Finite Group Databasebeta

← #46835357

Subgroups — #46835357

Order 34688.

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

normal subgroups only

This group has 228 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#31774523C211yesmaximal, characteristic
2#31774228C212yesmaximal, complemented
3#31773822C211yesmaximal, characteristic
4#31773635C212yesmaximal, complemented
5#323×C271C211yesmaximal, characteristic
6#22576139—21no—
7#22576139C2212yes—
8#22576137—22no—
9#22576137—22no—
10C2×C4⋊Dic271C2211yescharacteristic
11C2×C22⋊Dic271—21no—
12C2×C22⋊Dic271—21no—
13#22576092—22no—
14#22576085—22no—
15C271⋊#95—21no—
16C271⋊#95C2211yescharacteristic
17C2×C22⋊C4×C271C2211yescharacteristic
18C22⋊C8×C271C2212yes—
19C2×Dic542—21no—
20C2×Dic542D412yes—
21C22×Dic271—21no—
22C22×Dic271—21no—
23C22×Dic271—21no—
24C22×Dic271—21no—
25C4⋊Dic271—21no—
26C4⋊Dic271—22no—
27C4⋊Dic271D412yes—
28Dic271⋊C4—22no—
29Dic271⋊C4—22no—
30C22⋊Dic271—22no—
31C22⋊Dic271—22no—
32C22⋊Dic271—42no—
33C24×C271D411yescharacteristic
34C22×C4×C271D411yescharacteristic
35C22×C4×C271C2311yescharacteristic
36C2×C8×C271—22no—
37C22⋊C4×C271—22no—
38Dic542—42no—
39Dic542—81no—
40C2×Dic271—22no—
41C2×Dic271—22no—
42C2×Dic271—41no—
43C2×Dic271—41no—
44C2×Dic271—41no—
45C2×Dic271—41no—
46C2×Dic271—41no—
47C2×Dic271—41no—
48C2×Dic271—81no—
49C2×Dic271—81no—
50C23×C271—21no—
51C23×C271—21no—
52C23×C271—41no—
53C23×C271—41no—
54C23×C271SD1612yes—
55C23×C271C2×D411yescharacteristic
56C2×C4×C271—21no—
57C2×C4×C271—41no—
58C2×C4×C271—41no—
59C2×C4×C271C2×D412yes—
60C8×C271—42no—
61C22×C271—21no—
62C22×C271—21no—
63C22×C271—22no—
64C22×C271—42no—
65C22×C271—41no—
66C22×C271—41no—
67C22×C271—81no—
68C22×C271#12011yescharacteristic
69C22×C271C2×SD1611yescharacteristic
70C22×C271#13711yescharacteristic
71C4×C271—22no—
72C4×C271—41no—
73Dic271—42no—
74Dic271—42no—
75Dic271—81no—
76Dic271—81no—
77C2×C271—21no—
78C2×C271—21no—
79C2×C271—42no—
80C2×C271#44811yescharacteristic
81C2×C271#45511yescharacteristic
82C2×C271#45811yescharacteristic
83C271#160811yescharacteristic, complemented, Sylow
84#1608—2711nomaximal, Sylow
85#543—2711no—
86#483—2712no—
87#397—2711no—
88#328—2712no—
89#323D27111yescharacteristic
90C22⋊Q8—2712no—
91C22⋊Q8—5422no—
92C22⋊Q8—5422no—
93C22⋊Q8—5421no—
94C2×C4⋊C4—2711no—
95C2×C22⋊C4—5421no—
96C2×C22⋊C4—5421no—
97C2×C22⋊C4C2×D27111yescharacteristic
98#106—5422no—
99#103—5422no—
100C22⋊C8C2×D27112yes—
101#95—2711no—
102#95—5421no—
103C24D108411yescharacteristic
104C2×Q8—2712no—
105C2×Q8—5421no—
106C22×C4—5421no—
107C22×C4—5421no—
108C22×C4—5421no—
109C22×C4—5421no—
110C22×C4D108411yescharacteristic
111C22×C4C22×D27111yescharacteristic
112C2×C8—22no—
113C4⋊C4—2712no—
114C4⋊C4—5422no—
115C4⋊C4—5421no—
116C4⋊C4—5422no—
117C4⋊C4—5422no—
118C22⋊C4—22no—
119C22⋊C4—5422no—
120C22⋊C4—5422no—
121C22⋊C4—10842no—
122C23—21no—
123C23—21no—
124C23—41no—
125C23—41no—
126C23#1686464312yes—
127C23C2×D108411yescharacteristic
128Q8—10842no—
129Q8—21681no—
130C2×C4—21no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—5422no—
134C2×C4—5422no—
135C2×C4—10841no—
136C2×C4—10841no—
137C2×C4—10841no—
138C2×C4—10841no—
139C2×C4—10841no—
140C2×C4—10841no—
141C2×C4—21681no—
142C2×C4—21681no—
143C2×C4D4×D27112yes—
144C8—42no—
145C22—21no—
146C22—21no—
147C22—22no—
148C22—42no—
149C22—41no—
150C22—41no—
151C22—81no—
152C22#2257613011yescharacteristic
153C22C2×#1686464311yescharacteristic
154C22#2257619311yescharacteristic
155C4—22no—
156C4—41no—
157C4—10842no—
158C4—10842no—
159C4—21681no—
160C4—21681no—
161C2—21no—
162C2—21no—
163C2—42no—
164C2#3177403811yescharacteristic
165C2#3177407711yescharacteristic
166C2#3177408811yescharacteristic
subgroupG/H
centerC22#22576130
commutator, cosocleC22×C4×C271C23
FrattiniC22×C4C22×D271
Fitting#323×C271C2
socleC22×C271#120
prime radicalC24×C271D4
hypercenter#323D271
hypocenterC271#1608
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
2. #31774228C2semidirect
4. #31773635C2semidirect
83. C271#1608semidirect

Maximal subgroups

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

subgroupindex
#317745232
#317742282
#317738222
#317736352
#323×C2712
#1608271