Finite Group Databasebeta

← #10877459

Subgroups — #10877459

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
1C3×#2442C211yesmaximal, characteristic
2#73518C211yesmaximal, characteristic
3#71744C211yesmaximal, characteristic, complemented
4#71740C211yesmaximal, characteristic, complemented
5#71559C211yesmaximal, characteristic
6C2×#4529C211yesmaximal, characteristic
7#69614C211yesmaximal, characteristic, complemented
8#69614C211yesmaximal, characteristic, complemented
9#69613C211yesmaximal, characteristic, complemented
10#69613C211yesmaximal, characteristic, complemented
11#69560C211yesmaximal, characteristic, complemented
12#69547C211yesmaximal, characteristic, complemented
13#69514C211yesmaximal, characteristic
14#69501C211yesmaximal, characteristic
15C22⋊C4×C3⋊C8C211yesmaximal, characteristic
16#27699—31nomaximal, Sylow
17C2×C3×#125C2211yescharacteristic
18C22×C3×C4⋊C4C2211yescharacteristic
19C22×C4⋊Dic3C2211yescharacteristic
20C2×#924C2211yescharacteristic
21C23×C3⋊C8C2211yescharacteristic
22C3×#400—21no—
23C3×#398—21no—
24C3×#389C2211yescharacteristic
25C3×#389C2211yescharacteristic
26C3×#385—21no—
27C3×#382C2211yescharacteristic
28C3×#380—21no—
29C3×#380C2211yescharacteristic
30C3×C4×C22⋊C4C2211yescharacteristic
31C2×#830C2211yescharacteristic
32C2×#810C2211yescharacteristic
33C2×#810C2211yescharacteristic
34C2×#810C2211yescharacteristic
35C2×#810C2211yescharacteristic
36#4943—21no—
37#4943—21no—
38#4936C2211yescharacteristic
39#4936C2211yescharacteristic
40#4929C2211yescharacteristic
41#4929C2211yescharacteristic
42C2×C4×C3⋊C8C2211yescharacteristic
43C2×C4×C3⋊C8C2211yescharacteristic
44#4659—21no—
45#4659—21no—
46#4529C2211yescharacteristic
47#4529C2211yescharacteristic
48#4529C2211yescharacteristic
49#4529C2211yescharacteristic
50#4529C2211yescharacteristic
51#4529C2211yescharacteristic
52#4529C2211yescharacteristic
53#4529C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4524C2211yescharacteristic
59#4524C2211yescharacteristic
60#2503—31no—
61#2442D311yescharacteristic
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⋊C4—31no—
74C2×#340—31no—
75C23×C3×C4—21no—
76C23×C3×C4C2311yescharacteristic
77C23×Dic3—21no—
78C22×Dic6C2311yescharacteristic
79C3×#125—22no—
80C3×#125—41no—
81C2×C3×C4⋊C4—22no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—41no—
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4D411yescharacteristic
94C2×C3×C22⋊C4D411yescharacteristic
95C2×C3×C42C2311yescharacteristic
96C2×C3×C42C2311yescharacteristic
97C2×#217—21no—
98C2×C4⋊Dic3—22no—
99C2×C4⋊Dic3—22no—
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104#924—22no—
105#924—22no—
106C2×Dic3⋊C4—21no—
107C22×C3⋊C8—22no—
108C22×C3⋊C8—22no—
109C22×C3⋊C8—41no—
110C22×C3⋊C8C2311yescharacteristic
111C22×C3⋊C8C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113C22×C3⋊C8C2311yescharacteristic
114C3×#95—21no—
115C3×#95—21no—
116C3×#95—21no—
117C3×#95—21no—
118C3×#95—21no—
119C3×#95D411yescharacteristic
120C3×#95D411yescharacteristic
121C3⋊#95—21no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129#830—22no—
130#810—22no—
131#810—22no—
132#810—41no—
133#810—41no—
134C4×C3⋊C8—22no—
135C4×C3⋊C8—41no—
136C23×C8—31no—
137C2×#125C2×D311yescharacteristic
138C2×C22⋊Q8—31no—
139C22×C4⋊C4—31no—
140C22×C4⋊C4C2×D311yescharacteristic
141C2×#103—31no—
142C2×#103—31no—
143C2×#103—31no—
144C2×#103—31no—
145C2×C22⋊C8—31no—
146C2×C4×C8—31no—
147C2×C4×C8—31no—
148#400—21no—
149#400—61no—
150#400—61no—
151#398—21no—
152#391—31no—
153#391—31no—
154#389C2×D311yescharacteristic
155#389C2×D311yescharacteristic
156#385—21no—
157#382—31no—
158#382—31no—
159#382C2×D311yescharacteristic
160#381—61no—
161#381—61no—
162#380—21no—
163#380C2×D311yescharacteristic
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#336—31no—
178#336—31no—
179C24×C3C2×D411yescharacteristic
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—42no—
189C22×C3×C4—42no—
190C22×C3×C4—81no—
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4#4112yes—
197C22×C3×C4#4112yes—
198C22×C3×C4C2411yescharacteristic
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3—21no—
203C22×Dic3—42no—
204C22×Dic3—42no—
205C22×Dic3—42no—
206C2×Dic6—21no—
207C2×Dic6—21no—
208C2×Dic6—21no—
209C2×Dic6—21no—
210C2×Dic6—21no—
211C2×Dic6—21no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—42no—
220C3×C4⋊C4—42no—
221C3×C4⋊C4—42no—
222C3×C4⋊C4—81no—
223C3×C4⋊C4—81no—
224C3×C4⋊C4#4112yes—
225C3×C22⋊C4—22no—
226C3×C22⋊C4—22no—
227C3×C22⋊C4—42no—
228C3×C42—21no—
229C3×C42—21no—
230C3×C42—21no—
231C3×C42—21no—
232#217—42no—
233C4⋊Dic3—21no—
234C4⋊Dic3—21no—
235C4⋊Dic3—21no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—42no—
240Dic3⋊C4—42no—
241C2×C3⋊C8—21no—
242C2×C3⋊C8—21no—
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—44no—
253C2×C3⋊C8#4112yes—
254C22×Q8—31no—
255C23×C4—21no—
256C23×C4—61no—
257C23×C4C22×D311yescharacteristic
258C22×C8—31no—
259C22×C8—31no—
260C22×C8—31no—
261C22×C8—31no—
262C22×C8—62no—
263C22×C8—62no—
264C22×C8—121no—
265#125—22no—
266#125—41no—
267C22⋊Q8—62no—
268C22⋊Q8—62no—
269C2×C4⋊C4—22no—
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—41no—
279C2×C4⋊C4—62no—
280C2×C4⋊C4—62no—
281C2×C4⋊C4—61no—
282C2×C4⋊C4—61no—
283C2×C4⋊C4C22×D311yescharacteristic
284C2×C4⋊C4C22×D311yescharacteristic
285C2×C4⋊C4C22×D311yescharacteristic
286C2×C4⋊C4C22×D311yescharacteristic
287C2×C22⋊C4—21no—
288C2×C22⋊C4—61no—
289C2×C22⋊C4#6711yescharacteristic
290C2×C22⋊C4#6711yescharacteristic
291C2×C42C22×D311yescharacteristic
292C2×C42C22×D311yescharacteristic
293#103—62no—
294#103—62no—
295#103—121no—
296#103—121no—
297C22⋊C8—62no—
298C4×C8—62no—
299C4×C8—121no—
300#95—21no—
301#95—21no—
302#95—21no—
303#95—21no—
304#95—21no—
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#95#6711yescharacteristic
314#95#6711yescharacteristic
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—22no—
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—41no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—84no—
354C2×C3×C4#12412yes—
355C2×C3×C4C2×#4111yescharacteristic
356C2×C3×C4C2×#4111yescharacteristic
357C2×C3×C4C2×#4111yescharacteristic
358C2×C3×C4C2×#4111yescharacteristic
359C2×C3×C4H(5,𝔽2)11yescharacteristic
360C2×C3×C42−1+411yescharacteristic
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—41no—
373C2×Dic3—84no—
374Dic6—41no—
375Dic6—41no—
376Dic6—41no—
377Dic6—41no—
378C3⋊C8—22no—
379C3⋊C8—41no—
380C3⋊C8—41no—
381C3⋊C8—41no—
382C24C2×#6711yescharacteristic
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—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—61no—
400C22×C4—61no—
401C22×C4—61no—
402C22×C4—61no—
403C22×C4—81no—
404C22×C4—122no—
405C22×C4—122no—
406C22×C4—122no—
407C22×C4#23912yes—
408C22×C4#23912yes—
409C22×C4C2×#6711yescharacteristic
410C22×C4C2×#6711yescharacteristic
411C22×C4C2×#6711yescharacteristic
412C22×C4C2×#6711yescharacteristic
413C22×C4C2×#6711yescharacteristic
414C22×C4C23×D311yescharacteristic
415C2×C8—32no—
416C2×C8—61no—
417C2×C8—61no—
418C2×C8—61no—
419C2×C8—61no—
420C2×C8—61no—
421C2×C8—61no—
422C2×C8—61no—
423C2×C8—61no—
424C2×C8—61no—
425C2×C8—61no—
426C2×C8—61no—
427C2×C8—124no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—61no—
439C4⋊C4—61no—
440C4⋊C4—61no—
441C4⋊C4—61no—
442C4⋊C4—61no—
443C4⋊C4—61no—
444C4⋊C4—81no—
445C4⋊C4—81no—
446C4⋊C4—122no—
447C4⋊C4—122no—
448C4⋊C4#23712yes—
449C22⋊C4—22no—
450C22⋊C4—22no—
451C22⋊C4—42no—
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—42no—
476C3×C4—41no—
477C3×C4—81no—
478C3×C4—81no—
479C3×C4—81no—
480C3×C4#52911yescharacteristic
481C3×C4#53911yescharacteristic
482C3×C4#55111yescharacteristic
483C3×C4#55211yescharacteristic
484Dic3—81no—
485Dic3—81no—
486Dic3—81no—
487Dic3—81no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23#94712yes—
493C23#95012yes—
494C23#95012yes—
495C23C22×#6711yescharacteristic
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—22no—
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—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—84no—
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—244no—
544C2×C4#94712yes—
545C2×C4C2×#23711yescharacteristic
546C2×C4#100411yescharacteristic
547C2×C4C2×#23911yescharacteristic
548C2×C4C2×#23911yescharacteristic
549C2×C4D3×#4111yescharacteristic
550C2×C4#101011yescharacteristic
551C8—62no—
552C8—121no—
553C8—121no—
554C8—121no—
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#506814yes—
570C22#580611yescharacteristic
571C22C2×#94711yescharacteristic
572C22C2×#95011yescharacteristic
573C22C2×#95011yescharacteristic
574C22#582411yescharacteristic
575C22#583611yescharacteristic
576C22#583811yescharacteristic
577C4—22no—
578C4—41no—
579C4—42no—
580C4—41no—
581C4—41no—
582C4—41no—
583C4—81no—
584C4—81no—
585C4—81no—
586C4—241no—
587C4—241no—
588C4—241no—
589C4—241no—
590C4#562411yescharacteristic
591C4#563711yescharacteristic
592C4#564211yescharacteristic
593C4#564511yescharacteristic
594C3#2769911yescharacteristic, complemented, Sylow
595C2—24no—
596C2#8277111yescharacteristic
597C2C2×#506811yescharacteristic
598C2#8347111yescharacteristic
599C2#8349411yescharacteristic
600C2#8353411yescharacteristic
601C2#8359011yescharacteristic
602C2#8360611yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2442C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#2442D3
hypocenterC3#27699
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2442

Complements

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

NH ≅ G/Ntype
3. #71744C2semidirect
4. #71740C2semidirect
7. #69614C2semidirect
8. #69614C2semidirect
9. #69613C2semidirect
10. #69613C2semidirect
11. #69560C2semidirect
12. #69547C2semidirect
594. C3#27699semidirect

Maximal subgroups

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

subgroupindex
C3×#24422
#735182
#717442
#717402
#715592
C2×#45292
#696142
#696132
#695602
#695472
#695142
#695012
C22⋊C4×C3⋊C82
#276993