Finite Group Databasebeta

← #11796524

Subgroups — #11796524

Order 1152.

690 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 197 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
1C32×#536C211yesmaximal, characteristic
2C2×#63663C212yesmaximal, complemented
3#10589414C212yesmaximal, complemented
4#10589408C212yesmaximal, complemented
5#71872—34nomaximal
6C2×C32×#41C411yescharacteristic
7C22×C32×Q8C2211yescharacteristic
8C32×C4⋊Q8C411yescharacteristic
9C32×C4⋊Q8C2211yescharacteristic
10C32×#124C411yescharacteristic
11C32×#124C2211yescharacteristic
12C32×#123—22no—
13C32×C22⋊Q8—22no—
14C32×#117C411yescharacteristic
15C2×#3748—22no—
16#63663—22no—
17#63663C2214yes—
18C3×#536D314yes—
19C2×#836—38no—
20#4548—38no—
21#4542—38no—
22C32×#41—42no—
23C2×C32×Q8—42no—
24C2×C32×Q8C2×C411yescharacteristic
25C2×C32×Q8C2×C411yescharacteristic
26C2×C32×Q8D412yes—
27C2×C32×Q8C2311yescharacteristic
28C2×C32×D4—21no—
29C2×C32×D4C2×C411yescharacteristic
30C22×C32×C4—21no—
31C22×C32×C4C2×C411yescharacteristic
32C22×C32×C4D412yes—
33C32×C4⋊C4—21no—
34C32×C4⋊C4—22no—
35C32×C4⋊C4—22no—
36C32×C22⋊C4—21no—
37C32×C22⋊C4—22no—
38C32×C22⋊C4—22no—
39C32×C42C2×C411yescharacteristic
40C32×C42C2×C411yescharacteristic
41#3748—24no—
42#3748—42no—
43C2×C3⋊(C3⋊C8)—42no—
44#2133—91noSylow
45C2×C3×#41Dic314yes—
46C22×C3×Q8C2×D314yes—
47C3×C4⋊Q8Dic314yes—
48C3×C4⋊Q8C2×D314yes—
49C3×#124Dic314yes—
50C3×#124C2×D314yes—
51C3×#123—28no—
52C3×C22⋊Q8—28no—
53C3×#117Dic314yes—
54C2×#208—68no—
55#836—316no—
56#836—68no—
57C23×C32—21no—
58C23×C32C22⋊C411yescharacteristic
59C32×Q8—41no—
60C32×Q8—42no—
61C32×Q8—41no—
62C32×Q8—41no—
63C32×D4—41no—
64C32×D4—42no—
65C2×C32×C4—22no—
66C2×C32×C4—22no—
67C2×C32×C4—22no—
68C2×C32×C4—21no—
69C2×C32×C4—21no—
70C2×C32×C4—21no—
71C2×C32×C4—42no—
72C2×C32×C4C22⋊C412yes—
73C2×C32×C4C22⋊C411yescharacteristic
74C2×C32×C4C22×C411yescharacteristic
75C2×C32×C4C2×D412yes—
76C3⋊(C3⋊C8)—44no—
77#536C3⋊D311yescharacteristic
78C2×#101—92no—
79#356—92no—
80#355—92no—
81C3×#41—48no—
82C2×C3×Q8—48no—
83C2×C3×Q8C2×Dic314yes—
84C2×C3×Q8C2×Dic314yes—
85C2×C3×Q8#6718yes—
86C2×C3×Q8C22×D314yes—
87C2×C3×D4—24no—
88C2×C3×D4C2×Dic314yes—
89C22×C3×C4—24no—
90C22×C3×C4C2×Dic314yes—
91C22×C3×C4#6718yes—
92C3×C4⋊C4—24no—
93C3×C4⋊C4—28no—
94C3×C4⋊C4—28no—
95C3×C22⋊C4—24no—
96C3×C22⋊C4—28no—
97C3×C22⋊C4—28no—
98C3×C42C2×Dic314yes—
99C3×C42C2×Dic314yes—
100#208—616no—
101#208—128no—
102C2×C3⋊C8—128no—
103C22×C32—41no—
104C22×C32—42no—
105C22×C32—41no—
106C22×C32#9912yes—
107C22×C32C2×C22⋊C411yescharacteristic
108C32×C4—22no—
109C32×C4—21no—
110C32×C4—22no—
111C32×C4—21no—
112C32×C4—41no—
113C32×C4—41no—
114C32×C4—41no—
115C2×#41C3⋊Dic311yescharacteristic
116C22×Q8C2×C3⋊D311yescharacteristic
117C2×M16—182no—
118C4⋊Q8C3⋊Dic311yescharacteristic
119C4⋊Q8C2×C3⋊D311yescharacteristic
120#124C3⋊Dic311yescharacteristic
121#124C2×C3⋊D311yescharacteristic
122#123—22no—
123C22⋊Q8—22no—
124#117C3⋊Dic311yescharacteristic
125#101—94no—
126#101—182no—
127C23×C3—24no—
128C23×C3#21714yes—
129C3×Q8—44no—
130C3×Q8—48no—
131C3×Q8—44no—
132C3×Q8—44no—
133C3×D4—44no—
134C3×D4—48no—
135C2×C3×C4—28no—
136C2×C3×C4—28no—
137C2×C3×C4—28no—
138C2×C3×C4—24no—
139C2×C3×C4—24no—
140C2×C3×C4—24no—
141C2×C3×C4—48no—
142C2×C3×C4#21718yes—
143C2×C3×C4#21714yes—
144C2×C3×C4C22×Dic314yes—
145C2×C3×C4C2×#6718yes—
146C3⋊C8—1216no—
147C2×C32—21no—
148C2×C32—22no—
149C2×C32—81no—
150C2×C32C2×#9911yescharacteristic
151#41—42no—
152C2×Q8—42no—
153C2×Q8C2×C3⋊Dic311yescharacteristic
154C2×Q8C2×C3⋊Dic311yescharacteristic
155C2×Q8#63812yes—
156C2×Q8C22×C3⋊D311yescharacteristic
157C2×D4—21no—
158C2×D4C2×C3⋊Dic311yescharacteristic
159C22×C4—21no—
160C22×C4C2×C3⋊Dic311yescharacteristic
161C22×C4#63812yes—
162M16—184no—
163M16—362no—
164C2×C8—362no—
165C4⋊C4—21no—
166C4⋊C4—22no—
167C4⋊C4—22no—
168C22⋊C4—21no—
169C22⋊C4—22no—
170C22⋊C4—22no—
171C42C2×C3⋊Dic311yescharacteristic
172C42C2×C3⋊Dic311yescharacteristic
173C22×C3—44no—
174C22×C3—48no—
175C22×C3—44no—
176C22×C3#83418yes—
177C22×C3C2×#21714yes—
178C3×C4—28no—
179C3×C4—24no—
180C3×C4—28no—
181C3×C4—24no—
182C3×C4—44no—
183C3×C4—44no—
184C3×C4—44no—
185C32#213311yescharacteristic, complemented, Sylow
186C23—21no—
187C23#375711yescharacteristic
188Q8—41no—
189Q8—42no—
190Q8—41no—
191Q8—41no—
192D4—41no—
193D4—42no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—42no—
201C2×C4#375712yes—
202C2×C4#375711yescharacteristic
203C2×C4C22×C3⋊Dic311yescharacteristic
204C2×C4C2×#63812yes—
205C8—364no—
206C2×C3—24no—
207C2×C3—28no—
208C2×C3—84no—
209C2×C3C2×#83414yes—
210C22—41no—
211C22—42no—
212C22—41no—
213C22#6366112yes—
214C22C2×#375711yescharacteristic
215C4—22no—
216C4—21no—
217C4—22no—
218C4—21no—
219C4—41no—
220C4—41no—
221C4—41no—
222C3#7187214yescomplemented
223C2—21no—
224C2—22no—
225C2—81no—
226C2C2×#6366111yescharacteristic
subgroupG/H
centerC2C2×#63661
commutatorC2×C32×C4C22×C4
FrattiniC2×Q8C22×C3⋊D3
FittingC32×#536C2
socleC2×C32C2×#99
prime radicalC2×C32×#41C4
cosocleC2×C32×Q8C23
hypercenter#536C3⋊D3
hypocenterC32#2133
LCSG ▷ C2×C32×C4 ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C23 ◁ C22×Q8 ◁ #536

Complements

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

NH ≅ G/Ntype
2. C2×#63663C2semidirect
3. #10589414C2semidirect
4. #10589408C2semidirect
185. C32#2133semidirect
222. C3#71872semidirect

Maximal subgroups

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

subgroupindex
C32×#5362
C2×#636632
#105894142
#105894082
#718723