Finite Group Databasebeta

← #10765298

Subgroups — #10765298

Order 768.

598 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83352C211yesmaximal, characteristic
2#83331C211yesmaximal, characteristic, complemented
3#79013C211yesmaximal, characteristic
4C3×#1481C211yesmaximal, characteristic, complemented
5#70763C211yesmaximal, characteristic
6#70744C211yesmaximal, characteristic, complemented
7#70423C211yesmaximal, characteristic, complemented
8#68222C211yesmaximal, characteristic, complemented
9#68220C211yesmaximal, characteristic, complemented
10#68219C211yesmaximal, characteristic, complemented
11#68216C211yesmaximal, characteristic, complemented
12#68215C211yesmaximal, characteristic, complemented
13#68214C211yesmaximal, characteristic, complemented
14#68213C211yesmaximal, characteristic, complemented
15#68212C211yesmaximal, characteristic, complemented
16#17921—31nomaximal, Sylow
17C3×#533C2211yescharacteristic
18C4×#235C2211yescharacteristic
19C3×#431C2211yescharacteristic
20C2×C3×C4⋊C8C2211yescharacteristic
21C8×#67—21no—
22C8×#67—21no—
23#5094C411yescharacteristic
24#5090C411yescharacteristic
25#5087C411yescharacteristic
26#5083C411yescharacteristic
27#5047C411yescharacteristic
28#5047C411yescharacteristic
29#5045C411yescharacteristic
30#5045C411yescharacteristic
31C4×#216C2211yescharacteristic
32C4×#216C2211yescharacteristic
33C4×#215C2211yescharacteristic
34C4×#215C2211yescharacteristic
35#5024—21no—
36#5024—21no—
37#5024C2211yescharacteristic
38#5024C2211yescharacteristic
39Q8×C3⋊C8C2211yescharacteristic
40Q8×C3⋊C8C2211yescharacteristic
41#5000—21no—
42#5000—21no—
43#4926C2211yescharacteristic
44#4925C2211yescharacteristic
45#4925C2211yescharacteristic
46#4834C2211yescharacteristic
47#4834C2211yescharacteristic
48#4833—21no—
49D3×C4⋊C8—21no—
50#4830C2211yescharacteristic
51#4830C2211yescharacteristic
52C3×Q8⋊C8C2211yescharacteristic
53C3×Q8⋊C8C2211yescharacteristic
54C3×Q8⋊C8C2211yescharacteristic
55C3×Q8⋊C8C2211yescharacteristic
56#4480C2211yescharacteristic
57#4480C2211yescharacteristic
58#4479C2211yescharacteristic
59#4479C2211yescharacteristic
60#4459C2211yescharacteristic
61#4459C2211yescharacteristic
62#4456C2211yescharacteristic
63#4456C2211yescharacteristic
64#4455C2211yescharacteristic
65#4454C2211yescharacteristic
66#4454C2211yescharacteristic
67C8×C3⋊C8C2211yescharacteristic
68#3107—31no—
69#2988—31no—
70#2985—31no—
71#1587—31no—
72#1585—31no—
73#1584—31no—
74#1582—31no—
75#1579—31no—
76SD16⋊C8—31no—
77C8×Q16—31no—
78C8×SD16—31no—
79#1573—31no—
80#1488—31no—
81#1481D311yescharacteristic, complemented
82#1454—31no—
83C2×#235C2×C411yescharacteristic
84C22×C3×C8—21no—
85C3×C4⋊Q8C2×C411yescharacteristic
86C3×C4⋊Q8C2×C411yescharacteristic
87C3×#125—21no—
88C3×C22⋊Q8—21no—
89C3×C22⋊Q8C2×C411yescharacteristic
90C3×C22⋊Q8C2×C411yescharacteristic
91C3×C4×Q8—21no—
92C3×C4×Q8C2311yescharacteristic
93C3×C4×Q8C2311yescharacteristic
94C3×#117—21no—
95C2×C3×C42C2311yescharacteristic
96C2×#216C2×C411yescharacteristic
97C2×#216C2×C411yescharacteristic
98C2×#215C2×C411yescharacteristic
99C2×#215C2×C411yescharacteristic
100C4×#67—21no—
101C4×#67—21no—
102#926C2×C411yescharacteristic
103C2×D3×C8—21no—
104C2×D3×C8—21no—
105C4×D12C2×C411yescharacteristic
106C4×D12C2311yescharacteristic
107#872—21no—
108C42×D3—21no—
109C4×Dic6C2×C411yescharacteristic
110C4×Dic6C2311yescharacteristic
111C3×C4⋊C8—21no—
112C3×C4⋊C8—21no—
113C3×C4⋊C8C2311yescharacteristic
114C3×C4⋊C8C2311yescharacteristic
115C3×C22⋊C8—21no—
116C3×C4×C8C2311yescharacteristic
117C3×C4×C8C2311yescharacteristic
118#835C2×C411yescharacteristic
119#835C2×C411yescharacteristic
120#835C2×C411yescharacteristic
121#835C2×C411yescharacteristic
122#830—21no—
123#830C2×C411yescharacteristic
124#830C2×C411yescharacteristic
125#830D411yescharacteristic
126#830D411yescharacteristic
127(C2×D3)⋊C8—21no—
128(C2×D3)⋊C8—21no—
129Dic3⋊C8—21no—
130Dic3⋊C8—21no—
131C8×Dic3—21no—
132C8×Dic3—21no—
133#810C2×C411yescharacteristic
134#810C2×C411yescharacteristic
135#809C2×C411yescharacteristic
136#809C2×C411yescharacteristic
137#808C2×C411yescharacteristic
138#808C2×C411yescharacteristic
139#807C2×C411yescharacteristic
140#807C2×C411yescharacteristic
141C4⋊(C3⋊C8)—21no—
142C4⋊(C3⋊C8)—21no—
143C4⋊(C3⋊C8)—21no—
144C4⋊(C3⋊C8)—21no—
145C4⋊(C3⋊C8)C2×C411yescharacteristic
146C4⋊(C3⋊C8)C2×C411yescharacteristic
147C4⋊(C3⋊C8)D411yescharacteristic
148C4⋊(C3⋊C8)D411yescharacteristic
149C4⋊(C3⋊C8)C2311yescharacteristic
150C4⋊(C3⋊C8)C2311yescharacteristic
151#803—21no—
152#803—21no—
153#803—21no—
154C4×C3⋊C8—21no—
155C4×C3⋊C8—21no—
156C4×C3⋊C8C2311yescharacteristic
157C4×C3⋊C8C2311yescharacteristic
158C4×C3⋊C8C2311yescharacteristic
159C4×C3⋊C8C2311yescharacteristic
160#533C2×D311yescharacteristic
161C4×#41—31no—
162#485—31no—
163#485—31no—
164#475—31no—
165#474—31no—
166#462—31no—
167#461—31no—
168#452—31no—
169#452—31no—
170#446—31no—
171#446—31no—
172#446—61no—
173#446—61no—
174C8×Q8—31no—
175C8×Q8—31no—
176C8×Q8—31no—
177C8×Q8—31no—
178C4×Q16—31no—
179C4×Q16—31no—
180C4×SD16—31no—
181C4×SD16—31no—
182C8×D4—31no—
183C8×D4—31no—
184C8×D4—61no—
185C8×D4—61no—
186#433—61no—
187#433—61no—
188#432—31no—
189#431—31no—
190#431—31no—
191#431—61no—
192#431C2×D311yescharacteristic
193C2×C4⋊C8—61no—
194C2×C4⋊C8C2×D311yescharacteristic
195#335—31no—
196#335—31no—
197#334—31no—
198#334—31no—
199Q8⋊C8—31no—
200Q8⋊C8—31no—
201Q8⋊C8C2×D311yescharacteristic
202Q8⋊C8C2×D311yescharacteristic
203Q8⋊C8C2×D311yescharacteristic
204Q8⋊C8C2×D311yescharacteristic
205D4⋊C8—31no—
206D4⋊C8—31no—
207#322—31no—
208#322—31no—
209#322—31no—
210C82—31no—
211C2×C3×Q8—21no—
212C2×C3×Q8C22×C411yescharacteristic
213C2×C3×Q8C22×C411yescharacteristic
214C22×C3×C4C22×C411yescharacteristic
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4#4111yescharacteristic
217C2×#67—21no—
218#235—22no—
219#235—22no—
220C2×D12C22×C411yescharacteristic
221C2×C4×D3—21no—
222C2×C4×D3—21no—
223C2×C4×D3—21no—
224C2×Dic6C22×C411yescharacteristic
225C2×C3×C8—21no—
226C2×C3×C8—21no—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C2×C3×C8—81no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
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×C42—21no—
242C3×C42—21no—
243C3×C42C22×C411yescharacteristic
244C3×C42C2×D411yescharacteristic
245C3×C42#4111yescharacteristic
246C3×C42C2411yescharacteristic
247#217—21no—
248#216—22no—
249#216—22no—
250#215—22no—
251#215—22no—
252(C2×D3)⋊C4—21no—
253(C2×D3)⋊C4—21no—
254C4⋊Dic3C22×C411yescharacteristic
255C4⋊Dic3C22×C411yescharacteristic
256Dic3⋊C4—21no—
257Dic3⋊C4—21no—
258C4×Dic3—21no—
259C4×Dic3—21no—
260C4×Dic3—21no—
261C2×C3⋊C8—21no—
262C2×C3⋊C8—21no—
263C2×C3⋊C8C22×C411yescharacteristic
264C2×C3⋊C8C22×C411yescharacteristic
265C2×C3⋊C8C22×C411yescharacteristic
266C2×C3⋊C8C22×C411yescharacteristic
267C2×C3⋊C8C2×D411yescharacteristic
268C2×C3⋊C8C2×D411yescharacteristic
269C2×C3⋊C8C2×D411yescharacteristic
270C2×C3⋊C8C2×D411yescharacteristic
271D3×C8—81no—
272D3×C8—81no—
273C2×#41—31no—
274C2×Q16—31no—
275C2×Q16—31no—
276C2×SD16—31no—
277C2×SD16—31no—
278C22×C8—21no—
279C22×C8—61no—
280C22×C8—61no—
281C4⋊Q8C4×D311yescharacteristic
282C4⋊Q8C4×D311yescharacteristic
283#125—21no—
284C22⋊Q8—21no—
285C22⋊Q8C4×D311yescharacteristic
286C22⋊Q8C4×D311yescharacteristic
287C4×Q8—21no—
288C4×Q8—31no—
289C4×Q8—31no—
290C4×Q8C22×D311yescharacteristic
291C4×Q8C22×D311yescharacteristic
292C4×D4—31no—
293C4×D4—31no—
294C4×D4—61no—
295C4×D4—61no—
296#117—21no—
297#117—31no—
298#117—61no—
299C2×C42—61no—
300C2×C42C22×D311yescharacteristic
301#107—31no—
302#107—31no—
303#106—31no—
304#106—31no—
305C4⋊C8—21no—
306C4⋊C8—21no—
307C4⋊C8—31no—
308C4⋊C8—31no—
309C4⋊C8—31no—
310C4⋊C8—31no—
311C4⋊C8—31no—
312C4⋊C8—31no—
313C4⋊C8—61no—
314C4⋊C8—61no—
315C4⋊C8—61no—
316C4⋊C8—61no—
317C4⋊C8—61no—
318C4⋊C8—61no—
319C4⋊C8C22×D311yescharacteristic
320C4⋊C8C22×D311yescharacteristic
321#103—31no—
322#103—31no—
323#103—31no—
324#103—31no—
325#103—31no—
326#103—31no—
327#102—31no—
328#102—31no—
329C22⋊C8—21no—
330C22⋊C8—31no—
331C22⋊C8—31no—
332C22⋊C8—31no—
333C22⋊C8—31no—
334C22⋊C8—61no—
335C22⋊C8—61no—
336C22⋊C8—61no—
337#97—61no—
338#97—61no—
339#97—61no—
340C4×C8—31no—
341C4×C8—31no—
342C4×C8—31no—
343C4×C8—31no—
344C4×C8—61no—
345C4×C8—61no—
346C4×C8—61no—
347C4×C8—61no—
348C4×C8C22×D311yescharacteristic
349C4×C8C22×D311yescharacteristic
350C23×C3C4×D411yescharacteristic
351C22×D3—21no—
352C3×Q8—21no—
353C3×Q8—21no—
354C3×Q8—21no—
355C3×Q8—21no—
356C3×Q8—81no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—41no—
366C2×C3×C4C4×D411yescharacteristic
367C2×C3×C4C4×D411yescharacteristic
368C2×C3×C4C4×D411yescharacteristic
369C2×C3×C4C23×C411yescharacteristic
370C2×C3×C4C22×D411yescharacteristic
371C2×C3×C4C2×#4111yescharacteristic
372#67—24no—
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—21no—
376D12—21no—
377D12#13112yes—
378C4×D3—24no—
379C4×D3—41no—
380C4×D3—41no—
381C4×D3—81no—
382Dic6—21no—
383Dic6#13112yes—
384C3×C8—41no—
385C3×C8—41no—
386C3×C8—41no—
387C3×C8—41no—
388C3⋊C8—21no—
389C3⋊C8—21no—
390C3⋊C8—21no—
391C3⋊C8—21no—
392C3⋊C8—21no—
393C3⋊C8—21no—
394C3⋊C8—21no—
395C3⋊C8—21no—
396C3⋊C8—41no—
397C3⋊C8—41no—
398#41—62no—
399#41—62no—
400C2×Q8—21no—
401C2×Q8—31no—
402C2×Q8C2×C4×D311yescharacteristic
403C2×Q8C2×C4×D311yescharacteristic
404C2×D4—31no—
405C2×D4—61no—
406C22×C4—61no—
407C22×C4—61no—
408C22×C4—61no—
409C22×C4C2×C4×D311yescharacteristic
410C22×C4D3×D411yescharacteristic
411C22×C4#23711yescharacteristic
412Q16—62no—
413Q16—62no—
414SD16—62no—
415SD16—62no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—31no—
421C2×C8—31no—
422C2×C8—31no—
423C2×C8—31no—
424C2×C8—31no—
425C2×C8—31no—
426C2×C8—31no—
427C2×C8—31no—
428C2×C8—61no—
429C2×C8—61no—
430C2×C8—81no—
431C2×C8—241no—
432C2×C8—241no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—31no—
441C4⋊C4—31no—
442C4⋊C4—61no—
443C4⋊C4—61no—
444C4⋊C4C2×C4×D311yescharacteristic
445C4⋊C4C2×C4×D311yescharacteristic
446C22⋊C4—21no—
447C22⋊C4—21no—
448C22⋊C4—61no—
449C22⋊C4—61no—
450C22⋊C4—61no—
451C42—21no—
452C42—21no—
453C42—61no—
454C42—61no—
455C42—61no—
456C42C2×C4×D311yescharacteristic
457C42D3×D411yescharacteristic
458C42#23711yescharacteristic
459C42C23×D311yescharacteristic
460C22×C3—21no—
461C22×C3—21no—
462C22×C3—21no—
463C22×C3C2×C4×D411yescharacteristic
464C2×D3—22no—
465C2×D3—41no—
466C2×D3—41no—
467C3×C4—21no—
468C3×C4—21no—
469C3×C4—21no—
470C3×C4—21no—
471C3×C4—41no—
472C3×C4—41no—
473C3×C4—41no—
474C3×C4—41no—
475C3×C4C2×#13111yescharacteristic
476C3×C4C2×#13111yescharacteristic
477C3×C4#57711yescharacteristic
478C3×C4#57811yescharacteristic
479Dic3—22no—
480Dic3—41no—
481Dic3—41no—
482C23—61no—
483C23#67⋊C411yescharacteristic
484Q8—21no—
485Q8—21no—
486Q8—21no—
487Q8—21no—
488Q8—32no—
489Q8—61no—
490Q8—81no—
491D4—32no—
492D4—64no—
493D4—61no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—41no—
503C2×C4—64no—
504C2×C4—61no—
505C2×C4—61no—
506C2×C4—61no—
507C2×C4—121no—
508C2×C4—121no—
509C2×C4—241no—
510C2×C4#67⋊C411yescharacteristic
511C2×C4#67⋊C411yescharacteristic
512C2×C4#67⋊C411yescharacteristic
513C2×C4C22×C4×D311yescharacteristic
514C2×C4C2×D3×D411yescharacteristic
515C2×C4C2×#23711yescharacteristic
516C8—41no—
517C8—41no—
518C8—41no—
519C8—41no—
520C8—61no—
521C8—61no—
522C8—61no—
523C8—61no—
524C8—61no—
525C8—61no—
526C8—61no—
527C8—61no—
528C8—121no—
529C8—121no—
530C2×C3—41no—
531C2×C3#292911yescharacteristic
532C2×C3#294711yescharacteristic
533C2×C3#295711yescharacteristic
534D3—42no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—62no—
539C22—121no—
540C22—121no—
541C22C2×#67⋊C411yescharacteristic
542C4—21no—
543C4—21no—
544C4—21no—
545C4—21no—
546C4—41no—
547C4—41no—
548C4—41no—
549C4—41no—
550C4—62no—
551C4—121no—
552C4—121no—
553C4D3×#13111yescharacteristic
554C4D3×#13111yescharacteristic
555C4#576211yescharacteristic
556C4#576611yescharacteristic
557C3#1792111yescharacteristic, complemented, Sylow
558C2—41no—
559C2—122no—
560C2#7865511yescharacteristic
561C2#7877411yescharacteristic
562C2#7899011yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1481C2
socleC22×C3C2×C4×D4
prime radicalC2×#235C2×C4
cosocleC3×C42C24
hypercenter#1481D3
hypocenterC3#17921
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1481

Complements

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

NH ≅ G/Ntype
2. #83331C2semidirect
4. C3×#1481C2semidirect
6. #70744C2semidirect
7. #70423C2semidirect
8. #68222C2semidirect
9. #68220C2semidirect
10. #68219C2semidirect
11. #68216C2semidirect
12. #68215C2semidirect
13. #68214C2semidirect
14. #68213C2semidirect
15. #68212C2semidirect
81. #1481D3semidirect
557. C3#17921semidirect

Maximal subgroups

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

subgroupindex
#833522
#833312
#790132
C3×#14812
#707632
#707442
#704232
#682222
#682202
#682192
#682162
#682152
#682142
#682132
#682122
#179213