Finite Group Databasebeta

← #11790534

Subgroups — #11790534

Order 1152.

433 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 433 conjugacy classes of subgroups — too many to draw. Only its 124 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#10592821C211yesmaximal, characteristic
2C3×#5124C212yesmaximal
3C2×#63578C211yesmaximal, characteristic
4C2×#63578C211yesmaximal, characteristic
5#10589171C212yesmaximal
6#70725—32nomaximal
7C2×C32×M16C2211yescharacteristic
8#64135C2211yescharacteristic
9C22×C3⋊(C3⋊C8)C2211yescharacteristic
10C2×C3×#208C2212yes—
11C2×C3×#208C2212yes—
12#63642—21no—
13C8×C3⋊Dic3—21no—
14C3×#822—24no—
15#63578—21no—
16#63578—21no—
17#63578C414yescomplemented
18C3×#430—32no—
19#5124D312yes—
20C2×#822—32no—
21C2×#822—32no—
22#5122—32no—
23#5122—61no—
24#4559—32no—
25#4556—32no—
26C22×C32×C4C2311yescharacteristic
27C32×M16—21no—
28C32×M16D412yes—
29C2×C32×C8—21no—
30#3757Q812yes—
31C4⋊(C3⋊Dic3)—21no—
32C4×C3⋊Dic3—21no—
33C2×C3⋊(C3⋊C8)C2×C412yes—
34C2×C3⋊(C3⋊C8)D411yescharacteristic
35C2×C3⋊(C3⋊C8)D411yescharacteristic
36C2×C3⋊(C3⋊C8)Q812yes—
37C3×#208—22no—
38C3×#208—22no—
39C3×#208—22no—
40C3×#208C2×C414yes—
41C2×C3×C3⋊C8—22no—
42C2×C3×C3⋊C8—22no—
43#1953—91noSylow
44C2×C3×M16—21no—
45C2×C3×M16—32no—
46C2×C3×M16—32no—
47C2×C3×M16C2×D312yes—
48#926—32no—
49#926—61no—
50C2×#208C2×D312yes—
51C2×#208C2×D312yes—
52C22×C3⋊C8—32no—
53C22×C3⋊C8—61no—
54C3×#108—64no—
55#822—24no—
56#822—38no—
57#822—62no—
58#822—62no—
59#815—62no—
60#815—62no—
61C8×Dic3—62no—
62C8×Dic3—62no—
63C23×C32C2×Q811yescharacteristic
64C2×C32×C4C22×C411yescharacteristic
65C2×C32×C4C2×D411yescharacteristic
66C2×C32×C4C2×D411yescharacteristic
67C2×C32×C4C2×Q811yescharacteristic
68C2×C32×C4#4111yescharacteristic
69C2×C32×C4#4111yescharacteristic
70C2×C3⋊Dic3—22no—
71C32×C8—22no—
72C3⋊(C3⋊C8)C4⋊C412yes—
73C3⋊(C3⋊C8)C4⋊C412yes—
74C3×C3⋊C8—24no—
75C3×C3⋊C8—42no—
76#430—32no—
77C2×#108—91no—
78C2×#108—91no—
79#405—91no—
80#344—92no—
81C22×C3×C4—21no—
82C22×C3×C4C22×D312yes—
83C3×M16—22no—
84C3×M16—22no—
85C3×M16—22no—
86C3×M16—34no—
87C3×M16—62no—
88C3×M16—62no—
89C3×M16—62no—
90C3×M16#6714yes—
91C2×C3×C8—22no—
92C2×C3×C8—22no—
93C2×C3×C8—62no—
94C2×C3×C8—62no—
95#217—34no—
96#217—62no—
97C4⋊Dic3—62no—
98C4⋊Dic3—62no—
99C4×Dic3—62no—
100C4×Dic3—62no—
101#208—22no—
102#208—22no—
103#208—22no—
104#208C4×D314yes—
105C2×C3⋊C8—22no—
106C2×C3⋊C8—22no—
107C2×C3⋊C8—34no—
108C2×C3⋊C8—32no—
109C2×C3⋊C8—32no—
110C2×C3⋊C8—34no—
111C2×C3⋊C8—61no—
112C2×C3⋊C8—61no—
113C2×C3⋊C8—62no—
114C2×C3⋊C8—62no—
115C22×C32—41no—
116C22×C32C4×Q811yescharacteristic
117C22×C32C22⋊Q811yescharacteristic
118C22×C32C4⋊Q811yescharacteristic
119C32×C4C2×C4⋊C411yescharacteristic
120C32×C4C4×D411yescharacteristic
121C32×C4#12111yescharacteristic
122C32×C4#12511yescharacteristic
123C3⋊Dic3—42no—
124C2×M16—32no—
125C2×M16—32no—
126C2×M16D3211yescharacteristic
127C22×C8—91no—
128#117—91no—
129#108—64no—
130#108—94no—
131#108—181no—
132#108—181no—
133#97—181no—
134C4×C8—181no—
135C23×C3—21no—
136C23×C3D3×Q812yes—
137C2×C3×C4—21no—
138C2×C3×C4—21no—
139C2×C3×C4—21no—
140C2×C3×C4—21no—
141C2×C3×C4—21no—
142C2×C3×C4—21no—
143C2×C3×C4C2×C4×D312yes—
144C2×C3×C4#23512yes—
145C2×C3×C4D3×D412yes—
146C2×C3×C4#23712yes—
147C2×C3×C4D3×Q812yes—
148C2×C3×C4C2×#6712yes—
149C2×Dic3—64no—
150C2×Dic3—64no—
151C3×C8—24no—
152C3×C8—24no—
153C3×C8—64no—
154C3×C8—122no—
155C3⋊C8—24no—
156C3⋊C8—34no—
157C3⋊C8—34no—
158C3⋊C8—42no—
159C3⋊C8—62no—
160C3⋊C8—62no—
161C2×C32—21no—
162C2×C32—21no—
163C2×C32—21no—
164C2×C32#38911yescharacteristic
165C22×C4C2×D3211yescharacteristic
166M16—21no—
167M16—34no—
168M16—62no—
169M16—62no—
170M16—62no—
171M16#62512yes—
172C2×C8—21no—
173C2×C8—62no—
174C2×C8—62no—
175C2×C8—92no—
176C2×C8—92no—
177C2×C8—91no—
178C2×C8—91no—
179C4⋊C4—181no—
180C22⋊C4—92no—
181C42—181no—
182C22×C3—21no—
183C22×C3—21no—
184C22×C3—21no—
185C22×C3—42no—
186C22×C3—42no—
187C22×C3#88712yes—
188C22×C3#89612yes—
189C22×C3#94412yes—
190C3×C4—21no—
191C3×C4—21no—
192C3×C4—21no—
193C3×C4—21no—
194C3×C4#88912yes—
195C3×C4D3×C4⋊C412yes—
196C3×C4C4×#6712yes—
197C3×C4#93912yes—
198Dic3—124no—
199Dic3—124no—
200C32#195311yescharacteristic, complemented, Sylow
201C23#379711yescharacteristic
202C2×C4—182no—
203C2×C4#379711yescharacteristic
204C2×C4#379811yescharacteristic
205C2×C4C4×D3211yescharacteristic
206C2×C4#380511yescharacteristic
207C2×C4C2×#62511yescharacteristic
208C2×C4#381111yescharacteristic
209C8—22no—
210C8—64no—
211C8—92no—
212C8—92no—
213C8—122no—
214C2×C3—22no—
215C2×C3—22no—
216C2×C3—22no—
217C2×C3—22no—
218C2×C3—21no—
219C2×C3—41no—
220C2×C3—41no—
221C2×C3#497612yes—
222C22—41no—
223C22#6383811yescharacteristic
224C22#6386511yescharacteristic
225C22#6387311yescharacteristic
226C4—362no—
227C4#6387011yescharacteristic
228C4#6388311yescharacteristic
229C4C4×#62511yescharacteristic
230C4#6398011yescharacteristic
231C3—21no—
232C3#7072512yescomplemented
233C2—21no—
234C2—21no—
235C2—21no—
236C2#1059133711yescharacteristic
subgroupG/H
centerC4#63980
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C2×D32
FittingC2×C32×M16C22
socleC2×C32#389
prime radicalC23×C32C2×Q8
cosocleC22×C32×C4C23
hypercenterC2×M16D32
hypocenterC32#1953
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
17. #63578C4semidirect
200. C32#1953semidirect
232. C3#70725semidirect

Maximal subgroups

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

subgroupindex
#105928212
C3×#51242
C2×#635782
#105891712
#707253