Finite Group Databasebeta

← #46838019

Subgroups — #46838019

Order 34688.

234 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 65 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#31774468C211yesmaximal, characteristic
2#31773871C211yesmaximal, characteristic
3#31773688C214yesmaximal, complemented
4#413×C271C211yesmaximal, characteristic
5C22×Dic542C411yescharacteristic, complemented
6#22576158—21no—
7#22576147—21no—
8#22576141C412yescomplemented
9#22576141C2212yes—
10#22576139—21no—
11#22576139—21no—
12#22576119—21no—
13#22576119C411yescharacteristic, complemented
14#22576075—21no—
15#22576075C2212yes—
16C2×#41×C271C2211yescharacteristic
17C2×M16×C271—21no—
18#101×C271—21no—
19#100×C271C2212yes—
20C2×Dic542—21no—
21C2×Dic542—21no—
22C2×Dic542—41no—
23C2×Dic542—41no—
24C22×Dic271C2×C412yes—
25C4⋊Dic271—21no—
26C4⋊Dic271—41no—
27Dic271⋊C4—22no—
28Dic271⋊C4—22no—
29C22⋊Dic271—22no—
30C22⋊Dic271—22no—
31C22⋊Dic271—22no—
32C22⋊Dic271C2×C412yes—
33C4×Dic271—21no—
34C4×Dic271—41no—
35#41×C271—41no—
36#41×C271—41no—
37C2×Q8×C271D411yescharacteristic
38C2×D4×C271C2×C411yescharacteristic
39C2×D4×C271D411yescharacteristic
40C2×D4×C271C2311yescharacteristic
41C22×C4×C271C2×C411yescharacteristic
42C22×C4×C271D411yescharacteristic
43C22×C4×C271D411yescharacteristic
44M16×C271—22no—
45M16×C271—41no—
46C2×C8×C271—41no—
47Dic542—42no—
48Dic542—41no—
49Dic542—41no—
50C2×Dic271—22no—
51C2×Dic271—22no—
52C2×Dic271—42no—
53C2×Dic271—42no—
54C23×C271C22⋊C411yescharacteristic
55C23×C271C22×C411yescharacteristic
56C23×C271C2×D411yescharacteristic
57Q8×C271—41no—
58D4×C271—41no—
59D4×C271—41no—
60D4×C271—41no—
61C2×C4×C271—21no—
62C2×C4×C271—21no—
63C2×C4×C271—21no—
64C2×C4×C271—21no—
65C2×C4×C271—21no—
66C2×C4×C271—21no—
67C2×C4×C271C22⋊C411yescharacteristic
68C2×C4×C271C22⋊C411yescharacteristic
69C2×C4×C271C22⋊C411yescharacteristic
70C2×C4×C271C2×D411yescharacteristic
71C8×C271—42no—
72C22×C271—21no—
73C22×C271—21no—
74C22×C271—21no—
75C22×C271—41no—
76C22×C271—41no—
77C22×C271—41no—
78C22×C271C2×C22⋊C411yescharacteristic
79C4×C271—21no—
80C4×C271—21no—
81C4×C271—21no—
82C4×C271#9911yescharacteristic
83C4×C271#9911yescharacteristic
84Dic271—42no—
85Dic271—42no—
86Dic271—82no—
87C2×C271—21no—
88C2×C271—41no—
89C2×C271—41no—
90C2×C271—41no—
91C2×C271C2×#9911yescharacteristic
92C271#212311yescharacteristic, complemented, Sylow
93#2123—2711nomaximal, Sylow
94#536—2711no—
95#413D27111yescharacteristic
96#410—2711no—
97#352—2714no—
98C2×#41C2×D27111yescharacteristic
99C22×Q8—2711no—
100C2×M16—21no—
101C4⋊Q8—5421no—
102#124—5421no—
103#123—2712no—
104#123—2712no—
105C22⋊Q8—5421no—
106C22⋊Q8—5421no—
107#117—2711no—
108#117—5421no—
109#101—21no—
110#100C2×D27112yes—
111#99—2712no—
112#99—5421no—
113#41—41no—
114#41—41no—
115C2×Q8—5421no—
116C2×Q8—5421no—
117C2×Q8—10841no—
118C2×Q8—10841no—
119C2×Q8C22⋊D27111yescharacteristic
120C2×D4C4×D27111yescharacteristic
121C2×D4D108411yescharacteristic
122C2×D4C22×D27111yescharacteristic
123C22×C4—2712no—
124C22×C4C4×D27111yescharacteristic
125C22×C4C22⋊D27111yescharacteristic
126C22×C4D108411yescharacteristic
127M16—22no—
128M16—41no—
129C2×C8—41no—
130C4⋊C4—5422no—
131C4⋊C4—5421no—
132C4⋊C4—5422no—
133C4⋊C4—10841no—
134C22⋊C4—2712no—
135C22⋊C4—5422no—
136C22⋊C4—5422no—
137C22⋊C4—5422no—
138C42—5421no—
139C42—10841no—
140C23(C2×D271)⋊C411yescharacteristic
141C23C2×C4×D27111yescharacteristic
142C23C2×D108411yescharacteristic
143Q8—41no—
144Q8—10842no—
145Q8—10841no—
146Q8—10841no—
147D4—41no—
148D4—41no—
149D4—41no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—5422no—
157C2×C4—5422no—
158C2×C4—10842no—
159C2×C4—10842no—
160C2×C4(C2×D271)⋊C411yescharacteristic
161C2×C4(C2×D271)⋊C411yescharacteristic
162C2×C4(C2×D271)⋊C411yescharacteristic
163C2×C4C2×C22⋊D27111yescharacteristic
164C8—42no—
165C22—21no—
166C22—21no—
167C22—21no—
168C22—41no—
169C22—41no—
170C22—41no—
171C22C2×(C2×D271)⋊C411yescharacteristic
172C4—21no—
173C4—21no—
174C4—21no—
175C4—10842no—
176C4—10842no—
177C4—21682no—
178C4#2257607411yescharacteristic
179C4#2257607411yescharacteristic
180C2—21no—
181C2—41no—
182C2—41no—
183C2—41no—
184C2C2×#2257607411yescharacteristic
subgroupG/H
centerC2C2×#22576074
commutatorC23×C271C22×C4
FrattiniC2×D4C22×D271
Fitting#413×C271C2
socleC2×C271C2×#99
prime radicalC2×#41×C271C22
cosocleC2×D4×C271C23
hypercenter#413D271
hypocenterC271#2123
LCSG ▷ C23×C271 ▷ C22×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #413

Complements

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

NH ≅ G/Ntype
3. #31773688C2semidirect
5. C22×Dic542C4semidirect
8. #22576141C4semidirect
13. #22576119C4semidirect
92. C271#2123semidirect

Maximal subgroups

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

subgroupindex
#317744682
#317738712
#317736882
#413×C2712
#2123271