Finite Group Databasebeta

← #81189

Subgroups — #81189

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#5694C211yesmaximal, characteristic, complemented
2D4×D12C211yesmaximal, characteristic
3C3×#478C211yesmaximal, characteristic, complemented
4#5067C211yesmaximal, characteristic, complemented
5#5018C211yesmaximal, characteristic, complemented
6#4886C211yesmaximal, characteristic, complemented
7#4882C211yesmaximal, characteristic, complemented
8#4866C211yesmaximal, characteristic, complemented
9#4864C211yesmaximal, characteristic, complemented
10#4848C211yesmaximal, characteristic, complemented
11#4836C211yesmaximal, characteristic, complemented
12#4776C211yesmaximal, characteristic, complemented
13#4776C211yesmaximal, characteristic, complemented
14#4770C211yesmaximal, characteristic, complemented
15D3×#102C211yesmaximal, characteristic, complemented
16#3310—31nomaximal, Sylow
17C2×D3×D4C2211yescharacteristic
18C2×D3×D4C2211yescharacteristic
19C22×D12—21no—
20C3×#125C2211yescharacteristic, complemented
21C3×C4×D4C2211yescharacteristic
22C2×#214C2211yescharacteristic
23#930—21no—
24C2×#204C2211yescharacteristic, complemented
25C2×#204C2211yescharacteristic, complemented
26C2×ΓL(1,𝔽25)C2211yescharacteristic, complemented
27C2×D3×C8C2211yescharacteristic, complemented
28#898—21no—
29#897C2211yescharacteristic, complemented
30#897C2211yescharacteristic, complemented
31#895C2211yescharacteristic
32#895C2211yescharacteristic
33#894—21no—
34#892—21no—
35#892C2211yescharacteristic, complemented
36D3×C4⋊C4C2211yescharacteristic, complemented
37#890—21no—
38#882—21no—
39#882—21no—
40#874C2211yescharacteristic
41C4×D12C2211yescharacteristic
42C4×Dic6C2211yescharacteristic, 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#810C2211yescharacteristic, complemented
54#809C2211yescharacteristic, complemented
55#809C2211yescharacteristic, complemented
56#809C2211yescharacteristic, complemented
57#809C2211yescharacteristic, complemented
58#807C2211yescharacteristic, complemented
59#803C2211yescharacteristic, complemented
60#552—31no—
61D42—31no—
62#490—31no—
63#482—31no—
64#480—31no—
65#478D311yescharacteristic, complemented
66#468—31no—
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×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—44no—
90C2×D12D411yescharacteristic
91C2×D12C2311yescharacteristic
92C2×D12C2311yescharacteristic
93C2×C4×D3—21no—
94C2×C4×D3C2311yescharacteristic
95C2×C4×D3C2311yescharacteristic
96C2×Dic6C2311yescharacteristic
97C2×C3×C8C2311yescharacteristic
98C2×C3×C8C2311yescharacteristic
99C3×C4⋊C4—21no—
100C3×C4⋊C4—21no—
101C3×C4⋊C4C2311yescharacteristic
102C3×C4⋊C4C2311yescharacteristic
103C3×C4⋊C4C2311yescharacteristic
104C3×C22⋊C4—21no—
105C3×C42C2311yescharacteristic
106#214—22no—
107(C2×D3)⋊C4—21no—
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⋊Dic3—21no—
115C4⋊Dic3—21no—
116C4⋊Dic3D411yescharacteristic
117C4⋊Dic3C2311yescharacteristic
118Dic3⋊C4—21no—
119Dic3⋊C4—21no—
120C4×Dic3—21no—
121C4×Dic3—21no—
122C2×C3⋊C8C2311yescharacteristic
123C2×C3⋊C8C2311yescharacteristic
124#204—41no—
125#204—41no—
126ΓL(1,𝔽25)—41no—
127D3×C8—41no—
128C22×D4—31no—
129C22×D4—31no—
130C22×D4—61no—
131C2×SD16—31no—
132C2×SD16—31no—
133C2×SD16—31no—
134C2×M16—31no—
135C22×C8—31no—
136#127—31no—
137#126—61no—
138#125—61no—
139#125C2×D311yescharacteristic, complemented
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#107C2×D311yescharacteristic, complemented
156#106C2×D311yescharacteristic, complemented
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
167#102C2×D311yescharacteristic
168C22⋊C8—31no—
169C22⋊C8—31no—
170#97—31no—
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×D4—21no—
183C3×D4#4112yes—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—42no—
188C2×C3×C4C2×D411yescharacteristic
189C2×C3×C4C2×D411yescharacteristic
190C2×C3×C4C2411yescharacteristic
191#67—42no—
192#67—42no—
193C2×Dic3—21no—
194C2×Dic3—21no—
195C2×Dic3C2×D411yescharacteristic
196C2×Dic3C2×D411yescharacteristic
197D12—22no—
198D12—21no—
199D12—21no—
200D12—21no—
201D12—21no—
202D12—42no—
203D12—42no—
204D12—42no—
205D12—81no—
206C4×D3—21no—
207C4×D3—21no—
208C4×D3—21no—
209C4×D3—21no—
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—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—62no—
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—21no—
250C4⋊C4—31no—
251C4⋊C4—31no—
252C4⋊C4—61no—
253C4⋊C4—61no—
254C4⋊C4—61no—
255C4⋊C4—61no—
256C4⋊C4C22×D311yescharacteristic
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—61no—
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—61no—
311Q8—61no—
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#23912yes—
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×#23911yescharacteristic
381C4#100911yescharacteristic
382C3#331011yescharacteristic, complemented, Sylow
383C2—42no—
384C2—62no—
385C2—121no—
386C2—241no—
387C2—241no—
388C2#563011yescharacteristic
389C2#577011yescharacteristic
390C2D3×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#478C2
socleC22×C3C22×D4
prime radicalD4×D12C2
hypercenter#478D3
hypocenterC3#3310
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. #5694C2semidirect
3. C3×#478C2semidirect
4. #5067C2semidirect
5. #5018C2semidirect
6. #4886C2semidirect
7. #4882C2semidirect
8. #4866C2semidirect
9. #4864C2semidirect
10. #4848C2semidirect
11. #4836C2semidirect
12. #4776C2semidirect
13. #4776C2semidirect
14. #4770C2semidirect
15. D3×#102C2semidirect
20. C3×#125C22semidirect
24. C2×#204C22semidirect
25. C2×#204C22semidirect
26. C2×ΓL(1,𝔽25)C22semidirect
27. C2×D3×C8C22semidirect
29. #897C22semidirect
30. #897C22semidirect
35. #892C22semidirect
36. D3×C4⋊C4C22semidirect
42. C4×Dic6C22semidirect
43. C3×#107C22semidirect
44. C3×#106C22semidirect
45. C3×C4⋊C8C22semidirect
49. #821C22semidirect
50. #821C22semidirect
51. (C2×D3)⋊C8C22semidirect
52. (C2×D3)⋊C8C22semidirect
53. #810C22semidirect
54. #809C22semidirect
55. #809C22semidirect
56. #809C22semidirect
57. #809C22semidirect
58. #807C22semidirect
59. #803C22semidirect
65. #478D3semidirect
139. #125C2×D3semidirect
155. #107C2×D3semidirect
156. #106C2×D3semidirect
157. C4⋊C8C2×D3semidirect
382. C3#3310semidirect

Maximal subgroups

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

subgroupindex
#56942
D4×D122
C3×#4782
#50672
#50182
#48862
#48822
#48662
#48642
#48482
#48362
#47762
#47702
D3×#1022
#33103