Finite Group Databasebeta

← #36368894

Subgroups — #36368894

Order 22144.

246 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 68 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#26012512C211yesmaximal, characteristic, complemented
2#26012508C211yesmaximal, characteristic, complemented
3Q8⋊#13063443C212yesmaximal, complemented
4D4⋊#13063443C212yesmaximal, complemented
5#41×Dic173C211yesmaximal, characteristic
6#19149746—21no—
7C4⋊#13063443C2212yes—
8C22⋊#13063443—21no—
9C22⋊#13063443—21no—
10#19149734—21no—
11C4×#13063443C2212yes—
12Q8×Dic173C411yescharacteristic
13Q8×Dic173C2211yescharacteristic
14D4×Dic173—21no—
15D4×Dic173—21no—
16D4×Dic173C411yescharacteristic
17D4×Dic173C2211yescharacteristic
18#19149638—21no—
19#19149638C411yescharacteristic
20C2×C4×Dic173—21no—
21C2×C4×Dic173C2211yescharacteristic
22C2×#41×C173C411yescharacteristic
23C2×#13063443—22no—
24C2×#13063443—22no—
25C22×Dic173—21no—
26C22×Dic173—21no—
27C22×Dic173D411yescharacteristic
28C22×Dic173D411yescharacteristic
29C4⋊Dic173—21no—
30C4⋊Dic173—21no—
31C4⋊Dic173C2×C411yescharacteristic
32C4⋊Dic173C2×C411yescharacteristic
33C22⋊Dic173—21no—
34C22⋊Dic173—21no—
35C22⋊Dic173—21no—
36C4×Dic173—21no—
37C4×Dic173—21no—
38C4×Dic173—21no—
39C4×Dic173C2×C411yescharacteristic
40C4×Dic173D411yescharacteristic
41C4×Dic173D411yescharacteristic
42C4×Dic173C2311yescharacteristic
43#41×C173—22no—
44#41×C173—22no—
45C2×Q8×C173C2×C411yescharacteristic
46C2×D4×C173—21no—
47C2×D4×C173C2×C411yescharacteristic
48C22×C4×C173—21no—
49C22×C4×C173C2×C411yescharacteristic
50#13063443—42no—
51#13063443—42no—
52C2×Dic173—22no—
53C2×Dic173—21no—
54C2×Dic173—21no—
55C2×Dic173—41no—
56C2×Dic173—42no—
57C2×Dic173—42no—
58C2×Dic173C22⋊C411yescharacteristic
59C2×Dic173C22⋊C411yescharacteristic
60C2×Dic173C2×D411yescharacteristic
61C2×Dic173C2×D411yescharacteristic
62C23×C173—21no—
63C23×C173C22⋊C411yescharacteristic
64Q8×C173—21no—
65Q8×C173M1612yes—
66D4×C173—24no—
67D4×C173—21no—
68D4×C173M1612yes—
69C2×C4×C173—24no—
70C2×C4×C173—21no—
71C2×C4×C173—21no—
72C2×C4×C173—21no—
73C2×C4×C173C22⋊C411yescharacteristic
74C2×C4×C173C22×C411yescharacteristic
75C22×C173—21no—
76C22×C173—22no—
77C22×C173—21no—
78C22×C173—21no—
79C22×C173—41no—
80C22×C173—41no—
81C22×C173C2×C22⋊C411yescharacteristic
82C4×C173—21no—
83C4×C173—22no—
84C4×C173C2×M1611yescharacteristic
85C4×C173C2×M1611yescharacteristic
86Dic173—21no—
87Dic173—21no—
88Dic173—21no—
89Dic173—41no—
90Dic173—41no—
91Dic173#13512yes—
92C2×C173—41no—
93C2×C173—42no—
94C2×C173#40711yescharacteristic
95C2×C173#41611yescharacteristic
96C2×C173#42111yescharacteristic
97C173#148811yescharacteristic, complemented, Sylow
98#1488—1731nomaximal, Sylow
99C4×#41—1731no—
100#432—1731no—
101#431—1731no—
102Q8⋊C8—1732no—
103D4⋊C8—1732no—
104C2×#41#1062192411yescharacteristic
105C4×Q8—1731no—
106C4×Q8—1731no—
107C4×D4—3461no—
108C4×D4—3461no—
109C4×D4—1731no—
110C4×D4—1731no—
111#117—3461no—
112#117—1731no—
113C2×C42—3461no—
114C2×C42—1731no—
115C4⋊C8—3461no—
116C4⋊C8—1732no—
117C22⋊C8—3461no—
118C22⋊C8—3461no—
119#97—3461no—
120C4×C8—1732no—
121#41—22no—
122#41—22no—
123C2×Q8C2×#1062192411yescharacteristic
124C2×D4—21no—
125C2×D4C2×#1062192411yescharacteristic
126C22×C4—21no—
127C22×C4—3461no—
128C22×C4—3461no—
129C22×C4—1731no—
130C22×C4—1731no—
131C22×C4C2×#1062192411yescharacteristic
132C2×C8—3462no—
133C2×C8—3462no—
134C4⋊C4—3461no—
135C4⋊C4—3461no—
136C4⋊C4—1731no—
137C4⋊C4—1731no—
138C22⋊C4—3461no—
139C22⋊C4—3461no—
140C22⋊C4—3461no—
141C42—3461no—
142C42—3461no—
143C42—3461no—
144C42—1731no—
145C42—1731no—
146C42—1731no—
147C42—1731no—
148C23—21no—
149C23C22⋊#1062192411yescharacteristic
150Q8—21no—
151Q8#1458383512yes—
152D4—24no—
153D4—21no—
154D4#1458383512yes—
155C2×C4—24no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—3462no—
160C2×C4—3461no—
161C2×C4—3461no—
162C2×C4—1731no—
163C2×C4—1731no—
164C2×C4—1731no—
165C2×C4—1731no—
166C2×C4—6921no—
167C2×C4—6922no—
168C2×C4—6922no—
169C2×C4C22⋊#1062192411yescharacteristic
170C2×C4C22×#1062192411yescharacteristic
171C8—6922no—
172C8—6922no—
173C22—21no—
174C22—22no—
175C22—21no—
176C22—21no—
177C22—41no—
178C22—41no—
179C22C2×C22⋊#1062192411yescharacteristic
180C4—21no—
181C4—22no—
182C4—3461no—
183C4—3461no—
184C4—3461no—
185C4—1732no—
186C4—6921no—
187C4—6921no—
188C4C2×#1458383511yescharacteristic
189C4C2×#1458383511yescharacteristic
190C2—41no—
191C2—42no—
192C2#2601246111yescharacteristic
193C2#2601248111yescharacteristic
194C2#2601248811yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
Fitting, prime radicalC2×#41×C173C4
socleC22×C173C2×C22⋊C4
cosocleC4×Dic173C23
hypercenterC2×#41#10621924
hypocenterC173#1488
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. #26012512C2semidirect
2. #26012508C2semidirect
3. Q8⋊#13063443C2semidirect
4. D4⋊#13063443C2semidirect
97. C173#1488semidirect

Maximal subgroups

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

subgroupindex
#260125122
#260125082
Q8⋊#130634432
D4⋊#130634432
#41×Dic1732
#1488173