Finite Group Databasebeta

← #10830097

Subgroups — #10830097

Order 768.

1180 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5585C211yesmaximal, characteristic, complemented
2#82501C211yesmaximal, characteristic, complemented
3C2×#4843C211yesmaximal, characteristic
4C2×#4710C211yesmaximal, characteristic, complemented
5C3×#1985C211yesmaximal, characteristic, complemented
6#70486C211yesmaximal, characteristic, complemented
7#69823C211yesmaximal, characteristic, complemented
8#68675C212yesmaximal, complemented
9#68673C212yesmaximal, complemented
10#68670C212yesmaximal, complemented
11#68630C212yesmaximal, complemented
12#24198—31nomaximal, Sylow
13C2×C3×C4×Q8C2211yescharacteristic
14C2×#947—21no—
15C22×D24C2212yes—
16C22×#204C2211yescharacteristic
17#5585—24no—
18#5585—24no—
19C2×#894—22no—
20C2×#893—21no—
21C2×#875C2212yes—
22C2×#874C2211yescharacteristic
23C2×C4×D12C2212yes—
24C2×C4×D12C2211yescharacteristic
25C2×C4×Dic6C2211yescharacteristic
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×#97C2211yescharacteristic
28#5231—21no—
29C2×#821C2212yes—
30C2×#821C2212yes—
31C2×#821C2211yescharacteristic
32C2×#821C2212yes—
33C2×#818C2211yescharacteristic
34C2×#816C2211yescharacteristic
35C2×#816C2212yes—
36#4993—21no—
37#4989—22no—
38#4939C2212yes—
39#4936C2211yescharacteristic
40#4928C2212yes—
41#4843—24no—
42#4710—24no—
43C3×#336C2212yes—
44C3×#336C2212yes—
45#4551C2212yes—
46C2×#559—31no—
47C2×#459—31no—
48C2×#440—31no—
49#2580—31no—
50#2042—32no—
51#2012—32no—
52#1985D311yescharacteristic, complemented
53#1961—31no—
54#1879—32no—
55#1862—32no—
56#1805—31no—
57C22×C3×Q8D411yescharacteristic
58C22×D12—21no—
59C22×D12—21no—
60C22×D12C2312yes—
61C22×D12C2311yescharacteristic
62C22×C4×D3—22no—
63C22×C4×D3—21no—
64C22×Dic6C2311yescharacteristic
65C22×C3×C8C2312yes—
66C22×C3×C8C2311yescharacteristic
67C22×C3×C8C2311yescharacteristic
68C3×C4×Q8—22no—
69C3×C4×Q8—22no—
70C2×C3×C4⋊C4D412yes—
71C2×C3×C4⋊C4D411yescharacteristic
72C2×C3×C42C2312yes—
73C2×C3×C42C2311yescharacteristic
74#947—28no—
75C2×(C2×D3)⋊C4—22no—
76C2×(C2×D3)⋊C4—22no—
77C2×(C2×D3)⋊C4—22no—
78C2×(C2×D3)⋊C4—22no—
79C2×(C2×D3)⋊C4—21no—
80C2×(C2×D3)⋊C4—22no—
81C2×C4⋊Dic3C2311yescharacteristic
82C2×C4⋊Dic3C2312yes—
83C2×C4⋊Dic3C2311yescharacteristic
84C2×Dic3⋊C4—21no—
85C2×Dic3⋊C4—22no—
86C2×C4×Dic3—21no—
87C2×D24—24no—
88C2×D24—24no—
89C2×D24—42no—
90C2×#204—22no—
91C2×#204—22no—
92C2×#204—22no—
93#894—216no—
94#893—28no—
95#875—24no—
96#875—24no—
97#874—22no—
98#874D414yes—
99C4×D12—24no—
100C4×D12—22no—
101C4×D12—22no—
102C4×D12—24no—
103C4×Dic6—22no—
104C4×Dic6D414yes—
105C3×C4⋊C8—22no—
106C3×#97—22no—
107C3⋊#95—22no—
108C3⋊#95—21no—
109C3⋊#95—21no—
110#821—24no—
111#821—24no—
112#821—22no—
113#821—42no—
114#818—22no—
115#816—22no—
116#816—42no—
117C22×SD16—31no—
118C22×D8—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×Q8C2×D311yescharacteristic
127C2×C4×D4—31no—
128C2×C4×D4—32no—
129C2×C4×D4—61no—
130#459—64no—
131#440—64no—
132C2×#107—31no—
133C2×C4⋊C8C2×D311yescharacteristic
134C2×#103—31no—
135C2×#103—32no—
136C2×#102—32no—
137C2×#102—32no—
138C2×#102—31no—
139C2×#102—32no—
140C2×#97C2×D311yescharacteristic
141#399—61no—
142#393—61no—
143#390—32no—
144#389—32no—
145#385—62no—
146#382—31no—
147#340—32no—
148#336C2×D312yes—
149#336C2×D312yes—
150C23×D3—21no—
151C23×D3—22no—
152C2×C3×Q8—22no—
153C2×C3×Q8—42no—
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—22no—
162C22×Dic3—21no—
163C2×D12—21no—
164C2×D12—21no—
165C2×D12—22no—
166C2×D12—21no—
167C2×D12—22no—
168C2×D12—22no—
169C2×D12—21no—
170C2×D12—22no—
171C2×D12—24no—
172C2×D12—22no—
173C2×D12—21no—
174C2×D12—24no—
175C2×D12—44no—
176C2×D12—82no—
177C2×D12C2×D414yes—
178C2×D12#4112yes—
179C2×C4×D3—28no—
180C2×C4×D3—42no—
181C2×C4×D3—82no—
182C2×C4×D3—82no—
183C2×C4×D3—82no—
184C2×Dic6—21no—
185C2×Dic6—21no—
186C2×Dic6—21no—
187C2×Dic6—21no—
188C2×Dic6—21no—
189C2×Dic6C2×D412yes—
190C2×C3×C8—21no—
191C2×C3×C8—21no—
192C2×C3×C8—21no—
193C2×C3×C8—22no—
194C2×C3×C8—22no—
195C2×C3×C8—22no—
196C2×C3×C8—21no—
197C2×C3×C8—21no—
198C2×C3×C8#4112yes—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—24no—
201C3×C42—22no—
202C3×C42—22no—
203C3×C42C2×D412yes—
204C3×C42C2×D412yes—
205(C2×D3)⋊C4—28no—
206(C2×D3)⋊C4—28no—
207(C2×D3)⋊C4—28no—
208(C2×D3)⋊C4—44no—
209(C2×D3)⋊C4—42no—
210(C2×D3)⋊C4—82no—
211C4⋊Dic3—21no—
212C4⋊Dic3—21no—
213C4⋊Dic3—21no—
214C4⋊Dic3—22no—
215C4⋊Dic3—22no—
216C4⋊Dic3C2×D412yes—
217Dic3⋊C4—24no—
218Dic3⋊C4—82no—
219C4×Dic3—24no—
220D24—48no—
221#204—28no—
222C22×Q8—31no—
223C22×Q8D1211yescharacteristic
224C22×D4—32no—
225C22×D4—31no—
226C22×D4—61no—
227C22×D4—61no—
228C23×C4—62no—
229C23×C4—61no—
230C2×SD16—62no—
231C2×SD16—62no—
232C2×SD16—62no—
233C2×D8—64no—
234C2×D8—64no—
235C2×D8—122no—
236C22×C8C22×D312yes—
237C22×C8C22×D311yescharacteristic
238C22×C8C22×D311yescharacteristic
239#127—34no—
240#127—62no—
241#124—64no—
242#124—64no—
243#124—68no—
244#123—616no—
245C4×Q8—22no—
246C4×Q8—22no—
247C4×Q8—34no—
248C4×Q8—62no—
249C4×D4—64no—
250C4×D4—64no—
251C4×D4—68no—
252C4×D4—62no—
253C4×D4—62no—
254C2×C4⋊C4—31no—
255C2×C4⋊C4—31no—
256C2×C4⋊C4—32no—
257C2×C4⋊C4—61no—
258C2×C4⋊C4—62no—
259C2×C4⋊C4D1211yescharacteristic
260C2×C4⋊C4D1212yes—
261C2×C22⋊C4—62no—
262C2×C22⋊C4—62no—
263C2×C22⋊C4—62no—
264C2×C22⋊C4—61no—
265C2×C22⋊C4—62no—
266C2×C22⋊C4—62no—
267C2×C42—61no—
268C2×C42C22×D312yes—
269C2×C42C22×D311yescharacteristic
270#107—62no—
271C4⋊C8—22no—
272#103—62no—
273#103—122no—
274#102—62no—
275#102—64no—
276#102—64no—
277#102—122no—
278#97—22no—
279#95—61no—
280#95—61no—
281#95—62no—
282C23×C3C22×D411yescharacteristic
283C22×D3—24no—
284C22×D3—22no—
285C22×D3—41no—
286C22×D3—42no—
287C22×D3—42no—
288C22×D3—41no—
289C22×D3—42no—
290C22×D3—42no—
291C22×D3—42no—
292C22×D3—42no—
293C22×D3—42no—
294C22×D3—42no—
295C3×Q8—44no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4#12012yes—
305C2×C3×C4#12012yes—
306C2×C3×C4C22×D411yescharacteristic
307C2×C3×C4C22×D411yescharacteristic
308C2×C3×C4C2×#4111yescharacteristic
309C2×C3×C4C2×#4111yescharacteristic
310C2×C3×C4H(5,𝔽2)11yescharacteristic
311C2×C3×C42−1+411yescharacteristic
312C2×Dic3—22no—
313C2×Dic3—22no—
314C2×Dic3—41no—
315C2×Dic3—41no—
316C2×Dic3—42no—
317C2×Dic3—41no—
318C2×Dic3—42no—
319C2×Dic3—42no—
320C2×Dic3—42no—
321C2×Dic3—42no—
322D12—22no—
323D12—22no—
324D12—24no—
325D12—24no—
326D12—48no—
327D12—42no—
328D12—41no—
329D12—42no—
330D12—41no—
331D12—84no—
332C4×D3—88no—
333C4×D3—84no—
334Dic6—22no—
335Dic6—22no—
336Dic6—41no—
337Dic6—41no—
338C3×C8—22no—
339C3×C8—41no—
340C3×C8—42no—
341C24—61no—
342C24—62no—
343C2×Q8—22no—
344C2×Q8—32no—
345C2×Q8—42no—
346C2×Q8—61no—
347C2×Q8—61no—
348C2×Q8—61no—
349C2×Q8—61no—
350C2×Q8—61no—
351C2×D4—34no—
352C2×D4—32no—
353C2×D4—64no—
354C2×D4—64no—
355C2×D4—61no—
356C2×D4—61no—
357C2×D4—62no—
358C2×D4—62no—
359C2×D4—61no—
360C2×D4—62no—
361C2×D4—61no—
362C2×D4—62no—
363C2×D4—62no—
364C2×D4—61no—
365C2×D4—124no—
366C2×D4—242no—
367C22×C4—68no—
368C22×C4—61no—
369C22×C4—61no—
370C22×C4—62no—
371C22×C4—61no—
372C22×C4—122no—
373C22×C4—242no—
374C22×C4—242no—
375C22×C4—242no—
376C22×C4C2×D1212yes—
377C22×C4C2×D1212yes—
378C22×C4C2×D1211yescharacteristic
379C22×C4C2×D1211yescharacteristic
380C22×C4C23×D311yescharacteristic
381SD16—68no—
382D8—128no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—22no—
387C2×C8—21no—
388C2×C8—22no—
389C2×C8—21no—
390C2×C8—22no—
391C2×C8#23912yes—
392C4⋊C4—22no—
393C4⋊C4—24no—
394C4⋊C4—32no—
395C4⋊C4—64no—
396C4⋊C4—61no—
397C4⋊C4—61no—
398C4⋊C4—61no—
399C4⋊C4—62no—
400C4⋊C4—62no—
401C4⋊C4—242no—
402C22⋊C4—68no—
403C22⋊C4—68no—
404C22⋊C4—68no—
405C22⋊C4—124no—
406C22⋊C4—122no—
407C22⋊C4—242no—
408C42—22no—
409C42—22no—
410C42—64no—
411C42D3×D412yes—
412C42D3×D412yes—
413C22×C3C2×#12011yescharacteristic
414C22×C3#54811yescharacteristic
415C22×C3#55011yescharacteristic
416C22×C3#57512yes—
417C22×C3#57612yes—
418C2×D3—44no—
419C2×D3—42no—
420C2×D3—44no—
421C2×D3—44no—
422C2×D3—44no—
423C2×D3—82no—
424C2×D3—82no—
425C2×D3—82no—
426C2×D3—82no—
427C2×D3—82no—
428C2×D3—81no—
429C2×D3—81no—
430C3×C4—22no—
431C3×C4—22no—
432C3×C4—42no—
433C3×C4—42no—
434C3×C4#52511yescharacteristic
435C3×C4#53711yescharacteristic
436C3×C4#54611yescharacteristic
437C3×C4#54711yescharacteristic
438Dic3—42no—
439Dic3—42no—
440Dic3—82no—
441Dic3—81no—
442C23—64no—
443C23—62no—
444C23—121no—
445C23—122no—
446C23—122no—
447C23—122no—
448C23—122no—
449C23—122no—
450C23—122no—
451C23—122no—
452C23—122no—
453C23—121no—
454C23C22×D1211yescharacteristic
455Q8—44no—
456Q8—62no—
457Q8—62no—
458Q8—121no—
459Q8—121no—
460D4—62no—
461D4—64no—
462D4—64no—
463D4—62no—
464D4—128no—
465D4—121no—
466D4—122no—
467D4—121no—
468D4—122no—
469D4—244no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—62no—
479C2×C4—62no—
480C2×C4—122no—
481C2×C4—121no—
482C2×C4—121no—
483C2×C4—121no—
484C2×C4—122no—
485C2×C4—122no—
486C2×C4—122no—
487C2×C4—122no—
488C2×C4—244no—
489C2×C4—248no—
490C2×C4#88212yes—
491C2×C4#88212yes—
492C2×C4C2×D3×D411yescharacteristic
493C2×C4C2×D3×D411yescharacteristic
494C2×C4C2×#23911yescharacteristic
495C2×C4#100711yescharacteristic
496C2×C4D3×#4111yescharacteristic
497C2×C4#100911yescharacteristic
498C8—22no—
499C8—42no—
500C8—41no—
501C2×C3#240711yescharacteristic
502C2×C3#300511yescharacteristic
503C2×C3#300811yescharacteristic
504C2×C3#337012yes—
505C2×C3#340312yes—
506D3—82no—
507D3—84no—
508C22—124no—
509C22—122no—
510C22—124no—
511C22—124no—
512C22—124no—
513C22—241no—
514C22—241no—
515C22—242no—
516C22—242no—
517C22—242no—
518C22—242no—
519C22—242no—
520C22C2×#88211yescharacteristic
521C22#557611yescharacteristic
522C22#557711yescharacteristic
523C22#574812yes—
524C22#575212yes—
525C4—22no—
526C4—22no—
527C4—42no—
528C4—42no—
529C4—122no—
530C4—122no—
531C4—241no—
532C4—242no—
533C4#562711yescharacteristic
534C4#562911yescharacteristic
535C4#567611yescharacteristic
536C4#572611yescharacteristic
537C3#2419811yescharacteristic, complemented, Sylow
538C2—244no—
539C2—242no—
540C2#7661811yescharacteristic
541C2#7863811yescharacteristic
542C2#7867011yescharacteristic
543C2#7995612yes—
544C2#7997112yes—
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1985C2
socleC23×C3C22×D4
prime radicalC2×#4843C2
hypercenter#1985D3
hypocenterC3#24198
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #1985

Complements

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

NH ≅ G/Ntype
1. C2×#5585C2semidirect
2. #82501C2semidirect
4. C2×#4710C2semidirect
5. C3×#1985C2semidirect
6. #70486C2semidirect
7. #69823C2semidirect
8. #68675C2semidirect
9. #68673C2semidirect
10. #68670C2semidirect
11. #68630C2semidirect
52. #1985D3semidirect
537. C3#24198semidirect

Maximal subgroups

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

subgroupindex
C2×#55852
#825012
C2×#48432
C2×#47102
C3×#19852
#704862
#698232
#686752
#686732
#686702
#686302
#241983