Finite Group Databasebeta

← #27141663

Subgroups — #27141663

Order 12416.

584 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 101 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×#13095512C211yesmaximal, characteristic
2C2×#15690729C211yesmaximal, characteristic, complemented
3C2×#15690726C212yesmaximal, complemented
4#20260698C212yesmaximal, complemented
5C2×#101×C97C211yesmaximal, characteristic, complemented
6C2×#13095512—24no—
7C2×#13095512—24no—
8C2×#13095512—22no—
9C2×#13095512—22no—
10C22×D388C412yes—
11C22×D388C2211yescharacteristic
12C22×C4×D97C412yes—
13C22×C4×D97C2211yescharacteristic
14C2×#13095495C2212yes—
15#15690729—22no—
16#15690726—24no—
17#15690722—24no—
18C22×Q8×C97C2211yescharacteristic
19C2×M16×C97C2212yes—
20#101×C97—22no—
21C23×D97C2×C412yes—
22C23×D97C2×C411yescharacteristic
23#13095512—216no—
24#13095512—216no—
25C2×D388—24no—
26C2×D388—24no—
27C2×D388—24no—
28C2×D388—22no—
29C2×D388—21no—
30C2×D388—21no—
31C2×D388D414yes—
32C22×Dic97C2×C411yescharacteristic
33C2×C4×D97—24no—
34C2×C4×D97—24no—
35C2×C4×D97—22no—
36C2×C4×D97—21no—
37C2×C4×D97—21no—
38C2×C4×D97—24no—
39C2×C4×D97D414yes—
40#13095495—22no—
41#13095495—22no—
42C2×#11715895—22no—
43C2×Q8×C97—22no—
44C2×Q8×C97—22no—
45C2×Q8×C97D412yes—
46C2×Q8×C97D412yes—
47C22×C4×C97C2×C412yes—
48C22×C4×C97C2311yescharacteristic
49M16×C97—22no—
50M16×C97—22no—
51C2×C8×C97—22no—
52C23×C97C22×C411yescharacteristic
53C22×D97—21no—
54C22×D97—22no—
55C22×D97—22no—
56C22×D97—21no—
57C22×D97—42no—
58C22×D97—44no—
59C22×D97C22⋊C412yes—
60C22×D97C22⋊C414yes—
61Q8×C97—24no—
62Q8×C97—24no—
63C2×C4×C97—22no—
64C2×C4×C97—22no—
65C2×C4×C97C22⋊C414yes—
66C2×C4×C97C2×D412yes—
67C2×C4×C97C2×D412yes—
68C2×C4×C97C2×D411yescharacteristic
69C2×C4×C97C2×D411yescharacteristic
70C2×Dic97—21no—
71C2×Dic97—21no—
72C2×Dic97C22⋊C412yes—
73D388—216no—
74D388—24no—
75D388—24no—
76C4×D97—216no—
77C4×D97—24no—
78C4×D97—24no—
79C8×C97—42no—
80#11715895—42no—
81C22×C97—22no—
82C22×C97C2×C22⋊C411yescharacteristic
83C22×C97C2×C22⋊C411yescharacteristic
84C22×C97C2×C22⋊C411yescharacteristic
85C2×D97—22no—
86C2×D97—24no—
87C2×D97—24no—
88C2×D97—42no—
89C2×D97—42no—
90C2×D97—44no—
91C2×D97—44no—
92C2×D97—44no—
93C4×C97—24no—
94C4×C97#12012yes—
95C4×C97#12012yes—
96Dic97—22no—
97C2×C97—21no—
98C2×C97—21no—
99C2×C97#37911yescharacteristic
100C2×C97#41312yes—
101D97—42no—
102D97—44no—
103#1785—971nomaximal, Sylow
104C97#178511yescharacteristic, complemented, Sylow
105C22×#41—971no—
106C2×#101—971no—
107C2×#101D9711yescharacteristic, complemented
108C2×#100—972no—
109#407—972no—
110C2×#41—1944no—
111C2×#41—1944no—
112C2×#41—1942no—
113C2×#41—1942no—
114C22×Q8C2×D9711yescharacteristic
115C22×D4—972no—
116C22×D4—971no—
117C23×C4—971no—
118C23×C4—972no—
119C2×M16—972no—
120C2×M16C2×D9712yes—
121#101—22no—
122#101—1942no—
123#100—1944no—
124C22⋊C8—1944no—
125C24—972no—
126C24—971no—
127#41—19416no—
128#41—19416no—
129C2×Q8—22no—
130C2×Q8—22no—
131C2×Q8D38812yes—
132C2×Q8C22⋊D9712yes—
133C2×D4—974no—
134C2×D4—1944no—
135C2×D4—1944no—
136C2×D4—1944no—
137C2×D4—1942no—
138C2×D4—1941no—
139C2×D4—1941no—
140C22×C4—971no—
141C22×C4—974no—
142C22×C4—1944no—
143C22×C4—1944no—
144C22×C4—1944no—
145C22×C4—1942no—
146C22×C4—1941no—
147C22×C4—1941no—
148C22×C4C4×D9712yes—
149C22×C4C22×D9711yescharacteristic
150M16—22no—
151M16—22no—
152M16—1942no—
153M16—1942no—
154C2×C8—22no—
155C2×C8—1942no—
156C23—974no—
157C23—972no—
158C23—3884no—
159C23—3882no—
160C23—1941no—
161C23—1942no—
162C23—1942no—
163C23—1941no—
164C23C2×C4×D9711yescharacteristic
165Q8—24no—
166Q8—24no—
167D4—19416no—
168D4—1944no—
169D4—1944no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—972no—
173C2×C4—19416no—
174C2×C4—1941no—
175C2×C4—1941no—
176C2×C4—1944no—
177C2×C4—1944no—
178C2×C4#1309548814yes—
179C2×C4D4×D9712yes—
180C2×C4D4×D9712yes—
181C2×C4C2×C22⋊D9711yescharacteristic
182C2×C4C2×D38811yescharacteristic
183C8—42no—
184C8—3882no—
185C22—22no—
186C22—3884no—
187C22—3882no—
188C22—3884no—
189C22—3882no—
190C22—3884no—
191C22—1944no—
192C22—1944no—
193C22—1942no—
194C22C22⋊C4×D9711yescharacteristic
195C22C22⋊C4×D9711yescharacteristic
196C22C2×#1309548811yescharacteristic
197C4—24no—
198C4—1942no—
199C4#1569077812yes—
200C4#1569078012yes—
201C2—21no—
202C2—21no—
203C2—3884no—
204C2—3882no—
205C2#2026059011yescharacteristic
206C2#2026071812yes—
subgroupG/H
centerC22C22⋊C4×D97
commutatorC23×C97C22×C4
FrattiniC22×C4C22×D97
FittingC2×#101×C97C2
socleC22×C97C2×C22⋊C4
prime radicalC22×#13095512C2
cosocleC22×C4×C97C23
hypercenterC2×#101D97
hypocenterC97#1785
LCSG ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
2. C2×#15690729C2semidirect
3. C2×#15690726C2semidirect
4. #20260698C2semidirect
5. C2×#101×C97C2semidirect
104. C97#1785semidirect
107. C2×#101D97semidirect

Maximal subgroups

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

subgroupindex
C22×#130955122
C2×#156907292
C2×#156907262
#202606982
C2×#101×C972
#178597