Finite Group Databasebeta

← #44402973

Subgroups — #44402973

Order 29056.

696 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 155 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×D4×Dic227C211yesmaximal, characteristic, complemented
2C2×(C22⋊D227)⋊C4C212yesmaximal, complemented
3C2×#20300246C211yesmaximal, characteristic, complemented
4#28408962C211yesmaximal, characteristic, complemented
5#28408959C211yesmaximal, characteristic, complemented
6#28408955C212yesmaximal, complemented
7#28408947C212yesmaximal, complemented
8#28408940C211yesmaximal, characteristic
9#28408883C211yesmaximal, characteristic, complemented
10C22⋊C4×Dic227C212yesmaximal, complemented
11#394×C227C211yesmaximal, characteristic, complemented
12C22×C22⋊D227C2212yes—
13C23×Dic227C2212yes—
14C22×C4×D227C2211yescharacteristic
15D4×Dic227—24no—
16(C22⋊D227)⋊C4—28no—
17#20300246—24no—
18C2×C4⋊Dic227C2211yescharacteristic
19C2×C4⋊Dic227C2211yescharacteristic
20C2×Dic227⋊C4C2212yes—
21C2×Dic227⋊C4C2212yes—
22C2×C22⋊Dic227C2212yes—
23C2×C22⋊Dic227C2212yes—
24C2×(C2×D227)⋊C4C2211yescharacteristic
25C2×(C2×D227)⋊C4C2211yescharacteristic
26C2×(C2×D227)⋊C4C2212yes—
27C2×C4×Dic227C2211yescharacteristic
28C2×C4×Dic227C2212yes—
29C2×C4×Dic227C2211yescharacteristic
30C227⋊#95C2212yes—
31C227⋊#95C2212yes—
32C227⋊#95C2211yescharacteristic
33C22×D4×C227C2211yescharacteristic
34C2×C4⋊C4×C227C2211yescharacteristic
35C2×C22⋊C4×C227C2212yes—
36C2×C22⋊C4×C227C2212yes—
37#95×C227C2211yescharacteristic
38C23×D227C2311yescharacteristic
39C2×C22⋊D227—24no—
40C2×C22⋊D227—24no—
41C2×C22⋊D227—24no—
42C22×Dic227—24no—
43C22×Dic227—24no—
44C22×Dic227—24no—
45C22×Dic227C2312yes—
46C22×Dic227C2311yescharacteristic
47C22×Dic227C2311yescharacteristic
48C22×Dic227C2312yes—
49C22×Dic227C2311yescharacteristic
50C2×C4×D227—22no—
51C2×C4×D227—22no—
52C2×C4×D227D414yescomplemented
53C4⋊Dic227—22no—
54C4⋊Dic227—22no—
55Dic227⋊C4—24no—
56Dic227⋊C4—24no—
57C22⋊Dic227—24no—
58C22⋊Dic227—24no—
59(C2×D227)⋊C4—24no—
60(C2×D227)⋊C4—22no—
61(C2×D227)⋊C4—22no—
62C4×Dic227—22no—
63C4×Dic227—24no—
64C4×Dic227D414yescomplemented
65C24×C227C2312yes—
66C2×D4×C227—24no—
67C2×D4×C227—22no—
68C22×C4×C227C2312yes—
69C22×C4×C227C2311yescharacteristic
70C22×C4×C227C2311yescharacteristic
71C22×C4×C227C2311yescharacteristic
72C4⋊C4×C227—22no—
73C22⋊C4×C227—24no—
74C22⋊C4×C227—24no—
75C22×D227—21no—
76C22×D227—21no—
77C22×D227—22no—
78C22×D227—22no—
79C22×D227C2×D412yes—
80C22⋊D227—216no—
81C2×Dic227—22no—
82C2×Dic227—22no—
83C2×Dic227—22no—
84C2×Dic227—21no—
85C2×Dic227—22no—
86C2×Dic227—22no—
87C2×Dic227—22no—
88C2×Dic227—22no—
89C2×Dic227—48no—
90C2×Dic227C2×D412yes—
91C2×Dic227C2×D412yes—
92C2×Dic227C2×D412yes—
93C2×Dic227#4114yes—
94C2×Dic227#4112yes—
95C4×D227—28no—
96C23×C227—22no—
97C23×C227—22no—
98C23×C227—22no—
99C23×C227—22no—
100C23×C227—22no—
101C23×C227—22no—
102C23×C227#4114yes—
103C23×C227C2411yescharacteristic
104D4×C227—28no—
105C2×C4×C227—21no—
106C2×C4×C227—22no—
107C2×C4×C227—21no—
108C2×C4×C227—22no—
109C2×C4×C227—21no—
110C2×C4×C227—22no—
111C2×C4×C227—22no—
112C2×C4×C227—22no—
113C2×C4×C227C2×D412yes—
114C2×C4×C227C2×D412yes—
115C22×C227—24no—
116C22×C227—24no—
117C22×C227—24no—
118C22×C227—42no—
119C22×C227—42no—
120C22×C227—42no—
121C22×C227—42no—
122C22×C227C22×D411yescharacteristic
123C22×C227C22×D411yescharacteristic
124C22×C227C2×#4111yescharacteristic
125C22×C227C2×#4112yes—
126C22×C227C2×#4112yes—
127C2×D227—22no—
128C2×D227—24no—
129C2×D227—41no—
130C2×D227—42no—
131C2×D227—41no—
132C4×C227—22no—
133C4×C227—22no—
134C4×C227—41no—
135C4×C227—42no—
136Dic227—24no—
137Dic227—22no—
138Dic227—42no—
139Dic227—41no—
140Dic227#12114yes—
141C2×C227—44no—
142C2×C227C2×#12111yescharacteristic
143C2×C227#52511yescharacteristic
144C2×C227#52512yes—
145C2×C227#52912yes—
146C2×C227#53211yescharacteristic
147D227—42no—
148C227#239211yescharacteristic, complemented, Sylow
149#2392—2271nomaximal, Sylow
150C2×C4×D4—2271no—
151C2×C4×D4—2272no—
152C2×#117—2271no—
153#398—2271no—
154#397—2271no—
155#396—2272no—
156#394—2272no—
157#394D22711yescharacteristic, complemented
158#392—2271no—
159C4×C4⋊C4—2271no—
160C4×C22⋊C4—2272no—
161C22×D4—2272no—
162C22×D4C2×D22711yescharacteristic
163C23×C4—2272no—
164C23×C4—2271no—
165C4×D4—4548no—
166C4×D4—4544no—
167#117—4544no—
168C2×C4⋊C4—2271no—
169C2×C4⋊C4—2272no—
170C2×C4⋊C4—2271no—
171C2×C4⋊C4—2272no—
172C2×C4⋊C4C2×D22711yescharacteristic
173C2×C22⋊C4—2272no—
174C2×C22⋊C4—2272no—
175C2×C22⋊C4—2271no—
176C2×C22⋊C4—2271no—
177C2×C22⋊C4—2272no—
178C2×C22⋊C4C2×D22712yes—
179C2×C22⋊C4C2×D22712yes—
180C2×C42—2271no—
181C2×C42—2272no—
182C2×C42—2271no—
183#95—2272no—
184#95—2272no—
185#95—2271no—
186#95C2×D22711yescharacteristic
187C24—2271no—
188C24C22×D22712yes—
189C2×D4—24no—
190C2×D4—22no—
191C2×D4—4544no—
192C2×D4—4544no—
193C2×D4—4544no—
194C22×C4—4542no—
195C22×C4—4542no—
196C22×C4—4544no—
197C22×C4—4544no—
198C22×C4—4544no—
199C22×C4—2271no—
200C22×C4—2272no—
201C22×C4—2272no—
202C22×C4—2271no—
203C22×C4—2274no—
204C22×C4—2271no—
205C22×C4C22×D22711yescharacteristic
206C22×C4C22×D22712yes—
207C22×C4C22×D22711yescharacteristic
208C22×C4C22×D22711yescharacteristic
209C4⋊C4—22no—
210C4⋊C4—4544no—
211C4⋊C4—4542no—
212C4⋊C4—4544no—
213C4⋊C4—4542no—
214C22⋊C4—24no—
215C22⋊C4—24no—
216C22⋊C4—4544no—
217C22⋊C4—4542no—
218C22⋊C4—4544no—
219C22⋊C4—4544no—
220C22⋊C4—4542no—
221C42—4542no—
222C42—4544no—
223C42—2274no—
224C23—22no—
225C23—22no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—4541no—
231C23—4541no—
232C23—4542no—
233C23—4542no—
234C23—2272no—
235C23#1681388314yes—
236C23C23×D22711yescharacteristic
237D4—28no—
238D4—45416no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—9088no—
248C2×C4—4548no—
249C2×C4—4541no—
250C2×C4—4542no—
251C2×C4—4542no—
252C2×C4—4542no—
253C2×C4—4542no—
254C2×C4—4542no—
255C2×C4—4542no—
256C2×C4—4542no—
257C2×C4—2272no—
258C2×C4—2274no—
259C2×C4—2272no—
260C2×C4—2272no—
261C2×C4—2272no—
262C2×C4D4×D22712yes—
263C2×C4D4×D22712yes—
264C22—24no—
265C22—24no—
266C22—24no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22—42no—
271C22—9081no—
272C22—9082no—
273C22—9081no—
274C22—4542no—
275C22—4544no—
276C22C2×D4×D22711yescharacteristic
277C22C2×D4×D22711yescharacteristic
278C22#41×D22711yescharacteristic
279C22#41×D22712yes—
280C22C2×#1681388312yes—
281C4—22no—
282C4—22no—
283C4—41no—
284C4—42no—
285C4—9082no—
286C4—9081no—
287C4—4544no—
288C4—4542no—
289C4—2274no—
290C2—44no—
291C2—9082no—
292C2#121×D22711yescharacteristic
293C2#2840953811yescharacteristic
294C2#2840954212yes—
295C2#2840955912yes—
296C2#2840958111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D227
commutator, socle, cosocleC23×C227C24
Fitting#394×C227C2
prime radical#28408940C2
hypercenter#394D227
hypocenterC227#2392
LCSG ▷ C23×C227 ▷ C227
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×D4×Dic227C2semidirect
2. C2×(C22⋊D227)⋊C4C2semidirect
3. C2×#20300246C2semidirect
4. #28408962C2semidirect
5. #28408959C2semidirect
6. #28408955C2semidirect
7. #28408947C2semidirect
9. #28408883C2semidirect
10. C22⋊C4×Dic227C2semidirect
11. #394×C227C2semidirect
52. C2×C4×D227D4semidirect
64. C4×Dic227D4semidirect
148. C227#2392semidirect
157. #394D227semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic2272
C2×(C22⋊D227)⋊C42
C2×#203002462
#284089622
#284089592
#284089552
#284089472
#284089402
#284088832
C22⋊C4×Dic2272
#394×C2272
#2392227