Finite Group Databasebeta

← #41012422

Subgroups — #41012422

Order 27008.

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

normal subgroups only

This group has 236 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#27213772C211yesmaximal, characteristic
2#27213481C212yesmaximal, complemented
3#27212882C212yesmaximal, complemented
4#27212870C211yesmaximal, characteristic
5#399×C211C211yesmaximal, characteristic
6#20269956—21no—
7#20269956C2212yes—
8#20269954—22no—
9#20269954—22no—
10C2×C22⋊Dic211—21no—
11#20269910—22no—
12#20269900—22no—
13C22⋊(C211⋊C8)C2212yes—
14C2×C4⋊C4×C211—21no—
15C2×C4⋊C4×C211C2211yescharacteristic
16C2×C22⋊C4×C211—21no—
17C2×C22⋊C4×C211C2211yescharacteristic
18#95×C211C2211yescharacteristic
19C2×Dic422—21no—
20C2×Dic422D412yes—
21C22×Dic211—21no—
22C22×Dic211—21no—
23C2×C211⋊C8—22no—
24C4⋊Dic211—21no—
25C4⋊Dic211D412yes—
26Dic211⋊C4—22no—
27Dic211⋊C4—22no—
28C22⋊Dic211—22no—
29C22⋊Dic211—22no—
30C24×C211D411yescharacteristic
31C22×C4×C211—21no—
32C22×C4×C211—21no—
33C22×C4×C211D411yescharacteristic
34C22×C4×C211C2311yescharacteristic
35C4⋊C4×C211—22no—
36C4⋊C4×C211—42no—
37C22⋊C4×C211—22no—
38C22⋊C4×C211—42no—
39Dic422—42no—
40Dic422—81no—
41C2×Dic211—22no—
42C2×Dic211—22no—
43C2×Dic211—81no—
44C2×Dic211—81no—
45C211⋊C8—42no—
46C23×C211—21no—
47C23×C211—21no—
48C23×C211—41no—
49C23×C211—41no—
50C23×C211Q1612yes—
51C23×C211C2×D411yescharacteristic
52C2×C4×C211—22no—
53C2×C4×C211—21no—
54C2×C4×C211—41no—
55C2×C4×C211—41no—
56C2×C4×C211—41no—
57C2×C4×C211—41no—
58C2×C4×C211—41no—
59C2×C4×C211—41no—
60C2×C4×C211—41no—
61C2×C4×C211C2×D412yes—
62C22×C211—21no—
63C22×C211—21no—
64C22×C211—22no—
65C22×C211—42no—
66C22×C211—41no—
67C22×C211—41no—
68C22×C211—81no—
69C22×C211#12011yescharacteristic
70C22×C211C2×Q1611yescharacteristic
71C22×C211#13711yescharacteristic
72C4×C211—22no—
73C4×C211—42no—
74C4×C211—41no—
75C4×C211—81no—
76Dic211—42no—
77Dic211—42no—
78C2×C211—21no—
79C2×C211—21no—
80C2×C211—42no—
81C2×C211#45111yescharacteristic
82C2×C211#45311yescharacteristic
83C2×C211#45811yescharacteristic
84C211#161411yescharacteristic, complemented, Sylow
85#1614—2111nomaximal, Sylow
86#543—2111no—
87#484—2112no—
88#399D21111yescharacteristic
89#328—2112no—
90#323—2111no—
91C22⋊Q8—4222no—
92C22⋊Q8—4222no—
93C22⋊Q8—4221no—
94C22⋊Q8—2112no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4C2×D21111yescharacteristic
97C2×C22⋊C4—21no—
98C2×C22⋊C4—4221no—
99C2×C22⋊C4C2×D21111yescharacteristic
100#107—4222no—
101#103—4222no—
102C22⋊C8—2112no—
103#95C2×D21111yescharacteristic
104C24C22⋊D21111yescharacteristic
105C2×Q8—4221no—
106C2×Q8—2112no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—4221no—
110C22×C4—4221no—
111C22×C4C22⋊D21111yescharacteristic
112C22×C4C22×D21111yescharacteristic
113C2×C8—4222no—
114C4⋊C4—22no—
115C4⋊C4—42no—
116C4⋊C4—4222no—
117C4⋊C4—4221no—
118C4⋊C4—4222no—
119C4⋊C4—2112no—
120C22⋊C4—22no—
121C22⋊C4—42no—
122C22⋊C4—4222no—
123C22⋊C4—4222no—
124C23—21no—
125C23—21no—
126C23—41no—
127C23—41no—
128C23#1680995212yes—
129C23C2×C22⋊D21111yescharacteristic
130Q8—8442no—
131Q8—16881no—
132C2×C4—22no—
133C2×C4—21no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—16881no—
142C2×C4—16881no—
143C2×C4—4222no—
144C2×C4—4222no—
145C2×C4D4×D21112yes—
146C8—8442no—
147C22—21no—
148C22—21no—
149C22—22no—
150C22—42no—
151C22—41no—
152C22—41no—
153C22—81no—
154C22#2026994511yescharacteristic
155C22C2×#1680995211yescharacteristic
156C22#2027000711yescharacteristic
157C4—22no—
158C4—42no—
159C4—41no—
160C4—81no—
161C4—8442no—
162C4—8442no—
163C2—21no—
164C2—21no—
165C2—42no—
166C2#2721330111yescharacteristic
167C2#2721331311yescharacteristic
168C2#2721333511yescharacteristic
subgroupG/H
centerC22#20269945
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#399×C211C2
socleC22×C211#120
prime radicalC24×C211D4
hypercenter#399D211
hypocenterC211#1614
LCSG ▷ C22×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #399

Complements

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

NH ≅ G/Ntype
2. #27213481C2semidirect
3. #27212882C2semidirect
84. C211#1614semidirect

Maximal subgroups

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

subgroupindex
#272137722
#272134812
#272128822
#272128702
#399×C2112
#1614211