Finite Group Databasebeta

← #49131200

Subgroups — #49131200

Order 36224.

462 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#121×D283C211yesmaximal, characteristic
2#117×D283C211yesmaximal, characteristic, complemented
3#31826790C211yesmaximal, characteristic, complemented
4#31826789C211yesmaximal, characteristic, complemented
5#31826786C211yesmaximal, characteristic, complemented
6#31826785C211yesmaximal, characteristic, complemented
7#31826782C211yesmaximal, characteristic, complemented
8#31826780C211yesmaximal, characteristic, complemented
9#31826777C211yesmaximal, characteristic, complemented
10#31826506C211yesmaximal, characteristic, complemented
11#31826504C211yesmaximal, characteristic, complemented
12#31826468C211yesmaximal, characteristic, complemented
13#31826468C211yesmaximal, characteristic, complemented
14#31826427C211yesmaximal, characteristic, complemented
15#482×C283C211yesmaximal, characteristic, complemented
16C2×D4×D283—21no—
17C2×D4×D283C2211yescharacteristic
18C22×C4×D283C2211yescharacteristic
19C2×#17985375C2211yescharacteristic
20C2×#17985375C2211yescharacteristic
21#22583585C2211yescharacteristic
22#22583584C2211yescharacteristic
23#22583583—21no—
24#22583582—21no—
25#22583581C2211yescharacteristic
26#22583569C2211yescharacteristic, complemented
27#22583568C2211yescharacteristic
28#22583568C2211yescharacteristic
29#22583567—21no—
30#22583566—21no—
31C4⋊C4×D283C2211yescharacteristic
32C4⋊C4×D283C2211yescharacteristic, complemented
33C4⋊C4×D283C2211yescharacteristic, complemented
34C22⋊C4×D283—21no—
35C22⋊C4×D283—21no—
36C42×D283—21no—
37#22583544C2211yescharacteristic, complemented
38#22583543C2211yescharacteristic, complemented
39#22583543C2211yescharacteristic, complemented
40#22583542C2211yescharacteristic, complemented
41#22583541C2211yescharacteristic, complemented
42#22583541C2211yescharacteristic, complemented
43#22583532C2211yescharacteristic, complemented
44#22583532C2211yescharacteristic, complemented
45#22583531C2211yescharacteristic, complemented
46#22583531C2211yescharacteristic, complemented
47#22583530C2211yescharacteristic, complemented
48#22583530C2211yescharacteristic, complemented
49C22⋊(C283⋊C8)C2211yescharacteristic, complemented
50(C2×D283)⋊C8C2211yescharacteristic, complemented
51(C2×D283)⋊C8C2211yescharacteristic, complemented
52#121×C283C2211yescharacteristic
53#117×C283C2211yescharacteristic, complemented
54#107×C283C2211yescharacteristic, complemented
55#106×C283C2211yescharacteristic, complemented
56#102×C283C2211yescharacteristic, complemented
57#102×C283C2211yescharacteristic, complemented
58C22⋊C8×C283C2211yescharacteristic, complemented
59C23×D283—21no—
60C23×D283D411yescharacteristic
61C2×D1132—21no—
62C2×D1132C2311yescharacteristic
63C2×C22⋊D283—21no—
64C2×C22⋊D283—21no—
65C2×C22⋊D283—21no—
66D4×D283—22no—
67D4×D283—22no—
68D4×D283—44no—
69C22×Dic283D411yescharacteristic
70C2×C4×D283—22no—
71C2×C4×D283—22no—
72C2×C4×D283—21no—
73C2×C4×D283—21no—
74C2×C4×D283—21no—
75C2×C4×D283D411yescharacteristic
76C2×C4×D283D411yescharacteristic
77C2×C4×D283C2311yescharacteristic
78C2×C4×D283C2311yescharacteristic
79#17985375—22no—
80#17985375—22no—
81C2×C283⋊C8C2311yescharacteristic
82C2×C283⋊C8C2311yescharacteristic
83C4⋊Dic283C2311yescharacteristic
84C4⋊Dic283C2311yescharacteristic
85C4⋊Dic283C2311yescharacteristic
86Dic283⋊C4—21no—
87Dic283⋊C4—21no—
88Dic283⋊C4—21no—
89C22⋊Dic283—21no—
90C22⋊Dic283—21no—
91(C2×D283)⋊C4—21no—
92(C2×D283)⋊C4—21no—
93(C2×D283)⋊C4—21no—
94(C2×D283)⋊C4—21no—
95C4×Dic283—21no—
96C4×Dic283—21no—
97C4×Dic283—21no—
98C2×D4×C283—21no—
99C2×D4×C283C2311yescharacteristic
100C22×C4×C283C2311yescharacteristic
101C2×C8×C283C2311yescharacteristic
102C2×C8×C283C2311yescharacteristic
103C4⋊C4×C283C2311yescharacteristic
104C4⋊C4×C283C2311yescharacteristic
105C4⋊C4×C283C2311yescharacteristic
106C22⋊C4×C283—21no—
107C22⋊C4×C283—21no—
108C42×C283—21no—
109C22×D283—22no—
110C22×D283—22no—
111C22×D283—21no—
112C22×D283—41no—
113C22×D283—42no—
114C22×D283—42no—
115C22×D283—42no—
116C22×D283C2×D411yescharacteristic
117C22×D283C2×D411yescharacteristic
118D1132—21no—
119D1132—21no—
120D1132—42no—
121C22⋊D283—42no—
122C22⋊D283—42no—
123C22⋊D283—42no—
124C2×Dic283—22no—
125C2×Dic283—21no—
126C2×Dic283—21no—
127C2×Dic283—21no—
128C2×Dic283C2×D411yescharacteristic
129C2×Dic283C2×D411yescharacteristic
130C4×D283—22no—
131C4×D283—21no—
132C4×D283—21no—
133C4×D283—22no—
134C4×D283—42no—
135C4×D283—42no—
136C4×D283—42no—
137C4×D283#4112yes—
138C4×D283#4112yes—
139C283⋊C8—21no—
140C283⋊C8—21no—
141C23×C283—21no—
142C23×C283C2×D411yescharacteristic
143D4×C283—21no—
144D4×C283—21no—
145D4×C283—42no—
146C2×C4×C283—21no—
147C2×C4×C283—21no—
148C2×C4×C283—21no—
149C2×C4×C283—21no—
150C2×C4×C283—21no—
151C2×C4×C283C2×D411yescharacteristic
152C2×C4×C283C2411yescharacteristic
153C8×C283—21no—
154C8×C283—21no—
155C22×C283—21no—
156C22×C283—21no—
157C22×C283—21no—
158C22×C283—41no—
159C22×C283—41no—
160C22×C283—41no—
161C22×C283C22×D411yescharacteristic
162C2×D283—21no—
163C2×D283—21no—
164C2×D283—21no—
165C2×D283—21no—
166C2×D283—44no—
167C2×D283—41no—
168C2×D283—41no—
169C2×D283—41no—
170C2×D283—84no—
171C2×D283#12312yes—
172C2×D283AGL(1,ℤ/8)12yes—
173C4×C283—21no—
174C4×C283—41no—
175C4×C283—41no—
176C4×C283—41no—
177C4×C283C2×#4111yescharacteristic
178C4×C283C2×#4111yescharacteristic
179Dic283—21no—
180Dic283—41no—
181Dic283—41no—
182Dic283—41no—
183Dic283#12312yes—
184C2×C283—41no—
185C2×C283—81no—
186C2×C283C2×#12311yescharacteristic
187C2×C283C2×AGL(1,ℤ/8)11yescharacteristic
188C2×C283#57511yescharacteristic
189D283—22no—
190D283—22no—
191D283—81no—
192C283#309411yescharacteristic, complemented, Sylow
193#3094—2831nomaximal, Sylow
194C2×#121—2831no—
195C2×#117—2831no—
196#482—2831no—
197#482—2831no—
198#482—2831no—
199#482D28311yescharacteristic, complemented
200#481—2831no—
201#481—2831no—
202#480—2831no—
203#480—2831no—
204#428—2831no—
205#428—2831no—
206#418—2831no—
207#418—2831no—
208#407—2831no—
209C22×D4—2831no—
210C22×D4—5661no—
211C23×C4—2831no—
212C2×M16—2831no—
213C2×M16—2831no—
214#121—2831no—
215#121—2831no—
216#121—2831no—
217#121—5661no—
218#121—5661no—
219#121C2×D28311yescharacteristic
220#117—2831no—
221#117—2831no—
222#117—2831no—
223#117—5661no—
224#117—5661no—
225#117C2×D28311yescharacteristic, complemented
226C2×C4⋊C4—2831no—
227C2×C4⋊C4—2831no—
228C2×C4⋊C4—2831no—
229C2×C22⋊C4—5661no—
230C2×C22⋊C4—5661no—
231C2×C42—5661no—
232#107—2831no—
233#107—2831no—
234#107—2831no—
235#107C2×D28311yescharacteristic, complemented
236#106—2831no—
237#106—2831no—
238#106—2831no—
239#106C2×D28311yescharacteristic, complemented
240#102—2831no—
241#102—2831no—
242#102—2831no—
243#102—2831no—
244#102—2831no—
245#102—2831no—
246#102C2×D28311yescharacteristic, complemented
247#102C2×D28311yescharacteristic, complemented
248C22⋊C8—2831no—
249C22⋊C8—2831no—
250C22⋊C8—2831no—
251C22⋊C8C2×D28311yescharacteristic, complemented
252C24—2831no—
253C24—5661no—
254C2×D4—21no—
255C2×D4—2831no—
256C2×D4—5661no—
257C2×D4—5662no—
258C2×D4—5662no—
259C2×D4—5661no—
260C2×D4—5661no—
261C2×D4—5661no—
262C2×D4—11324no—
263C2×D4C22×D28311yescharacteristic
264C22×C4—2831no—
265C22×C4—2831no—
266C22×C4—2831no—
267C22×C4—2831no—
268C22×C4—2831no—
269C22×C4—5662no—
270C22×C4—5662no—
271C22×C4—5661no—
272C22×C4—5661no—
273C22×C4—5661no—
274C22×C4C22×D28311yescharacteristic
275M16—5662no—
276M16—5662no—
277C2×C8—2831no—
278C2×C8—2831no—
279C2×C8C22×D28311yescharacteristic
280C2×C8C22×D28311yescharacteristic
281C4⋊C4—2831no—
282C4⋊C4—2831no—
283C4⋊C4—2831no—
284C4⋊C4—5661no—
285C4⋊C4—5661no—
286C4⋊C4—5661no—
287C4⋊C4C22×D28311yescharacteristic
288C4⋊C4C22×D28311yescharacteristic
289C4⋊C4C22×D28311yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—5661no—
293C22⋊C4—5661no—
294C22⋊C4—5661no—
295C22⋊C4—5661no—
296C22⋊C4—5661no—
297C22⋊C4—5661no—
298C42—21no—
299C42—5661no—
300C42—5661no—
301C42—5661no—
302C23—21no—
303C23—2831no—
304C23—2831no—
305C23—5662no—
306C23—5662no—
307C23—5661no—
308C23—11321no—
309C23—11322no—
310C23—11322no—
311C23—11322no—
312C23D4×D28311yescharacteristic
313D4—21no—
314D4—21no—
315D4—42no—
316D4—5661no—
317D4—5661no—
318D4—11322no—
319D4—11322no—
320D4—11322no—
321D4—11322no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—2831no—
328C2×C4—2832no—
329C2×C4—2832no—
330C2×C4—2831no—
331C2×C4—5662no—
332C2×C4—5662no—
333C2×C4—5662no—
334C2×C4—5661no—
335C2×C4—5661no—
336C2×C4—5661no—
337C2×C4—5661no—
338C2×C4—5661no—
339C2×C4—11322no—
340C2×C4—11322no—
341C2×C4—11322no—
342C2×C4D4×D28311yescharacteristic
343C2×C4C23×D28311yescharacteristic
344C8—21no—
345C8—21no—
346C8—5661no—
347C8—5661no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—2832no—
355C22—2832no—
356C22—5661no—
357C22—5661no—
358C22—5661no—
359C22—5661no—
360C22—11324no—
361C22—11321no—
362C22—11321no—
363C22—11321no—
364C22—22644no—
365C22C2×D4×D28311yescharacteristic
366C4—21no—
367C4—41no—
368C4—41no—
369C4—41no—
370C4—2832no—
371C4—5661no—
372C4—11321no—
373C4—11321no—
374C4—11321no—
375C4#41×D28311yescharacteristic
376C4#41×D28311yescharacteristic
377C2—41no—
378C2—81no—
379C2—5662no—
380C2—5662no—
381C2—22641no—
382C2#123×D28311yescharacteristic
383C2AGL(1,ℤ/8)×D28311yescharacteristic
384C2#3182726211yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#482×C283C2
socleC22×C283C22×D4
prime radical#121×D283C2
hypercenter#482D283
hypocenterC283#3094
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #482

Complements

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

NH ≅ G/Ntype
2. #117×D283C2semidirect
3. #31826790C2semidirect
4. #31826789C2semidirect
5. #31826786C2semidirect
6. #31826785C2semidirect
7. #31826782C2semidirect
8. #31826780C2semidirect
9. #31826777C2semidirect
10. #31826506C2semidirect
11. #31826504C2semidirect
12. #31826468C2semidirect
13. #31826468C2semidirect
14. #31826427C2semidirect
15. #482×C283C2semidirect
26. #22583569C22semidirect
32. C4⋊C4×D283C22semidirect
33. C4⋊C4×D283C22semidirect
37. #22583544C22semidirect
38. #22583543C22semidirect
39. #22583543C22semidirect
40. #22583542C22semidirect
41. #22583541C22semidirect
42. #22583541C22semidirect
43. #22583532C22semidirect
44. #22583532C22semidirect
45. #22583531C22semidirect
46. #22583531C22semidirect
47. #22583530C22semidirect
48. #22583530C22semidirect
49. C22⋊(C283⋊C8)C22semidirect
50. (C2×D283)⋊C8C22semidirect
51. (C2×D283)⋊C8C22semidirect
53. #117×C283C22semidirect
54. #107×C283C22semidirect
55. #106×C283C22semidirect
56. #102×C283C22semidirect
57. #102×C283C22semidirect
58. C22⋊C8×C283C22semidirect
192. C283#3094semidirect
199. #482D283semidirect
225. #117C2×D283semidirect
235. #107C2×D283semidirect
239. #106C2×D283semidirect
246. #102C2×D283semidirect
247. #102C2×D283semidirect
251. C22⋊C8C2×D283semidirect

Maximal subgroups

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

subgroupindex
#121×D2832
#117×D2832
#318267902
#318267892
#318267862
#318267852
#318267822
#318267802
#318267772
#318265062
#318265042
#318264682
#318264272
#482×C2832
#3094283