Finite Group Databasebeta

← #83985

Subgroups — #83985

Order 384.

496 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 143 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×C3×#137C211yesmaximal, characteristic, complemented
2#5828C211yesmaximal, characteristic
3C2×#946C211yesmaximal, characteristic, complemented
4C2×D3×M16C211yesmaximal, characteristic, complemented
5#5232C211yesmaximal, characteristic, complemented
6#5225C211yesmaximal, characteristic, complemented
7#5189C212yesmaximal, complemented
8#5188C212yesmaximal, complemented
9#5174C212yesmaximal, complemented
10#5171C212yesmaximal, complemented
11#5121C211yesmaximal, characteristic, complemented
12#3056—31nomaximal, Sylow
13C2×C3×#41C2211yescharacteristic
14C22×C3×Q8C2211yescharacteristic
15C22×C4×D3C2211yescharacteristic
16C3×#137—24no—
17C2×C3×Q16C2212yescomplemented
18C2×C3×SD16C2212yescomplemented
19C2×C3×M16C2211yescharacteristic
20#946—21no—
21#946—21no—
22#946C2212yes—
23#946C2212yes—
24#946C2211yescharacteristic
25#946C2211yescharacteristic
26#938C2211yescharacteristic
27#938C2211yescharacteristic
28#937—21no—
29#935—21no—
30C4×#67—21no—
31C4×#67—21no—
32C2×(C2×D3)⋊C4—21no—
33#926C2211yescharacteristic
34C2×C4⋊Dic3C2211yescharacteristic
35C2×Dic3⋊C4—21no—
36C2×#208C2211yescharacteristic
37D3×M16—24no—
38C2×ΓL(1,𝔽25)C2212yes—
39C2×D3×C8C2212yes—
40#835C2212yescomplemented
41#835C2212yescomplemented
42#835C2212yescomplemented
43#832C2212yescomplemented
44#818C2212yescomplemented
45#817C2212yescomplemented
46C2×#137D311yescharacteristic, complemented
47C22×M16—31no—
48#525—31no—
49C2×C22⋊Q8—31no—
50#470—32no—
51#468—32no—
52#467—32no—
53#465—32no—
54#428—31no—
55#419—31no—
56#417—31no—
57C23×D3D411yescharacteristic
58C3×#41—22no—
59C3×#41—22no—
60C2×C3×Q8—21no—
61C2×C3×Q8—22no—
62C2×C3×Q8—22no—
63C2×C3×Q8C2312yes—
64C2×C3×Q8C2311yescharacteristic
65C2×C3×D4—21no—
66C2×C3×D4C2311yescharacteristic
67C22×C3×C4—21no—
68C22×C3×C4—21no—
69C22×C3×C4C2311yescharacteristic
70C2×#67—21no—
71C2×#67—21no—
72C22×Dic3—21no—
73C22×Dic3D411yescharacteristic
74C2×C4×D3—22no—
75C2×C4×D3—22no—
76C2×C4×D3D411yescharacteristic
77C2×C4×D3D411yescharacteristic
78C2×C4×D3C2311yescharacteristic
79C2×C4×D3C2311yescharacteristic
80C3×Q16—24no—
81C3×SD16—24no—
82C3×M16D414yescomplemented
83C2×C3×C8C2312yes—
84#217—21no—
85#217—21no—
86#217—21no—
87(C2×D3)⋊C4—22no—
88(C2×D3)⋊C4—22no—
89(C2×D3)⋊C4—21no—
90(C2×D3)⋊C4—21no—
91C4⋊Dic3—21no—
92C4⋊Dic3C2311yescharacteristic
93C4⋊Dic3C2312yes—
94C4⋊Dic3C2311yescharacteristic
95Dic3⋊C4—22no—
96Dic3⋊C4—22no—
97Dic3⋊C4—21no—
98Dic3⋊C4—21no—
99C4×Dic3—21no—
100#208—22no—
101C2×C3⋊C8C2312yes—
102ΓL(1,𝔽25)—24no—
103D3×C8—24no—
104C2×#41C2×D311yescharacteristic
105C22×Q8C2×D311yescharacteristic
106C23×C4—31no—
107#137—24no—
108C2×Q16C2×D312yescomplemented
109C2×SD16C2×D312yescomplemented
110C2×M16—31no—
111C2×M16—32no—
112C2×M16—64no—
113C2×M16C2×D311yescharacteristic
114C22×C8—32no—
115#123—61no—
116C22⋊Q8—32no—
117C22⋊Q8—32no—
118C22⋊Q8—31no—
119C22⋊Q8—31no—
120C22⋊Q8—61no—
121C22⋊Q8—61no—
122#121—31no—
123#121—31no—
124#120—61no—
125C4×D4—61no—
126C4×D4—61no—
127#117—31no—
128C2×C4⋊C4—31no—
129C2×C4⋊C4—61no—
130C2×C22⋊C4—61no—
131#107—32no—
132#106—32no—
133#103—32no—
134#103—32no—
135#103—32no—
136#102—32no—
137C23×C3—21no—
138C23×C3C2×D411yescharacteristic
139C22×D3—22no—
140C22×D3—22no—
141C22×D3—41no—
142C22×D3C2×D411yescharacteristic
143C22×D3C2×D411yescharacteristic
144C3×Q8—22no—
145C3×Q8—21no—
146C3×Q8—22no—
147C3×Q8—21no—
148C3×Q8—42no—
149C3×D4—21no—
150C3×D4—21no—
151C3×D4—42no—
152C2×C3×C4—21no—
153C2×C3×C4—22no—
154C2×C3×C4—42no—
155C2×C3×C4—42no—
156C2×C3×C4C2×D411yescharacteristic
157C2×C3×C4C2×D412yes—
158C2×C3×C4#4112yes—
159C2×C3×C4C2411yescharacteristic
160#67—42no—
161#67—42no—
162C2×Dic3—22no—
163C2×Dic3—21no—
164C2×Dic3—21no—
165C2×Dic3—22no—
166C2×Dic3—42no—
167C2×Dic3C2×D411yescharacteristic
168C2×Dic3C2×D411yescharacteristic
169C4×D3—22no—
170C4×D3—22no—
171C4×D3—21no—
172C4×D3—21no—
173C4×D3—21no—
174C4×D3—21no—
175C3×C8C2×D414yes—
176C3⋊C8—22no—
177C24—31no—
178#41—22no—
179#41—22no—
180C2×Q8—21no—
181C2×Q8—22no—
182C2×Q8—22no—
183C2×Q8C22×D312yes—
184C2×Q8C22×D311yescharacteristic
185C2×D4—21no—
186C2×D4—61no—
187C2×D4—61no—
188C2×D4C22×D311yescharacteristic
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—31no—
192C22×C4—31no—
193C22×C4—31no—
194C22×C4—31no—
195C22×C4—31no—
196C22×C4—62no—
197C22×C4—62no—
198C22×C4—61no—
199C22×C4C22×D311yescharacteristic
200Q16—24no—
201SD16—24no—
202M16—62no—
203M16—64no—
204M16#6714yescomplemented
205C2×C8—32no—
206C2×C8—64no—
207C2×C8C22×D312yes—
208C4⋊C4—32no—
209C4⋊C4—31no—
210C4⋊C4—31no—
211C4⋊C4—62no—
212C4⋊C4—62no—
213C4⋊C4—61no—
214C4⋊C4—61no—
215C4⋊C4—61no—
216C22⋊C4—62no—
217C22⋊C4—62no—
218C22⋊C4—61no—
219C22⋊C4—61no—
220C22⋊C4—61no—
221C22⋊C4—61no—
222C22⋊C4—61no—
223C42—61no—
224C22×C3—21no—
225C22×C3—21no—
226C22×C3—41no—
227C22×C3—41no—
228C22×C3—41no—
229C22×C3#12112yes—
230C22×C3C22×D411yescharacteristic
231C2×D3—21no—
232C2×D3—21no—
233C2×D3—21no—
234C2×D3—21no—
235C2×D3—21no—
236C2×D3—44no—
237C2×D3#13712yescomplemented
238C3×C4—41no—
239C3×C4—42no—
240C3×C4#12112yes—
241C3×C4C22×D411yescharacteristic
242C3×C4C2×#4111yescharacteristic
243Dic3—21no—
244Dic3—21no—
245Dic3—41no—
246Dic3—42no—
247Dic3—41no—
248C23—21no—
249C23—31no—
250C23—31no—
251C23—62no—
252C23—62no—
253C23—121no—
254C23D3×D411yescharacteristic
255Q8—22no—
256Q8—21no—
257Q8—22no—
258Q8—21no—
259Q8—42no—
260D4—21no—
261D4—21no—
262D4—42no—
263D4—122no—
264D4—122no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—31no—
268C2×C4—31no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—62no—
272C2×C4—62no—
273C2×C4—62no—
274C2×C4—61no—
275C2×C4—61no—
276C2×C4—61no—
277C2×C4—61no—
278C2×C4—61no—
279C2×C4—61no—
280C2×C4—62no—
281C2×C4—122no—
282C2×C4D3×D411yescharacteristic
283C2×C4#23712yes—
284C2×C4C2×#6712yes—
285C2×C4C23×D311yescharacteristic
286C8—62no—
287C8C2×#6714yes—
288C2×C3—22no—
289C2×C3—81no—
290C2×C3C2×#12111yescharacteristic
291C2×C3C2×#13711yescharacteristic
292C2×C3#57511yescharacteristic
293D3—22no—
294D3—41no—
295C22—21no—
296C22—21no—
297C22—32no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—61no—
302C22—61no—
303C22—61no—
304C22—61no—
305C22—61no—
306C22—124no—
307C22#93812yes—
308C22C2×D3×D411yescharacteristic
309C4—41no—
310C4—42no—
311C4—61no—
312C4—61no—
313C4—121no—
314C4—122no—
315C4—121no—
316C4#93812yes—
317C4C2×#23711yescharacteristic
318C4C22×#6711yescharacteristic
319C3#305611yescharacteristic, complemented, Sylow
320C2—22no—
321C2—62no—
322C2—81no—
323C2—121no—
324C2D3×#13711yescharacteristic
325C2#577711yescharacteristic
326C2C2×#93811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC2×C3×#137C2
socleC22×C3C22×D4
prime radical#5828C2
hypercenterC2×#137D3
hypocenterC3#3056
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×C3×#137C2semidirect
3. C2×#946C2semidirect
4. C2×D3×M16C2semidirect
5. #5232C2semidirect
6. #5225C2semidirect
7. #5189C2semidirect
8. #5188C2semidirect
9. #5174C2semidirect
10. #5171C2semidirect
11. #5121C2semidirect
17. C2×C3×Q16C22semidirect
18. C2×C3×SD16C22semidirect
40. #835C22semidirect
41. #835C22semidirect
42. #835C22semidirect
43. #832C22semidirect
44. #818C22semidirect
45. #817C22semidirect
46. C2×#137D3semidirect
82. C3×M16D4semidirect
108. C2×Q16C2×D3semidirect
109. C2×SD16C2×D3semidirect
204. M16#67semidirect
237. C2×D3#137semidirect
319. C3#3056semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1372
#58282
C2×#9462
C2×D3×M162
#52322
#52252
#51892
#51882
#51742
#51712
#51212
#30563