Finite Group Databasebeta

← #46846747

Subgroups — #46846747

Order 34688.

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

normal subgroups only

This group has 400 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#31774453C211yesmaximal, characteristic, complemented
2#124×D271C211yesmaximal, characteristic
3#31774258C212yesmaximal, complemented
4#31774255C212yesmaximal, complemented
5#31774254C211yesmaximal, characteristic, complemented
6#31774252C211yesmaximal, characteristic, complemented
7#31774251C211yesmaximal, characteristic, complemented
8#31773904C212yesmaximal, complemented
9#31773900C212yesmaximal
10C4×#16864637C211yesmaximal, characteristic, complemented
11#488×C271C211yesmaximal, characteristic, complemented
12C2×Q8×D271C2211yescharacteristic
13C2×D4×D271C2211yescharacteristic
14C2×#16864637C2212yes—
15#22576159C2212yescomplemented
16#22576152C2211yescharacteristic, complemented
17#22576149C2211yescharacteristic
18#22576148C2211yescharacteristic
19#22576147C2211yescharacteristic
20#22576146—22no—
21#22576138—22no—
22#22576125—22no—
23#22576118C2212yes—
24C22⋊C4×D271—22no—
25C42×D271C2211yescharacteristic
26#22576085C2212yescomplemented
27#22576084C2212yescomplemented
28#22576083C2212yescomplemented
29#22576082C2212yes—
30#22576081C2212yes—
31#22576080C2212yes—
32#22576070C2212yescomplemented
33C4×C271⋊C8C2211yescharacteristic, complemented
34#125×C271C2211yescharacteristic, complemented
35#124×C271C2211yescharacteristic
36#103×C271C2212yescomplemented
37#102×C271C2212yes—
38C4×C8×C271C2211yescharacteristic, complemented
39C23×D271—21no—
40C2×Dic542—22no—
41C2×Dic542—21no—
42C2×Dic542C2311yescharacteristic
43Q8×D271—24no—
44C2×D1084C2311yescharacteristic
45C2×C22⋊D271—21no—
46D4×D271—24no—
47C2×C4×D271—21no—
48C2×C4×D271D412yes—
49C2×C4×D271C2311yescharacteristic
50#16864637—24no—
51C2×C271⋊C8C2312yes—
52C4⋊Dic271C2312yes—
53Dic271⋊C4—22no—
54Dic271⋊C4—22no—
55C22⋊Dic271—22no—
56(C2×D271)⋊C4—22no—
57(C2×D271)⋊C4—22no—
58(C2×D271)⋊C4—22no—
59C4×Dic271—22no—
60C4×Dic271D412yes—
61C4×Dic271C2311yescharacteristic
62C2×Q8×C271C2311yescharacteristic
63C2×D4×C271C2311yescharacteristic
64C2×C8×C271C2312yes—
65C4⋊C4×C271—22no—
66C4⋊C4×C271C2312yes—
67C22⋊C4×C271—22no—
68C42×C271C2311yescharacteristic
69C22×D271—21no—
70C22×D271—44no—
71C22×D271—42no—
72C22×D271C2×D411yescharacteristic
73Dic542—22no—
74Dic542—44no—
75Dic542—42no—
76D1084—22no—
77C22⋊D271—42no—
78C2×Dic271—22no—
79C2×Dic271—21no—
80C2×Dic271C2×D412yes—
81C2×Dic271C2×D411yescharacteristic
82C4×D271—42no—
83C4×D271—42no—
84C4×D271#4114yes—
85C271⋊C8—22no—
86C23×C271—21no—
87Q8×C271—22no—
88D4×C271—22no—
89C2×C4×C271—21no—
90C2×C4×C271—22no—
91C2×C4×C271C2×D412yes—
92C2×C4×C271C2411yescharacteristic
93C8×C271—22no—
94C22×C271—41no—
95C22×C271—42no—
96C22×C271C22×D411yescharacteristic
97C2×D271—22no—
98C2×D271—41no—
99C2×D271—42no—
100C2×D271—84no—
101C2×D271#12412yes—
102C4×C271—22no—
103C4×C271—41no—
104C4×C271—42no—
105C4×C271C2×#4112yes—
106Dic271—22no—
107Dic271—41no—
108Dic271—42no—
109Dic271#12412yes—
110C2×C271—81no—
111C2×C271C2×#12411yescharacteristic
112C2×C271#57512yes—
113D271—22no—
114D271—81no—
115C271#314011yescharacteristic, complemented, Sylow
116#3140—2711nomaximal, Sylow
117#533—2711no—
118C2×#124—2711no—
119#489—2712no—
120#489—2712no—
121#488—2711no—
122#488—2711no—
123#488—2711no—
124#488D27111yescharacteristic, complemented
125#419—2712no—
126#418—2712no—
127C4×M16—2711no—
128C22×Q8—2711no—
129C22×D4—2711no—
130C2×M16—2712no—
131C4⋊Q8—2712no—
132#125—2711no—
133#125C2×D27111yescharacteristic, complemented
134#124—2711no—
135#124—2711no—
136#124—2711no—
137#124—5422no—
138#124C2×D27111yescharacteristic
139C22⋊Q8—5422no—
140C4×Q8—5422no—
141#117—2712no—
142C2×C22⋊C4—5422no—
143C2×C42—2711no—
144#103—2712no—
145#103—2712no—
146#103—2712no—
147#103C2×D27112yescomplemented
148#102—2712no—
149#102—2712no—
150#102—2712no—
151#102C2×D27112yes—
152#97—2712no—
153C4×C8—2711no—
154C4×C8C2×D27111yescharacteristic, complemented
155C24—5421no—
156C2×Q8—2711no—
157C2×Q8—5422no—
158C2×Q8—5421no—
159C2×Q8—5424no—
160C2×Q8C22×D27111yescharacteristic
161C2×D4—2711no—
162C2×D4—5424no—
163C2×D4—5421no—
164C2×D4C22×D27111yescharacteristic
165C22×C4—2711no—
166C22×C4—2712no—
167C22×C4—5421no—
168M16—5424no—
169C2×C8—2712no—
170C2×C8C22×D27112yes—
171C4⋊C4—22no—
172C4⋊C4—2712no—
173C4⋊C4—5422no—
174C4⋊C4—5422no—
175C4⋊C4C22×D27112yes—
176C22⋊C4—22no—
177C22⋊C4—5422no—
178C22⋊C4—5422no—
179C22⋊C4—5422no—
180C22⋊C4—5422no—
181C42—2712no—
182C42—2711no—
183C42—5422no—
184C42C22×D27111yescharacteristic
185C23—21no—
186C23—2711no—
187C23—5421no—
188C23—10842no—
189C23—10844no—
190Q8—22no—
191Q8—5422no—
192Q8—10842no—
193Q8—10844no—
194D4—22no—
195D4—5422no—
196D4—10842no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—2711no—
200C2×C4—2714no—
201C2×C4—2712no—
202C2×C4—5422no—
203C2×C4—5421no—
204C2×C4—10842no—
205C2×C4—10842no—
206C2×C4D4×D27112yes—
207C2×C4C23×D27111yescharacteristic
208C8—22no—
209C8—5422no—
210C22—41no—
211C22—42no—
212C22—2712no—
213C22—5422no—
214C22—10841no—
215C22—10842no—
216C22—21684no—
217C22C2×D4×D27111yescharacteristic
218C4—22no—
219C4—41no—
220C4—42no—
221C4—2712no—
222C4—5422no—
223C4—10841no—
224C4—10842no—
225C4#41×D27112yes—
226C2—81no—
227C2—5422no—
228C2—21681no—
229C2#124×D27111yescharacteristic
230C2#3177469412yes—
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#488×C271C2
socleC22×C271C22×D4
prime radicalC2×D4×D271C22
hypercenter#488D271
hypocenterC271#3140
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
1. #31774453C2semidirect
3. #31774258C2semidirect
4. #31774255C2semidirect
5. #31774254C2semidirect
6. #31774252C2semidirect
7. #31774251C2semidirect
8. #31773904C2semidirect
10. C4×#16864637C2semidirect
11. #488×C271C2semidirect
15. #22576159C22semidirect
16. #22576152C22semidirect
26. #22576085C22semidirect
27. #22576084C22semidirect
28. #22576083C22semidirect
32. #22576070C22semidirect
33. C4×C271⋊C8C22semidirect
34. #125×C271C22semidirect
36. #103×C271C22semidirect
38. C4×C8×C271C22semidirect
115. C271#3140semidirect
124. #488D271semidirect
133. #125C2×D271semidirect
147. #103C2×D271semidirect
154. C4×C8C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317744532
#124×D2712
#317742582
#317742552
#317742542
#317742522
#317742512
#317739042
#317739002
C4×#168646372
#488×C2712
#3140271