Finite Group Databasebeta

← C2×#81892 #11664805

Subgroups — C2×#81892 #11664805

Order 768.

1854 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5729C212yesmaximal
2C2×C3×#498C211yesmaximal, characteristic, complemented
3C2×#5081C212yesmaximal, complemented
4#81892C2116yesmaximal, direct product factor
5C2×#4861C212yesmaximal
6C2×#4860C211yesmaximal, characteristic
7C2×D3×#106C211yesmaximal, characteristic
8C2×#4856C212yesmaximal, complemented
9C2×#4858C211yesmaximal, characteristic, complemented
10C2×#4855C211yesmaximal, characteristic, complemented
11C2×#4689C211yesmaximal, characteristic
12C2×#4683C211yesmaximal, characteristic, complemented
13C2×#3163—31nomaximal, Sylow
14C2×C3×C4⋊Q8C2212yes—
15C2×#946—22no—
16C2×#944—22no—
17C22×ΓL(1,𝔽25)C2211yescharacteristic
18C22×D3×C8C2211yescharacteristic
19#5729—28no—
20#5729C22116yes—
21C2×#896—22no—
22C2×#892C2212yes—
23C2×D3×C4⋊C4C2212yes—
24C2×#890C2212yes—
25C2×#889—22no—
26C2×#888C2212yes—
27C2×#872C2211yescharacteristic
28C2×#870C2212yes—
29C3×#498C2218yescomplemented
30C2×C3×#106C2211yescharacteristic
31C2×C3×#106C2211yescharacteristic
32C2×C3×#106C2212yes—
33C2×C3×C4×C8C2211yescharacteristic
34C2×#818C2212yes—
35C2×#817C2211yescharacteristic
36C2×#817C2211yescharacteristic
37C2×#815C2211yescharacteristic
38C2×C8×Dic3C2211yescharacteristic
39#5081C22116yescomplemented
40C2×#808C2212yes—
41C2×#808C2212yes—
42C2×#807C2212yes—
43C2×#807C2212yes—
44C2×#803C2211yescharacteristic
45#4861C22116yes—
46#4860C2218yes—
47D3×#106C2218yes—
48#4858C2218yescomplemented
49#4856C22116yescomplemented
50#4855C2218yescomplemented
51#4689C2218yes—
52#4683C2218yescomplemented
53C2×#544—32no—
54#3163—316no—
55C2×#501—32no—
56C2×#501—32no—
57C2×#499—31no—
58C2×#499—31no—
59C2×#498—31no—
60C2×#498D311yescharacteristic, complemented
61C2×#428—32no—
62C2×#427—31no—
63C22×#106—31no—
64C2×#405—31no—
65C22×C3×Q8—22no—
66C22×C4×D3—22no—
67C22×C4×D3C2311yescharacteristic
68C22×Dic6—22no—
69C22×C3×C8C2311yescharacteristic
70C22×C3×C8C2311yescharacteristic
71C3×C4⋊Q8—24no—
72C3×C4⋊Q8C2318yes—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4C2312yes—
75C2×C3×C4⋊C4C2312yes—
76C2×C3×C42C2311yescharacteristic
77#946—216no—
78#944—216no—
79C2×(C2×D3)⋊C4—22no—
80C2×(C2×D3)⋊C4D412yes—
81C2×C4⋊Dic3—22no—
82C2×C4⋊Dic3C2312yes—
83C2×C4⋊Dic3C2312yes—
84C2×Dic3⋊C4—22no—
85C2×Dic3⋊C4—22no—
86C2×Dic3⋊C4—22no—
87C2×Dic3⋊C4—22no—
88C2×Dic3⋊C4—22no—
89C2×Dic3⋊C4D412yes—
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#896—216no—
99#892—24no—
100#892C2318yes—
101D3×C4⋊C4—24no—
102D3×C4⋊C4C2318yes—
103#890—24no—
104#890C2318yes—
105#889—216no—
106#888—24no—
107#888C2318yes—
108#872D414yes—
109#872C2314yes—
110#870—24no—
111#870C2318yes—
112C3×#106C2314yes—
113C3×#106C2318yes—
114C3×#106C2314yes—
115C3×C4×C8C2314yes—
116#818C2318yes—
117#817C2314yes—
118#817C2314yes—
119#815C2314yes—
120C8×Dic3C2314yes—
121#808C2318yes—
122#808C2318yes—
123#807C2318yes—
124#807C2318yes—
125#803C2314yes—
126C22×M16—31no—
127C23×C8—31no—
128#544—316no—
129#544—68no—
130C2×C4⋊Q8—32no—
131C2×C4⋊Q8—62no—
132C2×C4⋊Q8C2×D312yes—
133C2×#125—32no—
134C2×#125—32no—
135C2×#125—62no—
136C2×C22⋊Q8—62no—
137C2×C22⋊Q8—62no—
138C2×#117—31no—
139C2×#117—32no—
140C22×C4⋊C4—32no—
141#501—316no—
142#501—316no—
143#499—38no—
144#499—38no—
145#498—38no—
146#498C2×D318yescomplemented
147#428—316no—
148#427—38no—
149C2×#107—32no—
150C2×#107—32no—
151C2×#107—32no—
152C2×#106—31no—
153C2×#106—31no—
154C2×#106—32no—
155C2×#106—32no—
156C2×#106—38no—
157C2×#106C2×D311yescharacteristic
158C2×#106C2×D311yescharacteristic
159C2×#106C2×D312yes—
160#405—38no—
161C2×#97—31no—
162C2×#97—31no—
163C2×C4×C8—31no—
164C2×C4×C8C2×D311yescharacteristic
165C23×D3C2×D411yescharacteristic
166C2×C3×Q8—28no—
167C2×C3×Q8—84no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2411yescharacteristic
172C22×Dic3—22no—
173C22×Dic3—22no—
174C22×Dic3C2×D412yes—
175C22×Dic3C2×D411yescharacteristic
176C2×C4×D3—28no—
177C2×C4×D3—84no—
178C2×C4×D3C2×D412yes—
179C2×C4×D3C2×Q814yes—
180C2×C4×D3C2×Q814yes—
181C2×C4×D3C2412yes—
182C2×Dic6—28no—
183C2×Dic6—84no—
184C2×C3×C8C2×Q814yes—
185C2×C3×C8C2×Q814yes—
186C2×C3×C8C2412yes—
187C2×C3×C8C2412yes—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—28no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4C2414yes—
192C3×C4⋊C4C2414yes—
193C3×C42C2×D412yes—
194C3×C42C2412yes—
195(C2×D3)⋊C4—28no—
196(C2×D3)⋊C4C2×D418yes—
197C4⋊Dic3—22no—
198C4⋊Dic3—22no—
199C4⋊Dic3—28no—
200C4⋊Dic3C2414yes—
201C4⋊Dic3C2414yes—
202Dic3⋊C4—28no—
203Dic3⋊C4—28no—
204Dic3⋊C4—28no—
205Dic3⋊C4—28no—
206Dic3⋊C4—28no—
207Dic3⋊C4C2×D418yes—
208C4×Dic3—28no—
209C4×Dic3C2×D412yes—
210C4×Dic3C2412yes—
211C2×C3⋊C8C2×Q814yes—
212C2×C3⋊C8C2×Q814yes—
213C2×C3⋊C8C2412yes—
214C2×C3⋊C8C2412yes—
215ΓL(1,𝔽25)C2×Q8116yescomplemented
216D3×C8C2×Q8116yescomplemented
217C22×Q8—22no—
218C22×Q8—62no—
219C23×C4—31no—
220C23×C4—62no—
221C2×M16—38no—
222C2×M16—34no—
223C22×C8—31no—
224C22×C8—31no—
225C22×C8—38no—
226C22×C8—34no—
227C22×C8C22×D311yescharacteristic
228C22×C8C22×D311yescharacteristic
229C4⋊Q8—24no—
230C4⋊Q8—38no—
231C4⋊Q8—64no—
232C4⋊Q8—616no—
233C4⋊Q8C22×D318yes—
234#125—38no—
235#125—38no—
236#125—64no—
237#125—616no—
238#125—64no—
239C22⋊Q8—616no—
240C22⋊Q8—616no—
241#117—34no—
242#117—34no—
243#117—38no—
244#117—64no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—32no—
247C2×C4⋊C4—32no—
248C2×C4⋊C4—32no—
249C2×C4⋊C4—38no—
250C2×C4⋊C4—64no—
251C2×C4⋊C4—62no—
252C2×C4⋊C4—62no—
253C2×C4⋊C4—62no—
254C2×C4⋊C4—62no—
255C2×C4⋊C4—62no—
256C2×C4⋊C4—62no—
257C2×C4⋊C4C22×D312yes—
258C2×C4⋊C4C22×D312yes—
259C2×C22⋊C4—32no—
260C2×C22⋊C4—62no—
261C2×C42—31no—
262C2×C42—62no—
263C2×C42C22×D311yescharacteristic
264#107—38no—
265#107—38no—
266#107—38no—
267#106—34no—
268#106—34no—
269#106—38no—
270#106—38no—
271#106C22×D314yes—
272#106C22×D314yes—
273#106C22×D318yes—
274#97—34no—
275#97—34no—
276C4×C8—34no—
277C4×C8C22×D314yes—
278C23×C3C22×D411yescharacteristic
279C22×D3—24no—
280C22×D3C4⋊Q814yes—
281C22×D3C22×D412yes—
282C3×Q8—88no—
283C2×C3×C4—24no—
284C2×C3×C4—24no—
285C2×C3×C4—24no—
286C2×C3×C4—44no—
287C2×C3×C4—44no—
288C2×C3×C4C22×D414yes—
289C2×C3×C4C22×D411yescharacteristic
290C2×C3×C4C22×Q812yes—
291C2×C3×C4C22×Q812yes—
292C2×C3×C4C2511yescharacteristic
293C2×Dic3—24no—
294C2×Dic3—24no—
295C2×Dic3—24no—
296C2×Dic3—44no—
297C2×Dic3—44no—
298C2×Dic3C4⋊Q814yes—
299C2×Dic3C22×D414yes—
300C2×Dic3C22×D412yes—
301C4×D3—88no—
302C4×D3C4⋊Q818yes—
303C4×D3C22×Q814yes—
304C4×D3C22×Q814yes—
305Dic6—88no—
306C3×C8C22×Q814yes—
307C3×C8C22×Q814yes—
308C3⋊C8C22×Q814yes—
309C3⋊C8C22×Q814yes—
310C24—31no—
311C2×Q8—28no—
312C2×Q8—68no—
313C2×Q8—84no—
314C2×Q8—244no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—31no—
318C22×C4—32no—
319C22×C4—34no—
320C22×C4—34no—
321C22×C4—32no—
322C22×C4—32no—
323C22×C4—68no—
324C22×C4—62no—
325C22×C4—62no—
326C22×C4—244no—
327C22×C4D3×D412yes—
328C22×C4C23×D311yescharacteristic
329M16—316no—
330C2×C8—34no—
331C2×C8—34no—
332C2×C8—32no—
333C2×C8—32no—
334C2×C8—316no—
335C2×C8D3×Q814yes—
336C2×C8D3×Q814yes—
337C2×C8C23×D312yes—
338C2×C8C23×D312yes—
339C4⋊C4—28no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—38no—
343C4⋊C4—34no—
344C4⋊C4—34no—
345C4⋊C4—68no—
346C4⋊C4—68no—
347C4⋊C4—68no—
348C4⋊C4—68no—
349C4⋊C4—62no—
350C4⋊C4—62no—
351C4⋊C4—68no—
352C4⋊C4—68no—
353C4⋊C4C23×D314yes—
354C4⋊C4C23×D314yes—
355C22⋊C4—38no—
356C22⋊C4—68no—
357C42—32no—
358C42—32no—
359C42—68no—
360C42D3×D412yes—
361C42C23×D312yes—
362C22×C3C2×C4⋊Q812yes—
363C22×C3#57714yes—
364C22×C3C23×D411yescharacteristic
365C2×D3—28no—
366C2×D3—24no—
367C2×D3C2×C4⋊Q814yes—
368C3×C4—24no—
369C3×C4—44no—
370C3×C4—44no—
371C3×C4C2×C4⋊Q812yes—
372C3×C4C23×Q811yescharacteristic
373C3×C4C23×Q811yescharacteristic
374Dic3—24no—
375Dic3—44no—
376Dic3—44no—
377Dic3C2×C4⋊Q814yes—
378C23—32no—
379C23—34no—
380C23—64no—
381C23C2×D3×D411yescharacteristic
382Q8—88no—
383Q8—248no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—32no—
388C2×C4—34no—
389C2×C4—34no—
390C2×C4—34no—
391C2×C4—38no—
392C2×C4—34no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—64no—
396C2×C4—64no—
397C2×C4—64no—
398C2×C4—124no—
399C2×C4—124no—
400C2×C4—248no—
401C2×C4C2×D3×D414yes—
402C2×C4C2×D3×D411yescharacteristic
403C2×C4C2×D3×Q812yes—
404C2×C4C2×D3×Q812yes—
405C2×C4C24×D311yescharacteristic
406C8—34no—
407C8—34no—
408C8C2×D3×Q814yes—
409C8C2×D3×Q814yes—
410C2×C3#316314yescomplemented
411C2×C3C22×C4⋊Q811yescharacteristic
412C2×C3C2×#57712yes—
413D3—24no—
414C22—34no—
415C22—64no—
416C22—68no—
417C22D3×C4⋊Q812yes—
418C22#576214yes—
419C22C22×D3×D411yescharacteristic
420C4—24no—
421C4—34no—
422C4—44no—
423C4—44no—
424C4—64no—
425C4—124no—
426C4—124no—
427C4D3×C4⋊Q812yes—
428C4C22×D3×Q811yescharacteristic
429C4C22×D3×Q811yescharacteristic
430C3C2×#316311yescharacteristic, complemented, Sylow
431C2—64no—
432C2#8189214yesdirect product factor
433C2C2×D3×C4⋊Q811yescharacteristic
434C2C2×#576212yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#498C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenterC2×#498D3
hypocenterC3C2×#3163
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#498

Complements

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

NH ≅ G/Ntype
2. C2×C3×#498C2semidirect
3. C2×#5081C2semidirect
4. #81892C2direct (N × H)
8. C2×#4856C2semidirect
9. C2×#4858C2semidirect
10. C2×#4855C2semidirect
12. C2×#4683C2semidirect
29. C3×#498C22semidirect
39. #5081C22semidirect
48. #4858C22semidirect
49. #4856C22semidirect
50. #4855C22semidirect
52. #4683C22semidirect
60. C2×#498D3semidirect
94. C2×ΓL(1,𝔽25)Q8semidirect
96. C2×D3×C8Q8semidirect
146. #498C2×D3semidirect
215. ΓL(1,𝔽25)C2×Q8semidirect
216. D3×C8C2×Q8semidirect
410. C2×C3#3163semidirect
430. C3C2×#3163semidirect
432. C2#81892direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#57292
C2×C3×#4982
C2×#50812
#818922
C2×#48612
C2×#48602
C2×D3×#1062
C2×#48562
C2×#48582
C2×#48552
C2×#46892
C2×#46832
C2×#31633