Finite Group Databasebeta

← #3568⋊C9 #11919853

Subgroups — #3568⋊C9 #11919853

Order 1152.

366 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 32 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#10589566C211yesmaximal, characteristic, complemented
2C3×#3568C311yesmaximal, characteristic
3#126⋊C9—42nomaximal
4#120⋊C9—42nomaximal
5C2×C42⋊C9—41nomaximal
6C3×#563—31no—
7C3×H(3,𝔽4)C2×C311yescharacteristic
8C3×#556—32no—
9C3×#551—32no—
10C3×#542—32no—
11C3×#538—32no—
12C3×#529—31no—
13C22⋊(C22⋊C9)—42no—
14C42⋊C9—41no—
15C42⋊C9—42no—
16#3568C911yescharacteristic, complemented, Sylow
17C3×C4⋊Q8—32no—
18C3×#126—32no—
19C3×#126—32no—
20C3×#126—32no—
21C3×#126—32no—
22C3×#126A412yes—
23C3×#125—31no—
24C3×#124—32no—
25C3×#124—32no—
26C3×#124—31no—
27C3×#124—32no—
28C3×#123—32no—
29C3×#123—32no—
30C3×#123—32no—
31C3×C22⋊Q8—32no—
32C3×C22⋊Q8—32no—
33C3×C22⋊Q8—31no—
34C3×C22⋊Q8—32no—
35C3×#121—31no—
36C3×#121—32no—
37C3×#120—32no—
38C3×#120A412yes—
39C3×C4×Q8—31no—
40C3×C4×Q8—32no—
41C3×C4×D4—32no—
42C3×C4×D4—31no—
43C3×C4×D4—32no—
44C3×#117—32no—
45C3×#117—32no—
46C2×C3×C22⋊C4—32no—
47C2×C3×C42A411yescharacteristic
48C2×C22⋊C9—161no—
49#563—31no—
50H(3,𝔽4)C2×C911yescharacteristic, complemented
51#556—32no—
52#551—32no—
53#542—32no—
54#538—32no—
55#529—31no—
56C24×C3C2×A412yes—
57C2×C3×Q8—32no—
58C2×C3×Q8—31no—
59C2×C3×D4—31no—
60C2×C3×D4—32no—
61C2×C3×D4—32no—
62C22×C3×C4—32no—
63C22×C3×C4—32no—
64C22×C3×C4—31no—
65C3×C4⋊C4—32no—
66C3×C4⋊C4—32no—
67C3×C4⋊C4—32no—
68C3×C4⋊C4—32no—
69C3×C4⋊C4—31no—
70C3×C4⋊C4—32no—
71C3×C4⋊C4—31no—
72C3×C4⋊C4—32no—
73C3×C22⋊C4—32no—
74C3×C22⋊C4—32no—
75C3×C22⋊C4—32no—
76C3×C22⋊C4—32no—
77C3×C22⋊C4—32no—
78C3×C22⋊C4—32no—
79C3×C22⋊C4—32no—
80C3×C22⋊C4—32no—
81C3×C22⋊C4—32no—
82C3×C42—31no—
83C3×C42—32no—
84C3×C42—32no—
85C3×C42C2×A411yescharacteristic
86C3×C42C2×A412yes—
87C22⋊C9—161no—
88C22⋊C9—324no—
89C4⋊Q8—32no—
90#126—32no—
91#126—32no—
92#126—32no—
93#126—32no—
94#126C22⋊C912yescomplemented
95#125—31no—
96#124—32no—
97#124—32no—
98#124—31no—
99#124—32no—
100#123—32no—
101#123—32no—
102#123—32no—
103C22⋊Q8—31no—
104C22⋊Q8—32no—
105C22⋊Q8—32no—
106C22⋊Q8—32no—
107#121—31no—
108#121—32no—
109#120—32no—
110#120C22⋊C912yescomplemented
111C4×Q8—31no—
112C4×Q8—32no—
113C4×D4—32no—
114C4×D4—31no—
115C4×D4—32no—
116#117—32no—
117#117—32no—
118C2×C22⋊C4—32no—
119C2×C42C22⋊C911yescharacteristic, complemented
120C23×C3—32no—
121C23×C3—122no—
122C23×C3C22⋊A411yescharacteristic
123C3×Q8—122no—
124C3×Q8—121no—
125C3×D4—122no—
126C3×D4—121no—
127C3×D4—122no—
128C2×C3×C4—31no—
129C2×C3×C4—32no—
130C2×C3×C4—32no—
131C2×C3×C4—31no—
132C2×C3×C4—32no—
133C2×C3×C4—122no—
134C2×C3×C4—121no—
135C2×C3×C4—122no—
136C2×C9—641no—
137C24C2×C22⋊C912yes—
138C2×Q8—32no—
139C2×Q8—31no—
140C2×D4—31no—
141C2×D4—32no—
142C2×D4—32no—
143C22×C4—32no—
144C22×C4—32no—
145C22×C4—31no—
146C4⋊C4—32no—
147C4⋊C4—32no—
148C4⋊C4—32no—
149C4⋊C4—32no—
150C4⋊C4—31no—
151C4⋊C4—32no—
152C4⋊C4—31no—
153C4⋊C4—32no—
154C22⋊C4—32no—
155C22⋊C4—32no—
156C22⋊C4—32no—
157C22⋊C4—32no—
158C22⋊C4—32no—
159C22⋊C4—32no—
160C22⋊C4—32no—
161C22⋊C4—32no—
162C22⋊C4—32no—
163C42—31no—
164C42—32no—
165C42—32no—
166C42C2×C22⋊C911yescharacteristic
167C42C2×C22⋊C912yes—
168C22×C3—61no—
169C22×C3—62no—
170C22×C3—62no—
171C22×C3—62no—
172C22×C3—84no—
173C22×C3C2×C22⋊A411yescharacteristic
174C3×C4—61no—
175C3×C4—62no—
176C3×C4—62no—
177C3×C4—61no—
178C3×C4—62no—
179C9—641noSylow
180C23—32no—
181C23—122no—
182C23C22⋊(C22⋊C9)11yescharacteristic
183Q8—122no—
184Q8—121no—
185D4—122no—
186D4—121no—
187D4—122no—
188C2×C4—31no—
189C2×C4—32no—
190C2×C4—32no—
191C2×C4—31no—
192C2×C4—32no—
193C2×C4—122no—
194C2×C4—121no—
195C2×C4—122no—
196C2×C3—31no—
197C2×C3—41no—
198C2×C3—122no—
199C22—61no—
200C22—62no—
201C22—62no—
202C22—62no—
203C22—84no—
204C22C2×C22⋊(C22⋊C9)11yescharacteristic
205C4—61no—
206C4—62no—
207C4—62no—
208C4—61no—
209C4—62no—
210C3#3568⋊C311yescharacteristic
211C2—31no—
212C2—41no—
213C2—122no—
subgroupG/H
center, hypercenterC3#3568⋊C3
commutator, hypocenterH(3,𝔽4)C2×C9
Frattini, socleC22×C3C2×C22⋊A4
Fitting, prime radicalC3×#3568C3
cosocleC3×H(3,𝔽4)C2×C3
LCSG ▷ H(3,𝔽4)
UCSC1 ◁ C3

Complements

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

NH ≅ G/Ntype
1. #10589566C2semidirect
16. #3568C9semidirect
50. H(3,𝔽4)C2×C9semidirect
94. #126C22⋊C9semidirect
110. #120C22⋊C9semidirect
119. C2×C42C22⋊C9semidirect

Maximal subgroups

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

subgroupindex
#105895662
C3×#35683
#126⋊C94
#120⋊C94
C2×C42⋊C94