Finite Group Databasebeta

← #46845728

Subgroups — #46845728

Order 34688.

698 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 161 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#31774695C212yesmaximal, complemented
2#31774471C211yesmaximal, characteristic, complemented
3#31774080C214yesmaximal, complemented
4#31774070C214yesmaximal, complemented
5#31773914C212yesmaximal, complemented
6C2×#22576079C211yesmaximal, characteristic, complemented
7C2×2−1+4×C271C211yesmaximal, characteristic, complemented
8C2×#16864654C2211yescharacteristic
9#22576191—22no—
10#22576190C2214yescomplemented
11#22576186—22no—
12#22576185C2214yescomplemented
13#22576180—24no—
14C2×#16864638C2212yes—
15#22576158C2212yescomplemented
16#22576149C2212yescomplemented
17#22576143—22no—
18#22576136—22no—
19#22576119C2211yescharacteristic
20#22576087C2218yescomplemented
21#22576079C2214yescomplemented
222−1+4×C271—24no—
232−1+4×C271—22no—
242−1+4×C271C2214yes—
25C2×#41×C271—22no—
26C2×#41×C271—22no—
27C2×#41×C271C2212yes—
28C22×Q8×C271—22no—
29C22×Q8×C271C2211yescharacteristic
30#16864654—22no—
31#16864654—22no—
32C2×Dic542C2311yescharacteristic
33C2×D1084C2311yescharacteristic
34C2×C22⋊D271—21no—
35C2×C4×D271—21no—
36#16864644—24no—
37#16864642—24no—
38#16864641—24no—
39#16864639—24no—
40#16864638—22no—
41#16864638C2314yes—
42C2×C271⋊C8—22no—
43C4⋊Dic271—21no—
44Dic271⋊C4—24no—
45C22⋊Dic271—21no—
46(C2×D271)⋊C4—24no—
47C4×Dic271C2312yes—
48#41×C271—28no—
49#41×C271—28no—
50#41×C271—28no—
51#41×C271—28no—
52#41×C271—22no—
53#41×C271—22no—
54#41×C271D414yes—
55#41×C271C2314yes—
56C2×Q8×C271—22no—
57C2×Q8×C271—28no—
58C2×Q8×C271—28no—
59C2×Q8×C271—22no—
60C2×Q8×C271—22no—
61C2×Q8×C271D412yes—
62C2×Q8×C271D412yes—
63C2×Q8×C271C2312yes—
64C2×D4×C271—22no—
65C2×D4×C271—22no—
66C2×D4×C271D412yes—
67C22×C4×C271—24no—
68C22×C4×C271—22no—
69C22×C4×C271D412yes—
70C22×C4×C271C2311yescharacteristic
71C22×D271—21no—
72Dic542—22no—
73D1084—22no—
74C22⋊D271—41no—
75C22⋊D271—41no—
76C2×Dic271—22no—
77C2×Dic271—21no—
78C4×D271—42no—
79C271⋊C8—24no—
80C23×C271—22no—
81C23×C271C2×D411yescharacteristic
82Q8×C271—28no—
83Q8×C271—22no—
84Q8×C271—24no—
85Q8×C271—24no—
86Q8×C271C2×D414yes—
87D4×C271—28no—
88D4×C271—24no—
89D4×C271—24no—
90D4×C271—22no—
91D4×C271C2×D414yes—
92C2×C4×C271—216no—
93C2×C4×C271—22no—
94C2×C4×C271—22no—
95C2×C4×C271—22no—
96C2×C4×C271—24no—
97C2×C4×C271—24no—
98C2×C4×C271C2×D414yes—
99C2×C4×C271C2×D411yescharacteristic
100C2×C4×C271C2×D412yes—
101C2×C4×C271C2×D412yes—
102C2×C4×C271C2411yescharacteristic
103C22×C271—24no—
104C22×C271—41no—
105C22×C271—44no—
106C22×C271#12011yescharacteristic
107C22×C271#12011yescharacteristic
108C22×C271C22×D411yescharacteristic
109C2×D271—42no—
110C2×D271—41no—
111C4×C271—24no—
112C4×C271—24no—
113C4×C271#12012yes—
114C4×C271C22×D412yes—
115Dic271—42no—
116Dic271—41no—
117C2×C271—21no—
118C2×C271—21no—
119C2×C271—21no—
120C2×C271—44no—
121C2×C271C2×#12011yescharacteristic
122D271—81no—
123C271#302011yescharacteristic, complemented, Sylow
124#3020—2711nomaximal, Sylow
125C2×2−1+4D27111yescharacteristic, complemented
126#575—2712no—
127#536—2711no—
128#456—2714no—
129#454—2714no—
130#421—2712no—
131C2×#101—2711no—
1322−1+4—24no—
1332−1+4—22no—
1342−1+4C2×D27114yes—
135C2×#41—22no—
136C2×#41—22no—
137C2×#41—2711no—
138C2×#41C2×D27112yes—
139C22×Q8—22no—
140C22×Q8C2×D27111yescharacteristic
141#137—2714no—
142#137—5422no—
143AGL(1,ℤ/8)—2714no—
144AGL(1,ℤ/8)—5422no—
145#135—5424no—
146C2×M16—2712no—
147C4⋊Q8—2712no—
148#124—2712no—
149#123—5422no—
150C22⋊Q8—5422no—
151#117—2711no—
152#104—2718no—
153#101—2714no—
154#41—28no—
155#41—28no—
156#41—28no—
157#41—28no—
158#41—22no—
159#41—22no—
160#41—5422no—
161#41—5422no—
162#41C22⋊D27114yes—
163#41C22×D27114yes—
164C2×Q8—22no—
165C2×Q8—28no—
166C2×Q8—28no—
167C2×Q8—22no—
168C2×Q8—22no—
169C2×Q8—2711no—
170C2×Q8C22⋊D27112yes—
171C2×Q8C22⋊D27112yes—
172C2×Q8C22×D27112yes—
173C2×D4—22no—
174C2×D4—22no—
175C2×D4—2711no—
176C2×D4—5421no—
177C2×D4C22⋊D27112yes—
178C22×C4—24no—
179C22×C4—22no—
180C22×C4—5421no—
181C22×C4C22⋊D27112yes—
182C22×C4C22×D27111yescharacteristic
183Q16—5424no—
184SD16—5424no—
185SD16—5424no—
186D8—5424no—
187M16—2714no—
188M16—5422no—
189C2×C8—5422no—
190C4⋊C4—5421no—
191C4⋊C4—5424no—
192C22⋊C4—5421no—
193C22⋊C4—5424no—
194C42—2712no—
195C23—22no—
196C23—5421no—
197C23C2×C22⋊D27111yescharacteristic
198Q8—28no—
199Q8—22no—
200Q8—24no—
201Q8—24no—
202Q8—5422no—
203Q8C2×C22⋊D27114yes—
204D4—28no—
205D4—24no—
206D4—24no—
207D4—22no—
208D4—5422no—
209D4—10841no—
210D4—10841no—
211D4C2×C22⋊D27114yes—
212C2×C4—216no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—24no—
217C2×C4—24no—
218C2×C4—5422no—
219C2×C4—5421no—
220C2×C4—10842no—
221C2×C4C2×C22⋊D27114yes—
222C2×C4C2×C22⋊D27111yescharacteristic
223C2×C4C2×C22⋊D27112yes—
224C2×C4C2×C22⋊D27112yes—
225C2×C4C23×D27111yescharacteristic
226C8—5424no—
227C22—24no—
228C22—41no—
229C22—44no—
230C22—10842no—
231C22—10841no—
232C22#2257612911yescharacteristic
233C22#2257612911yescharacteristic
234C22C22×C22⋊D27111yescharacteristic
235C4—24no—
236C4—24no—
237C4—10842no—
238C4—10841no—
239C4#2257612912yes—
240C4C22×C22⋊D27112yes—
241C2—21no—
242C2—21no—
243C2—21no—
244C2—44no—
245C2—21681no—
246C2C2×#2257612911yescharacteristic
subgroupG/H
centerC2C2×#22576129
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
FittingC2×2−1+4×C271C2
socleC2×C271C2×#120
hypercenterC2×2−1+4D271
hypocenterC271#3020
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×2−1+4

Complements

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

NH ≅ G/Ntype
1. #31774695C2semidirect
2. #31774471C2semidirect
3. #31774080C2semidirect
4. #31774070C2semidirect
5. #31773914C2semidirect
6. C2×#22576079C2semidirect
7. C2×2−1+4×C271C2semidirect
10. #22576190C22semidirect
12. #22576185C22semidirect
15. #22576158C22semidirect
16. #22576149C22semidirect
20. #22576087C22semidirect
21. #22576079C22semidirect
123. C271#3020semidirect
125. C2×2−1+4D271semidirect

Maximal subgroups

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

subgroupindex
#317746952
#317744712
#317740802
#317740702
#317739142
C2×#225760792
C2×2−1+4×C2712
#3020271