Finite Group Databasebeta

← C2×#79632 #11675624

Subgroups — C2×#79632 #11675624

Order 768.

1842 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5570C211yesmaximal, characteristic, complemented
2C2×#5543C211yesmaximal, characteristic
3C2×C3×C4×SD16C211yesmaximal, characteristic
4C2×Dic3×SD16C212yesmaximal
5C2×#5021C211yesmaximal, characteristic, complemented
6C2×#5007C211yesmaximal, characteristic
7#79632C2116yesmaximal, direct product factor
8C2×#4858C212yesmaximal, complemented
9C2×#4799C212yesmaximal, complemented
10C2×#4763C212yesmaximal
11C2×C8×Dic6C211yesmaximal, characteristic, complemented
12C2×#4680C211yesmaximal, characteristic, complemented
13C2×#3381—31nomaximal, Sylow
14C22×C3×SD16C2211yescharacteristic
15C2×C3×C4×Q8C2211yescharacteristic
16C2×C3×C4×D4C2211yescharacteristic
17C2×Q8×Dic3C2212yes—
18C2×D4×Dic3C2212yes—
19C22×C4⋊Dic3—21no—
20C2×#924—21no—
21#5570—24no—
22#5570C2218yescomplemented
23#5543—24no—
24#5543C2218yes—
25C2×#890—22no—
26C2×#890C2212yes—
27C2×#890C2212yes—
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—
41Dic3×SD16C22116yes—
42C2×#817C2212yes—
43C2×#817C2212yes—
44C2×Dic3⋊C8C2212yes—
45C2×C8×Dic3C2212yes—
46#5021C2218yescomplemented
47#5007C2218yes—
48C2×#807C2212yes—
49C2×#807C2212yes—
50C2×C4⋊(C3⋊C8)C2211yescharacteristic
51#4858C22116yescomplemented
52#4799C22116yescomplemented
53#4763C22116yes—
54#4680C2218yescomplemented
55C8×Dic6C2218yescomplemented
56C2×#557—31no—
57C2×#554—31no—
58#3381—316no—
59C2×#499—32no—
60C2×#498—31no—
61C2×#479—32no—
62C2×#477—31no—
63C2×#478—32no—
64C2×#474—31no—
65C2×C8×Q8—31no—
66C2×C4×SD16—32no—
67C2×C4×SD16D311yescharacteristic
68C22×C3×Q8C2311yescharacteristic
69C22×C3×D4C2311yescharacteristic
70C23×C3×C4—21no—
71C23×Dic3—22no—
72C22×Dic6—21no—
73C22×Dic6—21no—
74C22×Dic6D411yescharacteristic
75C2×C3×SD16Q818yescomplemented
76C2×C3×SD16C2314yes—
77C22×C3×C8C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C3×C4×Q8—22no—
80C3×C4×Q8C2314yes—
81C3×C4×D4—22no—
82C3×C4×D4C2314yes—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C22⋊C4—21no—
87C2×C3×C42—21no—
88C2×C3×C42C2311yescharacteristic
89C2×#217—22no—
90Q8×Dic3—24no—
91Q8×Dic3C2318yes—
92D4×Dic3—24no—
93D4×Dic3C2318yes—
94C2×C4⋊Dic3—21no—
95C2×C4⋊Dic3—21no—
96C2×C4⋊Dic3—22no—
97C2×C4⋊Dic3—22no—
98C2×C4⋊Dic3—22no—
99C2×C4⋊Dic3—28no—
100C2×C4⋊Dic3D411yescharacteristic
101C2×C4⋊Dic3C2312yes—
102C2×C4⋊Dic3C2312yes—
103#924—28no—
104C2×Dic3⋊C4—22no—
105C2×Dic3⋊C4—22no—
106C2×Dic3⋊C4D412yes—
107C2×C4×Dic3—22no—
108C2×C4×Dic3C2312yes—
109C22×C3⋊C8C2312yes—
110#890—216no—
111#890—24no—
112#890—24no—
113#890C2318yes—
114#890C2318yes—
115#878—216no—
116#869—28no—
117#869—22no—
118#869—22no—
119#869C2314yes—
120#869C2314yes—
121C4×Dic6—28no—
122C4×Dic6D414yes—
123C4×Dic6C2314yes—
124C3×#106C2314yes—
125C3×#103C2314yes—
126C3×#102C2314yes—
127C3×C4×C8C2314yes—
128#835C2318yes—
129#832C2318yes—
130#817C2318yes—
131#817C2318yes—
132Dic3⋊C8C2318yes—
133C8×Dic3C2318yes—
134#807C2318yes—
135#807C2318yes—
136C4⋊(C3⋊C8)C2314yes—
137C22×SD16C2×D311yescharacteristic
138#557—38no—
139#557—64no—
140#554—38no—
141#554—64no—
142C2×C4⋊Q8—31no—
143C2×C4⋊Q8—31no—
144C2×C4⋊Q8—61no—
145C2×#125—32no—
146C2×#125—32no—
147C2×#125—62no—
148C2×C22⋊Q8—62no—
149C2×C22⋊Q8—61no—
150C2×C4×Q8—31no—
151C2×C4×Q8—32no—
152C2×C4×Q8—61no—
153C2×C4×Q8C2×D311yescharacteristic
154C2×C4×D4—32no—
155C2×C4×D4C2×D311yescharacteristic
156C22×C4⋊C4—61no—
157#499—316no—
158#498—38no—
159#479—316no—
160#478—316no—
161#477—38no—
162#474—38no—
163C8×Q8—38no—
164C4×SD16—316no—
165C4×SD16C2×D318yes—
166C2×#107—32no—
167C2×#107—32no—
168C2×#106—32no—
169C2×#106—32no—
170C2×#106C2×D311yescharacteristic
171C2×C4⋊C8—32no—
172C2×C4⋊C8—31no—
173C2×#103—32no—
174C2×#103C2×D311yescharacteristic
175C2×#102—32no—
176C2×#102C2×D311yescharacteristic
177C2×C4×C8—32no—
178C2×C4×C8C2×D311yescharacteristic
179C24×C3—21no—
180C2×C3×Q8—21no—
181C2×C3×Q8—22no—
182C2×C3×Q8C2×Q814yes—
183C2×C3×Q8C2412yes—
184C2×C3×D4—21no—
185C2×C3×D4—22no—
186C2×C3×D4C2×Q814yes—
187C2×C3×D4C2412yes—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—24no—
192C22×C3×C4—44no—
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2411yescharacteristic
196C22×Dic3—22no—
197C22×Dic3—22no—
198C22×Dic3—28no—
199C22×Dic3—84no—
200C22×Dic3C2×D412yes—
201C22×Dic3C2×D412yes—
202C2×Dic6—24no—
203C2×Dic6—24no—
204C2×Dic6—24no—
205C2×Dic6—44no—
206C2×Dic6—82no—
207C2×Dic6C2×D414yes—
208C3×SD16C2×Q8116yescomplemented
209C2×C3×C8—22no—
210C2×C3×C8C2×Q814yes—
211C2×C3×C8C2412yes—
212C2×C3×C8C2412yes—
213C3×C4⋊C4—24no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4C2412yes—
217C3×C4⋊C4C2412yes—
218C3×C22⋊C4—24no—
219C3×C42—24no—
220C3×C42C2×D412yes—
221C3×C42C2412yes—
222#217—28no—
223C4⋊Dic3—22no—
224C4⋊Dic3—22no—
225C4⋊Dic3—24no—
226C4⋊Dic3—24no—
227C4⋊Dic3—28no—
228C4⋊Dic3—28no—
229C4⋊Dic3—28no—
230C4⋊Dic3C2×D414yes—
231C4⋊Dic3C2414yes—
232C4⋊Dic3C2414yes—
233Dic3⋊C4—28no—
234Dic3⋊C4—28no—
235Dic3⋊C4C2×D418yes—
236C4×Dic3—28no—
237C4×Dic3C2×D414yes—
238C4×Dic3C2414yes—
239C2×C3⋊C8—24no—
240C2×C3⋊C8C2414yes—
241C22×Q8—31no—
242C22×Q8—61no—
243C22×Q8—61no—
244C22×Q8C22×D311yescharacteristic
245C22×D4C22×D311yescharacteristic
246C23×C4—21no—
247C23×C4—62no—
248C2×SD16Dic618yescomplemented
249C2×SD16C22×D314yes—
250C22×C8—32no—
251C22×C8C22×D311yescharacteristic
252C22×C8C22×D311yescharacteristic
253C4⋊Q8—34no—
254C4⋊Q8—34no—
255C4⋊Q8—68no—
256C4⋊Q8—62no—
257C4⋊Q8—62no—
258#125—38no—
259#125—38no—
260#125—64no—
261#125—64no—
262#125—616no—
263C22⋊Q8—616no—
264C22⋊Q8—68no—
265C4×Q8—22no—
266C4×Q8—34no—
267C4×Q8—34no—
268C4×Q8—38no—
269C4×Q8—68no—
270C4×Q8—64no—
271C4×Q8C22×D314yes—
272C4×D4—22no—
273C4×D4—38no—
274C4×D4—64no—
275C4×D4C22×D314yes—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—31no—
278C2×C4⋊C4—32no—
279C2×C4⋊C4—32no—
280C2×C4⋊C4—32no—
281C2×C4⋊C4—68no—
282C2×C4⋊C4—62no—
283C2×C4⋊C4—62no—
284C2×C4⋊C4—62no—
285C2×C4⋊C4—62no—
286C2×C4⋊C4—62no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4C22×D311yescharacteristic
290C2×C4⋊C4C22×D311yescharacteristic
291C2×C22⋊C4—21no—
292C2×C22⋊C4—62no—
293C2×C42—21no—
294C2×C42—32no—
295C2×C42—62no—
296C2×C42C22×D311yescharacteristic
297#107—38no—
298#107—38no—
299#106—38no—
300#106—38no—
301#106C22×D314yes—
302C4⋊C8—38no—
303C4⋊C8—34no—
304#103—38no—
305#103C22×D314yes—
306#102—38no—
307#102C22×D314yes—
308C4×C8—38no—
309C4×C8C22×D314yes—
310C23×C3—24no—
311C23×C3—22no—
312C23×C3—42no—
313C23×C3—42no—
314C23×C3C22×D411yescharacteristic
315C3×Q8—24no—
316C3×Q8—22no—
317C3×Q8C22×Q814yes—
318C3×D4—24no—
319C3×D4—22no—
320C3×D4C22×Q814yes—
321C2×C3×C4—24no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4—48no—
329C2×C3×C4#13514yes—
330C2×C3×C4C22×D412yes—
331C2×C3×C4C22×D411yescharacteristic
332C2×C3×C4C22×D412yes—
333C2×C3×C4C22×Q812yes—
334C2×C3×C42−1+412yes—
335C2×C3×C4C2511yescharacteristic
336C2×Dic3—24no—
337C2×Dic3—24no—
338C2×Dic3—24no—
339C2×Dic3—24no—
340C2×Dic3—44no—
341C2×Dic3—44no—
342C2×Dic3—88no—
343C2×Dic3C22×D414yes—
344C2×Dic3C22×D414yes—
345Dic6—28no—
346Dic6—48no—
347Dic6—84no—
348C3×C8—22no—
349C3×C8C22×Q814yes—
350C3⋊C8—24no—
351C24—21no—
352C2×Q8—21no—
353C2×Q8—22no—
354C2×Q8—34no—
355C2×Q8—64no—
356C2×Q8—64no—
357C2×Q8—64no—
358C2×Q8—124no—
359C2×Q8—242no—
360C2×Q8C2×Dic614yes—
361C2×Q8C23×D312yes—
362C2×D4—21no—
363C2×D4—22no—
364C2×D4C2×Dic614yes—
365C2×D4C23×D312yes—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—24no—
370C22×C4—32no—
371C22×C4—32no—
372C22×C4—44no—
373C22×C4—68no—
374C22×C4—62no—
375C22×C4—62no—
376C22×C4—244no—
377C22×C4D3×D411yescharacteristic
378C22×C4D3×D411yescharacteristic
379C22×C4C23×D311yescharacteristic
380SD16C2×Dic6116yescomplemented
381C2×C8—22no—
382C2×C8—34no—
383C2×C8—64no—
384C2×C8C2×Dic614yes—
385C2×C8C23×D312yes—
386C2×C8C23×D312yes—
387C4⋊C4—24no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—34no—
391C4⋊C4—38no—
392C4⋊C4—34no—
393C4⋊C4—34no—
394C4⋊C4—68no—
395C4⋊C4—68no—
396C4⋊C4—68no—
397C4⋊C4—68no—
398C4⋊C4—68no—
399C4⋊C4—62no—
400C4⋊C4—62no—
401C4⋊C4—64no—
402C4⋊C4—64no—
403C4⋊C4C23×D312yes—
404C4⋊C4C23×D312yes—
405C22⋊C4—24no—
406C22⋊C4—68no—
407C42—24no—
408C42—34no—
409C42—34no—
410C42—68no—
411C42D3×D412yes—
412C42C23×D312yes—
413C22×C3—24no—
414C22×C3—48no—
415C22×C3—42no—
416C22×C3—42no—
417C22×C3D4×Q812yes—
418C22×C3C2×#13512yes—
419C22×C3#57712yes—
420C22×C3C23×D411yescharacteristic
421C3×C4—24no—
422C3×C4—22no—
423C3×C4—42no—
424C3×C4—42no—
425C3×C4D4×Q812yes—
426C3×C4C2×#13514yes—
427C3×C4C23×Q811yescharacteristic
428C3×C4C2×2−1+411yescharacteristic
429Dic3—24no—
430Dic3—24no—
431Dic3—44no—
432Dic3—44no—
433C23—22no—
434C23—24no—
435C23—42no—
436C23—42no—
437C23C2×D3×D411yescharacteristic
438Q8—22no—
439Q8—24no—
440Q8—68no—
441Q8—128no—
442Q8—244no—
443Q8C22×Dic614yes—
444D4—22no—
445D4—24no—
446D4C22×Dic614yes—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—34no—
453C2×C4—34no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—48no—
457C2×C4—64no—
458C2×C4—64no—
459C2×C4—64no—
460C2×C4—64no—
461C2×C4—124no—
462C2×C4—124no—
463C2×C4—248no—
464C2×C4#91614yes—
465C2×C4C22×Dic612yes—
466C2×C4C2×D3×D412yes—
467C2×C4C2×D3×D412yes—
468C2×C4C2×D3×D411yescharacteristic
469C2×C4#101012yes—
470C2×C4C24×D311yescharacteristic
471C8—22no—
472C8—64no—
473C8C22×Dic614yes—
474C2×C3—44no—
475C2×C3#338114yescomplemented
476C2×C3C2×D4×Q811yescharacteristic
477C2×C3C22×#13511yescharacteristic
478C2×C3C2×#57711yescharacteristic
479C22—24no—
480C22—48no—
481C22—42no—
482C22—42no—
483C22D4×Dic612yes—
484C22C2×#91612yes—
485C22#577012yes—
486C22C22×D3×D411yescharacteristic
487C4—22no—
488C4—24no—
489C4—42no—
490C4—42no—
491C4—64no—
492C4—64no—
493C4—124no—
494C4—124no—
495C4D4×Dic612yes—
496C4C2×#91614yes—
497C4C23×Dic611yescharacteristic
498C4C2×#101011yescharacteristic
499C3C2×#338111yescharacteristic, complemented, Sylow
500C2—44no—
501C2#7963214yesdirect product factor
502C2C2×D4×Dic611yescharacteristic
503C2C22×#91611yescharacteristic
504C2C2×#577011yescharacteristic
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×#3381
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×#5021C2semidirect
7. #79632C2direct (N × H)
8. C2×#4858C2semidirect
9. C2×#4799C2semidirect
11. C2×C8×Dic6C2semidirect
12. C2×#4680C2semidirect
22. #5570C22semidirect
46. #5021C22semidirect
51. #4858C22semidirect
52. #4799C22semidirect
54. #4680C22semidirect
55. C8×Dic6C22semidirect
75. C2×C3×SD16Q8semidirect
208. C3×SD16C2×Q8semidirect
248. C2×SD16Dic6semidirect
380. SD16C2×Dic6semidirect
475. C2×C3#3381semidirect
499. C3C2×#3381semidirect
501. C2#79632direct (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×#55432
C2×C3×C4×SD162
C2×Dic3×SD162
C2×#50212
C2×#50072
#796322
C2×#48582
C2×#47992
C2×#47632
C2×C8×Dic62
C2×#46802
C2×#33813