Finite Group Databasebeta

← #27145071

Subgroups — #27145071

Order 12416.

864 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 864 conjugacy classes of subgroups — too many to draw. Only its 173 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×#13095511C211yesmaximal, characteristic
2C2×#15690793C212yesmaximal, complemented
3#20260655C212yesmaximal, complemented
4#20260652C212yesmaximal, complemented
5#20260614C212yesmaximal, complemented
6#20260606C214yesmaximal, complemented
7#20260598C211yesmaximal, characteristic, complemented
8C22×C4⋊C4×C97C211yesmaximal, characteristic, complemented
9C2×#13095511—28no—
10C2×#13095511—24no—
11C22×Dic194C2211yescharacteristic
12C22×D388C2211yescharacteristic
13C22×C22⋊D97C2212yes—
14C22×C4×D97C2212yes—
15#15690793—28no—
16C2×#13095490C2214yes—
17C2×C22⋊Dic97C2212yes—
18C2×#13095488C2214yes—
19C2×#13095488C2214yes—
20#15690717C2214yes—
21#15690717C2212yes—
22#15690717C2212yes—
23C23×C4×C97C2211yescharacteristic
24C23×C4×C97C2211yescharacteristic
25C23×C4×C97C2211yescharacteristic
26C2×C4⋊C4×C97—24no—
27C2×C4⋊C4×C97C2212yes—
28C2×C4⋊C4×C97C2212yes—
29C23×D97C2312yes—
30#13095511—232no—
31C2×Dic194—22no—
32C2×Dic194—24no—
33C2×D388—24no—
34C2×D388—22no—
35C2×C22⋊D97—24no—
36C2×C22⋊D97—24no—
37C2×C22⋊D97D418yes—
38C22×Dic97C2314yes—
39C22×Dic97C2312yes—
40C2×C4×D97—24no—
41C2×C4×D97—24no—
42C2×C4×D97—24no—
43#13095490—28no—
44C22⋊Dic97—24no—
45#13095488—28no—
46#13095488—28no—
47C24×C97C2311yescharacteristic
48C22×C4×C97—24no—
49C22×C4×C97—22no—
50C22×C4×C97—24no—
51C22×C4×C97—24no—
52C22×C4×C97—22no—
53C22×C4×C97D414yes—
54C22×C4×C97C2312yes—
55C22×C4×C97C2311yescharacteristic
56C22×C4×C97C2311yescharacteristic
57C22×C4×C97C2312yes—
58C4⋊C4×C97—24no—
59C4⋊C4×C97—24no—
60C23×C97—22no—
61C23×C97—22no—
62C23×C97C2×D414yes—
63C23×C97C2×D412yes—
64C23×C97C2411yescharacteristic
65C22×D97—22no—
66C22×D97—22no—
67C22×D97—24no—
68C22×D97—24no—
69C22×D97C2×D414yes—
70C2×C4×C97—28no—
71C2×C4×C97—28no—
72C2×C4×C97—22no—
73C2×C4×C97—22no—
74C2×C4×C97—22no—
75C2×C4×C97—22no—
76C2×C4×C97—22no—
77C2×C4×C97—24no—
78C2×C4×C97—44no—
79C2×C4×C97C2×D412yes—
80C2×C4×C97C2×D412yes—
81C2×C4×C97#4114yescomplemented
82C22⋊D97—216no—
83C2×Dic97—22no—
84C2×Dic97—22no—
85C2×Dic97—24no—
86C2×Dic97—24no—
87C2×Dic97—24no—
88C2×Dic97C2×D414yes—
89D388—28no—
90C4×D97—216no—
91Dic194—28no—
92C22×C97—22no—
93C22×C97—22no—
94C22×C97—28no—
95C22×C97#12014yes—
96C22×C97C22×D412yes—
97C22×C97C22×D411yescharacteristic
98C22×C97C2×#4112yes—
99C22×C972−1+412yes—
100C2×D97—24no—
101C2×D97—28no—
102C2×D97—42no—
103C2×D97—42no—
104C2×D97—44no—
105C4×C97—22no—
106C4×C97—22no—
107C4×C97—24no—
108C4×C97—42no—
109Dic97—24no—
110Dic97—44no—
111C2×C97—24no—
112C2×C97C2×#12011yescharacteristic
113C2×C97#52511yescharacteristic
114C2×C97#53611yescharacteristic
115C2×C97#54714yes—
116D97—44no—
117#2406—971nomaximal, Sylow
118C97#240611yescharacteristic, complemented, Sylow
119C22×#41—971no—
120C2×#123—972no—
121C22×C4⋊C4D9711yescharacteristic, complemented
122#395—972no—
123#394—972no—
124#386—972no—
125#385—974no—
126#381—971no—
127C2×#41—1948no—
128C2×#41—1944no—
129C22×Q8—971no—
130C22×D4—972no—
131C22×D4—971no—
132C23×C4—972no—
133C23×C4C2×D9711yescharacteristic
134C23×C4C2×D9711yescharacteristic
135C23×C4C2×D9711yescharacteristic
136#123—1948no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—974no—
139C2×C4⋊C4C2×D9712yes—
140C2×C4⋊C4C2×D9712yes—
141C2×C22⋊C4—972no—
142C2×C22⋊C4—974no—
143C2×C22⋊C4—974no—
144#95—972no—
145#95—972no—
146#95—974no—
147C24—972no—
148C24C22×D9711yescharacteristic
149#41—19432no—
150C2×Q8—1944no—
151C2×Q8—1942no—
152C2×D4—978no—
153C2×D4—1944no—
154C2×D4—1944no—
155C2×D4—1944no—
156C2×D4—1942no—
157C22×C4—24no—
158C22×C4—22no—
159C22×C4—24no—
160C22×C4—24no—
161C22×C4—22no—
162C22×C4—972no—
163C22×C4—974no—
164C22×C4—1944no—
165C22×C4—1944no—
166C22×C4—1944no—
167C22×C4C22⋊D9714yes—
168C22×C4C22×D9712yes—
169C22×C4C22×D9711yescharacteristic
170C22×C4C22×D9711yescharacteristic
171C22×C4C22×D9712yes—
172C4⋊C4—24no—
173C4⋊C4—24no—
174C4⋊C4—1948no—
175C22⋊C4—1948no—
176C22⋊C4—1944no—
177C22⋊C4—1948no—
178C23—22no—
179C23—22no—
180C23—974no—
181C23—1942no—
182C23—1942no—
183C23—1944no—
184C23—1944no—
185C23D4×D9714yes—
186C23C2×C22⋊D9712yes—
187C23C23×D9711yescharacteristic
188Q8—1948no—
189D4—1948no—
190D4—19416no—
191C2×C4—28no—
192C2×C4—28no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—24no—
198C2×C4—22no—
199C2×C4—44no—
200C2×C4—974no—
201C2×C4—19416no—
202C2×C4—1944no—
203C2×C4—1944no—
204C2×C4—1944no—
205C2×C4—1942no—
206C2×C4—1942no—
207C2×C4C2×C22⋊D9712yes—
208C2×C4C2×C22⋊D9712yes—
209C2×C4#1309551114yescomplemented
210C22—28no—
211C22—22no—
212C22—22no—
213C22—3884no—
214C22—3882no—
215C22—3882no—
216C22—1948no—
217C22—1944no—
218C22#1569077814yes—
219C22C2×D4×D9712yes—
220C22C22×C22⋊D9711yescharacteristic
221C22C2×#1309551112yes—
222C22#1569085812yes—
223C4—22no—
224C4—22no—
225C4—24no—
226C4—42no—
227C4—3884no—
228C4—1944no—
229C2—24no—
230C2—3884no—
231C2C2×#1569077811yescharacteristic
232C2#2026124611yescharacteristic
233C2#2026130911yescharacteristic
234C2#2026138614yes—
subgroupG/H
center, FrattiniC23C23×D97
commutator, socle, cosocleC23×C97C24
FittingC22×C4⋊C4×C97C2
prime radicalC22×#13095511C2
hypercenterC22×C4⋊C4D97
hypocenterC97#2406
LCSG ▷ C23×C97 ▷ C97
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#15690793C2semidirect
3. #20260655C2semidirect
4. #20260652C2semidirect
5. #20260614C2semidirect
6. #20260606C2semidirect
7. #20260598C2semidirect
8. C22×C4⋊C4×C97C2semidirect
81. C2×C4×C97#41semidirect
118. C97#2406semidirect
121. C22×C4⋊C4D97semidirect
209. C2×C4#13095511semidirect

Maximal subgroups

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

subgroupindex
C22×#130955112
C2×#156907932
#202606552
#202606522
#202606142
#202606062
#202605982
C22×C4⋊C4×C972
#240697