Finite Group Databasebeta

← #83485

Subgroups — #83485

Order 384.

462 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×D42C211yesmaximal, characteristic, complemented
2#5565C211yesmaximal, characteristic, complemented
3#5217C211yesmaximal, characteristic, complemented
4#5217C211yesmaximal, characteristic, complemented
5#5215C211yesmaximal, characteristic, complemented
6C2×#832C211yesmaximal, characteristic, complemented
7#5072C211yesmaximal, characteristic, complemented
8#5071C211yesmaximal, characteristic, complemented
9#5039C211yesmaximal, characteristic, complemented
10#5038C211yesmaximal, characteristic, complemented
11#5035C211yesmaximal, characteristic, complemented
12#5034C211yesmaximal, characteristic, complemented
13#5014C211yesmaximal, characteristic, complemented
14#5011C211yesmaximal, characteristic, complemented
15#5009C211yesmaximal, characteristic, complemented
16#3318—31nomaximal, Sylow
17C22×C3×D4—21no—
18C22×C3×D4C2211yescharacteristic
19C22×C3×D4C2211yescharacteristic
20C3×#127C2211yescharacteristic
21C3×#121—21no—
22C3×#121C2211yescharacteristic
23C3×#121C2211yescharacteristic
24C3×#120—21no—
25C3×#120—21no—
26C3×C4×D4C2211yescharacteristic
27C3×C4×D4C2211yescharacteristic
28#939—21no—
29C2×#213C2211yescharacteristic
30C2×#213C2211yescharacteristic, complemented
31C2×#213C2211yescharacteristic, complemented
32#930C2211yescharacteristic, complemented
33#930C2211yescharacteristic, complemented
34#926C2211yescharacteristic, complemented
35C2×C4⋊Dic3C2211yescharacteristic, complemented
36C2×#208C2211yescharacteristic, complemented
37C22×C3⋊C8C2211yescharacteristic, complemented
38#898—21no—
39#883—21no—
40#67⋊C4—21no—
41C4×D12C2211yescharacteristic, complemented
42#870C2211yescharacteristic, complemented
43#832C2211yescharacteristic
44#832C2211yescharacteristic
45#832C2211yescharacteristic
46#832C2211yescharacteristic, complemented
47#832C2211yescharacteristic, complemented
48#832C2211yescharacteristic, complemented
49#832C2211yescharacteristic, complemented
50#832C2211yescharacteristic, complemented
51#832C2211yescharacteristic, complemented
52#830C2211yescharacteristic, complemented
53#830C2211yescharacteristic, complemented
54#809C2211yescharacteristic, complemented
55#808C2211yescharacteristic, complemented
56#808C2211yescharacteristic, complemented
57#807C2211yescharacteristic, complemented
58C4⋊(C3⋊C8)C2211yescharacteristic, complemented
59#803C2211yescharacteristic, complemented
60#553—31no—
61D42D311yescharacteristic, complemented
62#490—31no—
63#482—31no—
64#481—31no—
65#478—31no—
66#469—31no—
67#466—31no—
68#459—31no—
69#447—31no—
70#447—31no—
71#442—31no—
72#435—31no—
73#418—31no—
74C2×#102—31no—
75C24×C3—21no—
76C24×C3—21no—
77C2×C3×D4—22no—
78C2×C3×D4—22no—
79C2×C3×D4—21no—
80C2×C3×D4—21no—
81C2×C3×D4—21no—
82C2×C3×D4—21no—
83C2×C3×D4—21no—
84C2×C3×D4—21no—
85C2×C3×D4—41no—
86C2×C3×D4—41no—
87C2×C3×D4—44no—
88C2×C3×D4D411yescharacteristic
89C2×C3×D4C2311yescharacteristic
90C2×C3×D4C2311yescharacteristic
91C2×C3×D4C2311yescharacteristic
92C22×C3×C4—21no—
93C22×C3×C4C2311yescharacteristic
94C22×C3×C4C2311yescharacteristic
95C2×#67—21no—
96C2×#67—21no—
97C22×Dic3—21no—
98C2×D12C2311yescharacteristic
99C2×C4×D3—21no—
100C3×C4⋊C4D411yescharacteristic
101C3×C4⋊C4C2311yescharacteristic
102C3×C22⋊C4—21no—
103C3×C22⋊C4—21no—
104C3×C22⋊C4—21no—
105C3×C22⋊C4D411yescharacteristic
106C3×C22⋊C4D411yescharacteristic
107C3×C42C2311yescharacteristic
108#217—21no—
109#213—22no—
110#213—41no—
111#213—41no—
112(C2×D3)⋊C4—21no—
113(C2×D3)⋊C4—21no—
114(C2×D3)⋊C4—21no—
115C4⋊Dic3—21no—
116C4⋊Dic3C2311yescharacteristic
117C4⋊Dic3C2311yescharacteristic
118C4⋊Dic3C2311yescharacteristic
119Dic3⋊C4—21no—
120Dic3⋊C4—21no—
121C4×Dic3—21no—
122#208—41no—
123C2×C3⋊C8—41no—
124C2×C3⋊C8C2311yescharacteristic
125C2×C3⋊C8C2311yescharacteristic
126C2×C3⋊C8C2311yescharacteristic
127C2×C3⋊C8C2311yescharacteristic
128C22×D4—21no—
129C22×D4C2×D311yescharacteristic
130C22×D4C2×D311yescharacteristic
131C2×D8—31no—
132C2×D8—31no—
133C2×D8—31no—
134C2×M16—31no—
135C22×C8—31no—
136#127C2×D311yescharacteristic
137#126—61no—
138#125—31no—
139#123—61no—
140#121—21no—
141#121—31no—
142#121—31no—
143#121—61no—
144#121C2×D311yescharacteristic
145#121C2×D311yescharacteristic
146#120—21no—
147#120—21no—
148C4×D4—31no—
149C4×D4—61no—
150C4×D4C2×D311yescharacteristic
151C4×D4C2×D311yescharacteristic
152#117—31no—
153C2×C4⋊C4—31no—
154#107—31no—
155#106—31no—
156#106—31no—
157C4⋊C8—31no—
158#102—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⋊C8—31no—
169C22⋊C8—31no—
170#97—31no—
171C23×C3—21no—
172C23×C3—21no—
173C23×C3—21no—
174C23×C3—42no—
175C23×C3—42no—
176C23×C3—42no—
177C23×C3—42no—
178C23×C3—81no—
179C23×C3—81no—
180C23×C3C2×D411yescharacteristic
181C23×C3C2×D411yescharacteristic
182C22×D3—21no—
183C3×D4—22no—
184C3×D4—21no—
185C3×D4—21no—
186C3×D4—21no—
187C3×D4—21no—
188C3×D4—21no—
189C3×D4—42no—
190C3×D4—42no—
191C3×D4—42no—
192C3×D4—42no—
193C3×D4—42no—
194C3×D4—81no—
195C3×D4#4112yes—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—42no—
202C2×C3×C4C2×D411yescharacteristic
203C2×C3×C4C2×D411yescharacteristic
204C2×C3×C4C2×D411yescharacteristic
205C2×C3×C4C2×D411yescharacteristic
206C2×C3×C4C2411yescharacteristic
207#67—42no—
208#67—81no—
209C2×Dic3—21no—
210C2×Dic3—21no—
211C2×Dic3—21no—
212C2×Dic3—81no—
213D12—21no—
214D12—21no—
215C4×D3—81no—
216C3⋊C8—21no—
217C3⋊C8—21no—
218C3⋊C8—21no—
219C3⋊C8—21no—
220C24—21no—
221C24—21no—
222C2×D4—22no—
223C2×D4—22no—
224C2×D4—21no—
225C2×D4—21no—
226C2×D4—21no—
227C2×D4—21no—
228C2×D4—21no—
229C2×D4—21no—
230C2×D4—31no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4—44no—
234C2×D4—61no—
235C2×D4—61no—
236C2×D4#6711yescharacteristic
237C2×D4C22×D311yescharacteristic
238C2×D4C22×D311yescharacteristic
239C2×D4C22×D311yescharacteristic
240C22×C4—21no—
241C22×C4—61no—
242C22×C4—61no—
243C22×C4C22×D311yescharacteristic
244C22×C4C22×D311yescharacteristic
245D8—62no—
246D8—121no—
247D8—121no—
248M16—121no—
249C2×C8—31no—
250C2×C8—31no—
251C2×C8—31no—
252C2×C8—31no—
253C2×C8—121no—
254C4⋊C4—31no—
255C4⋊C4—31no—
256C4⋊C4—31no—
257C4⋊C4—61no—
258C4⋊C4—61no—
259C4⋊C4—61no—
260C4⋊C4#6711yescharacteristic
261C4⋊C4C22×D311yescharacteristic
262C22⋊C4—21no—
263C22⋊C4—21no—
264C22⋊C4—21no—
265C22⋊C4—61no—
266C22⋊C4—61no—
267C22⋊C4—61no—
268C22⋊C4—61no—
269C22⋊C4#6711yescharacteristic
270C22⋊C4#6711yescharacteristic
271C42—61no—
272C42C22×D311yescharacteristic
273C22×C3—21no—
274C22×C3—21no—
275C22×C3—21no—
276C22×C3—22no—
277C22×C3—21no—
278C22×C3—21no—
279C22×C3—41no—
280C22×C3—41no—
281C22×C3—41no—
282C22×C3—41no—
283C22×C3—41no—
284C22×C3—41no—
285C22×C3—41no—
286C22×C3—41no—
287C22×C3—84no—
288C22×C3—84no—
289C22×C3AGL(1,ℤ/8)12yes—
290C22×C3C22×D411yescharacteristic
291C2×D3—41no—
292C2×D3—41no—
293C2×D3—41no—
294C3×C4—21no—
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—41no—
299C3×C4C2×#4111yescharacteristic
300C3×C4H(5,𝔽2)11yescharacteristic
301Dic3—41no—
302Dic3—41no—
303Dic3—41no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—42no—
308C23—42no—
309C23—42no—
310C23—42no—
311C23—61no—
312C23—81no—
313C23—81no—
314C23C2×#6711yescharacteristic
315C23C2×#6711yescharacteristic
316D4—22no—
317D4—21no—
318D4—21no—
319D4—21no—
320D4—21no—
321D4—21no—
322D4—42no—
323D4—42no—
324D4—42no—
325D4—42no—
326D4—42no—
327D4—61no—
328D4—61no—
329D4—81no—
330D4—122no—
331D4—241no—
332D4#23712yes—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—42no—
339C2×C4—61no—
340C2×C4—61no—
341C2×C4—61no—
342C2×C4—241no—
343C2×C4—241no—
344C2×C4C2×#6711yescharacteristic
345C2×C4C2×#6711yescharacteristic
346C2×C4C2×#6711yescharacteristic
347C2×C4C2×#6711yescharacteristic
348C2×C4C23×D311yescharacteristic
349C8—61no—
350C8—61no—
351C8—61no—
352C8—61no—
353C2×C3—22no—
354C2×C3—41no—
355C2×C3—42no—
356C2×C3—81no—
357C2×C3—81no—
358C2×C3#54611yescharacteristic
359C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
360C2×C3#57611yescharacteristic
361D3—81no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—22no—
367C22—21no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—84no—
377C22—84no—
378C22—121no—
379C22—121no—
380C22—121no—
381C22#93212yes—
382C22C22×#6711yescharacteristic
383C4—21no—
384C4—21no—
385C4—21no—
386C4—21no—
387C4—41no—
388C4—121no—
389C4—121no—
390C4—121no—
391C4C2×#23711yescharacteristic
392C4#100411yescharacteristic
393C3#331811yescharacteristic, complemented, Sylow
394C2—22no—
395C2—41no—
396C2—42no—
397C2—81no—
398C2—81no—
399C2—241no—
400C2C2×#93211yescharacteristic
401C2#580611yescharacteristic
402C2#583511yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×D42C2
socleC22×C3C22×D4
hypercenterD42D3
hypocenterC3#3318
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. C3×D42C2semidirect
2. #5565C2semidirect
3. #5217C2semidirect
4. #5217C2semidirect
5. #5215C2semidirect
6. C2×#832C2semidirect
7. #5072C2semidirect
8. #5071C2semidirect
9. #5039C2semidirect
10. #5038C2semidirect
11. #5035C2semidirect
12. #5034C2semidirect
13. #5014C2semidirect
14. #5011C2semidirect
15. #5009C2semidirect
30. C2×#213C22semidirect
31. C2×#213C22semidirect
32. #930C22semidirect
33. #930C22semidirect
34. #926C22semidirect
35. C2×C4⋊Dic3C22semidirect
36. C2×#208C22semidirect
37. C22×C3⋊C8C22semidirect
41. C4×D12C22semidirect
42. #870C22semidirect
46. #832C22semidirect
47. #832C22semidirect
48. #832C22semidirect
49. #832C22semidirect
50. #832C22semidirect
51. #832C22semidirect
52. #830C22semidirect
53. #830C22semidirect
54. #809C22semidirect
55. #808C22semidirect
56. #808C22semidirect
57. #807C22semidirect
58. C4⋊(C3⋊C8)C22semidirect
59. #803C22semidirect
61. D42D3semidirect
393. C3#3318semidirect

Maximal subgroups

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

subgroupindex
C3×D422
#55652
#52172
#52152
C2×#8322
#50722
#50712
#50392
#50382
#50352
#50342
#50142
#50112
#50092
#33183