Finite Group Databasebeta

← #46837529

Subgroups — #46837529

Order 34688.

466 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 67 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#31774523C211yesmaximal, characteristic, complemented
2#31773752C212yesmaximal
3#31773752C211yesmaximal, characteristic
4#31773663C212yesmaximal, complemented
5#342×C271C211yesmaximal, characteristic, complemented
6C24×D271C2211yescharacteristic
7#22576139—22no—
8#22576137—24no—
9C2×C22⋊Dic271C2211yescharacteristic
10C2×C22⋊Dic271C2212yes—
11C2×(C2×D271)⋊C4—24no—
12C2×(C2×D271)⋊C4—22no—
13C2×C22⋊C4×C271C2212yes—
14C2×C22⋊C4×C271C2211yescharacteristic
15C23×D271—22no—
16C23×D271—21no—
17C23×D271—21no—
18C23×D271—22no—
19C23×D271—44no—
20C23×D271Q812yescomplemented
21C2×Dic542—22no—
22C22×Dic271—21no—
23C22×Dic271D414yes—
24C4⋊Dic271—22no—
25Dic271⋊C4—24no—
26C22⋊Dic271—22no—
27C22⋊Dic271—22no—
28C22⋊Dic271—22no—
29(C2×D271)⋊C4—44no—
30(C2×D271)⋊C4—48no—
31C24×C271C2311yescharacteristic
32C22×C4×C271—22no—
33C22×C4×C271D412yes—
34C22⋊C4×C271—22no—
35C22⋊C4×C271—22no—
36C22⋊C4×C271—22no—
37C22×D271—28no—
38C22×D271—24no—
39C22×D271—24no—
40C22×D271—22no—
41C22×D271—44no—
42C22×D271—44no—
43C22×D271—48no—
44C22×D271—44no—
45C22×D271—42no—
46C22×D271—41no—
47C22×D271—42no—
48C22×D271—41no—
49Dic542—82no—
50C2×Dic271—24no—
51C2×Dic271—44no—
52C2×Dic271—42no—
53C2×Dic271—41no—
54C23×C271—42no—
55C23×C271C2×D412yes—
56C23×C271C2×D411yescharacteristic
57C23×C271C2×Q811yescharacteristic
58C23×C271#4111yescharacteristic
59C23×C271#4112yes—
60C2×C4×C271—22no—
61C2×C4×C271—42no—
62C2×C4×C271—44no—
63C2×C4×C271—42no—
64C22×C271—24no—
65C22×C271—22no—
66C22×C271—41no—
67C22×C271—41no—
68C22×C271—42no—
69C22×C271#12011yescharacteristic
70C22×C271C22⋊Q812yes—
71C22×C271C22⋊Q811yescharacteristic
72C22×C271#12412yes—
73C22×C271#12411yescharacteristic
74C2×D271—28no—
75C2×D271—24no—
76C2×D271—44no—
77C2×D271—48no—
78C2×D271—44no—
79C2×D271—42no—
80C2×D271—42no—
81C2×D271—44no—
82C4×C271—42no—
83C4×C271—82no—
84Dic271—44no—
85Dic271—81no—
86C2×C271—21no—
87C2×C271—22no—
88C2×C271—22no—
89C2×C271—21no—
90C2×C271#39311yescharacteristic
91C2×C271B(4,𝔽2)12yes—
92D271—44no—
93C271#202911yescharacteristic, complemented, Sylow
94#2029—2711nomaximal, Sylow
95#543—2711no—
96#379—2712no—
97#379—2711no—
98#342—2712no—
99#342D27111yescharacteristic, complemented
100C25—2711no—
101C22⋊Q8—5424no—
102C22⋊Q8—5422no—
103C2×C22⋊C4—2712no—
104C2×C22⋊C4—2711no—
105C2×C22⋊C4—5424no—
106C2×C22⋊C4—5422no—
107C2×C22⋊C4C2×D27112yes—
108C2×C22⋊C4C2×D27111yescharacteristic
109C24—2712no—
110C24—5422no—
111C24—5421no—
112C24—5421no—
113C24—5422no—
114C24—10844no—
115C24C22×D27111yescharacteristic
116C2×Q8—5422no—
117C22×C4—22no—
118C22×C4—2714no—
119C22×C4—5421no—
120C22×C4D108412yes—
121C4⋊C4—5424no—
122C4⋊C4—5422no—
123C22⋊C4—22no—
124C22⋊C4—22no—
125C22⋊C4—22no—
126C22⋊C4—5422no—
127C22⋊C4—5422no—
128C22⋊C4—5422no—
129C22⋊C4—10844no—
130C22⋊C4—10848no—
131C23—42no—
132C23—5428no—
133C23—5424no—
134C23—5424no—
135C23—5422no—
136C23—10844no—
137C23—10842no—
138C23—10844no—
139C23—10844no—
140C23—10848no—
141C23—10841no—
142C23—10842no—
143C23—10841no—
144C23D4×D27112yes—
145C23C2×D108411yescharacteristic
146C23Q8×D27111yescharacteristic
147C23#1686465311yescharacteristic
148C23#1686465412yes—
149Q8—21682no—
150C2×C4—22no—
151C2×C4—42no—
152C2×C4—44no—
153C2×C4—42no—
154C2×C4—5424no—
155C2×C4—10844no—
156C2×C4—10842no—
157C2×C4—10841no—
158C22—24no—
159C22—22no—
160C22—41no—
161C22—41no—
162C22—42no—
163C22—5428no—
164C22—5424no—
165C22—10848no—
166C22—10844no—
167C22—10844no—
168C22—10842no—
169C22—10842no—
170C22—10844no—
171C22#2257613011yescharacteristic
172C22#2257613812yes—
173C22#2257614011yescharacteristic
174C22#2257614612yes—
175C22#2257614811yescharacteristic
176C4—42no—
177C4—82no—
178C4—10844no—
179C4—21681no—
180C2—22no—
181C2—21no—
182C2—22no—
183C2—21no—
184C2—10844no—
185C2#3177381011yescharacteristic
186C2#3177408512yes—
subgroupG/H
centerC22#22576130
commutator, cosocleC24×C271C23
FrattiniC24C22×D271
Fitting#342×C271C2
socleC22×C271#120
prime radicalC24×D271C22
hypercenter#342D271
hypocenterC271#2029
LCSG ▷ C24×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
1. #31774523C2semidirect
4. #31773663C2semidirect
5. #342×C271C2semidirect
20. C23×D271Q8semidirect
93. C271#2029semidirect
99. #342D271semidirect

Maximal subgroups

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

subgroupindex
#317745232
#317737522
#317736632
#342×C2712
#2029271