Finite Group Databasebeta

← #80397

Subgroups — #80397

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#5717C211yesmaximal, characteristic
2#5581C211yesmaximal, characteristic, complemented
3C3×#460C211yesmaximal, characteristic, complemented
4#5174C211yesmaximal, characteristic, complemented
5#5171C211yesmaximal, characteristic, complemented
6#5169C211yesmaximal, characteristic, complemented
7#5169C211yesmaximal, characteristic, complemented
8#5071C211yesmaximal, characteristic, complemented
9#5026C211yesmaximal, characteristic, complemented
10#4847C211yesmaximal, characteristic, complemented
11#4836C211yesmaximal, characteristic, complemented
12#4779C211yesmaximal, characteristic, complemented
13#4774C211yesmaximal, characteristic, complemented
14#4770C211yesmaximal, characteristic, complemented
15D3×#102C211yesmaximal, characteristic, complemented
16#3310—31nomaximal, Sylow
17C2×D3×D4—21no—
18C2×D3×D4C2211yescharacteristic
19C2×D3×D4C2211yescharacteristic
20C2×C3×SD16C2211yescharacteristic, complemented
21C2×C3×SD16C2211yescharacteristic, complemented
22C3×#127C2211yescharacteristic
23C3×C4×Q8C2211yescharacteristic, complemented
24#946C2211yescharacteristic, complemented
25#946C2211yescharacteristic, complemented
26C2×#215C2211yescharacteristic
27#938—21no—
28#938C2211yescharacteristic
29#938C2211yescharacteristic
30#937—21no—
31#937—21no—
32C2×ΓL(1,𝔽25)C2211yescharacteristic, complemented
33C2×D3×C8C2211yescharacteristic, complemented
34#894—21no—
35#892C2211yescharacteristic, complemented
36D3×C4⋊C4C2211yescharacteristic, complemented
37#889—21no—
38#876—21no—
39C4×D12C2211yescharacteristic
40C4×D12C2211yescharacteristic
41#872—21no—
42#870C2211yescharacteristic, complemented
43C3×C4⋊C8C2211yescharacteristic, complemented
44C3×#102C2211yescharacteristic, complemented
45C3×#102C2211yescharacteristic, complemented
46#835C2211yescharacteristic, complemented
47#832C2211yescharacteristic, complemented
48#832C2211yescharacteristic, complemented
49#832C2211yescharacteristic, complemented
50#832C2211yescharacteristic, complemented
51#821C2211yescharacteristic
52#821C2211yescharacteristic
53(C2×D3)⋊C8C2211yescharacteristic, complemented
54(C2×D3)⋊C8C2211yescharacteristic, complemented
55#818C2211yescharacteristic, complemented
56#817C2211yescharacteristic, complemented
57#809C2211yescharacteristic
58#807C2211yescharacteristic, complemented
59#803C2211yescharacteristic, 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#460D311yescharacteristic, complemented
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×D4—21no—
80C2×C3×D4C2311yescharacteristic
81C2×C3×D4C2311yescharacteristic
82C2×#67—21no—
83C2×#67—21no—
84C2×#67—21no—
85C2×#67—21no—
86D3×D4—22no—
87D3×D4—22no—
88D3×D4—41no—
89D3×D4—41no—
90D3×D4—44no—
91C2×D12D411yescharacteristic
92C2×D12C2311yescharacteristic
93C2×C4×D3—21no—
94C2×C4×D3—21no—
95C2×C4×D3C2311yescharacteristic
96C2×C4×D3C2311yescharacteristic
97C3×SD16—41no—
98C3×SD16—41no—
99C2×C3×C8C2311yescharacteristic
100C2×C3×C8C2311yescharacteristic
101C3×C4⋊C4—21no—
102C3×C4⋊C4C2311yescharacteristic
103C3×C42—21no—
104C3×C42C2311yescharacteristic
105#217—21no—
106#217—21no—
107#215—22no—
108(C2×D3)⋊C4—21no—
109(C2×D3)⋊C4—21no—
110(C2×D3)⋊C4—21no—
111(C2×D3)⋊C4—21no—
112(C2×D3)⋊C4D411yescharacteristic
113(C2×D3)⋊C4D411yescharacteristic
114C4⋊Dic3D411yescharacteristic
115C4⋊Dic3C2311yescharacteristic
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×SD16C2×D311yescharacteristic, complemented
133C2×SD16C2×D311yescharacteristic, complemented
134C2×M16—31no—
135C22×C8—31no—
136#127C2×D311yescharacteristic
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×D4—31no—
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#102C2×D311yescharacteristic, complemented
167#102C2×D311yescharacteristic, complemented
168C22⋊C8—31no—
169C22⋊C8—31no—
170#97—31no—
171C23×C3—21no—
172C23×C3—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—21no—
186C3×D4—21no—
187C3×D4—21no—
188C3×D4—42no—
189C3×D4—81no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4C2×D411yescharacteristic
193C2×C3×C4C2×D411yescharacteristic
194C2×C3×C4C2411yescharacteristic
195#67—42no—
196#67—42no—
197#67—42no—
198#67—42no—
199C2×Dic3—21no—
200C2×Dic3—21no—
201C2×Dic3—21no—
202C2×Dic3C2×D411yescharacteristic
203C2×Dic3C2×D411yescharacteristic
204D12—22no—
205D12—21no—
206D12#4112yes—
207C4×D3—21no—
208C4×D3—21no—
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—21no—
222C2×D4—31no—
223C2×D4—31no—
224C2×D4—61no—
225C2×D4—61no—
226C2×D4—62no—
227C2×D4—62no—
228C2×D4—61no—
229C2×D4—61no—
230C2×D4—121no—
231C2×D4—121no—
232C2×D4—124no—
233C2×D4C22×D311yescharacteristic
234C2×D4C22×D311yescharacteristic
235C22×C4—31no—
236C22×C4—31no—
237C22×C4—61no—
238C22×C4—61no—
239SD16—41no—
240SD16—41no—
241SD16—62no—
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—31no—
253C4⋊C4—61no—
254C4⋊C4—61no—
255C4⋊C4—61no—
256C4⋊C4—61no—
257C4⋊C4—61no—
258C4⋊C4C22×D311yescharacteristic
259C22⋊C4—31no—
260C22⋊C4—31no—
261C22⋊C4—61no—
262C22⋊C4—61no—
263C22⋊C4—61no—
264C22⋊C4—61no—
265C22⋊C4—61no—
266C22⋊C4—61no—
267C42—21no—
268C42—61no—
269C42C22×D311yescharacteristic
270C22×C3—41no—
271C22×C3—41no—
272C22×C3—41no—
273C22×C3—41no—
274C22×C3—41no—
275C22×C3—41no—
276C22×C3C22×D411yescharacteristic
277C2×D3—21no—
278C2×D3—21no—
279C2×D3—21no—
280C2×D3—22no—
281C2×D3—21no—
282C2×D3—21no—
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×C4C2×#4111yescharacteristic
293C3×C4H(5,𝔽2)11yescharacteristic
294Dic3—21no—
295Dic3—21no—
296Dic3—41no—
297Dic3—41no—
298Dic3—41no—
299C23—21no—
300C23—21no—
301C23—31no—
302C23—31no—
303C23—61no—
304C23—122no—
305C23—122no—
306C23—122no—
307C23—122no—
308C23—241no—
309C23—241no—
310Q8—21no—
311Q8—21no—
312D4—21no—
313D4—21no—
314D4—21no—
315D4—21no—
316D4—32no—
317D4—42no—
318D4—62no—
319D4—61no—
320D4—81no—
321D4—122no—
322D4—122no—
323D4—122no—
324D4—122no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—31no—
328C2×C4—31no—
329C2×C4—61no—
330C2×C4—61no—
331C2×C4—61no—
332C2×C4—61no—
333C2×C4—61no—
334C2×C4—61no—
335C2×C4—61no—
336C2×C4—122no—
337C2×C4—122no—
338C2×C4D3×D411yescharacteristic
339C2×C4D3×D411yescharacteristic
340C2×C4C23×D311yescharacteristic
341C8—21no—
342C8—21no—
343C8—61no—
344C8—61no—
345C2×C3—81no—
346C2×C3—81no—
347C2×C3#54611yescharacteristic
348C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
349C2×C3#57711yescharacteristic
350D3—22no—
351D3—41no—
352D3—42no—
353C22—32no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—61no—
361C22—61no—
362C22—61no—
363C22—61no—
364C22—61no—
365C22—62no—
366C22—121no—
367C22—121no—
368C22—244no—
369C22—244no—
370C22C2×D3×D411yescharacteristic
371C4—21no—
372C4—21no—
373C4—41no—
374C4—41no—
375C4—61no—
376C4—61no—
377C4—121no—
378C4—121no—
379C4—121no—
380C4#100411yescharacteristic
381C4D3×#4111yescharacteristic
382C3#331011yescharacteristic, complemented, Sylow
383C2—62no—
384C2—81no—
385C2—81no—
386C2—121no—
387C2—122no—
388C2#561211yescharacteristic
389C2#576211yescharacteristic
390C2D3×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#460C2
socleC22×C3C22×D4
prime radical#5717C2
hypercenter#460D3
hypocenterC3#3310
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
2. #5581C2semidirect
3. C3×#460C2semidirect
4. #5174C2semidirect
5. #5171C2semidirect
6. #5169C2semidirect
7. #5169C2semidirect
8. #5071C2semidirect
9. #5026C2semidirect
10. #4847C2semidirect
11. #4836C2semidirect
12. #4779C2semidirect
13. #4774C2semidirect
14. #4770C2semidirect
15. D3×#102C2semidirect
20. C2×C3×SD16C22semidirect
21. C2×C3×SD16C22semidirect
23. C3×C4×Q8C22semidirect
24. #946C22semidirect
25. #946C22semidirect
32. C2×ΓL(1,𝔽25)C22semidirect
33. C2×D3×C8C22semidirect
35. #892C22semidirect
36. D3×C4⋊C4C22semidirect
42. #870C22semidirect
43. C3×C4⋊C8C22semidirect
44. C3×#102C22semidirect
45. C3×#102C22semidirect
46. #835C22semidirect
47. #832C22semidirect
48. #832C22semidirect
49. #832C22semidirect
50. #832C22semidirect
53. (C2×D3)⋊C8C22semidirect
54. (C2×D3)⋊C8C22semidirect
55. #818C22semidirect
56. #817C22semidirect
58. #807C22semidirect
59. #803C22semidirect
68. #460D3semidirect
132. C2×SD16C2×D3semidirect
133. C2×SD16C2×D3semidirect
148. C4×Q8C2×D3semidirect
157. C4⋊C8C2×D3semidirect
166. #102C2×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
#57172
#55812
C3×#4602
#51742
#51712
#51692
#50712
#50262
#48472
#48362
#47792
#47742
#47702
D3×#1022
#33103