Finite Group Databasebeta

← #27153170

Subgroups — #27153170

Order 12416.

370 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 109 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#20261338C212yesmaximal, complemented
2#20261038C212yesmaximal, complemented
3#20261033C212yesmaximal, complemented
4#20260986C212yesmaximal, complemented
5#20260971C211yesmaximal, characteristic, complemented
6#20260929C212yesmaximal, complemented
7#20260928C212yesmaximal, complemented
8#20260761C211yesmaximal, characteristic, complemented
9#466×C97C211yesmaximal, characteristic, complemented
10C2×D776C2211yescharacteristic
11C2×#13095496C2212yes—
12C2×#13095495C2211yescharacteristic
13#15690806C2212yes—
14#15690801C2212yes—
15#15690794—22no—
16#15690792—22no—
17#15690784C2212yes—
18#15690782—22no—
19#15690781—22no—
20D4×Dic97C2212yes—
21#15690772C2212yes—
22#15690769C2211yescharacteristic
23#15690744C2211yescharacteristic
24#15690741C2212yes—
25#15690739C2212yes—
26#15690739C2212yes—
27#15690732C2212yes—
28#15690731C2212yes—
29#15690730C2212yes—
30C2×D8×C97C2211yescharacteristic
31C22×C8×C97C2211yescharacteristic
32#121×C97C2212yes—
33#107×C97C2211yescharacteristic
34#102×C97C2212yes—
35C2×D388C2312yes—
36C2×C22⋊D97—22no—
37C2×C22⋊D97—22no—
38C2×C22⋊D97—22no—
39C22×Dic97—22no—
40C2×C4×D97—22no—
41D776—41no—
42#13095496—42no—
43#13095495—41no—
44C2×#11715895C2312yes—
45C4⋊Dic97—22no—
46C4⋊Dic97D411yescharacteristic
47C4⋊Dic97D411yescharacteristic
48C4⋊Dic97C2312yes—
49#13095490—22no—
50C22⋊Dic97—22no—
51C22⋊Dic97D412yes—
52#13095488—22no—
53#13095488—22no—
54C4×Dic97C2312yes—
55C2×D4×C97—22no—
56C2×D4×C97C2312yes—
57C22×C4×C97C2311yescharacteristic
58D8×C97—41no—
59C2×C8×C97—41no—
60C2×C8×C97C2311yescharacteristic
61C2×C8×C97C2311yescharacteristic
62C4⋊C4×C97C2312yes—
63C22⋊C4×C97—22no—
64C23×C97—22no—
65C23×C97C2×D411yescharacteristic
66C22×D97—22no—
67D4×C97—22no—
68D4×C97—22no—
69D4×C97—82no—
70C2×C4×C97—21no—
71C2×C4×C97—21no—
72C2×C4×C97—22no—
73C2×C4×C97C2×D411yescharacteristic
74C2×C4×C97C2411yescharacteristic
75C22⋊D97—82no—
76C22⋊D97—82no—
77C22⋊D97—82no—
78C2×Dic97—22no—
79C2×Dic97—82no—
80C2×Dic97C2×D412yes—
81C2×Dic97C2×D412yes—
82D388—22no—
83D388—22no—
84C4×D97—82no—
85C8×C97—21no—
86C8×C97—21no—
87#11715895—22no—
88C22×C97—21no—
89C22×C97—21no—
90C22×C97—21no—
91C22×C97—42no—
92C22×C97—42no—
93C22×C97—42no—
94C22×C97C22×D411yescharacteristic
95C2×D97—42no—
96C2×D97—42no—
97C2×D97—42no—
98C4×C97—21no—
99C4×C97—42no—
100C4×C97H(5,𝔽2)11yescharacteristic
101C4×C972−1+411yescharacteristic
102Dic97—22no—
103Dic97—22no—
104Dic97—42no—
105C2×C97—41no—
106C2×C97—82no—
107C2×C97#53711yescharacteristic
108C2×C97#57611yescharacteristic
109C2×C97#57611yescharacteristic
110D97—82no—
111#3259—971nomaximal, Sylow
112C97#325911yescharacteristic, complemented, Sylow
113#540—972no—
114#478—972no—
115#478—972no—
116#469—972no—
117#466—971no—
118#466D9711yescharacteristic, complemented
119#459—972no—
120#459—972no—
121#424—971no—
122C2×D8—972no—
123C2×D8—971no—
124C2×D8C2×D9711yescharacteristic
125C2×M16—971no—
126C22×C8C2×D9711yescharacteristic
127#127—972no—
128#125—972no—
129#123—1942no—
130#123—1942no—
131#121—972no—
132#121—1942no—
133#121—1942no—
134#121C2×D9712yes—
135C4×D4—972no—
136C4×D4—972no—
137#117—971no—
138#107—971no—
139#107C2×D9711yescharacteristic
140#106—972no—
141C4⋊C8—972no—
142C4⋊C8—972no—
143#102—972no—
144#102—972no—
145#102—972no—
146#102C2×D9712yes—
147C2×D4—22no—
148C2×D4—972no—
149C2×D4—1942no—
150C2×D4—1942no—
151C2×D4—1942no—
152C2×D4C22×D9712yes—
153C22×C4—1942no—
154C22×C4—1942no—
155C22×C4C22×D9711yescharacteristic
156D8—41no—
157D8—3881no—
158D8—3882no—
159M16—3881no—
160C2×C8—41no—
161C2×C8—972no—
162C2×C8C22×D9711yescharacteristic
163C2×C8C22×D9711yescharacteristic
164C4⋊C4—971no—
165C4⋊C4—971no—
166C4⋊C4—972no—
167C4⋊C4—1942no—
168C4⋊C4—1942no—
169C4⋊C4C22×D9712yes—
170C22⋊C4—22no—
171C22⋊C4—972no—
172C22⋊C4—1942no—
173C22⋊C4—1942no—
174C22⋊C4—1942no—
175C42—972no—
176C23—22no—
177C23—1942no—
178C23D4×D9711yescharacteristic
179D4—22no—
180D4—22no—
181D4—82no—
182D4—7762no—
183D4—7762no—
184D4—7762no—
185D4—1942no—
186D4—1942no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—7762no—
191C2×C4—7762no—
192C2×C4—972no—
193C2×C4—972no—
194C2×C4—1942no—
195C2×C4D4×D9711yescharacteristic
196C2×C4C23×D9711yescharacteristic
197C8—21no—
198C8—21no—
199C8—1942no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—42no—
204C22—42no—
205C22—42no—
206C22—3882no—
207C22—3882no—
208C22—3882no—
209C22C2×D4×D9711yescharacteristic
210C4—21no—
211C4—42no—
212C4—3882no—
213C4—1942no—
214C4—1942no—
215C4#1569085511yescharacteristic
216C4#1569085711yescharacteristic
217C2—41no—
218C2—82no—
219C2—7762no—
220C2#2026131211yescharacteristic
221C2#2026153611yescharacteristic
222C2#2026153711yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#466×C97C2
socleC22×C97C22×D4
hypercenter#466D97
hypocenterC97#3259
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. #20261338C2semidirect
2. #20261038C2semidirect
3. #20261033C2semidirect
4. #20260986C2semidirect
5. #20260971C2semidirect
6. #20260929C2semidirect
7. #20260928C2semidirect
8. #20260761C2semidirect
9. #466×C97C2semidirect
112. C97#3259semidirect
118. #466D97semidirect

Maximal subgroups

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

subgroupindex
#202613382
#202610382
#202610332
#202609862
#202609712
#202609292
#202609282
#202607612
#466×C972
#325997