Finite Group Databasebeta

← #80011

Subgroups — #80011

Order 384.

442 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5805C211yesmaximal, characteristic, complemented
2#5607C211yesmaximal, characteristic, complemented
3C3×#447C211yesmaximal, characteristic, complemented
4#5217C211yesmaximal, characteristic, complemented
5#5159C211yesmaximal, characteristic, complemented
6#5157C211yesmaximal, characteristic, complemented
7#5156C211yesmaximal, characteristic, complemented
8#5156C211yesmaximal, characteristic, complemented
9#5035C211yesmaximal, characteristic, complemented
10#4784C211yesmaximal, characteristic, complemented
11#4782C211yesmaximal, characteristic, complemented
12#4779C211yesmaximal, characteristic, complemented
13#4775C211yesmaximal, characteristic, complemented
14#4737C211yesmaximal, characteristic, complemented
15#4727C211yesmaximal, characteristic, complemented
16#3199—31nomaximal, Sylow
17C22×C3×D4C2211yescharacteristic, complemented
18C2×D3×D4C2211yescharacteristic, complemented
19C2×#235C2211yescharacteristic
20C2×C3×D8C2211yescharacteristic
21C2×C3×D8C2211yescharacteristic
22C3×#121C2211yescharacteristic
23#953—21no—
24#953—21no—
25#940—21no—
26#940—21no—
27#938C2211yescharacteristic
28#938C2211yescharacteristic
29#938C2211yescharacteristic, complemented
30#938C2211yescharacteristic, complemented
31#938C2211yescharacteristic, complemented
32#937—21no—
33#936—21no—
34C2×#213C2211yescharacteristic
35C2×#213C2211yescharacteristic
36#930C2211yescharacteristic
37#926C2211yescharacteristic, complemented
38C2×ΓL(1,𝔽25)C2211yescharacteristic, complemented
39C2×D3×C8C2211yescharacteristic, complemented
40#895C2211yescharacteristic, complemented
41#892C2211yescharacteristic, complemented
42#885—21no—
43#882—21no—
44C3×#102C2211yescharacteristic, complemented
45C3×#102C2211yescharacteristic, complemented
46C3×C22⋊C8C2211yescharacteristic, complemented
47#832C2211yescharacteristic, complemented
48#832C2211yescharacteristic, complemented
49#832C2211yescharacteristic, complemented
50#832C2211yescharacteristic, complemented
51#830C2211yescharacteristic, complemented
52#821C2211yescharacteristic, complemented
53#821C2211yescharacteristic, complemented
54(C2×D3)⋊C8C2211yescharacteristic, complemented
55(C2×D3)⋊C8C2211yescharacteristic, complemented
56#818C2211yescharacteristic, complemented
57#817C2211yescharacteristic, complemented
58#808C2211yescharacteristic, complemented
59#807C2211yescharacteristic, complemented
60#535—31no—
61#535—31no—
62#482—31no—
63#482—31no—
64#482—31no—
65#482—31no—
66#469—31no—
67#469—31no—
68#466—31no—
69#466—31no—
70#447—31no—
71#447—31no—
72#447—31no—
73#447D311yescharacteristic, complemented
74#408—31no—
75C24×C3—21no—
76C23×D3—21no—
77C2×C3×D4—21no—
78C2×C3×D4—21no—
79C2×C3×D4—41no—
80C2×C3×D4—41no—
81C2×C3×D4C2311yescharacteristic
82C2×C3×D4C2311yescharacteristic
83C2×C3×D4C2311yescharacteristic
84C22×C3×C4C2311yescharacteristic
85C2×#67—21no—
86C2×#67—21no—
87C2×#67—21no—
88C2×#67—21no—
89C2×#67—21no—
90C2×#67—21no—
91C2×#67—21no—
92C2×#67D411yescharacteristic
93C2×#67D411yescharacteristic
94D3×D4—41no—
95D3×D4—41no—
96#235—41no—
97#235—41no—
98C2×D12—21no—
99C2×D12D411yescharacteristic
100C2×D12C2311yescharacteristic
101C2×C4×D3C2311yescharacteristic
102C2×C4×D3C2311yescharacteristic
103C2×Dic6D411yescharacteristic
104C3×D8—41no—
105C3×D8—41no—
106C2×C3×C8C2311yescharacteristic
107C2×C3×C8C2311yescharacteristic
108C3×C4⋊C4C2311yescharacteristic
109C3×C22⋊C4—21no—
110#217—21no—
111#217—21no—
112#217—21no—
113#217—21no—
114#217—21no—
115#217—21no—
116#213—41no—
117#213—41no—
118(C2×D3)⋊C4—21no—
119(C2×D3)⋊C4—21no—
120(C2×D3)⋊C4—21no—
121C4⋊Dic3C2311yescharacteristic
122C4⋊Dic3C2311yescharacteristic
123C4⋊Dic3C2311yescharacteristic
124C4×Dic3—21no—
125C4×Dic3—21no—
126C2×C3⋊C8C2311yescharacteristic
127C2×C3⋊C8C2311yescharacteristic
128ΓL(1,𝔽25)—41no—
129D3×C8—41no—
130C2×#41—31no—
131C22×D4—31no—
132C22×D4C2×D311yescharacteristic, complemented
133C2×D8—31no—
134C2×D8—31no—
135C2×D8C2×D311yescharacteristic
136C2×D8C2×D311yescharacteristic
137C2×M16—31no—
138C22×C8—31no—
139#127—61no—
140#127—61no—
141#124—61no—
142#124—61no—
143#121—31no—
144#121—31no—
145#121—31no—
146#121—31no—
147#121—31no—
148#121—31no—
149#121—31no—
150#121C2×D311yescharacteristic
151#120—61no—
152#120—61no—
153#120—61no—
154#120—61no—
155#117—31no—
156#117—31no—
157#107—31no—
158#107—31no—
159#106—31no—
160#106—31no—
161#102—31no—
162#102—31no—
163#102—31no—
164#102—31no—
165#102—31no—
166#102—31no—
167#102C2×D311yescharacteristic, complemented
168#102C2×D311yescharacteristic, complemented
169C22⋊C8—31no—
170C22⋊C8—31no—
171C22⋊C8—31no—
172C22⋊C8C2×D311yescharacteristic, complemented
173C23×C3—21no—
174C23×C3—21no—
175C23×C3—21no—
176C23×C3—42no—
177C23×C3—81no—
178C23×C3—81no—
179C23×C3C2×D411yescharacteristic
180C22×D3—21no—
181C22×D3—42no—
182C22×D3—81no—
183C22×D3—81no—
184C22×D3C2×D411yescharacteristic
185C22×D3C2×D411yescharacteristic
186C3×D4—21no—
187C3×D4—21no—
188C3×D4—21no—
189C3×D4—21no—
190C3×D4—21no—
191C3×D4—21no—
192C3×D4—42no—
193C3×D4—81no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4C2×D411yescharacteristic
198C2×C3×C4C2411yescharacteristic
199#67—41no—
200#67—41no—
201#67—42no—
202#67—81no—
203#67—81no—
204#67—81no—
205#67—81no—
206#67—81no—
207#67—81no—
208C2×Dic3—21no—
209C2×Dic3—21no—
210C2×Dic3—21no—
211C2×Dic3C2×D411yescharacteristic
212C2×Dic3C2×D411yescharacteristic
213D12—21no—
214D12—21no—
215D12—41no—
216D12—81no—
217C4×D3—21no—
218C4×D3—21no—
219C4×D3—21no—
220C4×D3—21no—
221Dic6—41no—
222C3×C8—21no—
223C3×C8—21no—
224C3⋊C8—21no—
225C3⋊C8—21no—
226C24—21no—
227C24—61no—
228#41—121no—
229#41—121no—
230C2×Q8—31no—
231C2×D4—21no—
232C2×D4—21no—
233C2×D4—31no—
234C2×D4—31no—
235C2×D4—31no—
236C2×D4—31no—
237C2×D4—41no—
238C2×D4—41no—
239C2×D4—61no—
240C2×D4—61no—
241C2×D4—61no—
242C2×D4—61no—
243C2×D4—61no—
244C2×D4—61no—
245C2×D4—61no—
246C2×D4—61no—
247C2×D4—121no—
248C2×D4—121no—
249C2×D4C22×D311yescharacteristic
250C2×D4C22×D311yescharacteristic
251C2×D4C22×D311yescharacteristic
252C22×C4—31no—
253C22×C4—31no—
254C22×C4C22×D311yescharacteristic
255D8—41no—
256D8—41no—
257D8—121no—
258D8—121no—
259M16—121no—
260C2×C8—31no—
261C2×C8—31no—
262C2×C8—121no—
263C2×C8C22×D311yescharacteristic
264C2×C8C22×D311yescharacteristic
265C4⋊C4—31no—
266C4⋊C4—31no—
267C4⋊C4—31no—
268C4⋊C4C22×D311yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—61no—
271C22⋊C4—61no—
272C22⋊C4—61no—
273C22⋊C4—61no—
274C22⋊C4—61no—
275C22⋊C4—61no—
276C22⋊C4—61no—
277C22⋊C4—61no—
278C22⋊C4—61no—
279C42—61no—
280C42—61no—
281C22×C3—21no—
282C22×C3—21no—
283C22×C3—21no—
284C22×C3—41no—
285C22×C3—41no—
286C22×C3—41no—
287C22×C3—41no—
288C22×C3—41no—
289C22×C3—41no—
290C22×C3—41no—
291C22×C3—41no—
292C22×C3—41no—
293C22×C3—84no—
294C22×C3C22×D411yescharacteristic
295C2×D3—21no—
296C2×D3—21no—
297C2×D3—21no—
298C2×D3—21no—
299C2×D3—21no—
300C2×D3—21no—
301C2×D3—41no—
302C2×D3—41no—
303C2×D3—41no—
304C2×D3—84no—
305C3×C4—21no—
306C3×C4—41no—
307C3×C4H(5,𝔽2)11yescharacteristic
308C3×C4H(5,𝔽2)11yescharacteristic
309Dic3—21no—
310Dic3—21no—
311Dic3—41no—
312Dic3—41no—
313Dic3—41no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—31no—
318C23—31no—
319C23—42no—
320C23—61no—
321C23—81no—
322C23—81no—
323C23—122no—
324C23—241no—
325C23—241no—
326C23D3×D411yescharacteristic
327Q8—121no—
328D4—21no—
329D4—21no—
330D4—21no—
331D4—21no—
332D4—21no—
333D4—21no—
334D4—42no—
335D4—61no—
336D4—61no—
337D4—81no—
338D4—121no—
339D4—121no—
340D4—121no—
341D4—122no—
342D4—241no—
343D4—241no—
344D4—241no—
345D4—241no—
346D4—241no—
347D4—241no—
348D4—241no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—31no—
353C2×C4—31no—
354C2×C4—61no—
355C2×C4—61no—
356C2×C4—61no—
357C2×C4—61no—
358C2×C4—61no—
359C2×C4—61no—
360C2×C4—61no—
361C2×C4D3×D411yescharacteristic
362C2×C4C23×D311yescharacteristic
363C8—21no—
364C8—21no—
365C8—61no—
366C8—61no—
367C2×C3—41no—
368C2×C3—81no—
369C2×C3—81no—
370C2×C3—81no—
371C2×C3#53411yescharacteristic
372C2×C3#57611yescharacteristic
373C2×C3#57611yescharacteristic
374D3—41no—
375D3—41no—
376D3—81no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—61no—
390C22—61no—
391C22—61no—
392C22—61no—
393C22—61no—
394C22—61no—
395C22—84no—
396C22—121no—
397C22—121no—
398C22—121no—
399C22—244no—
400C22C2×D3×D411yescharacteristic
401C4—21no—
402C4—41no—
403C4—61no—
404C4—61no—
405C4—121no—
406C4—121no—
407C4—121no—
408C4#100411yescharacteristic
409C4#100411yescharacteristic
410C3#319911yescharacteristic, complemented, Sylow
411C2—41no—
412C2—81no—
413C2—81no—
414C2—81no—
415C2—121no—
416C2—121no—
417C2—241no—
418C2#558911yescharacteristic
419C2#575711yescharacteristic
420C2#577411yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#447C2
socleC22×C3C22×D4
hypercenter#447D3
hypocenterC3#3199
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #5805C2semidirect
2. #5607C2semidirect
3. C3×#447C2semidirect
4. #5217C2semidirect
5. #5159C2semidirect
6. #5157C2semidirect
7. #5156C2semidirect
8. #5156C2semidirect
9. #5035C2semidirect
10. #4784C2semidirect
11. #4782C2semidirect
12. #4779C2semidirect
13. #4775C2semidirect
14. #4737C2semidirect
15. #4727C2semidirect
17. C22×C3×D4C22semidirect
18. C2×D3×D4C22semidirect
29. #938C22semidirect
30. #938C22semidirect
31. #938C22semidirect
37. #926C22semidirect
38. C2×ΓL(1,𝔽25)C22semidirect
39. C2×D3×C8C22semidirect
40. #895C22semidirect
41. #892C22semidirect
44. C3×#102C22semidirect
45. C3×#102C22semidirect
46. C3×C22⋊C8C22semidirect
47. #832C22semidirect
48. #832C22semidirect
49. #832C22semidirect
50. #832C22semidirect
51. #830C22semidirect
52. #821C22semidirect
53. #821C22semidirect
54. (C2×D3)⋊C8C22semidirect
55. (C2×D3)⋊C8C22semidirect
56. #818C22semidirect
57. #817C22semidirect
58. #808C22semidirect
59. #807C22semidirect
73. #447D3semidirect
132. C22×D4C2×D3semidirect
167. #102C2×D3semidirect
168. #102C2×D3semidirect
172. C22⋊C8C2×D3semidirect
410. C3#3199semidirect

Maximal subgroups

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

subgroupindex
#58052
#56072
C3×#4472
#52172
#51592
#51572
#51562
#50352
#47842
#47822
#47792
#47752
#47372
#47272
#31993