Finite Group Databasebeta

← #80031

Subgroups — #80031

Order 384.

386 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 121 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
1C2×Q8×Dic3C211yesmaximal, characteristic, complemented
2#5600C211yesmaximal, characteristic
3C3×#448C211yesmaximal, characteristic
4#5228C211yesmaximal, characteristic, complemented
5#5166C212yesmaximal, complemented
6#5164C212yesmaximal, complemented
7#5034C211yesmaximal, characteristic
8#4796C212yesmaximal, complemented
9#4789C212yesmaximal, complemented
10#4722C211yesmaximal, characteristic, complemented
11#4719C211yesmaximal, characteristic, complemented
12#3117—31nomaximal, Sylow
13C22×C3×Q8C2211yescharacteristic
14C2×C3×SD16C2212yes—
15C3×#121C2211yescharacteristic
16Q8×Dic3—21no—
17Q8×Dic3—21no—
18Q8×Dic3C2212yes—
19Q8×Dic3C2212yes—
20C2×#216C2212yes—
21#936C2211yescharacteristic
22#935—21no—
23D4×Dic3—21no—
24D4×Dic3C2211yescharacteristic
25C2×C4⋊Dic3—21no—
26C2×C4⋊Dic3C2211yescharacteristic
27#924C2211yescharacteristic
28C2×C4×Dic3—21no—
29C2×C4×Dic3C2211yescharacteristic
30#890C2211yescharacteristic
31#887C2211yescharacteristic
32#879—21no—
33#877—21no—
34C3×#103C2212yes—
35C3×C22⋊C8C2211yescharacteristic
36#835C2212yes—
37#832C2212yes—
38#830C2211yescharacteristic
39#818C2212yes—
40#816C2212yes—
41#815C2212yes—
42Dic3⋊C8C2212yes—
43#808C2212yes—
44#529—31no—
45C2×C4×Q8—31no—
46#484—31no—
47#481—31no—
48#479—32no—
49#464—32no—
50#451—31no—
51#448D311yescharacteristic
52#441—32no—
53#440—32no—
54#432—31no—
55C2×C3×Q8—24no—
56C2×C3×Q8—21no—
57C2×C3×Q8C2312yes—
58C2×C3×D4—21no—
59C2×C3×D4C2311yescharacteristic
60C22×C3×C4—21no—
61C22×C3×C4C2311yescharacteristic
62C22×Dic3—21no—
63C22×Dic3—21no—
64C22×Dic3D411yescharacteristic
65C22×Dic3D411yescharacteristic
66C2×Dic6C2311yescharacteristic
67C3×SD16—24no—
68C2×C3×C8C2312yes—
69C3×C4⋊C4C2311yescharacteristic
70C3×C22⋊C4—21no—
71#217—21no—
72#217—21no—
73#217—21no—
74#217—21no—
75#216—24no—
76C4⋊Dic3—21no—
77C4⋊Dic3—21no—
78C4⋊Dic3—22no—
79C4⋊Dic3—22no—
80C4⋊Dic3C2312yes—
81C4⋊Dic3C2311yescharacteristic
82Dic3⋊C4—21no—
83Dic3⋊C4—21no—
84Dic3⋊C4—21no—
85C4×Dic3—21no—
86C4×Dic3—22no—
87C4×Dic3—22no—
88C4×Dic3D411yescharacteristic
89C4×Dic3D411yescharacteristic
90C4×Dic3C2311yescharacteristic
91C4×Dic3C2311yescharacteristic
92C2×C3⋊C8C2312yes—
93C22×Q8C2×D311yescharacteristic
94C2×Q16—32no—
95C2×SD16C2×D312yes—
96#126—61no—
97#125—31no—
98#124—31no—
99#123—61no—
100C22⋊Q8—31no—
101#121C2×D311yescharacteristic
102C4×Q8—32no—
103C4×Q8—32no—
104C4×Q8—31no—
105C4×Q8—61no—
106C4×Q8—61no—
107C4×D4—31no—
108C4×D4—61no—
109#117—61no—
110C2×C4⋊C4—31no—
111C2×C4⋊C4—61no—
112C2×C42—31no—
113C2×C42—61no—
114#107—32no—
115#106—32no—
116C4⋊C8—32no—
117#103—32no—
118#103—32no—
119#103C2×D312yes—
120#102—32no—
121C22⋊C8—31no—
122C22⋊C8C2×D311yescharacteristic
123#97—32no—
124C23×C3—21no—
125C23×C3C2×D411yescharacteristic
126C3×Q8—24no—
127C3×Q8—22no—
128C3×Q8#4114yes—
129C3×D4—22no—
130C3×D4—42no—
131C2×C3×C4—24no—
132C2×C3×C4—22no—
133C2×C3×C4—22no—
134C2×C3×C4—21no—
135C2×C3×C4C2×D411yescharacteristic
136C2×C3×C4C2411yescharacteristic
137C2×Dic3—22no—
138C2×Dic3—22no—
139C2×Dic3—21no—
140C2×Dic3—21no—
141C2×Dic3—41no—
142C2×Dic3—42no—
143C2×Dic3—42no—
144C2×Dic3C2×D411yescharacteristic
145C2×Dic3C2×D411yescharacteristic
146C2×Dic3C2×D411yescharacteristic
147C2×Dic3C2×D411yescharacteristic
148Dic6—22no—
149C3×C8—22no—
150C3⋊C8—22no—
151C2×Q8—24no—
152C2×Q8—21no—
153C2×Q8—31no—
154C2×Q8C22×D312yes—
155C2×D4—21no—
156C2×D4C22×D311yescharacteristic
157C22×C4—21no—
158C22×C4—31no—
159C22×C4—31no—
160C22×C4—61no—
161C22×C4—61no—
162C22×C4C22×D311yescharacteristic
163Q16—64no—
164SD16—24no—
165C2×C8—32no—
166C2×C8C22×D312yes—
167C4⋊C4—32no—
168C4⋊C4—31no—
169C4⋊C4—62no—
170C4⋊C4—62no—
171C4⋊C4—61no—
172C4⋊C4—61no—
173C4⋊C4—61no—
174C4⋊C4—61no—
175C4⋊C4—61no—
176C4⋊C4C22×D311yescharacteristic
177C22⋊C4—21no—
178C22⋊C4—61no—
179C22⋊C4—61no—
180C22⋊C4—61no—
181C22⋊C4—61no—
182C42—31no—
183C42—31no—
184C42—31no—
185C42—31no—
186C42—62no—
187C42—62no—
188C42—61no—
189C22×C3—21no—
190C22×C3—21no—
191C22×C3—41no—
192C22×C3—41no—
193C22×C3—41no—
194C22×C3#13712yes—
195C22×C3C22×D411yescharacteristic
196C3×C4—24no—
197C3×C4—21no—
198C3×C4—41no—
199C3×C4C2×#4112yes—
200Dic3—21no—
201Dic3—21no—
202Dic3—21no—
203Dic3—21no—
204Dic3—41no—
205Dic3—42no—
206Dic3—41no—
207C23—21no—
208C23D3×D411yescharacteristic
209Q8—24no—
210Q8—22no—
211Q8—62no—
212Q8#23714yes—
213D4—22no—
214D4—42no—
215C2×C4—24no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—31no—
220C2×C4—31no—
221C2×C4—31no—
222C2×C4—31no—
223C2×C4—62no—
224C2×C4—62no—
225C2×C4—61no—
226C2×C4—61no—
227C2×C4—121no—
228C2×C4—122no—
229C2×C4—122no—
230C2×C4D3×D411yescharacteristic
231C2×C4C23×D311yescharacteristic
232C8—22no—
233C8—62no—
234C2×C3—22no—
235C2×C3—81no—
236C2×C3#52511yescharacteristic
237C2×C3C2×#13711yescharacteristic
238C2×C3#57511yescharacteristic
239C22—21no—
240C22—21no—
241C22—41no—
242C22—41no—
243C22—41no—
244C22#91512yes—
245C22C2×D3×D411yescharacteristic
246C4—24no—
247C4—21no—
248C4—41no—
249C4—61no—
250C4—61no—
251C4—61no—
252C4—61no—
253C4—122no—
254C4—121no—
255C4—121no—
256C4C2×#23712yes—
257C3#311711yescharacteristic, complemented, Sylow
258C2—22no—
259C2—81no—
260C2#558611yescharacteristic
261C2C2×#91511yescharacteristic
262C2#577711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#448C2
socleC22×C3C22×D4
prime radicalC3×#121C22
hypercenter#448D3
hypocenterC3#3117
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C2×Q8×Dic3C2semidirect
4. #5228C2semidirect
5. #5166C2semidirect
6. #5164C2semidirect
8. #4796C2semidirect
9. #4789C2semidirect
10. #4722C2semidirect
11. #4719C2semidirect
257. C3#3117semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×Dic32
#56002
C3×#4482
#52282
#51662
#51642
#50342
#47962
#47892
#47222
#47192
#31173