Finite Group Databasebeta

← #81400

Subgroups — #81400

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#5607C211yesmaximal, characteristic, complemented
2#5525C211yesmaximal, characteristic, complemented
3C3×#482C211yesmaximal, characteristic, complemented
4#5036C211yesmaximal, characteristic, complemented
5#5003C211yesmaximal, characteristic, complemented
6#4885C211yesmaximal, characteristic, complemented
7#4883C211yesmaximal, characteristic, complemented
8#4866C211yesmaximal, characteristic, complemented
9#4865C211yesmaximal, characteristic, complemented
10#4782C211yesmaximal, characteristic, complemented
11#4775C211yesmaximal, characteristic, complemented
12#4773C211yesmaximal, characteristic, complemented
13#4773C211yesmaximal, characteristic, complemented
14#4732C211yesmaximal, characteristic, complemented
15#4727C211yesmaximal, characteristic, complemented
16#3199—31nomaximal, Sylow
17C2×D3×D4C2211yescharacteristic, complemented
18C2×#235C2211yescharacteristic
19C22×D12C2211yescharacteristic, complemented
20C3×#121C2211yescharacteristic
21C3×#117C2211yescharacteristic, complemented
22#940—21no—
23#938C2211yescharacteristic, complemented
24#937—21no—
25C2×#213C2211yescharacteristic
26C2×#213C2211yescharacteristic
27#930C2211yescharacteristic
28C2×D24C2211yescharacteristic
29C2×D24C2211yescharacteristic
30C2×ΓL(1,𝔽25)C2211yescharacteristic, complemented
31C2×D3×C8C2211yescharacteristic, complemented
32#895C2211yescharacteristic
33#895C2211yescharacteristic
34#895C2211yescharacteristic, complemented
35#895C2211yescharacteristic, complemented
36#895C2211yescharacteristic, complemented
37#892C2211yescharacteristic, complemented
38#885—21no—
39#882—21no—
40#882—21no—
41#882—21no—
42#875—21no—
43#874—21no—
44C3×#107C2211yescharacteristic, complemented
45C3×#106C2211yescharacteristic, complemented
46C3×#102C2211yescharacteristic, complemented
47C3×#102C2211yescharacteristic, complemented
48C3×C22⋊C8C2211yescharacteristic, complemented
49#830C2211yescharacteristic, complemented
50#821C2211yescharacteristic, complemented
51#821C2211yescharacteristic, complemented
52(C2×D3)⋊C8C2211yescharacteristic, complemented
53(C2×D3)⋊C8C2211yescharacteristic, complemented
54#809C2211yescharacteristic, complemented
55#809C2211yescharacteristic, complemented
56#809C2211yescharacteristic, complemented
57#809C2211yescharacteristic, complemented
58#808C2211yescharacteristic, complemented
59#807C2211yescharacteristic, complemented
60#535—31no—
61#535—31no—
62#482—31no—
63#482—31no—
64#482—31no—
65#482D311yescharacteristic, complemented
66#469—31no—
67#469—31no—
68#466—31no—
69#466—31no—
70#447—31no—
71#447—31no—
72#447—31no—
73#447—31no—
74#408—31no—
75C23×D3—21no—
76C23×D3—21no—
77C2×C3×D4—21no—
78C2×C3×D4C2311yescharacteristic
79C22×C3×C4C2311yescharacteristic
80C2×#67—21no—
81C2×#67—21no—
82C2×#67—21no—
83C2×#67D411yescharacteristic
84C2×#67D411yescharacteristic
85D3×D4—41no—
86D3×D4—41no—
87#235—41no—
88#235—41no—
89C2×D12—21no—
90C2×D12—21no—
91C2×D12—21no—
92C2×D12—21no—
93C2×D12—21no—
94C2×D12—21no—
95C2×D12—41no—
96C2×D12—41no—
97C2×D12D411yescharacteristic
98C2×D12C2311yescharacteristic
99C2×D12C2311yescharacteristic
100C2×D12C2311yescharacteristic
101C2×C4×D3C2311yescharacteristic
102C2×C4×D3C2311yescharacteristic
103C2×Dic6D411yescharacteristic
104C2×C3×C8C2311yescharacteristic
105C2×C3×C8C2311yescharacteristic
106C3×C4⋊C4C2311yescharacteristic
107C3×C4⋊C4C2311yescharacteristic
108C3×C4⋊C4C2311yescharacteristic
109C3×C22⋊C4—21no—
110C3×C22⋊C4—21no—
111C3×C42—21no—
112#217—21no—
113#213—41no—
114#213—41no—
115(C2×D3)⋊C4—21no—
116(C2×D3)⋊C4—21no—
117(C2×D3)⋊C4—21no—
118(C2×D3)⋊C4—21no—
119(C2×D3)⋊C4—21no—
120(C2×D3)⋊C4—21no—
121(C2×D3)⋊C4—21no—
122C4⋊Dic3C2311yescharacteristic
123C4×Dic3—21no—
124C2×C3⋊C8C2311yescharacteristic
125C2×C3⋊C8C2311yescharacteristic
126D24—41no—
127D24—41no—
128ΓL(1,𝔽25)—41no—
129D3×C8—41no—
130C2×#41—31no—
131C22×D4—31no—
132C22×D4—31no—
133C2×D8—31no—
134C2×D8—31no—
135C2×D8—31no—
136C2×D8—31no—
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#117C2×D311yescharacteristic, complemented
157#107—31no—
158#107C2×D311yescharacteristic, complemented
159#106—31no—
160#106C2×D311yescharacteristic, complemented
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×C3C2×D411yescharacteristic
175C22×D3—21no—
176C22×D3—21no—
177C22×D3—21no—
178C22×D3—42no—
179C22×D3—42no—
180C22×D3—81no—
181C22×D3—81no—
182C22×D3—81no—
183C22×D3—81no—
184C22×D3C2×D411yescharacteristic
185C22×D3C2×D411yescharacteristic
186C3×D4—21no—
187C3×D4—21no—
188C3×D4—81no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4C2×D411yescharacteristic
195C2×C3×C4C2411yescharacteristic
196#67—41no—
197#67—41no—
198#67—42no—
199#67—81no—
200#67—81no—
201C2×Dic3—21no—
202C2×Dic3C2×D411yescharacteristic
203C2×Dic3C2×D411yescharacteristic
204D12—21no—
205D12—21no—
206D12—21no—
207D12—21no—
208D12—21no—
209D12—21no—
210D12—41no—
211D12—42no—
212D12—81no—
213D12—81no—
214D12—81no—
215D12—81no—
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—61no—
227C24—61no—
228#41—121no—
229#41—121no—
230C2×Q8—31no—
231C2×D4—21no—
232C2×D4—31no—
233C2×D4—31no—
234C2×D4—31no—
235C2×D4—31no—
236C2×D4—31no—
237C2×D4—31no—
238C2×D4—61no—
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×D4—121no—
250C2×D4—121no—
251C2×D4C22×D311yescharacteristic
252C22×C4—31no—
253C22×C4—31no—
254C22×C4C22×D311yescharacteristic
255D8—121no—
256D8—121no—
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⋊C4C22×D311yescharacteristic
267C4⋊C4C22×D311yescharacteristic
268C4⋊C4C22×D311yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—21no—
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—21no—
280C42—61no—
281C22×C3—21no—
282C22×C3—21no—
283C22×C3—21no—
284C22×C3—41no—
285C22×C3—41no—
286C22×C3—41no—
287C22×C3C22×D411yescharacteristic
288C2×D3—21no—
289C2×D3—21no—
290C2×D3—21no—
291C2×D3—21no—
292C2×D3—21no—
293C2×D3—21no—
294C2×D3—41no—
295C2×D3—41no—
296C2×D3—41no—
297C2×D3—41no—
298C2×D3—41no—
299C2×D3—41no—
300C2×D3—41no—
301C2×D3—41no—
302C2×D3—41no—
303C2×D3—84no—
304C2×D3—84no—
305C3×C4—21no—
306C3×C4—41no—
307C3×C4—41no—
308C3×C4—41no—
309C3×C4H(5,𝔽2)11yescharacteristic
310C3×C4H(5,𝔽2)11yescharacteristic
311Dic3—21no—
312Dic3—21no—
313Dic3—41no—
314C23—21no—
315C23—31no—
316C23—31no—
317C23—61no—
318C23—61no—
319C23—61no—
320C23—122no—
321C23—122no—
322C23—241no—
323C23—241no—
324C23—241no—
325C23—241no—
326C23D3×D411yescharacteristic
327Q8—121no—
328D4—21no—
329D4—21no—
330D4—61no—
331D4—61no—
332D4—61no—
333D4—61no—
334D4—61no—
335D4—61no—
336D4—81no—
337D4—121no—
338D4—122no—
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—21no—
353C2×C4—21no—
354C2×C4—31no—
355C2×C4—31no—
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#53411yescharacteristic
370C2×C3#57611yescharacteristic
371C2×C3#57611yescharacteristic
372D3—41no—
373D3—41no—
374D3—81no—
375D3—81no—
376D3—81no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—61no—
384C22—61no—
385C22—61no—
386C22—61no—
387C22—61no—
388C22—61no—
389C22—121no—
390C22—121no—
391C22—121no—
392C22—121no—
393C22—121no—
394C22—121no—
395C22—121no—
396C22—121no—
397C22—121no—
398C22—244no—
399C22—244no—
400C22C2×D3×D411yescharacteristic
401C4—21no—
402C4—41no—
403C4—41no—
404C4—41no—
405C4—61no—
406C4—61no—
407C4—121no—
408C4#100911yescharacteristic
409C4#100911yescharacteristic
410C3#319911yescharacteristic, complemented, Sylow
411C2—41no—
412C2—81no—
413C2—121no—
414C2—121no—
415C2—241no—
416C2—241no—
417C2—241no—
418C2#565311yescharacteristic
419C2#576911yescharacteristic
420C2#577411yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#482C2
socleC22×C3C22×D4
hypercenter#482D3
hypocenterC3#3199
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #482

Complements

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

NH ≅ G/Ntype
1. #5607C2semidirect
2. #5525C2semidirect
3. C3×#482C2semidirect
4. #5036C2semidirect
5. #5003C2semidirect
6. #4885C2semidirect
7. #4883C2semidirect
8. #4866C2semidirect
9. #4865C2semidirect
10. #4782C2semidirect
11. #4775C2semidirect
12. #4773C2semidirect
13. #4773C2semidirect
14. #4732C2semidirect
15. #4727C2semidirect
17. C2×D3×D4C22semidirect
19. C22×D12C22semidirect
21. C3×#117C22semidirect
23. #938C22semidirect
30. C2×ΓL(1,𝔽25)C22semidirect
31. C2×D3×C8C22semidirect
34. #895C22semidirect
35. #895C22semidirect
36. #895C22semidirect
37. #892C22semidirect
44. C3×#107C22semidirect
45. C3×#106C22semidirect
46. C3×#102C22semidirect
47. C3×#102C22semidirect
48. C3×C22⋊C8C22semidirect
49. #830C22semidirect
50. #821C22semidirect
51. #821C22semidirect
52. (C2×D3)⋊C8C22semidirect
53. (C2×D3)⋊C8C22semidirect
54. #809C22semidirect
55. #809C22semidirect
56. #809C22semidirect
57. #809C22semidirect
58. #808C22semidirect
59. #807C22semidirect
65. #482D3semidirect
156. #117C2×D3semidirect
158. #107C2×D3semidirect
160. #106C2×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
#56072
#55252
C3×#4822
#50362
#50032
#48852
#48832
#48662
#48652
#47822
#47752
#47732
#47322
#47272
#31993