Finite Group Databasebeta

← #27142641

Subgroups — #27142641

Order 12416.

296 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 97 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×#15690746C211yesmaximal, characteristic, complemented
2#20260744C211yesmaximal, characteristic, complemented
3#20260740C211yesmaximal, characteristic
4C2×#15690729C211yesmaximal, characteristic, complemented
5C2×#15690727C211yesmaximal, characteristic
6#20260690C211yesmaximal, characteristic, complemented
7C22×SD16×C97C211yesmaximal, characteristic
8C2×#13095495C2211yescharacteristic
9C2×#13095495C2211yescharacteristic
10C2×#13095495C2211yescharacteristic
11#15690769C2211yescharacteristic
12#15690746—22no—
13#15690734—22no—
14#15690731—22no—
15#15690729—22no—
16#15690727—22no—
17#15690720—21no—
18C8×Dic97—21no—
19C22×Q8×C97C2211yescharacteristic
20C22×D4×C97C2211yescharacteristic
21C2×SD16×C97—21no—
22C2×SD16×C97—21no—
23C2×SD16×C97—22no—
24C2×SD16×C97C414yescomplemented
25C22×C8×C97C2211yescharacteristic
26#13095495—21no—
27#13095495—21no—
28#13095495—21no—
29#13095495—21no—
30#13095495—21no—
31#13095495D412yes—
32C2×#11715895—21no—
33C2×#11715895—21no—
34C2×#11715895—21no—
35C4⋊Dic97D412yes—
36C22⋊Dic97—21no—
37C4×Dic97—21no—
38C24×C97—21no—
39C2×Q8×C97—21no—
40C2×Q8×C97—21no—
41C2×Q8×C97—21no—
42C2×Q8×C97—21no—
43C2×Q8×C97—21no—
44C2×Q8×C97C2×C412yes—
45C2×D4×C97—21no—
46C2×D4×C97—21no—
47C2×D4×C97—21no—
48C2×D4×C97—21no—
49C2×D4×C97—21no—
50C2×D4×C97C2×C412yes—
51C22×C4×C97—21no—
52C22×C4×C97C2311yescharacteristic
53SD16×C97—24no—
54SD16×C97—24no—
55C2×C8×C97C2×C412yes—
56C2×C8×C97D411yescharacteristic
57C2×C8×C97D411yescharacteristic
58C2×C8×C97D412yes—
59C23×C97—22no—
60C23×C97—41no—
61C23×C97—41no—
62C23×C97—82no—
63C23×C97#4111yescharacteristic
64Q8×C97—22no—
65Q8×C97—22no—
66Q8×C97—41no—
67Q8×C97—41no—
68D4×C97—22no—
69D4×C97—22no—
70D4×C97—41no—
71D4×C97—41no—
72C2×C4×C97—22no—
73C2×C4×C97—41no—
74C2×C4×C97—41no—
75C2×C4×C97C22×C411yescharacteristic
76C2×C4×C97C2×D411yescharacteristic
77C2×C4×C97C2×D411yescharacteristic
78C2×C4×C97C2×D411yescharacteristic
79C2×C4×C97C2×D411yescharacteristic
80C2×C4×C97#4111yescharacteristic
81C2×Dic97—22no—
82C8×C97C22⋊C412yes—
83C8×C97C22⋊C412yes—
84#11715895—22no—
85#11715895—41no—
86#11715895—41no—
87C22×C97—22no—
88C22×C97—42no—
89C22×C97—82no—
90C22×C97—82no—
91C22×C97—82no—
92C22×C97C4×D411yescharacteristic
93C22×C97#12111yescharacteristic
94C22×C97#12411yescharacteristic
95C4×C97—42no—
96C4×C97C2×C22⋊C411yescharacteristic
97C4×C97C4×D411yescharacteristic
98C4×C97#12111yescharacteristic
99C4×C97#12711yescharacteristic
100Dic97—42no—
101C2×C97—21no—
102C2×C97—21no—
103C2×C97—82no—
104C2×C97#39011yescharacteristic
105C2×C97#47112yes—
106#1977—971nomaximal, Sylow
107C97#197711yescharacteristic, complemented, Sylow
108C22×SD16D9711yescharacteristic
109C2×#108—971no—
110#419—971no—
111#418—971no—
112C2×#101—971no—
113C2×#100—971no—
114#405—971no—
115C22×Q8C2×D9711yescharacteristic
116C22×D4C2×D9711yescharacteristic
117C2×SD16—21no—
118C2×SD16—21no—
119C2×SD16—22no—
120C2×SD16Dic9714yescomplemented
121C2×M16—971no—
122C2×M16—971no—
123C2×M16—971no—
124C22×C8C2×D9711yescharacteristic
125#117—971no—
126#108—1942no—
127#103—1942no—
128#102—1942no—
129#101—1942no—
130#100—1942no—
131#97—1941no—
132C4×C8—1941no—
133C24—21no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8C2×Dic9712yes—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4C2×Dic9712yes—
146C22×C4—21no—
147C22×C4C22×D9711yescharacteristic
148SD16—24no—
149SD16—24no—
150M16—972no—
151M16—1941no—
152M16—1941no—
153M16—1941no—
154M16—1941no—
155M16—1941no—
156C2×C8—1941no—
157C2×C8—1941no—
158C2×C8—1941no—
159C2×C8C2×Dic9712yes—
160C2×C8C22⋊D9711yescharacteristic
161C2×C8C22⋊D9711yescharacteristic
162C2×C8C22⋊D9712yes—
163C4⋊C4—972no—
164C22⋊C4—1941no—
165C42—1941no—
166C23—22no—
167C23—41no—
168C23—41no—
169C23—82no—
170C23#1309551011yescharacteristic
171Q8—22no—
172Q8—22no—
173Q8—41no—
174Q8—41no—
175D4—22no—
176D4—22no—
177D4—41no—
178D4—41no—
179C2×C4—22no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—1942no—
183C2×C4C22×Dic9711yescharacteristic
184C2×C4D4×D9711yescharacteristic
185C2×C4D4×D9711yescharacteristic
186C2×C4C2×C22⋊D9711yescharacteristic
187C2×C4C2×C22⋊D9711yescharacteristic
188C2×C4#1309551011yescharacteristic
189C8—3881no—
190C8—3881no—
191C8—1942no—
192C8C22⋊Dic9712yes—
193C8C22⋊Dic9712yes—
194C22—22no—
195C22—42no—
196C22—82no—
197C22—82no—
198C22—82no—
199C22D4×Dic9711yescharacteristic
200C22#1569078111yescharacteristic
201C22#1569079711yescharacteristic
202C4—42no—
203C4—3882no—
204C4C2×C22⋊Dic9711yescharacteristic
205C4D4×Dic9711yescharacteristic
206C4#1569078111yescharacteristic
207C4#1569080611yescharacteristic
208C2—21no—
209C2—21no—
210C2—82no—
211C2#2026063811yescharacteristic
212C2#2026099512yes—
subgroupG/H
centerC22#15690781
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
FittingC22×SD16×C97C2
socleC22×C97#121
prime radicalC22×D4×C97C22
cosocleC22×C4×C97C23
hypercenterC22×SD16D97
hypocenterC97#1977
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#15690746C2semidirect
2. #20260744C2semidirect
4. C2×#15690729C2semidirect
6. #20260690C2semidirect
24. C2×SD16×C97C4semidirect
107. C97#1977semidirect
120. C2×SD16Dic97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907462
#202607442
#202607402
C2×#156907292
C2×#156907272
#202606902
C22×SD16×C972
#197797