Finite Group Databasebeta

← #49131502

Subgroups — #49131502

Order 36224.

418 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31827069C211yesmaximal, characteristic, complemented
2#31826937C211yesmaximal, characteristic, complemented
3#31826788C211yesmaximal, characteristic, complemented
4#31826787C211yesmaximal, characteristic, complemented
5#31826766C211yesmaximal, characteristic, complemented
6#31826765C211yesmaximal, characteristic, complemented
7#31826657C211yesmaximal, characteristic, complemented
8#31826597C211yesmaximal, characteristic, complemented
9#31826597C211yesmaximal, characteristic, complemented
10#31826577C211yesmaximal, characteristic, complemented
11#31826574C211yesmaximal, characteristic, complemented
12D8×Dic283C211yesmaximal, characteristic, complemented
13#31826545C211yesmaximal, characteristic, complemented
14#31826525C211yesmaximal, characteristic, complemented
15#459×C283C211yesmaximal, characteristic, complemented
16C2×D4×D283C2211yescharacteristic
17C2×#17985377C2211yescharacteristic
18C2×#17985377C2211yescharacteristic
19#22583602C2211yescharacteristic, complemented
20#22583591—21no—
21#22583591—21no—
22#22583582—21no—
23#22583582—21no—
24#22583581C2211yescharacteristic
25#22583581C2211yescharacteristic
26C4×D1132C2211yescharacteristic
27D4×Dic283C2211yescharacteristic
28D4×Dic283C2211yescharacteristic, complemented
29D4×Dic283C2211yescharacteristic, complemented
30#22583572C2211yescharacteristic
31C4×C22⋊D283—21no—
32(C22⋊D283)⋊C4—21no—
33#22583568C2211yescharacteristic
34#22583567C2211yescharacteristic
35C2×(C2×D283)⋊C4—21no—
36C22⋊C4×D283—21no—
37#22583544C2211yescharacteristic, complemented
38#22583543C2211yescharacteristic, complemented
39#22583542C2211yescharacteristic, complemented
40#22583541C2211yescharacteristic, complemented
41C4⋊(C283⋊C8)C2211yescharacteristic, complemented
42#22583532C2211yescharacteristic
43#22583532C2211yescharacteristic
44#22583531C2211yescharacteristic
45#22583531C2211yescharacteristic
46#22583531C2211yescharacteristic, complemented
47#22583531C2211yescharacteristic, complemented
48(C2×D283)⋊C8C2211yescharacteristic
49(C2×D283)⋊C8C2211yescharacteristic
50#22583520C2211yescharacteristic, complemented
51C8×Dic283C2211yescharacteristic, complemented
52C2×D8×C283C2211yescharacteristic
53C2×D8×C283C2211yescharacteristic
54#127×C283C2211yescharacteristic, complemented
55C4×D4×C283C2211yescharacteristic
56C4⋊C8×C283C2211yescharacteristic, complemented
57#102×C283C2211yescharacteristic, complemented
58#102×C283C2211yescharacteristic, complemented
59C23×D283—21no—
60C2×D1132C2311yescharacteristic
61C2×C22⋊D283—21no—
62C2×C22⋊D283—21no—
63C2×C22⋊D283—21no—
64D4×D283—22no—
65D4×D283—22no—
66C22×Dic283—21no—
67C22×Dic283—21no—
68C22×Dic283—21no—
69C2×C4×D283—21no—
70C2×C4×D283—21no—
71C2×C4×D283C2311yescharacteristic
72#17985377—22no—
73#17985377—22no—
74C2×C283⋊C8C2311yescharacteristic
75C2×C283⋊C8C2311yescharacteristic
76C4⋊Dic283C2311yescharacteristic
77C4⋊Dic283C2311yescharacteristic
78C4⋊Dic283C2311yescharacteristic
79Dic283⋊C4—21no—
80Dic283⋊C4—21no—
81Dic283⋊C4D411yescharacteristic
82Dic283⋊C4D411yescharacteristic
83C22⋊Dic283—21no—
84C22⋊Dic283—21no—
85C22⋊Dic283—21no—
86C22⋊Dic283—21no—
87C22⋊Dic283—21no—
88(C2×D283)⋊C4—21no—
89(C2×D283)⋊C4—21no—
90(C2×D283)⋊C4—21no—
91(C2×D283)⋊C4D411yescharacteristic
92(C2×D283)⋊C4D411yescharacteristic
93C4×Dic283—21no—
94C4×Dic283C2311yescharacteristic
95C2×D4×C283—21no—
96C2×D4×C283—21no—
97C2×D4×C283C2311yescharacteristic
98C2×D4×C283C2311yescharacteristic
99C2×D4×C283C2311yescharacteristic
100C22×C4×C283—21no—
101D8×C283—22no—
102D8×C283—22no—
103C2×C8×C283C2311yescharacteristic
104C2×C8×C283C2311yescharacteristic
105C4⋊C4×C283C2311yescharacteristic
106C22⋊C4×C283—21no—
107C42×C283C2311yescharacteristic
108C22×D283—21no—
109C22×D283—24no—
110C22×D283—81no—
111C22×D283—81no—
112C22×D283C2×D411yescharacteristic
113D1132—21no—
114D1132#4112yes—
115C22⋊D283—24no—
116C22⋊D283—81no—
117C22⋊D283—81no—
118C2×Dic283—21no—
119C2×Dic283—21no—
120C2×Dic283—21no—
121C2×Dic283—81no—
122C2×Dic283—81no—
123C2×Dic283—81no—
124C2×Dic283C2×D411yescharacteristic
125C2×Dic283C2×D411yescharacteristic
126C2×Dic283C2×D411yescharacteristic
127C4×D283—21no—
128C4×D283—21no—
129C4×D283—81no—
130C4×D283—81no—
131C283⋊C8—21no—
132C283⋊C8—21no—
133C23×C283—21no—
134C23×C283—21no—
135C23×C283—21no—
136D4×C283—21no—
137D4×C283—21no—
138D4×C283—21no—
139D4×C283—21no—
140D4×C283—21no—
141D4×C283—81no—
142D4×C283—81no—
143D4×C283#4112yes—
144C2×C4×C283—21no—
145C2×C4×C283—81no—
146C2×C4×C283C2×D411yescharacteristic
147C2×C4×C283C2×D411yescharacteristic
148C2×C4×C283C2411yescharacteristic
149C8×C283—21no—
150C8×C283—21no—
151C22×C283—22no—
152C22×C283—41no—
153C22×C283—41no—
154C22×C283—41no—
155C22×C283—41no—
156C22×C283—41no—
157C22×C283—41no—
158C22×C283—41no—
159C22×C283—41no—
160C22×C283C22×D411yescharacteristic
161C2×D283—21no—
162C2×D283—22no—
163C2×D283—21no—
164C2×D283—21no—
165C2×D283—41no—
166C2×D283—41no—
167C2×D283—84no—
168C4×C283—21no—
169C4×C283—21no—
170C4×C283—41no—
171C4×C283C2×#4111yescharacteristic
172C4×C283C2×#4111yescharacteristic
173Dic283—21no—
174Dic283—21no—
175Dic283—21no—
176Dic283—41no—
177Dic283—41no—
178Dic283—41no—
179C2×C283—42no—
180C2×C283—81no—
181C2×C283—81no—
182C2×C283#52511yescharacteristic
183C2×C283#57611yescharacteristic
184C2×C283#57611yescharacteristic
185D283—41no—
186D283—42no—
187C283#312211yescharacteristic, complemented, Sylow
188#3122—2831nomaximal, Sylow
189#540—2831no—
190#518—2831no—
191#482—2831no—
192#482—2831no—
193#478—2831no—
194#478—2831no—
195#459—2831no—
196#459D28311yescharacteristic, complemented
197#447—2831no—
198#447—2831no—
199#442—2831no—
200#442—2831no—
201C4×D8—2831no—
202C4×D8—2831no—
203#433—2831no—
204C22×D4—2831no—
205C2×D8—2831no—
206C2×D8—2831no—
207C2×D8C2×D28311yescharacteristic
208C2×D8C2×D28311yescharacteristic
209#127C2×D28311yescharacteristic, complemented
210#125—2831no—
211#123—5661no—
212#123—5661no—
213#121—2831no—
214#121—2831no—
215#121—5661no—
216#121—5661no—
217C4×D4—2831no—
218C4×D4—2831no—
219C4×D4—2831no—
220C4×D4—2831no—
221C4×D4—2831no—
222C4×D4—5661no—
223C4×D4—5661no—
224C4×D4C2×D28311yescharacteristic
225#117—2831no—
226#117—2831no—
227C2×C22⋊C4—5661no—
228C2×C22⋊C4—5661no—
229#107—2831no—
230#107—2831no—
231#106—2831no—
232#106—2831no—
233C4⋊C8—2831no—
234C4⋊C8C2×D28311yescharacteristic, complemented
235#102—2831no—
236#102—2831no—
237#102—2831no—
238#102—2831no—
239#102—2831no—
240#102—2831no—
241#102C2×D28311yescharacteristic, complemented
242#102C2×D28311yescharacteristic, complemented
243C22⋊C8—2831no—
244C22⋊C8—2831no—
245#97—2831no—
246C4×C8—2831no—
247C24—5661no—
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—2831no—
251C2×D4—5661no—
252C2×D4—5662no—
253C2×D4—5662no—
254C2×D4—5661no—
255C2×D4—5661no—
256C2×D4C22×D28311yescharacteristic
257C2×D4C22×D28311yescharacteristic
258C2×D4C22×D28311yescharacteristic
259C22×C4—21no—
260C22×C4—2831no—
261C22×C4—5661no—
262C22×C4—5661no—
263C22×C4—5661no—
264C22×C4—5661no—
265C22×C4—5661no—
266D8—22no—
267D8—22no—
268D8—5662no—
269D8—5662no—
270C2×C8—2831no—
271C2×C8—2831no—
272C2×C8C22×D28311yescharacteristic
273C2×C8C22×D28311yescharacteristic
274C4⋊C4—2831no—
275C4⋊C4—2831no—
276C4⋊C4—2831no—
277C4⋊C4—2831no—
278C4⋊C4—2831no—
279C4⋊C4—5661no—
280C4⋊C4—5661no—
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—2831no—
294C42—5661no—
295C42C22×D28311yescharacteristic
296C23—21no—
297C23—21no—
298C23—21no—
299C23—2831no—
300C23—5664no—
301C23—5661no—
302C23—22641no—
303C23—22641no—
304D4—21no—
305D4—21no—
306D4—21no—
307D4—21no—
308D4—21no—
309D4—81no—
310D4—81no—
311D4—2832no—
312D4—5664no—
313D4—5661no—
314D4—22641no—
315D4—22641no—
316D4#1798539112yes—
317C2×C4—21no—
318C2×C4—81no—
319C2×C4—2831no—
320C2×C4—2831no—
321C2×C4—2831no—
322C2×C4—5661no—
323C2×C4—5661no—
324C2×C4—5661no—
325C2×C4—5661no—
326C2×C4—5661no—
327C2×C4—22641no—
328C2×C4—22641no—
329C2×C4—22641no—
330C2×C4—22641no—
331C2×C4—22641no—
332C2×C4D4×D28311yescharacteristic
333C2×C4D4×D28311yescharacteristic
334C2×C4C23×D28311yescharacteristic
335C8—21no—
336C8—21no—
337C8—5661no—
338C8—5661no—
339C22—22no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—5661no—
349C22—5662no—
350C22—5661no—
351C22—5661no—
352C22—11321no—
353C22—11321no—
354C22—22644no—
355C22C2×D4×D28311yescharacteristic
356C4—21no—
357C4—21no—
358C4—41no—
359C4—5661no—
360C4—5661no—
361C4—5661no—
362C4—11321no—
363C4—11321no—
364C4—11321no—
365C4#41×D28311yescharacteristic
366C4C2×#1798539111yescharacteristic
367C2—42no—
368C2—81no—
369C2—81no—
370C2—11321no—
371C2—11322no—
372C2#3182697511yescharacteristic
373C2#3182726611yescharacteristic
374C2#3182726911yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#459×C283C2
socleC22×C283C22×D4
hypercenter#459D283
hypocenterC283#3122
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #31827069C2semidirect
2. #31826937C2semidirect
3. #31826788C2semidirect
4. #31826787C2semidirect
5. #31826766C2semidirect
6. #31826765C2semidirect
7. #31826657C2semidirect
8. #31826597C2semidirect
9. #31826597C2semidirect
10. #31826577C2semidirect
11. #31826574C2semidirect
12. D8×Dic283C2semidirect
13. #31826545C2semidirect
14. #31826525C2semidirect
15. #459×C283C2semidirect
19. #22583602C22semidirect
28. D4×Dic283C22semidirect
29. D4×Dic283C22semidirect
37. #22583544C22semidirect
38. #22583543C22semidirect
39. #22583542C22semidirect
40. #22583541C22semidirect
41. C4⋊(C283⋊C8)C22semidirect
46. #22583531C22semidirect
47. #22583531C22semidirect
50. #22583520C22semidirect
51. C8×Dic283C22semidirect
54. #127×C283C22semidirect
56. C4⋊C8×C283C22semidirect
57. #102×C283C22semidirect
58. #102×C283C22semidirect
187. C283#3122semidirect
196. #459D283semidirect
209. #127C2×D283semidirect
234. C4⋊C8C2×D283semidirect
241. #102C2×D283semidirect
242. #102C2×D283semidirect

Maximal subgroups

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

subgroupindex
#318270692
#318269372
#318267882
#318267872
#318267662
#318267652
#318266572
#318265972
#318265772
#318265742
D8×Dic2832
#318265452
#318265252
#459×C2832
#3122283