Finite Group Databasebeta

← #81459

Subgroups — #81459

Order 384.

362 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 115 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#5624C211yesmaximal, characteristic
2C2×#888C211yesmaximal, characteristic, complemented
3C3×#484C211yesmaximal, characteristic, complemented
4#5044C211yesmaximal, characteristic, complemented
5#4968C211yesmaximal, characteristic
6#4890C212yesmaximal, complemented
7#4876C212yesmaximal, complemented
8#4794C212yesmaximal, complemented
9#4791C212yesmaximal, complemented
10#4735C211yesmaximal, characteristic
11#4719C211yesmaximal, characteristic, complemented
12#3239—31nomaximal, Sylow
13C22×Dic6—21no—
14C3×C22⋊Q8C2211yescharacteristic
15C2×C3×C4⋊C4C2211yescharacteristic
16#947C2211yescharacteristic
17Q8×Dic3C2211yescharacteristic
18#930C2211yescharacteristic
19C2×C4⋊Dic3C2211yescharacteristic
20C2×Dic3⋊C4—21no—
21C2×C4×Dic3C2211yescharacteristic
22#898—21no—
23#893C2211yescharacteristic
24#892—21no—
25#890C2211yescharacteristic
26#888—21no—
27#888—21no—
28#888C2212yes—
29#888C2212yes—
30#886—21no—
31#67⋊C4—21no—
32C3×#107C2212yes—
33C3×#103C2212yes—
34C3×C22⋊C8C2211yescharacteristic
35#835C2212yes—
36#830C2211yescharacteristic
37#821C2212yes—
38#817C2212yes—
39#815C2212yes—
40Dic3⋊C8C2212yes—
41#809C2212yes—
42#808C2212yes—
43#807C2212yes—
44C2×C4⋊Q8—31no—
45#529—31no—
46#501—32no—
47#489—32no—
48#484—31no—
49#484D311yescharacteristic, complemented
50#481—31no—
51#481—31no—
52#475—32no—
53#474—32no—
54#432—31no—
55C2×C3×Q8C2311yescharacteristic
56C22×C3×C4—21no—
57C22×C3×C4C2311yescharacteristic
58C2×#67—21no—
59C22×Dic3—21no—
60C22×Dic3D411yescharacteristic
61C22×Dic3D411yescharacteristic
62C2×D12C2311yescharacteristic
63C2×C4×D3—21no—
64C2×Dic6—22no—
65C2×Dic6—22no—
66C2×Dic6—44no—
67C2×C3×C8C2312yes—
68C3×C4⋊C4—21no—
69C3×C4⋊C4—21no—
70C3×C4⋊C4C2311yescharacteristic
71C3×C4⋊C4C2312yes—
72C3×C22⋊C4—21no—
73(C2×D3)⋊C4—21no—
74(C2×D3)⋊C4—21no—
75(C2×D3)⋊C4—21no—
76(C2×D3)⋊C4—21no—
77C4⋊Dic3—21no—
78C4⋊Dic3—21no—
79C4⋊Dic3—21no—
80C4⋊Dic3C2312yes—
81C4⋊Dic3C2311yescharacteristic
82Dic3⋊C4—21no—
83Dic3⋊C4—22no—
84Dic3⋊C4—22no—
85C4×Dic3—21no—
86C4×Dic3D411yescharacteristic
87C4×Dic3D411yescharacteristic
88C4×Dic3C2311yescharacteristic
89C4×Dic3C2311yescharacteristic
90C2×C3⋊C8C2312yes—
91C22×Q8—61no—
92C4⋊Q8—32no—
93C4⋊Q8—32no—
94C4⋊Q8—61no—
95C4⋊Q8—61no—
96#126—61no—
97#125—31no—
98#124—31no—
99#123—61no—
100C22⋊Q8C2×D311yescharacteristic
101#121—31no—
102C4×Q8—31no—
103C4×D4—31no—
104C4×D4—61no—
105#117—61no—
106C2×C4⋊C4—31no—
107C2×C4⋊C4—61no—
108C2×C4⋊C4C2×D311yescharacteristic
109C2×C42—31no—
110#107—32no—
111#107C2×D312yes—
112#106—32no—
113#106—32no—
114C4⋊C8—32no—
115#103—32no—
116#103C2×D312yes—
117#102—32no—
118#102—32no—
119C22⋊C8—31no—
120C22⋊C8C2×D311yescharacteristic
121#97—32no—
122C23×C3C2×D411yescharacteristic
123C22×D3—21no—
124C3×Q8—22no—
125C2×C3×C4—22no—
126C2×C3×C4—21no—
127C2×C3×C4—22no—
128C2×C3×C4—21no—
129C2×C3×C4—42no—
130C2×C3×C4C2×D411yescharacteristic
131C2×C3×C4C2411yescharacteristic
132#67—42no—
133C2×Dic3—22no—
134C2×Dic3—22no—
135C2×Dic3—21no—
136C2×Dic3—22no—
137C2×Dic3—42no—
138C2×Dic3C2×D411yescharacteristic
139C2×Dic3C2×D411yescharacteristic
140C2×Dic3C2×D411yescharacteristic
141C2×Dic3C2×D411yescharacteristic
142D12—22no—
143C4×D3—42no—
144Dic6—44no—
145Dic6—44no—
146C3×C8—22no—
147C3⋊C8—22no—
148C2×Q8—62no—
149C2×Q8—62no—
150C2×Q8—124no—
151C2×Q8C22×D311yescharacteristic
152C2×D4—31no—
153C2×D4—61no—
154C22×C4—21no—
155C22×C4—31no—
156C22×C4—31no—
157C22×C4—61no—
158C22×C4—61no—
159C22×C4C22×D311yescharacteristic
160C2×C8—32no—
161C2×C8C22×D312yes—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—32no—
165C4⋊C4—31no—
166C4⋊C4—62no—
167C4⋊C4—62no—
168C4⋊C4—61no—
169C4⋊C4—61no—
170C4⋊C4—61no—
171C4⋊C4—61no—
172C4⋊C4C22×D312yes—
173C4⋊C4C22×D311yescharacteristic
174C22⋊C4—21no—
175C22⋊C4—61no—
176C22⋊C4—61no—
177C22⋊C4—61no—
178C22⋊C4—61no—
179C42—31no—
180C42—31no—
181C42—31no—
182C42—31no—
183C42—61no—
184C22×C3—21no—
185C22×C3—21no—
186C22×C3#13712yes—
187C22×C3C22×D411yescharacteristic
188C2×D3—41no—
189C2×D3—41no—
190C2×D3—41no—
191C3×C4—21no—
192C3×C4—41no—
193C3×C4—42no—
194C3×C4—41no—
195C3×C42−1+412yes—
196Dic3—21no—
197Dic3—21no—
198Dic3—21no—
199Dic3—42no—
200Dic3—41no—
201Dic3AGL(1,ℤ/8)12yes—
202C23—61no—
203C23D3×D411yescharacteristic
204Q8—22no—
205Q8—124no—
206Q8—124no—
207D4—62no—
208D4—122no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—31no—
214C2×C4—31no—
215C2×C4—31no—
216C2×C4—31no—
217C2×C4—42no—
218C2×C4—62no—
219C2×C4—62no—
220C2×C4—61no—
221C2×C4—62no—
222C2×C4—122no—
223C2×C4—122no—
224C2×C4D3×D411yescharacteristic
225C2×C4C23×D311yescharacteristic
226C8—22no—
227C8—62no—
228C2×C3—22no—
229C2×C3#53611yescharacteristic
230C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
231C2×C3C2×#13711yescharacteristic
232D3—81no—
233C22—21no—
234C22—21no—
235C22—121no—
236C22—121no—
237C22—121no—
238C22#91812yes—
239C22C2×D3×D411yescharacteristic
240C4—21no—
241C4—32no—
242C4—41no—
243C4—42no—
244C4—41no—
245C4—61no—
246C4—61no—
247C4—61no—
248C4—121no—
249C4—122no—
250C4#101012yes—
251C3#323911yescharacteristic, complemented, Sylow
252C2—22no—
253C2—241no—
254C2#564811yescharacteristic
255C2C2×#91811yescharacteristic
256C2D3×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#484C2
socleC22×C3C22×D4
prime radical#930C22
hypercenter#484D3
hypocenterC3#3239
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
2. C2×#888C2semidirect
3. C3×#484C2semidirect
4. #5044C2semidirect
6. #4890C2semidirect
7. #4876C2semidirect
8. #4794C2semidirect
9. #4791C2semidirect
11. #4719C2semidirect
49. #484D3semidirect
251. C3#3239semidirect

Maximal subgroups

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

subgroupindex
#56242
C2×#8882
C3×#4842
#50442
#49682
#48902
#48762
#47942
#47912
#47352
#47192
#32393