Finite Group Databasebeta

← #79211

Subgroups — #79211

Order 384.

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

normal subgroups only

This group has 548 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×#908C211yesmaximal, characteristic, complemented
2C2×D3×M16C211yesmaximal, characteristic, complemented
3#5524C211yesmaximal, characteristic, complemented
4C2×#895C211yesmaximal, characteristic, complemented
5C3×#428C211yesmaximal, characteristic, complemented
6#5001C211yesmaximal, characteristic, complemented
7#4966C211yesmaximal, characteristic, complemented
8#4886C212yesmaximal, complemented
9#4883C212yesmaximal, complemented
10#4865C212yesmaximal, complemented
11#4864C212yesmaximal, complemented
12#3055—31nomaximal, Sylow
13C2×#235C2211yescharacteristic
14C22×D12—21no—
15C22×D12C2211yescharacteristic
16C22×C4×D3C2211yescharacteristic
17C2×C3×M16C2211yescharacteristic
18C3×#117C2211yescharacteristic
19C2×C3×C4⋊C4C2211yescharacteristic
20C2×(C2×D3)⋊C4—21no—
21C2×#208C2211yescharacteristic
22#908—24no—
23D3×M16—24no—
24C2×D24C2212yescomplemented
25C2×#204C2212yescomplemented
26C2×ΓL(1,𝔽25)C2212yescomplemented
27C2×D3×C8C2212yescomplemented
28#897C2211yescharacteristic, complemented
29#897C2211yescharacteristic, complemented
30#895—21no—
31#895—21no—
32#895C2212yes—
33#895C2212yes—
34#895C2211yescharacteristic, complemented
35#895C2211yescharacteristic, complemented
36#886—21no—
37#882—21no—
38C4×D12—21no—
39C4×D12—21no—
40C3×#107C2212yescomplemented
41C3×#106C2212yescomplemented
42#810C2212yescomplemented
43#809C2212yescomplemented
44#809C2212yescomplemented
45#809C2212yescomplemented
46C2×AGL(1,ℤ/8)—31no—
47C22×M16—31no—
48#525—31no—
49C2×#121—31no—
50#469—32no—
51#468—32no—
52#466—32no—
53#465—32no—
54#428D311yescharacteristic, complemented
55#418—31no—
56#417—31no—
57C23×D3—21no—
58C23×D3D411yescharacteristic
59C22×C3×C4—21no—
60C22×C3×C4C2311yescharacteristic
61C2×#67—21no—
62C22×Dic3D411yescharacteristic
63#235—22no—
64#235—22no—
65C2×D12—22no—
66C2×D12—22no—
67C2×D12—21no—
68C2×D12—21no—
69C2×D12—22no—
70C2×D12—22no—
71C2×D12—21no—
72C2×D12—44no—
73C2×D12C2312yes—
74C2×D12C2311yescharacteristic
75C2×C4×D3—22no—
76C2×C4×D3—22no—
77C2×C4×D3—21no—
78C2×C4×D3D411yescharacteristic
79C2×C4×D3D411yescharacteristic
80C2×C4×D3C2311yescharacteristic
81C2×C4×D3C2311yescharacteristic
82C2×Dic6C2311yescharacteristic
83C3×M16D414yescomplemented
84C2×C3×C8C2312yes—
85C3×C4⋊C4—21no—
86C3×C4⋊C4C2311yescharacteristic
87C3×C4⋊C4C2311yescharacteristic
88C3×C4⋊C4C2312yes—
89C3×C22⋊C4—21no—
90C3×C42—21no—
91(C2×D3)⋊C4—21no—
92(C2×D3)⋊C4—21no—
93(C2×D3)⋊C4—21no—
94(C2×D3)⋊C4—21no—
95(C2×D3)⋊C4—22no—
96(C2×D3)⋊C4—22no—
97C4⋊Dic3—21no—
98C4⋊Dic3—21no—
99#208—22no—
100C2×C3⋊C8C2312yes—
101D24—24no—
102#204—24no—
103ΓL(1,𝔽25)—24no—
104D3×C8—24no—
105C2×#41—31no—
106C22×D4—31no—
107C22×D4—61no—
108C23×C4—31no—
109AGL(1,ℤ/8)—64no—
110C2×SD16—32no—
111C2×D8—32no—
112C2×M16—31no—
113C2×M16—32no—
114C2×M16—64no—
115C2×M16C2×D311yescharacteristic
116C22×C8—32no—
117#123—61no—
118C22⋊Q8—31no—
119C22⋊Q8—31no—
120#121—32no—
121#121—32no—
122#121—31no—
123#121—31no—
124#121—61no—
125#121—61no—
126#120—61no—
127C4×D4—61no—
128C4×D4—61no—
129#117C2×D311yescharacteristic
130C2×C4⋊C4C2×D311yescharacteristic
131C2×C22⋊C4—61no—
132#107C2×D312yescomplemented
133#106C2×D312yescomplemented
134#103—32no—
135#102—32no—
136#102—32no—
137#102—32no—
138C23×C3C2×D411yescharacteristic
139C22×D3—22no—
140C22×D3—22no—
141C22×D3—21no—
142C22×D3—22no—
143C22×D3—41no—
144C22×D3—42no—
145C22×D3—42no—
146C22×D3—42no—
147C22×D3C2×D411yescharacteristic
148C22×D3C2×D411yescharacteristic
149C2×C3×C4—21no—
150C2×C3×C4—21no—
151C2×C3×C4—22no—
152C2×C3×C4—42no—
153C2×C3×C4C2×D412yes—
154C2×C3×C4C2×D411yescharacteristic
155C2×C3×C4#4112yes—
156C2×C3×C4C2411yescharacteristic
157#67—42no—
158C2×Dic3—22no—
159C2×Dic3—21no—
160C2×Dic3C2×D411yescharacteristic
161C2×Dic3C2×D411yescharacteristic
162D12—22no—
163D12—21no—
164D12—22no—
165D12—21no—
166D12—42no—
167D12—44no—
168D12—42no—
169D12—44no—
170D12—42no—
171C4×D3—22no—
172C4×D3—22no—
173C4×D3—21no—
174C4×D3—21no—
175C4×D3—21no—
176C4×D3—21no—
177C4×D3—42no—
178Dic6—21no—
179Dic6—21no—
180C3×C8C2×D414yescomplemented
181C3⋊C8—22no—
182C24—31no—
183C24—61no—
184#41—62no—
185#41—62no—
186C2×Q8—31no—
187C2×D4—31no—
188C2×D4—32no—
189C2×D4—61no—
190C2×D4—61no—
191C2×D4—62no—
192C2×D4—62no—
193C2×D4—61no—
194C2×D4—62no—
195C2×D4—61no—
196C2×D4—62no—
197C2×D4—124no—
198C22×C4—21no—
199C22×C4—31no—
200C22×C4—31no—
201C22×C4—31no—
202C22×C4—31no—
203C22×C4—31no—
204C22×C4—62no—
205C22×C4—62no—
206C22×C4—61no—
207C22×C4C22×D311yescharacteristic
208SD16—64no—
209D8—64no—
210M16—62no—
211M16—64no—
212M16D1214yescomplemented
213C2×C8—32no—
214C2×C8—64no—
215C2×C8C22×D312yes—
216C4⋊C4—21no—
217C4⋊C4—61no—
218C4⋊C4—61no—
219C4⋊C4C22×D311yescharacteristic
220C4⋊C4C22×D311yescharacteristic
221C4⋊C4C22×D312yes—
222C22⋊C4—21no—
223C22⋊C4—62no—
224C22⋊C4—62no—
225C22⋊C4—61no—
226C22⋊C4—61no—
227C22⋊C4—61no—
228C22⋊C4—61no—
229C42—21no—
230C22×C3—21no—
231C22×C3—21no—
232C22×C3#12112yes—
233C22×C3C22×D411yescharacteristic
234C2×D3—21no—
235C2×D3—21no—
236C2×D3—21no—
237C2×D3—21no—
238C2×D3—21no—
239C2×D3—44no—
240C2×D3—41no—
241C2×D3—42no—
242C2×D3—42no—
243C2×D3—41no—
244C2×D3—41no—
245C2×D3—42no—
246C2×D3—84no—
247C2×D3AGL(1,ℤ/8)12yescomplemented
248C3×C4—42no—
249C3×C4—41no—
250C3×C4—41no—
251C3×C4#12112yes—
252C3×C4C22×D411yescharacteristic
253C3×C4C2×#4111yescharacteristic
254Dic3—21no—
255Dic3—21no—
256Dic3—41no—
257C23—31no—
258C23—31no—
259C23—62no—
260C23—62no—
261C23—61no—
262C23—62no—
263C23—121no—
264C23—122no—
265C23—122no—
266C23—122no—
267C23D3×D411yescharacteristic
268Q8—61no—
269Q8—61no—
270D4—61no—
271D4—62no—
272D4—62no—
273D4—61no—
274D4—124no—
275D4—122no—
276D4—124no—
277D4—122no—
278D4—122no—
279D4—122no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—31no—
284C2×C4—31no—
285C2×C4—42no—
286C2×C4—62no—
287C2×C4—62no—
288C2×C4—62no—
289C2×C4—61no—
290C2×C4—61no—
291C2×C4—61no—
292C2×C4—61no—
293C2×C4—61no—
294C2×C4—122no—
295C2×C4C2×D1212yes—
296C2×C4D3×D411yescharacteristic
297C2×C4#23912yes—
298C2×C4C23×D311yescharacteristic
299C8—62no—
300C8C2×D1214yescomplemented
301C2×C3—22no—
302C2×C3C2×#12111yescharacteristic
303C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
304C2×C3#57511yescharacteristic
305D3—22no—
306D3—41no—
307D3—81no—
308D3—82no—
309C22—21no—
310C22—21no—
311C22—32no—
312C22—61no—
313C22—61no—
314C22—61no—
315C22—61no—
316C22—61no—
317C22—124no—
318C22—121no—
319C22—122no—
320C22—122no—
321C22—121no—
322C22—121no—
323C22—122no—
324C22—244no—
325C22#89512yes—
326C22C2×D3×D411yescharacteristic
327C4—42no—
328C4—41no—
329C4—41no—
330C4—61no—
331C4—61no—
332C4—121no—
333C4#89512yes—
334C4C22×D1211yescharacteristic
335C4C2×#23911yescharacteristic
336C3#305511yescharacteristic, complemented, Sylow
337C2—22no—
338C2—62no—
339C2—121no—
340C2—242no—
341C2—241no—
342C2C2×#89511yescharacteristic
343C2D3×AGL(1,ℤ/8)11yescharacteristic
344C2#577711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#428C2
socleC22×C3C22×D4
hypercenter#428D3
hypocenterC3#3055
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×#908C2semidirect
2. C2×D3×M16C2semidirect
3. #5524C2semidirect
4. C2×#895C2semidirect
5. C3×#428C2semidirect
6. #5001C2semidirect
7. #4966C2semidirect
8. #4886C2semidirect
9. #4883C2semidirect
10. #4865C2semidirect
11. #4864C2semidirect
24. C2×D24C22semidirect
25. C2×#204C22semidirect
26. C2×ΓL(1,𝔽25)C22semidirect
27. C2×D3×C8C22semidirect
28. #897C22semidirect
29. #897C22semidirect
34. #895C22semidirect
35. #895C22semidirect
40. C3×#107C22semidirect
41. C3×#106C22semidirect
42. #810C22semidirect
43. #809C22semidirect
44. #809C22semidirect
45. #809C22semidirect
54. #428D3semidirect
83. C3×M16D4semidirect
132. #107C2×D3semidirect
133. #106C2×D3semidirect
180. C3×C8C2×D4semidirect
212. M16D12semidirect
247. C2×D3AGL(1,ℤ/8)semidirect
300. C8C2×D12semidirect
336. C3#3055semidirect

Maximal subgroups

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

subgroupindex
C2×#9082
C2×D3×M162
#55242
C2×#8952
C3×#4282
#50012
#49662
#48862
#48832
#48652
#48642
#30553