Finite Group Databasebeta

← #28378387

Subgroups — #28378387

Order 13952.

216 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 60 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#20296042C211yesmaximal, characteristic, complemented
2#20296029C211yesmaximal, characteristic, complemented
3#20295876C212yesmaximal, complemented
4D4⋊#11724786C212yesmaximal, complemented
5#20295557C211yesmaximal, characteristic
6C2×#13182038—21no—
7#16812766—21no—
8C4⋊#11724786—21no—
9C4⋊#11724786C2212yes—
10C22⋊#11724786—21no—
11C4×#11724786C2212yes—
12#16812692—21no—
13#16812690C411yescharacteristic
14#16812686C411yescharacteristic
15#16812680—21no—
16#16812678C411yescharacteristic
17#16812664C2211yescharacteristic
18D4×Dic109—21no—
19D4×Dic109C2211yescharacteristic
20#16812655—21no—
21C2×C4×Dic109C2211yescharacteristic
22#121×C109C411yescharacteristic
23#13182038—42no—
24C2×#11724786—22no—
25C2×#11724786—22no—
26C2×Dic218C2×C411yescharacteristic
27C22×Dic109—21no—
28C22×Dic109D411yescharacteristic
29C22×Dic109D411yescharacteristic
30C4⋊Dic109—21no—
31C4⋊Dic109C2×C411yescharacteristic
32#13182008—21no—
33#13182008—21no—
34#13182008—21no—
35C22⋊Dic109—21no—
36C22⋊Dic109—21no—
37C22⋊Dic109—21no—
38C22⋊Dic109—21no—
39C4×Dic109—21no—
40C4×Dic109C2×C411yescharacteristic
41C4×Dic109D411yescharacteristic
42C4×Dic109D411yescharacteristic
43C4×Dic109C2311yescharacteristic
44C2×D4×C109—21no—
45C2×D4×C109C2×C411yescharacteristic
46C22×C4×C109C2×C411yescharacteristic
47C4⋊C4×C109C2×C411yescharacteristic
48C22⋊C4×C109—21no—
49C23×C109—21no—
50C23×C109C22⋊C411yescharacteristic
51D4×C109—22no—
52D4×C109—42no—
53C2×C4×C109—22no—
54C2×C4×C109—21no—
55C2×C4×C109C22⋊C411yescharacteristic
56C2×C4×C109C22×C411yescharacteristic
57C2×Dic109—21no—
58C2×Dic109—21no—
59C2×Dic109—21no—
60C2×Dic109—21no—
61C2×Dic109—41no—
62C2×Dic109—41no—
63C2×Dic109—41no—
64C2×Dic109C22⋊C411yescharacteristic
65C2×Dic109C22⋊C411yescharacteristic
66C2×Dic109C2×D411yescharacteristic
67C2×Dic109C2×D411yescharacteristic
68Dic218—22no—
69#11724786—42no—
70#11724786—42no—
71C22×C109—21no—
72C22×C109—21no—
73C22×C109—21no—
74C22×C109—41no—
75C22×C109—41no—
76C22×C109—41no—
77C22×C109C2×C22⋊C411yescharacteristic
78C4×C109—21no—
79C4×C109—41no—
80C4×C109#13112yes—
81Dic109—21no—
82Dic109—21no—
83Dic109—21no—
84Dic109—41no—
85Dic109—41no—
86Dic109AGL(1,ℤ/8)11yescharacteristic
87Dic109#13711yescharacteristic
88C2×C109—41no—
89C2×C109—81no—
90C2×C109#40811yescharacteristic
91C2×C109#41711yescharacteristic
92C2×C109#42111yescharacteristic
93#1495—1091nomaximal, Sylow
94C109#149511yescharacteristic, complemented, Sylow
95#529—1091no—
96#431—1091no—
97#423—1091no—
98Q8⋊C8—1092no—
99D4⋊C8—1092no—
100C2×M16—2181no—
101#126—2181no—
102#125—1091no—
103#124—1091no—
104#123—2181no—
105C22⋊Q8—1091no—
106#121#8913711yescharacteristic
107C4×Q8—1091no—
108C4×D4—1091no—
109C4×D4—2181no—
110#117—2181no—
111C2×C42—1091no—
112C4⋊C8—1092no—
113C4⋊C8—2181no—
114C4⋊C8—2181no—
115C22⋊C8—2181no—
116C4×C8—1092no—
117C2×Q8—1091no—
118C2×D4—21no—
119C2×D4C2×#8913711yescharacteristic
120C22×C4—1091no—
121C22×C4—1091no—
122C22×C4—2181no—
123C22×C4C2×#8913711yescharacteristic
124M16—4362no—
125C2×C8—2182no—
126C2×C8—2182no—
127C4⋊C4—1091no—
128C4⋊C4—2181no—
129C4⋊C4—2181no—
130C4⋊C4—2181no—
131C4⋊C4—2181no—
132C4⋊C4C2×#8913711yescharacteristic
133C22⋊C4—21no—
134C22⋊C4—2181no—
135C22⋊C4—2181no—
136C22⋊C4—2181no—
137C22⋊C4—2181no—
138C42—1091no—
139C42—1091no—
140C42—1091no—
141C42—1091no—
142C42—2181no—
143C23—21no—
144C23C22⋊#8913711yescharacteristic
145Q8—2182no—
146D4—22no—
147D4—42no—
148C2×C4—22no—
149C2×C4—21no—
150C2×C4—1091no—
151C2×C4—1091no—
152C2×C4—1091no—
153C2×C4—1091no—
154C2×C4—4361no—
155C2×C4—4361no—
156C2×C4—4361no—
157C2×C4—2181no—
158C2×C4—2181no—
159C2×C4—2181no—
160C2×C4—2181no—
161C2×C4C22⋊#8913711yescharacteristic
162C2×C4C22×#8913711yescharacteristic
163C8—4362no—
164C8—4362no—
165C22—21no—
166C22—21no—
167C22—21no—
168C22—41no—
169C22—41no—
170C22—41no—
171C22C2×C22⋊#8913711yescharacteristic
172C4—21no—
173C4—41no—
174C4—1091no—
175C4—1091no—
176C4—4361no—
177C4—4361no—
178C4—2181no—
179C4—2181no—
180C4—2181no—
181C4#1681278612yes—
182C2—41no—
183C2—81no—
184C2#2029599911yescharacteristic
185C2#2029601711yescharacteristic
186C2#2029602411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#89137
commutatorC2×C4×C109C22×C4
FrattiniC2×C4C22×#89137
Fitting, prime radical#121×C109C4
socleC22×C109C2×C22⋊C4
cosocleC4×Dic109C23
hypercenter#121#89137
hypocenterC109#1495
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
1. #20296042C2semidirect
2. #20296029C2semidirect
3. #20295876C2semidirect
4. D4⋊#11724786C2semidirect
94. C109#1495semidirect

Maximal subgroups

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

subgroupindex
#202960422
#202960292
#202958762
D4⋊#117247862
#202955572
#1495109