Finite Group Databasebeta

← #68352

Subgroups — #68352

Order 384.

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

normal subgroups only

This group has 228 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#5683C211yesmaximal, characteristic
2#5055C211yesmaximal, characteristic, complemented
3#4715C211yesmaximal, characteristic, complemented
4#4714C211yesmaximal, characteristic, complemented
5C3×#329C211yesmaximal, characteristic, complemented
6#4464C211yesmaximal, characteristic, complemented
7#4464C211yesmaximal, characteristic, complemented
8#1659—31nomaximal, Sylow
9C3×#124C2211yescharacteristic
10#947—21no—
11#937—21no—
12C2×Dic12—21no—
13C2×#204—21no—
14#885—21no—
15#885—21no—
16#882—21no—
17#882—21no—
18#875C2211yescharacteristic
19#875C2211yescharacteristic
20#869C2211yescharacteristic, complemented
21C3×#97C2211yescharacteristic, complemented
22C3×#97C2211yescharacteristic, complemented
23#835—21no—
24#832—21no—
25#821—21no—
26#816—21no—
27#803C2211yescharacteristic, complemented
28H(3,𝔽4)—31no—
29#497—31no—
30#489—31no—
31#489—31no—
32#329—31no—
33#329—31no—
34#329D311yescharacteristic, complemented
35C23×D3—21no—
36C2×C3×Q8D411yescharacteristic
37C2×C3×D4D411yescharacteristic
38C2×#67—21no—
39C2×#67—21no—
40C2×D12—21no—
41C2×D12D411yescharacteristic
42C2×D12D411yescharacteristic
43C2×Dic6—21no—
44C2×Dic6D411yescharacteristic
45C2×Dic6D411yescharacteristic
46C2×C3×C8—21no—
47C2×C3×C8—21no—
48C3×C22⋊C4—21no—
49C3×C22⋊C4—21no—
50C3×C42C2311yescharacteristic
51#217—21no—
52(C2×D3)⋊C4—21no—
53(C2×D3)⋊C4—21no—
54(C2×D3)⋊C4—21no—
55(C2×D3)⋊C4—21no—
56(C2×D3)⋊C4—21no—
57(C2×D3)⋊C4—21no—
58C4⋊Dic3—21no—
59C4⋊Dic3—21no—
60C4×Dic3—21no—
61C2×C3⋊C8—21no—
62Dic12—42no—
63#204—42no—
64C2×Q16—61no—
65C2×SD16—61no—
66C4⋊Q8—31no—
67#124—31no—
68#124—31no—
69#124—61no—
70#124—61no—
71#124—61no—
72#124C2×D311yescharacteristic
73#120—61no—
74#120—61no—
75#120—61no—
76#103—61no—
77#103—61no—
78#102—61no—
79#102—61no—
80#97—31no—
81#97C2×D311yescharacteristic, complemented
82#97C2×D311yescharacteristic, complemented
83C23×C3—21no—
84C22×D3—21no—
85C22×D3—21no—
86C22×D3—81no—
87C22×D3—81no—
88C22×D3—81no—
89C3×Q8—41no—
90C3×D4—41no—
91C2×C3×C4—21no—
92C2×C3×C4C2×D411yescharacteristic
93C2×C3×C4C2×D411yescharacteristic
94C2×C3×C4C2×D411yescharacteristic
95#67—81no—
96#67—81no—
97C2×Dic3—21no—
98C2×Dic3—21no—
99C2×Dic3—41no—
100D12—41no—
101D12—41no—
102D12—81no—
103Dic6—41no—
104Dic6—41no—
105Dic6—41no—
106Dic6—41no—
107C3×C8—22no—
108C3×C8—41no—
109C3⋊C8—41no—
110C24—61no—
111C2×Q8—31no—
112C2×Q8—31no—
113C2×Q8—61no—
114C2×Q8D1211yescharacteristic
115C2×D4—31no—
116C2×D4—31no—
117C2×D4—61no—
118C2×D4—61no—
119C2×D4—61no—
120C2×D4D1211yescharacteristic
121Q16—122no—
122SD16—122no—
123C2×C8—21no—
124C2×C8—21no—
125C2×C8—61no—
126C4⋊C4—61no—
127C4⋊C4—61no—
128C22⋊C4—21no—
129C22⋊C4—21no—
130C22⋊C4—61no—
131C22⋊C4—61no—
132C22⋊C4—61no—
133C22⋊C4—61no—
134C22⋊C4—61no—
135C22⋊C4—61no—
136C22⋊C4—61no—
137C42—61no—
138C42C22×D311yescharacteristic
139C22×C3—41no—
140C22×C3—41no—
141C22×C3—41no—
142C22×C3#12011yescharacteristic
143C2×D3—41no—
144C2×D3—41no—
145C2×D3—41no—
146C2×D3—41no—
147C2×D3—41no—
148C2×D3—41no—
149C2×D3—84no—
150C3×C4—21no—
151C3×C4—21no—
152C3×C4—21no—
153C3×C4—41no—
154Dic3—41no—
155Dic3—41no—
156Dic3—81no—
157C23—21no—
158C23—61no—
159C23—61no—
160C23—241no—
161C23—241no—
162C23—241no—
163Q8—41no—
164Q8—121no—
165Q8—121no—
166Q8—121no—
167Q8—121no—
168D4—41no—
169D4—121no—
170D4—121no—
171D4—241no—
172D4—241no—
173D4—241no—
174C2×C4—21no—
175C2×C4—61no—
176C2×C4—61no—
177C2×C4—121no—
178C2×C4C2×D1211yescharacteristic
179C2×C4D3×D411yescharacteristic
180C2×C4D3×D411yescharacteristic
181C8—22no—
182C8—41no—
183C8—121no—
184C2×C3—81no—
185C2×C3#45311yescharacteristic
186C2×C3#45511yescharacteristic
187C2×C3#45511yescharacteristic
188D3—81no—
189D3—81no—
190C22—41no—
191C22—41no—
192C22—41no—
193C22—121no—
194C22—121no—
195C22—121no—
196C22—121no—
197C22—121no—
198C22—121no—
199C22—244no—
200C22#88211yescharacteristic
201C4—21no—
202C4—21no—
203C4—21no—
204C4—41no—
205C4—121no—
206C4—121no—
207C4—241no—
208C3#165911yescharacteristic, complemented, Sylow
209C2—81no—
210C2—241no—
211C2—241no—
212C2#474611yescharacteristic
213C2#482211yescharacteristic
214C2#482511yescharacteristic
subgroupG/H
centerC22#882
commutator, cosocleC3×C42C23
FrattiniC42C22×D3
FittingC3×#329C2
socleC22×C3#120
prime radical#5683C2
hypercenter#329D3
hypocenterC3#1659
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #5055C2semidirect
3. #4715C2semidirect
4. #4714C2semidirect
5. C3×#329C2semidirect
6. #4464C2semidirect
7. #4464C2semidirect
20. #869C22semidirect
21. C3×#97C22semidirect
22. C3×#97C22semidirect
27. #803C22semidirect
34. #329D3semidirect
81. #97C2×D3semidirect
82. #97C2×D3semidirect
208. C3#1659semidirect

Maximal subgroups

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

subgroupindex
#56832
#50552
#47152
#47142
C3×#3292
#44642
#16593