Finite Group Databasebeta

← #34019192

Subgroups — #34019192

Order 18688.

670 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626762C211yesmaximal, characteristic, complemented
2D4×#11924037C211yesmaximal, characteristic, complemented
3#22625572C211yesmaximal, characteristic, complemented
4#22625564C211yesmaximal, characteristic, complemented
5#22625552C211yesmaximal, characteristic, complemented
6#22625546C211yesmaximal, characteristic, complemented
7#22625541C211yesmaximal, characteristic, complemented
8#22625532C211yesmaximal, characteristic, complemented
9#22625527C211yesmaximal, characteristic, complemented
10#22625514C211yesmaximal, characteristic, complemented
11#22625507C211yesmaximal, characteristic, complemented
12#22625286C211yesmaximal, characteristic, complemented
13#22625230C211yesmaximal, characteristic, complemented
14#22625148C211yesmaximal, characteristic, complemented
15#22619973C211yesmaximal, characteristic
16C22×#11924037—21no—
17#17989637C2211yescharacteristic, complemented
18#17989636C2211yescharacteristic, complemented
19#17989622—21no—
20#17989622C2211yescharacteristic
21#17989622C2211yescharacteristic
22#17989622C2211yescharacteristic, complemented
23#17989618—21no—
24#17989618—21no—
25#17989618—21no—
26D4×#10596833C2211yescharacteristic
27D4×#10596833C2211yescharacteristic
28D4×#10596833C2211yescharacteristic, complemented
29#17989611—21no—
30#17989603C2211yescharacteristic
31#17989603C2211yescharacteristic
32#17989590C2211yescharacteristic
33#17989557—21no—
34#17989551—21no—
35C4×#11924037C2211yescharacteristic
36#17989451C2211yescharacteristic, complemented
37#17989450C2211yescharacteristic, complemented
38#17989449C2211yescharacteristic, complemented
39#17989448C2211yescharacteristic, complemented
40#17989437C2211yescharacteristic, complemented
41#17989437C2211yescharacteristic, 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#17989118C2211yescharacteristic
53C2×D4×Dic73C2211yescharacteristic
54#17988923C411yescharacteristic
55#17988903C411yescharacteristic
56#17988885C411yescharacteristic
57#17988885C411yescharacteristic
58#17988800C411yescharacteristic
59#17988800C411yescharacteristic
60#17988738C411yescharacteristic
61#17988678C2211yescharacteristic
62#17988678C2211yescharacteristic
63D8×Dic73C2211yescharacteristic
64D8×Dic73C2211yescharacteristic
65#17988637C2211yescharacteristic
66#447×C73C411yescharacteristic
67C2×#11924037—21no—
68C2×#11924037—21no—
69C2×#11924037—21no—
70C2×#11924037—21no—
71C2×#11924037—44no—
72C2×#11924037D411yescharacteristic
73C2×#11924037D411yescharacteristic
74C22×#10596833—21no—
75C22×#10596833—21no—
76C22×#10596833—21no—
77#14556658—21no—
78#14556658—21no—
79C4⋊#10596833—21no—
80C4⋊#10596833—21no—
81C4⋊#10596833C2311yescharacteristic
82C4⋊#10596833C2311yescharacteristic
83C4⋊#10596833C2311yescharacteristic
84C4⋊#10596833C2311yescharacteristic
85C22⋊#10596833—21no—
86C22⋊#10596833—21no—
87C22⋊#10596833—21no—
88C22⋊#10596833—21no—
89C22⋊#10596833—21no—
90C22⋊#10596833—21no—
91C22⋊#10596833—21no—
92C22⋊#10596833—21no—
93#14556646—21no—
94#14556646D411yescharacteristic
95#14556646D411yescharacteristic
96C4×#10596833—21no—
97C4×#10596833—21no—
98C4×#10596833C2311yescharacteristic
99C4×#10596833C2311yescharacteristic
100C4×#10596833C2311yescharacteristic
101C4×#10596833C2311yescharacteristic
102C23×Dic73—21no—
103C2×#11924016C2×C411yescharacteristic
104C2×#11924016C2×C411yescharacteristic
105#14556584—21no—
106#14556582C2×C411yescharacteristic
107#14556578C2×C411yescharacteristic
108#14556572—21no—
109#14556570C2×C411yescharacteristic
110#14556556C2311yescharacteristic
111D4×Dic73—21no—
112D4×Dic73—21no—
113D4×Dic73—21no—
114D4×Dic73C2×C411yescharacteristic
115D4×Dic73C2×C411yescharacteristic
116D4×Dic73C2311yescharacteristic
117D4×Dic73C2311yescharacteristic
118D4×Dic73C2311yescharacteristic
119#14556547—21no—
120C2×C4⋊Dic73C2×C411yescharacteristic
121C2×C22⋊Dic73—21no—
122C2×C4×Dic73C2311yescharacteristic
123#14556525C2×C411yescharacteristic
124#14556525C2×C411yescharacteristic
125#14556522C2×C411yescharacteristic
126#14556522C2×C411yescharacteristic
127#14556520C2×C411yescharacteristic
128#14556520C2×C411yescharacteristic
129#14556516C2×C411yescharacteristic
130#14556516C2×C411yescharacteristic
131#14556512C2×C411yescharacteristic
132#14556512C2×C411yescharacteristic
133#14556512C2×C411yescharacteristic
134#14556512C2×C411yescharacteristic
135C22⋊#10596831C2×C411yescharacteristic
136C8×Dic73C2311yescharacteristic
137C8×Dic73C2311yescharacteristic
138C22×D4×C73C2×C411yescharacteristic
139C2×D8×C73C2×C411yescharacteristic
140C2×D8×C73C2×C411yescharacteristic
141#121×C73C2×C411yescharacteristic
142#102×C73C2×C411yescharacteristic
143#102×C73C2×C411yescharacteristic
144C22⋊C8×C73C2×C411yescharacteristic
145#11924037—22no—
146#11924037—41no—
147#11924037—42no—
148#11924037—42no—
149#11924037—42no—
150#11924037—42no—
151C2×#10596833—21no—
152C2×#10596833—21no—
153C2×#10596833—21no—
154C2×#10596833—21no—
155C2×#10596833—21no—
156C2×#10596833—21no—
157C2×#10596833—42no—
158C2×#10596833—42no—
159C2×#10596833—42no—
160C2×#10596833C2×D411yescharacteristic
161C2×#10596833C2×D411yescharacteristic
162C2×#10596833C2×D411yescharacteristic
163C2×#10596833C2×D411yescharacteristic
164C2×Dic146C22×C411yescharacteristic
165C22×Dic73—21no—
166C22×Dic73—21no—
167C22×Dic73—21no—
168C22×Dic73—21no—
169C22×Dic73—21no—
170C22×Dic73—21no—
171C22×Dic73—42no—
172C22×Dic73—42no—
173C22×Dic73C2×D411yescharacteristic
174C22×Dic73#4111yescharacteristic
175#11924016—22no—
176#11924016—41no—
177C2×#10596831C22×C411yescharacteristic
178C2×#10596831C22×C411yescharacteristic
179C4⋊Dic73—21no—
180C4⋊Dic73—21no—
181C4⋊Dic73C22×C411yescharacteristic
182C4⋊Dic73C22×C411yescharacteristic
183C4⋊Dic73C22×C411yescharacteristic
184#11924007—21no—
185#11924007—21no—
186#11924007—21no—
187C22⋊Dic73—21no—
188C22⋊Dic73—21no—
189C22⋊Dic73—21no—
190C22⋊Dic73—21no—
191C22⋊Dic73—21no—
192C22⋊Dic73—21no—
193C22⋊Dic73—21no—
194C22⋊Dic73—21no—
195C4×Dic73—21no—
196C4×Dic73C22×C411yescharacteristic
197C4×Dic73C2×D411yescharacteristic
198C4×Dic73#4111yescharacteristic
199C4×Dic73C2411yescharacteristic
200C24×C73—21no—
201C2×D4×C73—22no—
202C2×D4×C73—21no—
203C2×D4×C73—21no—
204C2×D4×C73—41no—
205C2×D4×C73C22×C411yescharacteristic
206C2×D4×C73C22×C411yescharacteristic
207C2×D4×C73C22×C411yescharacteristic
208C22×C4×C73C22×C411yescharacteristic
209D8×C73—22no—
210D8×C73—41no—
211C2×C8×C73C22×C411yescharacteristic
212C2×C8×C73C22×C411yescharacteristic
213C4⋊C4×C73C22×C411yescharacteristic
214C22⋊C4×C73—21no—
215C23×C73—21no—
216C23×C73—21no—
217C23×C73—21no—
218C23×C73—42no—
219C23×C73—42no—
220C23×C73—81no—
221C23×C73C4×D411yescharacteristic
222D4×C73—21no—
223D4×C73—21no—
224D4×C73—21no—
225D4×C73—21no—
226D4×C73—21no—
227D4×C73—42no—
228D4×C73—81no—
229D4×C73#13112yes—
230C2×C4×C73—21no—
231C2×C4×C73—21no—
232C2×C4×C73—21no—
233C2×C4×C73C4×D411yescharacteristic
234C2×C4×C73C23×C411yescharacteristic
235C2×Dic73—21no—
236C2×Dic73—21no—
237C2×Dic73—21no—
238C2×Dic73—21no—
239C2×Dic73—21no—
240C2×Dic73—21no—
241C2×Dic73—41no—
242C2×Dic73—42no—
243C2×Dic73—42no—
244C2×Dic73—42no—
245C2×Dic73—41no—
246C2×Dic73—41no—
247C2×Dic73—41no—
248C2×Dic73—41no—
249C2×Dic73—41no—
250C2×Dic73—41no—
251C2×Dic73C4×D411yescharacteristic
252C2×Dic73C4×D411yescharacteristic
253C2×Dic73C22×D411yescharacteristic
254C2×Dic73C2×#4111yescharacteristic
255Dic146—21no—
256Dic146—21no—
257#10596833—21no—
258#10596833—21no—
259#10596833—21no—
260#10596833—41no—
261#10596833—41no—
262#10596833—41no—
263#10596833—41no—
264#10596833—41no—
265#10596833—41no—
266#10596833AGL(1,ℤ/8)12yes—
267C8×C73—21no—
268C8×C73—21no—
269#10596831—21no—
270#10596831—21no—
271C22×C73—21no—
272C22×C73—22no—
273C22×C73—21no—
274C22×C73—21no—
275C22×C73—41no—
276C22×C73—41no—
277C22×C73—41no—
278C22×C73—41no—
279C22×C73—41no—
280C22×C73—41no—
281C22×C73—41no—
282C22×C73—41no—
283C22×C73—84no—
284C22×C73C2×C4×D411yescharacteristic
285C4×C73—21no—
286C4×C73—41no—
287C4×C73C2×#13111yescharacteristic
288C4×C73#56811yescharacteristic
289Dic73—21no—
290Dic73—21no—
291Dic73—21no—
292Dic73—41no—
293Dic73—41no—
294Dic73—41no—
295Dic73—41no—
296Dic73C2×AGL(1,ℤ/8)11yescharacteristic
297Dic73#57511yescharacteristic
298#18008—731nomaximal, Sylow
299C2×C73—41no—
300C2×C73—42no—
301C2×C73—81no—
302C2×C73—81no—
303C2×C73#293311yescharacteristic
304C2×C73C4×AGL(1,ℤ/8)11yescharacteristic
305C2×C73#295711yescharacteristic
306#3105—731no—
307#2969—731no—
308D4×M16—731no—
309#1592—731no—
310#1589—731no—
311#1586—731no—
312#1585—731no—
313#1583—731no—
314#1581—731no—
315#1580—731no—
316C8×SD16—731no—
317#1574—731no—
318#1495—731no—
319#1478—731no—
320#1453—731no—
321C73#1800811yescharacteristic, complemented, Sylow
322C22×M16—1461no—
323#529—731no—
324C2×C4×D4—731no—
325#484—731no—
326#481—731no—
327#478—731no—
328#478—731no—
329#463—731no—
330#463—731no—
331#450—731no—
332#447#6440811yescharacteristic
333#446—731no—
334C8×Q8—731no—
335C4×SD16—731no—
336C4×SD16—731no—
337C4×D8—731no—
338C4×D8—731no—
339#436—731no—
340#436—731no—
341#436—731no—
342#436—1461no—
343#435—1461no—
344#435—1461no—
345#435—1461no—
346C8×D4—731no—
347C8×D4—731no—
348C8×D4—731no—
349#433—1461no—
350#431—731no—
351#431—731no—
352#431—731no—
353#423—731no—
354#407—1461no—
355#405—1461no—
356C4×M16—731no—
357#335—731no—
358#335—731no—
359#334—731no—
360#334—731no—
361Q8⋊C8—731no—
362Q8⋊C8—731no—
363D4⋊C8—731no—
364D4⋊C8—731no—
365D4⋊C8—731no—
366D4⋊C8—731no—
367D4⋊C8—731no—
368D4⋊C8—731no—
369#322—731no—
370#322—731no—
371#322—731no—
372C82—731no—
373C22×D4C2×#6440811yescharacteristic
374C23×C4—1461no—
375C2×SD16—731no—
376C2×SD16—731no—
377C2×D8C2×#6440811yescharacteristic
378C2×D8C2×#6440811yescharacteristic
379C2×M16—2924no—
380C2×M16—731no—
381C2×M16—731no—
382C2×M16—1461no—
383C2×M16—1461no—
384C2×M16—1461no—
385C2×M16—1461no—
386C22×C8—1461no—
387C22×C8—1461no—
388C22×C8—1461no—
389#126—1461no—
390#125—731no—
391#124—731no—
392#123—1461no—
393C22⋊Q8—731no—
394#121C2×#6440811yescharacteristic
395C4×Q8—731no—
396C4×D4—731no—
397C4×D4—731no—
398C4×D4—731no—
399C4×D4—731no—
400C4×D4—731no—
401C4×D4—1461no—
402C4×D4—1461no—
403C4×D4—1461no—
404#117—1461no—
405C2×C4⋊C4—731no—
406C2×C22⋊C4—1461no—
407C2×C42—731no—
408#107—731no—
409#107—731no—
410#106—731no—
411#106—731no—
412C4⋊C8—731no—
413C4⋊C8—731no—
414C4⋊C8—731no—
415C4⋊C8—731no—
416C4⋊C8—731no—
417C4⋊C8—731no—
418C4⋊C8—1461no—
419C4⋊C8—1461no—
420C4⋊C8—1461no—
421C4⋊C8—1461no—
422#103—731no—
423#103—731no—
424#102—731no—
425#102—731no—
426#102—731no—
427#102—731no—
428#102C2×#6440811yescharacteristic
429#102C2×#6440811yescharacteristic
430C22⋊C8—731no—
431C22⋊C8—1461no—
432C22⋊C8—1461no—
433C22⋊C8—1461no—
434C22⋊C8—1461no—
435C22⋊C8—1461no—
436C22⋊C8—1461no—
437C22⋊C8—1461no—
438C22⋊C8—1461no—
439C22⋊C8C2×#6440811yescharacteristic
440#97—731no—
441#97—731no—
442#97—1461no—
443C4×C8—731no—
444C4×C8—731no—
445C4×C8—731no—
446C4×C8—731no—
447C4×C8—731no—
448C4×C8—731no—
449C4×C8—1461no—
450C4×C8—1461no—
451C24—21no—
452C2×Q8—731no—
453C2×D4—22no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—41no—
457C2×D4C22×#6440811yescharacteristic
458C2×D4C22×#6440811yescharacteristic
459C2×D4C22×#6440811yescharacteristic
460C22×C4—2922no—
461C22×C4—2922no—
462C22×C4—731no—
463C22×C4—731no—
464C22×C4—1461no—
465C22×C4—1461no—
466C22×C4—1461no—
467C22×C4—1461no—
468C22×C4—1461no—
469C22×C4—1461no—
470C22×C4C22×#6440811yescharacteristic
471SD16—2921no—
472SD16—1462no—
473D8—22no—
474D8—41no—
475M16—2922no—
476M16—2922no—
477M16—2922no—
478M16—2922no—
479M16—2921no—
480M16—1462no—
481C2×C8—2922no—
482C2×C8—2922no—
483C2×C8—2922no—
484C2×C8—731no—
485C2×C8—731no—
486C2×C8—731no—
487C2×C8—731no—
488C2×C8—731no—
489C2×C8—731no—
490C2×C8—1461no—
491C2×C8—1461no—
492C2×C8—1461no—
493C2×C8—1461no—
494C2×C8—1461no—
495C2×C8—1461no—
496C2×C8C22×#6440811yescharacteristic
497C2×C8C22×#6440811yescharacteristic
498C4⋊C4—731no—
499C4⋊C4—731no—
500C4⋊C4—731no—
501C4⋊C4—1461no—
502C4⋊C4—1461no—
503C4⋊C4—1461no—
504C4⋊C4—1461no—
505C4⋊C4—1461no—
506C4⋊C4C22×#6440811yescharacteristic
507C22⋊C4—21no—
508C22⋊C4—1461no—
509C22⋊C4—1461no—
510C22⋊C4—1461no—
511C22⋊C4—1461no—
512C22⋊C4—1461no—
513C22⋊C4—1461no—
514C22⋊C4—1461no—
515C22⋊C4—1461no—
516C42—731no—
517C42—731no—
518C42—731no—
519C42—731no—
520C42—1461no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—42no—
525C23—42no—
526C23—81no—
527C23D4×#6440811yescharacteristic
528Q8—1461no—
529Q8—1461no—
530D4—21no—
531D4—21no—
532D4—21no—
533D4—21no—
534D4—21no—
535D4—42no—
536D4—81no—
537D4#1455667812yes—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—2922no—
542C2×C4—2921no—
543C2×C4—2921no—
544C2×C4—2921no—
545C2×C4—2921no—
546C2×C4—2921no—
547C2×C4—2921no—
548C2×C4—2922no—
549C2×C4—2922no—
550C2×C4—2921no—
551C2×C4—731no—
552C2×C4—731no—
553C2×C4—731no—
554C2×C4—731no—
555C2×C4—1461no—
556C2×C4—1461no—
557C2×C4—1461no—
558C2×C4—1461no—
559C2×C4—1461no—
560C2×C4—1461no—
561C2×C4D4×#6440811yescharacteristic
562C2×C4C23×#6440811yescharacteristic
563C8—21no—
564C8—21no—
565C8—2921no—
566C8—2921no—
567C8—2921no—
568C8—2921no—
569C8—2921no—
570C8—2921no—
571C8—732no—
572C8—1461no—
573C8—1461no—
574C8—1461no—
575C8—1461no—
576C8—1461no—
577C22—21no—
578C22—22no—
579C22—21no—
580C22—21no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—84no—
590C22C2×D4×#6440811yescharacteristic
591C4—21no—
592C4—41no—
593C4—2921no—
594C4—2921no—
595C4—2921no—
596C4—2921no—
597C4—731no—
598C4—731no—
599C4—1461no—
600C4—1461no—
601C4—1461no—
602C4C2×#1455667811yescharacteristic
603C4#1798966511yescharacteristic
604C2—41no—
605C2—42no—
606C2—81no—
607C2—81no—
608C2#2262666311yescharacteristic
609C2AGL(1,ℤ/8)×#6440811yescharacteristic
610C2#2262670711yescharacteristic
subgroupG/H
centerC22C2×D4×#64408
commutatorC2×C4×C73C23×C4
FrattiniC2×C4C23×#64408
Fitting, prime radical#447×C73C4
socleC22×C73C2×C4×D4
cosocleC4×Dic73C24
hypercenter#447#64408
hypocenterC73#18008
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. #22626762C2semidirect
2. D4×#11924037C2semidirect
3. #22625572C2semidirect
4. #22625564C2semidirect
5. #22625552C2semidirect
6. #22625546C2semidirect
7. #22625541C2semidirect
8. #22625532C2semidirect
9. #22625527C2semidirect
10. #22625514C2semidirect
11. #22625507C2semidirect
12. #22625286C2semidirect
13. #22625230C2semidirect
14. #22625148C2semidirect
17. #17989637C22semidirect
18. #17989636C22semidirect
22. #17989622C22semidirect
28. D4×#10596833C22semidirect
36. #17989451C22semidirect
37. #17989450C22semidirect
38. #17989449C22semidirect
39. #17989448C22semidirect
40. #17989437C22semidirect
41. #17989437C22semidirect
46. D4⋊#10596833C22semidirect
47. D4⋊#10596833C22semidirect
48. #17989429C22semidirect
49. #17989429C22semidirect
50. #17989428C22semidirect
51. C8×#10596833C22semidirect
321. C73#18008semidirect

Maximal subgroups

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

subgroupindex
#226267622
D4×#119240372
#226255722
#226255642
#226255522
#226255462
#226255412
#226255322
#226255272
#226255142
#226255072
#226252862
#226252302
#226251482
#226199732
#1800873