Finite Group Databasebeta

← #46839630

Subgroups — #46839630

Order 34688.

560 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 143 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×#22576122C211yesmaximal, characteristic
2C2×#22576118C211yesmaximal, characteristic, complemented
3#31773832C211yesmaximal, characteristic, complemented
4#31773827C211yesmaximal, characteristic, complemented
5#31773826C212yesmaximal, complemented
6#31773824C211yesmaximal, characteristic, complemented
7#31773815C212yesmaximal
8#31773748C212yesmaximal, complemented
9#31773744C212yesmaximal, complemented
10#31773744C211yesmaximal, characteristic, complemented
11#400×C271C211yesmaximal, characteristic, complemented
12C22×D1084C2211yescharacteristic
13C22×C4×D271C2211yescharacteristic
14C22×C4×D271C2211yescharacteristic
15#22576122—24no—
16#22576118—24no—
17C2×C4⋊Dic271C2211yescharacteristic
18C2×Dic271⋊C4C2212yes—
19C2×Dic271⋊C4C2212yes—
20C2×(C2×D271)⋊C4C2212yes—
21C2×(C2×D271)⋊C4C2212yes—
22C2×(C2×D271)⋊C4C2212yes—
23C2×(C2×D271)⋊C4C2212yes—
24C2×(C2×D271)⋊C4C2211yescharacteristic
25C2×(C2×D271)⋊C4C2211yescharacteristic
26C2×C4×Dic271C2212yes—
27C2×C4×Dic271C2212yes—
28C2×C4×Dic271C2211yescharacteristic
29C271⋊#95C2212yes—
30C271⋊#95C2211yescharacteristic
31C271⋊#95C2212yes—
32C2×C4⋊C4×C271C2211yescharacteristic
33C2×C4⋊C4×C271C2211yescharacteristic
34#95×C271C2211yescharacteristic
35#95×C271C2212yes—
36#95×C271C2212yes—
37C23×D271C2311yescharacteristic
38C23×D271C2311yescharacteristic
39C2×D1084—22no—
40C2×D1084—22no—
41C2×D1084—22no—
42C22×Dic271C2311yescharacteristic
43C22×Dic271C2312yes—
44C22×Dic271C2312yes—
45C22×Dic271C2311yescharacteristic
46C2×C4×D271—24no—
47C2×C4×D271—22no—
48C2×C4×D271—24no—
49C2×C4×D271D414yes—
50C4⋊Dic271—22no—
51Dic271⋊C4—24no—
52Dic271⋊C4—24no—
53(C2×D271)⋊C4—24no—
54(C2×D271)⋊C4—22no—
55(C2×D271)⋊C4—24no—
56(C2×D271)⋊C4—24no—
57(C2×D271)⋊C4—24no—
58(C2×D271)⋊C4—22no—
59C4×Dic271—24no—
60C4×Dic271—22no—
61C4×Dic271—24no—
62C22×C4×C271C2311yescharacteristic
63C22×C4×C271C2312yes—
64C22×C4×C271C2312yes—
65C22×C4×C271C2311yescharacteristic
66C22×C4×C271C2311yescharacteristic
67C4⋊C4×C271—22no—
68C4⋊C4×C271—22no—
69C22×D271—21no—
70C22×D271—22no—
71C22×D271—22no—
72C22×D271—22no—
73C22×D271—22no—
74C22×D271—21no—
75C22×D271—21no—
76C22×D271—21no—
77C22×D271C2×D412yes—
78C22×D271#4112yes—
79D1084—28no—
80C2×Dic271—22no—
81C2×Dic271—22no—
82C2×Dic271—21no—
83C2×Dic271—22no—
84C2×Dic271—22no—
85C2×Dic271—22no—
86C2×Dic271—22no—
87C2×Dic271C2×D412yes—
88C2×Dic271#4114yes—
89C2×Dic271#4114yes—
90C4×D271—28no—
91C4×D271—44no—
92C23×C271C2411yescharacteristic
93C2×C4×C271—21no—
94C2×C4×C271—22no—
95C2×C4×C271—21no—
96C2×C4×C271—22no—
97C2×C4×C271—21no—
98C2×C4×C271—22no—
99C2×C4×C271—22no—
100C2×C4×C271—22no—
101C2×C4×C271—22no—
102C2×C4×C271—22no—
103C2×C4×C271—22no—
104C2×C4×C271C2×D412yes—
105C2×C4×C271#4112yes—
106C22×C271C22×D411yescharacteristic
107C22×C271C2×#4112yes—
108C22×C271C2×#4111yescharacteristic
109C22×C271C2×#4112yes—
110C22×C271C2×#4111yescharacteristic
111C2×D271—22no—
112C2×D271—22no—
113C2×D271—22no—
114C2×D271—24no—
115C2×D271—22no—
116C2×D271—41no—
117C2×D271—42no—
118C2×D271—42no—
119C2×D271—42no—
120C2×D271—41no—
121C4×C271—22no—
122C4×C271—22no—
123C4×C271—41no—
124C4×C271—42no—
125C4×C271—42no—
126Dic271—24no—
127Dic271—42no—
128Dic271—41no—
129Dic271#12314yes—
130C2×C271C2×#12311yescharacteristic
131C2×C271#52511yescharacteristic
132C2×C271#52912yes—
133C2×C271#52912yes—
134C2×C271#53211yescharacteristic
135D271—42no—
136D271—42no—
137C271#239511yescharacteristic, complemented, Sylow
138#2395—2711nomaximal, Sylow
139C2×C4×D4—2711no—
140C2×#117—2711no—
141#400D27111yescharacteristic, complemented
142#399—2711no—
143#398—2711no—
144#397—2711no—
145#397—2712no—
146#394—2712no—
147C4×C4⋊C4—2712no—
148C4×C22⋊C4—2712no—
149C4×C22⋊C4—2711no—
150C22×D4—2711no—
151C23×C4—2711no—
152C23×C4—2711no—
153C4×D4—5424no—
154#117—5424no—
155C2×C4⋊C4—2711no—
156C2×C4⋊C4—2712no—
157C2×C4⋊C4—2712no—
158C2×C4⋊C4C2×D27111yescharacteristic
159C2×C4⋊C4C2×D27111yescharacteristic
160C2×C22⋊C4—2712no—
161C2×C22⋊C4—2711no—
162C2×C22⋊C4—2711no—
163C2×C22⋊C4—2712no—
164C2×C22⋊C4—2712no—
165C2×C22⋊C4—2712no—
166C2×C42—2712no—
167C2×C42—2711no—
168C2×C42—2712no—
169#95—2712no—
170#95—2711no—
171#95—2712no—
172#95C2×D27112yes—
173#95C2×D27111yescharacteristic
174#95C2×D27112yes—
175C24—2711no—
176C24—2711no—
177C2×D4—5422no—
178C2×D4—5422no—
179C2×D4—5422no—
180C22×C4—2711no—
181C22×C4—2712no—
182C22×C4—2714no—
183C22×C4—2711no—
184C22×C4—2712no—
185C22×C4—5424no—
186C22×C4—5422no—
187C22×C4—5424no—
188C22×C4C22×D27111yescharacteristic
189C22×C4C22×D27112yes—
190C22×C4C22×D27112yes—
191C22×C4C22×D27111yescharacteristic
192C22×C4C22×D27111yescharacteristic
193C4⋊C4—22no—
194C4⋊C4—22no—
195C4⋊C4—5424no—
196C4⋊C4—5422no—
197C4⋊C4—5424no—
198C22⋊C4—5424no—
199C22⋊C4—5424no—
200C22⋊C4—5424no—
201C22⋊C4—5422no—
202C22⋊C4—5424no—
203C22⋊C4—5422no—
204C42—5424no—
205C42—5422no—
206C42—5424no—
207C23—2712no—
208C23—2712no—
209C23—5421no—
210C23—5422no—
211C23—5422no—
212C23—5422no—
213C23—5422no—
214C23—5421no—
215C23—5421no—
216C23—5421no—
217C23C23×D27111yescharacteristic
218D4—5428no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—2712no—
231C2×C4—2714no—
232C2×C4—2714no—
233C2×C4—5428no—
234C2×C4—5422no—
235C2×C4—5422no—
236C2×C4—5421no—
237C2×C4—5422no—
238C2×C4—5422no—
239C2×C4—5422no—
240C2×C4—5422no—
241C2×C4—10844no—
242C2×C4D4×D27112yes—
243C2×C4#1686465512yes—
244C22—5422no—
245C22—5422no—
246C22—5422no—
247C22—5424no—
248C22—5422no—
249C22—10841no—
250C22—10842no—
251C22—10842no—
252C22—10842no—
253C22—10841no—
254C22C2×D4×D27111yescharacteristic
255C22#41×D27112yes—
256C22#41×D27111yescharacteristic
257C22#41×D27112yes—
258C22C2×#1686465511yescharacteristic
259C4—22no—
260C4—22no—
261C4—41no—
262C4—42no—
263C4—42no—
264C4—2714no—
265C4—5424no—
266C4—10842no—
267C4—10841no—
268C2—10842no—
269C2—10842no—
270C2#123×D27111yescharacteristic
271C2#3177441011yescharacteristic
272C2#3177443012yes—
273C2#3177443212yes—
274C2#3177444611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
Fitting#400×C271C2
prime radicalC22×D1084C22
hypercenter#400D271
hypocenterC271#2395
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
2. C2×#22576118C2semidirect
3. #31773832C2semidirect
4. #31773827C2semidirect
5. #31773826C2semidirect
6. #31773824C2semidirect
8. #31773748C2semidirect
9. #31773744C2semidirect
10. #31773744C2semidirect
11. #400×C271C2semidirect
137. C271#2395semidirect
141. #400D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761222
C2×#225761182
#317738322
#317738272
#317738262
#317738242
#317738152
#317737482
#317737442
#400×C2712
#2395271