Finite Group Databasebeta

← #27154827

Subgroups — #27154827

Order 12416.

380 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 113 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#20261467C212yesmaximal, complemented
2#20261121C212yesmaximal, complemented
3#20260959C212yesmaximal, complemented
4#20260947C212yesmaximal, complemented
5#20260928C212yesmaximal, complemented
6#20260865C211yesmaximal, characteristic, complemented
7#20260836C212yesmaximal, complemented
8C4×D776C211yesmaximal, characteristic, complemented
9#495×C97C211yesmaximal, characteristic, complemented
10C2×#13095498C2212yes—
11C2×D776C2211yescharacteristic
12C2×#13095496C2212yes—
13#15690806C2212yes—
14#15690799C2212yes—
15#15690796—22no—
16#15690794—22no—
17#15690792—22no—
18Q8×Dic97C2212yes—
19C4×Dic194C2211yescharacteristic
20C4×D388C2212yes—
21D4×Dic97C2212yes—
22#15690770—22no—
23#15690744C2211yescharacteristic
24C4⋊#11715895C2211yescharacteristic
25#15690739C2212yes—
26#15690734C2212yes—
27#15690732C2212yes—
28#15690731C2212yes—
29#15690720C2212yes—
30C2×Q16×C97C2211yescharacteristic
31C2×SD16×C97C2212yes—
32C2×D8×C97C2211yescharacteristic
33#124×C97C2212yes—
34C4×C8×C97C2211yescharacteristic
35C2×Dic194D411yescharacteristic
36C2×D388C2312yes—
37C2×C22⋊D97—22no—
38C2×C22⋊D97—22no—
39C22×Dic97—22no—
40C2×C4×D97—22no—
41#13095498—42no—
42D776—22no—
43#13095496—42no—
44C2×#11715895C2312yes—
45C4⋊Dic97—22no—
46C4⋊Dic97D411yescharacteristic
47C4⋊Dic97C2312yes—
48#13095490D412yes—
49C22⋊Dic97—22no—
50#13095488—22no—
51#13095488—22no—
52#13095488—22no—
53C4×Dic97—22no—
54C4×Dic97C2312yes—
55C2×Q8×C97C2312yes—
56C2×D4×C97C2312yes—
57Q16×C97—41no—
58SD16×C97—24no—
59D8×C97—41no—
60C2×C8×C97C2311yescharacteristic
61C2×C8×C97C2311yescharacteristic
62C22⋊C4×C97—22no—
63C22⋊C4×C97—22no—
64C42×C97C2311yescharacteristic
65C23×C97—22no—
66C22×D97—22no—
67Q8×C97—22no—
68Q8×C97—22no—
69D4×C97—22no—
70D4×C97—22no—
71C2×C4×C97—22no—
72C2×C4×C97C2×D411yescharacteristic
73C2×C4×C97C2×D411yescharacteristic
74C2×C4×C97C2411yescharacteristic
75C22⋊D97—44no—
76C22⋊D97—82no—
77C2×Dic97—22no—
78C2×Dic97—82no—
79C2×Dic97C2×D412yes—
80C2×Dic97C2×D412yes—
81D388—22no—
82D388—22no—
83C4×D97—82no—
84Dic194—41no—
85C8×C97—21no—
86C8×C97#4112yes—
87#11715895—22no—
88C22×C97—42no—
89C22×C97—42no—
90C22×C97—42no—
91C22×C97C22×D411yescharacteristic
92C2×D97—42no—
93C2×D97—42no—
94C2×D97—42no—
95C4×C97—21no—
96C4×C97—21no—
97C4×C97—42no—
98C4×C97C2×#4111yescharacteristic
99C4×C97H(5,𝔽2)11yescharacteristic
100Dic97—22no—
101Dic97—22no—
102Dic97—42no—
103C2×C97—82no—
104C2×C97#55011yescharacteristic
105C2×C97#57511yescharacteristic
106C2×C97#57611yescharacteristic
107D97—82no—
108#3403—971nomaximal, Sylow
109C97#340311yescharacteristic, complemented, Sylow
110#559—972no—
111#496—972no—
112#495D9711yescharacteristic, complemented
113#464—972no—
114#463—972no—
115#459—972no—
116#446—971no—
117#440—972no—
118C4×D8—971no—
119C2×Q16C2×D9711yescharacteristic
120C2×SD16—972no—
121C2×SD16C2×D9712yes—
122C2×D8—972no—
123C2×D8—971no—
124C2×D8C2×D9711yescharacteristic
125#127—972no—
126#124—972no—
127#124—1942no—
128#124C2×D9712yes—
129#123—1942no—
130#123—1942no—
131C4×Q8—971no—
132C4×Q8—972no—
133C4×D4—972no—
134C4×D4—972no—
135C4×D4—1942no—
136#107—971no—
137C4⋊C8—972no—
138C4⋊C8—971no—
139#103—972no—
140#102—972no—
141#102—972no—
142#97—972no—
143C4×C8C2×D9711yescharacteristic
144C2×Q8—971no—
145C2×Q8C22×D9712yes—
146C2×D4—972no—
147C2×D4—1942no—
148C2×D4—1942no—
149C2×D4C22×D9712yes—
150C22×C4—1942no—
151C22×C4—1942no—
152Q16—41no—
153SD16—24no—
154SD16—3882no—
155D8—41no—
156D8—3882no—
157D8—1942no—
158C2×C8—972no—
159C2×C8C22×D9711yescharacteristic
160C2×C8C22×D9711yescharacteristic
161C4⋊C4—972no—
162C4⋊C4—971no—
163C4⋊C4—972no—
164C4⋊C4—1942no—
165C22⋊C4—22no—
166C22⋊C4—22no—
167C22⋊C4—1942no—
168C22⋊C4—1942no—
169C22⋊C4—1942no—
170C22⋊C4—1942no—
171C42—972no—
172C42—1942no—
173C42C22×D9711yescharacteristic
174C23—22no—
175C23—1942no—
176Q8—22no—
177Q8—22no—
178Q8—3881no—
179D4—22no—
180D4—22no—
181D4—7762no—
182D4—3884no—
183D4—1942no—
184D4—1942no—
185C2×C4—22no—
186C2×C4—7762no—
187C2×C4—7762no—
188C2×C4—972no—
189C2×C4—972no—
190C2×C4—1942no—
191C2×C4D4×D9711yescharacteristic
192C2×C4D4×D9711yescharacteristic
193C2×C4C23×D9711yescharacteristic
194C8—21no—
195C8—1942no—
196C8#1309551012yes—
197C22—42no—
198C22—42no—
199C22—42no—
200C22—3882no—
201C22—3882no—
202C22—3882no—
203C22C2×D4×D9711yescharacteristic
204C4—21no—
205C4—21no—
206C4—42no—
207C4—3882no—
208C4—1942no—
209C4—1942no—
210C4C2×#1309551011yescharacteristic
211C4#1569085511yescharacteristic
212C2—82no—
213C2—7762no—
214C2#2026140511yescharacteristic
215C2#2026153211yescharacteristic
216C2#2026153711yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#495×C97C2
socleC22×C97C22×D4
hypercenter#495D97
hypocenterC97#3403
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #495

Complements

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

NH ≅ G/Ntype
1. #20261467C2semidirect
2. #20261121C2semidirect
3. #20260959C2semidirect
4. #20260947C2semidirect
5. #20260928C2semidirect
6. #20260865C2semidirect
7. #20260836C2semidirect
8. C4×D776C2semidirect
9. #495×C97C2semidirect
109. C97#3403semidirect
112. #495D97semidirect

Maximal subgroups

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

subgroupindex
#202614672
#202611212
#202609592
#202609472
#202609282
#202608652
#202608362
C4×D7762
#495×C972
#340397