Finite Group Databasebeta

← #27142942

Subgroups — #27142942

Order 12416.

316 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 85 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
1C2×#15690790C211yesmaximal, characteristic
2C2×#15690743C211yesmaximal, characteristic, complemented
3C2×#15690735C211yesmaximal, characteristic, complemented
4C2×#15690733C211yesmaximal, characteristic, complemented
5C2×#15690722C211yesmaximal, characteristic
6#20260592C211yesmaximal, characteristic
7#340×C97C211yesmaximal, characteristic, complemented
8C22×Dic194C2211yescharacteristic
9C22×C4×D97C2211yescharacteristic
10C22×#11715895C2211yescharacteristic
11#15690790—22no—
12#15690790—22no—
13C2×C4⋊Dic97—21no—
14C2×C4⋊Dic97C2211yescharacteristic
15C2×#13095488—21no—
16C2×#13095488—21no—
17#15690743—22no—
18#15690735—22no—
19#15690733—22no—
20#15690722—22no—
21#15690717—21no—
22C22×C8×C97C2211yescharacteristic
23C2×C4⋊C4×C97C2211yescharacteristic
24C2×C4⋊C4×C97C2211yescharacteristic
25C23×D97D411yescharacteristic
26C2×Dic194—21no—
27C2×Dic194—21no—
28C2×Dic194—21no—
29C2×Dic194—21no—
30C2×Dic194—21no—
31C2×Dic194D412yes—
32C22×Dic97—21no—
33C22×Dic97—21no—
34C22×Dic97D411yescharacteristic
35C2×C4×D97—21no—
36C2×C4×D97—21no—
37C2×C4×D97—21no—
38C2×C4×D97—21no—
39C2×C4×D97—21no—
40C2×C4×D97—21no—
41C2×#11715895—21no—
42C2×#11715895—21no—
43C2×#11715895D412yes—
44C4⋊Dic97—24no—
45C4⋊Dic97—21no—
46C4⋊Dic97—21no—
47#13095488—24no—
48#13095488—42no—
49C22×C4×C97—21no—
50C22×C4×C97—21no—
51C22×C4×C97C2311yescharacteristic
52C2×C8×C97—21no—
53C2×C8×C97—21no—
54C2×C8×C97—21no—
55C4⋊C4×C97—21no—
56C4⋊C4×C97—21no—
57C4⋊C4×C97—21no—
58C4⋊C4×C97D412yes—
59C23×C97C2×D411yescharacteristic
60C22×D97—21no—
61C22×D97—21no—
62C22×D97—21no—
63C22×D97—21no—
64C22×D97—21no—
65C22×D97—21no—
66C22×D97Q1612yes—
67C2×C4×C97—22no—
68C2×C4×C97—41no—
69C2×C4×C97—41no—
70C2×C4×C97—41no—
71C2×C4×C97—41no—
72C2×C4×C97—41no—
73C2×C4×C97C2×D411yescharacteristic
74C2×C4×C97C2×D411yescharacteristic
75C2×C4×C97C2×D411yescharacteristic
76C2×C4×C97#4111yescharacteristic
77C2×C4×C97#4111yescharacteristic
78C2×C4×C97#4111yescharacteristic
79C2×Dic97—22no—
80C2×Dic97—21no—
81C2×Dic97—21no—
82C2×Dic97—21no—
83C2×Dic97—41no—
84C2×Dic97—41no—
85C2×Dic97—41no—
86C2×Dic97—41no—
87C2×Dic97—41no—
88C4×D97—41no—
89C4×D97—41no—
90C4×D97—41no—
91C4×D97—41no—
92Dic194—22no—
93Dic194—22no—
94Dic194—41no—
95Dic194—41no—
96C8×C97—41no—
97#11715895—22no—
98C22×C97#12011yescharacteristic
99C22×C97#12111yescharacteristic
100C22×C97#12311yescharacteristic
101C22×C97C2×Q1611yescharacteristic
102C22×C97#13511yescharacteristic
103C22×C97AGL(1,ℤ/8)11yescharacteristic
104C22×C97#13711yescharacteristic
105C2×D97—22no—
106C2×D97—22no—
107C2×D97—22no—
108C2×D97—41no—
109C2×D97—41no—
110C2×D97—41no—
111C2×D97—41no—
112C4×C97—42no—
113C4×C97—81no—
114C4×C97#12111yescharacteristic
115C4×C97#12111yescharacteristic
116C4×C97#12311yescharacteristic
117C4×C97#12411yescharacteristic
118Dic97—42no—
119Dic97—41no—
120Dic97—81no—
121C2×C97#39411yescharacteristic
122C2×C97#44911yescharacteristic
123C2×C97#45111yescharacteristic
124C2×C97#46711yescharacteristic
125C2×C97#46811yescharacteristic
126C2×C97#48411yescharacteristic
127C2×C97#48511yescharacteristic
128D97—42no—
129#2036—971nomaximal, Sylow
130C97#203611yescharacteristic, complemented, Sylow
131C2×C22⋊Q8—971no—
132C2×#107—971no—
133C2×#103—971no—
134C2×#103—971no—
135C2×C22⋊C8—971no—
136#380—971no—
137#340D9711yescharacteristic, complemented
138C22×Q8—971no—
139C23×C4—971no—
140C22×C8—971no—
141C22×C8C2×D9711yescharacteristic
142C22⋊Q8—1942no—
143C22⋊Q8—1942no—
144C2×C4⋊C4—971no—
145C2×C4⋊C4—1941no—
146C2×C4⋊C4C2×D9711yescharacteristic
147C2×C4⋊C4C2×D9711yescharacteristic
148C2×C22⋊C4—1941no—
149C2×C22⋊C4—1941no—
150#107—1942no—
151#103—1942no—
152#103—1942no—
153C22⋊C8—1942no—
154#95—1941no—
155C24—971no—
156C2×Q8—972no—
157C2×Q8—1941no—
158C2×Q8—1941no—
159C2×Q8—1941no—
160C2×Q8—1941no—
161C2×Q8—1941no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—971no—
165C22×C4—1941no—
166C22×C4—1941no—
167C22×C4—1941no—
168C22×C4—1941no—
169C22×C4—1941no—
170C22×C4—1941no—
171C22×C4—1941no—
172C22×C4—1941no—
173C22×C4C22×D9711yescharacteristic
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—972no—
178C2×C8—1941no—
179C2×C8—1941no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—1944no—
184C4⋊C4—1941no—
185C4⋊C4—1941no—
186C4⋊C4D38812yes—
187C22⋊C4—3882no—
188C22⋊C4—1944no—
189C23—972no—
190C23—1941no—
191C23—1941no—
192C23—1941no—
193C23—1941no—
194C23—1941no—
195C23—1941no—
196C23D4×D9711yescharacteristic
197Q8—3881no—
198Q8—3881no—
199Q8—1942no—
200Q8—1942no—
201C2×C4—22no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—3881no—
208C2×C4—3881no—
209C2×C4—3881no—
210C2×C4—3881no—
211C2×C4—3881no—
212C2×C4—3881no—
213C2×C4—3881no—
214C2×C4—3881no—
215C2×C4—3881no—
216C2×C4—1942no—
217C2×C4—1941no—
218C2×C4—1941no—
219C2×C4—1941no—
220C2×C4D4×D9711yescharacteristic
221C2×C4D4×D9711yescharacteristic
222C2×C4C2×D38811yescharacteristic
223C2×C4#1309551111yescharacteristic
224C2×C4#1309551111yescharacteristic
225C2×C4#1309551211yescharacteristic
226C8—41no—
227C8—1942no—
228C22—3881no—
229C22—3881no—
230C22—3881no—
231C22—3881no—
232C22—1942no—
233C22—1942no—
234C22—1942no—
235C22#1569078011yescharacteristic
236C22#1569078311yescharacteristic
237C22#1569079511yescharacteristic
238C22Q16×D9711yescharacteristic
239C22#1569083111yescharacteristic
240C22#1569083411yescharacteristic
241C22#1569083911yescharacteristic
242C4—42no—
243C4—81no—
244C4—7761no—
245C4—3881no—
246C4—3882no—
247C4#1569078311yescharacteristic
248C4#1569078511yescharacteristic
249C4#1569079511yescharacteristic
250C4#1569079811yescharacteristic
251C2—3882no—
252C2#2026065311yescharacteristic
253C2#2026088111yescharacteristic
254C2#2026089211yescharacteristic
255C2#2026097511yescharacteristic
256C2#2026098011yescharacteristic
257C2#2026107211yescharacteristic
258C2#2026107911yescharacteristic
subgroupG/H
centerC23D4×D97
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
Fitting#340×C97C2
socleC23×C97C2×D4
prime radicalC23×D97D4
hypercenter#340D97
hypocenterC97#2036
LCSG ▷ C22×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
2. C2×#15690743C2semidirect
3. C2×#15690735C2semidirect
4. C2×#15690733C2semidirect
7. #340×C97C2semidirect
130. C97#2036semidirect
137. #340D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907902
C2×#156907432
C2×#156907352
C2×#156907332
C2×#156907222
#202605922
#340×C972
#203697