Finite Group Databasebeta

← #10765675

Subgroups — #10765675

Order 768.

590 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2989C211yesmaximal, characteristic, complemented
2#82528C211yesmaximal, characteristic
3#79295C211yesmaximal, characteristic, complemented
4#71641C211yesmaximal, characteristic, complemented
5#70956C211yesmaximal, characteristic, complemented
6#70955C211yesmaximal, characteristic, complemented
7#70954C211yesmaximal, characteristic, complemented
8#70951C211yesmaximal, characteristic, complemented
9#70865C211yesmaximal, characteristic, complemented
10#70862C211yesmaximal, characteristic, complemented
11#70860C211yesmaximal, characteristic, complemented
12#70859C211yesmaximal, characteristic, complemented
13#70743C211yesmaximal, characteristic, complemented
14#70427C211yesmaximal, characteristic, complemented
15#70422C211yesmaximal, characteristic, complemented
16#17954—31nomaximal, Sylow
17C3×C4×#41C2211yescharacteristic
18#5479C2211yescharacteristic
19C3×#446C2211yescharacteristic
20C3×#446C2211yescharacteristic
21C3×#436C2211yescharacteristic
22C3×#436C2211yescharacteristic
23C3×#435—21no—
24C3×#435—21no—
25C3×#432—21no—
26C3×#431C2211yescharacteristic
27C3×#423—21no—
28C2×C3×C4⋊C8C2211yescharacteristic
29#5239C411yescharacteristic
30#5238C411yescharacteristic
31C4×#216C2211yescharacteristic
32#5028C411yescharacteristic
33C4×#215C2211yescharacteristic
34#5022C411yescharacteristic
35C4×#214C2211yescharacteristic
36#5016C411yescharacteristic
37C4×#213C2211yescharacteristic
38#5009C411yescharacteristic
39#5003C411yescharacteristic
40#4996C411yescharacteristic
41#4925C2211yescharacteristic
42#4924C2211yescharacteristic
43C4×#208C2211yescharacteristic
44#4840—21no—
45#4839—21no—
46#4729—21no—
47#4728—21no—
48#4688C2211yescharacteristic
49C8×D12C2211yescharacteristic
50#4679C2211yescharacteristic
51C8×Dic6C2211yescharacteristic
52#4535C2211yescharacteristic
53#4535C2211yescharacteristic
54D4⋊(C3⋊C8)C2211yescharacteristic
55D4⋊(C3⋊C8)C2211yescharacteristic
56#4484C2211yescharacteristic
57#4484C2211yescharacteristic
58#4483C2211yescharacteristic
59#4483C2211yescharacteristic
60#4479C2211yescharacteristic
61#4479C2211yescharacteristic
62#4479C2211yescharacteristic
63#4479C2211yescharacteristic
64#4455C2211yescharacteristic
65#4454C2211yescharacteristic
66#4454C2211yescharacteristic
67C8×C3⋊C8C2211yescharacteristic
68#3102—31no—
69#2993—31no—
70#2989D311yescharacteristic, complemented
71#1592—31no—
72#1592—31no—
73#1591—31no—
74#1589—31no—
75#1588—31no—
76#1585—31no—
77#1584—31no—
78#1581—31no—
79#1571—31no—
80#1489—31no—
81#1483—31no—
82#1453—31no—
83C2×C3×#41C2×C411yescharacteristic
84C2×C3×M16—21no—
85C2×C3×M16—21no—
86C22×C3×C8—21no—
87C3×C4×Q8C2×C411yescharacteristic
88C3×C4×Q8C2311yescharacteristic
89C3×C4×D4—21no—
90C3×C4×D4—21no—
91C3×C4×D4C2×C411yescharacteristic
92C3×C4×D4C2311yescharacteristic
93C3×#117—21no—
94C3×#117C2×C411yescharacteristic
95C2×C3×C42—21no—
96C2×C3×C42C2311yescharacteristic
97C2×#216C2×C411yescharacteristic
98C2×#215C2×C411yescharacteristic
99C2×#214C2×C411yescharacteristic
100C2×#213C2×C411yescharacteristic
101#930C2×C411yescharacteristic
102#929—21no—
103C4×#67—21no—
104#924C2×C411yescharacteristic
105C2×#208—21no—
106C2×#208—21no—
107C2×ΓL(1,𝔽25)—21no—
108C2×D3×C8—21no—
109#876—21no—
110#875C2×C411yescharacteristic
111C4×D12C2311yescharacteristic
112#872—21no—
113#870C2×C411yescharacteristic
114C4×Dic6C2311yescharacteristic
115C3×C4⋊C8—21no—
116C3×C4⋊C8—21no—
117C3×C4⋊C8—21no—
118C3×C4⋊C8D411yescharacteristic
119C3×C4⋊C8D411yescharacteristic
120C3×C4⋊C8C2311yescharacteristic
121C3×C4⋊C8C2311yescharacteristic
122C3×C22⋊C8—21no—
123C3×C22⋊C8—21no—
124C3×C22⋊C8D411yescharacteristic
125C3×C22⋊C8D411yescharacteristic
126C3×#97—21no—
127C3×#97—21no—
128C3×C4×C8C2311yescharacteristic
129C3×C4×C8C2311yescharacteristic
130#835C2×C411yescharacteristic
131#835C2×C411yescharacteristic
132#832C2×C411yescharacteristic
133#832C2×C411yescharacteristic
134#830C2×C411yescharacteristic
135#830C2×C411yescharacteristic
136(C2×D3)⋊C8—21no—
137(C2×D3)⋊C8—21no—
138#815—21no—
139Dic3⋊C8—21no—
140Dic3⋊C8—21no—
141C8×Dic3—21no—
142#810C2×C411yescharacteristic
143#810C2×C411yescharacteristic
144#809C2×C411yescharacteristic
145#809C2×C411yescharacteristic
146#808C2×C411yescharacteristic
147#808C2×C411yescharacteristic
148#807C2×C411yescharacteristic
149#807C2×C411yescharacteristic
150C4⋊(C3⋊C8)—21no—
151C4⋊(C3⋊C8)C2×C411yescharacteristic
152C4⋊(C3⋊C8)C2×C411yescharacteristic
153C4⋊(C3⋊C8)C2311yescharacteristic
154C4⋊(C3⋊C8)C2311yescharacteristic
155#803—21no—
156C4×C3⋊C8C2311yescharacteristic
157C4×C3⋊C8C2311yescharacteristic
158C4×C3⋊C8C2311yescharacteristic
159C4×C3⋊C8C2311yescharacteristic
160#529—31no—
161C4×#41C2×D311yescharacteristic
162#485—31no—
163#482—31no—
164#479—31no—
165#478—31no—
166#464—31no—
167#463—31no—
168#452—31no—
169#449—31no—
170#446—31no—
171#446C2×D311yescharacteristic
172#446C2×D311yescharacteristic
173C8×Q8—31no—
174C4×Q16—31no—
175C4×SD16—31no—
176C4×SD16—31no—
177C4×D8—31no—
178#436—31no—
179#436—61no—
180#436C2×D311yescharacteristic
181#436C2×D311yescharacteristic
182#435—21no—
183#435—21no—
184#435—61no—
185C8×D4—31no—
186#433—61no—
187#432—21no—
188#431—31no—
189#431C2×D311yescharacteristic
190#424—61no—
191#423—21no—
192#423—31no—
193C2×C4⋊C8C2×D311yescharacteristic
194C4×M16—31no—
195#335—31no—
196#335—31no—
197#335—31no—
198#335—31no—
199Q8⋊C8—31no—
200Q8⋊C8—31no—
201Q8⋊C8—31no—
202Q8⋊C8—31no—
203D4⋊C8—31no—
204D4⋊C8—31no—
205D4⋊C8—31no—
206D4⋊C8—31no—
207#322—31no—
208#322—31no—
209#322—31no—
210C82—31no—
211C3×#41—41no—
212C3×#41—41no—
213C2×C3×Q8C22×C411yescharacteristic
214C2×C3×D4—21no—
215C2×C3×D4C22×C411yescharacteristic
216C22×C3×C4—21no—
217C22×C3×C4—21no—
218C22×C3×C4—21no—
219C22×C3×C4C22×C411yescharacteristic
220C22×C3×C4C2×D411yescharacteristic
221C22×C3×C4#4111yescharacteristic
222C2×#67—21no—
223C2×D12C22×C411yescharacteristic
224C2×C4×D3—21no—
225C2×Dic6C22×C411yescharacteristic
226C3×M16—81no—
227C3×M16—81no—
228C2×C3×C8—21no—
229C2×C3×C8—21no—
230C2×C3×C8—81no—
231C2×C3×C8C2×D411yescharacteristic
232C2×C3×C8C2×D411yescharacteristic
233C2×C3×C8C2×D411yescharacteristic
234C2×C3×C8C2×D411yescharacteristic
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4C22×C411yescharacteristic
238C3×C4⋊C4C22×C411yescharacteristic
239C3×C22⋊C4—21no—
240C3×C22⋊C4—21no—
241C3×C22⋊C4—21no—
242C3×C42—21no—
243C3×C42—21no—
244C3×C42—21no—
245C3×C42C22×C411yescharacteristic
246C3×C42C2×D411yescharacteristic
247C3×C42#4111yescharacteristic
248C3×C42C2411yescharacteristic
249#217—21no—
250#216—41no—
251#215—41no—
252#214—41no—
253#213—41no—
254(C2×D3)⋊C4—21no—
255(C2×D3)⋊C4—21no—
256(C2×D3)⋊C4—21no—
257(C2×D3)⋊C4—21no—
258C4⋊Dic3C22×C411yescharacteristic
259C4⋊Dic3C22×C411yescharacteristic
260Dic3⋊C4—21no—
261Dic3⋊C4—21no—
262Dic3⋊C4—21no—
263Dic3⋊C4—21no—
264C4×Dic3—21no—
265#208—42no—
266#208—81no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8C22×C411yescharacteristic
272C2×C3⋊C8C22×C411yescharacteristic
273C2×C3⋊C8C22×C411yescharacteristic
274C2×C3⋊C8C22×C411yescharacteristic
275ΓL(1,𝔽25)—81no—
276D3×C8—81no—
277C2×#41C4×D311yescharacteristic
278C2×Q16—31no—
279C2×SD16—31no—
280C2×SD16—31no—
281C2×D8—31no—
282C2×M16—21no—
283C2×M16—21no—
284C2×M16—61no—
285C2×M16—61no—
286C2×M16—61no—
287C22×C8—21no—
288C22×C8—61no—
289#126—61no—
290#125—31no—
291#124—31no—
292#123—61no—
293C22⋊Q8—31no—
294#121—31no—
295C4×Q8—31no—
296C4×Q8C4×D311yescharacteristic
297C4×Q8C22×D311yescharacteristic
298C4×D4—21no—
299C4×D4—21no—
300C4×D4—31no—
301C4×D4—61no—
302C4×D4C4×D311yescharacteristic
303C4×D4C22×D311yescharacteristic
304#117—21no—
305#117—61no—
306#117C4×D311yescharacteristic
307C2×C42—21no—
308C2×C42C22×D311yescharacteristic
309#107—31no—
310#107—31no—
311#106—31no—
312#106—31no—
313C4⋊C8—21no—
314C4⋊C8—21no—
315C4⋊C8—21no—
316C4⋊C8—31no—
317C4⋊C8—31no—
318C4⋊C8—31no—
319C4⋊C8—31no—
320C4⋊C8—61no—
321C4⋊C8—61no—
322C4⋊C8—61no—
323C4⋊C8#6711yescharacteristic
324C4⋊C8#6711yescharacteristic
325C4⋊C8C22×D311yescharacteristic
326C4⋊C8C22×D311yescharacteristic
327#103—31no—
328#103—31no—
329#103—31no—
330#103—31no—
331#102—31no—
332#102—31no—
333#102—31no—
334#102—31no—
335C22⋊C8—21no—
336C22⋊C8—21no—
337C22⋊C8—31no—
338C22⋊C8—31no—
339C22⋊C8—61no—
340C22⋊C8—61no—
341C22⋊C8#6711yescharacteristic
342C22⋊C8#6711yescharacteristic
343#97—21no—
344#97—21no—
345#97—61no—
346#97—61no—
347C4×C8—31no—
348C4×C8—31no—
349C4×C8—31no—
350C4×C8—31no—
351C4×C8—61no—
352C4×C8C22×D311yescharacteristic
353C4×C8C22×D311yescharacteristic
354C23×C3—21no—
355C23×C3C4×D411yescharacteristic
356C22×D3—21no—
357C3×Q8—21no—
358C3×Q8—21no—
359C3×D4—21no—
360C3×D4—21no—
361C3×D4—42no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—41no—
370C2×C3×C4—42no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—81no—
374C2×C3×C4C4×D411yescharacteristic
375C2×C3×C4C4×D411yescharacteristic
376C2×C3×C4C4×D411yescharacteristic
377C2×C3×C4C23×C411yescharacteristic
378C2×C3×C4C22×D411yescharacteristic
379C2×C3×C4C2×#4111yescharacteristic
380#67—81no—
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—21no—
384D12—21no—
385D12—21no—
386C4×D3—41no—
387C4×D3—41no—
388Dic6—21no—
389Dic6—21no—
390C3×C8—21no—
391C3×C8—21no—
392C3×C8—21no—
393C3×C8—21no—
394C3×C8—41no—
395C3×C8—41no—
396C3⋊C8—22no—
397C3⋊C8—21no—
398C3⋊C8—21no—
399C3⋊C8—21no—
400C3⋊C8—21no—
401C3⋊C8—41no—
402C3⋊C8—41no—
403C3⋊C8—41no—
404#41—41no—
405#41—41no—
406C2×Q8—31no—
407C2×Q8C2×C4×D311yescharacteristic
408C2×D4—21no—
409C2×D4—31no—
410C2×D4—61no—
411C2×D4C2×C4×D311yescharacteristic
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—61no—
416C22×C4C2×C4×D311yescharacteristic
417C22×C4#23511yescharacteristic
418C22×C4C2×#6711yescharacteristic
419Q16—121no—
420SD16—121no—
421SD16—121no—
422D8—121no—
423M16—81no—
424M16—81no—
425M16—122no—
426M16—241no—
427M16—241no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—31no—
431C2×C8—31no—
432C2×C8—31no—
433C2×C8—31no—
434C2×C8—61no—
435C2×C8—61no—
436C2×C8—61no—
437C2×C8—61no—
438C2×C8—81no—
439C2×C8—241no—
440C2×C8C2×#6711yescharacteristic
441C2×C8C2×#6711yescharacteristic
442C2×C8C2×#6711yescharacteristic
443C2×C8C2×#6711yescharacteristic
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—31no—
447C4⋊C4—31no—
448C4⋊C4—61no—
449C4⋊C4—61no—
450C4⋊C4—61no—
451C4⋊C4—61no—
452C4⋊C4C2×C4×D311yescharacteristic
453C4⋊C4C2×C4×D311yescharacteristic
454C22⋊C4—21no—
455C22⋊C4—21no—
456C22⋊C4—21no—
457C22⋊C4—61no—
458C22⋊C4—61no—
459C22⋊C4—61no—
460C22⋊C4—61no—
461C22⋊C4—61no—
462C42—21no—
463C42—21no—
464C42—21no—
465C42—61no—
466C42C2×C4×D311yescharacteristic
467C42#23511yescharacteristic
468C42C2×#6711yescharacteristic
469C42C23×D311yescharacteristic
470C22×C3—21no—
471C22×C3—21no—
472C22×C3—21no—
473C22×C3—41no—
474C22×C3—41no—
475C22×C3—41no—
476C22×C3C2×C4×D411yescharacteristic
477C2×D3—41no—
478C2×D3—41no—
479C2×D3—41no—
480C3×C4—21no—
481C3×C4—21no—
482C3×C4—21no—
483C3×C4—21no—
484C3×C4—41no—
485C3×C4—41no—
486C3×C4—41no—
487C3×C4#56811yescharacteristic
488C3×C4#56811yescharacteristic
489C3×C4#57611yescharacteristic
490C3×C4#57811yescharacteristic
491Dic3—41no—
492Dic3—41no—
493Dic3—41no—
494C23—21no—
495C23—61no—
496C23C4×#6711yescharacteristic
497Q8—21no—
498Q8—21no—
499Q8—61no—
500Q8—61no—
501D4—21no—
502D4—21no—
503D4—42no—
504D4—61no—
505D4—61no—
506D4—241no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—61no—
519C2×C4—61no—
520C2×C4—61no—
521C2×C4—81no—
522C2×C4—121no—
523C2×C4—121no—
524C2×C4C4×#6711yescharacteristic
525C2×C4C4×#6711yescharacteristic
526C2×C4C4×#6711yescharacteristic
527C2×C4C22×C4×D311yescharacteristic
528C2×C4C2×#23511yescharacteristic
529C2×C4C22×#6711yescharacteristic
530C8—21no—
531C8—21no—
532C8—21no—
533C8—21no—
534C8—41no—
535C8—41no—
536C8—62no—
537C8—61no—
538C8—61no—
539C8—61no—
540C8—61no—
541C8—121no—
542C8—121no—
543C8—121no—
544C2×C3—41no—
545C2×C3—81no—
546C2×C3#293011yescharacteristic
547C2×C3#295111yescharacteristic
548C2×C3#295611yescharacteristic
549D3—81no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—121no—
557C22—121no—
558C22—121no—
559C22C2×C4×#6711yescharacteristic
560C4—21no—
561C4—21no—
562C4—21no—
563C4—21no—
564C4—41no—
565C4—41no—
566C4—41no—
567C4—121no—
568C4—121no—
569C4—121no—
570C4#574511yescharacteristic
571C4#574511yescharacteristic
572C4#583511yescharacteristic
573C4#583811yescharacteristic
574C3#1795411yescharacteristic, complemented, Sylow
575C2—41no—
576C2—81no—
577C2—241no—
578C2#8255711yescharacteristic
579C2#8375611yescharacteristic
580C2#8381111yescharacteristic
subgroupG/H
centerC2×C4C22×#67
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#2989C2
socleC22×C3C2×C4×D4
prime radical#5238C4
cosocleC3×C42C24
hypercenter#2989D3
hypocenterC3#17954
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ #431 ◁ #2989

Complements

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

NH ≅ G/Ntype
1. C3×#2989C2semidirect
3. #79295C2semidirect
4. #71641C2semidirect
5. #70956C2semidirect
6. #70955C2semidirect
7. #70954C2semidirect
8. #70951C2semidirect
9. #70865C2semidirect
10. #70862C2semidirect
11. #70860C2semidirect
12. #70859C2semidirect
13. #70743C2semidirect
14. #70427C2semidirect
15. #70422C2semidirect
70. #2989D3semidirect
574. C3#17954semidirect

Maximal subgroups

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

subgroupindex
C3×#29892
#825282
#792952
#716412
#709562
#709552
#709542
#709512
#708652
#708622
#708602
#708592
#707432
#704272
#704222
#179543