Finite Group Databasebeta

← #83496

Subgroups — #83496

Order 384.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#546C211yesmaximal, characteristic, complemented
2#5555C211yesmaximal, characteristic, complemented
3#5238C211yesmaximal, characteristic, complemented
4#5232C211yesmaximal, characteristic, complemented
5#5217C211yesmaximal, characteristic, complemented
6C2×#832C211yesmaximal, characteristic, complemented
7#5057C211yesmaximal, characteristic, complemented
8#5055C211yesmaximal, characteristic, complemented
9#5048C211yesmaximal, characteristic, complemented
10#5043C211yesmaximal, characteristic, complemented
11#5039C211yesmaximal, characteristic, complemented
12#5034C211yesmaximal, characteristic, complemented
13#5014C211yesmaximal, characteristic, complemented
14#5011C211yesmaximal, characteristic, complemented
15#5008C211yesmaximal, characteristic, complemented
16#3306—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18C22×C3×D4C2211yescharacteristic
19C2×#237—21no—
20C3×#124C2211yescharacteristic
21C3×#123—21no—
22C3×C22⋊Q8C2211yescharacteristic
23C3×#121—21no—
24C3×#121C2211yescharacteristic
25C3×#120—21no—
26C3×C4×D4C2211yescharacteristic
27C3×C4×D4C2211yescharacteristic
28C2×C3×C22⋊C4—21no—
29C2×#215C2211yescharacteristic, complemented
30C2×#213C2211yescharacteristic
31C2×#213C2211yescharacteristic
32#930C2211yescharacteristic
33#930C2211yescharacteristic, complemented
34C2×C4⋊Dic3C2211yescharacteristic
35C2×C4⋊Dic3C2211yescharacteristic, complemented
36C2×#208C2211yescharacteristic, complemented
37C22×C3⋊C8C2211yescharacteristic
38#897—21no—
39#886—21no—
40#886—21no—
41C4×D12C2211yescharacteristic, complemented
42#869C2211yescharacteristic, complemented
43#835C2211yescharacteristic, complemented
44#835C2211yescharacteristic, complemented
45#835C2211yescharacteristic, complemented
46#832C2211yescharacteristic
47#832C2211yescharacteristic
48#832C2211yescharacteristic
49#832C2211yescharacteristic
50#832C2211yescharacteristic
51#832C2211yescharacteristic
52#830C2211yescharacteristic
53#830C2211yescharacteristic, complemented
54#809C2211yescharacteristic, complemented
55#808C2211yescharacteristic, complemented
56#808C2211yescharacteristic, complemented
57#808C2211yescharacteristic, complemented
58C4⋊(C3⋊C8)C2211yescharacteristic, complemented
59#803C2211yescharacteristic, complemented
60#547—31no—
61#546D311yescharacteristic, complemented
62#489—31no—
63#483—31no—
64#481—31no—
65#476—31no—
66#469—31no—
67#465—31no—
68#463—31no—
69#449—31no—
70#447—31no—
71#442—31no—
72#435—31no—
73#417—31no—
74C2×#102—31no—
75C24×C3—21no—
76C3×#41—22no—
77C3×#41—41no—
78C2×C3×Q8C2311yescharacteristic
79C2×C3×D4—22no—
80C2×C3×D4—21no—
81C2×C3×D4—21no—
82C2×C3×D4—21no—
83C2×C3×D4—41no—
84C2×C3×D4D411yescharacteristic
85C2×C3×D4C2311yescharacteristic
86C2×C3×D4C2311yescharacteristic
87C22×C3×C4—21no—
88C22×C3×C4—21no—
89C22×C3×C4C2311yescharacteristic
90C22×C3×C4C2311yescharacteristic
91C2×#67—21no—
92C2×#67—21no—
93C22×Dic3—21no—
94C22×Dic3—21no—
95#237—44no—
96C2×D12C2311yescharacteristic
97C2×C4×D3—21no—
98C2×Dic6—21no—
99C3×C4⋊C4—21no—
100C3×C4⋊C4D411yescharacteristic
101C3×C4⋊C4C2311yescharacteristic
102C3×C22⋊C4—21no—
103C3×C22⋊C4—21no—
104C3×C22⋊C4—21no—
105C3×C22⋊C4—21no—
106C3×C22⋊C4—21no—
107C3×C22⋊C4D411yescharacteristic
108C3×C22⋊C4D411yescharacteristic
109C3×C42C2311yescharacteristic
110#215—41no—
111#213—22no—
112#213—41no—
113(C2×D3)⋊C4—21no—
114(C2×D3)⋊C4—21no—
115(C2×D3)⋊C4—21no—
116C4⋊Dic3—21no—
117C4⋊Dic3—21no—
118C4⋊Dic3—21no—
119C4⋊Dic3C2311yescharacteristic
120C4⋊Dic3C2311yescharacteristic
121C4⋊Dic3C2311yescharacteristic
122#208—41no—
123C2×C3⋊C8—41no—
124C2×C3⋊C8C2311yescharacteristic
125C2×C3⋊C8C2311yescharacteristic
126C2×C3⋊C8C2311yescharacteristic
127C2×C3⋊C8C2311yescharacteristic
128C2×#41—61no—
129C2×#41C2×D311yescharacteristic
130C22×D4C2×D311yescharacteristic
131C2×SD16—31no—
132C2×D8—31no—
133C2×D8—31no—
134C2×M16—31no—
135C22×C8—31no—
136C4⋊Q8—31no—
137#124C2×D311yescharacteristic
138#123—21no—
139#123—61no—
140#123—61no—
141C22⋊Q8—61no—
142C22⋊Q8C2×D311yescharacteristic
143#121—21no—
144#121—31no—
145#121—31no—
146#121C2×D311yescharacteristic
147#120—21no—
148C4×D4—31no—
149C4×D4C2×D311yescharacteristic
150C4×D4C2×D311yescharacteristic
151C2×C4⋊C4—31no—
152C2×C4⋊C4—31no—
153C2×C22⋊C4—21no—
154#106—31no—
155#106—31no—
156#106—31no—
157C4⋊C8—31no—
158#103—31no—
159#103—31no—
160#103—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—42no—
174C23×C3—42no—
175C23×C3—81no—
176C23×C3C2×D411yescharacteristic
177C23×C3C2×D411yescharacteristic
178C22×D3—21no—
179C3×Q8—21no—
180C3×Q8—21no—
181C3×D4—22no—
182C3×D4—21no—
183C3×D4—21no—
184C3×D4—21no—
185C3×D4—42no—
186C3×D4—42no—
187C3×D4—81no—
188C3×D4#4112yescomplemented
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—42no—
196C2×C3×C4—81no—
197C2×C3×C4C2×D411yescharacteristic
198C2×C3×C4C2×D411yescharacteristic
199C2×C3×C4C2×D411yescharacteristic
200C2×C3×C4C2×D411yescharacteristic
201C2×C3×C4C2411yescharacteristic
202#67—42no—
203#67—42no—
204C2×Dic3—21no—
205C2×Dic3—21no—
206C2×Dic3—21no—
207C2×Dic3—42no—
208C2×Dic3—42no—
209D12—21no—
210D12—21no—
211C4×D3—42no—
212Dic6—42no—
213C3⋊C8—21no—
214C3⋊C8—21no—
215C3⋊C8—21no—
216C3⋊C8—21no—
217C24—21no—
218#41—22no—
219#41—41no—
220#41—124no—
221C2×Q8—61no—
222C2×Q8C22×D311yescharacteristic
223C2×D4—22no—
224C2×D4—21no—
225C2×D4—21no—
226C2×D4—21no—
227C2×D4—31no—
228C2×D4—41no—
229C2×D4—61no—
230C2×D4—61no—
231C2×D4#6711yescharacteristic
232C2×D4C22×D311yescharacteristic
233C2×D4C22×D311yescharacteristic
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—61no—
237C22×C4—61no—
238C22×C4—61no—
239C22×C4C22×D311yescharacteristic
240C22×C4C22×D311yescharacteristic
241SD16—121no—
242D8—62no—
243D8—121no—
244M16—121no—
245C2×C8—31no—
246C2×C8—31no—
247C2×C8—31no—
248C2×C8—31no—
249C2×C8—121no—
250C4⋊C4—21no—
251C4⋊C4—31no—
252C4⋊C4—31no—
253C4⋊C4—31no—
254C4⋊C4—61no—
255C4⋊C4—61no—
256C4⋊C4—61no—
257C4⋊C4#6711yescharacteristic
258C4⋊C4C22×D311yescharacteristic
259C22⋊C4—21no—
260C22⋊C4—21no—
261C22⋊C4—21no—
262C22⋊C4—21no—
263C22⋊C4—21no—
264C22⋊C4—61no—
265C22⋊C4—61no—
266C22⋊C4—61no—
267C22⋊C4#6711yescharacteristic
268C22⋊C4#6711yescharacteristic
269C42C22×D311yescharacteristic
270C22×C3—21no—
271C22×C3—21no—
272C22×C3—21no—
273C22×C3—22no—
274C22×C3—21no—
275C22×C3—21no—
276C22×C3—41no—
277C22×C3—41no—
278C22×C3—41no—
279C22×C3—41no—
280C22×C3—41no—
281C22×C3—84no—
282C22×C3AGL(1,ℤ/8)12yes—
283C22×C3C22×D411yescharacteristic
284C2×D3—41no—
285C2×D3—41no—
286C2×D3—41no—
287C3×C4—21no—
288C3×C4—21no—
289C3×C4—21no—
290C3×C4—21no—
291C3×C4—41no—
292C3×C4—41no—
293C3×C4C2×#4111yescharacteristic
294C3×C4H(5,𝔽2)11yescharacteristic
295Dic3—41no—
296Dic3—41no—
297Dic3—41no—
298C23—21no—
299C23—21no—
300C23—42no—
301C23—42no—
302C23—61no—
303C23—81no—
304C23C2×#6711yescharacteristic
305C23C2×#6711yescharacteristic
306Q8—21no—
307Q8—21no—
308Q8—122no—
309D4—22no—
310D4—21no—
311D4—21no—
312D4—21no—
313D4—42no—
314D4—42no—
315D4—61no—
316D4—61no—
317D4—81no—
318D4—122no—
319D4—122no—
320D4#23712yescomplemented
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—42no—
328C2×C4—61no—
329C2×C4—61no—
330C2×C4—61no—
331C2×C4—81no—
332C2×C4—122no—
333C2×C4—122no—
334C2×C4—122no—
335C2×C4C2×#6711yescharacteristic
336C2×C4C2×#6711yescharacteristic
337C2×C4C2×#6711yescharacteristic
338C2×C4C2×#6711yescharacteristic
339C2×C4C23×D311yescharacteristic
340C8—61no—
341C8—61no—
342C8—61no—
343C8—61no—
344C2×C3—22no—
345C2×C3—41no—
346C2×C3—42no—
347C2×C3—81no—
348C2×C3#54611yescharacteristic
349C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
350C2×C3#57711yescharacteristic
351D3—81no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—22no—
356C22—21no—
357C22—21no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—84no—
364C22—121no—
365C22—121no—
366C22—121no—
367C22#94912yes—
368C22C22×#6711yescharacteristic
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—41no—
374C4—41no—
375C4—121no—
376C4—121no—
377C4—121no—
378C4C2×#23711yescharacteristic
379C4#100411yescharacteristic
380C3#330611yescharacteristic, complemented, Sylow
381C2—22no—
382C2—41no—
383C2—42no—
384C2—81no—
385C2—241no—
386C2#580611yescharacteristic
387C2C2×#94911yescharacteristic
388C2#583611yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#546C2
socleC22×C3C22×D4
hypercenter#546D3
hypocenterC3#3306
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. C3×#546C2semidirect
2. #5555C2semidirect
3. #5238C2semidirect
4. #5232C2semidirect
5. #5217C2semidirect
6. C2×#832C2semidirect
7. #5057C2semidirect
8. #5055C2semidirect
9. #5048C2semidirect
10. #5043C2semidirect
11. #5039C2semidirect
12. #5034C2semidirect
13. #5014C2semidirect
14. #5011C2semidirect
15. #5008C2semidirect
29. C2×#215C22semidirect
33. #930C22semidirect
35. C2×C4⋊Dic3C22semidirect
36. C2×#208C22semidirect
41. C4×D12C22semidirect
42. #869C22semidirect
43. #835C22semidirect
44. #835C22semidirect
45. #835C22semidirect
53. #830C22semidirect
54. #809C22semidirect
55. #808C22semidirect
56. #808C22semidirect
57. #808C22semidirect
58. C4⋊(C3⋊C8)C22semidirect
59. #803C22semidirect
61. #546D3semidirect
188. C3×D4#41semidirect
320. D4#237semidirect
380. C3#3306semidirect

Maximal subgroups

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

subgroupindex
C3×#5462
#55552
#52382
#52322
#52172
C2×#8322
#50572
#50552
#50482
#50432
#50392
#50342
#50142
#50112
#50082
#33063