Finite Group Databasebeta

← C2×#79977 #11675690

Subgroups — C2×#79977 #11675690

Order 768.

2006 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×D12C211yesmaximal, characteristic
2C2×#5567C211yesmaximal, characteristic, complemented
3C2×C3×#446C211yesmaximal, characteristic, complemented
4#79977C2116yesmaximal, direct product factor
5C2×#4850C212yesmaximal, complemented
6C2×#4851C211yesmaximal, characteristic, complemented
7C2×#4847C212yesmaximal
8C2×#4846C211yesmaximal, characteristic
9C2×#4710C212yesmaximal
10C2×#4702C212yesmaximal, complemented
11C2×C4×#204C211yesmaximal, characteristic
12C2×#4682C211yesmaximal, characteristic, complemented
13C2×#3386—31nomaximal, Sylow
14C22×D3×Q8—21no—
15C2×C3×C4×Q8C2211yescharacteristic
16C2×#944—21no—
17C22×#204C2211yescharacteristic
18Q8×D12—24no—
19Q8×D12C2218yes—
20#5567—24no—
21#5567C2218yescomplemented
22C2×#897—21no—
23C2×#897—22no—
24C2×#889—22no—
25C2×#887—21no—
26C2×C4×D12C2211yescharacteristic
27C2×C4×D12C2212yes—
28C2×#870C2212yes—
29C2×#869C2211yescharacteristic
30C2×#869C2211yescharacteristic
31C2×#869C2212yes—
32C2×C4×Dic6C2211yescharacteristic
33C2×C4×Dic6C2212yes—
34C3×#446C2218yescomplemented
35C2×C3×C4⋊C8C2211yescharacteristic
36C2×C3×C4⋊C8C2212yes—
37C2×C3×#97C2212yes—
38C2×C3×C4×C8C2211yescharacteristic
39C2×#821C2211yescharacteristic
40C2×#821C2212yes—
41C2×#818C2212yes—
42C2×#818C2212yes—
43C2×#818C2212yes—
44C2×#817C2211yescharacteristic
45C2×#817C2212yes—
46C2×#816C2211yescharacteristic
47C2×#816C2212yes—
48#4851C2218yescomplemented
49#4850C22116yescomplemented
50#4847C22116yes—
51#4846C2218yes—
52#4710C22116yes—
53#4702C22116yescomplemented
54C4×#204C2218yes—
55#4682C2218yescomplemented
56C2×#557—31no—
57C2×D4×Q8—31no—
58#3386—316no—
59C2×#501—32no—
60C2×#500—31no—
61C2×#477—32no—
62C2×#475—31no—
63C2×#478—32no—
64C2×#474—31no—
65C2×#446D311yescharacteristic, complemented
66C2×#440—32no—
67C2×C4×SD16—31no—
68C22×C3×Q8D411yescharacteristic
69C2×D3×Q8—28no—
70C2×D3×Q8—48no—
71C22×D12C2311yescharacteristic
72C22×C4×D3—22no—
73C22×C4×D3—21no—
74C22×Dic6—22no—
75C22×Dic6—21no—
76C22×Dic6—21no—
77C22×Dic6C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C22×C3×C8C2312yes—
80C22×C3×C8C2311yescharacteristic
81C3×C4×Q8D414yes—
82C3×C4×Q8C2314yes—
83C2×C3×C4⋊C4D411yescharacteristic
84C2×C3×C4⋊C4D412yes—
85C2×C3×C42C2311yescharacteristic
86C2×C3×C42C2312yes—
87#944—28no—
88C2×(C2×D3)⋊C4—22no—
89C2×(C2×D3)⋊C4—21no—
90C2×C4⋊Dic3—21no—
91C2×C4⋊Dic3—22no—
92C2×C4⋊Dic3—21no—
93C2×C4⋊Dic3C2312yes—
94C2×C4⋊Dic3C2311yescharacteristic
95C2×C4⋊Dic3C2312yes—
96C2×C4⋊Dic3C2311yescharacteristic
97C2×Dic3⋊C4—21no—
98C2×Dic3⋊C4—22no—
99C2×Dic3⋊C4—22no—
100C2×Dic3⋊C4—22no—
101C2×C4×Dic3—22no—
102C2×C4×Dic3—21no—
103C2×#204Q818yescomplemented
104C2×#204C2314yes—
105#897—28no—
106#897—216no—
107#889—216no—
108#887—28no—
109C4×D12—22no—
110C4×D12—24no—
111C4×D12C2314yes—
112C4×D12C2318yes—
113#870—24no—
114#870C2318yes—
115#869—22no—
116#869—22no—
117#869—24no—
118#869C2314yes—
119#869C2314yes—
120#869C2318yes—
121C4×Dic6—24no—
122C4×Dic6—22no—
123C4×Dic6C2314yes—
124C4×Dic6C2318yes—
125C3×C4⋊C8C2314yes—
126C3×C4⋊C8C2318yes—
127C3×#97C2318yes—
128C3×C4×C8C2314yes—
129#821C2314yes—
130#821C2318yes—
131#818C2318yes—
132#818C2318yes—
133#818C2318yes—
134#817C2314yes—
135#817C2318yes—
136#816C2314yes—
137#816C2318yes—
138C23×Q8—61no—
139C22×SD16—31no—
140#557—38no—
141#557—64no—
142D4×Q8—38no—
143D4×Q8—64no—
144C2×C4⋊Q8—31no—
145C2×C4⋊Q8—32no—
146C2×C4⋊Q8—31no—
147C2×C4⋊Q8—61no—
148C2×#125—32no—
149C2×#125—62no—
150C2×C22⋊Q8—61no—
151C2×C22⋊Q8—62no—
152C2×C4×Q8—31no—
153C2×C4×Q8—32no—
154C2×C4×Q8—61no—
155C2×C4×Q8C2×D311yescharacteristic
156C2×C4×D4—32no—
157C2×C4×D4—31no—
158#501—316no—
159#500—38no—
160#478—316no—
161#477—316no—
162#475—38no—
163#474—38no—
164#446C2×D318yescomplemented
165#440—316no—
166C4×SD16—38no—
167C2×#107—32no—
168C2×#107—32no—
169C2×#107—32no—
170C2×#106—32no—
171C2×#106—31no—
172C2×C4⋊C8C2×D312yes—
173C2×C4⋊C8C2×D311yescharacteristic
174C2×#103—32no—
175C2×#103—31no—
176C2×#102—31no—
177C2×#102—32no—
178C2×#97C2×D312yes—
179C2×C4×C8C2×D311yescharacteristic
180C23×D3—21no—
181C2×C3×Q8—24no—
182C2×C3×Q8C2×D414yes—
183C22×C3×C4C2×D412yes—
184C22×C3×C4C2×D412yes—
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2411yescharacteristic
188C22×Dic3—21no—
189C22×Dic3—21no—
190C22×Dic3—22no—
191C22×Dic3—21no—
192C22×Dic3—22no—
193D3×Q8—432no—
194C2×D12—21no—
195C2×D12—22no—
196C2×D12C2×Q814yes—
197C2×D12C2412yes—
198C2×C4×D3—28no—
199C2×C4×D3—24no—
200C2×C4×D3—48no—
201C2×C4×D3—44no—
202C2×Dic6—21no—
203C2×Dic6—22no—
204C2×Dic6—24no—
205C2×Dic6—28no—
206C2×Dic6—24no—
207C2×Dic6—48no—
208C2×Dic6—44no—
209C2×Dic6—44no—
210C2×Dic6C2×Q814yescomplemented
211C2×Dic6C2412yes—
212C2×C3×C8—22no—
213C2×C3×C8—24no—
214C2×C3×C8C2×Q814yescomplemented
215C2×C3×C8C2414yes—
216C2×C3×C8C2412yes—
217C2×C3×C8C2412yes—
218C3×C4⋊C4C2×D414yes—
219C3×C4⋊C4C2×D418yes—
220C3×C42C2×D412yes—
221C3×C42C2×D414yes—
222C3×C42C2412yes—
223C3×C42C2414yes—
224(C2×D3)⋊C4—28no—
225(C2×D3)⋊C4—24no—
226C4⋊Dic3—22no—
227C4⋊Dic3—21no—
228C4⋊Dic3—22no—
229C4⋊Dic3—21no—
230C4⋊Dic3—24no—
231C4⋊Dic3—28no—
232C4⋊Dic3—24no—
233C4⋊Dic3C2412yes—
234C4⋊Dic3C2414yes—
235C4⋊Dic3C2412yes—
236C4⋊Dic3C2414yes—
237Dic3⋊C4—24no—
238Dic3⋊C4—28no—
239Dic3⋊C4—28no—
240Dic3⋊C4—28no—
241C4×Dic3—28no—
242C4×Dic3—24no—
243#204C2×Q8116yescomplemented
244C22×Q8—31no—
245C22×Q8—68no—
246C22×Q8—61no—
247C22×Q8—62no—
248C22×Q8—61no—
249C22×Q8—128no—
250C22×Q8D1211yescharacteristic
251C22×D4—31no—
252C23×C4—62no—
253C23×C4—61no—
254C2×SD16—38no—
255C2×SD16—34no—
256C22×C8C22×D311yescharacteristic
257C22×C8C22×D312yes—
258C22×C8C22×D311yescharacteristic
259C4⋊Q8—34no—
260C4⋊Q8—38no—
261C4⋊Q8—34no—
262C4⋊Q8—68no—
263C4⋊Q8—64no—
264C4⋊Q8—62no—
265C4⋊Q8—62no—
266#125—38no—
267#125—616no—
268#125—64no—
269C22⋊Q8—68no—
270C22⋊Q8—616no—
271C4×Q8—34no—
272C4×Q8—38no—
273C4×Q8—68no—
274C4×Q8—64no—
275C4×Q8—62no—
276C4×Q8D1214yes—
277C4×Q8C22×D314yes—
278C4×D4—34no—
279C4×D4—38no—
280C4×D4—64no—
281C4×D4—62no—
282C2×C4⋊C4—32no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4—32no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—62no—
288C2×C4⋊C4—62no—
289C2×C4⋊C4—62no—
290C2×C4⋊C4—61no—
291C2×C4⋊C4—62no—
292C2×C4⋊C4—61no—
293C2×C4⋊C4D1211yescharacteristic
294C2×C4⋊C4D1212yes—
295C2×C22⋊C4—62no—
296C2×C22⋊C4—61no—
297C2×C42—62no—
298C2×C42—61no—
299C2×C42C22×D312yes—
300C2×C42C22×D311yescharacteristic
301#107—38no—
302#107—38no—
303#107—38no—
304#106—34no—
305#106—38no—
306C4⋊C8C22×D314yes—
307C4⋊C8C22×D318yes—
308#103—34no—
309#103—38no—
310#102—34no—
311#102—38no—
312#97C22×D318yes—
313C4×C8C22×D314yes—
314C23×C3C22×D411yescharacteristic
315C22×D3—24no—
316C22×D3—22no—
317C22×D3—42no—
318C22×D3—42no—
319C3×Q8—28no—
320C2×C3×C4—24no—
321C2×C3×C4—24no—
322C2×C3×C4—22no—
323C2×C3×C4AGL(1,ℤ/8)14yes—
324C2×C3×C4C22×D414yes—
325C2×C3×C4C22×D414yes—
326C2×C3×C4C22×D412yes—
327C2×C3×C4C22×D411yescharacteristic
328C2×C3×C4C22×D412yes—
329C2×C3×C4C22×Q812yes—
330C2×C3×C42−1+412yes—
331C2×C3×C4C2511yescharacteristic
332C2×Dic3—24no—
333C2×Dic3—22no—
334C2×Dic3—22no—
335C2×Dic3—24no—
336C2×Dic3—22no—
337C2×Dic3—24no—
338C2×Dic3—42no—
339C2×Dic3—44no—
340C2×Dic3—44no—
341C2×Dic3—42no—
342D12—24no—
343D12—22no—
344D12C22×Q814yes—
345C4×D3—416no—
346C4×D3—48no—
347Dic6—24no—
348Dic6—22no—
349Dic6—416no—
350Dic6—48no—
351Dic6—48no—
352Dic6C22×Q814yescomplemented
353C3×C8—24no—
354C3×C8—22no—
355C3×C8C22×Q814yescomplemented
356C24—61no—
357C2×Q8—24no—
358C2×Q8—34no—
359C2×Q8—32no—
360C2×Q8—61no—
361C2×Q8—64no—
362C2×Q8—68no—
363C2×Q8—64no—
364C2×Q8—62no—
365C2×Q8—128no—
366C2×Q8—1232no—
367C2×Q8—124no—
368C2×Q8—124no—
369C2×Q8C2×D1214yes—
370C2×D4—34no—
371C2×D4—32no—
372C2×D4—61no—
373C2×D4—62no—
374C22×C4—61no—
375C22×C4—61no—
376C22×C4—62no—
377C22×C4—62no—
378C22×C4—61no—
379C22×C4—68no—
380C22×C4—64no—
381C22×C4—128no—
382C22×C4—124no—
383C22×C4C2×D1212yes—
384C22×C4C2×D1211yescharacteristic
385C22×C4C2×D1212yes—
386C22×C4C2×D1211yescharacteristic
387C22×C4C23×D311yescharacteristic
388SD16—316no—
389C2×C8—22no—
390C2×C8—24no—
391C2×C8D3×Q814yescomplemented
392C2×C8C23×D312yes—
393C2×C8C23×D314yes—
394C2×C8C23×D312yes—
395C4⋊C4—32no—
396C4⋊C4—34no—
397C4⋊C4—32no—
398C4⋊C4—34no—
399C4⋊C4—62no—
400C4⋊C4—61no—
401C4⋊C4—61no—
402C4⋊C4—62no—
403C4⋊C4—64no—
404C4⋊C4—68no—
405C4⋊C4—68no—
406C4⋊C4—68no—
407C4⋊C4—64no—
408C4⋊C4—68no—
409C4⋊C4—64no—
410C4⋊C4C2×D1214yes—
411C4⋊C4C2×D1218yes—
412C22⋊C4—68no—
413C22⋊C4—64no—
414C42—68no—
415C42—64no—
416C42C2×D1214yes—
417C42C2×D1212yes—
418C42C23×D312yes—
419C42C23×D314yes—
420C22×C3D4×Q812yes—
421C22×C3C2×AGL(1,ℤ/8)12yes—
422C22×C3#57812yes—
423C22×C3C23×D411yescharacteristic
424C2×D3—24no—
425C2×D3—48no—
426C2×D3—42no—
427C2×D3—42no—
428C3×C4—24no—
429C3×C4—24no—
430C3×C4—22no—
431C3×C4D4×Q812yes—
432C3×C4C2×AGL(1,ℤ/8)14yes—
433C3×C4C23×Q811yescharacteristic
434C3×C4C2×2−1+411yescharacteristic
435Dic3—24no—
436Dic3—42no—
437Dic3—44no—
438Dic3—44no—
439Dic3—42no—
440C23—62no—
441C23—64no—
442C23—122no—
443C23—122no—
444C23C22×D1211yescharacteristic
445Q8—28no—
446Q8—34no—
447Q8—62no—
448Q8—64no—
449Q8—1216no—
450Q8—128no—
451Q8—128no—
452D4—34no—
453D4—62no—
454D4—64no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—62no—
459C2×C4—64no—
460C2×C4—62no—
461C2×C4—64no—
462C2×C4—64no—
463C2×C4—62no—
464C2×C4—1216no—
465C2×C4—124no—
466C2×C4—122no—
467C2×C4—124no—
468C2×C4—122no—
469C2×C4—128no—
470C2×C4#90814yes—
471C2×C4C22×D1214yes—
472C2×C4C22×D1212yes—
473C2×C4C22×D1214yes—
474C2×C4C22×D1212yes—
475C2×C4C22×D1211yescharacteristic
476C2×C4C2×D3×Q812yes—
477C2×C4#101012yes—
478C2×C4C24×D311yescharacteristic
479C8—22no—
480C8—24no—
481C8C2×D3×Q814yescomplemented
482C2×C3#338614yescomplemented
483C2×C3C2×D4×Q811yescharacteristic
484C2×C3C22×AGL(1,ℤ/8)11yescharacteristic
485C2×C3C2×#57811yescharacteristic
486D3—44no—
487C22—64no—
488C22—128no—
489C22—122no—
490C22—122no—
491C22Q8×D1212yes—
492C22C2×#90812yes—
493C22#575312yes—
494C22C23×D1211yescharacteristic
495C4—24no—
496C4—24no—
497C4—22no—
498C4—64no—
499C4—124no—
500C4—122no—
501C4—124no—
502C4—122no—
503C4Q8×D1212yes—
504C4C2×#90814yes—
505C4C22×D3×Q811yescharacteristic
506C4C2×#101011yescharacteristic
507C3C2×#338611yescharacteristic, complemented, Sylow
508C2—124no—
509C2#7997714yesdirect product factor
510C2C2×Q8×D1211yescharacteristic
511C2C22×#90811yescharacteristic
512C2C2×#575311yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#446C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenterC2×#446D3
hypocenterC3C2×#3386
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#446

Complements

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

NH ≅ G/Ntype
2. C2×#5567C2semidirect
3. C2×C3×#446C2semidirect
4. #79977C2direct (N × H)
5. C2×#4850C2semidirect
6. C2×#4851C2semidirect
10. C2×#4702C2semidirect
12. C2×#4682C2semidirect
21. #5567C22semidirect
34. C3×#446C22semidirect
48. #4851C22semidirect
49. #4850C22semidirect
53. #4702C22semidirect
55. #4682C22semidirect
65. C2×#446D3semidirect
103. C2×#204Q8semidirect
164. #446C2×D3semidirect
210. C2×Dic6C2×Q8semidirect
214. C2×C3×C8C2×Q8semidirect
243. #204C2×Q8semidirect
352. Dic6C22×Q8semidirect
355. C3×C8C22×Q8semidirect
391. C2×C8D3×Q8semidirect
481. C8C2×D3×Q8semidirect
482. C2×C3#3386semidirect
507. C3C2×#3386semidirect
509. C2#79977direct (N × H)

Maximal subgroups

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

subgroupindex
C2×Q8×D122
C2×#55672
C2×C3×#4462
#799772
C2×#48502
C2×#48512
C2×#48472
C2×#48462
C2×#47102
C2×#47022
C2×C4×#2042
C2×#46822
C2×#33863