Finite Group Databasebeta

← C2×#83628 #11675613

Subgroups — C2×#83628 #11675613

Order 768.

1918 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#554C211yesmaximal, characteristic, complemented
2C2×#5579C211yesmaximal, characteristic, complemented
3#83628C2116yesmaximal, direct product factor
4C2×#5084C211yesmaximal, characteristic, complemented
5C2×#5078C211yesmaximal, characteristic, complemented
6C2×#5066C212yesmaximal, complemented
7C2×#5065C212yesmaximal, complemented
8C2×Q8×C3⋊C8C211yesmaximal, characteristic, complemented
9C2×#5009C212yesmaximal, complemented
10C2×#5008C211yesmaximal, characteristic, complemented
11C2×C4×#213C211yesmaximal, characteristic
12C2×C4×#213C212yesmaximal
13C2×#3380—31nomaximal, Sylow
14C3×#554—24no—
15C3×#554C2218yescomplemented
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×#125C2212yes—
18C2×C3×C22⋊Q8—21no—
19C2×C3×C22⋊Q8—22no—
20C2×C3×C4×Q8C2211yescharacteristic
21C2×C3×C4×D4C2211yescharacteristic
22C2×C3×C4×D4C2212yes—
23C22×C3×C4⋊C4—21no—
24C2×#946—21no—
25C22×#213C2211yescharacteristic
26#5579—24no—
27#5579C2218yescomplemented
28C2×#896—22no—
29C2×D3×C4⋊C4—21no—
30C2×C4×D12C2211yescharacteristic
31C2×C4×D12C2212yes—
32C2×#870C2212yes—
33C2×#869C2211yescharacteristic
34C2×#832C2212yes—
35C2×#832C2211yescharacteristic
36#5084C2218yescomplemented
37#5078C2218yescomplemented
38#5066C22116yescomplemented
39#5065C22116yescomplemented
40Q8×C3⋊C8C2218yescomplemented
41C4×#213C22116yes—
42C4×#213C2218yes—
43#5009C22116yescomplemented
44#5008C2218yescomplemented
45C2×#809C2212yes—
46C2×#809C2211yescharacteristic
47C2×#808C2212yes—
48C2×#808C2212yes—
49C2×#807C2212yes—
50C2×#807C2212yes—
51C2×#807C2211yescharacteristic
52C2×C4⋊(C3⋊C8)C2211yescharacteristic
53C2×C4⋊(C3⋊C8)C2212yes—
54C2×C4×C3⋊C8C2211yescharacteristic
55C2×C4×C3⋊C8C2212yes—
56C2×#554—31no—
57C2×#554D311yescharacteristic, complemented
58#3380—316no—
59C2×#500—31no—
60C2×#499—32no—
61C2×#478—32no—
62C2×#478—32no—
63C2×#476—31no—
64C2×#476—31no—
65C2×C8×Q8—31no—
66C2×C4×D8—32no—
67C2×C4×D8—31no—
68C22×C3×Q8—21no—
69C22×C3×Q8D411yescharacteristic
70C22×C3×D4C2311yescharacteristic
71C23×C3×C4—21no—
72C23×C3×C4—22no—
73C22×D12C2311yescharacteristic
74C22×C4×D3—21no—
75C22×C4×D3—22no—
76C22×Dic6—21no—
77C3×C4⋊Q8—22no—
78C3×C4⋊Q8C2314yes—
79C3×#125—24no—
80C3×#125C2318yes—
81C3×C22⋊Q8—28no—
82C3×C22⋊Q8—216no—
83C3×C4×Q8D414yes—
84C3×C4×Q8C2314yes—
85C3×C4×D4—22no—
86C3×C4×D4—24no—
87C3×C4×D4C2314yes—
88C3×C4×D4C2318yes—
89C2×C3×C4⋊C4—22no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—22no—
92C2×C3×C4⋊C4—28no—
93C2×C3×C4⋊C4D411yescharacteristic
94C2×C3×C4⋊C4D412yes—
95C2×C3×C4⋊C4C2312yes—
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C22⋊C4—21no—
98C2×C3×C22⋊C4—22no—
99C2×C3×C42C2311yescharacteristic
100C2×C3×C42C2312yes—
101#946—28no—
102C2×#213Q818yescomplemented
103C2×#213C2314yes—
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×D12C2318yes—
119C4×D12C2314yes—
120#870—24no—
121#870C2318yes—
122#869—22no—
123#869C2314yes—
124#832C2318yes—
125#832C2314yes—
126#809C2314yes—
127#809C2318yes—
128#808C2318yes—
129#808C2318yes—
130#807C2318yes—
131#807C2314yes—
132#807C2318yes—
133C4⋊(C3⋊C8)C2314yes—
134C4⋊(C3⋊C8)C2318yes—
135C4×C3⋊C8C2318yes—
136C4×C3⋊C8C2314yes—
137C22×D8—31no—
138#554—24no—
139#554—38no—
140#554—64no—
141#554C2×D318yescomplemented
142C2×C4⋊Q8—31no—
143C2×C4⋊Q8C2×D311yescharacteristic
144C2×#125—32no—
145C2×#125C2×D312yes—
146C2×C22⋊Q8—21no—
147C2×C22⋊Q8—22no—
148C2×C22⋊Q8—61no—
149C2×C22⋊Q8—62no—
150C2×C4×Q8C2×D311yescharacteristic
151C2×C4×D4—32no—
152C2×C4×D4—31no—
153C2×C4×D4C2×D312yes—
154C2×C4×D4C2×D311yescharacteristic
155C22×C4⋊C4—21no—
156C22×C4⋊C4—61no—
157#500—38no—
158#499—316no—
159#478—316no—
160#478—316no—
161#476—38no—
162#476—38no—
163C8×Q8—38no—
164C4×D8—316no—
165C4×D8—38no—
166C2×#107—32no—
167C2×#107—31no—
168C2×#107—32no—
169C2×#106—32no—
170C2×#106—32no—
171C2×C4⋊C8—32no—
172C2×C4⋊C8—31no—
173C2×#102—31no—
174C2×#102—32no—
175C2×#102—31no—
176C2×#102—32no—
177C2×C4×C8—32no—
178C2×C4×C8—31no—
179C24×C3—21no—
180C23×D3—21no—
181C2×C3×Q8—24no—
182C2×C3×Q8—24no—
183C2×C3×Q8—82no—
184C2×C3×Q8C2×D414yes—
185C2×C3×D4—21no—
186C2×C3×D4—22no—
187C2×C3×D4C2×Q814yes—
188C2×C3×D4C2412yes—
189C22×C3×C4—21no—
190C22×C3×C4—22no—
191C22×C3×C4—24no—
192C22×C3×C4—28no—
193C22×C3×C4—44no—
194C22×C3×C4—84no—
195C22×C3×C4C2×D412yes—
196C22×C3×C4C2×D412yes—
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2411yescharacteristic
200C22×Dic3—21no—
201C22×Dic3—22no—
202C2×D12—21no—
203C2×D12—22no—
204C2×D12C2×Q814yes—
205C2×D12C2412yes—
206C2×C4×D3—24no—
207C2×C4×D3—28no—
208C2×C4×D3—44no—
209C2×C4×D3—84no—
210C2×Dic6—24no—
211C2×Dic6—82no—
212C3×C4⋊C4—28no—
213C3×C4⋊C4—24no—
214C3×C4⋊C4—28no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—22no—
217C3×C4⋊C4C2×D414yes—
218C3×C4⋊C4C2×D418yes—
219C3×C4⋊C4C2414yes—
220C3×C4⋊C4C2412yes—
221C3×C22⋊C4—24no—
222C3×C22⋊C4—28no—
223C3×C42C2×D412yes—
224C3×C42C2×D414yes—
225C3×C42C2412yes—
226C3×C42C2414yes—
227#213C2×Q8116yescomplemented
228(C2×D3)⋊C4—24no—
229(C2×D3)⋊C4—28no—
230C4⋊Dic3—22no—
231C4⋊Dic3—24no—
232C4⋊Dic3—21no—
233C4⋊Dic3C2414yes—
234C4⋊Dic3C2412yes—
235Dic3⋊C4—28no—
236Dic3⋊C4—28no—
237C2×C3⋊C8—22no—
238C2×C3⋊C8—24no—
239C2×C3⋊C8C2×Q814yes—
240C2×C3⋊C8C2412yes—
241C2×C3⋊C8C2414yes—
242C2×C3⋊C8C2412yes—
243C22×Q8—21no—
244C22×Q8—61no—
245C22×Q8#6711yescharacteristic
246C22×D4—31no—
247C22×D4C22×D311yescharacteristic
248C23×C4—21no—
249C23×C4—22no—
250C23×C4—61no—
251C23×C4—62no—
252C2×D8—38no—
253C2×D8—34no—
254C22×C8—31no—
255C22×C8—32no—
256C22×C8—31no—
257C4⋊Q8—22no—
258C4⋊Q8—34no—
259C4⋊Q8—62no—
260C4⋊Q8C22×D314yes—
261#125—24no—
262#125—38no—
263#125—64no—
264#125C22×D318yes—
265C22⋊Q8—28no—
266C22⋊Q8—216no—
267C22⋊Q8—68no—
268C22⋊Q8—616no—
269C4×Q8#6714yes—
270C4×Q8C22×D314yes—
271C4×D4—24no—
272C4×D4—22no—
273C4×D4—34no—
274C4×D4—38no—
275C4×D4—64no—
276C4×D4—62no—
277C4×D4C22×D314yes—
278C4×D4C22×D318yes—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—28no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—32no—
285C2×C4⋊C4—62no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—62no—
288C2×C4⋊C4—68no—
289C2×C4⋊C4#6711yescharacteristic
290C2×C4⋊C4#6712yes—
291C2×C4⋊C4C22×D311yescharacteristic
292C2×C4⋊C4C22×D312yes—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—61no—
296C2×C22⋊C4—62no—
297C2×C42C22×D312yes—
298C2×C42C22×D311yescharacteristic
299#107—34no—
300#107—38no—
301#107—38no—
302#106—38no—
303#106—38no—
304C4⋊C8—34no—
305C4⋊C8—38no—
306#102—38no—
307#102—34no—
308#102—38no—
309#102—34no—
310C4×C8—38no—
311C4×C8—34no—
312C23×C3—24no—
313C23×C3—22no—
314C23×C3—42no—
315C23×C3—42no—
316C23×C3C22×D411yescharacteristic
317C22×D3—24no—
318C22×D3—22no—
319C22×D3—42no—
320C22×D3—42no—
321C3×Q8—28no—
322C3×Q8—84no—
323C3×D4—24no—
324C3×D4—22no—
325C3×D4C22×Q814yes—
326C2×C3×C4—24no—
327C2×C3×C4—22no—
328C2×C3×C4—24no—
329C2×C3×C4—22no—
330C2×C3×C4—24no—
331C2×C3×C4—42no—
332C2×C3×C4—44no—
333C2×C3×C4—48no—
334C2×C3×C4—88no—
335C2×C3×C4#13514yes—
336C2×C3×C4C22×D414yes—
337C2×C3×C4C22×D414yes—
338C2×C3×C4C22×D412yes—
339C2×C3×C4C22×D411yescharacteristic
340C2×C3×C4C22×D412yes—
341C2×C3×C4C22×Q812yes—
342C2×C3×C42−1+412yes—
343C2×C3×C4C2511yescharacteristic
344C2×Dic3—22no—
345C2×Dic3—24no—
346C2×Dic3—42no—
347C2×Dic3—44no—
348D12—24no—
349D12—22no—
350D12C22×Q814yes—
351C4×D3—48no—
352C4×D3—88no—
353Dic6—84no—
354C3⋊C8—22no—
355C3⋊C8—24no—
356C3⋊C8C22×Q814yes—
357C24—21no—
358C24—61no—
359C2×Q8—24no—
360C2×Q8—24no—
361C2×Q8—64no—
362C2×Q8—82no—
363C2×Q8—242no—
364C2×Q8C2×#6714yes—
365C2×D4—22no—
366C2×D4—21no—
367C2×D4—34no—
368C2×D4—32no—
369C2×D4—62no—
370C2×D4—61no—
371C2×D4D3×Q814yes—
372C2×D4C23×D312yes—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—28no—
377C22×C4—44no—
378C22×C4—61no—
379C22×C4—62no—
380C22×C4—64no—
381C22×C4—68no—
382C22×C4—84no—
383C22×C4—124no—
384C22×C4—244no—
385C22×C4C2×#6712yes—
386C22×C4C2×#6711yescharacteristic
387C22×C4C2×#6712yes—
388C22×C4C2×#6711yescharacteristic
389C22×C4C23×D311yescharacteristic
390D8—316no—
391C2×C8—34no—
392C2×C8—32no—
393C2×C8—34no—
394C2×C8—32no—
395C2×C8—64no—
396C2×C8—62no—
397C4⋊C4—28no—
398C4⋊C4—24no—
399C4⋊C4—28no—
400C4⋊C4—21no—
401C4⋊C4—22no—
402C4⋊C4—32no—
403C4⋊C4—34no—
404C4⋊C4—61no—
405C4⋊C4—62no—
406C4⋊C4—68no—
407C4⋊C4—64no—
408C4⋊C4—68no—
409C4⋊C4C2×#6714yes—
410C4⋊C4C2×#6718yes—
411C4⋊C4C23×D312yes—
412C4⋊C4C23×D314yes—
413C22⋊C4—24no—
414C22⋊C4—28no—
415C22⋊C4—64no—
416C22⋊C4—68no—
417C42C2×#6714yes—
418C42C2×#6712yes—
419C42C23×D314yes—
420C42C23×D312yes—
421C22×C3—24no—
422C22×C3—48no—
423C22×C3—42no—
424C22×C3—42no—
425C22×C3D4×Q812yes—
426C22×C3C2×#13512yes—
427C22×C3#57612yes—
428C22×C3C23×D411yescharacteristic
429C2×D3—24no—
430C2×D3—48no—
431C2×D3—42no—
432C2×D3—42no—
433C3×C4—24no—
434C3×C4—22no—
435C3×C4—24no—
436C3×C4—42no—
437C3×C4—44no—
438C3×C4D4×Q812yes—
439C3×C4C2×#13514yes—
440C3×C4C23×Q811yescharacteristic
441C3×C4C2×2−1+411yescharacteristic
442Dic3—42no—
443Dic3—44no—
444C23—22no—
445C23—24no—
446C23—42no—
447C23—42no—
448C23—62no—
449C23—64no—
450C23—122no—
451C23—122no—
452C23C22×#6711yescharacteristic
453Q8—28no—
454Q8—84no—
455Q8—244no—
456D4—22no—
457D4—24no—
458D4—34no—
459D4—62no—
460D4—64no—
461D4C2×D3×Q814yes—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—48no—
470C2×C4—62no—
471C2×C4—64no—
472C2×C4—88no—
473C2×C4—122no—
474C2×C4—124no—
475C2×C4—128no—
476C2×C4—248no—
477C2×C4#95014yes—
478C2×C4C2×D3×Q812yes—
479C2×C4#100712yes—
480C2×C4C22×#6714yes—
481C2×C4C22×#6712yes—
482C2×C4C22×#6714yes—
483C2×C4C22×#6711yescharacteristic
484C2×C4C22×#6712yes—
485C2×C4C24×D311yescharacteristic
486C8—34no—
487C8—64no—
488C8—62no—
489C2×C3—44no—
490C2×C3#338014yescomplemented
491C2×C3C2×D4×Q811yescharacteristic
492C2×C3C22×#13511yescharacteristic
493C2×C3C2×#57611yescharacteristic
494D3—44no—
495C22—24no—
496C22—48no—
497C22—42no—
498C22—42no—
499C22—64no—
500C22—128no—
501C22—122no—
502C22—122no—
503C22Q8×#6712yes—
504C22C2×#95012yes—
505C22#583512yes—
506C22C23×#6711yescharacteristic
507C4—24no—
508C4—24no—
509C4—22no—
510C4—42no—
511C4—44no—
512C4—122no—
513C4—124no—
514C4Q8×#6712yes—
515C4C2×#95014yes—
516C4C22×D3×Q811yescharacteristic
517C4C2×#100711yescharacteristic
518C3C2×#338011yescharacteristic, complemented, Sylow
519C2—44no—
520C2—124no—
521C2#8362814yesdirect product factor
522C2C2×Q8×#6711yescharacteristic
523C2C22×#95011yescharacteristic
524C2C2×#583511yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#554C2
socleC23×C3C22×D4
prime radicalC22×#213C22
hypercenterC2×#554D3
hypocenterC3C2×#3380
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
1. C2×C3×#554C2semidirect
2. C2×#5579C2semidirect
3. #83628C2direct (N × H)
4. C2×#5084C2semidirect
5. C2×#5078C2semidirect
6. C2×#5066C2semidirect
7. C2×#5065C2semidirect
8. C2×Q8×C3⋊C8C2semidirect
9. C2×#5009C2semidirect
10. C2×#5008C2semidirect
15. C3×#554C22semidirect
27. #5579C22semidirect
36. #5084C22semidirect
37. #5078C22semidirect
38. #5066C22semidirect
39. #5065C22semidirect
40. Q8×C3⋊C8C22semidirect
43. #5009C22semidirect
44. #5008C22semidirect
57. C2×#554D3semidirect
102. C2×#213Q8semidirect
141. #554C2×D3semidirect
227. #213C2×Q8semidirect
490. C2×C3#3380semidirect
518. C3C2×#3380semidirect
521. C2#83628direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5542
C2×#55792
#836282
C2×#50842
C2×#50782
C2×#50662
C2×#50652
C2×Q8×C3⋊C82
C2×#50092
C2×#50082
C2×C4×#2132
C2×#33803