Finite Group Databasebeta

← #10877447

Subgroups — #10877447

Order 768.

730 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#73518C211yesmaximal, characteristic
2#73495C211yesmaximal, characteristic
3C3×C8×C22⋊C4C211yesmaximal, characteristic
4#71972C211yesmaximal, characteristic
5C2×#4551C211yesmaximal, characteristic
6#69930C211yesmaximal, characteristic
7#69924C211yesmaximal, characteristic
8#69778C211yesmaximal, characteristic, complemented
9#69767C211yesmaximal, characteristic, complemented
10#69767C211yesmaximal, characteristic, complemented
11#69763C211yesmaximal, characteristic, complemented
12#68690C211yesmaximal, characteristic, complemented
13#68644C211yesmaximal, characteristic, complemented
14#68644C211yesmaximal, characteristic, complemented
15#68638C211yesmaximal, characteristic, complemented
16#27699—31nomaximal, Sylow
17C23×C3×C8C2211yescharacteristic
18C22×C4⋊Dic3C2211yescharacteristic
19C22×C4⋊Dic3C2211yescharacteristic
20C2×#924C2211yescharacteristic
21C2×#870C2211yescharacteristic
22C2×C3×C22⋊C8C2211yescharacteristic
23C2×C3×C4×C8C2211yescharacteristic
24C2×C3×C4×C8C2211yescharacteristic
25C3×C4×C22⋊C4C2211yescharacteristic
26#5212C2211yescharacteristic
27C2×#816C2211yescharacteristic
28C2×#816C2211yescharacteristic
29C2×#816C2211yescharacteristic
30C2×#816C2211yescharacteristic
31#4946—21no—
32#4943—21no—
33#4943—21no—
34#4942—21no—
35#4936C2211yescharacteristic
36#4936C2211yescharacteristic
37#4936C2211yescharacteristic
38#4929C2211yescharacteristic
39#4929C2211yescharacteristic
40#4928C2211yescharacteristic
41#4928C2211yescharacteristic
42#4660—21no—
43#4659—21no—
44#4659—21no—
45#4657—21no—
46C3×#336C2211yescharacteristic
47C3×#336C2211yescharacteristic
48#4551C2211yescharacteristic
49#4551C2211yescharacteristic
50#4551C2211yescharacteristic
51#4551C2211yescharacteristic
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#2503—31no—
61#2442—31no—
62#2101—31no—
63#2101—31no—
64#2074—31no—
65#2026—31no—
66#1991—31no—
67#1933—31no—
68#1933—31no—
69#1892—31no—
70#1872—31no—
71#1835—31no—
72#1823—31no—
73C8×C22⋊C4D311yescharacteristic
74C2×#340—31no—
75C23×C3×C4C2311yescharacteristic
76C23×Dic3—21no—
77C23×Dic3—21no—
78C22×Dic6C2311yescharacteristic
79C22×C3×C8—22no—
80C22×C3×C8—22no—
81C22×C3×C8—41no—
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C2×C3×C22⋊C4D411yescharacteristic
87C2×C3×C22⋊C4D411yescharacteristic
88C2×C3×C42C2311yescharacteristic
89C2×C3×C42C2311yescharacteristic
90C2×#217—21no—
91C2×#217—21no—
92C2×C4⋊Dic3—22no—
93C2×C4⋊Dic3—22no—
94C2×C4⋊Dic3—22no—
95C2×C4⋊Dic3—21no—
96C2×C4⋊Dic3—21no—
97C2×C4⋊Dic3—41no—
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105#924—22no—
106#924—22no—
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112#870—22no—
113#870—41no—
114C3×C22⋊C8—22no—
115C3×C4×C8—22no—
116C3×C4×C8—41no—
117C3×#95D411yescharacteristic
118C3×#95D411yescharacteristic
119C3⋊#95—21no—
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—
130C3⋊#95—21no—
131C3⋊#95—21no—
132#816—22no—
133#816—22no—
134#816—41no—
135#816—41no—
136C23×C8C2×D311yescharacteristic
137C2×#125—31no—
138C2×C22⋊Q8—31no—
139C22×C4⋊C4—31no—
140C22×C4⋊C4—31no—
141C2×#103—31no—
142C2×#103—31no—
143C2×#103—31no—
144C2×#103—31no—
145C2×C22⋊C8C2×D311yescharacteristic
146C2×C4×C8C2×D311yescharacteristic
147C2×C4×C8C2×D311yescharacteristic
148#400—61no—
149#400—61no—
150#400—61no—
151#398—61no—
152#391—31no—
153#391—31no—
154#389—31no—
155#389—31no—
156#385—61no—
157#382—31no—
158#382—31no—
159#382—31no—
160#381—61no—
161#381—61no—
162#380—31no—
163#380—61no—
164C4×C22⋊C4C2×D311yescharacteristic
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#340—31no—
174#340—31no—
175#340—31no—
176#340—31no—
177#336C2×D311yescharacteristic
178#336C2×D311yescharacteristic
179C24×C3C2×D411yescharacteristic
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4#4112yes—
190C22×C3×C4#4112yes—
191C22×C3×C4C2411yescharacteristic
192C22×Dic3—21no—
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3—21no—
197C22×Dic3—21no—
198C22×Dic3—21no—
199C22×Dic3—21no—
200C22×Dic3—42no—
201C22×Dic3—42no—
202C22×Dic3—42no—
203C22×Dic3—42no—
204C22×Dic3—42no—
205C22×Dic3—81no—
206C2×Dic6—21no—
207C2×Dic6—21no—
208C2×Dic6—21no—
209C2×Dic6—21no—
210C2×Dic6—21no—
211C2×Dic6—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—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8—44no—
224C2×C3×C8#4112yes—
225C3×C22⋊C4—22no—
226C3×C22⋊C4—22no—
227C3×C42—21no—
228C3×C42—21no—
229C3×C42—21no—
230C3×C42—21no—
231#217—42no—
232#217—42no—
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—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—42no—
247C4⋊Dic3—42no—
248C4⋊Dic3#4112yes—
249Dic3⋊C4—42no—
250Dic3⋊C4—42no—
251Dic3⋊C4—42no—
252Dic3⋊C4—81no—
253Dic3⋊C4—81no—
254C22×Q8—31no—
255C23×C4—61no—
256C23×C4—61no—
257C23×C4C22×D311yescharacteristic
258C22×C8—22no—
259C22×C8—22no—
260C22×C8—41no—
261C22×C8C22×D311yescharacteristic
262C22×C8C22×D311yescharacteristic
263C22×C8C22×D311yescharacteristic
264C22×C8C22×D311yescharacteristic
265#125—62no—
266#125—121no—
267C22⋊Q8—62no—
268C22⋊Q8—62no—
269C2×C4⋊C4—31no—
270C2×C4⋊C4—31no—
271C2×C4⋊C4—31no—
272C2×C4⋊C4—31no—
273C2×C4⋊C4—31no—
274C2×C4⋊C4—31no—
275C2×C4⋊C4—31no—
276C2×C4⋊C4—62no—
277C2×C4⋊C4—62no—
278C2×C4⋊C4—62no—
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⋊C4—121no—
287C2×C22⋊C4—61no—
288C2×C22⋊C4—61no—
289C2×C22⋊C4D1211yescharacteristic
290C2×C22⋊C4D1211yescharacteristic
291C2×C42C22×D311yescharacteristic
292C2×C42C22×D311yescharacteristic
293#103—62no—
294#103—62no—
295#103—121no—
296#103—121no—
297C22⋊C8—22no—
298C4×C8—22no—
299C4×C8—41no—
300#95—61no—
301#95—61no—
302#95—61no—
303#95—61no—
304#95—61no—
305#95—61no—
306#95—61no—
307#95—61no—
308#95—61no—
309#95—61no—
310#95—61no—
311#95—61no—
312#95—61no—
313#95D1211yescharacteristic
314#95D1211yescharacteristic
315C23×C3—21no—
316C23×C3—21no—
317C23×C3—21no—
318C23×C3—21no—
319C23×C3#12412yes—
320C23×C3#13512yes—
321C23×C3#13512yes—
322C23×C3C22×D411yescharacteristic
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
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#12412yes—
342C2×C3×C4C2×#4111yescharacteristic
343C2×C3×C4C2×#4111yescharacteristic
344C2×C3×C4C2×#4111yescharacteristic
345C2×C3×C4C2×#4111yescharacteristic
346C2×C3×C4H(5,𝔽2)11yescharacteristic
347C2×C3×C42−1+411yescharacteristic
348C2×Dic3—22no—
349C2×Dic3—41no—
350C2×Dic3—41no—
351C2×Dic3—41no—
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×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—84no—
373C2×Dic3—84no—
374Dic6—41no—
375Dic6—41no—
376Dic6—41no—
377Dic6—41no—
378C3×C8—22no—
379C3×C8—41no—
380C3×C8—41no—
381C3×C8—41no—
382C24C2×D1211yescharacteristic
383C2×Q8—61no—
384C2×Q8—61no—
385C2×Q8—61no—
386C2×Q8—61no—
387C2×Q8—61no—
388C2×Q8—61no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—61no—
394C22×C4—61no—
395C22×C4—61no—
396C22×C4—61no—
397C22×C4—61no—
398C22×C4—61no—
399C22×C4—61no—
400C22×C4—61no—
401C22×C4—122no—
402C22×C4—122no—
403C22×C4—122no—
404C22×C4—122no—
405C22×C4—122no—
406C22×C4—241no—
407C22×C4C2×D1211yescharacteristic
408C22×C4C2×D1211yescharacteristic
409C22×C4C2×D1211yescharacteristic
410C22×C4C2×D1211yescharacteristic
411C22×C4C2×D1211yescharacteristic
412C22×C4#23512yes—
413C22×C4#23512yes—
414C22×C4C23×D311yescharacteristic
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—44no—
427C2×C8#23712yes—
428C4⋊C4—32no—
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—61no—
439C4⋊C4—61no—
440C4⋊C4—61no—
441C4⋊C4—61no—
442C4⋊C4—122no—
443C4⋊C4—122no—
444C4⋊C4—122no—
445C4⋊C4—122no—
446C4⋊C4—122no—
447C4⋊C4—241no—
448C4⋊C4—241no—
449C22⋊C4—22no—
450C22⋊C4—22no—
451C22⋊C4—122no—
452C22⋊C4—122no—
453C42—21no—
454C42—21no—
455C42—21no—
456C42—21no—
457C22×C3—22no—
458C22×C3—22no—
459C22×C3—22no—
460C22×C3—22no—
461C22×C3—22no—
462C22×C3—22no—
463C22×C3#48814yes—
464C22×C3C2×#12411yescharacteristic
465C22×C3#54611yescharacteristic
466C22×C3#54711yescharacteristic
467C22×C3C2×#13511yescharacteristic
468C22×C3C2×#13511yescharacteristic
469C22×C3#57711yescharacteristic
470C22×C3#57811yescharacteristic
471C3×C4—22no—
472C3×C4—41no—
473C3×C4—41no—
474C3×C4—41no—
475C3×C4—41no—
476C3×C4#52911yescharacteristic
477C3×C4#53911yescharacteristic
478C3×C4#55111yescharacteristic
479C3×C4#55211yescharacteristic
480Dic3—42no—
481Dic3—81no—
482Dic3—81no—
483Dic3—81no—
484Dic3—81no—
485Dic3—81no—
486Dic3—81no—
487Dic3—81no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23#87512yes—
493C23#90412yes—
494C23#90412yes—
495C23C22×D1211yescharacteristic
496Q8—121no—
497Q8—121no—
498Q8—121no—
499Q8—121no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—21no—
507C2×C4—21no—
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—62no—
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—121no—
528C2×C4—121no—
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—244no—
543C2×C4—244no—
544C2×C4#87512yes—
545C2×C4C2×#23511yescharacteristic
546C2×C4C2×#23511yescharacteristic
547C2×C4C2×#23711yescharacteristic
548C2×C4#100411yescharacteristic
549C2×C4D3×#4111yescharacteristic
550C2×C4#101011yescharacteristic
551C8—22no—
552C8—41no—
553C8—41no—
554C8—41no—
555C2×C3—24no—
556C2×C3#248811yescharacteristic
557C2×C3C2×#48811yescharacteristic
558C2×C3#313711yescharacteristic
559C2×C3#329611yescharacteristic
560C2×C3#330011yescharacteristic
561C2×C3#333611yescharacteristic
562C2×C3#333711yescharacteristic
563C22—22no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22#469714yes—
570C22C2×#87511yescharacteristic
571C22#555411yescharacteristic
572C22#555511yescharacteristic
573C22C2×#90411yescharacteristic
574C22C2×#90411yescharacteristic
575C22#575111yescharacteristic
576C22#575311yescharacteristic
577C4—22no—
578C4—41no—
579C4—41no—
580C4—41no—
581C4—41no—
582C4—122no—
583C4—241no—
584C4—241no—
585C4—241no—
586C4—241no—
587C4—241no—
588C4—241no—
589C4—241no—
590C4#549511yescharacteristic
591C4#553111yescharacteristic
592C4#553811yescharacteristic
593C4#556411yescharacteristic
594C3#2769911yescharacteristic, complemented, Sylow
595C2—24no—
596C2#7350411yescharacteristic
597C2C2×#469711yescharacteristic
598C2#7857511yescharacteristic
599C2#7941311yescharacteristic
600C2#7941411yescharacteristic
601C2#7944011yescharacteristic
602C2#7944911yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C8×C22⋊C4C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenterC8×C22⋊C4D3
hypocenterC3#27699
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4

Complements

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

NH ≅ G/Ntype
8. #69778C2semidirect
9. #69767C2semidirect
10. #69767C2semidirect
11. #69763C2semidirect
12. #68690C2semidirect
13. #68644C2semidirect
14. #68644C2semidirect
15. #68638C2semidirect
594. C3#27699semidirect

Maximal subgroups

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

subgroupindex
#735182
#734952
C3×C8×C22⋊C42
#719722
C2×#45512
#699302
#699242
#697782
#697672
#697632
#686902
#686442
#686382
#276993