Finite Group Databasebeta

← #68287

Subgroups — #68287

Order 384.

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

normal subgroups only

This group has 260 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#5040C211yesmaximal, characteristic, complemented
3#4739C211yesmaximal, characteristic, complemented
4#4673C211yesmaximal, characteristic, complemented
5C3×#327C211yesmaximal, characteristic, complemented
6#4468C211yesmaximal, characteristic, complemented
7#4452C211yesmaximal, characteristic, complemented
8#1606—31nomaximal, Sylow
9C22×Dic6C2211yescharacteristic
10C3×#121C2211yescharacteristic
11#939—21no—
12#937—21no—
13D4×Dic3—21no—
14C2×#214—21no—
15C2×(C2×D3)⋊C4—21no—
16C2×(C2×D3)⋊C4C2211yescharacteristic
17#924C2211yescharacteristic
18C2×Dic12—21no—
19#898—21no—
20#886—21no—
21#885—21no—
22C3×C22⋊C8C2211yescharacteristic
23C3×#95C2211yescharacteristic
24C3⋊#95—21no—
25#830C2211yescharacteristic
26#816—21no—
27#810—21no—
28#538—31no—
29#451—31no—
30#448—31no—
31#393—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×Dic6—21no—
46C2×Dic6—21no—
47C2×Dic6—42no—
48C2×Dic6D411yescharacteristic
49C2×C3×C8—21no—
50C3×C4⋊C4D411yescharacteristic
51C3×C22⋊C4—21no—
52#217—21no—
53#217—21no—
54#214—42no—
55(C2×D3)⋊C4—21no—
56(C2×D3)⋊C4—21no—
57(C2×D3)⋊C4—21no—
58(C2×D3)⋊C4—21no—
59(C2×D3)⋊C4—42no—
60C4⋊Dic3—21no—
61C4⋊Dic3D411yescharacteristic
62Dic3⋊C4—21no—
63C4×Dic3—21no—
64C2×C3⋊C8—21no—
65Dic12—42no—
66C22×Q8—31no—
67C2×Q16—61no—
68C2×SD16—61no—
69#126—61no—
70#124—61no—
71#123—61no—
72C22⋊Q8—31no—
73#121—61no—
74#121C2×D311yescharacteristic
75#120—61no—
76C4×D4—61no—
77C2×C22⋊C4—31no—
78C2×C22⋊C4—61no—
79#103—61no—
80#103—61no—
81C22⋊C8—31no—
82C22⋊C8C2×D311yescharacteristic
83#95—61no—
84#95C2×D311yescharacteristic
85C23×C3—21no—
86C23×C3C2×D411yescharacteristic
87C22×D3—21no—
88C22×D3—21no—
89C22×D3—21no—
90C22×D3—41no—
91C22×D3—41no—
92C3×D4—41no—
93C3×D4—41no—
94C3×D4—41no—
95C2×C3×C4—21no—
96C2×C3×C4—41no—
97C2×C3×C4—41no—
98C2×C3×C4—41no—
99C2×C3×C4—41no—
100C2×C3×C4C2×D411yescharacteristic
101C2×C3×C4C2×D411yescharacteristic
102#67—81no—
103#67—81no—
104C2×Dic3—22no—
105C2×Dic3—21no—
106C2×Dic3—21no—
107C2×Dic3—21no—
108C2×Dic3—41no—
109C2×Dic3—41no—
110C2×Dic3—41no—
111C2×Dic3—81no—
112Dic6—22no—
113Dic6—22no—
114Dic6—41no—
115Dic6—41no—
116Dic6—41no—
117C3×C8—41no—
118C3⋊C8—41no—
119C24—31no—
120C2×Q8—31no—
121C2×Q8—61no—
122C2×Q8—61no—
123C2×Q8—122no—
124C2×D4—21no—
125C2×D4—61no—
126C2×D4—61no—
127C2×D4D1211yescharacteristic
128C22×C4—21no—
129C22×C4—31no—
130C22×C4—61no—
131C22×C4—61no—
132C22×C4C22×D311yescharacteristic
133Q16—122no—
134SD16—122no—
135C2×C8—21no—
136C2×C8—61no—
137C4⋊C4—31no—
138C4⋊C4—61no—
139C4⋊C4—61no—
140C4⋊C4D1211yescharacteristic
141C22⋊C4—21no—
142C22⋊C4—61no—
143C22⋊C4—61no—
144C22⋊C4—61no—
145C22⋊C4—61no—
146C22⋊C4—61no—
147C22⋊C4—61no—
148C22⋊C4—122no—
149C42—61no—
150C22×C3—21no—
151C22×C3—21no—
152C22×C3—41no—
153C22×C3—41no—
154C22×C3—41no—
155C22×C3#12011yescharacteristic
156C22×C3#13511yescharacteristic
157C22×C3#13711yescharacteristic
158C2×D3—42no—
159C2×D3—41no—
160C2×D3—41no—
161C2×D3—41no—
162C2×D3—81no—
163C3×C4—21no—
164C3×C4—21no—
165C3×C4—41no—
166C3×C4—81no—
167Dic3—41no—
168Dic3—41no—
169Dic3—42no—
170Dic3—41no—
171C23—21no—
172C23—61no—
173C23—61no—
174C23—61no—
175C23—121no—
176C23—121no—
177C23D3×D411yescharacteristic
178Q8—62no—
179Q8—62no—
180Q8—121no—
181Q8—121no—
182Q8—121no—
183D4—41no—
184D4—41no—
185D4—41no—
186D4—241no—
187D4—241no—
188C2×C4—21no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—62no—
194C2×C4—61no—
195C2×C4—61no—
196C2×C4—61no—
197C2×C4—121no—
198C2×C4—121no—
199C2×C4—121no—
200C2×C4—241no—
201C2×C4C2×D1211yescharacteristic
202C2×C4D3×D411yescharacteristic
203C8—41no—
204C8—121no—
205C2×C3—21no—
206C2×C3—21no—
207C2×C3—81no—
208C2×C3#45211yescharacteristic
209C2×C3#45511yescharacteristic
210C2×C3B(4,𝔽2)11yescharacteristic
211D3—81no—
212C22—21no—
213C22—21no—
214C22—41no—
215C22—41no—
216C22—41no—
217C22—122no—
218C22—121no—
219C22—121no—
220C22—121no—
221C22—241no—
222C22#88211yescharacteristic
223C22#91211yescharacteristic
224C22#91511yescharacteristic
225C4—21no—
226C4—21no—
227C4—41no—
228C4—81no—
229C4—122no—
230C4—121no—
231C4—121no—
232C4—121no—
233C3#160611yescharacteristic, complemented, Sylow
234C2—21no—
235C2—21no—
236C2—81no—
237C2—241no—
238C2#474111yescharacteristic
239C2#478311yescharacteristic
240C2#482511yescharacteristic
subgroupG/H
centerC22#882
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#327C2
socleC22×C3#120
prime radical#5617C2
hypercenter#327D3
hypocenterC3#1606
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. #5040C2semidirect
3. #4739C2semidirect
4. #4673C2semidirect
5. C3×#327C2semidirect
6. #4468C2semidirect
7. #4452C2semidirect
33. #327D3semidirect
233. C3#1606semidirect

Maximal subgroups

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

subgroupindex
#56172
#50402
#47392
#46732
C3×#3272
#44682
#44522
#16063