Finite Group Databasebeta

← #10875972

Subgroups — #10875972

Order 768.

704 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#73694C211yesmaximal, characteristic
2C4⋊C4×Dic6C211yesmaximal, characteristic, complemented
3C3×#1917C211yesmaximal, characteristic, complemented
4#70486C211yesmaximal, characteristic, complemented
5#70484C211yesmaximal, characteristic
6#69818C211yesmaximal, characteristic
7#69813C211yesmaximal, characteristic, complemented
8#68686C211yesmaximal, characteristic
9#68686C211yesmaximal, characteristic
10#68686C211yesmaximal, characteristic
11#68683C211yesmaximal, characteristic, complemented
12#68644C211yesmaximal, characteristic, complemented
13#68643C211yesmaximal, characteristic, complemented
14#68635C211yesmaximal, characteristic
15#68621C211yesmaximal, characteristic, complemented
16#27572—31nomaximal, Sylow
17C22×#204C2211yescharacteristic
18C2×C4×Dic6C2211yescharacteristic
19C2×C4×Dic6C2211yescharacteristic
20C2×C4×Dic6C2211yescharacteristic
21C2×C3×C4⋊C8C2211yescharacteristic
22C2×C3×#97C2211yescharacteristic
23C3×C4×C4⋊C4C2211yescharacteristic
24C2×#821C2211yescharacteristic
25C2×#821C2211yescharacteristic
26C2×#821C2211yescharacteristic
27C2×#818C2211yescharacteristic
28C2×#816C2211yescharacteristic
29C2×#816C2211yescharacteristic
30C2×#816C2211yescharacteristic
31#4992—21no—
32#4990—21no—
33#4975—21no—
34Dic3×C4⋊C4—21no—
35#4939C2211yescharacteristic
36#4939C2211yescharacteristic
37#4939C2211yescharacteristic
38#4936C2211yescharacteristic
39#4935C2211yescharacteristic
40C4×C4⋊Dic3C2211yescharacteristic
41#4928C2211yescharacteristic
42#4928C2211yescharacteristic
43#4928C2211yescharacteristic
44#4675—21no—
45#4669—21no—
46#4647—21no—
47#4647—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#2513—31no—
61Q8×C4⋊C4—31no—
62#2101—31no—
63#2100—31no—
64#2071—31no—
65#2067—31no—
66#1936—31no—
67#1918—31no—
68#1917D311yescharacteristic, complemented
69#1880—31no—
70#1880—31no—
71#1880—31no—
72#1863—31no—
73#1805—31no—
74#1763—31no—
75C22×D12C2311yescharacteristic
76C22×Dic6C2311yescharacteristic
77C22×C3×C8C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C2×C3×C4⋊C4D411yescharacteristic
82C2×C3×C4⋊C4D411yescharacteristic
83C2×C3×C42C2311yescharacteristic
84C2×C3×C42C2311yescharacteristic
85C2×C3×C42C2311yescharacteristic
86C2×(C2×D3)⋊C4—21no—
87C2×(C2×D3)⋊C4—21no—
88C2×(C2×D3)⋊C4—21no—
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⋊Dic3C2311yescharacteristic
95C2×C4⋊Dic3C2311yescharacteristic
96C2×C4⋊Dic3C2311yescharacteristic
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×C4×Dic3—21no—
107C2×C4×Dic3—21no—
108C2×C4×Dic3—21no—
109C2×C4×Dic3—21no—
110C2×#204—22no—
111C2×#204—22no—
112C2×#204—22no—
113C4×Dic6—22no—
114C4×Dic6—22no—
115C4×Dic6—22no—
116C4×Dic6—22no—
117C4×Dic6—41no—
118C4×Dic6—41no—
119C3×C4⋊C8—41no—
120C3×#97—41no—
121C3×#95D411yescharacteristic
122C3×#95D411yescharacteristic
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129#821—22no—
130#821—41no—
131#821—41no—
132#818—22no—
133#816—22no—
134#816—22no—
135#816—22no—
136C22×SD16—31no—
137C2×C4×Q8—31no—
138C2×C4×Q8—31no—
139C2×C4×Q8—31no—
140C2×#107—31no—
141C2×C4⋊C8C2×D311yescharacteristic
142C2×#103—31no—
143C2×#103—31no—
144C2×#103—31no—
145C2×#102—31no—
146C2×#102—31no—
147C2×#102—31no—
148C2×#97C2×D311yescharacteristic
149#397—61no—
150#397—61no—
151#390—31no—
152#390—31no—
153#390—31no—
154#389—31no—
155#389—31no—
156#389—31no—
157#389—61no—
158#389—61no—
159#388—61no—
160#388—61no—
161#384—31no—
162#384—61no—
163#382—31no—
164C4×C4⋊C4—31no—
165C4×C4⋊C4—61no—
166C4×C4⋊C4C2×D311yescharacteristic
167#340—31no—
168#340—31no—
169#340—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#336C2×D311yescharacteristic
176#336C2×D311yescharacteristic
177#336C2×D311yescharacteristic
178#336C2×D311yescharacteristic
179C23×D3—21no—
180C22×C3×C4C2×D411yescharacteristic
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2411yescharacteristic
187C22×Dic3—21no—
188C22×Dic3—21no—
189C22×Dic3—21no—
190C22×Dic3—21no—
191C22×Dic3—21no—
192C22×Dic3—21no—
193C22×Dic3—21no—
194C2×D12—21no—
195C2×D12—21no—
196C2×D12—21no—
197C2×D12—21no—
198C2×D12—21no—
199C2×D12—21no—
200C2×Dic6—21no—
201C2×Dic6—21no—
202C2×Dic6—21no—
203C2×Dic6#4112yes—
204C2×Dic6#4112yes—
205C2×Dic6#4112yes—
206C2×C3×C8—21no—
207C2×C3×C8—21no—
208C2×C3×C8—21no—
209C2×C3×C8—21no—
210C2×C3×C8—21no—
211C2×C3×C8—21no—
212C2×C3×C8—21no—
213C2×C3×C8—21no—
214C2×C3×C8—21no—
215C2×C3×C8—21no—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C3×C4⋊C4—22no—
219C3×C4⋊C4—41no—
220C3×C42—21no—
221C3×C42—21no—
222C3×C42—21no—
223C3×C42—21no—
224C3×C42—21no—
225C3×C42—21no—
226(C2×D3)⋊C4—42no—
227(C2×D3)⋊C4—42no—
228(C2×D3)⋊C4—81no—
229(C2×D3)⋊C4—81no—
230(C2×D3)⋊C4—81no—
231(C2×D3)⋊C4—81no—
232C4⋊Dic3—21no—
233C4⋊Dic3—21no—
234C4⋊Dic3—21no—
235C4⋊Dic3—21no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—42no—
244C4⋊Dic3—81no—
245C4⋊Dic3#4112yes—
246Dic3⋊C4—24no—
247Dic3⋊C4—42no—
248Dic3⋊C4—42no—
249Dic3⋊C4—81no—
250Dic3⋊C4—81no—
251Dic3⋊C4—81no—
252C4×Dic3—24no—
253C4×Dic3—42no—
254C4×Dic3—42no—
255C4×Dic3—81no—
256#204—44no—
257C22×Q8—31no—
258C22×D4—31no—
259C2×SD16—62no—
260C2×SD16—62no—
261C2×SD16—62no—
262C22×C8C22×D311yescharacteristic
263C22×C8C22×D311yescharacteristic
264C22×C8C22×D311yescharacteristic
265C22×C8C22×D311yescharacteristic
266C4×Q8—62no—
267C4×Q8—62no—
268C4×Q8—62no—
269C4×Q8—62no—
270C4×Q8—121no—
271C4×Q8—121no—
272C2×C4⋊C4—31no—
273C2×C4⋊C4—31no—
274C2×C4⋊C4—31no—
275C2×C4⋊C4—31no—
276C2×C4⋊C4—31no—
277C2×C4⋊C4—31no—
278C2×C4⋊C4—61no—
279C2×C4⋊C4—61no—
280C2×C4⋊C4—61no—
281C2×C4⋊C4—61no—
282C2×C4⋊C4—61no—
283C2×C4⋊C4—61no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4D1211yescharacteristic
287C2×C4⋊C4D1211yescharacteristic
288C2×C22⋊C4—61no—
289C2×C22⋊C4—61no—
290C2×C22⋊C4—61no—
291C2×C22⋊C4—61no—
292C2×C22⋊C4—61no—
293C2×C22⋊C4—61no—
294C2×C42—61no—
295C2×C42—61no—
296C2×C42—61no—
297C2×C42—61no—
298C2×C42C22×D311yescharacteristic
299C2×C42C22×D311yescharacteristic
300C2×C42C22×D311yescharacteristic
301#107—62no—
302C4⋊C8—41no—
303#103—62no—
304#103—62no—
305#103—62no—
306#102—62no—
307#102—121no—
308#102—121no—
309#97—41no—
310#95—61no—
311#95—61no—
312#95—61no—
313#95—61no—
314#95—61no—
315#95—61no—
316#95D1211yescharacteristic
317#95D1211yescharacteristic
318C23×C3C22×D411yescharacteristic
319C22×D3—41no—
320C22×D3—41no—
321C22×D3—41no—
322C22×D3—41no—
323C22×D3—41no—
324C22×D3—41no—
325C22×D3—41no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
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#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—22no—
351C2×Dic3—22no—
352C2×Dic3—22no—
353C2×Dic3—22no—
354C2×Dic3—41no—
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—
371D12—41no—
372D12—41no—
373D12—41no—
374D12—41no—
375Dic6—22no—
376Dic6—22no—
377Dic6—22no—
378Dic6#12614yes—
379C3×C8—41no—
380C3×C8—41no—
381C3×C8—41no—
382C3×C8—41no—
383C24—61no—
384C2×Q8—32no—
385C2×Q8—32no—
386C2×Q8—32no—
387C2×Q8—61no—
388C2×Q8—61no—
389C2×Q8—61no—
390C2×D4—61no—
391C2×D4—61no—
392C2×D4—61no—
393C2×D4—61no—
394C2×D4—61no—
395C2×D4—61no—
396C22×C4—61no—
397C22×C4—61no—
398C22×C4—61no—
399C22×C4—61no—
400C22×C4—61no—
401C22×C4—61no—
402C22×C4—61no—
403C22×C4C2×D1211yescharacteristic
404C22×C4C2×D1211yescharacteristic
405C22×C4C2×D1211yescharacteristic
406C22×C4C2×D1211yescharacteristic
407C22×C4C2×D1211yescharacteristic
408C22×C4C2×D1211yescharacteristic
409C22×C4C23×D311yescharacteristic
410SD16—124no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—21no—
423C4⋊C4—22no—
424C4⋊C4—32no—
425C4⋊C4—41no—
426C4⋊C4—64no—
427C4⋊C4—61no—
428C4⋊C4—61no—
429C4⋊C4—61no—
430C4⋊C4—61no—
431C4⋊C4—61no—
432C4⋊C4—61no—
433C4⋊C4—61no—
434C4⋊C4—61no—
435C4⋊C4—61no—
436C4⋊C4—61no—
437C4⋊C4—61no—
438C4⋊C4—122no—
439C4⋊C4—122no—
440C4⋊C4—122no—
441C4⋊C4—241no—
442C4⋊C4—241no—
443C4⋊C4—241no—
444C4⋊C4—241no—
445C22⋊C4—122no—
446C22⋊C4—122no—
447C22⋊C4—241no—
448C22⋊C4—241no—
449C22⋊C4—241no—
450C22⋊C4—241no—
451C42—21no—
452C42—21no—
453C42—21no—
454C42—21no—
455C42—21no—
456C42—21no—
457C42—64no—
458C42—122no—
459C42—122no—
460C42—241no—
461C22×C3#52511yescharacteristic
462C22×C3#54611yescharacteristic
463C22×C3#54811yescharacteristic
464C22×C3C2×#13711yescharacteristic
465C22×C3#57511yescharacteristic
466C22×C3#57611yescharacteristic
467C22×C3#57711yescharacteristic
468C2×D3—81no—
469C2×D3—81no—
470C2×D3—81no—
471C2×D3—81no—
472C2×D3—81no—
473C2×D3—81no—
474C2×D3—81no—
475C3×C4—22no—
476C3×C4—41no—
477C3×C4—41no—
478C3×C4—41no—
479C3×C4—41no—
480C3×C4—41no—
481C3×C4C2×#12611yescharacteristic
482C3×C4#53911yescharacteristic
483C3×C4#55211yescharacteristic
484C3×C4#55311yescharacteristic
485Dic3—24no—
486Dic3—42no—
487Dic3—81no—
488Dic3—81no—
489Dic3—81no—
490Dic3—81no—
491Dic3—81no—
492C23—121no—
493C23—121no—
494C23—121no—
495C23—121no—
496C23—121no—
497C23—121no—
498C23—121no—
499C23C22×D1211yescharacteristic
500Q8—34no—
501Q8—62no—
502Q8—62no—
503Q8—62no—
504D4—121no—
505D4—121no—
506D4—121no—
507D4—121no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
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—62no—
526C2×C4—62no—
527C2×C4—62no—
528C2×C4—62no—
529C2×C4—121no—
530C2×C4—121no—
531C2×C4—121no—
532C2×C4—121no—
533C2×C4—121no—
534C2×C4—121no—
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#90912yes—
547C2×C4#100411yescharacteristic
548C2×C4#100711yescharacteristic
549C2×C4D3×#4111yescharacteristic
550C2×C4D3×#4111yescharacteristic
551C2×C4D3×#4111yescharacteristic
552C2×C4D3×#4111yescharacteristic
553C8—41no—
554C8—41no—
555C8—41no—
556C8—41no—
557C2×C3#248111yescharacteristic
558C2×C3#311711yescharacteristic
559C2×C3#311911yescharacteristic
560C2×C3#330911yescharacteristic
561C2×C3#332911yescharacteristic
562C2×C3#337111yescharacteristic
563C2×C3#337511yescharacteristic
564D3—161no—
565C22—241no—
566C22—241no—
567C22—241no—
568C22—241no—
569C22—241no—
570C22—241no—
571C22—241no—
572C22#552411yescharacteristic
573C22#555411yescharacteristic
574C22#557611yescharacteristic
575C22C2×#90911yescharacteristic
576C22#574811yescharacteristic
577C22#575111yescharacteristic
578C22#575211yescharacteristic
579C4—22no—
580C4—41no—
581C4—41no—
582C4—41no—
583C4—41no—
584C4—41no—
585C4—64no—
586C4—122no—
587C4—241no—
588C4—241no—
589C4—241no—
590C4—241no—
591C4—241no—
592C4#563811yescharacteristic
593C4#566111yescharacteristic
594C4#569311yescharacteristic
595C4D3×#12611yescharacteristic
596C3#2757211yescharacteristic, complemented, Sylow
597C2—481no—
598C2#7502011yescharacteristic
599C2#7935711yescharacteristic
600C2#7938411yescharacteristic
601C2#7951011yescharacteristic
602C2#7956411yescharacteristic
603C2#7998211yescharacteristic
604C2#7998411yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1917C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1917D3
hypocenterC3#27572
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. C4⋊C4×Dic6C2semidirect
3. C3×#1917C2semidirect
4. #70486C2semidirect
7. #69813C2semidirect
11. #68683C2semidirect
12. #68644C2semidirect
13. #68643C2semidirect
15. #68621C2semidirect
68. #1917D3semidirect
596. C3#27572semidirect

Maximal subgroups

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

subgroupindex
#736942
C4⋊C4×Dic62
C3×#19172
#704862
#704842
#698182
#698132
#686862
#686832
#686442
#686432
#686352
#686212
#275723