Finite Group Databasebeta

← C2×#31770791 #46820268

Subgroups — C2×#31770791 #46820268

Order 34432.

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×#22575498C211yesmaximal, characteristic, complemented
2C2×#22575422C212yesmaximal, complemented
3#31770791C218yesmaximal, direct product factor
4C2×#22575368C211yesmaximal, characteristic, complemented
5C2×#22575367C212yesmaximal, complemented
6C2×#120×C269C211yesmaximal, characteristic, complemented
7#22575498—24no—
8#22575498—22no—
9#22575498C2214yescomplemented
10C2×#16864402—21no—
11C2×#16864401—22no—
12C22×C22⋊D269—22no—
13C22×C22⋊D269—21no—
14C22×C22⋊D269C2212yes—
15C2×D4×D269—22no—
16#22575422—24no—
17#22575422C2218yescomplemented
18C2×C22⋊Dic269—22no—
19C2×C22⋊Dic269C2212yescomplemented
20#22575368C2214yescomplemented
21#22575367C2218yescomplemented
22C25×C269C2211yescharacteristic, complemented
23C22×D4×C269—21no—
24C22×D4×C269C2211yescharacteristic
25#120×C269—22no—
26#120×C269C2214yescomplemented
27C2×C22⋊C4×C269—21no—
28C2×C22⋊C4×C269C2211yescharacteristic, complemented
29C23×D269—21no—
30C23×D269D412yescomplemented
31#16864402—24no—
32#16864402—24no—
33#16864401—28no—
34#16864401—28no—
35C2×Dic538—21no—
36C2×D1076—21no—
37C2×C22⋊D269—22no—
38C2×C22⋊D269—24no—
39C2×C22⋊D269—28no—
40C2×C22⋊D269—22no—
41C2×C22⋊D269—22no—
42C2×C22⋊D269—22no—
43C2×C22⋊D269—22no—
44C2×C22⋊D269—24no—
45C2×C22⋊D269—48no—
46C2×C22⋊D269D414yescomplemented
47C2×C22⋊D269C2314yes—
48D4×D269—28no—
49D4×D269—28no—
50C22×Dic269—21no—
51C22×Dic269—22no—
52C22×Dic269D412yescomplemented
53C2×C4×D269—22no—
54C22⋊Dic269—28no—
55C22⋊Dic269—22no—
56C22⋊Dic269C2314yescomplemented
57C24×C269—22no—
58C24×C269—21no—
59C24×C269—22no—
60C24×C269—21no—
61C24×C269—44no—
62C24×C269D411yescharacteristic, complemented
63C24×C269C2311yescharacteristic
64C24×C269C2312yescomplemented
65C2×D4×C269—22no—
66C2×D4×C269—22no—
67C2×D4×C269—24no—
68C2×D4×C269—44no—
69C2×D4×C269D412yescomplemented
70C2×D4×C269C2312yes—
71C22×C4×C269—21no—
72C22×C4×C269D411yescharacteristic, complemented
73C22⋊C4×C269—21no—
74C22⋊C4×C269—24no—
75C22⋊C4×C269C2312yescomplemented
76C22×D269—22no—
77C22×D269—24no—
78C22×D269—22no—
79C22×D269—44no—
80C22×D269—82no—
81C22×D269C2×D414yescomplemented
82Dic538—24no—
83D1076—24no—
84C22⋊D269—28no—
85C22⋊D269—24no—
86C22⋊D269—24no—
87C22⋊D269—24no—
88C22⋊D269—24no—
89C22⋊D269—416no—
90C2×Dic269—24no—
91C2×Dic269—22no—
92C2×Dic269—22no—
93C2×Dic269—24no—
94C2×Dic269—44no—
95C2×Dic269C2×D414yescomplemented
96C4×D269—28no—
97C23×C269—24no—
98C23×C269—28no—
99C23×C269—21no—
100C23×C269—22no—
101C23×C269—24no—
102C23×C269—21no—
103C23×C269—42no—
104C23×C269—44no—
105C23×C269—42no—
106C23×C269—44no—
107C23×C269—44no—
108C23×C269—48no—
109C23×C269—42no—
110C23×C269—41no—
111C23×C269—42no—
112C23×C269—41no—
113C23×C269C2×D412yes—
114C23×C269C2×D412yes—
115C23×C269C2×D411yescharacteristic
116C23×C269C2×D411yescharacteristic
117C23×C269C2×D412yescomplemented
118C23×C269C2411yescharacteristic
119D4×C269—24no—
120D4×C269—24no—
121D4×C269—48no—
122C2×C4×C269—21no—
123C2×C4×C269—21no—
124C2×C4×C269—22no—
125C2×C4×C269—42no—
126C2×C4×C269C2×D412yescomplemented
127C22×C269—22no—
128C22×C269—24no—
129C22×C269—24no—
130C22×C269—28no—
131C22×C269—22no—
132C22×C269—42no—
133C22×C269—41no—
134C22×C269—41no—
135C22×C269—42no—
136C22×C269—44no—
137C22×C269—42no—
138C22×C269—48no—
139C22×C269—44no—
140C22×C269—42no—
141C22×C269—44no—
142C22×C269—42no—
143C22×C269—81no—
144C22×C269#12011yescharacteristic
145C22×C269#12012yes—
146C22×C269#12011yescharacteristic
147C22×C269C22×D412yes—
148C22×C269C22×D411yescharacteristic
149C2×D269—24no—
150C2×D269—44no—
151C2×D269—44no—
152C2×D269—82no—
153C2×D269—84no—
154C4×C269—22no—
155C4×C269—42no—
156Dic269—24no—
157Dic269—44no—
158C2×C269—22no—
159C2×C269—21no—
160C2×C269—22no—
161C2×C269—21no—
162C2×C269—42no—
163C2×C269—44no—
164C2×C269B(4,𝔽2)12yescomplemented
165C2×C269C2×#12011yescharacteristic
166D269—44no—
167C269C2×B(4,𝔽2)11yescharacteristic, complemented, Sylow
168C2×B(4,𝔽2)—2691nomaximal, Sylow
169C2×H(5,𝔽2)—2691no—
170C2×#120—2692no—
171C2×#120D26911yescharacteristic, complemented
172B(4,𝔽2)—2698no—
173C2×#99—2691no—
174C2×#99—2692no—
175C25C2×D26911yescharacteristic, complemented
176H(5,𝔽2)—2694no—
177H(5,𝔽2)—5384no—
178H(5,𝔽2)—5382no—
179C2×#41—5382no—
180C2×#41—5381no—
181C22×D4—21no—
182C22×D4—2692no—
183C22×D4—5382no—
184C22×D4—5381no—
185C22×D4—5382no—
186C22×D4C2×D26911yescharacteristic
187#120—22no—
188#120—2698no—
189#120—5384no—
190#120C2×D26914yescomplemented
191C2×C22⋊C4—21no—
192C2×C22⋊C4—2692no—
193C2×C22⋊C4—5382no—
194C2×C22⋊C4C2×D26911yescharacteristic, complemented
195#99—2694no—
196#99—2698no—
197C24—22no—
198C24—21no—
199C24—22no—
200C24—21no—
201C24—44no—
202C24—2692no—
203C24—5381no—
204C24C22⋊D26911yescharacteristic, complemented
205C24C22×D26911yescharacteristic
206C24C22×D26912yescomplemented
207#41—5384no—
208#41—5388no—
209#41—5388no—
210#41—5384no—
211C2×Q8—5381no—
212C2×D4—22no—
213C2×D4—22no—
214C2×D4—24no—
215C2×D4—44no—
216C2×D4—2694no—
217C2×D4—2694no—
218C2×D4—5381no—
219C2×D4—5388no—
220C2×D4—5388no—
221C2×D4—5382no—
222C2×D4—5384no—
223C2×D4—5382no—
224C2×D4—5382no—
225C2×D4—5382no—
226C2×D4—5382no—
227C2×D4—5384no—
228C2×D4—5388no—
229C2×D4—10768no—
230C2×D4C22⋊D26912yescomplemented
231C2×D4C22×D26912yes—
232C22×C4—21no—
233C22×C4—2692no—
234C22×C4—5382no—
235C22×C4—5381no—
236C22×C4—5382no—
237C22×C4C22⋊D26911yescharacteristic, complemented
238C22⋊C4—21no—
239C22⋊C4—24no—
240C22⋊C4—2694no—
241C22⋊C4—5388no—
242C22⋊C4—5382no—
243C22⋊C4C22×D26912yescomplemented
244C23—24no—
245C23—28no—
246C23—21no—
247C23—24no—
248C23—22no—
249C23—21no—
250C23—42no—
251C23—44no—
252C23—42no—
253C23—44no—
254C23—44no—
255C23—48no—
256C23—42no—
257C23—41no—
258C23—42no—
259C23—41no—
260C23—2694no—
261C23—5384no—
262C23—5382no—
263C23—5382no—
264C23—10764no—
265C23—21522no—
266C23D4×D26912yes—
267C23D4×D26912yes—
268C23C2×C22⋊D26911yescharacteristic
269C23C2×C22⋊D26911yescharacteristic
270C23C2×C22⋊D26912yescomplemented
271C23C23×D26911yescharacteristic
272Q8—5384no—
273D4—24no—
274D4—24no—
275D4—48no—
276D4—5384no—
277D4—5388no—
278D4—5384no—
279D4—5384no—
280D4—5384no—
281D4—5384no—
282D4—107616no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—42no—
287C2×C4—2694no—
288C2×C4—5388no—
289C2×C4—5384no—
290C2×C4—5382no—
291C2×C4—5382no—
292C2×C4—5384no—
293C2×C4—10764no—
294C2×C4C2×C22⋊D26912yescomplemented
295C22—22no—
296C22—24no—
297C22—24no—
298C22—28no—
299C22—22no—
300C22—42no—
301C22—41no—
302C22—41no—
303C22—42no—
304C22—42no—
305C22—44no—
306C22—44no—
307C22—48no—
308C22—44no—
309C22—42no—
310C22—42no—
311C22—81no—
312C22—5384no—
313C22—10764no—
314C22—10764no—
315C22—21522no—
316C22—21524no—
317C22#2257542111yescharacteristic
318C22#2257542112yes—
319C22#2257542111yescharacteristic
320C22C2×D4×D26912yes—
321C22C22×C22⋊D26911yescharacteristic
322C4—22no—
323C4—42no—
324C4—5384no—
325C4—10764no—
326C2—21no—
327C2—22no—
328C2—21no—
329C2—22no—
330C2—44no—
331C2—42no—
332C2—10764no—
333C2#3177079112yesdirect product factor
334C2C2×#2257542111yescharacteristic
subgroupG/H
centerC22#22575421
commutator, cosocleC23×C269C24
FrattiniC23C23×D269
FittingC2×#120×C269C2
socleC22×C269#120
hypercenterC2×#120D269
hypocenterC269C2×B(4,𝔽2)
LCSG ▷ C23×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C24 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C2×#22575498C2semidirect
2. C2×#22575422C2semidirect
3. #31770791C2direct (N × H)
4. C2×#22575368C2semidirect
5. C2×#22575367C2semidirect
6. C2×#120×C269C2semidirect
9. #22575498C22semidirect
17. #22575422C22semidirect
19. C2×C22⋊Dic269C22semidirect
20. #22575368C22semidirect
21. #22575367C22semidirect
22. C25×C269C22semidirect
26. #120×C269C22semidirect
28. C2×C22⋊C4×C269C22semidirect
30. C23×D269D4semidirect
46. C2×C22⋊D269D4semidirect
52. C22×Dic269D4semidirect
56. C22⋊Dic269C23semidirect
62. C24×C269D4semidirect
64. C24×C269C23semidirect
69. C2×D4×C269D4semidirect
72. C22×C4×C269D4semidirect
75. C22⋊C4×C269C23semidirect
81. C22×D269C2×D4semidirect
95. C2×Dic269C2×D4semidirect
117. C23×C269C2×D4semidirect
126. C2×C4×C269C2×D4semidirect
164. C2×C269B(4,𝔽2)semidirect
167. C269C2×B(4,𝔽2)semidirect
171. C2×#120D269semidirect
175. C25C2×D269semidirect
190. #120C2×D269semidirect
194. C2×C22⋊C4C2×D269semidirect
204. C24C22⋊D269semidirect
206. C24C22×D269semidirect
230. C2×D4C22⋊D269semidirect
237. C22×C4C22⋊D269semidirect
243. C22⋊C4C22×D269semidirect
270. C23C2×C22⋊D269semidirect
294. C2×C4C2×C22⋊D269semidirect
333. C2#31770791direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#225754982
C2×#225754222
#317707912
C2×#225753682
C2×#225753672
C2×#120×C2692
C2×B(4,𝔽2)269