Finite Group Databasebeta

← #41012020

Subgroups — #41012020

Order 27008.

248 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 59 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#124×D211C211yesmaximal, characteristic
2#27213201C212yesmaximal, complemented
3#27212886C211yesmaximal, characteristic, complemented
4#27212885C212yesmaximal, complemented
5#329×C211C211yesmaximal, characteristic, complemented
6C2×Q8×D211C411yescharacteristic
7C2×D4×D211C411yescharacteristic
8#20269966C411yescharacteristic
9#20269965C2211yescharacteristic
10#20269964C411yescharacteristic
11#20269963—22no—
12C22⋊C4×D211—22no—
13C42×D211C2211yescharacteristic
14Dic211⋊C8—22no—
15(C2×D211)⋊C8—22no—
16#20269888C2212yescomplemented
17#124×C211C2211yescharacteristic
18#97×C211C2212yescomplemented
19C23×D211—21no—
20C2×Dic422—21no—
21C2×Dic422C2×C411yescharacteristic
22Q8×D211—42no—
23C2×D844C2×C411yescharacteristic
24C2×C22⋊D211—21no—
25D4×D211—42no—
26C2×C4×D211—21no—
27C2×C4×D211C2×C411yescharacteristic
28C2×C4×D211D412yes—
29C2×C211⋊C8—22no—
30C22⋊Dic211—22no—
31(C2×D211)⋊C4—22no—
32(C2×D211)⋊C4—22no—
33C4×Dic211C2×C411yescharacteristic
34C4×Dic211D412yes—
35C2×Q8×C211C2×C411yescharacteristic
36C2×D4×C211C2×C411yescharacteristic
37C2×C8×C211—22no—
38C22⋊C4×C211—22no—
39C42×C211C2311yescharacteristic
40C22×D211—21no—
41C22×D211—44no—
42C22×D211—42no—
43C22×D211C22⋊C411yescharacteristic
44Dic422—42no—
45Dic422—41no—
46D844—41no—
47C22⋊D211—42no—
48C2×Dic211—21no—
49C2×Dic211C22⋊C412yes—
50C2×Dic211C22⋊C411yescharacteristic
51C4×D211—22no—
52C4×D211—42no—
53C4×D211—42no—
54C211⋊C8—42no—
55C23×C211—21no—
56Q8×C211—41no—
57D4×C211—41no—
58C2×C4×C211—21no—
59C2×C4×C211C22×C411yescharacteristic
60C2×C4×C211C2×D412yes—
61C8×C211—42no—
62C22×C211—41no—
63C22×C211—42no—
64C22×C211C2×C22⋊C411yescharacteristic
65C2×D211—22no—
66C2×D211—41no—
67C2×D211—42no—
68C2×D211—84no—
69C2×D211#10012yes—
70C4×C211—21no—
71C4×C211—22no—
72C4×C211—41no—
73Dic211—21no—
74Dic211—22no—
75Dic211—41no—
76C2×C211—81no—
77C2×C211C2×#10011yescharacteristic
78C2×C211#42112yes—
79D211—22no—
80D211—81no—
81C211#153311yescharacteristic, complemented, Sylow
82#1533—2111nomaximal, Sylow
83C2×#124—2111no—
84#432—2112no—
85#329—2111no—
86#329—2112no—
87#329D21111yescharacteristic, complemented
88C22×Q8—2111no—
89C22×D4—2111no—
90#124—4222no—
91#124—2111no—
92#124—2111no—
93#124—2111no—
94#124C2×D21111yescharacteristic
95C2×C22⋊C4—4222no—
96C2×C42—2111no—
97C4⋊C8—4222no—
98C22⋊C8—4222no—
99#97—2112no—
100#97C2×D21112yescomplemented
101C24—4221no—
102C2×Q8—8442no—
103C2×Q8—4221no—
104C2×Q8—2111no—
105C2×Q8C4×D21111yescharacteristic
106C2×D4—8442no—
107C2×D4—4221no—
108C2×D4—2111no—
109C2×D4C4×D21111yescharacteristic
110C22×C4—4221no—
111C22×C4—2112no—
112C22×C4—2111no—
113C2×C8—22no—
114C2×C8—4222no—
115C22⋊C4—22no—
116C22⋊C4—4222no—
117C22⋊C4—4222no—
118C22⋊C4—4222no—
119C42—2111no—
120C42—2112no—
121C42C22×D21111yescharacteristic
122C23—21no—
123C23—8444no—
124C23—8442no—
125C23—4221no—
126C23—2111no—
127Q8—41no—
128Q8—8441no—
129Q8—8442no—
130D4—41no—
131D4—8441no—
132D4—8442no—
133C2×C4—21no—
134C2×C4—8442no—
135C2×C4—8442no—
136C2×C4—4222no—
137C2×C4—4221no—
138C2×C4—2112no—
139C2×C4—2111no—
140C2×C4C2×C4×D21111yescharacteristic
141C2×C4D4×D21112yes—
142C8—42no—
143C8—8442no—
144C22—41no—
145C22—42no—
146C22—8441no—
147C22—8442no—
148C22—16884no—
149C22—4222no—
150C22—2112no—
151C22C22⋊C4×D21111yescharacteristic
152C4—21no—
153C4—22no—
154C4—41no—
155C4—8441no—
156C4—4221no—
157C4—4222no—
158C2—81no—
159C2—16881no—
160C2—4222no—
161C2#100×D21111yescharacteristic
162C2#2721316112yes—
subgroupG/H
centerC22C22⋊C4×D211
commutatorC2×C4×C211C22×C4
FrattiniC42C22×D211
Fitting#329×C211C2
socleC22×C211C2×C22⋊C4
prime radicalC2×D4×D211C4
cosocleC42×C211C23
hypercenter#329D211
hypocenterC211#1533
LCSG ▷ C2×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #27213201C2semidirect
3. #27212886C2semidirect
4. #27212885C2semidirect
5. #329×C211C2semidirect
16. #20269888C22semidirect
18. #97×C211C22semidirect
81. C211#1533semidirect
87. #329D211semidirect
100. #97C2×D211semidirect

Maximal subgroups

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

subgroupindex
#124×D2112
#272132012
#272128862
#272128852
#329×C2112
#1533211