Finite Group Databasebeta

← #10875987

Subgroups — #10875987

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
1C4⋊C4×D12C211yesmaximal, characteristic
2#73690C211yesmaximal, characteristic, complemented
3C3×#1917C211yesmaximal, characteristic, complemented
4#70488C211yesmaximal, characteristic, complemented
5#70479C211yesmaximal, characteristic
6#69817C211yesmaximal, characteristic, complemented
7#69812C211yesmaximal, characteristic
8#68672C211yesmaximal, characteristic
9#68672C211yesmaximal, characteristic
10#68672C211yesmaximal, characteristic
11#68671C211yesmaximal, characteristic
12#68643C211yesmaximal, characteristic, complemented
13#68642C211yesmaximal, characteristic, complemented
14#68636C211yesmaximal, characteristic, complemented
15#68623C211yesmaximal, characteristic, complemented
16#27573—31nomaximal, Sylow
17C22×#204C2211yescharacteristic
18C2×D3×C4⋊C4—21no—
19C2×C4×D12C2211yescharacteristic
20C2×C4×D12C2211yescharacteristic
21C2×C4×D12C2211yescharacteristic
22C2×C3×C4⋊C8C2211yescharacteristic
23C2×C3×#97C2211yescharacteristic
24C3×C4×C4⋊C4C2211yescharacteristic
25C2×#821C2211yescharacteristic
26C2×#821C2211yescharacteristic
27C2×#821C2211yescharacteristic
28C2×#817C2211yescharacteristic
29C2×#816C2211yescharacteristic
30C2×#816C2211yescharacteristic
31C2×#816C2211yescharacteristic
32#4987—21no—
33#4986—21no—
34#4980—21no—
35#4935C2211yescharacteristic
36#4935C2211yescharacteristic
37C4×C4⋊Dic3C2211yescharacteristic
38#4929C2211yescharacteristic
39#4929C2211yescharacteristic
40#4929C2211yescharacteristic
41#4928C2211yescharacteristic
42#4928C2211yescharacteristic
43#4928C2211yescharacteristic
44#4668—21no—
45#4668—21no—
46#4660—21no—
47#4654—21no—
48C3×#336C2211yescharacteristic
49C3×#336C2211yescharacteristic
50C3×#336C2211yescharacteristic
51C3×#336C2211yescharacteristic
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#2511—31no—
61D4×C4⋊C4—31no—
62#2100—31no—
63#2099—31no—
64#2072—31no—
65#2066—31no—
66#1935—31no—
67#1920—31no—
68#1917D311yescharacteristic, complemented
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
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C2×C3×C4⋊C4D411yescharacteristic
85C2×C3×C4⋊C4D411yescharacteristic
86C2×C3×C42C2311yescharacteristic
87C2×C3×C42C2311yescharacteristic
88C2×C3×C42C2311yescharacteristic
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—21no—
92C2×C4⋊Dic3—21no—
93C2×C4⋊Dic3—21no—
94C2×C4⋊Dic3—21no—
95C2×C4⋊Dic3—21no—
96C2×C4⋊Dic3C2311yescharacteristic
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×C4×Dic3—21no—
106C2×#204—22no—
107C2×#204—22no—
108C2×#204—22no—
109D3×C4⋊C4—44no—
110C4×D12—22no—
111C4×D12—22no—
112C4×D12—22no—
113C4×D12—22no—
114C4×D12—41no—
115C4×D12—41no—
116C3×C4⋊C8—41no—
117C3×#97—41no—
118C3×#95D411yescharacteristic
119C3×#95D411yescharacteristic
120C3⋊#95—21no—
121C3⋊#95—21no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130#821—22no—
131#821—22no—
132#821—22no—
133#817—22no—
134#816—22no—
135#816—41no—
136#816—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⋊C8C2×D311yescharacteristic
144C2×#103—31no—
145C2×#103—31no—
146C2×#103—31no—
147C2×#102—31no—
148C2×#102—31no—
149C2×#102—31no—
150C2×#97C2×D311yescharacteristic
151#400—61no—
152#400—61no—
153#391—31no—
154#391—31no—
155#391—31no—
156#389—31no—
157#389—31no—
158#389—31no—
159#387—61no—
160#387—61no—
161#385—61no—
162#385—61no—
163#384—31no—
164#384—31no—
165#380—61no—
166C4×C4⋊C4—31no—
167C4×C4⋊C4C2×D311yescharacteristic
168#340—31no—
169#340—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#336C2×D311yescharacteristic
177#336C2×D311yescharacteristic
178#336C2×D311yescharacteristic
179#336C2×D311yescharacteristic
180C23×D3—21no—
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2411yescharacteristic
188C22×Dic3—21no—
189C22×Dic3—21no—
190C22×Dic3—21no—
191C22×Dic3—21no—
192C22×Dic3—21no—
193C22×Dic3—21no—
194C22×Dic3—21no—
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—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C2×C3×C8—21no—
227C2×C3×C8—21no—
228C3×C4⋊C4—22no—
229C3×C4⋊C4—22no—
230C3×C42—21no—
231C3×C42—21no—
232C3×C42—21no—
233C3×C42—21no—
234C3×C42—21no—
235C3×C42—21no—
236(C2×D3)⋊C4—24no—
237(C2×D3)⋊C4—42no—
238(C2×D3)⋊C4—42no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—42no—
251C4⋊Dic3—42no—
252C4⋊Dic3—81no—
253C4⋊Dic3—81no—
254C4⋊Dic3#4112yes—
255Dic3⋊C4—42no—
256Dic3⋊C4—42no—
257Dic3⋊C4—81no—
258C4×Dic3—81no—
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×C8C22×D311yescharacteristic
269C22×C8C22×D311yescharacteristic
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—31no—
279C2×C4⋊C4—31no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4—61no—
290C2×C4⋊C4—61no—
291C2×C4⋊C4—124no—
292C2×C4⋊C4D1211yescharacteristic
293C2×C4⋊C4D1211yescharacteristic
294C2×C22⋊C4—61no—
295C2×C22⋊C4—61no—
296C2×C22⋊C4—61no—
297C2×C42—61no—
298C2×C42C22×D311yescharacteristic
299C2×C42C22×D311yescharacteristic
300C2×C42C22×D311yescharacteristic
301#106—62no—
302C4⋊C8—41no—
303#103—62no—
304#103—121no—
305#103—121no—
306#102—62no—
307#102—62no—
308#102—62no—
309#97—41no—
310#95—61no—
311#95—61no—
312#95—61no—
313#95—61no—
314#95—61no—
315#95—61no—
316#95—61no—
317#95—61no—
318#95—61no—
319#95—61no—
320#95D1211yescharacteristic
321#95D1211yescharacteristic
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—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4AGL(1,ℤ/8)12yes—
347C2×C3×C4#13712yes—
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—22no—
355C2×Dic3—41no—
356C2×Dic3—41no—
357C2×Dic3—41no—
358C2×Dic3—41no—
359C2×Dic3—41no—
360C2×Dic3—41no—
361C2×Dic3—41no—
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—
371C2×Dic3—41no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—41no—
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—64no—
404C22×C4—61no—
405C22×C4—61no—
406C22×C4—61no—
407C22×C4—61no—
408C22×C4—61no—
409C22×C4—61no—
410C22×C4—61no—
411C22×C4—122no—
412C22×C4—122no—
413C22×C4—122no—
414C22×C4—122no—
415C22×C4—122no—
416C22×C4—122no—
417C22×C4—241no—
418C22×C4—241no—
419C22×C4C2×D1211yescharacteristic
420C22×C4C2×D1211yescharacteristic
421C22×C4C2×D1211yescharacteristic
422C22×C4C2×D1211yescharacteristic
423C22×C4C2×D1211yescharacteristic
424C22×C4C2×D1211yescharacteristic
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—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—32no—
442C4⋊C4—61no—
443C4⋊C4—61no—
444C4⋊C4—61no—
445C4⋊C4—61no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—61no—
449C4⋊C4—61no—
450C4⋊C4—61no—
451C4⋊C4—61no—
452C4⋊C4—61no—
453C4⋊C4—122no—
454C4⋊C4—122no—
455C4⋊C4—122no—
456C4⋊C4—122no—
457C4⋊C4—241no—
458C4⋊C4—241no—
459C4⋊C4—241no—
460C22⋊C4—64no—
461C22⋊C4—122no—
462C22⋊C4—122no—
463C42—21no—
464C42—21no—
465C42—21no—
466C42—21no—
467C42—21no—
468C42—21no—
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—22no—
486C3×C4—41no—
487C3×C4—41no—
488C3×C4—41no—
489C3×C4—41no—
490C3×C4C2×#12611yescharacteristic
491C3×C4#53911yescharacteristic
492C3×C4#55211yescharacteristic
493C3×C4#55311yescharacteristic
494Dic3—42no—
495Dic3—81no—
496Dic3—81no—
497Dic3—81no—
498Dic3—81no—
499Dic3—81no—
500Dic3—81no—
501C23—62no—
502C23—62no—
503C23—62no—
504C23—121no—
505C23—121no—
506C23—121no—
507C23—121no—
508C23C22×D1211yescharacteristic
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—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—62no—
534C2×C4—128no—
535C2×C4—121no—
536C2×C4—121no—
537C2×C4—121no—
538C2×C4—121no—
539C2×C4—121no—
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—121no—
550C2×C4—121no—
551C2×C4—121no—
552C2×C4—121no—
553C2×C4—121no—
554C2×C4—121no—
555C2×C4—244no—
556C2×C4—244no—
557C2×C4#90812yes—
558C2×C4#90912yes—
559C2×C4#100411yescharacteristic
560C2×C4#100711yescharacteristic
561C2×C4D3×#4111yescharacteristic
562C2×C4D3×#4111yescharacteristic
563C2×C4D3×#4111yescharacteristic
564C2×C4D3×#4111yescharacteristic
565C8—41no—
566C8—41no—
567C8—41no—
568C8—41no—
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#552411yescharacteristic
585C22#555411yescharacteristic
586C22#557611yescharacteristic
587C22C2×#90811yescharacteristic
588C22C2×#90911yescharacteristic
589C22#575111yescharacteristic
590C22#575111yescharacteristic
591C4—22no—
592C4—22no—
593C4—41no—
594C4—41no—
595C4—41no—
596C4—41no—
597C4—122no—
598C4—241no—
599C4—241no—
600C4—241no—
601C4—241no—
602C4—241no—
603C4—241no—
604C4#563811yescharacteristic
605C4#566111yescharacteristic
606C4#569311yescharacteristic
607C4D3×#12611yescharacteristic
608C3#2757311yescharacteristic, complemented, Sylow
609C2—124no—
610C2#7502011yescharacteristic
611C2#7934411yescharacteristic
612C2#7938911yescharacteristic
613C2#7950111yescharacteristic
614C2#7955811yescharacteristic
615C2#7997211yescharacteristic
616C2#7997311yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1917C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1917D3
hypocenterC3#27573
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1917

Complements

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

NH ≅ G/Ntype
2. #73690C2semidirect
3. C3×#1917C2semidirect
4. #70488C2semidirect
6. #69817C2semidirect
12. #68643C2semidirect
13. #68642C2semidirect
14. #68636C2semidirect
15. #68623C2semidirect
68. #1917D3semidirect
608. C3#27573semidirect

Maximal subgroups

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

subgroupindex
C4⋊C4×D122
#736902
C3×#19172
#704882
#704792
#698172
#698122
#686722
#686712
#686432
#686422
#686362
#686232
#275733