Finite Group Databasebeta

← #27151866

Subgroups — #27151866

Order 12416.

380 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 133 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#20261292C211yesmaximal, characteristic
2#20261065C212yesmaximal
3#20261020C212yesmaximal, complemented
4#20260945C212yesmaximal, complemented
5#20260891C212yesmaximal
6#20260840C214yesmaximal, complemented
7#20260792C211yesmaximal, characteristic
8C2×C4×Q8×C97C211yesmaximal, characteristic
9C2×#13095501C2214yes—
10#15690810C2211yescharacteristic
11#15690810C2211yescharacteristic
12#15690802—21no—
13#15690789—21no—
14#15690789C2212yes—
15C4×Dic194—21no—
16C4×Dic194C2212yes—
17#15690769—21no—
18#15690741C2214yes—
19C4⋊#11715895C2212yes—
20#15690734C2214yes—
21#15690733C2214yes—
22C22⋊#11715895C2212yes—
23#15690721C2212yes—
24C22×Q8×C97C2211yescharacteristic
25C4×Q8×C97—21no—
26C4×Q8×C97—21no—
27C4×Q8×C97C2212yes—
28C4×Q8×C97C2212yes—
29C2×C4⋊C4×C97—21no—
30C2×C4⋊C4×C97C2211yescharacteristic
31C2×C42×C97—21no—
32C2×C42×C97C2211yescharacteristic
33C2×Dic194—21no—
34C2×Dic194C2312yes—
35#13095501—28no—
36C2×#11715895C2314yes—
37C4⋊Dic97—21no—
38C4⋊Dic97—21no—
39C4⋊Dic97—21no—
40C4⋊Dic97C2312yes—
41#13095490—22no—
42#13095490—22no—
43C22⋊Dic97—22no—
44C4×Dic97—22no—
45C2×Q8×C97—24no—
46C2×Q8×C97—21no—
47C2×Q8×C97C2312yes—
48C22×C4×C97—21no—
49C22×C4×C97—21no—
50C22×C4×C97D411yescharacteristic
51C22×C4×C97D411yescharacteristic
52C22×C4×C97C2311yescharacteristic
53C4⋊C4×C97—22no—
54C4⋊C4×C97—22no—
55C4⋊C4×C97—21no—
56C4⋊C4×C97C2312yes—
57C42×C97—22no—
58C42×C97—22no—
59C42×C97D411yescharacteristic
60C42×C97D411yescharacteristic
61C42×C97C2311yescharacteristic
62C42×C97C2311yescharacteristic
63C23×C97C2×D411yescharacteristic
64Q8×C97—24no—
65Q8×C97—22no—
66Q8×C97#4114yes—
67C2×C4×C97—24no—
68C2×C4×C97—22no—
69C2×C4×C97—22no—
70C2×C4×C97—22no—
71C2×C4×C97—22no—
72C2×C4×C97—22no—
73C2×C4×C97—42no—
74C2×C4×C97C2×D411yescharacteristic
75C2×C4×C97C2×D411yescharacteristic
76C2×C4×C97C2×D411yescharacteristic
77C2×C4×C97C2×D411yescharacteristic
78C2×C4×C97C2×D411yescharacteristic
79C2×C4×C97C2411yescharacteristic
80C2×Dic97—22no—
81C2×Dic97—22no—
82Dic194—24no—
83Dic194—42no—
84#11715895—24no—
85C22×C97—21no—
86C22×C97—21no—
87C22×C97#13712yes—
88C22×C97C22×D411yescharacteristic
89C4×C97—24no—
90C4×C97—21no—
91C4×C97—21no—
92C4×C97—21no—
93C4×C97—21no—
94C4×C97—42no—
95C4×C97#13712yes—
96C4×C97C2×#4112yes—
97Dic97—42no—
98Dic97—42no—
99C2×C97—22no—
100C2×C97#52511yescharacteristic
101C2×C97C2×#13711yescharacteristic
102C2×C97C2×#13711yescharacteristic
103#3113—971nomaximal, Sylow
104C97#311311yescharacteristic, complemented, Sylow
105#533—971no—
106C2×C4×Q8D9711yescharacteristic
107#483—972no—
108#475—972no—
109#462—972no—
110#451—972no—
111#441—974no—
112#432—971no—
113C22×Q8C2×D9711yescharacteristic
114C2×Q16—974no—
115C4⋊Q8—971no—
116C4⋊Q8—971no—
117#125—1941no—
118C22⋊Q8—972no—
119C22⋊Q8—1941no—
120C4×Q8—21no—
121C4×Q8—21no—
122C4×Q8—972no—
123C4×Q8—1941no—
124C4×Q8C2×D9712yes—
125C4×Q8C2×D9712yes—
126#117—1941no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4C2×D9711yescharacteristic
129C2×C42—21no—
130C2×C42C2×D9711yescharacteristic
131#106—974no—
132C4⋊C8—972no—
133#103—974no—
134#103—974no—
135C22⋊C8—972no—
136#97—972no—
137C2×Q8—24no—
138C2×Q8—21no—
139C2×Q8—972no—
140C2×Q8—1941no—
141C2×Q8C22×D9712yes—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4C22⋊D9711yescharacteristic
145C22×C4C22⋊D9711yescharacteristic
146C22×C4C22×D9711yescharacteristic
147Q16—1948no—
148C2×C8—974no—
149C4⋊C4—22no—
150C4⋊C4—22no—
151C4⋊C4—21no—
152C4⋊C4—972no—
153C4⋊C4—1942no—
154C4⋊C4—1942no—
155C4⋊C4—1941no—
156C4⋊C4—1941no—
157C4⋊C4—1941no—
158C4⋊C4C22×D9712yes—
159C22⋊C4—1942no—
160C42—22no—
161C42—22no—
162C42—1942no—
163C42C22⋊D9711yescharacteristic
164C42C22⋊D9711yescharacteristic
165C42C22×D9711yescharacteristic
166C42C22×D9711yescharacteristic
167C23C2×C22⋊D9711yescharacteristic
168Q8—24no—
169Q8—22no—
170Q8—3882no—
171Q8—1944no—
172Q8#1309551114yes—
173C2×C4—24no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—42no—
180C2×C4—1942no—
181C2×C4—1942no—
182C2×C4C2×C22⋊D9711yescharacteristic
183C2×C4C2×C22⋊D9711yescharacteristic
184C2×C4C2×C22⋊D9711yescharacteristic
185C2×C4C2×C22⋊D9711yescharacteristic
186C2×C4C2×C22⋊D9711yescharacteristic
187C2×C4C23×D9711yescharacteristic
188C8—1944no—
189C22—21no—
190C22—21no—
191C22#1569084012yes—
192C22C22×C22⋊D9711yescharacteristic
193C4—24no—
194C4—21no—
195C4—21no—
196C4—21no—
197C4—21no—
198C4—42no—
199C4—3882no—
200C4—3882no—
201C4#1569084112yes—
202C4C2×#1309551112yes—
203C2—22no—
204C2#2026124611yescharacteristic
205C2C2×#1569084011yescharacteristic
206C2C2×#1569084111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×C4×Q8×C97C2
socleC22×C97C22×D4
prime radicalC23×C97C2×D4
hypercenterC2×C4×Q8D97
hypocenterC97#3113
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
3. #20261020C2semidirect
4. #20260945C2semidirect
6. #20260840C2semidirect
104. C97#3113semidirect

Maximal subgroups

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

subgroupindex
#202612922
#202610652
#202610202
#202609452
#202608912
#202608402
#202607922
C2×C4×Q8×C972
#311397