Finite Group Databasebeta

← #27142562

Subgroups — #27142562

Order 12416.

360 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 87 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#20261281C211yesmaximal, characteristic
2#20260758C211yesmaximal, characteristic, complemented
3C2×#15690738C211yesmaximal, characteristic, complemented
4C2×#15690737C211yesmaximal, characteristic, complemented
5C2×#15690736C211yesmaximal, characteristic, complemented
6C42×Dic97C211yesmaximal, characteristic, complemented
7C2×#104×C97C211yesmaximal, characteristic, complemented
8C2×#13095511C2211yescharacteristic
9C2×#13095495C2211yescharacteristic
10#15690808C411yescharacteristic, complemented
11#15690806C411yescharacteristic, complemented
12#15690799C411yescharacteristic, complemented
13#15690797C411yescharacteristic, complemented
14#15690782—22no—
15C4×C22⋊D97—22no—
16C2×C4×Dic97—22no—
17C2×C4×Dic97C2211yescharacteristic
18C2×C4×Dic97C2211yescharacteristic
19#15690739—22no—
20#15690738—22no—
21#15690737—22no—
22#15690736—22no—
23C2×#41×C97C2211yescharacteristic
24C2×M16×C97C2211yescharacteristic
25C2×C42×C97C2211yescharacteristic
26#104×C97—22no—
27#13095511—21no—
28#13095511—21no—
29#13095511—21no—
30#13095511—21no—
31C2×Dic194C2×C411yescharacteristic
32C2×D388C2×C411yescharacteristic
33C2×C22⋊D97—21no—
34C2×C22⋊D97—22no—
35C22×Dic97—21no—
36C22×Dic97D412yes—
37C2×C4×D97—21no—
38#13095495—21no—
39#13095495—21no—
40C2×#11715895—21no—
41#13095490—22no—
42C22⋊Dic97—22no—
43#13095488—22no—
44C4×Dic97—28no—
45C4×Dic97—21no—
46C4×Dic97C2×C411yescharacteristic
47C4×Dic97C2×C411yescharacteristic
48C4×Dic97D412yes—
49C4×Dic97D412yescomplemented
50#41×C97—21no—
51#41×C97—21no—
52#41×C97—21no—
53#41×C97—21no—
54C2×Q8×C97C2×C411yescharacteristic
55C2×D4×C97—21no—
56C2×D4×C97C2×C411yescharacteristic
57C22×C4×C97—21no—
58C22×C4×C97—21no—
59C22×C4×C97C2311yescharacteristic
60M16×C97—21no—
61M16×C97—21no—
62C2×C8×C97—21no—
63C42×C97—21no—
64C42×C97D412yescomplemented
65C23×C97—21no—
66C23×C97C2×D411yescharacteristic
67C22×D97—21no—
68Q8×C97—21no—
69Q8×C97—21no—
70D4×C97—21no—
71D4×C97—21no—
72D4×C97—41no—
73D4×C97—41no—
74C2×C4×C97—24no—
75C2×C4×C97—22no—
76C2×C4×C97—21no—
77C2×C4×C97—41no—
78C2×C4×C97—41no—
79C2×C4×C97C22×C411yescharacteristic
80C2×C4×C97C2×D411yescharacteristic
81C2×C4×C97C2×D411yescharacteristic
82C2×C4×C97C2×D411yescharacteristic
83C2×C4×C97#4111yescharacteristic
84C2×C4×C97#4111yescharacteristic
85C22⋊D97—41no—
86C22⋊D97—41no—
87C22⋊D97—44no—
88C2×Dic97—24no—
89C2×Dic97—22no—
90C2×Dic97—24no—
91C2×Dic97—21no—
92C2×Dic97C22⋊C412yes—
93C2×Dic97C22⋊C412yes—
94D388—21no—
95D388—21no—
96C4×D97—41no—
97C4×D97—41no—
98Dic194—21no—
99Dic194—21no—
100C8×C97—41no—
101#11715895—41no—
102C22×C97—22no—
103C22×C97—41no—
104C22×C97—42no—
105C22×C97C2×C22⋊C411yescharacteristic
106C22×C97#12111yescharacteristic
107C22×C97#12111yescharacteristic
108C2×D97—41no—
109C2×D97—42no—
110C4×C97—24no—
111C4×C97—41no—
112C4×C97C4×D411yescharacteristic
113C4×C97C4×D411yescharacteristic
114C4×C97#12411yescharacteristic
115C4×C97#12711yescharacteristic
116Dic97—24no—
117Dic97—22no—
118Dic97—41no—
119Dic97#10414yescomplemented
120C2×C97—21no—
121C2×C97—21no—
122C2×C97—81no—
123C2×C97#39011yescharacteristic
124C2×C97C2×#10412yes—
125D97—81no—
126#1962—971nomaximal, Sylow
127C97#196211yescharacteristic, complemented, Sylow
128#532—971no—
129#423—971no—
130C2×#104—971no—
131C2×#104—971no—
132C2×#104—971no—
133C2×#104D9711yescharacteristic, complemented
134C43—971no—
135C2×#41—971no—
136C2×#41C2×D9711yescharacteristic
137C2×M16—971no—
138C2×M16C2×D9711yescharacteristic
139C4⋊Q8—971no—
140#127—971no—
141#124—971no—
142#124—971no—
143#121—1942no—
144C4×D4—1942no—
145C2×C42—971no—
146C2×C42—971no—
147C2×C42—1942no—
148C2×C42C2×D9711yescharacteristic
149C4⋊C8—1942no—
150#104—22no—
151#104—1942no—
152#104—1942no—
153#104—1942no—
154#41—21no—
155#41—21no—
156#41—21no—
157#41—21no—
158#41—1941no—
159#41—1941no—
160#41—1941no—
161#41—1941no—
162C2×Q8—971no—
163C2×Q8C4×D9711yescharacteristic
164C2×D4—21no—
165C2×D4—971no—
166C2×D4—1942no—
167C2×D4—1941no—
168C2×D4C4×D9711yescharacteristic
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—972no—
172C22×C4—1941no—
173C22×C4—1941no—
174C22×C4C22×D9711yescharacteristic
175M16—21no—
176M16—21no—
177M16—1941no—
178M16—1941no—
179C2×C8—21no—
180C2×C8—1941no—
181C4⋊C4—1942no—
182C22⋊C4—1942no—
183C22⋊C4—1942no—
184C42—21no—
185C42—971no—
186C42—971no—
187C42—972no—
188C42—972no—
189C42—1948no—
190C42—1941no—
191C42C22⋊D9712yescomplemented
192C23—21no—
193C23—1941no—
194C23D4×D9711yescharacteristic
195Q8—21no—
196Q8—21no—
197Q8—1941no—
198Q8—1941no—
199D4—21no—
200D4—21no—
201D4—41no—
202D4—41no—
203D4—3884no—
204D4—3881no—
205D4—3881no—
206D4—1941no—
207D4—1941no—
208C2×C4—24no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—972no—
214C2×C4—972no—
215C2×C4—3881no—
216C2×C4—3881no—
217C2×C4—1944no—
218C2×C4—1941no—
219C2×C4—1944no—
220C2×C4—1942no—
221C2×C4C2×C4×D9711yescharacteristic
222C2×C4D4×D9711yescharacteristic
223C2×C4D4×D9711yescharacteristic
224C2×C4C2×C22⋊D9711yescharacteristic
225C2×C4#1309551011yescharacteristic
226C2×C4#1309551111yescharacteristic
227C8—41no—
228C8—3881no—
229C22—22no—
230C22—41no—
231C22—42no—
232C22—3881no—
233C22—3882no—
234C22C22⋊C4×D9711yescharacteristic
235C22#1569078211yescharacteristic
236C22#1569078311yescharacteristic
237C4—24no—
238C4—41no—
239C4—974no—
240C4—3881no—
241C4—1944no—
242C4—1942no—
243C4#1569077011yescharacteristic
244C4C4×C22⋊D9711yescharacteristic
245C4#1569079611yescharacteristic
246C4#1569080611yescharacteristic
247C2—21no—
248C2—21no—
249C2—81no—
250C2—7761no—
251C2#2026063511yescharacteristic
252C2#104×D9712yes—
subgroupG/H
centerC2×C4D4×D97
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
FittingC2×#104×C97C2
socleC22×C97C2×C22⋊C4
prime radical#20261281C2
cosocleC22×C4×C97C23
hypercenterC2×#104D97
hypocenterC97#1962
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
2. #20260758C2semidirect
3. C2×#15690738C2semidirect
4. C2×#15690737C2semidirect
5. C2×#15690736C2semidirect
6. C42×Dic97C2semidirect
7. C2×#104×C97C2semidirect
10. #15690808C4semidirect
11. #15690806C4semidirect
12. #15690799C4semidirect
13. #15690797C4semidirect
49. C4×Dic97D4semidirect
64. C42×C97D4semidirect
119. Dic97#104semidirect
127. C97#1962semidirect
133. C2×#104D97semidirect
191. C42C22⋊D97semidirect

Maximal subgroups

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

subgroupindex
#202612812
#202607582
C2×#156907382
C2×#156907372
C2×#156907362
C42×Dic972
C2×#104×C972
#196297