Finite Group Databasebeta

← #27150111

Subgroups — #27150111

Order 12416.

504 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 177 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×#15690809C211yesmaximal, characteristic
2C2×#15690808C211yesmaximal, characteristic
3#20260642C211yesmaximal, characteristic
4#20260641C211yesmaximal, characteristic
5#20260640C211yesmaximal, characteristic
6#20260639C214yesmaximal
7#20260599C214yesmaximal
8C42×Dic97C211yesmaximal, characteristic
9#391×C97C211yesmaximal, characteristic
10C22×Dic194C2212yes—
11C22×Dic194C2211yescharacteristic
12#15690809—24no—
13#15690808—24no—
14C2×C4⋊Dic97C2211yescharacteristic
15C2×#13095490C2212yes—
16C2×#13095490C2212yes—
17C2×#13095490C2212yes—
18C2×C4×Dic97C2212yes—
19C2×C4×Dic97C2211yescharacteristic
20C2×C4×Dic97C2211yescharacteristic
21C2×C4×Dic97C2212yes—
22#15690717C2214yes—
23#15690717C2214yes—
24#15690717C2214yes—
25C22×Q8×C97C2211yescharacteristic
26C2×C4⋊C4×C97C2211yescharacteristic
27C2×C42×C97C2211yescharacteristic
28#95×C97C2214yes—
29C2×Dic194—22no—
30C2×Dic194—22no—
31C2×Dic194—22no—
32C2×Dic194—24no—
33C2×Dic194—28no—
34C22×Dic97C2312yes—
35C22×Dic97C2312yes—
36C22×Dic97C2312yes—
37C22×Dic97C2311yescharacteristic
38C22×Dic97C2311yescharacteristic
39C4⋊Dic97—22no—
40#13095490—24no—
41#13095490—24no—
42#13095490—24no—
43C4×Dic97—22no—
44C4×Dic97—24no—
45C4×Dic97D414yes—
46C4×Dic97Q818yescomplemented
47C2×Q8×C97—22no—
48C2×Q8×C97—22no—
49C2×Q8×C97—22no—
50C22×C4×C97C2312yes—
51C22×C4×C97C2312yes—
52C22×C4×C97C2312yes—
53C22×C4×C97C2311yescharacteristic
54C4⋊C4×C97—22no—
55C42×C97—22no—
56C23×C97C2411yescharacteristic
57Q8×C97—44no—
58C2×C4×C97—22no—
59C2×C4×C97—22no—
60C2×C4×C97—22no—
61C2×C4×C97—22no—
62C2×C4×C97—22no—
63C2×C4×C97—22no—
64C2×C4×C97—24no—
65C2×C4×C97C2×D412yes—
66C2×C4×C97C2×Q814yes—
67C2×C4×C97#4112yes—
68C2×C4×C97#4112yes—
69C2×Dic97—22no—
70C2×Dic97—22no—
71C2×Dic97—22no—
72C2×Dic97—22no—
73C2×Dic97—24no—
74C2×Dic97—22no—
75C2×Dic97—22no—
76C2×Dic97C2×D412yes—
77C2×Dic97C2×D412yes—
78C2×Dic97C2×Q814yes—
79C2×Dic97C2×Q814yes—
80C2×Dic97#4114yes—
81Dic194—44no—
82Dic194—48no—
83C22×C97C22×D411yescharacteristic
84C22×C97C22×Q812yes—
85C22×C97C2×#4112yes—
86C22×C97C2×#4111yescharacteristic
87C22×C97C2×#4111yescharacteristic
88C4×C97—24no—
89C4×C97—42no—
90C4×C97—42no—
91C4×C97#12414yes—
92Dic97—24no—
93Dic97—42no—
94Dic97—42no—
95Dic97#12414yes—
96Dic97C4⋊Q814yes—
97C2×C97C2×#12411yescharacteristic
98C2×C97C2×#12411yescharacteristic
99C2×C97C2×C4⋊Q811yescharacteristic
100C2×C97#53312yes—
101C2×C97#53312yes—
102#2862—971nomaximal, Sylow
103C97#286211yescharacteristic, complemented, Sylow
104C2×C4⋊Q8—971no—
105C2×C4⋊Q8—971no—
106#391—971no—
107#391—971no—
108#391—971no—
109#391—974no—
110#391D9711yescharacteristic
111#382—974no—
112C43—971no—
113C22×Q8—972no—
114C22×Q8—971no—
115C22×Q8C2×D9711yescharacteristic
116C4⋊Q8—1944no—
117C4⋊Q8—1944no—
118C2×C4⋊C4—972no—
119C2×C4⋊C4—972no—
120C2×C4⋊C4—972no—
121C2×C4⋊C4—971no—
122C2×C4⋊C4C2×D9711yescharacteristic
123C2×C42—971no—
124C2×C42—972no—
125C2×C42—972no—
126C2×C42—971no—
127C2×C42C2×D9711yescharacteristic
128#95—974no—
129#95—974no—
130#95—974no—
131#95C2×D9714yes—
132C2×Q8—22no—
133C2×Q8—22no—
134C2×Q8—22no—
135C2×Q8—1944no—
136C2×Q8—1948no—
137C2×Q8—1942no—
138C2×Q8—1942no—
139C2×Q8—1942no—
140C22×C4—971no—
141C22×C4—971no—
142C22×C4—972no—
143C22×C4—972no—
144C22×C4—972no—
145C22×C4C22×D9712yes—
146C22×C4C22×D9712yes—
147C22×C4C22×D9712yes—
148C22×C4C22×D9711yescharacteristic
149C4⋊C4—22no—
150C4⋊C4—1944no—
151C4⋊C4—1944no—
152C4⋊C4—1944no—
153C4⋊C4—1942no—
154C42—22no—
155C42—974no—
156C42—978no—
157C42—1944no—
158C42—1942no—
159C23C23×D9711yescharacteristic
160Q8—44no—
161Q8—3888no—
162Q8—3884no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—24no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—972no—
171C2×C4—972no—
172C2×C4—974no—
173C2×C4—974no—
174C2×C4—974no—
175C2×C4—1942no—
176C2×C4—1942no—
177C2×C4—1944no—
178C2×C4—1942no—
179C2×C4—1942no—
180C2×C4—1942no—
181C2×C4—1942no—
182C2×C4D4×D9712yes—
183C2×C4Q8×D9714yes—
184C2×C4#1309551012yes—
185C2×C4#1309551112yes—
186C22C2×D4×D9711yescharacteristic
187C22C2×Q8×D9712yes—
188C22#41×D9712yes—
189C22C2×#1309551011yescharacteristic
190C22C2×#1309551111yescharacteristic
191C4—24no—
192C4—42no—
193C4—42no—
194C4—974no—
195C4—974no—
196C4—3882no—
197C4—3882no—
198C4—1944no—
199C4#1569079614yes—
200C2#124×D9711yescharacteristic
201C2C2×#1569079611yescharacteristic
202C2C4⋊Q8×D9711yescharacteristic
203C2#2026128712yes—
204C2#2026128912yes—
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, prime radical, cosocleC23×C97C24
Fitting#391×C97C2
hypercenter#391D97
hypocenterC97#2862
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
46. C4×Dic97Q8semidirect
103. C97#2862semidirect

Maximal subgroups

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

subgroupindex
C2×#156908092
C2×#156908082
#202606422
#202606412
#202606402
#202606392
#202605992
C42×Dic972
#391×C972
#286297