Finite Group Databasebeta

← #34059986

Subgroups — #34059986

Order 19072.

264 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 92 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#23750641C211yesmaximal, characteristic, complemented
2C2×C4⋊#11924753C211yesmaximal, characteristic, complemented
3Q8⋊#11924753C212yesmaximal, complemented
4D4⋊#11924753C212yesmaximal, complemented
5#41×Dic149C211yesmaximal, characteristic
6C22×#11924753—21no—
7C4⋊#11924753—21no—
8C4⋊#11924753—21no—
9C4⋊#11924753C2212yes—
10C22⋊#11924753—21no—
11C4×#11924753C2212yes—
12Q8×Dic149C411yescharacteristic
13Q8×Dic149C2211yescharacteristic
14D4×Dic149—21no—
15D4×Dic149—21no—
16D4×Dic149C411yescharacteristic
17D4×Dic149C2211yescharacteristic
18#18011870—21no—
19#18011870C411yescharacteristic
20C2×C4×Dic149—21no—
21C2×C4×Dic149C2211yescharacteristic
22C2×#41×C149C411yescharacteristic
23C2×#11924753—22no—
24C2×#11924753—22no—
25C2×#11924753—42no—
26C22×Dic149—21no—
27C22×Dic149—21no—
28C22×Dic149D411yescharacteristic
29C22×Dic149D411yescharacteristic
30C4⋊Dic149—21no—
31C4⋊Dic149—21no—
32C4⋊Dic149C2×C411yescharacteristic
33C4⋊Dic149C2×C411yescharacteristic
34C22⋊Dic149—21no—
35C22⋊Dic149—21no—
36C22⋊Dic149—21no—
37C4×Dic149—21no—
38C4×Dic149—21no—
39C4×Dic149—21no—
40C4×Dic149C2×C411yescharacteristic
41C4×Dic149D411yescharacteristic
42C4×Dic149D411yescharacteristic
43C4×Dic149C2311yescharacteristic
44#41×C149—22no—
45#41×C149C814yescomplemented
46C2×Q8×C149C2×C411yescharacteristic
47C2×D4×C149—21no—
48C2×D4×C149C2×C411yescharacteristic
49C22×C4×C149—21no—
50C22×C4×C149C2×C411yescharacteristic
51#11924753—42no—
52#11924753—42no—
53C2×Dic149—21no—
54C2×Dic149—21no—
55C2×Dic149—21no—
56C2×Dic149—21no—
57C2×Dic149—41no—
58C2×Dic149—42no—
59C2×Dic149—41no—
60C2×Dic149—41no—
61C2×Dic149C22⋊C411yescharacteristic
62C2×Dic149C22⋊C411yescharacteristic
63C2×Dic149C2×D411yescharacteristic
64C2×Dic149C2×D411yescharacteristic
65C23×C149—21no—
66C23×C149C22⋊C411yescharacteristic
67Q8×C149—21no—
68Q8×C149C2×C812yescomplemented
69D4×C149—24no—
70D4×C149—21no—
71D4×C149C2×C812yescomplemented
72C2×C4×C149—21no—
73C2×C4×C149—24no—
74C2×C4×C149C22⋊C411yescharacteristic
75C2×C4×C149C2×C812yes—
76C2×C4×C149M1612yes—
77C2×C4×C149C22×C411yescharacteristic
78C22×C149—21no—
79C22×C149—21no—
80C22×C149—22no—
81C22×C149—41no—
82C22×C149—41no—
83C22×C149C22⋊C812yes—
84C22×C149C2×C22⋊C411yescharacteristic
85C4×C149—22no—
86C4×C149C22⋊C812yes—
87C4×C149C22×C811yescharacteristic
88C4×C149C2×M1611yescharacteristic
89Dic149—21no—
90Dic149—21no—
91Dic149—21no—
92Dic149—41no—
93Dic149—41no—
94Dic149AGL(1,ℤ/8)11yescharacteristic
95Dic149#13711yescharacteristic
96C2×C149—22no—
97C2×C149—42no—
98C2×C149C2×C22⋊C811yescharacteristic
99C2×C149#41711yescharacteristic
100C2×C149#42111yescharacteristic
101C149#147711yescharacteristic, complemented, Sylow
102#1477—1491nomaximal, Sylow
103C4×#41—1491no—
104#431—1491no—
105C2×C4⋊C8—1491no—
106Q8⋊C8—1492no—
107D4⋊C8—1492no—
108C2×#41#1059701011yescharacteristic
109C22×C8—2981no—
110C4×Q8—1491no—
111C4×Q8—1491no—
112C4×D4—2981no—
113C4×D4—2981no—
114C4×D4—1491no—
115C4×D4—1491no—
116#117—2981no—
117#117—1491no—
118C2×C42—2981no—
119C2×C42—1491no—
120C4⋊C8—2981no—
121C4⋊C8—2981no—
122C4⋊C8—1492no—
123C22⋊C8—2981no—
124C4×C8—1492no—
125#41—22no—
126#41#1192475314yescomplemented
127C2×Q8C2×#1059701011yescharacteristic
128C2×D4—21no—
129C2×D4C2×#1059701011yescharacteristic
130C22×C4—21no—
131C22×C4—2981no—
132C22×C4—2981no—
133C22×C4—1491no—
134C22×C4—1491no—
135C22×C4C2×#1059701011yescharacteristic
136C2×C8—2982no—
137C2×C8—2982no—
138C2×C8—5962no—
139C4⋊C4—2981no—
140C4⋊C4—2981no—
141C4⋊C4—1491no—
142C4⋊C4—1491no—
143C22⋊C4—2981no—
144C22⋊C4—2981no—
145C22⋊C4—2981no—
146C42—2981no—
147C42—2981no—
148C42—2981no—
149C42—1491no—
150C42—1491no—
151C42—1491no—
152C42—1491no—
153C23—21no—
154C23C22⋊#1059701011yescharacteristic
155Q8—21no—
156Q8C2×#1192475312yescomplemented
157D4—24no—
158D4—21no—
159D4C2×#1192475312yescomplemented
160C2×C4—21no—
161C2×C4—24no—
162C2×C4—2981no—
163C2×C4—2981no—
164C2×C4—2981no—
165C2×C4—2981no—
166C2×C4—5961no—
167C2×C4—5962no—
168C2×C4—5961no—
169C2×C4—5961no—
170C2×C4—1491no—
171C2×C4—1491no—
172C2×C4—1491no—
173C2×C4—1491no—
174C2×C4C22⋊#1059701011yescharacteristic
175C2×C4C2×#1192475312yes—
176C2×C4#1455874512yes—
177C2×C4C22×#1059701011yescharacteristic
178C8—5962no—
179C8—5962no—
180C22—21no—
181C22—21no—
182C22—22no—
183C22—41no—
184C22—41no—
185C22C22⋊#1192475312yes—
186C22C2×C22⋊#1059701011yescharacteristic
187C4—22no—
188C4—2981no—
189C4—2981no—
190C4—2981no—
191C4—5961no—
192C4—5961no—
193C4—1491no—
194C4—1491no—
195C4C22⋊#1192475312yes—
196C4C22×#1192475311yescharacteristic
197C4C2×#1455874511yescharacteristic
198C2—22no—
199C2—42no—
200C2C2×C22⋊#1192475311yescharacteristic
201C2#2375061611yescharacteristic
202C2#2375062211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10597010
commutatorC4×C149C22×C8
FrattiniC2×C4C22×#10597010
Fitting, prime radicalC2×#41×C149C4
socleC22×C149C2×C22⋊C4
cosocleC4×Dic149C23
hypercenterC2×#41#10597010
hypocenterC149#1477
LCSG ▷ C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. #23750641C2semidirect
2. C2×C4⋊#11924753C2semidirect
3. Q8⋊#11924753C2semidirect
4. D4⋊#11924753C2semidirect
45. #41×C149C8semidirect
68. Q8×C149C2×C8semidirect
71. D4×C149C2×C8semidirect
101. C149#1477semidirect
126. #41#11924753semidirect
156. Q8C2×#11924753semidirect
159. D4C2×#11924753semidirect

Maximal subgroups

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

subgroupindex
#237506412
C2×C4⋊#119247532
Q8⋊#119247532
D4⋊#119247532
#41×Dic1492
#1477149