Finite Group Databasebeta

← #27142170

Subgroups — #27142170

Order 12416.

360 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 113 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
1C22×#13095495C211yesmaximal, characteristic
2C2×#15690734C212yesmaximal, complemented
3#20260595C211yesmaximal, characteristic
4#20260497C212yesmaximal, complemented
5C2×C22⋊Q8×C97C211yesmaximal, characteristic
6C2×#13095495—24no—
7C2×#13095495—22no—
8C22×#11715895C2212yes—
9C2×C4⋊Dic97C2212yes—
10C2×C22⋊Dic97—21no—
11#15690734—24no—
12#15690717—21no—
13C22×Q8×C97C2211yescharacteristic
14C23×C4×C97C2211yescharacteristic
15C22⋊Q8×C97—22no—
16C22⋊Q8×C97C414yescomplemented
17C2×C4⋊C4×C97—21no—
18C2×C4⋊C4×C97C2211yescharacteristic
19C2×C22⋊C4×C97—21no—
20C22×Dic97—22no—
21#13095495—28no—
22C2×#11715895—22no—
23C2×#11715895—22no—
24C2×#11715895D414yes—
25C4⋊Dic97—22no—
26C4⋊Dic97—22no—
27C22⋊Dic97—42no—
28C24×C97D411yescharacteristic
29C2×Q8×C97—21no—
30C2×Q8×C97—22no—
31C2×Q8×C97—22no—
32C2×Q8×C97C2×C412yes—
33C22×C4×C97—22no—
34C22×C4×C97—22no—
35C22×C4×C97—21no—
36C22×C4×C97—21no—
37C22×C4×C97C2×C412yes—
38C22×C4×C97D411yescharacteristic
39C22×C4×C97D412yes—
40C22×C4×C97C2311yescharacteristic
41C4⋊C4×C97—24no—
42C4⋊C4×C97—21no—
43C4⋊C4×C97C2×C412yes—
44C22⋊C4×C97—24no—
45C23×C97—22no—
46C23×C97—21no—
47C23×C97—21no—
48C23×C97—22no—
49C23×C97C22⋊C412yes—
50C23×C97C2×D411yescharacteristic
51Q8×C97—24no—
52Q8×C97—41no—
53Q8×C97—41no—
54C2×C4×C97—22no—
55C2×C4×C97—24no—
56C2×C4×C97—22no—
57C2×C4×C97—22no—
58C2×C4×C97—22no—
59C2×C4×C97—22no—
60C2×C4×C97—42no—
61C2×C4×C97—41no—
62C2×C4×C97—41no—
63C2×C4×C97C22⋊C412yes—
64C2×C4×C97C22×C411yescharacteristic
65C2×C4×C97C2×D412yes—
66C2×C4×C97C2×D411yescharacteristic
67C2×C4×C97#4112yes—
68C2×Dic97—42no—
69C2×Dic97—42no—
70C2×Dic97—42no—
71#11715895—24no—
72C22×C97—22no—
73C22×C97—22no—
74C22×C97—22no—
75C22×C97—42no—
76C22×C97—42no—
77C22×C97C2×C22⋊C411yescharacteristic
78C22×C97#12112yes—
79C22×C97AGL(1,ℤ/8)11yescharacteristic
80C22×C97#13711yescharacteristic
81C22×C97#13712yes—
82C4×C97—22no—
83C4×C97—42no—
84C4×C97—42no—
85C4×C97C4×D412yes—
86C4×C97#12011yescharacteristic
87C4×C97#12311yescharacteristic
88Dic97—82no—
89C2×C97—42no—
90C2×C97#38611yescharacteristic
91C2×C97#41711yescharacteristic
92C2×C97#41911yescharacteristic
93C2×C97#46812yes—
94C2×C97#47012yes—
95#1894—971nomaximal, Sylow
96C97#189411yescharacteristic, complemented, Sylow
97C22×M16—971no—
98C2×C22⋊Q8D9711yescharacteristic
99C2×#103—972no—
100#380—971no—
101#340—972no—
102C22×Q8C2×D9711yescharacteristic
103C23×C4C2×D9711yescharacteristic
104C2×M16—1944no—
105C2×M16—1942no—
106C22×C8—972no—
107C22⋊Q8—22no—
108C22⋊Q8Dic9714yescomplemented
109C2×C4⋊C4—21no—
110C2×C4⋊C4—972no—
111C2×C4⋊C4C2×D9711yescharacteristic
112C2×C22⋊C4—21no—
113C2×C22⋊C4—1941no—
114#103—1944no—
115#95—1941no—
116C24C22⋊D9711yescharacteristic
117C2×Q8—21no—
118C2×Q8—22no—
119C2×Q8—22no—
120C2×Q8C2×Dic9712yes—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—1942no—
126C22×C4C2×Dic9712yes—
127C22×C4C22⋊D9711yescharacteristic
128C22×C4C22⋊D9712yes—
129C22×C4C22×D9711yescharacteristic
130M16—1948no—
131C2×C8—974no—
132C2×C8—1942no—
133C2×C8—1942no—
134C4⋊C4—24no—
135C4⋊C4—21no—
136C4⋊C4—1942no—
137C4⋊C4—1942no—
138C4⋊C4C2×Dic9712yes—
139C22⋊C4—24no—
140C22⋊C4—3882no—
141C23—22no—
142C23—21no—
143C23—21no—
144C23—22no—
145C23C22⋊Dic9712yes—
146C23C2×C22⋊D9711yescharacteristic
147Q8—24no—
148Q8—41no—
149Q8—41no—
150C2×C4—22no—
151C2×C4—24no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—42no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—3882no—
160C2×C4—3882no—
161C2×C4—3882no—
162C2×C4C22⋊Dic9712yes—
163C2×C4C22×Dic9711yescharacteristic
164C2×C4D4×D9712yes—
165C2×C4C2×C22⋊D9711yescharacteristic
166C2×C4#1309551012yes—
167C8—1944no—
168C22—22no—
169C22—22no—
170C22—22no—
171C22—42no—
172C22—42no—
173C22C2×C22⋊Dic9711yescharacteristic
174C22#1569078212yes—
175C22#1569083511yescharacteristic
176C22#1569084011yescharacteristic
177C22#1569084112yes—
178C4—22no—
179C4—42no—
180C4—42no—
181C4—7762no—
182C4D4×Dic9712yes—
183C4#1569077811yescharacteristic
184C4#1569079111yescharacteristic
185C2—42no—
186C2#2026061511yescharacteristic
187C2#2026073611yescharacteristic
188C2#2026074411yescharacteristic
189C2#2026098312yes—
190C2#2026099112yes—
subgroupG/H
centerC23C2×C22⋊D97
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
FittingC2×C22⋊Q8×C97C2
socleC23×C97C2×D4
prime radicalC24×C97D4
cosocleC22×C4×C97C23
hypercenterC2×C22⋊Q8D97
hypocenterC97#1894
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#15690734C2semidirect
4. #20260497C2semidirect
16. C22⋊Q8×C97C4semidirect
96. C97#1894semidirect
108. C22⋊Q8Dic97semidirect

Maximal subgroups

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

subgroupindex
C22×#130954952
C2×#156907342
#202605952
#202604972
C2×C22⋊Q8×C972
#189497