Finite Group Databasebeta

← #41014563

Subgroups — #41014563

Order 27008.

466 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 67 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, complemented
2#27213001C212yesmaximal
3#27213001C211yesmaximal, characteristic
4#27212912C212yesmaximal, complemented
5#342×C211C211yesmaximal, characteristic, complemented
6C24×D211C2211yescharacteristic
7#20269956—22no—
8#20269954—24no—
9C2×C22⋊Dic211C2211yescharacteristic
10C2×C22⋊Dic211C2212yes—
11C2×(C2×D211)⋊C4—24no—
12C2×(C2×D211)⋊C4—22no—
13C2×C22⋊C4×C211C2212yes—
14C2×C22⋊C4×C211C2211yescharacteristic
15C23×D211—22no—
16C23×D211—21no—
17C23×D211—21no—
18C23×D211—22no—
19C23×D211—44no—
20C23×D211Q812yescomplemented
21C2×Dic422—22no—
22C22×Dic211—21no—
23C22×Dic211D414yes—
24C4⋊Dic211—22no—
25Dic211⋊C4—24no—
26C22⋊Dic211—22no—
27C22⋊Dic211—22no—
28C22⋊Dic211—22no—
29(C2×D211)⋊C4—44no—
30(C2×D211)⋊C4—48no—
31C24×C211C2311yescharacteristic
32C22×C4×C211—22no—
33C22×C4×C211D412yes—
34C22⋊C4×C211—22no—
35C22⋊C4×C211—22no—
36C22⋊C4×C211—22no—
37C22×D211—28no—
38C22×D211—24no—
39C22×D211—24no—
40C22×D211—22no—
41C22×D211—44no—
42C22×D211—44no—
43C22×D211—48no—
44C22×D211—44no—
45C22×D211—42no—
46C22×D211—41no—
47C22×D211—42no—
48C22×D211—41no—
49Dic422—82no—
50C2×Dic211—24no—
51C2×Dic211—44no—
52C2×Dic211—42no—
53C2×Dic211—41no—
54C23×C211—42no—
55C23×C211C2×D412yes—
56C23×C211C2×D411yescharacteristic
57C23×C211C2×Q811yescharacteristic
58C23×C211#4111yescharacteristic
59C23×C211#4112yes—
60C2×C4×C211—22no—
61C2×C4×C211—42no—
62C2×C4×C211—44no—
63C2×C4×C211—42no—
64C22×C211—24no—
65C22×C211—22no—
66C22×C211—41no—
67C22×C211—41no—
68C22×C211—42no—
69C22×C211#12011yescharacteristic
70C22×C211C22⋊Q812yes—
71C22×C211C22⋊Q811yescharacteristic
72C22×C211#12412yes—
73C22×C211#12411yescharacteristic
74C2×D211—28no—
75C2×D211—24no—
76C2×D211—44no—
77C2×D211—48no—
78C2×D211—44no—
79C2×D211—42no—
80C2×D211—42no—
81C2×D211—44no—
82C4×C211—42no—
83C4×C211—82no—
84Dic211—44no—
85Dic211—81no—
86C2×C211—21no—
87C2×C211—22no—
88C2×C211—22no—
89C2×C211—21no—
90C2×C211#39311yescharacteristic
91C2×C211B(4,𝔽2)12yes—
92D211—44no—
93C211#202911yescharacteristic, complemented, Sylow
94#2029—2111nomaximal, Sylow
95#543—2111no—
96#379—2112no—
97#379—2111no—
98#342—2112no—
99#342D21111yescharacteristic, complemented
100C25—2111no—
101C22⋊Q8—4224no—
102C22⋊Q8—4222no—
103C2×C22⋊C4—4224no—
104C2×C22⋊C4—4222no—
105C2×C22⋊C4—2111no—
106C2×C22⋊C4—2112no—
107C2×C22⋊C4C2×D21112yes—
108C2×C22⋊C4C2×D21111yescharacteristic
109C24—8444no—
110C24—4221no—
111C24—4222no—
112C24—4222no—
113C24—4221no—
114C24—2112no—
115C24C22×D21111yescharacteristic
116C2×Q8—4222no—
117C22×C4—22no—
118C22×C4—4221no—
119C22×C4—2114no—
120C22×C4D84412yes—
121C4⋊C4—4224no—
122C4⋊C4—4222no—
123C22⋊C4—22no—
124C22⋊C4—22no—
125C22⋊C4—22no—
126C22⋊C4—8444no—
127C22⋊C4—8448no—
128C22⋊C4—4222no—
129C22⋊C4—4222no—
130C22⋊C4—4222no—
131C23—42no—
132C23—8444no—
133C23—8442no—
134C23—8444no—
135C23—8448no—
136C23—8444no—
137C23—8441no—
138C23—8442no—
139C23—8441no—
140C23—4228no—
141C23—4224no—
142C23—4224no—
143C23—4222no—
144C23D4×D21112yes—
145C23C2×D84411yescharacteristic
146C23Q8×D21111yescharacteristic
147C23#1680996111yescharacteristic
148C23#1680996212yes—
149Q8—16882no—
150C2×C4—22no—
151C2×C4—42no—
152C2×C4—44no—
153C2×C4—42no—
154C2×C4—8444no—
155C2×C4—8442no—
156C2×C4—8441no—
157C2×C4—4224no—
158C22—24no—
159C22—22no—
160C22—41no—
161C22—41no—
162C22—42no—
163C22—8448no—
164C22—8444no—
165C22—8444no—
166C22—8442no—
167C22—8442no—
168C22—8444no—
169C22—4228no—
170C22—4224no—
171C22#2026994711yescharacteristic
172C22#2026995512yes—
173C22#2026995711yescharacteristic
174C22#2026996312yes—
175C22#2026996511yescharacteristic
176C4—42no—
177C4—82no—
178C4—8444no—
179C4—16881no—
180C2—22no—
181C2—21no—
182C2—22no—
183C2—21no—
184C2—8444no—
185C2#2721305911yescharacteristic
186C2#2721333412yes—
subgroupG/H
centerC22#20269947
commutator, cosocleC24×C211C23
FrattiniC24C22×D211
Fitting#342×C211C2
socleC22×C211#120
prime radicalC24×D211C22
hypercenter#342D211
hypocenterC211#2029
LCSG ▷ C24×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
1. #27213772C2semidirect
4. #27212912C2semidirect
5. #342×C211C2semidirect
20. C23×D211Q8semidirect
93. C211#2029semidirect
99. #342D211semidirect

Maximal subgroups

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

subgroupindex
#272137722
#272130012
#272129122
#342×C2112
#2029211