Finite Group Databasebeta

← #46835339

Subgroups — #46835339

Order 34688.

238 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 51 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#31774526C211yesmaximal, characteristic
2#31774055C211yesmaximal, characteristic
3#31774052C211yesmaximal, characteristic
4#31773818C211yesmaximal, characteristic
5#31773633C211yesmaximal, characteristic
6#31773622C211yesmaximal, characteristic
7#328×C271C211yesmaximal, characteristic
8C22×Dic542C2211yescharacteristic
9C2×Dic1084—21no—
10C2×#16864644—21no—
11#22576159—21no—
12#22576158—21no—
13#22576150—21no—
14#22576150—21no—
15#22576139C2211yescharacteristic
16#22576137—21no—
17#22576116—21no—
18C2×C4⋊Dic271—21no—
19C2×Dic271⋊C4C2211yescharacteristic
20#22576085—21no—
21#22576083—21no—
22C22⋊(C271⋊C8)C2211yescharacteristic
23C271⋊#95—21no—
24C22⋊Q8×C271C2211yescharacteristic
25C22⋊C8×C271C2211yescharacteristic
26#95×C271C2211yescharacteristic
27C2×Dic542—21no—
28C2×Dic542—21no—
29C2×Dic542—21no—
30C2×Dic542—42no—
31C2×Dic542D411yescharacteristic
32C22×Dic271—21no—
33C22×Dic271D411yescharacteristic
34C22×Dic271D411yescharacteristic
35Dic1084—42no—
36#16864644—42no—
37C2×C271⋊C8—21no—
38C4⋊Dic271—21no—
39C4⋊Dic271—42no—
40C4⋊Dic271D411yescharacteristic
41Dic271⋊C4—21no—
42Dic271⋊C4—21no—
43Dic271⋊C4—21no—
44Dic271⋊C4—21no—
45Dic271⋊C4—21no—
46Dic271⋊C4—21no—
47Dic271⋊C4—21no—
48C22⋊Dic271—21no—
49C4×Dic271—21no—
50C4×Dic271—21no—
51C2×Q8×C271D411yescharacteristic
52C22×C4×C271—21no—
53C22×C4×C271C2311yescharacteristic
54C2×C8×C271—21no—
55C4⋊C4×C271—21no—
56C4⋊C4×C271D411yescharacteristic
57C22⋊C4×C271—21no—
58Dic542—22no—
59Dic542—22no—
60Dic542—41no—
61Dic542—41no—
62Dic542—41no—
63Dic542—81no—
64C2×Dic271—21no—
65C2×Dic271—21no—
66C2×Dic271—22no—
67C2×Dic271—21no—
68C2×Dic271—21no—
69C2×Dic271—21no—
70C2×Dic271—41no—
71C2×Dic271—41no—
72C2×Dic271—41no—
73C2×Dic271Q1612yescomplemented
74C271⋊C8—41no—
75C23×C271C2×D411yescharacteristic
76Q8×C271—41no—
77C2×C4×C271—22no—
78C2×C4×C271—21no—
79C2×C4×C271—21no—
80C2×C4×C271—41no—
81C2×C4×C271—41no—
82C2×C4×C271—41no—
83C2×C4×C271C2×D411yescharacteristic
84C2×C4×C271C2×D411yescharacteristic
85C8×C271—41no—
86C22×C271—21no—
87C22×C271—21no—
88C22×C271#12011yescharacteristic
89C22×C271C2×Q1611yescharacteristic
90C22×C271#13711yescharacteristic
91C4×C271—21no—
92C4×C271—21no—
93C4×C271—42no—
94C4×C271—41no—
95C4×C271—41no—
96Dic271—22no—
97Dic271—41no—
98Dic271—42no—
99Dic271—41no—
100Dic271—41no—
101C2×C271—21no—
102C2×C271—21no—
103C2×C271#45111yescharacteristic
104C2×C271#45611yescharacteristic
105C2×C271B(4,𝔽2)11yescharacteristic
106C271#160511yescharacteristic, complemented, Sylow
107#1605—2711nomaximal, Sylow
108#544—2711no—
109#451—2711no—
110#451—2711no—
111#395—2711no—
112#328—2711no—
113#328D27111yescharacteristic
114#324—2711no—
115C22×Q8—2711no—
116C2×Q16—5421no—
117C2×Q16—5421no—
118C4⋊Q8—5421no—
119C4⋊Q8—5421no—
120#125—5421no—
121#125—5421no—
122C22⋊Q8—2711no—
123C22⋊Q8—5421no—
124C22⋊Q8C2×D27111yescharacteristic
125#117—5421no—
126C2×C4⋊C4—2711no—
127C2×C4⋊C4—5421no—
128#103—5421no—
129#103—5421no—
130C22⋊C8—2711no—
131C22⋊C8C2×D27111yescharacteristic
132#95—5421no—
133#95C2×D27111yescharacteristic
134C2×Q8—2711no—
135C2×Q8—5421no—
136C2×Q8—5421no—
137C2×Q8—5421no—
138C2×Q8—10842no—
139C2×Q8D108411yescharacteristic
140C22×C4—21no—
141C22×C4—2711no—
142C22×C4—2711no—
143C22×C4—5421no—
144C22×C4C22×D27111yescharacteristic
145Q16—10842no—
146Q16—10842no—
147C2×C8—21no—
148C2×C8—5421no—
149C4⋊C4—21no—
150C4⋊C4—2711no—
151C4⋊C4—5421no—
152C4⋊C4—5421no—
153C4⋊C4—5421no—
154C4⋊C4—5421no—
155C4⋊C4—5421no—
156C4⋊C4—5421no—
157C4⋊C4—5421no—
158C4⋊C4—5421no—
159C4⋊C4—10842no—
160C4⋊C4D108411yescharacteristic
161C22⋊C4—21no—
162C22⋊C4—5421no—
163C42—5421no—
164C42—5421no—
165C23D4×D27111yescharacteristic
166Q8—41no—
167Q8—5422no—
168Q8—5422no—
169Q8—10841no—
170Q8—10841no—
171Q8—10841no—
172Q8—21681no—
173C2×C4—22no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—2712no—
180C2×C4—5421no—
181C2×C4—5421no—
182C2×C4—5422no—
183C2×C4—5421no—
184C2×C4—5421no—
185C2×C4—5421no—
186C2×C4—10841no—
187C2×C4—10841no—
188C2×C4—10841no—
189C2×C4D4×D27111yescharacteristic
190C2×C4C2×D108411yescharacteristic
191C8—41no—
192C8—10841no—
193C22—21no—
194C22—21no—
195C22#2257613011yescharacteristic
196C22Q16×D27111yescharacteristic
197C22#2257618911yescharacteristic
198C4—21no—
199C4—21no—
200C4—42no—
201C4—41no—
202C4—41no—
203C4—5422no—
204C4—10841no—
205C4—10842no—
206C4—10841no—
207C4—10841no—
208C2—21no—
209C2—21no—
210C2#3177405411yescharacteristic
211C2#3177408111yescharacteristic
212C2#3177408511yescharacteristic
subgroupG/H
centerC22#22576130
commutator, cosocleC22×C4×C271C23
FrattiniC22×C4C22×D271
Fitting#328×C271C2
socleC22×C271#120
prime radicalC23×C271C2×D4
hypercenter#328D271
hypocenterC271#1605
LCSG ▷ C22×C4×C271 ▷ C22×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
73. C2×Dic271Q16semidirect
106. C271#1605semidirect

Maximal subgroups

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

subgroupindex
#317745262
#317740552
#317740522
#317738182
#317736332
#317736222
#328×C2712
#1605271