Finite Group Databasebeta

← #79501

Subgroups — #79501

Order 384.

450 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×D12C211yesmaximal, characteristic
2#5541C211yesmaximal, characteristic, complemented
3C3×#435C211yesmaximal, characteristic, complemented
4#5134C211yesmaximal, characteristic, complemented
5#5131C211yesmaximal, characteristic, complemented
6#5114C211yesmaximal, characteristic, complemented
7C2×#821C211yesmaximal, characteristic, complemented
8#4848C211yesmaximal, characteristic, complemented
9#4847C211yesmaximal, characteristic, complemented
10#4737C211yesmaximal, characteristic, complemented
11#4736C211yesmaximal, characteristic, complemented
12#4731C211yesmaximal, characteristic, complemented
13#4731C211yesmaximal, characteristic, complemented
14#4713C211yesmaximal, characteristic, complemented
15#4710C211yesmaximal, characteristic, complemented
16#3310—31nomaximal, Sylow
17C2×D3×D4—21no—
18C22×D12C2211yescharacteristic
19C22×D12C2211yescharacteristic
20C2×C3×M16C2211yescharacteristic, complemented
21C22×C3×C8C2211yescharacteristic, complemented
22C3×C4×D4C2211yescharacteristic
23#935—21no—
24#930C2211yescharacteristic
25#930C2211yescharacteristic
26#926C2211yescharacteristic, complemented
27C2×C4⋊Dic3C2211yescharacteristic, complemented
28#924C2211yescharacteristic, complemented
29#924C2211yescharacteristic, complemented
30C2×#204C2211yescharacteristic
31C2×#204C2211yescharacteristic, complemented
32C2×#204C2211yescharacteristic, complemented
33#895—21no—
34#889—21no—
35#882—21no—
36#882—21no—
37#879—21no—
38#877—21no—
39#874C2211yescharacteristic
40C4×D12C2211yescharacteristic
41#870C2211yescharacteristic, complemented
42C4×Dic6C2211yescharacteristic, complemented
43C3×C4⋊C8C2211yescharacteristic, complemented
44C3×C22⋊C8C2211yescharacteristic, complemented
45C3×C22⋊C8C2211yescharacteristic, complemented
46C3×#97C2211yescharacteristic, complemented
47#821C2211yescharacteristic
48#821C2211yescharacteristic
49#821C2211yescharacteristic
50#821C2211yescharacteristic, complemented
51#821C2211yescharacteristic, complemented
52#821C2211yescharacteristic, complemented
53#821C2211yescharacteristic, complemented
54#821C2211yescharacteristic, complemented
55#821C2211yescharacteristic, complemented
56#818C2211yescharacteristic, complemented
57#818C2211yescharacteristic, complemented
58#817C2211yescharacteristic, complemented
59#816C2211yescharacteristic, complemented
60#552—31no—
61D42—31no—
62#490—31no—
63#482—31no—
64#480—31no—
65#478—31no—
66#468—31no—
67#465—31no—
68#460—31no—
69#450—31no—
70#450—31no—
71#440—31no—
72#435D311yescharacteristic, complemented
73#418—31no—
74C2×#102—31no—
75C23×D3—21no—
76C23×D3—21no—
77C2×C3×D4D411yescharacteristic
78C22×C3×C4C2311yescharacteristic
79C22×C3×C4C2311yescharacteristic
80C2×#67—21no—
81C2×#67—21no—
82C22×Dic3—21no—
83D3×D4—44no—
84C2×D12—22no—
85C2×D12—22no—
86C2×D12—21no—
87C2×D12—21no—
88C2×D12—21no—
89C2×D12—21no—
90C2×D12—41no—
91C2×D12—41no—
92C2×D12C2311yescharacteristic
93C2×D12C2311yescharacteristic
94C2×D12C2311yescharacteristic
95C2×C4×D3—21no—
96C2×Dic6C2311yescharacteristic
97C3×M16—41no—
98C2×C3×C8—41no—
99C2×C3×C8C2311yescharacteristic
100C2×C3×C8C2311yescharacteristic
101C2×C3×C8C2311yescharacteristic
102C2×C3×C8C2311yescharacteristic
103C3×C4⋊C4D411yescharacteristic
104C3×C22⋊C4D411yescharacteristic
105C3×C22⋊C4D411yescharacteristic
106C3×C42C2311yescharacteristic
107#217—21no—
108#217—21no—
109#217—21no—
110(C2×D3)⋊C4—21no—
111(C2×D3)⋊C4—21no—
112(C2×D3)⋊C4—21no—
113C4⋊Dic3—21no—
114C4⋊Dic3C2311yescharacteristic
115C4⋊Dic3C2311yescharacteristic
116C4⋊Dic3C2311yescharacteristic
117C4⋊Dic3C2311yescharacteristic
118Dic3⋊C4—21no—
119Dic3⋊C4—21no—
120Dic3⋊C4—21no—
121Dic3⋊C4—21no—
122Dic3⋊C4—21no—
123C4×Dic3—21no—
124C4×Dic3—21no—
125#204—22no—
126#204—41no—
127#204—41no—
128C22×D4—31no—
129C22×D4—31no—
130C22×D4—61no—
131C2×SD16—31no—
132C2×SD16—31no—
133C2×SD16—31no—
134C2×M16C2×D311yescharacteristic, complemented
135C22×C8C2×D311yescharacteristic, complemented
136#127—31no—
137#126—61no—
138#125—31no—
139#125—61no—
140#123—61no—
141C22⋊Q8—31no—
142C22⋊Q8—31no—
143#121—31no—
144#121—31no—
145#121—61no—
146#120—61no—
147#120—61no—
148C4×Q8—31no—
149C4×D4—31no—
150C4×D4C2×D311yescharacteristic
151#117—31no—
152#117—61no—
153C2×C4⋊C4—31no—
154#107—31no—
155#107—31no—
156#106—31no—
157C4⋊C8C2×D311yescharacteristic, complemented
158#103—31no—
159#102—31no—
160#102—31no—
161#102—31no—
162#102—31no—
163#102—31no—
164#102—31no—
165#102—31no—
166#102—31no—
167#102—31no—
168C22⋊C8C2×D311yescharacteristic, complemented
169C22⋊C8C2×D311yescharacteristic, complemented
170#97C2×D311yescharacteristic, complemented
171C23×C3C2×D411yescharacteristic
172C23×C3C2×D411yescharacteristic
173C22×D3—21no—
174C22×D3—21no—
175C22×D3—21no—
176C22×D3—42no—
177C22×D3—42no—
178C22×D3—42no—
179C22×D3—42no—
180C22×D3—81no—
181C22×D3—81no—
182C3×D4—22no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4C2×D411yescharacteristic
188C2×C3×C4C2×D411yescharacteristic
189C2×C3×C4C2×D411yescharacteristic
190C2×C3×C4C2×D411yescharacteristic
191C2×C3×C4C2411yescharacteristic
192#67—42no—
193#67—42no—
194C2×Dic3—21no—
195C2×Dic3—21no—
196C2×Dic3—21no—
197C2×Dic3—21no—
198C2×Dic3—21no—
199C2×Dic3—42no—
200D12—21no—
201D12—21no—
202D12—21no—
203D12—21no—
204D12—21no—
205D12—42no—
206D12—42no—
207D12—42no—
208D12—81no—
209D12#4112yes—
210C4×D3—42no—
211Dic6—21no—
212Dic6—21no—
213C3×C8—21no—
214C3×C8—21no—
215C3×C8—21no—
216C3×C8—21no—
217C24—61no—
218C24—61no—
219C2×Q8—31no—
220C2×D4—31no—
221C2×D4—31no—
222C2×D4—31no—
223C2×D4—62no—
224C2×D4—62no—
225C2×D4—61no—
226C2×D4—61no—
227C2×D4—61no—
228C2×D4—61no—
229C2×D4—61no—
230C2×D4—61no—
231C2×D4—121no—
232C2×D4—121no—
233C2×D4—124no—
234C2×D4D1211yescharacteristic
235C22×C4—61no—
236C22×C4—61no—
237C22×C4C22×D311yescharacteristic
238C22×C4C22×D311yescharacteristic
239SD16—62no—
240SD16—121no—
241SD16—121no—
242M16—41no—
243C2×C8—41no—
244C2×C8C22×D311yescharacteristic
245C2×C8C22×D311yescharacteristic
246C2×C8C22×D311yescharacteristic
247C2×C8C22×D311yescharacteristic
248C4⋊C4—31no—
249C4⋊C4—31no—
250C4⋊C4—31no—
251C4⋊C4—31no—
252C4⋊C4—61no—
253C4⋊C4—61no—
254C4⋊C4—61no—
255C4⋊C4—61no—
256C4⋊C4—61no—
257C4⋊C4—61no—
258C4⋊C4D1211yescharacteristic
259C22⋊C4—61no—
260C22⋊C4—61no—
261C22⋊C4—61no—
262C22⋊C4—61no—
263C22⋊C4—61no—
264C22⋊C4—61no—
265C22⋊C4D1211yescharacteristic
266C22⋊C4D1211yescharacteristic
267C42—61no—
268C42—61no—
269C42C22×D311yescharacteristic
270C22×C3—21no—
271C22×C3—21no—
272C22×C3—21no—
273C22×C3—21no—
274C22×C3—21no—
275C22×C3AGL(1,ℤ/8)12yes—
276C22×C3C22×D411yescharacteristic
277C2×D3—22no—
278C2×D3—41no—
279C2×D3—41no—
280C2×D3—41no—
281C2×D3—41no—
282C2×D3—41no—
283C2×D3—41no—
284C2×D3—41no—
285C2×D3—41no—
286C2×D3—84no—
287C2×D3—84no—
288C3×C4—21no—
289C3×C4—21no—
290C3×C4—21no—
291C3×C4—21no—
292C3×C4C2×#4111yescharacteristic
293C3×C4H(5,𝔽2)11yescharacteristic
294Dic3—41no—
295Dic3—41no—
296Dic3—41no—
297Dic3—41no—
298Dic3—41no—
299C23—61no—
300C23—61no—
301C23—61no—
302C23—122no—
303C23—122no—
304C23—122no—
305C23—122no—
306C23—241no—
307C23—241no—
308C23C2×D1211yescharacteristic
309C23C2×D1211yescharacteristic
310Q8—61no—
311Q8—61no—
312D4—22no—
313D4—32no—
314D4—61no—
315D4—61no—
316D4—61no—
317D4—61no—
318D4—61no—
319D4—122no—
320D4—122no—
321D4—122no—
322D4—122no—
323D4—122no—
324D4—241no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—61no—
330C2×C4—61no—
331C2×C4—61no—
332C2×C4—61no—
333C2×C4—61no—
334C2×C4—122no—
335C2×C4—122no—
336C2×C4C2×D1211yescharacteristic
337C2×C4C2×D1211yescharacteristic
338C2×C4C2×D1211yescharacteristic
339C2×C4C2×D1211yescharacteristic
340C2×C4C23×D311yescharacteristic
341C8—21no—
342C8—21no—
343C8—21no—
344C8—21no—
345C2×C3—22no—
346C2×C3—41no—
347C2×C3#54611yescharacteristic
348C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
349C2×C3#57711yescharacteristic
350D3—42no—
351D3—81no—
352D3—81no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—62no—
359C22—121no—
360C22—121no—
361C22—121no—
362C22—121no—
363C22—121no—
364C22—121no—
365C22—121no—
366C22—121no—
367C22—244no—
368C22—244no—
369C22#90812yes—
370C22C22×D1211yescharacteristic
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—121no—
376C4—121no—
377C4—121no—
378C4—121no—
379C4—121no—
380C4#100411yescharacteristic
381C4D3×#4111yescharacteristic
382C3#331011yescharacteristic, complemented, Sylow
383C2—22no—
384C2—41no—
385C2—122no—
386C2—241no—
387C2—241no—
388C2#555411yescharacteristic
389C2C2×#90811yescharacteristic
390C2#575111yescharacteristic
subgroupG/H
centerC22C22×D12
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#435C2
socleC22×C3C22×D4
prime radicalD4×D12C2
hypercenter#435D3
hypocenterC3#3310
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
2. #5541C2semidirect
3. C3×#435C2semidirect
4. #5134C2semidirect
5. #5131C2semidirect
6. #5114C2semidirect
7. C2×#821C2semidirect
8. #4848C2semidirect
9. #4847C2semidirect
10. #4737C2semidirect
11. #4736C2semidirect
12. #4731C2semidirect
13. #4731C2semidirect
14. #4713C2semidirect
15. #4710C2semidirect
20. C2×C3×M16C22semidirect
21. C22×C3×C8C22semidirect
26. #926C22semidirect
27. C2×C4⋊Dic3C22semidirect
28. #924C22semidirect
29. #924C22semidirect
31. C2×#204C22semidirect
32. C2×#204C22semidirect
41. #870C22semidirect
42. C4×Dic6C22semidirect
43. C3×C4⋊C8C22semidirect
44. C3×C22⋊C8C22semidirect
45. C3×C22⋊C8C22semidirect
46. C3×#97C22semidirect
50. #821C22semidirect
51. #821C22semidirect
52. #821C22semidirect
53. #821C22semidirect
54. #821C22semidirect
55. #821C22semidirect
56. #818C22semidirect
57. #818C22semidirect
58. #817C22semidirect
59. #816C22semidirect
72. #435D3semidirect
134. C2×M16C2×D3semidirect
135. C22×C8C2×D3semidirect
157. C4⋊C8C2×D3semidirect
168. C22⋊C8C2×D3semidirect
169. C22⋊C8C2×D3semidirect
170. #97C2×D3semidirect
382. C3#3310semidirect

Maximal subgroups

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

subgroupindex
D4×D122
#55412
C3×#4352
#51342
#51312
#51142
C2×#8212
#48482
#48472
#47372
#47362
#47312
#47132
#47102
#33103