Finite Group Databasebeta

← C2×#79731 #11670356

Subgroups — C2×#79731 #11670356

Order 768.

2794 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#918C212yesmaximal, complemented
2C2×Q8×D12C211yesmaximal, characteristic, complemented
3C2×#5577C211yesmaximal, characteristic
4C2×C3×C4×Q16C211yesmaximal, characteristic, complemented
5C2×#5031C211yesmaximal, characteristic, complemented
6C2×#5027C211yesmaximal, characteristic, complemented
7#79731C2116yesmaximal, direct product factor
8C2×#4886C212yesmaximal, complemented
9C2×#4808C212yesmaximal, complemented
10C2×#4806C212yesmaximal, complemented
11C2×#4693C211yesmaximal, characteristic, complemented
12C2×#4688C211yesmaximal, characteristic, complemented
13C2×#3278—31nomaximal, Sylow
14C22×#239C2212yes—
15C22×D3×Q8C2212yes—
16C22×C3×Q16C2211yescharacteristic
17C2×C3×C4×Q8C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C22×#216C2212yes—
20C22×#215C2212yes—
21C2×#918—216no—
22C2×#918C22116yescomplemented
23C22×#204C2212yes—
24C22×#204C2212yes—
25C22×ΓL(1,𝔽25)C2212yes—
26#5577—24no—
27#5577C2218yes—
28Q8×D12—24no—
29Q8×D12C2218yescomplemented
30C2×#897—22no—
31C2×#897C2212yes—
32C2×#895—22no—
33C2×#895C2212yes—
34C2×#874—21no—
35C2×#874C2211yescharacteristic
36C2×C4×D12—21no—
37C2×C4×D12—21no—
38C2×C4×D12C2211yescharacteristic
39C2×#869—21no—
40C2×#869C2211yescharacteristic
41C3×C4×Q16C2218yescomplemented
42C2×C3×#107C2211yescharacteristic
43C2×C3×#103C2211yescharacteristic
44C2×C3×#103C2211yescharacteristic
45C2×C3×C4×C8C2211yescharacteristic
46C2×(C2×D3)⋊C8C2212yes—
47#5031C2218yescomplemented
48#5027C2218yescomplemented
49C2×#810C2212yes—
50C2×#809C2212yes—
51C2×C4⋊(C3⋊C8)C2211yescharacteristic
52#4886C22116yescomplemented
53#4808C22116yescomplemented
54#4806C22116yescomplemented
55#4693C2218yescomplemented
56#4688C2218yescomplemented
57C22×#137—32no—
58C2×#550—31no—
59C2×D4×Q8—31no—
60#3278—316no—
61C2×#492—31no—
62C2×#468—32no—
63C2×#462—31no—
64C2×#460—31no—
65C2×#452—32no—
66C2×#448—32no—
67C2×C4×Q16D311yescharacteristic, complemented
68C2×#436—31no—
69C22×C3×Q8C2311yescharacteristic
70C22×C3×Q8C2311yescharacteristic
71C2×#239—24no—
72C2×#239—28no—
73C2×#239—44no—
74C2×#239C2318yes—
75C2×D3×Q8—24no—
76C2×D3×Q8—28no—
77C2×D3×Q8—44no—
78C2×D3×Q8C2318yes—
79C22×D12—22no—
80C22×D12—21no—
81C22×D12—22no—
82C22×D12—21no—
83C22×D12D411yescharacteristic
84C22×D12C2312yes—
85C22×C4×D3—22no—
86C22×C4×D3—22no—
87C22×C4×D3C2312yes—
88C22×Dic6—22no—
89C22×Dic6C2312yes—
90C2×C3×Q16D418yescomplemented
91C2×C3×Q16C2314yes—
92C22×C3×C8C2311yescharacteristic
93C22×C3×C8C2311yescharacteristic
94C3×C4×Q8—22no—
95C3×C4×Q8—22no—
96C3×C4×Q8C2314yes—
97C3×C4×Q8C2314yes—
98C2×C3×C4⋊C4—21no—
99C2×C3×C4⋊C4—21no—
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C4⋊C4C2311yescharacteristic
102C2×C3×C42—21no—
103C2×C3×C42—21no—
104C2×C3×C42C2311yescharacteristic
105C2×#216—28no—
106C2×#216C2318yes—
107C2×#215—28no—
108C2×#215C2318yes—
109C2×(C2×D3)⋊C4—22no—
110C2×(C2×D3)⋊C4—22no—
111C2×(C2×D3)⋊C4D412yes—
112C2×C4⋊Dic3—21no—
113C2×C4⋊Dic3—21no—
114C2×C4⋊Dic3—22no—
115C2×C4⋊Dic3D411yescharacteristic
116C22×C3⋊C8C2312yes—
117#918—264no—
118C2×#204—28no—
119C2×#204—44no—
120C2×#204C2318yes—
121C2×#204C2318yes—
122C2×ΓL(1,𝔽25)—28no—
123C2×ΓL(1,𝔽25)C2318yes—
124#897—24no—
125#897—216no—
126#897C2318yes—
127#895—24no—
128#895—216no—
129#895C2318yes—
130#874—28no—
131#874—22no—
132#874C2314yes—
133C4×D12—28no—
134C4×D12—28no—
135C4×D12D414yes—
136C4×D12C2314yes—
137#869—22no—
138#869—28no—
139#869C2314yes—
140C3×#107C2314yes—
141C3×#103C2314yes—
142C3×#103C2314yes—
143C3×C4×C8C2314yes—
144(C2×D3)⋊C8C2318yes—
145#810C2318yes—
146#809C2318yes—
147C4⋊(C3⋊C8)C2314yes—
148C22×#41—32no—
149C23×Q8—32no—
150C2×#137—316no—
151C2×#137—616no—
152C22×Q16—32no—
153C22×Q16C2×D311yescharacteristic
154C22×SD16—32no—
155C22×SD16—32no—
156C22×SD16—32no—
157C22×M16—32no—
158#550—38no—
159#550—64no—
160D4×Q8—38no—
161D4×Q8—64no—
162C2×C4⋊Q8—31no—
163C2×C4⋊Q8—61no—
164C2×#127—31no—
165C2×#127—61no—
166C2×C22⋊Q8—32no—
167C2×C22⋊Q8—62no—
168C2×#121—32no—
169C2×#121—62no—
170C2×C4×Q8C2×D311yescharacteristic
171C2×C4×Q8C2×D311yescharacteristic
172C2×C4×D4—31no—
173C2×C4×D4—61no—
174C2×C4×D4—61no—
175#492—38no—
176#468—316no—
177#462—38no—
178#460—38no—
179#452—316no—
180#448—316no—
181C4×Q16C2×D318yescomplemented
182#436—38no—
183C2×#107C2×D311yescharacteristic
184C2×C4⋊C8—31no—
185C2×#103—32no—
186C2×#103C2×D311yescharacteristic
187C2×#103C2×D311yescharacteristic
188C2×#102—32no—
189C2×C22⋊C8—32no—
190C2×C4×C8C2×D311yescharacteristic
191C23×D3—22no—
192C23×D3C2×D412yes—
193C2×C3×Q8—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8—22no—
196C2×C3×Q8—22no—
197C2×C3×Q8C2×D414yes—
198C2×C3×Q8C2×D414yes—
199C2×C3×Q8C2412yes—
200C2×C3×Q8C2412yes—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2411yescharacteristic
208C22×Dic3—22no—
209C22×Dic3C2×D412yes—
210#239—216no—
211#239—48no—
212#239—416no—
213D3×Q8—216no—
214D3×Q8—48no—
215D3×Q8—416no—
216C2×D12—24no—
217C2×D12—24no—
218C2×D12—24no—
219C2×D12—28no—
220C2×D12—24no—
221C2×D12—22no—
222C2×D12—28no—
223C2×D12—24no—
224C2×D12—48no—
225C2×D12—44no—
226C2×D12—82no—
227C2×D12—84no—
228C2×D12C2×D414yes—
229C2×D12C2414yes—
230C2×C4×D3—24no—
231C2×C4×D3—24no—
232C2×C4×D3—28no—
233C2×C4×D3—28no—
234C2×C4×D3—48no—
235C2×C4×D3—48no—
236C2×C4×D3C2×D414yes—
237C2×C4×D3C2414yes—
238C2×Dic6—24no—
239C2×Dic6—24no—
240C2×Dic6—22no—
241C2×Dic6—28no—
242C2×Dic6—48no—
243C2×Dic6C2414yes—
244C3×Q16C2×D4116yescomplemented
245C2×C3×C8—22no—
246C2×C3×C8C2×D414yes—
247C2×C3×C8C2412yes—
248C2×C3×C8C2412yes—
249C3×C4⋊C4—24no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—24no—
252C3×C4⋊C4—21no—
253C3×C4⋊C4C2412yes—
254C3×C4⋊C4C2412yes—
255C3×C42—24no—
256C3×C42—24no—
257C3×C42C2×D412yes—
258C3×C42C2412yes—
259#216—216no—
260#215—216no—
261(C2×D3)⋊C4—28no—
262(C2×D3)⋊C4—28no—
263(C2×D3)⋊C4C2×D418yes—
264C4⋊Dic3—24no—
265C4⋊Dic3—24no—
266C4⋊Dic3—28no—
267C4⋊Dic3C2×D414yes—
268C2×C3⋊C8—24no—
269C2×C3⋊C8C2414yes—
270#204—216no—
271#204—48no—
272ΓL(1,𝔽25)—216no—
273C2×#41—38no—
274C2×#41—64no—
275C2×#41—68no—
276C2×#41—124no—
277C22×Q8—38no—
278C22×Q8—32no—
279C22×Q8—62no—
280C22×Q8—64no—
281C22×Q8—68no—
282C22×Q8—124no—
283C22×Q8C22×D311yescharacteristic
284C22×Q8C22×D311yescharacteristic
285C22×D4—31no—
286C22×D4—32no—
287C22×D4—62no—
288C22×D4—62no—
289C22×D4—61no—
290C22×D4—61no—
291C23×C4—32no—
292C23×C4—62no—
293C23×C4—62no—
294#137—664no—
295C2×Q16—38no—
296C2×Q16—68no—
297C2×Q16D1218yescomplemented
298C2×Q16C22×D314yes—
299C2×SD16—38no—
300C2×SD16—38no—
301C2×SD16—38no—
302C2×SD16—68no—
303C2×SD16—68no—
304C2×SD16—124no—
305C2×M16—38no—
306C2×M16—68no—
307C22×C8—32no—
308C22×C8C22×D311yescharacteristic
309C22×C8C22×D311yescharacteristic
310C4⋊Q8—34no—
311C4⋊Q8—68no—
312C4⋊Q8—62no—
313#127—34no—
314#127—68no—
315#127—62no—
316C22⋊Q8—38no—
317C22⋊Q8—616no—
318C22⋊Q8—64no—
319#121—38no—
320#121—616no—
321#121—64no—
322C4×Q8—22no—
323C4×Q8—22no—
324C4×Q8C22×D314yes—
325C4×Q8C22×D314yes—
326C4×D4—34no—
327C4×D4—34no—
328C4×D4—68no—
329C4×D4—68no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—31no—
333C2×C4⋊C4—62no—
334C2×C4⋊C4—61no—
335C2×C4⋊C4—61no—
336C2×C4⋊C4C22×D311yescharacteristic
337C2×C4⋊C4C22×D311yescharacteristic
338C2×C22⋊C4—32no—
339C2×C22⋊C4—62no—
340C2×C22⋊C4—62no—
341C2×C42—21no—
342C2×C42—21no—
343C2×C42C22×D311yescharacteristic
344#107C22×D314yes—
345C4⋊C8—34no—
346#103—38no—
347#103C22×D314yes—
348#103C22×D314yes—
349#102—38no—
350C22⋊C8—38no—
351C4×C8C22×D314yes—
352C23×C3C22×D411yescharacteristic
353C22×D3—24no—
354C22×D3—24no—
355C22×D3—24no—
356C22×D3—24no—
357C22×D3—44no—
358C22×D3—44no—
359C22×D3—44no—
360C22×D3C22×D414yes—
361C3×Q8—24no—
362C3×Q8—24no—
363C3×Q8—22no—
364C3×Q8—22no—
365C3×Q8C22×D414yes—
366C3×Q8C22×D414yes—
367C2×C3×C4—24no—
368C2×C3×C4—24no—
369C2×C3×C4—22no—
370C2×C3×C4—22no—
371C2×C3×C4—22no—
372C2×C3×C4—22no—
373C2×C3×C4—22no—
374C2×C3×C4—42no—
375C2×C3×C4—42no—
376C2×C3×C4#13714yescomplemented
377C2×C3×C4C22×D412yes—
378C2×C3×C4C22×D412yes—
379C2×C3×C4C22×D411yescharacteristic
380C2×C3×C4C22×D412yes—
381C2×C3×C4H(5,𝔽2)12yes—
382C2×C3×C4C2511yescharacteristic
383C2×Dic3—24no—
384C2×Dic3—24no—
385C2×Dic3—44no—
386C2×Dic3C22×D414yes—
387D12—28no—
388D12—24no—
389D12—24no—
390D12—416no—
391D12—44no—
392D12—48no—
393D12—84no—
394D12—88no—
395C4×D3—28no—
396C4×D3—24no—
397C4×D3—24no—
398C4×D3—416no—
399C4×D3—416no—
400Dic6—24no—
401Dic6—24no—
402Dic6—416no—
403Dic6—44no—
404C3×C8—22no—
405C3×C8C22×D414yes—
406C3⋊C8—24no—
407C24—32no—
408C24—62no—
409#41—616no—
410#41—128no—
411#41—1216no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—34no—
417C2×Q8—68no—
418C2×Q8—62no—
419C2×Q8—616no—
420C2×Q8—64no—
421C2×Q8—64no—
422C2×Q8—128no—
423C2×Q8—128no—
424C2×Q8—1216no—
425C2×Q8C2×D1214yes—
426C2×Q8C2×D1214yes—
427C2×Q8C23×D312yes—
428C2×Q8C23×D312yes—
429C2×D4—34no—
430C2×D4—34no—
431C2×D4—68no—
432C2×D4—68no—
433C2×D4—64no—
434C2×D4—64no—
435C2×D4—64no—
436C2×D4—62no—
437C2×D4—64no—
438C2×D4—64no—
439C2×D4—128no—
440C2×D4—124no—
441C2×D4—242no—
442C2×D4—244no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—32no—
448C22×C4—34no—
449C22×C4—34no—
450C22×C4—68no—
451C22×C4—68no—
452C22×C4—64no—
453C22×C4—64no—
454C22×C4—62no—
455C22×C4—128no—
456C22×C4—128no—
457C22×C4D3×D411yescharacteristic
458C22×C4D3×D411yescharacteristic
459C22×C4C23×D311yescharacteristic
460Q16—616no—
461Q16C2×D12116yescomplemented
462SD16—616no—
463SD16—616no—
464SD16—128no—
465M16—616no—
466C2×C8—22no—
467C2×C8—34no—
468C2×C8—64no—
469C2×C8C2×D1214yes—
470C2×C8C23×D312yes—
471C2×C8C23×D312yes—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—34no—
477C4⋊C4—68no—
478C4⋊C4—64no—
479C4⋊C4—64no—
480C4⋊C4C23×D312yes—
481C4⋊C4C23×D312yes—
482C22⋊C4—38no—
483C22⋊C4—68no—
484C22⋊C4—68no—
485C42—24no—
486C42—24no—
487C42D3×D412yes—
488C42C23×D312yes—
489C22×C3D4212yes—
490C22×C3C2×#13712yes—
491C22×C3#57712yes—
492C22×C3C23×D411yescharacteristic
493C2×D3—24no—
494C2×D3—24no—
495C2×D3—24no—
496C2×D3—48no—
497C2×D3—44no—
498C2×D3—44no—
499C2×D3—44no—
500C2×D3—88no—
501C3×C4—24no—
502C3×C4—24no—
503C3×C4—22no—
504C3×C4—42no—
505C3×C4—42no—
506C3×C4D4212yes—
507C3×C4C2×#13714yescomplemented
508C3×C4C23×D411yescharacteristic
509C3×C4C2×H(5,𝔽2)11yescharacteristic
510Dic3—24no—
511Dic3—44no—
512C23—34no—
513C23—64no—
514C23—64no—
515C23—64no—
516C23—64no—
517C23—124no—
518C23—124no—
519C23—124no—
520C23C2×D3×D411yescharacteristic
521Q8—22no—
522Q8—22no—
523Q8—24no—
524Q8—24no—
525Q8—64no—
526Q8—64no—
527Q8—1216no—
528Q8—124no—
529Q8C22×D1214yes—
530Q8C22×D1214yes—
531D4—64no—
532D4—64no—
533D4—68no—
534D4—1216no—
535D4—128no—
536D4—124no—
537D4—244no—
538D4—248no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—34no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—64no—
550C2×C4—64no—
551C2×C4—64no—
552C2×C4—68no—
553C2×C4—64no—
554C2×C4—1216no—
555C2×C4—1216no—
556C2×C4—124no—
557C2×C4#91814yescomplemented
558C2×C4C22×D1212yes—
559C2×C4C2×D3×D412yes—
560C2×C4C2×D3×D412yes—
561C2×C4C2×D3×D411yescharacteristic
562C2×C4#100912yes—
563C2×C4C24×D311yescharacteristic
564C8—22no—
565C8—64no—
566C8C22×D1214yes—
567C2×C3#327814yescomplemented
568C2×C3C2×D4211yescharacteristic
569C2×C3C22×#13711yescharacteristic
570C2×C3C2×#57711yescharacteristic
571D3—44no—
572D3—84no—
573C22—64no—
574C22—64no—
575C22—64no—
576C22—128no—
577C22—124no—
578C22—124no—
579C22—124no—
580C22—248no—
581C22D4×D1212yes—
582C22C2×#91812yes—
583C22#577012yes—
584C22C22×D3×D411yescharacteristic
585C4—24no—
586C4—24no—
587C4—22no—
588C4—42no—
589C4—42no—
590C4—64no—
591C4—124no—
592C4D4×D1212yes—
593C4C2×#91814yescomplemented
594C4C23×D1211yescharacteristic
595C4C2×#100911yescharacteristic
596C3C2×#327811yescharacteristic, complemented, Sylow
597C2—124no—
598C2—244no—
599C2#7973114yesdirect product factor
600C2C2×D4×D1211yescharacteristic
601C2C22×#91811yescharacteristic
602C2C2×#577011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×C4×Q16C2
socleC23×C3C22×D4
prime radicalC2×#5577C2
hypercenterC2×C4×Q16D3
hypocenterC3C2×#3278
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×Q16

Complements

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

NH ≅ G/Ntype
1. C22×#918C2semidirect
2. C2×Q8×D12C2semidirect
4. C2×C3×C4×Q16C2semidirect
5. C2×#5031C2semidirect
6. C2×#5027C2semidirect
7. #79731C2direct (N × H)
8. C2×#4886C2semidirect
9. C2×#4808C2semidirect
10. C2×#4806C2semidirect
11. C2×#4693C2semidirect
12. C2×#4688C2semidirect
22. C2×#918C22semidirect
29. Q8×D12C22semidirect
41. C3×C4×Q16C22semidirect
47. #5031C22semidirect
48. #5027C22semidirect
52. #4886C22semidirect
53. #4808C22semidirect
54. #4806C22semidirect
55. #4693C22semidirect
56. #4688C22semidirect
67. C2×C4×Q16D3semidirect
90. C2×C3×Q16D4semidirect
181. C4×Q16C2×D3semidirect
244. C3×Q16C2×D4semidirect
297. C2×Q16D12semidirect
376. C2×C3×C4#137semidirect
461. Q16C2×D12semidirect
507. C3×C4C2×#137semidirect
557. C2×C4#918semidirect
567. C2×C3#3278semidirect
593. C4C2×#918semidirect
596. C3C2×#3278semidirect
599. C2#79731direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#9182
C2×Q8×D122
C2×#55772
C2×C3×C4×Q162
C2×#50312
C2×#50272
#797312
C2×#48862
C2×#48082
C2×#48062
C2×#46932
C2×#46882
C2×#32783