Finite Group Databasebeta

← #27154822

Subgroups — #27154822

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#20261468C212yesmaximal, complemented
2#20261123C212yesmaximal, complemented
3#20261119C211yesmaximal, characteristic, complemented
4#20260956C212yesmaximal, complemented
5#20260950C212yesmaximal, complemented
6#20260929C212yesmaximal, complemented
7#20260865C211yesmaximal, characteristic, complemented
8#20260834C212yesmaximal, complemented
9C4×D8×C97C211yesmaximal, characteristic, complemented
10C2×Dic388C2211yescharacteristic
11C2×#13095500C2212yes—
12C2×#13095499C2212yes—
13C2×D776C2211yescharacteristic
14C2×#13095496C2212yes—
15#15690806C2212yes—
16#15690798C2212yes—
17#15690797C2212yes—
18#15690796—22no—
19#15690793—22no—
20#15690792—22no—
21C4×Dic194C2211yescharacteristic
22#15690775C2212yes—
23#15690772C2212yes—
24C4×C22⋊D97—22no—
25C4⋊#11715895C2211yescharacteristic
26#15690739C2212yes—
27#15690733C2212yes—
28#15690730C2212yes—
29#15690720C2212yes—
30C2×D8×C97C2211yescharacteristic
31C4×D4×C97C2212yes—
32#107×C97C2211yescharacteristic
33#102×C97C2212yes—
34C4×C8×C97C2211yescharacteristic
35C2×Dic194D411yescharacteristic
36C2×Dic194C2312yes—
37C2×D388C2312yes—
38C2×C22⋊D97—22no—
39C2×C22⋊D97—22no—
40C2×C4×D97—22no—
41Dic388—41no—
42#13095500—24no—
43#13095499—42no—
44D776—41no—
45#13095496—42no—
46C2×#11715895C2312yes—
47C4⋊Dic97D411yescharacteristic
48#13095490—22no—
49#13095490D412yes—
50C22⋊Dic97—22no—
51C22⋊Dic97—22no—
52#13095488—22no—
53#13095488—22no—
54#13095488—22no—
55C4×Dic97—22no—
56C4×Dic97C2312yes—
57C2×D4×C97C2312yes—
58C22×C4×C97—22no—
59D8×C97—22no—
60C2×C8×C97C2311yescharacteristic
61C2×C8×C97C2311yescharacteristic
62C4⋊C4×C97C2312yes—
63C22⋊C4×C97—22no—
64C42×C97C2311yescharacteristic
65C23×C97—22no—
66C22×D97—22no—
67D4×C97—22no—
68D4×C97—22no—
69C2×C4×C97—22no—
70C2×C4×C97—82no—
71C2×C4×C97C2×D411yescharacteristic
72C2×C4×C97C2×D411yescharacteristic
73C2×C4×C97C2411yescharacteristic
74C22⋊D97—44no—
75C22⋊D97—82no—
76C2×Dic97—22no—
77C2×Dic97C2×D412yes—
78C2×Dic97C2×D412yes—
79D388—22no—
80D388—22no—
81C4×D97—82no—
82Dic194—22no—
83Dic194—22no—
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#495—971no—
113#464—972no—
114#463—972no—
115#459—972no—
116#446—971no—
117#440—972no—
118C4×D8D9711yescharacteristic, complemented
119C2×Q16—971no—
120C2×SD16—972no—
121C2×SD16—972no—
122C2×D8—972no—
123C2×D8—971no—
124C2×D8C2×D9711yescharacteristic
125#127—972no—
126#124—972no—
127#124—972no—
128#124—1942no—
129#123—1942no—
130#123—1942no—
131C4×Q8—971no—
132C4×Q8—972no—
133C4×D4—972no—
134C4×D4—1942no—
135C4×D4C2×D9712yes—
136#107C2×D9711yescharacteristic
137C4⋊C8—972no—
138C4⋊C8—971no—
139#103—972no—
140#102—972no—
141#102C2×D9712yes—
142#97—972no—
143C4×C8C2×D9711yescharacteristic
144C2×Q8—971no—
145C2×Q8—972no—
146C2×D4—972no—
147C2×D4—1942no—
148C2×D4—1942no—
149C2×D4C22×D9712yes—
150C22×C4—22no—
151C22×C4—1942no—
152Q16—3881no—
153SD16—3882no—
154SD16—1944no—
155D8—22no—
156D8—3881no—
157D8—3882no—
158C2×C8—972no—
159C2×C8C22×D9711yescharacteristic
160C2×C8C22×D9711yescharacteristic
161C4⋊C4—972no—
162C4⋊C4—971no—
163C4⋊C4—1942no—
164C4⋊C4C22×D9712yes—
165C22⋊C4—22no—
166C22⋊C4—1942no—
167C22⋊C4—1942no—
168C22⋊C4—1942no—
169C22⋊C4—1942no—
170C22⋊C4—1942no—
171C42—972no—
172C42—1942no—
173C42C22×D9711yescharacteristic
174C23—22no—
175C23—1942no—
176Q8—3881no—
177Q8—1942no—
178Q8—1942no—
179D4—22no—
180D4—22no—
181D4—7762no—
182D4—3884no—
183D4—1942no—
184D4—1942no—
185C2×C4—22no—
186C2×C4—82no—
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#1309551112yes—
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×#1309551111yescharacteristic
211C4#1569085611yescharacteristic
212C2—82no—
213C2—7762no—
214C2#2026140611yescharacteristic
215C2#2026153211yescharacteristic
216C2#2026153611yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC4×D8×C97C2
socleC22×C97C22×D4
hypercenterC4×D8D97
hypocenterC97#3403
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. #20261468C2semidirect
2. #20261123C2semidirect
3. #20261119C2semidirect
4. #20260956C2semidirect
5. #20260950C2semidirect
6. #20260929C2semidirect
7. #20260865C2semidirect
8. #20260834C2semidirect
9. C4×D8×C97C2semidirect
109. C97#3403semidirect
118. C4×D8D97semidirect

Maximal subgroups

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

subgroupindex
#202614682
#202611232
#202611192
#202609562
#202609502
#202609292
#202608652
#202608342
C4×D8×C972
#340397