Finite Group Databasebeta

← #81863

Subgroups — #81863

Order 384.

366 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 119 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#5728C211yesmaximal, characteristic, complemented
2#5680C211yesmaximal, characteristic
3C3×#497C211yesmaximal, characteristic, complemented
4#5188C212yesmaximal, complemented
5#5184C212yesmaximal, complemented
6#5171C212yesmaximal
7#5164C212yesmaximal, complemented
8#5082C211yesmaximal, characteristic, complemented
9#5055C211yesmaximal, characteristic, complemented
10D3×#97C211yesmaximal, characteristic, complemented
11#4702C211yesmaximal, characteristic, complemented
12#3181—31nomaximal, Sylow
13C2×C3×Q16C2212yes—
14C2×C3×SD16C2212yes—
15C3×C4⋊Q8C2211yescharacteristic
16C3×#124C2211yescharacteristic
17#946C2211yescharacteristic
18#946C2212yes—
19Q8×Dic3C2212yes—
20Q8×Dic3C2211yescharacteristic
21#938C2211yescharacteristic
22D4×Dic3C2211yescharacteristic
23C2×D3×C8C2212yes—
24#897—21no—
25#892—21no—
26#890—21no—
27#887—21no—
28#883—21no—
29#67⋊C4—21no—
30#879—21no—
31#877—21no—
32C42×D3C2211yescharacteristic
33#870C2211yescharacteristic
34#869C2211yescharacteristic
35C3×#97C2211yescharacteristic
36#835C2212yes—
37#835C2212yes—
38#835C2212yes—
39#832C2212yes—
40#818C2212yes—
41#817C2212yes—
42#815C2212yes—
43#803C2211yescharacteristic
44#533—31no—
45#529—31no—
46#501—31no—
47#497D311yescharacteristic, complemented
48#491—31no—
49#489—31no—
50#467—32no—
51#465—32no—
52#441—32no—
53#440—32no—
54C2×#97—31no—
55C2×C3×Q8C2311yescharacteristic
56C2×C3×Q8C2312yes—
57C2×C3×D4C2311yescharacteristic
58C2×#67—21no—
59C22×Dic3—21no—
60C2×C4×D3D411yescharacteristic
61C2×C4×D3D411yescharacteristic
62C2×C4×D3C2311yescharacteristic
63C2×Dic6—21no—
64C3×Q16—24no—
65C3×SD16—24no—
66C2×C3×C8C2312yes—
67C3×C4⋊C4—21no—
68C3×C4⋊C4—21no—
69C3×C22⋊C4—21no—
70C3×C22⋊C4—21no—
71C3×C42C2311yescharacteristic
72#217—21no—
73#217—21no—
74(C2×D3)⋊C4—21no—
75(C2×D3)⋊C4—22no—
76C4⋊Dic3—21no—
77C4⋊Dic3—21no—
78C4⋊Dic3—22no—
79C4⋊Dic3C2311yescharacteristic
80C4⋊Dic3C2312yes—
81C4⋊Dic3C2311yescharacteristic
82Dic3⋊C4—21no—
83Dic3⋊C4—21no—
84Dic3⋊C4—21no—
85Dic3⋊C4—22no—
86C4×Dic3—21no—
87C4×Dic3—22no—
88C4×Dic3D411yescharacteristic
89C4×Dic3D411yescharacteristic
90C4×Dic3C2311yescharacteristic
91C2×C3⋊C8C2312yes—
92D3×C8—24no—
93C2×Q16C2×D312yes—
94C2×SD16C2×D312yes—
95C22×C8—32no—
96C4⋊Q8—31no—
97C4⋊Q8C2×D311yescharacteristic
98#126—61no—
99#125—31no—
100#125—61no—
101#124C2×D311yescharacteristic
102#123—61no—
103C22⋊Q8—31no—
104C22⋊Q8—32no—
105C22⋊Q8—61no—
106#121—31no—
107C4×Q8—31no—
108C4×Q8—32no—
109C4×Q8—61no—
110C4×D4—31no—
111C4×D4—61no—
112#117—61no—
113#117—61no—
114C2×C42—31no—
115#107—32no—
116#106—32no—
117#103—32no—
118#103—32no—
119#103—32no—
120#102—32no—
121#97—32no—
122#97—31no—
123#97C2×D311yescharacteristic
124C23×C3—21no—
125C22×D3C2×D411yescharacteristic
126C3×Q8—22no—
127C3×Q8—22no—
128C3×Q8—22no—
129C3×D4—22no—
130C2×C3×C4—21no—
131C2×C3×C4—22no—
132C2×C3×C4C2×D411yescharacteristic
133C2×C3×C4C2×D411yescharacteristic
134C2×C3×C4C2411yescharacteristic
135#67—42no—
136C2×Dic3—21no—
137C2×Dic3—22no—
138C2×Dic3—21no—
139C2×Dic3—42no—
140C2×Dic3C2×D411yescharacteristic
141C2×Dic3C2×D411yescharacteristic
142C2×Dic3C2×D411yescharacteristic
143C4×D3—22no—
144C4×D3—22no—
145C4×D3—41no—
146Dic6—42no—
147C3×C8#4114yes—
148C3⋊C8—22no—
149C2×Q8—61no—
150C2×Q8C22×D311yescharacteristic
151C2×Q8C22×D312yes—
152C2×D4—61no—
153C2×D4C22×D311yescharacteristic
154C22×C4—31no—
155C22×C4—31no—
156C22×C4—31no—
157C22×C4—61no—
158Q16—24no—
159SD16—24no—
160C2×C8—32no—
161C2×C8—64no—
162C2×C8C22×D312yes—
163C4⋊C4—21no—
164C4⋊C4—21no—
165C4⋊C4—32no—
166C4⋊C4—31no—
167C4⋊C4—31no—
168C4⋊C4—61no—
169C4⋊C4—62no—
170C4⋊C4—61no—
171C4⋊C4—62no—
172C4⋊C4—61no—
173C4⋊C4—61no—
174C4⋊C4—61no—
175C22⋊C4—21no—
176C22⋊C4—21no—
177C22⋊C4—62no—
178C22⋊C4—61no—
179C22⋊C4—61no—
180C22⋊C4—61no—
181C42—31no—
182C42—31no—
183C42—31no—
184C42—61no—
185C42—62no—
186C42C22×D311yescharacteristic
187C22×C3—41no—
188C22×C3—41no—
189C22×C3—41no—
190C22×C3C22×D411yescharacteristic
191C2×D3—21no—
192C2×D3—21no—
193C2×D3#13712yes—
194C3×C4—21no—
195C3×C4—21no—
196C3×C4—41no—
197C3×C4—42no—
198C3×C4C2×#4112yes—
199Dic3—21no—
200Dic3—21no—
201Dic3—21no—
202Dic3—41no—
203Dic3—42no—
204Dic3—41no—
205C23—21no—
206C23—31no—
207Q8—22no—
208Q8—22no—
209Q8—22no—
210Q8—122no—
211D4—22no—
212D4—122no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—31no—
216C2×C4—31no—
217C2×C4—31no—
218C2×C4—62no—
219C2×C4—62no—
220C2×C4—61no—
221C2×C4—62no—
222C2×C4—61no—
223C2×C4—121no—
224C2×C4—122no—
225C2×C4D3×D411yescharacteristic
226C2×C4D3×D411yescharacteristic
227C2×C4C23×D311yescharacteristic
228C8—62no—
229C8#23714yes—
230C2×C3—81no—
231C2×C3#53211yescharacteristic
232C2×C3C2×#13711yescharacteristic
233C2×C3#57511yescharacteristic
234D3—22no—
235C22—32no—
236C22—41no—
237C22—41no—
238C22—41no—
239C22—61no—
240C22—61no—
241C22C2×D3×D411yescharacteristic
242C4—21no—
243C4—21no—
244C4—41no—
245C4—42no—
246C4—61no—
247C4—61no—
248C4—61no—
249C4—121no—
250C4—121no—
251C4—122no—
252C4C2×#23712yes—
253C3#318111yescharacteristic, complemented, Sylow
254C2—62no—
255C2—81no—
256C2#571611yescharacteristic
257C2D3×#13711yescharacteristic
258C2#577711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#497C2
socleC22×C3C22×D4
prime radical#938C22
hypercenter#497D3
hypocenterC3#3181
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. #5728C2semidirect
3. C3×#497C2semidirect
4. #5188C2semidirect
5. #5184C2semidirect
7. #5164C2semidirect
8. #5082C2semidirect
9. #5055C2semidirect
10. D3×#97C2semidirect
11. #4702C2semidirect
47. #497D3semidirect
253. C3#3181semidirect

Maximal subgroups

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

subgroupindex
#57282
#56802
C3×#4972
#51882
#51842
#51712
#51642
#50822
#50552
D3×#972
#47022
#31813