Finite Group Databasebeta

← #38767257

Subgroups — #38767257

Order 25472.

212 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 65 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#27164555C211yesmaximal, characteristic
2#27164070C211yesmaximal, characteristic, complemented
3#27164069C211yesmaximal, characteristic, complemented
4Q8⋊(C199⋊C8)C211yesmaximal, characteristic, complemented
5#27163747C211yesmaximal, characteristic, complemented
6#27163745C211yesmaximal, characteristic, complemented
7Q8⋊C8×C199C211yesmaximal, characteristic, complemented
8#20263950—21no—
9#20263946C411yescharacteristic
10#20263944C411yescharacteristic
11#20263940—21no—
12#20263931C411yescharacteristic
13#20263930C411yescharacteristic
14C4×D796—21no—
15C4×D796C2211yescharacteristic
16#20263912—21no—
17C42×D199C2211yescharacteristic
18C4⋊(C199⋊C8)C2211yescharacteristic
19Dic199⋊C8—21no—
20(C2×D199)⋊C8—21no—
21(C2×D199)⋊C8—21no—
22C4×C199⋊C8C2211yescharacteristic
23C8×Dic199—21no—
24C4×Q8×C199C2211yescharacteristic
25C4⋊C8×C199C2211yescharacteristic
26C4×C8×C199C2211yescharacteristic
27C2×D796—21no—
28C2×D796C2×C411yescharacteristic
29C2×C4×D199—21no—
30C2×C4×D199C2×C411yescharacteristic
31C2×C4×D199D411yescharacteristic
32C2×C4×D199D411yescharacteristic
33C2×C199⋊C8—21no—
34C2×C199⋊C8—21no—
35C4⋊Dic199—21no—
36C4⋊Dic199C2×C411yescharacteristic
37Dic199⋊C4—21no—
38Dic199⋊C4—21no—
39(C2×D199)⋊C4—21no—
40(C2×D199)⋊C4—21no—
41(C2×D199)⋊C4—21no—
42(C2×D199)⋊C4—21no—
43(C2×D199)⋊C4—21no—
44C4×Dic199C2×C411yescharacteristic
45C4×Dic199D411yescharacteristic
46C4×Dic199D411yescharacteristic
47C2×Q8×C199C2×C411yescharacteristic
48C2×C8×C199—21no—
49C2×C8×C199—21no—
50C4⋊C4×C199—21no—
51C4⋊C4×C199C2×C411yescharacteristic
52C42×C199—21no—
53C42×C199C2311yescharacteristic
54C22×D199—21no—
55C22×D199C22⋊C411yescharacteristic
56D796—21no—
57D796—21no—
58D796—42no—
59C2×Dic199—21no—
60C2×Dic199C22⋊C411yescharacteristic
61C2×Dic199C22⋊C411yescharacteristic
62C2×Dic199C22⋊C411yescharacteristic
63C4×D199—21no—
64C4×D199—21no—
65C4×D199—21no—
66C4×D199—21no—
67C4×D199—41no—
68C4×D199—41no—
69C4×D199—41no—
70C199⋊C8—41no—
71C199⋊C8—41no—
72Q8×C199—21no—
73Q8×C199—21no—
74C2×C4×C199—21no—
75C2×C4×C199—21no—
76C2×C4×C199C22×C411yescharacteristic
77C2×C4×C199C2×D411yescharacteristic
78C2×C4×C199C2×D411yescharacteristic
79C8×C199—41no—
80C8×C199—41no—
81C22×C199C2×C22⋊C411yescharacteristic
82C2×D199—21no—
83C2×D199—21no—
84C2×D199—21no—
85C2×D199—41no—
86C2×D199—41no—
87C2×D199—41no—
88C4×C199—21no—
89C4×C199—41no—
90C4×C199—41no—
91C4×C199#13111yescharacteristic
92C4×C199#13111yescharacteristic
93C4×C199#13511yescharacteristic
94C4×C199#13511yescharacteristic
95Dic199—21no—
96Dic199—21no—
97Dic199—21no—
98Dic199—41no—
99C2×C199#40811yescharacteristic
100C2×C199#41611yescharacteristic
101C2×C199#42111yescharacteristic
102D199—41no—
103D199—81no—
104C199#148211yescharacteristic, complemented, Sylow
105#1482—1991nomaximal, Sylow
106#529—1991no—
107#431—1991no—
108#431—1991no—
109Q8⋊C8—1991no—
110Q8⋊C8D19911yescharacteristic, complemented
111D4⋊C8—1991no—
112D4⋊C8—1991no—
113#126—3981no—
114#125—1991no—
115#124—1991no—
116#123—3981no—
117C22⋊Q8—1991no—
118#121—1991no—
119C4×Q8C2×D19911yescharacteristic
120C4×D4—3981no—
121C4×D4—1991no—
122#117—3981no—
123C2×C42—1991no—
124C4⋊C8—3981no—
125C4⋊C8—1991no—
126C4⋊C8C2×D19911yescharacteristic
127C22⋊C8—3981no—
128C22⋊C8—3981no—
129C4×C8—3981no—
130C4×C8—1991no—
131C4×C8C2×D19911yescharacteristic
132C2×Q8C4×D19911yescharacteristic
133C2×D4—3981no—
134C2×D4—1991no—
135C22×C4—3981no—
136C22×C4—1991no—
137C22×C4—1991no—
138C22×C4—1991no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—3981no—
142C2×C8—3981no—
143C4⋊C4—21no—
144C4⋊C4—3981no—
145C4⋊C4—3981no—
146C4⋊C4—3981no—
147C4⋊C4—1991no—
148C4⋊C4C4×D19911yescharacteristic
149C22⋊C4—3981no—
150C22⋊C4—3981no—
151C22⋊C4—3981no—
152C22⋊C4—3981no—
153C22⋊C4—3981no—
154C42—21no—
155C42—1991no—
156C42—1991no—
157C42—1991no—
158C42C22×D19911yescharacteristic
159C23—3981no—
160C23—1991no—
161Q8—21no—
162Q8—21no—
163D4—7962no—
164D4—3981no—
165D4—3981no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—7961no—
169C2×C4—7961no—
170C2×C4—7961no—
171C2×C4—3981no—
172C2×C4—3981no—
173C2×C4—3981no—
174C2×C4—3981no—
175C2×C4—3981no—
176C2×C4—1991no—
177C2×C4—1991no—
178C2×C4—1991no—
179C2×C4C2×C4×D19911yescharacteristic
180C2×C4D4×D19911yescharacteristic
181C2×C4D4×D19911yescharacteristic
182C8—41no—
183C8—41no—
184C8—7961no—
185C8—7961no—
186C22—7961no—
187C22—7961no—
188C22—7961no—
189C22—3981no—
190C22—3981no—
191C22—3981no—
192C22C22⋊C4×D19911yescharacteristic
193C4—21no—
194C4—41no—
195C4—41no—
196C4—7961no—
197C4—3981no—
198C4—3981no—
199C4—3981no—
200C4#2026396111yescharacteristic
201C4#2026396311yescharacteristic
202C4#2026397611yescharacteristic
203C4#2026397811yescharacteristic
204C2—7961no—
205C2—15921no—
206C2#2716398311yescharacteristic
207C2#2716401311yescharacteristic
208C2#2716403311yescharacteristic
subgroupG/H
centerC2×C4D4×D199
commutatorC2×C4×C199C22×C4
FrattiniC42C22×D199
FittingQ8⋊C8×C199C2
socleC22×C199C2×C22⋊C4
prime radical#20263930C4
cosocleC42×C199C23
hypercenterQ8⋊C8D199
hypocenterC199#1482
LCSG ▷ C2×C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2×C4 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. #27164070C2semidirect
3. #27164069C2semidirect
4. Q8⋊(C199⋊C8)C2semidirect
5. #27163747C2semidirect
6. #27163745C2semidirect
7. Q8⋊C8×C199C2semidirect
104. C199#1482semidirect
110. Q8⋊C8D199semidirect

Maximal subgroups

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

subgroupindex
#271645552
#271640702
#271640692
Q8⋊(C199⋊C8)2
#271637472
#271637452
Q8⋊C8×C1992
#1482199