Finite Group Databasebeta

← #80422

Subgroups — #80422

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#5721C211yesmaximal, characteristic, complemented
2#5542C211yesmaximal, characteristic
3C3×#461C211yesmaximal, characteristic
4#5167C212yesmaximal
5#5166C212yesmaximal
6#5092C211yesmaximal, characteristic, complemented
7C4×#214C211yesmaximal, characteristic
8#4849C211yesmaximal, characteristic, complemented
9#4831C211yesmaximal, characteristic, complemented
10#4795C212yesmaximal, complemented
11#4789C212yesmaximal, complemented
12#3398—31nomaximal, Sylow
13C2×C3×SD16C2212yes—
14C3×C4⋊Q8C2211yescharacteristic
15C3×C4×D4C2211yescharacteristic
16Q8×Dic3C2212yes—
17#944C2212yes—
18C2×#216C2212yes—
19C2×#216C2212yes—
20#936C2212yes—
21C2×#214C2211yescharacteristic
22#926—21no—
23#924—21no—
24C2×Dic12C2212yes—
25#890—21no—
26#889—22no—
27#887—21no—
28#887C2212yes—
29#879—22no—
30#869C2211yescharacteristic
31C4×Dic6C2211yescharacteristic
32C4×Dic6C2211yescharacteristic
33C3×C4⋊C8C2211yescharacteristic
34C3×#103C2212yes—
35#832C2211yescharacteristic
36#816C2212yes—
37#815C2212yes—
38Dic3⋊C8C2212yes—
39#810C2211yescharacteristic
40#808C2211yescharacteristic
41C4×C3⋊C8C2211yescharacteristic
42#557—31no—
43#556—31no—
44#497—32no—
45#494—31no—
46#464—32no—
47#462—31no—
48#462—32no—
49#461D311yescharacteristic
50#446—31no—
51#441—32no—
52C4×SD16—31no—
53C2×C3×Q8C2312yes—
54C2×C3×D4C2311yescharacteristic
55C22×C3×C4—21no—
56C2×Dic6D411yescharacteristic
57C2×Dic6C2311yescharacteristic
58C2×Dic6C2312yes—
59C3×SD16—42no—
60C2×C3×C8C2312yes—
61C3×C4⋊C4—21no—
62C3×C4⋊C4—21no—
63C3×C4⋊C4C2311yescharacteristic
64C3×C22⋊C4—21no—
65C3×C42C2311yescharacteristic
66#217—22no—
67#217—22no—
68#216—24no—
69#216—42no—
70#214—22no—
71C4⋊Dic3—21no—
72C4⋊Dic3—21no—
73C4⋊Dic3—22no—
74C4⋊Dic3D411yescharacteristic
75C4⋊Dic3C2311yescharacteristic
76Dic3⋊C4—21no—
77Dic3⋊C4—22no—
78Dic3⋊C4—22no—
79Dic3⋊C4—22no—
80Dic3⋊C4D412yes—
81C4×Dic3—21no—
82C4×Dic3—22no—
83C4×Dic3—22no—
84C4×Dic3C2312yes—
85C2×C3⋊C8C2311yescharacteristic
86C2×C3⋊C8C2311yescharacteristic
87Dic12—42no—
88C2×Q16—32no—
89C2×Q16—32no—
90C2×Q16—32no—
91C2×SD16—31no—
92C2×SD16C2×D312yes—
93C4⋊Q8—32no—
94C4⋊Q8—31no—
95C4⋊Q8C2×D311yescharacteristic
96#126—62no—
97#125—62no—
98#125—61no—
99#124—32no—
100C22⋊Q8—61no—
101C4×Q8—31no—
102C4×Q8—32no—
103C4×Q8—32no—
104C4×Q8—31no—
105C4×Q8—61no—
106C4×D4C2×D311yescharacteristic
107#117—61no—
108#106—31no—
109C4⋊C8—32no—
110C4⋊C8C2×D311yescharacteristic
111#103—32no—
112#103—31no—
113#103C2×D312yes—
114#102—31no—
115#97—32no—
116C4×C8—31no—
117C23×C3—21no—
118C3×Q8—22no—
119C3×Q8—22no—
120C3×D4—21no—
121C3×D4—21no—
122C2×C3×C4—21no—
123C2×C3×C4—22no—
124C2×C3×C4—42no—
125C2×C3×C4C2×D411yescharacteristic
126C2×C3×C4C2×D411yescharacteristic
127C2×C3×C4C2411yescharacteristic
128C2×Dic3—21no—
129C2×Dic3—21no—
130C2×Dic3—22no—
131C2×Dic3C2×D412yes—
132C2×Dic3C2×D412yes—
133Dic6—22no—
134Dic6—21no—
135Dic6—22no—
136Dic6—21no—
137Dic6—22no—
138C3×C8—22no—
139C3⋊C8—21no—
140C3⋊C8#4112yes—
141C2×Q8—31no—
142C2×Q8—32no—
143C2×Q8—31no—
144C2×Q8C22×D312yes—
145C2×D4C22×D311yescharacteristic
146C22×C4—21no—
147Q16—64no—
148Q16—122no—
149Q16—122no—
150SD16—42no—
151SD16—62no—
152C2×C8—31no—
153C2×C8—31no—
154C2×C8C22×D312yes—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—31no—
158C4⋊C4—32no—
159C4⋊C4—31no—
160C4⋊C4—62no—
161C4⋊C4—62no—
162C4⋊C4—62no—
163C4⋊C4—62no—
164C4⋊C4—61no—
165C4⋊C4—61no—
166C4⋊C4—61no—
167C4⋊C4C22×D311yescharacteristic
168C22⋊C4—21no—
169C22⋊C4—62no—
170C22⋊C4—62no—
171C42—32no—
172C42—62no—
173C42—62no—
174C42—61no—
175C42C22×D311yescharacteristic
176C22×C3—41no—
177C22×C3—41no—
178C22×C3—41no—
179C22×C3C22×D411yescharacteristic
180C3×C4—21no—
181C3×C4—42no—
182C3×C4—41no—
183C3×C4#13712yes—
184C3×C4C2×#4111yescharacteristic
185C3×C4H(5,𝔽2)11yescharacteristic
186Dic3—22no—
187Dic3—22no—
188Dic3—41no—
189Dic3—42no—
190Dic3—41no—
191C23—21no—
192Q8—22no—
193Q8—22no—
194Q8—62no—
195Q8—61no—
196Q8—62no—
197Q8—61no—
198Q8—62no—
199D4—21no—
200D4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—32no—
204C2×C4—32no—
205C2×C4—42no—
206C2×C4—61no—
207C2×C4—62no—
208C2×C4—61no—
209C2×C4D3×D411yescharacteristic
210C2×C4D3×D411yescharacteristic
211C2×C4C23×D311yescharacteristic
212C8—22no—
213C8—32no—
214C8—61no—
215C2×C3—81no—
216C2×C3#55011yescharacteristic
217C2×C3C2×#13711yescharacteristic
218C2×C3#57811yescharacteristic
219C22—41no—
220C22—41no—
221C22—41no—
222C22C2×D3×D411yescharacteristic
223C4—21no—
224C4—41no—
225C4—42no—
226C4—62no—
227C4—62no—
228C4—121no—
229C4—122no—
230C4—121no—
231C4#91512yes—
232C4#100411yescharacteristic
233C4D3×#4111yescharacteristic
234C3#339811yescharacteristic, complemented, Sylow
235C2—81no—
236C2#559911yescharacteristic
237C2C2×#91511yescharacteristic
238C2#577911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#461C2
socleC22×C3C22×D4
prime radicalC2×C3×D4C23
hypercenter#461D3
hypocenterC3#3398
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #5721C2semidirect
6. #5092C2semidirect
8. #4849C2semidirect
9. #4831C2semidirect
10. #4795C2semidirect
11. #4789C2semidirect
234. C3#3398semidirect

Maximal subgroups

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

subgroupindex
#57212
#55422
C3×#4612
#51672
#51662
#50922
C4×#2142
#48492
#48312
#47952
#47892
#33983