Finite Group Databasebeta

← #46839624

Subgroups — #46839624

Order 34688.

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

normal subgroups only

This group has 560 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
1C2×C4×C22⋊D271C211yesmaximal, characteristic
2C2×#22576119C211yesmaximal, characteristic, complemented
3#31773837C211yesmaximal, characteristic, complemented
4#31773834C211yesmaximal, characteristic, complemented
5#31773829C211yesmaximal, characteristic, complemented
6#31773824C211yesmaximal, characteristic, complemented
7#31773822C212yesmaximal, complemented
8#31773813C212yesmaximal
9C4×C4⋊Dic271C212yesmaximal, complemented
10C4×(C2×D271)⋊C4C212yesmaximal, complemented
11C4×C22⋊C4×C271C211yesmaximal, characteristic, complemented
12C22×C22⋊D271C2211yescharacteristic
13C22×C4×D271C2211yescharacteristic
14C4×C22⋊D271—24no—
15#22576119—24no—
16C2×C4⋊Dic271C2212yes—
17C2×C4⋊Dic271C2212yes—
18C2×C4⋊Dic271C2211yescharacteristic
19C2×C4⋊Dic271C2211yescharacteristic
20C2×Dic271⋊C4C2211yescharacteristic
21C2×C22⋊Dic271C2212yes—
22C2×C22⋊Dic271C2211yescharacteristic
23C2×(C2×D271)⋊C4C2212yes—
24C2×(C2×D271)⋊C4C2212yes—
25C2×(C2×D271)⋊C4C2211yescharacteristic
26C2×C4×Dic271C2212yes—
27C2×C4×Dic271C2211yescharacteristic
28C271⋊#95C2212yes—
29C271⋊#95C2212yes—
30C271⋊#95C2212yes—
31C271⋊#95C2212yes—
32C23×C4×C271C2211yescharacteristic
33C2×C22⋊C4×C271C2212yes—
34C2×C42×C271C2212yes—
35#95×C271C2211yescharacteristic
36#95×C271C2211yescharacteristic
37C23×D271C2311yescharacteristic
38C2×C22⋊D271—22no—
39C2×C22⋊D271—22no—
40C2×C22⋊D271—22no—
41C22×Dic271C2311yescharacteristic
42C22×Dic271C2311yescharacteristic
43C22×Dic271C2312yes—
44C22×Dic271C2312yes—
45C22×Dic271C2311yescharacteristic
46C2×C4×D271—22no—
47C2×C4×D271—24no—
48C4⋊Dic271—24no—
49C4⋊Dic271—22no—
50C4⋊Dic271—22no—
51C4⋊Dic271—24no—
52Dic271⋊C4—22no—
53C22⋊Dic271—24no—
54C22⋊Dic271—22no—
55(C2×D271)⋊C4—24no—
56(C2×D271)⋊C4—22no—
57(C2×D271)⋊C4—24no—
58C4×Dic271—24no—
59C4×Dic271—22no—
60C24×C271C2311yescharacteristic
61C22×C4×C271—24no—
62C22×C4×C271D414yes—
63C22×C4×C271C2312yes—
64C22×C4×C271C2311yescharacteristic
65C22×C4×C271C2312yes—
66C22×C4×C271C2311yescharacteristic
67C22⋊C4×C271—24no—
68C42×C271—24no—
69C22×D271—21no—
70C22×D271—22no—
71C22×D271—22no—
72C22×D271—21no—
73C22×D271#4112yes—
74C22⋊D271—28no—
75C2×Dic271—21no—
76C2×Dic271—21no—
77C2×Dic271—21no—
78C2×Dic271—22no—
79C2×Dic271—22no—
80C2×Dic271—21no—
81C2×Dic271—22no—
82C2×Dic271—22no—
83C2×Dic271—22no—
84C2×Dic271—22no—
85C2×Dic271—22no—
86C2×Dic271—22no—
87C2×Dic271#4112yes—
88C4×D271—44no—
89C23×C271—22no—
90C23×C271—22no—
91C23×C271—21no—
92C23×C271—21no—
93C23×C271C2×D412yes—
94C23×C271C2411yescharacteristic
95C2×C4×C271—22no—
96C2×C4×C271—22no—
97C2×C4×C271—22no—
98C2×C4×C271—28no—
99C2×C4×C271—22no—
100C2×C4×C271—22no—
101C2×C4×C271—22no—
102C2×C4×C271C2×D412yes—
103C2×C4×C271C2×D412yes—
104C2×C4×C271#4114yes—
105C2×C4×C271#4114yes—
106C22×C271—22no—
107C22×C271—22no—
108C22×C271—22no—
109C22×C271—42no—
110C22×C271—42no—
111C22×C271C22×D411yescharacteristic
112C22×C271C2×#4111yescharacteristic
113C22×C271C2×#4111yescharacteristic
114C22×C271C2×#4112yes—
115C22×C271C2×#4112yes—
116C2×D271—22no—
117C2×D271—24no—
118C2×D271—41no—
119C2×D271—42no—
120C2×D271—41no—
121C4×C271—22no—
122C4×C271—24no—
123C4×C271—42no—
124C4×C271#12314yes—
125Dic271—22no—
126Dic271—41no—
127Dic271—42no—
128Dic271—41no—
129Dic271—42no—
130C2×C271—42no—
131C2×C271C2×#12311yescharacteristic
132C2×C271#52511yescharacteristic
133C2×C271#52912yes—
134C2×C271#52912yes—
135C2×C271#53211yescharacteristic
136D271—42no—
137C271#239511yescharacteristic, complemented, Sylow
138#2395—2711nomaximal, Sylow
139C2×C4×D4—2711no—
140C2×#117—2711no—
141#400—2711no—
142#399—2711no—
143#398—2711no—
144#397—2712no—
145#397—2711no—
146#394—2712no—
147C4×C4⋊C4—2712no—
148C4×C22⋊C4—2712no—
149C4×C22⋊C4D27111yescharacteristic, complemented
150C22×D4—2711no—
151C23×C4—2711no—
152C23×C4C2×D27111yescharacteristic
153C4×D4—5424no—
154#117—5424no—
155C2×C4⋊C4—2711no—
156C2×C4⋊C4—2711no—
157C2×C4⋊C4—2711no—
158C2×C4⋊C4—2712no—
159C2×C4⋊C4—2712no—
160C2×C22⋊C4—2712no—
161C2×C22⋊C4—2711no—
162C2×C22⋊C4—2711no—
163C2×C22⋊C4—2712no—
164C2×C22⋊C4—2712no—
165C2×C22⋊C4C2×D27112yes—
166C2×C42—2712no—
167C2×C42—2711no—
168C2×C42C2×D27112yes—
169#95—2712no—
170#95—2712no—
171#95—2712no—
172#95—2712no—
173#95C2×D27111yescharacteristic
174#95C2×D27111yescharacteristic
175C24—2711no—
176C24C22×D27111yescharacteristic
177C2×D4—5422no—
178C2×D4—5422no—
179C2×D4—5422no—
180C22×C4—24no—
181C22×C4—2711no—
182C22×C4—2711no—
183C22×C4—2711no—
184C22×C4—2712no—
185C22×C4—2712no—
186C22×C4—5422no—
187C22×C4—5424no—
188C22×C4D108414yes—
189C22×C4C22×D27112yes—
190C22×C4C22×D27111yescharacteristic
191C22×C4C22×D27112yes—
192C22×C4C22×D27111yescharacteristic
193C4⋊C4—5424no—
194C4⋊C4—5422no—
195C4⋊C4—5422no—
196C4⋊C4—5422no—
197C4⋊C4—5424no—
198C22⋊C4—24no—
199C22⋊C4—5424no—
200C22⋊C4—5424no—
201C22⋊C4—5422no—
202C22⋊C4—5422no—
203C22⋊C4—5424no—
204C42—24no—
205C42—5424no—
206C42—5422no—
207C23—21no—
208C23—21no—
209C23—22no—
210C23—22no—
211C23—2712no—
212C23—5421no—
213C23—5421no—
214C23—5422no—
215C23—5422no—
216C23C2×D108412yes—
217C23C23×D27111yescharacteristic
218D4—5428no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—28no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—2712no—
227C2×C4—5421no—
228C2×C4—5421no—
229C2×C4—5421no—
230C2×C4—5422no—
231C2×C4—5422no—
232C2×C4—5421no—
233C2×C4—5422no—
234C2×C4—5422no—
235C2×C4—5422no—
236C2×C4—5422no—
237C2×C4—5422no—
238C2×C4—5422no—
239C2×C4—10844no—
240C2×C4C2×D108412yes—
241C2×C4C2×D108412yes—
242C2×C4#1686465314yes—
243C2×C4#1686465414yes—
244C22—22no—
245C22—22no—
246C22—22no—
247C22—42no—
248C22—42no—
249C22—5422no—
250C22—5424no—
251C22—10841no—
252C22—10842no—
253C22—10841no—
254C22C22×D108411yescharacteristic
255C22#41×D27111yescharacteristic
256C22#41×D27111yescharacteristic
257C22C2×#1686465312yes—
258C22C2×#1686465412yes—
259C4—22no—
260C4—24no—
261C4—42no—
262C4—5422no—
263C4—10841no—
264C4—10842no—
265C4—10841no—
266C4—10842no—
267C4#2257614414yes—
268C2—42no—
269C2—10842no—
270C2C2×#2257614411yescharacteristic
271C2#3177440911yescharacteristic
272C2#3177442712yes—
273C2#3177442912yes—
274C2#3177444711yescharacteristic
subgroupG/H
centerC22×C4C22×D271
commutator, socle, cosocleC23×C271C24
FrattiniC23C23×D271
FittingC4×C22⋊C4×C271C2
prime radicalC22×C22⋊D271C22
hypercenterC4×C22⋊C4D271
hypocenterC271#2395
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C22×C4 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#22576119C2semidirect
3. #31773837C2semidirect
4. #31773834C2semidirect
5. #31773829C2semidirect
6. #31773824C2semidirect
7. #31773822C2semidirect
9. C4×C4⋊Dic271C2semidirect
10. C4×(C2×D271)⋊C4C2semidirect
11. C4×C22⋊C4×C271C2semidirect
137. C271#2395semidirect
149. C4×C22⋊C4D271semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊D2712
C2×#225761192
#317738372
#317738342
#317738292
#317738242
#317738222
#317738132
C4×C4⋊Dic2712
C4×(C2×D271)⋊C42
C4×C22⋊C4×C2712
#2395271