Finite Group Databasebeta

← #10823450

Subgroups — #10823450

Order 768.

1424 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5477C211yesmaximal, characteristic
2C4×#924C211yesmaximal, characteristic, complemented
3C2×#4710C212yesmaximal, complemented
4C3×#1749C211yesmaximal, characteristic, complemented
5#69962C212yesmaximal, complemented
6#69934C212yesmaximal, complemented
7#69812C212yesmaximal, complemented
8#68689C211yesmaximal, characteristic, complemented
9#68676C211yesmaximal, characteristic, complemented
10#68632C211yesmaximal, characteristic, complemented
11#68629C211yesmaximal, characteristic, complemented
12#23576—31nomaximal, Sylow
13C22×#235—21no—
14C22×C3×C42C2211yescharacteristic
15C2×#930—21no—
16C2×#930C2212yes—
17C2×#924C2211yescharacteristic
18C22×#204C2212yes—
19#5477—28no—
20C2×#875—21no—
21C2×#874C2211yescharacteristic
22C2×C4×D12—21no—
23C2×C4×D12C2212yes—
24C2×#869C2211yescharacteristic
25C2×C4×Dic6C2211yescharacteristic
26C2×C3×C22⋊C8C2212yes—
27C2×C3×#97C2212yes—
28#5212C2211yescharacteristic
29#5210—21no—
30C4×#217—21no—
31C2×#821C2212yes—
32C2×#821C2212yes—
33C2×#821C2212yes—
34C2×#818C2212yes—
35C2×#816C2212yes—
36C4×C4⋊Dic3C2211yescharacteristic
37#4933—21no—
38C4×Dic3⋊C4—21no—
39#4929C2211yescharacteristic
40#4928C2211yescharacteristic
41#4710—28no—
42C3×#336C2211yescharacteristic
43C3×#336C2211yescharacteristic
44#4551C2212yes—
45#4551C2212yes—
46C2×#532—31no—
47C2×#440—32no—
48C4×C22⋊Q8—31no—
49#2057—31no—
50#2041—31no—
51#2033—32no—
52#2024—31no—
53#2014—31no—
54#2000—32no—
55#1762—32no—
56#1749D311yescharacteristic, complemented
57C23×C3×C4D411yescharacteristic
58C23×C3×C4D411yescharacteristic
59C23×C3×C4C2311yescharacteristic
60C22×#67—22no—
61C2×#235—28no—
62C2×#235—44no—
63C2×#235—44no—
64C22×D12—21no—
65C22×D12—21no—
66C22×D12C2312yes—
67C22×C4×D3—22no—
68C22×Dic6—21no—
69C22×Dic6C2311yescharacteristic
70C22×C3×C8C2312yes—
71C22×C3×C8C2312yes—
72C2×C3×C42—24no—
73C2×C3×C42D411yescharacteristic
74C2×C3×C42D411yescharacteristic
75C2×C3×C42C2311yescharacteristic
76C2×C3×C42C2311yescharacteristic
77C2×#217—21no—
78C2×#217—21no—
79#930—28no—
80#930—24no—
81#930—24no—
82C2×(C2×D3)⋊C4—22no—
83C2×(C2×D3)⋊C4—22no—
84C2×C4⋊Dic3—21no—
85C2×C4⋊Dic3C2311yescharacteristic
86C2×C4⋊Dic3C2311yescharacteristic
87C2×C4⋊Dic3C2312yes—
88C2×C4⋊Dic3C2311yescharacteristic
89#924—22no—
90#924—22no—
91C2×Dic3⋊C4—21no—
92C2×Dic3⋊C4—21no—
93C2×Dic3⋊C4—21no—
94C2×C4×Dic3—21no—
95C2×C4×Dic3—21no—
96C2×#204—24no—
97C2×#204—24no—
98C2×#204—24no—
99#875—28no—
100#874—24no—
101C4×D12—28no—
102C4×D12—24no—
103C4×D12D418yes—
104#869—24no—
105C4×Dic6—24no—
106C3×C22⋊C8—24no—
107C3×#97—24no—
108C3⋊#95—21no—
109C3⋊#95—21no—
110C3⋊#95—21no—
111C3⋊#95—21no—
112C3⋊#95—21no—
113#821—24no—
114#821—24no—
115#821—24no—
116#818—24no—
117#816—24no—
118C22×#41—61no—
119C22×SD16—32no—
120#532—68no—
121C2×C4⋊Q8—31no—
122C2×#127—31no—
123C2×#124—61no—
124C2×C22⋊Q8—31no—
125C2×#121—32no—
126C2×#121—61no—
127C2×C4×Q8—31no—
128C2×C4×D4—32no—
129C2×C4×D4—61no—
130C22×C42C2×D311yescharacteristic
131#440—68no—
132C2×#107—32no—
133C2×#103—32no—
134C2×#102—32no—
135C2×#102—32no—
136C2×#102—32no—
137C2×C22⋊C8C2×D312yes—
138C2×#97C2×D312yes—
139#391—31no—
140#389—31no—
141#387—61no—
142#385—61no—
143#380—31no—
144C4×C4⋊C4—31no—
145C4×C4⋊C4—61no—
146C4×C22⋊C4—61no—
147#340—32no—
148#340—32no—
149#336C2×D311yescharacteristic
150#336C2×D311yescharacteristic
151C24×C3C2×D411yescharacteristic
152C23×D3—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—21no—
161C22×C3×C4—21no—
162C22×C3×C4—41no—
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2411yescharacteristic
169C2×#67—28no—
170C2×#67—44no—
171C2×#67—44no—
172C22×Dic3—21no—
173C22×Dic3—21no—
174C22×Dic3—22no—
175C22×Dic3—21no—
176C22×Dic3—21no—
177#235—432no—
178C2×D12—22no—
179C2×D12—22no—
180C2×D12—22no—
181C2×D12—22no—
182C2×D12—24no—
183C2×D12—24no—
184C2×D12—22no—
185C2×D12—42no—
186C2×D12—42no—
187C2×D12—42no—
188C2×D12—42no—
189C2×D12C2×D414yes—
190C2×C4×D3—28no—
191C2×C4×D3—44no—
192C2×C4×D3—44no—
193C2×Dic6—22no—
194C2×Dic6—22no—
195C2×Dic6—24no—
196C2×Dic6—21no—
197C2×Dic6—42no—
198C2×Dic6—42no—
199C2×Dic6#4112yescomplemented
200C2×C3×C8—22no—
201C2×C3×C8—22no—
202C2×C3×C8—22no—
203C2×C3×C8—22no—
204C2×C3×C8—22no—
205C2×C3×C8#4114yescomplemented
206C3×C42—22no—
207C3×C42—22no—
208C3×C42—22no—
209C3×C42C2×D414yes—
210#217—24no—
211#217—42no—
212(C2×D3)⋊C4—28no—
213(C2×D3)⋊C4—28no—
214C4⋊Dic3—24no—
215C4⋊Dic3—22no—
216C4⋊Dic3—21no—
217C4⋊Dic3—22no—
218C4⋊Dic3—22no—
219C4⋊Dic3C2×D414yes—
220C4⋊Dic3#4112yescomplemented
221Dic3⋊C4—24no—
222Dic3⋊C4—42no—
223Dic3⋊C4—42no—
224C4×Dic3—42no—
225C4×Dic3—42no—
226#204—216no—
227C2×#41—68no—
228C2×#41—124no—
229C2×#41—124no—
230C22×Q8—31no—
231C22×Q8—61no—
232C22×D4—32no—
233C22×D4—62no—
234C22×D4—61no—
235C22×D4—61no—
236C23×C4—62no—
237C23×C4D1211yescharacteristic
238C23×C4D1211yescharacteristic
239C23×C4C22×D311yescharacteristic
240C2×SD16—64no—
241C2×SD16—64no—
242C2×SD16—64no—
243C22×C8C22×D312yes—
244C22×C8C22×D312yes—
245C4⋊Q8—64no—
246#127—64no—
247#124—68no—
248C22⋊Q8—62no—
249C22⋊Q8—62no—
250#121—64no—
251#121—64no—
252#121—68no—
253C4×Q8—64no—
254C4×D4—38no—
255C4×D4—68no—
256C4×D4—64no—
257C2×C4⋊C4—32no—
258C2×C4⋊C4—31no—
259C2×C4⋊C4—31no—
260C2×C4⋊C4—31no—
261C2×C4⋊C4—61no—
262C2×C4⋊C4—61no—
263C2×C4⋊C4—61no—
264C2×C4⋊C4—61no—
265C2×C22⋊C4—61no—
266C2×C22⋊C4—62no—
267C2×C22⋊C4—62no—
268C2×C22⋊C4—61no—
269C2×C42—24no—
270C2×C42—61no—
271C2×C42—61no—
272C2×C42D1211yescharacteristic
273C2×C42D1211yescharacteristic
274C2×C42C22×D311yescharacteristic
275C2×C42C22×D311yescharacteristic
276#107—64no—
277#103—64no—
278#102—64no—
279#102—64no—
280#102—64no—
281C22⋊C8—24no—
282#97—24no—
283#95—61no—
284#95—61no—
285#95—61no—
286#95—61no—
287#95—61no—
288C23×C3—21no—
289C23×C3—21no—
290C23×C3—21no—
291C23×C3—21no—
292C23×C3—21no—
293C23×C3—21no—
294C23×C3AGL(1,ℤ/8)12yes—
295C23×C3C22×D411yescharacteristic
296C22×D3—24no—
297C22×D3—42no—
298C22×D3—42no—
299C22×D3—42no—
300C22×D3—42no—
301C22×D3—42no—
302C22×D3—42no—
303C2×C3×C4—28no—
304C2×C3×C4—21no—
305C2×C3×C4—21no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—21no—
315C2×C3×C4—44no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4#12012yes—
319C2×C3×C4#12012yes—
320C2×C3×C4AGL(1,ℤ/8)12yes—
321C2×C3×C4#13712yes—
322C2×C3×C4C22×D412yes—
323C2×C3×C4C2×#4112yes—
324C2×C3×C4C2×#4111yescharacteristic
325C2×C3×C4C2×#4111yescharacteristic
326#67—416no—
327C2×Dic3—24no—
328C2×Dic3—22no—
329C2×Dic3—22no—
330C2×Dic3—41no—
331C2×Dic3—41no—
332C2×Dic3—41no—
333C2×Dic3—42no—
334C2×Dic3—42no—
335C2×Dic3—41no—
336C2×Dic3—41no—
337C2×Dic3—41no—
338C2×Dic3—41no—
339C2×Dic3—41no—
340C2×Dic3—41no—
341C2×Dic3—41no—
342D12—24no—
343D12—24no—
344D12—48no—
345D12—48no—
346D12—42no—
347D12—42no—
348C4×D3—416no—
349Dic6—24no—
350Dic6—48no—
351Dic6—42no—
352C3×C8—24no—
353C3×C8—42no—
354C24—62no—
355C24C2×D1211yescharacteristic
356#41—1232no—
357C2×Q8—32no—
358C2×Q8—64no—
359C2×Q8—61no—
360C2×Q8—62no—
361C2×Q8—62no—
362C2×Q8—122no—
363C2×Q8—122no—
364C2×D4—34no—
365C2×D4—64no—
366C2×D4—64no—
367C2×D4—68no—
368C2×D4—62no—
369C2×D4—62no—
370C2×D4—62no—
371C2×D4—62no—
372C2×D4—62no—
373C2×D4—122no—
374C2×D4—122no—
375C2×D4—124no—
376C2×D4—124no—
377C2×D4—122no—
378C2×D4—122no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—41no—
389C22×C4—61no—
390C22×C4—61no—
391C22×C4—62no—
392C22×C4—68no—
393C22×C4—61no—
394C22×C4—61no—
395C22×C4—124no—
396C22×C4—124no—
397C22×C4C2×D1211yescharacteristic
398C22×C4C2×D1211yescharacteristic
399C22×C4C2×D1211yescharacteristic
400C22×C4C2×D1211yescharacteristic
401C22×C4C2×D1211yescharacteristic
402C22×C4C23×D311yescharacteristic
403SD16—616no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8#23514yescomplemented
410C4⋊C4—34no—
411C4⋊C4—32no—
412C4⋊C4—64no—
413C4⋊C4—64no—
414C4⋊C4—62no—
415C4⋊C4—61no—
416C4⋊C4—62no—
417C4⋊C4—62no—
418C4⋊C4—122no—
419C4⋊C4—122no—
420C22⋊C4—68no—
421C22⋊C4—64no—
422C22⋊C4—68no—
423C22⋊C4—122no—
424C42—22no—
425C42—22no—
426C42—22no—
427C42—122no—
428C42—122no—
429C42D3×D414yes—
430C22×C3—22no—
431C22×C3—22no—
432C22×C3—22no—
433C22×C3—41no—
434C22×C3—41no—
435C22×C3—41no—
436C22×C3—41no—
437C22×C3C2×#12011yescharacteristic
438C22×C3#52511yescharacteristic
439C22×C3#53211yescharacteristic
440C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
441C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
442C22×C3C2×#13711yescharacteristic
443C22×C3#57511yescharacteristic
444C2×D3—44no—
445C2×D3—44no—
446C2×D3—44no—
447C2×D3—82no—
448C2×D3—82no—
449C2×D3—82no—
450C2×D3—82no—
451C3×C4—22no—
452C3×C4—22no—
453C3×C4—22no—
454C3×C4—22no—
455C3×C4—41no—
456C3×C4#52512yes—
457C3×C4#53212yes—
458Dic3—42no—
459Dic3—42no—
460Dic3—44no—
461Dic3—81no—
462Dic3—81no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—64no—
470C23—122no—
471C23—122no—
472C23—122no—
473C23—122no—
474C23—122no—
475C23—122no—
476C23#90812yes—
477C23C22×D1211yescharacteristic
478Q8—64no—
479Q8—128no—
480Q8—122no—
481D4—64no—
482D4—64no—
483D4—128no—
484D4—128no—
485D4—1216no—
486D4—122no—
487D4—122no—
488C2×C4—28no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—64no—
504C2×C4—62no—
505C2×C4—62no—
506C2×C4—1216no—
507C2×C4—121no—
508C2×C4—121no—
509C2×C4—121no—
510C2×C4—122no—
511C2×C4—122no—
512C2×C4—121no—
513C2×C4—121no—
514C2×C4—121no—
515C2×C4—121no—
516C2×C4—121no—
517C2×C4—121no—
518C2×C4—121no—
519C2×C4#88212yes—
520C2×C4#88212yes—
521C2×C4#90812yes—
522C2×C4#90912yes—
523C2×C4C2×#23512yes—
524C2×C4C2×D3×D412yes—
525C2×C4D3×#4111yescharacteristic
526C2×C4D3×#4111yescharacteristic
527C8—24no—
528C8—42no—
529C2×C3—42no—
530C2×C3#238811yescharacteristic
531C2×C3#300311yescharacteristic
532C2×C3#300711yescharacteristic
533C2×C3#311211yescharacteristic
534C2×C3#311511yescharacteristic
535C2×C3#318011yescharacteristic
536C2×C3#318211yescharacteristic
537D3—84no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—41no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22—124no—
546C22—124no—
547C22—124no—
548C22—242no—
549C22—242no—
550C22—242no—
551C22—242no—
552C22#547711yescharacteristic
553C22C2×#88211yescharacteristic
554C22#552411yescharacteristic
555C22C2×#90811yescharacteristic
556C22C2×#90811yescharacteristic
557C22C2×#90911yescharacteristic
558C22#574811yescharacteristic
559C4—22no—
560C4—22no—
561C4—22no—
562C4—22no—
563C4—41no—
564C4—124no—
565C4—122no—
566C4—122no—
567C4—241no—
568C4—241no—
569C4#554712yes—
570C4#554812yes—
571C3#2357611yescharacteristic, complemented, Sylow
572C2—42no—
573C2—244no—
574C2#7358011yescharacteristic
575C2#7850011yescharacteristic
576C2#7853311yescharacteristic
577C2#7863711yescharacteristic
578C2#7864011yescharacteristic
579C2#7934711yescharacteristic
580C2#7935511yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1749C2
socleC23×C3C22×D4
prime radicalC2×#5477C2
hypercenter#1749D3
hypocenterC3#23576
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1749

Complements

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

NH ≅ G/Ntype
2. C4×#924C2semidirect
3. C2×#4710C2semidirect
4. C3×#1749C2semidirect
5. #69962C2semidirect
6. #69934C2semidirect
7. #69812C2semidirect
8. #68689C2semidirect
9. #68676C2semidirect
10. #68632C2semidirect
11. #68629C2semidirect
56. #1749D3semidirect
199. C2×Dic6#41semidirect
205. C2×C3×C8#41semidirect
220. C4⋊Dic3#41semidirect
409. C2×C8#235semidirect
571. C3#23576semidirect

Maximal subgroups

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

subgroupindex
C2×#54772
C4×#9242
C2×#47102
C3×#17492
#699622
#699342
#698122
#686892
#686762
#686322
#686292
#235763