Finite Group Databasebeta

← #68323

Subgroups — #68323

Order 384.

240 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 49 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#5644C211yesmaximal, characteristic
2#5044C211yesmaximal, characteristic, complemented
3#4735C211yesmaximal, characteristic, complemented
4#4677C211yesmaximal, characteristic, complemented
5C3×#328C211yesmaximal, characteristic, complemented
6#4468C211yesmaximal, characteristic, complemented
7#4451C211yesmaximal, characteristic, complemented
8#1618—31nomaximal, Sylow
9C3×C22⋊Q8C2211yescharacteristic
10#947—21no—
11#930C2211yescharacteristic
12C2×(C2×D3)⋊C4—21no—
13C2×(C2×D3)⋊C4C2211yescharacteristic
14C2×C4⋊Dic3—21no—
15C2×C4⋊Dic3C2211yescharacteristic
16#898—21no—
17#895—21no—
18#894—21no—
19#882—21no—
20#67⋊C4—21no—
21C3×C22⋊C8C2211yescharacteristic
22C3×#95C2211yescharacteristic
23#835—21no—
24#830C2211yescharacteristic
25#821—21no—
26#817—21no—
27#807—21no—
28#538—31no—
29#484—31no—
30#481—31no—
31#399—31no—
32#328D311yescharacteristic, complemented
33#327—31no—
34#323—31no—
35C23×D3D411yescharacteristic
36C2×C3×Q8D411yescharacteristic
37C22×C3×C4—21no—
38C22×C3×C4C2311yescharacteristic
39C2×#67—21no—
40C22×Dic3—21no—
41C22×Dic3D411yescharacteristic
42C2×D12—21no—
43C2×D12—21no—
44C2×D12D411yescharacteristic
45C2×C4×D3—21no—
46C2×C3×C8—21no—
47C3×C4⋊C4—21no—
48C3×C4⋊C4D411yescharacteristic
49C3×C22⋊C4—21no—
50(C2×D3)⋊C4—21no—
51(C2×D3)⋊C4—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—42no—
57C4⋊Dic3—21no—
58C4⋊Dic3—21no—
59C4⋊Dic3—42no—
60C4⋊Dic3D411yescharacteristic
61Dic3⋊C4—21no—
62C4×Dic3—21no—
63C2×C3⋊C8—21no—
64#126—61no—
65#124—61no—
66#123—61no—
67C22⋊Q8C2×D311yescharacteristic
68#121—31no—
69#121—61no—
70#120—61no—
71C4×D4—61no—
72C2×C4⋊C4—31no—
73C2×C4⋊C4—61no—
74C2×C22⋊C4—31no—
75C2×C22⋊C4—61no—
76#107—61no—
77#106—61no—
78#103—61no—
79#102—61no—
80C22⋊C8—31no—
81C22⋊C8C2×D311yescharacteristic
82#95C2×D311yescharacteristic
83C23×C3C2×D411yescharacteristic
84C22×D3—21no—
85C22×D3—21no—
86C22×D3—21no—
87C22×D3—21no—
88C22×D3—41no—
89C22×D3—41no—
90C3×Q8—41no—
91C2×C3×C4—22no—
92C2×C3×C4—21no—
93C2×C3×C4—21no—
94C2×C3×C4—41no—
95C2×C3×C4—41no—
96C2×C3×C4—41no—
97C2×C3×C4C2×D411yescharacteristic
98C2×C3×C4C2×D411yescharacteristic
99#67—41no—
100#67—41no—
101C2×Dic3—21no—
102C2×Dic3—21no—
103C2×Dic3—41no—
104C2×Dic3—41no—
105C2×Dic3—41no—
106C2×Dic3—41no—
107D12—41no—
108D12—81no—
109D12—81no—
110C4×D3—81no—
111C3×C8—41no—
112C3⋊C8—41no—
113C24—31no—
114C2×Q8D1211yescharacteristic
115C2×D4—31no—
116C2×D4—61no—
117C2×D4—61no—
118C2×D4—61no—
119C22×C4—21no—
120C22×C4—31no—
121C22×C4—61no—
122C22×C4—61no—
123C22×C4C22×D311yescharacteristic
124C2×C8—21no—
125C2×C8—61no—
126C4⋊C4—21no—
127C4⋊C4—31no—
128C4⋊C4—61no—
129C4⋊C4—61no—
130C4⋊C4—61no—
131C4⋊C4—122no—
132C4⋊C4D1211yescharacteristic
133C22⋊C4—21no—
134C22⋊C4—61no—
135C22⋊C4—61no—
136C22⋊C4—61no—
137C22⋊C4—61no—
138C22⋊C4—61no—
139C22⋊C4—61no—
140C22⋊C4—122no—
141C42—61no—
142C22×C3—21no—
143C22×C3—21no—
144C22×C3#12011yescharacteristic
145C22×C3#13511yescharacteristic
146C22×C3#13711yescharacteristic
147C2×D3—42no—
148C2×D3—41no—
149C2×D3—41no—
150C2×D3—41no—
151C2×D3—41no—
152C2×D3—41no—
153C2×D3—41no—
154C2×D3—81no—
155C3×C4—21no—
156C3×C4—21no—
157C3×C4—42no—
158C3×C4—41no—
159C3×C4—41no—
160Dic3—41no—
161Dic3—41no—
162Dic3—81no—
163C23—61no—
164C23—61no—
165C23—61no—
166C23—61no—
167C23—121no—
168C23—121no—
169C23D3×D411yescharacteristic
170Q8—41no—
171D4—121no—
172D4—121no—
173D4—121no—
174D4—241no—
175D4—241no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—61no—
183C2×C4—61no—
184C2×C4—121no—
185C2×C4—121no—
186C2×C4—121no—
187C2×C4—121no—
188C2×C4—241no—
189C2×C4C2×D1211yescharacteristic
190C2×C4D3×D411yescharacteristic
191C8—41no—
192C8—121no—
193C2×C3—21no—
194C2×C3—21no—
195C2×C3#45211yescharacteristic
196C2×C3#45311yescharacteristic
197C2×C3#45811yescharacteristic
198D3—81no—
199D3—81no—
200C22—21no—
201C22—21no—
202C22—122no—
203C22—121no—
204C22—121no—
205C22—121no—
206C22—121no—
207C22—121no—
208C22—121no—
209C22—241no—
210C22#88211yescharacteristic
211C22#91611yescharacteristic
212C22#91811yescharacteristic
213C4—21no—
214C4—21no—
215C4—42no—
216C4—41no—
217C4—41no—
218C4—121no—
219C4—121no—
220C4—241no—
221C3#161811yescharacteristic, complemented, Sylow
222C2—21no—
223C2—21no—
224C2—241no—
225C2—241no—
226C2#474211yescharacteristic
227C2#480811yescharacteristic
228C2#482211yescharacteristic
subgroupG/H
centerC22#882
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#328C2
socleC22×C3#120
prime radical#5644C2
hypercenter#328D3
hypocenterC3#1618
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. #5044C2semidirect
3. #4735C2semidirect
4. #4677C2semidirect
5. C3×#328C2semidirect
6. #4468C2semidirect
7. #4451C2semidirect
32. #328D3semidirect
221. C3#1618semidirect

Maximal subgroups

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

subgroupindex
#56442
#50442
#47352
#46772
C3×#3282
#44682
#44512
#16183