Finite Group Databasebeta

← #10876043

Subgroups — #10876043

Order 768.

696 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2351C211yesmaximal, characteristic, complemented
2#73694C211yesmaximal, characteristic
3#71633C211yesmaximal, characteristic
4#71633C211yesmaximal, characteristic
5#71633C211yesmaximal, characteristic
6#71631C211yesmaximal, characteristic, complemented
7#71629C211yesmaximal, characteristic, complemented
8#71627C211yesmaximal, characteristic, complemented
9#69629C211yesmaximal, characteristic
10#69617C211yesmaximal, characteristic
11#69613C211yesmaximal, characteristic, complemented
12#69602C211yesmaximal, characteristic, complemented
13#69557C211yesmaximal, characteristic
14#69542C211yesmaximal, characteristic, complemented
15#69483C211yesmaximal, characteristic, complemented
16#27577—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×C4×Q8C2211yescharacteristic
20C22×#215C2211yescharacteristic
21C3×#389—21no—
22C3×#389—21no—
23C3×#389C2211yescharacteristic
24C3×#389C2211yescharacteristic
25C3×#382—21no—
26C3×#382C2211yescharacteristic
27C3×C4×C4⋊C4—21no—
28C3×C4×C4⋊C4C2211yescharacteristic
29C2×#835C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×#835C2211yescharacteristic
32#4992—21no—
33#4990—21no—
34C2×#809C2211yescharacteristic
35C2×#809C2211yescharacteristic
36C2×#809C2211yescharacteristic
37C2×#808C2211yescharacteristic
38#4939C2211yescharacteristic
39#4939C2211yescharacteristic
40#4939C2211yescharacteristic
41#4936C2211yescharacteristic
42C4×C4⋊Dic3C2211yescharacteristic
43#4928C2211yescharacteristic
44C2×C4⋊(C3⋊C8)C2211yescharacteristic
45C2×#803C2211yescharacteristic
46#4675—21no—
47#4669—21no—
48#4529C2211yescharacteristic
49#4529C2211yescharacteristic
50#4529C2211yescharacteristic
51#4529C2211yescharacteristic
52#4529C2211yescharacteristic
53#4529C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4524C2211yescharacteristic
57#4524C2211yescharacteristic
58#4524C2211yescharacteristic
59#4524C2211yescharacteristic
60#2513—31no—
61#2351D311yescharacteristic, complemented
62#2101—31no—
63#2101—31no—
64#2073—31no—
65#2065—31no—
66#1936—31no—
67#1920—31no—
68#1917—31no—
69#1880—31no—
70#1880—31no—
71#1880—31no—
72#1865—31no—
73#1805—31no—
74#1763—31no—
75C22×C3×Q8C2311yescharacteristic
76C22×D12C2311yescharacteristic
77C3×C4×Q8—22no—
78C3×C4×Q8—22no—
79C3×C4×Q8—22no—
80C3×C4×Q8—22no—
81C3×C4×Q8—41no—
82C3×C4×Q8—41no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4D411yescharacteristic
89C2×C3×C4⋊C4D411yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C42—21no—
94C2×C3×C42—21no—
95C2×C3×C42—21no—
96C2×C3×C42C2311yescharacteristic
97C2×C3×C42C2311yescharacteristic
98C2×C3×C42C2311yescharacteristic
99C2×#215—22no—
100C2×#215—22no—
101C2×#215—22no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4—21no—
106C2×(C2×D3)⋊C4—21no—
107C2×(C2×D3)⋊C4—21no—
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×Dic3⋊C4—21no—
112C2×C4×Dic3—21no—
113C22×C3⋊C8C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117C3×#95—21no—
118C3×#95—21no—
119C3×#95—21no—
120C3×#95—21no—
121C3×#95D411yescharacteristic
122C3×#95D411yescharacteristic
123#835—22no—
124#835—22no—
125#835—22no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130#809—22no—
131#809—41no—
132#809—41no—
133#808—22no—
134C4⋊(C3⋊C8)—41no—
135#803—41no—
136C22×SD16—31no—
137C2×C4×Q8C2×D311yescharacteristic
138C2×C4×Q8C2×D311yescharacteristic
139C2×C4×Q8C2×D311yescharacteristic
140C2×#106—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#397—61no—
150#397—61no—
151#390—31no—
152#390—31no—
153#390—31no—
154#389—21no—
155#389—21no—
156#389—31no—
157#389C2×D311yescharacteristic
158#389C2×D311yescharacteristic
159#388—61no—
160#388—61no—
161#382—21no—
162#382—31no—
163#382C2×D311yescharacteristic
164C4×C4⋊C4—21no—
165C4×C4⋊C4—31no—
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#336—31no—
176#336—31no—
177#336—31no—
178#336—31no—
179C23×D3—21no—
180C2×C3×Q8—21no—
181C2×C3×Q8—21no—
182C2×C3×Q8—21no—
183C2×C3×Q8#4112yes—
184C2×C3×Q8#4112yes—
185C2×C3×Q8#4112yes—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2411yescharacteristic
197C22×Dic3—21no—
198C22×Dic3—21no—
199C22×Dic3—21no—
200C2×D12—21no—
201C2×D12—21no—
202C2×D12—21no—
203C2×D12—21no—
204C2×D12—21no—
205C2×D12—21no—
206C3×C4⋊C4—24no—
207C3×C4⋊C4—21no—
208C3×C4⋊C4—21no—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—21no—
211C3×C4⋊C4—21no—
212C3×C4⋊C4—41no—
213C3×C4⋊C4—41no—
214C3×C4⋊C4—42no—
215C3×C4⋊C4—42no—
216C3×C4⋊C4—42no—
217C3×C4⋊C4—42no—
218C3×C4⋊C4#4112yes—
219C3×C42—24no—
220C3×C42—21no—
221C3×C42—21no—
222C3×C42—21no—
223C3×C42—21no—
224C3×C42—21no—
225C3×C42—21no—
226C3×C42—42no—
227C3×C42—42no—
228#215—44no—
229(C2×D3)⋊C4—42no—
230(C2×D3)⋊C4—42no—
231(C2×D3)⋊C4—81no—
232(C2×D3)⋊C4—81no—
233(C2×D3)⋊C4—81no—
234(C2×D3)⋊C4—81no—
235C4⋊Dic3—21no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241Dic3⋊C4—81no—
242C4×Dic3—81no—
243C2×C3⋊C8—21no—
244C2×C3⋊C8—21no—
245C2×C3⋊C8—21no—
246C2×C3⋊C8—21no—
247C2×C3⋊C8—21no—
248C2×C3⋊C8—21no—
249C2×C3⋊C8—21no—
250C2×C3⋊C8—21no—
251C2×C3⋊C8—21no—
252C2×C3⋊C8—21no—
253C2×C3⋊C8—21no—
254C2×C3⋊C8—21no—
255C22×Q8C22×D311yescharacteristic
256C22×D4—31no—
257C2×SD16—62no—
258C2×SD16—62no—
259C2×SD16—62no—
260C22×C8—31no—
261C22×C8—31no—
262C22×C8—31no—
263C22×C8—31no—
264C4×Q8—22no—
265C4×Q8—22no—
266C4×Q8—22no—
267C4×Q8—22no—
268C4×Q8—41no—
269C4×Q8—41no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—31no—
276C2×C4⋊C4—31no—
277C2×C4⋊C4—31no—
278C2×C4⋊C4—61no—
279C2×C4⋊C4#6711yescharacteristic
280C2×C4⋊C4#6711yescharacteristic
281C2×C4⋊C4C22×D311yescharacteristic
282C2×C4⋊C4C22×D311yescharacteristic
283C2×C4⋊C4C22×D311yescharacteristic
284C2×C22⋊C4—61no—
285C2×C22⋊C4—61no—
286C2×C22⋊C4—61no—
287C2×C22⋊C4—61no—
288C2×C22⋊C4—61no—
289C2×C22⋊C4—61no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—61no—
294C2×C42C22×D311yescharacteristic
295C2×C42C22×D311yescharacteristic
296C2×C42C22×D311yescharacteristic
297#106—62no—
298C4⋊C8—121no—
299#103—62no—
300#103—62no—
301#103—62no—
302#102—62no—
303#102—121no—
304#102—121no—
305#97—121no—
306#95—21no—
307#95—21no—
308#95—21no—
309#95—21no—
310#95—61no—
311#95—61no—
312#95—61no—
313#95—61no—
314#95#6711yescharacteristic
315#95#6711yescharacteristic
316C23×C3C22×D411yescharacteristic
317C22×D3—41no—
318C22×D3—41no—
319C22×D3—41no—
320C22×D3—41no—
321C22×D3—41no—
322C22×D3—41no—
323C22×D3—41no—
324C3×Q8—22no—
325C3×Q8—22no—
326C3×Q8—22no—
327C3×Q8#12614yes—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—22no—
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×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4C2×#4111yescharacteristic
359C2×C3×C4C2×#4111yescharacteristic
360C2×C3×C4C2×#4111yescharacteristic
361C2×C3×C4C2×#4111yescharacteristic
362C2×C3×C4H(5,𝔽2)11yescharacteristic
363C2×C3×C42−1+411yescharacteristic
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—
373D12—41no—
374D12—41no—
375D12—41no—
376D12—41no—
377C3⋊C8—41no—
378C3⋊C8—41no—
379C3⋊C8—41no—
380C3⋊C8—41no—
381C24—61no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8#23512yes—
386C2×Q8#23712yes—
387C2×Q8#23912yes—
388C2×D4—61no—
389C2×D4—61no—
390C2×D4—61no—
391C2×D4—61no—
392C2×D4—61no—
393C2×D4—61no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—61no—
399C22×C4—61no—
400C22×C4—61no—
401C22×C4C2×#6711yescharacteristic
402C22×C4C2×#6711yescharacteristic
403C22×C4C2×#6711yescharacteristic
404C22×C4C2×#6711yescharacteristic
405C22×C4C2×#6711yescharacteristic
406C22×C4C2×#6711yescharacteristic
407C22×C4C23×D311yescharacteristic
408SD16—124no—
409C2×C8—61no—
410C2×C8—61no—
411C2×C8—61no—
412C2×C8—61no—
413C2×C8—61no—
414C2×C8—61no—
415C2×C8—61no—
416C2×C8—61no—
417C2×C8—61no—
418C2×C8—61no—
419C2×C8—61no—
420C2×C8—61no—
421C4⋊C4—24no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—61no—
434C4⋊C4—61no—
435C4⋊C4—61no—
436C4⋊C4—61no—
437C4⋊C4—61no—
438C4⋊C4—61no—
439C4⋊C4—241no—
440C4⋊C4#23512yes—
441C22⋊C4—122no—
442C22⋊C4—122no—
443C22⋊C4—241no—
444C22⋊C4—241no—
445C22⋊C4—241no—
446C22⋊C4—241no—
447C42—24no—
448C42—21no—
449C42—21no—
450C42—21no—
451C42—21no—
452C42—21no—
453C42—21no—
454C42—42no—
455C42—42no—
456C42—241no—
457C22×C3#52511yescharacteristic
458C22×C3#54611yescharacteristic
459C22×C3#54811yescharacteristic
460C22×C3#57511yescharacteristic
461C22×C3#57511yescharacteristic
462C22×C3#57711yescharacteristic
463C22×C3#57711yescharacteristic
464C2×D3—81no—
465C2×D3—81no—
466C2×D3—81no—
467C2×D3—81no—
468C2×D3—81no—
469C2×D3—81no—
470C2×D3—81no—
471C3×C4—24no—
472C3×C4—41no—
473C3×C4—42no—
474C3×C4—41no—
475C3×C4—41no—
476C3×C4—41no—
477C3×C4—41no—
478C3×C4—41no—
479C3×C4—81no—
480C3×C4—81no—
481C3×C4C2×#12611yescharacteristic
482C3×C4#53911yescharacteristic
483C3×C4#55211yescharacteristic
484C3×C4#55311yescharacteristic
485Dic3—81no—
486Dic3—81no—
487Dic3—81no—
488C23—121no—
489C23—121no—
490C23—121no—
491C23—121no—
492C23—121no—
493C23—121no—
494C23—121no—
495C23C22×#6711yescharacteristic
496Q8—22no—
497Q8—22no—
498Q8—22no—
499Q8#89814yes—
500D4—121no—
501D4—121no—
502D4—121no—
503D4—121no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—22no—
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—21no—
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—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×C4C2×#23511yescharacteristic
544C2×C4C2×#23511yescharacteristic
545C2×C4C2×#23711yescharacteristic
546C2×C4#100411yescharacteristic
547C2×C4C2×#23911yescharacteristic
548C2×C4#100711yescharacteristic
549C8—121no—
550C8—121no—
551C8—121no—
552C8—121no—
553C2×C3#248111yescharacteristic
554C2×C3#311111yescharacteristic
555C2×C3#312611yescharacteristic
556C2×C3#332111yescharacteristic
557C2×C3#332611yescharacteristic
558C2×C3#337111yescharacteristic
559C2×C3#337111yescharacteristic
560D3—161no—
561C22—241no—
562C22—241no—
563C22—241no—
564C22—241no—
565C22—241no—
566C22—241no—
567C22—241no—
568C22#579111yescharacteristic
569C22#580611yescharacteristic
570C22#581711yescharacteristic
571C22#582211yescharacteristic
572C22#582211yescharacteristic
573C22#583711yescharacteristic
574C22#583711yescharacteristic
575C4—24no—
576C4—41no—
577C4—41no—
578C4—41no—
579C4—41no—
580C4—41no—
581C4—42no—
582C4—41no—
583C4—81no—
584C4—81no—
585C4—241no—
586C4—241no—
587C4—241no—
588C4C2×#89811yescharacteristic
589C4#551311yescharacteristic
590C4#551411yescharacteristic
591C4#551511yescharacteristic
592C3#2757711yescharacteristic, complemented, Sylow
593C2—481no—
594C2#8217511yescharacteristic
595C2#8253011yescharacteristic
596C2#8255311yescharacteristic
597C2#8353911yescharacteristic
598C2#8354011yescharacteristic
599C2#8369111yescharacteristic
600C2#8369111yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2351C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#2351D3
hypocenterC3#27577
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2351

Complements

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

NH ≅ G/Ntype
1. C3×#2351C2semidirect
6. #71631C2semidirect
7. #71629C2semidirect
8. #71627C2semidirect
11. #69613C2semidirect
12. #69602C2semidirect
14. #69542C2semidirect
15. #69483C2semidirect
61. #2351D3semidirect
592. C3#27577semidirect

Maximal subgroups

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

subgroupindex
C3×#23512
#736942
#716332
#716312
#716292
#716272
#696292
#696172
#696132
#696022
#695572
#695422
#694832
#275773