Finite Group Databasebeta

← C2×#81716 #11664483

Subgroups — C2×#81716 #11664483

Order 768.

2182 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5716C211yesmaximal, characteristic
2C2×D3×#125C211yesmaximal, characteristic, complemented
3C2×C3×#490C211yesmaximal, characteristic, complemented
4C2×#5071C211yesmaximal, characteristic, complemented
5C2×#5069C211yesmaximal, characteristic, complemented
6#81716C2116yesmaximal, direct product factor
7C2×#4771C214yesmaximal, complemented
8C2×#4768C214yesmaximal, complemented
9C2×#4703C211yesmaximal, characteristic, complemented
10C2×#4707C211yesmaximal, characteristic, complemented
11C2×#3142—31nomaximal, Sylow
12C22×#237C2212yes—
13C2×C3×#127C2211yescharacteristic
14C2×C3×#125C2211yescharacteristic
15C2×#938—22no—
16C2×#936C2212yes—
17C2×D4×Dic3—22no—
18C22×ΓL(1,𝔽25)C2212yes—
19#5716—24no—
20#5716C2218yes—
21D3×#125—24no—
22D3×#125C2218yescomplemented
23C2×D3×C4⋊C4—22no—
24C2×D3×C4⋊C4C2212yes—
25C2×#890C2212yes—
26C2×#889—22no—
27C2×C42×D3C2211yescharacteristic
28C2×#870C2211yescharacteristic
29C2×#869C2211yescharacteristic
30C3×#490C2218yescomplemented
31C2×C3×#102C2214yes—
32C2×C3×#97C2211yescharacteristic
33C2×#832C2214yes—
34C2×#816C2214yes—
35C2×C8×Dic3C2212yes—
36#5071C2218yescomplemented
37#5069C2218yescomplemented
38C2×#810C2214yes—
39C2×#803C2211yescharacteristic
40#4771C22132yescomplemented
41#4768C22132yescomplemented
42#4707C2218yescomplemented
43#4703C2218yescomplemented
44C2×#532—31no—
45C22×#125—31no—
46#3142—316no—
47C2×#491—31no—
48C2×#491—31no—
49C2×#490—31no—
50C2×#490D311yescharacteristic, complemented
51C2×#488—34no—
52C2×#417—34no—
53C2×C4×M16—31no—
54C22×C3×D4—22no—
55C22×C3×D4C2312yes—
56C22×#67—22no—
57C23×Dic3—22no—
58C2×#237—24no—
59C2×#237—216no—
60C2×#237C2318yes—
61C22×C4×D3—22no—
62C22×C4×D3D412yes—
63C22×C4×D3C2311yescharacteristic
64C22×Dic6C2312yes—
65C22×C3×C8C2312yes—
66C3×#127—22no—
67C3×#127C2314yes—
68C3×#125—22no—
69C3×#125C2314yes—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4C2312yes—
72C2×C3×C42C2311yescharacteristic
73C2×#217—24no—
74#938—216no—
75#936—24no—
76#936C2318yes—
77D4×Dic3—216no—
78C2×C4⋊Dic3—22no—
79C2×C4⋊Dic3—22no—
80C2×C4⋊Dic3C2312yes—
81C2×Dic3⋊C4—22no—
82C2×Dic3⋊C4—22no—
83C2×Dic3⋊C4—22no—
84C2×C4×Dic3D412yes—
85C2×C4×Dic3C2311yescharacteristic
86C22×C3⋊C8C2312yes—
87C2×ΓL(1,𝔽25)—28no—
88C2×ΓL(1,𝔽25)C2318yes—
89D3×C4⋊C4—24no—
90D3×C4⋊C4—216no—
91D3×C4⋊C4C2318yes—
92#890—24no—
93#890C2318yes—
94#889—216no—
95C42×D3D414yes—
96C42×D3C2314yes—
97#870—22no—
98#870C2314yes—
99#869—22no—
100#869C2314yes—
101C3×#102C23116yes—
102C3×#97C2314yes—
103#832C23116yes—
104#816C23116yes—
105C8×Dic3C2318yes—
106#810C23116yes—
107#803C2314yes—
108C22×#41—32no—
109C22×M16—32no—
110#532—38no—
111#532—64no—
112C2×C4⋊Q8—31no—
113C2×#127C2×D311yescharacteristic
114C2×#125—38no—
115C2×#125—32no—
116C2×#125—31no—
117C2×#125—62no—
118C2×#125—64no—
119C2×#125C2×D311yescharacteristic
120C2×#124—32no—
121C2×#121—62no—
122C2×C4×D4—62no—
123C22×C4⋊C4—32no—
124C22×C4⋊C4—62no—
125C22×C42—31no—
126#491—38no—
127#491—38no—
128#490—38no—
129#490C2×D318yescomplemented
130#488—332no—
131#417—332no—
132C2×#103—34no—
133C2×#103—34no—
134C2×#102—34no—
135C2×#102C2×D314yes—
136C4×M16—38no—
137C2×#97—31no—
138C2×#97C2×D311yescharacteristic
139C2×C4×C8—32no—
140C24×C3—22no—
141C23×D3C2×D411yescharacteristic
142C2×C3×D4—28no—
143C2×C3×D4—22no—
144C2×C3×D4—28no—
145C2×C3×D4—48no—
146C2×C3×D4C2414yes—
147C22×C3×C4—22no—
148C22×C3×C4C2×D412yes—
149C22×C3×C4C2411yescharacteristic
150C2×#67—28no—
151C2×#67—48no—
152C22×Dic3—22no—
153C22×Dic3—22no—
154C22×Dic3—28no—
155C22×Dic3—48no—
156C22×Dic3C2×D412yes—
157C22×Dic3C2×D411yescharacteristic
158#237—232no—
159#237—416no—
160C2×C4×D3—28no—
161C2×C4×D3—44no—
162C2×C4×D3—48no—
163C2×C4×D3C2×D418yes—
164C2×C4×D3C2×D412yes—
165C2×C4×D3#4118yes—
166C2×C4×D3C2412yes—
167C2×Dic6—28no—
168C2×Dic6—22no—
169C2×Dic6C2414yes—
170C2×C3×C8—24no—
171C2×C3×C8C2414yes—
172C3×C4⋊C4—28no—
173C3×C4⋊C4—22no—
174C3×C4⋊C4C2414yes—
175C3×C42C2×D412yes—
176C3×C42C2412yes—
177#217—216no—
178C4⋊Dic3—28no—
179C4⋊Dic3—22no—
180C4⋊Dic3—28no—
181C4⋊Dic3C2414yes—
182Dic3⋊C4—28no—
183Dic3⋊C4—28no—
184Dic3⋊C4—28no—
185C4×Dic3C2×D412yes—
186C4×Dic3C2×D418yes—
187C4×Dic3C2412yes—
188C2×C3⋊C8—24no—
189C2×C3⋊C8C2414yes—
190ΓL(1,𝔽25)—216no—
191C2×#41—38no—
192C2×#41—64no—
193C2×#41—616no—
194C22×Q8—32no—
195C22×D4—22no—
196C22×D4—62no—
197C22×D4C22×D312yes—
198C23×C4—31no—
199C23×C4—32no—
200C23×C4—62no—
201C23×C4—62no—
202C2×M16—38no—
203C2×M16—68no—
204C22×C8—32no—
205C22×C8C22×D312yes—
206C4⋊Q8—34no—
207C4⋊Q8—62no—
208#127—22no—
209#127C22×D314yes—
210#125—22no—
211#125—38no—
212#125—34no—
213#125—616no—
214#125—64no—
215#125—62no—
216#125C22×D314yes—
217#124—38no—
218#124—64no—
219#121—616no—
220C4×D4—616no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—32no—
223C2×C4⋊C4—38no—
224C2×C4⋊C4—64no—
225C2×C4⋊C4—616no—
226C2×C4⋊C4—62no—
227C2×C4⋊C4—62no—
228C2×C4⋊C4—62no—
229C2×C4⋊C4—62no—
230C2×C4⋊C4—62no—
231C2×C4⋊C4C22×D312yes—
232C2×C22⋊C4—64no—
233C2×C42—32no—
234C2×C42—31no—
235C2×C42—34no—
236C2×C42—34no—
237C2×C42C22×D311yescharacteristic
238#103—316no—
239#103—316no—
240#102—316no—
241#102C22×D3116yes—
242#97—34no—
243#97C22×D314yes—
244C4×C8—38no—
245C23×C3—24no—
246C23×C3—44no—
247C23×C3—48no—
248C23×C3C22×D411yescharacteristic
249C22×D3—24no—
250C22×D3#12414yes—
251C22×D3C22×D412yes—
252C3×D4—28no—
253C3×D4—416no—
254C3×D4—44no—
255C2×C3×C4—24no—
256C2×C3×C4—24no—
257C2×C3×C4—44no—
258C2×C3×C4C22×D414yes—
259C2×C3×C4C22×D411yescharacteristic
260C2×C3×C4C2×#4114yes—
261C2×C3×C4C2511yescharacteristic
262#67—416no—
263C2×Dic3—24no—
264C2×Dic3—24no—
265C2×Dic3—24no—
266C2×Dic3—44no—
267C2×Dic3—44no—
268C2×Dic3—416no—
269C2×Dic3#12414yes—
270C2×Dic3C22×D414yes—
271C2×Dic3C22×D412yes—
272C4×D3—48no—
273C4×D3—416no—
274C4×D3#12418yes—
275C4×D3C2×#4118yes—
276Dic6—28no—
277Dic6—44no—
278C3×C8—24no—
279C3⋊C8—24no—
280C24—22no—
281C24—31no—
282#41—632no—
283#41—1216no—
284C2×Q8—34no—
285C2×Q8—62no—
286C2×Q8—68no—
287C2×D4—28no—
288C2×D4—22no—
289C2×D4—28no—
290C2×D4—48no—
291C2×D4—68no—
292C2×D4—128no—
293C2×D4C23×D314yes—
294C22×C4—22no—
295C22×C4—31no—
296C22×C4—32no—
297C22×C4—38no—
298C22×C4—32no—
299C22×C4—32no—
300C22×C4—38no—
301C22×C4—68no—
302C22×C4—68no—
303C22×C4—62no—
304C22×C4—62no—
305C22×C4—124no—
306C22×C4—128no—
307C22×C4—128no—
308C22×C4D3×D412yes—
309C22×C4C23×D311yescharacteristic
310M16—616no—
311C2×C8—24no—
312C2×C8—34no—
313C2×C8—64no—
314C2×C8C23×D314yes—
315C4⋊C4—28no—
316C4⋊C4—22no—
317C4⋊C4—34no—
318C4⋊C4—68no—
319C4⋊C4—68no—
320C4⋊C4—68no—
321C4⋊C4—62no—
322C4⋊C4—68no—
323C4⋊C4—68no—
324C4⋊C4C23×D314yes—
325C22⋊C4—616no—
326C42—38no—
327C42—32no—
328C42—32no—
329C42D3×D412yes—
330C42C23×D312yes—
331C22×C3—44no—
332C22×C3—48no—
333C22×C3—88no—
334C22×C3C2×#12412yes—
335C22×C3#57514yes—
336C22×C3C23×D411yescharacteristic
337C2×D3—28no—
338C2×D3—24no—
339C2×D3C2×#12414yes—
340C3×C4—24no—
341C3×C4—44no—
342C3×C4C2×#12412yes—
343C3×C4C22×#4112yes—
344Dic3—24no—
345Dic3—44no—
346Dic3—44no—
347Dic3C2×#12414yes—
348C23—24no—
349C23—34no—
350C23—32no—
351C23—44no—
352C23—48no—
353C23—64no—
354C23C2×D3×D411yescharacteristic
355Q8—68no—
356Q8—124no—
357D4—28no—
358D4—416no—
359D4—44no—
360D4—1216no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—34no—
364C2×C4—32no—
365C2×C4—38no—
366C2×C4—38no—
367C2×C4—34no—
368C2×C4—44no—
369C2×C4—64no—
370C2×C4—64no—
371C2×C4—64no—
372C2×C4—128no—
373C2×C4—1216no—
374C2×C4—1216no—
375C2×C4—124no—
376C2×C4—124no—
377C2×C4C2×D3×D414yes—
378C2×C4C2×D3×D411yescharacteristic
379C2×C4D3×#4114yes—
380C2×C4C24×D311yescharacteristic
381C8—24no—
382C8—64no—
383C2×C3—84no—
384C2×C3#314214yescomplemented
385C2×C3C22×#12411yescharacteristic
386C2×C3C2×#57512yes—
387D3—24no—
388C22—34no—
389C22—44no—
390C22—48no—
391C22—68no—
392C22—64no—
393C22—88no—
394C22D3×#12412yes—
395C22#577314yes—
396C22C22×D3×D411yescharacteristic
397C4—24no—
398C4—34no—
399C4—44no—
400C4—64no—
401C4—124no—
402C4—124no—
403C4D3×#12412yes—
404C4C2×D3×#4112yes—
405C3C2×#314211yescharacteristic, complemented, Sylow
406C2—64no—
407C2—84no—
408C2#8171614yesdirect product factor
409C2C2×D3×#12411yescharacteristic
410C2C2×#577312yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#490C2
socleC23×C3C22×D4
prime radicalC2×#5716C2
hypercenterC2×#490D3
hypocenterC3C2×#3142
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#490

Complements

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

NH ≅ G/Ntype
2. C2×D3×#125C2semidirect
3. C2×C3×#490C2semidirect
4. C2×#5071C2semidirect
5. C2×#5069C2semidirect
6. #81716C2direct (N × H)
7. C2×#4771C2semidirect
8. C2×#4768C2semidirect
9. C2×#4703C2semidirect
10. C2×#4707C2semidirect
22. D3×#125C22semidirect
30. C3×#490C22semidirect
36. #5071C22semidirect
37. #5069C22semidirect
40. #4771C22semidirect
41. #4768C22semidirect
42. #4707C22semidirect
43. #4703C22semidirect
50. C2×#490D3semidirect
129. #490C2×D3semidirect
384. C2×C3#3142semidirect
405. C3C2×#3142semidirect
408. C2#81716direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#57162
C2×D3×#1252
C2×C3×#4902
C2×#50712
C2×#50692
#817162
C2×#47712
C2×#47682
C2×#47032
C2×#47072
C2×#31423