Finite Group Databasebeta

← #28379059

Subgroups — #28379059

Order 13952.

338 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 74 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×#13182038C211yesmaximal, characteristic
2C2×#16812760C212yesmaximal, complemented
3#20295996C211yesmaximal, characteristic
4#20295906C212yesmaximal, complemented
5C2×#16812676C211yesmaximal, characteristic
6C2×#13182038—24no—
7C2×#13182038—21no—
8C2×#13182038C2212yes—
9C2×#13182038C2212yes—
10C22×#11724786—21no—
11#16812760—24no—
12C22⋊#11724786—21no—
13C22⋊#11724786—21no—
14C22×Dic218C2211yescharacteristic
15C23×Dic109C2211yescharacteristic
16#16812676—21no—
17#16812676—21no—
18#16812676—21no—
19#16812676—21no—
20C2×C4⋊Dic109C411yescharacteristic
21C2×#13182008C411yescharacteristic
22C2×#13182008C2211yescharacteristic
23C2×C22⋊Dic109C411yescharacteristic
24C2×C22⋊C4×C109C411yescharacteristic
25#13182038—24no—
26#13182038—22no—
27#13182038—22no—
28#13182038—22no—
29#13182038D414yes—
30C2×#11724786—24no—
31C2×#11724786—22no—
32C2×#11724786—22no—
33C2×Dic218—21no—
34C2×Dic218—21no—
35C2×Dic218—22no—
36C2×Dic218—22no—
37C22×Dic109—22no—
38C22×Dic109—22no—
39C22×Dic109C2×C411yescharacteristic
40C22×Dic109C2×C411yescharacteristic
41C22×Dic109C2×C411yescharacteristic
42C22×Dic109D411yescharacteristic
43C22×Dic109D411yescharacteristic
44C22×Dic109D411yescharacteristic
45C22×Dic109D411yescharacteristic
46C22×Dic109C2311yescharacteristic
47C4⋊Dic109—21no—
48C4⋊Dic109—21no—
49#13182008—21no—
50#13182008—21no—
51#13182008—21no—
52#13182008—21no—
53C22⋊Dic109—21no—
54C22⋊Dic109—21no—
55C24×C109C2×C411yescharacteristic
56C22×C4×C109C2×C411yescharacteristic
57C22×C4×C109C2×C411yescharacteristic
58C22⋊C4×C109—21no—
59C22⋊C4×C109—21no—
60C23×C109—22no—
61C23×C109—22no—
62C23×C109—41no—
63C23×C109C22⋊C411yescharacteristic
64C23×C109C22⋊C411yescharacteristic
65C23×C109C22×C411yescharacteristic
66C2×C4×C109—21no—
67C2×C4×C109—21no—
68C2×C4×C109—22no—
69C2×C4×C109—41no—
70C2×Dic109—22no—
71C2×Dic109—21no—
72C2×Dic109—21no—
73C2×Dic109—22no—
74C2×Dic109—22no—
75C2×Dic109—21no—
76C2×Dic109—21no—
77C2×Dic109—21no—
78C2×Dic109—21no—
79C2×Dic109—22no—
80C2×Dic109—41no—
81C2×Dic109C22⋊C411yescharacteristic
82C2×Dic109C22⋊C411yescharacteristic
83C2×Dic109C2×D412yes—
84C2×Dic109C2×D411yescharacteristic
85C2×Dic109C2×D411yescharacteristic
86C2×Dic109#4112yes—
87Dic218—41no—
88Dic218—42no—
89Dic218—41no—
90#11724786—24no—
91#11724786—42no—
92C22×C109—21no—
93C22×C109—21no—
94C22×C109—21no—
95C22×C109—21no—
96C22×C109—21no—
97C22×C109—21no—
98C22×C109—21no—
99C22×C109—44no—
100C22×C109#10112yescomplemented
101C22×C109C2×C22⋊C411yescharacteristic
102C22×C109C4×D412yes—
103C4×C109—41no—
104C4×C109—41no—
105Dic109—21no—
106Dic109—21no—
107Dic109—41no—
108Dic109—41no—
109Dic109#12011yescharacteristic
110Dic109#12112yes—
111Dic109#12311yescharacteristic
112C2×C109—22no—
113C2×C109—22no—
114C2×C109—41no—
115C2×C109#38611yescharacteristic
116C2×C109C2×#10111yescharacteristic
117C2×C109#41311yescharacteristic
118#1901—1091nomaximal, Sylow
119C109#190111yescharacteristic, complemented, Sylow
120C22×M16—1091no—
121C2×C22⋊Q8—1091no—
122C2×#101—1092no—
123#407—1091no—
124#343—1092no—
125C22×Q8—1091no—
126C23×C4—1091no—
127C2×M16—1092no—
128C2×M16—1092no—
129C2×M16—2184no—
130C2×M16—2181no—
131C22×C8—2181no—
132C22⋊Q8—2181no—
133C22⋊Q8—2181no—
134C22⋊Q8—2181no—
135C22⋊Q8—2181no—
136C2×C4⋊C4—1091no—
137C2×C4⋊C4—1091no—
138C2×C4⋊C4—1091no—
139C2×C22⋊C4—1091no—
140C2×C22⋊C4#8913711yescharacteristic
141#101—2184no—
142C22⋊C8—2181no—
143C22⋊C8—2181no—
144C24C2×#8913711yescharacteristic
145C2×Q8—2182no—
146C2×Q8—2181no—
147C2×Q8—2181no—
148C2×Q8—2182no—
149C22×C4—1091no—
150C22×C4—1091no—
151C22×C4—1091no—
152C22×C4—1091no—
153C22×C4—1091no—
154C22×C4—1091no—
155C22×C4—1091no—
156C22×C4—1091no—
157C22×C4—2182no—
158C22×C4—2182no—
159C22×C4C2×#8913711yescharacteristic
160C22×C4C2×#8913711yescharacteristic
161M16—1094no—
162M16—2182no—
163M16—2182no—
164M16—2182no—
165M16—2184no—
166C2×C8—2182no—
167C2×C8—2182no—
168C2×C8—2184no—
169C4⋊C4—2181no—
170C4⋊C4—2181no—
171C4⋊C4—2181no—
172C4⋊C4—2181no—
173C4⋊C4—2181no—
174C4⋊C4—2181no—
175C22⋊C4—21no—
176C22⋊C4—21no—
177C22⋊C4—2181no—
178C22⋊C4—2181no—
179C23—22no—
180C23—22no—
181C23—41no—
182C23C22⋊#8913711yescharacteristic
183C23C22⋊#8913711yescharacteristic
184C23C22×#8913711yescharacteristic
185Q8—4362no—
186Q8—4361no—
187Q8—4361no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—41no—
192C2×C4—1091no—
193C2×C4—1091no—
194C2×C4—1092no—
195C2×C4—1092no—
196C2×C4—1091no—
197C2×C4—1091no—
198C2×C4—4361no—
199C2×C4—2181no—
200C2×C4—2181no—
201C2×C4—2182no—
202C2×C4—2182no—
203C2×C4—2181no—
204C2×C4—2181no—
205C2×C4—2181no—
206C2×C4—2181no—
207C2×C4—2182no—
208C2×C4—2182no—
209C8—4362no—
210C8—2184no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—21no—
216C22—21no—
217C22—21no—
218C22—44no—
219C22#1681276012yescomplemented
220C22C2×C22⋊#8913711yescharacteristic
221C22D4×#8913712yes—
222C4—41no—
223C4—41no—
224C4—1092no—
225C4—1091no—
226C4—1091no—
227C4—4361no—
228C4—4361no—
229C4—2181no—
230C4—2181no—
231C2—22no—
232C2—22no—
233C2—41no—
234C2#2029597411yescharacteristic
235C2C2×#1681276011yescharacteristic
236C2#2029601311yescharacteristic
subgroupG/H
centerC22C2×C22⋊#89137
commutatorC23×C109C22×C4
FrattiniC23C22×#89137
FittingC2×C22⋊C4×C109C4
socleC22×C109C2×C22⋊C4
prime radicalC24×C109C2×C4
cosocleC22×Dic109C23
hypercenterC2×C22⋊C4#89137
hypocenterC109#1901
LCSG ▷ C23×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C24 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#16812760C2semidirect
4. #20295906C2semidirect
100. C22×C109#101semidirect
119. C109#1901semidirect
219. C22#16812760semidirect

Maximal subgroups

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

subgroupindex
C22×#131820382
C2×#168127602
#202959962
#202959062
C2×#168126762
#1901109