Finite Group Databasebeta

← #80563

Subgroups — #80563

Order 384.

414 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5676C211yesmaximal, characteristic, complemented
2#5563C211yesmaximal, characteristic, complemented
3C3×#463C211yesmaximal, characteristic, complemented
4#5174C211yesmaximal, characteristic, complemented
5#5172C211yesmaximal, characteristic, complemented
6#5157C211yesmaximal, characteristic, complemented
7#5156C211yesmaximal, characteristic, complemented
8#5055C211yesmaximal, characteristic, complemented
9#5014C211yesmaximal, characteristic, complemented
10#4846C211yesmaximal, characteristic, complemented
11#4836C211yesmaximal, characteristic, complemented
12#4814C211yesmaximal, characteristic, complemented
13#4804C211yesmaximal, characteristic, complemented
14#4784C211yesmaximal, characteristic, complemented
15#4770C211yesmaximal, characteristic, complemented
16#3319—31nomaximal, Sylow
17C2×#239C2211yescharacteristic
18C2×D3×D4C2211yescharacteristic
19C2×C3×SD16C2211yescharacteristic, complemented
20C2×C3×D8C2211yescharacteristic
21C3×#124C2211yescharacteristic
22C3×C4×D4C2211yescharacteristic, complemented
23#946C2211yescharacteristic
24#938C2211yescharacteristic
25#938C2211yescharacteristic
26#938C2211yescharacteristic, complemented
27C2×#213C2211yescharacteristic
28#930—21no—
29#926—21no—
30C2×ΓL(1,𝔽25)C2211yescharacteristic
31C2×D3×C8C2211yescharacteristic, complemented
32#892C2211yescharacteristic
33#892C2211yescharacteristic, complemented
34#885—21no—
35#885—21no—
36#884—21no—
37#883—21no—
38#882—21no—
39D3×C22⋊C4—21no—
40C4×D12C2211yescharacteristic
41C4×D12C2211yescharacteristic
42#869C2211yescharacteristic, complemented
43C3×C4⋊C8C2211yescharacteristic, complemented
44C3×#103C2211yescharacteristic, complemented
45C3×#102C2211yescharacteristic
46#835C2211yescharacteristic, complemented
47#835C2211yescharacteristic, complemented
48#832C2211yescharacteristic
49#832C2211yescharacteristic
50#832C2211yescharacteristic, complemented
51#821C2211yescharacteristic
52#821C2211yescharacteristic
53(C2×D3)⋊C8C2211yescharacteristic
54(C2×D3)⋊C8C2211yescharacteristic, complemented
55#818C2211yescharacteristic, complemented
56#818C2211yescharacteristic, complemented
57#809C2211yescharacteristic
58#808C2211yescharacteristic, complemented
59#803C2211yescharacteristic, complemented
60#555—31no—
61#546—31no—
62#489—31no—
63#485—31no—
64#482—31no—
65#474—31no—
66#468—31no—
67#466—31no—
68#463D311yescharacteristic, complemented
69#449—31no—
70#447—31no—
71#442—31no—
72#435—31no—
73#418—31no—
74#416—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—
84#239—22no—
85#239—41no—
86D3×D4—22no—
87D3×D4—41no—
88C2×D12—21no—
89C2×D12D411yescharacteristic
90C2×D12C2311yescharacteristic
91C2×C4×D3—21no—
92C2×C4×D3—21no—
93C2×C4×D3C2311yescharacteristic
94C2×C4×D3C2311yescharacteristic
95C2×Dic6—21no—
96C3×SD16—41no—
97C3×D8—41no—
98C2×C3×C8C2311yescharacteristic
99C2×C3×C8C2311yescharacteristic
100C3×C4⋊C4C2311yescharacteristic
101C3×C22⋊C4—21no—
102C3×C22⋊C4—21no—
103C3×C22⋊C4—21no—
104C3×C42C2311yescharacteristic
105#217—21no—
106#217—21no—
107#217—21no—
108#213—22no—
109(C2×D3)⋊C4—21no—
110(C2×D3)⋊C4—21no—
111(C2×D3)⋊C4—21no—
112(C2×D3)⋊C4—21no—
113(C2×D3)⋊C4D411yescharacteristic
114(C2×D3)⋊C4D411yescharacteristic
115C4⋊Dic3—21no—
116C4⋊Dic3D411yescharacteristic
117C4⋊Dic3C2311yescharacteristic
118C4⋊Dic3C2311yescharacteristic
119C4⋊Dic3C2311yescharacteristic
120Dic3⋊C4—21no—
121C4×Dic3—21no—
122C4×Dic3—21no—
123C2×C3⋊C8C2311yescharacteristic
124C2×C3⋊C8C2311yescharacteristic
125ΓL(1,𝔽25)—41no—
126D3×C8—41no—
127C2×#41—31no—
128C22×D4—31no—
129C2×SD16C2×D311yescharacteristic, complemented
130C2×D8—31no—
131C2×D8C2×D311yescharacteristic
132C2×M16—31no—
133C22×C8—31no—
134C4⋊Q8—31no—
135#124—61no—
136#124—61no—
137#124C2×D311yescharacteristic
138#123—61no—
139C22⋊Q8—31no—
140#121—31no—
141#121—31no—
142#121—31no—
143#121—61no—
144#121—61no—
145#120—61no—
146C4×D4—31no—
147C4×D4—31no—
148C4×D4C2×D311yescharacteristic, complemented
149#117—31no—
150#117—31no—
151#117—61no—
152C2×C22⋊C4—61no—
153#107—31no—
154#107—31no—
155#106—31no—
156C4⋊C8C2×D311yescharacteristic, complemented
157#103—31no—
158#103—31no—
159#103C2×D311yescharacteristic, complemented
160#102—31no—
161#102—31no—
162#102—31no—
163#102—31no—
164#102—31no—
165#102—31no—
166#102C2×D311yescharacteristic
167C22⋊C8—31no—
168C22⋊C8—31no—
169#97—31no—
170C23×C3—21no—
171C23×C3—21no—
172C22×D3—21no—
173C22×D3—42no—
174C22×D3—42no—
175C22×D3—81no—
176C22×D3C2×D411yescharacteristic
177C22×D3C2×D411yescharacteristic
178C3×Q8—21no—
179C3×Q8—21no—
180C3×D4—21no—
181C3×D4—21no—
182C3×D4—21no—
183C3×D4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—42no—
187C2×C3×C4C2×D411yescharacteristic
188C2×C3×C4C2×D411yescharacteristic
189C2×C3×C4C2411yescharacteristic
190#67—42no—
191#67—81no—
192#67—81no—
193#67—81no—
194C2×Dic3—21no—
195C2×Dic3—21no—
196C2×Dic3—21no—
197C2×Dic3C2×D411yescharacteristic
198C2×Dic3C2×D411yescharacteristic
199D12—21no—
200D12—41no—
201D12—42no—
202D12#4112yes—
203C4×D3—21no—
204C4×D3—21no—
205C4×D3—21no—
206C4×D3—21no—
207C4×D3—42no—
208C4×D3—81no—
209Dic6—81no—
210C3×C8—21no—
211C3×C8—21no—
212C3⋊C8—21no—
213C3⋊C8—21no—
214C24—61no—
215#41—62no—
216#41—121no—
217C2×Q8—61no—
218C2×Q8C22×D311yescharacteristic
219C2×D4—31no—
220C2×D4—31no—
221C2×D4—61no—
222C2×D4—61no—
223C2×D4—62no—
224C2×D4—61no—
225C2×D4—61no—
226C2×D4—61no—
227C2×D4—121no—
228C2×D4C22×D311yescharacteristic
229C2×D4C22×D311yescharacteristic
230C22×C4—21no—
231C22×C4—31no—
232C22×C4—31no—
233C22×C4—61no—
234C22×C4—61no—
235SD16—41no—
236D8—41no—
237D8—62no—
238M16—121no—
239C2×C8—31no—
240C2×C8—31no—
241C2×C8—121no—
242C2×C8C22×D311yescharacteristic
243C2×C8C22×D311yescharacteristic
244C4⋊C4—31no—
245C4⋊C4—31no—
246C4⋊C4—31no—
247C4⋊C4—31no—
248C4⋊C4—61no—
249C4⋊C4—61no—
250C4⋊C4C22×D311yescharacteristic
251C22⋊C4—21no—
252C22⋊C4—21no—
253C22⋊C4—21no—
254C22⋊C4—31no—
255C22⋊C4—31no—
256C22⋊C4—61no—
257C22⋊C4—61no—
258C22⋊C4—61no—
259C22⋊C4—61no—
260C22⋊C4—61no—
261C22⋊C4—61no—
262C22⋊C4—61no—
263C42—61no—
264C42—61no—
265C42C22×D311yescharacteristic
266C22×C3—41no—
267C22×C3—41no—
268C22×C3—41no—
269C22×C3—41no—
270C22×C3—41no—
271C22×C3—41no—
272C22×C3C22×D411yescharacteristic
273C2×D3—21no—
274C2×D3—21no—
275C2×D3—22no—
276C2×D3—21no—
277C2×D3—21no—
278C2×D3—21no—
279C2×D3—21no—
280C2×D3—41no—
281C2×D3—41no—
282C2×D3—84no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—41no—
286C3×C4—41no—
287C3×C4C2×#4111yescharacteristic
288C3×C4H(5,𝔽2)11yescharacteristic
289Dic3—21no—
290Dic3—21no—
291Dic3—41no—
292Dic3—41no—
293Dic3—41no—
294C23—21no—
295C23—21no—
296C23—31no—
297C23—31no—
298C23—61no—
299C23—122no—
300C23—122no—
301C23—241no—
302Q8—21no—
303Q8—21no—
304Q8—241no—
305D4—21no—
306D4—21no—
307D4—21no—
308D4—21no—
309D4—32no—
310D4—61no—
311D4—121no—
312D4—122no—
313D4—122no—
314D4—241no—
315D4—241no—
316D4—241no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—31no—
320C2×C4—31no—
321C2×C4—42no—
322C2×C4—61no—
323C2×C4—61no—
324C2×C4—61no—
325C2×C4—61no—
326C2×C4—61no—
327C2×C4—61no—
328C2×C4—61no—
329C2×C4—122no—
330C2×C4—241no—
331C2×C4D3×D411yescharacteristic
332C2×C4D3×D411yescharacteristic
333C2×C4C23×D311yescharacteristic
334C8—21no—
335C8—21no—
336C8—61no—
337C8—61no—
338C2×C3—81no—
339C2×C3—81no—
340C2×C3#54611yescharacteristic
341C2×C3#57511yescharacteristic
342C2×C3#57611yescharacteristic
343D3—41no—
344D3—41no—
345D3—42no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—61no—
353C22—61no—
354C22—61no—
355C22—61no—
356C22—61no—
357C22—61no—
358C22—62no—
359C22—121no—
360C22—121no—
361C22—244no—
362C22C2×D3×D411yescharacteristic
363C4—21no—
364C4—21no—
365C4—41no—
366C4—41no—
367C4—61no—
368C4—61no—
369C4—121no—
370C4—121no—
371C4—121no—
372C4#100411yescharacteristic
373C4D3×#4111yescharacteristic
374C3#331911yescharacteristic, complemented, Sylow
375C2—81no—
376C2—81no—
377C2—121no—
378C2—121no—
379C2—122no—
380C2#561211yescharacteristic
381C2#576911yescharacteristic
382C2#577311yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#463C2
socleC22×C3C22×D4
hypercenter#463D3
hypocenterC3#3319
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. #5676C2semidirect
2. #5563C2semidirect
3. C3×#463C2semidirect
4. #5174C2semidirect
5. #5172C2semidirect
6. #5157C2semidirect
7. #5156C2semidirect
8. #5055C2semidirect
9. #5014C2semidirect
10. #4846C2semidirect
11. #4836C2semidirect
12. #4814C2semidirect
13. #4804C2semidirect
14. #4784C2semidirect
15. #4770C2semidirect
19. C2×C3×SD16C22semidirect
22. C3×C4×D4C22semidirect
26. #938C22semidirect
31. C2×D3×C8C22semidirect
33. #892C22semidirect
42. #869C22semidirect
43. C3×C4⋊C8C22semidirect
44. C3×#103C22semidirect
46. #835C22semidirect
47. #835C22semidirect
50. #832C22semidirect
54. (C2×D3)⋊C8C22semidirect
55. #818C22semidirect
56. #818C22semidirect
58. #808C22semidirect
59. #803C22semidirect
68. #463D3semidirect
129. C2×SD16C2×D3semidirect
148. C4×D4C2×D3semidirect
156. C4⋊C8C2×D3semidirect
159. #103C2×D3semidirect
374. C3#3319semidirect

Maximal subgroups

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

subgroupindex
#56762
#55632
C3×#4632
#51742
#51722
#51572
#51562
#50552
#50142
#48462
#48362
#48142
#48042
#47842
#47702
#33193