Finite Group Databasebeta

← #27144450

Subgroups — #27144450

Order 12416.

612 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 197 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×#15690799C211yesmaximal, characteristic
2C2×Q8×Dic97C211yesmaximal, characteristic, complemented
3C2×#15690772C211yesmaximal, characteristic
4#20260637C211yesmaximal, characteristic
5#20260624C214yesmaximal, complemented
6#20260609C214yesmaximal, complemented
7#20260604C211yesmaximal, characteristic, complemented
8#20260581C211yesmaximal, characteristic, complemented
9#391×C97C211yesmaximal, characteristic, complemented
10C22×D388C2211yescharacteristic
11C22×C4×D97C2212yes—
12#15690799C418yescomplemented
13Q8×Dic97—24no—
14#15690772—24no—
15C2×C4⋊Dic97C2212yes—
16C2×C4⋊Dic97C2211yescharacteristic
17C2×#13095490C2214yes—
18C2×#13095488C2212yes—
19C2×#13095488C2214yes—
20C2×#13095488C2214yes—
21C2×C4×Dic97C2212yes—
22C2×C4×Dic97C2211yescharacteristic
23C2×C4×Dic97C2211yescharacteristic
24#15690717C2212yes—
25#15690717C2212yes—
26C22×Q8×C97C2211yescharacteristic
27C2×C4⋊C4×C97C2211yescharacteristic
28C2×C42×C97C2211yescharacteristic
29#95×C97C2214yes—
30C23×D97C2312yes—
31C2×D388—22no—
32C2×D388—22no—
33C2×D388C2×C414yes—
34C22×Dic97C2312yes—
35C22×Dic97C2312yes—
36C22×Dic97C2311yescharacteristic
37C22×Dic97C2311yescharacteristic
38C2×C4×D97—24no—
39C2×C4×D97—44no—
40C4⋊Dic97—22no—
41C4⋊Dic97—24no—
42#13095490—44no—
43#13095488—24no—
44#13095488—44no—
45#13095488C2×C4116yescomplemented
46C4×Dic97—24no—
47C4×Dic97C2×C414yescomplemented
48C4×Dic97D414yes—
49C2×Q8×C97—22no—
50C2×Q8×C97—22no—
51C2×Q8×C97C2×C414yescomplemented
52C22×C4×C97C2312yes—
53C22×C4×C97C2312yes—
54C22×C4×C97C2312yes—
55C22×C4×C97C2311yescharacteristic
56C4⋊C4×C97—22no—
57C42×C97—22no—
58C23×C97C2411yescharacteristic
59C22×D97—22no—
60C22×D97—22no—
61C22×D97—24no—
62C22×D97—24no—
63C22×D97C22×C414yes—
64Q8×C97—28no—
65C2×C4×C97—22no—
66C2×C4×C97—22no—
67C2×C4×C97—22no—
68C2×C4×C97—22no—
69C2×C4×C97—22no—
70C2×C4×C97—24no—
71C2×C4×C97—22no—
72C2×C4×C97C22×C414yes—
73C2×C4×C97C22×C412yes—
74C2×C4×C97C2×D412yes—
75C2×C4×C97#4112yes—
76C2×Dic97—22no—
77C2×Dic97—22no—
78C2×Dic97—22no—
79C2×Dic97—22no—
80C2×Dic97—22no—
81C2×Dic97—24no—
82C2×Dic97C22×C414yes—
83C2×Dic97C2×D412yes—
84C2×Dic97C2×D412yes—
85D388—28no—
86C4×D97—48no—
87C22×C97C23×C411yescharacteristic
88C22×C97C22×D411yescharacteristic
89C22×C97C2×#4111yescharacteristic
90C22×C97H(5,𝔽2)12yes—
91C22×C972−1+412yes—
92C2×D97—24no—
93C2×D97—28no—
94C2×D97—42no—
95C2×D97—42no—
96C2×D97—44no—
97C4×C97—24no—
98C4×C97—42no—
99C4×C97—42no—
100C4×C97C4×D414yes—
101Dic97—24no—
102Dic97—22no—
103Dic97—22no—
104Dic97—42no—
105C2×C97C2×C4×D411yescharacteristic
106C2×C97#52012yes—
107C2×C97#53511yescharacteristic
108C2×C97#53611yescharacteristic
109C2×C97#54212yes—
110D97—44no—
111#2326—971nomaximal, Sylow
112C97#232611yescharacteristic, complemented, Sylow
113C2×#124—971no—
114C2×C4×Q8—971no—
115C2×C4×D4—971no—
116#391D9711yescharacteristic, complemented
117#390—971no—
118#388—974no—
119#385—974no—
120#384—971no—
121#376—971no—
122C22×Q8C2×D9711yescharacteristic
123C22×D4—971no—
124C23×C4—972no—
125#124—978no—
126C4×Q8—1944no—
127C4×D4—1944no—
128C2×C4⋊C4—974no—
129C2×C4⋊C4—971no—
130C2×C4⋊C4—972no—
131C2×C4⋊C4C2×D9711yescharacteristic
132C2×C22⋊C4—974no—
133C2×C22⋊C4—974no—
134C2×C22⋊C4—972no—
135C2×C42—971no—
136C2×C42—971no—
137C2×C42—972no—
138C2×C42C2×D9711yescharacteristic
139#95—972no—
140#95—972no—
141#95C2×D9714yes—
142C24—972no—
143C2×Q8—22no—
144C2×Q8—22no—
145C2×Q8C4×D9714yescomplemented
146C2×D4—974no—
147C2×D4—1942no—
148C2×D4—1942no—
149C22×C4—971no—
150C22×C4—972no—
151C22×C4—971no—
152C22×C4—972no—
153C22×C4—3884no—
154C22×C4—1944no—
155C22×C4C22×D9712yes—
156C22×C4C22×D9712yes—
157C22×C4C22×D9712yes—
158C22×C4C22×D9711yescharacteristic
159C4⋊C4—22no—
160C4⋊C4—3884no—
161C4⋊C4—1944no—
162C4⋊C4—1942no—
163C22⋊C4—9716no—
164C22⋊C4—3884no—
165C22⋊C4—1944no—
166C42—22no—
167C42—974no—
168C42—974no—
169C42—1944no—
170C23—974no—
171C23—1942no—
172C23—1942no—
173C23—1944no—
174C23—1944no—
175C23C23×D9711yescharacteristic
176Q8—28no—
177D4—1948no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—24no—
184C2×C4—22no—
185C2×C4—974no—
186C2×C4—972no—
187C2×C4—972no—
188C2×C4—3888no—
189C2×C4—1942no—
190C2×C4—1944no—
191C2×C4—1942no—
192C2×C4—1942no—
193C2×C4—1942no—
194C2×C4—1942no—
195C2×C4C2×C4×D9714yes—
196C2×C4C2×C4×D9712yes—
197C2×C4D4×D9712yes—
198C2×C4#1309551012yes—
199C22—3882no—
200C22—3882no—
201C22—3884no—
202C22—1944no—
203C22—1948no—
204C22C22×C4×D9711yescharacteristic
205C22C2×D4×D9711yescharacteristic
206C22C2×#1309551011yescharacteristic
207C22#1569085612yes—
208C22#1569085812yes—
209C4—24no—
210C4—42no—
211C4—42no—
212C4—3882no—
213C4—1944no—
214C4—1942no—
215C4—1942no—
216C4#1569077014yes—
217C2—3884no—
218C2C2×#1569077011yescharacteristic
219C2#2026121712yes—
220C2#2026130311yescharacteristic
221C2#2026131011yescharacteristic
222C2#2026135812yes—
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting#391×C97C2
socle, cosocleC23×C97C24
prime radicalC22×D388C22
hypercenter#391D97
hypocenterC97#2326
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
2. C2×Q8×Dic97C2semidirect
5. #20260624C2semidirect
6. #20260609C2semidirect
7. #20260604C2semidirect
8. #20260581C2semidirect
9. #391×C97C2semidirect
12. #15690799C4semidirect
45. #13095488C2×C4semidirect
47. C4×Dic97C2×C4semidirect
51. C2×Q8×C97C2×C4semidirect
112. C97#2326semidirect
116. #391D97semidirect
145. C2×Q8C4×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907992
C2×Q8×Dic972
C2×#156907722
#202606372
#202606242
#202606092
#202606042
#202605812
#391×C972
#232697