Finite Group Databasebeta

← #80821

Subgroups — #80821

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
1#5639C211yesmaximal, characteristic, complemented
2#5609C211yesmaximal, characteristic
3C3×#468C211yesmaximal, characteristic, complemented
4#5169C211yesmaximal, characteristic, complemented
5#5163C211yesmaximal, characteristic, complemented
6#5131C211yesmaximal, characteristic, complemented
7#5127C211yesmaximal, characteristic, complemented
8#5046C211yesmaximal, characteristic, complemented
9#5035C211yesmaximal, characteristic, complemented
10#4891C211yesmaximal, characteristic, complemented
11#4882C211yesmaximal, characteristic, complemented
12#4817C211yesmaximal, characteristic, complemented
13#4807C211yesmaximal, characteristic, complemented
14D3×#102C211yesmaximal, characteristic, complemented
15#4764C211yesmaximal, characteristic, complemented
16#3310—31nomaximal, Sylow
17C2×D3×D4—21no—
18C2×D3×D4C2211yescharacteristic
19C22×D12C2211yescharacteristic
20C2×C3×SD16C2211yescharacteristic, complemented
21C2×C3×M16C2211yescharacteristic, complemented
22C3×C22⋊Q8C2211yescharacteristic, complemented
23C3×#121C2211yescharacteristic
24#946C2211yescharacteristic, complemented
25Q8×Dic3C2211yescharacteristic, complemented
26C2×#215C2211yescharacteristic
27C2×#215C2211yescharacteristic, complemented
28#940C2211yescharacteristic
29#937—21no—
30C4×#67C2211yescharacteristic
31#926C2211yescharacteristic, complemented
32C2×D3×C8C2211yescharacteristic, complemented
33#895C2211yescharacteristic
34#893C2211yescharacteristic
35#892—21no—
36D3×C4⋊C4C2211yescharacteristic, complemented
37#890C2211yescharacteristic, complemented
38#889—21no—
39#884—21no—
40#883—21no—
41#882—21no—
42#879—21no—
43C3×#107C2211yescharacteristic, complemented
44C3×#103C2211yescharacteristic, complemented
45C3×#102C2211yescharacteristic, complemented
46#832C2211yescharacteristic, complemented
47#832C2211yescharacteristic, complemented
48#830C2211yescharacteristic, complemented
49#821C2211yescharacteristic
50#821C2211yescharacteristic
51#821C2211yescharacteristic, complemented
52#821C2211yescharacteristic, complemented
53(C2×D3)⋊C8C2211yescharacteristic, complemented
54#815C2211yescharacteristic, complemented
55Dic3⋊C8C2211yescharacteristic, complemented
56#809C2211yescharacteristic
57#809C2211yescharacteristic, complemented
58#808C2211yescharacteristic, complemented
59#807C2211yescharacteristic, complemented
60#552—31no—
61D42—31no—
62#490—31no—
63#482—31no—
64#480—31no—
65#478—31no—
66#468D311yescharacteristic, complemented
67#465—31no—
68#460—31no—
69#450—31no—
70#450—31no—
71#440—31no—
72#435—31no—
73#418—31no—
74C2×#102—31no—
75C23×D3—21no—
76C23×D3—21no—
77C2×C3×Q8C2311yescharacteristic
78C2×C3×D4—21no—
79C2×C3×D4C2311yescharacteristic
80C22×C3×C4C2311yescharacteristic
81C2×#67—21no—
82C2×#67—21no—
83C2×#67—21no—
84C2×#67D411yescharacteristic
85D3×D4—22no—
86D3×D4—41no—
87D3×D4—44no—
88C2×D12—22no—
89C2×D12—21no—
90C2×D12—21no—
91C2×D12—41no—
92C2×D12C2311yescharacteristic
93C2×D12C2311yescharacteristic
94C2×C4×D3—21no—
95C2×C4×D3—21no—
96C2×C4×D3C2311yescharacteristic
97C3×SD16—41no—
98C3×M16—41no—
99C2×C3×C8C2311yescharacteristic
100C2×C3×C8C2311yescharacteristic
101C3×C4⋊C4—21no—
102C3×C4⋊C4C2311yescharacteristic
103C3×C4⋊C4C2311yescharacteristic
104C3×C22⋊C4—21no—
105C3×C22⋊C4—21no—
106#217—21no—
107#217D411yescharacteristic
108#215—22no—
109#215—41no—
110(C2×D3)⋊C4—21no—
111(C2×D3)⋊C4—21no—
112(C2×D3)⋊C4—21no—
113(C2×D3)⋊C4D411yescharacteristic
114C4⋊Dic3—21no—
115C4⋊Dic3—21no—
116C4⋊Dic3C2311yescharacteristic
117C4⋊Dic3C2311yescharacteristic
118Dic3⋊C4—21no—
119Dic3⋊C4—21no—
120Dic3⋊C4—21no—
121Dic3⋊C4D411yescharacteristic
122C4×Dic3—21no—
123C4×Dic3—21no—
124C4×Dic3C2311yescharacteristic
125C2×C3⋊C8C2311yescharacteristic
126C2×C3⋊C8C2311yescharacteristic
127D3×C8—41no—
128C22×D4—31no—
129C22×D4—31no—
130C22×D4—61no—
131C2×SD16—31no—
132C2×SD16—31no—
133C2×SD16C2×D311yescharacteristic, complemented
134C2×M16C2×D311yescharacteristic, complemented
135C22×C8—31no—
136#127—31no—
137#126—61no—
138#125—31no—
139#125—61no—
140#123—61no—
141C22⋊Q8—31no—
142C22⋊Q8C2×D311yescharacteristic, complemented
143#121—31no—
144#121—61no—
145#121C2×D311yescharacteristic
146#120—61no—
147#120—61no—
148C4×Q8—31no—
149C4×D4—31no—
150C4×D4—31no—
151#117—31no—
152#117—61no—
153C2×C4⋊C4—31no—
154#107—31no—
155#107C2×D311yescharacteristic, complemented
156#106—31no—
157C4⋊C8—31no—
158#103C2×D311yescharacteristic, complemented
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#102C2×D311yescharacteristic, complemented
168C22⋊C8—31no—
169C22⋊C8—31no—
170#97—31no—
171C23×C3—21no—
172C23×C3C2×D411yescharacteristic
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
182C3×Q8—21no—
183C3×Q8—21no—
184C3×D4—21no—
185C3×D4—21no—
186C3×D4—42no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4C2×D411yescharacteristic
193C2×C3×C4C2411yescharacteristic
194#67—22no—
195#67—42no—
196#67—42no—
197#67—81no—
198C2×Dic3—21no—
199C2×Dic3—21no—
200C2×Dic3C2×D411yescharacteristic
201C2×Dic3C2×D411yescharacteristic
202C2×Dic3C2×D411yescharacteristic
203D12—21no—
204D12—21no—
205D12—21no—
206D12—42no—
207D12—42no—
208D12#4112yes—
209C4×D3—21no—
210C4×D3—21no—
211C4×D3—42no—
212C4×D3—42no—
213C3×C8—21no—
214C3×C8—21no—
215C3⋊C8—21no—
216C3⋊C8—21no—
217C24—61no—
218C24—61no—
219C2×Q8C22×D311yescharacteristic
220C2×D4—21no—
221C2×D4—31no—
222C2×D4—31no—
223C2×D4—31no—
224C2×D4—62no—
225C2×D4—61no—
226C2×D4—61no—
227C2×D4—61no—
228C2×D4—61no—
229C2×D4—62no—
230C2×D4—61no—
231C2×D4—121no—
232C2×D4—121no—
233C2×D4—124no—
234C2×D4C22×D311yescharacteristic
235C22×C4—31no—
236C22×C4—61no—
237C22×C4—61no—
238C22×C4C22×D311yescharacteristic
239SD16—41no—
240SD16—62no—
241SD16—121no—
242M16—41no—
243C2×C8—31no—
244C2×C8—31no—
245C2×C8—121no—
246C2×C8C22×D311yescharacteristic
247C2×C8C22×D311yescharacteristic
248C4⋊C4—21no—
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⋊C4C22×D311yescharacteristic
258C4⋊C4C22×D311yescharacteristic
259C22⋊C4—21no—
260C22⋊C4—21no—
261C22⋊C4—31no—
262C22⋊C4—31no—
263C22⋊C4—61no—
264C22⋊C4—61no—
265C22⋊C4—61no—
266C22⋊C4—61no—
267C42—31no—
268C42—61no—
269C42—61no—
270C22×C3—21no—
271C22×C3—21no—
272C22×C3—21no—
273C22×C3—41no—
274C22×C3—41no—
275C22×C3—41no—
276C22×C3C22×D411yescharacteristic
277C2×D3—21no—
278C2×D3—21no—
279C2×D3—22no—
280C2×D3—41no—
281C2×D3—41no—
282C2×D3—41no—
283C2×D3—41no—
284C2×D3—41no—
285C2×D3—84no—
286C2×D3—84no—
287C2×D3AGL(1,ℤ/8)12yes—
288C3×C4—21no—
289C3×C4—41no—
290C3×C4—41no—
291C3×C4—41no—
292C3×C4C2×#4111yescharacteristic
293C3×C4H(5,𝔽2)11yescharacteristic
294Dic3—21no—
295Dic3—21no—
296Dic3—21no—
297Dic3—41no—
298Dic3—41no—
299C23—21no—
300C23—31no—
301C23—61no—
302C23—61no—
303C23—122no—
304C23—122no—
305C23—122no—
306C23—122no—
307C23—241no—
308C23—241no—
309C23D3×D411yescharacteristic
310Q8—21no—
311Q8—21no—
312D4—21no—
313D4—21no—
314D4—32no—
315D4—42no—
316D4—62no—
317D4—61no—
318D4—61no—
319D4—61no—
320D4—122no—
321D4—122no—
322D4—122no—
323D4—122no—
324D4—241no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—31no—
331C2×C4—31no—
332C2×C4—31no—
333C2×C4—61no—
334C2×C4—61no—
335C2×C4—61no—
336C2×C4—61no—
337C2×C4—122no—
338C2×C4—122no—
339C2×C4D3×D411yescharacteristic
340C2×C4C23×D311yescharacteristic
341C8—21no—
342C8—21no—
343C8—61no—
344C8—61no—
345C2×C3—41no—
346C2×C3—81no—
347C2×C3#54611yescharacteristic
348C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
349C2×C3#57711yescharacteristic
350D3—22no—
351D3—42no—
352D3—81no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—32no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—61no—
361C22—61no—
362C22—62no—
363C22—121no—
364C22—121no—
365C22—121no—
366C22—121no—
367C22—121no—
368C22—244no—
369C22—244no—
370C22C2×D3×D411yescharacteristic
371C4—21no—
372C4—41no—
373C4—41no—
374C4—41no—
375C4—61no—
376C4—61no—
377C4—61no—
378C4—121no—
379C4—121no—
380C4#100411yescharacteristic
381C4D3×#4111yescharacteristic
382C3#331011yescharacteristic, complemented, Sylow
383C2—41no—
384C2—62no—
385C2—81no—
386C2—122no—
387C2—241no—
388C2#561011yescharacteristic
389C2D3×AGL(1,ℤ/8)11yescharacteristic
390C2#577811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#468C2
socleC22×C3C22×D4
prime radical#5609C2
hypercenter#468D3
hypocenterC3#3310
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. #5639C2semidirect
3. C3×#468C2semidirect
4. #5169C2semidirect
5. #5163C2semidirect
6. #5131C2semidirect
7. #5127C2semidirect
8. #5046C2semidirect
9. #5035C2semidirect
10. #4891C2semidirect
11. #4882C2semidirect
12. #4817C2semidirect
13. #4807C2semidirect
14. D3×#102C2semidirect
15. #4764C2semidirect
20. C2×C3×SD16C22semidirect
21. C2×C3×M16C22semidirect
22. C3×C22⋊Q8C22semidirect
24. #946C22semidirect
25. Q8×Dic3C22semidirect
27. C2×#215C22semidirect
31. #926C22semidirect
32. C2×D3×C8C22semidirect
36. D3×C4⋊C4C22semidirect
37. #890C22semidirect
43. C3×#107C22semidirect
44. C3×#103C22semidirect
45. C3×#102C22semidirect
46. #832C22semidirect
47. #832C22semidirect
48. #830C22semidirect
51. #821C22semidirect
52. #821C22semidirect
53. (C2×D3)⋊C8C22semidirect
54. #815C22semidirect
55. Dic3⋊C8C22semidirect
57. #809C22semidirect
58. #808C22semidirect
59. #807C22semidirect
66. #468D3semidirect
133. C2×SD16C2×D3semidirect
134. C2×M16C2×D3semidirect
142. C22⋊Q8C2×D3semidirect
155. #107C2×D3semidirect
158. #103C2×D3semidirect
167. #102C2×D3semidirect
382. C3#3310semidirect

Maximal subgroups

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

subgroupindex
#56392
#56092
C3×#4682
#51692
#51632
#51312
#51272
#50462
#50352
#48912
#48822
#48172
#48072
D3×#1022
#47642
#33103