Finite Group Databasebeta

← #10876073

Subgroups — #10876073

Order 768.

738 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2349C211yesmaximal, characteristic
2#73696C211yesmaximal, characteristic, complemented
3#71612C211yesmaximal, characteristic
4#71612C211yesmaximal, characteristic
5#71612C211yesmaximal, characteristic
6#71609C211yesmaximal, characteristic
7#71607C211yesmaximal, characteristic
8#71605C211yesmaximal, characteristic
9#69634C211yesmaximal, characteristic, complemented
10#69616C211yesmaximal, characteristic, complemented
11#69613C211yesmaximal, characteristic, complemented
12#69601C211yesmaximal, characteristic, complemented
13#69560C211yesmaximal, characteristic, complemented
14#69542C211yesmaximal, characteristic, complemented
15#69483C211yesmaximal, characteristic, complemented
16#27579—31nomaximal, Sylow
17C2×C3×C4×D4C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C2×C3×C4×D4C2211yescharacteristic
20C22×#214C2211yescharacteristic
21C3×#389C2211yescharacteristic
22C3×#389C2211yescharacteristic
23C3×#385—21no—
24C3×#385—21no—
25C3×#382C2211yescharacteristic
26C3×#380—21no—
27C3×C4×C4⋊C4C2211yescharacteristic
28C2×C3×#95—21no—
29C2×#832C2211yescharacteristic
30C2×#832C2211yescharacteristic
31C2×#832C2211yescharacteristic
32#4985—21no—
33#4978—21no—
34C2×#810C2211yescharacteristic
35C2×#810C2211yescharacteristic
36C2×#810C2211yescharacteristic
37C2×#808C2211yescharacteristic
38#4936C2211yescharacteristic
39C4×C4⋊Dic3C2211yescharacteristic
40#4929C2211yescharacteristic
41#4929C2211yescharacteristic
42#4929C2211yescharacteristic
43#4928C2211yescharacteristic
44C2×C4⋊(C3⋊C8)C2211yescharacteristic
45C2×#803C2211yescharacteristic
46C3⋊#401—21no—
47#4651—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#2514—31no—
61#2349D311yescharacteristic
62#2101—31no—
63#2101—31no—
64#2074—31no—
65#2066—31no—
66#1935—31no—
67#1919—31no—
68#1917—31no—
69#1877—31no—
70#1877—31no—
71#1877—31no—
72#1864—31no—
73#1804—31no—
74#1762—31no—
75C22×C3×D4C2311yescharacteristic
76C23×C3×C4—21no—
77C23×C3×C4—21no—
78C23×C3×C4—21no—
79C22×Dic6C2311yescharacteristic
80C3×C4×D4—22no—
81C3×C4×D4—22no—
82C3×C4×D4—22no—
83C3×C4×D4—22no—
84C3×C4×D4—41no—
85C3×C4×D4—41no—
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×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C42C2311yescharacteristic
97C2×C3×C42C2311yescharacteristic
98C2×C3×C42C2311yescharacteristic
99C2×#214—22no—
100C2×#214—22no—
101C2×#214—22no—
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×Dic3⋊C4—21no—
106C2×C4×Dic3—21no—
107C22×C3⋊C8C2311yescharacteristic
108C22×C3⋊C8C2311yescharacteristic
109C22×C3⋊C8C2311yescharacteristic
110C22×C3⋊C8C2311yescharacteristic
111C3×#95—21no—
112C3×#95—21no—
113C3×#95—21no—
114C3×#95—21no—
115C3×#95D411yescharacteristic
116C3×#95D411yescharacteristic
117#832—22no—
118#832—22no—
119#832—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⋊#95—21no—
129C3⋊#95—21no—
130#810—22no—
131#810—41no—
132#810—41no—
133#808—22no—
134C4⋊(C3⋊C8)—41no—
135#803—41no—
136C22×SD16—31no—
137C2×C4×D4C2×D311yescharacteristic
138C2×C4×D4C2×D311yescharacteristic
139C2×C4×D4C2×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#401—61no—
150#400—61no—
151#391—31no—
152#391—31no—
153#391—31no—
154#389—31no—
155#389C2×D311yescharacteristic
156#389C2×D311yescharacteristic
157#387—61no—
158#385—21no—
159#385—21no—
160#383—61no—
161#382—31no—
162#382C2×D311yescharacteristic
163#380—21no—
164C4×C4⋊C4—31no—
165C4×C4⋊C4C2×D311yescharacteristic
166C2×#95—21no—
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—
179C24×C3—21no—
180C2×C3×D4—21no—
181C2×C3×D4—21no—
182C2×C3×D4—21no—
183C2×C3×D4#4112yes—
184C2×C3×D4#4112yes—
185C2×C3×D4#4112yes—
186C22×C3×C4—21no—
187C22×C3×C4—24no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—42no—
191C22×C3×C4—42no—
192C22×C3×C4—42no—
193C22×C3×C4—42no—
194C22×C3×C4—81no—
195C22×C3×C4—81no—
196C22×C3×C4—81no—
197C22×C3×C4—81no—
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2411yescharacteristic
205C22×Dic3—21no—
206C22×Dic3—21no—
207C22×Dic3—21no—
208C22×Dic3—21no—
209C2×Dic6—21no—
210C2×Dic6—21no—
211C2×Dic6—21no—
212C2×Dic6—21no—
213C2×Dic6—21no—
214C2×Dic6—21no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—41no—
221C3×C4⋊C4—41no—
222C3×C4⋊C4—81no—
223C3×C4⋊C4—81no—
224C3×C4⋊C4#4112yes—
225C3×C22⋊C4—24no—
226C3×C22⋊C4—42no—
227C3×C22⋊C4—42no—
228C3×C42—21no—
229C3×C42—21no—
230C3×C42—21no—
231C3×C42—21no—
232C3×C42—21no—
233C3×C42—21no—
234#214—44no—
235C4⋊Dic3—21no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241Dic3⋊C4—42no—
242C4×Dic3—42no—
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×Q8—31no—
256C22×D4C22×D311yescharacteristic
257C23×C4—21no—
258C23×C4—21no—
259C23×C4—21no—
260C2×SD16—62no—
261C2×SD16—62no—
262C2×SD16—62no—
263C22×C8—31no—
264C22×C8—31no—
265C22×C8—31no—
266C22×C8—31no—
267C4×D4—22no—
268C4×D4—22no—
269C4×D4—22no—
270C4×D4—22no—
271C4×D4—41no—
272C4×D4—41no—
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—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C42—61no—
288C2×C42C22×D311yescharacteristic
289C2×C42C22×D311yescharacteristic
290C2×C42C22×D311yescharacteristic
291#106—62no—
292C4⋊C8—121no—
293#103—62no—
294#103—121no—
295#103—121no—
296#102—62no—
297#102—62no—
298#102—62no—
299#97—121no—
300#95—21no—
301#95—21no—
302#95—21no—
303#95—21no—
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#95—61no—
314#95#6711yescharacteristic
315#95#6711yescharacteristic
316C23×C3—22no—
317C23×C3—22no—
318C23×C3—22no—
319C23×C3—41no—
320C23×C3—41no—
321C23×C3—41no—
322C23×C3—41no—
323C23×C3C22×D411yescharacteristic
324C3×D4—22no—
325C3×D4—22no—
326C3×D4—22no—
327C3×D4#12614yes—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—22no—
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—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
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—41no—
354C2×C3×C4—41no—
355C2×C3×C4—84no—
356C2×C3×C4—84no—
357C2×C3×C4—84no—
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—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376Dic6—41no—
377Dic6—41no—
378Dic6—41no—
379Dic6—41no—
380C3⋊C8—41no—
381C3⋊C8—41no—
382C3⋊C8—41no—
383C3⋊C8—41no—
384C24—21no—
385C2×Q8—61no—
386C2×Q8—61no—
387C2×Q8—61no—
388C2×Q8—61no—
389C2×Q8—61no—
390C2×Q8—61no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4#23512yes—
395C2×D4#23712yes—
396C2×D4#23912yes—
397C22×C4—21no—
398C22×C4—24no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—61no—
406C22×C4—61no—
407C22×C4—61no—
408C22×C4—61no—
409C22×C4—81no—
410C22×C4—81no—
411C22×C4—81no—
412C22×C4—81no—
413C22×C4C2×#6711yescharacteristic
414C22×C4C2×#6711yescharacteristic
415C22×C4C2×#6711yescharacteristic
416C22×C4C2×#6711yescharacteristic
417C22×C4C2×#6711yescharacteristic
418C22×C4C2×#6711yescharacteristic
419C22×C4C23×D311yescharacteristic
420SD16—124no—
421C2×C8—61no—
422C2×C8—61no—
423C2×C8—61no—
424C2×C8—61no—
425C2×C8—61no—
426C2×C8—61no—
427C2×C8—61no—
428C2×C8—61no—
429C2×C8—61no—
430C2×C8—61no—
431C2×C8—61no—
432C2×C8—61no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—41no—
439C4⋊C4—41no—
440C4⋊C4—61no—
441C4⋊C4—61no—
442C4⋊C4—61no—
443C4⋊C4—61no—
444C4⋊C4—61no—
445C4⋊C4—61no—
446C4⋊C4—81no—
447C4⋊C4—81no—
448C4⋊C4—122no—
449C4⋊C4#23512yes—
450C22⋊C4—24no—
451C22⋊C4—42no—
452C22⋊C4—42no—
453C42—21no—
454C42—21no—
455C42—21no—
456C42—21no—
457C42—21no—
458C42—21no—
459C42—122no—
460C22×C3—24no—
461C22×C3—42no—
462C22×C3—42no—
463C22×C3—42no—
464C22×C3—42no—
465C22×C3—42no—
466C22×C3—42no—
467C22×C3#52511yescharacteristic
468C22×C3#54611yescharacteristic
469C22×C3#54811yescharacteristic
470C22×C3#57511yescharacteristic
471C22×C3#57511yescharacteristic
472C22×C3#57711yescharacteristic
473C22×C3#57711yescharacteristic
474C3×C4—41no—
475C3×C4—41no—
476C3×C4—42no—
477C3×C4—41no—
478C3×C4—41no—
479C3×C4—41no—
480C3×C4—41no—
481C3×C4—81no—
482C3×C4—81no—
483C3×C4C2×#12611yescharacteristic
484C3×C4#53911yescharacteristic
485C3×C4#55211yescharacteristic
486C3×C4#55311yescharacteristic
487Dic3—81no—
488Dic3—81no—
489Dic3—81no—
490Dic3—81no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23C22×#6711yescharacteristic
499Q8—121no—
500Q8—121no—
501Q8—121no—
502Q8—121no—
503D4—22no—
504D4—22no—
505D4—22no—
506D4#89814yes—
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—84no—
535C2×C4—84no—
536C2×C4—84no—
537C2×C4—121no—
538C2×C4—121no—
539C2×C4—121no—
540C2×C4—121no—
541C2×C4—121no—
542C2×C4—121no—
543C2×C4—121no—
544C2×C4—121no—
545C2×C4—121no—
546C2×C4—121no—
547C2×C4—121no—
548C2×C4—121no—
549C2×C4C2×#23511yescharacteristic
550C2×C4C2×#23511yescharacteristic
551C2×C4C2×#23711yescharacteristic
552C2×C4#100411yescharacteristic
553C2×C4C2×#23911yescharacteristic
554C2×C4#100711yescharacteristic
555C8—121no—
556C8—121no—
557C8—121no—
558C8—121no—
559C2×C3—44no—
560C2×C3#248111yescharacteristic
561C2×C3#311111yescharacteristic
562C2×C3#312311yescharacteristic
563C2×C3#331111yescharacteristic
564C2×C3#332311yescharacteristic
565C2×C3#337611yescharacteristic
566C2×C3#337611yescharacteristic
567C22—24no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—42no—
574C22#579111yescharacteristic
575C22#580611yescharacteristic
576C22#581711yescharacteristic
577C22#582211yescharacteristic
578C22#582211yescharacteristic
579C22#583611yescharacteristic
580C22#583611yescharacteristic
581C4—41no—
582C4—41no—
583C4—41no—
584C4—41no—
585C4—41no—
586C4—41no—
587C4—42no—
588C4—81no—
589C4—81no—
590C4—241no—
591C4—241no—
592C4—241no—
593C4—241no—
594C4C2×#89811yescharacteristic
595C4#551311yescharacteristic
596C4#551411yescharacteristic
597C4#551511yescharacteristic
598C3#2757911yescharacteristic, complemented, Sylow
599C2—44no—
600C2#8217511yescharacteristic
601C2#8253011yescharacteristic
602C2#8254511yescharacteristic
603C2#8349711yescharacteristic
604C2#8350111yescharacteristic
605C2#8364511yescharacteristic
606C2#8364511yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2349C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#2349D3
hypocenterC3#27579
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2349

Complements

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

NH ≅ G/Ntype
2. #73696C2semidirect
9. #69634C2semidirect
10. #69616C2semidirect
11. #69613C2semidirect
12. #69601C2semidirect
13. #69560C2semidirect
14. #69542C2semidirect
15. #69483C2semidirect
598. C3#27579semidirect

Maximal subgroups

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

subgroupindex
C3×#23492
#736962
#716122
#716092
#716072
#716052
#696342
#696162
#696132
#696012
#695602
#695422
#694832
#275793