Finite Group Databasebeta

← #41012031

Subgroups — #41012031

Order 27008.

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

normal subgroups only

This group has 238 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#27213695C211yesmaximal, characteristic
2#27213201C212yesmaximal, complemented
3#27212886C211yesmaximal, characteristic, complemented
4#27212885C212yesmaximal, complemented
5#329×C211C211yesmaximal, characteristic, complemented
6C2×#16809963C411yescharacteristic
7C2×#16809961C411yescharacteristic
8#20269975C411yescharacteristic
9#20269973C411yescharacteristic
10#20269965C2211yescharacteristic
11#20269950—22no—
12(C22⋊D211)⋊C4—22no—
13C42×D211C2211yescharacteristic
14Dic211⋊C8—22no—
15(C2×D211)⋊C8—22no—
16#20269888C2212yes—
17#124×C211C2211yescharacteristic
18#97×C211C2212yes—
19#16809963—42no—
20#16809961—42no—
21C2×Dic422C2×C411yescharacteristic
22C2×D844—21no—
23C2×D844C2×C411yescharacteristic
24C2×C22⋊D211—21no—
25C2×C22⋊D211—22no—
26C22×Dic211—21no—
27C2×C4×D211—21no—
28C2×C4×D211C2×C411yescharacteristic
29C2×C4×D211D412yes—
30C2×C211⋊C8—22no—
31Dic211⋊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×D211C22⋊C411yescharacteristic
42Dic422—41no—
43D844—41no—
44D844—42no—
45C22⋊D211—44no—
46C22⋊D211—42no—
47C2×Dic211—21no—
48C2×Dic211—42no—
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—21no—
66C2×D211—22no—
67C2×D211—41no—
68C2×D211—42no—
69C4×C211—21no—
70C4×C211—22no—
71C4×C211—41no—
72Dic211—22no—
73Dic211—41no—
74Dic211#10012yes—
75C2×C211—81no—
76C2×C211C2×#10011yescharacteristic
77C2×C211#42112yes—
78D211—41no—
79D211—81no—
80C211#153611yescharacteristic, complemented, Sylow
81#1536—2111nomaximal, Sylow
82#532—2111no—
83#432—2112no—
84#329—2111no—
85#329—2112no—
86#329D21111yescharacteristic, complemented
87C2×#41—2111no—
88C2×#41—2111no—
89C4⋊Q8—2111no—
90#127—2111no—
91#124—2111no—
92#124C2×D21111yescharacteristic
93#121—4222no—
94C4×D4—4222no—
95C2×C42—2111no—
96C4⋊C8—4222no—
97C22⋊C8—4222no—
98#97—2112no—
99#97C2×D21112yes—
100#41—8442no—
101#41—8442no—
102C2×Q8—2111no—
103C2×Q8C4×D21111yescharacteristic
104C2×D4—4221no—
105C2×D4—4221no—
106C2×D4—4222no—
107C2×D4—2111no—
108C2×D4C4×D21111yescharacteristic
109C22×C4—4221no—
110C22×C4—4221no—
111C22×C4—2112no—
112C22×C4—2111no—
113C2×C8—22no—
114C2×C8—4222no—
115C4⋊C4—4222no—
116C22⋊C4—22no—
117C22⋊C4—4222no—
118C42—2111no—
119C42—2112no—
120C42C22×D21111yescharacteristic
121C23—21no—
122C23—4221no—
123C23—2111no—
124Q8—41no—
125Q8—8441no—
126D4—41no—
127D4—8441no—
128D4—8442no—
129D4—8442no—
130D4—8444no—
131C2×C4—21no—
132C2×C4—8442no—
133C2×C4—8442no—
134C2×C4—8442no—
135C2×C4—4222no—
136C2×C4—4221no—
137C2×C4—2112no—
138C2×C4—2111no—
139C2×C4C2×C4×D21111yescharacteristic
140C2×C4D4×D21112yes—
141C8—42no—
142C8—8442no—
143C22—41no—
144C22—42no—
145C22—8441no—
146C22—8442no—
147C22—4221no—
148C22—4222no—
149C22C22⋊C4×D21111yescharacteristic
150C4—21no—
151C4—22no—
152C4—41no—
153C4—8441no—
154C4—4222no—
155C4—2112no—
156C2—81no—
157C2—8441no—
158C2—16881no—
159C2#100×D21111yescharacteristic
160C2#2721316112yes—
subgroupG/H
centerC22C22⋊C4×D211
commutatorC2×C4×C211C22×C4
FrattiniC42C22×D211
Fitting#329×C211C2
socleC22×C211C2×C22⋊C4
prime radical#27213695C2
cosocleC42×C211C23
hypercenter#329D211
hypocenterC211#1536
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
80. C211#1536semidirect
86. #329D211semidirect

Maximal subgroups

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

subgroupindex
#272136952
#272132012
#272128862
#272128852
#329×C2112
#1536211