Finite Group Databasebeta

← #81191

Subgroups — #81191

Order 384.

462 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5695C211yesmaximal, characteristic, complemented
2D4×D12C211yesmaximal, characteristic, complemented
3C3×#478C211yesmaximal, characteristic, complemented
4#5067C211yesmaximal, characteristic, complemented
5#5014C211yesmaximal, characteristic, complemented
6#4885C211yesmaximal, characteristic, complemented
7#4883C211yesmaximal, characteristic, complemented
8#4865C211yesmaximal, characteristic, complemented
9#4863C211yesmaximal, characteristic, complemented
10#4843C211yesmaximal, characteristic, complemented
11#4836C211yesmaximal, characteristic, complemented
12#4773C211yesmaximal, characteristic, complemented
13#4773C211yesmaximal, characteristic, complemented
14#4770C211yesmaximal, characteristic, complemented
15D3×#102C211yesmaximal, characteristic, complemented
16#3318—31nomaximal, Sylow
17C2×D3×D4C2211yescharacteristic
18C2×D3×D4C2211yescharacteristic
19C22×D12—21no—
20C3×#125C2211yescharacteristic, complemented
21C3×C4×D4C2211yescharacteristic
22C2×#213C2211yescharacteristic
23#930—21no—
24C2×D24C2211yescharacteristic, complemented
25C2×D24C2211yescharacteristic, complemented
26C2×ΓL(1,𝔽25)C2211yescharacteristic, complemented
27C2×D3×C8C2211yescharacteristic, complemented
28#898—21no—
29#895—21no—
30#895C2211yescharacteristic
31#895C2211yescharacteristic
32#895C2211yescharacteristic, complemented
33#895C2211yescharacteristic, complemented
34#894—21no—
35#893—21no—
36#892C2211yescharacteristic, complemented
37D3×C4⋊C4C2211yescharacteristic, complemented
38#882—21no—
39#882—21no—
40#874C2211yescharacteristic
41C4×D12C2211yescharacteristic
42C4×D12C2211yescharacteristic, complemented
43C3×#107C2211yescharacteristic, complemented
44C3×#106C2211yescharacteristic, complemented
45C3×C4⋊C8C2211yescharacteristic, complemented
46C3×#102C2211yescharacteristic
47C3×#102C2211yescharacteristic
48#832C2211yescharacteristic
49#821C2211yescharacteristic, complemented
50#821C2211yescharacteristic, complemented
51(C2×D3)⋊C8C2211yescharacteristic, complemented
52(C2×D3)⋊C8C2211yescharacteristic, complemented
53#809C2211yescharacteristic, complemented
54#809C2211yescharacteristic, complemented
55#809C2211yescharacteristic, complemented
56#809C2211yescharacteristic, complemented
57#809C2211yescharacteristic, complemented
58#808C2211yescharacteristic, complemented
59#803C2211yescharacteristic, complemented
60#553—31no—
61D42—31no—
62#490—31no—
63#482—31no—
64#481—31no—
65#478D311yescharacteristic, complemented
66#469—31no—
67#466—31no—
68#459—31no—
69#447—31no—
70#447—31no—
71#442—31no—
72#435—31no—
73#418—31no—
74C2×#102—31no—
75C23×D3—21no—
76C23×D3—21no—
77C2×C3×D4C2311yescharacteristic
78C22×C3×C4—21no—
79C2×#67—21no—
80C2×#67—21no—
81D3×D4—22no—
82D3×D4—22no—
83D3×D4—41no—
84D3×D4—41no—
85C2×D12—21no—
86C2×D12—21no—
87C2×D12—21no—
88C2×D12—21no—
89C2×D12—21no—
90C2×D12—21no—
91C2×D12—44no—
92C2×D12D411yescharacteristic
93C2×D12C2311yescharacteristic
94C2×D12C2311yescharacteristic
95C2×D12C2311yescharacteristic
96C2×C4×D3—21no—
97C2×C4×D3—21no—
98C2×C4×D3C2311yescharacteristic
99C2×C4×D3C2311yescharacteristic
100C2×C3×C8C2311yescharacteristic
101C2×C3×C8C2311yescharacteristic
102C3×C4⋊C4—21no—
103C3×C4⋊C4—21no—
104C3×C4⋊C4C2311yescharacteristic
105C3×C4⋊C4C2311yescharacteristic
106C3×C4⋊C4C2311yescharacteristic
107C3×C22⋊C4—21no—
108C3×C42C2311yescharacteristic
109#213—22no—
110(C2×D3)⋊C4—21no—
111(C2×D3)⋊C4—21no—
112(C2×D3)⋊C4—21no—
113(C2×D3)⋊C4—21no—
114(C2×D3)⋊C4—21no—
115(C2×D3)⋊C4—21no—
116(C2×D3)⋊C4D411yescharacteristic
117(C2×D3)⋊C4D411yescharacteristic
118C4⋊Dic3D411yescharacteristic
119C4⋊Dic3C2311yescharacteristic
120Dic3⋊C4—21no—
121C4×Dic3—21no—
122C2×C3⋊C8C2311yescharacteristic
123C2×C3⋊C8C2311yescharacteristic
124D24—41no—
125D24—41no—
126ΓL(1,𝔽25)—41no—
127D3×C8—41no—
128C22×D4—31no—
129C22×D4—31no—
130C22×D4—61no—
131C2×D8—31no—
132C2×D8—31no—
133C2×D8—31no—
134C2×M16—31no—
135C22×C8—31no—
136#127—31no—
137#126—61no—
138#125C2×D311yescharacteristic, complemented
139#123—61no—
140#121—31no—
141#121—31no—
142#121—31no—
143#121—31no—
144#121—61no—
145#121—61no—
146#120—61no—
147#120—61no—
148C4×D4—31no—
149C4×D4—31no—
150C4×D4—61no—
151C4×D4C2×D311yescharacteristic
152#117—31no—
153C2×C4⋊C4—31no—
154#107C2×D311yescharacteristic, complemented
155#106—31no—
156#106C2×D311yescharacteristic, complemented
157C4⋊C8C2×D311yescharacteristic, complemented
158#102—31no—
159#102—31no—
160#102—31no—
161#102—31no—
162#102—31no—
163#102—31no—
164#102—31no—
165#102—31no—
166#102C2×D311yescharacteristic
167#102C2×D311yescharacteristic
168C22⋊C8—31no—
169C22⋊C8—31no—
170#97—31no—
171C23×C3—21no—
172C22×D3—21no—
173C22×D3—21no—
174C22×D3—21no—
175C22×D3—42no—
176C22×D3—42no—
177C22×D3—42no—
178C22×D3—42no—
179C22×D3—81no—
180C22×D3—81no—
181C22×D3C2×D411yescharacteristic
182C22×D3C2×D411yescharacteristic
183C3×D4—21no—
184C3×D4#4112yes—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—42no—
189C2×C3×C4C2×D411yescharacteristic
190C2×C3×C4C2×D411yescharacteristic
191C2×C3×C4C2411yescharacteristic
192#67—42no—
193#67—42no—
194C2×Dic3—21no—
195C2×Dic3C2×D411yescharacteristic
196C2×Dic3C2×D411yescharacteristic
197D12—22no—
198D12—21no—
199D12—21no—
200D12—21no—
201D12—21no—
202D12—21no—
203D12—21no—
204D12—42no—
205D12—42no—
206D12—42no—
207D12—42no—
208D12—81no—
209D12—81no—
210C4×D3—21no—
211C4×D3—21no—
212C4×D3—21no—
213C4×D3—21no—
214C4×D3—81no—
215C4×D3—81no—
216C3×C8—21no—
217C3×C8—21no—
218C3⋊C8—21no—
219C3⋊C8—21no—
220C24—61no—
221C24—61no—
222C2×D4—31no—
223C2×D4—31no—
224C2×D4—31no—
225C2×D4—31no—
226C2×D4—61no—
227C2×D4—61no—
228C2×D4—61no—
229C2×D4—61no—
230C2×D4—61no—
231C2×D4—61no—
232C2×D4—62no—
233C2×D4—62no—
234C2×D4—61no—
235C2×D4—61no—
236C2×D4—124no—
237C2×D4—121no—
238C2×D4—121no—
239C2×D4C22×D311yescharacteristic
240C22×C4—21no—
241C22×C4—31no—
242C22×C4—31no—
243C22×C4—61no—
244C22×C4—61no—
245D8—62no—
246D8—121no—
247D8—121no—
248M16—121no—
249C2×C8—31no—
250C2×C8—31no—
251C2×C8—121no—
252C2×C8C22×D311yescharacteristic
253C2×C8C22×D311yescharacteristic
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—31no—
257C4⋊C4—31no—
258C4⋊C4—61no—
259C4⋊C4C22×D311yescharacteristic
260C4⋊C4C22×D311yescharacteristic
261C4⋊C4C22×D311yescharacteristic
262C22⋊C4—21no—
263C22⋊C4—31no—
264C22⋊C4—31no—
265C22⋊C4—61no—
266C22⋊C4—61no—
267C22⋊C4—61no—
268C22⋊C4—61no—
269C22⋊C4—61no—
270C22⋊C4—61no—
271C42—61no—
272C42C22×D311yescharacteristic
273C22×C3—22no—
274C22×C3—41no—
275C22×C3—41no—
276C22×C3C22×D411yescharacteristic
277C2×D3—21no—
278C2×D3—21no—
279C2×D3—21no—
280C2×D3—21no—
281C2×D3—21no—
282C2×D3—41no—
283C2×D3—41no—
284C2×D3—41no—
285C2×D3—41no—
286C2×D3—41no—
287C2×D3—41no—
288C2×D3—41no—
289C2×D3—41no—
290C2×D3—41no—
291C2×D3—84no—
292C2×D3—84no—
293C2×D3AGL(1,ℤ/8)12yes—
294C3×C4—21no—
295C3×C4—21no—
296C3×C4—41no—
297C3×C4—41no—
298C3×C4—41no—
299C3×C4C2×#4111yescharacteristic
300C3×C4H(5,𝔽2)11yescharacteristic
301Dic3—21no—
302Dic3—21no—
303Dic3—41no—
304C23—21no—
305C23—31no—
306C23—31no—
307C23—61no—
308C23—61no—
309C23—61no—
310C23—122no—
311C23—122no—
312C23—122no—
313C23—122no—
314C23—241no—
315C23—241no—
316D4—21no—
317D4—62no—
318D4—61no—
319D4—61no—
320D4—61no—
321D4—61no—
322D4—61no—
323D4—61no—
324D4—122no—
325D4—122no—
326D4—122no—
327D4—122no—
328D4—122no—
329D4—122no—
330D4—241no—
331D4—241no—
332D4#23912yes—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—31no—
337C2×C4—31no—
338C2×C4—42no—
339C2×C4—61no—
340C2×C4—61no—
341C2×C4—61no—
342C2×C4—61no—
343C2×C4—61no—
344C2×C4—241no—
345C2×C4—241no—
346C2×C4D3×D411yescharacteristic
347C2×C4D3×D411yescharacteristic
348C2×C4C23×D311yescharacteristic
349C8—21no—
350C8—21no—
351C8—61no—
352C8—61no—
353C2×C3—42no—
354C2×C3#54611yescharacteristic
355C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
356C2×C3#57611yescharacteristic
357D3—22no—
358D3—41no—
359D3—81no—
360D3—81no—
361D3—81no—
362C22—22no—
363C22—32no—
364C22—41no—
365C22—41no—
366C22—61no—
367C22—61no—
368C22—61no—
369C22—61no—
370C22—61no—
371C22—121no—
372C22—121no—
373C22—121no—
374C22—121no—
375C22—121no—
376C22—121no—
377C22—121no—
378C22—121no—
379C22—121no—
380C22—244no—
381C22—244no—
382C22C2×D3×D411yescharacteristic
383C4—21no—
384C4—21no—
385C4—41no—
386C4—41no—
387C4—41no—
388C4—61no—
389C4—61no—
390C4—121no—
391C4C2×#23911yescharacteristic
392C4#100911yescharacteristic
393C3#331811yescharacteristic, complemented, Sylow
394C2—42no—
395C2—62no—
396C2—121no—
397C2—241no—
398C2—241no—
399C2—241no—
400C2#563011yescharacteristic
401C2#576911yescharacteristic
402C2D3×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#478C2
socleC22×C3C22×D4
hypercenter#478D3
hypocenterC3#3318
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #5695C2semidirect
2. D4×D12C2semidirect
3. C3×#478C2semidirect
4. #5067C2semidirect
5. #5014C2semidirect
6. #4885C2semidirect
7. #4883C2semidirect
8. #4865C2semidirect
9. #4863C2semidirect
10. #4843C2semidirect
11. #4836C2semidirect
12. #4773C2semidirect
13. #4773C2semidirect
14. #4770C2semidirect
15. D3×#102C2semidirect
20. C3×#125C22semidirect
24. C2×D24C22semidirect
25. C2×D24C22semidirect
26. C2×ΓL(1,𝔽25)C22semidirect
27. C2×D3×C8C22semidirect
32. #895C22semidirect
33. #895C22semidirect
36. #892C22semidirect
37. D3×C4⋊C4C22semidirect
42. C4×D12C22semidirect
43. C3×#107C22semidirect
44. C3×#106C22semidirect
45. C3×C4⋊C8C22semidirect
49. #821C22semidirect
50. #821C22semidirect
51. (C2×D3)⋊C8C22semidirect
52. (C2×D3)⋊C8C22semidirect
53. #809C22semidirect
54. #809C22semidirect
55. #809C22semidirect
56. #809C22semidirect
57. #809C22semidirect
58. #808C22semidirect
59. #803C22semidirect
65. #478D3semidirect
138. #125C2×D3semidirect
154. #107C2×D3semidirect
156. #106C2×D3semidirect
157. C4⋊C8C2×D3semidirect
393. C3#3318semidirect

Maximal subgroups

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

subgroupindex
#56952
D4×D122
C3×#4782
#50672
#50142
#48852
#48832
#48652
#48632
#48432
#48362
#47732
#47702
D3×#1022
#33183