Finite Group Databasebeta

← #79216

Subgroups — #79216

Order 384.

414 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 125 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#5524C211yesmaximal, characteristic
2C2×#892C211yesmaximal, characteristic, complemented
3C3×#428C211yesmaximal, characteristic, complemented
4#5133C211yesmaximal, characteristic, complemented
5#5126C211yesmaximal, characteristic, complemented
6#4995C211yesmaximal, characteristic, complemented
7#4966C211yesmaximal, characteristic, complemented
8#4888C212yesmaximal, complemented
9#4885C212yesmaximal, complemented
10#4868C212yesmaximal, complemented
11#4866C212yesmaximal, complemented
12#3093—31nomaximal, Sylow
13C2×#235C2211yescharacteristic
14C22×C4×D3C2211yescharacteristic
15C2×C3×M16C2211yescharacteristic
16C3×#117C2211yescharacteristic
17C2×C3×C4⋊C4C2211yescharacteristic
18C2×(C2×D3)⋊C4—21no—
19C2×C4⋊Dic3C2211yescharacteristic
20C2×C4×Dic3—21no—
21C2×#208C2211yescharacteristic
22#897C2211yescharacteristic
23#897C2211yescharacteristic
24#895C2211yescharacteristic
25#895C2211yescharacteristic
26#892—21no—
27#892—21no—
28#892C2212yes—
29#892C2212yes—
30#886—21no—
31#882—21no—
32C4×D12—21no—
33C4×D12—21no—
34C3×#107C2212yescomplemented
35C3×#106C2212yescomplemented
36#821C2212yescomplemented
37(C2×D3)⋊C8C2212yes—
38(C2×D3)⋊C8C2212yes—
39#816C2212yescomplemented
40#810C2212yescomplemented
41#809C2212yescomplemented
42#808C2212yescomplemented
43#807C2212yescomplemented
44#525—31no—
45C2×#117—31no—
46#485—32no—
47#485—32no—
48#482—32no—
49#482—32no—
50#428—31no—
51#428D311yescharacteristic, complemented
52#417—31no—
53#417—31no—
54#407—31no—
55C23×D3D411yescharacteristic
56C22×C3×C4—21no—
57C22×C3×C4C2311yescharacteristic
58C2×#67—21no—
59C22×Dic3—21no—
60C22×Dic3D411yescharacteristic
61#235—22no—
62#235—22no—
63C2×D12—21no—
64C2×D12—21no—
65C2×D12C2311yescharacteristic
66C2×C4×D3—22no—
67C2×C4×D3—22no—
68C2×C4×D3—21no—
69C2×C4×D3D411yescharacteristic
70C2×C4×D3D411yescharacteristic
71C2×C4×D3C2311yescharacteristic
72C2×C4×D3C2311yescharacteristic
73C2×Dic6C2311yescharacteristic
74C3×M16—22no—
75C2×C3×C8C2312yes—
76C3×C4⋊C4—21no—
77C3×C4⋊C4C2311yescharacteristic
78C3×C4⋊C4C2311yescharacteristic
79C3×C4⋊C4C2312yes—
80C3×C22⋊C4—21no—
81C3×C42—21no—
82(C2×D3)⋊C4—21no—
83(C2×D3)⋊C4—21no—
84(C2×D3)⋊C4—22no—
85(C2×D3)⋊C4—22no—
86(C2×D3)⋊C4—21no—
87(C2×D3)⋊C4—21no—
88C4⋊Dic3—21no—
89C4⋊Dic3—21no—
90C4⋊Dic3—21no—
91C4⋊Dic3C2312yes—
92C4×Dic3—22no—
93C4×Dic3—22no—
94#208—22no—
95C2×C3⋊C8C2312yes—
96C2×#41—31no—
97C23×C4—31no—
98C2×M16—31no—
99C2×M16C2×D311yescharacteristic
100#123—61no—
101C22⋊Q8—31no—
102C22⋊Q8—31no—
103#121—31no—
104#121—31no—
105#120—61no—
106C4×D4—61no—
107C4×D4—61no—
108#117—32no—
109#117—32no—
110#117—61no—
111#117—61no—
112#117C2×D311yescharacteristic
113C2×C4⋊C4—31no—
114C2×C4⋊C4C2×D311yescharacteristic
115C2×C22⋊C4—61no—
116C2×C42—61no—
117#107—32no—
118#107C2×D312yescomplemented
119#106—32no—
120#106C2×D312yescomplemented
121#103—32no—
122#103—32no—
123#102—32no—
124#102—32no—
125C22⋊C8—32no—
126C22⋊C8—32no—
127C23×C3C2×D411yescharacteristic
128C22×D3—22no—
129C22×D3—21no—
130C22×D3—41no—
131C22×D3—41no—
132C22×D3C2×D411yescharacteristic
133C22×D3C2×D411yescharacteristic
134C2×C3×C4—22no—
135C2×C3×C4—21no—
136C2×C3×C4—21no—
137C2×C3×C4—42no—
138C2×C3×C4C2×D411yescharacteristic
139C2×C3×C4#4112yes—
140C2×C3×C4#4112yes—
141C2×C3×C4C2411yescharacteristic
142#67—42no—
143C2×Dic3—22no—
144C2×Dic3—22no—
145C2×Dic3—21no—
146C2×Dic3—42no—
147C2×Dic3C2×D411yescharacteristic
148C2×Dic3C2×D411yescharacteristic
149D12—21no—
150D12—21no—
151D12—42no—
152D12—42no—
153C4×D3—22no—
154C4×D3—22no—
155C4×D3—21no—
156C4×D3—21no—
157C4×D3—21no—
158C4×D3—21no—
159C4×D3—42no—
160Dic6—21no—
161Dic6—21no—
162C3×C8—22no—
163C3⋊C8—22no—
164C24—31no—
165#41—62no—
166#41—62no—
167C2×Q8—31no—
168C2×D4—31no—
169C2×D4—61no—
170C2×D4—61no—
171C2×D4—61no—
172C22×C4—21no—
173C22×C4—31no—
174C22×C4—31no—
175C22×C4—31no—
176C22×C4—31no—
177C22×C4—31no—
178C22×C4—62no—
179C22×C4—62no—
180C22×C4—61no—
181C22×C4—61no—
182C22×C4C22×D311yescharacteristic
183M16—22no—
184M16—62no—
185C2×C8—32no—
186C2×C8C22×D312yes—
187C4⋊C4—21no—
188C4⋊C4—32no—
189C4⋊C4—61no—
190C4⋊C4—61no—
191C4⋊C4—61no—
192C4⋊C4C22×D312yes—
193C4⋊C4C22×D311yescharacteristic
194C4⋊C4C22×D311yescharacteristic
195C22⋊C4—21no—
196C22⋊C4—61no—
197C22⋊C4—61no—
198C22⋊C4—62no—
199C22⋊C4—62no—
200C22⋊C4—61no—
201C22⋊C4—61no—
202C42—21no—
203C42—62no—
204C42—62no—
205C22×C3—21no—
206C22×C3—21no—
207C22×C3#12312yes—
208C22×C3C22×D411yescharacteristic
209C2×D3—21no—
210C2×D3—21no—
211C2×D3—21no—
212C2×D3—21no—
213C2×D3—21no—
214C2×D3—21no—
215C2×D3—44no—
216C2×D3—41no—
217C2×D3—41no—
218C2×D3—41no—
219C3×C4—41no—
220C3×C4—41no—
221C3×C4—42no—
222C3×C4#12312yes—
223C3×C4C2×#4111yescharacteristic
224C3×C4C2×#4111yescharacteristic
225Dic3—21no—
226Dic3—21no—
227Dic3—41no—
228Dic3—42no—
229C23—31no—
230C23—31no—
231C23—62no—
232C23—61no—
233C23—121no—
234C23—121no—
235C23D3×D411yescharacteristic
236Q8—61no—
237Q8—61no—
238D4—61no—
239D4—61no—
240D4—122no—
241D4—122no—
242D4—122no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—31no—
247C2×C4—31no—
248C2×C4—42no—
249C2×C4—62no—
250C2×C4—62no—
251C2×C4—62no—
252C2×C4—61no—
253C2×C4—61no—
254C2×C4—61no—
255C2×C4—61no—
256C2×C4—62no—
257C2×C4—61no—
258C2×C4—122no—
259C2×C4—122no—
260C2×C4#23512yes—
261C2×C4D3×D411yescharacteristic
262C2×C4#23912yes—
263C2×C4C23×D311yescharacteristic
264C8—22no—
265C8—62no—
266C2×C3—22no—
267C2×C3C2×#12311yescharacteristic
268C2×C3#57511yescharacteristic
269C2×C3#57511yescharacteristic
270D3—41no—
271D3—41no—
272D3—81no—
273C22—21no—
274C22—21no—
275C22—61no—
276C22—61no—
277C22—61no—
278C22—61no—
279C22—61no—
280C22—61no—
281C22—124no—
282C22—121no—
283C22—121no—
284C22—121no—
285C22#89412yes—
286C22C2×D3×D411yescharacteristic
287C4—42no—
288C4—41no—
289C4—41no—
290C4—61no—
291C4—61no—
292C4—121no—
293C4—122no—
294C4#89412yes—
295C4C2×#23511yescharacteristic
296C4C2×#23911yescharacteristic
297C3#309311yescharacteristic, complemented, Sylow
298C2—22no—
299C2—121no—
300C2—121no—
301C2—241no—
302C2C2×#89411yescharacteristic
303C2#577311yescharacteristic
304C2#577711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#428C2
socleC22×C3C22×D4
prime radical#5524C2
hypercenter#428D3
hypocenterC3#3093
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
2. C2×#892C2semidirect
3. C3×#428C2semidirect
4. #5133C2semidirect
5. #5126C2semidirect
6. #4995C2semidirect
7. #4966C2semidirect
8. #4888C2semidirect
9. #4885C2semidirect
10. #4868C2semidirect
11. #4866C2semidirect
34. C3×#107C22semidirect
35. C3×#106C22semidirect
36. #821C22semidirect
39. #816C22semidirect
40. #810C22semidirect
41. #809C22semidirect
42. #808C22semidirect
43. #807C22semidirect
51. #428D3semidirect
118. #107C2×D3semidirect
120. #106C2×D3semidirect
297. C3#3093semidirect

Maximal subgroups

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

subgroupindex
#55242
C2×#8922
C3×#4282
#51332
#51262
#49952
#49662
#48882
#48852
#48682
#48662
#30933