Finite Group Databasebeta

← #27157686

Subgroups — #27157686

Order 12416.

360 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 88 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×C4⋊#87604C211yesmaximal, characteristic
2C2×#15690878C212yesmaximal, complemented
3#20261677C211yesmaximal, characteristic
4#20261609C212yesmaximal, complemented
5C22⋊C8×D97C211yesmaximal, characteristic
6C23×#87604—21no—
7C2×C4⋊#87604—24no—
8C2×C4⋊#87604—21no—
9C2×C4⋊#87604C2211yescharacteristic
10C2×C4⋊#87604C2211yescharacteristic
11C2×C4⋊#87604C2212yes—
12C2×C22⋊#87604—21no—
13#15690878—24no—
14(C2×#87604)⋊C4—21no—
15C22×C4×D97C2211yescharacteristic
16C2×C8×D97C2212yes—
17C22⋊#11715895C411yescharacteristic, complemented
18#15690722C412yescomplemented
19C22⋊C8×C97C411yescharacteristic, complemented
20C22×#87604—24no—
21C22×#87604—22no—
22C22×#87604—21no—
23C22×#87604—21no—
24C22×#87604—42no—
25C22×#87604—42no—
26C4⋊#87604—24no—
27C4⋊#87604—22no—
28C4⋊#87604—22no—
29C4⋊#87604—22no—
30C4⋊#87604D414yes—
31C22⋊#87604—42no—
32C23×D97D411yescharacteristic
33C22×Dic97Q811yescharacteristic
34C2×C4×D97—22no—
35C2×C4×D97—22no—
36C2×C4×D97C2×C411yescharacteristic
37C2×C4×D97D411yescharacteristic
38C2×C4×D97Q811yescharacteristic
39C2×C4×D97C2311yescharacteristic
40C2×#11715895C2×C412yescomplemented
41C8×D97—22no—
42C8×D97—22no—
43C22×C4×C97C2×C411yescharacteristic
44C2×C8×C97C2×C412yescomplemented
45C23×C97C4⋊C411yescharacteristic
46C22×D97—21no—
47C22×D97—21no—
48C22×D97—21no—
49C22×D97—21no—
50C22×D97C4⋊C411yescharacteristic
51C22×D97SD1612yes—
52C22×D97SD1612yes—
53C22×D97C2×D411yescharacteristic
54C2×#87604—24no—
55C2×#87604—48no—
56C2×#87604—42no—
57C2×#87604—42no—
58C2×#87604—41no—
59C2×#87604—41no—
60C2×#87604—41no—
61C2×#87604—41no—
62C2×#87604—41no—
63C2×#87604—41no—
64C2×C4×C97—22no—
65C2×C4×C97C4⋊C411yescharacteristic
66C2×C4×C97C22×C411yescharacteristic
67C2×Dic97—22no—
68C2×Dic97C4⋊C411yescharacteristic
69C2×Dic97C2×Q811yescharacteristic
70C4×D97—21no—
71C4×D97—21no—
72C4×D97—22no—
73C4×D97—22no—
74C4×D97C2×D412yes—
75C4×D97#4112yes—
76C8×C97—22no—
77#11715895—22no—
78C22×C97—21no—
79C22×C97—21no—
80C22×C97#10612yescomplemented
81C22×C97C2×C4⋊C411yescharacteristic
82C2×D97—21no—
83C2×D97—21no—
84C2×D97—22no—
85C2×D97—22no—
86C2×D97—22no—
87C2×D97—22no—
88C2×D97#10612yescomplemented
89C2×D97#12011yescharacteristic
90C2×D97#12311yescharacteristic
91C2×D97C2×SD1611yescharacteristic
92C2×D97C2×SD1611yescharacteristic
93C2×D97AGL(1,ℤ/8)11yescharacteristic
94C2×D97#13711yescharacteristic
95#87604—44no—
96#87604—81no—
97#87604—81no—
98C4×C97—21no—
99C4×C97C4×D412yes—
100Dic97—21no—
101Dic97C22⋊Q812yes—
102C2×C97—22no—
103C2×C97#38511yescharacteristic
104C2×C97C2×#10611yescharacteristic
105C2×C97#42811yescharacteristic
106D97—22no—
107D97#44811yescharacteristic
108D97#45011yescharacteristic
109D97#48111yescharacteristic
110D97#48311yescharacteristic
111#1853—971nomaximal, Sylow
112C97#185311yescharacteristic, complemented, Sylow
113C22×C4⋊C4—971no—
114C2×#106—972no—
115C2×C22⋊C8—971no—
116#380—971no—
117#340—972no—
118C23×C4—971no—
119C23×C4—1941no—
120C22×C8—972no—
121C2×C4⋊C4—972no—
122C2×C4⋊C4—971no—
123C2×C4⋊C4—971no—
124C2×C4⋊C4—1944no—
125C2×C4⋊C4—1941no—
126C2×C22⋊C4—1941no—
127#106—1944no—
128C22⋊C8—971no—
129C22⋊C8—972no—
130C22⋊C8#8760411yescharacteristic, complemented
131#95—1941no—
132C24—971no—
133C22×C4—971no—
134C22×C4—971no—
135C22×C4—971no—
136C22×C4—971no—
137C22×C4—971no—
138C22×C4—3882no—
139C22×C4—3882no—
140C22×C4—1942no—
141C22×C4—1942no—
142C22×C4—1941no—
143C22×C4—1941no—
144C22×C4—1942no—
145C22×C4—1944no—
146C22×C4C2×#8760411yescharacteristic
147C2×C8—972no—
148C2×C8—1942no—
149C2×C8—1942no—
150C2×C8C2×#8760412yescomplemented
151C4⋊C4—974no—
152C4⋊C4—1942no—
153C4⋊C4—1942no—
154C4⋊C4—1942no—
155C4⋊C4—1944no—
156C22⋊C4—3882no—
157C23—971no—
158C23—972no—
159C23—972no—
160C23—971no—
161C23—1941no—
162C23—1941no—
163C23—1941no—
164C23—1941no—
165C23C4⋊#8760411yescharacteristic
166C2×C4—22no—
167C2×C4—971no—
168C2×C4—972no—
169C2×C4—972no—
170C2×C4—971no—
171C2×C4—3881no—
172C2×C4—3881no—
173C2×C4—3881no—
174C2×C4—3881no—
175C2×C4—3881no—
176C2×C4—3881no—
177C2×C4—3882no—
178C2×C4—3882no—
179C2×C4—3888no—
180C2×C4—1942no—
181C2×C4—1941no—
182C2×C4—1941no—
183C2×C4—1942no—
184C2×C4—1944no—
185C2×C4—1942no—
186C2×C4C4⋊#8760411yescharacteristic
187C2×C4C22×#8760411yescharacteristic
188C8—22no—
189C8—1942no—
190C22—21no—
191C22—21no—
192C22—971no—
193C22—971no—
194C22—971no—
195C22—971no—
196C22—971no—
197C22—971no—
198C22—972no—
199C22—1942no—
200C22—1942no—
201C22—1942no—
202C22—1942no—
203C22—1941no—
204C22—1941no—
205C22#1569087812yescomplemented
206C22C2×C4⋊#8760411yescharacteristic
207C4—21no—
208C4—7761no—
209C4—7761no—
210C4—972no—
211C4—3884no—
212C4—1941no—
213C4D4×#8760412yes—
214C2—22no—
215C2—971no—
216C2—971no—
217C2—971no—
218C2—971no—
219C2—1942no—
220C2#2026168211yescharacteristic
221C2C2×#1569087811yescharacteristic
222C2#2026174611yescharacteristic
subgroupG/H
centerC22C2×C4⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC22⋊C8×C97C4
socleC22×C97C2×C4⋊C4
prime radicalC23×D97D4
cosocleC2×C4×D97C23
hypercenterC22⋊C8#87604
hypocenterC97#1853
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#15690878C2semidirect
4. #20261609C2semidirect
17. C22⋊#11715895C4semidirect
18. #15690722C4semidirect
19. C22⋊C8×C97C4semidirect
40. C2×#11715895C2×C4semidirect
44. C2×C8×C97C2×C4semidirect
80. C22×C97#106semidirect
88. C2×D97#106semidirect
112. C97#1853semidirect
130. C22⋊C8#87604semidirect
150. C2×C8C2×#87604semidirect
205. C22#15690878semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊#876042
C2×#156908782
#202616772
#202616092
C22⋊C8×D972
#185397