Finite Group Databasebeta

← #10829949

Subgroups — #10829949

Order 768.

1214 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5585C211yesmaximal, characteristic, complemented
2#82500C211yesmaximal, characteristic, complemented
3C2×#4843C211yesmaximal, characteristic
4C2×C4×#204C211yesmaximal, characteristic, complemented
5C3×#1984C211yesmaximal, characteristic, complemented
6#70485C211yesmaximal, characteristic, complemented
7#69772C211yesmaximal, characteristic, complemented
8#68675C212yesmaximal, complemented
9#68673C212yesmaximal, complemented
10#68671C212yesmaximal, complemented
11#68630C212yesmaximal, complemented
12#24186—31nomaximal, Sylow
13C2×C3×C4×Q8C2211yescharacteristic
14C2×#947—21no—
15C2×#946—21no—
16C22×D24C2212yes—
17C22×#204C2211yescharacteristic
18#5585—24no—
19#5585—24no—
20C2×#894—22no—
21C2×#893—21no—
22C2×#875C2212yes—
23C2×#874C2211yescharacteristic
24C2×C4×D12C2212yes—
25C2×C4×D12C2211yescharacteristic
26C2×C4×Dic6C2211yescharacteristic
27C2×C3×C4⋊C8C2211yescharacteristic
28C2×C3×C4×C8C2211yescharacteristic
29C2×#821C2212yes—
30C2×#821C2212yes—
31C2×#821C2211yescharacteristic
32C2×#821C2212yes—
33C2×#817C2211yescharacteristic
34C2×#816C2211yescharacteristic
35C2×#816C2212yes—
36#4994—22no—
37#4987—21no—
38#4939C2212yes—
39#4935C2211yescharacteristic
40#4928C2212yes—
41#4843—24no—
42C4×#204—24no—
43C3×#336C2212yes—
44C3×#336C2212yes—
45#4551C2212yes—
46C2×#559—31no—
47C2×#459—31no—
48C2×C4×SD16—31no—
49#2538—31no—
50#2042—32no—
51#2012—32no—
52#1984D311yescharacteristic, complemented
53#1957—31no—
54#1879—32no—
55#1864—32no—
56#1803—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⋊C4D411yescharacteristic
71C2×C3×C4⋊C4D412yes—
72C2×C3×C42C2312yes—
73C2×C3×C42C2311yescharacteristic
74#947—28no—
75#946—44no—
76C2×(C2×D3)⋊C4—22no—
77C2×(C2×D3)⋊C4—22no—
78C2×(C2×D3)⋊C4—22no—
79C2×(C2×D3)⋊C4—22no—
80C2×(C2×D3)⋊C4—21no—
81C2×(C2×D3)⋊C4—22no—
82C2×C4⋊Dic3—21no—
83C2×C4⋊Dic3—21no—
84C2×C4⋊Dic3C2311yescharacteristic
85C2×C4⋊Dic3C2312yes—
86C2×C4⋊Dic3C2311yescharacteristic
87C2×Dic3⋊C4—21no—
88C2×Dic3⋊C4—22no—
89C2×C4×Dic3—21no—
90C2×D24—24no—
91C2×D24—24no—
92C2×D24—42no—
93C2×#204—22no—
94C2×#204—22no—
95C2×#204—22no—
96#894—216no—
97#893—28no—
98#875—24no—
99#875—24no—
100#874—22no—
101#874D414yes—
102C4×D12—24no—
103C4×D12—22no—
104C4×D12—22no—
105C4×D12—24no—
106C4×Dic6—22no—
107C4×Dic6D414yes—
108C3×C4⋊C8—22no—
109C3×C4×C8—22no—
110C3⋊#95—22no—
111#821—24no—
112#821—24no—
113#821—22no—
114#821—42no—
115#817—22no—
116#816—22no—
117#816—42no—
118C22×SD16—31no—
119C22×D8—32no—
120#559—64no—
121#559—64no—
122C2×#127—31no—
123C2×#124—32no—
124C2×#124—61no—
125C2×#123—62no—
126C2×C22⋊Q8—61no—
127C2×C4×Q8—31no—
128C2×C4×Q8C2×D311yescharacteristic
129C2×C4×D4—32no—
130C2×C4×D4—31no—
131C2×C4×D4—61no—
132#459—64no—
133C4×SD16—64no—
134C2×#106—31no—
135C2×C4⋊C8C2×D311yescharacteristic
136C2×#103—31no—
137C2×#103—32no—
138C2×#102—32no—
139C2×#102—32no—
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—22no—
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—21no—
163C22×Dic3—21no—
164C22×Dic3—22no—
165C2×D12—21no—
166C2×D12—21no—
167C2×D12—22no—
168C2×D12—21no—
169C2×D12—22no—
170C2×D12—22no—
171C2×D12—21no—
172C2×D12—22no—
173C2×D12—24no—
174C2×D12—22no—
175C2×D12—21no—
176C2×D12—24no—
177C2×D12—42no—
178C2×D12—42no—
179C2×D12—81no—
180C2×D12—81no—
181C2×D12C2×D414yes—
182C2×D12#4112yes—
183C2×C4×D3—28no—
184C2×C4×D3—42no—
185C2×C4×D3—42no—
186C2×C4×D3—42no—
187C2×C4×D3—82no—
188C2×C4×D3—82no—
189C2×Dic6—21no—
190C2×Dic6—21no—
191C2×Dic6—21no—
192C2×Dic6—21no—
193C2×Dic6—21no—
194C2×Dic6C2×D412yes—
195C2×C3×C8—21no—
196C2×C3×C8—21no—
197C2×C3×C8—21no—
198C2×C3×C8—22no—
199C2×C3×C8—22no—
200C2×C3×C8—22no—
201C2×C3×C8—21no—
202C2×C3×C8—21no—
203C2×C3×C8#4112yes—
204C3×C4⋊C4—22no—
205C3×C4⋊C4—24no—
206C3×C42—22no—
207C3×C42—22no—
208C3×C42C2×D412yes—
209C3×C42C2×D412yes—
210(C2×D3)⋊C4—28no—
211(C2×D3)⋊C4—28no—
212(C2×D3)⋊C4—28no—
213(C2×D3)⋊C4—44no—
214(C2×D3)⋊C4—42no—
215(C2×D3)⋊C4—82no—
216C4⋊Dic3—21no—
217C4⋊Dic3—21no—
218C4⋊Dic3—21no—
219C4⋊Dic3—22no—
220C4⋊Dic3—22no—
221C4⋊Dic3—42no—
222C4⋊Dic3—81no—
223C4⋊Dic3C2×D412yes—
224Dic3⋊C4—24no—
225Dic3⋊C4—44no—
226C4×Dic3—24no—
227D24—48no—
228#204—28no—
229C22×Q8—31no—
230C22×Q8D1211yescharacteristic
231C22×D4—32no—
232C22×D4—31no—
233C22×D4—61no—
234C22×D4—61no—
235C23×C4—62no—
236C23×C4—61no—
237C2×SD16—62no—
238C2×SD16—62no—
239C2×SD16—62no—
240C2×D8—64no—
241C2×D8—64no—
242C2×D8—122no—
243C22×C8C22×D312yes—
244C22×C8C22×D311yescharacteristic
245C22×C8C22×D311yescharacteristic
246#127—34no—
247#127—62no—
248#124—64no—
249#124—64no—
250#124—68no—
251#123—616no—
252C22⋊Q8—124no—
253C4×Q8—22no—
254C4×Q8—22no—
255C4×Q8—34no—
256C4×Q8—62no—
257C4×D4—64no—
258C4×D4—64no—
259C4×D4—68no—
260C4×D4—62no—
261C4×D4—62no—
262C2×C4⋊C4—31no—
263C2×C4⋊C4—31no—
264C2×C4⋊C4—32no—
265C2×C4⋊C4—61no—
266C2×C4⋊C4—61no—
267C2×C4⋊C4—62no—
268C2×C4⋊C4—61no—
269C2×C4⋊C4D1211yescharacteristic
270C2×C4⋊C4D1212yes—
271C2×C22⋊C4—62no—
272C2×C22⋊C4—62no—
273C2×C22⋊C4—62no—
274C2×C22⋊C4—61no—
275C2×C22⋊C4—62no—
276C2×C22⋊C4—62no—
277C2×C42—61no—
278C2×C42C22×D312yes—
279C2×C42C22×D311yescharacteristic
280#106—62no—
281C4⋊C8—22no—
282#103—62no—
283#103—122no—
284#102—62no—
285#102—64no—
286#102—64no—
287#102—122no—
288C4×C8—22no—
289#95—62no—
290C23×C3C22×D411yescharacteristic
291C22×D3—24no—
292C22×D3—22no—
293C22×D3—41no—
294C22×D3—42no—
295C22×D3—42no—
296C22×D3—41no—
297C22×D3—41no—
298C22×D3—42no—
299C22×D3—41no—
300C22×D3—42no—
301C22×D3—41no—
302C22×D3—42no—
303C22×D3—42no—
304C22×D3—41no—
305C3×Q8—44no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4#12012yes—
315C2×C3×C4#12012yes—
316C2×C3×C4#13512yes—
317C2×C3×C4AGL(1,ℤ/8)12yes—
318C2×C3×C4C22×D411yescharacteristic
319C2×C3×C4C22×D411yescharacteristic
320C2×C3×C4C2×#4111yescharacteristic
321C2×C3×C4C2×#4111yescharacteristic
322C2×C3×C4H(5,𝔽2)11yescharacteristic
323C2×C3×C42−1+411yescharacteristic
324C2×Dic3—22no—
325C2×Dic3—22no—
326C2×Dic3—41no—
327C2×Dic3—41no—
328C2×Dic3—41no—
329C2×Dic3—41no—
330C2×Dic3—41no—
331C2×Dic3—42no—
332C2×Dic3—41no—
333C2×Dic3—41no—
334C2×Dic3—42no—
335C2×Dic3—42no—
336D12—22no—
337D12—22no—
338D12—24no—
339D12—24no—
340D12—42no—
341D12—41no—
342D12—42no—
343D12—41no—
344D12—48no—
345D12—84no—
346C4×D3—88no—
347C4×D3—84no—
348Dic6—22no—
349Dic6—22no—
350Dic6—41no—
351Dic6—41no—
352C3×C8—22no—
353C3×C8—41no—
354C3×C8—42no—
355C24—61no—
356C24—62no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—32no—
361C2×Q8—61no—
362C2×Q8—61no—
363C2×Q8—61no—
364C2×Q8—61no—
365C2×Q8—61no—
366C2×D4—34no—
367C2×D4—32no—
368C2×D4—64no—
369C2×D4—64no—
370C2×D4—61no—
371C2×D4—61no—
372C2×D4—62no—
373C2×D4—62no—
374C2×D4—61no—
375C2×D4—62no—
376C2×D4—61no—
377C2×D4—62no—
378C2×D4—62no—
379C2×D4—61no—
380C2×D4—122no—
381C2×D4—122no—
382C2×D4—241no—
383C2×D4—241no—
384C22×C4—68no—
385C22×C4—61no—
386C22×C4—61no—
387C22×C4—61no—
388C22×C4—62no—
389C22×C4—122no—
390C22×C4—122no—
391C22×C4—122no—
392C22×C4—242no—
393C22×C4—242no—
394C22×C4C2×D1212yes—
395C22×C4C2×D1212yes—
396C22×C4C2×D1211yescharacteristic
397C22×C4C2×D1211yescharacteristic
398C22×C4C23×D311yescharacteristic
399SD16—68no—
400D8—128no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—22no—
405C2×C8—21no—
406C2×C8—22no—
407C2×C8—21no—
408C2×C8—22no—
409C2×C8#23912yes—
410C4⋊C4—22no—
411C4⋊C4—24no—
412C4⋊C4—32no—
413C4⋊C4—64no—
414C4⋊C4—61no—
415C4⋊C4—61no—
416C4⋊C4—61no—
417C4⋊C4—62no—
418C4⋊C4—62no—
419C4⋊C4—124no—
420C4⋊C4—122no—
421C4⋊C4—241no—
422C22⋊C4—68no—
423C22⋊C4—68no—
424C22⋊C4—68no—
425C22⋊C4—124no—
426C22⋊C4—122no—
427C22⋊C4—242no—
428C42—22no—
429C42—22no—
430C42—64no—
431C42D3×D412yes—
432C42D3×D412yes—
433C22×C3C2×#12011yescharacteristic
434C22×C3#54811yescharacteristic
435C22×C3#55011yescharacteristic
436C22×C3C2×#13511yescharacteristic
437C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
438C22×C3#57611yescharacteristic
439C22×C3#57711yescharacteristic
440C2×D3—42no—
441C2×D3—42no—
442C2×D3—42no—
443C2×D3—44no—
444C2×D3—44no—
445C2×D3—44no—
446C2×D3—82no—
447C2×D3—82no—
448C2×D3—81no—
449C2×D3—82no—
450C2×D3—81no—
451C2×D3—82no—
452C2×D3—81no—
453C2×D3—81no—
454C3×C4—22no—
455C3×C4—42no—
456C3×C4—42no—
457C3×C4#44914yes—
458C3×C4#52511yescharacteristic
459C3×C4#53711yescharacteristic
460C3×C4#54611yescharacteristic
461C3×C4#54711yescharacteristic
462Dic3—42no—
463Dic3—42no—
464Dic3—82no—
465Dic3—81no—
466C23—64no—
467C23—62no—
468C23—121no—
469C23—122no—
470C23—121no—
471C23—121no—
472C23—122no—
473C23—122no—
474C23—121no—
475C23—121no—
476C23—122no—
477C23—122no—
478C23—122no—
479C23—121no—
480C23C22×D1211yescharacteristic
481Q8—44no—
482Q8—62no—
483Q8—62no—
484Q8—121no—
485Q8—121no—
486D4—62no—
487D4—64no—
488D4—64no—
489D4—62no—
490D4—121no—
491D4—128no—
492D4—122no—
493D4—121no—
494D4—122no—
495D4—244no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—62no—
505C2×C4—62no—
506C2×C4—121no—
507C2×C4—121no—
508C2×C4—121no—
509C2×C4—121no—
510C2×C4—121no—
511C2×C4—121no—
512C2×C4—122no—
513C2×C4—121no—
514C2×C4—122no—
515C2×C4—122no—
516C2×C4—248no—
517C2×C4—244no—
518C2×C4#88212yes—
519C2×C4#88212yes—
520C2×C4#90412yes—
521C2×C4#90812yes—
522C2×C4C2×D3×D411yescharacteristic
523C2×C4C2×D3×D411yescharacteristic
524C2×C4C2×#23911yescharacteristic
525C2×C4#100711yescharacteristic
526C2×C4D3×#4111yescharacteristic
527C2×C4#100911yescharacteristic
528C8—22no—
529C8—42no—
530C8—41no—
531C2×C3#240711yescharacteristic
532C2×C3C2×#44911yescharacteristic
533C2×C3#301211yescharacteristic
534C2×C3#336111yescharacteristic
535C2×C3#337311yescharacteristic
536C2×C3#339411yescharacteristic
537C2×C3#339911yescharacteristic
538D3—82no—
539D3—84no—
540C22—124no—
541C22—122no—
542C22—122no—
543C22—124no—
544C22—124no—
545C22—122no—
546C22—241no—
547C22—241no—
548C22—242no—
549C22—242no—
550C22—241no—
551C22—241no—
552C22—242no—
553C22—242no—
554C22C2×#88211yescharacteristic
555C22#557611yescharacteristic
556C22#557711yescharacteristic
557C22C2×#90411yescharacteristic
558C22C2×#90811yescharacteristic
559C22#575111yescharacteristic
560C22#575211yescharacteristic
561C4—22no—
562C4—42no—
563C4—42no—
564C4—122no—
565C4—122no—
566C4—241no—
567C4—242no—
568C4#473414yes—
569C4#562711yescharacteristic
570C4#562911yescharacteristic
571C4#567611yescharacteristic
572C4#572611yescharacteristic
573C3#2418611yescharacteristic, complemented, Sylow
574C2—244no—
575C2—242no—
576C2#7661811yescharacteristic
577C2C2×#473411yescharacteristic
578C2#7867211yescharacteristic
579C2#7991311yescharacteristic
580C2#7992711yescharacteristic
581C2#7995211yescharacteristic
582C2#7997311yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1984C2
socleC23×C3C22×D4
prime radicalC2×#4843C2
hypercenter#1984D3
hypocenterC3#24186
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×#5585C2semidirect
2. #82500C2semidirect
4. C2×C4×#204C2semidirect
5. C3×#1984C2semidirect
6. #70485C2semidirect
7. #69772C2semidirect
8. #68675C2semidirect
9. #68673C2semidirect
10. #68671C2semidirect
11. #68630C2semidirect
52. #1984D3semidirect
573. C3#24186semidirect

Maximal subgroups

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

subgroupindex
C2×#55852
#825002
C2×#48432
C2×C4×#2042
C3×#19842
#704852
#697722
#686752
#686732
#686712
#686302
#241863