Finite Group Databasebeta

← #36368892

Subgroups — #36368892

Order 22144.

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

normal subgroups only

This group has 248 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
2#26012495C211yesmaximal, characteristic
3#26012341C212yesmaximal, complemented
4Q8⋊#13063443C212yesmaximal, complemented
5C4×Q8×D173C211yesmaximal, characteristic
6C2×#14583836—21no—
7C4⋊#13063443—21no—
8C4⋊#13063443—21no—
9C4⋊#13063443C2212yes—
10#19149736—21no—
11C4×#13063443C2212yes—
12C2×Q8×D173C411yescharacteristic
13Q8×Dic173C2211yescharacteristic
14C4×Dic346—21no—
15C4×Dic346C411yescharacteristic
16#19149644—21no—
17#19149644C2211yescharacteristic
18C4⋊C4×D173—21no—
19C4⋊C4×D173C411yescharacteristic
20C42×D173—21no—
21C42×D173C2211yescharacteristic
22C4×Q8×C173C411yescharacteristic
23#14583836—42no—
24C2×#13063443—22no—
25C2×#13063443—22no—
26C2×Dic346—21no—
27C2×Dic346C2×C411yescharacteristic
28Q8×D173—22no—
29Q8×D173—22no—
30C2×C4×D173—21no—
31C2×C4×D173—21no—
32C2×C4×D173C2×C411yescharacteristic
33C2×C4×D173D411yescharacteristic
34C2×C4×D173D411yescharacteristic
35C4⋊Dic173—21no—
36C4⋊Dic173C2×C411yescharacteristic
37#14583805—21no—
38#14583805—21no—
39#14583805—21no—
40C4×Dic173—21no—
41C4×Dic173—21no—
42C4×Dic173—21no—
43C4×Dic173D411yescharacteristic
44C4×Dic173D411yescharacteristic
45C4×Dic173C2311yescharacteristic
46C2×Q8×C173C2×C411yescharacteristic
47C4⋊C4×C173—21no—
48C4⋊C4×C173C2×C411yescharacteristic
49C42×C173—21no—
50C42×C173C2×C411yescharacteristic
51#13063443—42no—
52#13063443—42no—
53C22×D173C22⋊C411yescharacteristic
54Dic346—24no—
55Dic346—21no—
56Dic346M1612yescomplemented
57C2×Dic173—21no—
58C2×Dic173—21no—
59C2×Dic173C22⋊C411yescharacteristic
60C2×Dic173C2×D411yescharacteristic
61C2×Dic173C2×D411yescharacteristic
62C4×D173—24no—
63C4×D173—21no—
64C4×D173—21no—
65C4×D173—41no—
66C4×D173—42no—
67C4×D173SD1612yes—
68C4×D173Q1612yes—
69Q8×C173—21no—
70Q8×C173M1612yescomplemented
71C2×C4×C173—21no—
72C2×C4×C173—21no—
73C2×C4×C173C22⋊C411yescharacteristic
74C2×C4×C173C22⋊C411yescharacteristic
75C2×C4×C173C22×C411yescharacteristic
76C22×C173C2×C22⋊C411yescharacteristic
77C2×D173—21no—
78C2×D173—21no—
79C2×D173#10312yes—
80C4×C173—22no—
81C4×C173—21no—
82C4×C173—41no—
83C4×C173#10312yes—
84C4×C173C2×M1611yescharacteristic
85C4×C173C2×M1611yescharacteristic
86Dic173—22no—
87Dic173—21no—
88Dic173—21no—
89Dic173—41no—
90Dic173C2×SD1611yescharacteristic
91Dic173C2×Q1611yescharacteristic
92C2×C173#40711yescharacteristic
93C2×C173C2×#10311yescharacteristic
94C2×C173#42111yescharacteristic
95D173—22no—
96C173#148711yescharacteristic, complemented, Sylow
97#1487—1731nomaximal, Sylow
98C2×C4×Q8—1731no—
99#431—1731no—
100#423—1731no—
101Q8⋊C8—1732no—
102Q8⋊C8—1732no—
103C22×Q8—1731no—
104C2×M16—3461no—
105C4×Q8—3461no—
106C4×Q8—3461no—
107C4×Q8—1731no—
108C4×Q8—1731no—
109C4×Q8—1731no—
110C4×Q8#1062192411yescharacteristic
111C2×C4⋊C4—3461no—
112C2×C4⋊C4—1731no—
113C2×C42—3461no—
114C2×C42—1731no—
115C4⋊C8—3461no—
116C4⋊C8—3461no—
117C4⋊C8—1732no—
118C22⋊C8—3461no—
119C4×C8—1732no—
120C2×Q8—3461no—
121C2×Q8—3462no—
122C2×Q8—3462no—
123C2×Q8—1731no—
124C2×Q8C2×#1062192411yescharacteristic
125C22×C4—3461no—
126C22×C4—3461no—
127C22×C4—1731no—
128C22×C4—1731no—
129C22×C4—1731no—
130M16—6922no—
131C2×C8—3462no—
132C2×C8—3462no—
133C4⋊C4—21no—
134C4⋊C4—3461no—
135C4⋊C4—3461no—
136C4⋊C4—3461no—
137C4⋊C4—3461no—
138C4⋊C4—1731no—
139C4⋊C4C2×#1062192411yescharacteristic
140C42—21no—
141C42—3461no—
142C42—3461no—
143C42—3461no—
144C42—1731no—
145C42—1731no—
146C42—1731no—
147C42C2×#1062192411yescharacteristic
148C23—1731no—
149Q8—21no—
150Q8—3464no—
151Q8—3461no—
152Q8—1732no—
153Q8#1458383612yescomplemented
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—3464no—
157C2×C4—3461no—
158C2×C4—3461no—
159C2×C4—3461no—
160C2×C4—3461no—
161C2×C4—1731no—
162C2×C4—1731no—
163C2×C4—1731no—
164C2×C4—1732no—
165C2×C4—1732no—
166C2×C4—6921no—
167C2×C4—6922no—
168C2×C4C22⋊#1062192411yescharacteristic
169C2×C4C22⋊#1062192411yescharacteristic
170C2×C4C22×#1062192411yescharacteristic
171C8—6922no—
172C8—6922no—
173C22—3461no—
174C22—3461no—
175C22—1732no—
176C22C2×C22⋊#1062192411yescharacteristic
177C4—22no—
178C4—21no—
179C4—41no—
180C4—3462no—
181C4—3461no—
182C4—3461no—
183C4—1731no—
184C4—1731no—
185C4—6921no—
186C4#1914974212yes—
187C4D173⋊M1611yescharacteristic
188C4C2×#1458383611yescharacteristic
189C2—3462no—
190C2#2601246211yescharacteristic
191C2C2×#1914974211yescharacteristic
192C2#2601248811yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
FittingC4×Q8×C173C4
socleC22×C173C2×C22⋊C4
prime radicalC22×D173C22⋊C4
cosocleC4×Dic173C23
hypercenterC4×Q8#10621924
hypocenterC173#1487
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
3. #26012341C2semidirect
4. Q8⋊#13063443C2semidirect
56. Dic346M16semidirect
70. Q8×C173M16semidirect
96. C173#1487semidirect
153. Q8#14583836semidirect

Maximal subgroups

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

subgroupindex
#260125102
#260124952
#260123412
Q8⋊#130634432
C4×Q8×D1732
#1487173