Finite Group Databasebeta

← #46846198

Subgroups — #46846198

Order 34688.

398 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 133 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#31774525C211yesmaximal, characteristic, complemented
2#31774371C211yesmaximal, characteristic
3#31774200C212yesmaximal, complemented
4#31774196C212yesmaximal, complemented
5#31774186C212yesmaximal, complemented
6#31774176C212yesmaximal, complemented
7#31773940C211yesmaximal, characteristic, complemented
8#31773923C211yesmaximal, characteristic, complemented
9#31773901C211yesmaximal, characteristic
10C2×#22576082C211yesmaximal, characteristic
11#427×C271C211yesmaximal, characteristic, complemented
12C22×D1084C2211yescharacteristic
13C2×#16864638C2211yescharacteristic
14#22576160—21no—
15#22576159C2212yescomplemented
16#22576152—21no—
17#22576151C2212yescomplemented
18#22576137—22no—
19C4×D1084—21no—
20C4×D1084—21no—
21#22576122C2212yes—
22#22576122C2212yes—
23#22576119C2211yescharacteristic
24C2×C4⋊Dic271C2211yescharacteristic
25C22⋊C4×D271—22no—
26#22576093C2212yescomplemented
27#22576091C2212yescomplemented
28Dic271⋊C8C2212yescomplemented
29Dic271⋊C8C2212yescomplemented
30#22576082C2212yes—
31#22576082C2212yes—
32#22576080C2212yes—
33#22576080C2212yes—
34C22×C8×C271C2211yescharacteristic
35#117×C271C2211yescharacteristic
36#117×C271C2211yescharacteristic
37#107×C271C2211yescharacteristic, complemented
38#107×C271C2211yescharacteristic, complemented
39#106×C271C2211yescharacteristic, complemented
40#106×C271C2211yescharacteristic, complemented
41C23×D271—21no—
42C2×Dic542—22no—
43C2×D1084—21no—
44C2×D1084—22no—
45C2×D1084Q814yescomplemented
46C2×D1084C2312yes—
47C22×Dic271—21no—
48C2×C4×D271—22no—
49C2×C4×D271—22no—
50#16864638—22no—
51C2×C271⋊C8C2312yes—
52C4⋊Dic271—22no—
53C4⋊Dic271—21no—
54C4⋊Dic271D411yescharacteristic
55C4⋊Dic271D411yescharacteristic
56C4⋊Dic271C2312yes—
57Dic271⋊C4—22no—
58Dic271⋊C4—22no—
59C22⋊Dic271D412yes—
60(C2×D271)⋊C4—22no—
61(C2×D271)⋊C4—22no—
62C4×Dic271C2312yes—
63C22×C4×C271C2311yescharacteristic
64C2×C8×C271—22no—
65C2×C8×C271C2311yescharacteristic
66C2×C8×C271C2311yescharacteristic
67C4⋊C4×C271C2311yescharacteristic
68C4⋊C4×C271C2311yescharacteristic
69C4⋊C4×C271C2311yescharacteristic
70C4⋊C4×C271C2311yescharacteristic
71C22⋊C4×C271—21no—
72C22⋊C4×C271—21no—
73C42×C271—21no—
74C42×C271—21no—
75C22×D271—22no—
76C22×D271—24no—
77C22×D271—81no—
78C22×D271—81no—
79Dic542—82no—
80D1084—24no—
81D1084—22no—
82D1084C2×Q814yes—
83C2×Dic271—22no—
84C2×Dic271—81no—
85C2×Dic271C2×D412yes—
86C2×Dic271C2×D412yes—
87C4×D271—82no—
88C4×D271—82no—
89C271⋊C8—22no—
90C23×C271C2×D411yescharacteristic
91C2×C4×C271—21no—
92C2×C4×C271—21no—
93C2×C4×C271—21no—
94C2×C4×C271—21no—
95C2×C4×C271C2×D411yescharacteristic
96C2×C4×C271C2×Q812yes—
97C2×C4×C271#4112yes—
98C2×C4×C271C2411yescharacteristic
99C8×C271—21no—
100C8×C271—21no—
101C22×C271—21no—
102C22×C271—21no—
103C22×C271C22⋊Q812yes—
104C22×C271C22×D411yescharacteristic
105C2×D271—24no—
106C2×D271—42no—
107C2×D271—42no—
108C2×D271—84no—
109C4×C271—41no—
110C4×C271—41no—
111C4×C271—41no—
112C4×C271—41no—
113C4×C271C22⋊Q812yes—
114C4×C271C22×Q811yescharacteristic
115C4×C271C2×#4111yescharacteristic
116Dic271—22no—
117Dic271—22no—
118Dic271—42no—
119C2×C271—22no—
120C2×C271C2×C22⋊Q811yescharacteristic
121C2×C271#57611yescharacteristic
122C2×C271#57711yescharacteristic
123D271—44no—
124C271#308111yescharacteristic, complemented, Sylow
125#3081—2711nomaximal, Sylow
126#544—2711no—
127#518—2711no—
128#478—2712no—
129#478—2712no—
130#476—2712no—
131#474—2712no—
132#428—2711no—
133#427D27111yescharacteristic, complemented
134#424—2711no—
135#418—2711no—
136C2×#102—2711no—
137C22×D4—2711no—
138C2×M16—2711no—
139C22×C8C2×D27111yescharacteristic
140C4⋊Q8—2712no—
141C4⋊Q8—5421no—
142#125—2712no—
143#125—5421no—
144C22⋊Q8—5422no—
145C4×D4—2712no—
146C4×D4—2712no—
147C4×D4—5421no—
148C4×D4—5421no—
149#117—2711no—
150#117C2×D27111yescharacteristic
151#117C2×D27111yescharacteristic
152C2×C4⋊C4—2711no—
153C2×C22⋊C4—5422no—
154#107—2712no—
155#107C2×D27111yescharacteristic, complemented
156#107C2×D27111yescharacteristic, complemented
157#106—2712no—
158#106C2×D27111yescharacteristic, complemented
159#106C2×D27111yescharacteristic, complemented
160C4⋊C8—2712no—
161C4⋊C8—2712no—
162#102—2712no—
163#102—2712no—
164#102—2712no—
165#102—2712no—
166C24—5421no—
167C2×Q8—5422no—
168C2×D4—2714no—
169C2×D4—2712no—
170C2×D4—5421no—
171C2×D4—5422no—
172C22×C4—5422no—
173C22×C4—5422no—
174C22×C4—5421no—
175C22×C4C22×D27111yescharacteristic
176M16—5422no—
177C2×C8—22no—
178C2×C8—2712no—
179C2×C8C22×D27111yescharacteristic
180C2×C8C22×D27111yescharacteristic
181C4⋊C4—2711no—
182C4⋊C4—2711no—
183C4⋊C4—2712no—
184C4⋊C4—5422no—
185C4⋊C4—5422no—
186C4⋊C4—5422no—
187C4⋊C4—5421no—
188C4⋊C4C22×D27111yescharacteristic
189C4⋊C4C22×D27111yescharacteristic
190C4⋊C4C22×D27111yescharacteristic
191C4⋊C4C22×D27111yescharacteristic
192C22⋊C4—21no—
193C22⋊C4—21no—
194C22⋊C4—2712no—
195C22⋊C4—5422no—
196C22⋊C4—5422no—
197C42—21no—
198C42—21no—
199C42—2712no—
200C23—5422no—
201C23—5424no—
202C23—21681no—
203C23—21681no—
204C23D4×D27111yescharacteristic
205Q8—21682no—
206D4—2714no—
207D4—5424no—
208D4—5422no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—2712no—
214C2×C4—2712no—
215C2×C4—5422no—
216C2×C4—21682no—
217C2×C4—21682no—
218C2×C4—21681no—
219C2×C4D4×D27111yescharacteristic
220C2×C4Q8×D27112yes—
221C2×C4#1686465412yes—
222C2×C4C23×D27111yescharacteristic
223C8—21no—
224C8—21no—
225C8—5422no—
226C22—21no—
227C22—21no—
228C22—5424no—
229C22—10842no—
230C22—10842no—
231C22—21684no—
232C22#2257613812yes—
233C22C2×D4×D27111yescharacteristic
234C4—41no—
235C4—41no—
236C4—41no—
237C4—41no—
238C4—5422no—
239C4—5422no—
240C4—10842no—
241C4#2257613812yes—
242C4C2×Q8×D27111yescharacteristic
243C4C2×#1686465411yescharacteristic
244C2—22no—
245C2—10844no—
246C2C2×#2257613811yescharacteristic
247C2#3177469911yescharacteristic
248C2#3177470411yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#427×C271C2
socleC22×C271C22×D4
prime radicalC22×D1084C22
hypercenter#427D271
hypocenterC271#3081
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #427

Complements

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

NH ≅ G/Ntype
1. #31774525C2semidirect
3. #31774200C2semidirect
4. #31774196C2semidirect
5. #31774186C2semidirect
6. #31774176C2semidirect
7. #31773940C2semidirect
8. #31773923C2semidirect
11. #427×C271C2semidirect
15. #22576159C22semidirect
17. #22576151C22semidirect
26. #22576093C22semidirect
27. #22576091C22semidirect
28. Dic271⋊C8C22semidirect
29. Dic271⋊C8C22semidirect
37. #107×C271C22semidirect
38. #107×C271C22semidirect
39. #106×C271C22semidirect
40. #106×C271C22semidirect
45. C2×D1084Q8semidirect
124. C271#3081semidirect
133. #427D271semidirect
155. #107C2×D271semidirect
156. #107C2×D271semidirect
158. #106C2×D271semidirect
159. #106C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317745252
#317743712
#317742002
#317741962
#317741862
#317741762
#317739402
#317739232
#317739012
C2×#225760822
#427×C2712
#3081271