Finite Group Databasebeta

← #80473

Subgroups — #80473

Order 384.

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

normal subgroups only

This group has 350 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#5732C211yesmaximal, characteristic
2#5567C211yesmaximal, characteristic, complemented
3C3×#462C211yesmaximal, characteristic, complemented
4#5190C212yesmaximal
5#5182C212yesmaximal, complemented
6#5092C211yesmaximal, characteristic, complemented
7C4×#215C211yesmaximal, characteristic
8#4841C211yesmaximal, characteristic
9#4831C211yesmaximal, characteristic, complemented
10#4798C212yesmaximal
11#4789C212yesmaximal, complemented
12#3398—31nomaximal, Sylow
13C2×C3×Q16C2212yes—
14C3×C4⋊Q8C2211yescharacteristic
15C3×C4×Q8C2211yescharacteristic
16#947C2212yes—
17Q8×Dic3C2212yes—
18#944C2212yes—
19C2×#216C2212yes—
20C2×#216C2212yes—
21C2×#215C2211yescharacteristic
22C2×#204C2212yes—
23#898—22no—
24#897—21no—
25#892—21no—
26#889—22no—
27#887C2212yes—
28C4×D12C2211yescharacteristic
29#870—21no—
30#869C2211yescharacteristic
31C4×Dic6—21no—
32C4×Dic6C2211yescharacteristic
33C3×C4⋊C8C2211yescharacteristic
34C3×#103C2212yes—
35#835C2211yescharacteristic
36#816C2212yes—
37#815C2212yes—
38Dic3⋊C8C2212yes—
39#809C2211yescharacteristic
40#808C2211yescharacteristic
41C4×C3⋊C8C2211yescharacteristic
42#557—31no—
43#556—31no—
44#497—32no—
45#494—31no—
46#464—32no—
47#462—32no—
48#462D311yescharacteristic, complemented
49#461—31no—
50#446—31no—
51#441—32no—
52C4×SD16—31no—
53C2×C3×Q8C2311yescharacteristic
54C2×C3×Q8C2312yes—
55C2×D12C2311yescharacteristic
56C2×C4×D3—21no—
57C2×Dic6D411yescharacteristic
58C2×Dic6C2312yes—
59C3×Q16—42no—
60C2×C3×C8C2312yes—
61C3×C4⋊C4—21no—
62C3×C4⋊C4—21no—
63C3×C4⋊C4—21no—
64C3×C4⋊C4C2311yescharacteristic
65C3×C42—21no—
66C3×C42C2311yescharacteristic
67#216—24no—
68#216—42no—
69#215—22no—
70(C2×D3)⋊C4—21no—
71(C2×D3)⋊C4—22no—
72(C2×D3)⋊C4—22no—
73C4⋊Dic3—21no—
74C4⋊Dic3—21no—
75C4⋊Dic3—22no—
76C4⋊Dic3D411yescharacteristic
77C4⋊Dic3C2311yescharacteristic
78Dic3⋊C4—22no—
79Dic3⋊C4—22no—
80Dic3⋊C4—22no—
81Dic3⋊C4D412yes—
82C4×Dic3—22no—
83C4×Dic3—22no—
84C4×Dic3C2312yes—
85C2×C3⋊C8C2311yescharacteristic
86C2×C3⋊C8C2311yescharacteristic
87#204—42no—
88C2×Q16—32no—
89C2×Q16—32no—
90C2×Q16C2×D312yes—
91C2×SD16—31no—
92C2×SD16—32no—
93C4⋊Q8—32no—
94C4⋊Q8—31no—
95C4⋊Q8C2×D311yescharacteristic
96#126—62no—
97#125—61no—
98#125—62no—
99#124—32no—
100C22⋊Q8—61no—
101C4×Q8—31no—
102C4×Q8—32no—
103C4×Q8—32no—
104C4×Q8—61no—
105C4×Q8C2×D311yescharacteristic
106C4×D4—31no—
107#117—61no—
108#106—31no—
109C4⋊C8—32no—
110C4⋊C8C2×D311yescharacteristic
111#103—31no—
112#103—32no—
113#103C2×D312yes—
114#102—31no—
115#97—32no—
116C4×C8—31no—
117C22×D3—21no—
118C3×Q8—21no—
119C3×Q8—22no—
120C3×Q8—22no—
121C3×Q8—21no—
122C2×C3×C4—21no—
123C2×C3×C4—22no—
124C2×C3×C4—21no—
125C2×C3×C4C2×D411yescharacteristic
126C2×C3×C4C2×D411yescharacteristic
127C2×C3×C4C2411yescharacteristic
128C2×Dic3—21no—
129C2×Dic3—22no—
130C2×Dic3C2×D412yes—
131C2×Dic3C2×D412yes—
132D12—21no—
133D12—21no—
134C4×D3—42no—
135Dic6—22no—
136Dic6—22no—
137Dic6—22no—
138C3×C8—22no—
139C3⋊C8—21no—
140C3⋊C8#4112yes—
141C2×Q8—31no—
142C2×Q8—32no—
143C2×Q8C22×D311yescharacteristic
144C2×Q8C22×D312yes—
145C2×D4—31no—
146C22×C4—61no—
147Q16—42no—
148Q16—64no—
149Q16—122no—
150SD16—62no—
151SD16—122no—
152C2×C8—31no—
153C2×C8—31no—
154C2×C8C22×D312yes—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—31no—
159C4⋊C4—32no—
160C4⋊C4—31no—
161C4⋊C4—62no—
162C4⋊C4—62no—
163C4⋊C4—62no—
164C4⋊C4—62no—
165C4⋊C4—61no—
166C4⋊C4—61no—
167C4⋊C4C22×D311yescharacteristic
168C22⋊C4—62no—
169C22⋊C4—62no—
170C22⋊C4—61no—
171C42—21no—
172C42—32no—
173C42—62no—
174C42—62no—
175C42C22×D311yescharacteristic
176C22×C3C22×D411yescharacteristic
177C2×D3—41no—
178C2×D3—41no—
179C2×D3—41no—
180C3×C4—21no—
181C3×C4—42no—
182C3×C4—41no—
183C3×C4—41no—
184C3×C4#13712yes—
185C3×C4C2×#4111yescharacteristic
186C3×C4H(5,𝔽2)11yescharacteristic
187Dic3—22no—
188Dic3—22no—
189Dic3—42no—
190Dic3—41no—
191C23—61no—
192Q8—21no—
193Q8—22no—
194Q8—21no—
195Q8—22no—
196Q8—62no—
197Q8—62no—
198Q8—62no—
199D4—61no—
200D4—61no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—32no—
205C2×C4—32no—
206C2×C4—61no—
207C2×C4—62no—
208C2×C4—122no—
209C2×C4D3×D411yescharacteristic
210C2×C4D3×D411yescharacteristic
211C2×C4C23×D311yescharacteristic
212C8—22no—
213C8—32no—
214C8—61no—
215C2×C3#55011yescharacteristic
216C2×C3C2×#13711yescharacteristic
217C2×C3#57811yescharacteristic
218D3—81no—
219C22—121no—
220C22—121no—
221C22—121no—
222C22C2×D3×D411yescharacteristic
223C4—21no—
224C4—41no—
225C4—41no—
226C4—42no—
227C4—62no—
228C4—62no—
229C4—121no—
230C4—122no—
231C4#91812yes—
232C4#100411yescharacteristic
233C4D3×#4111yescharacteristic
234C3#339811yescharacteristic, complemented, Sylow
235C2—241no—
236C2#559911yescharacteristic
237C2C2×#91811yescharacteristic
238C2#577911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#462C2
socleC22×C3C22×D4
prime radicalC2×D12C23
hypercenter#462D3
hypocenterC3#3398
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
2. #5567C2semidirect
3. C3×#462C2semidirect
5. #5182C2semidirect
6. #5092C2semidirect
9. #4831C2semidirect
11. #4789C2semidirect
48. #462D3semidirect
234. C3#3398semidirect

Maximal subgroups

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

subgroupindex
#57322
#55672
C3×#4622
#51902
#51822
#50922
C4×#2152
#48412
#48312
#47982
#47892
#33983