Finite Group Databasebeta

← C3×#3567⋊C3 #11919850

Subgroups — C3×#3567⋊C3 #11919850

Order 1152.

402 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 38 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
1C3×#564⋊C3C211yesmaximal, characteristic, complemented
2C3×#3567C311yesmaximal, characteristic, complemented
3#3567⋊C3C313yesmaximal, direct product factor
4C3×#127⋊C3—42nomaximal
5C3×#126⋊C3—42nomaximal
6C2×C3×C42⋊C3—41nomaximal
7C3×#564C2×C311yescharacteristic, complemented
8C3×#562—31no—
9C3×#553—32no—
10C3×#552—32no—
11C3×#540—32no—
12C3×#539—32no—
13C3×#529—31no—
14#564⋊C3C2×C313yescomplemented
15C3×C42⋊C3—41no—
16C3×C42⋊C3—42no—
17C3×C42⋊C3—42no—
18#3567C3211yescharacteristic, complemented, Sylow
19C3×#127A412yes—
20C3×#126—32no—
21C3×#126—32no—
22C3×#126A412yes—
23C3×#125—32no—
24C3×#125—31no—
25C3×#125—32no—
26C3×#125—32no—
27C3×#124—31no—
28C3×#123—32no—
29C3×#123—32no—
30C3×#123—32no—
31C3×#123—32no—
32C3×#123—32no—
33C3×C22⋊Q8—32no—
34C3×C22⋊Q8—32no—
35C3×C22⋊Q8—31no—
36C3×#121—32no—
37C3×#121—32no—
38C3×#121—31no—
39C3×#121—32no—
40C3×#121—32no—
41C3×C4×Q8—31no—
42C3×C4×D4—32no—
43C3×C4×D4—31no—
44C3×C4×D4—32no—
45C3×C4×D4—32no—
46C3×#117—32no—
47C3×#117—32no—
48C2×C3×C4⋊C4—32no—
49C2×C3×C42A411yescharacteristic
50#127⋊C3—46no—
51#126⋊C3—46no—
52C2×C42⋊C3—43no—
53C2×C3×A4—161no—
54#564C2×C3211yescharacteristic, complemented
55#562—31no—
56#553—32no—
57#552—32no—
58#540—32no—
59#539—32no—
60#529—31no—
61C2×C3×Q8—31no—
62C2×C3×D4—32no—
63C2×C3×D4—32no—
64C2×C3×D4—31no—
65C2×C3×D4—32no—
66C22×C3×C4—32no—
67C22×C3×C4—32no—
68C22×C3×C4—32no—
69C22×C3×C4—31no—
70C3×C4⋊C4—32no—
71C3×C4⋊C4—32no—
72C3×C4⋊C4—32no—
73C3×C4⋊C4—32no—
74C3×C4⋊C4—32no—
75C3×C4⋊C4—32no—
76C3×C4⋊C4—32no—
77C3×C4⋊C4—32no—
78C3×C4⋊C4—31no—
79C3×C4⋊C4—31no—
80C3×C22⋊C4—32no—
81C3×C22⋊C4—32no—
82C3×C22⋊C4—32no—
83C3×C22⋊C4—32no—
84C3×C22⋊C4—32no—
85C3×C22⋊C4—32no—
86C3×C22⋊C4—32no—
87C3×C42—31no—
88C3×C42—32no—
89C3×C42C2×A411yescharacteristic
90C3×C42C2×A412yes—
91C3×C42C2×A412yes—
92C42⋊C3—43no—
93C42⋊C3—46no—
94C42⋊C3—46no—
95C3×A4—161no—
96#127C3×A412yes—
97#126—32no—
98#126—32no—
99#126C3×A412yes—
100#125—32no—
101#125—31no—
102#125—32no—
103#125—32no—
104#124—31no—
105#123—32no—
106#123—32no—
107#123—32no—
108#123—32no—
109#123—32no—
110C22⋊Q8—31no—
111C22⋊Q8—32no—
112C22⋊Q8—32no—
113#121—31no—
114#121—32no—
115#121—32no—
116#121—32no—
117#121—32no—
118C4×Q8—31no—
119C4×D4—32no—
120C4×D4—31no—
121C4×D4—32no—
122C4×D4—32no—
123#117—32no—
124#117—32no—
125C2×C4⋊C4—32no—
126C2×C42C3×A411yescharacteristic
127C23×C3—32no—
128C23×C3C22⋊A411yescharacteristic
129C2×A4—163no—
130C3×Q8—121no—
131C3×D4—122no—
132C3×D4—122no—
133C3×D4—122no—
134C3×D4—121no—
135C2×C3×C4—31no—
136C2×C3×C4—31no—
137C2×C3×C4—32no—
138C2×C3×C4—32no—
139C2×C3×C4—32no—
140C2×C3×C4—122no—
141C2×C3×C4—121no—
142C2×C3×C4—122no—
143C2×C3×C4—122no—
144C2×C32—641no—
145C2×Q8—31no—
146C2×D4—32no—
147C2×D4—32no—
148C2×D4—31no—
149C2×D4—32no—
150C22×C4—32no—
151C22×C4—32no—
152C22×C4—32no—
153C22×C4—31no—
154C4⋊C4—32no—
155C4⋊C4—32no—
156C4⋊C4—32no—
157C4⋊C4—32no—
158C4⋊C4—32no—
159C4⋊C4—32no—
160C4⋊C4—32no—
161C4⋊C4—32no—
162C4⋊C4—31no—
163C4⋊C4—31no—
164C22⋊C4—32no—
165C22⋊C4—32no—
166C22⋊C4—32no—
167C22⋊C4—32no—
168C22⋊C4—32no—
169C22⋊C4—32no—
170C22⋊C4—32no—
171C42—31no—
172C42—32no—
173C42C2×C3×A411yescharacteristic
174C42C2×C3×A412yes—
175C42C2×C3×A412yes—
176C22×C3—61no—
177C22×C3—62no—
178C22×C3—62no—
179C22×C3—62no—
180C22×C3C2×C22⋊A411yescharacteristic
181A4—163no—
182C3×C4—61no—
183C3×C4—62no—
184C3×C4—62no—
185C3×C4—62no—
186C3×C4—61no—
187C32—641noSylow
188C23—32no—
189C23C3×C22⋊A411yescharacteristic
190Q8—121no—
191D4—122no—
192D4—122no—
193D4—122no—
194D4—121no—
195C2×C4—31no—
196C2×C4—31no—
197C2×C4—32no—
198C2×C4—32no—
199C2×C4—32no—
200C2×C4—122no—
201C2×C4—121no—
202C2×C4—122no—
203C2×C4—122no—
204C2×C3—31no—
205C2×C3—41no—
206C2×C3—122no—
207C2×C3—643no—
208C22—61no—
209C22—62no—
210C22—62no—
211C22—62no—
212C22C2×C3×C22⋊A411yescharacteristic
213C4—61no—
214C4—62no—
215C4—62no—
216C4—62no—
217C4—61no—
218C3—643no—
219C3#3567⋊C311yescharacteristic, direct product factor
220C2—31no—
221C2—41no—
222C2—122no—
subgroupG/H
center, hypercenterC3#3567⋊C3
commutator, cosocle, hypocenter#564C2×C32
FrattiniC22C2×C3×C22⋊A4
FittingC3×#3567C3
socleC22×C3C2×C22⋊A4
LCSG ▷ #564
UCSC1 ◁ C3

Complements

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

NH ≅ G/Ntype
1. C3×#564⋊C3C2semidirect
2. C3×#3567C3semidirect
3. #3567⋊C3C3direct (N × H)
7. C3×#564C2×C3semidirect
14. #564⋊C3C2×C3semidirect
18. #3567C32semidirect
54. #564C2×C32semidirect
219. C3#3567⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#564⋊C32
C3×#35673
#3567⋊C33
C3×#127⋊C34
C3×#126⋊C34
C2×C3×C42⋊C34