Finite Group Databasebeta

← #27141053

Subgroups — #27141053

Order 12416.

208 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 49 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#20261145C211yesmaximal, characteristic, complemented
2#20261130C211yesmaximal, characteristic, complemented
3#20261094C211yesmaximal, characteristic, complemented
4#20260477C211yesmaximal, characteristic, complemented
5#20260475C211yesmaximal, characteristic, complemented
6#20260475C211yesmaximal, characteristic, complemented
7#562×C97C211yesmaximal, characteristic
8C2×#13095501—21no—
9C2×#13095499—21no—
10#15690810C2211yescharacteristic
11#15690743—21no—
12#15690741—21no—
13#15690734—21no—
14#15690731—21no—
15#15690721C2211yescharacteristic
16#15690721C2211yescharacteristic
17#15690721C2211yescharacteristic
18#125×C97C2211yescharacteristic
19#124×C97C2211yescharacteristic
20#124×C97C2211yescharacteristic
21#123×C97—21no—
22#123×C97—21no—
23C22⋊Q8×C97—21no—
24C22⋊Q8×C97—21no—
25#121×C97—21no—
26#121×C97—21no—
27C2×Dic194—21no—
28#13095501—42no—
29#13095499—42no—
30C2×#11715895—21no—
31C2×#11715895—21no—
32C2×#11715895—21no—
33C4⋊Dic97—21no—
34C4⋊Dic97—21no—
35C2×Q8×C97D411yescharacteristic
36C2×Q8×C97D411yescharacteristic
37C2×D4×C97—21no—
38C2×D4×C97—21no—
39C2×D4×C97D411yescharacteristic
40C2×D4×C97D411yescharacteristic
41C22×C4×C97—21no—
42C22×C4×C97—21no—
43C4⋊C4×C97—21no—
44C4⋊C4×C97—21no—
45C4⋊C4×C97—21no—
46C4⋊C4×C97—21no—
47C4⋊C4×C97D411yescharacteristic
48C4⋊C4×C97D411yescharacteristic
49C22⋊C4×C97—21no—
50C22⋊C4×C97—21no—
51C22⋊C4×C97—21no—
52C22⋊C4×C97—21no—
53C22⋊C4×C97—21no—
54C22⋊C4×C97—21no—
55C42×C97C2311yescharacteristic
56C23×C97—21no—
57C23×C97—21no—
58Q8×C97—41no—
59Q8×C97—41no—
60D4×C97—41no—
61D4×C97—41no—
62D4×C97—81no—
63D4×C97—81no—
64C2×C4×C97—21no—
65C2×C4×C97—21no—
66C2×C4×C97—21no—
67C2×C4×C97—21no—
68C2×C4×C97—81no—
69C2×C4×C97—81no—
70C2×C4×C97C2×D411yescharacteristic
71C2×C4×C97C2×D411yescharacteristic
72C2×C4×C97C2×D411yescharacteristic
73C2×Dic97—41no—
74Dic194—41no—
75Dic194—41no—
76#11715895—22no—
77#11715895—22no—
78#11715895—41no—
79C22×C97—41no—
80C22×C97—41no—
81C22×C97—41no—
82C22×C97—41no—
83C22×C97—41no—
84C22×C97—41no—
85C22×C97#12011yescharacteristic
86C4×C97—21no—
87C4×C97—21no—
88C4×C97—21no—
89C4×C97—41no—
90C4×C97—41no—
91C4×C97—41no—
92C4×C97—41no—
93Dic97—81no—
94C2×C97—81no—
95C2×C97—81no—
96C2×C97#45311yescharacteristic
97C2×C97#45511yescharacteristic
98C2×C97#45611yescharacteristic
99#1661—971nomaximal, Sylow
100C97#166111yescharacteristic, complemented, Sylow
101#562D9711yescharacteristic
102#501—971no—
103#497—971no—
104#489—971no—
105#330—971no—
106#329—971no—
107#329—971no—
108C2×Q16—1941no—
109C2×SD16—1941no—
110C4⋊Q8—971no—
111#125C2×D9711yescharacteristic
112#124C2×D9711yescharacteristic
113#124C2×D9711yescharacteristic
114#123—21no—
115#123—21no—
116C22⋊Q8—21no—
117C22⋊Q8—21no—
118#121—21no—
119#121—21no—
120#107—1941no—
121#106—1941no—
122#103—1941no—
123#102—1941no—
124#97—971no—
125#97—971no—
126#97—971no—
127C2×Q8—1941no—
128C2×Q8C22⋊D9711yescharacteristic
129C2×Q8C22⋊D9711yescharacteristic
130C2×D4—21no—
131C2×D4—21no—
132C2×D4C22⋊D9711yescharacteristic
133C2×D4C22⋊D9711yescharacteristic
134C22×C4—21no—
135C22×C4—21no—
136Q16—3882no—
137SD16—3882no—
138C2×C8—1941no—
139C2×C8—1941no—
140C2×C8—1941no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—21no—
144C4⋊C4—21no—
145C4⋊C4—1941no—
146C4⋊C4—1941no—
147C4⋊C4C22⋊D9711yescharacteristic
148C4⋊C4C22⋊D9711yescharacteristic
149C22⋊C4—21no—
150C22⋊C4—21no—
151C22⋊C4—21no—
152C22⋊C4—21no—
153C22⋊C4—21no—
154C22⋊C4—21no—
155C42C22×D9711yescharacteristic
156C23—21no—
157C23—21no—
158Q8—41no—
159Q8—41no—
160Q8—3881no—
161Q8—3881no—
162D4—41no—
163D4—41no—
164D4—81no—
165D4—81no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—81no—
171C2×C4—81no—
172C2×C4—3881no—
173C2×C4C2×C22⋊D9711yescharacteristic
174C2×C4C2×C22⋊D9711yescharacteristic
175C2×C4C2×C22⋊D9711yescharacteristic
176C8—3881no—
177C8—1942no—
178C8—1942no—
179C22—41no—
180C22—41no—
181C22—41no—
182C22—41no—
183C22—41no—
184C22—41no—
185C22#1569077911yescharacteristic
186C4—21no—
187C4—21no—
188C4—21no—
189C4—41no—
190C4—41no—
191C4—41no—
192C4—41no—
193C4—7761no—
194C2—81no—
195C2—81no—
196C2#2026090311yescharacteristic
197C2#2026091311yescharacteristic
198C2#2026091811yescharacteristic
subgroupG/H
centerC22#15690779
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting, prime radical#562×C97C2
socleC22×C97#120
hypercenter#562D97
hypocenterC97#1661
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #562

Complements

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

NH ≅ G/Ntype
1. #20261145C2semidirect
2. #20261130C2semidirect
3. #20261094C2semidirect
4. #20260477C2semidirect
5. #20260475C2semidirect
6. #20260475C2semidirect
100. C97#1661semidirect

Maximal subgroups

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

subgroupindex
#202611452
#202611302
#202610942
#202604772
#202604752
#562×C972
#166197