Finite Group Databasebeta

← #46846885

Subgroups — #46846885

Order 34688.

348 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 137 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#31774453C211yesmaximal, characteristic
2#31774452C211yesmaximal, characteristic
3#31774307C211yesmaximal, characteristic, complemented
4#31774306C211yesmaximal, characteristic, complemented
5#31774305C212yesmaximal, complemented
6#31774304C212yesmaximal, complemented
7#31774303C211yesmaximal, characteristic, complemented
8#31773934C212yesmaximal
9#31773932C212yesmaximal
10#97×D271C211yesmaximal, characteristic
11#501×C271C211yesmaximal, characteristic, complemented
12C2×C8×D271C2212yes—
13#22576160C2211yescharacteristic
14#22576159C2211yescharacteristic
15#22576159C2211yescharacteristic
16#22576152C2211yescharacteristic
17#22576151C2212yes—
18#22576140—21no—
19#22576138—21no—
20#22576138—21no—
21#22576136—21no—
22Q8×Dic271—21no—
23#22576125—21no—
24#22576125—21no—
25#22576125—21no—
26#22576118C2211yescharacteristic
27#22576118C2212yes—
28#22576118C2211yescharacteristic
29C42×D271C2211yescharacteristic
30#22576094C2212yes—
31#22576093C2212yes—
32#22576093C2212yes—
33#22576092C2212yes—
34#22576091C2212yes—
35#22576091C2212yes—
36#22576071C2211yescharacteristic
37#22576070C2212yes—
38C4⋊Q8×C271C2211yescharacteristic
39#125×C271C2211yescharacteristic
40#107×C271C2212yes—
41#106×C271C2212yes—
42#97×C271C2211yescharacteristic
43C2×Dic542—21no—
44C2×Dic542—21no—
45C2×Dic542—21no—
46C2×C4×D271D411yescharacteristic
47C2×C4×D271D411yescharacteristic
48C2×C4×D271C2311yescharacteristic
49C2×C271⋊C8C2312yes—
50C8×D271Q818yescomplemented
51C4⋊Dic271—22no—
52C4⋊Dic271—21no—
53C4⋊Dic271C2312yes—
54C4⋊Dic271C2311yescharacteristic
55C4⋊Dic271C2311yescharacteristic
56Dic271⋊C4—22no—
57Dic271⋊C4—21no—
58Dic271⋊C4—21no—
59Dic271⋊C4—21no—
60Dic271⋊C4—21no—
61(C2×D271)⋊C4—22no—
62(C2×D271)⋊C4—21no—
63(C2×D271)⋊C4—21no—
64C4×Dic271—21no—
65C4×Dic271—21no—
66C4×Dic271—22no—
67C4×Dic271D411yescharacteristic
68C4×Dic271D411yescharacteristic
69C4×Dic271C2311yescharacteristic
70C2×Q8×C271—21no—
71C2×C8×C271C2312yes—
72C4⋊C4×C271—21no—
73C4⋊C4×C271—21no—
74C4⋊C4×C271—21no—
75C4⋊C4×C271C2312yes—
76C4⋊C4×C271C2311yescharacteristic
77C4⋊C4×C271C2311yescharacteristic
78C42×C271C2311yescharacteristic
79C22×D271C2×D411yescharacteristic
80Dic542—42no—
81Dic542—42no—
82Dic542—42no—
83C2×Dic271—21no—
84C2×Dic271—21no—
85C2×Dic271—22no—
86C2×Dic271C2×D411yescharacteristic
87C2×Dic271C2×D411yescharacteristic
88C2×Dic271C2×D411yescharacteristic
89C4×D271—41no—
90C4×D271—41no—
91C4×D271C2×Q814yes—
92C271⋊C8C2×Q814yes—
93Q8×C271—42no—
94C2×C4×C271—21no—
95C2×C4×C271—21no—
96C2×C4×C271—22no—
97C2×C4×C271C2×D411yescharacteristic
98C2×C4×C271C2×D411yescharacteristic
99C2×C4×C271C2411yescharacteristic
100C8×C271C2×Q814yes—
101C22×C271C22×D411yescharacteristic
102C2×D271—21no—
103C2×D271—21no—
104C2×D271C4⋊Q812yes—
105C4×C271—21no—
106C4×C271—21no—
107C4×C271—42no—
108C4×C271—41no—
109C4×C271—41no—
110C4×C271C22×Q812yes—
111Dic271—21no—
112Dic271—21no—
113Dic271—41no—
114Dic271—41no—
115Dic271—42no—
116Dic271C4⋊Q812yes—
117C2×C271C2×C4⋊Q811yescharacteristic
118C2×C271#57511yescharacteristic
119C2×C271#57511yescharacteristic
120D271—22no—
121C271#316211yescharacteristic, complemented, Sylow
122#3162—2711nomaximal, Sylow
123#533—2711no—
124#533—2711no—
125#501—2711no—
126#501—2711no—
127#501—2712no—
128#501—2712no—
129#501—2711no—
130#501D27111yescharacteristic, complemented
131#427—2712no—
132#427—2712no—
133C2×#97—2711no—
134C22×C8—2712no—
135C4⋊Q8—2711no—
136C4⋊Q8—2711no—
137C4⋊Q8—2711no—
138C4⋊Q8C2×D27111yescharacteristic
139#125—2712no—
140#125—2711no—
141#125C2×D27111yescharacteristic
142C22⋊Q8—5421no—
143C22⋊Q8—5421no—
144C22⋊Q8—5421no—
145C22⋊Q8—5421no—
146C4×Q8—5421no—
147C4×Q8—5421no—
148C4×Q8—5421no—
149C4×Q8—5421no—
150#117—2711no—
151#117—2712no—
152#117—2711no—
153C2×C42—2711no—
154#107—2712no—
155#107—2712no—
156#107—2712no—
157#107C2×D27112yes—
158#106—2712no—
159#106—2712no—
160#106—2712no—
161#106C2×D27112yes—
162#97—2712no—
163#97—2711no—
164#97C2×D27111yescharacteristic
165C2×Q8—21no—
166C2×Q8—5421no—
167C2×Q8—5421no—
168C2×Q8—5421no—
169C22×C4—2711no—
170C22×C4—2711no—
171C22×C4—2711no—
172C2×C8—2712no—
173C2×C8—2718no—
174C2×C8C22×D27112yes—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—2711no—
179C4⋊C4—2712no—
180C4⋊C4—2711no—
181C4⋊C4—5422no—
182C4⋊C4—5421no—
183C4⋊C4—5421no—
184C4⋊C4—5422no—
185C4⋊C4—5421no—
186C4⋊C4—5421no—
187C4⋊C4—5421no—
188C4⋊C4C22×D27111yescharacteristic
189C4⋊C4C22×D27112yes—
190C4⋊C4C22×D27111yescharacteristic
191C22⋊C4—5421no—
192C22⋊C4—5421no—
193C22⋊C4—5422no—
194C42—2711no—
195C42—2711no—
196C42—2711no—
197C42—5421no—
198C42—5421no—
199C42—5422no—
200C42C22×D27111yescharacteristic
201C23—2711no—
202Q8—42no—
203Q8—10842no—
204Q8—10842no—
205Q8—10842no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—2711no—
210C2×C4—2714no—
211C2×C4—2711no—
212C2×C4—2711no—
213C2×C4—5421no—
214C2×C4—5422no—
215C2×C4—5421no—
216C2×C4—10841no—
217C2×C4—10841no—
218C2×C4D4×D27111yescharacteristic
219C2×C4D4×D27111yescharacteristic
220C2×C4C23×D27111yescharacteristic
221C8—2714no—
222C8Q8×D27114yes—
223C22—2712no—
224C22—5421no—
225C22—5421no—
226C22C2×D4×D27111yescharacteristic
227C4—21no—
228C4—21no—
229C4—42no—
230C4—41no—
231C4—41no—
232C4—2712no—
233C4—5421no—
234C4—5421no—
235C4—10841no—
236C4—10842no—
237C4—10841no—
238C4C2×Q8×D27112yes—
239C2—5422no—
240C2C4⋊Q8×D27111yescharacteristic
241C2#3177469311yescharacteristic
242C2#3177469611yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#501×C271C2
socleC22×C271C22×D4
prime radicalC22×D271C2×D4
hypercenter#501D271
hypocenterC271#3162
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
3. #31774307C2semidirect
4. #31774306C2semidirect
5. #31774305C2semidirect
6. #31774304C2semidirect
7. #31774303C2semidirect
11. #501×C271C2semidirect
50. C8×D271Q8semidirect
121. C271#3162semidirect
130. #501D271semidirect

Maximal subgroups

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

subgroupindex
#317744532
#317744522
#317743072
#317743062
#317743052
#317743042
#317743032
#317739342
#317739322
#97×D2712
#501×C2712
#3162271