Finite Group Databasebeta

← #27140784

Subgroups — #27140784

Order 12416.

240 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 53 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#20261068C211yesmaximal, characteristic, complemented
2#20261051C211yesmaximal, characteristic, complemented
3#20260671C211yesmaximal, characteristic, complemented
4#20260472C211yesmaximal, characteristic, complemented
5#20260469C211yesmaximal, characteristic, complemented
6#20260459C211yesmaximal, characteristic, complemented
7#538×C97C211yesmaximal, characteristic
8C2×C4⋊Dic97C2211yescharacteristic
9C2×#13095490—21no—
10C2×C22⋊Dic97—21no—
11#15690743—21no—
12#15690741—21no—
13#15690734—21no—
14#15690731—21no—
15C22⋊#11715895C2211yescharacteristic
16C22⋊#11715895C2211yescharacteristic
17#15690717C2211yescharacteristic
18#126×C97—21no—
19#124×C97—21no—
20#123×C97—21no—
21C22⋊Q8×C97C2211yescharacteristic
22#121×C97—21no—
23#121×C97C2211yescharacteristic
24#120×C97—21no—
25C4×D4×C97—21no—
26C2×C22⋊C4×C97C2211yescharacteristic
27C22×Dic97—21no—
28C22×Dic97—21no—
29C2×#11715895—21no—
30C2×#11715895—21no—
31C4⋊Dic97—21no—
32C4⋊Dic97—21no—
33#13095490—42no—
34C22⋊Dic97—42no—
35C24×C97D411yescharacteristic
36C2×Q8×C97D411yescharacteristic
37C2×D4×C97—21no—
38C2×D4×C97—21no—
39C2×D4×C97—21no—
40C2×D4×C97D411yescharacteristic
41C22×C4×C97—21no—
42C22×C4×C97D411yescharacteristic
43C22×C4×C97C2311yescharacteristic
44C4⋊C4×C97—21no—
45C4⋊C4×C97—21no—
46C4⋊C4×C97D411yescharacteristic
47C4⋊C4×C97D411yescharacteristic
48C22⋊C4×C97—21no—
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×C97—21no—
56C23×C97—21no—
57C23×C97—21no—
58C23×C97—21no—
59C23×C97—21no—
60C23×C97—41no—
61C23×C97—41no—
62C23×C97C2×D411yescharacteristic
63Q8×C97—41no—
64D4×C97—41no—
65D4×C97—41no—
66D4×C97—41no—
67D4×C97—81no—
68D4×C97—81no—
69C2×C4×C97—21no—
70C2×C4×C97—21no—
71C2×C4×C97—21no—
72C2×C4×C97—21no—
73C2×C4×C97—41no—
74C2×C4×C97—41no—
75C2×C4×C97—81no—
76C2×C4×C97C2×D411yescharacteristic
77C2×C4×C97C2×D411yescharacteristic
78C2×Dic97—22no—
79C2×Dic97—41no—
80C2×Dic97—41no—
81C2×Dic97—41no—
82C2×Dic97—41no—
83C2×Dic97—41no—
84#11715895—41no—
85#11715895—41no—
86C22×C97—21no—
87C22×C97—21no—
88C22×C97—42no—
89C22×C97—41no—
90C22×C97—41no—
91C22×C97—41no—
92C22×C97—41no—
93C22×C97—41no—
94C22×C97—41no—
95C22×C97—81no—
96C22×C97#12011yescharacteristic
97C22×C97#13511yescharacteristic
98C22×C97#13711yescharacteristic
99C4×C97—21no—
100C4×C97—21no—
101C4×C97—41no—
102C4×C97—41no—
103C4×C97—41no—
104C4×C97—41no—
105Dic97—42no—
106Dic97—81no—
107C2×C97—21no—
108C2×C97—21no—
109C2×C97—81no—
110C2×C97—81no—
111C2×C97#45211yescharacteristic
112C2×C97#45311yescharacteristic
113C2×C97#45811yescharacteristic
114#1618—971nomaximal, Sylow
115C97#161811yescharacteristic, complemented, Sylow
116#538D9711yescharacteristic
117#484—971no—
118#481—971no—
119#399—971no—
120#328—971no—
121#327—971no—
122#323—971no—
123#126—21no—
124#124—21no—
125#123—21no—
126C22⋊Q8C2×D9711yescharacteristic
127#121—21no—
128#121C2×D9711yescharacteristic
129#120—21no—
130C4×D4—21no—
131C2×C4⋊C4—971no—
132C2×C4⋊C4—1941no—
133C2×C22⋊C4—1941no—
134C2×C22⋊C4C2×D9711yescharacteristic
135#107—1941no—
136#106—1941no—
137#103—1941no—
138#102—1941no—
139C22⋊C8—971no—
140C22⋊C8—971no—
141#95—971no—
142C24C22⋊D9711yescharacteristic
143C2×Q8C22⋊D9711yescharacteristic
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4C22⋊D9711yescharacteristic
148C22×C4—21no—
149C22×C4—1941no—
150C22×C4—1941no—
151C22×C4C22⋊D9711yescharacteristic
152C22×C4C22×D9711yescharacteristic
153C2×C8—1941no—
154C2×C8—1941no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—3882no—
158C4⋊C4—1941no—
159C4⋊C4—1941no—
160C4⋊C4C22⋊D9711yescharacteristic
161C4⋊C4C22⋊D9711yescharacteristic
162C22⋊C4—21no—
163C22⋊C4—21no—
164C22⋊C4—21no—
165C22⋊C4—21no—
166C22⋊C4—21no—
167C22⋊C4—21no—
168C22⋊C4—21no—
169C22⋊C4—3882no—
170C42—21no—
171C23—21no—
172C23—21no—
173C23—21no—
174C23—21no—
175C23—41no—
176C23—41no—
177C23C2×C22⋊D9711yescharacteristic
178Q8—41no—
179D4—41no—
180D4—41no—
181D4—41no—
182D4—81no—
183D4—81no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—81no—
191C2×C4—3881no—
192C2×C4—3881no—
193C2×C4—3881no—
194C2×C4—3881no—
195C2×C4—3881no—
196C2×C4—1942no—
197C2×C4C2×C22⋊D9711yescharacteristic
198C2×C4C2×C22⋊D9711yescharacteristic
199C8—3881no—
200C8—3881no—
201C22—21no—
202C22—21no—
203C22—42no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22—41no—
208C22—41no—
209C22—41no—
210C22—81no—
211C22#1569077911yescharacteristic
212C22#1569083011yescharacteristic
213C22#1569084011yescharacteristic
214C4—21no—
215C4—21no—
216C4—41no—
217C4—41no—
218C4—41no—
219C4—41no—
220C4—7761no—
221C4—3882no—
222C2—21no—
223C2—21no—
224C2—81no—
225C2—81no—
226C2#2026089711yescharacteristic
227C2#2026090311yescharacteristic
228C2#2026092511yescharacteristic
subgroupG/H
centerC22#15690779
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
Fitting, prime radical#538×C97C2
socleC22×C97#120
hypercenter#538D97
hypocenterC97#1618
LCSG ▷ C22×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #538

Complements

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

NH ≅ G/Ntype
1. #20261068C2semidirect
2. #20261051C2semidirect
3. #20260671C2semidirect
4. #20260472C2semidirect
5. #20260469C2semidirect
6. #20260459C2semidirect
115. C97#1618semidirect

Maximal subgroups

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

subgroupindex
#202610682
#202610512
#202606712
#202604722
#202604692
#202604592
#538×C972
#161897