Finite Group Databasebeta

← #10830952

Subgroups — #10830952

Order 768.

1384 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#216C211yesmaximal, characteristic, complemented
2#84068C211yesmaximal, characteristic
3C2×#4816C212yesmaximal, complemented
4C2×#4730C211yesmaximal, characteristic
5#74017C211yesmaximal, characteristic
6C3×#1787C211yesmaximal, characteristic, complemented
7#70285C212yesmaximal, complemented
8#70273C212yesmaximal, complemented
9#70259C212yesmaximal, complemented
10#69963C211yesmaximal, characteristic, complemented
11#69511C211yesmaximal, characteristic, complemented
12#24273—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C23×Dic6C2211yescharacteristic
15C2×#946C2212yes—
16C22×#216—28no—
17C22×#216—24no—
18C22×#216C2212yes—
19C22×#216C2212yes—
20C2×C4×#67C2211yescharacteristic
21C23×C3⋊C8C2211yescharacteristic
22C22×ΓL(1,𝔽25)C2211yescharacteristic
23C2×#897C2212yes—
24C2×C3×#103C2212yes—
25C2×C3×#103C2212yes—
26C2×C3×C22⋊C8C2211yescharacteristic
27C3×#380C2211yescharacteristic
28#5231—21no—
29#5230—21no—
30#5229C2212yes—
31#5213—21no—
32#5212C2211yescharacteristic
33#5210—21no—
34C2×(C2×D3)⋊C8C2211yescharacteristic
35C2×#816C2212yes—
36C2×#816C2212yes—
37C2×#815C2211yescharacteristic
38C2×Dic3⋊C8C2211yescharacteristic
39#4973C2212yes—
40C2×#808C2212yes—
41#4958—21no—
42#4950—21no—
43#4816—28no—
44#4730—24no—
45#4673—21no—
46#4656—21no—
47#4529C2212yes—
48C23×Q16—31no—
49C2×#470—32no—
50C2×#435—31no—
51#2409—31no—
52#2409—31no—
53#2056—32no—
54#2001—32no—
55#1971—32no—
56#1807—31no—
57#1787—31no—
58#1787D311yescharacteristic, complemented
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
68C23×Dic3—21no—
69C22×C4×D3C2311yescharacteristic
70C22×Dic6—22no—
71C22×Dic6—24no—
72C22×Dic6—22no—
73C22×Dic6—21no—
74C22×Dic6—42no—
75C22×Dic6C2312yes—
76C22×C3×C8C2311yescharacteristic
77C22×C3×C8C2311yescharacteristic
78C2×C3×C4⋊C4C2312yes—
79C2×C3×C22⋊C4—21no—
80C2×#217—21no—
81C2×#217D411yescharacteristic
82#946—24no—
83#946—42no—
84C2×#216—232no—
85C2×#216—24no—
86C2×#216—24no—
87C2×#216—24no—
88C2×#216—24no—
89C2×#216—24no—
90C2×#216D418yescomplemented
91C4×#67—22no—
92C4×#67—22no—
93C2×(C2×D3)⋊C4—22no—
94C2×(C2×D3)⋊C4—22no—
95C2×(C2×D3)⋊C4D411yescharacteristic
96C2×C4⋊Dic3—22no—
97C2×C4⋊Dic3C2312yes—
98C2×Dic3⋊C4—22no—
99C2×Dic3⋊C4D411yescharacteristic
100C2×C4×Dic3C2311yescharacteristic
101C22×C3⋊C8—24no—
102C22×C3⋊C8D414yes—
103C22×C3⋊C8C2311yescharacteristic
104C22×C3⋊C8C2311yescharacteristic
105C2×ΓL(1,𝔽25)—22no—
106C2×ΓL(1,𝔽25)—24no—
107#897—24no—
108#897—42no—
109C3×#103—24no—
110C3×#103—42no—
111C3×C22⋊C8—22no—
112C3×#95—21no—
113C3⋊#95—21no—
114C3⋊#95—22no—
115C3⋊#95—22no—
116C3⋊#95—22no—
117C3⋊#95—22no—
118(C2×D3)⋊C8—22no—
119#816—24no—
120#816—42no—
121#815—22no—
122Dic3⋊C8—22no—
123#808—24no—
124C23×Q8—31no—
125C23×Q8C2×D311yescharacteristic
126C22×Q16—32no—
127C22×Q16—32no—
128C22×Q16—64no—
129C22×Q16—68no—
130C22×M16—31no—
131C23×C8—31no—
132C2×C22⋊Q8—32no—
133C2×C22⋊Q8—32no—
134C2×C4×D4—31no—
135#470—68no—
136#435—64no—
137C2×#106—32no—
138C2×C4⋊C8—31no—
139C2×#103—32no—
140C2×#103—32no—
141C2×#103C2×D312yes—
142C2×#103C2×D312yes—
143C2×C22⋊C8—31no—
144C2×C22⋊C8C2×D311yescharacteristic
145C2×#97—31no—
146#395—61no—
147#395—61no—
148#393—61no—
149#393—61no—
150#391—32no—
151#391—32no—
152#386—61no—
153#386—61no—
154#385—61no—
155#385—61no—
156#380—31no—
157#380C2×D311yescharacteristic
158#340—32no—
159C24×C3C2×D411yescharacteristic
160C23×D3C2×D411yescharacteristic
161C2×C3×Q8—22no—
162C2×C3×Q8—28no—
163C2×C3×Q8—22no—
164C2×C3×Q8—22no—
165C2×C3×Q8—22no—
166C2×C3×Q8—22no—
167C2×C3×Q8—44no—
168C2×C3×Q8—48no—
169C2×C3×Q8—44no—
170C2×C3×Q8—42no—
171C2×C3×Q8C2×D414yes—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—44no—
177C22×C3×C4—42no—
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4C2×D412yes—
180C22×C3×C4#4112yes—
181C22×C3×C4C2411yescharacteristic
182C2×#67—22no—
183C2×#67—22no—
184C2×#67—22no—
185C22×Dic3—22no—
186C22×Dic3—22no—
187C22×Dic3—44no—
188C22×Dic3—42no—
189C22×Dic3C2×D411yescharacteristic
190C22×Dic3C2×D411yescharacteristic
191C22×Dic3C2×D411yescharacteristic
192C2×C4×D3—21no—
193C2×C4×D3—22no—
194C2×C4×D3—21no—
195C2×C4×D3—22no—
196C2×Dic6—28no—
197C2×Dic6—22no—
198C2×Dic6—22no—
199C2×Dic6—22no—
200C2×Dic6—22no—
201C2×Dic6—22no—
202C2×Dic6—48no—
203C2×Dic6—44no—
204C2×Dic6—44no—
205C2×Dic6—42no—
206C2×Dic6C2×D414yes—
207C2×C3×C8—21no—
208C2×C3×C8—21no—
209C2×C3×C8—22no—
210C2×C3×C8—22no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4—22no—
213C3×C22⋊C4—42no—
214#217—22no—
215#217—42no—
216#216—216no—
217#216—216no—
218(C2×D3)⋊C4—22no—
219(C2×D3)⋊C4—44no—
220(C2×D3)⋊C4—44no—
221C4⋊Dic3—22no—
222C4⋊Dic3—22no—
223C4⋊Dic3—44no—
224Dic3⋊C4—22no—
225Dic3⋊C4—44no—
226C4×Dic3—21no—
227C4×Dic3—21no—
228C2×C3⋊C8—22no—
229C2×C3⋊C8—28no—
230C2×C3⋊C8C2×D412yes—
231C2×C3⋊C8C2×D414yes—
232C2×C3⋊C8C2×D412yes—
233ΓL(1,𝔽25)—44no—
234C22×Q8—21no—
235C22×Q8—24no—
236C22×Q8—22no—
237C22×Q8—22no—
238C22×Q8—32no—
239C22×Q8—42no—
240C22×Q8—64no—
241C22×Q8—61no—
242C22×Q8—62no—
243C22×Q8—62no—
244C22×Q8—122no—
245C22×Q8C22×D312yes—
246C22×D4—31no—
247C23×C4—21no—
248C23×C4—31no—
249C23×C4—61no—
250C23×C4C22×D311yescharacteristic
251C2×Q16—38no—
252C2×Q16—64no—
253C2×Q16—64no—
254C2×Q16—64no—
255C2×Q16—64no—
256C2×Q16—632no—
257C2×Q16—64no—
258C2×M16—62no—
259C2×M16—64no—
260C22×C8—31no—
261C22×C8—34no—
262C22×C8—31no—
263C22×C8—64no—
264C22×C8C22×D311yescharacteristic
265C22×C8C22×D311yescharacteristic
266C22⋊Q8—64no—
267C22⋊Q8—64no—
268C22⋊Q8—122no—
269C22⋊Q8—122no—
270C4×D4—62no—
271C4×D4—62no—
272C2×C4⋊C4—31no—
273C2×C4⋊C4—32no—
274C2×C4⋊C4—62no—
275C2×C4⋊C4—62no—
276C2×C4⋊C4C22×D312yes—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—31no—
279C2×C22⋊C4—31no—
280C2×C22⋊C4—62no—
281C2×C22⋊C4—62no—
282C2×C22⋊C4—61no—
283C2×C42—31no—
284#106—64no—
285C4⋊C8—62no—
286#103—24no—
287#103—42no—
288#103—64no—
289#103—122no—
290C22⋊C8—22no—
291C22⋊C8—62no—
292#97—62no—
293#95—21no—
294#95—62no—
295#95—62no—
296#95—62no—
297#95—62no—
298#95—61no—
299C23×C3—22no—
300C23×C3—22no—
301C23×C3#12112yes—
302C23×C3#13712yes—
303C23×C3#13712yes—
304C23×C3C22×D411yescharacteristic
305C22×D3—21no—
306C22×D3—22no—
307C22×D3—21no—
308C22×D3—21no—
309C22×D3—22no—
310C3×Q8—24no—
311C3×Q8—24no—
312C3×Q8—48no—
313C3×Q8—42no—
314C3×Q8—42no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—24no—
319C2×C3×C4—24no—
320C2×C3×C4—48no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4#12112yes—
327C2×C3×C4C22×D411yescharacteristic
328C2×C3×C4C22×D412yes—
329C2×C3×C4C2×#4111yescharacteristic
330C2×C3×C4H(5,𝔽2)12yes—
331#67—44no—
332C2×Dic3—21no—
333C2×Dic3—21no—
334C2×Dic3—21no—
335C2×Dic3—22no—
336C2×Dic3—24no—
337C2×Dic3—21no—
338C2×Dic3—21no—
339C2×Dic3—22no—
340C2×Dic3—48no—
341C2×Dic3—42no—
342C2×Dic3—42no—
343C2×Dic3—42no—
344C2×Dic3—42no—
345C2×Dic3—42no—
346C4×D3—41no—
347C4×D3—41no—
348C4×D3—42no—
349Dic6—24no—
350Dic6—24no—
351Dic6—48no—
352Dic6—42no—
353Dic6—42no—
354C3×C8—41no—
355C3×C8—41no—
356C3⋊C8—22no—
357C3⋊C8#12014yes—
358C24—31no—
359C24D3×D411yescharacteristic
360C2×Q8—22no—
361C2×Q8—28no—
362C2×Q8—22no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—22no—
366C2×Q8—34no—
367C2×Q8—44no—
368C2×Q8—48no—
369C2×Q8—44no—
370C2×Q8—42no—
371C2×Q8—68no—
372C2×Q8—62no—
373C2×Q8—62no—
374C2×Q8—62no—
375C2×Q8—62no—
376C2×Q8—62no—
377C2×Q8—124no—
378C2×Q8—128no—
379C2×Q8—124no—
380C2×Q8—122no—
381C2×Q8D3×D414yes—
382C2×D4—62no—
383C2×D4—62no—
384C2×D4—62no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—31no—
390C22×C4—31no—
391C22×C4—31no—
392C22×C4—44no—
393C22×C4—42no—
394C22×C4—62no—
395C22×C4—61no—
396C22×C4—61no—
397C22×C4—62no—
398C22×C4—62no—
399C22×C4—62no—
400C22×C4—124no—
401C22×C4—122no—
402C22×C4#23512yes—
403C22×C4D3×D411yescharacteristic
404C22×C4D3×D412yes—
405C22×C4C23×D311yescharacteristic
406Q16—616no—
407Q16—616no—
408M16—124no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—32no—
414C2×C8—34no—
415C2×C8—32no—
416C2×C8—68no—
417C2×C8—62no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—62no—
421C4⋊C4—62no—
422C4⋊C4—62no—
423C4⋊C4—124no—
424C4⋊C4—124no—
425C22⋊C4—42no—
426C22⋊C4—62no—
427C22⋊C4—62no—
428C22⋊C4—124no—
429C22⋊C4—124no—
430C22⋊C4—122no—
431C42—61no—
432C42—61no—
433C22×C3—22no—
434C22×C3—24no—
435C22×C3—24no—
436C22×C3—22no—
437C22×C3#47014yescomplemented
438C22×C3C2×#12111yescharacteristic
439C22×C3D4212yes—
440C22×C3C2×#13711yescharacteristic
441C22×C3C2×#13711yescharacteristic
442C22×C3#57812yes—
443C2×D3—41no—
444C2×D3—42no—
445C2×D3—42no—
446C2×D3—41no—
447C2×D3—41no—
448C3×C4—22no—
449C3×C4—44no—
450C3×C4—82no—
451C3×C4C2×#12011yescharacteristic
452C3×C4#53511yescharacteristic
453C3×C4#54612yes—
454Dic3—44no—
455Dic3—41no—
456Dic3—41no—
457Dic3—41no—
458Dic3—82no—
459C23—22no—
460C23—22no—
461C23—61no—
462C23—61no—
463C23—61no—
464C23—62no—
465C23—62no—
466C23#88412yes—
467C23#91512yes—
468C23#91812yes—
469C23C2×D3×D411yescharacteristic
470Q8—24no—
471Q8—24no—
472Q8—48no—
473Q8—42no—
474Q8—42no—
475Q8—64no—
476Q8—64no—
477Q8—128no—
478Q8—122no—
479Q8—122no—
480D4—124no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—48no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—61no—
493C2×C4—61no—
494C2×C4—61no—
495C2×C4—61no—
496C2×C4—61no—
497C2×C4—62no—
498C2×C4—64no—
499C2×C4—62no—
500C2×C4—128no—
501C2×C4—122no—
502C2×C4—122no—
503C2×C4—122no—
504C2×C4—121no—
505C2×C4—121no—
506C2×C4—122no—
507C2×C4—122no—
508C2×C4—122no—
509C2×C4#88412yes—
510C2×C4C2×#23511yescharacteristic
511C2×C4C2×D3×D412yes—
512C2×C4C2×D3×D411yescharacteristic
513C2×C4#100412yes—
514C8—34no—
515C8—41no—
516C8—41no—
517C8—62no—
518C2×C3—24no—
519C2×C3#240811yescharacteristic
520C2×C3C2×#47011yescharacteristic
521C2×C3#305911yescharacteristic
522C2×C3#327012yes—
523C2×C3#327012yes—
524D3—81no—
525C22—24no—
526C22—22no—
527C22—22no—
528C22—24no—
529C22—122no—
530C22—121no—
531C22—121no—
532C22—122no—
533C22—121no—
534C22#481614yescomplemented
535C22C2×#88411yescharacteristic
536C22#559012yes—
537C22C2×#91511yescharacteristic
538C22C2×#91811yescharacteristic
539C22#577912yes—
540C4—22no—
541C4—44no—
542C4—82no—
543C4—121no—
544C4—124no—
545C4—121no—
546C4—121no—
547C4—242no—
548C4#549012yes—
549C4D3×#12011yescharacteristic
550C4#559211yescharacteristic
551C3#2427311yescharacteristic, complemented, Sylow
552C2—24no—
553C2—241no—
554C2#7382411yescharacteristic
555C2C2×#481611yescharacteristic
556C2#7896411yescharacteristic
557C2#8006412yes—
558C2#8020812yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1787C2
socleC23×C3C22×D4
prime radicalC22×#67D4
hypercenter#1787D3
hypocenterC3#24273
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1787

Complements

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

NH ≅ G/Ntype
1. C23×#216C2semidirect
3. C2×#4816C2semidirect
6. C3×#1787C2semidirect
7. #70285C2semidirect
8. #70273C2semidirect
9. #70259C2semidirect
10. #69963C2semidirect
11. #69511C2semidirect
58. #1787D3semidirect
90. C2×#216D4semidirect
437. C22×C3#470semidirect
534. C22#4816semidirect
551. C3#24273semidirect

Maximal subgroups

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

subgroupindex
C23×#2162
#840682
C2×#48162
C2×#47302
#740172
C3×#17872
#702852
#702732
#702592
#699632
#695112
#242733