Finite Group Databasebeta

← #41014646

Subgroups — #41014646

Order 27008.

382 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 71 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
1C2×#20269892C212yesmaximal, complemented
2#27213026C212yesmaximal, complemented
3#27213018C211yesmaximal, characteristic, complemented
4#27212913C211yesmaximal, characteristic, complemented
5#534×C211C211yesmaximal, characteristic
6C23×Dic211C2211yescharacteristic
7C2×Dic211⋊C4—21no—
8C2×C22⋊Dic211—22no—
9C2×C22⋊Dic211C2212yes—
10C2×C22⋊Dic211C2211yescharacteristic
11#20269892—24no—
12C211⋊#95—21no—
13C211⋊#95—22no—
14C22×D4×C211C2212yes—
15#123×C211—22no—
16#121×C211—21no—
17#121×C211—21no—
18#120×C211—22no—
19C2×C22⋊C4×C211C2211yescharacteristic
20C22×Dic211—22no—
21C22×Dic211—21no—
22C22×Dic211—21no—
23C22×Dic211—22no—
24C22×Dic211—42no—
25Dic211⋊C4—42no—
26C22⋊Dic211—22no—
27C22⋊Dic211—21no—
28C22⋊Dic211—22no—
29C22⋊Dic211—44no—
30C22⋊Dic211D412yescomplemented
31C24×C211D412yes—
32C24×C211C2311yescharacteristic
33C2×D4×C211—22no—
34C2×D4×C211—22no—
35C2×D4×C211—22no—
36C2×D4×C211—22no—
37C2×D4×C211—22no—
38C2×D4×C211—22no—
39C2×D4×C211—21no—
40C2×D4×C211—21no—
41C22×C4×C211D412yes—
42C22×C4×C211D411yescharacteristic
43C22×C4×C211D411yescharacteristic
44C4⋊C4×C211—21no—
45C4⋊C4×C211—21no—
46C22⋊C4×C211—22no—
47C22⋊C4×C211—22no—
48C22⋊C4×C211—22no—
49C2×Dic211—24no—
50C2×Dic211—22no—
51C2×Dic211—22no—
52C2×Dic211—41no—
53C2×Dic211—41no—
54C2×Dic211—41no—
55C2×Dic211—42no—
56C2×Dic211—41no—
57C2×Dic211—42no—
58C2×Dic211—42no—
59C2×Dic211—42no—
60C23×C211—22no—
61C23×C211—22no—
62C23×C211—22no—
63C23×C211—41no—
64C23×C211—41no—
65C23×C211—42no—
66C23×C211—42no—
67C23×C211C2×D411yescharacteristic
68C23×C211C2×D412yes—
69C23×C211C2×D411yescharacteristic
70C23×C211#4111yescharacteristic
71C23×C211#4112yes—
72D4×C211—42no—
73D4×C211—42no—
74D4×C211—42no—
75D4×C211—42no—
76D4×C211—81no—
77D4×C211—81no—
78C2×C4×C211—21no—
79C2×C4×C211—21no—
80C2×C4×C211—22no—
81C2×C4×C211—22no—
82C2×C4×C211—22no—
83C2×C4×C211—41no—
84C2×C4×C211—41no—
85C22×C211—21no—
86C22×C211—22no—
87C22×C211—21no—
88C22×C211—22no—
89C22×C211—41no—
90C22×C211—41no—
91C22×C211—42no—
92C22×C211—44no—
93C22×C211—42no—
94C22×C211—42no—
95C22×C211—42no—
96C22×C211—82no—
97C22×C211#12011yescharacteristic
98C22×C211#12111yescharacteristic
99C22×C211#12112yes—
100C22×C211#12312yes—
101C22×C211#12411yescharacteristic
102C4×C211—42no—
103C4×C211—41no—
104C4×C211—41no—
105Dic211—42no—
106Dic211—42no—
107Dic211—82no—
108C2×C211—21no—
109C2×C211—22no—
110C2×C211—22no—
111C2×C211—21no—
112C2×C211—82no—
113C2×C211#39411yescharacteristic
114C2×C211B(4,𝔽2)11yescharacteristic
115C2×C211#45811yescharacteristic
116C211#204411yescharacteristic, complemented, Sylow
117#2044—2111nomaximal, Sylow
118#534D21111yescharacteristic
119C2×#99—2112no—
120#386—2112no—
121#385—2111no—
122#342—2111no—
123C22×D4C2×D21112yes—
124C23×C4—2111no—
125#123—22no—
126#121—21no—
127#121—21no—
128#120—22no—
129C2×C4⋊C4—4221no—
130C2×C22⋊C4—4222no—
131C2×C22⋊C4—2112no—
132C2×C22⋊C4—2111no—
133C2×C22⋊C4C2×D21111yescharacteristic
134#99—4224no—
135#95—4221no—
136#95—4222no—
137C24C22⋊D21112yes—
138C24C22×D21111yescharacteristic
139C2×D4—22no—
140C2×D4—22no—
141C2×D4—22no—
142C2×D4—22no—
143C2×D4—22no—
144C2×D4—22no—
145C2×D4—21no—
146C2×D4—21no—
147C22×C4—8442no—
148C22×C4—4222no—
149C22×C4—4222no—
150C22×C4—4221no—
151C22×C4—4221no—
152C22×C4C22⋊D21112yes—
153C22×C4C22⋊D21111yescharacteristic
154C22×C4C22⋊D21111yescharacteristic
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—8442no—
158C22⋊C4—22no—
159C22⋊C4—22no—
160C22⋊C4—22no—
161C22⋊C4—8444no—
162C22⋊C4—4222no—
163C22⋊C4—4222no—
164C22⋊C4—4221no—
165C22⋊C4—2112no—
166C23—22no—
167C23—22no—
168C23—22no—
169C23—41no—
170C23—41no—
171C23—42no—
172C23—42no—
173C23D4×D21111yescharacteristic
174C23C2×C22⋊D21111yescharacteristic
175C23C2×C22⋊D21112yes—
176C23#1680996111yescharacteristic
177C23#1680996112yes—
178D4—42no—
179D4—42no—
180D4—42no—
181D4—42no—
182D4—81no—
183D4—81no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—8441no—
192C2×C4—8441no—
193C2×C4—8442no—
194C2×C4—8441no—
195C2×C4—8441no—
196C2×C4—8442no—
197C2×C4—8442no—
198C2×C4—8442no—
199C2×C4—4224no—
200C2×C4—4222no—
201C2×C4—4222no—
202C22—21no—
203C22—22no—
204C22—21no—
205C22—22no—
206C22—41no—
207C22—41no—
208C22—42no—
209C22—44no—
210C22—42no—
211C22—42no—
212C22—42no—
213C22—82no—
214C22#2026994611yescharacteristic
215C22#2026994811yescharacteristic
216C22#2026994912yes—
217C22#2026995812yes—
218C22#2026996411yescharacteristic
219C4—42no—
220C4—41no—
221C4—41no—
222C4—8442no—
223C4—8442no—
224C4—16882no—
225C2—21no—
226C2—22no—
227C2—22no—
228C2—21no—
229C2—82no—
230C2#2721306011yescharacteristic
231C2#2721333311yescharacteristic
232C2#2721333611yescharacteristic
subgroupG/H
centerC22#20269946
commutator, cosocleC24×C211C23
FrattiniC24C22×D211
Fitting, prime radical#534×C211C2
socleC22×C211#120
hypercenter#534D211
hypocenterC211#2044
LCSG ▷ C24×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C24 ◁ #534

Complements

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

NH ≅ G/Ntype
1. C2×#20269892C2semidirect
2. #27213026C2semidirect
3. #27213018C2semidirect
4. #27212913C2semidirect
30. C22⋊Dic211D4semidirect
116. C211#2044semidirect

Maximal subgroups

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

subgroupindex
C2×#202698922
#272130262
#272130182
#272129132
#534×C2112
#2044211