Finite Group Databasebeta

← C2×#31774085 #46845749

Subgroups — C2×#31774085 #46845749

Order 34688.

1018 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 1018 conjugacy classes of subgroups — too many to draw. Only its 149 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22576205C211yesmaximal, characteristic, complemented
2C2×#22576130C212yesmaximal, complemented
3C2×#22576128C211yesmaximal, characteristic, complemented
4#31774085C218yesmaximal, direct product factor
5C2×#22576074C212yesmaximal, complemented
6C2×#99×C271C211yesmaximal, characteristic, complemented
7C24×D271C2211yescharacteristic, complemented
8#22576205—24no—
9#22576205—22no—
10#22576205C2214yescomplemented
11C2×#16864654—21no—
12C2×#16864653—22no—
13C22×D1084—22no—
14C22×C22⋊D271—21no—
15C22×C22⋊D271—21no—
16C22×C22⋊D271C2212yes—
17C2×D4×D271—22no—
18#22576130—24no—
19#22576130C2218yescomplemented
20#22576128—22no—
21#22576128C2214yescomplemented
22C2×C22⋊Dic271C2211yescharacteristic, complemented
23C2×(C2×D271)⋊C4—22no—
24C2×(C2×D271)⋊C4—21no—
25#22576074C2218yescomplemented
26C22×D4×C271C2211yescharacteristic
27C2×C22⋊C4×C271C2212yescomplemented
28#99×C271C2214yescomplemented
29C23×D271—21no—
30C23×D271—21no—
31C23×D271—22no—
32C23×D271—21no—
33C23×D271—22no—
34C23×D271—44no—
35C23×D271D412yescomplemented
36C23×D271C2312yescomplemented
37#16864654—24no—
38#16864654—24no—
39#16864653—28no—
40#16864653—28no—
41C2×Dic542—21no—
42C2×D1084—21no—
43C2×D1084—28no—
44C2×D1084—48no—
45C2×C22⋊D271—22no—
46C2×C22⋊D271—24no—
47C2×C22⋊D271—22no—
48C2×C22⋊D271—22no—
49C2×C22⋊D271—22no—
50C2×C22⋊D271—22no—
51C2×C22⋊D271—24no—
52C2×C22⋊D271—24no—
53C2×C22⋊D271—44no—
54C2×C22⋊D271D414yescomplemented
55C2×C22⋊D271C2314yes—
56D4×D271—28no—
57D4×D271—28no—
58C22×Dic271—21no—
59C22×Dic271—21no—
60C22×Dic271D412yescomplemented
61C2×C4×D271—22no—
62C22⋊Dic271—21no—
63C22⋊Dic271C2312yescomplemented
64(C2×D271)⋊C4—28no—
65(C2×D271)⋊C4—24no—
66C24×C271D411yescharacteristic, complemented
67C24×C271C2311yescharacteristic
68C2×D4×C271—22no—
69C2×D4×C271—22no—
70C2×D4×C271D412yescomplemented
71C2×D4×C271C2312yes—
72C22×C4×C271—22no—
73C22×C4×C271D411yescharacteristic, complemented
74C22⋊C4×C271—22no—
75C22⋊C4×C271C2314yescomplemented
76C22×D271—24no—
77C22×D271—28no—
78C22×D271—22no—
79C22×D271—22no—
80C22×D271—24no—
81C22×D271—24no—
82C22×D271—22no—
83C22×D271—44no—
84C22×D271—44no—
85C22×D271—48no—
86C22×D271—44no—
87C22×D271—42no—
88C22×D271—41no—
89C22×D271—42no—
90C22×D271—41no—
91C22×D271—44no—
92C22×D271—82no—
93C22×D271C2×D414yescomplemented
94Dic542—24no—
95D1084—24no—
96D1084—416no—
97C22⋊D271—28no—
98C22⋊D271—24no—
99C22⋊D271—24no—
100C22⋊D271—24no—
101C22⋊D271—24no—
102C22⋊D271—48no—
103C2×Dic271—24no—
104C2×Dic271—22no—
105C2×Dic271—22no—
106C2×Dic271—22no—
107C2×Dic271—42no—
108C2×Dic271C2×D414yescomplemented
109C4×D271—28no—
110C23×C271—21no—
111C23×C271—21no—
112C23×C271—42no—
113C23×C271—42no—
114C23×C271C2×D412yes—
115C23×C271C2×D412yes—
116C23×C271C2×D411yescharacteristic
117C23×C271C2×D411yescharacteristic
118C23×C271C2×D412yescomplemented
119C23×C271C2411yescharacteristic
120D4×C271—24no—
121D4×C271—24no—
122C2×C4×C271—21no—
123C2×C4×C271—21no—
124C2×C4×C271—24no—
125C2×C4×C271—44no—
126C2×C4×C271C2×D412yescomplemented
127C22×C271—24no—
128C22×C271—22no—
129C22×C271—22no—
130C22×C271—42no—
131C22×C271—41no—
132C22×C271—41no—
133C22×C271—42no—
134C22×C271—42no—
135C22×C271—81no—
136C22×C271#12011yescharacteristic
137C22×C271#12012yes—
138C22×C271#12011yescharacteristic
139C22×C271C22×D412yes—
140C22×C271C22×D411yescharacteristic
141C2×D271—24no—
142C2×D271—28no—
143C2×D271—24no—
144C2×D271—42no—
145C2×D271—44no—
146C2×D271—48no—
147C2×D271—44no—
148C2×D271—42no—
149C2×D271—44no—
150C2×D271—44no—
151C2×D271—44no—
152C2×D271—82no—
153C2×D271—84no—
154C4×C271—22no—
155C4×C271—44no—
156Dic271—24no—
157Dic271—42no—
158C2×C271—21no—
159C2×C271—22no—
160C2×C271—22no—
161C2×C271—21no—
162C2×C271—42no—
163C2×C271B(4,𝔽2)12yescomplemented
164C2×C271C2×#12011yescharacteristic
165D271—44no—
166D271—44no—
167C271C2×B(4,𝔽2)11yescharacteristic, complemented, Sylow
168C2×B(4,𝔽2)—2711nomaximal, Sylow
169C2×H(5,𝔽2)—2711no—
170C2×#120—2711no—
171C2×#120—2712no—
172B(4,𝔽2)—2718no—
173C2×#99—2712no—
174C2×#99D27111yescharacteristic, complemented
175C25—2711no—
176H(5,𝔽2)—2714no—
177H(5,𝔽2)—5424no—
178H(5,𝔽2)—5422no—
179C2×#41—5422no—
180C2×#41—5421no—
181C22×D4—2712no—
182C22×D4—5422no—
183C22×D4—5422no—
184C22×D4—5421no—
185C22×D4—5421no—
186C22×D4C2×D27111yescharacteristic
187#120—2714no—
188#120—2718no—
189#120—5424no—
190#120—5422no—
191C2×C22⋊C4—2711no—
192C2×C22⋊C4—5422no—
193C2×C22⋊C4—5421no—
194C2×C22⋊C4C2×D27112yescomplemented
195#99—2718no—
196#99C2×D27114yescomplemented
197C24—2712no—
198C24—2712no—
199C24—5421no—
200C24—5422no—
201C24—5421no—
202C24—5422no—
203C24—5421no—
204C24—10844no—
205C24D108411yescharacteristic, complemented
206C24C22×D27111yescharacteristic
207#41—5424no—
208#41—5428no—
209#41—5428no—
210#41—5424no—
211C2×Q8—5421no—
212C2×D4—22no—
213C2×D4—22no—
214C2×D4—2714no—
215C2×D4—2714no—
216C2×D4—5421no—
217C2×D4—5428no—
218C2×D4—5428no—
219C2×D4—5428no—
220C2×D4—5422no—
221C2×D4—5424no—
222C2×D4—5422no—
223C2×D4—5422no—
224C2×D4—5422no—
225C2×D4—5422no—
226C2×D4—5424no—
227C2×D4—5424no—
228C2×D4—10848no—
229C2×D4—10844no—
230C2×D4D108412yescomplemented
231C2×D4C22×D27112yes—
232C22×C4—22no—
233C22×C4—2712no—
234C22×C4—5422no—
235C22×C4—5421no—
236C22×C4—5421no—
237C22×C4D108411yescharacteristic, complemented
238C22⋊C4—22no—
239C22⋊C4—2712no—
240C22⋊C4—5428no—
241C22⋊C4—5421no—
242C22⋊C4—5424no—
243C22⋊C4C22×D27114yescomplemented
244C23—21no—
245C23—21no—
246C23—42no—
247C23—42no—
248C23—2714no—
249C23—5428no—
250C23—5424no—
251C23—5424no—
252C23—5422no—
253C23—5424no—
254C23—5422no—
255C23—5422no—
256C23—10844no—
257C23—10842no—
258C23—10844no—
259C23—10844no—
260C23—10848no—
261C23—10844no—
262C23—10841no—
263C23—10842no—
264C23—10841no—
265C23—21682no—
266C23D4×D27112yes—
267C23D4×D27112yes—
268C23C2×D108411yescharacteristic
269C23C2×D108411yescharacteristic
270C23C2×D108412yescomplemented
271C23C23×D27111yescharacteristic
272Q8—5424no—
273D4—24no—
274D4—24no—
275D4—5424no—
276D4—5428no—
277D4—5424no—
278D4—5424no—
279D4—5424no—
280D4—5424no—
281D4—108416no—
282D4—10848no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—24no—
286C2×C4—44no—
287C2×C4—2714no—
288C2×C4—5428no—
289C2×C4—5424no—
290C2×C4—5422no—
291C2×C4—5422no—
292C2×C4—5422no—
293C2×C4—10842no—
294C2×C4C2×D108412yescomplemented
295C22—24no—
296C22—22no—
297C22—22no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22—42no—
302C22—42no—
303C22—81no—
304C22—5428no—
305C22—5424no—
306C22—5424no—
307C22—10844no—
308C22—10844no—
309C22—10842no—
310C22—10844no—
311C22—10848no—
312C22—10844no—
313C22—10844no—
314C22—10842no—
315C22—21682no—
316C22—21684no—
317C22#2257613011yescharacteristic
318C22#2257613012yes—
319C22#2257613011yescharacteristic
320C22C2×D4×D27112yes—
321C22C22×D108411yescharacteristic
322C4—22no—
323C4—44no—
324C4—5424no—
325C4—10842no—
326C2—22no—
327C2—21no—
328C2—21no—
329C2—22no—
330C2—42no—
331C2—10844no—
332C2—10844no—
333C2#3177408512yesdirect product factor
334C2C2×#2257613011yescharacteristic
subgroupG/H
centerC22#22576130
commutator, cosocleC23×C271C24
FrattiniC23C23×D271
FittingC2×#99×C271C2
socleC22×C271#120
hypercenterC2×#99D271
hypocenterC271C2×B(4,𝔽2)
LCSG ▷ C23×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
1. C2×#22576205C2semidirect
2. C2×#22576130C2semidirect
3. C2×#22576128C2semidirect
4. #31774085C2direct (N × H)
5. C2×#22576074C2semidirect
6. C2×#99×C271C2semidirect
7. C24×D271C22semidirect
10. #22576205C22semidirect
19. #22576130C22semidirect
21. #22576128C22semidirect
22. C2×C22⋊Dic271C22semidirect
25. #22576074C22semidirect
27. C2×C22⋊C4×C271C22semidirect
28. #99×C271C22semidirect
35. C23×D271D4semidirect
36. C23×D271C23semidirect
54. C2×C22⋊D271D4semidirect
60. C22×Dic271D4semidirect
63. C22⋊Dic271C23semidirect
66. C24×C271D4semidirect
70. C2×D4×C271D4semidirect
73. C22×C4×C271D4semidirect
75. C22⋊C4×C271C23semidirect
93. C22×D271C2×D4semidirect
108. C2×Dic271C2×D4semidirect
118. C23×C271C2×D4semidirect
126. C2×C4×C271C2×D4semidirect
163. C2×C271B(4,𝔽2)semidirect
167. C271C2×B(4,𝔽2)semidirect
174. C2×#99D271semidirect
194. C2×C22⋊C4C2×D271semidirect
196. #99C2×D271semidirect
205. C24D1084semidirect
230. C2×D4D1084semidirect
237. C22×C4D1084semidirect
243. C22⋊C4C22×D271semidirect
270. C23C2×D1084semidirect
294. C2×C4C2×D1084semidirect
333. C2#31774085direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#225762052
C2×#225761302
C2×#225761282
#317740852
C2×#225760742
C2×#99×C2712
C2×B(4,𝔽2)271