Finite Group Databasebeta

← C2×#28409218 #44409117

Subgroups — C2×#28409218 #44409117

Order 29056.

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×#20300333C211yesmaximal, characteristic, complemented
2C2×#20300257C212yesmaximal, complemented
3#28409218C218yesmaximal, direct product factor
4C2×#20300203C211yesmaximal, characteristic, complemented
5C2×#20300202C212yesmaximal, complemented
6C2×#120×C227C211yesmaximal, characteristic, complemented
7#20300333—24no—
8#20300333—22no—
9#20300333C2214yescomplemented
10C2×#16813884—21no—
11C2×#16813883—22no—
12C22×C22⋊D227—22no—
13C22×C22⋊D227—21no—
14C22×C22⋊D227C2212yes—
15C2×D4×D227—22no—
16#20300257—24no—
17#20300257C2218yescomplemented
18C2×C22⋊Dic227—22no—
19C2×C22⋊Dic227C2212yescomplemented
20#20300203C2214yescomplemented
21#20300202C2218yescomplemented
22C25×C227C2211yescharacteristic, complemented
23C22×D4×C227—21no—
24C22×D4×C227C2211yescharacteristic
25#120×C227—22no—
26#120×C227C2214yescomplemented
27C2×C22⋊C4×C227—21no—
28C2×C22⋊C4×C227C2211yescharacteristic, complemented
29C23×D227—21no—
30C23×D227D412yescomplemented
31#16813884—24no—
32#16813884—24no—
33#16813883—28no—
34#16813883—28no—
35C2×Dic454—21no—
36C2×D908—21no—
37C2×C22⋊D227—22no—
38C2×C22⋊D227—24no—
39C2×C22⋊D227—28no—
40C2×C22⋊D227—22no—
41C2×C22⋊D227—22no—
42C2×C22⋊D227—22no—
43C2×C22⋊D227—22no—
44C2×C22⋊D227—24no—
45C2×C22⋊D227—48no—
46C2×C22⋊D227D414yescomplemented
47C2×C22⋊D227C2314yes—
48D4×D227—28no—
49D4×D227—28no—
50C22×Dic227—21no—
51C22×Dic227—22no—
52C22×Dic227D412yescomplemented
53C2×C4×D227—22no—
54C22⋊Dic227—28no—
55C22⋊Dic227—22no—
56C22⋊Dic227C2314yescomplemented
57C24×C227—22no—
58C24×C227—21no—
59C24×C227—21no—
60C24×C227—22no—
61C24×C227—44no—
62C24×C227D411yescharacteristic, complemented
63C24×C227C2311yescharacteristic
64C24×C227C2312yescomplemented
65C2×D4×C227—22no—
66C2×D4×C227—22no—
67C2×D4×C227—24no—
68C2×D4×C227—44no—
69C2×D4×C227D412yescomplemented
70C2×D4×C227C2312yes—
71C22×C4×C227—21no—
72C22×C4×C227D411yescharacteristic, complemented
73C22⋊C4×C227—21no—
74C22⋊C4×C227—24no—
75C22⋊C4×C227C2312yescomplemented
76C22×D227—22no—
77C22×D227—24no—
78C22×D227—22no—
79C22×D227—44no—
80C22×D227—82no—
81C22×D227C2×D414yescomplemented
82Dic454—24no—
83D908—24no—
84C22⋊D227—28no—
85C22⋊D227—24no—
86C22⋊D227—24no—
87C22⋊D227—24no—
88C22⋊D227—24no—
89C22⋊D227—416no—
90C2×Dic227—24no—
91C2×Dic227—22no—
92C2×Dic227—22no—
93C2×Dic227—24no—
94C2×Dic227—44no—
95C2×Dic227C2×D414yescomplemented
96C4×D227—28no—
97C23×C227—24no—
98C23×C227—28no—
99C23×C227—21no—
100C23×C227—22no—
101C23×C227—24no—
102C23×C227—21no—
103C23×C227—42no—
104C23×C227—44no—
105C23×C227—42no—
106C23×C227—44no—
107C23×C227—44no—
108C23×C227—48no—
109C23×C227—42no—
110C23×C227—41no—
111C23×C227—42no—
112C23×C227—41no—
113C23×C227C2×D412yes—
114C23×C227C2×D411yescharacteristic
115C23×C227C2×D412yes—
116C23×C227C2×D411yescharacteristic
117C23×C227C2×D412yescomplemented
118C23×C227C2411yescharacteristic
119D4×C227—24no—
120D4×C227—24no—
121D4×C227—48no—
122C2×C4×C227—21no—
123C2×C4×C227—21no—
124C2×C4×C227—22no—
125C2×C4×C227—42no—
126C2×C4×C227C2×D412yescomplemented
127C22×C227—22no—
128C22×C227—24no—
129C22×C227—24no—
130C22×C227—28no—
131C22×C227—22no—
132C22×C227—42no—
133C22×C227—41no—
134C22×C227—41no—
135C22×C227—42no—
136C22×C227—44no—
137C22×C227—42no—
138C22×C227—48no—
139C22×C227—44no—
140C22×C227—42no—
141C22×C227—44no—
142C22×C227—42no—
143C22×C227—81no—
144C22×C227#12011yescharacteristic
145C22×C227#12012yes—
146C22×C227#12011yescharacteristic
147C22×C227C22×D412yes—
148C22×C227C22×D411yescharacteristic
149C2×D227—24no—
150C2×D227—44no—
151C2×D227—44no—
152C2×D227—82no—
153C2×D227—84no—
154C4×C227—22no—
155C4×C227—42no—
156Dic227—24no—
157Dic227—44no—
158C2×C227—22no—
159C2×C227—21no—
160C2×C227—22no—
161C2×C227—21no—
162C2×C227—42no—
163C2×C227—44no—
164C2×C227B(4,𝔽2)12yescomplemented
165C2×C227C2×#12011yescharacteristic
166D227—44no—
167C227C2×B(4,𝔽2)11yescharacteristic, complemented, Sylow
168C2×B(4,𝔽2)—2271nomaximal, Sylow
169C2×H(5,𝔽2)—2271no—
170C2×#120—2272no—
171C2×#120D22711yescharacteristic, complemented
172B(4,𝔽2)—2278no—
173C2×#99—2271no—
174C2×#99—2272no—
175C25C2×D22711yescharacteristic, complemented
176H(5,𝔽2)—4544no—
177H(5,𝔽2)—4542no—
178H(5,𝔽2)—2274no—
179C2×#41—4542no—
180C2×#41—4541no—
181C22×D4—21no—
182C22×D4—4542no—
183C22×D4—4542no—
184C22×D4—4541no—
185C22×D4—2272no—
186C22×D4C2×D22711yescharacteristic
187#120—22no—
188#120—4544no—
189#120—2278no—
190#120C2×D22714yescomplemented
191C2×C22⋊C4—21no—
192C2×C22⋊C4—4542no—
193C2×C22⋊C4—2272no—
194C2×C22⋊C4C2×D22711yescharacteristic, complemented
195#99—2274no—
196#99—2278no—
197C24—22no—
198C24—21no—
199C24—22no—
200C24—21no—
201C24—44no—
202C24—4541no—
203C24—2272no—
204C24C22⋊D22711yescharacteristic, complemented
205C24C22×D22711yescharacteristic
206C24C22×D22712yescomplemented
207#41—4544no—
208#41—4548no—
209#41—4548no—
210#41—4544no—
211C2×Q8—4541no—
212C2×D4—22no—
213C2×D4—22no—
214C2×D4—24no—
215C2×D4—44no—
216C2×D4—9088no—
217C2×D4—4541no—
218C2×D4—4548no—
219C2×D4—4548no—
220C2×D4—4548no—
221C2×D4—4542no—
222C2×D4—4544no—
223C2×D4—4542no—
224C2×D4—4542no—
225C2×D4—4542no—
226C2×D4—4542no—
227C2×D4—4544no—
228C2×D4—2274no—
229C2×D4—2274no—
230C2×D4C22⋊D22712yescomplemented
231C2×D4C22×D22712yes—
232C22×C4—21no—
233C22×C4—4542no—
234C22×C4—4542no—
235C22×C4—4541no—
236C22×C4—2272no—
237C22×C4C22⋊D22711yescharacteristic, complemented
238C22⋊C4—21no—
239C22⋊C4—24no—
240C22⋊C4—4542no—
241C22⋊C4—4548no—
242C22⋊C4—2274no—
243C22⋊C4C22×D22712yescomplemented
244C23—24no—
245C23—28no—
246C23—24no—
247C23—22no—
248C23—21no—
249C23—21no—
250C23—42no—
251C23—44no—
252C23—42no—
253C23—44no—
254C23—44no—
255C23—48no—
256C23—42no—
257C23—41no—
258C23—42no—
259C23—41no—
260C23—18162no—
261C23—9084no—
262C23—4544no—
263C23—4542no—
264C23—4542no—
265C23—2274no—
266C23D4×D22712yes—
267C23D4×D22712yes—
268C23C2×C22⋊D22711yescharacteristic
269C23C2×C22⋊D22711yescharacteristic
270C23C2×C22⋊D22712yescomplemented
271C23C23×D22711yescharacteristic
272Q8—4544no—
273D4—24no—
274D4—24no—
275D4—48no—
276D4—90816no—
277D4—4544no—
278D4—4548no—
279D4—4544no—
280D4—4544no—
281D4—4544no—
282D4—4544no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—42no—
287C2×C4—9084no—
288C2×C4—4548no—
289C2×C4—4544no—
290C2×C4—4544no—
291C2×C4—4542no—
292C2×C4—4542no—
293C2×C4—2274no—
294C2×C4C2×C22⋊D22712yescomplemented
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—18162no—
313C22—18164no—
314C22—9084no—
315C22—9084no—
316C22—4544no—
317C22#2030025611yescharacteristic
318C22#2030025612yes—
319C22#2030025611yescharacteristic
320C22C2×D4×D22712yes—
321C22C22×C22⋊D22711yescharacteristic
322C4—22no—
323C4—42no—
324C4—9084no—
325C4—4544no—
326C2—21no—
327C2—22no—
328C2—21no—
329C2—22no—
330C2—44no—
331C2—42no—
332C2—9084no—
333C2#2840921812yesdirect product factor
334C2C2×#2030025611yescharacteristic
subgroupG/H
centerC22#20300256
commutator, cosocleC23×C227C24
FrattiniC23C23×D227
FittingC2×#120×C227C2
socleC22×C227#120
hypercenterC2×#120D227
hypocenterC227C2×B(4,𝔽2)
LCSG ▷ C23×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C22 ◁ C24 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C2×#20300333C2semidirect
2. C2×#20300257C2semidirect
3. #28409218C2direct (N × H)
4. C2×#20300203C2semidirect
5. C2×#20300202C2semidirect
6. C2×#120×C227C2semidirect
9. #20300333C22semidirect
17. #20300257C22semidirect
19. C2×C22⋊Dic227C22semidirect
20. #20300203C22semidirect
21. #20300202C22semidirect
22. C25×C227C22semidirect
26. #120×C227C22semidirect
28. C2×C22⋊C4×C227C22semidirect
30. C23×D227D4semidirect
46. C2×C22⋊D227D4semidirect
52. C22×Dic227D4semidirect
56. C22⋊Dic227C23semidirect
62. C24×C227D4semidirect
64. C24×C227C23semidirect
69. C2×D4×C227D4semidirect
72. C22×C4×C227D4semidirect
75. C22⋊C4×C227C23semidirect
81. C22×D227C2×D4semidirect
95. C2×Dic227C2×D4semidirect
117. C23×C227C2×D4semidirect
126. C2×C4×C227C2×D4semidirect
164. C2×C227B(4,𝔽2)semidirect
167. C227C2×B(4,𝔽2)semidirect
171. C2×#120D227semidirect
175. C25C2×D227semidirect
190. #120C2×D227semidirect
194. C2×C22⋊C4C2×D227semidirect
204. C24C22⋊D227semidirect
206. C24C22×D227semidirect
230. C2×D4C22⋊D227semidirect
237. C22×C4C22⋊D227semidirect
243. C22⋊C4C22×D227semidirect
270. C23C2×C22⋊D227semidirect
294. C2×C4C2×C22⋊D227semidirect
333. C2#28409218direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#203003332
C2×#203002572
#284092182
C2×#203002032
C2×#203002022
C2×#120×C2272
C2×B(4,𝔽2)227