Finite Group Databasebeta

← #46835284

Subgroups — #46835284

Order 34688.

368 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 53 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, complemented
2#31774033C212yesmaximal, complemented
3#31773806C211yesmaximal, characteristic, complemented
4#31773632C212yesmaximal, complemented
5#323×C271C211yesmaximal, characteristic, complemented
6C22×D1084C2211yescharacteristic, complemented
7C22×C22⋊D271—21no—
8C22×C22⋊D271—21no—
9C2×D2168—22no—
10#22576144—22no—
11#22576134—21no—
12#22576134C2212yes—
13#22576130—22no—
14C2×(C2×D271)⋊C4—21no—
15#22576082—22no—
16C271⋊#95C2211yescharacteristic, complemented
17C2×C22⋊C4×C271C2211yescharacteristic
18C22⋊C8×C271C2212yescomplemented
19C23×D271—21no—
20C23×D271—21no—
21C2×D1084—21no—
22C2×D1084—22no—
23C2×D1084—42no—
24C2×D1084D412yescomplemented
25C2×C22⋊D271—22no—
26C2×C22⋊D271—22no—
27C2×C22⋊D271—42no—
28C2×C22⋊D271—42no—
29C2×C22⋊D271—42no—
30C2×C22⋊D271—42no—
31C2×C22⋊D271—42no—
32C22×Dic271—21no—
33C22×Dic271—21no—
34D2168—44no—
35C4⋊Dic271—21no—
36C4⋊Dic271D412yescomplemented
37(C2×D271)⋊C4—22no—
38(C2×D271)⋊C4—22no—
39(C2×D271)⋊C4—42no—
40C24×C271D411yescharacteristic
41C22×C4×C271D411yescharacteristic
42C22×C4×C271C2311yescharacteristic
43C2×C8×C271—22no—
44C22⋊C4×C271—22no—
45C22×D271—22no—
46C22×D271—22no—
47C22×D271—41no—
48C22×D271—41no—
49C22×D271—42no—
50C22×D271—41no—
51C22×D271—41no—
52C22×D271—41no—
53C22×D271—41no—
54C22×D271—81no—
55C22×D271—81no—
56D1084—24no—
57D1084—42no—
58D1084—42no—
59D1084—81no—
60C22⋊D271—44no—
61C22⋊D271—48no—
62C22⋊D271—42no—
63C22⋊D271—42no—
64C2×Dic271—22no—
65C2×Dic271—22no—
66C2×Dic271—41no—
67C2×Dic271—42no—
68C2×Dic271—41no—
69C23×C271—21no—
70C23×C271—21no—
71C23×C271—41no—
72C23×C271—41no—
73C23×C271D812yescomplemented
74C23×C271C2×D411yescharacteristic
75C2×C4×C271—21no—
76C2×C4×C271—41no—
77C2×C4×C271—41no—
78C2×C4×C271C2×D412yes—
79C8×C271—42no—
80C22×C271—21no—
81C22×C271—21no—
82C22×C271—22no—
83C22×C271—42no—
84C22×C271—41no—
85C22×C271—41no—
86C22×C271—81no—
87C22×C271#12011yescharacteristic
88C22×C271C2×D811yescharacteristic
89C22×C271AGL(1,ℤ/8)11yescharacteristic
90C2×D271—42no—
91C2×D271—42no—
92C2×D271—42no—
93C2×D271—42no—
94C2×D271—42no—
95C2×D271—42no—
96C2×D271—81no—
97C2×D271—81no—
98C2×D271—84no—
99C2×D271—81no—
100C2×D271—81no—
101C4×C271—22no—
102C4×C271—41no—
103Dic271—42no—
104Dic271—42no—
105C2×C271—21no—
106C2×C271—21no—
107C2×C271—42no—
108C2×C271#44711yescharacteristic
109C2×C271#45311yescharacteristic
110C2×C271B(4,𝔽2)11yescharacteristic
111D271—82no—
112D271—82no—
113C271#159511yescharacteristic, complemented, Sylow
114#1595—2711nomaximal, Sylow
115#534—2711no—
116#447—2712no—
117#392—2711no—
118#327—2712no—
119#323D27111yescharacteristic, complemented
120C22×D4—2711no—
121C22×D4—5421no—
122C22×D4—5421no—
123C2×D8—5422no—
124#123—5422no—
125#121—2712no—
126#121—5421no—
127#120—5422no—
128C2×C22⋊C4—5421no—
129C2×C22⋊C4C2×D27111yescharacteristic
130#102—5422no—
131C22⋊C8C2×D27112yescomplemented
132#95—2711no—
133C24—5421no—
134C24—5421no—
135C24D108411yescharacteristic
136C2×D4—2712no—
137C2×D4—5421no—
138C2×D4—5422no—
139C2×D4—5422no—
140C2×D4—5422no—
141C2×D4—10842no—
142C2×D4—10842no—
143C2×D4—10842no—
144C2×D4—10842no—
145C2×D4—10842no—
146C2×D4—10842no—
147C22×C4—5421no—
148C22×C4—5421no—
149C22×C4D108411yescharacteristic
150C22×C4C22×D27111yescharacteristic
151D8—10844no—
152C2×C8—22no—
153C4⋊C4—2712no—
154C4⋊C4—5421no—
155C22⋊C4—22no—
156C22⋊C4—5422no—
157C22⋊C4—5422no—
158C22⋊C4—10842no—
159C23—21no—
160C23—21no—
161C23—41no—
162C23—41no—
163C23—5422no—
164C23—5422no—
165C23—10841no—
166C23—10841no—
167C23—10841no—
168C23—10842no—
169C23—10841no—
170C23—10841no—
171C23—10841no—
172C23—21681no—
173C23—21681no—
174C23D216812yescomplemented
175C23C2×D108411yescharacteristic
176D4—5424no—
177D4—10842no—
178D4—10842no—
179D4—10844no—
180D4—10848no—
181D4—10842no—
182D4—10842no—
183D4—21681no—
184C2×C4—21no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—5422no—
188C2×C4—5422no—
189C2×C4—10841no—
190C2×C4—10842no—
191C2×C4—10841no—
192C2×C4D4×D27112yes—
193C8—42no—
194C22—21no—
195C22—21no—
196C22—22no—
197C22—42no—
198C22—41no—
199C22—41no—
200C22—81no—
201C22—10842no—
202C22—10842no—
203C22—10842no—
204C22—10842no—
205C22—10842no—
206C22—10842no—
207C22—21681no—
208C22—21681no—
209C22—21684no—
210C22—21681no—
211C22—21681no—
212C22#2257613011yescharacteristic
213C22C2×D216811yescharacteristic
214C22#2257618811yescharacteristic
215C4—22no—
216C4—41no—
217C4—10842no—
218C4—10842no—
219C2—21no—
220C2—21no—
221C2—42no—
222C2—21682no—
223C2—21682no—
224C2#3177403311yescharacteristic
225C2#3177406711yescharacteristic
226C2#3177408511yescharacteristic
subgroupG/H
centerC22#22576130
commutator, cosocleC22×C4×C271C23
FrattiniC22×C4C22×D271
Fitting#323×C271C2
socleC22×C271#120
hypercenter#323D271
hypocenterC271#1595
LCSG ▷ C22×C4×C271 ▷ C22×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
1. #31774459C2semidirect
2. #31774033C2semidirect
3. #31773806C2semidirect
4. #31773632C2semidirect
5. #323×C271C2semidirect
6. C22×D1084C22semidirect
16. C271⋊#95C22semidirect
18. C22⋊C8×C271C22semidirect
24. C2×D1084D4semidirect
36. C4⋊Dic271D4semidirect
73. C23×C271D8semidirect
113. C271#1595semidirect
119. #323D271semidirect
131. C22⋊C8C2×D271semidirect
174. C23D2168semidirect

Maximal subgroups

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

subgroupindex
#317744592
#317740332
#317738062
#317736322
#323×C2712
#1595271