Finite Group Databasebeta

← #46837496

Subgroups — #46837496

Order 34688.

430 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 75 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#31774459C211yesmaximal, characteristic
2C2×#22576144C212yesmaximal, complemented
3C2×#22576074C212yesmaximal, complemented
4#31773758C211yesmaximal, characteristic, complemented
5#342×C271C211yesmaximal, characteristic, complemented
6C22×C22⋊D271C2212yes—
7C23×Dic271C2211yescharacteristic
8#22576144—22no—
9#22576144—24no—
10#22576144—24no—
11#22576134—22no—
12#22576130—22no—
13C2×C4⋊Dic271—22no—
14C2×C22⋊Dic271C2211yescharacteristic
15C2×(C2×D271)⋊C4—22no—
16#22576074—24no—
17C271⋊#95—22no—
18C2×C22⋊C4×C271C2211yescharacteristic
19C2×C22⋊C4×C271C2212yes—
20C23×D271D412yes—
21C2×D1084—22no—
22C2×C22⋊D271—22no—
23C2×C22⋊D271—22no—
24C2×C22⋊D271—22no—
25C2×C22⋊D271—22no—
26C2×C22⋊D271—22no—
27C2×C22⋊D271—22no—
28C22×Dic271—22no—
29C22×Dic271—21no—
30C22×Dic271—22no—
31C22×Dic271—21no—
32C22×Dic271—21no—
33C22×Dic271D412yes—
34C22×Dic271D412yescomplemented
35C4⋊Dic271—22no—
36C4⋊Dic271—28no—
37C22⋊Dic271—22no—
38(C2×D271)⋊C4—24no—
39(C2×D271)⋊C4—28no—
40C24×C271C2311yescharacteristic
41C22×C4×C271—22no—
42C22×C4×C271D412yes—
43C22⋊C4×C271—22no—
44C22⋊C4×C271—22no—
45C22⋊C4×C271D414yescomplemented
46C22×D271—22no—
47C22×D271—22no—
48C22×D271—22no—
49C22×D271—44no—
50D1084—82no—
51C22⋊D271—42no—
52C22⋊D271—42no—
53C22⋊D271—42no—
54C22⋊D271—42no—
55C2×Dic271—24no—
56C2×Dic271—22no—
57C2×Dic271—22no—
58C2×Dic271—22no—
59C2×Dic271—22no—
60C2×Dic271—41no—
61C2×Dic271—42no—
62C2×Dic271—41no—
63C2×Dic271—42no—
64C2×Dic271—41no—
65C23×C271—42no—
66C23×C271C2×D412yes—
67C23×C271C2×D411yescharacteristic
68C23×C271C2×D411yescharacteristic
69C23×C271C2×D412yes—
70C23×C271#4111yescharacteristic
71C2×C4×C271—22no—
72C2×C4×C271—24no—
73C2×C4×C271—44no—
74C2×C4×C271—42no—
75C22×C271—22no—
76C22×C271—24no—
77C22×C271—41no—
78C22×C271—42no—
79C22×C271—41no—
80C22×C271#12012yes—
81C22×C271#12011yescharacteristic
82C22×C271#12111yescharacteristic
83C22×C271#12112yes—
84C22×C271#12711yescharacteristic
85C2×D271—44no—
86C2×D271—42no—
87C2×D271—44no—
88C2×D271—82no—
89C4×C271—44no—
90C4×C271—42no—
91Dic271—42no—
92Dic271—42no—
93Dic271—81no—
94C2×C271—21no—
95C2×C271—22no—
96C2×C271—22no—
97C2×C271—21no—
98C2×C271#39211yescharacteristic
99C2×C271#45812yes—
100D271—82no—
101C271#202211yescharacteristic, complemented, Sylow
102#2022—2711nomaximal, Sylow
103#534—2711no—
104C2×#123—2712no—
105C2×#99—2712no—
106#381—2711no—
107#342D27111yescharacteristic, complemented
108C22×D4—2712no—
109C23×C4—2711no—
110#123—5424no—
111#123—5422no—
112#123—5424no—
113#121—5422no—
114#120—5422no—
115C2×C4⋊C4—5422no—
116C2×C22⋊C4—2711no—
117C2×C22⋊C4—5422no—
118C2×C22⋊C4C2×D27111yescharacteristic
119C2×C22⋊C4C2×D27112yes—
120#99—5424no—
121#95—5422no—
122C24—2712no—
123C24C22×D27111yescharacteristic
124C2×D4—5422no—
125C2×D4—5422no—
126C2×D4—5422no—
127C2×D4—5422no—
128C2×D4—5422no—
129C2×D4—5422no—
130C2×D4—5422no—
131C22×C4—22no—
132C22×C4—2712no—
133C22×C4—2712no—
134C22×C4—5422no—
135C22×C4—5421no—
136C22×C4—5422no—
137C22×C4—5421no—
138C22×C4—5421no—
139C22×C4D108412yes—
140C4⋊C4—5428no—
141C4⋊C4—5422no—
142C22⋊C4—22no—
143C22⋊C4—22no—
144C22⋊C4—5428no—
145C22⋊C4—5424no—
146C22⋊C4—5422no—
147C22⋊C4D108414yescomplemented
148C23—42no—
149C23—5422no—
150C23—5422no—
151C23—5422no—
152C23—10844no—
153C23D4×D27112yes—
154C23D4×D27111yescharacteristic
155C23C2×D108411yescharacteristic
156C23C2×D108412yes—
157C23#1686465511yescharacteristic
158D4—10842no—
159D4—10842no—
160D4—10842no—
161D4—10842no—
162D4—21682no—
163C2×C4—24no—
164C2×C4—22no—
165C2×C4—44no—
166C2×C4—42no—
167C2×C4—5424no—
168C2×C4—5422no—
169C2×C4—5422no—
170C2×C4—5422no—
171C2×C4—5422no—
172C2×C4—10841no—
173C2×C4—10842no—
174C2×C4—10841no—
175C2×C4—10842no—
176C2×C4—10841no—
177C22—22no—
178C22—24no—
179C22—41no—
180C22—42no—
181C22—41no—
182C22—10844no—
183C22—10842no—
184C22—10844no—
185C22—21682no—
186C22#2257613012yes—
187C22#2257613011yescharacteristic
188C22#2257613511yescharacteristic
189C22#2257613512yes—
190C22#2257615711yescharacteristic
191C4—44no—
192C4—42no—
193C4—10842no—
194C4—10842no—
195C4—21681no—
196C2—22no—
197C2—21no—
198C2—22no—
199C2—21no—
200C2—21682no—
201C2#3177380711yescharacteristic
202C2#3177408812yes—
subgroupG/H
centerC22#22576130
commutator, cosocleC24×C271C23
FrattiniC24C22×D271
Fitting#342×C271C2
socleC22×C271#120
prime radical#31774459C2
hypercenter#342D271
hypocenterC271#2022
LCSG ▷ C24×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
2. C2×#22576144C2semidirect
3. C2×#22576074C2semidirect
4. #31773758C2semidirect
5. #342×C271C2semidirect
34. C22×Dic271D4semidirect
45. C22⋊C4×C271D4semidirect
101. C271#2022semidirect
107. #342D271semidirect
147. C22⋊C4D1084semidirect

Maximal subgroups

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

subgroupindex
#317744592
C2×#225761442
C2×#225760742
#317737582
#342×C2712
#2022271