Finite Group Databasebeta

← #10830950

Subgroups — #10830950

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
2C3×#2409C211yesmaximal, characteristic
3C2×#5052C212yesmaximal, complemented
4#82349C211yesmaximal, characteristic
5C2×#5011C211yesmaximal, characteristic
6#72132C211yesmaximal, characteristic, complemented
7#71700C212yesmaximal, complemented
8#69681C212yesmaximal, complemented
9#69678C212yesmaximal, complemented
10#69525C211yesmaximal, characteristic, complemented
11#69511C211yesmaximal, characteristic, complemented
12#24273—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C23×Dic6C2211yescharacteristic
15C2×C3×C22⋊Q8C2212yes—
16C2×C3×C4×D4C2211yescharacteristic
17C22×#216—28no—
18C22×#216—24no—
19C22×#216C2212yes—
20C22×#216C2212yes—
21C2×#924C2212yes—
22C22×#208C2211yescharacteristic
23C23×C3⋊C8C2211yescharacteristic
24C3×#395—21no—
25C3×#393—21no—
26C3×#391C2212yes—
27C3×#386—21no—
28C3×#385—21no—
29C3×#380C2211yescharacteristic
30C2×#835C2212yes—
31C2×#835C2212yes—
32#5220—21no—
33#5212C2211yescharacteristic
34C2×#830C2211yescharacteristic
35C2×#830C2211yescharacteristic
36#5052—28no—
37#5011—24no—
38#4979—21no—
39C2×#810C2212yes—
40C2×#810C2212yes—
41C2×#808C2212yes—
42#4947—21no—
43#4946—21no—
44#4929C2212yes—
45C2×C4⋊(C3⋊C8)C2211yescharacteristic
46C2×#803C2211yescharacteristic
47#4529C2212yes—
48C23×Q16—31no—
49C2×#470—32no—
50C2×#435—31no—
51#2409—31no—
52#2409D311yescharacteristic
53#2056—32no—
54#2001—32no—
55#1971—32no—
56#1807—31no—
57#1787—31no—
58#1787—31no—
59C22×C3×Q8—21no—
60C22×C3×Q8—24no—
61C22×C3×Q8—22no—
62C22×C3×Q8—22no—
63C22×C3×Q8—42no—
64C22×C3×Q8C2312yes—
65C22×C3×D4D411yescharacteristic
66C23×C3×C4—21no—
67C23×C3×C4C2311yescharacteristic
68C23×C3×C4C2311yescharacteristic
69C23×Dic3—21no—
70C22×Dic6—22no—
71C22×Dic6—24no—
72C22×Dic6—22no—
73C22×Dic6—21no—
74C22×Dic6—42no—
75C22×Dic6C2312yes—
76C3×C22⋊Q8—24no—
77C3×C22⋊Q8—42no—
78C3×C4×D4—22no—
79C3×C4×D4—22no—
80C2×C3×C4⋊C4—22no—
81C2×C3×C4⋊C4D411yescharacteristic
82C2×C3×C4⋊C4C2312yes—
83C2×C3×C22⋊C4—21no—
84C2×C3×C22⋊C4—22no—
85C2×C3×C22⋊C4D411yescharacteristic
86C2×C3×C22⋊C4D411yescharacteristic
87C2×C3×C42C2311yescharacteristic
88C2×#217—21no—
89C2×#217—22no—
90C2×#216—232no—
91C2×#216—24no—
92C2×#216—24no—
93C2×#216—24no—
94C2×#216—24no—
95C2×#216—24no—
96C2×#216D418yescomplemented
97C2×C4⋊Dic3C2312yes—
98#924—24no—
99#924—42no—
100C2×Dic3⋊C4—22no—
101C2×#208—22no—
102C2×#208—24no—
103C22×C3⋊C8—24no—
104C22×C3⋊C8D414yes—
105C22×C3⋊C8C2311yescharacteristic
106C22×C3⋊C8C2311yescharacteristic
107C22×C3⋊C8C2311yescharacteristic
108C22×C3⋊C8C2311yescharacteristic
109C3×#95—21no—
110C3×#95—22no—
111C3×#95—22no—
112#835—24no—
113#835—42no—
114C3⋊#95—22no—
115C3⋊#95—22no—
116C3⋊#95—21no—
117#830—22no—
118#830—22no—
119#810—24no—
120#810—42no—
121#808—24no—
122C4⋊(C3⋊C8)—22no—
123#803—22no—
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⋊Q8C2×D312yes—
134C2×C4×D4C2×D311yescharacteristic
135#470—68no—
136#435—64no—
137C2×#106—32no—
138C2×C4⋊C8—31no—
139C2×#103—32no—
140C2×#103—32no—
141C2×#103—32no—
142C2×#103—32no—
143C2×C22⋊C8—31no—
144C2×C22⋊C8—31no—
145C2×#97—31no—
146#395—21no—
147#395—61no—
148#393—21no—
149#393—61no—
150#391—32no—
151#391C2×D312yes—
152#386—21no—
153#386—61no—
154#385—21no—
155#385—61no—
156#380—31no—
157#380C2×D311yescharacteristic
158#340—32no—
159C24×C3C2×D411yescharacteristic
160C24×C3C2×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—
172C2×C3×D4—22no—
173C2×C3×D4—22no—
174C2×C3×D4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—21no—
179C22×C3×C4—22no—
180C22×C3×C4—21no—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—44no—
184C22×C3×C4—42no—
185C22×C3×C4C2×D412yes—
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4#4112yes—
191C22×C3×C4C2411yescharacteristic
192C22×Dic3—22no—
193C22×Dic3—22no—
194C22×Dic3—44no—
195C22×Dic3—42no—
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—
207C3×C4⋊C4—22no—
208C3×C4⋊C4—22no—
209C3×C4⋊C4—22no—
210C3×C4⋊C4—44no—
211C3×C22⋊C4—22no—
212C3×C22⋊C4—22no—
213C3×C22⋊C4—44no—
214C3×C22⋊C4—42no—
215C3×C42—21no—
216C3×C42—21no—
217#217—44no—
218#217—42no—
219#216—216no—
220#216—216no—
221C4⋊Dic3—22no—
222C4⋊Dic3—22no—
223Dic3⋊C4—44no—
224#208—44no—
225C2×C3⋊C8—22no—
226C2×C3⋊C8—28no—
227C2×C3⋊C8—21no—
228C2×C3⋊C8—21no—
229C2×C3⋊C8—22no—
230C2×C3⋊C8—22no—
231C2×C3⋊C8C2×D412yes—
232C2×C3⋊C8C2×D412yes—
233C2×C3⋊C8C2×D414yes—
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#6711yescharacteristic
247C23×C4—21no—
248C23×C4—61no—
249C23×C4C22×D311yescharacteristic
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—31no—
264C22×C8—31no—
265C22×C8—64no—
266C22⋊Q8—24no—
267C22⋊Q8—42no—
268C22⋊Q8—64no—
269C22⋊Q8—122no—
270C4×D4—22no—
271C4×D4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—32no—
274C2×C4⋊C4—62no—
275C2×C4⋊C4#6711yescharacteristic
276C2×C4⋊C4C22×D312yes—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—62no—
280C2×C22⋊C4—61no—
281C2×C22⋊C4#6711yescharacteristic
282C2×C22⋊C4#6711yescharacteristic
283C2×C42C22×D311yescharacteristic
284#106—64no—
285C4⋊C8—62no—
286#103—64no—
287#103—64no—
288#103—122no—
289#103—122no—
290C22⋊C8—62no—
291C22⋊C8—62no—
292#97—62no—
293#95—21no—
294#95—22no—
295#95—22no—
296#95—61no—
297#95—62no—
298#95—62no—
299C23×C3—22no—
300C23×C3—22no—
301C23×C3—21no—
302C23×C3—22no—
303C23×C3—21no—
304C23×C3—21no—
305C23×C3—22no—
306C23×C3#12112yes—
307C23×C3#13712yes—
308C23×C3#13712yes—
309C23×C3C22×D411yescharacteristic
310C3×Q8—24no—
311C3×Q8—24no—
312C3×Q8—48no—
313C3×Q8—42no—
314C3×Q8—42no—
315C3×D4—44no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—24no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—22no—
324C2×C3×C4—24no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—22no—
328C2×C3×C4—48no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4—42no—
337C2×C3×C4#12112yes—
338C2×C3×C4C22×D411yescharacteristic
339C2×C3×C4C22×D412yes—
340C2×C3×C4C2×#4111yescharacteristic
341C2×C3×C4H(5,𝔽2)12yes—
342C2×Dic3—24no—
343C2×Dic3—48no—
344C2×Dic3—42no—
345C2×Dic3—42no—
346C2×Dic3—42no—
347C2×Dic3—42no—
348C2×Dic3—42no—
349Dic6—24no—
350Dic6—24no—
351Dic6—48no—
352Dic6—42no—
353Dic6—42no—
354C3⋊C8—22no—
355C3⋊C8—41no—
356C3⋊C8—41no—
357C3⋊C8#12014yes—
358C24C2×#6711yescharacteristic
359C24C2×#6711yescharacteristic
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—22no—
383C2×D4—22no—
384C2×D4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—44no—
394C22×C4—42no—
395C22×C4—62no—
396C22×C4—62no—
397C22×C4—124no—
398C22×C4—122no—
399C22×C4D3×D412yes—
400C22×C4#23712yes—
401C22×C4C2×#6711yescharacteristic
402C22×C4C2×#6711yescharacteristic
403C22×C4C2×#6711yescharacteristic
404C22×C4C2×#6711yescharacteristic
405C22×C4C23×D311yescharacteristic
406Q16—616no—
407Q16—616no—
408M16—124no—
409C2×C8—32no—
410C2×C8—34no—
411C2×C8—32no—
412C2×C8—68no—
413C2×C8—62no—
414C2×C8—61no—
415C2×C8—61no—
416C2×C8—62no—
417C2×C8—62no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—44no—
422C4⋊C4—62no—
423C4⋊C4—62no—
424C4⋊C4—124no—
425C22⋊C4—22no—
426C22⋊C4—22no—
427C22⋊C4—44no—
428C22⋊C4—42no—
429C22⋊C4—124no—
430C22⋊C4—122no—
431C42—21no—
432C42—21no—
433C22×C3—22no—
434C22×C3—24no—
435C22×C3—24no—
436C22×C3—22no—
437C22×C3—41no—
438C22×C3—42no—
439C22×C3—42no—
440C22×C3—41no—
441C22×C3—41no—
442C22×C3#47014yescomplemented
443C22×C3C2×#12111yescharacteristic
444C22×C3D4212yes—
445C22×C3C2×#13711yescharacteristic
446C22×C3C2×#13711yescharacteristic
447C22×C3#57812yes—
448C3×C4—22no—
449C3×C4—41no—
450C3×C4—44no—
451C3×C4—41no—
452C3×C4—41no—
453C3×C4—82no—
454C3×C4C2×#12011yescharacteristic
455C3×C4#53511yescharacteristic
456C3×C4#54612yes—
457Dic3—44no—
458Dic3—82no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—22no—
466C23#93912yes—
467C23#94212yes—
468C23#95112yes—
469C23C22×#6711yescharacteristic
470Q8—24no—
471Q8—24no—
472Q8—48no—
473Q8—42no—
474Q8—42no—
475Q8—64no—
476Q8—64no—
477Q8—128no—
478Q8—122no—
479Q8—122no—
480D4—44no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—48no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—64no—
503C2×C4—128no—
504C2×C4—122no—
505C2×C4—122no—
506C2×C4—122no—
507C2×C4—122no—
508C2×C4—122no—
509C2×C4#93912yes—
510C2×C4C2×D3×D412yes—
511C2×C4C2×D3×D411yescharacteristic
512C2×C4C2×#23711yescharacteristic
513C2×C4#100412yes—
514C8—34no—
515C8—62no—
516C8—121no—
517C8—121no—
518C2×C3—24no—
519C2×C3—81no—
520C2×C3#240811yescharacteristic
521C2×C3C2×#47011yescharacteristic
522C2×C3#305911yescharacteristic
523C2×C3#327012yes—
524C2×C3#327012yes—
525C22—24no—
526C22—24no—
527C22—22no—
528C22—22no—
529C22—41no—
530C22—42no—
531C22—42no—
532C22—41no—
533C22—41no—
534C22#505214yescomplemented
535C22D4×#6712yes—
536C22C2×#93911yescharacteristic
537C22C2×#94211yescharacteristic
538C22C2×#95111yescharacteristic
539C22#583812yes—
540C4—22no—
541C4—41no—
542C4—41no—
543C4—44no—
544C4—41no—
545C4—82no—
546C4—124no—
547C4—242no—
548C4D3×#12011yescharacteristic
549C4#559112yes—
550C4#559211yescharacteristic
551C3#2427311yescharacteristic, complemented, Sylow
552C2—24no—
553C2—81no—
554C2#8258211yescharacteristic
555C2C2×#505211yescharacteristic
556C2#8339911yescharacteristic
557C2#8354612yes—
558C2#8356512yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2409C2
socleC23×C3C22×D4
prime radicalC22×C3×D4D4
hypercenter#2409D3
hypocenterC3#24273
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2409

Complements

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

NH ≅ G/Ntype
1. C23×#216C2semidirect
3. C2×#5052C2semidirect
6. #72132C2semidirect
7. #71700C2semidirect
8. #69681C2semidirect
9. #69678C2semidirect
10. #69525C2semidirect
11. #69511C2semidirect
96. C2×#216D4semidirect
442. C22×C3#470semidirect
534. C22#5052semidirect
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
C3×#24092
C2×#50522
#823492
C2×#50112
#721322
#717002
#696812
#696782
#695252
#695112
#242733