Finite Group Databasebeta

← #46834865

Subgroups — #46834865

Order 34688.

334 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 65 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
1#31774407C211yesmaximal, characteristic
2#31773859C211yesmaximal, characteristic, complemented
3#95×D271C211yesmaximal, characteristic, complemented
4#31773635C211yesmaximal, characteristic, complemented
5#31773633C211yesmaximal, characteristic, complemented
6#31773631C211yesmaximal, characteristic, complemented
7#327×C271C211yesmaximal, characteristic, complemented
8C2×#16864653C411yescharacteristic, complemented
9C22×C4×D271—21no—
10C22×C4×D271C2211yescharacteristic
11C2×#16864637—21no—
12#22576142—21no—
13#22576140C411yescharacteristic, complemented
14#22576139C2211yescharacteristic
15#22576131C411yescharacteristic, complemented
16#22576128—21no—
17D4×Dic271—21no—
18(C22⋊D271)⋊C4—21no—
19#22576118C411yescharacteristic, complemented
20C22⋊(C271⋊C8)C2211yescharacteristic
21(C2×D271)⋊C8—21no—
22C271⋊#95—21no—
23C271⋊#95C2211yescharacteristic
24#121×C271C2211yescharacteristic
25C22⋊C8×C271C2211yescharacteristic
26#95×C271C2211yescharacteristic
27C23×D271D411yescharacteristic
28#16864653—42no—
29C2×Dic542C2×C411yescharacteristic, complemented
30C2×C22⋊D271—21no—
31C2×C22⋊D271—21no—
32C22×Dic271—21no—
33C22×Dic271—21no—
34C22×Dic271D411yescharacteristic
35C2×C4×D271—24no—
36C2×C4×D271—42no—
37C2×C4×D271—42no—
38C2×C4×D271—42no—
39C2×C4×D271C2×C411yescharacteristic
40C2×C4×D271C2×C411yescharacteristic
41C2×C4×D271D411yescharacteristic
42C2×C4×D271D411yescharacteristic
43#16864637—42no—
44C2×C271⋊C8—21no—
45C4⋊Dic271—21no—
46C4⋊Dic271C2×C411yescharacteristic, complemented
47Dic271⋊C4—21no—
48C22⋊Dic271—21no—
49C22⋊Dic271—21no—
50(C2×D271)⋊C4—21no—
51(C2×D271)⋊C4—21no—
52C4×Dic271—21no—
53C2×D4×C271—21no—
54C2×D4×C271C2×C411yescharacteristic, complemented
55C22×C4×C271—21no—
56C22×C4×C271C2311yescharacteristic
57C2×C8×C271—21no—
58C4⋊C4×C271C2×C411yescharacteristic, complemented
59C22⋊C4×C271—21no—
60C22×D271—21no—
61C22×D271—21no—
62C22×D271—22no—
63C22×D271—22no—
64C22×D271C22⋊C411yescharacteristic
65C22×D271C22⋊C411yescharacteristic
66Dic542—41no—
67C22⋊D271—42no—
68C22⋊D271—42no—
69C2×Dic271—22no—
70C2×Dic271—21no—
71C2×Dic271—21no—
72C2×Dic271—41no—
73C2×Dic271—42no—
74C2×Dic271—41no—
75C2×Dic271—41no—
76C2×Dic271C22⋊C411yescharacteristic
77C2×Dic271C22⋊C411yescharacteristic
78C4×D271—22no—
79C4×D271—22no—
80C4×D271—41no—
81C4×D271—41no—
82C4×D271—48no—
83C271⋊C8—41no—
84C23×C271—21no—
85C23×C271C2×D411yescharacteristic
86D4×C271—41no—
87D4×C271—41no—
88D4×C271—41no—
89C2×C4×C271—22no—
90C2×C4×C271—21no—
91C2×C4×C271—41no—
92C2×C4×C271—41no—
93C2×C4×C271—41no—
94C2×C4×C271C22×C411yescharacteristic
95C2×C4×C271C2×D411yescharacteristic
96C8×C271—41no—
97C22×C271—21no—
98C22×C271—21no—
99C22×C271—41no—
100C22×C271—41no—
101C22×C271—41no—
102C22×C271C2×C22⋊C411yescharacteristic
103C22×C271AGL(1,ℤ/8)11yescharacteristic
104C22×C271#13711yescharacteristic
105C2×D271—24no—
106C2×D271—24no—
107C2×D271—21no—
108C2×D271—21no—
109C2×D271—21no—
110C2×D271—21no—
111C2×D271#9912yes—
112C2×D271#10412yes—
113C4×C271—21no—
114C4×C271—21no—
115C4×C271—42no—
116C4×C271—41no—
117Dic271—21no—
118Dic271—21no—
119Dic271—42no—
120Dic271—41no—
121Dic271—41no—
122C2×C271—21no—
123C2×C271—21no—
124C2×C271—81no—
125C2×C271C2×#9911yescharacteristic
126C2×C271#41711yescharacteristic
127C2×C271C2×#10411yescharacteristic
128D271—22no—
129D271—22no—
130C271#150411yescharacteristic, complemented, Sylow
131#1504—2711nomaximal, Sylow
132#525—2711no—
133#407—2711no—
134C2×#95—2711no—
135#328—2711no—
136#328—2711no—
137#327—2711no—
138#327D27111yescharacteristic, complemented
139C2×#41—2711no—
140C23×C4—2711no—
141C23×C4—5421no—
142C2×M16—5421no—
143#123—5421no—
144C22⋊Q8—2711no—
145C22⋊Q8—2711no—
146#121—2711no—
147#121C2×D27111yescharacteristic
148#120—5421no—
149C4×D4—5421no—
150C4×D4—5421no—
151#117—2711no—
152C22⋊C8—2711no—
153C22⋊C8—5421no—
154C22⋊C8C2×D27111yescharacteristic
155#95—2711no—
156#95—5421no—
157#95C2×D27111yescharacteristic
158C24—2711no—
159#41—10842no—
160C2×Q8—2711no—
161C2×D4—21no—
162C2×D4—5421no—
163C2×D4—5421no—
164C2×D4C4×D27111yescharacteristic, complemented
165C22×C4—21no—
166C22×C4—2711no—
167C22×C4—2711no—
168C22×C4—2711no—
169C22×C4—2711no—
170C22×C4—2711no—
171C22×C4—5424no—
172C22×C4—5421no—
173C22×C4—5421no—
174C22×C4—10842no—
175C22×C4—10842no—
176C22×C4—10842no—
177C22×C4C22×D27111yescharacteristic
178M16—10842no—
179C2×C8—21no—
180C2×C8—5421no—
181C4⋊C4—2711no—
182C4⋊C4—5421no—
183C4⋊C4—5421no—
184C4⋊C4C4×D27111yescharacteristic, complemented
185C22⋊C4—21no—
186C22⋊C4—5421no—
187C22⋊C4—5421no—
188C22⋊C4—5421no—
189C22⋊C4—5421no—
190C42—5421no—
191C23—21no—
192C23—2711no—
193C23—2711no—
194C23—5422no—
195C23—5422no—
196C23—5421no—
197C23—5421no—
198C23D4×D27111yescharacteristic
199Q8—10841no—
200D4—41no—
201D4—41no—
202D4—41no—
203D4—10842no—
204D4—10842no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—2711no—
211C2×C4—2711no—
212C2×C4—5422no—
213C2×C4—5422no—
214C2×C4—5422no—
215C2×C4—5421no—
216C2×C4—5421no—
217C2×C4—10841no—
218C2×C4—10841no—
219C2×C4—10848no—
220C2×C4—10841no—
221C2×C4—10842no—
222C2×C4—10841no—
223C2×C4—10841no—
224C2×C4C2×C4×D27111yescharacteristic
225C2×C4D4×D27111yescharacteristic
226C8—41no—
227C8—10841no—
228C22—21no—
229C22—21no—
230C22—41no—
231C22—41no—
232C22—41no—
233C22—2712no—
234C22—2712no—
235C22—5424no—
236C22—5424no—
237C22—5421no—
238C22—5421no—
239C22—5421no—
240C22—5421no—
241C22C22⋊C4×D27111yescharacteristic
242C22#2257618411yescharacteristic
243C22#2257618911yescharacteristic
244C4—21no—
245C4—21no—
246C4—42no—
247C4—41no—
248C4—5421no—
249C4—5421no—
250C4—10842no—
251C4—10841no—
252C4—10841no—
253C2—21no—
254C2—21no—
255C2—81no—
256C2—5422no—
257C2—5422no—
258C2#99×D27111yescharacteristic
259C2#3177389511yescharacteristic
260C2#104×D27111yescharacteristic
subgroupG/H
centerC22C22⋊C4×D271
commutatorC2×C4×C271C22×C4
FrattiniC22×C4C22×D271
Fitting#327×C271C2
socleC22×C271C2×C22⋊C4
prime radical#31774407C2
cosocleC22×C4×C271C23
hypercenter#327D271
hypocenterC271#1504
LCSG ▷ C2×C4×C271 ▷ C22×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #31773859C2semidirect
3. #95×D271C2semidirect
4. #31773635C2semidirect
5. #31773633C2semidirect
6. #31773631C2semidirect
7. #327×C271C2semidirect
8. C2×#16864653C4semidirect
13. #22576140C4semidirect
15. #22576131C4semidirect
19. #22576118C4semidirect
29. C2×Dic542C2×C4semidirect
46. C4⋊Dic271C2×C4semidirect
54. C2×D4×C271C2×C4semidirect
58. C4⋊C4×C271C2×C4semidirect
130. C271#1504semidirect
138. #327D271semidirect
164. C2×D4C4×D271semidirect
184. C4⋊C4C4×D271semidirect

Maximal subgroups

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

subgroupindex
#317744072
#317738592
#95×D2712
#317736352
#317736332
#317736312
#327×C2712
#1504271