Finite Group Databasebeta

← C2×#87023 #11703446

Subgroups — C2×#87023 #11703446

Order 768.

7854 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#557C211yesmaximal, characteristic, complemented
2#87023C2132yesmaximal, direct product factor
3C2×#5726C213yesmaximal, complemented
4C2×#5725C213yesmaximal, complemented
5C2×#5695C216yesmaximal, complemented
6C2×#5697C216yesmaximal, complemented
7C2×#5580C213yesmaximal, complemented
8C2×#5577C212yesmaximal, complemented
9C2×#5577C211yesmaximal, characteristic, complemented
10C2×#5577C213yesmaximal, complemented
11C2×#5574C212yesmaximal, complemented
12C2×C4×#239C211yesmaximal, characteristic, complemented
13C2×#3517—31nomaximal, Sylow
14C22×#239C2211yescharacteristic
15C22×#239C2212yes—
16C22×#239C2216yes—
17C3×#557C22116yescomplemented
18C2×C3×C4⋊Q8C2213yes—
19C2×C3×#125C2216yes—
20C2×C3×C4×Q8C2211yescharacteristic
21C2×C3×C4×Q8C2212yes—
22C2×C3×C4×Q8C2213yes—
23C2×#947C2213yes—
24C2×#946C2213yes—
25C2×Q8×Dic3C2211yescharacteristic
26#5726C22148yescomplemented
27#5725C22148yescomplemented
28#5697C22196yescomplemented
29#5695C22196yescomplemented
30#5580C22148yescomplemented
31#5577C22148yescomplemented
32#5577C22132yescomplemented
33#5577C22116yescomplemented
34#5574C22132yescomplemented
35C4×#239C22116yescomplemented
36C2×#898C2216yes—
37C2×#895C2213yes—
38C2×#895C22112yes—
39C2×#895C2216yes—
40C2×#895C2216yes—
41C2×#895C2216yes—
42C2×#895C2216yes—
43C2×#894C2216yes—
44C2×#894C22112yes—
45C2×#893C2213yes—
46C2×#893C2216yes—
47C2×#893C2216yes—
48C2×#892C2213yes—
49C2×D3×C4⋊C4C2216yes—
50C2×#890C2211yescharacteristic
51C2×#874C2216yes—
52C2×#874C2213yes—
53C2×C4×D12C2213yes—
54C2×C4×D12C2216yes—
55C2×C4×D12C2216yes—
56C2×C4×D12C2213yes—
57C2×#872C2216yes—
58C2×C42×D3C2213yes—
59#3517—332no—
60C2×#557D311yescharacteristic, complemented
61C2×#553—36no—
62C2×#550—33no—
63C2×#550—31no—
64C2×#550—32no—
65C2×#547—33no—
66C2×#540—36no—
67C2×#532—33no—
68C2×#529—33no—
69C2×#520—32no—
70C2×C4×#41—31no—
71C22×C3×Q8C2313yes—
72C22×C3×Q8C2311yescharacteristic
73C2×#239—28no—
74C2×#239—216no—
75C2×#239—424no—
76C2×#239C2318yes—
77C2×#239C23116yes—
78C2×#239C23148yes—
79C22×D12C2313yes—
80C22×D12C2313yes—
81C22×D12C2316yes—
82C22×D12C23112yes—
83C22×D12C2316yes—
84C22×D12C2316yes—
85C22×C4×D3C2313yes—
86C22×C4×D3C2313yes—
87C22×C4×D3C2316yes—
88C22×C4×D3C23112yes—
89C22×C4×D3C2313yes—
90C3×C4⋊Q8C23124yes—
91C3×#125C23148yes—
92C3×C4×Q8C2318yes—
93C3×C4×Q8C23116yes—
94C3×C4×Q8C23124yes—
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C4⋊C4C2316yes—
97C2×C3×C4⋊C4C2313yes—
98C2×C3×C4⋊C4C2316yes—
99C2×C3×C4⋊C4C2316yes—
100C2×C3×C42C2313yes—
101C2×C3×C42C2316yes—
102#947C23124yes—
103#946C23124yes—
104Q8×Dic3C2318yes—
105C2×(C2×D3)⋊C4C2316yes—
106C2×(C2×D3)⋊C4C2316yes—
107C2×(C2×D3)⋊C4C2316yes—
108C2×(C2×D3)⋊C4C2316yes—
109C2×(C2×D3)⋊C4C2316yes—
110C2×(C2×D3)⋊C4C2316yes—
111C2×C4⋊Dic3C2312yes—
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2313yes—
114C2×Dic3⋊C4C2312yes—
115C2×Dic3⋊C4C2316yes—
116C2×C4×Dic3C2311yescharacteristic
117C2×C4×Dic3C2313yes—
118C2×C4×Dic3C2313yes—
119#898C23148yes—
120#895C23196yes—
121#895C23148yes—
122#895C23148yes—
123#895C23148yes—
124#895C23148yes—
125#895C23124yes—
126#894C23196yes—
127#894C23148yes—
128#893C23148yes—
129#893C23124yes—
130#893C23148yes—
131#892C23124yes—
132D3×C4⋊C4C23148yes—
133#890C2318yes—
134#874C23148yes—
135#874C23124yes—
136C4×D12C23148yes—
137C4×D12C23148yes—
138C4×D12C23124yes—
139C4×D12C23124yes—
140#872C23148yes—
141C42×D3C23124yes—
142C22×#41—31no—
143C22×#41—32no—
144C22×#41—36no—
145#557C2×D3116yescomplemented
146#553—396no—
147#550—348no—
148#550—316no—
149#550—332no—
150#547—348no—
151#540—396no—
152#532—348no—
153C2×C4⋊Q8C2×D313yes—
154C2×#127—33no—
155C2×#127—36no—
156#529—348no—
157C2×#126—36no—
158C2×#125—31no—
159C2×#125C2×D316yes—
160C2×#124—33no—
161C2×#123—36no—
162C2×#123—312no—
163C2×C22⋊Q8—33no—
164C2×#121—33no—
165C2×#121—36no—
166C2×#121—36no—
167C2×#121—36no—
168C2×#121—312no—
169C2×#121—36no—
170#520—332no—
171C4×#41—316no—
172C2×C4×Q8—31no—
173C2×C4×Q8C2×D311yescharacteristic
174C2×C4×Q8C2×D312yes—
175C2×C4×Q8C2×D313yes—
176C2×C4×D4—33no—
177C2×C4×D4—36no—
178C2×C4×D4—36no—
179C2×C4×D4—33no—
180C2×C4×D4—36no—
181C2×C4×D4—33no—
182C2×C4×D4—36no—
183C2×#117—33no—
184C2×#117—36no—
185C22×C4⋊C4—36no—
186C22×C42—33no—
187C23×D3C2413yes—
188C23×D3C2416yes—
189C2×C3×Q8—212no—
190C2×C3×Q8#4118yescomplemented
191C2×C3×Q8C24112yes—
192C2×C3×Q8C2414yes—
193C22×C3×C4C2413yes—
194C22×C3×C4C2412yes—
195C22×C3×C4C2416yes—
196C22×C3×C4C2413yes—
197C22×C3×C4C2411yescharacteristic
198C22×Dic3C2411yescharacteristic
199C22×Dic3C2411yescharacteristic
200C22×Dic3C2413yes—
201C22×Dic3C2412yes—
202#239—232no—
203#239—264no—
204#239—496no—
205C2×D12—212no—
206C2×D12—224no—
207C2×D12—212no—
208C2×D12—412no—
209C2×D12—412no—
210C2×D12—424no—
211C2×D12C24124yes—
212C2×D12C24112yes—
213C2×D12C24112yes—
214C2×D12C24124yes—
215C2×D12C24124yes—
216C2×D12C24148yes—
217C2×C4×D3—212no—
218C2×C4×D3—224no—
219C2×C4×D3—212no—
220C2×C4×D3—46no—
221C2×C4×D3—424no—
222C2×C4×D3C24112yes—
223C2×C4×D3C24112yes—
224C2×C4×D3C24124yes—
225C2×C4×D3C24112yes—
226C2×C4×D3C24148yes—
227C3×C4⋊C4C2414yes—
228C3×C4⋊C4C24124yes—
229C3×C4⋊C4C24112yes—
230C3×C4⋊C4C24124yes—
231C3×C4⋊C4C24124yes—
232C3×C42C24112yes—
233C3×C42C24124yes—
234(C2×D3)⋊C4C24124yes—
235(C2×D3)⋊C4C24124yes—
236(C2×D3)⋊C4C24124yes—
237(C2×D3)⋊C4C24124yes—
238(C2×D3)⋊C4C24124yes—
239(C2×D3)⋊C4C24124yes—
240C4⋊Dic3C2418yes—
241C4⋊Dic3C2414yes—
242C4⋊Dic3C24112yes—
243Dic3⋊C4C2418yes—
244Dic3⋊C4C24124yes—
245C4×Dic3C2414yes—
246C4×Dic3C24112yes—
247C4×Dic3C24112yes—
248C2×#41—38no—
249C2×#41—316no—
250C2×#41—348no—
251C2×#41—68no—
252C2×#41—616no—
253C2×#41—1224no—
254C22×Q8C22×D313yes—
255C22×Q8C22×D311yescharacteristic
256C22×D4—33no—
257C22×D4—36no—
258C22×D4—33no—
259C22×D4—36no—
260C22×D4—36no—
261C22×D4—312no—
262C23×C4—36no—
263C23×C4—33no—
264C23×C4—33no—
265C23×C4—312no—
266C23×C4—33no—
267C4⋊Q8C22×D3124yes—
268#127—324no—
269#127—348no—
270#126—348no—
271#125—38no—
272#125C22×D3148yes—
273#124—324no—
274#123—348no—
275#123—396no—
276C22⋊Q8—324no—
277#121—324no—
278#121—348no—
279#121—348no—
280#121—348no—
281#121—396no—
282#121—348no—
283C4×Q8—38no—
284C4×Q8C22×D318yes—
285C4×Q8C22×D3116yes—
286C4×Q8C22×D3124yes—
287C4×D4—324no—
288C4×D4—348no—
289C4×D4—348no—
290C4×D4—324no—
291C4×D4—348no—
292C4×D4—324no—
293C4×D4—348no—
294#117—324no—
295#117—348no—
296C2×C4⋊C4—33no—
297C2×C4⋊C4—36no—
298C2×C4⋊C4—32no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—32no—
301C2×C4⋊C4—348no—
302C2×C4⋊C4C22×D316yes—
303C2×C4⋊C4C22×D313yes—
304C2×C4⋊C4C22×D316yes—
305C2×C4⋊C4C22×D311yescharacteristic
306C2×C4⋊C4C22×D316yes—
307C2×C22⋊C4—36no—
308C2×C22⋊C4—36no—
309C2×C22⋊C4—36no—
310C2×C22⋊C4—36no—
311C2×C22⋊C4—36no—
312C2×C22⋊C4—36no—
313C2×C42—33no—
314C2×C42—33no—
315C2×C42—31no—
316C2×C42—324no—
317C2×C42C22×D316yes—
318C2×C42C22×D313yes—
319C23×C3C2511yescharacteristic
320C22×D3—26no—
321C22×D3—212no—
322C22×D3—26no—
323C22×D3—212no—
324C22×D3—26no—
325C22×D3—212no—
326C22×D3C2516yes—
327C22×D3C25112yes—
328C3×Q8—224no—
329C3×Q8C2×#41116yescomplemented
330C2×C3×C4—212no—
331C2×C3×C4—26no—
332C2×C3×C4—24no—
333C2×C3×C4C2×#41112yes—
334C2×C3×C4H(5,𝔽2)14yes—
335C2×C3×C4C2516yes—
336C2×C3×C4C2514yes—
337C2×C3×C4C25112yes—
338C2×C3×C4C2516yes—
339C2×C3×C4C2512yes—
340C2×Dic3—22no—
341C2×Dic3—24no—
342C2×Dic3—26no—
343C2×Dic3—22no—
344C2×Dic3C2512yes—
345C2×Dic3C2512yes—
346C2×Dic3C2516yes—
347C2×Dic3C2514yes—
348D12—248no—
349D12—224no—
350D12—224no—
351D12—424no—
352D12—424no—
353D12—448no—
354C4×D3—224no—
355C4×D3—248no—
356C4×D3—224no—
357C4×D3—412no—
358C4×D3—448no—
359C24—33no—
360C24—36no—
361#41—632no—
362#41—664no—
363#41—1296no—
364C2×Q8—212no—
365C2×Q8#23918yescomplemented
366C2×Q8C23×D3112yes—
367C2×Q8C23×D314yes—
368C2×D4—312no—
369C2×D4—324no—
370C2×D4—312no—
371C2×D4—324no—
372C2×D4—324no—
373C2×D4—348no—
374C2×D4—612no—
375C2×D4—624no—
376C2×D4—612no—
377C2×D4—1224no—
378C2×D4—1212no—
379C2×D4—1212no—
380C22×C4—31no—
381C22×C4—31no—
382C22×C4—33no—
383C22×C4—32no—
384C22×C4—312no—
385C22×C4—312no—
386C22×C4—324no—
387C22×C4—312no—
388C22×C4—348no—
389C22×C4—624no—
390C22×C4—612no—
391C22×C4—612no—
392C22×C4—1224no—
393C22×C4—126no—
394C22×C4C23×D313yes—
395C22×C4C23×D312yes—
396C22×C4C23×D316yes—
397C22×C4C23×D313yes—
398C22×C4C23×D311yescharacteristic
399C4⋊C4—312no—
400C4⋊C4—324no—
401C4⋊C4—38no—
402C4⋊C4—34no—
403C4⋊C4—38no—
404C4⋊C4C23×D3124yes—
405C4⋊C4C23×D3112yes—
406C4⋊C4C23×D3124yes—
407C4⋊C4C23×D314yes—
408C4⋊C4C23×D3124yes—
409C22⋊C4—324no—
410C22⋊C4—324no—
411C22⋊C4—324no—
412C22⋊C4—324no—
413C22⋊C4—324no—
414C22⋊C4—324no—
415C42—34no—
416C42—312no—
417C42—312no—
418C42C23×D3112yes—
419C42C23×D3124yes—
420C22×C3C22×#4112yes—
421C22×C3C2×H(5,𝔽2)12yes—
422C22×C3#58512yes—
423C22×C3C2611yescharacteristic
424C2×D3—26no—
425C2×D3—212no—
426C2×D3—26no—
427C2×D3—212no—
428C2×D3—26no—
429C2×D3—212no—
430C2×D3—412no—
431C2×D3—424no—
432C3×C4—212no—
433C3×C4—26no—
434C3×C4—24no—
435C3×C4C22×#41112yes—
436C3×C4C2×H(5,𝔽2)14yes—
437Dic3—22no—
438Dic3—24no—
439Dic3—26no—
440Dic3—22no—
441C23—36no—
442C23—312no—
443C23—66no—
444C23—612no—
445C23—66no—
446C23—612no—
447C23—66no—
448C23—612no—
449C23C24×D311yescharacteristic
450Q8—224no—
451Q8C2×#239116yescomplemented
452D4—624no—
453D4—648no—
454D4—624no—
455D4—1248no—
456D4—1224no—
457D4—1224no—
458C2×C4—212no—
459C2×C4—26no—
460C2×C4—24no—
461C2×C4—32no—
462C2×C4—32no—
463C2×C4—36no—
464C2×C4—34no—
465C2×C4—62no—
466C2×C4—64no—
467C2×C4—66no—
468C2×C4—62no—
469C2×C4—624no—
470C2×C4—648no—
471C2×C4—624no—
472C2×C4—1248no—
473C2×C4—1212no—
474C2×C4C2×#239112yes—
475C2×C4#100914yes—
476C2×C4C24×D316yes—
477C2×C4C24×D314yes—
478C2×C4C24×D3112yes—
479C2×C4C24×D316yes—
480C2×C4C24×D312yes—
481C2×C3#351714yescomplemented
482C2×C3C23×#4111yescharacteristic
483C2×C3C22×H(5,𝔽2)11yescharacteristic
484C2×C3C2×#58511yescharacteristic
485D3—46no—
486D3—412no—
487C22—66no—
488C22—612no—
489C22—66no—
490C22—612no—
491C22—66no—
492C22—612no—
493C22—1212no—
494C22—1224no—
495C22C22×#23912yes—
496C22C2×#100912yes—
497C22#596812yes—
498C22C25×D311yescharacteristic
499C4—212no—
500C4—26no—
501C4—24no—
502C4—62no—
503C4—64no—
504C4—66no—
505C4—62no—
506C4C22×#239112yes—
507C4C2×#100914yes—
508C3C2×#351711yescharacteristic, complemented, Sylow
509C2—126no—
510C2—1212no—
511C2#8702314yesdirect product factor
512C2C23×#23911yescharacteristic
513C2C22×#100911yescharacteristic
514C2C2×#596811yescharacteristic
subgroupG/H
centerC23C24×D3
commutator, cosocleC22×C3C26
FrattiniC22C25×D3
FittingC2×C3×#557C2
socleC23×C3C25
hypercenterC2×#557D3
hypocenterC3C2×#3517
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. #87023C2direct (N × H)
3. C2×#5726C2semidirect
4. C2×#5725C2semidirect
5. C2×#5695C2semidirect
6. C2×#5697C2semidirect
7. C2×#5580C2semidirect
8. C2×#5577C2semidirect
9. C2×#5577C2semidirect
10. C2×#5577C2semidirect
11. C2×#5574C2semidirect
12. C2×C4×#239C2semidirect
17. C3×#557C22semidirect
26. #5726C22semidirect
27. #5725C22semidirect
28. #5697C22semidirect
29. #5695C22semidirect
30. #5580C22semidirect
31. #5577C22semidirect
32. #5577C22semidirect
33. #5577C22semidirect
34. #5574C22semidirect
35. C4×#239C22semidirect
60. C2×#557D3semidirect
145. #557C2×D3semidirect
190. C2×C3×Q8#41semidirect
329. C3×Q8C2×#41semidirect
365. C2×Q8#239semidirect
451. Q8C2×#239semidirect
481. C2×C3#3517semidirect
508. C3C2×#3517semidirect
511. C2#87023direct (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
#870232
C2×#57262
C2×#57252
C2×#56952
C2×#56972
C2×#55802
C2×#55772
C2×#55742
C2×C4×#2392
C2×#35173