Finite Group Databasebeta

← C2×#83706 #11675672

Subgroups — C2×#83706 #11675672

Order 768.

1854 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×Q82C211yesmaximal, characteristic, complemented
2C2×#5579C211yesmaximal, characteristic
3#83706C2116yesmaximal, direct product factor
4C2×#5084C211yesmaximal, characteristic
5C2×#5087C212yesmaximal
6C2×#5081C212yesmaximal, complemented
7C2×#5078C211yesmaximal, characteristic, complemented
8C2×#5026C212yesmaximal
9C2×#5024C211yesmaximal, characteristic, complemented
10C2×#5022C212yesmaximal, complemented
11C2×#5021C211yesmaximal, characteristic, complemented
12C2×C4×#215C211yesmaximal, characteristic
13C2×#3385—31nomaximal, Sylow
14C3×Q82—24no—
15C3×Q82C2218yescomplemented
16C2×C3×C4⋊Q8—21no—
17C2×C3×C4⋊Q8—22no—
18C2×C3×C4⋊Q8C2211yescharacteristic
19C2×C3×C4⋊Q8C2212yes—
20C2×C3×C4×Q8—21no—
21C2×C3×C4×Q8C2212yes—
22C2×C3×C4×Q8C2211yescharacteristic
23C2×C3×C4×Q8C2211yescharacteristic
24C2×#946—21no—
25C22×#215C2211yescharacteristic
26#5579—24no—
27#5579C2218yes—
28C2×#896—22no—
29C2×D3×C4⋊C4—21no—
30C2×C4×D12C2212yes—
31C2×C4×D12C2211yescharacteristic
32C2×#870C2212yes—
33C2×#869C2211yescharacteristic
34C2×#835C2212yes—
35C2×#835C2211yescharacteristic
36#5087C22116yes—
37#5084C2218yes—
38#5081C22116yescomplemented
39#5078C2218yescomplemented
40#5026C22116yes—
41C4×#215C2218yes—
42#5024C2218yescomplemented
43#5022C22116yescomplemented
44#5021C2218yescomplemented
45C2×#809C2212yes—
46C2×#809C2211yescharacteristic
47C2×#808C2211yescharacteristic
48C2×#808C2212yes—
49C2×#807C2212yes—
50C2×#807C2212yes—
51C2×#807C2212yes—
52C2×C4⋊(C3⋊C8)C2212yes—
53C2×C4⋊(C3⋊C8)C2211yescharacteristic
54C2×#803C2212yes—
55C2×C4×C3⋊C8C2211yescharacteristic
56C2×Q82D311yescharacteristic, complemented
57C2×#554—31no—
58#3385—316no—
59C2×#501—32no—
60C2×#500—31no—
61C2×#479—32no—
62C2×#477—31no—
63C2×#476—31no—
64C2×#474—32no—
65C2×#446—31no—
66C2×#440—32no—
67C2×C4×SD16—31no—
68C22×C3×Q8—22no—
69C22×C3×Q8—21no—
70C22×C3×Q8D411yescharacteristic
71C22×C3×Q8C2311yescharacteristic
72C22×D12C2311yescharacteristic
73C22×C4×D3—21no—
74C22×C4×D3—22no—
75C22×Dic6—21no—
76C3×C4⋊Q8—28no—
77C3×C4⋊Q8—216no—
78C3×C4⋊Q8—22no—
79C3×C4⋊Q8—24no—
80C3×C4⋊Q8C2314yes—
81C3×C4⋊Q8C2318yes—
82C3×C4×Q8—28no—
83C3×C4×Q8—22no—
84C3×C4×Q8—24no—
85C3×C4×Q8D414yes—
86C3×C4×Q8C2314yes—
87C3×C4×Q8C2318yes—
88C3×C4×Q8C2314yes—
89C2×C3×C4⋊C4—22no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—22no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4D411yescharacteristic
94C2×C3×C4⋊C4D412yes—
95C2×C3×C4⋊C4C2312yes—
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C42—22no—
98C2×C3×C42—21no—
99C2×C3×C42C2311yescharacteristic
100C2×C3×C42C2312yes—
101#946—28no—
102C2×#215Q818yescomplemented
103C2×#215C2314yes—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4—22no—
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3C2312yes—
108C2×C4⋊Dic3C2311yescharacteristic
109C2×Dic3⋊C4—22no—
110C2×Dic3⋊C4—22no—
111C22×C3⋊C8C2311yescharacteristic
112C22×C3⋊C8C2312yes—
113C22×C3⋊C8C2311yescharacteristic
114#896—216no—
115D3×C4⋊C4—28no—
116C4×D12—24no—
117C4×D12—22no—
118C4×D12C2314yes—
119C4×D12C2318yes—
120#870—24no—
121#870C2318yes—
122#869—22no—
123#869C2314yes—
124#835C2318yes—
125#835C2314yes—
126#809C2314yes—
127#809C2318yes—
128#808C2318yes—
129#808C2314yes—
130#807C2318yes—
131#807C2318yes—
132#807C2318yes—
133C4⋊(C3⋊C8)C2314yes—
134C4⋊(C3⋊C8)C2318yes—
135#803C2318yes—
136C4×C3⋊C8C2314yes—
137C22×SD16—31no—
138Q82—24no—
139Q82C2×D318yescomplemented
140#554—38no—
141#554—64no—
142C2×C4⋊Q8—21no—
143C2×C4⋊Q8—22no—
144C2×C4⋊Q8—31no—
145C2×C4⋊Q8C2×D311yescharacteristic
146C2×C4⋊Q8C2×D312yes—
147C2×#125—32no—
148C2×C22⋊Q8—61no—
149C2×C22⋊Q8—62no—
150C2×C4×Q8—21no—
151C2×C4×Q8C2×D311yescharacteristic
152C2×C4×Q8C2×D312yes—
153C2×C4×Q8C2×D311yescharacteristic
154C2×C4×D4—31no—
155C2×C4×D4—32no—
156C22×C4⋊C4—61no—
157#501—316no—
158#500—38no—
159#479—316no—
160#477—38no—
161#476—38no—
162#474—316no—
163#446—38no—
164#440—316no—
165C4×SD16—38no—
166C2×#107—32no—
167C2×#107—32no—
168C2×#107—32no—
169C2×#106—32no—
170C2×#106—31no—
171C2×C4⋊C8—31no—
172C2×C4⋊C8—32no—
173C2×#103—32no—
174C2×#103—31no—
175C2×#102—31no—
176C2×#102—32no—
177C2×#97—32no—
178C2×C4×C8—31no—
179C23×D3—21no—
180C2×C3×Q8—24no—
181C2×C3×Q8—21no—
182C2×C3×Q8—28no—
183C2×C3×Q8—24no—
184C2×C3×Q8—22no—
185C2×C3×Q8—82no—
186C2×C3×Q8—84no—
187C2×C3×Q8C2×D414yes—
188C2×C3×Q8C2×Q814yes—
189C2×C3×Q8C2412yes—
190C22×C3×C4—21no—
191C22×C3×C4—22no—
192C22×C3×C4—21no—
193C22×C3×C4C2×D412yes—
194C22×C3×C4C2×D412yes—
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2411yescharacteristic
198C22×Dic3—21no—
199C22×Dic3—22no—
200C2×D12—21no—
201C2×D12—22no—
202C2×D12C2×Q814yes—
203C2×D12C2412yes—
204C2×C4×D3—24no—
205C2×C4×D3—28no—
206C2×C4×D3—44no—
207C2×C4×D3—84no—
208C2×Dic6—24no—
209C2×Dic6—82no—
210C3×C4⋊C4—28no—
211C3×C4⋊C4—24no—
212C3×C4⋊C4—28no—
213C3×C4⋊C4—24no—
214C3×C4⋊C4—22no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4C2×D414yes—
217C3×C4⋊C4C2×D418yes—
218C3×C4⋊C4C2412yes—
219C3×C4⋊C4C2414yes—
220C3×C42—28no—
221C3×C42—24no—
222C3×C42C2×D412yes—
223C3×C42C2×D414yes—
224C3×C42C2412yes—
225C3×C42C2414yes—
226#215C2×Q8116yescomplemented
227(C2×D3)⋊C4—24no—
228(C2×D3)⋊C4—28no—
229C4⋊Dic3—22no—
230C4⋊Dic3—24no—
231C4⋊Dic3—21no—
232C4⋊Dic3C2414yes—
233C4⋊Dic3C2412yes—
234Dic3⋊C4—28no—
235Dic3⋊C4—28no—
236C2×C3⋊C8—22no—
237C2×C3⋊C8—24no—
238C2×C3⋊C8C2×Q814yes—
239C2×C3⋊C8C2412yes—
240C2×C3⋊C8C2414yes—
241C2×C3⋊C8C2412yes—
242C22×Q8—21no—
243C22×Q8—22no—
244C22×Q8—61no—
245C22×Q8#6711yescharacteristic
246C22×Q8C22×D311yescharacteristic
247C22×D4—31no—
248C23×C4—61no—
249C23×C4—62no—
250C2×SD16—38no—
251C2×SD16—34no—
252C22×C8—31no—
253C22×C8—31no—
254C22×C8—32no—
255C4⋊Q8—28no—
256C4⋊Q8—216no—
257C4⋊Q8—24no—
258C4⋊Q8—22no—
259C4⋊Q8—34no—
260C4⋊Q8—62no—
261C4⋊Q8C22×D318yes—
262C4⋊Q8C22×D314yes—
263#125—38no—
264#125—64no—
265C22⋊Q8—68no—
266C22⋊Q8—616no—
267C4×Q8—28no—
268C4×Q8—24no—
269C4×Q8—22no—
270C4×Q8#6714yes—
271C4×Q8C22×D314yes—
272C4×Q8C22×D318yes—
273C4×Q8C22×D314yes—
274C4×D4—34no—
275C4×D4—38no—
276C4×D4—64no—
277C4×D4—62no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—32no—
284C2×C4⋊C4—62no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4—62no—
287C2×C4⋊C4—68no—
288C2×C4⋊C4#6711yescharacteristic
289C2×C4⋊C4#6712yes—
290C2×C4⋊C4C22×D311yescharacteristic
291C2×C4⋊C4C22×D312yes—
292C2×C22⋊C4—61no—
293C2×C22⋊C4—62no—
294C2×C42—22no—
295C2×C42—21no—
296C2×C42C22×D312yes—
297C2×C42C22×D311yescharacteristic
298#107—38no—
299#107—38no—
300#107—38no—
301#106—34no—
302#106—38no—
303C4⋊C8—38no—
304C4⋊C8—34no—
305#103—34no—
306#103—38no—
307#102—34no—
308#102—38no—
309#97—38no—
310C4×C8—34no—
311C23×C3C22×D411yescharacteristic
312C22×D3—24no—
313C22×D3—22no—
314C22×D3—42no—
315C22×D3—42no—
316C3×Q8—24no—
317C3×Q8—28no—
318C3×Q8—22no—
319C3×Q8—84no—
320C3×Q8—88no—
321C3×Q8C22×Q814yes—
322C2×C3×C4—24no—
323C2×C3×C4—24no—
324C2×C3×C4—22no—
325C2×C3×C4—24no—
326C2×C3×C4—22no—
327C2×C3×C4—24no—
328C2×C3×C4—22no—
329C2×C3×C4—44no—
330C2×C3×C4—42no—
331C2×C3×C4#13714yes—
332C2×C3×C4C22×D414yes—
333C2×C3×C4C22×D414yes—
334C2×C3×C4C22×D412yes—
335C2×C3×C4C22×D411yescharacteristic
336C2×C3×C4C22×D412yes—
337C2×C3×C4C22×Q812yes—
338C2×C3×C42−1+412yes—
339C2×C3×C4C2511yescharacteristic
340C2×Dic3—22no—
341C2×Dic3—24no—
342C2×Dic3—42no—
343C2×Dic3—44no—
344D12—24no—
345D12—22no—
346D12C22×Q814yes—
347C4×D3—48no—
348C4×D3—88no—
349Dic6—84no—
350C3⋊C8—22no—
351C3⋊C8—24no—
352C3⋊C8C22×Q814yes—
353C24—61no—
354C2×Q8—24no—
355C2×Q8—22no—
356C2×Q8—21no—
357C2×Q8—24no—
358C2×Q8—28no—
359C2×Q8—64no—
360C2×Q8—84no—
361C2×Q8—82no—
362C2×Q8—242no—
363C2×Q8D3×Q814yes—
364C2×Q8C2×#6714yes—
365C2×Q8C23×D312yes—
366C2×D4—34no—
367C2×D4—32no—
368C2×D4—62no—
369C2×D4—61no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—61no—
374C22×C4—62no—
375C22×C4—64no—
376C22×C4—68no—
377C22×C4—124no—
378C22×C4—244no—
379C22×C4C2×#6712yes—
380C22×C4C2×#6711yescharacteristic
381C22×C4C2×#6712yes—
382C22×C4C2×#6711yescharacteristic
383C22×C4C23×D311yescharacteristic
384SD16—316no—
385C2×C8—34no—
386C2×C8—32no—
387C2×C8—32no—
388C2×C8—34no—
389C2×C8—62no—
390C2×C8—64no—
391C4⋊C4—28no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—28no—
395C4⋊C4—21no—
396C4⋊C4—22no—
397C4⋊C4—32no—
398C4⋊C4—34no—
399C4⋊C4—61no—
400C4⋊C4—62no—
401C4⋊C4—68no—
402C4⋊C4—64no—
403C4⋊C4—68no—
404C4⋊C4C2×#6714yes—
405C4⋊C4C2×#6718yes—
406C4⋊C4C23×D312yes—
407C4⋊C4C23×D314yes—
408C22⋊C4—64no—
409C22⋊C4—68no—
410C42—28no—
411C42—24no—
412C42C2×#6714yes—
413C42C2×#6712yes—
414C42C23×D312yes—
415C42C23×D314yes—
416C22×C3D4×Q812yes—
417C22×C3C2×#13712yes—
418C22×C3#57612yes—
419C22×C3C23×D411yescharacteristic
420C2×D3—24no—
421C2×D3—48no—
422C2×D3—42no—
423C2×D3—42no—
424C3×C4—24no—
425C3×C4—24no—
426C3×C4—22no—
427C3×C4—24no—
428C3×C4—44no—
429C3×C4—42no—
430C3×C4D4×Q812yes—
431C3×C4C2×#13714yes—
432C3×C4C23×Q811yescharacteristic
433C3×C4C2×2−1+411yescharacteristic
434Dic3—42no—
435Dic3—44no—
436C23—62no—
437C23—64no—
438C23—122no—
439C23—122no—
440C23C22×#6711yescharacteristic
441Q8—22no—
442Q8—28no—
443Q8—24no—
444Q8—88no—
445Q8—84no—
446Q8—244no—
447Q8C2×D3×Q814yes—
448D4—34no—
449D4—62no—
450D4—64no—
451C2×C4—22no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—62no—
461C2×C4—64no—
462C2×C4—122no—
463C2×C4—124no—
464C2×C4—128no—
465C2×C4—248no—
466C2×C4#94214yes—
467C2×C4C2×D3×Q812yes—
468C2×C4#100712yes—
469C2×C4C22×#6714yes—
470C2×C4C22×#6712yes—
471C2×C4C22×#6714yes—
472C2×C4C22×#6712yes—
473C2×C4C22×#6711yescharacteristic
474C2×C4C24×D311yescharacteristic
475C8—34no—
476C8—62no—
477C8—64no—
478C2×C3#338514yescomplemented
479C2×C3C2×D4×Q811yescharacteristic
480C2×C3C22×#13711yescharacteristic
481C2×C3C2×#57611yescharacteristic
482D3—44no—
483C22—64no—
484C22—122no—
485C22—122no—
486C22—128no—
487C22C2×#94212yes—
488C22Q8×#6712yes—
489C22#583512yes—
490C22C23×#6711yescharacteristic
491C4—24no—
492C4—24no—
493C4—24no—
494C4—22no—
495C4—42no—
496C4—44no—
497C4—122no—
498C4—124no—
499C4C2×#94214yes—
500C4Q8×#6712yes—
501C4C22×D3×Q811yescharacteristic
502C4C2×#100711yescharacteristic
503C3C2×#338511yescharacteristic, complemented, Sylow
504C2—124no—
505C2#8370614yesdirect product factor
506C2C22×#94211yescharacteristic
507C2C2×Q8×#6711yescharacteristic
508C2C2×#583511yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×Q82C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenterC2×Q82D3
hypocenterC3C2×#3385
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×Q82

Complements

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

NH ≅ G/Ntype
1. C2×C3×Q82C2semidirect
3. #83706C2direct (N × H)
6. C2×#5081C2semidirect
7. C2×#5078C2semidirect
9. C2×#5024C2semidirect
10. C2×#5022C2semidirect
11. C2×#5021C2semidirect
15. C3×Q82C22semidirect
38. #5081C22semidirect
39. #5078C22semidirect
42. #5024C22semidirect
43. #5022C22semidirect
44. #5021C22semidirect
56. C2×Q82D3semidirect
102. C2×#215Q8semidirect
139. Q82C2×D3semidirect
226. #215C2×Q8semidirect
478. C2×C3#3385semidirect
503. C3C2×#3385semidirect
505. C2#83706direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×Q822
C2×#55792
#837062
C2×#50842
C2×#50872
C2×#50812
C2×#50782
C2×#50262
C2×#50242
C2×#50222
C2×#50212
C2×C4×#2152
C2×#33853