Finite Group Databasebeta

← #46845213

Subgroups — #46845213

Order 34688.

528 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 175 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×D8×D271C211yesmaximal, characteristic
2#31774369C212yesmaximal, complemented
3#31774010C211yesmaximal, characteristic, complemented
4#31774009C211yesmaximal, characteristic, complemented
5#31774008C211yesmaximal, characteristic, complemented
6C4×#16864639C212yesmaximal, complemented
7#31773977C212yesmaximal, complemented
8#31773932C211yesmaximal, characteristic, complemented
9#31773900C212yesmaximal, complemented
10#31773854C211yesmaximal, characteristic, complemented
11#442×C271C211yesmaximal, characteristic, complemented
12C2×D4×D271C2212yes—
13C2×D2168C2211yescharacteristic
14C2×#16864639C2212yes—
15D8×D271C418yescomplemented
16C2×#16864637C2211yescharacteristic
17C2×C8×D271C2211yescharacteristic
18C4×D1084C2212yescomplemented
19D4×Dic271C2212yescomplemented
20#22576122C2212yescomplemented
21C4×C22⋊D271—22no—
22(C22⋊D271)⋊C4—22no—
23#22576118C2212yes—
24#22576117C2211yescharacteristic
25C2×(C2×D271)⋊C4—22no—
26C22⋊C4×D271—22no—
27#22576093C2212yescomplemented
28#22576092C2211yescharacteristic, complemented
29#22576082C2212yescomplemented
30#22576081C2212yescomplemented
31#22576080C2212yescomplemented
32#22576070C2211yescharacteristic, complemented
33C4×C271⋊C8C2211yescharacteristic, complemented
34C8×Dic271C2211yescharacteristic, complemented
35C2×D8×C271C2211yescharacteristic
36C4×D4×C271C2212yescomplemented
37#106×C271C2211yescharacteristic, complemented
38#102×C271C2212yescomplemented
39#97×C271C2211yescharacteristic, complemented
40C23×D271—22no—
41C2×D1084C2312yes—
42C2×C22⋊D271—22no—
43D4×D271—24no—
44D4×D271C2×C418yescomplemented
45C22×Dic271—22no—
46C2×C4×D271—22no—
47C2×C4×D271—22no—
48C2×C4×D271C2311yescharacteristic
49D2168C2×C414yescomplemented
50#16864639C2×C418yescomplemented
51#16864637—22no—
52C2×C271⋊C8C2311yescharacteristic
53C2×C271⋊C8C2311yescharacteristic
54C8×D271C2×C414yescomplemented
55C4⋊Dic271C2312yes—
56Dic271⋊C4D412yes—
57C22⋊Dic271—22no—
58(C2×D271)⋊C4—22no—
59(C2×D271)⋊C4—22no—
60(C2×D271)⋊C4—22no—
61(C2×D271)⋊C4D412yes—
62C4×Dic271—22no—
63C4×Dic271C2311yescharacteristic
64C2×D4×C271C2312yes—
65C22×C4×C271—22no—
66D8×C271C2×C414yescomplemented
67C2×C8×C271C2311yescharacteristic
68C2×C8×C271C2311yescharacteristic
69C4⋊C4×C271C2312yes—
70C22⋊C4×C271—22no—
71C42×C271C2311yescharacteristic
72C22×D271—22no—
73C22×D271—28no—
74C22×D271—82no—
75C22×D271—82no—
76C22×D271C2×D411yescharacteristic
77D1084—22no—
78D1084C22×C414yes—
79C22⋊D271—28no—
80C2×Dic271—22no—
81C2×Dic271—82no—
82C2×Dic271C2×D412yes—
83C2×Dic271C2×D411yescharacteristic
84C4×D271—82no—
85C4×D271—82no—
86C4×D271C22×C412yes—
87C4×D271#4112yes—
88C271⋊C8—21no—
89C271⋊C8C22×C412yes—
90C23×C271—22no—
91D4×C271—22no—
92D4×C271C22×C414yes—
93C2×C4×C271—22no—
94C2×C4×C271—82no—
95C2×C4×C271C2×D412yes—
96C2×C4×C271C2411yescharacteristic
97C8×C271—21no—
98C8×C271C22×C412yes—
99C22×C271—24no—
100C22×C271—42no—
101C22×C271—42no—
102C22×C271C22×D411yescharacteristic
103C2×D271—22no—
104C2×D271—24no—
105C2×D271—42no—
106C2×D271—42no—
107C2×D271—88no—
108C2×D271C4×D412yes—
109C4×C271—22no—
110C4×C271—42no—
111C4×C271C23×C411yescharacteristic
112C4×C271C2×#4111yescharacteristic
113Dic271—22no—
114Dic271—42no—
115Dic271C4×D412yes—
116C2×C271—44no—
117C2×C271C2×C4×D411yescharacteristic
118C2×C271#57612yes—
119D271—22no—
120D271—44no—
121C271#294811yescharacteristic, complemented, Sylow
122#2948—2711nomaximal, Sylow
123C22×D8—2711no—
124#518—2712no—
125#442—2711no—
126#442—2711no—
127#442—2711no—
128#442D27111yescharacteristic, complemented
129C4×D8—2712no—
130C4×D8—2712no—
131#427—2711no—
132#418—2712no—
133#405—2711no—
134C22×D4—2712no—
135C2×D8—2712no—
136C2×D8—2718no—
137C2×D8—2711no—
138C2×D8C2×D27111yescharacteristic
139C2×M16—2711no—
140C22×C8—2711no—
141C4×D4—2712no—
142C4×D4—2712no—
143C4×D4—2712no—
144C4×D4—5422no—
145C4×D4—5422no—
146C4×D4C2×D27112yescomplemented
147#117—2711no—
148#117—2712no—
149C2×C22⋊C4—5422no—
150C2×C22⋊C4—5422no—
151#107—2712no—
152#106—2711no—
153#106C2×D27111yescharacteristic, complemented
154#102—2712no—
155#102—2712no—
156#102—2712no—
157#102C2×D27112yescomplemented
158#97—2711no—
159#97C2×D27111yescharacteristic, complemented
160C4×C8—2711no—
161C4×C8—2711no—
162C24—5422no—
163C2×D4—2718no—
164C2×D4—2712no—
165C2×D4—5424no—
166C2×D4—5422no—
167C2×D4C22×D27112yes—
168C22×C4—22no—
169C22×C4—2711no—
170C22×C4—5422no—
171C22×C4—5422no—
172C22×C4—5422no—
173D8—2718no—
174D8—2714no—
175D8C4×D27114yescomplemented
176M16—5422no—
177C2×C8—2711no—
178C2×C8—2714no—
179C2×C8—2711no—
180C2×C8C22×D27111yescharacteristic
181C2×C8C22×D27111yescharacteristic
182C4⋊C4—2712no—
183C4⋊C4—2712no—
184C4⋊C4C22×D27112yes—
185C22⋊C4—22no—
186C22⋊C4—2712no—
187C22⋊C4—5422no—
188C22⋊C4—5422no—
189C22⋊C4—5422no—
190C22⋊C4—5422no—
191C42—2711no—
192C42—5422no—
193C42C22×D27111yescharacteristic
194C23—22no—
195C23—2711no—
196C23—5428no—
197C23—5422no—
198C23—21682no—
199C23—21682no—
200D4—22no—
201D4—2714no—
202D4—5428no—
203D4—5422no—
204D4C2×C4×D27114yes—
205C2×C4—22no—
206C2×C4—82no—
207C2×C4—2711no—
208C2×C4—2712no—
209C2×C4—2712no—
210C2×C4—2712no—
211C2×C4—5422no—
212C2×C4—21682no—
213C2×C4—21682no—
214C2×C4—21682no—
215C2×C4D4×D27112yes—
216C2×C4C23×D27111yescharacteristic
217C8—21no—
218C8—2712no—
219C8—5421no—
220C8C2×C4×D27112yes—
221C22—24no—
222C22—42no—
223C22—42no—
224C22—2712no—
225C22—5422no—
226C22—5424no—
227C22—10842no—
228C22—10842no—
229C22—21688no—
230C22C2×D4×D27111yescharacteristic
231C4—22no—
232C4—42no—
233C4—2712no—
234C4—5422no—
235C4—10842no—
236C4C22×C4×D27111yescharacteristic
237C4#41×D27111yescharacteristic
238C2—44no—
239C2—5422no—
240C2—10844no—
241C2C4×D4×D27111yescharacteristic
242C2#3177470112yes—
subgroupG/H
centerC22C2×D4×D271
commutatorC4×C271C23×C4
FrattiniC2×C4C23×D271
Fitting#442×C271C2
socleC22×C271C22×D4
prime radicalC2×D8×D271C2
cosocleC2×C4×C271C24
hypercenter#442D271
hypocenterC271#2948
LCSG ▷ C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
2. #31774369C2semidirect
3. #31774010C2semidirect
4. #31774009C2semidirect
5. #31774008C2semidirect
6. C4×#16864639C2semidirect
7. #31773977C2semidirect
8. #31773932C2semidirect
9. #31773900C2semidirect
10. #31773854C2semidirect
11. #442×C271C2semidirect
15. D8×D271C4semidirect
18. C4×D1084C22semidirect
19. D4×Dic271C22semidirect
20. #22576122C22semidirect
27. #22576093C22semidirect
28. #22576092C22semidirect
29. #22576082C22semidirect
30. #22576081C22semidirect
31. #22576080C22semidirect
32. #22576070C22semidirect
33. C4×C271⋊C8C22semidirect
34. C8×Dic271C22semidirect
36. C4×D4×C271C22semidirect
37. #106×C271C22semidirect
38. #102×C271C22semidirect
39. #97×C271C22semidirect
44. D4×D271C2×C4semidirect
49. D2168C2×C4semidirect
50. #16864639C2×C4semidirect
54. C8×D271C2×C4semidirect
66. D8×C271C2×C4semidirect
121. C271#2948semidirect
128. #442D271semidirect
146. C4×D4C2×D271semidirect
153. #106C2×D271semidirect
157. #102C2×D271semidirect
159. #97C2×D271semidirect
175. D8C4×D271semidirect

Maximal subgroups

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

subgroupindex
C2×D8×D2712
#317743692
#317740102
#317740092
#317740082
C4×#168646392
#317739772
#317739322
#317739002
#317738542
#442×C2712
#2948271