Finite Group Databasebeta

← #10875992

Subgroups — #10875992

Order 768.

782 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#76202C211yesmaximal, characteristic
2#74332C211yesmaximal, characteristic, complemented
3C3×#2099C211yesmaximal, characteristic, complemented
4#69634C211yesmaximal, characteristic, complemented
5#69584C211yesmaximal, characteristic
6#68914C211yesmaximal, characteristic, complemented
7#68896C211yesmaximal, characteristic
8#68896C211yesmaximal, characteristic
9#68895C211yesmaximal, characteristic
10#68855C211yesmaximal, characteristic, complemented
11#68843C211yesmaximal, characteristic, complemented
12#68819C211yesmaximal, characteristic, complemented
13#68782C211yesmaximal, characteristic
14#68672C211yesmaximal, characteristic
15#68614C211yesmaximal, characteristic, complemented
16#27573—31nomaximal, Sylow
17C22×#204C2211yescharacteristic
18C2×#893C2211yescharacteristic
19C2×#893C2211yescharacteristic
20C2×D3×C4⋊C4—21no—
21C2×C4×D12C2211yescharacteristic
22C3×#389C2211yescharacteristic
23C3×#384C2211yescharacteristic
24C2×#821C2211yescharacteristic
25C2×#816C2211yescharacteristic
26C2×#815C2211yescharacteristic
27#4989—21no—
28#4986—21no—
29#4985—21no—
30#4984C2211yescharacteristic
31#4984C2211yescharacteristic
32#4980—21no—
33#4978—21no—
34#4975C2211yescharacteristic
35Dic3×C4⋊C4C2211yescharacteristic
36#4973C2211yescharacteristic
37#4973C2211yescharacteristic
38C2×#810C2211yescharacteristic
39C2×#810C2211yescharacteristic
40C2×#809C2211yescharacteristic
41C2×#809C2211yescharacteristic
42C2×#808C2211yescharacteristic
43C4×Dic3⋊C4C2211yescharacteristic
44#4929C2211yescharacteristic
45C2×C4⋊(C3⋊C8)C2211yescharacteristic
46#4668—21no—
47#4667—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—
61D4×C4⋊C4—31no—
62#2100—31no—
63#2099D311yescharacteristic, complemented
64#2072—31no—
65#2066—31no—
66#1935—31no—
67#1920—31no—
68#1917—31no—
69#1877—31no—
70#1877—31no—
71#1877—31no—
72#1864—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⋊C4—21no—
98C2×Dic3⋊C4—21no—
99C2×Dic3⋊C4D411yescharacteristic
100C2×Dic3⋊C4D411yescharacteristic
101C2×C4×Dic3—21no—
102C2×C4×Dic3C2311yescharacteristic
103C2×C4×Dic3C2311yescharacteristic
104C22×C3⋊C8C2311yescharacteristic
105C22×C3⋊C8C2311yescharacteristic
106C2×#204—22no—
107C2×#204—22no—
108C2×#204—22no—
109#893—22no—
110#893—22no—
111#893—22no—
112#893—41no—
113D3×C4⋊C4—44no—
114C4×D12—22no—
115C4×D12—41no—
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⋊#95—21no—
126C3⋊#95D411yescharacteristic
127C3⋊#95D411yescharacteristic
128#821—22no—
129#816—41no—
130#815—41no—
131#810—22no—
132#810—41no—
133#809—22no—
134#809—22no—
135#808—22no—
136C4⋊(C3⋊C8)—41no—
137C22×SD16—31no—
138C2×C4×D4—31no—
139C2×C4×D4—31no—
140C2×C4×D4—31no—
141C22×C4⋊C4—61no—
142C2×#106—31no—
143C2×C4⋊C8—31no—
144C2×#103—31no—
145C2×#103—31no—
146C2×#103—31no—
147C2×#102—31no—
148C2×#102—31no—
149C2×#102—31no—
150C2×#97—31no—
151#400—61no—
152#400—61no—
153#391—31no—
154#391—31no—
155#391—31no—
156#389—31no—
157#389—31no—
158#389C2×D311yescharacteristic
159#387—61no—
160#387—61no—
161#385—61no—
162#385—61no—
163#384—31no—
164#384C2×D311yescharacteristic
165#380—61no—
166C4×C4⋊C4—31no—
167C4×C4⋊C4—31no—
168#340—31no—
169#340—31no—
170#340—31no—
171#340—31no—
172#340C2×D311yescharacteristic
173#340C2×D311yescharacteristic
174#340C2×D311yescharacteristic
175#340C2×D311yescharacteristic
176#336—31no—
177#336—31no—
178#336—31no—
179#336C2×D311yescharacteristic
180C23×D3—21no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2411yescharacteristic
188C22×Dic3—21no—
189C22×Dic3—21no—
190C22×Dic3—21no—
191C22×Dic3C2×D411yescharacteristic
192C22×Dic3C2×D411yescharacteristic
193C22×Dic3C2×D411yescharacteristic
194C22×Dic3C2×D411yescharacteristic
195C2×D12—21no—
196C2×D12—21no—
197C2×D12—21no—
198C2×D12#4112yes—
199C2×D12#4112yes—
200C2×D12#4112yes—
201C2×C4×D3—24no—
202C2×C4×D3—42no—
203C2×C4×D3—42no—
204C2×C4×D3—42no—
205C2×C4×D3—42no—
206C2×C4×D3—42no—
207C2×C4×D3—42no—
208C2×C4×D3—81no—
209C2×C4×D3—81no—
210C2×Dic6—21no—
211C2×Dic6—21no—
212C2×Dic6—21no—
213C2×Dic6—21no—
214C2×Dic6—21no—
215C2×Dic6—21no—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—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—21no—
229C3×C4⋊C4—42no—
230C3×C4⋊C4—42no—
231C3×C4⋊C4—81no—
232C3×C4⋊C4#4112yes—
233C3×C42—21no—
234C3×C42—21no—
235(C2×D3)⋊C4—24no—
236(C2×D3)⋊C4—42no—
237(C2×D3)⋊C4—42no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—42no—
243C4⋊Dic3—81no—
244Dic3⋊C4—22no—
245Dic3⋊C4—22no—
246Dic3⋊C4—42no—
247Dic3⋊C4—81no—
248C4×Dic3—21no—
249C4×Dic3—21no—
250C4×Dic3—21no—
251C4×Dic3—21no—
252C4×Dic3—81no—
253C2×C3⋊C8—21no—
254C2×C3⋊C8—21no—
255C2×C3⋊C8—21no—
256C2×C3⋊C8—21no—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
259#204—44no—
260C22×Q8—31no—
261C22×D4—31no—
262C23×C4—61no—
263C23×C4—61no—
264C23×C4—61no—
265C2×SD16—62no—
266C2×SD16—62no—
267C2×SD16—62no—
268C22×C8—31no—
269C22×C8—31no—
270C22×C8C22×D311yescharacteristic
271C22×C8C22×D311yescharacteristic
272C4×D4—62no—
273C4×D4—62no—
274C4×D4—62no—
275C4×D4—62no—
276C4×D4—121no—
277C4×D4—121no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4—124no—
290C2×C4⋊C4C22×D311yescharacteristic
291C2×C4⋊C4C22×D311yescharacteristic
292C2×C4⋊C4C22×D311yescharacteristic
293C2×C4⋊C4C22×D311yescharacteristic
294C2×C22⋊C4—61no—
295C2×C22⋊C4—61no—
296C2×C22⋊C4—61no—
297C2×C42—31no—
298C2×C42—31no—
299C2×C42—61no—
300C2×C42C22×D311yescharacteristic
301#106—62no—
302C4⋊C8—121no—
303#103—62no—
304#103—121no—
305#103—121no—
306#102—62no—
307#102—62no—
308#102—62no—
309#97—121no—
310#95—21no—
311#95—31no—
312#95—31no—
313#95—61no—
314#95—61no—
315#95—61no—
316#95—61no—
317#95—61no—
318#95—61no—
319#95—61no—
320#95—61no—
321#95—61no—
322C23×C3C22×D411yescharacteristic
323C22×D3—22no—
324C22×D3—22no—
325C22×D3—22no—
326C22×D3—41no—
327C22×D3—41no—
328C22×D3—41no—
329C22×D3—41no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—22no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
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—41no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4AGL(1,ℤ/8)12yes—
348C2×C3×C4C2×#4111yescharacteristic
349C2×C3×C4C2×#4111yescharacteristic
350C2×C3×C4C2×#4111yescharacteristic
351C2×C3×C4C2×#4111yescharacteristic
352C2×C3×C4H(5,𝔽2)11yescharacteristic
353C2×C3×C42−1+411yescharacteristic
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—21no—
363C2×Dic3—21no—
364C2×Dic3—21no—
365C2×Dic3—41no—
366C2×Dic3—41no—
367C2×Dic3—41no—
368C2×Dic3—41no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3#13712yes—
375D12—22no—
376D12—22no—
377D12—22no—
378D12#12614yes—
379C4×D3—48no—
380C4×D3—84no—
381C4×D3—84no—
382Dic6—41no—
383Dic6—41no—
384Dic6—41no—
385Dic6—41no—
386C3×C8—41no—
387C3×C8—41no—
388C3⋊C8—41no—
389C3⋊C8—41no—
390C24—61no—
391C2×Q8—61no—
392C2×Q8—61no—
393C2×Q8—61no—
394C2×Q8—61no—
395C2×Q8—61no—
396C2×Q8—61no—
397C2×D4—32no—
398C2×D4—32no—
399C2×D4—32no—
400C2×D4—61no—
401C2×D4—61no—
402C2×D4—61no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—31no—
408C22×C4—31no—
409C22×C4—31no—
410C22×C4—31no—
411C22×C4—64no—
412C22×C4—61no—
413C22×C4—61no—
414C22×C4—61no—
415C22×C4—122no—
416C22×C4—122no—
417C22×C4—122no—
418C22×C4—122no—
419C22×C4—122no—
420C22×C4—122no—
421C22×C4—241no—
422C22×C4—241no—
423C22×C4D3×D411yescharacteristic
424C22×C4D3×D411yescharacteristic
425C22×C4C23×D311yescharacteristic
426SD16—124no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—61no—
434C2×C8—61no—
435C2×C8—61no—
436C2×C8—61no—
437C2×C8—61no—
438C2×C8—61no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—62no—
449C4⋊C4—62no—
450C4⋊C4—61no—
451C4⋊C4—61no—
452C4⋊C4—61no—
453C4⋊C4—61no—
454C4⋊C4—81no—
455C4⋊C4—122no—
456C4⋊C4—122no—
457C4⋊C4—241no—
458C4⋊C4—241no—
459C4⋊C4#23912yes—
460C22⋊C4—64no—
461C22⋊C4—122no—
462C22⋊C4—122no—
463C42—21no—
464C42—21no—
465C42—61no—
466C42—61no—
467C42—61no—
468C42—61no—
469C42—241no—
470C22×C3#52511yescharacteristic
471C22×C3#54611yescharacteristic
472C22×C3#54811yescharacteristic
473C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
474C22×C3C2×#13711yescharacteristic
475C22×C3#57711yescharacteristic
476C22×C3#57711yescharacteristic
477C2×D3—24no—
478C2×D3—42no—
479C2×D3—42no—
480C2×D3—42no—
481C2×D3—42no—
482C2×D3—42no—
483C2×D3—42no—
484C3×C4—22no—
485C3×C4—41no—
486C3×C4—42no—
487C3×C4—81no—
488C3×C4—81no—
489C3×C4—81no—
490C3×C4C2×#12611yescharacteristic
491C3×C4#53911yescharacteristic
492C3×C4#55211yescharacteristic
493C3×C4#55311yescharacteristic
494Dic3—22no—
495Dic3—41no—
496Dic3—41no—
497Dic3—41no—
498Dic3—81no—
499Dic3—81no—
500Dic3—81no—
501C23—62no—
502C23—62no—
503C23—62no—
504C23—121no—
505C23—121no—
506C23—121no—
507C23—121no—
508C23C2×D3×D411yescharacteristic
509Q8—121no—
510Q8—121no—
511Q8—121no—
512Q8—121no—
513D4—34no—
514D4—62no—
515D4—62no—
516D4—62no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—32no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—61no—
536C2×C4—61no—
537C2×C4—61no—
538C2×C4—61no—
539C2×C4—61no—
540C2×C4—61no—
541C2×C4—61no—
542C2×C4—61no—
543C2×C4—61no—
544C2×C4—61no—
545C2×C4—61no—
546C2×C4—128no—
547C2×C4—121no—
548C2×C4—121no—
549C2×C4—121no—
550C2×C4—121no—
551C2×C4—121no—
552C2×C4—121no—
553C2×C4—121no—
554C2×C4—121no—
555C2×C4—121no—
556C2×C4—244no—
557C2×C4—244no—
558C2×C4#91112yes—
559C2×C4C2×#23911yescharacteristic
560C2×C4D3×#4111yescharacteristic
561C2×C4D3×#4111yescharacteristic
562C2×C4D3×#4111yescharacteristic
563C2×C4#100911yescharacteristic
564C2×C4#101011yescharacteristic
565C8—41no—
566C8—41no—
567C8—121no—
568C8—121no—
569C2×C3#248111yescharacteristic
570C2×C3#311011yescharacteristic
571C2×C3#312511yescharacteristic
572C2×C3#331011yescharacteristic
573C2×C3#333111yescharacteristic
574C2×C3#337211yescharacteristic
575C2×C3#337311yescharacteristic
576D3—44no—
577C22—64no—
578C22—122no—
579C22—122no—
580C22—122no—
581C22—122no—
582C22—122no—
583C22—122no—
584C22#562711yescharacteristic
585C22#565411yescharacteristic
586C22#567711yescharacteristic
587C22C2×#91111yescharacteristic
588C22#576211yescharacteristic
589C22D3×#13711yescharacteristic
590C22#577811yescharacteristic
591C4—22no—
592C4—41no—
593C4—42no—
594C4—62no—
595C4—81no—
596C4—81no—
597C4—81no—
598C4—121no—
599C4—121no—
600C4—121no—
601C4—241no—
602C4—241no—
603C4—241no—
604C4#569411yescharacteristic
605C4#569511yescharacteristic
606C4D3×#12611yescharacteristic
607C4#570711yescharacteristic
608C3#2757311yescharacteristic, complemented, Sylow
609C2—124no—
610C2#7814511yescharacteristic
611C2#8095011yescharacteristic
612C2#8100111yescharacteristic
613C2#8130311yescharacteristic
614C2#8144711yescharacteristic
615C2#8158711yescharacteristic
616C2#8173211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2099C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#2099D3
hypocenterC3#27573
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2099

Complements

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

NH ≅ G/Ntype
2. #74332C2semidirect
3. C3×#2099C2semidirect
4. #69634C2semidirect
6. #68914C2semidirect
10. #68855C2semidirect
11. #68843C2semidirect
12. #68819C2semidirect
15. #68614C2semidirect
63. #2099D3semidirect
608. C3#27573semidirect

Maximal subgroups

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

subgroupindex
#762022
#743322
C3×#20992
#696342
#695842
#689142
#688962
#688952
#688552
#688432
#688192
#687822
#686722
#686142
#275733