Finite Group Databasebeta

← C2×#79672 #11670334

Subgroups — C2×#79672 #11670334

Order 768.

2742 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#915C212yesmaximal, complemented
2C2×Q8×D12C211yesmaximal, characteristic, complemented
3C2×#5555C211yesmaximal, characteristic
4C2×C3×C4×SD16C211yesmaximal, characteristic, complemented
5C2×#5031C211yesmaximal, characteristic, complemented
6C2×#5019C211yesmaximal, characteristic, complemented
7#79672C2116yesmaximal, direct product factor
8C2×#4867C212yesmaximal, complemented
9C2×#4809C212yesmaximal, complemented
10C2×#4783C212yesmaximal, complemented
11C2×#4699C211yesmaximal, characteristic, complemented
12C2×#4688C211yesmaximal, characteristic, complemented
13C2×#3276—31nomaximal, Sylow
14C22×D3×Q8C2212yes—
15C22×#237C2212yes—
16C22×C3×SD16C2211yescharacteristic
17C2×C3×C4×Q8C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C22×#216C2212yes—
20C22×#214C2212yes—
21C2×#930—21no—
22C22×C4⋊Dic3—21no—
23C2×#915—216no—
24C2×#915C22116yescomplemented
25C22×Dic12C2212yes—
26C22×Dic12C2212yes—
27C22×ΓL(1,𝔽25)C2212yes—
28Q8×D12—24no—
29Q8×D12C2218yescomplemented
30#5555—24no—
31#5555C2218yes—
32C2×#897—22no—
33C2×#897C2212yes—
34C2×#897C2212yes—
35C2×#886—22no—
36C2×C4×D12—21no—
37C2×C4×D12C2211yescharacteristic
38C2×#869—21no—
39C2×#869C2211yescharacteristic
40C2×#869C2211yescharacteristic
41C3×C4×SD16C2218yescomplemented
42C2×C3×#106C2211yescharacteristic
43C2×C3×#103C2211yescharacteristic
44C2×C3×#102C2211yescharacteristic
45C2×C3×C4×C8C2211yescharacteristic
46C2×(C2×D3)⋊C8C2212yes—
47#5031C2218yescomplemented
48#5019C2218yescomplemented
49C2×#810C2212yes—
50C2×#810C2212yes—
51C2×C4⋊(C3⋊C8)C2211yescharacteristic
52#4867C22116yescomplemented
53#4809C22116yescomplemented
54#4783C22116yescomplemented
55#4699C2218yescomplemented
56#4688C2218yescomplemented
57C22×#137—32no—
58C2×D4×Q8—31no—
59C2×#547—31no—
60#3276—316no—
61C2×#494—31no—
62C2×#470—32no—
63C2×#462—31no—
64C2×#461—31no—
65C2×#452—32no—
66C2×#451—32no—
67C2×C4×SD16D311yescharacteristic, complemented
68C2×#436—31no—
69C22×C3×Q8C2311yescharacteristic
70C22×C3×D4C2311yescharacteristic
71C23×C3×C4—21no—
72C22×#67—22no—
73C23×Dic3—22no—
74C2×D3×Q8—24no—
75C2×D3×Q8—28no—
76C2×D3×Q8—44no—
77C2×D3×Q8C2318yes—
78C2×#237—28no—
79C2×#237—24no—
80C2×#237—44no—
81C2×#237C2318yes—
82C22×D12D411yescharacteristic
83C22×C4×D3—22no—
84C22×C4×D3C2312yes—
85C22×Dic6—22no—
86C22×Dic6C2312yes—
87C22×Dic6C2312yes—
88C2×C3×SD16D418yescomplemented
89C2×C3×SD16C2314yes—
90C22×C3×C8C2311yescharacteristic
91C22×C3×C8C2311yescharacteristic
92C3×C4×Q8—22no—
93C3×C4×Q8C2314yes—
94C3×C4×D4—22no—
95C3×C4×D4C2314yes—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C22⋊C4—21no—
100C2×C3×C42—21no—
101C2×C3×C42C2311yescharacteristic
102C2×#216—28no—
103C2×#216C2318yes—
104C2×#214—28no—
105C2×#214C2318yes—
106#930—28no—
107C2×(C2×D3)⋊C4—22no—
108C2×(C2×D3)⋊C4—22no—
109C2×(C2×D3)⋊C4D412yes—
110C2×C4⋊Dic3—21no—
111C2×C4⋊Dic3—21no—
112C2×C4⋊Dic3—21no—
113C2×C4⋊Dic3—21no—
114C2×C4⋊Dic3—22no—
115C2×C4⋊Dic3—22no—
116C2×C4⋊Dic3—28no—
117C2×C4⋊Dic3D411yescharacteristic
118C22×C3⋊C8C2312yes—
119#915—264no—
120C2×Dic12—28no—
121C2×Dic12—44no—
122C2×Dic12C2318yes—
123C2×Dic12C2318yes—
124C2×ΓL(1,𝔽25)—28no—
125C2×ΓL(1,𝔽25)C2318yes—
126#897—24no—
127#897—24no—
128#897—216no—
129#897C2318yes—
130#897C2318yes—
131#886—216no—
132C4×D12—28no—
133C4×D12D414yes—
134C4×D12C2314yes—
135#869—22no—
136#869—22no—
137#869—28no—
138#869C2314yes—
139#869C2314yes—
140C3×#106C2314yes—
141C3×#103C2314yes—
142C3×#102C2314yes—
143C3×C4×C8C2314yes—
144(C2×D3)⋊C8C2318yes—
145#810C2318yes—
146#810C2318yes—
147C4⋊(C3⋊C8)C2314yes—
148C22×#41—32no—
149C23×Q8—32no—
150C2×#137—316no—
151C2×#137—616no—
152C22×Q16—32no—
153C22×Q16—32no—
154C22×Q16—32no—
155C22×SD16—32no—
156C22×SD16C2×D311yescharacteristic
157C22×M16—32no—
158D4×Q8—38no—
159D4×Q8—64no—
160#547—38no—
161#547—64no—
162C2×C4⋊Q8—31no—
163C2×C4⋊Q8—31no—
164C2×C4⋊Q8—61no—
165C2×#123—62no—
166C2×C22⋊Q8—32no—
167C2×C22⋊Q8—32no—
168C2×C22⋊Q8—62no—
169C2×#121—61no—
170C2×C4×Q8C2×D311yescharacteristic
171C2×C4×D4—31no—
172C2×C4×D4—61no—
173C2×C4×D4C2×D311yescharacteristic
174C22×C4⋊C4—61no—
175#494—38no—
176#470—316no—
177#462—38no—
178#461—38no—
179#452—316no—
180#451—316no—
181C4×SD16C2×D318yescomplemented
182#436—38no—
183C2×#106C2×D311yescharacteristic
184C2×C4⋊C8—31no—
185C2×#103—32no—
186C2×#103—32no—
187C2×#103C2×D311yescharacteristic
188C2×#102C2×D311yescharacteristic
189C2×C22⋊C8—32no—
190C2×C4×C8C2×D311yescharacteristic
191C24×C3—21no—
192C23×D3C2×D412yes—
193C2×C3×Q8—21no—
194C2×C3×Q8—22no—
195C2×C3×Q8C2×D414yes—
196C2×C3×Q8C2412yes—
197C2×C3×D4—21no—
198C2×C3×D4—22no—
199C2×C3×D4C2×D414yes—
200C2×C3×D4C2412yes—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—24no—
205C22×C3×C4—44no—
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2411yescharacteristic
209C2×#67—28no—
210C2×#67—48no—
211C22×Dic3—22no—
212C22×Dic3—22no—
213C22×Dic3—28no—
214C22×Dic3—48no—
215C22×Dic3C2×D412yes—
216D3×Q8—216no—
217D3×Q8—48no—
218D3×Q8—416no—
219#237—216no—
220#237—48no—
221#237—416no—
222C2×D12—24no—
223C2×D12C2×D414yes—
224C2×C4×D3—24no—
225C2×C4×D3—24no—
226C2×C4×D3—28no—
227C2×C4×D3—48no—
228C2×C4×D3C2×D414yes—
229C2×C4×D3C2414yes—
230C2×Dic6—24no—
231C2×Dic6—24no—
232C2×Dic6—24no—
233C2×Dic6—24no—
234C2×Dic6—22no—
235C2×Dic6—22no—
236C2×Dic6—28no—
237C2×Dic6—48no—
238C2×Dic6C2414yes—
239C2×Dic6C2414yes—
240C3×SD16C2×D4116yescomplemented
241C2×C3×C8—22no—
242C2×C3×C8C2×D414yes—
243C2×C3×C8C2412yes—
244C2×C3×C8C2412yes—
245C3×C4⋊C4—24no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4C2412yes—
249C3×C4⋊C4C2412yes—
250C3×C22⋊C4—24no—
251C3×C42—24no—
252C3×C42C2×D412yes—
253C3×C42C2412yes—
254#216—216no—
255#214—216no—
256(C2×D3)⋊C4—28no—
257(C2×D3)⋊C4—28no—
258(C2×D3)⋊C4C2×D418yes—
259C4⋊Dic3—24no—
260C4⋊Dic3—24no—
261C4⋊Dic3—24no—
262C4⋊Dic3—24no—
263C4⋊Dic3—28no—
264C4⋊Dic3—28no—
265C4⋊Dic3C2×D414yes—
266C2×C3⋊C8—24no—
267C2×C3⋊C8C2414yes—
268Dic12—216no—
269Dic12—48no—
270ΓL(1,𝔽25)—216no—
271C2×#41—38no—
272C2×#41—64no—
273C2×#41—68no—
274C2×#41—124no—
275C22×Q8—38no—
276C22×Q8—32no—
277C22×Q8—32no—
278C22×Q8—62no—
279C22×Q8—64no—
280C22×Q8—68no—
281C22×Q8—124no—
282C22×Q8C22×D311yescharacteristic
283C22×D4—31no—
284C22×D4—62no—
285C22×D4C22×D311yescharacteristic
286C23×C4—21no—
287C23×C4—32no—
288C23×C4—62no—
289C23×C4—62no—
290#137—664no—
291C2×Q16—38no—
292C2×Q16—38no—
293C2×Q16—38no—
294C2×Q16—68no—
295C2×Q16—68no—
296C2×Q16—124no—
297C2×SD16—38no—
298C2×SD16—68no—
299C2×SD16D1218yescomplemented
300C2×SD16C22×D314yes—
301C2×M16—38no—
302C2×M16—68no—
303C22×C8—32no—
304C22×C8C22×D311yescharacteristic
305C22×C8C22×D311yescharacteristic
306C4⋊Q8—34no—
307C4⋊Q8—34no—
308C4⋊Q8—68no—
309C4⋊Q8—62no—
310C4⋊Q8—62no—
311#123—616no—
312C22⋊Q8—38no—
313C22⋊Q8—38no—
314C22⋊Q8—616no—
315C22⋊Q8—64no—
316C22⋊Q8—64no—
317#121—68no—
318C4×Q8—22no—
319C4×Q8C22×D314yes—
320C4×D4—22no—
321C4×D4—34no—
322C4×D4—34no—
323C4×D4—68no—
324C4×D4C22×D314yes—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—31no—
327C2×C4⋊C4—68no—
328C2×C4⋊C4—62no—
329C2×C4⋊C4—62no—
330C2×C4⋊C4—61no—
331C2×C4⋊C4—61no—
332C2×C4⋊C4—61no—
333C2×C4⋊C4—61no—
334C2×C4⋊C4C22×D311yescharacteristic
335C2×C4⋊C4C22×D311yescharacteristic
336C2×C22⋊C4—21no—
337C2×C22⋊C4—32no—
338C2×C22⋊C4—62no—
339C2×C22⋊C4—62no—
340C2×C42—21no—
341C2×C42C22×D311yescharacteristic
342#106C22×D314yes—
343C4⋊C8—34no—
344#103—38no—
345#103—38no—
346#103C22×D314yes—
347#102C22×D314yes—
348C22⋊C8—38no—
349C4×C8C22×D314yes—
350C23×C3—24no—
351C23×C3—22no—
352C23×C3—42no—
353C23×C3—42no—
354C23×C3C22×D411yescharacteristic
355C22×D3—24no—
356C22×D3—24no—
357C22×D3—24no—
358C22×D3C22×D414yes—
359C3×Q8—24no—
360C3×Q8—22no—
361C3×Q8C22×D414yes—
362C3×D4—24no—
363C3×D4—22no—
364C3×D4C22×D414yes—
365C2×C3×C4—24no—
366C2×C3×C4—22no—
367C2×C3×C4—22no—
368C2×C3×C4—22no—
369C2×C3×C4—22no—
370C2×C3×C4—42no—
371C2×C3×C4—42no—
372C2×C3×C4—48no—
373C2×C3×C4#13714yescomplemented
374C2×C3×C4C22×D412yes—
375C2×C3×C4C22×D412yes—
376C2×C3×C4C22×D411yescharacteristic
377C2×C3×C4C22×D412yes—
378C2×C3×C4H(5,𝔽2)12yes—
379C2×C3×C4C2511yescharacteristic
380#67—416no—
381C2×Dic3—24no—
382C2×Dic3—24no—
383C2×Dic3—24no—
384C2×Dic3—416no—
385C2×Dic3—44no—
386C2×Dic3—44no—
387C2×Dic3C22×D414yes—
388D12—28no—
389C4×D3—28no—
390C4×D3—24no—
391C4×D3—24no—
392C4×D3—416no—
393Dic6—24no—
394Dic6—24no—
395Dic6—24no—
396Dic6—24no—
397Dic6—416no—
398Dic6—44no—
399Dic6—44no—
400C3×C8—22no—
401C3×C8C22×D414yes—
402C3⋊C8—24no—
403C24—21no—
404C24—32no—
405#41—616no—
406#41—128no—
407#41—1216no—
408C2×Q8—21no—
409C2×Q8—22no—
410C2×Q8—34no—
411C2×Q8—34no—
412C2×Q8—68no—
413C2×Q8—62no—
414C2×Q8—62no—
415C2×Q8—616no—
416C2×Q8—64no—
417C2×Q8—64no—
418C2×Q8—64no—
419C2×Q8—64no—
420C2×Q8—128no—
421C2×Q8—128no—
422C2×Q8—1216no—
423C2×Q8C2×D1214yes—
424C2×Q8C23×D312yes—
425C2×D4—21no—
426C2×D4—22no—
427C2×D4—34no—
428C2×D4—68no—
429C2×D4—64no—
430C2×D4—128no—
431C2×D4C2×D1214yes—
432C2×D4C23×D312yes—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—24no—
437C22×C4—32no—
438C22×C4—34no—
439C22×C4—34no—
440C22×C4—44no—
441C22×C4—68no—
442C22×C4—68no—
443C22×C4—64no—
444C22×C4—64no—
445C22×C4—62no—
446C22×C4—62no—
447C22×C4—128no—
448C22×C4—128no—
449C22×C4D3×D411yescharacteristic
450C22×C4D3×D411yescharacteristic
451C22×C4C23×D311yescharacteristic
452Q16—616no—
453Q16—616no—
454Q16—128no—
455SD16—616no—
456SD16C2×D12116yescomplemented
457M16—616no—
458C2×C8—22no—
459C2×C8—34no—
460C2×C8—64no—
461C2×C8C2×D1214yes—
462C2×C8C23×D312yes—
463C2×C8C23×D312yes—
464C4⋊C4—24no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—34no—
468C4⋊C4—68no—
469C4⋊C4—68no—
470C4⋊C4—64no—
471C4⋊C4—64no—
472C4⋊C4—64no—
473C4⋊C4—64no—
474C4⋊C4C23×D312yes—
475C4⋊C4C23×D312yes—
476C22⋊C4—24no—
477C22⋊C4—38no—
478C22⋊C4—68no—
479C22⋊C4—68no—
480C42—24no—
481C42D3×D412yes—
482C42C23×D312yes—
483C22×C3—24no—
484C22×C3—48no—
485C22×C3—42no—
486C22×C3—42no—
487C22×C3D4212yes—
488C22×C3C2×#13712yes—
489C22×C3#57812yes—
490C22×C3C23×D411yescharacteristic
491C2×D3—24no—
492C2×D3—24no—
493C2×D3—24no—
494C2×D3—48no—
495C3×C4—24no—
496C3×C4—22no—
497C3×C4—42no—
498C3×C4—42no—
499C3×C4D4212yes—
500C3×C4C2×#13714yescomplemented
501C3×C4C23×D411yescharacteristic
502C3×C4C2×H(5,𝔽2)11yescharacteristic
503Dic3—24no—
504Dic3—44no—
505Dic3—44no—
506C23—22no—
507C23—24no—
508C23—34no—
509C23—42no—
510C23—42no—
511C23—64no—
512C23—64no—
513C23—64no—
514C23C2×D3×D411yescharacteristic
515Q8—22no—
516Q8—24no—
517Q8—64no—
518Q8—64no—
519Q8—64no—
520Q8—64no—
521Q8—1216no—
522Q8—124no—
523Q8—124no—
524Q8C22×D1214yes—
525D4—22no—
526D4—24no—
527D4—68no—
528D4—1216no—
529D4C22×D1214yes—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—34no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—48no—
539C2×C4—64no—
540C2×C4—64no—
541C2×C4—64no—
542C2×C4—68no—
543C2×C4—64no—
544C2×C4—64no—
545C2×C4—1216no—
546C2×C4—1216no—
547C2×C4—124no—
548C2×C4—124no—
549C2×C4#91514yescomplemented
550C2×C4C22×D1212yes—
551C2×C4C2×D3×D412yes—
552C2×C4C2×D3×D412yes—
553C2×C4C2×D3×D411yescharacteristic
554C2×C4#100912yes—
555C2×C4C24×D311yescharacteristic
556C8—22no—
557C8—64no—
558C8C22×D1214yes—
559C2×C3—44no—
560C2×C3#327614yescomplemented
561C2×C3C2×D4211yescharacteristic
562C2×C3C22×#13711yescharacteristic
563C2×C3C2×#57811yescharacteristic
564D3—44no—
565C22—24no—
566C22—48no—
567C22—42no—
568C22—42no—
569C22—64no—
570C22—64no—
571C22—64no—
572C22—128no—
573C22D4×D1212yes—
574C22C2×#91512yes—
575C22#577112yes—
576C22C22×D3×D411yescharacteristic
577C4—24no—
578C4—22no—
579C4—42no—
580C4—42no—
581C4—64no—
582C4—124no—
583C4—124no—
584C4D4×D1212yes—
585C4C2×#91514yescomplemented
586C4C23×D1211yescharacteristic
587C4C2×#100911yescharacteristic
588C3C2×#327611yescharacteristic, complemented, Sylow
589C2—44no—
590C2—124no—
591C2#7967214yesdirect product factor
592C2C2×D4×D1211yescharacteristic
593C2C22×#91511yescharacteristic
594C2C2×#577111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×C4×SD16C2
socleC23×C3C22×D4
prime radicalC2×#5555C2
hypercenterC2×C4×SD16D3
hypocenterC3C2×#3276
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×SD16

Complements

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

NH ≅ G/Ntype
1. C22×#915C2semidirect
2. C2×Q8×D12C2semidirect
4. C2×C3×C4×SD16C2semidirect
5. C2×#5031C2semidirect
6. C2×#5019C2semidirect
7. #79672C2direct (N × H)
8. C2×#4867C2semidirect
9. C2×#4809C2semidirect
10. C2×#4783C2semidirect
11. C2×#4699C2semidirect
12. C2×#4688C2semidirect
24. C2×#915C22semidirect
29. Q8×D12C22semidirect
41. C3×C4×SD16C22semidirect
47. #5031C22semidirect
48. #5019C22semidirect
52. #4867C22semidirect
53. #4809C22semidirect
54. #4783C22semidirect
55. #4699C22semidirect
56. #4688C22semidirect
67. C2×C4×SD16D3semidirect
88. C2×C3×SD16D4semidirect
181. C4×SD16C2×D3semidirect
240. C3×SD16C2×D4semidirect
299. C2×SD16D12semidirect
373. C2×C3×C4#137semidirect
456. SD16C2×D12semidirect
500. C3×C4C2×#137semidirect
549. C2×C4#915semidirect
560. C2×C3#3276semidirect
585. C4C2×#915semidirect
588. C3C2×#3276semidirect
591. C2#79672direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#9152
C2×Q8×D122
C2×#55552
C2×C3×C4×SD162
C2×#50312
C2×#50192
#796722
C2×#48672
C2×#48092
C2×#47832
C2×#46992
C2×#46882
C2×#32763