Finite Group Databasebeta

← #79800

Subgroups — #79800

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#5581C211yesmaximal, characteristic, complemented
2D4×D12C211yesmaximal, characteristic
3C3×#440C211yesmaximal, characteristic, complemented
4#5169C211yesmaximal, characteristic, complemented
5#5169C211yesmaximal, characteristic, complemented
6#5027C211yesmaximal, characteristic, complemented
7#5009C211yesmaximal, characteristic, complemented
8#4885C211yesmaximal, characteristic, complemented
9#4882C211yesmaximal, characteristic, complemented
10#4812C211yesmaximal, characteristic, complemented
11#4807C211yesmaximal, characteristic, complemented
12#4770C211yesmaximal, characteristic, complemented
13D3×#102C211yesmaximal, characteristic, complemented
14#4713C211yesmaximal, characteristic, complemented
15#4706C211yesmaximal, characteristic, complemented
16#3310—31nomaximal, Sylow
17C2×D3×D4C2211yescharacteristic
18C2×D3×D4C2211yescharacteristic
19C22×D12—21no—
20C2×C3×SD16C2211yescharacteristic
21C3×C4×Q8C2211yescharacteristic, complemented
22C3×C4×D4C2211yescharacteristic
23#946C2211yescharacteristic, complemented
24#946C2211yescharacteristic, complemented
25C2×#215C2211yescharacteristic, complemented
26C2×#215C2211yescharacteristic, complemented
27#930—21no—
28C2×ΓL(1,𝔽25)C2211yescharacteristic, complemented
29C2×D3×C8C2211yescharacteristic, complemented
30#895C2211yescharacteristic
31#895C2211yescharacteristic
32#894—21no—
33#892C2211yescharacteristic, complemented
34D3×C4⋊C4C2211yescharacteristic, complemented
35#889—21no—
36#882—21no—
37#882—21no—
38#876—21no—
39#874C2211yescharacteristic
40C4×D12C2211yescharacteristic
41#872—21no—
42#870C2211yescharacteristic, complemented
43C3×#107C2211yescharacteristic, complemented
44C3×#103C2211yescharacteristic, complemented
45C3×#102C2211yescharacteristic
46C3×#97C2211yescharacteristic, complemented
47#832C2211yescharacteristic
48#832C2211yescharacteristic
49#821C2211yescharacteristic, complemented
50#821C2211yescharacteristic, complemented
51#821C2211yescharacteristic, complemented
52#821C2211yescharacteristic, complemented
53(C2×D3)⋊C8C2211yescharacteristic, complemented
54(C2×D3)⋊C8C2211yescharacteristic, complemented
55#809C2211yescharacteristic, complemented
56#809C2211yescharacteristic, complemented
57#808C2211yescharacteristic, complemented
58#807C2211yescharacteristic, complemented
59C4⋊(C3⋊C8)C2211yescharacteristic, 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#440D311yescharacteristic, complemented
72#435—31no—
73#418—31no—
74C2×#102—31no—
75C23×D3—21no—
76C23×D3—21no—
77C2×C3×Q8C2311yescharacteristic
78C2×C3×D4C2311yescharacteristic
79C22×C3×C4—21no—
80C2×#67—21no—
81C2×#67—21no—
82D3×D4—22no—
83D3×D4—22no—
84D3×D4—41no—
85D3×D4—41no—
86C2×D12—21no—
87C2×D12—21no—
88C2×D12—21no—
89C2×D12—21no—
90C2×D12—44no—
91C2×D12D411yescharacteristic
92C2×D12C2311yescharacteristic
93C2×D12C2311yescharacteristic
94C2×C4×D3—21no—
95C2×C4×D3C2311yescharacteristic
96C2×C4×D3C2311yescharacteristic
97C3×SD16—22no—
98C2×C3×C8C2311yescharacteristic
99C2×C3×C8C2311yescharacteristic
100C3×C4⋊C4—21no—
101C3×C4⋊C4C2311yescharacteristic
102C3×C4⋊C4C2311yescharacteristic
103C3×C22⋊C4—21no—
104C3×C42—21no—
105C3×C42C2311yescharacteristic
106#215—41no—
107#215—41no—
108(C2×D3)⋊C4—21no—
109(C2×D3)⋊C4—21no—
110(C2×D3)⋊C4—21no—
111(C2×D3)⋊C4—21no—
112(C2×D3)⋊C4—21no—
113(C2×D3)⋊C4D411yescharacteristic
114(C2×D3)⋊C4D411yescharacteristic
115C4⋊Dic3D411yescharacteristic
116C4⋊Dic3C2311yescharacteristic
117C4⋊Dic3C2311yescharacteristic
118Dic3⋊C4—21no—
119Dic3⋊C4—21no—
120Dic3⋊C4—21no—
121Dic3⋊C4—21no—
122Dic3⋊C4—21no—
123C4×Dic3—21no—
124C2×C3⋊C8C2311yescharacteristic
125C2×C3⋊C8C2311yescharacteristic
126ΓL(1,𝔽25)—41no—
127D3×C8—41no—
128C22×D4—31no—
129C22×D4—31no—
130C22×D4—61no—
131C2×SD16—31no—
132C2×SD16—31no—
133C2×SD16C2×D311yescharacteristic
134C2×M16—31no—
135C22×C8—31no—
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×Q8C2×D311yescharacteristic, complemented
149C4×D4—31no—
150C4×D4C2×D311yescharacteristic
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
168C22⋊C8—31no—
169C22⋊C8—31no—
170#97C2×D311yescharacteristic, complemented
171C23×C3—21no—
172C22×D3—21no—
173C22×D3—21no—
174C22×D3—42no—
175C22×D3—42no—
176C22×D3—42no—
177C22×D3—42no—
178C22×D3—81no—
179C22×D3—81no—
180C22×D3C2×D411yescharacteristic
181C22×D3C2×D411yescharacteristic
182C3×Q8—21no—
183C3×Q8—21no—
184C3×D4—21no—
185C3×D4#4112yes—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—42no—
190C2×C3×C4C2×D411yescharacteristic
191C2×C3×C4C2×D411yescharacteristic
192C2×C3×C4C2411yescharacteristic
193#67—42no—
194#67—42no—
195C2×Dic3—21no—
196C2×Dic3—21no—
197C2×Dic3C2×D411yescharacteristic
198C2×Dic3C2×D411yescharacteristic
199D12—22no—
200D12—21no—
201D12—21no—
202D12—21no—
203D12—21no—
204D12—42no—
205D12—42no—
206D12—42no—
207D12—81no—
208C4×D3—21no—
209C4×D3—21no—
210C4×D3—21no—
211C4×D3—21no—
212C4×D3—42no—
213C3×C8—21no—
214C3×C8—21no—
215C3⋊C8—21no—
216C3⋊C8—21no—
217C24—61no—
218C24—61no—
219C2×Q8C22×D311yescharacteristic
220C2×D4—31no—
221C2×D4—31no—
222C2×D4—31no—
223C2×D4—61no—
224C2×D4—61no—
225C2×D4—61no—
226C2×D4—61no—
227C2×D4—62no—
228C2×D4—62no—
229C2×D4—61no—
230C2×D4—61no—
231C2×D4—124no—
232C2×D4—121no—
233C2×D4—121no—
234C2×D4C22×D311yescharacteristic
235C22×C4—21no—
236C22×C4—31no—
237C22×C4—31no—
238C22×C4—61no—
239SD16—22no—
240SD16—121no—
241SD16—121no—
242M16—121no—
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—31no—
261C22⋊C4—31no—
262C22⋊C4—61no—
263C22⋊C4—61no—
264C22⋊C4—61no—
265C22⋊C4—61no—
266C22⋊C4—61no—
267C42—21no—
268C42—61no—
269C42C22×D311yescharacteristic
270C22×C3—22no—
271C22×C3—41no—
272C22×C3—41no—
273C22×C3C22×D411yescharacteristic
274C2×D3—21no—
275C2×D3—21no—
276C2×D3—21no—
277C2×D3—21no—
278C2×D3—21no—
279C2×D3—41no—
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—21no—
290C3×C4—41no—
291C3×C4—41no—
292C3×C4—41no—
293C3×C4C2×#4111yescharacteristic
294C3×C4H(5,𝔽2)11yescharacteristic
295Dic3—21no—
296Dic3—21no—
297Dic3—41no—
298Dic3—41no—
299C23—21no—
300C23—31no—
301C23—31no—
302C23—61no—
303C23—61no—
304C23—122no—
305C23—122no—
306C23—122no—
307C23—122no—
308C23—241no—
309C23—241no—
310Q8—21no—
311Q8—21no—
312D4—21no—
313D4—62no—
314D4—61no—
315D4—61no—
316D4—61no—
317D4—61no—
318D4—122no—
319D4—122no—
320D4—122no—
321D4—122no—
322D4—122no—
323D4—241no—
324D4#23512yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—31no—
329C2×C4—31no—
330C2×C4—42no—
331C2×C4—61no—
332C2×C4—61no—
333C2×C4—61no—
334C2×C4—61no—
335C2×C4—61no—
336C2×C4—61no—
337C2×C4—122no—
338C2×C4D3×D411yescharacteristic
339C2×C4D3×D411yescharacteristic
340C2×C4C23×D311yescharacteristic
341C8—21no—
342C8—21no—
343C8—61no—
344C8—61no—
345C2×C3—42no—
346C2×C3#54611yescharacteristic
347C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
348C2×C3#57711yescharacteristic
349D3—22no—
350D3—41no—
351D3—81no—
352D3—81no—
353C22—22no—
354C22—32no—
355C22—41no—
356C22—41no—
357C22—61no—
358C22—61no—
359C22—61no—
360C22—61no—
361C22—61no—
362C22—121no—
363C22—121no—
364C22—121no—
365C22—121no—
366C22—121no—
367C22—121no—
368C22—244no—
369C22—244no—
370C22C2×D3×D411yescharacteristic
371C4—21no—
372C4—21no—
373C4—41no—
374C4—41no—
375C4—41no—
376C4—61no—
377C4—61no—
378C4—121no—
379C4—121no—
380C4C2×#23511yescharacteristic
381C4#100911yescharacteristic
382C3#331011yescharacteristic, complemented, Sylow
383C2—42no—
384C2—62no—
385C2—121no—
386C2—241no—
387C2—241no—
388C2#555011yescharacteristic
389C2D3×AGL(1,ℤ/8)11yescharacteristic
390C2#577811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#440C2
socleC22×C3C22×D4
prime radicalD4×D12C2
hypercenter#440D3
hypocenterC3#3310
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. #5581C2semidirect
3. C3×#440C2semidirect
4. #5169C2semidirect
5. #5169C2semidirect
6. #5027C2semidirect
7. #5009C2semidirect
8. #4885C2semidirect
9. #4882C2semidirect
10. #4812C2semidirect
11. #4807C2semidirect
12. #4770C2semidirect
13. D3×#102C2semidirect
14. #4713C2semidirect
15. #4706C2semidirect
21. C3×C4×Q8C22semidirect
23. #946C22semidirect
24. #946C22semidirect
25. C2×#215C22semidirect
26. C2×#215C22semidirect
28. C2×ΓL(1,𝔽25)C22semidirect
29. C2×D3×C8C22semidirect
33. #892C22semidirect
34. D3×C4⋊C4C22semidirect
42. #870C22semidirect
43. C3×#107C22semidirect
44. C3×#103C22semidirect
46. C3×#97C22semidirect
49. #821C22semidirect
50. #821C22semidirect
51. #821C22semidirect
52. #821C22semidirect
53. (C2×D3)⋊C8C22semidirect
54. (C2×D3)⋊C8C22semidirect
55. #809C22semidirect
56. #809C22semidirect
57. #808C22semidirect
58. #807C22semidirect
59. C4⋊(C3⋊C8)C22semidirect
71. #440D3semidirect
148. C4×Q8C2×D3semidirect
155. #107C2×D3semidirect
158. #103C2×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
#55812
D4×D122
C3×#4402
#51692
#50272
#50092
#48852
#48822
#48122
#48072
#47702
D3×#1022
#47132
#47062
#33103