Finite Group Databasebeta

← #80544

Subgroups — #80544

Order 384.

420 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5674C211yesmaximal, characteristic, complemented
2#5562C211yesmaximal, characteristic, complemented
3C3×#463C211yesmaximal, characteristic, complemented
4#5176C211yesmaximal, characteristic, complemented
5#5172C211yesmaximal, characteristic, complemented
6#5156C211yesmaximal, characteristic, complemented
7#5151C211yesmaximal, characteristic, complemented
8#5057C211yesmaximal, characteristic, complemented
9#5015C211yesmaximal, characteristic, complemented
10#4843C211yesmaximal, characteristic, complemented
11#4840C211yesmaximal, characteristic, complemented
12#4810C211yesmaximal, characteristic, complemented
13#4800C211yesmaximal, characteristic, complemented
14#4773C211yesmaximal, characteristic, complemented
15#4764C211yesmaximal, characteristic, complemented
16#3220—31nomaximal, Sylow
17C2×#239C2211yescharacteristic
18C2×D3×D4C2211yescharacteristic
19C2×C3×SD16C2211yescharacteristic, complemented
20C2×C3×D8C2211yescharacteristic
21C3×#124C2211yescharacteristic
22C3×C4×D4C2211yescharacteristic, complemented
23#947C2211yescharacteristic
24C2×#215C2211yescharacteristic, complemented
25#940C2211yescharacteristic
26#940C2211yescharacteristic, complemented
27#938C2211yescharacteristic
28C2×#213C2211yescharacteristic
29C2×#213C2211yescharacteristic
30C2×#213C2211yescharacteristic, complemented
31#930—21no—
32#929—21no—
33C2×D24C2211yescharacteristic
34C2×D24C2211yescharacteristic, complemented
35#895C2211yescharacteristic
36#890C2211yescharacteristic, complemented
37#884—21no—
38#884—21no—
39#884—21no—
40#883—21no—
41#882—21no—
42#882—21no—
43#874C2211yescharacteristic
44C4×D12C2211yescharacteristic, complemented
45#872C2211yescharacteristic
46C3×C4⋊C8C2211yescharacteristic, complemented
47C3×#103C2211yescharacteristic, complemented
48C3×#102C2211yescharacteristic
49#835C2211yescharacteristic, complemented
50#832C2211yescharacteristic
51#821C2211yescharacteristic
52#821C2211yescharacteristic, complemented
53(C2×D3)⋊C8C2211yescharacteristic
54(C2×D3)⋊C8C2211yescharacteristic
55#815C2211yescharacteristic, complemented
56C8×Dic3C2211yescharacteristic, complemented
57#809C2211yescharacteristic
58#809C2211yescharacteristic, complemented
59C4⋊(C3⋊C8)C2211yescharacteristic, complemented
60#540—31no—
61#535—31no—
62#496—31no—
63#493—31no—
64#490—31no—
65#488—31no—
66#463—31no—
67#463D311yescharacteristic, complemented
68#459—31no—
69#459—31no—
70#449—31no—
71#449—31no—
72#447—31no—
73#447—31no—
74#433—31no—
75C23×D3—21no—
76C2×C3×Q8C2311yescharacteristic
77C2×C3×D4C2311yescharacteristic
78C2×C3×D4C2311yescharacteristic
79C22×C3×C4—21no—
80C2×#67—21no—
81C2×#67—21no—
82C2×#67—21no—
83C2×#67—21no—
84C2×#67—21no—
85C2×#67—21no—
86#239—41no—
87#239—41no—
88D3×D4—41no—
89D3×D4—41no—
90C2×D12—21no—
91C2×D12—21no—
92C2×D12—21no—
93C2×D12—21no—
94C2×D12C2311yescharacteristic
95C2×D12C2311yescharacteristic
96C2×D12C2311yescharacteristic
97C2×C4×D3—21no—
98C2×C4×D3—21no—
99C2×C4×D3C2311yescharacteristic
100C3×SD16—41no—
101C3×D8—41no—
102C2×C3×C8C2311yescharacteristic
103C2×C3×C8C2311yescharacteristic
104C3×C4⋊C4C2311yescharacteristic
105C3×C22⋊C4—21no—
106C3×C22⋊C4—21no—
107C3×C22⋊C4—21no—
108C3×C42C2311yescharacteristic
109#217—21no—
110#215—41no—
111#213—41no—
112#213—41no—
113#213—41no—
114(C2×D3)⋊C4—21no—
115(C2×D3)⋊C4—21no—
116(C2×D3)⋊C4—21no—
117(C2×D3)⋊C4—21no—
118(C2×D3)⋊C4D411yescharacteristic
119(C2×D3)⋊C4D411yescharacteristic
120C4⋊Dic3—21no—
121C4⋊Dic3C2311yescharacteristic
122Dic3⋊C4—21no—
123Dic3⋊C4D411yescharacteristic
124Dic3⋊C4D411yescharacteristic
125C4×Dic3C2311yescharacteristic
126C2×C3⋊C8C2311yescharacteristic
127C2×C3⋊C8C2311yescharacteristic
128D24—41no—
129D24—41no—
130C2×#41—31no—
131C22×D4—31no—
132C2×SD16—31no—
133C2×SD16C2×D311yescharacteristic, complemented
134C2×D8—31no—
135C2×D8—31no—
136C2×D8—31no—
137C2×D8—31no—
138C2×D8—31no—
139C2×D8C2×D311yescharacteristic
140#127—31no—
141#127—31no—
142#127—31no—
143#125—31no—
144#124—31no—
145#124C2×D311yescharacteristic
146#123—61no—
147#123—61no—
148#121—31no—
149#121—31no—
150#121—61no—
151#121—61no—
152#121—61no—
153#121—61no—
154#120—61no—
155#120—61no—
156C4×D4—31no—
157C4×D4C2×D311yescharacteristic, complemented
158#117—31no—
159C4⋊C8—31no—
160C4⋊C8C2×D311yescharacteristic, complemented
161#103—31no—
162#103C2×D311yescharacteristic, complemented
163#102—31no—
164#102—31no—
165#102—31no—
166#102—31no—
167#102—31no—
168#102C2×D311yescharacteristic
169C22⋊C8—31no—
170C22⋊C8—31no—
171#97—31no—
172C4×C8—31no—
173C23×C3—21no—
174C23×C3—21no—
175C22×D3—21no—
176C22×D3—21no—
177C22×D3—21no—
178C22×D3—42no—
179C22×D3—81no—
180C22×D3—81no—
181C22×D3C2×D411yescharacteristic
182C3×Q8—21no—
183C3×Q8—21no—
184C3×D4—21no—
185C3×D4—21no—
186C3×D4—21no—
187C3×D4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—81no—
191C2×C3×C4C2×D411yescharacteristic
192C2×C3×C4C2×D411yescharacteristic
193C2×C3×C4C2411yescharacteristic
194#67—42no—
195#67—81no—
196#67—81no—
197#67—81no—
198#67—81no—
199#67—81no—
200C2×Dic3—21no—
201C2×Dic3C2×D411yescharacteristic
202C2×Dic3C2×D411yescharacteristic
203C2×Dic3C2×D411yescharacteristic
204D12—21no—
205D12—21no—
206D12—21no—
207D12—21no—
208D12—21no—
209D12—21no—
210D12—42no—
211D12—81no—
212D12—81no—
213D12—81no—
214C4×D3—21no—
215C4×D3—21no—
216C4×D3—42no—
217C4×D3—81no—
218C3×C8—21no—
219C3×C8—21no—
220C3⋊C8—21no—
221C3⋊C8—21no—
222C24—61no—
223#41—121no—
224#41—121no—
225C2×Q8C22×D311yescharacteristic
226C2×D4—31no—
227C2×D4—31no—
228C2×D4—31no—
229C2×D4—61no—
230C2×D4—61no—
231C2×D4—61no—
232C2×D4—61no—
233C2×D4—61no—
234C2×D4—61no—
235C2×D4—61no—
236C2×D4—61no—
237C2×D4—61no—
238C2×D4—61no—
239C2×D4—121no—
240C2×D4—121no—
241C2×D4C22×D311yescharacteristic
242C2×D4C22×D311yescharacteristic
243C22×C4—21no—
244C22×C4—31no—
245C22×C4—61no—
246C22×C4—61no—
247SD16—41no—
248SD16—121no—
249D8—41no—
250D8—121no—
251D8—121no—
252D8—121no—
253D8—121no—
254D8—121no—
255C2×C8—31no—
256C2×C8—31no—
257C2×C8C22×D311yescharacteristic
258C2×C8C22×D311yescharacteristic
259C4⋊C4—31no—
260C4⋊C4—31no—
261C4⋊C4—31no—
262C4⋊C4—61no—
263C4⋊C4—61no—
264C4⋊C4C22×D311yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4—31no—
269C22⋊C4—31no—
270C22⋊C4—61no—
271C22⋊C4—61no—
272C22⋊C4—61no—
273C22⋊C4—61no—
274C22⋊C4—61no—
275C42—31no—
276C42C22×D311yescharacteristic
277C22×C3—41no—
278C22×C3—41no—
279C22×C3—41no—
280C22×C3—41no—
281C22×C3—41no—
282C22×C3—41no—
283C22×C3C22×D411yescharacteristic
284C2×D3—21no—
285C2×D3—21no—
286C2×D3—21no—
287C2×D3—41no—
288C2×D3—41no—
289C2×D3—41no—
290C2×D3—41no—
291C2×D3—41no—
292C2×D3—41no—
293C2×D3—41no—
294C2×D3—41no—
295C2×D3—41no—
296C2×D3—84no—
297C3×C4—21no—
298C3×C4—21no—
299C3×C4—41no—
300C3×C4—41no—
301C3×C4H(5,𝔽2)11yescharacteristic
302C3×C4H(5,𝔽2)11yescharacteristic
303Dic3—21no—
304Dic3—21no—
305Dic3—21no—
306Dic3—41no—
307C23—21no—
308C23—21no—
309C23—31no—
310C23—61no—
311C23—61no—
312C23—61no—
313C23—122no—
314C23—241no—
315C23—241no—
316Q8—21no—
317Q8—21no—
318D4—21no—
319D4—21no—
320D4—21no—
321D4—21no—
322D4—61no—
323D4—61no—
324D4—61no—
325D4—61no—
326D4—61no—
327D4—61no—
328D4—122no—
329D4—122no—
330D4—241no—
331D4—241no—
332D4—241no—
333D4—241no—
334D4—241no—
335D4—241no—
336D4—241no—
337D4—241no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—31no—
341C2×C4—31no—
342C2×C4—31no—
343C2×C4—61no—
344C2×C4—61no—
345C2×C4—61no—
346C2×C4—81no—
347C2×C4—122no—
348C2×C4—241no—
349C2×C4D3×D411yescharacteristic
350C2×C4D3×D411yescharacteristic
351C2×C4C23×D311yescharacteristic
352C8—21no—
353C8—21no—
354C8—61no—
355C8—61no—
356C2×C3—81no—
357C2×C3—81no—
358C2×C3#53511yescharacteristic
359C2×C3#57611yescharacteristic
360C2×C3#57611yescharacteristic
361D3—41no—
362D3—81no—
363D3—81no—
364D3—81no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—61no—
372C22—61no—
373C22—61no—
374C22—121no—
375C22—121no—
376C22—121no—
377C22—121no—
378C22—121no—
379C22—121no—
380C22—121no—
381C22—121no—
382C22—121no—
383C22—244no—
384C22C2×D3×D411yescharacteristic
385C4—21no—
386C4—21no—
387C4—41no—
388C4—41no—
389C4—61no—
390C4—61no—
391C4—61no—
392C4—121no—
393C4#100411yescharacteristic
394C4#100911yescharacteristic
395C3#322011yescharacteristic, complemented, Sylow
396C2—81no—
397C2—81no—
398C2—121no—
399C2—241no—
400C2—241no—
401C2—241no—
402C2#560711yescharacteristic
403C2#576911yescharacteristic
404C2#577411yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#463C2
socleC22×C3C22×D4
hypercenter#463D3
hypocenterC3#3220
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. #5674C2semidirect
2. #5562C2semidirect
3. C3×#463C2semidirect
4. #5176C2semidirect
5. #5172C2semidirect
6. #5156C2semidirect
7. #5151C2semidirect
8. #5057C2semidirect
9. #5015C2semidirect
10. #4843C2semidirect
11. #4840C2semidirect
12. #4810C2semidirect
13. #4800C2semidirect
14. #4773C2semidirect
15. #4764C2semidirect
19. C2×C3×SD16C22semidirect
22. C3×C4×D4C22semidirect
24. C2×#215C22semidirect
26. #940C22semidirect
30. C2×#213C22semidirect
34. C2×D24C22semidirect
36. #890C22semidirect
44. C4×D12C22semidirect
46. C3×C4⋊C8C22semidirect
47. C3×#103C22semidirect
49. #835C22semidirect
52. #821C22semidirect
55. #815C22semidirect
56. C8×Dic3C22semidirect
58. #809C22semidirect
59. C4⋊(C3⋊C8)C22semidirect
67. #463D3semidirect
133. C2×SD16C2×D3semidirect
157. C4×D4C2×D3semidirect
160. C4⋊C8C2×D3semidirect
162. #103C2×D3semidirect
395. C3#3220semidirect

Maximal subgroups

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

subgroupindex
#56742
#55622
C3×#4632
#51762
#51722
#51562
#51512
#50572
#50152
#48432
#48402
#48102
#48002
#47732
#47642
#32203