Finite Group Databasebeta

← #27142566

Subgroups — #27142566

Order 12416.

320 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 83 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
6#20260581C211yesmaximal, 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×Dic97C2211yescharacteristic
17C2×C4×Dic97C2211yescharacteristic
18#15690739—22no—
19#15690738—22no—
20#15690737—22no—
21#15690736—22no—
22#15690717—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—21no—
45C4×Dic97C2×C411yescharacteristic
46C4×Dic97C2×C411yescharacteristic
47C4×Dic97D412yes—
48C4×Dic97D412yescomplemented
49#41×C97—21no—
50#41×C97—21no—
51#41×C97—21no—
52#41×C97—21no—
53C2×Q8×C97C2×C411yescharacteristic
54C2×D4×C97—21no—
55C2×D4×C97C2×C411yescharacteristic
56C22×C4×C97—21no—
57C22×C4×C97—21no—
58C22×C4×C97C2311yescharacteristic
59M16×C97—21no—
60M16×C97—21no—
61C2×C8×C97—21no—
62C42×C97—21no—
63C42×C97D412yescomplemented
64C23×C97—21no—
65C23×C97C2×D411yescharacteristic
66C22×D97—21no—
67Q8×C97—21no—
68Q8×C97—21no—
69D4×C97—21no—
70D4×C97—21no—
71D4×C97—41no—
72D4×C97—41no—
73C2×C4×C97—21no—
74C2×C4×C97—22no—
75C2×C4×C97—42no—
76C2×C4×C97—41no—
77C2×C4×C97—41no—
78C2×C4×C97C22×C411yescharacteristic
79C2×C4×C97C2×D411yescharacteristic
80C2×C4×C97C2×D411yescharacteristic
81C2×C4×C97C2×D411yescharacteristic
82C2×C4×C97#4111yescharacteristic
83C2×C4×C97#4111yescharacteristic
84C22⋊D97—44no—
85C22⋊D97—41no—
86C22⋊D97—41no—
87C2×Dic97—21no—
88C2×Dic97—22no—
89C2×Dic97—42no—
90C2×Dic97—42no—
91C2×Dic97C22⋊C412yes—
92C2×Dic97C22⋊C412yes—
93D388—21no—
94D388—21no—
95C4×D97—41no—
96C4×D97—41no—
97Dic194—21no—
98Dic194—21no—
99C8×C97—41no—
100#11715895—41no—
101C22×C97—22no—
102C22×C97—41no—
103C22×C97—42no—
104C22×C97C2×C22⋊C411yescharacteristic
105C22×C97#12111yescharacteristic
106C22×C97#12111yescharacteristic
107C2×D97—41no—
108C2×D97—42no—
109C4×C97—42no—
110C4×C97—41no—
111C4×C97C4×D411yescharacteristic
112C4×C97C4×D411yescharacteristic
113C4×C97#12411yescharacteristic
114C4×C97#12711yescharacteristic
115Dic97—22no—
116Dic97—22no—
117Dic97—42no—
118Dic97—41no—
119C2×C97—21no—
120C2×C97—21no—
121C2×C97—81no—
122C2×C97#39011yescharacteristic
123C2×C97#42112yes—
124D97—81no—
125#1963—971nomaximal, Sylow
126C97#196311yescharacteristic, complemented, Sylow
127#532—971no—
128#423—971no—
129C2×#104—971no—
130C2×#104—971no—
131C2×#104—971no—
132C2×#104D9711yescharacteristic, complemented
133#376—971no—
134C2×#41—971no—
135C2×#41C2×D9711yescharacteristic
136C2×M16—971no—
137C2×M16C2×D9711yescharacteristic
138C4⋊Q8—971no—
139#127—971no—
140#124—971no—
141#124—971no—
142#121—1942no—
143C4×D4—1942no—
144C2×C42—971no—
145C2×C42—971no—
146C2×C42C2×D9711yescharacteristic
147C4⋊C8—1942no—
148#104—22no—
149#104—1942no—
150#104—1942no—
151#104—1942no—
152#95—1942no—
153#41—21no—
154#41—21no—
155#41—21no—
156#41—21no—
157#41—1941no—
158#41—1941no—
159#41—1941no—
160#41—1941no—
161C2×Q8—971no—
162C2×Q8C4×D9711yescharacteristic
163C2×D4—21no—
164C2×D4—971no—
165C2×D4—1942no—
166C2×D4—1941no—
167C2×D4C4×D9711yescharacteristic
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—972no—
171C22×C4—1941no—
172C22×C4—1941no—
173C22×C4C22×D9711yescharacteristic
174M16—21no—
175M16—21no—
176M16—1941no—
177M16—1941no—
178C2×C8—21no—
179C2×C8—1941no—
180C4⋊C4—1942no—
181C22⋊C4—1942no—
182C22⋊C4—1942no—
183C42—21no—
184C42—971no—
185C42—971no—
186C42—972no—
187C42—972no—
188C42—1941no—
189C42C22⋊D9712yescomplemented
190C23—21no—
191C23—1941no—
192C23D4×D9711yescharacteristic
193Q8—21no—
194Q8—21no—
195Q8—1941no—
196Q8—1941no—
197D4—21no—
198D4—21no—
199D4—41no—
200D4—41no—
201D4—3881no—
202D4—3881no—
203D4—3884no—
204D4—1941no—
205D4—1941no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—42no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—972no—
212C2×C4—972no—
213C2×C4—3882no—
214C2×C4—3881no—
215C2×C4—3881no—
216C2×C4—3882no—
217C2×C4—1941no—
218C2×C4—1942no—
219C2×C4C2×C4×D9711yescharacteristic
220C2×C4D4×D9711yescharacteristic
221C2×C4D4×D9711yescharacteristic
222C2×C4C2×C22⋊D9711yescharacteristic
223C2×C4#1309551011yescharacteristic
224C2×C4#1309551111yescharacteristic
225C8—41no—
226C8—3881no—
227C22—22no—
228C22—41no—
229C22—42no—
230C22—3881no—
231C22—3882no—
232C22C22⋊C4×D9711yescharacteristic
233C22#1569078211yescharacteristic
234C22#1569078311yescharacteristic
235C4—42no—
236C4—41no—
237C4—3882no—
238C4—3881no—
239C4—1942no—
240C4—1942no—
241C4#1569077011yescharacteristic
242C4C4×C22⋊D9711yescharacteristic
243C4#1569079611yescharacteristic
244C4#1569080611yescharacteristic
245C2—21no—
246C2—21no—
247C2—81no—
248C2—7761no—
249C2#2026063511yescharacteristic
250C2#2026075012yes—
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#1963
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. #20260581C2semidirect
7. C2×#104×C97C2semidirect
10. #15690808C4semidirect
11. #15690806C4semidirect
12. #15690799C4semidirect
13. #15690797C4semidirect
48. C4×Dic97D4semidirect
63. C42×C97D4semidirect
126. C97#1963semidirect
132. C2×#104D97semidirect
189. 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
#202605812
C2×#104×C972
#196397