Finite Group Databasebeta

← #10827623

Subgroups — #10827623

Order 768.

1228 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5529C211yesmaximal, characteristic
2C2×C3×#419C211yesmaximal, characteristic, complemented
3#84068C211yesmaximal, characteristic
4C2×#5228C211yesmaximal, characteristic, complemented
5#71996C211yesmaximal, characteristic
6#71989C211yesmaximal, characteristic, complemented
7#71575C211yesmaximal, characteristic, complemented
8#70281C212yesmaximal, complemented
9#70273C212yesmaximal, complemented
10#70268C212yesmaximal, complemented
11#70232C212yesmaximal, complemented
12#23983—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C22×C3×M16C2211yescharacteristic
15C2×C3×#117C2211yescharacteristic
16C2×#946C2212yes—
17C22×#216C2212yes—
18C2×C4×#67C2211yescharacteristic
19C2×#924C2211yescharacteristic
20#5529—24no—
21#5529—24no—
22C2×#897C2211yescharacteristic
23C2×D3×C4⋊C4C2211yescharacteristic
24C2×#890C2211yescharacteristic
25C2×#887C2211yescharacteristic
26C2×#883—21no—
27C2×#879—21no—
28C2×#872—21no—
29C2×#870—21no—
30C3×#419—24no—
31C2×C3×#103C2212yes—
32C2×C3×#103C2212yes—
33#5231—21no—
34#5230—21no—
35#5229C2212yes—
36#5228—24no—
37C2×#835C2212yes—
38#5213—21no—
39#5212C2211yescharacteristic
40#5210—21no—
41C2×#830C2211yescharacteristic
42C2×(C2×D3)⋊C8C2212yes—
43C2×#816C2212yes—
44#4551C2212yes—
45#4550C2212yes—
46#4529C2212yes—
47#4529C2212yes—
48C2×#552—31no—
49C2×#451—31no—
50C2×#419D311yescharacteristic, complemented
51#2409—31no—
52#2026—32no—
53#2001—32no—
54#1894—31no—
55#1878—32no—
56#1872—32no—
57#1870—31no—
58#1825—31no—
59C22×C3×Q8—21no—
60C22×C3×Q8—24no—
61C22×C3×Q8—22no—
62C22×C3×Q8—22no—
63C22×C3×Q8—42no—
64C22×C3×Q8C2312yes—
65C23×C3×C4—21no—
66C23×C3×C4C2311yescharacteristic
67C22×#67D411yescharacteristic
68C22×C4×D3—21no—
69C22×C4×D3C2311yescharacteristic
70C22×Dic6C2311yescharacteristic
71C2×C3×M16—22no—
72C2×C3×M16—42no—
73C22×C3×C8C2312yes—
74C3×#117—22no—
75C3×#117D414yes—
76C2×C3×C4⋊C4C2311yescharacteristic
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C22⋊C4—21no—
79C2×C3×C42—21no—
80C2×#217—21no—
81C2×#217—21no—
82C2×#217D411yescharacteristic
83#946—24no—
84#946—42no—
85C2×#216—24no—
86C2×#216—24no—
87C2×#216—42no—
88C4×#67—22no—
89C4×#67—22no—
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—22no—
92C2×(C2×D3)⋊C4D411yescharacteristic
93C2×C4⋊Dic3—21no—
94C2×C4⋊Dic3—21no—
95C2×C4⋊Dic3C2312yes—
96C2×C4⋊Dic3C2311yescharacteristic
97#924—22no—
98#924D414yes—
99C2×Dic3⋊C4—21no—
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—22no—
104C2×Dic3⋊C4D411yescharacteristic
105C2×C4×Dic3—21no—
106C2×C4×Dic3C2311yescharacteristic
107C22×C3⋊C8C2312yes—
108#897—22no—
109#897—22no—
110D3×C4⋊C4—22no—
111D3×C4⋊C4—22no—
112#890—22no—
113#890—22no—
114#887—22no—
115#887—22no—
116#883—28no—
117#879—28no—
118#872—28no—
119#870—28no—
120C3×#103—24no—
121C3×#103—24no—
122#835—24no—
123C3⋊#95—21no—
124C3⋊#95—22no—
125C3⋊#95—22no—
126#830—22no—
127(C2×D3)⋊C8—42no—
128#816—42no—
129C23×Q8C2×D311yescharacteristic
130C22×Q16—32no—
131C22×M16C2×D311yescharacteristic
132#552—64no—
133#552—64no—
134C2×#126—61no—
135C2×#125—31no—
136C2×#125—61no—
137C2×#123—61no—
138C2×C22⋊Q8—31no—
139C2×C22⋊Q8—32no—
140C2×C22⋊Q8—31no—
141C2×C4×Q8—31no—
142C2×C4×D4—31no—
143C2×#117—61no—
144C2×#117C2×D311yescharacteristic
145C22×C4⋊C4—31no—
146#451—64no—
147#419—24no—
148C2×#103—32no—
149C2×#103—32no—
150C2×#103C2×D312yes—
151C2×#103C2×D312yes—
152C2×C22⋊C8—31no—
153C2×C22⋊C8—32no—
154#395—61no—
155#393—61no—
156#391—32no—
157#386—61no—
158#385—61no—
159#380—31no—
160#340—32no—
161#340—32no—
162#340—32no—
163#336—32no—
164C24×C3C2×D411yescharacteristic
165C23×D3C2×D411yescharacteristic
166C2×C3×Q8—22no—
167C2×C3×Q8—28no—
168C2×C3×Q8—22no—
169C2×C3×Q8—22no—
170C2×C3×Q8—22no—
171C2×C3×Q8—22no—
172C2×C3×Q8—44no—
173C2×C3×Q8—48no—
174C2×C3×Q8—44no—
175C2×C3×Q8—42no—
176C2×C3×Q8#4114yes—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—21no—
181C22×C3×C4—21no—
182C22×C3×C4—44no—
183C22×C3×C4—42no—
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2×D412yes—
186C22×C3×C4C2×D412yes—
187C22×C3×C4C2411yescharacteristic
188C2×#67—22no—
189C2×#67—22no—
190C2×#67—41no—
191C22×Dic3—21no—
192C22×Dic3—21no—
193C22×Dic3—22no—
194C22×Dic3C2×D411yescharacteristic
195C22×Dic3C2×D411yescharacteristic
196C22×Dic3C2×D411yescharacteristic
197C2×C4×D3—24no—
198C2×C4×D3—21no—
199C2×C4×D3—21no—
200C2×C4×D3—22no—
201C2×C4×D3—22no—
202C2×C4×D3—42no—
203C2×C4×D3—42no—
204C2×Dic6—22no—
205C2×Dic6—22no—
206C2×Dic6—21no—
207C2×Dic6C2×D412yes—
208C3×M16—44no—
209C2×C3×C8—22no—
210C2×C3×C8—22no—
211C2×C3×C8—22no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4C2×D412yes—
215C3×C4⋊C4C2×D412yes—
216C3×C22⋊C4—24no—
217C3×C42—24no—
218#217—22no—
219#217—24no—
220#217—42no—
221#216—48no—
222(C2×D3)⋊C4—24no—
223(C2×D3)⋊C4—22no—
224(C2×D3)⋊C4—44no—
225C4⋊Dic3—24no—
226C4⋊Dic3—24no—
227C4⋊Dic3—21no—
228C4⋊Dic3—22no—
229C4⋊Dic3—22no—
230C4⋊Dic3C2×D412yes—
231Dic3⋊C4—24no—
232Dic3⋊C4—24no—
233Dic3⋊C4—24no—
234Dic3⋊C4—22no—
235Dic3⋊C4—24no—
236Dic3⋊C4—44no—
237C4×Dic3—24no—
238C4×Dic3—21no—
239C4×Dic3—21no—
240C2×C3⋊C8—22no—
241C2×C3⋊C8—22no—
242C2×C3⋊C8—22no—
243C22×Q8—21no—
244C22×Q8—24no—
245C22×Q8—22no—
246C22×Q8—22no—
247C22×Q8—31no—
248C22×Q8—42no—
249C22×Q8C22×D312yes—
250C22×D4—31no—
251C23×C4—21no—
252C23×C4—31no—
253C23×C4—61no—
254C23×C4C22×D311yescharacteristic
255C2×Q16—64no—
256C2×Q16—64no—
257C2×Q16—122no—
258C2×M16—22no—
259C2×M16—42no—
260C22×C8—32no—
261C22×C8C22×D312yes—
262#126—68no—
263#125—68no—
264#125—62no—
265#125—62no—
266#123—68no—
267C22⋊Q8—34no—
268C22⋊Q8—62no—
269C22⋊Q8—62no—
270C22⋊Q8—62no—
271C22⋊Q8—64no—
272C22⋊Q8—122no—
273C4×Q8—62no—
274C4×Q8—62no—
275C4×D4—62no—
276C4×D4—62no—
277#117—22no—
278#117—68no—
279#117#6714yes—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—32no—
283C2×C4⋊C4—62no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4—61no—
290C2×C4⋊C4—62no—
291C2×C4⋊C4—62no—
292C2×C4⋊C4C22×D311yescharacteristic
293C2×C4⋊C4C22×D311yescharacteristic
294C2×C22⋊C4—21no—
295C2×C22⋊C4—31no—
296C2×C22⋊C4—31no—
297C2×C22⋊C4—61no—
298C2×C22⋊C4—61no—
299C2×C22⋊C4—62no—
300C2×C22⋊C4—61no—
301C2×C42—21no—
302C2×C42—31no—
303C2×C42—61no—
304#103—24no—
305#103—24no—
306#103—64no—
307#103—122no—
308C22⋊C8—62no—
309C22⋊C8—122no—
310#95—62no—
311#95—62no—
312#95—61no—
313C23×C3—22no—
314C23×C3—21no—
315C23×C3—21no—
316C23×C3—22no—
317C23×C3#12012yes—
318C23×C3C22×D411yescharacteristic
319C22×D3—21no—
320C22×D3—22no—
321C22×D3—21no—
322C22×D3—22no—
323C22×D3#13712yes—
324C3×Q8—24no—
325C3×Q8—24no—
326C3×Q8—48no—
327C3×Q8—42no—
328C3×Q8—42no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—24no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—24no—
336C2×C3×C4—48no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—42no—
342C2×C3×C4#12012yes—
343C2×C3×C4C22×D411yescharacteristic
344C2×C3×C4C22×D411yescharacteristic
345C2×C3×C4C2×#4112yes—
346C2×C3×C4H(5,𝔽2)12yes—
347#67—44no—
348C2×Dic3—21no—
349C2×Dic3—21no—
350C2×Dic3—22no—
351C2×Dic3—21no—
352C2×Dic3—22no—
353C2×Dic3—22no—
354C2×Dic3—21no—
355C2×Dic3—21no—
356C2×Dic3—22no—
357C2×Dic3—42no—
358C2×Dic3—41no—
359C2×Dic3—41no—
360C2×Dic3—42no—
361C2×Dic3—42no—
362C2×Dic3—42no—
363C4×D3—48no—
364C4×D3—41no—
365C4×D3—41no—
366C4×D3—42no—
367Dic6—24no—
368Dic6—41no—
369Dic6—41no—
370C3×C8—42no—
371C3⋊C8—42no—
372C24—31no—
373C24D3×D411yescharacteristic
374C2×Q8—22no—
375C2×Q8—28no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—32no—
381C2×Q8—44no—
382C2×Q8—48no—
383C2×Q8—44no—
384C2×Q8—42no—
385C2×Q8—61no—
386C2×Q8—62no—
387C2×Q8—62no—
388C2×Q8#23514yes—
389C2×D4—62no—
390C2×D4—62no—
391C2×D4—121no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—31no—
398C22×C4—31no—
399C22×C4—31no—
400C22×C4—44no—
401C22×C4—42no—
402C22×C4—64no—
403C22×C4—61no—
404C22×C4—61no—
405C22×C4—62no—
406C22×C4—61no—
407C22×C4—62no—
408C22×C4—61no—
409C22×C4—62no—
410C22×C4—122no—
411C22×C4—122no—
412C22×C4D3×D411yescharacteristic
413C22×C4D3×D412yes—
414C22×C4C2×#6712yes—
415C22×C4C23×D311yescharacteristic
416Q16—128no—
417M16—44no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—62no—
422C2×C8—62no—
423C2×C8—62no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—32no—
427C4⋊C4—64no—
428C4⋊C4—64no—
429C4⋊C4—64no—
430C4⋊C4—64no—
431C4⋊C4—64no—
432C4⋊C4—62no—
433C4⋊C4—64no—
434C4⋊C4—61no—
435C4⋊C4—62no—
436C4⋊C4—62no—
437C4⋊C4—124no—
438C4⋊C4C2×#6712yes—
439C4⋊C4C2×#6712yes—
440C22⋊C4—24no—
441C22⋊C4—64no—
442C22⋊C4—62no—
443C22⋊C4—64no—
444C22⋊C4—62no—
445C22⋊C4—124no—
446C22⋊C4—122no—
447C42—24no—
448C42—64no—
449C42—61no—
450C42—61no—
451C22×C3—22no—
452C22×C3—22no—
453C22×C3—22no—
454C22×C3—42no—
455C22×C3—42no—
456C22×C3C2×#12011yescharacteristic
457C22×C3#54612yes—
458C22×C3C2×#13711yescharacteristic
459C22×C3#57511yescharacteristic
460C22×C3#57812yes—
461C2×D3—22no—
462C2×D3—24no—
463C2×D3—41no—
464C2×D3—41no—
465C2×D3—42no—
466C3×C4—22no—
467C3×C4—44no—
468C3×C4—42no—
469C3×C4—42no—
470C3×C4#52511yescharacteristic
471C3×C4#53511yescharacteristic
472C3×C4#54612yes—
473Dic3—42no—
474Dic3—41no—
475Dic3—42no—
476Dic3—41no—
477Dic3—41no—
478Dic3—82no—
479C23—22no—
480C23—21no—
481C23—21no—
482C23—22no—
483C23—32no—
484C23—62no—
485C23—61no—
486C23—62no—
487C23—61no—
488C23#93712yes—
489C23C2×D3×D411yescharacteristic
490Q8—24no—
491Q8—24no—
492Q8—48no—
493Q8—42no—
494Q8—42no—
495Q8—64no—
496Q8—121no—
497Q8—121no—
498D4—124no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—48no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—62no—
513C2×C4—61no—
514C2×C4—61no—
515C2×C4—62no—
516C2×C4—61no—
517C2×C4—62no—
518C2×C4—61no—
519C2×C4—62no—
520C2×C4—61no—
521C2×C4—128no—
522C2×C4—122no—
523C2×C4—121no—
524C2×C4—121no—
525C2×C4—122no—
526C2×C4—121no—
527C2×C4—121no—
528C2×C4—122no—
529C2×C4—122no—
530C2×C4—122no—
531C2×C4#93712yes—
532C2×C4C2×#23512yes—
533C2×C4C2×D3×D411yescharacteristic
534C2×C4#100412yes—
535C2×C4C22×#6711yescharacteristic
536C8—42no—
537C8—122no—
538C2×C3—42no—
539C2×C3#240411yescharacteristic
540C2×C3#300711yescharacteristic
541C2×C3#301111yescharacteristic
542C2×C3#331212yes—
543C2×C3#332712yes—
544D3—42no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—42no—
549C22—42no—
550C22—62no—
551C22—64no—
552C22—121no—
553C22—122no—
554C22—121no—
555C22#549012yes—
556C22D3×#13711yescharacteristic
557C22#577711yescharacteristic
558C22#577912yes—
559C22C2×#93711yescharacteristic
560C4—22no—
561C4—42no—
562C4—42no—
563C4—44no—
564C4—121no—
565C4—122no—
566C4—121no—
567C4—121no—
568C4—122no—
569C4—242no—
570C4#549012yes—
571C4#579111yescharacteristic
572C4#580511yescharacteristic
573C3#2398311yescharacteristic, complemented, Sylow
574C2—42no—
575C2—122no—
576C2#7896612yes—
577C2#7897112yes—
578C2#8212511yescharacteristic
579C2#8398111yescharacteristic
580C2#8398311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#419C2
socleC23×C3C22×D4
prime radicalC22×#67D4
hypercenterC2×#419D3
hypocenterC3#23983
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#419

Complements

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

NH ≅ G/Ntype
2. C2×C3×#419C2semidirect
4. C2×#5228C2semidirect
6. #71989C2semidirect
7. #71575C2semidirect
8. #70281C2semidirect
9. #70273C2semidirect
10. #70268C2semidirect
11. #70232C2semidirect
50. C2×#419D3semidirect
573. C3#23983semidirect

Maximal subgroups

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

subgroupindex
C2×#55292
C2×C3×#4192
#840682
C2×#52282
#719962
#719892
#715752
#702812
#702732
#702682
#702322
#239833