Finite Group Databasebeta

← #10876037

Subgroups — #10876037

Order 768.

746 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#76203C211yesmaximal, characteristic
2#74332C211yesmaximal, characteristic, complemented
3C3×#2100C211yesmaximal, characteristic, complemented
4#69635C211yesmaximal, characteristic, complemented
5#69584C211yesmaximal, characteristic
6#68915C211yesmaximal, characteristic, complemented
7#68896C211yesmaximal, characteristic
8#68896C211yesmaximal, characteristic
9#68894C211yesmaximal, characteristic
10#68855C211yesmaximal, characteristic, complemented
11#68845C211yesmaximal, characteristic, complemented
12#68815C211yesmaximal, characteristic, complemented
13#68782C211yesmaximal, characteristic
14#68672C211yesmaximal, characteristic
15#68614C211yesmaximal, characteristic, complemented
16#27576—31nomaximal, Sylow
17C22×#204C2211yescharacteristic
18C2×#893C2211yescharacteristic
19C2×#893C2211yescharacteristic
20C2×C4×D12C2211yescharacteristic
21C3×#389C2211yescharacteristic
22C3×#384C2211yescharacteristic
23C2×#821C2211yescharacteristic
24C2×#816C2211yescharacteristic
25C2×#815C2211yescharacteristic
26#4989—21no—
27#4987—21no—
28#4986—21no—
29#4985—21no—
30#4984C2211yescharacteristic
31#4984C2211yescharacteristic
32#4980—21no—
33#4978—21no—
34#4977C2211yescharacteristic
35Dic3×C4⋊C4C2211yescharacteristic
36#4973C2211yescharacteristic
37#4973C2211yescharacteristic
38C2×#810C2211yescharacteristic
39C2×#810C2211yescharacteristic
40C2×#809C2211yescharacteristic
41C2×#809C2211yescharacteristic
42C2×#807C2211yescharacteristic
43C4×Dic3⋊C4C2211yescharacteristic
44#4929C2211yescharacteristic
45C2×C4⋊(C3⋊C8)C2211yescharacteristic
46#4668—21no—
47D3×#95—21no—
48C3×#340C2211yescharacteristic
49C3×#340C2211yescharacteristic
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52C3×#336C2211yescharacteristic
53#4551C2211yescharacteristic
54#4550C2211yescharacteristic
55#4550C2211yescharacteristic
56#4550C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2511—31no—
61#2349—31no—
62#2101—31no—
63#2100D311yescharacteristic, complemented
64#2074—31no—
65#2068—31no—
66#1935—31no—
67#1918—31no—
68#1917—31no—
69#1877—31no—
70#1877—31no—
71#1877—31no—
72#1862—31no—
73#1802—31no—
74#1762—31no—
75C22×D12C2311yescharacteristic
76C22×C4×D3—21no—
77C22×C4×D3—21no—
78C22×C4×D3—21no—
79C22×Dic6C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C42C2311yescharacteristic
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—21no—
93C2×C4⋊Dic3—21no—
94C2×C4⋊Dic3—21no—
95C2×C4⋊Dic3C2311yescharacteristic
96C2×C4⋊Dic3C2311yescharacteristic
97C2×Dic3⋊C4D411yescharacteristic
98C2×Dic3⋊C4D411yescharacteristic
99C2×C4×Dic3—21no—
100C2×C4×Dic3C2311yescharacteristic
101C2×C4×Dic3C2311yescharacteristic
102C22×C3⋊C8C2311yescharacteristic
103C22×C3⋊C8C2311yescharacteristic
104C2×#204—22no—
105C2×#204—22no—
106C2×#204—22no—
107#893—22no—
108#893—22no—
109#893—22no—
110#893—41no—
111C4×D12—22no—
112C4×D12—41no—
113C3×#95—21no—
114C3⋊#95—21no—
115C3⋊#95—21no—
116C3⋊#95—21no—
117C3⋊#95—21no—
118C3⋊#95—21no—
119C3⋊#95—21no—
120C3⋊#95—21no—
121C3⋊#95—21no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95D411yescharacteristic
126C3⋊#95D411yescharacteristic
127#821—22no—
128#816—41no—
129#815—41no—
130#810—22no—
131#810—41no—
132#809—22no—
133#809—22no—
134#807—22no—
135C4⋊(C3⋊C8)—41no—
136C22×SD16—31no—
137C2×C4×D4—31no—
138C2×C4×D4—31no—
139C2×C4×D4—31no—
140C2×#107—31no—
141C2×C4⋊C8—31no—
142C2×#103—31no—
143C2×#103—31no—
144C2×#103—31no—
145C2×#102—31no—
146C2×#102—31no—
147C2×#102—31no—
148C2×#97—31no—
149#400—61no—
150#400—61no—
151#391—31no—
152#391—31no—
153#391—31no—
154#389—31no—
155#389—31no—
156#389C2×D311yescharacteristic
157#387—61no—
158#387—61no—
159#385—61no—
160#385—61no—
161#384C2×D311yescharacteristic
162#382—31no—
163#380—61no—
164C4×C4⋊C4—31no—
165C4×C4⋊C4—31no—
166C2×#95—61no—
167#340—31no—
168#340—31no—
169#340—31no—
170#340—31no—
171#340C2×D311yescharacteristic
172#340C2×D311yescharacteristic
173#340C2×D311yescharacteristic
174#340C2×D311yescharacteristic
175#336—31no—
176#336—31no—
177#336—31no—
178#336C2×D311yescharacteristic
179C23×D3—21no—
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2411yescharacteristic
187C22×Dic3—21no—
188C22×Dic3—21no—
189C22×Dic3—21no—
190C22×Dic3C2×D411yescharacteristic
191C22×Dic3C2×D411yescharacteristic
192C22×Dic3C2×D411yescharacteristic
193C22×Dic3C2×D411yescharacteristic
194C2×D12—21no—
195C2×D12—21no—
196C2×D12—21no—
197C2×D12#4112yes—
198C2×D12#4112yes—
199C2×D12#4112yes—
200C2×C4×D3—24no—
201C2×C4×D3—42no—
202C2×C4×D3—42no—
203C2×C4×D3—42no—
204C2×C4×D3—42no—
205C2×C4×D3—81no—
206C2×C4×D3—81no—
207C2×C4×D3—81no—
208C2×C4×D3—81no—
209C2×Dic6—21no—
210C2×Dic6—21no—
211C2×Dic6—21no—
212C2×Dic6—21no—
213C2×Dic6—21no—
214C2×Dic6—21no—
215C2×C3×C8—21no—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—42no—
229C3×C4⋊C4—81no—
230C3×C4⋊C4—81no—
231C3×C4⋊C4#4112yes—
232C3×C42—21no—
233C3×C42—21no—
234(C2×D3)⋊C4—24no—
235(C2×D3)⋊C4—42no—
236(C2×D3)⋊C4—42no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—81no—
242C4⋊Dic3—81no—
243Dic3⋊C4—22no—
244Dic3⋊C4—41no—
245C4×Dic3—21no—
246C4×Dic3—21no—
247C4×Dic3—21no—
248C4×Dic3—21no—
249C4×Dic3—81no—
250C2×C3⋊C8—21no—
251C2×C3⋊C8—21no—
252C2×C3⋊C8—21no—
253C2×C3⋊C8—21no—
254C2×C3⋊C8—21no—
255C2×C3⋊C8—21no—
256#204—44no—
257C22×Q8—31no—
258C22×D4—31no—
259C23×C4—61no—
260C23×C4—61no—
261C23×C4—61no—
262C2×SD16—62no—
263C2×SD16—62no—
264C2×SD16—62no—
265C22×C8—31no—
266C22×C8—31no—
267C22×C8C22×D311yescharacteristic
268C22×C8C22×D311yescharacteristic
269C4×D4—62no—
270C4×D4—62no—
271C4×D4—62no—
272C4×D4—62no—
273C4×D4—121no—
274C4×D4—121no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—31no—
279C2×C4⋊C4—31no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—61no—
283C2×C4⋊C4—61no—
284C2×C4⋊C4C22×D311yescharacteristic
285C2×C4⋊C4C22×D311yescharacteristic
286C2×C4⋊C4C22×D311yescharacteristic
287C2×C4⋊C4C22×D311yescharacteristic
288C2×C22⋊C4—61no—
289C2×C22⋊C4—61no—
290C2×C22⋊C4—61no—
291C2×C42—31no—
292C2×C42—31no—
293C2×C42—61no—
294C2×C42C22×D311yescharacteristic
295#107—62no—
296C4⋊C8—121no—
297#103—62no—
298#103—121no—
299#103—121no—
300#102—62no—
301#102—62no—
302#102—62no—
303#97—121no—
304#95—21no—
305#95—31no—
306#95—31no—
307#95—61no—
308#95—61no—
309#95—61no—
310#95—61no—
311#95—61no—
312#95—61no—
313#95—61no—
314#95—61no—
315#95—61no—
316#95—61no—
317#95—61no—
318C23×C3C22×D411yescharacteristic
319C22×D3—22no—
320C22×D3—22no—
321C22×D3—22no—
322C22×D3—41no—
323C22×D3—41no—
324C22×D3—41no—
325C22×D3—41no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—22no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4#13712yes—
344C2×C3×C4C2×#4111yescharacteristic
345C2×C3×C4C2×#4111yescharacteristic
346C2×C3×C4C2×#4111yescharacteristic
347C2×C3×C4C2×#4111yescharacteristic
348C2×C3×C4H(5,𝔽2)11yescharacteristic
349C2×C3×C42−1+411yescharacteristic
350C2×Dic3—21no—
351C2×Dic3—21no—
352C2×Dic3—21no—
353C2×Dic3—21no—
354C2×Dic3—21no—
355C2×Dic3—21no—
356C2×Dic3—21no—
357C2×Dic3—21no—
358C2×Dic3—21no—
359C2×Dic3—21no—
360C2×Dic3—21no—
361C2×Dic3—21no—
362C2×Dic3—41no—
363C2×Dic3—41no—
364C2×Dic3—41no—
365C2×Dic3—41no—
366C2×Dic3—41no—
367C2×Dic3—41no—
368C2×Dic3—41no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371D12—22no—
372D12—22no—
373D12—22no—
374D12#12614yes—
375C4×D3—84no—
376C4×D3—84no—
377C4×D3—84no—
378Dic6—41no—
379Dic6—41no—
380Dic6—41no—
381Dic6—41no—
382C3×C8—41no—
383C3×C8—41no—
384C3⋊C8—41no—
385C3⋊C8—41no—
386C24—61no—
387C2×Q8—61no—
388C2×Q8—61no—
389C2×Q8—61no—
390C2×Q8—61no—
391C2×Q8—61no—
392C2×Q8—61no—
393C2×D4—32no—
394C2×D4—32no—
395C2×D4—32no—
396C2×D4—61no—
397C2×D4—61no—
398C2×D4—61no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—31no—
404C22×C4—31no—
405C22×C4—31no—
406C22×C4—31no—
407C22×C4—64no—
408C22×C4—61no—
409C22×C4—61no—
410C22×C4—61no—
411C22×C4—122no—
412C22×C4—122no—
413C22×C4—122no—
414C22×C4—122no—
415C22×C4—241no—
416C22×C4—241no—
417C22×C4—241no—
418C22×C4—241no—
419C22×C4D3×D411yescharacteristic
420C22×C4D3×D411yescharacteristic
421C22×C4C23×D311yescharacteristic
422SD16—124no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—61no—
430C2×C8—61no—
431C2×C8—61no—
432C2×C8—61no—
433C2×C8—61no—
434C2×C8—61no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—42no—
443C4⋊C4—62no—
444C4⋊C4—61no—
445C4⋊C4—61no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—81no—
449C4⋊C4—81no—
450C4⋊C4—121no—
451C4⋊C4—241no—
452C4⋊C4—241no—
453C4⋊C4#23912yes—
454C22⋊C4—64no—
455C22⋊C4—122no—
456C22⋊C4—122no—
457C42—21no—
458C42—21no—
459C42—61no—
460C42—61no—
461C42—61no—
462C42—61no—
463C42—241no—
464C22×C3#52511yescharacteristic
465C22×C3#54611yescharacteristic
466C22×C3#54811yescharacteristic
467C22×C3C2×#13711yescharacteristic
468C22×C3#57511yescharacteristic
469C22×C3#57611yescharacteristic
470C22×C3#57711yescharacteristic
471C2×D3—24no—
472C2×D3—42no—
473C2×D3—42no—
474C2×D3—42no—
475C2×D3—42no—
476C2×D3—42no—
477C2×D3—42no—
478C3×C4—22no—
479C3×C4—42no—
480C3×C4—41no—
481C3×C4—81no—
482C3×C4—81no—
483C3×C4—81no—
484C3×C4C2×#12611yescharacteristic
485C3×C4#53911yescharacteristic
486C3×C4#55211yescharacteristic
487C3×C4#55311yescharacteristic
488Dic3—41no—
489Dic3—41no—
490Dic3—41no—
491Dic3—41no—
492Dic3—81no—
493Dic3—81no—
494Dic3—81no—
495C23—62no—
496C23—62no—
497C23—62no—
498C23—121no—
499C23—121no—
500C23—121no—
501C23—121no—
502C23C2×D3×D411yescharacteristic
503Q8—121no—
504Q8—121no—
505Q8—121no—
506Q8—121no—
507D4—34no—
508D4—62no—
509D4—62no—
510D4—62no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—61no—
529C2×C4—61no—
530C2×C4—61no—
531C2×C4—61no—
532C2×C4—61no—
533C2×C4—61no—
534C2×C4—61no—
535C2×C4—61no—
536C2×C4—61no—
537C2×C4—61no—
538C2×C4—61no—
539C2×C4—61no—
540C2×C4—121no—
541C2×C4—121no—
542C2×C4—121no—
543C2×C4—121no—
544C2×C4—121no—
545C2×C4—121no—
546C2×C4—121no—
547C2×C4—121no—
548C2×C4—121no—
549C2×C4—244no—
550C2×C4—244no—
551C2×C4—244no—
552C2×C4#91812yes—
553C2×C4C2×#23911yescharacteristic
554C2×C4D3×#4111yescharacteristic
555C2×C4D3×#4111yescharacteristic
556C2×C4D3×#4111yescharacteristic
557C2×C4#100911yescharacteristic
558C2×C4#101011yescharacteristic
559C8—41no—
560C8—41no—
561C8—121no—
562C8—121no—
563C2×C3#248111yescharacteristic
564C2×C3#311611yescharacteristic
565C2×C3#311811yescharacteristic
566C2×C3#331411yescharacteristic
567C2×C3#332311yescharacteristic
568C2×C3#337011yescharacteristic
569C2×C3#337211yescharacteristic
570D3—44no—
571C22—64no—
572C22—122no—
573C22—122no—
574C22—122no—
575C22—122no—
576C22—122no—
577C22—122no—
578C22#562711yescharacteristic
579C22#565411yescharacteristic
580C22#567711yescharacteristic
581C22#576211yescharacteristic
582C22C2×#91811yescharacteristic
583C22#577311yescharacteristic
584C22#577411yescharacteristic
585C4—22no—
586C4—42no—
587C4—41no—
588C4—81no—
589C4—81no—
590C4—81no—
591C4—121no—
592C4—121no—
593C4—121no—
594C4—121no—
595C4—241no—
596C4—241no—
597C4—241no—
598C4#569411yescharacteristic
599C4#569511yescharacteristic
600C4D3×#12611yescharacteristic
601C4#570711yescharacteristic
602C3#2757611yescharacteristic, complemented, Sylow
603C2—124no—
604C2#7814511yescharacteristic
605C2#8104811yescharacteristic
606C2#8109511yescharacteristic
607C2#8133311yescharacteristic
608C2#8148611yescharacteristic
609C2#8162311yescharacteristic
610C2#8171011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2100C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#2100D3
hypocenterC3#27576
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2100

Complements

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

NH ≅ G/Ntype
2. #74332C2semidirect
3. C3×#2100C2semidirect
4. #69635C2semidirect
6. #68915C2semidirect
10. #68855C2semidirect
11. #68845C2semidirect
12. #68815C2semidirect
15. #68614C2semidirect
63. #2100D3semidirect
602. C3#27576semidirect

Maximal subgroups

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

subgroupindex
#762032
#743322
C3×#21002
#696352
#695842
#689152
#688962
#688942
#688552
#688452
#688152
#687822
#686722
#686142
#275763