Finite Group Databasebeta

← C2×#86784 #11703357

Subgroups — C2×#86784 #11703357

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×#1010C212yesmaximal, complemented
2C2×C3×#550C211yesmaximal, characteristic, complemented
3C2×#5831C212yesmaximal, complemented
4#86784C2132yesmaximal, direct product factor
5C2×#5718C213yesmaximal, complemented
6C2×#5716C213yesmaximal, complemented
7C2×#5608C216yesmaximal, complemented
8C2×#5598C216yesmaximal, complemented
9C2×Q8×D12C211yesmaximal, characteristic, complemented
10C2×C4×D3×Q8C211yesmaximal, characteristic, complemented
11C2×#5548C213yesmaximal, complemented
12C2×D4×Dic6C213yesmaximal, complemented
13C2×#3497—31nomaximal, Sylow
14C22×C3×#41C2212yes—
15C2×#1010—232no—
16C2×#1010C22132yescomplemented
17C22×D3×Q8C2212yes—
18C22×D3×Q8C2211yescharacteristic
19C22×#237C2216yes—
20C22×#237C2216yes—
21C22×#235C2216yes—
22C23×Dic6C2216yes—
23C3×#550C22116yescomplemented
24C2×C3×#127C2213yes—
25C2×C3×#121C2216yes—
26C2×C3×C4×Q8C2211yescharacteristic
27C2×C3×C4×D4C2213yes—
28#5831C22132yescomplemented
29C2×Q8×Dic3C2211yescharacteristic
30C2×#940C2213yes—
31C2×#939C22112yes—
32C2×#938C2213yes—
33C2×#938C2216yes—
34C2×#938C2216yes—
35C2×#936C2216yes—
36C2×D4×Dic3C2216yes—
37C2×D4×Dic3C2216yes—
38C2×C4×#67C2216yes—
39C2×#924C2216yes—
40#5718C22148yescomplemented
41#5716C22148yescomplemented
42#5608C22196yescomplemented
43#5598C22196yescomplemented
44Q8×D12C22116yescomplemented
45C4×D3×Q8C22116yescomplemented
46#5548C22148yescomplemented
47D4×Dic6C22148yescomplemented
48C2×#897C2216yes—
49C2×#895C2211yescharacteristic
50C2×D3×C4⋊C4C2213yes—
51C2×#888C2213yes—
52C2×#887C2213yes—
53C2×#885C2216yes—
54C2×#884C2216yes—
55C2×#878C22112yes—
56C2×C4×D12C2213yes—
57C2×C42×D3C2213yes—
58C2×#869C2213yes—
59C2×C4×Dic6C2213yes—
60C22×2−1+4—32no—
61#3497—332no—
62C2×#550—33no—
63C2×#550D311yescharacteristic, complemented
64C2×D4×Q8—31no—
65C2×D4×Q8—33no—
66C2×#548—32no—
67C2×#548—36no—
68C2×#537—36no—
69C2×#532—33no—
70C2×#532—33no—
71C22×C4×Q8—31no—
72C2×C3×#41—216no—
73C2×C3×#41C23116yes—
74C22×C3×Q8C2311yescharacteristic
75C22×C3×D4C2316yes—
76C22×C3×D4C2313yes—
77C22×C3×D4C2316yes—
78C23×C3×C4C2316yes—
79C22×#67C2316yes—
80C22×#67C2316yes—
81C22×#67C2316yes—
82C23×Dic3C23112yes—
83#1010—2256no—
84C2×D3×Q8—216no—
85C2×D3×Q8D4116yescomplemented
86C2×D3×Q8C23116yes—
87C2×D3×Q8C2318yes—
88C2×#237—248no—
89C2×#237—248no—
90C2×#237C23148yes—
91C2×#237C23148yes—
92C2×#235—248no—
93C2×#235C23148yes—
94C22×D12C2312yes—
95C22×D12C2311yescharacteristic
96C22×C4×D3C2316yes—
97C22×C4×D3C2311yescharacteristic
98C22×C4×D3C2313yes—
99C22×C4×D3C2313yes—
100C22×Dic6—248no—
101C22×Dic6C2316yes—
102C22×Dic6C2316yes—
103C22×Dic6C23148yes—
104C22×Dic6C2313yes—
105C3×#127C23124yes—
106C3×#121C23148yes—
107C3×C4×Q8C2318yes—
108C3×C4×D4C23124yes—
109C2×C3×C4⋊C4C2313yes—
110C2×C3×C4⋊C4C2311yescharacteristic
111C2×C3×C22⋊C4C2316yes—
112C2×C3×C42C2313yes—
113C2×#217C23112yes—
114C2×#217C2316yes—
115Q8×Dic3C2318yes—
116#940C23124yes—
117#939C23196yes—
118#938C23124yes—
119#938C23148yes—
120#938C23148yes—
121#936C23148yes—
122D4×Dic3C23148yes—
123D4×Dic3C23148yes—
124C4×#67C23148yes—
125C2×(C2×D3)⋊C4C2316yes—
126C2×(C2×D3)⋊C4C2312yes—
127C2×C4⋊Dic3C2313yes—
128C2×C4⋊Dic3C2313yes—
129C2×C4⋊Dic3C2316yes—
130#924C23148yes—
131C2×Dic3⋊C4C2316yes—
132C2×Dic3⋊C4C2316yes—
133C2×C4×Dic3C2313yes—
134C2×C4×Dic3C2313yes—
135C2×C4×Dic3C2313yes—
136#897C23148yes—
137#895C2318yes—
138D3×C4⋊C4C23124yes—
139#888C23124yes—
140#887C23124yes—
141#885C23148yes—
142#884C23148yes—
143#878C23196yes—
144C4×D12C23124yes—
145C42×D3C23124yes—
146#869C23124yes—
147C4×Dic6C23124yes—
148C2×2−1+4—332no—
149C2×2−1+4—632no—
150C22×#41—36no—
151C22×#41—36no—
152C22×#41—36no—
153C22×#41C2×D312yes—
154C23×Q8—31no—
155C23×Q8—32no—
156C23×Q8—36no—
157#550—348no—
158#550C2×D3116yescomplemented
159D4×Q8—316no—
160D4×Q8—348no—
161#548—332no—
162#548—396no—
163#537—396no—
164#532—348no—
165#532—348no—
166C2×C4⋊Q8—33no—
167C2×C4⋊Q8—33no—
168C2×#127—33no—
169C2×#127C2×D313yes—
170C2×#124—36no—
171C2×#124—36no—
172C2×C22⋊Q8—36no—
173C2×C22⋊Q8—36no—
174C2×C22⋊Q8—312no—
175C2×#121—33no—
176C2×#121—36no—
177C2×#121—36no—
178C2×#121—312no—
179C2×#121—31no—
180C2×#121—36no—
181C2×#121C2×D316yes—
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×C3C2416yes—
195C23×D3C2412yes—
196C23×D3C2411yescharacteristic
197C3×#41—264no—
198C2×C3×Q8C2×D418yes—
199C2×C3×Q8C2414yes—
200C2×C3×D4—224no—
201C2×C3×D4—212no—
202C2×C3×D4—412no—
203C2×C3×D4C24112yes—
204C2×C3×D4C24124yes—
205C2×C3×D4C24124yes—
206C22×C3×C4—224no—
207C22×C3×C4C2413yes—
208C22×C3×C4C2412yes—
209C22×C3×C4C24124yes—
210C22×C3×C4C2413yes—
211C22×C3×C4C2411yescharacteristic
212C2×#67—224no—
213C2×#67—224no—
214C2×#67—412no—
215C2×#67C24124yes—
216C2×#67C24124yes—
217C2×#67C24124yes—
218C22×Dic3—248no—
219C22×Dic3C24148yes—
220C22×Dic3C2411yescharacteristic
221C22×Dic3C2413yes—
222C22×Dic3C2416yes—
223C22×Dic3C2413yes—
224D3×Q8—264no—
225D3×Q8C2×D4164yescomplemented
226#237—2192no—
227#237—2192no—
228#235—2192no—
229C2×D12—28no—
230C2×D12—24no—
231C2×D12C2414yes—
232C2×D12C2418yes—
233C2×C4×D3—224no—
234C2×C4×D3—24no—
235C2×C4×D3—46no—
236C2×C4×D3C2×D4124yes—
237C2×C4×D3C2414yes—
238C2×C4×D3C24124yes—
239C2×C4×D3C24112yes—
240C2×C4×D3C24112yes—
241C2×Dic6—224no—
242C2×Dic6—224no—
243C2×Dic6—2192no—
244C2×Dic6C2×D4124yes—
245C2×Dic6C24124yes—
246C2×Dic6C24124yes—
247C2×Dic6C24112yes—
248C3×C4⋊C4C24112yes—
249C3×C4⋊C4C2414yes—
250C3×C22⋊C4C24124yes—
251C3×C42C24112yes—
252#217C24148yes—
253#217C24124yes—
254(C2×D3)⋊C4C24124yes—
255(C2×D3)⋊C4C2418yes—
256C4⋊Dic3C24112yes—
257C4⋊Dic3C24112yes—
258C4⋊Dic3C24124yes—
259Dic3⋊C4C24124yes—
260Dic3⋊C4C24124yes—
261C4×Dic3C24112yes—
262C4×Dic3C24112yes—
263C4×Dic3C24112yes—
2642−1+4—6256no—
265C2×#41—216no—
266C2×#41—348no—
267C2×#41—348no—
268C2×#41—348no—
269C2×#41—648no—
270C2×#41—648no—
271C2×#41—648no—
272C2×#41C22×D3116yes—
273C22×Q8—316no—
274C22×Q8—38no—
275C22×Q8—316no—
276C22×Q8—33no—
277C22×Q8—36no—
278C22×Q8—36no—
279C22×Q8—348no—
280C22×Q8—648no—
281C22×Q8—616no—
282C22×Q8C22×D311yescharacteristic
283C22×D4—36no—
284C22×D4—36no—
285C22×D4—36no—
286C22×D4—31no—
287C22×D4—32no—
288C22×D4C22×D313yes—
289C22×D4C22×D316yes—
290C22×D4C22×D316yes—
291C23×C4—33no—
292C23×C4—312no—
293C23×C4—36no—
294C23×C4—31no—
295C23×C4—33no—
296C23×C4C22×D316yes—
297C4⋊Q8—324no—
298C4⋊Q8—324no—
299#127—324no—
300#127C22×D3124yes—
301#124—348no—
302#124—348no—
303C22⋊Q8—348no—
304C22⋊Q8—348no—
305C22⋊Q8—396no—
306#121—324no—
307#121—348no—
308#121—348no—
309#121—396no—
310#121—38no—
311#121—348no—
312#121C22×D3148yes—
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—33no—
325C2×C4⋊C4—36no—
326C2×C4⋊C4—36no—
327C2×C4⋊C4—324no—
328C2×C4⋊C4C22×D313yes—
329C2×C4⋊C4C22×D311yescharacteristic
330C2×C22⋊C4—36no—
331C2×C22⋊C4—312no—
332C2×C22⋊C4—32no—
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—212no—
341C23×C3—212no—
342C23×C3—212no—
343C23×C3C25112yes—
344C23×C3C2511yescharacteristic
345C22×D3—22no—
346C22×D3—22no—
347C22×D3—24no—
348C22×D3—24no—
349C22×D3—24no—
350C22×D3C22×D414yes—
351C22×D3C2514yes—
352C22×D3C2512yes—
353C3×Q8C22×D4116yes—
354C3×D4—248no—
355C3×D4—224no—
356C3×D4—424no—
357C2×C3×C4—24no—
358C2×C3×C4—26no—
359C2×C3×C4—248no—
360C2×C3×C4C22×D4112yes—
361C2×C3×C42−1+414yescomplemented
362C2×C3×C4C2516yes—
363C2×C3×C4C2514yes—
364C2×C3×C4C2516yes—
365C2×C3×C4C2512yes—
366#67—248no—
367#67—248no—
368#67—424no—
369C2×Dic3—212no—
370C2×Dic3—22no—
371C2×Dic3—26no—
372C2×Dic3—296no—
373C2×Dic3C22×D4112yes—
374C2×Dic3C2512yes—
375C2×Dic3C2516yes—
376C2×Dic3C25112yes—
377C2×Dic3C2516yes—
378D12—216no—
379D12—28no—
380C4×D3—248no—
381C4×D3—28no—
382C4×D3—412no—
383C4×D3C22×D4148yes—
384Dic6—248no—
385Dic6—248no—
386Dic6C22×D4148yes—
387C24—31no—
388C24—32no—
389C24C23×D316yes—
390#41—264no—
391#41—6192no—
392#41—6192no—
393#41—6192no—
394C2×Q8—364no—
395C2×Q8—324no—
396C2×Q8—312no—
397C2×Q8—324no—
398C2×Q8—324no—
399C2×Q8—624no—
400C2×Q8—6192no—
401C2×Q8—664no—
402C2×Q8—624no—
403C2×Q8D3×D418yes—
404C2×Q8C23×D314yes—
405C2×D4—224no—
406C2×D4—212no—
407C2×D4—324no—
408C2×D4—324no—
409C2×D4—324no—
410C2×D4—34no—
411C2×D4—38no—
412C2×D4—412no—
413C2×D4—68no—
414C2×D4—624no—
415C2×D4—624no—
416C2×D4—64no—
417C2×D4—1212no—
418C2×D4C23×D3112yes—
419C2×D4C23×D3124yes—
420C2×D4C23×D3124yes—
421C22×C4—224no—
422C22×C4—33no—
423C22×C4—324no—
424C22×C4—312no—
425C22×C4—348no—
426C22×C4—31no—
427C22×C4—33no—
428C22×C4—36no—
429C22×C4—34no—
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×D312yes—
439C22×C4C23×D313yes—
440C22×C4C23×D311yescharacteristic
441C4⋊C4—324no—
442C4⋊C4—312no—
443C4⋊C4—312no—
444C4⋊C4—324no—
445C4⋊C4—324no—
446C4⋊C4C23×D3112yes—
447C4⋊C4C23×D314yes—
448C22⋊C4—324no—
449C22⋊C4—348no—
450C22⋊C4—38no—
451C22⋊C4—324no—
452C22⋊C4C23×D3124yes—
453C42—312no—
454C42—312no—
455C42—312no—
456C42C23×D3112yes—
457C22×C3—212no—
458C22×C3—212no—
459C22×C3—212no—
460C22×C3—424no—
461C22×C3C23×D412yes—
462C22×C3C2×2−1+412yes—
463C22×C3#58512yes—
464C22×C3C2611yescharacteristic
465C2×D3—22no—
466C2×D3—22no—
467C2×D3—28no—
468C2×D3—24no—
469C2×D3—24no—
470C2×D3—24no—
471C2×D3—48no—
472C2×D3C23×D414yes—
473C3×C4—24no—
474C3×C4—26no—
475C3×C4C23×D4112yes—
476C3×C4C2×2−1+414yescomplemented
477Dic3—212no—
478Dic3—22no—
479Dic3—26no—
480Dic3C23×D4112yes—
481C23—212no—
482C23—212no—
483C23—212no—
484C23—34no—
485C23—32no—
486C23—34no—
487C23—62no—
488C23—62no—
489C23—64no—
490C23—64no—
491C23—64no—
492C23C24×D3112yes—
493C23C24×D311yescharacteristic
494Q8—348no—
495Q8—648no—
496Q8—648no—
497Q8C2×D3×D4116yes—
498D4—248no—
499D4—224no—
500D4—424no—
501D4—616no—
502D4—648no—
503D4—648no—
504D4—68no—
505D4—1224no—
506C2×C4—24no—
507C2×C4—26no—
508C2×C4—248no—
509C2×C4—312no—
510C2×C4—36no—
511C2×C4—348no—
512C2×C4—32no—
513C2×C4—36no—
514C2×C4—312no—
515C2×C4—696no—
516C2×C4—612no—
517C2×C4—62no—
518C2×C4—66no—
519C2×C4—648no—
520C2×C4—68no—
521C2×C4—1212no—
522C2×C4C2×D3×D4112yes—
523C2×C4#101014yescomplemented
524C2×C4C24×D316yes—
525C2×C4C24×D314yes—
526C2×C4C24×D316yes—
527C2×C4C24×D312yes—
528C2×C3—412no—
529C2×C3#349714yescomplemented
530C2×C3C24×D411yescharacteristic
531C2×C3C22×2−1+411yescharacteristic
532C2×C3C2×#58511yescharacteristic
533D3—24no—
534D3—44no—
535C22—212no—
536C22—212no—
537C22—212no—
538C22—34no—
539C22—424no—
540C22—68no—
541C22—62no—
542C22—62no—
543C22—64no—
544C22—64no—
545C22—64no—
546C22—128no—
547C22C22×D3×D412yes—
548C22C2×#101012yes—
549C22#596612yes—
550C22C25×D311yescharacteristic
551C4—24no—
552C4—26no—
553C4—312no—
554C4—612no—
555C4—62no—
556C4—66no—
557C4C22×D3×D4112yes—
558C4C2×#101014yescomplemented
559C3C2×#349711yescharacteristic, complemented, Sylow
560C2—412no—
561C2—64no—
562C2—124no—
563C2#8678414yesdirect product factor
564C2C23×D3×D411yescharacteristic
565C2C22×#101011yescharacteristic
566C2C2×#596611yescharacteristic
subgroupG/H
centerC23C24×D3
commutator, cosocleC22×C3C26
FrattiniC22C25×D3
FittingC2×C3×#550C2
socleC23×C3C25
hypercenterC2×#550D3
hypocenterC3C2×#3497
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#550

Complements

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

NH ≅ G/Ntype
1. C22×#1010C2semidirect
2. C2×C3×#550C2semidirect
3. C2×#5831C2semidirect
4. #86784C2direct (N × H)
5. C2×#5718C2semidirect
6. C2×#5716C2semidirect
7. C2×#5608C2semidirect
8. C2×#5598C2semidirect
9. C2×Q8×D12C2semidirect
10. C2×C4×D3×Q8C2semidirect
11. C2×#5548C2semidirect
12. C2×D4×Dic6C2semidirect
16. C2×#1010C22semidirect
23. C3×#550C22semidirect
28. #5831C22semidirect
40. #5718C22semidirect
41. #5716C22semidirect
42. #5608C22semidirect
43. #5598C22semidirect
44. Q8×D12C22semidirect
45. C4×D3×Q8C22semidirect
46. #5548C22semidirect
47. D4×Dic6C22semidirect
63. C2×#550D3semidirect
85. C2×D3×Q8D4semidirect
158. #550C2×D3semidirect
225. D3×Q8C2×D4semidirect
361. C2×C3×C42−1+4semidirect
476. C3×C4C2×2−1+4semidirect
523. C2×C4#1010semidirect
529. C2×C3#3497semidirect
558. C4C2×#1010semidirect
559. C3C2×#3497semidirect
563. C2#86784direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#10102
C2×C3×#5502
C2×#58312
#867842
C2×#57182
C2×#57162
C2×#56082
C2×#55982
C2×Q8×D122
C2×C4×D3×Q82
C2×#55482
C2×D4×Dic62
C2×#34973