Finite Group Databasebeta

← #36368863

Subgroups — #36368863

Order 22144.

312 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 78 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#26012509C211yesmaximal, characteristic, complemented
3#26012339C212yesmaximal, complemented
4D4⋊#13063443C212yesmaximal, complemented
5C4×D4×D173C211yesmaximal, characteristic
6#19149746—21no—
7C4⋊#13063443C2212yescomplemented
8#19149736—21no—
9#19149736—21no—
10C4×#13063443—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
23C2×#13063443—22no—
24C2×#13063443—22no—
25C23×D173—21no—
26C2×D692C2×C411yescharacteristic
27C2×C22⋊D173—21no—
28D4×D173—22no—
29D4×D173C814yescomplemented
30C22×Dic173—21no—
31C2×C4×D173—21no—
32C2×C4×D173—21no—
33C2×C4×D173—21no—
34C2×C4×D173—44no—
35C2×C4×D173C2×C411yescharacteristic
36C2×C4×D173D411yescharacteristic
37C2×C4×D173D411yescharacteristic
38C4⋊Dic173C2×C411yescharacteristic
39#14583805—21no—
40C22⋊Dic173—21no—
41#14583803—21no—
42#14583803—21no—
43C4×Dic173D411yescharacteristic
44C4×Dic173D411yescharacteristic
45C4×Dic173C2311yescharacteristic
46C2×D4×C173C2×C411yescharacteristic
47C22×C4×C173—21no—
48C4⋊C4×C173C2×C411yescharacteristic
49C22⋊C4×C173—21no—
50C42×C173C2×C411yescharacteristic
51#13063443—42no—
52#13063443—42no—
53C22×D173—21no—
54C22×D173—24no—
55C22×D173—42no—
56C22×D173—42no—
57C22×D173C22⋊C411yescharacteristic
58D692—21no—
59D692C2×C812yes—
60C22⋊D173—24no—
61C2×Dic173—21no—
62C2×Dic173—42no—
63C2×Dic173C22⋊C411yescharacteristic
64C2×Dic173C2×D411yescharacteristic
65C2×Dic173C2×D411yescharacteristic
66C4×D173—22no—
67C4×D173—41no—
68C4×D173—42no—
69C4×D173—42no—
70C4×D173—42no—
71C4×D173C2×C812yes—
72C4×D173M1612yes—
73C23×C173—21no—
74D4×C173—21no—
75D4×C173C2×C812yes—
76C2×C4×C173—21no—
77C2×C4×C173—42no—
78C2×C4×C173C22⋊C411yescharacteristic
79C2×C4×C173C22⋊C411yescharacteristic
80C2×C4×C173C22×C411yescharacteristic
81C22×C173—22no—
82C22×C173—41no—
83C22×C173—41no—
84C22×C173C2×C22⋊C411yescharacteristic
85C2×D173—21no—
86C2×D173—21no—
87C2×D173—22no—
88C2×D173—48no—
89C2×D173—41no—
90C2×D173—41no—
91C2×D173C22⋊C812yes—
92C4×C173—21no—
93C4×C173—21no—
94C4×C173—41no—
95C4×C173C22×C811yescharacteristic
96C4×C173C2×M1611yescharacteristic
97Dic173—21no—
98Dic173—41no—
99Dic173C22⋊C812yes—
100Dic173AGL(1,ℤ/8)12yes—
101C2×C173—42no—
102C2×C173C2×C22⋊C811yescharacteristic
103C2×C173#41811yescharacteristic
104C2×C173#42111yescharacteristic
105D173—22no—
106D173—42no—
107C173#147811yescharacteristic, complemented, Sylow
108#1478—1731nomaximal, Sylow
109C2×C4×D4—1731no—
110#431—1731no—
111#431—1731no—
112D4⋊C8—1732no—
113D4⋊C8—1732no—
114C22×D4—1731no—
115C23×C4—3461no—
116C4×D4—3461no—
117C4×D4—3461no—
118C4×D4—1731no—
119C4×D4—1731no—
120C4×D4—1731no—
121C4×D4#1062192411yescharacteristic
122C2×C4⋊C4—1731no—
123C2×C22⋊C4—3461no—
124C2×C42—1731no—
125C4⋊C8—3461no—
126C4⋊C8—1732no—
127C22⋊C8—3461no—
128C22⋊C8—3461no—
129C4×C8—3461no—
130C4×C8—1732no—
131C24—3461no—
132C2×D4—3462no—
133C2×D4—3461no—
134C2×D4—1734no—
135C2×D4—1731no—
136C2×D4C2×#1062192411yescharacteristic
137C22×C4—21no—
138C22×C4—3461no—
139C22×C4—3461no—
140C22×C4—3461no—
141C22×C4—3461no—
142C22×C4—1731no—
143C22×C4—1731no—
144C22×C4—1731no—
145C22×C4—6924no—
146C2×C8—3462no—
147C2×C8—3462no—
148C4⋊C4—3461no—
149C4⋊C4—1731no—
150C4⋊C4C2×#1062192411yescharacteristic
151C22⋊C4—21no—
152C22⋊C4—3461no—
153C22⋊C4—3461no—
154C22⋊C4—3461no—
155C42—1731no—
156C42—1731no—
157C42—1731no—
158C42C2×#1062192411yescharacteristic
159C23—21no—
160C23—3464no—
161C23—3461no—
162C23—1731no—
163C23—6922no—
164C23—6922no—
165D4—21no—
166D4—3464no—
167D4—3461no—
168D4—1732no—
169D4D173⋊C812yes—
170C2×C4—21no—
171C2×C4—42no—
172C2×C4—3462no—
173C2×C4—3461no—
174C2×C4—1731no—
175C2×C4—1731no—
176C2×C4—1731no—
177C2×C4—1732no—
178C2×C4—1732no—
179C2×C4—6921no—
180C2×C4—6922no—
181C2×C4—6922no—
182C2×C4—6922no—
183C2×C4—6922no—
184C2×C4C22⋊#1062192411yescharacteristic
185C2×C4C22⋊#1062192411yescharacteristic
186C2×C4C22×#1062192411yescharacteristic
187C8—6922no—
188C8—6922no—
189C22—22no—
190C22—41no—
191C22—41no—
192C22—3461no—
193C22—3461no—
194C22—3462no—
195C22—1732no—
196C22—6928no—
197C22—6921no—
198C22—6921no—
199C22C2×C22⋊#1062192411yescharacteristic
200C4—21no—
201C4—21no—
202C4—41no—
203C4—3461no—
204C4—1732no—
205C4—1732no—
206C4—6921no—
207C4C2×D173⋊C811yescharacteristic
208C4D173⋊M1611yescharacteristic
209C2—42no—
210C2—3462no—
211C2—6922no—
212C2D173⋊(C22⋊C8)11yescharacteristic
213C2#2601248311yescharacteristic
214C2#2601248811yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC4×C173C22×C8
FrattiniC2×C4C22×#10621924
FittingC4×D4×C173C4
socleC22×C173C2×C22⋊C4
prime radicalC2×D4×D173C4
cosocleC4×Dic173C23
hypercenterC4×D4#10621924
hypocenterC173#1478
LCSG ▷ 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. #26012509C2semidirect
3. #26012339C2semidirect
4. D4⋊#13063443C2semidirect
7. C4⋊#13063443C22semidirect
11. C4×#13063443C22semidirect
29. D4×D173C8semidirect
107. C173#1478semidirect

Maximal subgroups

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

subgroupindex
#260125102
#260125092
#260123392
D4⋊#130634432
C4×D4×D1732
#1478173