Finite Group Databasebeta

← #10853247

Subgroups — #10853247

Order 768.

704 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#74327C211yesmaximal, characteristic
2#73768C211yesmaximal, characteristic
3C3×#1947C211yesmaximal, characteristic, complemented
4#70633C211yesmaximal, characteristic
5#70631C211yesmaximal, characteristic
6#70621C211yesmaximal, characteristic, complemented
7#70618C211yesmaximal, characteristic, complemented
8#70478C211yesmaximal, characteristic
9#70464C211yesmaximal, characteristic, complemented
10#69572C211yesmaximal, characteristic, complemented
11#69459C211yesmaximal, characteristic, complemented
12#68871C211yesmaximal, characteristic
13#68867C211yesmaximal, characteristic
14#68818C211yesmaximal, characteristic, complemented
15#68802C211yesmaximal, characteristic, complemented
16#26018—31nomaximal, Sylow
17C22×ΓL(1,𝔽25)C2211yescharacteristic
18C2×#892C2211yescharacteristic
19C2×D3×C4⋊C4C2211yescharacteristic
20C2×#890C2211yescharacteristic
21C2×C3×#107C2211yescharacteristic
22C2×C3×#107C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C3×#384C2211yescharacteristic
25C3×C4×C4⋊C4C2211yescharacteristic
26C2×(C2×D3)⋊C8C2211yescharacteristic
27C2×#817C2211yescharacteristic
28C2×Dic3⋊C8C2211yescharacteristic
29#4994—21no—
30#4993—21no—
31#4987C2211yescharacteristic
32#4987C2211yescharacteristic
33#4980—21no—
34#4976—21no—
35Dic3×C4⋊C4C2211yescharacteristic
36#4972C2211yescharacteristic
37#4972C2211yescharacteristic
38C2×#808C2211yescharacteristic
39C4×(C2×D3)⋊C4—21no—
40C4×(C2×D3)⋊C4C2211yescharacteristic
41#4933—21no—
42#4928C2211yescharacteristic
43#4928C2211yescharacteristic
44#4667—21no—
45#4652—21no—
46C3×#340C2211yescharacteristic
47C3×#340C2211yescharacteristic
48#4551C2211yescharacteristic
49#4551C2211yescharacteristic
50#4551C2211yescharacteristic
51#4550C2211yescharacteristic
52#4529C2211yescharacteristic
53#4529C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60#2441—31no—
61#2345—31no—
62#2100—31no—
63#2088—31no—
64#2086—31no—
65#2059—31no—
66#1989—31no—
67#1947D311yescharacteristic, complemented
68#1920—31no—
69#1919—31no—
70#1854—31no—
71#1851—31no—
72#1823—31no—
73#1809—31no—
74#1736—31no—
75C22×C4×D3—21no—
76C22×C4×D3C2311yescharacteristic
77C22×C3×C8C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C2×C3×C4⋊C4—21no—
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)⋊C4D411yescharacteristic
91C2×(C2×D3)⋊C4D411yescharacteristic
92C2×C4⋊Dic3—21no—
93C2×C4⋊Dic3C2311yescharacteristic
94C2×C4⋊Dic3C2311yescharacteristic
95C2×C4⋊Dic3C2311yescharacteristic
96C2×C4⋊Dic3C2311yescharacteristic
97C2×Dic3⋊C4—21no—
98C2×Dic3⋊C4—21no—
99C2×Dic3⋊C4—21no—
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—21no—
103C2×C4×Dic3—21no—
104C2×C4×Dic3—21no—
105C2×C4×Dic3C2311yescharacteristic
106C22×C3⋊C8C2311yescharacteristic
107C22×C3⋊C8C2311yescharacteristic
108C2×ΓL(1,𝔽25)—22no—
109C2×ΓL(1,𝔽25)—22no—
110C2×ΓL(1,𝔽25)—41no—
111#892—22no—
112#892—41no—
113D3×C4⋊C4—22no—
114D3×C4⋊C4Q814yes—
115#890—22no—
116#890—41no—
117C3×#107—41no—
118C3×#107—41no—
119C3×C4⋊C8—22no—
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⋊#95D411yescharacteristic
129C3⋊#95D411yescharacteristic
130(C2×D3)⋊C8—22no—
131#817—22no—
132Dic3⋊C8—41no—
133#808—22no—
134C22×M16—31no—
135C2×#125—31no—
136C2×#117—31no—
137C22×C4⋊C4—31no—
138C2×#107C2×D311yescharacteristic
139C2×#107C2×D311yescharacteristic
140C2×#106—31no—
141C2×#106—31no—
142C2×C4⋊C8—31no—
143C2×C4⋊C8C2×D311yescharacteristic
144C2×C22⋊C8—31no—
145#399—61no—
146#396—61no—
147#389—31no—
148#389—31no—
149#389—31no—
150#389—31no—
151#389—61no—
152#387—61no—
153#387—61no—
154#387—61no—
155#384C2×D311yescharacteristic
156#380—31no—
157#380—31no—
158#380—61no—
159C4×C4⋊C4—31no—
160C4×C4⋊C4C2×D311yescharacteristic
161C4×C22⋊C4—31no—
162C4×C22⋊C4—61no—
163#340—31no—
164#340—31no—
165#340—31no—
166#340—31no—
167#340—31no—
168#340—31no—
169#340—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340C2×D311yescharacteristic
174#340C2×D311yescharacteristic
175#336—31no—
176#336—31no—
177C23×D3C2×D411yescharacteristic
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×Dic3—21no—
189C22×Dic3C2×D411yescharacteristic
190C22×Dic3C2×D411yescharacteristic
191C22×Dic3C2×D411yescharacteristic
192C2×C4×D3—21no—
193C2×C4×D3—21no—
194C2×C4×D3—21no—
195C2×C4×D3—21no—
196C2×C4×D3—24no—
197C2×C4×D3—81no—
198C2×C4×D3—81no—
199C2×C4×D3C2×Q812yes—
200C2×C4×D3#4112yes—
201C2×C3×C8—21no—
202C2×C3×C8—21no—
203C2×C3×C8—21no—
204C2×C3×C8—21no—
205C2×C3×C8—21no—
206C2×C3×C8#4112yes—
207C3×C4⋊C4—21no—
208C3×C4⋊C4—21no—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—21no—
211C3×C4⋊C4—21no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—42no—
215C3×C4⋊C4—81no—
216C3×C4⋊C4C2×Q812yes—
217C3×C42—21no—
218C3×C42—21no—
219C3×C42—81no—
220(C2×D3)⋊C4—22no—
221(C2×D3)⋊C4—41no—
222(C2×D3)⋊C4—42no—
223(C2×D3)⋊C4—42no—
224(C2×D3)⋊C4—81no—
225C4⋊Dic3—21no—
226C4⋊Dic3—21no—
227C4⋊Dic3—21no—
228C4⋊Dic3—21no—
229C4⋊Dic3—21no—
230C4⋊Dic3—21no—
231C4⋊Dic3—42no—
232C4⋊Dic3C2×Q812yes—
233C4⋊Dic3#4112yes—
234Dic3⋊C4—24no—
235Dic3⋊C4—42no—
236Dic3⋊C4—42no—
237Dic3⋊C4—81no—
238Dic3⋊C4—81no—
239Dic3⋊C4—81no—
240C4×Dic3—21no—
241C4×Dic3—21no—
242C4×Dic3—42no—
243C4×Dic3—81no—
244C2×C3⋊C8—21no—
245C2×C3⋊C8—21no—
246C2×C3⋊C8—21no—
247C2×C3⋊C8—21no—
248C2×C3⋊C8—21no—
249C2×C3⋊C8—21no—
250ΓL(1,𝔽25)—44no—
251C23×C4—31no—
252C23×C4—61no—
253C2×M16—62no—
254C2×M16—62no—
255C2×M16—121no—
256C22×C8—31no—
257C22×C8—31no—
258C22×C8C22×D311yescharacteristic
259C22×C8C22×D311yescharacteristic
260#125—62no—
261#125—121no—
262#117—62no—
263#117—121no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—31no—
267C2×C4⋊C4—31no—
268C2×C4⋊C4—34no—
269C2×C4⋊C4—31no—
270C2×C4⋊C4—31no—
271C2×C4⋊C4—62no—
272C2×C4⋊C4—61no—
273C2×C4⋊C4—61no—
274C2×C4⋊C4—61no—
275C2×C4⋊C4—61no—
276C2×C4⋊C4—61no—
277C2×C4⋊C4—61no—
278C2×C4⋊C4—61no—
279C2×C4⋊C4C22×D311yescharacteristic
280C2×C4⋊C4C22×D311yescharacteristic
281C2×C4⋊C4C22×D311yescharacteristic
282C2×C4⋊C4C22×D311yescharacteristic
283C2×C22⋊C4—31no—
284C2×C22⋊C4—31no—
285C2×C22⋊C4—61no—
286C2×C22⋊C4—61no—
287C2×C22⋊C4—61no—
288C2×C42—21no—
289C2×C42—31no—
290C2×C42—61no—
291C2×C42—61no—
292C2×C42C22×D311yescharacteristic
293#107—41no—
294#107—41no—
295#106—62no—
296#106—62no—
297C4⋊C8—22no—
298C4⋊C8—121no—
299C22⋊C8—62no—
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—21no—
312C22×D3—21no—
313C22×D3—21no—
314C22×D3—21no—
315C22×D3—21no—
316C22×D3—21no—
317C22×D3C22⋊Q812yes—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—22no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—41no—
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#13712yes—
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—22no—
345C2×Dic3—21no—
346C2×Dic3—21no—
347C2×Dic3—22no—
348C2×Dic3—21no—
349C2×Dic3—21no—
350C2×Dic3—21no—
351C2×Dic3—21no—
352C2×Dic3—41no—
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×Dic3—41no—
362C2×Dic3C22⋊Q812yes—
363C4×D3—22no—
364C4×D3—22no—
365C4×D3—22no—
366C4×D3—22no—
367C4×D3—84no—
368C3×C8—22no—
369C3×C8—41no—
370C3⋊C8—41no—
371C3⋊C8—41no—
372C24—31no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—31no—
378C22×C4—32no—
379C22×C4—32no—
380C22×C4—31no—
381C22×C4—31no—
382C22×C4—64no—
383C22×C4—61no—
384C22×C4—61no—
385C22×C4—61no—
386C22×C4—61no—
387C22×C4—61no—
388C22×C4—61no—
389C22×C4—61no—
390C22×C4—61no—
391C22×C4—241no—
392C22×C4—241no—
393C22×C4D3×D411yescharacteristic
394C22×C4D3×D411yescharacteristic
395C22×C4C23×D311yescharacteristic
396M16—124no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—61no—
403C2×C8—61no—
404C2×C8—61no—
405C2×C8—61no—
406C2×C8—61no—
407C2×C8—61no—
408C2×C8#23512yes—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—32no—
417C4⋊C4—32no—
418C4⋊C4—42no—
419C4⋊C4—64no—
420C4⋊C4—61no—
421C4⋊C4—61no—
422C4⋊C4—61no—
423C4⋊C4—61no—
424C4⋊C4—61no—
425C4⋊C4—61no—
426C4⋊C4—81no—
427C4⋊C4—122no—
428C4⋊C4—122no—
429C4⋊C4—122no—
430C4⋊C4—241no—
431C4⋊C4—241no—
432C4⋊C4—241no—
433C4⋊C4D3×Q812yes—
434C22⋊C4—62no—
435C22⋊C4—121no—
436C22⋊C4—122no—
437C22⋊C4—122no—
438C22⋊C4—241no—
439C42—21no—
440C42—21no—
441C42—61no—
442C42—61no—
443C42—81no—
444C42—122no—
445C42—241no—
446C22×C3C2×C22⋊Q811yescharacteristic
447C22×C3#54611yescharacteristic
448C22×C3#54711yescharacteristic
449C22×C3C2×#13711yescharacteristic
450C22×C3#57511yescharacteristic
451C22×C3#57611yescharacteristic
452C22×C3#57711yescharacteristic
453C2×D3—22no—
454C2×D3—22no—
455C2×D3—22no—
456C2×D3—41no—
457C2×D3—41no—
458C2×D3—41no—
459C2×D3—41no—
460C3×C4—22no—
461C3×C4—42no—
462C3×C4—41no—
463C3×C4—81no—
464C3×C4—81no—
465C3×C4—81no—
466C3×C4#53311yescharacteristic
467C3×C4#54411yescharacteristic
468C3×C4#55111yescharacteristic
469C3×C4#55211yescharacteristic
470Dic3—22no—
471Dic3—41no—
472Dic3—42no—
473Dic3—42no—
474Dic3—41no—
475Dic3—81no—
476Dic3—81no—
477C23—32no—
478C23—61no—
479C23—61no—
480C23—61no—
481C23—61no—
482C23—61no—
483C23—61no—
484C23C2×D3×D411yescharacteristic
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—22no—
491C2×C4—32no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—61no—
504C2×C4—61no—
505C2×C4—62no—
506C2×C4—62no—
507C2×C4—62no—
508C2×C4—62no—
509C2×C4—61no—
510C2×C4—61no—
511C2×C4—61no—
512C2×C4—61no—
513C2×C4—61no—
514C2×C4—61no—
515C2×C4—62no—
516C2×C4—62no—
517C2×C4—121no—
518C2×C4—121no—
519C2×C4—121no—
520C2×C4—121no—
521C2×C4—121no—
522C2×C4—121no—
523C2×C4—121no—
524C2×C4—121no—
525C2×C4—121no—
526C2×C4—121no—
527C2×C4—244no—
528C2×C4#91812yes—
529C2×C4C2×#23511yescharacteristic
530C2×C4#100411yescharacteristic
531C2×C4C2×D3×Q811yescharacteristic
532C2×C4#100711yescharacteristic
533C2×C4D3×#4111yescharacteristic
534C2×C4D3×#4111yescharacteristic
535C8—22no—
536C8—41no—
537C8—121no—
538C8—121no—
539C2×C3#243811yescharacteristic
540C2×C3#307811yescharacteristic
541C2×C3#308111yescharacteristic
542C2×C3#330911yescharacteristic
543C2×C3#333111yescharacteristic
544C2×C3#334611yescharacteristic
545C2×C3#335111yescharacteristic
546D3—42no—
547C22—62no—
548C22—62no—
549C22—62no—
550C22—121no—
551C22—121no—
552C22—121no—
553C22—121no—
554C22#550711yescharacteristic
555C22#561211yescharacteristic
556C22D3×C22⋊Q811yescharacteristic
557C22#575711yescharacteristic
558C22C2×#91811yescharacteristic
559C22#577311yescharacteristic
560C22#577811yescharacteristic
561C4—22no—
562C4—42no—
563C4—41no—
564C4—62no—
565C4—81no—
566C4—81no—
567C4—81no—
568C4—122no—
569C4—121no—
570C4—122no—
571C4—121no—
572C4—241no—
573C4—241no—
574C4#555611yescharacteristic
575C4#557811yescharacteristic
576C4#569311yescharacteristic
577C4#572911yescharacteristic
578C3#2601811yescharacteristic, complemented, Sylow
579C2—122no—
580C2#7622211yescharacteristic
581C2#7915911yescharacteristic
582C2#7923111yescharacteristic
583C2#8035411yescharacteristic
584C2#8049611yescharacteristic
585C2#8096711yescharacteristic
586C2#8110811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1947C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#1947D3
hypocenterC3#26018
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1947

Complements

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

NH ≅ G/Ntype
3. C3×#1947C2semidirect
6. #70621C2semidirect
7. #70618C2semidirect
9. #70464C2semidirect
10. #69572C2semidirect
11. #69459C2semidirect
14. #68818C2semidirect
15. #68802C2semidirect
67. #1947D3semidirect
578. C3#26018semidirect

Maximal subgroups

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

subgroupindex
#743272
#737682
C3×#19472
#706332
#706312
#706212
#706182
#704782
#704642
#695722
#694592
#688712
#688672
#688182
#688022
#260183