Finite Group Databasebeta

← #49133298

Subgroups — #49133298

Order 36224.

402 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31827183C211yesmaximal, characteristic, complemented
2#31827109C211yesmaximal, characteristic
3#31826823C211yesmaximal, characteristic, complemented
4#31826806C211yesmaximal, characteristic, complemented
5#31826788C211yesmaximal, characteristic, complemented
6#31826753C211yesmaximal, characteristic, complemented
7#31826723C211yesmaximal, characteristic, complemented
8#31826693C211yesmaximal, characteristic, complemented
9#31826686C211yesmaximal, characteristic, complemented
10#31826628C211yesmaximal, characteristic, complemented
11#31826616C211yesmaximal, characteristic, complemented
12#31826562C211yesmaximal, characteristic, complemented
13#31826534C211yesmaximal, characteristic, complemented
14#31826458C211yesmaximal, characteristic, complemented
15#418×C283C211yesmaximal, characteristic, complemented
16C2×#17985391C2211yescharacteristic
17C2×Dic1132C2211yescharacteristic, complemented
18C2×#17985380C2211yescharacteristic
19C2×#17985380C2211yescharacteristic
20C2×C8×D283C2211yescharacteristic, complemented
21#22583609C2211yescharacteristic, complemented
22#22583604—21no—
23#22583603—21no—
24#22583597C2211yescharacteristic
25#22583596—21no—
26#22583591—21no—
27#22583590C2211yescharacteristic, complemented
28#22583589C2211yescharacteristic
29#22583589C2211yescharacteristic
30#22583582—21no—
31#22583581C2211yescharacteristic
32#22583579—21no—
33C4×Dic566—21no—
34#22583575C2211yescharacteristic, complemented
35D4×Dic283C2211yescharacteristic
36C4×C22⋊D283C2211yescharacteristic
37#22583568C2211yescharacteristic, complemented
38C2×C22⋊Dic283—21no—
39#22583543C2211yescharacteristic, complemented
40#22583542C2211yescharacteristic, complemented
41#22583541C2211yescharacteristic, complemented
42Dic283⋊C8C2211yescharacteristic, complemented
43#22583535C2211yescharacteristic, complemented
44#22583535C2211yescharacteristic, complemented
45#22583533C2211yescharacteristic, complemented
46#22583533C2211yescharacteristic, complemented
47#22583531C2211yescharacteristic
48#22583531C2211yescharacteristic
49C22⋊(C283⋊C8)C2211yescharacteristic
50(C2×D283)⋊C8C2211yescharacteristic, complemented
51#22583520C2211yescharacteristic, complemented
52C22×D4×C283C2211yescharacteristic
53C2×M16×C283C2211yescharacteristic
54#117×C283C2211yescharacteristic
55#102×C283C2211yescharacteristic
56#102×C283C2211yescharacteristic
57#102×C283C2211yescharacteristic
58#102×C283C2211yescharacteristic
59#17985391—22no—
60#17985391—41no—
61C2×Dic566—21no—
62C2×Dic566C2311yescharacteristic
63C2×Dic566C2311yescharacteristic
64C2×C22⋊D283—21no—
65C2×C22⋊D283—21no—
66C2×C22⋊D283D411yescharacteristic
67C22×Dic283—21no—
68C22×Dic283—21no—
69C2×C4×D283C2311yescharacteristic
70Dic1132—41no—
71#17985380—22no—
72#17985380—41no—
73C2×C283⋊C8C2311yescharacteristic
74C2×C283⋊C8C2311yescharacteristic
75C8×D283—41no—
76C4⋊Dic283—21no—
77C4⋊Dic283C2311yescharacteristic
78C4⋊Dic283C2311yescharacteristic
79Dic283⋊C4—21no—
80Dic283⋊C4—21no—
81Dic283⋊C4—21no—
82Dic283⋊C4—21no—
83Dic283⋊C4D411yescharacteristic
84C22⋊Dic283—21no—
85C22⋊Dic283—21no—
86C22⋊Dic283—21no—
87C22⋊Dic283—21no—
88C22⋊Dic283—21no—
89C22⋊Dic283—21no—
90C22⋊Dic283D411yescharacteristic
91(C2×D283)⋊C4—21no—
92(C2×D283)⋊C4—21no—
93(C2×D283)⋊C4D411yescharacteristic
94C4×Dic283—21no—
95C4×Dic283—21no—
96C4×Dic283C2311yescharacteristic
97C24×C283—21no—
98C2×D4×C283—22no—
99C2×D4×C283—21no—
100C2×D4×C283—41no—
101C2×D4×C283C2311yescharacteristic
102C2×D4×C283C2311yescharacteristic
103C22×C4×C283C2311yescharacteristic
104M16×C283—41no—
105C2×C8×C283C2311yescharacteristic
106C2×C8×C283C2311yescharacteristic
107C4⋊C4×C283C2311yescharacteristic
108C4⋊C4×C283C2311yescharacteristic
109C22⋊C4×C283—21no—
110C42×C283—21no—
111C22×D283C2×D411yescharacteristic
112Dic566—21no—
113Dic566—21no—
114Dic566—21no—
115Dic566—21no—
116Dic566—42no—
117C22⋊D283—41no—
118C22⋊D283—42no—
119C22⋊D283—81no—
120C2×Dic283—21no—
121C2×Dic283—21no—
122C2×Dic283—21no—
123C2×Dic283—21no—
124C2×Dic283—42no—
125C2×Dic283—81no—
126C2×Dic283C2×D411yescharacteristic
127C2×Dic283C2×D411yescharacteristic
128C2×Dic283C2×D411yescharacteristic
129C4×D283—21no—
130C4×D283—21no—
131C283⋊C8—21no—
132C283⋊C8—21no—
133C23×C283—21no—
134C23×C283—21no—
135C23×C283—42no—
136C23×C283—42no—
137C23×C283—81no—
138C23×C283C2×D411yescharacteristic
139D4×C283—21no—
140D4×C283—21no—
141D4×C283—21no—
142D4×C283—42no—
143D4×C283#4112yes—
144C2×C4×C283—21no—
145C2×C4×C283—21no—
146C2×C4×C283—21no—
147C2×C4×C283—21no—
148C2×C4×C283C2×D411yescharacteristic
149C2×C4×C283C2411yescharacteristic
150C8×C283—21no—
151C8×C283—21no—
152C22×C283—21no—
153C22×C283—22no—
154C22×C283—21no—
155C22×C283—21no—
156C22×C283—41no—
157C22×C283—41no—
158C22×C283—41no—
159C22×C283—41no—
160C22×C283—41no—
161C22×C283—84no—
162C22×C283C22×D411yescharacteristic
163C2×D283—21no—
164C2×D283—21no—
165C2×D283—21no—
166C4×C283—21no—
167C4×C283—41no—
168C4×C283—41no—
169C4×C283C2×#4111yescharacteristic
170C4×C283H(5,𝔽2)11yescharacteristic
171Dic283—21no—
172Dic283—21no—
173Dic283—21no—
174Dic283—41no—
175Dic283—41no—
176Dic283—41no—
177Dic283—41no—
178C2×C283—41no—
179C2×C283—42no—
180C2×C283—81no—
181C2×C283#54611yescharacteristic
182C2×C283#57511yescharacteristic
183C2×C283#57711yescharacteristic
184D283—41no—
185C283#331111yescharacteristic, complemented, Sylow
186#3311—2831nomaximal, Sylow
187#556—2831no—
188#546—2831no—
189#489—2831no—
190#485—2831no—
191#482—2831no—
192#476—2831no—
193#470—2831no—
194#465—2831no—
195#464—2831no—
196#452—2831no—
197#450—2831no—
198#440—2831no—
199#435—2831no—
200#418D28311yescharacteristic, complemented
201#416—2831no—
202C2×#41—2831no—
203C22×D4C2×D28311yescharacteristic
204C2×Q16—2831no—
205C2×SD16—2831no—
206C2×SD16—2831no—
207C2×M16C2×D28311yescharacteristic
208C22×C8—2831no—
209C4⋊Q8—2831no—
210#126—5661no—
211#126—5661no—
212#124—2831no—
213#124—5661no—
214#123—5661no—
215C22⋊Q8—2831no—
216C22⋊Q8—2831no—
217C22⋊Q8—2831no—
218#121—2831no—
219#121—5661no—
220#120—5661no—
221C4×Q8—2831no—
222C4×Q8—5661no—
223C4×D4—2831no—
224C4×D4—2831no—
225#117—2831no—
226#117C2×D28311yescharacteristic
227C2×C22⋊C4—5661no—
228#107—2831no—
229#106—2831no—
230#106—2831no—
231C4⋊C8—2831no—
232#103—2831no—
233#103—2831no—
234#103—2831no—
235#103—2831no—
236#102—2831no—
237#102—2831no—
238#102C2×D28311yescharacteristic
239#102C2×D28311yescharacteristic
240#102C2×D28311yescharacteristic
241#102C2×D28311yescharacteristic
242C22⋊C8—2831no—
243C22⋊C8—2831no—
244#97—2831no—
245C24—21no—
246#41—5662no—
247#41—11321no—
248C2×Q8—2831no—
249C2×Q8—2831no—
250C2×Q8—5661no—
251C2×D4—22no—
252C2×D4—21no—
253C2×D4—41no—
254C2×D4—2831no—
255C2×D4—5661no—
256C2×D4—5661no—
257C2×D4C22×D28311yescharacteristic
258C2×D4C22×D28311yescharacteristic
259C22×C4—2831no—
260C22×C4—5661no—
261C22×C4—5661no—
262C22×C4C22×D28311yescharacteristic
263Q16—11321no—
264SD16—5662no—
265SD16—11321no—
266M16—41no—
267C2×C8—2831no—
268C2×C8—2831no—
269C2×C8—11321no—
270C2×C8C22×D28311yescharacteristic
271C2×C8C22×D28311yescharacteristic
272C4⋊C4—2831no—
273C4⋊C4—2831no—
274C4⋊C4—2831no—
275C4⋊C4—5661no—
276C4⋊C4—5661no—
277C4⋊C4—5661no—
278C4⋊C4—5661no—
279C4⋊C4—5661no—
280C4⋊C4C22×D28311yescharacteristic
281C4⋊C4C22×D28311yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—2831no—
284C22⋊C4—2831no—
285C22⋊C4—5661no—
286C22⋊C4—5661no—
287C22⋊C4—5661no—
288C22⋊C4—5661no—
289C22⋊C4—5661no—
290C22⋊C4—5661no—
291C22⋊C4—5661no—
292C22⋊C4—5661no—
293C42—21no—
294C42—2831no—
295C42—5661no—
296C42—5661no—
297C23—21no—
298C23—21no—
299C23—42no—
300C23—42no—
301C23—81no—
302C23—2831no—
303C23D4×D28311yescharacteristic
304Q8—5661no—
305Q8—5661no—
306Q8—5661no—
307Q8—5661no—
308Q8—11322no—
309D4—21no—
310D4—21no—
311D4—21no—
312D4—42no—
313D4—11321no—
314D4—11322no—
315D4—22641no—
316D4#1798539212yes—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—2831no—
322C2×C4—2831no—
323C2×C4—2831no—
324C2×C4—5661no—
325C2×C4—5661no—
326C2×C4—5661no—
327C2×C4—5661no—
328C2×C4—5661no—
329C2×C4—5661no—
330C2×C4—11322no—
331C2×C4—22641no—
332C2×C4D4×D28311yescharacteristic
333C2×C4C23×D28311yescharacteristic
334C8—21no—
335C8—21no—
336C8—5661no—
337C8—5661no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—22no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—84no—
348C22—5661no—
349C22—5661no—
350C22—5661no—
351C22C2×D4×D28311yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—5661no—
356C4—5661no—
357C4—5661no—
358C4—11321no—
359C4—11321no—
360C4—11321no—
361C4—11321no—
362C4C2×#1798539211yescharacteristic
363C4#2258365511yescharacteristic
364C2—41no—
365C2—42no—
366C2—81no—
367C2—11321no—
368C2#3182710511yescharacteristic
369C2#3182726111yescharacteristic
370C2#3182727511yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#418×C283C2
socleC22×C283C22×D4
prime radical#31827109C2
hypercenter#418D283
hypocenterC283#3311
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. #31827183C2semidirect
3. #31826823C2semidirect
4. #31826806C2semidirect
5. #31826788C2semidirect
6. #31826753C2semidirect
7. #31826723C2semidirect
8. #31826693C2semidirect
9. #31826686C2semidirect
10. #31826628C2semidirect
11. #31826616C2semidirect
12. #31826562C2semidirect
13. #31826534C2semidirect
14. #31826458C2semidirect
15. #418×C283C2semidirect
17. C2×Dic1132C22semidirect
20. C2×C8×D283C22semidirect
21. #22583609C22semidirect
27. #22583590C22semidirect
34. #22583575C22semidirect
37. #22583568C22semidirect
39. #22583543C22semidirect
40. #22583542C22semidirect
41. #22583541C22semidirect
42. Dic283⋊C8C22semidirect
43. #22583535C22semidirect
44. #22583535C22semidirect
45. #22583533C22semidirect
46. #22583533C22semidirect
50. (C2×D283)⋊C8C22semidirect
51. #22583520C22semidirect
185. C283#3311semidirect
200. #418D283semidirect

Maximal subgroups

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

subgroupindex
#318271832
#318271092
#318268232
#318268062
#318267882
#318267532
#318267232
#318266932
#318266862
#318266282
#318266162
#318265622
#318265342
#318264582
#418×C2832
#3311283