Finite Group Databasebeta

← #41012356

Subgroups — #41012356

Order 27008.

256 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 51 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#27213771C211yesmaximal, characteristic
2#27213304C211yesmaximal, characteristic, complemented
3#27213301C211yesmaximal, characteristic, complemented
4#27213059C211yesmaximal, characteristic
5#27212882C211yesmaximal, characteristic, complemented
6#27212869C211yesmaximal, characteristic
7#328×C211C211yesmaximal, characteristic, complemented
8C22×Dic422C2211yescharacteristic
9C2×Dic844—21no—
10C2×#16809952—21no—
11#20269957—21no—
12#20269956C2211yescharacteristic
13#20269955—21no—
14#20269955—21no—
15#20269954—21no—
16#20269953—21no—
17C2×(C2×D211)⋊C4—21no—
18C2×(C2×D211)⋊C4C2211yescharacteristic
19C22⋊C4×D211—21no—
20#20269902—21no—
21#20269900—21no—
22C22⋊(C211⋊C8)C2211yescharacteristic
23C211⋊#95—21no—
24C22⋊Q8×C211C2211yescharacteristic
25C22⋊C8×C211C2211yescharacteristic
26#95×C211C2211yescharacteristic
27C23×D211D411yescharacteristic
28C2×Dic422—21no—
29C2×Dic422—21no—
30C2×Dic422—21no—
31C2×Dic422—42no—
32C2×Dic422D411yescharacteristic
33C22×Dic211—21no—
34C22×Dic211D411yescharacteristic
35C2×C4×D211—21no—
36C2×C4×D211—21no—
37Dic844—42no—
38#16809952—42no—
39C2×C211⋊C8—21no—
40C4⋊Dic211—21no—
41C4⋊Dic211D411yescharacteristic
42Dic211⋊C4—21no—
43Dic211⋊C4—21no—
44Dic211⋊C4—21no—
45C22⋊Dic211—21no—
46(C2×D211)⋊C4—21no—
47(C2×D211)⋊C4—21no—
48(C2×D211)⋊C4—21no—
49(C2×D211)⋊C4—21no—
50(C2×D211)⋊C4—42no—
51C2×Q8×C211D411yescharacteristic
52C22×C4×C211—21no—
53C22×C4×C211C2311yescharacteristic
54C2×C8×C211—21no—
55C4⋊C4×C211—21no—
56C4⋊C4×C211D411yescharacteristic
57C22⋊C4×C211—21no—
58C22×D211—21no—
59C22×D211—21no—
60C22×D211—41no—
61C22×D211—41no—
62C22×D211Q1612yescomplemented
63Dic422—22no—
64Dic422—22no—
65Dic422—41no—
66Dic422—41no—
67Dic422—41no—
68Dic422—81no—
69C2×Dic211—22no—
70C2×Dic211—21no—
71C2×Dic211—21no—
72C2×Dic211—21no—
73C2×Dic211—41no—
74C2×Dic211—41no—
75C2×Dic211—41no—
76C4×D211—81no—
77C4×D211—81no—
78C211⋊C8—41no—
79C23×C211C2×D411yescharacteristic
80Q8×C211—41no—
81C2×C4×C211—21no—
82C2×C4×C211—21no—
83C2×C4×C211—41no—
84C2×C4×C211—41no—
85C2×C4×C211—41no—
86C2×C4×C211—41no—
87C2×C4×C211C2×D411yescharacteristic
88C2×C4×C211C2×D411yescharacteristic
89C8×C211—41no—
90C22×C211—21no—
91C22×C211—21no—
92C22×C211#12011yescharacteristic
93C22×C211C2×Q1611yescharacteristic
94C22×C211#13711yescharacteristic
95C2×D211—22no—
96C2×D211—41no—
97C2×D211—41no—
98C2×D211—42no—
99C2×D211—81no—
100C4×C211—21no—
101C4×C211—21no—
102C4×C211—41no—
103C4×C211—41no—
104C4×C211—81no—
105Dic211—41no—
106Dic211—41no—
107Dic211—41no—
108Dic211—42no—
109C2×C211—21no—
110C2×C211—21no—
111C2×C211#45111yescharacteristic
112C2×C211#45511yescharacteristic
113C2×C211B(4,𝔽2)11yescharacteristic
114D211—42no—
115C211#160211yescharacteristic, complemented, Sylow
116#1602—2111nomaximal, Sylow
117#543—2111no—
118#451—2111no—
119#451—2111no—
120#393—2111no—
121#328—2111no—
122#328D21111yescharacteristic, complemented
123#323—2111no—
124C22×Q8—2111no—
125C2×Q16—4221no—
126C2×Q16—4221no—
127C22⋊Q8—4221no—
128C22⋊Q8—4221no—
129C22⋊Q8—4221no—
130C22⋊Q8—4221no—
131C22⋊Q8—4221no—
132C22⋊Q8—2111no—
133C22⋊Q8C2×D21111yescharacteristic
134C2×C22⋊C4—4221no—
135C2×C22⋊C4—4221no—
136C2×C22⋊C4—2111no—
137#103—4221no—
138#103—4221no—
139C22⋊C8—2111no—
140C22⋊C8C2×D21111yescharacteristic
141#95—4221no—
142#95C2×D21111yescharacteristic
143C24—2111no—
144C2×Q8—8442no—
145C2×Q8—4221no—
146C2×Q8—4221no—
147C2×Q8—4221no—
148C2×Q8—2111no—
149C2×Q8D84411yescharacteristic
150C22×C4—21no—
151C22×C4—4221no—
152C22×C4—4221no—
153C22×C4—4221no—
154C22×C4—2111no—
155C22×C4C22×D21111yescharacteristic
156Q16—8442no—
157Q16—8442no—
158C2×C8—21no—
159C2×C8—4221no—
160C4⋊C4—21no—
161C4⋊C4—4221no—
162C4⋊C4—4221no—
163C4⋊C4—4221no—
164C4⋊C4—4221no—
165C4⋊C4—2111no—
166C4⋊C4D84411yescharacteristic
167C22⋊C4—21no—
168C22⋊C4—8442no—
169C22⋊C4—4221no—
170C22⋊C4—4221no—
171C22⋊C4—4221no—
172C22⋊C4—4221no—
173C22⋊C4—4221no—
174C23—8441no—
175C23—8441no—
176C23—4221no—
177C23—4221no—
178C23—2112no—
179C23D4×D21111yescharacteristic
180Q8—41no—
181Q8—8441no—
182Q8—8441no—
183Q8—8441no—
184Q8—16881no—
185Q8—4222no—
186Q8—4222no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—8441no—
194C2×C4—8441no—
195C2×C4—8441no—
196C2×C4—16881no—
197C2×C4—16881no—
198C2×C4—4222no—
199C2×C4—4221no—
200C2×C4—4221no—
201C2×C4—4221no—
202C2×C4D4×D21111yescharacteristic
203C2×C4C2×D84411yescharacteristic
204C8—41no—
205C8—8441no—
206C22—21no—
207C22—21no—
208C22—8442no—
209C22—8441no—
210C22—8441no—
211C22—16881no—
212C22—4222no—
213C22#2026994711yescharacteristic
214C22Q16×D21111yescharacteristic
215C22#2027000611yescharacteristic
216C4—21no—
217C4—21no—
218C4—41no—
219C4—41no—
220C4—81no—
221C4—8441no—
222C4—8441no—
223C4—8442no—
224C4—8441no—
225C2—21no—
226C2—21no—
227C2—8442no—
228C2#2721330311yescharacteristic
229C2#2721332511yescharacteristic
230C2#2721333411yescharacteristic
subgroupG/H
centerC22#20269947
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#328×C211C2
socleC22×C211#120
prime radicalC23×D211D4
hypercenter#328D211
hypocenterC211#1602
LCSG ▷ C22×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. #27213304C2semidirect
3. #27213301C2semidirect
5. #27212882C2semidirect
7. #328×C211C2semidirect
62. C22×D211Q16semidirect
115. C211#1602semidirect
122. #328D211semidirect

Maximal subgroups

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

subgroupindex
#272137712
#272133042
#272133012
#272130592
#272128822
#272128692
#328×C2112
#1602211