Finite Group Databasebeta

← #38767314

Subgroups — #38767314

Order 25472.

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

normal subgroups only

This group has 216 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#27164069C211yesmaximal, characteristic, complemented
3#27164040C211yesmaximal, characteristic, complemented
4Q8⋊(C199⋊C8)C211yesmaximal, characteristic, complemented
5#27163747C211yesmaximal, characteristic, complemented
6#27163745C211yesmaximal, characteristic, complemented
7Q8⋊C8×C199C211yesmaximal, characteristic, complemented
8C2×#15691869—21no—
9#20263950—21no—
10#20263946C411yescharacteristic
11#20263944C411yescharacteristic
12#20263940—21no—
13#20263931C411yescharacteristic
14#20263930C411yescharacteristic
15C4×D796—21no—
16C4×D796C2211yescharacteristic
17#20263912—21no—
18C42×D199C2211yescharacteristic
19C4⋊(C199⋊C8)C2211yescharacteristic
20Dic199⋊C8—21no—
21Dic199⋊C8—21no—
22(C2×D199)⋊C8—21no—
23C4×C199⋊C8C2211yescharacteristic
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
33#15691869—42no—
34C2×C199⋊C8—21no—
35C2×C199⋊C8—21no—
36C4⋊Dic199—21no—
37C4⋊Dic199C2×C411yescharacteristic
38Dic199⋊C4—21no—
39Dic199⋊C4—21no—
40(C2×D199)⋊C4—21no—
41(C2×D199)⋊C4—21no—
42(C2×D199)⋊C4—21no—
43(C2×D199)⋊C4—21no—
44(C2×D199)⋊C4—21no—
45C4×Dic199C2×C411yescharacteristic
46C4×Dic199D411yescharacteristic
47C4×Dic199D411yescharacteristic
48C2×Q8×C199C2×C411yescharacteristic
49C2×C8×C199—21no—
50C2×C8×C199—21no—
51C4⋊C4×C199—21no—
52C4⋊C4×C199C2×C411yescharacteristic
53C42×C199—21no—
54C42×C199C2311yescharacteristic
55C22×D199—21no—
56C22×D199C22⋊C411yescharacteristic
57D796—21no—
58D796—21no—
59D796—42no—
60C2×Dic199—21no—
61C2×Dic199C22⋊C411yescharacteristic
62C2×Dic199C22⋊C411yescharacteristic
63C2×Dic199C22⋊C411yescharacteristic
64C4×D199—21no—
65C4×D199—21no—
66C4×D199—21no—
67C4×D199—21no—
68C4×D199—41no—
69C4×D199—41no—
70C4×D199—41no—
71C199⋊C8—41no—
72C199⋊C8—41no—
73Q8×C199—21no—
74Q8×C199—21no—
75C2×C4×C199—21no—
76C2×C4×C199—21no—
77C2×C4×C199C22×C411yescharacteristic
78C2×C4×C199C2×D411yescharacteristic
79C2×C4×C199C2×D411yescharacteristic
80C8×C199—41no—
81C8×C199—41no—
82C22×C199C2×C22⋊C411yescharacteristic
83C2×D199—21no—
84C2×D199—21no—
85C2×D199—21no—
86C2×D199—41no—
87C2×D199—41no—
88C2×D199—41no—
89C4×C199—21no—
90C4×C199—41no—
91C4×C199—41no—
92C4×C199#13111yescharacteristic
93C4×C199#13111yescharacteristic
94C4×C199AGL(1,ℤ/8)11yescharacteristic
95C4×C199#13711yescharacteristic
96Dic199—21no—
97Dic199—21no—
98Dic199—21no—
99Dic199—41no—
100C2×C199#40811yescharacteristic
101C2×C199#41711yescharacteristic
102C2×C199#42111yescharacteristic
103D199—41no—
104D199—81no—
105C199#149511yescharacteristic, complemented, Sylow
106#1495—1991nomaximal, Sylow
107#529—1991no—
108#431—1991no—
109#423—1991no—
110Q8⋊C8—1991no—
111Q8⋊C8D19911yescharacteristic, complemented
112D4⋊C8—1991no—
113D4⋊C8—1991no—
114C2×M16—3981no—
115#126—3981no—
116#125—1991no—
117#124—1991no—
118#123—3981no—
119C22⋊Q8—1991no—
120#121—1991no—
121C4×Q8C2×D19911yescharacteristic
122C4×D4—3981no—
123C4×D4—1991no—
124#117—3981no—
125C2×C42—1991no—
126C4⋊C8—3981no—
127C4⋊C8—3981no—
128C4⋊C8—1991no—
129C4⋊C8C2×D19911yescharacteristic
130C22⋊C8—3981no—
131C4×C8—1991no—
132C4×C8C2×D19911yescharacteristic
133C2×Q8C4×D19911yescharacteristic
134C2×D4—3981no—
135C2×D4—1991no—
136C22×C4—3981no—
137C22×C4—1991no—
138C22×C4—1991no—
139C22×C4—1991no—
140M16—7962no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—3981no—
144C2×C8—3981no—
145C4⋊C4—21no—
146C4⋊C4—3981no—
147C4⋊C4—3981no—
148C4⋊C4—3981no—
149C4⋊C4—1991no—
150C4⋊C4C4×D19911yescharacteristic
151C22⋊C4—3981no—
152C22⋊C4—3981no—
153C22⋊C4—3981no—
154C22⋊C4—3981no—
155C22⋊C4—3981no—
156C42—21no—
157C42—1991no—
158C42—1991no—
159C42—1991no—
160C42C22×D19911yescharacteristic
161C23—3981no—
162C23—1991no—
163Q8—21no—
164Q8—21no—
165D4—7962no—
166D4—3981no—
167D4—3981no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—7961no—
171C2×C4—7961no—
172C2×C4—7961no—
173C2×C4—3981no—
174C2×C4—3981no—
175C2×C4—3981no—
176C2×C4—3981no—
177C2×C4—3981no—
178C2×C4—1991no—
179C2×C4—1991no—
180C2×C4—1991no—
181C2×C4C2×C4×D19911yescharacteristic
182C2×C4D4×D19911yescharacteristic
183C2×C4D4×D19911yescharacteristic
184C8—41no—
185C8—41no—
186C8—7961no—
187C8—7961no—
188C22—7961no—
189C22—7961no—
190C22—7961no—
191C22—3981no—
192C22—3981no—
193C22—3981no—
194C22C22⋊C4×D19911yescharacteristic
195C4—21no—
196C4—41no—
197C4—41no—
198C4—7961no—
199C4—3981no—
200C4—3981no—
201C4—3981no—
202C4#2026396111yescharacteristic
203C4#2026396311yescharacteristic
204C4#2026398211yescharacteristic
205C4#2026398711yescharacteristic
206C2—7961no—
207C2—15921no—
208C2#2716398311yescharacteristic
209C2#2716401811yescharacteristic
210C2#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#1495
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. #27164069C2semidirect
3. #27164040C2semidirect
4. Q8⋊(C199⋊C8)C2semidirect
5. #27163747C2semidirect
6. #27163745C2semidirect
7. Q8⋊C8×C199C2semidirect
105. C199#1495semidirect
111. Q8⋊C8D199semidirect

Maximal subgroups

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

subgroupindex
#271645552
#271640692
#271640402
Q8⋊(C199⋊C8)2
#271637472
#271637452
Q8⋊C8×C1992
#1495199