Finite Group Databasebeta

← C2×#87024 #11703401

Subgroups — C2×#87024 #11703401

Order 768.

7022 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#557C211yesmaximal, characteristic, complemented
2#87024C2132yesmaximal, direct product factor
3C2×#5732C213yesmaximal, complemented
4C2×#5728C213yesmaximal, complemented
5C2×#5694C216yesmaximal, complemented
6C2×#5698C216yesmaximal, complemented
7C2×Q8×D12C212yesmaximal, complemented
8C2×Q8×D12C211yesmaximal, characteristic, complemented
9C2×#5580C213yesmaximal
10C2×#5572C212yesmaximal, complemented
11C2×C4×D3×Q8C211yesmaximal, characteristic, complemented
12C2×#5570C213yesmaximal, complemented
13C2×#3523—31nomaximal, Sylow
14C22×D3×Q8C2211yescharacteristic
15C22×D3×Q8C2212yes—
16C3×#557C22116yescomplemented
17C2×C3×C4⋊Q8C2213yes—
18C2×C3×#125C2216yes—
19C2×C3×C4×Q8C2213yes—
20C2×C3×C4×Q8C2211yescharacteristic
21C2×C3×C4×Q8C2212yes—
22C2×#947C2213yes—
23C2×#946C2213yes—
24C2×Q8×Dic3C2216yes—
25C2×Q8×Dic3C2211yescharacteristic
26#5732C22148yescomplemented
27#5728C22148yescomplemented
28#5698C22196yescomplemented
29#5694C22196yescomplemented
30#5580C22148yes—
31Q8×D12C22132yescomplemented
32Q8×D12C22116yescomplemented
33#5572C22132yescomplemented
34C4×D3×Q8C22116yescomplemented
35#5570C22148yescomplemented
36C2×#898C2216yes—
37C2×#898C22112yes—
38C2×#897C2216yes—
39C2×#897C2216yes—
40C2×#897C2216yes—
41C2×#895C2211yescharacteristic
42C2×#894C2216yes—
43C2×#892C2216yes—
44C2×#892C2216yes—
45C2×D3×C4⋊C4C2213yes—
46C2×#890C2213yes—
47C2×#890C22112yes—
48C2×#890C2216yes—
49C2×#887C2213yes—
50C2×#887C2216yes—
51C2×C4×D12C2216yes—
52C2×C4×D12C2213yes—
53C2×#872C2216yes—
54C2×C42×D3C2213yes—
55C2×#869C2216yes—
56C2×#869C2213yes—
57C2×C4×Dic6C2213yes—
58C2×C4×Dic6C2216yes—
59#3523—332no—
60C2×#557—33no—
61C2×#557D311yescharacteristic, complemented
62C2×#556—33no—
63C2×#552—36no—
64C2×D4×Q8—32no—
65C2×D4×Q8—31no—
66C2×#541—36no—
67C2×#533—33no—
68C2×#529—33no—
69C2×#519—32no—
70C22×C4×Q8—31no—
71C22×C3×Q8C2313yes—
72C22×C3×Q8C2311yescharacteristic
73C2×D3×Q8—28no—
74C2×D3×Q8—216no—
75C2×D3×Q8C2318yes—
76C2×D3×Q8C23116yes—
77C22×D12C2312yes—
78C22×D12C2311yescharacteristic
79C22×C4×D3C2313yes—
80C22×C4×D3C2316yes—
81C22×C4×D3C2311yescharacteristic
82C22×C4×D3C2313yes—
83C22×Dic6C2313yes—
84C22×Dic6C2316yes—
85C3×C4⋊Q8C23124yes—
86C3×#125C23148yes—
87C3×C4×Q8C23124yes—
88C3×C4×Q8C2318yes—
89C3×C4×Q8C23116yes—
90C2×C3×C4⋊C4C2313yes—
91C2×C3×C4⋊C4C2316yes—
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2316yes—
94C2×C3×C4⋊C4C2316yes—
95C2×C3×C42C2313yes—
96C2×C3×C42C2316yes—
97#947C23124yes—
98#946C23124yes—
99Q8×Dic3C23148yes—
100Q8×Dic3C2318yes—
101C2×(C2×D3)⋊C4C2312yes—
102C2×(C2×D3)⋊C4C2316yes—
103C2×(C2×D3)⋊C4C2316yes—
104C2×(C2×D3)⋊C4C2316yes—
105C2×(C2×D3)⋊C4C2316yes—
106C2×C4⋊Dic3C2313yes—
107C2×C4⋊Dic3C2316yes—
108C2×C4⋊Dic3C2313yes—
109C2×C4⋊Dic3C2316yes—
110C2×C4⋊Dic3C23112yes—
111C2×Dic3⋊C4C2316yes—
112C2×Dic3⋊C4C2316yes—
113C2×Dic3⋊C4C2316yes—
114C2×C4×Dic3C2313yes—
115C2×C4×Dic3C23112yes—
116C2×C4×Dic3C2313yes—
117C2×C4×Dic3C2313yes—
118#898C23196yes—
119#898C23148yes—
120#897C23148yes—
121#897C23148yes—
122#897C23148yes—
123#895C2318yes—
124#894C23148yes—
125#892C23148yes—
126#892C23148yes—
127D3×C4⋊C4C23124yes—
128#890C23196yes—
129#890C23148yes—
130#890C23124yes—
131#887C23124yes—
132#887C23148yes—
133C4×D12C23148yes—
134C4×D12C23124yes—
135#872C23148yes—
136C42×D3C23124yes—
137#869C23148yes—
138#869C23124yes—
139C4×Dic6C23148yes—
140C4×Dic6C23124yes—
141C23×Q8—31no—
142C23×Q8—32no—
143#557—348no—
144#557C2×D3116yescomplemented
145#556—348no—
146#552—396no—
147D4×Q8—332no—
148D4×Q8—316no—
149#541—396no—
150#533—348no—
151C2×C4⋊Q8—33no—
152C2×C4⋊Q8—36no—
153C2×C4⋊Q8C2×D313yes—
154#529—348no—
155C2×#126—36no—
156C2×#126—312no—
157C2×#125—33no—
158C2×#125—36no—
159C2×#125—312no—
160C2×#125C2×D316yes—
161C2×#124—33no—
162C2×#123—36no—
163C2×C22⋊Q8—33no—
164C2×C22⋊Q8—36no—
165C2×C22⋊Q8—36no—
166C2×C22⋊Q8—36no—
167C2×#121—31no—
168#519—332no—
169C2×C4×Q8—31no—
170C2×C4×Q8—36no—
171C2×C4×Q8—316no—
172C2×C4×Q8—33no—
173C2×C4×Q8—36no—
174C2×C4×Q8—33no—
175C2×C4×Q8—36no—
176C2×C4×Q8C2×D311yescharacteristic
177C2×C4×Q8C2×D312yes—
178C2×C4×Q8C2×D313yes—
179C2×C4×D4—36no—
180C2×C4×D4—33no—
181C2×#117—36no—
182C2×#117—36no—
183C2×#117—36no—
184C22×C4⋊C4—33no—
185C22×C42—33no—
186C23×D3C2412yes—
187C23×D3C2411yescharacteristic
188C2×C3×Q8—212no—
189C2×C3×Q8#4118yes—
190C2×C3×Q8C24112yes—
191C2×C3×Q8C2414yes—
192C22×C3×C4C2416yes—
193C22×C3×C4C2413yes—
194C22×C3×C4C2412yes—
195C22×C3×C4C2413yes—
196C22×C3×C4C2411yescharacteristic
197C22×Dic3C2411yescharacteristic
198C22×Dic3C2413yes—
199C22×Dic3C2413yes—
200C22×Dic3C2416yes—
201D3×Q8—232no—
202D3×Q8—264no—
203C2×D12—28no—
204C2×D12—24no—
205C2×D12C2418yes—
206C2×D12C2414yes—
207C2×C4×D3—212no—
208C2×C4×D3—212no—
209C2×C4×D3—224no—
210C2×C4×D3—24no—
211C2×C4×D3C2414yes—
212C2×C4×D3C24112yes—
213C2×C4×D3C24112yes—
214C2×C4×D3C24124yes—
215C2×Dic6—212no—
216C2×Dic6—224no—
217C2×Dic6C24124yes—
218C2×Dic6C24112yes—
219C3×C4⋊C4C24112yes—
220C3×C4⋊C4C24124yes—
221C3×C4⋊C4C2414yes—
222C3×C4⋊C4C24124yes—
223C3×C4⋊C4C24124yes—
224C3×C42C24112yes—
225C3×C42C24124yes—
226(C2×D3)⋊C4C2418yes—
227(C2×D3)⋊C4C24124yes—
228(C2×D3)⋊C4C24124yes—
229(C2×D3)⋊C4C24124yes—
230(C2×D3)⋊C4C24124yes—
231C4⋊Dic3C24112yes—
232C4⋊Dic3C24124yes—
233C4⋊Dic3C24112yes—
234C4⋊Dic3C24124yes—
235C4⋊Dic3C24148yes—
236Dic3⋊C4C24124yes—
237Dic3⋊C4C24124yes—
238Dic3⋊C4C24124yes—
239C4×Dic3C24112yes—
240C4×Dic3C24148yes—
241C4×Dic3C24112yes—
242C4×Dic3C24112yes—
243C22×Q8—38no—
244C22×Q8—316no—
245C22×Q8—33no—
246C22×Q8—36no—
247C22×Q8—68no—
248C22×Q8—616no—
249C22×Q8C22×D313yes—
250C22×Q8C22×D311yescharacteristic
251C22×D4—32no—
252C22×D4—31no—
253C23×C4—33no—
254C23×C4—33no—
255C23×C4—36no—
256C23×C4—31no—
257C4⋊Q8—324no—
258C4⋊Q8—348no—
259C4⋊Q8C22×D3124yes—
260#126—348no—
261#126—396no—
262#125—324no—
263#125—348no—
264#125—396no—
265#125C22×D3148yes—
266#124—324no—
267#123—348no—
268C22⋊Q8—324no—
269C22⋊Q8—348no—
270C22⋊Q8—348no—
271C22⋊Q8—348no—
272#121—38no—
273C4×Q8—38no—
274C4×Q8—348no—
275C4×Q8—324no—
276C4×Q8—348no—
277C4×Q8—324no—
278C4×Q8—348no—
279C4×Q8C22×D318yes—
280C4×Q8C22×D3116yes—
281C4×Q8C22×D3124yes—
282C4×D4—348no—
283C4×D4—324no—
284#117—348no—
285#117—348no—
286#117—348no—
287C2×C4⋊C4—33no—
288C2×C4⋊C4—36no—
289C2×C4⋊C4—36no—
290C2×C4⋊C4—36no—
291C2×C4⋊C4—36no—
292C2×C4⋊C4—33no—
293C2×C4⋊C4—36no—
294C2×C4⋊C4—312no—
295C2×C4⋊C4—324no—
296C2×C4⋊C4C22×D313yes—
297C2×C4⋊C4C22×D316yes—
298C2×C4⋊C4C22×D316yes—
299C2×C4⋊C4C22×D311yescharacteristic
300C2×C4⋊C4C22×D316yes—
301C2×C22⋊C4—32no—
302C2×C22⋊C4—36no—
303C2×C22⋊C4—36no—
304C2×C22⋊C4—36no—
305C2×C22⋊C4—36no—
306C2×C42—33no—
307C2×C42—312no—
308C2×C42—33no—
309C2×C42—33no—
310C2×C42—324no—
311C2×C42C22×D316yes—
312C2×C42C22×D313yes—
313C23×C3C2511yescharacteristic
314C22×D3—22no—
315C22×D3—22no—
316C22×D3—24no—
317C22×D3—24no—
318C22×D3—24no—
319C22×D32−1+414yes—
320C22×D3C2514yes—
321C22×D3C2512yes—
322C3×Q8—224no—
323C3×Q8C2×#41116yes—
324C2×C3×C4—26no—
325C2×C3×C4—24no—
326C2×C3×C4—212no—
327C2×C3×C4C2×#41112yes—
328C2×C3×C42−1+414yes—
329C2×C3×C4C25112yes—
330C2×C3×C4C2516yes—
331C2×C3×C4C2514yes—
332C2×C3×C4C2516yes—
333C2×C3×C4C2512yes—
334C2×Dic3—26no—
335C2×Dic3—212no—
336C2×Dic3—22no—
337C2×Dic3—26no—
338C2×Dic3C2512yes—
339C2×Dic3C2516yes—
340C2×Dic3C2516yes—
341C2×Dic3C25112yes—
342D12—216no—
343D12—28no—
344C4×D3—224no—
345C4×D3—224no—
346C4×D3—248no—
347C4×D3—28no—
348Dic6—248no—
349Dic6—224no—
350C24—31no—
351C24—32no—
352C2×Q8—212no—
353C2×Q8—312no—
354C2×Q8—324no—
355C2×Q8—632no—
356C2×Q8—664no—
357C2×Q8—612no—
358C2×Q8—624no—
359C2×Q8#23918yes—
360C2×Q8C23×D3112yes—
361C2×Q8C23×D314yes—
362C2×D4—38no—
363C2×D4—34no—
364C2×D4—68no—
365C2×D4—64no—
366C22×C4—31no—
367C22×C4—33no—
368C22×C4—36no—
369C22×C4—33no—
370C22×C4—34no—
371C22×C4—312no—
372C22×C4—312no—
373C22×C4—324no—
374C22×C4—612no—
375C22×C4—612no—
376C22×C4—624no—
377C22×C4—64no—
378C22×C4C23×D316yes—
379C22×C4C23×D313yes—
380C22×C4C23×D312yes—
381C22×C4C23×D313yes—
382C22×C4C23×D311yescharacteristic
383C4⋊C4—312no—
384C4⋊C4—324no—
385C4⋊C4—324no—
386C4⋊C4—324no—
387C4⋊C4—324no—
388C4⋊C4—312no—
389C4⋊C4—324no—
390C4⋊C4—348no—
391C4⋊C4C23×D3112yes—
392C4⋊C4C23×D3124yes—
393C4⋊C4C23×D3124yes—
394C4⋊C4C23×D314yes—
395C4⋊C4C23×D3124yes—
396C22⋊C4—38no—
397C22⋊C4—324no—
398C22⋊C4—324no—
399C22⋊C4—324no—
400C22⋊C4—324no—
401C42—312no—
402C42—312no—
403C42—312no—
404C42—348no—
405C42C23×D3112yes—
406C42C23×D3124yes—
407C22×C3C22×#4112yes—
408C22×C3C2×2−1+412yes—
409C22×C3C2×2−1+412yes—
410C22×C3C2611yescharacteristic
411C2×D3—22no—
412C2×D3—22no—
413C2×D3—28no—
414C2×D3—24no—
415C2×D3—24no—
416C2×D3—24no—
417C2×D3—48no—
418C2×D3C2×2−1+414yes—
419C3×C4—26no—
420C3×C4—24no—
421C3×C4—212no—
422C3×C4C22×#41112yes—
423C3×C4C2×2−1+414yes—
424Dic3—26no—
425Dic3—212no—
426Dic3—22no—
427Dic3—26no—
428C23—34no—
429C23—32no—
430C23—34no—
431C23—62no—
432C23—62no—
433C23—64no—
434C23—64no—
435C23—64no—
436C23C24×D311yescharacteristic
437Q8—224no—
438Q8—624no—
439Q8—648no—
440Q8C2×#239116yes—
441D4—616no—
442D4—68no—
443C2×C4—212no—
444C2×C4—26no—
445C2×C4—24no—
446C2×C4—32no—
447C2×C4—36no—
448C2×C4—36no—
449C2×C4—312no—
450C2×C4—624no—
451C2×C4—612no—
452C2×C4—66no—
453C2×C4—62no—
454C2×C4—66no—
455C2×C4—624no—
456C2×C4—648no—
457C2×C4—68no—
458C2×C4C2×#239112yes—
459C2×C4#101014yes—
460C2×C4C24×D3112yes—
461C2×C4C24×D316yes—
462C2×C4C24×D314yes—
463C2×C4C24×D316yes—
464C2×C4C24×D312yes—
465C2×C3#352314yescomplemented
466C2×C3C23×#4111yescharacteristic
467C2×C3C22×2−1+411yescharacteristic
468C2×C3C22×2−1+411yescharacteristic
469D3—24no—
470D3—44no—
471C22—34no—
472C22—62no—
473C22—62no—
474C22—68no—
475C22—64no—
476C22—64no—
477C22—64no—
478C22—128no—
479C22C22×#23912yes—
480C22C2×#101012yes—
481C22D3×2−1+412yes—
482C22C25×D311yescharacteristic
483C4—212no—
484C4—26no—
485C4—24no—
486C4—66no—
487C4—612no—
488C4—62no—
489C4—66no—
490C4C22×#239112yes—
491C4C2×#101014yes—
492C3C2×#352311yescharacteristic, complemented, Sylow
493C2—64no—
494C2—124no—
495C2#8702414yesdirect product factor
496C2C23×#23911yescharacteristic
497C2C22×#101011yescharacteristic
498C2C2×D3×2−1+411yescharacteristic
subgroupG/H
centerC23C24×D3
commutator, cosocleC22×C3C26
FrattiniC22C25×D3
FittingC2×C3×#557C2
socleC23×C3C25
prime radicalC2×#895C22
hypercenterC2×#557D3
hypocenterC3C2×#3523
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
1. C2×C3×#557C2semidirect
2. #87024C2direct (N × H)
3. C2×#5732C2semidirect
4. C2×#5728C2semidirect
5. C2×#5694C2semidirect
6. C2×#5698C2semidirect
7. C2×Q8×D12C2semidirect
8. C2×Q8×D12C2semidirect
10. C2×#5572C2semidirect
11. C2×C4×D3×Q8C2semidirect
12. C2×#5570C2semidirect
16. C3×#557C22semidirect
26. #5732C22semidirect
27. #5728C22semidirect
28. #5698C22semidirect
29. #5694C22semidirect
31. Q8×D12C22semidirect
32. Q8×D12C22semidirect
33. #5572C22semidirect
34. C4×D3×Q8C22semidirect
35. #5570C22semidirect
61. C2×#557D3semidirect
144. #557C2×D3semidirect
465. C2×C3#3523semidirect
492. C3C2×#3523semidirect
495. C2#87024direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5572
#870242
C2×#57322
C2×#57282
C2×#56942
C2×#56982
C2×Q8×D122
C2×#55802
C2×#55722
C2×C4×D3×Q82
C2×#55702
C2×#35233