Finite Group Databasebeta

← C2×#81919 #11664839

Subgroups — C2×#81919 #11664839

Order 768.

1902 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×#125C211yesmaximal, characteristic
2C2×#5688C211yesmaximal, characteristic
3C2×C3×#499C211yesmaximal, characteristic, complemented
4C2×#5065C211yesmaximal, characteristic, complemented
5C2×#5065C211yesmaximal, characteristic, complemented
6#81919C2116yesmaximal, direct product factor
7C2×#4881C212yesmaximal
8C2×#4876C212yesmaximal, complemented
9C2×#4861C212yesmaximal
10C2×#4856C212yesmaximal, complemented
11C2×C4×ΓL(1,𝔽25)C211yesmaximal, characteristic
12C2×#4683C211yesmaximal, characteristic, complemented
13C2×#3166—31nomaximal, Sylow
14C2×C3×#125C2211yescharacteristic
15C2×C3×#125C2211yescharacteristic
16C22×ΓL(1,𝔽25)C2211yescharacteristic
17C22×ΓL(1,𝔽25)C2211yescharacteristic
18#5688—24no—
19#5688C2218yes—
20D3×#125—24no—
21D3×#125C2218yes—
22C2×#896—22no—
23C2×#892C2212yes—
24C2×D3×C4⋊C4—22no—
25C2×D3×C4⋊C4C2212yes—
26C2×#890C2212yes—
27C2×#889—22no—
28C2×#888C2212yes—
29C2×#887—22no—
30C2×C42×D3C2211yescharacteristic
31C2×#870C2211yescharacteristic
32C2×#870C2211yescharacteristic
33C3×#499C2218yescomplemented
34C2×C3×#107C2212yes—
35C2×C3×#106C2212yes—
36C2×C3×C4×C8C2211yescharacteristic
37C2×#818C2212yes—
38C2×#817C2212yes—
39C2×#815C2211yescharacteristic
40C2×#815C2211yescharacteristic
41#5065C2218yescomplemented
42#5065C2218yescomplemented
43C2×#808C2212yes—
44C2×#808C2212yes—
45C2×#807C2212yes—
46C2×#807C2212yes—
47C2×C4×C3⋊C8C2211yescharacteristic
48#4881C22116yes—
49#4876C22116yescomplemented
50#4861C22116yes—
51#4856C22116yescomplemented
52C4×ΓL(1,𝔽25)C2218yes—
53#4683C2218yescomplemented
54C2×#533—31no—
55C22×#125—31no—
56#3166—316no—
57C2×#501—32no—
58C2×#501—32no—
59C2×#499—31no—
60C2×#499—31no—
61C2×#499—31no—
62C2×#499D311yescharacteristic, complemented
63C2×#428—32no—
64C2×#428—32no—
65C2×C4×M16—31no—
66C22×C4×D3—22no—
67C22×C4×D3D412yes—
68C22×C4×D3C2311yescharacteristic
69C22×Dic6—22no—
70C22×C3×C8C2311yescharacteristic
71C22×C3×C8C2311yescharacteristic
72C3×#125—22no—
73C3×#125—22no—
74C3×#125C2314yes—
75C3×#125C2314yes—
76C2×C3×C4⋊C4—22no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4C2312yes—
79C2×C3×C4⋊C4C2312yes—
80C2×C3×C42C2311yescharacteristic
81C2×(C2×D3)⋊C4—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⋊C4—22no—
89C2×Dic3⋊C4—22no—
90C2×C4×Dic3—22no—
91C2×C4×Dic3D412yes—
92C2×C4×Dic3C2311yescharacteristic
93C22×C3⋊C8C2311yescharacteristic
94C22×C3⋊C8C2311yescharacteristic
95C2×ΓL(1,𝔽25)Q818yescomplemented
96C2×ΓL(1,𝔽25)Q818yescomplemented
97C2×ΓL(1,𝔽25)C2314yes—
98C2×ΓL(1,𝔽25)C2314yes—
99#896—216no—
100#892—24no—
101#892C2318yes—
102D3×C4⋊C4—24no—
103D3×C4⋊C4—216no—
104D3×C4⋊C4C2318yes—
105#890—24no—
106#890C2318yes—
107#889—216no—
108#888—24no—
109#888C2318yes—
110#887—216no—
111C42×D3D414yes—
112C42×D3C2314yes—
113#870—22no—
114#870—22no—
115#870C2314yes—
116#870C2314yes—
117C3×#107C2318yes—
118C3×#106C2318yes—
119C3×C4×C8C2314yes—
120#818C2318yes—
121#817C2318yes—
122#815C2314yes—
123#815C2314yes—
124#808C2318yes—
125#808C2318yes—
126#807C2318yes—
127#807C2318yes—
128C4×C3⋊C8C2314yes—
129C22×M16—31no—
130C22×M16—31no—
131#533—38no—
132#533—64no—
133C2×C4⋊Q8—32no—
134C2×#125—38no—
135C2×#125—32no—
136C2×#125—31no—
137C2×#125—31no—
138C2×#125—62no—
139C2×#125—64no—
140C2×#125C2×D311yescharacteristic
141C2×#125C2×D311yescharacteristic
142C2×C22⋊Q8—62no—
143C2×C4×Q8—62no—
144C2×#117—32no—
145C22×C4⋊C4—32no—
146C22×C4⋊C4—62no—
147C22×C42—31no—
148#501—316no—
149#501—316no—
150#499—38no—
151#499—38no—
152#499—38no—
153#499C2×D318yescomplemented
154#428—316no—
155#428—316no—
156C2×#107—32no—
157C2×#107—32no—
158C2×#107—32no—
159C2×#107C2×D312yes—
160C2×#106—32no—
161C2×#106—32no—
162C2×#106—32no—
163C2×#106C2×D312yes—
164C4×M16—38no—
165C2×#97—31no—
166C2×#97—31no—
167C2×C4×C8—31no—
168C2×C4×C8C2×D311yescharacteristic
169C23×D3C2×D411yescharacteristic
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4C2×D412yes—
173C22×C3×C4C2411yescharacteristic
174C22×Dic3—22no—
175C22×Dic3—22no—
176C22×Dic3C2×D412yes—
177C22×Dic3C2×D411yescharacteristic
178C2×C4×D3—28no—
179C2×C4×D3—44no—
180C2×C4×D3—48no—
181C2×C4×D3C2×D418yes—
182C2×C4×D3C2×D412yes—
183C2×C4×D3C2×Q814yes—
184C2×C4×D3C2×Q814yes—
185C2×C4×D3C2412yes—
186C2×Dic6—28no—
187C2×Dic6—48no—
188C2×C3×C8C2×Q814yes—
189C2×C3×C8C2×Q814yes—
190C2×C3×C8C2412yes—
191C2×C3×C8C2412yes—
192C3×C4⋊C4—28no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4—28no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4C2414yes—
197C3×C4⋊C4C2414yes—
198C3×C42C2×D412yes—
199C3×C42C2412yes—
200(C2×D3)⋊C4—28no—
201C4⋊Dic3—22no—
202C4⋊Dic3—22no—
203C4⋊Dic3—28no—
204C4⋊Dic3C2414yes—
205C4⋊Dic3C2414yes—
206Dic3⋊C4—28no—
207Dic3⋊C4—28no—
208Dic3⋊C4—28no—
209Dic3⋊C4—28no—
210Dic3⋊C4—28no—
211C4×Dic3—28no—
212C4×Dic3C2×D412yes—
213C4×Dic3C2×D418yes—
214C4×Dic3C2412yes—
215C2×C3⋊C8C2×Q814yes—
216C2×C3⋊C8C2×Q814yes—
217C2×C3⋊C8C2412yes—
218C2×C3⋊C8C2412yes—
219ΓL(1,𝔽25)C2×Q8116yescomplemented
220ΓL(1,𝔽25)C2×Q8116yescomplemented
221C22×Q8—62no—
222C23×C4—31no—
223C23×C4—32no—
224C23×C4—62no—
225C2×M16—38no—
226C2×M16—38no—
227C2×M16—34no—
228C2×M16—34no—
229C22×C8—31no—
230C22×C8—31no—
231C22×C8C22×D311yescharacteristic
232C22×C8C22×D311yescharacteristic
233C4⋊Q8—38no—
234C4⋊Q8—64no—
235#125—22no—
236#125—22no—
237#125—38no—
238#125—34no—
239#125—34no—
240#125—616no—
241#125—64no—
242#125—62no—
243#125—62no—
244#125C22×D314yes—
245#125C22×D314yes—
246C22⋊Q8—616no—
247C4×Q8—616no—
248#117—38no—
249#117—64no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—32no—
253C2×C4⋊C4—32no—
254C2×C4⋊C4—38no—
255C2×C4⋊C4—64no—
256C2×C4⋊C4—616no—
257C2×C4⋊C4—62no—
258C2×C4⋊C4—62no—
259C2×C4⋊C4—62no—
260C2×C4⋊C4—62no—
261C2×C4⋊C4—62no—
262C2×C4⋊C4—62no—
263C2×C4⋊C4C22×D312yes—
264C2×C4⋊C4C22×D312yes—
265C2×C22⋊C4—62no—
266C2×C42—32no—
267C2×C42—31no—
268C2×C42—34no—
269C2×C42—34no—
270C2×C42—62no—
271C2×C42C22×D311yescharacteristic
272#107—38no—
273#107—38no—
274#107—38no—
275#107C22×D318yes—
276#106—38no—
277#106—38no—
278#106—38no—
279#106C22×D318yes—
280#97—34no—
281#97—34no—
282C4×C8—34no—
283C4×C8C22×D314yes—
284C23×C3C22×D411yescharacteristic
285C22×D3—24no—
286C22×D3C4⋊Q814yes—
287C22×D3C22×D412yes—
288C2×C3×C4—24no—
289C2×C3×C4—24no—
290C2×C3×C4—24no—
291C2×C3×C4—44no—
292C2×C3×C4—44no—
293C2×C3×C4C22×D414yes—
294C2×C3×C4C22×D411yescharacteristic
295C2×C3×C4C22×Q812yes—
296C2×C3×C4C22×Q812yes—
297C2×C3×C4C2511yescharacteristic
298C2×Dic3—24no—
299C2×Dic3—24no—
300C2×Dic3—24no—
301C2×Dic3—44no—
302C2×Dic3—44no—
303C2×Dic3C4⋊Q814yes—
304C2×Dic3C22×D414yes—
305C2×Dic3C22×D412yes—
306C4×D3—48no—
307C4×D3—416no—
308C4×D3C4⋊Q818yes—
309C4×D3C22×Q814yes—
310C4×D3C22×Q814yes—
311Dic6—416no—
312C3×C8C22×Q814yes—
313C3×C8C22×Q814yes—
314C3⋊C8C22×Q814yes—
315C3⋊C8C22×Q814yes—
316C24—31no—
317C2×Q8—68no—
318C2×Q8—128no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—31no—
322C22×C4—32no—
323C22×C4—34no—
324C22×C4—34no—
325C22×C4—32no—
326C22×C4—32no—
327C22×C4—38no—
328C22×C4—68no—
329C22×C4—62no—
330C22×C4—62no—
331C22×C4—124no—
332C22×C4—128no—
333C22×C4D3×D412yes—
334C22×C4C23×D311yescharacteristic
335M16—316no—
336M16—316no—
337C2×C8—34no—
338C2×C8—34no—
339C2×C8—32no—
340C2×C8—32no—
341C2×C8D3×Q814yes—
342C2×C8D3×Q814yes—
343C2×C8C23×D312yes—
344C2×C8C23×D312yes—
345C4⋊C4—28no—
346C4⋊C4—28no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—34no—
350C4⋊C4—34no—
351C4⋊C4—68no—
352C4⋊C4—68no—
353C4⋊C4—68no—
354C4⋊C4—68no—
355C4⋊C4—62no—
356C4⋊C4—62no—
357C4⋊C4—68no—
358C4⋊C4—68no—
359C4⋊C4C23×D314yes—
360C4⋊C4C23×D314yes—
361C22⋊C4—68no—
362C42—38no—
363C42—32no—
364C42—32no—
365C42—68no—
366C42D3×D412yes—
367C42C23×D312yes—
368C22×C3C2×C4⋊Q812yes—
369C22×C3#57514yes—
370C22×C3C23×D411yescharacteristic
371C2×D3—28no—
372C2×D3—24no—
373C2×D3C2×C4⋊Q814yes—
374C3×C4—24no—
375C3×C4—44no—
376C3×C4—44no—
377C3×C4C2×C4⋊Q812yes—
378C3×C4C23×Q811yescharacteristic
379C3×C4C23×Q811yescharacteristic
380Dic3—24no—
381Dic3—44no—
382Dic3—44no—
383Dic3C2×C4⋊Q814yes—
384C23—32no—
385C23—34no—
386C23—64no—
387C23C2×D3×D411yescharacteristic
388Q8—1216no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—32no—
393C2×C4—34no—
394C2×C4—34no—
395C2×C4—34no—
396C2×C4—38no—
397C2×C4—34no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—64no—
401C2×C4—64no—
402C2×C4—64no—
403C2×C4—128no—
404C2×C4—1216no—
405C2×C4—124no—
406C2×C4—124no—
407C2×C4C2×D3×D414yes—
408C2×C4C2×D3×D411yescharacteristic
409C2×C4C2×D3×Q812yes—
410C2×C4C2×D3×Q812yes—
411C2×C4C24×D311yescharacteristic
412C8—34no—
413C8—34no—
414C8C2×D3×Q814yes—
415C8C2×D3×Q814yes—
416C2×C3#316614yescomplemented
417C2×C3C22×C4⋊Q811yescharacteristic
418C2×C3C2×#57512yes—
419D3—24no—
420C22—34no—
421C22—64no—
422C22—68no—
423C22D3×C4⋊Q812yes—
424C22#576814yes—
425C22C22×D3×D411yescharacteristic
426C4—24no—
427C4—34no—
428C4—44no—
429C4—44no—
430C4—64no—
431C4—124no—
432C4—124no—
433C4D3×C4⋊Q812yes—
434C4C22×D3×Q811yescharacteristic
435C4C22×D3×Q811yescharacteristic
436C3C2×#316611yescharacteristic, complemented, Sylow
437C2—64no—
438C2#8191914yesdirect product factor
439C2C2×D3×C4⋊Q811yescharacteristic
440C2C2×#576812yes—
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×#3166
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
3. C2×C3×#499C2semidirect
4. C2×#5065C2semidirect
5. C2×#5065C2semidirect
6. #81919C2direct (N × H)
8. C2×#4876C2semidirect
10. C2×#4856C2semidirect
12. C2×#4683C2semidirect
33. C3×#499C22semidirect
41. #5065C22semidirect
42. #5065C22semidirect
49. #4876C22semidirect
51. #4856C22semidirect
53. #4683C22semidirect
62. C2×#499D3semidirect
95. C2×ΓL(1,𝔽25)Q8semidirect
96. C2×ΓL(1,𝔽25)Q8semidirect
153. #499C2×D3semidirect
219. ΓL(1,𝔽25)C2×Q8semidirect
220. ΓL(1,𝔽25)C2×Q8semidirect
416. C2×C3#3166semidirect
436. C3C2×#3166semidirect
438. C2#81919direct (N × H)

Maximal subgroups

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

subgroupindex
C2×D3×#1252
C2×#56882
C2×C3×#4992
C2×#50652
#819192
C2×#48812
C2×#48762
C2×#48612
C2×#48562
C2×C4×ΓL(1,𝔽25)2
C2×#46832
C2×#31663