Finite Group Databasebeta

← #36368889

Subgroups — #36368889

Order 22144.

312 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 76 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#26012510C211yesmaximal, characteristic, complemented
2#26012495C211yesmaximal, characteristic, complemented
3#26012339C212yesmaximal, complemented
4D4⋊#13063443C212yesmaximal, complemented
5C4×D4×D173C211yesmaximal, characteristic
6C2×#14583836—21no—
7C4⋊#13063443—21no—
8C4⋊#13063443—21no—
9C4⋊#13063443C2212yescomplemented
10#19149736—21no—
11C4×#13063443C2212yescomplemented
12C2×D4×D173C411yescharacteristic
13C22×C4×D173—21no—
14C4×D692C411yescharacteristic
15D4×Dic173C2211yescharacteristic
16#19149641C2211yescharacteristic
17C4×C22⋊D173—21no—
18#19149639—21no—
19C4⋊C4×D173C411yescharacteristic
20C22⋊C4×D173—21no—
21C42×D173C2211yescharacteristic
22C4×D4×C173C411yescharacteristic
23#14583836—42no—
24C2×#13063443—22no—
25C2×#13063443—22no—
26C23×D173—21no—
27C2×D692C2×C411yescharacteristic
28C2×C22⋊D173—21no—
29D4×D173—22no—
30D4×D173—22no—
31C22×Dic173—21no—
32C2×C4×D173—21no—
33C2×C4×D173—21no—
34C2×C4×D173—21no—
35C2×C4×D173—42no—
36C2×C4×D173—42no—
37C2×C4×D173C2×C411yescharacteristic
38C2×C4×D173D411yescharacteristic
39C2×C4×D173D411yescharacteristic
40C4⋊Dic173C2×C411yescharacteristic
41#14583805—21no—
42C22⋊Dic173—21no—
43#14583803—21no—
44#14583803—21no—
45C4×Dic173D411yescharacteristic
46C4×Dic173D411yescharacteristic
47C4×Dic173C2311yescharacteristic
48C2×D4×C173C2×C411yescharacteristic
49C22×C4×C173—21no—
50C4⋊C4×C173C2×C411yescharacteristic
51C22⋊C4×C173—21no—
52C42×C173C2×C411yescharacteristic
53#13063443—42no—
54#13063443—42no—
55C22×D173—24no—
56C22×D173—21no—
57C22×D173—42no—
58C22×D173—42no—
59C22×D173C22⋊C411yescharacteristic
60D692—21no—
61D692M1612yescomplemented
62C22⋊D173—24no—
63C2×Dic173—21no—
64C2×Dic173—41no—
65C2×Dic173—41no—
66C2×Dic173C22⋊C411yescharacteristic
67C2×Dic173C2×D411yescharacteristic
68C2×Dic173C2×D411yescharacteristic
69C4×D173—21no—
70C4×D173—21no—
71C4×D173—41no—
72C4×D173—42no—
73C4×D173—42no—
74C4×D173—41no—
75C4×D173—41no—
76C4×D173D812yes—
77C4×D173SD1612yes—
78C23×C173—21no—
79D4×C173—21no—
80D4×C173M1612yescomplemented
81C2×C4×C173—21no—
82C2×C4×C173—42no—
83C2×C4×C173C22⋊C411yescharacteristic
84C2×C4×C173C22⋊C411yescharacteristic
85C2×C4×C173C22×C411yescharacteristic
86C22×C173—22no—
87C22×C173—41no—
88C22×C173—41no—
89C22×C173C2×C22⋊C411yescharacteristic
90C2×D173—21no—
91C2×D173—21no—
92C2×D173—22no—
93C2×D173—48no—
94C2×D173—41no—
95C2×D173—41no—
96C2×D173#10212yes—
97C4×C173—21no—
98C4×C173—41no—
99C4×C173#10212yes—
100C4×C173C2×M1611yescharacteristic
101C4×C173C2×M1611yescharacteristic
102Dic173—21no—
103Dic173—21no—
104Dic173—41no—
105Dic173C2×D811yescharacteristic
106Dic173C2×SD1611yescharacteristic
107C2×C173—42no—
108C2×C173#40711yescharacteristic
109C2×C173C2×#10211yescharacteristic
110C2×C173#42111yescharacteristic
111D173—22no—
112D173—42no—
113C173#148611yescharacteristic, complemented, Sylow
114#1486—1731nomaximal, Sylow
115C2×C4×D4—1731no—
116#431—1731no—
117#423—1731no—
118D4⋊C8—1732no—
119D4⋊C8—1732no—
120C22×D4—1731no—
121C23×C4—3461no—
122C2×M16—3461no—
123C4×D4—3461no—
124C4×D4—3461no—
125C4×D4—1731no—
126C4×D4—1731no—
127C4×D4—1731no—
128C4×D4#1062192411yescharacteristic
129C2×C4⋊C4—1731no—
130C2×C22⋊C4—3461no—
131C2×C42—1731no—
132C4⋊C8—3461no—
133C4⋊C8—3461no—
134C4⋊C8—1732no—
135C22⋊C8—3461no—
136C4×C8—1732no—
137C24—3461no—
138C2×D4—3462no—
139C2×D4—3462no—
140C2×D4—3461no—
141C2×D4—1731no—
142C2×D4C2×#1062192411yescharacteristic
143C22×C4—21no—
144C22×C4—3461no—
145C22×C4—3461no—
146C22×C4—3461no—
147C22×C4—3461no—
148C22×C4—1731no—
149C22×C4—1731no—
150C22×C4—1731no—
151C22×C4—6922no—
152C22×C4—6922no—
153M16—6922no—
154C2×C8—3462no—
155C2×C8—3462no—
156C4⋊C4—3461no—
157C4⋊C4—1731no—
158C4⋊C4C2×#1062192411yescharacteristic
159C22⋊C4—21no—
160C22⋊C4—3461no—
161C22⋊C4—3461no—
162C22⋊C4—3461no—
163C42—1731no—
164C42—1731no—
165C42—1731no—
166C42C2×#1062192411yescharacteristic
167C23—21no—
168C23—3464no—
169C23—3461no—
170C23—1731no—
171C23—6922no—
172C23—6922no—
173D4—21no—
174D4—3464no—
175D4—3461no—
176D4—1732no—
177D4#1458383612yescomplemented
178C2×C4—21no—
179C2×C4—42no—
180C2×C4—3461no—
181C2×C4—3461no—
182C2×C4—3461no—
183C2×C4—1731no—
184C2×C4—1731no—
185C2×C4—1731no—
186C2×C4—1732no—
187C2×C4—1732no—
188C2×C4—6921no—
189C2×C4—6921no—
190C2×C4—6921no—
191C2×C4—6922no—
192C2×C4—6921no—
193C2×C4—6921no—
194C2×C4—6922no—
195C2×C4C22⋊#1062192411yescharacteristic
196C2×C4C22⋊#1062192411yescharacteristic
197C2×C4C22×#1062192411yescharacteristic
198C8—6922no—
199C8—6922no—
200C22—22no—
201C22—41no—
202C22—41no—
203C22—3461no—
204C22—3461no—
205C22—3462no—
206C22—1732no—
207C22—6928no—
208C22—6921no—
209C22—6921no—
210C22C2×C22⋊#1062192411yescharacteristic
211C4—21no—
212C4—41no—
213C4—3461no—
214C4—3461no—
215C4—1731no—
216C4—1731no—
217C4—6921no—
218C4#1914974112yes—
219C4D173⋊M1611yescharacteristic
220C4C2×#1458383611yescharacteristic
221C2—42no—
222C2—3462no—
223C2—6922no—
224C2#2601246211yescharacteristic
225C2C2×#1914974111yescharacteristic
226C2#2601248811yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
FittingC4×D4×C173C4
socleC22×C173C2×C22⋊C4
prime radicalC2×D4×D173C4
cosocleC4×Dic173C23
hypercenterC4×D4#10621924
hypocenterC173#1486
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. #26012510C2semidirect
2. #26012495C2semidirect
3. #26012339C2semidirect
4. D4⋊#13063443C2semidirect
9. C4⋊#13063443C22semidirect
11. C4×#13063443C22semidirect
61. D692M16semidirect
80. D4×C173M16semidirect
113. C173#1486semidirect
177. D4#14583836semidirect

Maximal subgroups

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

subgroupindex
#260125102
#260124952
#260123392
D4⋊#130634432
C4×D4×D1732
#1486173