Finite Group Databasebeta

← #27142279

Subgroups — #27142279

Order 12416.

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

normal subgroups only

This group has 380 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
1#20261210C211yesmaximal, characteristic, complemented
2C2×#15690725C211yesmaximal, characteristic, complemented
3C2×#15690725C212yesmaximal, complemented
4#20260501C212yesmaximal, complemented
5#534×C97C211yesmaximal, characteristic
6D4×Dic97—24no—
7#15690769—21no—
8C2×C22⋊Dic97—21no—
9C2×C22⋊Dic97C2212yes—
10C2×C22⋊Dic97C2212yes—
11#15690725—24no—
12#15690725—22no—
13C22×D4×C97C2211yescharacteristic
14C22×D4×C97C2211yescharacteristic
15#123×C97—21no—
16#123×C97C412yescomplemented
17#121×C97—22no—
18#120×C97—21no—
19#120×C97C412yescomplemented
20C2×C22⋊C4×C97C2211yescharacteristic
21C22×Dic97—22no—
22C22×Dic97—22no—
23C22×Dic97—22no—
24C4⋊Dic97—22no—
25C22⋊Dic97—22no—
26C22⋊Dic97—22no—
27C22⋊Dic97—22no—
28C22⋊Dic97—22no—
29C22⋊Dic97D414yescomplemented
30C4×Dic97—22no—
31C24×C97C2×C411yescharacteristic, complemented
32C24×C97D411yescharacteristic
33C24×C97C2311yescharacteristic
34C2×D4×C97—22no—
35C2×D4×C97—22no—
36C2×D4×C97—21no—
37C2×D4×C97—21no—
38C2×D4×C97—22no—
39C2×D4×C97—22no—
40C2×D4×C97—22no—
41C2×D4×C97C2×C412yes—
42C2×D4×C97D412yes—
43C22×C4×C97—21no—
44C22×C4×C97C2×C411yescharacteristic, complemented
45C22×C4×C97D411yescharacteristic
46C4⋊C4×C97—22no—
47C22⋊C4×C97—22no—
48C22⋊C4×C97—21no—
49C22⋊C4×C97—22no—
50C22⋊C4×C97C2×C412yescomplemented
51C23×C97—21no—
52C23×C97—21no—
53C23×C97—21no—
54C23×C97—22no—
55C23×C97—81no—
56C23×C97—81no—
57C23×C97—81no—
58C23×C97C22⋊C412yes—
59C23×C97C22×C411yescharacteristic
60C23×C97C2×D412yes—
61C23×C97C2×D411yescharacteristic
62C23×C97C2×D411yescharacteristic
63C23×C97#4112yes—
64D4×C97—24no—
65D4×C97—24no—
66D4×C97—41no—
67D4×C97—42no—
68D4×C97—41no—
69D4×C97—82no—
70C2×C4×C97—21no—
71C2×C4×C97—21no—
72C2×C4×C97—22no—
73C2×C4×C97—21no—
74C2×C4×C97—22no—
75C2×C4×C97—81no—
76C2×C4×C97C22⋊C412yes—
77C2×Dic97—24no—
78C2×Dic97—42no—
79C2×Dic97—82no—
80C2×Dic97—82no—
81C2×Dic97—82no—
82C22×C97—22no—
83C22×C97—42no—
84C22×C97—42no—
85C22×C97—41no—
86C22×C97—41no—
87C22×C97—41no—
88C22×C97—42no—
89C22×C97—42no—
90C22×C97—42no—
91C22×C97—41no—
92C22×C97—81no—
93C22×C97—82no—
94C22×C97C2×C22⋊C411yescharacteristic
95C22×C97C4×D412yes—
96C22×C97#12011yescharacteristic
97C22×C97#12112yes—
98C22×C97#12311yescharacteristic
99C4×C97—22no—
100C4×C97—42no—
101C4×C97—41no—
102Dic97—44no—
103Dic97—82no—
104C2×C97—21no—
105C2×C97—22no—
106C2×C97—21no—
107C2×C97—22no—
108C2×C97—21no—
109C2×C97—42no—
110C2×C97—81no—
111C2×C97#38611yescharacteristic
112#1913—971nomaximal, Sylow
113C97#191311yescharacteristic, complemented, Sylow
114#534D9711yescharacteristic
115#518—971no—
116C2×#99—972no—
117C2×#99—971no—
118#342—972no—
119C22×D4C2×D9711yescharacteristic
120C22×D4C2×D9711yescharacteristic
121#123—21no—
122#123Dic9712yescomplemented
123#121—22no—
124#120—21no—
125#120Dic9712yescomplemented
126C4×D4—1944no—
127#117—1941no—
128C2×C22⋊C4—972no—
129C2×C22⋊C4—972no—
130C2×C22⋊C4—1941no—
131C2×C22⋊C4C2×D9711yescharacteristic
132#99—1944no—
133#99—1942no—
134C24C2×Dic9711yescharacteristic, complemented
135C24C22⋊D9711yescharacteristic
136C24C22×D9711yescharacteristic
137C2×D4—22no—
138C2×D4—22no—
139C2×D4—22no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—22no—
143C2×D4—22no—
144C2×D4C2×Dic9712yes—
145C2×D4C22⋊D9712yes—
146C22×C4—21no—
147C22×C4—1942no—
148C22×C4—1942no—
149C22×C4—1942no—
150C22×C4C2×Dic9711yescharacteristic, complemented
151C22×C4C22⋊D9711yescharacteristic
152C4⋊C4—22no—
153C4⋊C4—1942no—
154C22⋊C4—22no—
155C22⋊C4—21no—
156C22⋊C4—22no—
157C22⋊C4—974no—
158C22⋊C4—1942no—
159C22⋊C4—1942no—
160C22⋊C4—1942no—
161C22⋊C4—1942no—
162C22⋊C4C2×Dic9712yescomplemented
163C42—1942no—
164C23—21no—
165C23—21no—
166C23—21no—
167C23—22no—
168C23—81no—
169C23—81no—
170C23—81no—
171C23C22⋊Dic9712yes—
172C23C22×Dic9711yescharacteristic
173C23D4×D9712yes—
174C23C2×C22⋊D9711yescharacteristic
175C23C2×C22⋊D9711yescharacteristic
176C23#1309551012yes—
177D4—24no—
178D4—24no—
179D4—41no—
180D4—42no—
181D4—41no—
182D4—82no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—81no—
189C2×C4—7762no—
190C2×C4—7762no—
191C2×C4—7762no—
192C2×C4—3882no—
193C2×C4—1944no—
194C2×C4C22⋊Dic9712yes—
195C22—22no—
196C22—41no—
197C22—42no—
198C22—42no—
199C22—41no—
200C22—41no—
201C22—42no—
202C22—42no—
203C22—42no—
204C22—41no—
205C22—81no—
206C22—82no—
207C22C2×C22⋊Dic9711yescharacteristic
208C22D4×Dic9712yes—
209C22#1569077811yescharacteristic
210C22#1569078212yes—
211C22#1569079111yescharacteristic
212C4—22no—
213C4—42no—
214C4—41no—
215C4—7762no—
216C4—3884no—
217C2—21no—
218C2—22no—
219C2—21no—
220C2—22no—
221C2—21no—
222C2—42no—
223C2—81no—
224C2#2026061511yescharacteristic
subgroupG/H
centerC2#20260615
commutatorC23×C97C22×C4
FrattiniC24C22×D97
Fitting, prime radical#534×C97C2
socleC2×C97#386
cosocleC24×C97C23
hypercenter#534D97
hypocenterC97#1913
LCSG ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C24 ◁ #534

Complements

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

NH ≅ G/Ntype
1. #20261210C2semidirect
2. C2×#15690725C2semidirect
3. C2×#15690725C2semidirect
4. #20260501C2semidirect
16. #123×C97C4semidirect
19. #120×C97C4semidirect
29. C22⋊Dic97D4semidirect
31. C24×C97C2×C4semidirect
44. C22×C4×C97C2×C4semidirect
50. C22⋊C4×C97C2×C4semidirect
113. C97#1913semidirect
122. #123Dic97semidirect
125. #120Dic97semidirect
134. C24C2×Dic97semidirect
150. C22×C4C2×Dic97semidirect
162. C22⋊C4C2×Dic97semidirect

Maximal subgroups

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

subgroupindex
#202612102
C2×#156907252
#202605012
#534×C972
#191397