Finite Group Databasebeta

← #68318

Subgroups — #68318

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
2#5050C211yesmaximal, characteristic, complemented
3#4739C211yesmaximal, characteristic, complemented
4#4673C211yesmaximal, characteristic
5C3×#328C211yesmaximal, characteristic, complemented
6#4468C211yesmaximal, characteristic
7#4452C211yesmaximal, characteristic, complemented
8#1602—31nomaximal, Sylow
9C22×Dic6C2211yescharacteristic
10C3×C22⋊Q8C2211yescharacteristic
11#946—21no—
12C2×#216—21no—
13C2×(C2×D3)⋊C4—21no—
14C2×(C2×D3)⋊C4C2211yescharacteristic
15#924C2211yescharacteristic
16C2×Dic12—21no—
17#897—21no—
18#896—21no—
19#896—21no—
20D3×C22⋊C4—21no—
21#878—21no—
22C3×C22⋊C8C2211yescharacteristic
23C3×#95C2211yescharacteristic
24C3⋊#95—21no—
25#830C2211yescharacteristic
26#816—21no—
27#810—21no—
28#543—31no—
29#451—31no—
30#451—31no—
31#393—31no—
32#328—31no—
33#328D311yescharacteristic, complemented
34#323—31no—
35C23×D3D411yescharacteristic
36C2×C3×Q8D411yescharacteristic
37C22×C3×C4—21no—
38C22×C3×C4C2311yescharacteristic
39C22×Dic3—21no—
40C22×Dic3D411yescharacteristic
41C2×C4×D3—21no—
42C2×C4×D3—21no—
43C2×Dic6—21no—
44C2×Dic6—21no—
45C2×Dic6—21no—
46C2×Dic6—42no—
47C2×Dic6D411yescharacteristic
48C2×C3×C8—21no—
49C3×C4⋊C4—21no—
50C3×C4⋊C4D411yescharacteristic
51C3×C22⋊C4—21no—
52#217—21no—
53#216—42no—
54(C2×D3)⋊C4—21no—
55(C2×D3)⋊C4—21no—
56(C2×D3)⋊C4—21no—
57(C2×D3)⋊C4—21no—
58(C2×D3)⋊C4—42no—
59C4⋊Dic3—21no—
60C4⋊Dic3D411yescharacteristic
61Dic3⋊C4—21no—
62Dic3⋊C4—21no—
63Dic3⋊C4—21no—
64C2×C3⋊C8—21no—
65Dic12—42no—
66C22×Q8—31no—
67C2×Q16—61no—
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—61no—
80#103—61no—
81C22⋊C8—31no—
82C22⋊C8C2×D311yescharacteristic
83#95—61no—
84#95C2×D311yescharacteristic
85C23×C3C2×D411yescharacteristic
86C22×D3—21no—
87C22×D3—21no—
88C22×D3—41no—
89C22×D3—41no—
90C22×D3Q1612yescomplemented
91C3×Q8—41no—
92C2×C3×C4—21no—
93C2×C3×C4—21no—
94C2×C3×C4—41no—
95C2×C3×C4—41no—
96C2×C3×C4—41no—
97C2×C3×C4—41no—
98C2×C3×C4C2×D411yescharacteristic
99C2×C3×C4C2×D411yescharacteristic
100C2×Dic3—22no—
101C2×Dic3—21no—
102C2×Dic3—21no—
103C2×Dic3—21no—
104C2×Dic3—41no—
105C2×Dic3—41no—
106C2×Dic3—41no—
107C4×D3—81no—
108C4×D3—81no—
109Dic6—22no—
110Dic6—22no—
111Dic6—41no—
112Dic6—41no—
113Dic6—41no—
114Dic6—81no—
115C3×C8—41no—
116C3⋊C8—41no—
117C24—31no—
118C2×Q8—31no—
119C2×Q8—61no—
120C2×Q8—61no—
121C2×Q8—61no—
122C2×Q8—122no—
123C2×Q8D1211yescharacteristic
124C22×C4—21no—
125C22×C4—31no—
126C22×C4—61no—
127C22×C4—61no—
128C22×C4—61no—
129C22×C4C22×D311yescharacteristic
130Q16—122no—
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⋊C4D1211yescharacteristic
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—81no—
163Dic3—41no—
164Dic3—41no—
165Dic3—42no—
166Dic3—41no—
167C23—32no—
168C23—61no—
169C23—61no—
170C23—121no—
171C23—121no—
172C23D3×D411yescharacteristic
173Q8—41no—
174Q8—62no—
175Q8—62no—
176Q8—121no—
177Q8—121no—
178Q8—121no—
179Q8—241no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—62no—
187C2×C4—61no—
188C2×C4—61no—
189C2×C4—61no—
190C2×C4—121no—
191C2×C4—121no—
192C2×C4—121no—
193C2×C4—241no—
194C2×C4—241no—
195C2×C4C2×D1211yescharacteristic
196C2×C4D3×D411yescharacteristic
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—
212C22#88211yescharacteristic
213C22#91511yescharacteristic
214C22D3×Q1611yescharacteristic
215C4—21no—
216C4—21no—
217C4—41no—
218C4—41no—
219C4—81no—
220C4—122no—
221C4—121no—
222C4—121no—
223C4—121no—
224C3#160211yescharacteristic, complemented, Sylow
225C2—21no—
226C2—21no—
227C2—122no—
228C2#474111yescharacteristic
229C2#480911yescharacteristic
230C2#482511yescharacteristic
subgroupG/H
centerC22#882
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#328C2
socleC22×C3#120
prime radicalC23×D3D4
hypercenter#328D3
hypocenterC3#1602
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. #5050C2semidirect
3. #4739C2semidirect
5. C3×#328C2semidirect
7. #4452C2semidirect
33. #328D3semidirect
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
#50502
#47392
#46732
C3×#3282
#44682
#44522
#16023