Finite Group Databasebeta

← #10831700

Subgroups — #10831700

Order 768.

1296 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D4×Q8C211yesmaximal, characteristic, complemented
2C2×#5084C211yesmaximal, characteristic, complemented
3#82501C211yesmaximal, characteristic, complemented
4C2×#5014C211yesmaximal, characteristic
5#71792C212yesmaximal, complemented
6#71788C211yesmaximal, characteristic, complemented
7#71783C212yesmaximal, complemented
8#71635C211yesmaximal, characteristic, complemented
9#71613C212yesmaximal
10#71607C212yesmaximal, complemented
11#71605C211yesmaximal, characteristic, complemented
12#24338—31nomaximal, Sylow
13C23×C3×Q8—21no—
14C3×D4×Q8—24no—
15C3×D4×Q8—24no—
16C2×C3×C4⋊Q8C2212yes—
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×C22⋊Q8—22no—
19C2×C3×C22⋊Q8—21no—
20C2×C3×C4×Q8C2211yescharacteristic
21C2×C3×C4×D4C2212yes—
22C2×C3×C4×D4C2211yescharacteristic
23C22×#213C2211yescharacteristic
24C2×C4×D12C2211yescharacteristic
25#5231—21no—
26C2×#832C2211yescharacteristic
27C2×#832C2212yes—
28#5084—24no—
29#5014—24no—
30#4993—21no—
31#4989—22no—
32C2×#809C2212yes—
33C2×#809C2211yescharacteristic
34C2×#808C2212yes—
35C2×#808C2211yescharacteristic
36#4939C2212yes—
37#4936C2211yescharacteristic
38#4928C2212yes—
39C2×C4⋊(C3⋊C8)C2211yescharacteristic
40C2×#803C2211yescharacteristic
41#4529C2212yes—
42#4529C2212yes—
43#4529C2212yes—
44#4524C2212yes—
45#4524C2212yes—
46C2×D4×Q8D311yescharacteristic, complemented
47C2×#476—31no—
48C2×#442—31no—
49#2580—31no—
50#2058—32no—
51#2023—32no—
52#1993—31no—
53#1985—31no—
54#1879—32no—
55#1864—32no—
56#1804—31no—
57C22×C3×Q8—28no—
58C22×C3×Q8—21no—
59C22×C3×Q8—22no—
60C22×C3×Q8—48no—
61C22×C3×Q8D411yescharacteristic
62C22×C3×D4C2311yescharacteristic
63C23×C3×C4—22no—
64C23×C3×C4—21no—
65C22×D12C2311yescharacteristic
66C22×C4×D3—21no—
67C3×C4⋊Q8—24no—
68C3×C4⋊Q8—24no—
69C3×C4⋊Q8—22no—
70C3×C4⋊Q8D414yes—
71C3×C22⋊Q8—28no—
72C3×C22⋊Q8—216no—
73C3×C4×Q8—22no—
74C3×C4×Q8—22no—
75C3×C4×D4—24no—
76C3×C4×D4—24no—
77C3×C4×D4—22no—
78C3×C4×D4D414yes—
79C2×C3×C4⋊C4—22no—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4D412yes—
82C2×C3×C4⋊C4D411yescharacteristic
83C2×C3×C4⋊C4C2312yes—
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C22⋊C4—22no—
86C2×C3×C22⋊C4—21no—
87C2×C3×C42C2312yes—
88C2×C3×C42C2311yescharacteristic
89C2×#213—22no—
90C2×#213—22no—
91C2×#213Q814yescomplemented
92C2×(C2×D3)⋊C4—22no—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—22no—
95C2×C4⋊Dic3C2311yescharacteristic
96C2×C4⋊Dic3C2312yes—
97C2×Dic3⋊C4—22no—
98C22×C3⋊C8C2311yescharacteristic
99C22×C3⋊C8C2312yes—
100C22×C3⋊C8C2311yescharacteristic
101C4×D12—22no—
102C4×D12—22no—
103#832—22no—
104#832—24no—
105C3⋊#95—22no—
106C3⋊#95—21no—
107C3⋊#95—21no—
108#809—22no—
109#809—24no—
110#808—22no—
111#808—24no—
112C4⋊(C3⋊C8)—22no—
113#803—22no—
114C23×Q8—21no—
115C22×D8—31no—
116D4×Q8—24no—
117D4×Q8—24no—
118C2×C4⋊Q8C2×D312yes—
119C2×C4⋊Q8C2×D311yescharacteristic
120C2×C22⋊Q8—22no—
121C2×C22⋊Q8—21no—
122C2×C4×Q8C2×D311yescharacteristic
123C2×C4×D4—31no—
124C2×C4×D4C2×D312yes—
125C2×C4×D4C2×D311yescharacteristic
126#476—64no—
127#442—64no—
128C2×#106—31no—
129C2×#106—32no—
130C2×C4⋊C8—31no—
131C2×#102—32no—
132C2×#102—31no—
133C2×#102—32no—
134C2×#102—31no—
135C2×#97—31no—
136#399—61no—
137#393—61no—
138#390—32no—
139#389—32no—
140#385—62no—
141#382—31no—
142#340—32no—
143#340—32no—
144#340—32no—
145#336—32no—
146#336—32no—
147C24×C3—21no—
148C23×D3—21no—
149C2×C3×Q8—28no—
150C2×C3×Q8—22no—
151C2×C3×Q8—24no—
152C2×C3×Q8—24no—
153C2×C3×Q8—44no—
154C2×C3×Q8—44no—
155C2×C3×Q8—432no—
156C2×C3×Q8—44no—
157C2×C3×D4—21no—
158C2×C3×D4—21no—
159C2×C3×D4—21no—
160C2×C3×D4C2×D412yes—
161C2×C3×D4C2×Q812yes—
162C2×C3×D4#4112yes—
163C22×C3×C4—28no—
164C22×C3×C4—22no—
165C22×C3×C4—21no—
166C22×C3×C4—24no—
167C22×C3×C4—44no—
168C22×C3×C4—44no—
169C22×C3×C4—44no—
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D412yes—
174C22×C3×C4C2411yescharacteristic
175C22×Dic3—22no—
176C22×Dic3—21no—
177C2×D12—21no—
178C2×D12—21no—
179C2×D12—21no—
180C2×D12—21no—
181C2×D12—21no—
182C2×D12C2×Q812yescomplemented
183C2×C4×D3—42no—
184C2×C4×D3—82no—
185C3×C4⋊C4—28no—
186C3×C4⋊C4—24no—
187C3×C4⋊C4—24no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—21no—
191C3×C4⋊C4C2×D414yes—
192C3×C4⋊C4C2×D412yes—
193C3×C22⋊C4—28no—
194C3×C22⋊C4—24no—
195C3×C42—22no—
196C3×C42—22no—
197C3×C42C2×D412yes—
198C3×C42C2×D412yes—
199#213—28no—
200(C2×D3)⋊C4—42no—
201(C2×D3)⋊C4—44no—
202(C2×D3)⋊C4—82no—
203C4⋊Dic3—21no—
204C4⋊Dic3—21no—
205C4⋊Dic3—22no—
206C4⋊Dic3—22no—
207Dic3⋊C4—82no—
208C2×C3⋊C8—21no—
209C2×C3⋊C8—21no—
210C2×C3⋊C8—21no—
211C2×C3⋊C8—22no—
212C2×C3⋊C8—22no—
213C2×C3⋊C8—21no—
214C2×C3⋊C8—21no—
215C2×C3⋊C8—22no—
216C2×C3⋊C8C2×Q812yescomplemented
217C22×Q8—28no—
218C22×Q8—22no—
219C22×Q8—21no—
220C22×Q8—48no—
221C22×Q8#6711yescharacteristic
222C22×D4—31no—
223C22×D4C22×D311yescharacteristic
224C23×C4—22no—
225C23×C4—21no—
226C23×C4—61no—
227C2×D8—34no—
228C2×D8—62no—
229C2×D8—62no—
230C22×C8—31no—
231C22×C8—31no—
232C22×C8—32no—
233C4⋊Q8—24no—
234C4⋊Q8—24no—
235C4⋊Q8—22no—
236C4⋊Q8#6714yes—
237C22⋊Q8—28no—
238C22⋊Q8—216no—
239C4×Q8—22no—
240C4×Q8—22no—
241C4×D4—24no—
242C4×D4—24no—
243C4×D4—22no—
244C4×D4—62no—
245C4×D4—62no—
246C4×D4#6714yes—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—31no—
250C2×C4⋊C4—32no—
251C2×C4⋊C4—62no—
252C2×C4⋊C4#6712yes—
253C2×C4⋊C4#6711yescharacteristic
254C2×C4⋊C4C22×D312yes—
255C2×C4⋊C4C22×D311yescharacteristic
256C2×C22⋊C4—22no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—61no—
259C2×C22⋊C4—62no—
260C2×C22⋊C4—62no—
261C2×C42C22×D312yes—
262C2×C42C22×D311yescharacteristic
263#106—62no—
264#106—64no—
265C4⋊C8—62no—
266#102—64no—
267#102—62no—
268#102—62no—
269#102—64no—
270#97—62no—
271#95—61no—
272#95—61no—
273#95—62no—
274C23×C3—22no—
275C23×C3—22no—
276C23×C3—22no—
277C23×C3—42no—
278C23×C3—42no—
279C23×C3C22×D411yescharacteristic
280C22×D3—22no—
281C22×D3—41no—
282C22×D3—41no—
283C22×D3—42no—
284C22×D3—42no—
285C3×Q8—28no—
286C3×Q8—48no—
287C3×Q8—416no—
288C3×D4—22no—
289C3×D4—22no—
290C3×D4—22no—
291C3×D4C22⋊Q814yes—
292C2×C3×C4—22no—
293C2×C3×C4—22no—
294C2×C3×C4—22no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—24no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—48no—
302C2×C3×C4—416no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4#12012yes—
307C2×C3×C4#12012yes—
308C2×C3×C4AGL(1,ℤ/8)14yes—
309C2×C3×C4C22×D411yescharacteristic
310C2×C3×C4C22×D411yescharacteristic
311C2×C3×C4C22×Q811yescharacteristic
312C2×C3×C4C2×#4111yescharacteristic
313C2×C3×C42−1+411yescharacteristic
314C2×C3×C42−1+411yescharacteristic
315C2×Dic3—41no—
316C2×Dic3—42no—
317C2×Dic3—42no—
318C2×Dic3—42no—
319C2×Dic3—42no—
320D12—22no—
321D12—22no—
322D12—41no—
323D12—41no—
324C4×D3—84no—
325C3⋊C8—22no—
326C3⋊C8—41no—
327C3⋊C8—42no—
328C24—21no—
329C24—61no—
330C2×Q8—28no—
331C2×Q8—24no—
332C2×Q8—22no—
333C2×Q8—24no—
334C2×Q8—44no—
335C2×Q8—44no—
336C2×Q8—432no—
337C2×Q8—44no—
338C2×D4—21no—
339C2×D4—21no—
340C2×D4—21no—
341C2×D4—32no—
342C2×D4—61no—
343C2×D4—61no—
344C2×D4—61no—
345C2×D4—61no—
346C2×D4—61no—
347C2×D4D3×Q812yes—
348C2×D4#23912yes—
349C2×D4C2×#6712yes—
350C22×C4—28no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—24no—
354C22×C4—44no—
355C22×C4—44no—
356C22×C4—44no—
357C22×C4—62no—
358C22×C4—61no—
359C22×C4—122no—
360C22×C4—242no—
361C22×C4C2×#6712yes—
362C22×C4C2×#6712yes—
363C22×C4C2×#6711yescharacteristic
364C22×C4C2×#6711yescharacteristic
365C22×C4C23×D311yescharacteristic
366D8—68no—
367C2×C8—32no—
368C2×C8—61no—
369C2×C8—61no—
370C2×C8—62no—
371C2×C8—61no—
372C2×C8—61no—
373C2×C8—62no—
374C2×C8—61no—
375C2×C8—62no—
376C4⋊C4—28no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—21no—
382C4⋊C4—62no—
383C4⋊C4—62no—
384C4⋊C4—61no—
385C4⋊C4—61no—
386C4⋊C4—242no—
387C4⋊C4C2×#6714yes—
388C4⋊C4C2×#6712yes—
389C22⋊C4—28no—
390C22⋊C4—24no—
391C22⋊C4—124no—
392C22⋊C4—122no—
393C22⋊C4—242no—
394C42—22no—
395C42—22no—
396C42C2×#6712yes—
397C42C2×#6712yes—
398C22×C3—24no—
399C22×C3—44no—
400C22×C3—44no—
401C22×C3—42no—
402C22×C3—42no—
403C22×C3C2×#12011yescharacteristic
404C22×C3#54811yescharacteristic
405C22×C3D4×Q811yescharacteristic
406C22×C3C2×AGL(1,ℤ/8)12yes—
407C22×C3#57712yes—
408C2×D3—44no—
409C2×D3—42no—
410C2×D3—82no—
411C2×D3—81no—
412C2×D3—81no—
413C3×C4—22no—
414C3×C4—24no—
415C3×C4—22no—
416C3×C4—44no—
417C3×C4—42no—
418C3×C4—42no—
419C3×C4C2×C22⋊Q811yescharacteristic
420C3×C4#53611yescharacteristic
421C3×C4#54811yescharacteristic
422C3×C4D4×Q811yescharacteristic
423Dic3—82no—
424Dic3—81no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—42no—
429C23—42no—
430C23—62no—
431C23—122no—
432C23—121no—
433C23—122no—
434C23—121no—
435C23C22×#6711yescharacteristic
436Q8—28no—
437Q8—416no—
438Q8—48no—
439D4—22no—
440D4—22no—
441D4—22no—
442D4—62no—
443D4—62no—
444D4—121no—
445D4—121no—
446D4#94614yes—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—22no—
456C2×C4—416no—
457C2×C4—48no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—122no—
462C2×C4—122no—
463C2×C4—122no—
464C2×C4—122no—
465C2×C4—121no—
466C2×C4—244no—
467C2×C4#93214yes—
468C2×C4#95312yes—
469C2×C4#95312yes—
470C2×C4C2×D3×Q811yescharacteristic
471C2×C4C2×#23911yescharacteristic
472C2×C4#100711yescharacteristic
473C2×C4#100711yescharacteristic
474C2×C4C22×#6711yescharacteristic
475C2×C4C22×#6711yescharacteristic
476C8—62no—
477C8—121no—
478C8—122no—
479C2×C3—44no—
480C2×C3#240911yescharacteristic
481C2×C3#300311yescharacteristic
482C2×C3#301011yescharacteristic
483C2×C3#337312yes—
484C2×C3#338412yes—
485D3—82no—
486C22—24no—
487C22—42no—
488C22—44no—
489C22—44no—
490C22—42no—
491C22—122no—
492C22—124no—
493C22—241no—
494C22—242no—
495C22—241no—
496C22C2×#93212yes—
497C22Q8×#6711yescharacteristic
498C22#581711yescharacteristic
499C22#583712yes—
500C22C2×#95311yescharacteristic
501C4—24no—
502C4—22no—
503C4—22no—
504C4—44no—
505C4—42no—
506C4—42no—
507C4—242no—
508C4—241no—
509C4C2×#94611yescharacteristic
510C4Q8×#6711yescharacteristic
511C4#581611yescharacteristic
512C4#581711yescharacteristic
513C3#2433811yescharacteristic, complemented, Sylow
514C2—44no—
515C2—242no—
516C2#8356412yes—
517C2#8356912yes—
518C2#8403811yescharacteristic
519C2#8406811yescharacteristic
520C2#8411311yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×D4×Q8C2
socleC23×C3C22×D4
prime radicalC22×#213C22
hypercenterC2×D4×Q8D3
hypocenterC3#24338
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×D4×Q8C2semidirect
2. C2×#5084C2semidirect
3. #82501C2semidirect
5. #71792C2semidirect
6. #71788C2semidirect
7. #71783C2semidirect
8. #71635C2semidirect
10. #71607C2semidirect
11. #71605C2semidirect
46. C2×D4×Q8D3semidirect
91. C2×#213Q8semidirect
182. C2×D12C2×Q8semidirect
216. C2×C3⋊C8C2×Q8semidirect
513. C3#24338semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D4×Q82
C2×#50842
#825012
C2×#50142
#717922
#717882
#717832
#716352
#716132
#716072
#716052
#243383