Finite Group Databasebeta

← #79212

Subgroups — #79212

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×#909C211yesmaximal, characteristic, complemented
2C2×D3×M16C211yesmaximal, characteristic, complemented
3#5524C211yesmaximal, characteristic
4C2×#897C211yesmaximal, characteristic, complemented
5C3×#428C211yesmaximal, characteristic, complemented
6#5004C211yesmaximal, characteristic, complemented
7#4966C211yesmaximal, characteristic, complemented
8#4887C212yesmaximal, complemented
9#4886C212yesmaximal, complemented
10#4867C212yesmaximal, complemented
11#4864C212yesmaximal, complemented
12#3056—31nomaximal, Sylow
13C2×#235C2211yescharacteristic
14C22×C4×D3C2211yescharacteristic
15C22×Dic6C2211yescharacteristic
16C2×C3×M16C2211yescharacteristic
17C3×#117C2211yescharacteristic
18C2×C3×C4⋊C4C2211yescharacteristic
19C2×(C2×D3)⋊C4—21no—
20C2×C4⋊Dic3—21no—
21C2×#208C2211yescharacteristic
22#909—24no—
23D3×M16—24no—
24C2×Dic12C2212yescomplemented
25C2×#204C2212yescomplemented
26C2×ΓL(1,𝔽25)C2212yes—
27C2×D3×C8C2212yes—
28#897—21no—
29#897—21no—
30#897C2212yes—
31#897C2211yescharacteristic
32#897C2212yes—
33#897C2211yescharacteristic
34#895C2211yescharacteristic
35#895C2211yescharacteristic
36#886—21no—
37#882—21no—
38C4×D12—21no—
39C4×D12—21no—
40C3×#107C2212yescomplemented
41C3×#106C2212yescomplemented
42#810C2212yescomplemented
43#810C2212yescomplemented
44#810C2212yescomplemented
45#809C2212yescomplemented
46C2×#137—31no—
47C22×M16—31no—
48#525—31no—
49C2×C22⋊Q8—31no—
50#470—32no—
51#468—32no—
52#467—32no—
53#465—32no—
54#428D311yescharacteristic, complemented
55#419—31no—
56#417—31no—
57C23×D3D411yescharacteristic
58C22×C3×C4—21no—
59C22×C3×C4C2311yescharacteristic
60C2×#67—21no—
61C22×Dic3—21no—
62C22×Dic3D411yescharacteristic
63#235—22no—
64#235—22no—
65C2×D12—21no—
66C2×D12—21no—
67C2×D12C2311yescharacteristic
68C2×C4×D3—22no—
69C2×C4×D3—22no—
70C2×C4×D3—21no—
71C2×C4×D3D411yescharacteristic
72C2×C4×D3D411yescharacteristic
73C2×C4×D3C2311yescharacteristic
74C2×C4×D3C2311yescharacteristic
75C2×Dic6—22no—
76C2×Dic6—22no—
77C2×Dic6—21no—
78C2×Dic6C2312yes—
79C2×Dic6C2311yescharacteristic
80C3×M16D414yescomplemented
81C2×C3×C8C2312yes—
82C3×C4⋊C4—21no—
83C3×C4⋊C4C2312yes—
84C3×C4⋊C4C2311yescharacteristic
85C3×C4⋊C4C2311yescharacteristic
86C3×C22⋊C4—21no—
87C3×C42—21no—
88(C2×D3)⋊C4—21no—
89(C2×D3)⋊C4—21no—
90(C2×D3)⋊C4—22no—
91(C2×D3)⋊C4—22no—
92(C2×D3)⋊C4—21no—
93(C2×D3)⋊C4—21no—
94C4⋊Dic3—21no—
95C4⋊Dic3—21no—
96C4⋊Dic3—22no—
97C4⋊Dic3—22no—
98#208—22no—
99C2×C3⋊C8C2312yes—
100Dic12—24no—
101#204—24no—
102ΓL(1,𝔽25)—24no—
103D3×C8—24no—
104C2×#41—31no—
105C22×Q8—31no—
106C23×C4—31no—
107#137—64no—
108C2×Q16—32no—
109C2×SD16—32no—
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#117C2×D311yescharacteristic
128C2×C4⋊C4—61no—
129C2×C4⋊C4C2×D311yescharacteristic
130C2×C22⋊C4—61no—
131#107C2×D312yescomplemented
132#106C2×D312yescomplemented
133#103—32no—
134#103—32no—
135#103—32no—
136#102—32no—
137C23×C3C2×D411yescharacteristic
138C22×D3—22no—
139C22×D3—22no—
140C22×D3—21no—
141C22×D3—41no—
142C22×D3C2×D411yescharacteristic
143C22×D3C2×D411yescharacteristic
144C2×C3×C4—22no—
145C2×C3×C4—21no—
146C2×C3×C4—21no—
147C2×C3×C4—42no—
148C2×C3×C4C2×D412yes—
149C2×C3×C4C2×D411yescharacteristic
150C2×C3×C4#4112yes—
151C2×C3×C4C2411yescharacteristic
152#67—42no—
153C2×Dic3—22no—
154C2×Dic3—22no—
155C2×Dic3—21no—
156C2×Dic3—42no—
157C2×Dic3C2×D411yescharacteristic
158C2×Dic3C2×D411yescharacteristic
159D12—21no—
160D12—21no—
161D12—42no—
162D12—42no—
163C4×D3—22no—
164C4×D3—22no—
165C4×D3—21no—
166C4×D3—21no—
167C4×D3—21no—
168C4×D3—21no—
169C4×D3—42no—
170Dic6—22no—
171Dic6—21no—
172Dic6—22no—
173Dic6—21no—
174Dic6—42no—
175C3×C8C2×D414yes—
176C3⋊C8—22no—
177C24—31no—
178#41—62no—
179#41—62no—
180C2×Q8—32no—
181C2×Q8—31no—
182C2×Q8—61no—
183C2×Q8—62no—
184C2×Q8—62no—
185C2×D4—31no—
186C2×D4—61no—
187C2×D4—61no—
188C2×D4—61no—
189C22×C4—21no—
190C22×C4—31no—
191C22×C4—31no—
192C22×C4—31no—
193C22×C4—31no—
194C22×C4—31no—
195C22×C4—62no—
196C22×C4—62no—
197C22×C4—61no—
198C22×C4—61no—
199C22×C4C22×D311yescharacteristic
200Q16—64no—
201SD16—64no—
202M16—62no—
203M16—64no—
204M16D1214yescomplemented
205C2×C8—32no—
206C2×C8—64no—
207C2×C8C22×D312yes—
208C4⋊C4—21no—
209C4⋊C4—62no—
210C4⋊C4—62no—
211C4⋊C4—61no—
212C4⋊C4—61no—
213C4⋊C4C22×D312yes—
214C4⋊C4C22×D311yescharacteristic
215C4⋊C4C22×D311yescharacteristic
216C22⋊C4—21no—
217C22⋊C4—61no—
218C22⋊C4—61no—
219C22⋊C4—62no—
220C22⋊C4—62no—
221C22⋊C4—61no—
222C22⋊C4—61no—
223C42—21no—
224C22×C3—21no—
225C22×C3—21no—
226C22×C3#12112yes—
227C22×C3C22×D411yescharacteristic
228C2×D3—21no—
229C2×D3—21no—
230C2×D3—21no—
231C2×D3—21no—
232C2×D3—21no—
233C2×D3—44no—
234C2×D3—41no—
235C2×D3—41no—
236C2×D3—41no—
237C2×D3#13712yescomplemented
238C3×C4—41no—
239C3×C4—41no—
240C3×C4—42no—
241C3×C4#12112yes—
242C3×C4C22×D411yescharacteristic
243C3×C4C2×#4111yescharacteristic
244Dic3—21no—
245Dic3—21no—
246Dic3—42no—
247Dic3—41no—
248C23—31no—
249C23—31no—
250C23—62no—
251C23—62no—
252C23—61no—
253C23—121no—
254C23D3×D411yescharacteristic
255Q8—62no—
256Q8—61no—
257Q8—62no—
258Q8—61no—
259Q8—122no—
260D4—61no—
261D4—61no—
262D4—122no—
263D4—122no—
264D4—122no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—31no—
269C2×C4—31no—
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—62no—
280C2×C4—122no—
281C2×C4—122no—
282C2×C4C2×D1212yes—
283C2×C4D3×D411yescharacteristic
284C2×C4#23912yes—
285C2×C4C23×D311yescharacteristic
286C8—62no—
287C8C2×D1214yes—
288C2×C3—22no—
289C2×C3C2×#12111yescharacteristic
290C2×C3C2×#13711yescharacteristic
291C2×C3#57511yescharacteristic
292D3—22no—
293D3—41no—
294D3—81no—
295C22—21no—
296C22—21no—
297C22—32no—
298C22—61no—
299C22—61no—
300C22—61no—
301C22—61no—
302C22—61no—
303C22—124no—
304C22—121no—
305C22—121no—
306C22—121no—
307C22#89512yes—
308C22C2×D3×D411yescharacteristic
309C4—42no—
310C4—41no—
311C4—41no—
312C4—61no—
313C4—61no—
314C4—121no—
315C4—122no—
316C4#89512yes—
317C4C22×D1211yescharacteristic
318C4C2×#23911yescharacteristic
319C3#305611yescharacteristic, complemented, Sylow
320C2—22no—
321C2—62no—
322C2—121no—
323C2—241no—
324C2C2×#89511yescharacteristic
325C2#577311yescharacteristic
326C2D3×#13711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#428C2
socleC22×C3C22×D4
prime radical#5524C2
hypercenter#428D3
hypocenterC3#3056
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. C2×#909C2semidirect
2. C2×D3×M16C2semidirect
4. C2×#897C2semidirect
5. C3×#428C2semidirect
6. #5004C2semidirect
7. #4966C2semidirect
8. #4887C2semidirect
9. #4886C2semidirect
10. #4867C2semidirect
11. #4864C2semidirect
24. C2×Dic12C22semidirect
25. C2×#204C22semidirect
40. C3×#107C22semidirect
41. C3×#106C22semidirect
42. #810C22semidirect
43. #810C22semidirect
44. #810C22semidirect
45. #809C22semidirect
54. #428D3semidirect
80. C3×M16D4semidirect
131. #107C2×D3semidirect
132. #106C2×D3semidirect
204. M16D12semidirect
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×#9092
C2×D3×M162
#55242
C2×#8972
C3×#4282
#50042
#49662
#48872
#48862
#48672
#48642
#30563