Finite Group Databasebeta

← #46849086

Subgroups — #46849086

Order 34688.

372 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 117 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#31774621C211yesmaximal, characteristic, complemented
2D4×Dic542C211yesmaximal, characteristic, complemented
3#31774258C212yesmaximal, complemented
4#31774247C211yesmaximal, characteristic, complemented
5#31774190C212yesmaximal, complemented
6#31774178C211yesmaximal, characteristic, complemented
7#31774114C212yesmaximal
8#31774024C211yesmaximal, characteristic, complemented
9#31773998C212yesmaximal, complemented
10C4×#16864639C211yesmaximal, characteristic
11#461×C271C211yesmaximal, characteristic, complemented
12C22×Dic542—21no—
13C2×#16864639C2211yescharacteristic
14#22576160C2211yescharacteristic
15#22576159C2212yes—
16#22576155—22no—
17#22576149C2212yes—
18#22576140—21no—
19#22576139—21no—
20#22576137—22no—
21Q8×Dic271C2212yes—
22C4×Dic542C2211yescharacteristic
23C4×D1084C2211yescharacteristic
24D4×Dic271C2212yes—
25#22576118—21no—
26#22576094C2212yes—
27#22576093C2211yescharacteristic
28Dic271⋊C8C2212yes—
29#22576084C2212yes—
30#22576084C2212yes—
31#22576082C2212yes—
32#22576081C2211yescharacteristic
33#22576080C2211yescharacteristic
34#22576070C2212yes—
35C4×C271⋊C8C2211yescharacteristic
36C2×SD16×C271C2212yes—
37C4⋊Q8×C271C2211yescharacteristic
38C4×D4×C271C2211yescharacteristic
39C4⋊C8×C271C2211yescharacteristic
40#103×C271C2212yes—
41C2×Dic542—22no—
42C2×Dic542—21no—
43C2×Dic542—44no—
44C2×Dic542D411yescharacteristic
45C2×D1084C2311yescharacteristic
46C22×Dic271—22no—
47C2×C4×D271—21no—
48#16864639—22no—
49C2×C271⋊C8C2311yescharacteristic
50C2×C271⋊C8C2311yescharacteristic
51C4⋊Dic271—22no—
52C4⋊Dic271—21no—
53C4⋊Dic271D411yescharacteristic
54C4⋊Dic271C2312yes—
55C4⋊Dic271C2311yescharacteristic
56Dic271⋊C4—22no—
57Dic271⋊C4D412yes—
58C22⋊Dic271—22no—
59(C2×D271)⋊C4—22no—
60(C2×D271)⋊C4—22no—
61(C2×D271)⋊C4—21no—
62C4×Dic271—22no—
63C4×Dic271C2312yes—
64C2×Q8×C271C2312yes—
65C2×D4×C271C2311yescharacteristic
66C22×C4×C271—21no—
67SD16×C271—24no—
68C2×C8×C271C2312yes—
69C4⋊C4×C271—21no—
70C4⋊C4×C271—21no—
71C4⋊C4×C271C2311yescharacteristic
72C22⋊C4×C271—21no—
73C42×C271C2311yescharacteristic
74C22×D271—21no—
75Dic542—22no—
76Dic542—44no—
77Dic542—42no—
78D1084—21no—
79D1084—21no—
80C2×Dic271—21no—
81C2×Dic271—22no—
82C2×Dic271—44no—
83C2×Dic271C2×D412yes—
84C2×Dic271C2×D412yes—
85C4×D271—42no—
86C271⋊C8—21no—
87C271⋊C8—21no—
88C23×C271—21no—
89Q8×C271—22no—
90Q8×C271—22no—
91D4×C271—21no—
92D4×C271#4112yes—
93C2×C4×C271—21no—
94C2×C4×C271—22no—
95C2×C4×C271—42no—
96C2×C4×C271C2×D411yescharacteristic
97C2×C4×C271C2×D411yescharacteristic
98C2×C4×C271C2411yescharacteristic
99C8×C271—22no—
100C22×C271—22no—
101C22×C271—41no—
102C22×C271—41no—
103C22×C271C22×D411yescharacteristic
104C2×D271—41no—
105C2×D271—41no—
106C2×D271—41no—
107C4×C271—21no—
108C4×C271—42no—
109C4×C271—41no—
110C4×C271AGL(1,ℤ/8)12yes—
111C4×C271C2×#4111yescharacteristic
112C4×C2712−1+411yescharacteristic
113Dic271—22no—
114Dic271—22no—
115Dic271—41no—
116Dic271—42no—
117C2×C271—42no—
118C2×C271#54811yescharacteristic
119C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
120C2×C271#57811yescharacteristic
121D271—81no—
122C271#336711yescharacteristic, complemented, Sylow
123#3367—2711nomaximal, Sylow
124#556—2711no—
125D4×Q8—2711no—
126#489—2712no—
127#487—2711no—
128#477—2712no—
129#474—2711no—
130#463—2712no—
131#461D27111yescharacteristic, complemented
132#446—2711no—
133#440—2712no—
134C4×D8—2711no—
135C22×Q8—5421no—
136C2×SD16C2×D27112yes—
137C2×D8—2711no—
138C4⋊Q8—2712no—
139C4⋊Q8—2711no—
140C4⋊Q8C2×D27111yescharacteristic
141#126—5422no—
142#124—2712no—
143C22⋊Q8—5422no—
144C22⋊Q8—5421no—
145C22⋊Q8—5421no—
146C4×Q8—2711no—
147C4×Q8—2712no—
148C4×D4—2712no—
149C4×D4—2711no—
150C4×D4C2×D27111yescharacteristic
151#117—5421no—
152#107—2712no—
153#107—2711no—
154C4⋊C8—2712no—
155C4⋊C8C2×D27111yescharacteristic
156#103—2712no—
157#103—2712no—
158#103C2×D27112yes—
159#102—2711no—
160#102—2712no—
161#102—2711no—
162#97—2712no—
163C4×C8—2711no—
164C2×Q8—2711no—
165C2×Q8—5422no—
166C2×Q8—5421no—
167C2×Q8—10844no—
168C2×Q8C22×D27112yes—
169C2×D4—2711no—
170C2×D4C22×D27111yescharacteristic
171C22×C4—21no—
172C22×C4—5422no—
173C22×C4—5421no—
174SD16—24no—
175D8—5422no—
176C2×C8—2711no—
177C2×C8—2711no—
178C2×C8C22×D27112yes—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—2711no—
182C4⋊C4—2712no—
183C4⋊C4—2712no—
184C4⋊C4—2711no—
185C4⋊C4—5422no—
186C4⋊C4—5422no—
187C4⋊C4—5421no—
188C4⋊C4C22×D27111yescharacteristic
189C22⋊C4—21no—
190C22⋊C4—5422no—
191C22⋊C4—5422no—
192C22⋊C4—5422no—
193C22⋊C4—5421no—
194C42—2712no—
195C42—5422no—
196C42C22×D27111yescharacteristic
197C23—21no—
198C23—5421no—
199Q8—22no—
200Q8—22no—
201Q8—5422no—
202Q8—10844no—
203Q8—10842no—
204D4—21no—
205D4—5421no—
206D4—5421no—
207D4#1686465312yes—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—42no—
211C2×C4—2712no—
212C2×C4—2712no—
213C2×C4—5421no—
214C2×C4—5422no—
215C2×C4—10844no—
216C2×C4—10842no—
217C2×C4D4×D27111yescharacteristic
218C2×C4D4×D27111yescharacteristic
219C2×C4C23×D27111yescharacteristic
220C8—22no—
221C8—5421no—
222C8—5421no—
223C22—22no—
224C22—41no—
225C22—41no—
226C22—10841no—
227C22—10841no—
228C22—10841no—
229C22C2×D4×D27111yescharacteristic
230C4—21no—
231C4—42no—
232C4—41no—
233C4—5422no—
234C4—5422no—
235C4—10842no—
236C4—10841no—
237C4#2257618712yes—
238C4C2×#1686465311yescharacteristic
239C4#2257620711yescharacteristic
240C2—42no—
241C2—21681no—
242C2#3177455311yescharacteristic
243C2C2×#2257618711yescharacteristic
244C2#3177471311yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#461×C271C2
socleC22×C271C22×D4
prime radicalC2×#16864639C22
hypercenter#461D271
hypocenterC271#3367
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #31774621C2semidirect
2. D4×Dic542C2semidirect
3. #31774258C2semidirect
4. #31774247C2semidirect
5. #31774190C2semidirect
6. #31774178C2semidirect
8. #31774024C2semidirect
9. #31773998C2semidirect
11. #461×C271C2semidirect
122. C271#3367semidirect
131. #461D271semidirect

Maximal subgroups

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

subgroupindex
#317746212
D4×Dic5422
#317742582
#317742472
#317741902
#317741782
#317741142
#317740242
#317739982
C4×#168646392
#461×C2712
#3367271