Finite Group Databasebeta

← #68351

Subgroups — #68351

Order 384.

230 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 45 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#5684C211yesmaximal, characteristic, complemented
2#5058C211yesmaximal, characteristic, complemented
3#4713C211yesmaximal, characteristic, complemented
4#4712C211yesmaximal, characteristic, complemented
5C3×#329C211yesmaximal, characteristic, complemented
6#4465C211yesmaximal, characteristic, complemented
7#4464C211yesmaximal, characteristic, complemented
8#1656—31nomaximal, Sylow
9C3×#124C2211yescharacteristic
10#946—21no—
11C2×#215—21no—
12#939—21no—
13C2×#213—21no—
14C2×D24—21no—
15C2×#204—21no—
16#886—21no—
17#884—21no—
18#883—21no—
19#878—21no—
20#875C2211yescharacteristic
21#874C2211yescharacteristic
22#870C2211yescharacteristic
23C3×#97C2211yescharacteristic
24C3×#97C2211yescharacteristic
25#821—21no—
26#821—21no—
27#803C2211yescharacteristic
28#562—31no—
29#496—31no—
30#496—31no—
31#490—31no—
32#330—31no—
33#329—31no—
34#329D311yescharacteristic, complemented
35C2×C3×Q8D411yescharacteristic
36C2×C3×D4D411yescharacteristic
37C2×#67—21no—
38C2×#67—21no—
39C22×Dic3—21no—
40C2×D12—21no—
41C2×D12—21no—
42C2×D12—21no—
43C2×D12D411yescharacteristic
44C2×C4×D3—21no—
45C2×Dic6D411yescharacteristic
46C2×C3×C8—21no—
47C2×C3×C8—21no—
48C3×C22⋊C4—21no—
49C3×C22⋊C4—21no—
50C3×C42C2311yescharacteristic
51#217—21no—
52#215—42no—
53#213—42no—
54(C2×D3)⋊C4—21no—
55(C2×D3)⋊C4—21no—
56(C2×D3)⋊C4—21no—
57C4⋊Dic3—21no—
58C4⋊Dic3D411yescharacteristic
59C4⋊Dic3D411yescharacteristic
60Dic3⋊C4—21no—
61Dic3⋊C4—21no—
62Dic3⋊C4—21no—
63C2×C3⋊C8—21no—
64D24—42no—
65#204—42no—
66C2×SD16—61no—
67C2×SD16—61no—
68C2×D8—61no—
69C2×D8—61no—
70#127—31no—
71#125—31no—
72#124—31no—
73#124C2×D311yescharacteristic
74#123—61no—
75#123—61no—
76C22⋊Q8—61no—
77C22⋊Q8—61no—
78#121—61no—
79#121—61no—
80#102—61no—
81#102—61no—
82#97—31no—
83#97C2×D311yescharacteristic
84#97C2×D311yescharacteristic
85C23×C3—21no—
86C22×D3—21no—
87C22×D3—41no—
88C3×Q8—41no—
89C3×D4—41no—
90C2×C3×C4—21no—
91C2×C3×C4C2×D411yescharacteristic
92C2×C3×C4C2×D411yescharacteristic
93C2×C3×C4C2×D411yescharacteristic
94#67—81no—
95#67—81no—
96C2×Dic3—21no—
97C2×Dic3—21no—
98C2×Dic3—21no—
99C2×Dic3—81no—
100D12—41no—
101D12—41no—
102D12—41no—
103D12—41no—
104D12—41no—
105D12—42no—
106C4×D3—81no—
107Dic6—41no—
108C3×C8—22no—
109C3×C8—41no—
110C3⋊C8—22no—
111C2×Q8—31no—
112C2×Q8D1211yescharacteristic
113C2×D4—31no—
114C2×D4—61no—
115C2×D4—61no—
116C2×D4—61no—
117C2×D4—61no—
118C2×D4—61no—
119C2×D4D1211yescharacteristic
120C22×C4—61no—
121C22×C4—61no—
122SD16—122no—
123SD16—122no—
124D8—122no—
125D8—122no—
126C2×C8—21no—
127C2×C8—21no—
128C2×C8—61no—
129C4⋊C4—31no—
130C4⋊C4—31no—
131C4⋊C4—61no—
132C4⋊C4—61no—
133C4⋊C4—61no—
134C4⋊C4—61no—
135C22⋊C4—21no—
136C22⋊C4—21no—
137C22⋊C4—61no—
138C22⋊C4—61no—
139C22⋊C4—61no—
140C22⋊C4—61no—
141C42C22×D311yescharacteristic
142C22×C3—41no—
143C22×C3—41no—
144C22×C3—41no—
145C22×C3#12011yescharacteristic
146C2×D3—41no—
147C2×D3—41no—
148C2×D3—41no—
149C2×D3—81no—
150C2×D3—81no—
151C2×D3—81no—
152C3×C4—21no—
153C3×C4—21no—
154C3×C4—21no—
155C3×C4—41no—
156Dic3—41no—
157Dic3—41no—
158Dic3—41no—
159C23—21no—
160C23—61no—
161C23—121no—
162Q8—41no—
163Q8—121no—
164D4—41no—
165D4—121no—
166D4—121no—
167D4—121no—
168D4—121no—
169D4—121no—
170D4—122no—
171D4—241no—
172D4—241no—
173C2×C4—21no—
174C2×C4—61no—
175C2×C4—61no—
176C2×C4—61no—
177C2×C4—241no—
178C2×C4—241no—
179C2×C4C2×D1211yescharacteristic
180C2×C4D3×D411yescharacteristic
181C2×C4D3×D411yescharacteristic
182C8—22no—
183C8—41no—
184C8—62no—
185C2×C3—81no—
186C2×C3#45311yescharacteristic
187C2×C3#45311yescharacteristic
188C2×C3#45411yescharacteristic
189D3—81no—
190D3—161no—
191C22—41no—
192C22—41no—
193C22—41no—
194C22—121no—
195C22—121no—
196C22—121no—
197C22—241no—
198C22—241no—
199C22—241no—
200C22#88211yescharacteristic
201C4—21no—
202C4—21no—
203C4—21no—
204C4—41no—
205C4—121no—
206C4—121no—
207C4—121no—
208C3#165611yescharacteristic, complemented, Sylow
209C2—81no—
210C2—241no—
211C2—481no—
212C2#474711yescharacteristic
213C2#482211yescharacteristic
214C2#482711yescharacteristic
subgroupG/H
centerC22#882
commutator, cosocleC3×C42C23
FrattiniC42C22×D3
FittingC3×#329C2
socleC22×C3#120
hypercenter#329D3
hypocenterC3#1656
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
1. #5684C2semidirect
2. #5058C2semidirect
3. #4713C2semidirect
4. #4712C2semidirect
5. C3×#329C2semidirect
6. #4465C2semidirect
7. #4464C2semidirect
34. #329D3semidirect
208. C3#1656semidirect

Maximal subgroups

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

subgroupindex
#56842
#50582
#47132
#47122
C3×#3292
#44652
#44642
#16563