Finite Group Databasebeta

← C2×#79635 #11675686

Subgroups — C2×#79635 #11675686

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×#5570C211yesmaximal, characteristic, complemented
2C2×D4×Dic6C211yesmaximal, characteristic
3C2×C3×C4×SD16C211yesmaximal, characteristic
4C2×#5164C212yesmaximal
5C2×#5020C211yesmaximal, characteristic, complemented
6C2×#5007C211yesmaximal, characteristic
7#79635C2116yesmaximal, direct product factor
8C2×#4856C212yesmaximal, complemented
9C2×#4793C212yesmaximal, complemented
10C2×#4766C212yesmaximal
11C2×#4679C211yesmaximal, characteristic, complemented
12C2×#4682C211yesmaximal, characteristic, complemented
13C2×#3386—31nomaximal, Sylow
14C23×Dic6—21no—
15C22×C3×SD16C2211yescharacteristic
16C2×C3×C4×Q8C2211yescharacteristic
17C2×C3×C4×D4C2211yescharacteristic
18C2×Q8×Dic3C2212yes—
19C2×D4×Dic3C2212yes—
20C2×#924—21no—
21#5570—24no—
22#5570C2218yescomplemented
23D4×Dic6—24no—
24D4×Dic6C2218yes—
25C2×#890—22no—
26C2×#890C2212yes—
27C2×#888C2212yes—
28C2×#878—22no—
29C2×#869—21no—
30C2×#869C2211yescharacteristic
31C2×#869C2211yescharacteristic
32C2×C4×Dic6—21no—
33C2×C4×Dic6C2211yescharacteristic
34C3×C4×SD16C2218yes—
35C2×C3×#106C2211yescharacteristic
36C2×C3×#103C2211yescharacteristic
37C2×C3×#102C2211yescharacteristic
38C2×C3×C4×C8C2211yescharacteristic
39C2×#835C2212yes—
40C2×#832C2212yes—
41#5164C22116yes—
42C2×#818C2212yes—
43C2×#818C2212yes—
44C2×#815C2212yes—
45C2×Dic3⋊C8C2212yes—
46#5020C2218yescomplemented
47#5007C2218yes—
48C2×#808C2212yes—
49C2×#807C2212yes—
50C2×C4⋊(C3⋊C8)C2211yescharacteristic
51#4856C22116yescomplemented
52#4793C22116yescomplemented
53#4766C22116yes—
54#4682C2218yescomplemented
55#4679C2218yescomplemented
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×#446—31no—
66C2×#440—32no—
67C2×C4×SD16D311yescharacteristic
68C22×C3×Q8C2311yescharacteristic
69C22×C3×D4C2311yescharacteristic
70C23×C3×C4—21no—
71C23×Dic3—22no—
72C22×Dic6—21no—
73C22×Dic6—21no—
74C22×Dic6—22no—
75C22×Dic6—28no—
76C22×Dic6—48no—
77C22×Dic6D411yescharacteristic
78C2×C3×SD16Q818yescomplemented
79C2×C3×SD16C2314yes—
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C3×C4×Q8—22no—
83C3×C4×Q8C2314yes—
84C3×C4×D4—22no—
85C3×C4×D4C2314yes—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C22⋊C4—21no—
90C2×C3×C42—21no—
91C2×C3×C42C2311yescharacteristic
92C2×#217—22no—
93Q8×Dic3—24no—
94Q8×Dic3C2318yes—
95D4×Dic3—24no—
96D4×Dic3C2318yes—
97C2×C4⋊Dic3—21no—
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3—22no—
100C2×C4⋊Dic3—22no—
101C2×C4⋊Dic3D411yescharacteristic
102C2×C4⋊Dic3C2312yes—
103C2×C4⋊Dic3C2312yes—
104#924—28no—
105C2×Dic3⋊C4—22no—
106C2×Dic3⋊C4—22no—
107C2×Dic3⋊C4D412yes—
108C2×C4×Dic3—22no—
109C2×C4×Dic3C2312yes—
110C22×C3⋊C8C2312yes—
111#890—216no—
112#890—24no—
113#890C2318yes—
114#888—24no—
115#888C2318yes—
116#878—216no—
117#869—28no—
118#869—22no—
119#869—22no—
120#869C2314yes—
121#869C2314yes—
122C4×Dic6—28no—
123C4×Dic6D414yes—
124C4×Dic6C2314yes—
125C3×#106C2314yes—
126C3×#103C2314yes—
127C3×#102C2314yes—
128C3×C4×C8C2314yes—
129#835C2318yes—
130#832C2318yes—
131#818C2318yes—
132#818C2318yes—
133#815C2318yes—
134Dic3⋊C8C2318yes—
135#808C2318yes—
136#807C2318yes—
137C4⋊(C3⋊C8)C2314yes—
138C23×Q8—61no—
139C22×SD16C2×D311yescharacteristic
140#557—38no—
141#557—64no—
142D4×Q8—38no—
143D4×Q8—64no—
144C2×C4⋊Q8—32no—
145C2×C4⋊Q8—31no—
146C2×C4⋊Q8—31no—
147C2×C4⋊Q8—61no—
148C2×#125—32no—
149C2×#125—62no—
150C2×C22⋊Q8—62no—
151C2×C22⋊Q8—61no—
152C2×C4×Q8—31no—
153C2×C4×Q8—32no—
154C2×C4×Q8—61no—
155C2×C4×Q8C2×D311yescharacteristic
156C2×C4×D4—32no—
157C2×C4×D4C2×D311yescharacteristic
158#501—316no—
159#500—38no—
160#478—316no—
161#477—316no—
162#475—38no—
163#474—38no—
164#446—38no—
165#440—316no—
166C4×SD16C2×D318yes—
167C2×#107—32no—
168C2×#107—32no—
169C2×#107—32no—
170C2×#106—32no—
171C2×#106C2×D311yescharacteristic
172C2×C4⋊C8—32no—
173C2×C4⋊C8—31no—
174C2×#103—32no—
175C2×#103C2×D311yescharacteristic
176C2×#102—32no—
177C2×#102C2×D311yescharacteristic
178C2×#97—32no—
179C2×C4×C8C2×D311yescharacteristic
180C24×C3—21no—
181C2×C3×Q8—21no—
182C2×C3×Q8—22no—
183C2×C3×Q8C2×Q814yescomplemented
184C2×C3×Q8C2412yes—
185C2×C3×D4—21no—
186C2×C3×D4—22no—
187C2×C3×D4C2×Q814yes—
188C2×C3×D4C2412yes—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—24no—
193C22×C3×C4—44no—
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2411yescharacteristic
197C22×Dic3—22no—
198C22×Dic3—22no—
199C22×Dic3—28no—
200C22×Dic3—48no—
201C22×Dic3C2×D412yes—
202C22×Dic3C2×D412yes—
203C2×Dic6—24no—
204C2×Dic6—24no—
205C2×Dic6—24no—
206C2×Dic6—28no—
207C2×Dic6—432no—
208C2×Dic6—48no—
209C2×Dic6—44no—
210C2×Dic6—44no—
211C2×Dic6C2×D414yes—
212C3×SD16C2×Q8116yescomplemented
213C2×C3×C8—22no—
214C2×C3×C8C2×Q814yescomplemented
215C2×C3×C8C2412yes—
216C2×C3×C8C2412yes—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—24no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4C2412yes—
221C3×C4⋊C4C2412yes—
222C3×C22⋊C4—24no—
223C3×C42—24no—
224C3×C42C2×D412yes—
225C3×C42C2412yes—
226#217—28no—
227C4⋊Dic3—22no—
228C4⋊Dic3—22no—
229C4⋊Dic3—24no—
230C4⋊Dic3—24no—
231C4⋊Dic3—28no—
232C4⋊Dic3—28no—
233C4⋊Dic3C2×D414yes—
234C4⋊Dic3C2414yes—
235C4⋊Dic3C2414yes—
236Dic3⋊C4—28no—
237Dic3⋊C4—28no—
238Dic3⋊C4C2×D418yes—
239C4×Dic3—28no—
240C4×Dic3C2×D414yes—
241C4×Dic3C2414yes—
242C2×C3⋊C8—24no—
243C2×C3⋊C8C2414yes—
244C22×Q8—31no—
245C22×Q8—62no—
246C22×Q8—68no—
247C22×Q8—61no—
248C22×Q8—61no—
249C22×Q8—128no—
250C22×Q8C22×D311yescharacteristic
251C22×D4C22×D311yescharacteristic
252C23×C4—21no—
253C23×C4—62no—
254C2×SD16Dic618yescomplemented
255C2×SD16C22×D314yes—
256C22×C8—32no—
257C22×C8C22×D311yescharacteristic
258C22×C8C22×D311yescharacteristic
259C4⋊Q8—38no—
260C4⋊Q8—34no—
261C4⋊Q8—34no—
262C4⋊Q8—68no—
263C4⋊Q8—64no—
264C4⋊Q8—62no—
265C4⋊Q8—62no—
266#125—38no—
267#125—64no—
268#125—616no—
269C22⋊Q8—616no—
270C22⋊Q8—68no—
271C4×Q8—22no—
272C4×Q8—34no—
273C4×Q8—34no—
274C4×Q8—38no—
275C4×Q8—68no—
276C4×Q8—64no—
277C4×Q8C22×D314yes—
278C4×D4—22no—
279C4×D4—38no—
280C4×D4—64no—
281C4×D4C22×D314yes—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—32no—
285C2×C4⋊C4—32no—
286C2×C4⋊C4—32no—
287C2×C4⋊C4—62no—
288C2×C4⋊C4—62no—
289C2×C4⋊C4—62no—
290C2×C4⋊C4—62no—
291C2×C4⋊C4—61no—
292C2×C4⋊C4—61no—
293C2×C4⋊C4C22×D311yescharacteristic
294C2×C4⋊C4C22×D311yescharacteristic
295C2×C22⋊C4—21no—
296C2×C22⋊C4—62no—
297C2×C42—21no—
298C2×C42—32no—
299C2×C42—62no—
300C2×C42C22×D311yescharacteristic
301#107—38no—
302#107—38no—
303#107—38no—
304#106—38no—
305#106C22×D314yes—
306C4⋊C8—38no—
307C4⋊C8—34no—
308#103—38no—
309#103C22×D314yes—
310#102—38no—
311#102C22×D314yes—
312#97—38no—
313C4×C8C22×D314yes—
314C23×C3—24no—
315C23×C3—22no—
316C23×C3—42no—
317C23×C3—42no—
318C23×C3C22×D411yescharacteristic
319C3×Q8—24no—
320C3×Q8—22no—
321C3×Q8C22×Q814yescomplemented
322C3×D4—24no—
323C3×D4—22no—
324C3×D4C22×Q814yes—
325C2×C3×C4—24no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×C3×C4—48no—
333C2×C3×C4AGL(1,ℤ/8)14yes—
334C2×C3×C4C22×D412yes—
335C2×C3×C4C22×D411yescharacteristic
336C2×C3×C4C22×D412yes—
337C2×C3×C4C22×Q812yes—
338C2×C3×C42−1+412yes—
339C2×C3×C4C2511yescharacteristic
340C2×Dic3—24no—
341C2×Dic3—24no—
342C2×Dic3—24no—
343C2×Dic3—24no—
344C2×Dic3—416no—
345C2×Dic3—44no—
346C2×Dic3—44no—
347C2×Dic3C22×D414yes—
348C2×Dic3C22×D414yes—
349Dic6—28no—
350Dic6—416no—
351Dic6—48no—
352Dic6—48no—
353C3×C8—22no—
354C3×C8C22×Q814yescomplemented
355C3⋊C8—24no—
356C24—21no—
357C2×Q8—21no—
358C2×Q8—22no—
359C2×Q8—34no—
360C2×Q8—68no—
361C2×Q8—64no—
362C2×Q8—64no—
363C2×Q8—64no—
364C2×Q8—128no—
365C2×Q8—1232no—
366C2×Q8—124no—
367C2×Q8—124no—
368C2×Q8C2×Dic614yescomplemented
369C2×Q8C23×D312yes—
370C2×D4—21no—
371C2×D4—22no—
372C2×D4C2×Dic614yes—
373C2×D4C23×D312yes—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—24no—
378C22×C4—32no—
379C22×C4—32no—
380C22×C4—44no—
381C22×C4—68no—
382C22×C4—62no—
383C22×C4—62no—
384C22×C4—128no—
385C22×C4D3×D411yescharacteristic
386C22×C4D3×D411yescharacteristic
387C22×C4C23×D311yescharacteristic
388SD16C2×Dic6116yescomplemented
389C2×C8—22no—
390C2×C8—34no—
391C2×C8—64no—
392C2×C8C2×Dic614yescomplemented
393C2×C8C23×D312yes—
394C2×C8C23×D312yes—
395C4⋊C4—24no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—34no—
399C4⋊C4—38no—
400C4⋊C4—34no—
401C4⋊C4—34no—
402C4⋊C4—68no—
403C4⋊C4—68no—
404C4⋊C4—68no—
405C4⋊C4—68no—
406C4⋊C4—62no—
407C4⋊C4—62no—
408C4⋊C4—64no—
409C4⋊C4—64no—
410C4⋊C4C23×D312yes—
411C4⋊C4C23×D312yes—
412C22⋊C4—24no—
413C22⋊C4—68no—
414C42—24no—
415C42—34no—
416C42—34no—
417C42—68no—
418C42D3×D412yes—
419C42C23×D312yes—
420C22×C3—24no—
421C22×C3—48no—
422C22×C3—42no—
423C22×C3—42no—
424C22×C3D4×Q812yes—
425C22×C3C2×AGL(1,ℤ/8)12yes—
426C22×C3#57812yes—
427C22×C3C23×D411yescharacteristic
428C3×C4—24no—
429C3×C4—22no—
430C3×C4—42no—
431C3×C4—42no—
432C3×C4D4×Q812yes—
433C3×C4C2×AGL(1,ℤ/8)14yes—
434C3×C4C23×Q811yescharacteristic
435C3×C4C2×2−1+411yescharacteristic
436Dic3—24no—
437Dic3—24no—
438Dic3—44no—
439Dic3—44no—
440C23—22no—
441C23—24no—
442C23—42no—
443C23—42no—
444C23C2×D3×D411yescharacteristic
445Q8—22no—
446Q8—24no—
447Q8—68no—
448Q8—1216no—
449Q8—128no—
450Q8—128no—
451Q8C22×Dic614yescomplemented
452D4—22no—
453D4—24no—
454D4C22×Dic614yes—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—34no—
461C2×C4—34no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—48no—
465C2×C4—64no—
466C2×C4—64no—
467C2×C4—64no—
468C2×C4—64no—
469C2×C4—1216no—
470C2×C4—124no—
471C2×C4—124no—
472C2×C4#91414yes—
473C2×C4C22×Dic612yes—
474C2×C4C2×D3×D412yes—
475C2×C4C2×D3×D412yes—
476C2×C4C2×D3×D411yescharacteristic
477C2×C4#101012yes—
478C2×C4C24×D311yescharacteristic
479C8—22no—
480C8—64no—
481C8C22×Dic614yescomplemented
482C2×C3—44no—
483C2×C3#338614yescomplemented
484C2×C3C2×D4×Q811yescharacteristic
485C2×C3C22×AGL(1,ℤ/8)11yescharacteristic
486C2×C3C2×#57811yescharacteristic
487C22—24no—
488C22—48no—
489C22—42no—
490C22—42no—
491C22D4×Dic612yes—
492C22C2×#91412yes—
493C22#577112yes—
494C22C22×D3×D411yescharacteristic
495C4—22no—
496C4—24no—
497C4—42no—
498C4—42no—
499C4—64no—
500C4—64no—
501C4—124no—
502C4—124no—
503C4D4×Dic612yes—
504C4C2×#91414yes—
505C4C23×Dic611yescharacteristic
506C4C2×#101011yescharacteristic
507C3C2×#338611yescharacteristic, complemented, Sylow
508C2—44no—
509C2#7963514yesdirect product factor
510C2C2×D4×Dic611yescharacteristic
511C2C22×#91411yescharacteristic
512C2C2×#577111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×C4×SD16C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenterC2×C4×SD16D3
hypocenterC3C2×#3386
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. C2×#5570C2semidirect
5. C2×#5020C2semidirect
7. #79635C2direct (N × H)
8. C2×#4856C2semidirect
9. C2×#4793C2semidirect
11. C2×#4679C2semidirect
12. C2×#4682C2semidirect
22. #5570C22semidirect
46. #5020C22semidirect
51. #4856C22semidirect
52. #4793C22semidirect
54. #4682C22semidirect
55. #4679C22semidirect
78. C2×C3×SD16Q8semidirect
183. C2×C3×Q8C2×Q8semidirect
212. C3×SD16C2×Q8semidirect
214. C2×C3×C8C2×Q8semidirect
254. C2×SD16Dic6semidirect
321. C3×Q8C22×Q8semidirect
354. C3×C8C22×Q8semidirect
368. C2×Q8C2×Dic6semidirect
388. SD16C2×Dic6semidirect
392. C2×C8C2×Dic6semidirect
451. Q8C22×Dic6semidirect
481. C8C22×Dic6semidirect
483. C2×C3#3386semidirect
507. C3C2×#3386semidirect
509. C2#79635direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#55702
C2×D4×Dic62
C2×C3×C4×SD162
C2×#51642
C2×#50202
C2×#50072
#796352
C2×#48562
C2×#47932
C2×#47662
C2×#46792
C2×#46822
C2×#33863