Finite Group Databasebeta

← #27142639

Subgroups — #27142639

Order 12416.

296 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 93 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×#13095500C211yesmaximal, characteristic
2C2×#15690746C211yesmaximal, characteristic, complemented
3#20260743C211yesmaximal, characteristic, complemented
4#20260739C211yesmaximal, characteristic
5C2×#15690728C211yesmaximal, characteristic, complemented
6C2×#15690726C211yesmaximal, characteristic
7#405×C97C211yesmaximal, characteristic, complemented
8C22×Dic194C2211yescharacteristic
9C22×D388C2211yescharacteristic
10C2×#13095500—22no—
11C2×#13095500—21no—
12C2×#13095500—21no—
13C2×#13095500C414yescomplemented
14C2×#13095495C2211yescharacteristic
15C2×#13095495C2211yescharacteristic
16#15690746—22no—
17#15690733—22no—
18#15690730—22no—
19#15690728—22no—
20#15690726—22no—
21C2×M16×C97C2211yescharacteristic
22C22×C8×C97C2211yescharacteristic
23#117×C97C2211yescharacteristic
24#97×C97—21no—
25C4×C8×C97—21no—
26C23×D97—21no—
27C2×Dic194—21no—
28C2×Dic194—21no—
29C2×Dic194—21no—
30C2×Dic194—21no—
31C2×Dic194—21no—
32C2×Dic194C2×C412yes—
33C2×D388—21no—
34C2×D388—21no—
35C2×D388—21no—
36C2×D388—21no—
37C2×D388—21no—
38C2×D388C2×C412yes—
39C22×Dic97—21no—
40#13095500—24no—
41#13095500—24no—
42#13095495—21no—
43#13095495—21no—
44#13095495—21no—
45#13095495—21no—
46C2×#11715895—21no—
47C2×#11715895—21no—
48C22×C4×C97C2311yescharacteristic
49M16×C97—21no—
50M16×C97D412yes—
51C2×C8×C97—21no—
52C2×C8×C97C2×C412yes—
53C2×C8×C97D412yes—
54C2×C8×C97D411yescharacteristic
55C2×C8×C97D411yescharacteristic
56C4⋊C4×C97D412yes—
57C22⋊C4×C97—21no—
58C42×C97—21no—
59C23×C97#4111yescharacteristic
60C22×D97—22no—
61C22×D97—41no—
62C22×D97—41no—
63C22×D97—82no—
64C2×C4×C97—22no—
65C2×C4×C97C22×C411yescharacteristic
66C2×C4×C97C2×D411yescharacteristic
67C2×C4×C97C2×D411yescharacteristic
68C2×C4×C97C2×D411yescharacteristic
69C2×C4×C97C2×D411yescharacteristic
70C2×C4×C97#4111yescharacteristic
71C2×Dic97—22no—
72C2×Dic97—41no—
73C2×Dic97—41no—
74D388—22no—
75D388—22no—
76D388—41no—
77D388—41no—
78Dic194—22no—
79Dic194—22no—
80Dic194—41no—
81Dic194—41no—
82C8×C97—22no—
83C8×C97C22⋊C412yes—
84C8×C97C22⋊C412yes—
85#11715895—41no—
86#11715895—41no—
87C22×C97—22no—
88C22×C97C4×D411yescharacteristic
89C22×C97#12111yescharacteristic
90C22×C97#12411yescharacteristic
91C2×D97—42no—
92C2×D97—82no—
93C2×D97—82no—
94C2×D97—82no—
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#39011yescharacteristic
104C2×C97#47112yes—
105D97—82no—
106#1977—971nomaximal, Sylow
107C97#197711yescharacteristic, complemented, Sylow
108C22×SD16—971no—
109C2×#108—971no—
110#419—971no—
111#418—971no—
112C2×#101—971no—
113C2×#100—971no—
114#405D9711yescharacteristic, complemented
115C22×Q8—971no—
116C22×D4—971no—
117C2×SD16—974no—
118C2×SD16—1942no—
119C2×SD16—1941no—
120C2×SD16—1941no—
121C2×M16—971no—
122C2×M16—971no—
123C2×M16C2×D9711yescharacteristic
124C22×C8C2×D9711yescharacteristic
125#117C2×D9711yescharacteristic
126#108—1942no—
127#103—1942no—
128#102—1942no—
129#101—1942no—
130#100—1942no—
131#97—21no—
132C4×C8—21no—
133C24—1941no—
134C2×Q8—972no—
135C2×Q8—1941no—
136C2×Q8—1941no—
137C2×Q8—1941no—
138C2×Q8—1941no—
139C2×Q8—1941no—
140C2×D4—972no—
141C2×D4—1941no—
142C2×D4—1941no—
143C2×D4—1941no—
144C2×D4—1941no—
145C2×D4—1941no—
146C22×C4—1941no—
147C22×C4C22×D9711yescharacteristic
148SD16—1944no—
149SD16—1944no—
150M16—21no—
151M16—1941no—
152M16—1941no—
153M16—1941no—
154M16—1941no—
155M16D38812yes—
156C2×C8—21no—
157C2×C8—1941no—
158C2×C8—1941no—
159C2×C8C4×D9712yes—
160C2×C8D38812yes—
161C2×C8C22⋊D9711yescharacteristic
162C2×C8C22⋊D9711yescharacteristic
163C4⋊C4D38812yes—
164C22⋊C4—21no—
165C42—21no—
166C23—7762no—
167C23—3881no—
168C23—3881no—
169C23—1942no—
170C23#1309551111yescharacteristic
171Q8—3881no—
172Q8—3881no—
173Q8—1942no—
174Q8—1942no—
175D4—3881no—
176D4—3881no—
177D4—1942no—
178D4—1942no—
179C2×C4—22no—
180C2×C4—3881no—
181C2×C4—3881no—
182C2×C4—1942no—
183C2×C4C2×C4×D9711yescharacteristic
184C2×C4C2×C22⋊D9711yescharacteristic
185C2×C4C2×D38811yescharacteristic
186C2×C4C2×D38811yescharacteristic
187C2×C4C2×D38811yescharacteristic
188C2×C4#1309551111yescharacteristic
189C8—22no—
190C8—3881no—
191C8—3881no—
192C8#1309548812yes—
193C8#1309548812yes—
194C22—22no—
195C22—7762no—
196C22—7762no—
197C22—7762no—
198C22—3882no—
199C22C4×D38811yescharacteristic
200C22#1569078411yescharacteristic
201C22#1569079811yescharacteristic
202C4—42no—
203C4—3882no—
204C4C2×#1309548811yescharacteristic
205C4C4×D38811yescharacteristic
206C4#1569078411yescharacteristic
207C4#1569080711yescharacteristic
208C2—21no—
209C2—21no—
210C2—7762no—
211C2#2026063711yescharacteristic
212C2#2026099812yes—
subgroupG/H
centerC22#15690784
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#405×C97C2
socleC22×C97#121
prime radicalC22×D388C22
cosocleC22×C4×C97C23
hypercenter#405D97
hypocenterC97#1977
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #405

Complements

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

NH ≅ G/Ntype
2. C2×#15690746C2semidirect
3. #20260743C2semidirect
5. C2×#15690728C2semidirect
7. #405×C97C2semidirect
13. C2×#13095500C4semidirect
107. C97#1977semidirect
114. #405D97semidirect

Maximal subgroups

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

subgroupindex
C22×#130955002
C2×#156907462
#202607432
#202607392
C2×#156907282
C2×#156907262
#405×C972
#197797