Finite Group Databasebeta

← #68291

Subgroups — #68291

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#5617C211yesmaximal, characteristic
2#5034C211yesmaximal, characteristic, complemented
3#4722C211yesmaximal, characteristic, complemented
4#4677C211yesmaximal, characteristic, complemented
5C3×#327C211yesmaximal, characteristic, complemented
6#4468C211yesmaximal, characteristic, complemented
7#4452C211yesmaximal, characteristic, complemented
8#1618—31nomaximal, Sylow
9C3×#121C2211yescharacteristic
10#939—21no—
11#937—21no—
12D4×Dic3—21no—
13C2×(C2×D3)⋊C4—21no—
14C2×(C2×D3)⋊C4C2211yescharacteristic
15C2×C4⋊Dic3—21no—
16C2×C4⋊Dic3C2211yescharacteristic
17#924C2211yescharacteristic
18#898—21no—
19#886—21no—
20#885—21no—
21C3×C22⋊C8C2211yescharacteristic
22C3×#95C2211yescharacteristic
23#832—21no—
24#830C2211yescharacteristic
25#818—21no—
26#816—21no—
27#808—21no—
28#538—31no—
29#484—31no—
30#481—31no—
31#399—31no—
32#328—31no—
33#327D311yescharacteristic, complemented
34#323—31no—
35C23×D3D411yescharacteristic
36C2×C3×D4—21no—
37C2×C3×D4D411yescharacteristic
38C22×C3×C4—21no—
39C22×C3×C4C2311yescharacteristic
40C2×#67—21no—
41C2×#67—21no—
42C22×Dic3—21no—
43C22×Dic3—21no—
44C22×Dic3D411yescharacteristic
45C2×Dic6D411yescharacteristic
46C2×C3×C8—21no—
47C3×C4⋊C4D411yescharacteristic
48C3×C22⋊C4—21no—
49#217—21no—
50#217—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—42no—
56C4⋊Dic3—21no—
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⋊Q8—31no—
68#121—61no—
69#121C2×D311yescharacteristic
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×C3—21no—
84C23×C3C2×D411yescharacteristic
85C22×D3—21no—
86C22×D3—21no—
87C22×D3—21no—
88C22×D3—41no—
89C22×D3—41no—
90C3×D4—41no—
91C3×D4—41no—
92C3×D4—41no—
93C2×C3×C4—22no—
94C2×C3×C4—21no—
95C2×C3×C4—41no—
96C2×C3×C4—41no—
97C2×C3×C4—41no—
98C2×C3×C4C2×D411yescharacteristic
99C2×C3×C4C2×D411yescharacteristic
100#67—81no—
101#67—81no—
102C2×Dic3—21no—
103C2×Dic3—21no—
104C2×Dic3—21no—
105C2×Dic3—41no—
106C2×Dic3—41no—
107C2×Dic3—41no—
108C2×Dic3—41no—
109C2×Dic3—81no—
110Dic6—41no—
111C3×C8—41no—
112C3⋊C8—41no—
113C24—31no—
114C2×Q8—31no—
115C2×D4—21no—
116C2×D4—61no—
117C2×D4—61no—
118C2×D4D1211yescharacteristic
119C22×C4—21no—
120C22×C4—31no—
121C22×C4—61no—
122C22×C4—61no—
123C22×C4C22×D311yescharacteristic
124C2×C8—21no—
125C2×C8—61no—
126C4⋊C4—31no—
127C4⋊C4—61no—
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—41no—
145C22×C3—41no—
146C22×C3—41no—
147C22×C3#12011yescharacteristic
148C22×C3#13511yescharacteristic
149C22×C3#13711yescharacteristic
150C2×D3—42no—
151C2×D3—41no—
152C2×D3—41no—
153C2×D3—41no—
154C2×D3—81no—
155C3×C4—21no—
156C3×C4—21no—
157C3×C4—42no—
158C3×C4—41no—
159Dic3—41no—
160Dic3—41no—
161Dic3—41no—
162Dic3—81no—
163C23—21no—
164C23—61no—
165C23—61no—
166C23—61no—
167C23—121no—
168C23—121no—
169C23D3×D411yescharacteristic
170Q8—121no—
171D4—41no—
172D4—41no—
173D4—41no—
174D4—241no—
175D4—241no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—61no—
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—81no—
196C2×C3#45211yescharacteristic
197C2×C3#45311yescharacteristic
198C2×C3#45811yescharacteristic
199D3—81no—
200C22—21no—
201C22—21no—
202C22—41no—
203C22—41no—
204C22—41no—
205C22—122no—
206C22—121no—
207C22—121no—
208C22—121no—
209C22—241no—
210C22#88211yescharacteristic
211C22#91211yescharacteristic
212C22#91511yescharacteristic
213C4—21no—
214C4—21no—
215C4—42no—
216C4—41no—
217C4—121no—
218C4—121no—
219C4—121no—
220C4—241no—
221C3#161811yescharacteristic, complemented, Sylow
222C2—21no—
223C2—21no—
224C2—81no—
225C2—241no—
226C2#474211yescharacteristic
227C2#478311yescharacteristic
228C2#482211yescharacteristic
subgroupG/H
centerC22#882
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#327C2
socleC22×C3#120
prime radical#5617C2
hypercenter#327D3
hypocenterC3#1618
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #5034C2semidirect
3. #4722C2semidirect
4. #4677C2semidirect
5. C3×#327C2semidirect
6. #4468C2semidirect
7. #4452C2semidirect
33. #327D3semidirect
221. C3#1618semidirect

Maximal subgroups

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

subgroupindex
#56172
#50342
#47222
#46772
C3×#3272
#44682
#44522
#16183