Finite Group Databasebeta

← #10829993

Subgroups — #10829993

Order 768.

1146 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5585C211yesmaximal, characteristic
2#82482C211yesmaximal, characteristic, complemented
3C2×#4848C211yesmaximal, characteristic
4C2×C4×Dic12C211yesmaximal, characteristic, complemented
5C3×#1984C211yesmaximal, characteristic, complemented
6#70485C211yesmaximal, characteristic, complemented
7#69772C211yesmaximal, characteristic
8#68677C212yesmaximal, complemented
9#68672C212yesmaximal, complemented
10#68670C212yesmaximal, complemented
11#68630C212yesmaximal, complemented
12#24190—31nomaximal, Sylow
13C2×C3×C4×Q8C2211yescharacteristic
14C2×#947—21no—
15C22×Dic12C2211yescharacteristic
16C22×#204C2212yes—
17#5585—24no—
18#5585—24no—
19C2×#894—22no—
20C2×#893—21no—
21C2×#875C2212yes—
22C2×#874C2211yescharacteristic
23C2×C4×D12C2212yes—
24C2×C4×Dic6C2211yescharacteristic
25C2×C4×Dic6C2211yescharacteristic
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×C4×C8C2211yescharacteristic
28#5229—21no—
29C2×#821C2212yes—
30C2×#821C2212yes—
31C2×#821C2212yes—
32C2×#818C2211yescharacteristic
33C2×#816C2211yescharacteristic
34C2×#816C2211yescharacteristic
35C2×#816C2212yes—
36#4982—22no—
37#4972—21no—
38#4935C2211yescharacteristic
39#4929C2212yes—
40#4928C2212yes—
41#4848—24no—
42C4×Dic12—24no—
43C3×#336C2212yes—
44C3×#336C2212yes—
45#4551C2212yes—
46C2×#559—31no—
47C2×#460—31no—
48C2×C4×Q16—31no—
49#2582—31no—
50#2042—32no—
51#2015—32no—
52#1984D311yescharacteristic, complemented
53#1957—31no—
54#1877—32no—
55#1862—32no—
56#1803—31no—
57C22×C3×Q8D411yescharacteristic
58C22×D12—21no—
59C22×D12—21no—
60C22×D12C2312yes—
61C22×C4×D3—22no—
62C22×Dic6C2311yescharacteristic
63C22×Dic6C2311yescharacteristic
64C22×C3×C8C2312yes—
65C22×C3×C8C2311yescharacteristic
66C22×C3×C8C2311yescharacteristic
67C3×C4×Q8—22no—
68C3×C4×Q8—22no—
69C2×C3×C4⋊C4D411yescharacteristic
70C2×C3×C4⋊C4D412yes—
71C2×C3×C42C2312yes—
72C2×C3×C42C2311yescharacteristic
73#947—28no—
74C2×(C2×D3)⋊C4—22no—
75C2×(C2×D3)⋊C4—22no—
76C2×(C2×D3)⋊C4—22no—
77C2×C4⋊Dic3—21no—
78C2×C4⋊Dic3—21no—
79C2×C4⋊Dic3C2311yescharacteristic
80C2×C4⋊Dic3C2312yes—
81C2×C4⋊Dic3C2311yescharacteristic
82C2×Dic3⋊C4—21no—
83C2×Dic3⋊C4—21no—
84C2×Dic3⋊C4—22no—
85C2×C4×Dic3—21no—
86C2×C4×Dic3—21no—
87C2×Dic12—22no—
88C2×Dic12—22no—
89C2×Dic12—22no—
90C2×#204—24no—
91C2×#204—24no—
92C2×#204—42no—
93#894—216no—
94#893—28no—
95#875—24no—
96#875—24no—
97#874—22no—
98#874D414yes—
99C4×D12—24no—
100C4×D12—24no—
101C4×Dic6—22no—
102C4×Dic6—22no—
103C4×Dic6—22no—
104C4×Dic6D414yes—
105C3×C4⋊C8—22no—
106C3×C4×C8—22no—
107C3⋊#95—22no—
108C3⋊#95—22no—
109C3⋊#95—22no—
110#821—24no—
111#821—24no—
112#821—42no—
113#818—22no—
114#816—22no—
115#816—22no—
116#816—42no—
117C22×Q16—31no—
118C22×SD16—32no—
119#559—64no—
120#559—64no—
121C2×#127—31no—
122C2×#124—32no—
123C2×#124—61no—
124C2×#123—62no—
125C2×C4×Q8—31no—
126C2×C4×Q8—31no—
127C2×C4×Q8C2×D311yescharacteristic
128C2×C4×D4—32no—
129C2×C4×D4—61no—
130#460—64no—
131C4×Q16—64no—
132C2×#107—31no—
133C2×C4⋊C8C2×D311yescharacteristic
134C2×#103—31no—
135C2×#103—31no—
136C2×#103—32no—
137C2×#102—32no—
138C2×#102—32no—
139C2×#102—32no—
140C2×C4×C8C2×D311yescharacteristic
141#398—62no—
142#391—32no—
143#391—61no—
144#389—32no—
145#389—61no—
146#384—31no—
147#340—32no—
148#336C2×D312yes—
149#336C2×D312yes—
150C23×D3—22no—
151C2×C3×Q8—22no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C22×C3×C4C2×D412yes—
155C22×C3×C4C2×D412yes—
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2411yescharacteristic
159C22×Dic3—21no—
160C22×Dic3—21no—
161C22×Dic3—21no—
162C22×Dic3—22no—
163C22×Dic3—21no—
164C2×D12—22no—
165C2×D12—22no—
166C2×D12—22no—
167C2×D12—22no—
168C2×D12—24no—
169C2×D12—22no—
170C2×D12—24no—
171C2×D12—42no—
172C2×D12—42no—
173C2×D12—81no—
174C2×D12—81no—
175C2×D12C2×D414yes—
176C2×C4×D3—28no—
177C2×C4×D3—82no—
178C2×C4×D3—82no—
179C2×Dic6—21no—
180C2×Dic6—21no—
181C2×Dic6—21no—
182C2×Dic6—21no—
183C2×Dic6—21no—
184C2×Dic6—21no—
185C2×Dic6—21no—
186C2×Dic6—21no—
187C2×Dic6—21no—
188C2×Dic6—21no—
189C2×Dic6C2×D412yes—
190C2×Dic6#4112yes—
191C2×C3×C8—21no—
192C2×C3×C8—21no—
193C2×C3×C8—21no—
194C2×C3×C8—22no—
195C2×C3×C8—22no—
196C2×C3×C8—22no—
197C2×C3×C8—21no—
198C2×C3×C8—21no—
199C2×C3×C8#4112yes—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—24no—
202C3×C42—22no—
203C3×C42—22no—
204C3×C42C2×D412yes—
205C3×C42C2×D412yes—
206(C2×D3)⋊C4—28no—
207(C2×D3)⋊C4—28no—
208(C2×D3)⋊C4—28no—
209C4⋊Dic3—21no—
210C4⋊Dic3—21no—
211C4⋊Dic3—21no—
212C4⋊Dic3—22no—
213C4⋊Dic3—22no—
214C4⋊Dic3—42no—
215C4⋊Dic3—81no—
216C4⋊Dic3C2×D412yes—
217Dic3⋊C4—24no—
218Dic3⋊C4—42no—
219Dic3⋊C4—82no—
220C4×Dic3—24no—
221C4×Dic3—42no—
222Dic12—28no—
223#204—48no—
224C22×Q8—31no—
225C22×Q8—31no—
226C22×Q8D1211yescharacteristic
227C22×D4—32no—
228C22×D4—61no—
229C22×D4—61no—
230C23×C4—62no—
231C2×Q16—62no—
232C2×Q16—62no—
233C2×Q16—62no—
234C2×SD16—64no—
235C2×SD16—64no—
236C2×SD16—122no—
237C22×C8C22×D312yes—
238C22×C8C22×D311yescharacteristic
239C22×C8C22×D311yescharacteristic
240#127—34no—
241#127—62no—
242#124—64no—
243#124—64no—
244#124—68no—
245#123—616no—
246C4×Q8—22no—
247C4×Q8—22no—
248C4×Q8—34no—
249C4×Q8—62no—
250C4×Q8—62no—
251C4×Q8—62no—
252C4×D4—64no—
253C4×D4—64no—
254C4×D4—68no—
255C2×C4⋊C4—31no—
256C2×C4⋊C4—31no—
257C2×C4⋊C4—32no—
258C2×C4⋊C4—61no—
259C2×C4⋊C4—61no—
260C2×C4⋊C4—61no—
261C2×C4⋊C4—61no—
262C2×C4⋊C4—62no—
263C2×C4⋊C4D1211yescharacteristic
264C2×C4⋊C4D1212yes—
265C2×C22⋊C4—62no—
266C2×C22⋊C4—62no—
267C2×C22⋊C4—62no—
268C2×C42—61no—
269C2×C42—61no—
270C2×C42C22×D312yes—
271C2×C42C22×D311yescharacteristic
272#107—62no—
273C4⋊C8—22no—
274#103—62no—
275#103—62no—
276#103—122no—
277#102—64no—
278#102—64no—
279#102—122no—
280C4×C8—22no—
281#95—62no—
282#95—62no—
283#95—62no—
284C23×C3C22×D411yescharacteristic
285C22×D3—24no—
286C22×D3—42no—
287C22×D3—42no—
288C22×D3—42no—
289C22×D3—42no—
290C22×D3—42no—
291C22×D3—42no—
292C3×Q8—44no—
293C2×C3×C4—21no—
294C2×C3×C4—21no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4#12012yes—
302C2×C3×C4#12012yes—
303C2×C3×C4#13512yes—
304C2×C3×C4#13712yes—
305C2×C3×C4C22×D411yescharacteristic
306C2×C3×C4C22×D411yescharacteristic
307C2×C3×C4C2×#4111yescharacteristic
308C2×C3×C4C2×#4111yescharacteristic
309C2×C3×C4H(5,𝔽2)11yescharacteristic
310C2×C3×C42−1+411yescharacteristic
311C2×Dic3—22no—
312C2×Dic3—22no—
313C2×Dic3—22no—
314C2×Dic3—41no—
315C2×Dic3—41no—
316C2×Dic3—41no—
317C2×Dic3—41no—
318C2×Dic3—41no—
319C2×Dic3—41no—
320C2×Dic3—41no—
321C2×Dic3—42no—
322C2×Dic3—42no—
323C2×Dic3—41no—
324C2×Dic3—42no—
325C2×Dic3—41no—
326D12—24no—
327D12—24no—
328D12—42no—
329D12—42no—
330D12—48no—
331D12—84no—
332C4×D3—88no—
333Dic6—22no—
334Dic6—22no—
335Dic6—22no—
336Dic6—22no—
337Dic6—41no—
338Dic6—41no—
339Dic6—41no—
340Dic6—41no—
341C3×C8—22no—
342C3×C8—41no—
343C3×C8—42no—
344C24—62no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—32no—
349C2×Q8—32no—
350C2×Q8—61no—
351C2×Q8—61no—
352C2×Q8—61no—
353C2×Q8—61no—
354C2×Q8—61no—
355C2×Q8—61no—
356C2×Q8—61no—
357C2×Q8—61no—
358C2×Q8—61no—
359C2×Q8—61no—
360C2×D4—34no—
361C2×D4—64no—
362C2×D4—64no—
363C2×D4—62no—
364C2×D4—62no—
365C2×D4—62no—
366C2×D4—62no—
367C2×D4—62no—
368C2×D4—122no—
369C2×D4—122no—
370C2×D4—241no—
371C2×D4—241no—
372C22×C4—68no—
373C22×C4—61no—
374C22×C4—61no—
375C22×C4—61no—
376C22×C4—62no—
377C22×C4—61no—
378C22×C4—242no—
379C22×C4—242no—
380C22×C4C2×D1212yes—
381C22×C4C2×D1212yes—
382C22×C4C2×D1211yescharacteristic
383C22×C4C2×D1211yescharacteristic
384C22×C4C23×D311yescharacteristic
385Q16—68no—
386SD16—128no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—22no—
391C2×C8—21no—
392C2×C8—22no—
393C2×C8—21no—
394C2×C8—22no—
395C2×C8#23912yes—
396C4⋊C4—22no—
397C4⋊C4—24no—
398C4⋊C4—32no—
399C4⋊C4—64no—
400C4⋊C4—61no—
401C4⋊C4—61no—
402C4⋊C4—61no—
403C4⋊C4—62no—
404C4⋊C4—62no—
405C4⋊C4—122no—
406C4⋊C4—122no—
407C4⋊C4—242no—
408C4⋊C4—241no—
409C22⋊C4—68no—
410C22⋊C4—68no—
411C22⋊C4—68no—
412C42—22no—
413C42—22no—
414C42—64no—
415C42—122no—
416C42D3×D412yes—
417C42D3×D412yes—
418C22×C3C2×#12011yescharacteristic
419C22×C3#54811yescharacteristic
420C22×C3#55011yescharacteristic
421C22×C3C2×#13511yescharacteristic
422C22×C3C2×#13711yescharacteristic
423C22×C3#57611yescharacteristic
424C22×C3#57711yescharacteristic
425C2×D3—44no—
426C2×D3—44no—
427C2×D3—44no—
428C2×D3—82no—
429C2×D3—82no—
430C2×D3—82no—
431C2×D3—82no—
432C3×C4—22no—
433C3×C4—42no—
434C3×C4—42no—
435C3×C4#45214yes—
436C3×C4#52511yescharacteristic
437C3×C4#53711yescharacteristic
438C3×C4#54611yescharacteristic
439C3×C4#54711yescharacteristic
440Dic3—42no—
441Dic3—42no—
442Dic3—42no—
443Dic3—82no—
444Dic3—81no—
445C23—64no—
446C23—122no—
447C23—122no—
448C23—122no—
449C23—122no—
450C23—122no—
451C23—122no—
452C23C22×D1211yescharacteristic
453Q8—44no—
454Q8—62no—
455Q8—62no—
456Q8—62no—
457Q8—62no—
458Q8—121no—
459Q8—121no—
460Q8—121no—
461Q8—121no—
462D4—64no—
463D4—64no—
464D4—128no—
465D4—122no—
466D4—122no—
467D4—244no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—62no—
477C2×C4—62no—
478C2×C4—62no—
479C2×C4—121no—
480C2×C4—121no—
481C2×C4—121no—
482C2×C4—121no—
483C2×C4—121no—
484C2×C4—121no—
485C2×C4—122no—
486C2×C4—121no—
487C2×C4—121no—
488C2×C4—122no—
489C2×C4—121no—
490C2×C4—122no—
491C2×C4—248no—
492C2×C4#88212yes—
493C2×C4#88212yes—
494C2×C4#90412yes—
495C2×C4#90912yes—
496C2×C4C2×D3×D411yescharacteristic
497C2×C4C2×D3×D411yescharacteristic
498C2×C4C2×#23911yescharacteristic
499C2×C4#100711yescharacteristic
500C2×C4D3×#4111yescharacteristic
501C2×C4#100911yescharacteristic
502C8—22no—
503C8—42no—
504C8—41no—
505C2×C3#240711yescharacteristic
506C2×C3C2×#45211yescharacteristic
507C2×C3#301211yescharacteristic
508C2×C3#336511yescharacteristic
509C2×C3#337511yescharacteristic
510C2×C3#339611yescharacteristic
511C2×C3#340011yescharacteristic
512D3—84no—
513C22—124no—
514C22—124no—
515C22—124no—
516C22—242no—
517C22—242no—
518C22—242no—
519C22—242no—
520C22C2×#88211yescharacteristic
521C22#557611yescharacteristic
522C22#557711yescharacteristic
523C22C2×#90411yescharacteristic
524C22C2×#90911yescharacteristic
525C22#575111yescharacteristic
526C22#575211yescharacteristic
527C4—22no—
528C4—42no—
529C4—42no—
530C4—122no—
531C4—122no—
532C4—122no—
533C4—242no—
534C4—241no—
535C4#473314yes—
536C4#562711yescharacteristic
537C4#562911yescharacteristic
538C4#567611yescharacteristic
539C4#572611yescharacteristic
540C3#2419011yescharacteristic, complemented, Sylow
541C2—244no—
542C2#7661811yescharacteristic
543C2C2×#473311yescharacteristic
544C2#7867211yescharacteristic
545C2#7991511yescharacteristic
546C2#7993511yescharacteristic
547C2#7995311yescharacteristic
548C2#7998411yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1984C2
socleC23×C3C22×D4
prime radicalC2×#874C22
hypercenter#1984D3
hypocenterC3#24190
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #1984

Complements

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

NH ≅ G/Ntype
2. #82482C2semidirect
4. C2×C4×Dic12C2semidirect
5. C3×#1984C2semidirect
6. #70485C2semidirect
8. #68677C2semidirect
9. #68672C2semidirect
10. #68670C2semidirect
11. #68630C2semidirect
52. #1984D3semidirect
540. C3#24190semidirect

Maximal subgroups

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

subgroupindex
C2×#55852
#824822
C2×#48482
C2×C4×Dic122
C3×#19842
#704852
#697722
#686772
#686722
#686702
#686302
#241903