Finite Group Databasebeta

← C2×#86760 #11703359

Subgroups — C2×#86760 #11703359

Order 768.

11210 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#1007C212yesmaximal, complemented
2C2×C3×D4×Q8C211yesmaximal, characteristic, complemented
3C2×#5817C212yesmaximal, complemented
4#86760C2132yesmaximal, direct product factor
5C2×#5727C213yesmaximal, complemented
6C2×#5725C213yesmaximal, complemented
7C2×#5633C216yesmaximal, complemented
8C2×#5629C216yesmaximal, complemented
9C2×C4×D3×Q8C211yesmaximal, characteristic, complemented
10C2×#5577C211yesmaximal, characteristic, complemented
11C2×#5548C213yesmaximal, complemented
12C2×#5553C213yesmaximal, complemented
13C2×#3497—31nomaximal, Sylow
14C23×C3×Q8C2212yes—
15C2×#1007—232no—
16C2×#1007C22132yescomplemented
17C22×#239C2212yes—
18C22×#239C2216yes—
19C22×D3×Q8C2216yes—
20C22×D3×Q8C2211yescharacteristic
21C22×#235C2216yes—
22C22×#235C2216yes—
23C3×D4×Q8C22116yescomplemented
24C2×C3×C4⋊Q8C2213yes—
25C2×C3×C22⋊Q8C2216yes—
26C2×C3×C4×Q8C2211yescharacteristic
27C2×C3×C4×D4C2213yes—
28#5817C22132yescomplemented
29C2×#947C2216yes—
30C2×#946C2216yes—
31C2×Q8×Dic3C2211yescharacteristic
32C2×#940C2213yes—
33C2×#938C2211yescharacteristic
34C2×#930C2216yes—
35C2×C4×#67C2216yes—
36#5727C22148yescomplemented
37#5725C22148yescomplemented
38#5633C22196yescomplemented
39#5629C22196yescomplemented
40#5577C22116yescomplemented
41C4×D3×Q8C22116yescomplemented
42#5553C22148yescomplemented
43#5548C22148yescomplemented
44C2×#896C22112yes—
45C2×#895C2213yes—
46C2×#895C2216yes—
47C2×#895C2216yes—
48C2×#893C2216yes—
49C2×#893C2216yes—
50C2×D3×C4⋊C4C2213yes—
51C2×#888C2213yes—
52C2×#887C2213yes—
53C2×#885C2216yes—
54C2×#884C2216yes—
55C2×#884C22112yes—
56C2×#874C2213yes—
57C2×C4×D12C2213yes—
58C2×C42×D3C2213yes—
59C2×C4×Dic6C2213yes—
60C22×2−1+4—32no—
61#3497—332no—
62C2×#550—31no—
63C2×#550—33no—
64C2×D4×Q8—33no—
65C2×D4×Q8D311yescharacteristic, complemented
66C2×#548—32no—
67C2×#548—36no—
68C2×#537—36no—
69C2×#532—33no—
70C2×#532—33no—
71C22×C4×Q8—31no—
72C22×C3×Q8—216no—
73C22×C3×Q8C2316yes—
74C22×C3×Q8C23116yes—
75C22×C3×Q8C2311yescharacteristic
76C22×C3×D4C2311yescharacteristic
77C23×C3×C4C2316yes—
78C22×#67C2312yes—
79C22×#67C2316yes—
80C22×#67C2316yes—
81#1007—2256no—
82C2×#239—216no—
83C2×#239—248no—
84C2×#239C23148yes—
85C2×#239C23116yes—
86C2×D3×Q8—248no—
87C2×D3×Q8D4116yescomplemented
88C2×D3×Q8C23148yes—
89C2×D3×Q8C2318yes—
90C2×#235—248no—
91C2×#235—248no—
92C2×#235C23148yes—
93C2×#235C23148yes—
94C22×D12C2316yes—
95C22×D12C2316yes—
96C22×D12C2313yes—
97C22×D12C2316yes—
98C22×C4×D3C2316yes—
99C22×C4×D3C23112yes—
100C22×C4×D3C2311yescharacteristic
101C22×C4×D3C2313yes—
102C22×C4×D3C2313yes—
103C22×Dic6C2316yes—
104C22×Dic6C2313yes—
105C3×C4⋊Q8C23124yes—
106C3×C22⋊Q8C23148yes—
107C3×C4×Q8C2318yes—
108C3×C4×D4C23124yes—
109C2×C3×C4⋊C4C2313yes—
110C2×C3×C4⋊C4C2313yes—
111C2×C3×C4⋊C4C2316yes—
112C2×C3×C22⋊C4C2316yes—
113C2×C3×C42C2313yes—
114C2×#217C2312yes—
115#947C23148yes—
116#946C23148yes—
117Q8×Dic3C2318yes—
118#940C23124yes—
119#938C2318yes—
120#930C23148yes—
121C4×#67C23148yes—
122C2×(C2×D3)⋊C4C2316yes—
123C2×(C2×D3)⋊C4C23112yes—
124C2×(C2×D3)⋊C4C2316yes—
125C2×C4⋊Dic3C2313yes—
126C2×C4⋊Dic3C2311yescharacteristic
127C2×Dic3⋊C4C2316yes—
128C2×Dic3⋊C4C2316yes—
129C2×C4×Dic3C2313yes—
130C2×C4×Dic3C2313yes—
131C2×C4×Dic3C2313yes—
132#896C23196yes—
133#895C23148yes—
134#895C23124yes—
135#895C23148yes—
136#893C23148yes—
137#893C23148yes—
138D3×C4⋊C4C23124yes—
139#888C23124yes—
140#887C23124yes—
141#885C23148yes—
142#884C23196yes—
143#884C23148yes—
144#874C23124yes—
145C4×D12C23124yes—
146C42×D3C23124yes—
147C4×Dic6C23124yes—
148C2×2−1+4—332no—
149C2×2−1+4—632no—
150C22×#41—36no—
151C22×#41—32no—
152C22×#41—36no—
153C22×#41—36no—
154C23×Q8—31no—
155C23×Q8—36no—
156C23×Q8C2×D312yes—
157#550—316no—
158#550—348no—
159D4×Q8—348no—
160D4×Q8C2×D3116yescomplemented
161#548—332no—
162#548—396no—
163#537—396no—
164#532—348no—
165#532—348no—
166C2×C4⋊Q8—33no—
167C2×C4⋊Q8C2×D313yes—
168C2×#127—33no—
169C2×#127—33no—
170C2×#124—36no—
171C2×#124—36no—
172C2×C22⋊Q8—36no—
173C2×C22⋊Q8—312no—
174C2×C22⋊Q8C2×D316yes—
175C2×#121—31no—
176C2×#121—36no—
177C2×#121—33no—
178C2×#121—36no—
179C2×#121—36no—
180C2×#121—36no—
181C2×#121—312no—
182C2×C4×Q8—31no—
183C2×C4×Q8—316no—
184C2×C4×Q8—33no—
185C2×C4×Q8—33no—
186C2×C4×Q8C2×D311yescharacteristic
187C2×C4×D4—36no—
188C2×C4×D4—36no—
189C2×C4×D4—36no—
190C2×C4×D4—33no—
191C2×C4×D4C2×D313yes—
192C22×C4⋊C4—33no—
193C22×C42—33no—
194C24×C3C2412yes—
195C23×D3C2416yes—
196C23×D3C2411yescharacteristic
197C2×C3×Q8—264no—
198C2×C3×Q8—224no—
199C2×C3×Q8C2×D418yes—
200C2×C3×Q8C24124yes—
201C2×C3×Q8C2414yes—
202C2×C3×D4—24no—
203C2×C3×D4C2414yes—
204C22×C3×C4—224no—
205C22×C3×C4C2413yes—
206C22×C3×C4C2416yes—
207C22×C3×C4C24124yes—
208C22×C3×C4C2413yes—
209C22×C3×C4C2411yescharacteristic
210C2×#67—28no—
211C2×#67—224no—
212C2×#67—412no—
213C2×#67C2418yes—
214C2×#67C24124yes—
215C2×#67C24124yes—
216C22×Dic3C2411yescharacteristic
217C22×Dic3C2413yes—
218C22×Dic3C2412yes—
219C22×Dic3C2413yes—
220#239—264no—
221#239—2192no—
222D3×Q8—2192no—
223D3×Q8C2×D4164yescomplemented
224#235—2192no—
225#235—2192no—
226C2×D12—224no—
227C2×D12—224no—
228C2×D12—212no—
229C2×D12—412no—
230C2×D12C24112yes—
231C2×D12C24124yes—
232C2×D12C24124yes—
233C2×D12C24124yes—
234C2×C4×D3—224no—
235C2×C4×D3—24no—
236C2×C4×D3—248no—
237C2×C4×D3—46no—
238C2×C4×D3C2×D4124yes—
239C2×C4×D3C2414yes—
240C2×C4×D3C24124yes—
241C2×C4×D3C24148yes—
242C2×C4×D3C24112yes—
243C2×C4×D3C24112yes—
244C2×Dic6—224no—
245C2×Dic6C2×D4124yes—
246C2×Dic6C24124yes—
247C2×Dic6C24112yes—
248C3×C4⋊C4C24112yes—
249C3×C4⋊C4C24112yes—
250C3×C4⋊C4C24124yes—
251C3×C22⋊C4C24124yes—
252C3×C42C24112yes—
253#217C2418yes—
254(C2×D3)⋊C4C24124yes—
255(C2×D3)⋊C4C24148yes—
256(C2×D3)⋊C4C24124yes—
257C4⋊Dic3C24112yes—
258C4⋊Dic3C2414yes—
259Dic3⋊C4C24124yes—
260Dic3⋊C4C24124yes—
261C4×Dic3C24112yes—
262C4×Dic3C24112yes—
263C4×Dic3C24112yes—
2642−1+4—6256no—
265C2×#41—348no—
266C2×#41—316no—
267C2×#41—348no—
268C2×#41—348no—
269C2×#41—648no—
270C2×#41—648no—
271C2×#41—648no—
272C2×#41—616no—
273C22×Q8—216no—
274C22×Q8—316no—
275C22×Q8—38no—
276C22×Q8—348no—
277C22×Q8—33no—
278C22×Q8—36no—
279C22×Q8—648no—
280C22×Q8C22×D316yes—
281C22×Q8C22×D3116yes—
282C22×Q8C22×D311yescharacteristic
283C22×D4—32no—
284C22×D4—36no—
285C22×D4—36no—
286C22×D4—33no—
287C22×D4—36no—
288C22×D4—36no—
289C22×D4—36no—
290C22×D4C22×D311yescharacteristic
291C23×C4—33no—
292C23×C4—312no—
293C23×C4—36no—
294C23×C4—31no—
295C23×C4—33no—
296C23×C4C22×D316yes—
297C4⋊Q8—324no—
298C4⋊Q8C22×D3124yes—
299#127—324no—
300#127—324no—
301#124—348no—
302#124—348no—
303C22⋊Q8—348no—
304C22⋊Q8—396no—
305C22⋊Q8C22×D3148yes—
306#121—38no—
307#121—348no—
308#121—324no—
309#121—348no—
310#121—348no—
311#121—348no—
312#121—396no—
313C4×Q8—38no—
314C4×Q8—324no—
315C4×Q8—324no—
316C4×Q8C22×D318yes—
317C4×D4—348no—
318C4×D4—348no—
319C4×D4—348no—
320C4×D4—324no—
321C4×D4C22×D3124yes—
322C2×C4⋊C4—36no—
323C2×C4⋊C4—33no—
324C2×C4⋊C4—31no—
325C2×C4⋊C4—36no—
326C2×C4⋊C4—324no—
327C2×C4⋊C4C22×D313yes—
328C2×C4⋊C4C22×D313yes—
329C2×C4⋊C4C22×D316yes—
330C2×C22⋊C4—32no—
331C2×C22⋊C4—36no—
332C2×C22⋊C4—312no—
333C2×C22⋊C4—36no—
334C2×C22⋊C4C22×D316yes—
335C2×C42—33no—
336C2×C42—33no—
337C2×C42—33no—
338C2×C42—324no—
339C2×C42C22×D313yes—
340C23×C3—24no—
341C23×C3—24no—
342C23×C3—24no—
343C23×C3C2514yes—
344C23×C3C2511yescharacteristic
345C22×D3—22no—
346C22×D3—22no—
347C22×D3—212no—
348C22×D3—212no—
349C22×D3—212no—
350C22×D3C22×D414yes—
351C22×D3C25112yes—
352C22×D3C2512yes—
353C3×Q8—248no—
354C3×Q8C22×D4116yes—
355C3×D4—28no—
356C2×C3×C4—212no—
357C2×C3×C4—26no—
358C2×C3×C4—248no—
359C2×C3×C4C22×D4112yes—
360C2×C3×C42−1+414yescomplemented
361C2×C3×C4C2516yes—
362C2×C3×C4C25112yes—
363C2×C3×C4C2516yes—
364C2×C3×C4C2512yes—
365#67—216no—
366#67—248no—
367#67—424no—
368C2×Dic3—24no—
369C2×Dic3—22no—
370C2×Dic3—26no—
371C2×Dic3C22×D4112yes—
372C2×Dic3C2512yes—
373C2×Dic3C2516yes—
374C2×Dic3C2514yes—
375C2×Dic3C2516yes—
376D12—248no—
377D12—248no—
378D12—224no—
379D12—424no—
380C4×D3—248no—
381C4×D3—28no—
382C4×D3—296no—
383C4×D3—412no—
384C4×D3C22×D4148yes—
385Dic6—248no—
386Dic6C22×D4148yes—
387C24—31no—
388C24—36no—
389C24C23×D312yes—
390#41—6192no—
391#41—6192no—
392#41—6192no—
393#41—664no—
394C2×Q8—264no—
395C2×Q8—224no—
396C2×Q8—364no—
397C2×Q8—324no—
398C2×Q8—312no—
399C2×Q8—324no—
400C2×Q8—624no—
401C2×Q8—6192no—
402C2×Q8D3×D418yes—
403C2×Q8C23×D3124yes—
404C2×Q8C23×D314yes—
405C2×D4—24no—
406C2×D4—38no—
407C2×D4—324no—
408C2×D4—324no—
409C2×D4—312no—
410C2×D4—324no—
411C2×D4—324no—
412C2×D4—324no—
413C2×D4—624no—
414C2×D4—624no—
415C2×D4—68no—
416C2×D4—624no—
417C2×D4—612no—
418C2×D4—1212no—
419C2×D4—1212no—
420C2×D4C23×D314yes—
421C22×C4—224no—
422C22×C4—33no—
423C22×C4—324no—
424C22×C4—312no—
425C22×C4—31no—
426C22×C4—33no—
427C22×C4—32no—
428C22×C4—34no—
429C22×C4—348no—
430C22×C4—324no—
431C22×C4—312no—
432C22×C4—648no—
433C22×C4—624no—
434C22×C4—64no—
435C22×C4—126no—
436C22×C4C23×D3124yes—
437C22×C4C23×D313yes—
438C22×C4C23×D316yes—
439C22×C4C23×D313yes—
440C22×C4C23×D311yescharacteristic
441C4⋊C4—324no—
442C4⋊C4—312no—
443C4⋊C4—34no—
444C4⋊C4—324no—
445C4⋊C4C23×D3112yes—
446C4⋊C4C23×D3112yes—
447C4⋊C4C23×D3124yes—
448C22⋊C4—38no—
449C22⋊C4—324no—
450C22⋊C4—348no—
451C22⋊C4—324no—
452C22⋊C4C23×D3124yes—
453C42—312no—
454C42—312no—
455C42—312no—
456C42C23×D3112yes—
457C22×C3—24no—
458C22×C3—24no—
459C22×C3—24no—
460C22×C3—48no—
461C22×C3C23×D412yes—
462C22×C3C2×2−1+412yes—
463C22×C3#58512yes—
464C22×C3C2611yescharacteristic
465C2×D3—22no—
466C2×D3—22no—
467C2×D3—28no—
468C2×D3—212no—
469C2×D3—212no—
470C2×D3—212no—
471C2×D3—424no—
472C2×D3C23×D414yes—
473C3×C4—212no—
474C3×C4—26no—
475C3×C4C23×D4112yes—
476C3×C4C2×2−1+414yescomplemented
477Dic3—24no—
478Dic3—22no—
479Dic3—26no—
480Dic3C23×D4112yes—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—34no—
485C23—32no—
486C23—312no—
487C23—62no—
488C23—62no—
489C23—612no—
490C23—612no—
491C23—612no—
492C23C24×D314yes—
493C23C24×D311yescharacteristic
494Q8—248no—
495Q8—348no—
496Q8—648no—
497Q8C2×D3×D4116yes—
498D4—28no—
499D4—648no—
500D4—648no—
501D4—616no—
502D4—648no—
503D4—624no—
504D4—1224no—
505D4—1224no—
506C2×C4—212no—
507C2×C4—26no—
508C2×C4—248no—
509C2×C4—312no—
510C2×C4—36no—
511C2×C4—348no—
512C2×C4—32no—
513C2×C4—36no—
514C2×C4—34no—
515C2×C4—696no—
516C2×C4—64no—
517C2×C4—62no—
518C2×C4—66no—
519C2×C4—648no—
520C2×C4—68no—
521C2×C4—1212no—
522C2×C4C2×D3×D4112yes—
523C2×C4#100714yescomplemented
524C2×C4C24×D316yes—
525C2×C4C24×D3112yes—
526C2×C4C24×D316yes—
527C2×C4C24×D312yes—
528C2×C3—44no—
529C2×C3#349714yescomplemented
530C2×C3C24×D411yescharacteristic
531C2×C3C22×2−1+411yescharacteristic
532C2×C3C2×#58511yescharacteristic
533D3—24no—
534D3—412no—
535C22—24no—
536C22—24no—
537C22—24no—
538C22—34no—
539C22—48no—
540C22—62no—
541C22—62no—
542C22—68no—
543C22—612no—
544C22—612no—
545C22—612no—
546C22—1224no—
547C22C22×D3×D412yes—
548C22C2×#100712yes—
549C22#596812yes—
550C22C25×D311yescharacteristic
551C4—212no—
552C4—26no—
553C4—312no—
554C4—64no—
555C4—62no—
556C4—66no—
557C4C22×D3×D4112yes—
558C4C2×#100714yescomplemented
559C3C2×#349711yescharacteristic, complemented, Sylow
560C2—44no—
561C2—64no—
562C2—1212no—
563C2#8676014yesdirect product factor
564C2C23×D3×D411yescharacteristic
565C2C22×#100711yescharacteristic
566C2C2×#596811yescharacteristic
subgroupG/H
centerC23C24×D3
commutator, cosocleC22×C3C26
FrattiniC22C25×D3
FittingC2×C3×D4×Q8C2
socleC23×C3C25
hypercenterC2×D4×Q8D3
hypocenterC3C2×#3497
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×D4×Q8

Complements

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

NH ≅ G/Ntype
1. C22×#1007C2semidirect
2. C2×C3×D4×Q8C2semidirect
3. C2×#5817C2semidirect
4. #86760C2direct (N × H)
5. C2×#5727C2semidirect
6. C2×#5725C2semidirect
7. C2×#5633C2semidirect
8. C2×#5629C2semidirect
9. C2×C4×D3×Q8C2semidirect
10. C2×#5577C2semidirect
11. C2×#5548C2semidirect
12. C2×#5553C2semidirect
16. C2×#1007C22semidirect
23. C3×D4×Q8C22semidirect
28. #5817C22semidirect
36. #5727C22semidirect
37. #5725C22semidirect
38. #5633C22semidirect
39. #5629C22semidirect
40. #5577C22semidirect
41. C4×D3×Q8C22semidirect
42. #5553C22semidirect
43. #5548C22semidirect
65. C2×D4×Q8D3semidirect
87. C2×D3×Q8D4semidirect
160. D4×Q8C2×D3semidirect
223. D3×Q8C2×D4semidirect
360. C2×C3×C42−1+4semidirect
476. C3×C4C2×2−1+4semidirect
523. C2×C4#1007semidirect
529. C2×C3#3497semidirect
558. C4C2×#1007semidirect
559. C3C2×#3497semidirect
563. C2#86760direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#10072
C2×C3×D4×Q82
C2×#58172
#867602
C2×#57272
C2×#57252
C2×#56332
C2×#56292
C2×C4×D3×Q82
C2×#55772
C2×#55482
C2×#55532
C2×#34973