Finite Group Databasebeta

← #27142036

Subgroups — #27142036

Order 12416.

376 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 93 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
1C22×#13095495C211yesmaximal, characteristic, complemented
2#20261246C211yesmaximal, characteristic
3C2×#15690738C212yesmaximal, complemented
4#20260495C212yesmaximal, complemented
5C4×C22⋊C4×C97C211yesmaximal, characteristic, complemented
6C2×#13095511C2211yescharacteristic
7C2×#13095495—24no—
8C2×#13095495—21no—
9C2×#13095495C2212yes—
10C22×#11715895—21no—
11#15690793—21no—
12#15690789C411yescharacteristic, complemented
13#15690789C411yescharacteristic, complemented
14#15690784C411yescharacteristic, complemented
15#15690784C411yescharacteristic, complemented
16#15690779—21no—
17C4×C22⋊D97—21no—
18C4×C22⋊D97—21no—
19#15690769C2211yescharacteristic
20#15690738—24no—
21C23×C4×C97C2211yescharacteristic
22C2×C22⋊C4×C97—21no—
23C2×C42×C97C2212yes—
24#95×C97—21no—
25#13095511—22no—
26#13095511—22no—
27C2×Dic194C2×C411yescharacteristic
28C2×D388C2×C411yescharacteristic
29C2×C22⋊D97—21no—
30C2×C22⋊D97—21no—
31C2×C22⋊D97—21no—
32C2×C4×D97—21no—
33#13095495—24no—
34#13095495—22no—
35#13095495D414yescomplemented
36C2×#11715895—24no—
37C2×#11715895—22no—
38C4⋊Dic97C2×C411yescharacteristic
39C4⋊Dic97C2×C411yescharacteristic
40#13095490—21no—
41#13095490—21no—
42C22⋊Dic97—21no—
43C22⋊Dic97—21no—
44C22⋊Dic97—21no—
45#13095488—21no—
46#13095488—21no—
47C4×Dic97—21no—
48C24×C97D411yescharacteristic
49C22×C4×C97—22no—
50C22×C4×C97—22no—
51C22×C4×C97—22no—
52C22×C4×C97C2×C411yescharacteristic
53C22×C4×C97C2×C411yescharacteristic
54C22×C4×C97D411yescharacteristic
55C22×C4×C97D411yescharacteristic
56C22×C4×C97D411yescharacteristic
57C22×C4×C97C2311yescharacteristic
58C22⋊C4×C97—24no—
59C42×C97—22no—
60C42×C97—22no—
61C23×C97—22no—
62C23×C97—22no—
63C23×C97—41no—
64C23×C97C22⋊C411yescharacteristic
65C23×C97C22⋊C411yescharacteristic
66C23×C97C2×D411yescharacteristic
67C22×D97—21no—
68C2×C4×C97—22no—
69C2×C4×C97—22no—
70C2×C4×C97—22no—
71C2×C4×C97—21no—
72C2×C4×C97—21no—
73C2×C4×C97—21no—
74C2×C4×C97—21no—
75C2×C4×C97—24no—
76C2×C4×C97—42no—
77C2×C4×C97—42no—
78C2×C4×C97C22⋊C411yescharacteristic
79C2×C4×C97C22⋊C411yescharacteristic
80C2×C4×C97C22×C411yescharacteristic
81C2×C4×C97C2×D412yes—
82C2×C4×C97C2×D411yescharacteristic
83C2×C4×C97#4112yes—
84C22⋊D97—42no—
85C22⋊D97—42no—
86C22⋊D97—42no—
87C2×Dic97—21no—
88C2×Dic97—21no—
89C2×Dic97—21no—
90D388—22no—
91C4×D97—41no—
92C4×D97—41no—
93Dic194—22no—
94#11715895—24no—
95C22×C97—21no—
96C22×C97—21no—
97C22×C97—21no—
98C22×C97—21no—
99C22×C97—21no—
100C22×C97—21no—
101C22×C97—21no—
102C22×C97—44no—
103C22×C97#10412yescomplemented
104C22×C97C2×C22⋊C411yescharacteristic
105C22×C97#12112yes—
106C2×D97—41no—
107C2×D97—41no—
108C2×D97—41no—
109C4×C97—21no—
110C4×C97—21no—
111C4×C97—44no—
112C4×C97C4×D412yes—
113C4×C97#12011yescharacteristic
114C4×C97#12311yescharacteristic
115Dic97—41no—
116Dic97—41no—
117Dic97—41no—
118C2×C97—22no—
119C2×C97—22no—
120C2×C97—41no—
121C2×C97#38611yescharacteristic
122C2×C97C2×#10411yescharacteristic
123C2×C97#42111yescharacteristic
124D97—81no—
125#1871—971nomaximal, Sylow
126C97#187111yescharacteristic, complemented, Sylow
127C22×M16—971no—
128#525—971no—
129C2×#104—972no—
130C4×C22⋊C4D9711yescharacteristic, complemented
131#339—972no—
132C2×#41—971no—
133C23×C4C2×D9711yescharacteristic
134C2×M16—972no—
135C2×M16—1944no—
136C2×M16—1941no—
137C22×C8—1941no—
138#123—1941no—
139C22⋊Q8—971no—
140C22⋊Q8—971no—
141#121—971no—
142#121—971no—
143#120—1941no—
144C4×D4—1941no—
145C4×D4—1941no—
146#117—971no—
147C2×C22⋊C4—21no—
148C2×C42C2×D9712yes—
149#104—1944no—
150#95—21no—
151C24C22⋊D9711yescharacteristic
152#41—1942no—
153#41—1942no—
154C2×Q8—971no—
155C2×D4—971no—
156C2×D4—1941no—
157C2×D4—1941no—
158C2×D4—1941no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—1941no—
163C22×C4C4×D9711yescharacteristic
164C22×C4C4×D9711yescharacteristic
165C22×C4D38811yescharacteristic
166C22×C4D38811yescharacteristic
167C22×C4C22⋊D9711yescharacteristic
168C22×C4C22×D9711yescharacteristic
169M16—974no—
170M16—1944no—
171M16—1942no—
172C2×C8—1944no—
173C2×C8—1942no—
174C4⋊C4—971no—
175C4⋊C4—971no—
176C4⋊C4—1941no—
177C4⋊C4—1941no—
178C22⋊C4—24no—
179C22⋊C4—1941no—
180C22⋊C4—1941no—
181C22⋊C4—1941no—
182C22⋊C4—1941no—
183C22⋊C4—1941no—
184C42—22no—
185C42—22no—
186C42—1941no—
187C23—22no—
188C23—22no—
189C23—41no—
190C23—1941no—
191C23#1309548811yescharacteristic
192C23#1309548811yescharacteristic
193C23C2×C22⋊D9711yescharacteristic
194Q8—1942no—
195D4—3882no—
196D4—3882no—
197D4—3882no—
198D4—1942no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—24no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—3881no—
210C2×C4—3881no—
211C2×C4—1941no—
212C2×C4—1941no—
213C2×C4—1941no—
214C2×C4#1309548811yescharacteristic
215C2×C4#1309548811yescharacteristic
216C2×C4C2×C4×D9711yescharacteristic
217C2×C4D4×D9712yes—
218C2×C4C2×D38811yescharacteristic
219C2×C4#1309551012yes—
220C8—1944no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—44no—
229C22—3881no—
230C22—3881no—
231C22—3881no—
232C22#1569073812yescomplemented
233C22C2×#1309548811yescharacteristic
234C22#1569078212yes—
235C4—21no—
236C4—21no—
237C4—44no—
238C4—3881no—
239C4—3881no—
240C4—3881no—
241C4#1569077012yes—
242C4#1569078011yescharacteristic
243C4#1569079411yescharacteristic
244C2—22no—
245C2—22no—
246C2—41no—
247C2—7761no—
248C2#2026061211yescharacteristic
249C2C2×#1569073811yescharacteristic
250C2#2026075311yescharacteristic
subgroupG/H
centerC2×C4C2×D388
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
FittingC4×C22⋊C4×C97C2
socleC22×C97C2×C22⋊C4
prime radical#20261246C2
cosocleC22×C4×C97C23
hypercenterC4×C22⋊C4D97
hypocenterC97#1871
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#13095495C2semidirect
3. C2×#15690738C2semidirect
4. #20260495C2semidirect
5. C4×C22⋊C4×C97C2semidirect
12. #15690789C4semidirect
13. #15690789C4semidirect
14. #15690784C4semidirect
15. #15690784C4semidirect
35. #13095495D4semidirect
103. C22×C97#104semidirect
126. C97#1871semidirect
130. C4×C22⋊C4D97semidirect
232. C22#15690738semidirect

Maximal subgroups

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

subgroupindex
C22×#130954952
#202612462
C2×#156907382
#202604952
C4×C22⋊C4×C972
#187197