Finite Group Databasebeta

← #68316

Subgroups — #68316

Order 384.

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

normal subgroups only

This group has 316 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, complemented
2#5046C211yesmaximal, characteristic, complemented
3#4731C211yesmaximal, characteristic, complemented
4#4671C211yesmaximal, characteristic
5C3×#328C211yesmaximal, characteristic, complemented
6#4468C211yesmaximal, characteristic, complemented
7#4452C211yesmaximal, characteristic, complemented
8#1596—31nomaximal, Sylow
9C22×D12—21no—
10C22×D12C2211yescharacteristic
11C3×C22⋊Q8C2211yescharacteristic
12#946—21no—
13C2×#215—21no—
14C2×(C2×D3)⋊C4—21no—
15C2×(C2×D3)⋊C4C2211yescharacteristic
16#924C2211yescharacteristic
17C2×#204—21no—
18#897—21no—
19#896—21no—
20#896—21no—
21D3×C22⋊C4—21no—
22#878—21no—
23C3×C22⋊C8C2211yescharacteristic
24C3×#95C2211yescharacteristic
25#830C2211yescharacteristic
26#821—21no—
27#809—21no—
28#543—31no—
29#450—31no—
30#450—31no—
31#392—31no—
32#328—31no—
33#328D311yescharacteristic, complemented
34#323—31no—
35C23×D3—21no—
36C23×D3D411yescharacteristic
37C2×C3×Q8D411yescharacteristic, complemented
38C22×C3×C4—21no—
39C22×C3×C4C2311yescharacteristic
40C22×Dic3D411yescharacteristic
41C2×D12—21no—
42C2×D12—21no—
43C2×D12—42no—
44C2×D12—42no—
45C2×D12—42no—
46C2×D12—42no—
47C2×C4×D3—21no—
48C2×C4×D3—21no—
49C2×Dic6—21no—
50C2×Dic6D411yescharacteristic, complemented
51C2×C3×C8—21no—
52C3×C4⋊C4—21no—
53C3×C4⋊C4D411yescharacteristic, complemented
54C3×C22⋊C4—21no—
55#217—21no—
56#215—42no—
57(C2×D3)⋊C4—21no—
58(C2×D3)⋊C4—21no—
59(C2×D3)⋊C4—21no—
60(C2×D3)⋊C4—21no—
61(C2×D3)⋊C4—42no—
62C4⋊Dic3—21no—
63C4⋊Dic3D411yescharacteristic, complemented
64Dic3⋊C4—21no—
65Dic3⋊C4—21no—
66Dic3⋊C4—21no—
67C2×C3⋊C8—21no—
68#204—42no—
69C22×D4—31no—
70C22×D4—61no—
71C2×SD16—61no—
72C2×SD16—61no—
73C22⋊Q8—31no—
74C22⋊Q8—61no—
75C22⋊Q8—61no—
76C22⋊Q8—61no—
77C22⋊Q8—61no—
78C22⋊Q8—61no—
79C22⋊Q8C2×D311yescharacteristic
80C2×C22⋊C4—31no—
81C2×C22⋊C4—61no—
82C2×C22⋊C4—61no—
83#102—61no—
84#102—61no—
85C22⋊C8—31no—
86C22⋊C8C2×D311yescharacteristic
87#95C2×D311yescharacteristic
88C23×C3C2×D411yescharacteristic
89C22×D3—21no—
90C22×D3—21no—
91C22×D3—22no—
92C22×D3—41no—
93C22×D3—41no—
94C22×D3—41no—
95C22×D3—41no—
96C22×D3—41no—
97C22×D3—41no—
98C22×D3—41no—
99C22×D3—41no—
100C22×D3SD1612yescomplemented
101C3×Q8—41no—
102C2×C3×C4—22no—
103C2×C3×C4—21no—
104C2×C3×C4—21no—
105C2×C3×C4—41no—
106C2×C3×C4—41no—
107C2×C3×C4—41no—
108C2×C3×C4C2×D411yescharacteristic
109C2×C3×C4C2×D411yescharacteristic
110C2×Dic3—21no—
111C2×Dic3—21no—
112C2×Dic3—21no—
113C2×Dic3—41no—
114D12—22no—
115D12—22no—
116D12—41no—
117D12—41no—
118D12—48no—
119C4×D3—81no—
120C4×D3—81no—
121Dic6—41no—
122Dic6—81no—
123C3×C8—41no—
124C3⋊C8—41no—
125C24—31no—
126C24—61no—
127C2×Q8—31no—
128C2×Q8—61no—
129C2×Q8D1211yescharacteristic, complemented
130C2×D4—61no—
131C2×D4—61no—
132C2×D4—122no—
133C2×D4—122no—
134C2×D4—122no—
135C2×D4—122no—
136C22×C4—21no—
137C22×C4—31no—
138C22×C4—61no—
139C22×C4—61no—
140C22×C4C22×D311yescharacteristic
141SD16—122no—
142SD16—122no—
143C2×C8—21no—
144C2×C8—61no—
145C4⋊C4—21no—
146C4⋊C4—31no—
147C4⋊C4—61no—
148C4⋊C4—61no—
149C4⋊C4—61no—
150C4⋊C4—61no—
151C4⋊C4D1211yescharacteristic, complemented
152C22⋊C4—21no—
153C22⋊C4—61no—
154C22⋊C4—61no—
155C22⋊C4—61no—
156C22⋊C4—61no—
157C22⋊C4—61no—
158C22⋊C4—122no—
159C22×C3—21no—
160C22×C3—21no—
161C22×C3#12011yescharacteristic
162C22×C3C2×SD1611yescharacteristic
163C22×C3#13711yescharacteristic
164C2×D3—22no—
165C2×D3—41no—
166C2×D3—41no—
167C2×D3—42no—
168C2×D3—42no—
169C2×D3—42no—
170C2×D3—42no—
171C2×D3—81no—
172C2×D3—81no—
173C2×D3—81no—
174C2×D3—81no—
175C2×D3—81no—
176C3×C4—21no—
177C3×C4—21no—
178C3×C4—42no—
179C3×C4—41no—
180C3×C4—41no—
181Dic3—41no—
182Dic3—41no—
183Dic3—41no—
184C23—32no—
185C23—62no—
186C23—61no—
187C23—61no—
188C23—121no—
189C23—121no—
190C23—121no—
191C23—121no—
192C23—121no—
193C23—121no—
194C23—121no—
195C23—121no—
196C23D3×D411yescharacteristic
197Q8—41no—
198Q8—121no—
199Q8—241no—
200D4—62no—
201D4—62no—
202D4—121no—
203D4—121no—
204D4—128no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—61no—
212C2×C4—61no—
213C2×C4—61no—
214C2×C4—121no—
215C2×C4—241no—
216C2×C4—241no—
217C2×C4C2×D1211yescharacteristic
218C2×C4D3×D411yescharacteristic
219C8—41no—
220C8—121no—
221C2×C3—21no—
222C2×C3—21no—
223C2×C3#44811yescharacteristic
224C2×C3#45311yescharacteristic
225C2×C3B(4,𝔽2)11yescharacteristic
226D3—42no—
227D3—82no—
228C22—21no—
229C22—21no—
230C22—62no—
231C22—122no—
232C22—122no—
233C22—122no—
234C22—121no—
235C22—121no—
236C22—122no—
237C22—241no—
238C22—241no—
239C22—241no—
240C22—241no—
241C22—241no—
242C22#88211yescharacteristic
243C22D3×SD1611yescharacteristic
244C22#91811yescharacteristic
245C4—21no—
246C4—21no—
247C4—42no—
248C4—41no—
249C4—41no—
250C4—121no—
251C4—121no—
252C4—121no—
253C3#159611yescharacteristic, complemented, Sylow
254C2—21no—
255C2—21no—
256C2—122no—
257C2—242no—
258C2#474111yescharacteristic
259C2#480611yescharacteristic
260C2#482211yescharacteristic
subgroupG/H
centerC22#882
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#328C2
socleC22×C3#120
prime radical#4671C2
hypercenter#328D3
hypocenterC3#1596
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
1. #5634C2semidirect
2. #5046C2semidirect
3. #4731C2semidirect
5. C3×#328C2semidirect
6. #4468C2semidirect
7. #4452C2semidirect
33. #328D3semidirect
37. C2×C3×Q8D4semidirect
50. C2×Dic6D4semidirect
53. C3×C4⋊C4D4semidirect
63. C4⋊Dic3D4semidirect
100. C22×D3SD16semidirect
129. C2×Q8D12semidirect
151. C4⋊C4D12semidirect
253. C3#1596semidirect

Maximal subgroups

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

subgroupindex
#56342
#50462
#47312
#46712
C3×#3282
#44682
#44522
#15963