Finite Group Databasebeta

← #80044

Subgroups — #80044

Order 384.

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

normal subgroups only

This group has 420 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
1C2×#946C211yesmaximal, characteristic, complemented
2#5606C211yesmaximal, characteristic
3C3×#448C211yesmaximal, characteristic, complemented
4#5228C211yesmaximal, characteristic, complemented
5#5173C212yesmaximal, complemented
6#5171C212yesmaximal, complemented
7#5033C211yesmaximal, characteristic, complemented
8#4815C212yesmaximal, complemented
9#4813C212yesmaximal, complemented
10D3×C22⋊C8C211yesmaximal, characteristic, complemented
11#4721C211yesmaximal, characteristic, complemented
12#3190—31nomaximal, Sylow
13C22×C3×Q8C2211yescharacteristic
14C2×#237C2211yescharacteristic
15C22×C4×D3C2211yescharacteristic
16C2×C3×SD16C2212yescomplemented
17C3×#121C2211yescharacteristic
18#946—21no—
19#946—21no—
20#946C2212yes—
21#946C2212yes—
22C2×#216C2212yescomplemented
23#938C2211yescharacteristic
24#937—21no—
25D4×Dic3—21no—
26C2×(C2×D3)⋊C4—21no—
27C2×C4⋊Dic3C2211yescharacteristic
28#924C2211yescharacteristic
29C2×Dic3⋊C4—21no—
30C2×D3×C8C2212yes—
31#897C2211yescharacteristic
32#892C2211yescharacteristic
33#883—21no—
34#67⋊C4—21no—
35C3×#103C2212yescomplemented
36C3×C22⋊C8C2211yescharacteristic
37#835C2212yescomplemented
38#832C2212yescomplemented
39#830C2211yescharacteristic
40(C2×D3)⋊C8C2212yes—
41#817C2212yescomplemented
42#816C2212yescomplemented
43#807C2212yescomplemented
44#525—31no—
45C2×C22⋊Q8—31no—
46#485—32no—
47#483—31no—
48#480—31no—
49#467—32no—
50#465—32no—
51#452—32no—
52#451—31no—
53#448D311yescharacteristic, complemented
54C2×C22⋊C8—31no—
55C23×D3D411yescharacteristic
56C2×C3×Q8—21no—
57C2×C3×Q8—42no—
58C2×C3×Q8C2312yes—
59C2×C3×D4—21no—
60C2×C3×D4C2311yescharacteristic
61C22×C3×C4—21no—
62C22×C3×C4C2311yescharacteristic
63C2×#67—21no—
64C2×#67—21no—
65C22×Dic3—21no—
66C22×Dic3—21no—
67C22×Dic3D411yescharacteristic
68#237—42no—
69C2×C4×D3—42no—
70C2×C4×D3D411yescharacteristic
71C2×C4×D3D411yescharacteristic
72C2×C4×D3C2311yescharacteristic
73C2×C4×D3C2311yescharacteristic
74C2×Dic6C2311yescharacteristic
75C3×SD16—42no—
76C2×C3×C8C2312yes—
77C3×C4⋊C4C2311yescharacteristic
78C3×C22⋊C4—21no—
79#217—21no—
80#217—21no—
81#216—42no—
82(C2×D3)⋊C4—21no—
83(C2×D3)⋊C4—21no—
84(C2×D3)⋊C4—22no—
85(C2×D3)⋊C4—22no—
86C4⋊Dic3—21no—
87C4⋊Dic3—21no—
88C4⋊Dic3C2312yes—
89C4⋊Dic3C2311yescharacteristic
90Dic3⋊C4—21no—
91Dic3⋊C4—22no—
92Dic3⋊C4—22no—
93C4×Dic3—21no—
94C2×C3⋊C8C2312yes—
95D3×C8—42no—
96C2×#41—31no—
97C22×Q8C2×D311yescharacteristic
98C23×C4—31no—
99C2×Q16—32no—
100C2×SD16C2×D312yescomplemented
101C22×C8—32no—
102#123—61no—
103C22⋊Q8—32no—
104C22⋊Q8—32no—
105C22⋊Q8—31no—
106C22⋊Q8—31no—
107C22⋊Q8—61no—
108C22⋊Q8—61no—
109#121—31no—
110#121C2×D311yescharacteristic
111#120—61no—
112C4×D4—61no—
113C4×D4—61no—
114#117—31no—
115C2×C4⋊C4—31no—
116C2×C4⋊C4—61no—
117C2×C22⋊C4—61no—
118#107—32no—
119#106—32no—
120#103—32no—
121#103—32no—
122#103C2×D312yescomplemented
123#102—32no—
124C22⋊C8—31no—
125C22⋊C8—32no—
126C22⋊C8C2×D311yescharacteristic
127C23×C3—21no—
128C23×C3C2×D411yescharacteristic
129C22×D3—22no—
130C22×D3—22no—
131C22×D3—41no—
132C22×D3C2×D411yescharacteristic
133C22×D3C2×D411yescharacteristic
134C3×Q8—22no—
135C3×Q8—22no—
136C3×Q8—42no—
137C3×D4—22no—
138C3×D4—42no—
139C2×C3×C4—22no—
140C2×C3×C4—21no—
141C2×C3×C4—22no—
142C2×C3×C4—42no—
143C2×C3×C4C2×D411yescharacteristic
144C2×C3×C4C2411yescharacteristic
145#67—42no—
146#67—42no—
147C2×Dic3—21no—
148C2×Dic3—22no—
149C2×Dic3—21no—
150C2×Dic3—41no—
151C2×Dic3—42no—
152C2×Dic3—42no—
153C2×Dic3C2×D411yescharacteristic
154C2×Dic3C2×D411yescharacteristic
155C4×D3—22no—
156C4×D3—22no—
157C4×D3—41no—
158C4×D3—41no—
159Dic6—22no—
160C3×C8—22no—
161C3⋊C8—22no—
162C24—31no—
163#41—122no—
164C2×Q8—21no—
165C2×Q8—31no—
166C2×Q8—42no—
167C2×Q8C22×D312yes—
168C2×D4—21no—
169C2×D4—61no—
170C2×D4—61no—
171C2×D4C22×D311yescharacteristic
172C22×C4—21no—
173C22×C4—31no—
174C22×C4—31no—
175C22×C4—31no—
176C22×C4—31no—
177C22×C4—31no—
178C22×C4—61no—
179C22×C4—61no—
180C22×C4—122no—
181C22×C4C22×D311yescharacteristic
182Q16—122no—
183SD16—42no—
184C2×C8—32no—
185C2×C8—122no—
186C2×C8C22×D312yes—
187C4⋊C4—32no—
188C4⋊C4—31no—
189C4⋊C4—62no—
190C4⋊C4—62no—
191C4⋊C4—61no—
192C4⋊C4—61no—
193C4⋊C4—61no—
194C4⋊C4C22×D311yescharacteristic
195C22⋊C4—21no—
196C22⋊C4—62no—
197C22⋊C4—62no—
198C22⋊C4—61no—
199C22⋊C4—61no—
200C22⋊C4—61no—
201C22⋊C4—61no—
202C42—61no—
203C22×C3—21no—
204C22×C3—21no—
205C22×C3—41no—
206C22×C3—41no—
207C22×C3—41no—
208C22×C3#13512yes—
209C22×C3C22×D411yescharacteristic
210C2×D3—21no—
211C2×D3—21no—
212C2×D3—21no—
213C2×D3—21no—
214C2×D3—21no—
215C2×D3—44no—
216C2×D3#13712yes—
217C3×C4—21no—
218C3×C4—42no—
219C3×C4—41no—
220C3×C4H(5,𝔽2)12yes—
221Dic3—21no—
222Dic3—21no—
223Dic3—41no—
224Dic3—42no—
225Dic3—41no—
226C23—21no—
227C23—31no—
228C23—31no—
229C23—62no—
230C23—62no—
231C23—121no—
232C23D3×D411yescharacteristic
233Q8—22no—
234Q8—22no—
235Q8—42no—
236Q8—62no—
237D4—22no—
238D4—42no—
239D4—122no—
240D4—122no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—31no—
245C2×C4—31no—
246C2×C4—42no—
247C2×C4—62no—
248C2×C4—62no—
249C2×C4—61no—
250C2×C4—62no—
251C2×C4—61no—
252C2×C4—121no—
253C2×C4—121no—
254C2×C4—121no—
255C2×C4—122no—
256C2×C4—122no—
257C2×C4D3×D411yescharacteristic
258C2×C4C23×D311yescharacteristic
259C8—22no—
260C8—62no—
261C2×C3—22no—
262C2×C3—81no—
263C2×C3#53411yescharacteristic
264C2×C3C2×#13511yescharacteristic
265C2×C3C2×#13711yescharacteristic
266D3—22no—
267D3—41no—
268C22—21no—
269C22—21no—
270C22—32no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—61no—
275C22—61no—
276C22—61no—
277C22—61no—
278C22—61no—
279C22—124no—
280C22#91612yes—
281C22C2×D3×D411yescharacteristic
282C4—21no—
283C4—41no—
284C4—42no—
285C4—61no—
286C4—61no—
287C4—122no—
288C4—121no—
289C4—121no—
290C4#100412yes—
291C3#319011yescharacteristic, complemented, Sylow
292C2—22no—
293C2—62no—
294C2—81no—
295C2—121no—
296C2#558911yescharacteristic
297C2C2×#91611yescharacteristic
298C2D3×#13711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#448C2
socleC22×C3C22×D4
prime radical#5606C2
hypercenter#448D3
hypocenterC3#3190
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×#946C2semidirect
3. C3×#448C2semidirect
4. #5228C2semidirect
5. #5173C2semidirect
6. #5171C2semidirect
7. #5033C2semidirect
8. #4815C2semidirect
9. #4813C2semidirect
10. D3×C22⋊C8C2semidirect
11. #4721C2semidirect
16. C2×C3×SD16C22semidirect
22. C2×#216C22semidirect
35. C3×#103C22semidirect
37. #835C22semidirect
38. #832C22semidirect
41. #817C22semidirect
42. #816C22semidirect
43. #807C22semidirect
53. #448D3semidirect
100. C2×SD16C2×D3semidirect
122. #103C2×D3semidirect
291. C3#3190semidirect

Maximal subgroups

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

subgroupindex
C2×#9462
#56062
C3×#4482
#52282
#51732
#51712
#50332
#48152
#48132
D3×C22⋊C82
#47212
#31903