Finite Group Databasebeta

← C2×#81955 #11675693

Subgroups — C2×#81955 #11675693

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×#5727C211yesmaximal, characteristic
2C2×#5721C211yesmaximal, characteristic, complemented
3C2×C3×#500C211yesmaximal, characteristic, complemented
4C2×#5094C211yesmaximal, characteristic, complemented
5C2×#5087C211yesmaximal, characteristic
6#81955C2116yesmaximal, direct product factor
7C2×#4875C212yesmaximal, complemented
8C2×#4891C212yesmaximal
9C2×#4876C212yesmaximal, complemented
10C2×#4889C212yesmaximal
11C2×C4×#204C211yesmaximal, characteristic
12C2×#4679C211yesmaximal, characteristic, complemented
13C2×#3386—31nomaximal, Sylow
14C22×D3×Q8—21no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×#946—21no—
18C2×Q8×Dic3—21no—
19C2×#944—21no—
20C22×#204C2211yescharacteristic
21#5727—24no—
22#5727C2218yes—
23#5721—24no—
24#5721C2218yescomplemented
25C2×#896—22no—
26C2×#893C2212yes—
27C2×#890C2212yes—
28C2×#889—22no—
29C2×#888C2212yes—
30C2×#887C2212yes—
31C2×C4×D12C2211yescharacteristic
32C2×C4×Dic6C2211yescharacteristic
33C2×C4×Dic6C2211yescharacteristic
34C3×#500C2218yescomplemented
35C2×C3×#107C2212yes—
36C2×C3×#107C2212yes—
37C2×C3×C4×C8C2211yescharacteristic
38C2×#821C2211yescharacteristic
39C2×#817C2211yescharacteristic
40C2×#816C2211yescharacteristic
41C2×#815C2212yes—
42C2×Dic3⋊C8C2212yes—
43#5094C2218yescomplemented
44#5087C2218yes—
45C2×#810C2212yes—
46C2×#809C2212yes—
47C2×#808C2212yes—
48C2×#807C2212yes—
49C2×C4⋊(C3⋊C8)C2211yescharacteristic
50#4891C22116yes—
51#4889C22116yes—
52#4876C22116yescomplemented
53#4875C22116yescomplemented
54C4×#204C2218yes—
55#4679C2218yescomplemented
56C2×#557—31no—
57C2×D4×Q8—31no—
58#3386—316no—
59C2×#501—32no—
60C2×#500D311yescharacteristic, complemented
61C2×#477—32no—
62C2×#475—31no—
63C2×#478—32no—
64C2×#474—31no—
65C2×#446—31no—
66C2×#440—32no—
67C2×C4×SD16—31no—
68C22×C3×Q8—21no—
69C22×C3×Q8—21no—
70C2×D3×Q8—28no—
71C2×D3×Q8—48no—
72C22×D12C2311yescharacteristic
73C22×C4×D3—21no—
74C22×C4×D3—22no—
75C22×Dic6—22no—
76C22×Dic6D411yescharacteristic
77C22×Dic6C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C3×C4⋊Q8—22no—
81C3×C4⋊Q8—22no—
82C3×C4⋊Q8C2314yes—
83C3×C4⋊Q8C2314yes—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4C2312yes—
87C2×C3×C4⋊C4C2312yes—
88C2×C3×C42C2311yescharacteristic
89#946—28no—
90Q8×Dic3—28no—
91#944—28no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—22no—
94C2×C4⋊Dic3—22no—
95C2×C4⋊Dic3D411yescharacteristic
96C2×C4⋊Dic3C2311yescharacteristic
97C2×C4⋊Dic3C2311yescharacteristic
98C2×Dic3⋊C4—21no—
99C2×Dic3⋊C4—22no—
100C2×Dic3⋊C4—22no—
101C2×Dic3⋊C4—22no—
102C2×Dic3⋊C4D412yes—
103C2×C4×Dic3—21no—
104C2×C4×Dic3—22no—
105C2×C4×Dic3C2312yes—
106C22×C3⋊C8C2312yes—
107C2×#204Q818yescomplemented
108C2×#204C2314yes—
109#896—216no—
110#893—24no—
111#893C2318yes—
112#890—24no—
113#890C2318yes—
114#889—216no—
115#888—24no—
116#888C2318yes—
117#887—24no—
118#887C2318yes—
119C4×D12—22no—
120C4×D12C2314yes—
121C4×Dic6—22no—
122C4×Dic6D414yes—
123C4×Dic6C2314yes—
124C4×Dic6C2314yes—
125C3×#107C2318yes—
126C3×#107C2318yes—
127C3×C4×C8C2314yes—
128#821C2314yes—
129#817C2314yes—
130#816C2314yes—
131#815C2318yes—
132Dic3⋊C8C2318yes—
133#810C2318yes—
134#809C2318yes—
135#808C2318yes—
136#807C2318yes—
137C4⋊(C3⋊C8)C2314yes—
138C23×Q8—61no—
139C22×SD16—31no—
140#557—38no—
141#557—64no—
142D4×Q8—38no—
143D4×Q8—64no—
144C2×C4⋊Q8—32no—
145C2×C4⋊Q8—61no—
146C2×C4⋊Q8C2×D311yescharacteristic
147C2×C4⋊Q8C2×D311yescharacteristic
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—31no—
155C2×C4×Q8—61no—
156C2×C4×D4—32no—
157C2×C4×D4—31no—
158#501—316no—
159#500C2×D318yescomplemented
160#478—316no—
161#477—316no—
162#475—38no—
163#474—38no—
164#446—38no—
165#440—316no—
166C4×SD16—38no—
167C2×#107—32no—
168C2×#107C2×D312yes—
169C2×#107C2×D312yes—
170C2×#106—31no—
171C2×#106—32no—
172C2×C4⋊C8—32no—
173C2×C4⋊C8—31no—
174C2×#103—31no—
175C2×#103—32no—
176C2×#102—31no—
177C2×#102—32no—
178C2×#97—32no—
179C2×C4×C8C2×D311yescharacteristic
180C23×D3—21no—
181C2×C3×Q8—24no—
182C2×C3×Q8—24no—
183C2×C3×Q8—44no—
184C2×C3×Q8—44no—
185C22×C3×C4—22no—
186C22×C3×C4—22no—
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2411yescharacteristic
190C22×Dic3—21no—
191C22×Dic3—21no—
192C22×Dic3—21no—
193C22×Dic3C2×D412yes—
194C22×Dic3C2×D412yes—
195D3×Q8—432no—
196C2×D12—21no—
197C2×D12—22no—
198C2×D12C2×Q814yes—
199C2×D12C2412yes—
200C2×C4×D3—24no—
201C2×C4×D3—28no—
202C2×C4×D3—48no—
203C2×C4×D3—44no—
204C2×Dic6—24no—
205C2×Dic6—21no—
206C2×Dic6—22no—
207C2×Dic6—28no—
208C2×Dic6—48no—
209C2×Dic6C2×D414yes—
210C2×Dic6C2×Q814yescomplemented
211C2×Dic6C2412yes—
212C2×C3×C8—22no—
213C2×C3×C8C2×Q814yescomplemented
214C2×C3×C8C2412yes—
215C2×C3×C8C2412yes—
216C3×C4⋊C4—22no—
217C3×C4⋊C4—22no—
218C3×C4⋊C4—24no—
219C3×C4⋊C4—24no—
220C3×C4⋊C4C2414yes—
221C3×C4⋊C4C2414yes—
222C3×C42C2×D412yes—
223C3×C42C2412yes—
224(C2×D3)⋊C4—24no—
225(C2×D3)⋊C4—28no—
226C4⋊Dic3—21no—
227C4⋊Dic3—21no—
228C4⋊Dic3—28no—
229C4⋊Dic3C2×D414yes—
230C4⋊Dic3C2412yes—
231C4⋊Dic3C2412yes—
232Dic3⋊C4—24no—
233Dic3⋊C4—28no—
234Dic3⋊C4—28no—
235Dic3⋊C4—28no—
236Dic3⋊C4C2×D418yes—
237C4×Dic3—24no—
238C4×Dic3—28no—
239C4×Dic3C2×D414yes—
240C4×Dic3C2414yes—
241C2×C3⋊C8—24no—
242C2×C3⋊C8C2414yes—
243#204C2×Q8116yescomplemented
244C22×Q8—21no—
245C22×Q8—21no—
246C22×Q8—31no—
247C22×Q8—31no—
248C22×Q8—62no—
249C22×Q8—68no—
250C22×Q8—128no—
251C22×D4—31no—
252C23×C4—62no—
253C23×C4—61no—
254C2×SD16—38no—
255C2×SD16—34no—
256C22×C8—32no—
257C22×C8C22×D311yescharacteristic
258C22×C8C22×D311yescharacteristic
259C4⋊Q8—22no—
260C4⋊Q8—22no—
261C4⋊Q8—38no—
262C4⋊Q8—64no—
263C4⋊Q8—68no—
264C4⋊Q8C22×D314yes—
265C4⋊Q8C22×D314yes—
266#125—38no—
267#125—64no—
268#125—616no—
269C22⋊Q8—616no—
270C22⋊Q8—68no—
271C4×Q8—34no—
272C4×Q8—34no—
273C4×Q8—38no—
274C4×Q8—34no—
275C4×Q8—64no—
276C4×Q8—68no—
277C4×Q8—62no—
278C4×D4—38no—
279C4×D4—34no—
280C4×D4—64no—
281C4×D4—62no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4—32no—
286C2×C4⋊C4—31no—
287C2×C4⋊C4—31no—
288C2×C4⋊C4—62no—
289C2×C4⋊C4—62no—
290C2×C4⋊C4—62no—
291C2×C4⋊C4—62no—
292C2×C4⋊C4—61no—
293C2×C4⋊C4C22×D312yes—
294C2×C4⋊C4C22×D312yes—
295C2×C22⋊C4—62no—
296C2×C22⋊C4—61no—
297C2×C42—32no—
298C2×C42—62no—
299C2×C42—61no—
300C2×C42C22×D311yescharacteristic
301#107—38no—
302#107C22×D318yes—
303#107C22×D318yes—
304#106—34no—
305#106—38no—
306C4⋊C8—38no—
307C4⋊C8—34no—
308#103—34no—
309#103—38no—
310#102—34no—
311#102—38no—
312#97—38no—
313C4×C8C22×D314yes—
314C23×C3C22×D411yescharacteristic
315C22×D3—24no—
316C22×D3—22no—
317C22×D3—42no—
318C22×D3—42no—
319C3×Q8—48no—
320C3×Q8—48no—
321C2×C3×C4—22no—
322C2×C3×C4—24no—
323C2×C3×C4—24no—
324C2×C3×C4—44no—
325C2×C3×C4—44no—
326C2×C3×C4AGL(1,ℤ/8)14yes—
327C2×C3×C4C22×D412yes—
328C2×C3×C4C22×D411yescharacteristic
329C2×C3×C4C22×D412yes—
330C2×C3×C4C22×Q812yes—
331C2×C3×C42−1+412yes—
332C2×C3×C4C2511yescharacteristic
333C2×Dic3—24no—
334C2×Dic3—24no—
335C2×Dic3—24no—
336C2×Dic3—22no—
337C2×Dic3—22no—
338C2×Dic3—22no—
339C2×Dic3—42no—
340C2×Dic3—42no—
341C2×Dic3C22×D414yes—
342C2×Dic3C22×D414yes—
343D12—24no—
344D12—22no—
345D12C22×Q814yes—
346C4×D3—416no—
347C4×D3—48no—
348Dic6—24no—
349Dic6—28no—
350Dic6—22no—
351Dic6—416no—
352Dic6C22×Q814yescomplemented
353C3×C8—22no—
354C3×C8C22×Q814yescomplemented
355C3⋊C8—24no—
356C24—61no—
357C2×Q8—24no—
358C2×Q8—24no—
359C2×Q8—34no—
360C2×Q8—34no—
361C2×Q8—32no—
362C2×Q8—44no—
363C2×Q8—44no—
364C2×Q8—68no—
365C2×Q8—64no—
366C2×Q8—61no—
367C2×Q8—62no—
368C2×Q8—128no—
369C2×Q8—1232no—
370C2×D4—34no—
371C2×D4—32no—
372C2×D4—61no—
373C2×D4—62no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—32no—
377C22×C4—32no—
378C22×C4—68no—
379C22×C4—61no—
380C22×C4—61no—
381C22×C4—61no—
382C22×C4—64no—
383C22×C4—128no—
384C22×C4—124no—
385C22×C4D3×D411yescharacteristic
386C22×C4D3×D411yescharacteristic
387C22×C4C23×D311yescharacteristic
388SD16—316no—
389C2×C8—22no—
390C2×C8—34no—
391C2×C8—64no—
392C2×C8D3×Q814yescomplemented
393C2×C8C23×D312yes—
394C2×C8C23×D312yes—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—34no—
400C4⋊C4—38no—
401C4⋊C4—32no—
402C4⋊C4—32no—
403C4⋊C4—68no—
404C4⋊C4—68no—
405C4⋊C4—68no—
406C4⋊C4—68no—
407C4⋊C4—61no—
408C4⋊C4—61no—
409C4⋊C4—64no—
410C4⋊C4C23×D314yes—
411C4⋊C4C23×D314yes—
412C22⋊C4—68no—
413C22⋊C4—64no—
414C42—34no—
415C42—34no—
416C42—68no—
417C42—64no—
418C42D3×D412yes—
419C42C23×D312yes—
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—22no—
429C3×C4—44no—
430C3×C4—44no—
431C3×C4D4×Q812yes—
432C3×C4C2×AGL(1,ℤ/8)14yes—
433C3×C4C23×Q811yescharacteristic
434C3×C4C2×2−1+411yescharacteristic
435Dic3—24no—
436Dic3—24no—
437Dic3—24no—
438Dic3—42no—
439Dic3—42no—
440C23—62no—
441C23—64no—
442C23—122no—
443C23—122no—
444C23C2×D3×D411yescharacteristic
445Q8—34no—
446Q8—48no—
447Q8—48no—
448Q8—62no—
449Q8—64no—
450Q8—68no—
451Q8—1216no—
452D4—34no—
453D4—62no—
454D4—64no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—34no—
459C2×C4—34no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—62no—
463C2×C4—64no—
464C2×C4—64no—
465C2×C4—64no—
466C2×C4—62no—
467C2×C4—62no—
468C2×C4—1216no—
469C2×C4—122no—
470C2×C4—122no—
471C2×C4—128no—
472C2×C4#91114yes—
473C2×C4C2×D3×D412yes—
474C2×C4C2×D3×D411yescharacteristic
475C2×C4C2×D3×D412yes—
476C2×C4C2×D3×Q812yes—
477C2×C4#100712yes—
478C2×C4C24×D311yescharacteristic
479C8—22no—
480C8—64no—
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—
491C22#572712yes—
492C22C2×#91112yes—
493C22#576612yes—
494C22C22×D3×D411yescharacteristic
495C4—22no—
496C4—44no—
497C4—44no—
498C4—64no—
499C4—64no—
500C4—64no—
501C4—122no—
502C4—122no—
503C4#572712yes—
504C4C2×#91114yes—
505C4C22×D3×Q811yescharacteristic
506C4C2×#100711yescharacteristic
507C3C2×#338611yescharacteristic, complemented, Sylow
508C2—124no—
509C2#8195514yesdirect product factor
510C2C2×#572711yescharacteristic
511C2C22×#91111yescharacteristic
512C2C2×#576611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#500C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenterC2×#500D3
hypocenterC3C2×#3386
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#500

Complements

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

NH ≅ G/Ntype
2. C2×#5721C2semidirect
3. C2×C3×#500C2semidirect
4. C2×#5094C2semidirect
6. #81955C2direct (N × H)
7. C2×#4875C2semidirect
9. C2×#4876C2semidirect
12. C2×#4679C2semidirect
24. #5721C22semidirect
34. C3×#500C22semidirect
43. #5094C22semidirect
52. #4876C22semidirect
53. #4875C22semidirect
55. #4679C22semidirect
60. C2×#500D3semidirect
107. C2×#204Q8semidirect
159. #500C2×D3semidirect
210. C2×Dic6C2×Q8semidirect
213. C2×C3×C8C2×Q8semidirect
243. #204C2×Q8semidirect
352. Dic6C22×Q8semidirect
354. C3×C8C22×Q8semidirect
392. C2×C8D3×Q8semidirect
481. C8C2×D3×Q8semidirect
482. C2×C3#3386semidirect
507. C3C2×#3386semidirect
509. C2#81955direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#57272
C2×#57212
C2×C3×#5002
C2×#50942
C2×#50872
#819552
C2×#48752
C2×#48912
C2×#48762
C2×#48892
C2×C4×#2042
C2×#46792
C2×#33863