Finite Group Databasebeta

← #46835363

Subgroups — #46835363

Order 34688.

216 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 49 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#31774494C211yesmaximal, characteristic
2#31774228C211yesmaximal, characteristic, complemented
3#31774211C211yesmaximal, characteristic
4#31773827C211yesmaximal, characteristic, complemented
5#31773635C211yesmaximal, characteristic, complemented
6#31773632C211yesmaximal, characteristic
7#324×C271C211yesmaximal, characteristic, complemented
8#22576155—21no—
9#22576150—21no—
10#22576145—21no—
11#22576143—21no—
12#22576139C2211yescharacteristic
13#22576138—21no—
14#22576134C2211yescharacteristic
15C4×C22⋊D271—21no—
16C2×C4⋊Dic271C2211yescharacteristic
17C2×Dic271⋊C4—21no—
18#22576094—21no—
19#22576092—21no—
20#22576085—21no—
21#22576082—21no—
22C271⋊#95—21no—
23C271⋊#95C2211yescharacteristic
24C2×C4⋊C4×C271C2211yescharacteristic
25C22⋊C8×C271C2211yescharacteristic
26C22⋊C8×C271C2211yescharacteristic
27C2×Dic542D411yescharacteristic
28C2×D1084D411yescharacteristic
29C2×C22⋊D271—21no—
30C22×Dic271—21no—
31C22×Dic271—21no—
32C2×C4×D271—21no—
33C4⋊Dic271—21no—
34C4⋊Dic271—21no—
35C4⋊Dic271D411yescharacteristic
36C4⋊Dic271D411yescharacteristic
37Dic271⋊C4—21no—
38Dic271⋊C4—21no—
39Dic271⋊C4—21no—
40Dic271⋊C4—21no—
41Dic271⋊C4—42no—
42C22⋊Dic271—21no—
43(C2×D271)⋊C4—21no—
44(C2×D271)⋊C4—21no—
45(C2×D271)⋊C4—21no—
46(C2×D271)⋊C4—21no—
47C4×Dic271—21no—
48C22×C4×C271D411yescharacteristic
49C22×C4×C271D411yescharacteristic
50C22×C4×C271C2311yescharacteristic
51C2×C8×C271—21no—
52C2×C8×C271—21no—
53C4⋊C4×C271—21no—
54C4⋊C4×C271—21no—
55C22×D271—21no—
56Dic542—41no—
57D1084—41no—
58C22⋊D271—41no—
59C22⋊D271—41no—
60C2×Dic271—22no—
61C2×Dic271—21no—
62C2×Dic271—21no—
63C2×Dic271—21no—
64C2×Dic271—41no—
65C2×Dic271—41no—
66C2×Dic271—41no—
67C2×Dic271—41no—
68C2×Dic271—41no—
69C4×D271—81no—
70C23×C271C2×D411yescharacteristic
71C2×C4×C271—21no—
72C2×C4×C271—21no—
73C2×C4×C271—21no—
74C2×C4×C271—21no—
75C2×C4×C271—41no—
76C2×C4×C271—41no—
77C2×C4×C271C2×D411yescharacteristic
78C2×C4×C271C2×D411yescharacteristic
79C8×C271—41no—
80C8×C271—41no—
81C22×C271—21no—
82C22×C271—21no—
83C22×C271#12011yescharacteristic
84C22×C271#13511yescharacteristic
85C22×C271AGL(1,ℤ/8)11yescharacteristic
86C2×D271—41no—
87C2×D271—41no—
88C2×D271—41no—
89C4×C271—21no—
90C4×C271—21no—
91C4×C271—41no—
92C4×C271—41no—
93Dic271—42no—
94Dic271—41no—
95Dic271—41no—
96Dic271—41no—
97Dic271—81no—
98C2×C271—21no—
99C2×C271—21no—
100C2×C271#44911yescharacteristic
101C2×C271#45611yescharacteristic
102C2×C271#45811yescharacteristic
103D271—81no—
104C271#160911yescharacteristic, complemented, Sylow
105#1609—2711nomaximal, Sylow
106#539—2711no—
107#483—2711no—
108#480—2711no—
109#398—2711no—
110#328—2711no—
111#327—2711no—
112#324D27111yescharacteristic, complemented
113#126—5421no—
114#125—5421no—
115#123—5421no—
116#123—5421no—
117C22⋊Q8—2711no—
118C22⋊Q8—5421no—
119#121—2711no—
120C4×D4—5421no—
121C2×C4⋊C4—2711no—
122C2×C4⋊C4—5421no—
123C2×C4⋊C4C2×D27111yescharacteristic
124#107—5421no—
125#106—5421no—
126#103—5421no—
127#102—5421no—
128C22⋊C8C2×D27111yescharacteristic
129C22⋊C8C2×D27111yescharacteristic
130#95—2711no—
131#95—5421no—
132C2×Q8—2711no—
133C2×D4—2711no—
134C2×D4—5421no—
135C22×C4—5421no—
136C22×C4—5421no—
137C22×C4—5421no—
138C22×C4D108411yescharacteristic
139C22×C4D108411yescharacteristic
140C22×C4C22×D27111yescharacteristic
141C2×C8—21no—
142C2×C8—21no—
143C4⋊C4—21no—
144C4⋊C4—21no—
145C4⋊C4—2711no—
146C4⋊C4—2711no—
147C4⋊C4—5421no—
148C4⋊C4—5421no—
149C4⋊C4—5421no—
150C4⋊C4—5421no—
151C4⋊C4—5421no—
152C4⋊C4—5421no—
153C4⋊C4—10842no—
154C22⋊C4—5421no—
155C22⋊C4—5421no—
156C22⋊C4—5421no—
157C22⋊C4—5421no—
158C22⋊C4—5421no—
159C42—5421no—
160C23—5421no—
161C23C2×D108411yescharacteristic
162Q8—10841no—
163D4—10841no—
164D4—10841no—
165D4—10841no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—5422no—
173C2×C4—5421no—
174C2×C4—5421no—
175C2×C4—5421no—
176C2×C4—10841no—
177C2×C4—10841no—
178C2×C4—10841no—
179C2×C4—10841no—
180C2×C4—10841no—
181C2×C4—21681no—
182C2×C4D4×D27111yescharacteristic
183C2×C4D4×D27111yescharacteristic
184C8—41no—
185C8—41no—
186C22—21no—
187C22—21no—
188C22—10841no—
189C22—10841no—
190C22—10841no—
191C22#2257613011yescharacteristic
192C22#2257618211yescharacteristic
193C22#2257618811yescharacteristic
194C4—21no—
195C4—21no—
196C4—41no—
197C4—41no—
198C4—10842no—
199C4—10841no—
200C4—10841no—
201C4—10841no—
202C4—21681no—
203C2—21no—
204C2—21no—
205C2—21681no—
206C2#3177404411yescharacteristic
207C2#3177408211yescharacteristic
208C2#3177408811yescharacteristic
subgroupG/H
centerC22#22576130
commutator, cosocleC22×C4×C271C23
FrattiniC22×C4C22×D271
Fitting#324×C271C2
socleC22×C271#120
prime radical#22576134C22
hypercenter#324D271
hypocenterC271#1609
LCSG ▷ C22×C4×C271 ▷ C22×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #31774228C2semidirect
4. #31773827C2semidirect
5. #31773635C2semidirect
7. #324×C271C2semidirect
104. C271#1609semidirect
112. #324D271semidirect

Maximal subgroups

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

subgroupindex
#317744942
#317742282
#317742112
#317738272
#317736352
#317736322
#324×C2712
#1609271