Finite Group Databasebeta

← #46835369

Subgroups — #46835369

Order 34688.

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

normal subgroups only

This group has 254 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#31774478C211yesmaximal, characteristic
2#31774211C212yesmaximal, complemented
3#31773827C211yesmaximal, characteristic, complemented
4#31773632C212yesmaximal, complemented
5#324×C271C211yesmaximal, characteristic, complemented
6C2×#16864654—21no—
7#22576140—22no—
8#22576135—22no—
9#22576134—21no—
10#22576134C2212yes—
11C2×C4⋊Dic271C2211yescharacteristic
12C2×Dic271⋊C4—21no—
13#22576094—22no—
14#22576082—22no—
15C271⋊#95—21no—
16C271⋊#95C2211yescharacteristic
17C2×C4⋊C4×C271C2211yescharacteristic
18C22⋊C8×C271C2212yes—
19#16864654—42no—
20#16864654—42no—
21C2×Dic542—21no—
22C2×D1084—21no—
23C2×D1084—21no—
24C2×D1084D412yes—
25C2×C22⋊D271—22no—
26C22×Dic271—21no—
27C22×Dic271—21no—
28C2×C4×D271—22no—
29C4⋊Dic271—21no—
30C4⋊Dic271—22no—
31C4⋊Dic271D412yes—
32Dic271⋊C4—42no—
33(C2×D271)⋊C4—22no—
34(C2×D271)⋊C4—22no—
35C22×C4×C271D411yescharacteristic
36C22×C4×C271D411yescharacteristic
37C22×C4×C271C2311yescharacteristic
38C2×C8×C271—22no—
39C4⋊C4×C271—22no—
40C22×D271—22no—
41Dic542—42no—
42D1084—42no—
43D1084—42no—
44D1084—81no—
45C22⋊D271—42no—
46C22⋊D271—42no—
47C2×Dic271—22no—
48C2×Dic271—22no—
49C2×Dic271—41no—
50C2×Dic271—41no—
51C2×Dic271—41no—
52C2×Dic271—41no—
53C2×Dic271—41no—
54C4×D271—44no—
55C23×C271C2×D411yescharacteristic
56C2×C4×C271—21no—
57C2×C4×C271—21no—
58C2×C4×C271—21no—
59C2×C4×C271—41no—
60C2×C4×C271—41no—
61C2×C4×C271D812yes—
62C2×C4×C271C2×D412yes—
63C8×C271—42no—
64C22×C271—21no—
65C22×C271—21no—
66C22×C271#12011yescharacteristic
67C22×C271C2×D811yescharacteristic
68C22×C271AGL(1,ℤ/8)11yescharacteristic
69C2×D271—42no—
70C2×D271—42no—
71C2×D271—42no—
72C4×C271—22no—
73C4×C271—22no—
74C4×C271—41no—
75Dic271—42no—
76Dic271—42no—
77Dic271—81no—
78C2×C271—21no—
79C2×C271—21no—
80C2×C271#44711yescharacteristic
81C2×C271#45611yescharacteristic
82C2×C271#45811yescharacteristic
83D271—82no—
84C271#161011yescharacteristic, complemented, Sylow
85#1610—2711nomaximal, Sylow
86#537—2711no—
87#480—2712no—
88#398—2711no—
89#327—2712no—
90#324D27111yescharacteristic, complemented
91C2×#41—5421no—
92C22⋊Q8—5422no—
93#121—2712no—
94#121—5422no—
95#121—5421no—
96C2×C4⋊C4—2711no—
97C2×C4⋊C4—5421no—
98C2×C4⋊C4C2×D27111yescharacteristic
99#107—5422no—
100#102—5422no—
101C22⋊C8C2×D27112yes—
102#95—2711no—
103#95—5421no—
104#41—10842no—
105#41—10842no—
106C2×Q8—5421no—
107C2×D4—2712no—
108C2×D4—5421no—
109C2×D4—5421no—
110C2×D4—5422no—
111C22×C4—5422no—
112C22×C4—5421no—
113C22×C4—5421no—
114C22×C4D108411yescharacteristic
115C22×C4D108411yescharacteristic
116C22×C4C22×D27111yescharacteristic
117C2×C8—22no—
118C4⋊C4—22no—
119C4⋊C4—2712no—
120C4⋊C4—5421no—
121C4⋊C4—5422no—
122C4⋊C4—10842no—
123C22⋊C4—5422no—
124C22⋊C4—5422no—
125C23—5422no—
126C23C2×D108411yescharacteristic
127Q8—10842no—
128D4—10842no—
129D4—10842no—
130D4—10842no—
131D4—10842no—
132D4—21681no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—5422no—
139C2×C4—5422no—
140C2×C4—10844no—
141C2×C4—10841no—
142C2×C4—10841no—
143C2×C4—10841no—
144C2×C4—10841no—
145C2×C4—10841no—
146C2×C4D216812yes—
147C2×C4D4×D27112yes—
148C8—42no—
149C22—21no—
150C22—21no—
151C22—10842no—
152C22—10842no—
153C22—10842no—
154C22#2257613011yescharacteristic
155C22C2×D216811yescharacteristic
156C22#2257618811yescharacteristic
157C4—22no—
158C4—22no—
159C4—41no—
160C4—10842no—
161C4—10842no—
162C4—21681no—
163C2—21no—
164C2—21no—
165C2—21682no—
166C2#3177403311yescharacteristic
167C2#3177408211yescharacteristic
168C2#3177408811yescharacteristic
subgroupG/H
centerC22#22576130
commutator, cosocleC22×C4×C271C23
FrattiniC22×C4C22×D271
Fitting#324×C271C2
socleC22×C271#120
prime radical#31774478C2
hypercenter#324D271
hypocenterC271#1610
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. #31774211C2semidirect
3. #31773827C2semidirect
4. #31773632C2semidirect
5. #324×C271C2semidirect
84. C271#1610semidirect
90. #324D271semidirect

Maximal subgroups

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

subgroupindex
#317744782
#317742112
#317738272
#317736322
#324×C2712
#1610271