Finite Group Databasebeta

← #10853357

Subgroups — #10853357

Order 768.

706 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#76061C211yesmaximal, characteristic, complemented
2#73646C211yesmaximal, characteristic
3C3×#1921C211yesmaximal, characteristic, complemented
4#70639C211yesmaximal, characteristic
5#70616C211yesmaximal, characteristic, complemented
6#69629C211yesmaximal, characteristic
7#69490C211yesmaximal, characteristic, complemented
8#68899C211yesmaximal, characteristic
9#68893C211yesmaximal, characteristic
10#68848C211yesmaximal, characteristic, complemented
11#68819C211yesmaximal, characteristic, complemented
12#68801C211yesmaximal, characteristic, complemented
13#68780C211yesmaximal, characteristic
14#68678C211yesmaximal, characteristic
15#68617C211yesmaximal, characteristic, complemented
16#26028—31nomaximal, Sylow
17C2×#893C2211yescharacteristic
18C2×#890C2211yescharacteristic
19C2×#869C2211yescharacteristic
20C2×C3×#107C2211yescharacteristic
21C3×#389C2211yescharacteristic
22C3×C4×C4⋊C4C2211yescharacteristic
23C2×#821C2211yescharacteristic
24C2×#821C2211yescharacteristic
25C2×Dic3⋊C8C2211yescharacteristic
26C2×Dic3⋊C8C2211yescharacteristic
27C2×C8×Dic3C2211yescharacteristic
28#4988C2211yescharacteristic
29#4984—21no—
30#4984C2211yescharacteristic
31#4982—21no—
32#4977C2211yescharacteristic
33#4977C2211yescharacteristic
34Dic3×C4⋊C4C2211yescharacteristic
35Dic3×C4⋊C4C2211yescharacteristic
36C2×#809C2211yescharacteristic
37C2×#809C2211yescharacteristic
38C2×#808C2211yescharacteristic
39#4940—21no—
40#4939C2211yescharacteristic
41C4×(C2×D3)⋊C4—21no—
42C4×Dic3⋊C4C2211yescharacteristic
43C2×C4⋊(C3⋊C8)C2211yescharacteristic
44#4669—21no—
45#4664—21no—
46#4656—21no—
47#4646—21no—
48C3×#340C2211yescharacteristic
49C3×#340C2211yescharacteristic
50C3×#340C2211yescharacteristic
51C3×#336C2211yescharacteristic
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4550C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2550—31no—
61#2357—31no—
62#2067—31no—
63#2065—31no—
64#2025—31no—
65#2023—31no—
66#1989—31no—
67#1988—31no—
68#1949—31no—
69#1939—31no—
70#1921D311yescharacteristic, complemented
71#1920—31no—
72#1835—31no—
73#1804—31no—
74C4×#102—31no—
75C22×D12C2311yescharacteristic
76C22×C4×D3—21no—
77C22×Dic6—21no—
78C22×C3×C8C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C42—21no—
86C2×C3×C42C2311yescharacteristic
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⋊Dic3—21no—
95C2×C4⋊Dic3C2311yescharacteristic
96C2×C4⋊Dic3C2311yescharacteristic
97C2×C4⋊Dic3C2311yescharacteristic
98C2×Dic3⋊C4—21no—
99C2×Dic3⋊C4D411yescharacteristic
100C2×Dic3⋊C4D411yescharacteristic
101C2×C4×Dic3—21no—
102C2×C4×Dic3—21no—
103C2×C4×Dic3C2311yescharacteristic
104C2×C4×Dic3C2311yescharacteristic
105C22×C3⋊C8C2311yescharacteristic
106C22×C3⋊C8C2311yescharacteristic
107#893—22no—
108#893Q814yescomplemented
109#890—22no—
110#890—41no—
111#869—22no—
112#869—41no—
113C3×#107—22no—
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⋊#95D411yescharacteristic
124C3⋊#95D411yescharacteristic
125#821—22no—
126#821—41no—
127Dic3⋊C8—22no—
128Dic3⋊C8—41no—
129C8×Dic3—22no—
130#809—22no—
131#809—41no—
132#808—22no—
133C4⋊(C3⋊C8)—41no—
134C2×C4⋊Q8—31no—
135C2×#125—31no—
136C2×C4×D4—31no—
137C2×#107C2×D311yescharacteristic
138C2×#106—31no—
139C2×C4⋊C8—31no—
140C2×C4⋊C8—31no—
141C2×C4⋊C8—31no—
142C2×#102—31no—
143C2×#102—31no—
144C2×#102—31no—
145C2×#102—31no—
146C2×C4×C8—31no—
147#398—61no—
148#395—61no—
149#391—61no—
150#390—31no—
151#390—31no—
152#389—31no—
153#389—61no—
154#389C2×D311yescharacteristic
155#388—61no—
156#388—61no—
157#382—31no—
158#382—31no—
159#381—61no—
160C4×C4⋊C4—31no—
161C4×C4⋊C4—31no—
162C4×C4⋊C4—31no—
163C4×C4⋊C4C2×D311yescharacteristic
164C4×C22⋊C4—61no—
165#340—31no—
166#340—31no—
167#340—31no—
168#340—31no—
169#340—31no—
170#340—31no—
171#340—31no—
172#340C2×D311yescharacteristic
173#340C2×D311yescharacteristic
174#340C2×D311yescharacteristic
175#336—31no—
176#336C2×D311yescharacteristic
177C23×D3—21no—
178C22×C3×C4—21no—
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2411yescharacteristic
185C22×Dic3—21no—
186C22×Dic3—21no—
187C22×Dic3—21no—
188C22×Dic3C2×D411yescharacteristic
189C22×Dic3C2×D411yescharacteristic
190C22×Dic3C2×D411yescharacteristic
191C22×Dic3C2×D411yescharacteristic
192C2×D12—21no—
193C2×D12—21no—
194C2×D12—21no—
195C2×D12—21no—
196C2×D12—21no—
197C2×D12C2×Q812yes—
198C2×C4×D3—24no—
199C2×C4×D3—81no—
200C2×C4×D3—81no—
201C2×Dic6—42no—
202C2×Dic6—42no—
203C2×Dic6—81no—
204C2×C3×C8—21no—
205C2×C3×C8—21no—
206C2×C3×C8—21no—
207C2×C3×C8—21no—
208C2×C3×C8—21no—
209C2×C3×C8—21no—
210C3×C4⋊C4—21no—
211C3×C4⋊C4—21no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—42no—
217C3×C4⋊C4C2×Q812yes—
218C3×C4⋊C4#4112yes—
219C3×C42—21no—
220C3×C42—21no—
221C3×C42—81no—
222(C2×D3)⋊C4—24no—
223(C2×D3)⋊C4—42no—
224(C2×D3)⋊C4—42no—
225(C2×D3)⋊C4—81no—
226(C2×D3)⋊C4—81no—
227C4⋊Dic3—21no—
228C4⋊Dic3—21no—
229C4⋊Dic3—21no—
230C4⋊Dic3—21no—
231C4⋊Dic3—21no—
232C4⋊Dic3—42no—
233C4⋊Dic3—42no—
234C4⋊Dic3—81no—
235C4⋊Dic3#4112yes—
236Dic3⋊C4—22no—
237Dic3⋊C4—41no—
238Dic3⋊C4—42no—
239C4×Dic3—21no—
240C4×Dic3—21no—
241C4×Dic3—81no—
242C4×Dic3—81no—
243C4×Dic3C2×Q812yes—
244C4×Dic3#4112yes—
245C2×C3⋊C8—21no—
246C2×C3⋊C8—21no—
247C2×C3⋊C8—21no—
248C2×C3⋊C8—21no—
249C2×C3⋊C8—21no—
250C2×C3⋊C8—21no—
251C22×Q8—61no—
252C22×D4—31no—
253C23×C4—61no—
254C22×C8—31no—
255C22×C8—31no—
256C22×C8C22×D311yescharacteristic
257C22×C8C22×D311yescharacteristic
258C4⋊Q8—62no—
259C4⋊Q8—121no—
260#125—62no—
261#125—121no—
262C4×D4—34no—
263C4×D4—62no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—31no—
266C2×C4⋊C4—31no—
267C2×C4⋊C4—31no—
268C2×C4⋊C4—31no—
269C2×C4⋊C4—31no—
270C2×C4⋊C4—61no—
271C2×C4⋊C4—61no—
272C2×C4⋊C4—61no—
273C2×C4⋊C4—61no—
274C2×C4⋊C4C22×D311yescharacteristic
275C2×C4⋊C4C22×D311yescharacteristic
276C2×C4⋊C4C22×D311yescharacteristic
277C2×C4⋊C4C22×D311yescharacteristic
278C2×C22⋊C4—61no—
279C2×C22⋊C4—61no—
280C2×C22⋊C4—61no—
281C2×C22⋊C4—61no—
282C2×C22⋊C4—61no—
283C2×C42—21no—
284C2×C42—31no—
285C2×C42—31no—
286C2×C42—61no—
287C2×C42—61no—
288C2×C42C22×D311yescharacteristic
289#107—22no—
290#106—62no—
291C4⋊C8—62no—
292C4⋊C8—121no—
293C4⋊C8—121no—
294#102—62no—
295#102—62no—
296#102—121no—
297#102—121no—
298C4×C8—62no—
299#95—21no—
300#95—21no—
301#95—31no—
302#95—31no—
303#95—61no—
304#95—61no—
305#95—61no—
306#95—61no—
307#95—61no—
308#95—61no—
309#95—61no—
310C23×C3C22×D411yescharacteristic
311C22×D3—22no—
312C22×D3—41no—
313C22×D3—41no—
314C22×D3—41no—
315C22×D3—41no—
316C22×D3—41no—
317C22×D3—41no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—22no—
321C2×C3×C4—21no—
322C2×C3×C4—22no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—41no—
326C2×C3×C4—41no—
327C2×C3×C4—41no—
328C2×C3×C4—41no—
329C2×C3×C4—41no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
335C2×C3×C4#13512yes—
336C2×C3×C4C22×Q811yescharacteristic
337C2×C3×C4C2×#4111yescharacteristic
338C2×C3×C4C2×#4111yescharacteristic
339C2×C3×C4C2×#4111yescharacteristic
340C2×C3×C4H(5,𝔽2)11yescharacteristic
341C2×C3×C42−1+411yescharacteristic
342C2×Dic3—21no—
343C2×Dic3—21no—
344C2×Dic3—21no—
345C2×Dic3—21no—
346C2×Dic3—21no—
347C2×Dic3—21no—
348C2×Dic3—22no—
349C2×Dic3—21no—
350C2×Dic3—21no—
351C2×Dic3—21no—
352C2×Dic3—21no—
353C2×Dic3—41no—
354C2×Dic3—41no—
355C2×Dic3—41no—
356C2×Dic3—41no—
357C2×Dic3—41no—
358C2×Dic3—41no—
359C2×Dic3—41no—
360C2×Dic3—41no—
361C2×Dic3C22⋊Q812yes—
362C2×Dic3C22⋊Q812yes—
363D12—22no—
364D12—22no—
365D12—41no—
366D12—41no—
367C4×D3—84no—
368Dic6—84no—
369C3×C8—41no—
370C3×C8—41no—
371C3⋊C8—41no—
372C3⋊C8—41no—
373C24—61no—
374C2×Q8—122no—
375C2×Q8—122no—
376C2×Q8—241no—
377C2×D4—32no—
378C2×D4—61no—
379C2×D4—61no—
380C2×D4—61no—
381C2×D4—61no—
382C2×D4—61no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—31no—
388C22×C4—31no—
389C22×C4—31no—
390C22×C4—31no—
391C22×C4—64no—
392C22×C4—61no—
393C22×C4—61no—
394C22×C4—61no—
395C22×C4—241no—
396C22×C4—241no—
397C22×C4D3×D411yescharacteristic
398C22×C4D3×D411yescharacteristic
399C22×C4C23×D311yescharacteristic
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8—21no—
406C2×C8—61no—
407C2×C8—61no—
408C2×C8—61no—
409C2×C8—61no—
410C2×C8—61no—
411C2×C8—61no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—32no—
419C4⋊C4—42no—
420C4⋊C4—62no—
421C4⋊C4—61no—
422C4⋊C4—61no—
423C4⋊C4—61no—
424C4⋊C4—61no—
425C4⋊C4—61no—
426C4⋊C4—121no—
427C4⋊C4—122no—
428C4⋊C4—122no—
429C4⋊C4—122no—
430C4⋊C4—241no—
431C4⋊C4#23512yes—
432C4⋊C4D3×Q812yes—
433C22⋊C4—64no—
434C22⋊C4—122no—
435C22⋊C4—122no—
436C22⋊C4—241no—
437C22⋊C4—241no—
438C42—21no—
439C42—21no—
440C42—32no—
441C42—32no—
442C42—61no—
443C42—61no—
444C42—81no—
445C42—241no—
446C42—241no—
447C22×C3C2×C22⋊Q811yescharacteristic
448C22×C3#54611yescharacteristic
449C22×C3#54811yescharacteristic
450C22×C3C2×#13511yescharacteristic
451C22×C3#57511yescharacteristic
452C22×C3#57611yescharacteristic
453C22×C3#57711yescharacteristic
454C2×D3—42no—
455C2×D3—42no—
456C2×D3—42no—
457C2×D3—81no—
458C2×D3—81no—
459C2×D3—81no—
460C2×D3—81no—
461C3×C4—22no—
462C3×C4—42no—
463C3×C4—42no—
464C3×C4—41no—
465C3×C4—81no—
466C3×C4—81no—
467C3×C4#53311yescharacteristic
468C3×C4#54411yescharacteristic
469C3×C4#55511yescharacteristic
470C3×C4#55611yescharacteristic
471Dic3—22no—
472Dic3—22no—
473Dic3—42no—
474Dic3—41no—
475Dic3—41no—
476Dic3—81no—
477Dic3—81no—
478C23—62no—
479C23—121no—
480C23—121no—
481C23—121no—
482C23—121no—
483C23—121no—
484C23—121no—
485C23C2×D3×D411yescharacteristic
486Q8—244no—
487D4—62no—
488D4—62no—
489D4—121no—
490D4—121no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—32no—
499C2×C4—32no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—61no—
511C2×C4—61no—
512C2×C4—61no—
513C2×C4—61no—
514C2×C4—61no—
515C2×C4—61no—
516C2×C4—62no—
517C2×C4—61no—
518C2×C4—61no—
519C2×C4—61no—
520C2×C4—61no—
521C2×C4—121no—
522C2×C4—121no—
523C2×C4—121no—
524C2×C4—121no—
525C2×C4—121no—
526C2×C4—121no—
527C2×C4—121no—
528C2×C4—121no—
529C2×C4—244no—
530C2×C4#91912yes—
531C2×C4C2×#23511yescharacteristic
532C2×C4C2×D3×Q811yescharacteristic
533C2×C4D3×#4111yescharacteristic
534C2×C4D3×#4111yescharacteristic
535C2×C4#100911yescharacteristic
536C2×C4#101011yescharacteristic
537C8—41no—
538C8—41no—
539C8—121no—
540C8—121no—
541C2×C3#243911yescharacteristic
542C2×C3#307611yescharacteristic
543C2×C3#308111yescharacteristic
544C2×C3#329711yescharacteristic
545C2×C3#332611yescharacteristic
546C2×C3#336511yescharacteristic
547C2×C3#337011yescharacteristic
548D3—82no—
549C22—122no—
550C22—122no—
551C22—122no—
552C22—241no—
553C22—241no—
554C22—241no—
555C22—241no—
556C22#555211yescharacteristic
557C22D3×C22⋊Q811yescharacteristic
558C22#565411yescharacteristic
559C22C2×#91911yescharacteristic
560C22#577011yescharacteristic
561C22#577311yescharacteristic
562C22#577411yescharacteristic
563C4—22no—
564C4—41no—
565C4—42no—
566C4—42no—
567C4—62no—
568C4—62no—
569C4—81no—
570C4—81no—
571C4—121no—
572C4—121no—
573C4—122no—
574C4—241no—
575C4—241no—
576C4#553211yescharacteristic
577C4#557811yescharacteristic
578C4#568911yescharacteristic
579C4#570211yescharacteristic
580C3#2602811yescharacteristic, complemented, Sylow
581C2—242no—
582C2#7498311yescharacteristic
583C2#7970111yescharacteristic
584C2#7985211yescharacteristic
585C2#8110711yescharacteristic
586C2#8116811yescharacteristic
587C2#8141011yescharacteristic
588C2#8148511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1921C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1921D3
hypocenterC3#26028
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1921

Complements

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

NH ≅ G/Ntype
1. #76061C2semidirect
3. C3×#1921C2semidirect
5. #70616C2semidirect
7. #69490C2semidirect
10. #68848C2semidirect
11. #68819C2semidirect
12. #68801C2semidirect
15. #68617C2semidirect
70. #1921D3semidirect
108. #893Q8semidirect
580. C3#26028semidirect

Maximal subgroups

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

subgroupindex
#760612
#736462
C3×#19212
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