Finite Group Databasebeta

← C2×#81918 #11664818

Subgroups — C2×#81918 #11664818

Order 768.

1854 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5689C212yesmaximal
2C2×C3×#499C211yesmaximal, characteristic, complemented
3C2×#5064C212yesmaximal, complemented
4#81918C2116yesmaximal, direct product factor
5C2×#4880C211yesmaximal, characteristic
6C2×D3×#107C211yesmaximal, characteristic
7C2×#4878C211yesmaximal, characteristic, complemented
8C2×#4874C211yesmaximal, characteristic, complemented
9C2×#4861C212yesmaximal
10C2×#4856C212yesmaximal, complemented
11C2×#4689C211yesmaximal, characteristic
12C2×#4682C211yesmaximal, characteristic, complemented
13C2×#3164—31nomaximal, Sylow
14C2×C3×#125C2212yes—
15C22×ΓL(1,𝔽25)C2211yescharacteristic
16C22×D3×C8C2211yescharacteristic
17#5689—28no—
18#5689C22116yes—
19C2×#897—22no—
20C2×#896—22no—
21C2×#892C2212yes—
22C2×D3×C4⋊C4C2212yes—
23C2×#890C2212yes—
24C2×#889—22no—
25C2×#888—22no—
26C2×#888C2212yes—
27C2×#872C2211yescharacteristic
28C2×#869C2212yes—
29C3×#499C2218yescomplemented
30C2×C3×#107C2211yescharacteristic
31C2×C3×#107C2211yescharacteristic
32C2×C3×#106C2212yes—
33C2×C3×C4×C8C2211yescharacteristic
34C2×#818C2212yes—
35C2×#818C2211yescharacteristic
36C2×#818C2211yescharacteristic
37C2×#815C2211yescharacteristic
38C2×C8×Dic3C2211yescharacteristic
39#5064C22116yescomplemented
40C2×#808C2212yes—
41C2×#808C2212yes—
42C2×#807C2212yes—
43C2×#807C2212yes—
44C2×#803C2211yescharacteristic
45#4880C2218yes—
46D3×#107C2218yes—
47#4878C2218yescomplemented
48#4874C2218yescomplemented
49#4861C22116yes—
50#4856C22116yescomplemented
51#4689C2218yes—
52#4682C2218yescomplemented
53C2×#544—32no—
54#3164—316no—
55C2×#501—32no—
56C2×#501—32no—
57C2×#500—31no—
58C2×#500—31no—
59C2×#499—31no—
60C2×#499D311yescharacteristic, complemented
61C2×#428—32no—
62C2×#427—31no—
63C22×#107—31no—
64C2×#405—31no—
65C22×C4×D3—22no—
66C22×C4×D3C2311yescharacteristic
67C22×Dic6—22no—
68C22×Dic6—22no—
69C22×C3×C8C2311yescharacteristic
70C22×C3×C8C2311yescharacteristic
71C3×#125—24no—
72C3×#125C2318yes—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4C2312yes—
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C42C2311yescharacteristic
78C2×(C2×D3)⋊C4—22no—
79C2×(C2×D3)⋊C4D411yescharacteristic
80C2×(C2×D3)⋊C4D411yescharacteristic
81C2×C4⋊Dic3—22no—
82C2×C4⋊Dic3—22no—
83C2×C4⋊Dic3C2312yes—
84C2×C4⋊Dic3C2312yes—
85C2×Dic3⋊C4—22no—
86C2×Dic3⋊C4—22no—
87C2×Dic3⋊C4—22no—
88C2×Dic3⋊C4D411yescharacteristic
89C2×Dic3⋊C4D411yescharacteristic
90C2×C4×Dic3—22no—
91C2×C4×Dic3C2311yescharacteristic
92C22×C3⋊C8C2311yescharacteristic
93C22×C3⋊C8C2311yescharacteristic
94C2×ΓL(1,𝔽25)Q818yescomplemented
95C2×ΓL(1,𝔽25)C2314yes—
96C2×D3×C8Q818yescomplemented
97C2×D3×C8C2314yes—
98#897—216no—
99#896—216no—
100#892—24no—
101#892C2318yes—
102D3×C4⋊C4—24no—
103D3×C4⋊C4C2318yes—
104#890—24no—
105#890C2318yes—
106#889—216no—
107#888—216no—
108#888—24no—
109#888C2318yes—
110#872D414yes—
111#872C2314yes—
112#869—24no—
113#869C2318yes—
114C3×#107C2314yes—
115C3×#107C2314yes—
116C3×#106C2318yes—
117C3×C4×C8C2314yes—
118#818C2314yes—
119#818C2314yes—
120#818C2318yes—
121#815C2314yes—
122C8×Dic3C2314yes—
123#808C2318yes—
124#808C2318yes—
125#807C2318yes—
126#807C2318yes—
127#803C2314yes—
128C22×M16—31no—
129C23×C8—31no—
130#544—316no—
131#544—68no—
132C2×C4⋊Q8—32no—
133C2×C4⋊Q8—32no—
134C2×C4⋊Q8—62no—
135C2×#125—32no—
136C2×#125—62no—
137C2×#125C2×D312yes—
138C2×C22⋊Q8—62no—
139C2×C22⋊Q8—62no—
140C2×#117—31no—
141C2×#117—32no—
142C22×C4⋊C4—32no—
143#501—316no—
144#501—316no—
145#500—38no—
146#500—38no—
147#499—38no—
148#499C2×D318yescomplemented
149#428—316no—
150#427—38no—
151C2×#107—31no—
152C2×#107—31no—
153C2×#107—32no—
154C2×#107—32no—
155C2×#107—32no—
156C2×#107—38no—
157C2×#107C2×D311yescharacteristic
158C2×#107C2×D311yescharacteristic
159C2×#106—32no—
160C2×#106—32no—
161C2×#106C2×D312yes—
162#405—38no—
163C2×#97—31no—
164C2×#97—31no—
165C2×C4×C8—31no—
166C2×C4×C8C2×D311yescharacteristic
167C23×D3C2×D411yescharacteristic
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2411yescharacteristic
173C22×Dic3—22no—
174C22×Dic3—22no—
175C22×Dic3C2×D411yescharacteristic
176C22×Dic3C2×D411yescharacteristic
177C22×Dic3C2×D411yescharacteristic
178C2×C4×D3—28no—
179C2×C4×D3—84no—
180C2×C4×D3C2×D412yes—
181C2×C4×D3C2×Q814yes—
182C2×C4×D3C2×Q814yes—
183C2×C4×D3C2412yes—
184C2×Dic6—28no—
185C2×Dic6—28no—
186C2×Dic6—84no—
187C2×Dic6—84no—
188C2×C3×C8C2×Q814yes—
189C2×C3×C8C2×Q814yes—
190C2×C3×C8C2412yes—
191C2×C3×C8C2412yes—
192C3×C4⋊C4—22no—
193C3×C4⋊C4—28no—
194C3×C4⋊C4—28no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4C2414yes—
197C3×C4⋊C4C2414yes—
198C3×C42C2×D412yes—
199C3×C42C2412yes—
200(C2×D3)⋊C4—28no—
201(C2×D3)⋊C4C2×D414yes—
202(C2×D3)⋊C4C2×D414yes—
203C4⋊Dic3—22no—
204C4⋊Dic3—22no—
205C4⋊Dic3—28no—
206C4⋊Dic3—28no—
207C4⋊Dic3C2414yes—
208C4⋊Dic3C2414yes—
209Dic3⋊C4—28no—
210Dic3⋊C4—28no—
211Dic3⋊C4—28no—
212Dic3⋊C4C2×D414yes—
213Dic3⋊C4C2×D414yes—
214C4×Dic3—28no—
215C4×Dic3C2×D412yes—
216C4×Dic3C2412yes—
217C2×C3⋊C8C2×Q814yes—
218C2×C3⋊C8C2×Q814yes—
219C2×C3⋊C8C2412yes—
220C2×C3⋊C8C2412yes—
221ΓL(1,𝔽25)C2×Q8116yescomplemented
222D3×C8C2×Q8116yescomplemented
223C22×Q8—62no—
224C22×Q8—62no—
225C23×C4—31no—
226C23×C4—62no—
227C2×M16—38no—
228C2×M16—34no—
229C22×C8—31no—
230C22×C8—31no—
231C22×C8—38no—
232C22×C8—34no—
233C22×C8C22×D311yescharacteristic
234C22×C8C22×D311yescharacteristic
235C4⋊Q8—38no—
236C4⋊Q8—38no—
237C4⋊Q8—64no—
238C4⋊Q8—616no—
239C4⋊Q8—64no—
240#125—24no—
241#125—38no—
242#125—64no—
243#125—616no—
244#125C22×D318yes—
245C22⋊Q8—616no—
246C22⋊Q8—616no—
247#117—34no—
248#117—34no—
249#117—38no—
250#117—64no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—31no—
254C2×C4⋊C4—31no—
255C2×C4⋊C4—32no—
256C2×C4⋊C4—32no—
257C2×C4⋊C4—38no—
258C2×C4⋊C4—64no—
259C2×C4⋊C4—62no—
260C2×C4⋊C4—62no—
261C2×C4⋊C4—62no—
262C2×C4⋊C4—62no—
263C2×C4⋊C4—62no—
264C2×C4⋊C4C22×D312yes—
265C2×C4⋊C4C22×D312yes—
266C2×C22⋊C4—31no—
267C2×C22⋊C4—31no—
268C2×C22⋊C4—62no—
269C2×C42—31no—
270C2×C42—62no—
271C2×C42C22×D311yescharacteristic
272#107—34no—
273#107—34no—
274#107—38no—
275#107—38no—
276#107—38no—
277#107C22×D314yes—
278#107C22×D314yes—
279#106—38no—
280#106—38no—
281#106C22×D318yes—
282#97—34no—
283#97—34no—
284C4×C8—34no—
285C4×C8C22×D314yes—
286C23×C3C22×D411yescharacteristic
287C22×D3—22no—
288C22×D3—22no—
289C22×D3C4⋊Q814yes—
290C22×D3C22×D412yes—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4—24no—
294C2×C3×C4—24no—
295C2×C3×C4—44no—
296C2×C3×C4—44no—
297C2×C3×C4C22×D412yes—
298C2×C3×C4C22×D412yes—
299C2×C3×C4C22×D411yescharacteristic
300C2×C3×C4C22×Q812yes—
301C2×C3×C4C22×Q812yes—
302C2×C3×C4C2511yescharacteristic
303C2×Dic3—22no—
304C2×Dic3—22no—
305C2×Dic3—24no—
306C2×Dic3—24no—
307C2×Dic3—44no—
308C2×Dic3—44no—
309C2×Dic3C4⋊Q814yes—
310C2×Dic3C22×D412yes—
311C2×Dic3C22×D412yes—
312C2×Dic3C22×D412yes—
313C4×D3—88no—
314C4×D3C4⋊Q818yes—
315C4×D3C22×Q814yes—
316C4×D3C22×Q814yes—
317Dic6—88no—
318Dic6—88no—
319C3×C8C22×Q814yes—
320C3×C8C22×Q814yes—
321C3⋊C8C22×Q814yes—
322C3⋊C8C22×Q814yes—
323C24—31no—
324C2×Q8—68no—
325C2×Q8—68no—
326C2×Q8—244no—
327C2×Q8—244no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—31no—
331C22×C4—32no—
332C22×C4—34no—
333C22×C4—34no—
334C22×C4—32no—
335C22×C4—31no—
336C22×C4—31no—
337C22×C4—68no—
338C22×C4—62no—
339C22×C4—62no—
340C22×C4—244no—
341C22×C4D3×D411yescharacteristic
342C22×C4D3×D411yescharacteristic
343C22×C4C23×D311yescharacteristic
344M16—316no—
345C2×C8—34no—
346C2×C8—34no—
347C2×C8—32no—
348C2×C8—32no—
349C2×C8—316no—
350C2×C8D3×Q814yes—
351C2×C8D3×Q814yes—
352C2×C8C23×D312yes—
353C2×C8C23×D312yes—
354C4⋊C4—28no—
355C4⋊C4—28no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—34no—
359C4⋊C4—34no—
360C4⋊C4—34no—
361C4⋊C4—34no—
362C4⋊C4—68no—
363C4⋊C4—68no—
364C4⋊C4—68no—
365C4⋊C4—68no—
366C4⋊C4—62no—
367C4⋊C4—62no—
368C4⋊C4—68no—
369C4⋊C4C23×D314yes—
370C4⋊C4C23×D314yes—
371C22⋊C4—34no—
372C22⋊C4—34no—
373C22⋊C4—68no—
374C42—32no—
375C42—32no—
376C42—68no—
377C42D3×D412yes—
378C42C23×D312yes—
379C22×C3C2×C4⋊Q812yes—
380C22×C3#57612yes—
381C22×C3#57812yes—
382C22×C3C23×D411yescharacteristic
383C2×D3—28no—
384C2×D3—22no—
385C2×D3—22no—
386C2×D3C2×C4⋊Q814yes—
387C3×C4—22no—
388C3×C4—22no—
389C3×C4—44no—
390C3×C4—44no—
391C3×C4C2×C4⋊Q812yes—
392C3×C4C23×Q811yescharacteristic
393C3×C4C23×Q811yescharacteristic
394Dic3—22no—
395Dic3—22no—
396Dic3—44no—
397Dic3—44no—
398Dic3C2×C4⋊Q814yes—
399C23—32no—
400C23—34no—
401C23—62no—
402C23—62no—
403C23C2×D3×D411yescharacteristic
404Q8—248no—
405Q8—248no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—32no—
411C2×C4—34no—
412C2×C4—34no—
413C2×C4—34no—
414C2×C4—38no—
415C2×C4—32no—
416C2×C4—32no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—62no—
420C2×C4—62no—
421C2×C4—64no—
422C2×C4—64no—
423C2×C4—124no—
424C2×C4—124no—
425C2×C4—248no—
426C2×C4C2×D3×D412yes—
427C2×C4C2×D3×D412yes—
428C2×C4C2×D3×D411yescharacteristic
429C2×C4C2×D3×Q812yes—
430C2×C4C2×D3×Q812yes—
431C2×C4C24×D311yescharacteristic
432C8—34no—
433C8—34no—
434C8C2×D3×Q814yes—
435C8C2×D3×Q814yes—
436C2×C3#316414yescomplemented
437C2×C3C22×C4⋊Q811yescharacteristic
438C2×C3C2×#57611yescharacteristic
439C2×C3C2×#57811yescharacteristic
440D3—24no—
441C22—34no—
442C22—62no—
443C22—62no—
444C22—68no—
445C22D3×C4⋊Q812yes—
446C22#576912yes—
447C22#577112yes—
448C22C22×D3×D411yescharacteristic
449C4—22no—
450C4—22no—
451C4—34no—
452C4—44no—
453C4—44no—
454C4—62no—
455C4—62no—
456C4—124no—
457C4—124no—
458C4D3×C4⋊Q812yes—
459C4C22×D3×Q811yescharacteristic
460C4C22×D3×Q811yescharacteristic
461C3C2×#316411yescharacteristic, complemented, Sylow
462C2—64no—
463C2#8191814yesdirect product factor
464C2C2×D3×C4⋊Q811yescharacteristic
465C2C2×#576911yescharacteristic
466C2C2×#577111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#499C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenterC2×#499D3
hypocenterC3C2×#3164
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#499

Complements

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

NH ≅ G/Ntype
2. C2×C3×#499C2semidirect
3. C2×#5064C2semidirect
4. #81918C2direct (N × H)
7. C2×#4878C2semidirect
8. C2×#4874C2semidirect
10. C2×#4856C2semidirect
12. C2×#4682C2semidirect
29. C3×#499C22semidirect
39. #5064C22semidirect
47. #4878C22semidirect
48. #4874C22semidirect
50. #4856C22semidirect
52. #4682C22semidirect
60. C2×#499D3semidirect
94. C2×ΓL(1,𝔽25)Q8semidirect
96. C2×D3×C8Q8semidirect
148. #499C2×D3semidirect
221. ΓL(1,𝔽25)C2×Q8semidirect
222. D3×C8C2×Q8semidirect
436. C2×C3#3164semidirect
461. C3C2×#3164semidirect
463. C2#81918direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#56892
C2×C3×#4992
C2×#50642
#819182
C2×#48802
C2×D3×#1072
C2×#48782
C2×#48742
C2×#48612
C2×#48562
C2×#46892
C2×#46822
C2×#31643