—731no—
375D4⋊C8—731no—
376D4⋊C8—731no—
377D4⋊C8—731no—
378D4⋊C8—731no—
379#322—731no—
380#322—731no—
381#322—731no—
382C82—731no—
383C2×#41—1461no—
384C22×D4C2×#6440811yescharacteristic
385C23×C4—1461no—
386C2×D8—731no—
387C2×D8—731no—
388C2×D8C2×#6440811yescharacteristic
389C2×D8C2×#6440811yescharacteristic
390C2×M16—1461no—
391C2×M16—1461no—
392C2×M16—1461no—
393C22×C8—1461no—
394C22×C8—1461no—
395C22×C8—1461no—
396C22×C8—1461no—
397C22×C8—1461no—
398C22×C8—1461no—
399C4⋊Q8—731no—
400#127—731no—
401#124—1461no—
402#121—731no—
403#121—1461no—
404#121C2×#6440811yescharacteristic
405C4×D4—731no—
406C4×D4—731no—
407C4×D4—731no—
408C4×D4—731no—
409C4×D4—731no—
410C4×D4—731no—
411C4×D4—1461no—
412C4×D4—1461no—
413C4×D4—1461no—
414C2×C4⋊C4—731no—
415C2×C22⋊C4—1461no—
416C2×C42—731no—
417#107—731no—
418#107—731no—
419#107—731no—
420#107—731no—
421C4⋊C8—731no—
422C4⋊C8—731no—
423C4⋊C8—731no—
424C4⋊C8—731no—
425C4⋊C8—731no—
426C4⋊C8—731no—
427C4⋊C8—731no—
428C4⋊C8—731no—
429#102—731no—
430#102—731no—
431#102—731no—
432#102—731no—
433#102—731no—
434#102—731no—
435#102C2×#6440811yescharacteristic
436#102C2×#6440811yescharacteristic
437C22⋊C8—731no—
438C22⋊C8—731no—
439C22⋊C8—731no—
440C22⋊C8—1461no—
441C22⋊C8—1461no—
442C22⋊C8—1461no—
443C22⋊C8—1461no—
444C22⋊C8—1461no—
445C22⋊C8—1461no—
446C22⋊C8—1461no—
447C22⋊C8—1461no—
448C22⋊C8—1461no—
449C22⋊C8—1461no—
450C22⋊C8C2×#6440811yescharacteristic
451#97—1461no—
452#97—1461no—
453C4×C8—731no—
454C4×C8—731no—
455C4×C8—731no—
456C4×C8—731no—
457C4×C8—731no—
458C4×C8—731no—
459C4×C8—1461no—
460C24—21no—
461#41—5841no—
462#41—5841no—
463C2×Q8—1461no—
464C2×D4—22no—
465C2×D4—22no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—731no—
469C2×D4—1461no—
470C2×D4—1461no—
471C2×D4—1461no—
472C2×D4C22×#6440811yescharacteristic
473C2×D4C22×#6440811yescharacteristic
474C2×D4C22×#6440811yescharacteristic
475C22×C4—5841no—
476C22×C4—5841no—
477C22×C4—731no—
478C22×C4—731no—
479C22×C4—1461no—
480C22×C4—1461no—
481C22×C4—1461no—
482C22×C4—1461no—
483C22×C4—1461no—
484C22×C4—1461no—
485C22×C4—1461no—
486C22×C4C22×#6440811yescharacteristic
487D8—22no—
488D8—22no—
489D8—1462no—
490D8—1462no—
491M16—2922no—
492M16—5841no—
493M16—5841no—
494C2×C8—5841no—
495C2×C8—5841no—
496C2×C8—5841no—
497C2×C8—5841no—
498C2×C8—5841no—
499C2×C8—5841no—
500C2×C8—731no—
501C2×C8—731no—
502C2×C8—731no—
503C2×C8—731no—
504C2×C8—731no—
505C2×C8—731no—
506C2×C8—1461no—
507C2×C8—1461no—
508C2×C8—1461no—
509C2×C8—1461no—
510C2×C8—1461no—
511C2×C8—1461no—
512C2×C8C22×#6440811yescharacteristic
513C2×C8C22×#6440811yescharacteristic
514C4⋊C4—731no—
515C4⋊C4—731no—
516C4⋊C4—731no—
517C4⋊C4—1461no—
518C4⋊C4—1461no—
519C4⋊C4C22×#6440811yescharacteristic
520C22⋊C4—21no—
521C22⋊C4—1461no—
522C22⋊C4—1461no—
523C22⋊C4—1461no—
524C22⋊C4—1461no—
525C22⋊C4—1461no—
526C22⋊C4—1461no—
527C22⋊C4—1461no—
528C42—731no—
529C42—731no—
530C42—731no—
531C42—731no—
532C23—24no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—81no—
537C23—81no—
538C23—1461no—
539C23D4×#6440811yescharacteristic
540Q8—5841no—
541D4—24no—
542D4—21no—
543D4—21no—
544D4—21no—
545D4—21no—
546D4—81no—
547D4—5841no—
548D4—5841no—
549D4—5841no—
550D4—1461no—
551D4—1461no—
552D4#1455667812yes—
553D4#1455667812yes—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—2921no—
558C2×C4—2921no—
559C2×C4—2921no—
560C2×C4—2921no—
561C2×C4—2921no—
562C2×C4—2921no—
563C2×C4—2921no—
564C2×C4—2921no—
565C2×C4—2921no—
566C2×C4—5841no—
567C2×C4—5841no—
568C2×C4—5841no—
569C2×C4—731no—
570C2×C4—731no—
571C2×C4—731no—
572C2×C4—731no—
573C2×C4—1461no—
574C2×C4—1461no—
575C2×C4—1461no—
576C2×C4—1461no—
577C2×C4—1461no—
578C2×C4D4×#6440811yescharacteristic
579C2×C4C23×#6440811yescharacteristic
580C8—21no—
581C8—21no—
582C8—2921no—
583C8—2921no—
584C8—2921no—
585C8—2921no—
586C8—2921no—
587C8—1461no—
588C8—1461no—
589C8—1461no—
590C8—1461no—
591C8—1462no—
592C8—1461no—
593C8—1461no—
594C22—21no—
595C22—22no—
596C22—22no—
597C22—21no—
598C22—21no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—84no—
607C22—2921no—
608C22—2921no—
609C22—2921no—
610C22C2×D4×#6440811yescharacteristic
611C4—21no—
612C4—41no—
613C4—2921no—
614C4—2921no—
615C4—2921no—
616C4—731no—
617C4—731no—
618C4—1461no—
619C4—1461no—
620C4—1461no—
621C4C2×#1455667811yescharacteristic
622C4C2×#1455667811yescharacteristic
623C2—41no—
624C2—42no—
625C2—42no—
626C2—81no—
627C2—5841no—
628C2#2262664611yescharacteristic
629C2#2262669011yescharacteristic
630C2#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#17922
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. #22626821C2semidirect
2. #22626778C2semidirect
3. #22625552C2semidirect
4. #22625550C2semidirect
5. #22625532C2semidirect
6. #22625530C2semidirect
7. #22625528C2semidirect
8. #22625527C2semidirect
9. #22625511C2semidirect
10. D8×#10596833C2semidirect
11. #22625502C2semidirect
12. #22625270C2semidirect
13. #22625250C2semidirect
14. #22625148C2semidirect
16. #17989622C22semidirect
17. #17989621C22semidirect
20. #17989615C22semidirect
27. D4×#10596833C22semidirect
36. #17989451C22semidirect
37. #17989450C22semidirect
38. #17989449C22semidirect
39. #17989448C22semidirect
40. #17989435C22semidirect
41. #17989435C22semidirect
46. D4⋊#10596833C22semidirect
47. D4⋊#10596833C22semidirect
48. #17989429C22semidirect
49. #17989429C22semidirect
50. #17989428C22semidirect
51. C8×#10596833C22semidirect
331. C73#17922semidirect

Maximal subgroups

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

subgroupindex
#226268212
#226267782
#226255522
#226255502
#226255322
#226255302
#226255282
#226255272
#226255112
D8×#105968332
#226255022
#226252702
#226252502
#226251482
#226199262
#1792273