Finite Group Databasebeta

← C2×#79308 #11666409

Subgroups — C2×#79308 #11666409

Order 768.

2218 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5477C211yesmaximal, characteristic, complemented
2C2×#5479C211yesmaximal, characteristic, complemented
3C2×C3×#431C211yesmaximal, characteristic, complemented
4#79308C2116yesmaximal, direct product factor
5C2×#4849C211yesmaximal, characteristic, complemented
6C2×#4848C211yesmaximal, characteristic, complemented
7C2×#4841C211yesmaximal, characteristic, complemented
8C2×#4843C211yesmaximal, characteristic, complemented
9C2×#4733C212yesmaximal, complemented
10C2×#4734C212yesmaximal, complemented
11C2×#4696C212yesmaximal, complemented
12C2×#4697C212yesmaximal, complemented
13C2×#3212—31nomaximal, Sylow
14C22×#235C2212yes—
15C22×C3×C42C2211yescharacteristic
16C2×#930—21no—
17C2×#930—21no—
18C2×#930C2211yescharacteristic
19C2×#929—21no—
20C2×C4×#67—21no—
21C2×#924C2211yescharacteristic
22C22×Dic12C2212yes—
23C22×D24C2212yes—
24C22×#204C2212yes—
25C22×#204C2212yes—
26#5479—24no—
27#5479C2218yescomplemented
28C2×#876—21no—
29#5477—24no—
30#5477C2218yescomplemented
31C2×#875C2212yes—
32C2×#875C2211yescharacteristic
33C2×#874C2211yescharacteristic
34C2×C4×D12—21no—
35C2×C4×D12—21no—
36C2×C4×D12C2211yescharacteristic
37C2×#872—21no—
38C2×#870C2211yescharacteristic
39C2×#869C2211yescharacteristic
40C2×C4×Dic6C2211yescharacteristic
41C3×#431C2218yescomplemented
42C2×C3×C4⋊C8C2211yescharacteristic
43C2×C3×C4⋊C8C2211yescharacteristic
44C2×C3×C22⋊C8C2212yes—
45C2×C3×C4×C8C2212yes—
46C2×#821C2212yes—
47C2×#821C2212yes—
48C2×#816C2212yes—
49C2×#816C2212yes—
50#4849C2218yescomplemented
51#4848C2218yescomplemented
52#4843C2218yescomplemented
53#4841C2218yescomplemented
54#4734C22116yescomplemented
55#4733C22116yescomplemented
56#4697C22116yescomplemented
57#4696C22116yescomplemented
58C2×#532—31no—
59C2×#529—31no—
60#3212—316no—
61C2×#495—32no—
62C2×#488—32no—
63C2×#462—31no—
64C2×#460—31no—
65C2×#461—31no—
66C2×#459—31no—
67C2×#452—32no—
68C2×#449—32no—
69C2×#431D311yescharacteristic, complemented
70C23×C3×C4D411yescharacteristic
71C23×C3×C4D411yescharacteristic
72C23×C3×C4C2311yescharacteristic
73C22×#67—22no—
74C22×#67—21no—
75C2×#235—24no—
76C2×#235—44no—
77C2×#235—44no—
78C2×#235C2318yes—
79C22×D12—21no—
80C22×D12—21no—
81C22×D12C2312yes—
82C22×D12C2311yescharacteristic
83C22×C4×D3—22no—
84C22×C4×D3—21no—
85C22×Dic6C2312yes—
86C22×Dic6C2311yescharacteristic
87C22×C3×C8C2312yes—
88C22×C3×C8C2312yes—
89C2×C3×C42D411yescharacteristic
90C2×C3×C42D411yescharacteristic
91C2×C3×C42D414yes—
92C2×C3×C42C2311yescharacteristic
93C2×C3×C42C2311yescharacteristic
94C2×C3×C42C2314yes—
95C2×#217—21no—
96#930—28no—
97#930—28no—
98#930—22no—
99#930C2314yes—
100#929—28no—
101C4×#67—28no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—22no—
105C2×(C2×D3)⋊C4—22no—
106C2×(C2×D3)⋊C4—21no—
107C2×(C2×D3)⋊C4—21no—
108C2×C4⋊Dic3—21no—
109C2×C4⋊Dic3—21no—
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112#924—22no—
113#924C2314yes—
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—21no—
116C2×Dic3⋊C4—21no—
117C2×Dic3⋊C4—21no—
118C2×C4×Dic3—21no—
119C2×Dic12—44no—
120C2×Dic12C2318yes—
121C2×D24—44no—
122C2×D24C2318yes—
123C2×#204—44no—
124C2×#204—44no—
125C2×#204C2318yes—
126C2×#204C2318yes—
127#876—28no—
128#875—24no—
129#875—22no—
130#875C2314yes—
131#875C2318yes—
132#874—22no—
133#874C2314yes—
134C4×D12—28no—
135C4×D12—28no—
136C4×D12—22no—
137C4×D12C2314yes—
138#872—28no—
139#870—22no—
140#870C2314yes—
141#869—22no—
142#869C2314yes—
143C4×Dic6—22no—
144C4×Dic6C2314yes—
145C3×C4⋊C8C2314yes—
146C3×C4⋊C8C2314yes—
147C3×C22⋊C8C2318yes—
148C3×C4×C8C2318yes—
149#821C2318yes—
150#821C2318yes—
151#816C2318yes—
152#816C2318yes—
153C22×#41—32no—
154C22×Q16—32no—
155C22×SD16—32no—
156C22×SD16—32no—
157C22×D8—32no—
158#532—38no—
159#532—64no—
160C2×C4⋊Q8—31no—
161C2×#127—31no—
162#529—38no—
163#529—64no—
164C2×#126—61no—
165C2×#125—31no—
166C2×#124—32no—
167C2×#124—31no—
168C2×#123—61no—
169C2×C22⋊Q8—31no—
170C2×#121—31no—
171C2×#121—61no—
172C2×#121—61no—
173C2×C4×Q8—31no—
174C2×C4×D4—31no—
175C2×C4×D4—61no—
176C2×C4×D4—61no—
177C2×C4×D4—61no—
178C2×#117—61no—
179C22×C42C2×D311yescharacteristic
180#495—316no—
181#488—316no—
182#462—38no—
183#461—38no—
184#460—38no—
185#459—38no—
186#452—316no—
187#449—316no—
188#431C2×D318yescomplemented
189C2×C4⋊C8C2×D311yescharacteristic
190C2×C4⋊C8C2×D311yescharacteristic
191C2×#103—32no—
192C2×#103—32no—
193C2×#102—32no—
194C2×#102—32no—
195C2×C22⋊C8C2×D312yes—
196C2×C4×C8C2×D312yes—
197C24×C3C2×D411yescharacteristic
198C23×D3—21no—
199C23×D3—22no—
200C22×C3×C4—24no—
201C22×C3×C4—22no—
202C22×C3×C4—22no—
203C22×C3×C4—42no—
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D412yes—
209C22×C3×C4C2×D414yes—
210C22×C3×C4C2×D414yes—
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2412yes—
213C22×C3×C4C2411yescharacteristic
214C2×#67—24no—
215C2×#67—28no—
216C2×#67—48no—
217C2×#67—44no—
218C22×Dic3—21no—
219C22×Dic3—21no—
220C22×Dic3—21no—
221C22×Dic3—22no—
222#235—48no—
223#235—48no—
224#235—416no—
225C2×D12—22no—
226C2×D12—24no—
227C2×D12—22no—
228C2×D12—24no—
229C2×D12—24no—
230C2×D12—24no—
231C2×D12—21no—
232C2×D12—22no—
233C2×D12—44no—
234C2×D12—44no—
235C2×D12C2414yes—
236C2×D12C2412yes—
237C2×C4×D3—24no—
238C2×C4×D3—28no—
239C2×C4×D3—48no—
240C2×C4×D3—44no—
241C2×Dic6—22no—
242C2×Dic6—24no—
243C2×Dic6—22no—
244C2×Dic6—24no—
245C2×Dic6—21no—
246C2×Dic6—22no—
247C2×Dic6C2414yes—
248C2×Dic6C2412yes—
249C2×C3×C8—24no—
250C2×C3×C8—24no—
251C2×C3×C8C2414yes—
252C2×C3×C8C2414yes—
253C3×C42C2×D412yes—
254C3×C42C2×D412yes—
255C3×C42C2×D414yes—
256C3×C42C2×D414yes—
257C3×C42C2412yes—
258C3×C42C2412yes—
259#217—24no—
260(C2×D3)⋊C4—24no—
261(C2×D3)⋊C4—24no—
262(C2×D3)⋊C4—28no—
263(C2×D3)⋊C4—28no—
264(C2×D3)⋊C4—24no—
265(C2×D3)⋊C4—24no—
266C4⋊Dic3—21no—
267C4⋊Dic3—24no—
268C4⋊Dic3—24no—
269C4⋊Dic3—21no—
270C4⋊Dic3C2412yes—
271C4⋊Dic3C2412yes—
272Dic3⋊C4—24no—
273Dic3⋊C4—24no—
274Dic3⋊C4—24no—
275Dic3⋊C4—24no—
276C4×Dic3—24no—
277Dic12—48no—
278D24—48no—
279#204—48no—
280#204—48no—
281C2×#41—38no—
282C2×#41—64no—
283C2×#41—124no—
284C2×#41—124no—
285C22×Q8—31no—
286C22×Q8—32no—
287C22×D4—31no—
288C22×D4—32no—
289C22×D4—61no—
290C22×D4—62no—
291C22×D4—61no—
292C22×D4—61no—
293C23×C4—61no—
294C23×C4—62no—
295C23×C4D1211yescharacteristic
296C23×C4D1211yescharacteristic
297C23×C4C22×D311yescharacteristic
298C2×Q16—38no—
299C2×Q16—124no—
300C2×SD16—38no—
301C2×SD16—38no—
302C2×SD16—124no—
303C2×SD16—124no—
304C2×D8—38no—
305C2×D8—124no—
306C22×C8C22×D312yes—
307C22×C8C22×D312yes—
308C4⋊Q8—34no—
309C4⋊Q8—62no—
310#127—34no—
311#127—62no—
312#126—68no—
313#125—34no—
314#125—62no—
315#124—34no—
316#124—38no—
317#124—62no—
318#124—64no—
319#123—68no—
320C22⋊Q8—34no—
321C22⋊Q8—62no—
322#121—34no—
323#121—62no—
324#121—68no—
325#121—68no—
326C4×Q8—34no—
327C4×Q8—62no—
328C4×D4—34no—
329C4×D4—68no—
330C4×D4—62no—
331C4×D4—68no—
332C4×D4—68no—
333#117—68no—
334C2×C4⋊C4—31no—
335C2×C4⋊C4—31no—
336C2×C4⋊C4—61no—
337C2×C4⋊C4—61no—
338C2×C4⋊C4—61no—
339C2×C4⋊C4—61no—
340C2×C4⋊C4—61no—
341C2×C4⋊C4—61no—
342C2×C22⋊C4—61no—
343C2×C22⋊C4—61no—
344C2×C22⋊C4—61no—
345C2×C22⋊C4—61no—
346C2×C22⋊C4—62no—
347C2×C22⋊C4—62no—
348C2×C22⋊C4—61no—
349C2×C42—61no—
350C2×C42D1211yescharacteristic
351C2×C42D1211yescharacteristic
352C2×C42D1214yes—
353C2×C42C22×D311yescharacteristic
354C2×C42C22×D311yescharacteristic
355C2×C42C22×D314yes—
356C4⋊C8C22×D314yes—
357C4⋊C8C22×D314yes—
358#103—38no—
359#103—38no—
360#102—38no—
361#102—38no—
362C22⋊C8C22×D318yes—
363C4×C8C22×D318yes—
364C23×C3—22no—
365C23×C3—22no—
366C23×C3—22no—
367C23×C3C22×D412yes—
368C23×C3C22×D411yescharacteristic
369C22×D3—22no—
370C22×D3—24no—
371C22×D3—44no—
372C22×D3—42no—
373C22×D3—42no—
374C22×D3—44no—
375C22×D3—44no—
376C22×D3—42no—
377C2×C3×C4—22no—
378C2×C3×C4—22no—
379C2×C3×C4—22no—
380C2×C3×C4—22no—
381C2×C3×C4—24no—
382C2×C3×C4—28no—
383C2×C3×C4—22no—
384C2×C3×C4—22no—
385C2×C3×C4—44no—
386C2×C3×C4#13514yes—
387C2×C3×C4C22×D412yes—
388C2×C3×C4C22×D412yes—
389C2×C3×C4C22×D412yes—
390C2×C3×C4C22×D412yes—
391C2×C3×C4C22×D411yescharacteristic
392C2×C3×C4C22×D412yes—
393C2×C3×C4H(5,𝔽2)12yes—
394C2×C3×C4H(5,𝔽2)12yes—
395C2×C3×C4C2511yescharacteristic
396#67—416no—
397#67—48no—
398C2×Dic3—22no—
399C2×Dic3—22no—
400C2×Dic3—22no—
401C2×Dic3—24no—
402C2×Dic3—44no—
403C2×Dic3—42no—
404C2×Dic3—42no—
405C2×Dic3—42no—
406D12—22no—
407D12—24no—
408D12—22no—
409D12—24no—
410D12—48no—
411D12—42no—
412D12—44no—
413D12—48no—
414C4×D3—416no—
415C4×D3—48no—
416Dic6—22no—
417Dic6—24no—
418Dic6—22no—
419Dic6—24no—
420Dic6—42no—
421Dic6—44no—
422C3×C8—24no—
423C3×C8—24no—
424C24—61no—
425C24—62no—
426C24C2×D1211yescharacteristic
427#41—128no—
428#41—128no—
429#41—1216no—
430C2×Q8—34no—
431C2×Q8—32no—
432C2×Q8—61no—
433C2×Q8—62no—
434C2×Q8—62no—
435C2×Q8—64no—
436C2×Q8—62no—
437C2×Q8—64no—
438C2×D4—34no—
439C2×D4—32no—
440C2×D4—64no—
441C2×D4—68no—
442C2×D4—61no—
443C2×D4—64no—
444C2×D4—64no—
445C2×D4—62no—
446C2×D4—62no—
447C2×D4—64no—
448C2×D4—62no—
449C2×D4—64no—
450C2×D4—124no—
451C2×D4—128no—
452C2×D4—124no—
453C2×D4—124no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—42no—
458C22×C4—61no—
459C22×C4—61no—
460C22×C4—61no—
461C22×C4—62no—
462C22×C4—64no—
463C22×C4—68no—
464C22×C4—128no—
465C22×C4—124no—
466C22×C4C2×D1212yes—
467C22×C4C2×D1214yes—
468C22×C4C2×D1211yescharacteristic
469C22×C4C2×D1211yescharacteristic
470C22×C4C2×D1211yescharacteristic
471C22×C4C2×D1211yescharacteristic
472C22×C4C2×D1214yes—
473C22×C4C2×D1211yescharacteristic
474C22×C4C23×D312yes—
475C22×C4C23×D311yescharacteristic
476Q16—128no—
477SD16—128no—
478SD16—128no—
479D8—128no—
480C2×C8—24no—
481C2×C8—24no—
482C2×C8C23×D314yes—
483C2×C8C23×D314yes—
484C4⋊C4—32no—
485C4⋊C4—32no—
486C4⋊C4—61no—
487C4⋊C4—61no—
488C4⋊C4—64no—
489C4⋊C4—64no—
490C4⋊C4—64no—
491C4⋊C4—64no—
492C4⋊C4—64no—
493C4⋊C4—64no—
494C22⋊C4—64no—
495C22⋊C4—64no—
496C22⋊C4—64no—
497C22⋊C4—64no—
498C22⋊C4—68no—
499C22⋊C4—68no—
500C22⋊C4—64no—
501C42—64no—
502C42C2×D1214yes—
503C42C2×D1214yes—
504C42C2×D1212yes—
505C42C2×D1212yes—
506C42C23×D312yes—
507C42C23×D312yes—
508C22×C3—22no—
509C22×C3—22no—
510C22×C3—22no—
511C22×C3—44no—
512C22×C3#53512yes—
513C22×C3C2×#13512yes—
514C22×C3#57512yes—
515C22×C3C23×D411yescharacteristic
516C2×D3—44no—
517C2×D3—42no—
518C2×D3—42no—
519C2×D3—44no—
520C2×D3—44no—
521C2×D3—42no—
522C2×D3—88no—
523C2×D3—84no—
524C3×C4—22no—
525C3×C4—22no—
526C3×C4—22no—
527C3×C4—22no—
528C3×C4#53512yes—
529C3×C4C2×#13514yes—
530C3×C4C2×H(5,𝔽2)11yescharacteristic
531C3×C4C2×H(5,𝔽2)11yescharacteristic
532Dic3—44no—
533Dic3—42no—
534Dic3—42no—
535Dic3—42no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—62no—
540C23—64no—
541C23—122no—
542C23—124no—
543C23—124no—
544C23—122no—
545C23—124no—
546C23—122no—
547C23C22×D1212yes—
548C23C22×D1211yescharacteristic
549Q8—62no—
550Q8—64no—
551Q8—62no—
552Q8—64no—
553Q8—124no—
554Q8—122no—
555D4—62no—
556D4—64no—
557D4—62no—
558D4—64no—
559D4—128no—
560D4—1216no—
561D4—128no—
562D4—128no—
563D4—124no—
564D4—122no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—28no—
572C2×C4—24no—
573C2×C4—44no—
574C2×C4—62no—
575C2×C4—62no—
576C2×C4—62no—
577C2×C4—64no—
578C2×C4—1216no—
579C2×C4—122no—
580C2×C4—122no—
581C2×C4—124no—
582C2×C4—122no—
583C2×C4—128no—
584C2×C4#90414yes—
585C2×C4C22×D1212yes—
586C2×C4C22×D1212yes—
587C2×C4C22×D1212yes—
588C2×C4C22×D1212yes—
589C2×C4C22×D1212yes—
590C2×C4C22×D1211yescharacteristic
591C2×C4#100912yes—
592C2×C4#100912yes—
593C2×C4C24×D311yescharacteristic
594C8—24no—
595C8—24no—
596C2×C3—42no—
597C2×C3#321214yescomplemented
598C2×C3C2×#53511yescharacteristic
599C2×C3C22×#13511yescharacteristic
600C2×C3C2×#57511yescharacteristic
601D3—84no—
602D3—82no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—44no—
607C22—122no—
608C22—124no—
609C22—124no—
610C22—122no—
611C22—124no—
612C22—122no—
613C22—244no—
614C22—248no—
615C22#552512yes—
616C22C2×#90412yes—
617C22#574812yes—
618C22C23×D1211yescharacteristic
619C4—22no—
620C4—22no—
621C4—22no—
622C4—22no—
623C4—122no—
624C4—122no—
625C4—124no—
626C4—122no—
627C4#552512yes—
628C4C2×#90414yes—
629C4C2×#100911yescharacteristic
630C4C2×#100911yescharacteristic
631C3C2×#321211yescharacteristic, complemented, Sylow
632C2—42no—
633C2—242no—
634C2—244no—
635C2#7930814yesdirect product factor
636C2C2×#552511yescharacteristic
637C2C22×#90411yescharacteristic
638C2C2×#574811yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#431C2
socleC23×C3C22×D4
hypercenterC2×#431D3
hypocenterC3C2×#3212
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#431

Complements

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

NH ≅ G/Ntype
1. C2×#5477C2semidirect
2. C2×#5479C2semidirect
3. C2×C3×#431C2semidirect
4. #79308C2direct (N × H)
5. C2×#4849C2semidirect
6. C2×#4848C2semidirect
7. C2×#4841C2semidirect
8. C2×#4843C2semidirect
9. C2×#4733C2semidirect
10. C2×#4734C2semidirect
11. C2×#4696C2semidirect
12. C2×#4697C2semidirect
27. #5479C22semidirect
30. #5477C22semidirect
41. C3×#431C22semidirect
50. #4849C22semidirect
51. #4848C22semidirect
52. #4843C22semidirect
53. #4841C22semidirect
54. #4734C22semidirect
55. #4733C22semidirect
56. #4697C22semidirect
57. #4696C22semidirect
69. C2×#431D3semidirect
188. #431C2×D3semidirect
597. C2×C3#3212semidirect
631. C3C2×#3212semidirect
635. C2#79308direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#54772
C2×#54792
C2×C3×#4312
#793082
C2×#48492
C2×#48482
C2×#48412
C2×#48432
C2×#47332
C2×#47342
C2×#46962
C2×#46972
C2×#32123