Finite Group Databasebeta

← C2×#81136 #11675625

Subgroups — C2×#81136 #11675625

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×#5730C211yesmaximal, characteristic
2C2×#5570C211yesmaximal, characteristic, complemented
3C2×C3×#477C211yesmaximal, characteristic, complemented
4C2×#5080C211yesmaximal, characteristic, complemented
5C2×C4×#215C211yesmaximal, characteristic
6#81136C2116yesmaximal, direct product factor
7C2×#4891C212yesmaximal
8C2×#4878C212yesmaximal, complemented
9C2×#4846C211yesmaximal, characteristic
10C2×#4830C211yesmaximal, characteristic, complemented
11C2×#4796C212yesmaximal, complemented
12C2×#4788C212yesmaximal
13C2×#3381—31nomaximal, Sylow
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C4×Q8C2211yescharacteristic
16C2×#946—22no—
17C2×Q8×Dic3C2212yes—
18C22×#215C2211yescharacteristic
19#5730—24no—
20#5730C2218yes—
21#5570—24no—
22#5570C2218yescomplemented
23C2×#897—21no—
24C2×#893C2212yes—
25C2×D3×C4⋊C4—21no—
26C2×#890—22no—
27C2×#890C2212yes—
28C2×#890C2212yes—
29C2×C4×D12C2211yescharacteristic
30C2×#869—21no—
31C2×#869C2211yescharacteristic
32C2×C4×Dic6—21no—
33C2×C4×Dic6C2211yescharacteristic
34C3×#477C2218yescomplemented
35C2×C3×#107C2212yes—
36C2×C3×C4⋊C8C2211yescharacteristic
37C2×C3×#103C2212yes—
38C2×#835C2211yescharacteristic
39C2×#821C2212yes—
40C2×#818C2212yes—
41C2×Dic3⋊C8C2212yes—
42C2×C8×Dic3C2212yes—
43#5080C2218yescomplemented
44C4×#215C2218yes—
45C2×#809C2211yescharacteristic
46C2×#808C2212yes—
47C2×#808C2212yes—
48C2×#808C2211yescharacteristic
49C2×C4×C3⋊C8C2211yescharacteristic
50#4891C22116yes—
51#4878C22116yescomplemented
52#4846C2218yes—
53#4830C2218yescomplemented
54#4796C22116yescomplemented
55#4788C22116yes—
56C2×#557—31no—
57C2×#554—31no—
58#3381—316no—
59C2×#499—32no—
60C2×#498—31no—
61C2×#479—32no—
62C2×#477D311yescharacteristic, complemented
63C2×#478—32no—
64C2×#474—31no—
65C2×C8×Q8—31no—
66C2×C4×SD16—32no—
67C2×C4×SD16—31no—
68C22×C3×Q8—21no—
69C22×C3×Q8C2311yescharacteristic
70C22×D12C2311yescharacteristic
71C22×C4×D3—21no—
72C22×C4×D3—22no—
73C22×Dic6—21no—
74C22×Dic6D411yescharacteristic
75C22×C3×C8C2312yes—
76C3×C4⋊Q8—22no—
77C3×C4⋊Q8C2314yes—
78C3×C4×Q8—22no—
79C3×C4×Q8C2314yes—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4C2312yes—
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C42—21no—
85C2×C3×C42C2311yescharacteristic
86#946—216no—
87Q8×Dic3—24no—
88Q8×Dic3C2318yes—
89C2×#215Q818yescomplemented
90C2×#215C2314yes—
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—22no—
93C2×C4⋊Dic3—21no—
94C2×C4⋊Dic3—22no—
95C2×C4⋊Dic3—22no—
96C2×C4⋊Dic3—22no—
97C2×C4⋊Dic3D411yescharacteristic
98C2×C4⋊Dic3C2312yes—
99C2×C4⋊Dic3C2311yescharacteristic
100C2×Dic3⋊C4—22no—
101C2×Dic3⋊C4—22no—
102C2×Dic3⋊C4D412yes—
103C2×C4×Dic3—22no—
104C2×C4×Dic3C2312yes—
105C22×C3⋊C8C2311yescharacteristic
106C22×C3⋊C8C2311yescharacteristic
107#897—28no—
108#893—24no—
109#893C2318yes—
110D3×C4⋊C4—28no—
111#890—216no—
112#890—24no—
113#890—24no—
114#890C2318yes—
115#890C2318yes—
116C4×D12—22no—
117C4×D12C2314yes—
118#869—28no—
119#869—22no—
120#869C2314yes—
121C4×Dic6—28no—
122C4×Dic6D414yes—
123C4×Dic6C2314yes—
124C3×#107C2318yes—
125C3×C4⋊C8C2314yes—
126C3×#103C2318yes—
127#835C2314yes—
128#821C2318yes—
129#818C2318yes—
130Dic3⋊C8C2318yes—
131C8×Dic3C2318yes—
132#809C2314yes—
133#808C2314yes—
134#808C2318yes—
135#808C2318yes—
136C4×C3⋊C8C2314yes—
137C22×SD16—31no—
138#557—38no—
139#557—64no—
140#554—38no—
141#554—64no—
142C2×C4⋊Q8—31no—
143C2×C4⋊Q8—61no—
144C2×C4⋊Q8C2×D311yescharacteristic
145C2×#125—32no—
146C2×#125—32no—
147C2×#125—62no—
148C2×C22⋊Q8—61no—
149C2×C22⋊Q8—62no—
150C2×C4×Q8—31no—
151C2×C4×Q8—32no—
152C2×C4×Q8—61no—
153C2×C4×Q8C2×D311yescharacteristic
154C2×C4×D4—32no—
155C2×C4×D4—31no—
156C22×C4⋊C4—61no—
157#499—316no—
158#498—38no—
159#479—316no—
160#478—316no—
161#477C2×D318yescomplemented
162#474—38no—
163C8×Q8—38no—
164C4×SD16—38no—
165C4×SD16—316no—
166C2×#107—32no—
167C2×#107C2×D312yes—
168C2×#106—32no—
169C2×#106—32no—
170C2×#106—31no—
171C2×C4⋊C8—32no—
172C2×C4⋊C8C2×D311yescharacteristic
173C2×#103—31no—
174C2×#103C2×D312yes—
175C2×#102—32no—
176C2×#102—31no—
177C2×C4×C8—32no—
178C2×C4×C8—31no—
179C23×D3—21no—
180C2×C3×Q8—21no—
181C2×C3×Q8—24no—
182C2×C3×Q8—22no—
183C2×C3×Q8—82no—
184C2×C3×Q8C2×Q814yes—
185C2×C3×Q8C2412yes—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—22no—
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2411yescharacteristic
192C22×Dic3—22no—
193C22×Dic3—21no—
194C22×Dic3C2×D412yes—
195C22×Dic3C2×D412yes—
196C2×D12—21no—
197C2×D12—22no—
198C2×D12C2×Q814yes—
199C2×D12C2412yes—
200C2×C4×D3—24no—
201C2×C4×D3—28no—
202C2×C4×D3—44no—
203C2×C4×D3—84no—
204C2×Dic6—24no—
205C2×Dic6—24no—
206C2×Dic6—44no—
207C2×Dic6C2×D414yes—
208C2×C3×C8—24no—
209C2×C3×C8C2414yes—
210C3×C4⋊C4—24no—
211C3×C4⋊C4—21no—
212C3×C4⋊C4—22no—
213C3×C4⋊C4—24no—
214C3×C4⋊C4C2414yes—
215C3×C4⋊C4C2412yes—
216C3×C42—24no—
217C3×C42C2×D412yes—
218C3×C42C2412yes—
219#215C2×Q8116yescomplemented
220(C2×D3)⋊C4—24no—
221(C2×D3)⋊C4—28no—
222C4⋊Dic3—22no—
223C4⋊Dic3—24no—
224C4⋊Dic3—21no—
225C4⋊Dic3—28no—
226C4⋊Dic3—28no—
227C4⋊Dic3—28no—
228C4⋊Dic3C2×D414yes—
229C4⋊Dic3C2414yes—
230C4⋊Dic3C2412yes—
231Dic3⋊C4—28no—
232Dic3⋊C4—28no—
233Dic3⋊C4C2×D418yes—
234C4×Dic3—28no—
235C4×Dic3C2×D414yes—
236C4×Dic3C2414yes—
237C2×C3⋊C8—22no—
238C2×C3⋊C8C2×Q814yes—
239C2×C3⋊C8C2412yes—
240C2×C3⋊C8C2412yes—
241C22×Q8—21no—
242C22×Q8—31no—
243C22×Q8—61no—
244C22×Q8C22×D311yescharacteristic
245C22×D4—31no—
246C23×C4—62no—
247C23×C4—61no—
248C2×SD16—38no—
249C2×SD16—34no—
250C22×C8—31no—
251C22×C8—31no—
252C22×C8C22×D312yes—
253C4⋊Q8—22no—
254C4⋊Q8—34no—
255C4⋊Q8—68no—
256C4⋊Q8—62no—
257C4⋊Q8C22×D314yes—
258#125—38no—
259#125—38no—
260#125—64no—
261#125—64no—
262#125—616no—
263C22⋊Q8—68no—
264C22⋊Q8—616no—
265C4×Q8—22no—
266C4×Q8—34no—
267C4×Q8—34no—
268C4×Q8—38no—
269C4×Q8—68no—
270C4×Q8—64no—
271C4×Q8C22×D314yes—
272C4×D4—38no—
273C4×D4—34no—
274C4×D4—64no—
275C4×D4—62no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—31no—
279C2×C4⋊C4—32no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—32no—
282C2×C4⋊C4—68no—
283C2×C4⋊C4—62no—
284C2×C4⋊C4—62no—
285C2×C4⋊C4—62no—
286C2×C4⋊C4—62no—
287C2×C4⋊C4—62no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4C22×D312yes—
290C2×C4⋊C4C22×D311yescharacteristic
291C2×C22⋊C4—62no—
292C2×C22⋊C4—61no—
293C2×C42—21no—
294C2×C42—32no—
295C2×C42—62no—
296C2×C42C22×D311yescharacteristic
297#107—38no—
298#107C22×D318yes—
299#106—38no—
300#106—34no—
301#106—38no—
302C4⋊C8—38no—
303C4⋊C8C22×D314yes—
304#103—34no—
305#103C22×D318yes—
306#102—38no—
307#102—34no—
308C4×C8—38no—
309C4×C8—34no—
310C23×C3C22×D411yescharacteristic
311C22×D3—24no—
312C22×D3—22no—
313C22×D3—42no—
314C22×D3—42no—
315C3×Q8—24no—
316C3×Q8—22no—
317C3×Q8—84no—
318C3×Q8C22×Q814yes—
319C2×C3×C4—24no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—24no—
324C2×C3×C4—44no—
325C2×C3×C4—42no—
326C2×C3×C4#13514yes—
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—44no—
338C2×Dic3—42no—
339C2×Dic3C22×D414yes—
340C2×Dic3C22×D414yes—
341D12—24no—
342D12—22no—
343D12C22×Q814yes—
344C4×D3—48no—
345C4×D3—88no—
346Dic6—28no—
347Dic6—48no—
348C3×C8—24no—
349C3⋊C8—22no—
350C3⋊C8C22×Q814yes—
351C24—61no—
352C2×Q8—21no—
353C2×Q8—24no—
354C2×Q8—22no—
355C2×Q8—34no—
356C2×Q8—64no—
357C2×Q8—64no—
358C2×Q8—82no—
359C2×Q8—124no—
360C2×Q8D3×Q814yes—
361C2×Q8C23×D312yes—
362C2×D4—34no—
363C2×D4—32no—
364C2×D4—61no—
365C2×D4—62no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—32no—
370C22×C4—32no—
371C22×C4—68no—
372C22×C4—62no—
373C22×C4—61no—
374C22×C4—64no—
375C22×C4—124no—
376C22×C4—244no—
377C22×C4D3×D411yescharacteristic
378C22×C4D3×D411yescharacteristic
379C22×C4C23×D311yescharacteristic
380SD16—316no—
381C2×C8—24no—
382C2×C8—34no—
383C2×C8—32no—
384C2×C8—32no—
385C2×C8—62no—
386C2×C8C23×D314yes—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—22no—
390C4⋊C4—21no—
391C4⋊C4—34no—
392C4⋊C4—38no—
393C4⋊C4—34no—
394C4⋊C4—32no—
395C4⋊C4—68no—
396C4⋊C4—68no—
397C4⋊C4—68no—
398C4⋊C4—68no—
399C4⋊C4—68no—
400C4⋊C4—61no—
401C4⋊C4—62no—
402C4⋊C4—64no—
403C4⋊C4C23×D312yes—
404C4⋊C4C23×D314yes—
405C22⋊C4—68no—
406C22⋊C4—64no—
407C42—24no—
408C42—34no—
409C42—34no—
410C42—68no—
411C42D3×D412yes—
412C42C23×D312yes—
413C22×C3D4×Q812yes—
414C22×C3C2×#13512yes—
415C22×C3#57712yes—
416C22×C3C23×D411yescharacteristic
417C2×D3—24no—
418C2×D3—48no—
419C2×D3—42no—
420C2×D3—42no—
421C3×C4—24no—
422C3×C4—22no—
423C3×C4—44no—
424C3×C4—42no—
425C3×C4D4×Q812yes—
426C3×C4C2×#13514yes—
427C3×C4C23×Q811yescharacteristic
428C3×C4C2×2−1+411yescharacteristic
429Dic3—24no—
430Dic3—24no—
431Dic3—44no—
432Dic3—42no—
433C23—64no—
434C23—62no—
435C23—122no—
436C23—122no—
437C23C2×D3×D411yescharacteristic
438Q8—24no—
439Q8—22no—
440Q8—68no—
441Q8—84no—
442Q8—128no—
443Q8C2×D3×Q814yes—
444D4—34no—
445D4—64no—
446D4—62no—
447C2×C4—24no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—34no—
453C2×C4—34no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—64no—
457C2×C4—64no—
458C2×C4—64no—
459C2×C4—62no—
460C2×C4—122no—
461C2×C4—124no—
462C2×C4—128no—
463C2×C4—248no—
464C2×C4#91914yes—
465C2×C4C2×D3×D412yes—
466C2×C4C2×D3×D411yescharacteristic
467C2×C4C2×D3×D412yes—
468C2×C4C2×D3×Q812yes—
469C2×C4#101012yes—
470C2×C4C24×D311yescharacteristic
471C8—24no—
472C8—34no—
473C8—62no—
474C2×C3#338114yescomplemented
475C2×C3C2×D4×Q811yescharacteristic
476C2×C3C22×#13511yescharacteristic
477C2×C3C2×#57711yescharacteristic
478D3—44no—
479C22—64no—
480C22—122no—
481C22—122no—
482C22—128no—
483C22#563212yes—
484C22C2×#91912yes—
485C22#577512yes—
486C22C22×D3×D411yescharacteristic
487C4—22no—
488C4—24no—
489C4—44no—
490C4—42no—
491C4—64no—
492C4—64no—
493C4—122no—
494C4—124no—
495C4#563212yes—
496C4C2×#91914yes—
497C4C22×D3×Q811yescharacteristic
498C4C2×#101011yescharacteristic
499C3C2×#338111yescharacteristic, complemented, Sylow
500C2—124no—
501C2#8113614yesdirect product factor
502C2C2×#563211yescharacteristic
503C2C22×#91911yescharacteristic
504C2C2×#577511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#477C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenterC2×#477D3
hypocenterC3C2×#3381
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#477

Complements

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

NH ≅ G/Ntype
2. C2×#5570C2semidirect
3. C2×C3×#477C2semidirect
4. C2×#5080C2semidirect
6. #81136C2direct (N × H)
8. C2×#4878C2semidirect
10. C2×#4830C2semidirect
11. C2×#4796C2semidirect
22. #5570C22semidirect
34. C3×#477C22semidirect
43. #5080C22semidirect
51. #4878C22semidirect
53. #4830C22semidirect
54. #4796C22semidirect
62. C2×#477D3semidirect
89. C2×#215Q8semidirect
161. #477C2×D3semidirect
219. #215C2×Q8semidirect
474. C2×C3#3381semidirect
499. C3C2×#3381semidirect
501. C2#81136direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#57302
C2×#55702
C2×C3×#4772
C2×#50802
C2×C4×#2152
#811362
C2×#48912
C2×#48782
C2×#48462
C2×#48302
C2×#47962
C2×#47882
C2×#33813