Finite Group Databasebeta

← #10829938

Subgroups — #10829938

Order 768.

1150 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5584C211yesmaximal, characteristic, complemented
2#82500C211yesmaximal, characteristic, complemented
3C2×#4841C211yesmaximal, characteristic, complemented
4C2×C4×D24C211yesmaximal, characteristic
5C3×#1984C211yesmaximal, characteristic, complemented
6#70479C211yesmaximal, characteristic, complemented
7#69771C211yesmaximal, characteristic, complemented
8#68686C212yesmaximal, complemented
9#68683C212yesmaximal, complemented
10#68675C212yesmaximal
11#68630C212yesmaximal, complemented
12#24185—31nomaximal, Sylow
13C2×C3×C4×Q8C2211yescharacteristic
14C2×#946—21no—
15C2×#946—21no—
16C22×D24C2211yescharacteristic
17C22×#204C2212yes—
18#5584—24no—
19#5584—24no—
20C2×#898—22no—
21C2×#892—21no—
22C2×#875C2212yes—
23C2×C4×D12C2211yescharacteristic
24C2×C4×D12C2211yescharacteristic
25C2×#869C2211yescharacteristic
26C2×C4×Dic6C2212yes—
27C2×C3×C4⋊C8C2211yescharacteristic
28C2×C3×C4×C8C2211yescharacteristic
29C2×#821C2211yescharacteristic
30C2×#821C2211yescharacteristic
31C2×#821C2212yes—
32C2×#818C2211yescharacteristic
33C2×#816C2212yes—
34C2×#816C2212yes—
35C2×#816C2212yes—
36#4994—22no—
37#4987—21no—
38#4939C2212yes—
39#4935C2211yescharacteristic
40#4928C2212yes—
41#4841—24no—
42C4×D24—24no—
43C3×#336C2212yes—
44C3×#336C2212yes—
45#4551C2212yes—
46C2×#556—31no—
47C2×#461—31no—
48C2×C4×D8—31no—
49#2538—31no—
50#2042—32no—
51#2012—32no—
52#1984D311yescharacteristic, complemented
53#1956—31no—
54#1880—32no—
55#1863—32no—
56#1802—31no—
57C22×C3×Q8D411yescharacteristic
58C22×D12C2311yescharacteristic
59C22×D12C2311yescharacteristic
60C22×C4×D3—21no—
61C22×C4×D3—21no—
62C22×Dic6C2312yes—
63C22×C3×C8C2312yes—
64C22×C3×C8C2311yescharacteristic
65C22×C3×C8C2311yescharacteristic
66C3×C4×Q8—22no—
67C3×C4×Q8—22no—
68C2×C3×C4⋊C4D411yescharacteristic
69C2×C3×C4⋊C4D412yes—
70C2×C3×C42C2312yes—
71C2×C3×C42C2311yescharacteristic
72#946—28no—
73#946—44no—
74C2×(C2×D3)⋊C4—21no—
75C2×(C2×D3)⋊C4—22no—
76C2×(C2×D3)⋊C4—22no—
77C2×(C2×D3)⋊C4—22no—
78C2×(C2×D3)⋊C4—21no—
79C2×(C2×D3)⋊C4—22no—
80C2×C4⋊Dic3—21no—
81C2×C4⋊Dic3—21no—
82C2×C4⋊Dic3—21no—
83C2×C4⋊Dic3—21no—
84C2×C4⋊Dic3C2311yescharacteristic
85C2×C4⋊Dic3C2312yes—
86C2×C4⋊Dic3C2311yescharacteristic
87C2×Dic3⋊C4—22no—
88C2×Dic3⋊C4—22no—
89C2×C4×Dic3—22no—
90C2×D24—22no—
91C2×D24—22no—
92C2×D24—22no—
93C2×#204—24no—
94C2×#204—24no—
95C2×#204—42no—
96#898—216no—
97#892—28no—
98#875—24no—
99#875—24no—
100C4×D12—22no—
101C4×D12—22no—
102C4×D12—22no—
103C4×D12D414yes—
104#869—22no—
105#869D414yes—
106C4×Dic6—24no—
107C4×Dic6—24no—
108C3×C4⋊C8—22no—
109C3×C4×C8—22no—
110C3⋊#95—22no—
111#821—22no—
112#821—22no—
113#821—42no—
114#818—22no—
115#816—24no—
116#816—24no—
117#816—42no—
118C22×SD16—32no—
119C22×D8—31no—
120#556—64no—
121#556—64no—
122C2×C4⋊Q8—31no—
123C2×#126—62no—
124C2×#124—32no—
125C2×C22⋊Q8—61no—
126C2×C22⋊Q8—61no—
127C2×C4×Q8—32no—
128C2×C4×Q8C2×D311yescharacteristic
129C2×C4×D4—31no—
130C2×C4×D4—31no—
131C2×#117—61no—
132#461—64no—
133C4×D8—64no—
134C2×#107—31no—
135C2×C4⋊C8C2×D311yescharacteristic
136C2×#103—32no—
137C2×#103—32no—
138C2×#103—32no—
139C2×#102—31no—
140C2×#102—31no—
141C2×#102—32no—
142C2×C4×C8C2×D311yescharacteristic
143#396—62no—
144#390—32no—
145#389—32no—
146#384—31no—
147#380—61no—
148#340—32no—
149#336C2×D312yes—
150#336C2×D312yes—
151C23×D3—21no—
152C23×D3—21no—
153C2×C3×Q8—22no—
154C2×C3×Q8—22no—
155C2×C3×Q8—22no—
156C22×C3×C4C2×D412yes—
157C22×C3×C4C2×D412yes—
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2411yescharacteristic
161C22×Dic3—21no—
162C22×Dic3—22no—
163C22×Dic3—21no—
164C22×Dic3—22no—
165C2×D12—21no—
166C2×D12—21no—
167C2×D12—21no—
168C2×D12—21no—
169C2×D12—21no—
170C2×D12—21no—
171C2×D12—21no—
172C2×D12—21no—
173C2×D12—21no—
174C2×D12—21no—
175C2×D12C2×D412yes—
176C2×D12#4112yes—
177C2×C4×D3—24no—
178C2×C4×D3—42no—
179C2×C4×D3—42no—
180C2×C4×D3—42no—
181C2×C4×D3—42no—
182C2×C4×D3—42no—
183C2×Dic6—22no—
184C2×Dic6—22no—
185C2×Dic6—22no—
186C2×Dic6—22no—
187C2×Dic6—22no—
188C2×Dic6C2×D414yes—
189C2×C3×C8—21no—
190C2×C3×C8—21no—
191C2×C3×C8—22no—
192C2×C3×C8—21no—
193C2×C3×C8—22no—
194C2×C3×C8—21no—
195C2×C3×C8—22no—
196C2×C3×C8—21no—
197C2×C3×C8#4112yes—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—24no—
200C3×C42—22no—
201C3×C42—22no—
202C3×C42C2×D412yes—
203C3×C42C2×D412yes—
204(C2×D3)⋊C4—28no—
205(C2×D3)⋊C4—28no—
206(C2×D3)⋊C4—24no—
207(C2×D3)⋊C4—44no—
208(C2×D3)⋊C4—42no—
209(C2×D3)⋊C4—82no—
210C4⋊Dic3—24no—
211C4⋊Dic3—24no—
212C4⋊Dic3—21no—
213C4⋊Dic3—21no—
214C4⋊Dic3—21no—
215C4⋊Dic3—22no—
216C4⋊Dic3—22no—
217C4⋊Dic3—42no—
218C4⋊Dic3—81no—
219C4⋊Dic3C2×D412yes—
220Dic3⋊C4—28no—
221Dic3⋊C4—44no—
222C4×Dic3—28no—
223D24—28no—
224#204—48no—
225C22×Q8—32no—
226C22×Q8D1211yescharacteristic
227C22×D4—31no—
228C22×D4—31no—
229C23×C4—61no—
230C23×C4—61no—
231C2×SD16—64no—
232C2×SD16—64no—
233C2×SD16—122no—
234C2×D8—62no—
235C2×D8—62no—
236C2×D8—62no—
237C22×C8C22×D312yes—
238C22×C8C22×D311yescharacteristic
239C22×C8C22×D311yescharacteristic
240C4⋊Q8—34no—
241C4⋊Q8—62no—
242#126—616no—
243#124—64no—
244#124—64no—
245C22⋊Q8—68no—
246C22⋊Q8—124no—
247C4×Q8—22no—
248C4×Q8—22no—
249C4×Q8—64no—
250C4×Q8—64no—
251C4×D4—34no—
252C4×D4—62no—
253C4×D4—62no—
254C4×D4—62no—
255#117—68no—
256C2×C4⋊C4—31no—
257C2×C4⋊C4—31no—
258C2×C4⋊C4—32no—
259C2×C4⋊C4—61no—
260C2×C4⋊C4—62no—
261C2×C4⋊C4—61no—
262C2×C4⋊C4—61no—
263C2×C4⋊C4—62no—
264C2×C4⋊C4—61no—
265C2×C4⋊C4D1211yescharacteristic
266C2×C4⋊C4D1212yes—
267C2×C22⋊C4—62no—
268C2×C22⋊C4—61no—
269C2×C22⋊C4—62no—
270C2×C22⋊C4—61no—
271C2×C22⋊C4—62no—
272C2×C22⋊C4—62no—
273C2×C42—62no—
274C2×C42C22×D312yes—
275C2×C42C22×D311yescharacteristic
276#107—62no—
277C4⋊C8—22no—
278#103—64no—
279#103—64no—
280#103—122no—
281#102—62no—
282#102—62no—
283#102—122no—
284C4×C8—22no—
285#95—62no—
286C23×C3C22×D411yescharacteristic
287C22×D3—22no—
288C22×D3—22no—
289C22×D3—41no—
290C22×D3—41no—
291C22×D3—41no—
292C22×D3—41no—
293C22×D3—41no—
294C22×D3—41no—
295C22×D3—41no—
296C22×D3—41no—
297C22×D3—41no—
298C22×D3—41no—
299C22×D3—41no—
300C22×D3—41no—
301C3×Q8—44no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4#12012yes—
311C2×C3×C4#12012yes—
312C2×C3×C4#13512yes—
313C2×C3×C4AGL(1,ℤ/8)12yes—
314C2×C3×C4C22×D411yescharacteristic
315C2×C3×C4C22×D411yescharacteristic
316C2×C3×C4C2×#4111yescharacteristic
317C2×C3×C4C2×#4111yescharacteristic
318C2×C3×C4H(5,𝔽2)11yescharacteristic
319C2×C3×C42−1+411yescharacteristic
320C2×Dic3—22no—
321C2×Dic3—24no—
322C2×Dic3—42no—
323C2×Dic3—42no—
324C2×Dic3—41no—
325C2×Dic3—41no—
326C2×Dic3—41no—
327C2×Dic3—42no—
328C2×Dic3—41no—
329C2×Dic3—41no—
330C2×Dic3—42no—
331C2×Dic3—42no—
332D12—22no—
333D12—22no—
334D12—22no—
335D12—22no—
336D12—41no—
337D12—41no—
338D12—41no—
339D12—41no—
340C4×D3—48no—
341C4×D3—84no—
342Dic6—24no—
343Dic6—24no—
344Dic6—42no—
345Dic6—42no—
346C3×C8—22no—
347C3×C8—41no—
348C3×C8—42no—
349C24—61no—
350C24—61no—
351C2×Q8—22no—
352C2×Q8—22no—
353C2×Q8—22no—
354C2×Q8—34no—
355C2×Q8—62no—
356C2×Q8—62no—
357C2×Q8—62no—
358C2×Q8—62no—
359C2×Q8—62no—
360C2×D4—32no—
361C2×D4—32no—
362C2×D4—61no—
363C2×D4—61no—
364C2×D4—61no—
365C2×D4—61no—
366C2×D4—61no—
367C2×D4—61no—
368C2×D4—61no—
369C2×D4—61no—
370C2×D4—61no—
371C2×D4—61no—
372C22×C4—61no—
373C22×C4—62no—
374C22×C4—64no—
375C22×C4—61no—
376C22×C4—62no—
377C22×C4—122no—
378C22×C4—122no—
379C22×C4—122no—
380C22×C4—122no—
381C22×C4—122no—
382C22×C4C2×D1212yes—
383C22×C4C2×D1212yes—
384C22×C4C2×D1211yescharacteristic
385C22×C4C2×D1211yescharacteristic
386C22×C4C23×D311yescharacteristic
387SD16—128no—
388D8—68no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—22no—
393C2×C8—21no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—21no—
397C2×C8#23912yes—
398C4⋊C4—22no—
399C4⋊C4—24no—
400C4⋊C4—32no—
401C4⋊C4—68no—
402C4⋊C4—64no—
403C4⋊C4—64no—
404C4⋊C4—61no—
405C4⋊C4—61no—
406C4⋊C4—61no—
407C4⋊C4—62no—
408C4⋊C4—62no—
409C4⋊C4—124no—
410C4⋊C4—122no—
411C4⋊C4—241no—
412C22⋊C4—68no—
413C22⋊C4—68no—
414C22⋊C4—64no—
415C22⋊C4—124no—
416C22⋊C4—122no—
417C22⋊C4—242no—
418C42—22no—
419C42—22no—
420C42—68no—
421C42D3×D412yes—
422C42D3×D412yes—
423C22×C3C2×#12011yescharacteristic
424C22×C3#54811yescharacteristic
425C22×C3#55011yescharacteristic
426C22×C3C2×#13511yescharacteristic
427C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
428C22×C3#57711yescharacteristic
429C22×C3#57811yescharacteristic
430C2×D3—42no—
431C2×D3—42no—
432C2×D3—42no—
433C2×D3—42no—
434C2×D3—42no—
435C2×D3—42no—
436C2×D3—81no—
437C2×D3—81no—
438C2×D3—81no—
439C2×D3—81no—
440C2×D3—81no—
441C2×D3—81no—
442C2×D3—81no—
443C2×D3—81no—
444C3×C4—22no—
445C3×C4—42no—
446C3×C4—42no—
447C3×C4#44914yes—
448C3×C4#52511yescharacteristic
449C3×C4#53711yescharacteristic
450C3×C4#54611yescharacteristic
451C3×C4#54711yescharacteristic
452Dic3—44no—
453Dic3—42no—
454Dic3—82no—
455Dic3—81no—
456C23—62no—
457C23—62no—
458C23—121no—
459C23—121no—
460C23—121no—
461C23—121no—
462C23—121no—
463C23—121no—
464C23—121no—
465C23—121no—
466C23—121no—
467C23—121no—
468C23—121no—
469C23—121no—
470C23C22×D1211yescharacteristic
471Q8—44no—
472Q8—64no—
473Q8—64no—
474Q8—122no—
475Q8—122no—
476D4—62no—
477D4—62no—
478D4—62no—
479D4—62no—
480D4—121no—
481D4—121no—
482D4—121no—
483D4—121no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—62no—
493C2×C4—64no—
494C2×C4—121no—
495C2×C4—121no—
496C2×C4—122no—
497C2×C4—122no—
498C2×C4—128no—
499C2×C4—121no—
500C2×C4—121no—
501C2×C4—122no—
502C2×C4—121no—
503C2×C4—122no—
504C2×C4—122no—
505C2×C4—244no—
506C2×C4#88212yes—
507C2×C4#88212yes—
508C2×C4#90412yes—
509C2×C4#90812yes—
510C2×C4C2×D3×D411yescharacteristic
511C2×C4C2×D3×D411yescharacteristic
512C2×C4C2×#23911yescharacteristic
513C2×C4#100711yescharacteristic
514C2×C4D3×#4111yescharacteristic
515C2×C4#100911yescharacteristic
516C8—22no—
517C8—42no—
518C8—41no—
519C2×C3#240711yescharacteristic
520C2×C3C2×#44911yescharacteristic
521C2×C3#301311yescharacteristic
522C2×C3#336011yescharacteristic
523C2×C3#336711yescharacteristic
524C2×C3#339311yescharacteristic
525C2×C3#339711yescharacteristic
526D3—82no—
527D3—82no—
528C22—122no—
529C22—122no—
530C22—122no—
531C22—122no—
532C22—122no—
533C22—122no—
534C22—241no—
535C22—241no—
536C22—241no—
537C22—241no—
538C22—241no—
539C22—241no—
540C22—241no—
541C22—241no—
542C22C2×#88211yescharacteristic
543C22#557611yescharacteristic
544C22#557711yescharacteristic
545C22C2×#90411yescharacteristic
546C22C2×#90811yescharacteristic
547C22#575111yescharacteristic
548C22#575311yescharacteristic
549C4—22no—
550C4—42no—
551C4—42no—
552C4—122no—
553C4—124no—
554C4—241no—
555C4—242no—
556C4#473414yes—
557C4#562711yescharacteristic
558C4#562911yescharacteristic
559C4#567611yescharacteristic
560C4#572611yescharacteristic
561C3#2418511yescharacteristic, complemented, Sylow
562C2—242no—
563C2—242no—
564C2#7661811yescharacteristic
565C2C2×#473411yescharacteristic
566C2#7868011yescharacteristic
567C2#7991211yescharacteristic
568C2#7992611yescharacteristic
569C2#7995011yescharacteristic
570C2#7997011yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1984C2
socleC23×C3C22×D4
prime radicalC22×D24C22
hypercenter#1984D3
hypocenterC3#24185
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
1. C2×#5584C2semidirect
2. #82500C2semidirect
3. C2×#4841C2semidirect
5. C3×#1984C2semidirect
6. #70479C2semidirect
7. #69771C2semidirect
8. #68686C2semidirect
9. #68683C2semidirect
11. #68630C2semidirect
52. #1984D3semidirect
561. C3#24185semidirect

Maximal subgroups

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

subgroupindex
C2×#55842
#825002
C2×#48412
C2×C4×D242
C3×#19842
#704792
#697712
#686862
#686832
#686752
#686302
#241853