Finite Group Databasebeta

← #46846616

Subgroups — #46846616

Order 34688.

394 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 125 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#31774509C211yesmaximal, characteristic, complemented
2#31774238C212yesmaximal, complemented
3#31774197C212yesmaximal, complemented
4#31774102C212yesmaximal, complemented
5#31774048C212yesmaximal
6#31773997C212yesmaximal, complemented
7C4×#16864642C212yesmaximal, complemented
8#31773958C211yesmaximal, characteristic, complemented
9#518×C271C211yesmaximal, characteristic
10C2×#16864642C2212yes—
11C2×#16864642C2212yes—
12#22576160C2211yescharacteristic, complemented
13#22576153—22no—
14#22576152C2211yescharacteristic, complemented
15#22576150—22no—
16#22576139C2212yes—
17C4×Dic542C2212yescomplemented
18#22576093C2212yescomplemented
19#22576091C2212yescomplemented
20C4⋊(C271⋊C8)C2212yescomplemented
21#22576083C2212yescomplemented
22#22576083C2212yescomplemented
23#22576081C2212yes—
24#22576081C2212yes—
25C22⋊(C271⋊C8)C2212yes—
26#22576071C2211yescharacteristic, complemented
27C4×C271⋊C8C2211yescharacteristic, complemented
28C22×D4×C271C2211yescharacteristic
29C4×D4×C271—22no—
30C4×D4×C271C2212yes—
31C4×D4×C271C2212yes—
32#117×C271C2211yescharacteristic
33#117×C271C2211yescharacteristic
34C2×C22⋊C4×C271—22no—
35C2×Dic542C2312yes—
36#16864642—24no—
37#16864642—24no—
38C2×C271⋊C8C2312yes—
39C2×C271⋊C8C2312yes—
40C4⋊Dic271—22no—
41C4⋊Dic271C2312yes—
42Dic271⋊C4—22no—
43Dic271⋊C4—22no—
44Dic271⋊C4—22no—
45C22⋊Dic271—22no—
46C4×Dic271—22no—
47C24×C271—21no—
48C2×D4×C271—24no—
49C2×D4×C271—21no—
50C2×D4×C271C2312yes—
51C22×C4×C271—22no—
52C22×C4×C271—22no—
53C22×C4×C271C2311yescharacteristic
54C4⋊C4×C271D412yes—
55C4⋊C4×C271C2312yes—
56C22⋊C4×C271—21no—
57C22⋊C4×C271—22no—
58C22⋊C4×C271—22no—
59C22⋊C4×C271D412yes—
60C42×C271—21no—
61C42×C271C2311yescharacteristic
62C42×C271C2311yescharacteristic
63Dic542—22no—
64Dic542—22no—
65C2×Dic271—22no—
66C2×Dic271—22no—
67C271⋊C8—22no—
68C271⋊C8—22no—
69C23×C271—24no—
70C23×C271—22no—
71C23×C271—82no—
72C23×C271C2×D411yescharacteristic
73D4×C271—24no—
74D4×C271—22no—
75D4×C271#4114yes—
76C2×C4×C271—22no—
77C2×C4×C271—22no—
78C2×C4×C271—82no—
79C2×C4×C271—82no—
80C2×C4×C271C2×D411yescharacteristic
81C2×C4×C271C2×D412yes—
82C2×C4×C271C2×D412yes—
83C2×C4×C271C2411yescharacteristic
84C22×C271—21no—
85C22×C271—24no—
86C22×C271—22no—
87C22×C271—42no—
88C22×C271—42no—
89C22×C271—84no—
90C22×C271C22×D411yescharacteristic
91C4×C271—21no—
92C4×C271—22no—
93C4×C271—22no—
94C4×C271—42no—
95C4×C271C2×#4112yes—
96Dic271—42no—
97Dic271—42no—
98C2×C271—41no—
99C2×C271—44no—
100C2×C271#52511yescharacteristic
101C2×C271#57712yes—
102C271#312311yescharacteristic, complemented, Sylow
103#3123—2711nomaximal, Sylow
104#541—2711no—
105#518D27111yescharacteristic
106#485—2712no—
107#478—2712no—
108#461—2712no—
109#450—2712no—
110#440—2712no—
111C4×SD16—2712no—
112#433—2711no—
113C22×D4C2×D27111yescharacteristic
114C2×SD16—2712no—
115C2×SD16—2712no—
116C4⋊Q8—2711no—
117#126—5422no—
118#125—2711no—
119#125—5422no—
120C22⋊Q8—2712no—
121C4×Q8—2712no—
122C4×D4—22no—
123C4×D4C2×D27112yes—
124C4×D4C2×D27112yes—
125#117C2×D27111yescharacteristic
126#117C2×D27111yescharacteristic
127C2×C22⋊C4—22no—
128#107—2712no—
129#106—2712no—
130C4⋊C8—2712no—
131#103—2712no—
132#103—2712no—
133#102—2712no—
134#102—2712no—
135C22⋊C8—2712no—
136#97—2711no—
137C4×C8—2711no—
138C24—21no—
139C2×Q8—2712no—
140C2×D4—24no—
141C2×D4—21no—
142C2×D4C22×D27112yes—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4C22×D27111yescharacteristic
146SD16—5424no—
147SD16—5424no—
148C2×C8—2712no—
149C2×C8—2712no—
150C4⋊C4—2712no—
151C4⋊C4—5422no—
152C4⋊C4—5422no—
153C4⋊C4—5422no—
154C4⋊C4—5422no—
155C4⋊C4C22⋊D27112yes—
156C4⋊C4C22×D27112yes—
157C22⋊C4—21no—
158C22⋊C4—22no—
159C22⋊C4—22no—
160C22⋊C4—5422no—
161C22⋊C4C22⋊D27112yes—
162C42—21no—
163C42—5422no—
164C42C22×D27111yescharacteristic
165C42C22×D27111yescharacteristic
166C23—24no—
167C23—22no—
168C23—82no—
169C23C2×C22⋊D27111yescharacteristic
170Q8—5422no—
171Q8—5422no—
172D4—24no—
173D4—22no—
174D4#1686465414yes—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—82no—
178C2×C4—82no—
179C2×C4—5422no—
180C2×C4—5422no—
181C2×C4C2×C22⋊D27111yescharacteristic
182C2×C4C2×C22⋊D27112yes—
183C2×C4C2×C22⋊D27112yes—
184C2×C4C23×D27111yescharacteristic
185C8—5422no—
186C8—5422no—
187C22—21no—
188C22—24no—
189C22—22no—
190C22—42no—
191C22—42no—
192C22—84no—
193C22C22×C22⋊D27111yescharacteristic
194C4—21no—
195C4—22no—
196C4—22no—
197C4—42no—
198C4—10842no—
199C4—10842no—
200C4C2×#1686465412yes—
201C2—41no—
202C2—44no—
203C2#3177440811yescharacteristic
204C2#3177470812yes—
subgroupG/H
centerC22C22×C22⋊D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#518×C271C2
socleC22×C271C22×D4
prime radicalC22×D4×C271C22
hypercenter#518D271
hypocenterC271#3123
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ #117 ◁ #518

Complements

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

NH ≅ G/Ntype
1. #31774509C2semidirect
2. #31774238C2semidirect
3. #31774197C2semidirect
4. #31774102C2semidirect
6. #31773997C2semidirect
7. C4×#16864642C2semidirect
8. #31773958C2semidirect
12. #22576160C22semidirect
14. #22576152C22semidirect
17. C4×Dic542C22semidirect
18. #22576093C22semidirect
19. #22576091C22semidirect
20. C4⋊(C271⋊C8)C22semidirect
21. #22576083C22semidirect
22. #22576083C22semidirect
26. #22576071C22semidirect
27. C4×C271⋊C8C22semidirect
102. C271#3123semidirect

Maximal subgroups

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

subgroupindex
#317745092
#317742382
#317741972
#317741022
#317740482
#317739972
C4×#168646422
#317739582
#518×C2712
#3123271