Finite Group Databasebeta

← #10845942

Subgroups — #10845942

Order 768.

1154 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83188C211yesmaximal, characteristic
2C2×#5075C211yesmaximal, characteristic, complemented
3C2×#4778C212yesmaximal, complemented
4C2×C8×D12C211yesmaximal, characteristic, complemented
5#74369C211yesmaximal, characteristic, complemented
6C3×#1957C211yesmaximal, characteristic, complemented
7#70207C212yesmaximal, complemented
8#70205C212yesmaximal, complemented
9#70202C212yesmaximal, complemented
10C4×#816C211yesmaximal, characteristic, complemented
11#69568C211yesmaximal, characteristic, complemented
12#25387—31nomaximal, Sylow
13C2×C3×#127C2211yescharacteristic
14C2×#940—21no—
15C2×#938C2212yes—
16C2×#938C2212yes—
17C22×#214C2212yes—
18C22×D3×C8C2212yes—
19C2×#897C2212yes—
20C2×C4×D12C2211yescharacteristic
21C2×C4×Dic6C2211yescharacteristic
22C2×C3×#102C2212yes—
23C2×C3×#102C2212yes—
24C2×C3×C4×C8C2211yescharacteristic
25C3×#384C2211yescharacteristic
26#5223—22no—
27#5221—21no—
28C2×#832C2212yes—
29C2×(C2×D3)⋊C8C2212yes—
30C2×#816C2211yescharacteristic
31C2×#816C2211yescharacteristic
32#5075—24no—
33#4994—22no—
34#4988—21no—
35#4987C2212yes—
36#4984—21no—
37C2×#808C2212yes—
38C4×C4⋊Dic3C2211yescharacteristic
39C2×C4⋊(C3⋊C8)C2211yescharacteristic
40#4778—28no—
41C8×D12—24no—
42#4551C2211yescharacteristic
43#4551C2211yescharacteristic
44#4529C2212yes—
45C2×#465—32no—
46C2×#460—31no—
47C2×C8×D4—31no—
48#2541—31no—
49#2538—31no—
50#2034—32no—
51#2000—32no—
52#1957D311yescharacteristic, complemented
53#1948—31no—
54#1893—32no—
55C4×#103—31no—
56C22×C3×D4—21no—
57C22×C3×D4—21no—
58C22×C3×D4C2312yes—
59C22×#67—22no—
60C22×#67—22no—
61C22×D12D411yescharacteristic
62C22×C4×D3C2312yes—
63C22×Dic6C2311yescharacteristic
64C22×C3×C8C2311yescharacteristic
65C22×C3×C8C2311yescharacteristic
66C3×#127—22no—
67C3×#127—22no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4C2312yes—
71C2×C3×C42C2311yescharacteristic
72C2×#217—22no—
73C2×#217—22no—
74#940—44no—
75#938—24no—
76#938—24no—
77#938—42no—
78#938—42no—
79C2×#214—24no—
80C2×#214—24no—
81C2×#214—42no—
82C2×(C2×D3)⋊C4—22no—
83C2×(C2×D3)⋊C4—22no—
84C2×(C2×D3)⋊C4D412yes—
85C2×C4⋊Dic3—22no—
86C2×C4⋊Dic3D411yescharacteristic
87C2×C4⋊Dic3C2311yescharacteristic
88C2×C4⋊Dic3C2311yescharacteristic
89C2×C4⋊Dic3C2311yescharacteristic
90C2×Dic3⋊C4—21no—
91C2×C4×Dic3—21no—
92C2×C4×Dic3—21no—
93C22×C3⋊C8C2312yes—
94C2×D3×C8—24no—
95C2×D3×C8—24no—
96C2×D3×C8—42no—
97#897—24no—
98#897—42no—
99C4×D12—22no—
100C4×D12—22no—
101C4×Dic6—22no—
102C4×Dic6—22no—
103C3×#102—24no—
104C3×#102—42no—
105C3×C4×C8—22no—
106#832—24no—
107C3⋊#95—21no—
108C3⋊#95—22no—
109(C2×D3)⋊C8—24no—
110#816—22no—
111#816D414yescomplemented
112#808—24no—
113C4⋊(C3⋊C8)D414yescomplemented
114C22×SD16—32no—
115C23×C8—32no—
116C2×#127—61no—
117C2×#127C2×D311yescharacteristic
118C2×C22⋊Q8—32no—
119C2×#121—32no—
120C2×#121—32no—
121C2×C4×Q8—31no—
122C2×C4×D4—31no—
123#465—68no—
124#460—64no—
125C8×D4—64no—
126C2×#106—32no—
127C2×C4⋊C8—31no—
128C2×#103—31no—
129C2×#103—31no—
130C2×#102—32no—
131C2×#102C2×D312yes—
132C2×#102C2×D312yes—
133C2×C22⋊C8—32no—
134C2×C4×C8C2×D311yescharacteristic
135#396—62no—
136#392—62no—
137#390—61no—
138#390—61no—
139#389—61no—
140#384C2×D311yescharacteristic
141#380—32no—
142C4×C4⋊C4—31no—
143#340—31no—
144#340—31no—
145#340—32no—
146C24×C3—22no—
147C23×D3C2×D412yes—
148C2×C3×D4—22no—
149C2×C3×D4—22no—
150C2×C3×D4—22no—
151C2×C3×D4—22no—
152C2×C3×D4—22no—
153C2×C3×D4—22no—
154C2×C3×D4—42no—
155C2×C3×D4—42no—
156C2×C3×D4—42no—
157C2×C3×D4—42no—
158C2×C3×D4—81no—
159C2×C3×D4—81no—
160C22×C3×C4—22no—
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2411yescharacteristic
164C2×#67—44no—
165C2×#67—44no—
166C2×#67—44no—
167C2×#67—44no—
168C2×#67—82no—
169C2×#67—82no—
170C22×Dic3—21no—
171C22×Dic3—21no—
172C22×Dic3—21no—
173C22×Dic3—21no—
174C22×Dic3C2×D412yes—
175C2×D12—22no—
176C2×D12—22no—
177C2×D12SD1614yescomplemented
178C2×C4×D3—22no—
179C2×C4×D3—22no—
180C2×C4×D3—22no—
181C2×C4×D3—22no—
182C2×C4×D3—22no—
183C2×C4×D3—22no—
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—21no—
194C2×C3×C8—21no—
195C2×C3×C8C2×D412yes—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—42no—
199C3×C4⋊C4—81no—
200C3×C42C2×D412yes—
201C3×C42#4112yes—
202#217—44no—
203#217—44no—
204#214—48no—
205(C2×D3)⋊C4—24no—
206(C2×D3)⋊C4—44no—
207(C2×D3)⋊C4—82no—
208C4⋊Dic3—22no—
209C4⋊Dic3—21no—
210C4⋊Dic3—21no—
211C4⋊Dic3—21no—
212C4⋊Dic3—21no—
213C4⋊Dic3—44no—
214C4⋊Dic3C2×D412yes—
215C4⋊Dic3#4112yes—
216Dic3⋊C4—42no—
217C4×Dic3—42no—
218C4×Dic3—42no—
219C2×C3⋊C8—22no—
220C2×C3⋊C8—22no—
221C2×C3⋊C8C2×D414yes—
222D3×C8—48no—
223C22×Q8—31no—
224C22×D4—21no—
225C22×D4—21no—
226C22×D4—31no—
227C22×D4—62no—
228C22×D4—62no—
229C22×D4C22×D312yes—
230C23×C4—32no—
231C2×SD16—64no—
232C2×SD16—64no—
233C2×SD16—122no—
234C22×C8—32no—
235C22×C8—64no—
236C22×C8—64no—
237C22×C8—122no—
238C22×C8C22×D311yescharacteristic
239C22×C8C22×D311yescharacteristic
240#127—22no—
241#127—22no—
242#127—124no—
243C22⋊Q8—64no—
244C22⋊Q8—122no—
245#121—64no—
246#121—64no—
247#121—122no—
248#121—122no—
249C4×Q8—62no—
250C4×Q8—62no—
251C4×D4—62no—
252C4×D4—62no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—31no—
256C2×C4⋊C4—31no—
257C2×C4⋊C4—31no—
258C2×C4⋊C4—31no—
259C2×C4⋊C4—62no—
260C2×C4⋊C4—61no—
261C2×C4⋊C4C22×D312yes—
262C2×C22⋊C4—32no—
263C2×C22⋊C4—62no—
264C2×C22⋊C4—62no—
265C2×C22⋊C4—62no—
266C2×C22⋊C4—62no—
267C2×C42—61no—
268C2×C42—61no—
269C2×C42C22×D311yescharacteristic
270#106—64no—
271C4⋊C8—34no—
272#103—34no—
273#103—62no—
274#102—24no—
275#102—42no—
276#102—64no—
277C22⋊C8—64no—
278C4×C8—22no—
279#95—62no—
280#95—61no—
281C23×C3—42no—
282C23×C3—42no—
283C23×C3—42no—
284C23×C3—42no—
285C23×C3—42no—
286C23×C3—42no—
287C23×C3—42no—
288C23×C3C22×D411yescharacteristic
289C22×D3—22no—
290C22×D3—22no—
291C22×D3—22no—
292C22×D3—22no—
293C22×D3—22no—
294C22×D3—22no—
295C22×D3C2×SD1614yes—
296C3×D4—42no—
297C3×D4—42no—
298C3×D4—42no—
299C3×D4—42no—
300C3×D4—84no—
301C3×D4—84no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C2×C3×C4#12112yes—
308C2×C3×C4#12112yes—
309C2×C3×C4C2×SD1612yes—
310C2×C3×C4#13512yes—
311C2×C3×C4C22×D411yescharacteristic
312C2×C3×C4C22×D411yescharacteristic
313C2×C3×C4C2×#4111yescharacteristic
314C2×C3×C4C2×#4111yescharacteristic
315C2×C3×C4H(5,𝔽2)11yescharacteristic
316C2×C3×C4H(5,𝔽2)11yescharacteristic
317#67—88no—
318#67—88no—
319C2×Dic3—22no—
320C2×Dic3—22no—
321C2×Dic3—22no—
322C2×Dic3—22no—
323C2×Dic3—22no—
324C2×Dic3—22no—
325C2×Dic3—41no—
326C2×Dic3—41no—
327C2×Dic3—41no—
328C2×Dic3—41no—
329C2×Dic3—41no—
330C2×Dic3—41no—
331C2×Dic3—41no—
332C2×Dic3—41no—
333C2×Dic3—41no—
334D12—28no—
335C4×D3—42no—
336C4×D3—42no—
337C4×D3—42no—
338C4×D3—42no—
339Dic6—22no—
340Dic6—22no—
341Dic6—41no—
342Dic6—41no—
343C3×C8—22no—
344C3×C8—41no—
345C3⋊C8—24no—
346C24—22no—
347C24—32no—
348C2×Q8—32no—
349C2×Q8—61no—
350C2×Q8—61no—
351C2×Q8—61no—
352C2×Q8—61no—
353C2×Q8—61no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—34no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—42no—
364C2×D4—42no—
365C2×D4—62no—
366C2×D4—62no—
367C2×D4—81no—
368C2×D4—81no—
369C2×D4—124no—
370C2×D4—124no—
371C2×D4—124no—
372C2×D4—124no—
373C2×D4—242no—
374C2×D4—242no—
375C22×C4—22no—
376C22×C4—32no—
377C22×C4—61no—
378C22×C4—61no—
379C22×C4—61no—
380C22×C4—62no—
381C22×C4—62no—
382C22×C4—62no—
383C22×C4—62no—
384C22×C4—62no—
385C22×C4—62no—
386C22×C4—61no—
387C22×C4D3×D411yescharacteristic
388C22×C4D3×D411yescharacteristic
389C22×C4C23×D311yescharacteristic
390SD16—128no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—34no—
397C2×C8—62no—
398C2×C8—62no—
399C2×C8—128no—
400C2×C8D3×D412yes—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—32no—
404C4⋊C4—32no—
405C4⋊C4—42no—
406C4⋊C4—62no—
407C4⋊C4—61no—
408C4⋊C4—61no—
409C4⋊C4—61no—
410C4⋊C4—61no—
411C4⋊C4—81no—
412C4⋊C4—124no—
413C4⋊C4—122no—
414C22⋊C4—64no—
415C22⋊C4—124no—
416C22⋊C4—124no—
417C22⋊C4—124no—
418C22⋊C4—242no—
419C42—122no—
420C42—122no—
421C42#23512yes—
422C42D3×D412yes—
423C22×C3—82no—
424C22×C3—82no—
425C22×C3—82no—
426C22×C3—82no—
427C22×C3—82no—
428C22×C3—82no—
429C22×C3—82no—
430C22×C3C2×#12111yescharacteristic
431C22×C3D4211yescharacteristic
432C22×C3#54611yescharacteristic
433C22×C3C22×SD1611yescharacteristic
434C22×C3C2×#13511yescharacteristic
435C22×C3#57611yescharacteristic
436C22×C3#57711yescharacteristic
437C2×D3—24no—
438C2×D3—24no—
439C2×D3—24no—
440C2×D3—42no—
441C2×D3—42no—
442C2×D3—42no—
443C2×D3—42no—
444C3×C4—22no—
445C3×C4—82no—
446C3×C4#46514yescomplemented
447C3×C4#53211yescharacteristic
448C3×C4#54011yescharacteristic
449C3×C4D4211yescharacteristic
450C3×C4#55011yescharacteristic
451Dic3—42no—
452Dic3—42no—
453Dic3—42no—
454Dic3—42no—
455Dic3—81no—
456C23—34no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—62no—
465C23—62no—
466C23—62no—
467C23—62no—
468C23—62no—
469C23—62no—
470C23C2×D3×D411yescharacteristic
471Q8—62no—
472Q8—62no—
473Q8—121no—
474Q8—121no—
475D4—42no—
476D4—42no—
477D4—42no—
478D4—42no—
479D4—68no—
480D4—84no—
481D4—84no—
482D4—248no—
483D4—248no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—62no—
490C2×C4—62no—
491C2×C4—62no—
492C2×C4—62no—
493C2×C4—62no—
494C2×C4—62no—
495C2×C4—121no—
496C2×C4—121no—
497C2×C4—121no—
498C2×C4—121no—
499C2×C4—121no—
500C2×C4—121no—
501C2×C4—122no—
502C2×C4—122no—
503C2×C4—122no—
504C2×C4—122no—
505C2×C4—121no—
506C2×C4—121no—
507C2×C4—121no—
508C2×C4#88412yes—
509C2×C4#88412yes—
510C2×C4#91212yes—
511C2×C4D3×SD1612yes—
512C2×C4C2×#23511yescharacteristic
513C2×C4C2×D3×D411yescharacteristic
514C2×C4C2×D3×D411yescharacteristic
515C2×C4#100411yescharacteristic
516C2×C4D3×#4111yescharacteristic
517C2×C4#100911yescharacteristic
518C8—22no—
519C8—41no—
520C8—64no—
521C2×C3—162no—
522C2×C3#242811yescharacteristic
523C2×C3C2×#46511yescharacteristic
524C2×C3#306211yescharacteristic
525C2×C3D4×SD1611yescharacteristic
526C2×C3#328711yescharacteristic
527C2×C3#329511yescharacteristic
528C2×C3#330411yescharacteristic
529D3—44no—
530C22—64no—
531C22—64no—
532C22—64no—
533C22—82no—
534C22—82no—
535C22—82no—
536C22—82no—
537C22—82no—
538C22—82no—
539C22—82no—
540C22—122no—
541C22—122no—
542C22—122no—
543C22—122no—
544C22C2×#88411yescharacteristic
545C22#567611yescharacteristic
546C22#571711yescharacteristic
547C22C2×#91211yescharacteristic
548C22#575711yescharacteristic
549C22C2×D3×SD1611yescharacteristic
550C22#576211yescharacteristic
551C4—22no—
552C4—82no—
553C4—122no—
554C4—122no—
555C4—122no—
556C4—122no—
557C4—241no—
558C4#477814yescomplemented
559C4#554811yescharacteristic
560C4#556211yescharacteristic
561C4#559911yescharacteristic
562C4#560911yescharacteristic
563C3#2538711yescharacteristic, complemented, Sylow
564C2—124no—
565C2—162no—
566C2#7644411yescharacteristic
567C2C2×#477811yescharacteristic
568C2#7875211yescharacteristic
569C2#8157711yescharacteristic
570C2#8157811yescharacteristic
571C2#8177111yescharacteristic
572C2#8179011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1957C2
socleC23×C3C22×D4
prime radical#83188C2
hypercenter#1957D3
hypocenterC3#25387
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1957

Complements

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

NH ≅ G/Ntype
2. C2×#5075C2semidirect
3. C2×#4778C2semidirect
4. C2×C8×D12C2semidirect
5. #74369C2semidirect
6. C3×#1957C2semidirect
7. #70207C2semidirect
8. #70205C2semidirect
9. #70202C2semidirect
10. C4×#816C2semidirect
11. #69568C2semidirect
52. #1957D3semidirect
111. #816D4semidirect
113. C4⋊(C3⋊C8)D4semidirect
177. C2×D12SD16semidirect
446. C3×C4#465semidirect
558. C4#4778semidirect
563. C3#25387semidirect

Maximal subgroups

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

subgroupindex
#831882
C2×#50752
C2×#47782
C2×C8×D122
#743692
C3×#19572
#702072
#702052
#702022
C4×#8162
#695682
#253873