Finite Group Databasebeta

← #70980

Subgroups — #70980

Order 384.

256 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 51 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#5634C211yesmaximal, characteristic
2C3×#451C211yesmaximal, characteristic, complemented
3#5231C211yesmaximal, characteristic
4#5228C211yesmaximal, characteristic, complemented
5#4538C211yesmaximal, characteristic, complemented
6#4468C211yesmaximal, characteristic
7#4467C211yesmaximal, characteristic, complemented
8#1602—31nomaximal, Sylow
9C22×C3×Q8C2211yescharacteristic
10C2×C3×Q16—21no—
11C3×C22⋊Q8C2211yescharacteristic
12#946—21no—
13C2×#216—21no—
14C2×(C2×D3)⋊C4—21no—
15C2×(C2×D3)⋊C4C2211yescharacteristic
16#924C2211yescharacteristic
17#897—21no—
18#896—21no—
19#896—21no—
20D3×C22⋊C4—21no—
21#878—21no—
22C3×#103—21no—
23C3×C22⋊C8C2211yescharacteristic
24#835—21no—
25C3⋊#95—21no—
26C3⋊#95C2211yescharacteristic
27#830C2211yescharacteristic
28#543—31no—
29#451—31no—
30#451D311yescharacteristic, complemented
31#393—31no—
32#328—31no—
33#328—31no—
34#323—31no—
35C23×D3D411yescharacteristic
36C2×C3×Q8—21no—
37C2×C3×Q8—21no—
38C2×C3×Q8—42no—
39C2×C3×Q8D411yescharacteristic
40C22×C3×C4—21no—
41C22×C3×C4C2311yescharacteristic
42C22×Dic3—21no—
43C22×Dic3D411yescharacteristic
44C2×C4×D3—21no—
45C2×C4×D3—21no—
46C2×Dic6—21no—
47C2×Dic6D411yescharacteristic
48C3×Q16—42no—
49C2×C3×C8—21no—
50C3×C4⋊C4—21no—
51C3×C4⋊C4D411yescharacteristic
52C3×C22⋊C4—21no—
53#217—21no—
54#216—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—
63Dic3⋊C4—21no—
64Dic3⋊C4—21no—
65C2×C3⋊C8—21no—
66C22×Q8C2×D311yescharacteristic
67C2×Q16—21no—
68C2×Q16—61no—
69C22⋊Q8—31no—
70C22⋊Q8—61no—
71C22⋊Q8—61no—
72C22⋊Q8—61no—
73C22⋊Q8—61no—
74C22⋊Q8—61no—
75C22⋊Q8C2×D311yescharacteristic
76C2×C22⋊C4—31no—
77C2×C22⋊C4—61no—
78C2×C22⋊C4—61no—
79#103—21no—
80#103—61no—
81C22⋊C8—31no—
82C22⋊C8C2×D311yescharacteristic
83#95—31no—
84#95—61no—
85C23×C3C2×D411yescharacteristic
86C22×D3—21no—
87C22×D3—21no—
88C22×D3—41no—
89C22×D3—41no—
90C22×D3Q1612yescomplemented
91C3×Q8—22no—
92C3×Q8—22no—
93C3×Q8—41no—
94C3×Q8—41no—
95C3×Q8—41no—
96C2×C3×C4—22no—
97C2×C3×C4—21no—
98C2×C3×C4—21no—
99C2×C3×C4—41no—
100C2×C3×C4—41no—
101C2×C3×C4—41no—
102C2×C3×C4C2×D411yescharacteristic
103C2×C3×C4C2×D411yescharacteristic
104C2×Dic3—21no—
105C2×Dic3—21no—
106C2×Dic3—21no—
107C2×Dic3—41no—
108C2×Dic3—41no—
109C2×Dic3—41no—
110C2×Dic3—41no—
111C4×D3—81no—
112C4×D3—81no—
113Dic6—41no—
114Dic6—81no—
115C3×C8—41no—
116C3⋊C8—41no—
117C24—31no—
118C2×Q8—21no—
119C2×Q8—21no—
120C2×Q8—31no—
121C2×Q8—42no—
122C2×Q8—61no—
123C2×Q8#6711yescharacteristic
124C22×C4—21no—
125C22×C4—31no—
126C22×C4—61no—
127C22×C4—61no—
128C22×C4—61no—
129C22×C4C22×D311yescharacteristic
130Q16—42no—
131Q16—122no—
132C2×C8—21no—
133C2×C8—61no—
134C4⋊C4—21no—
135C4⋊C4—31no—
136C4⋊C4—61no—
137C4⋊C4—61no—
138C4⋊C4—61no—
139C4⋊C4—61no—
140C4⋊C4#6711yescharacteristic
141C22⋊C4—21no—
142C22⋊C4—61no—
143C22⋊C4—61no—
144C22⋊C4—61no—
145C22⋊C4—61no—
146C22⋊C4—61no—
147C22⋊C4—122no—
148C22×C3—21no—
149C22×C3—21no—
150C22×C3#12011yescharacteristic
151C22×C3C2×Q1611yescharacteristic
152C22×C3#13711yescharacteristic
153C2×D3—22no—
154C2×D3—41no—
155C2×D3—41no—
156C2×D3—42no—
157C2×D3—81no—
158C3×C4—21no—
159C3×C4—21no—
160C3×C4—41no—
161C3×C4—41no—
162C3×C4—42no—
163Dic3—41no—
164Dic3—41no—
165Dic3—41no—
166Dic3—81no—
167C23—32no—
168C23—61no—
169C23—61no—
170C23—121no—
171C23—121no—
172C23D3×D411yescharacteristic
173Q8—22no—
174Q8—22no—
175Q8—41no—
176Q8—41no—
177Q8—41no—
178Q8—121no—
179Q8—241no—
180C2×C4—22no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—61no—
187C2×C4—61no—
188C2×C4—61no—
189C2×C4—121no—
190C2×C4—121no—
191C2×C4—121no—
192C2×C4—121no—
193C2×C4—241no—
194C2×C4—241no—
195C2×C4D3×D411yescharacteristic
196C2×C4C2×#6711yescharacteristic
197C8—41no—
198C8—121no—
199C2×C3—21no—
200C2×C3—21no—
201C2×C3#45111yescharacteristic
202C2×C3#45511yescharacteristic
203C2×C3B(4,𝔽2)11yescharacteristic
204D3—42no—
205C22—21no—
206C22—21no—
207C22—62no—
208C22—122no—
209C22—121no—
210C22—121no—
211C22—241no—
212C22D3×Q1611yescharacteristic
213C22#91811yescharacteristic
214C22#93711yescharacteristic
215C4—21no—
216C4—21no—
217C4—41no—
218C4—42no—
219C4—41no—
220C4—121no—
221C4—121no—
222C4—121no—
223C4—241no—
224C3#160211yescharacteristic, complemented, Sylow
225C2—21no—
226C2—21no—
227C2—122no—
228C2#503211yescharacteristic
229C2#518611yescharacteristic
230C2#520211yescharacteristic
subgroupG/H
centerC22#937
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#451C2
socleC22×C3#120
prime radicalC23×D3D4
hypercenter#451D3
hypocenterC3#1602
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
2. C3×#451C2semidirect
4. #5228C2semidirect
5. #4538C2semidirect
7. #4467C2semidirect
30. #451D3semidirect
90. C22×D3Q16semidirect
224. C3#1602semidirect

Maximal subgroups

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

subgroupindex
#56342
C3×#4512
#52312
#52282
#45382
#44682
#44672
#16023