Finite Group Databasebeta

← #49133105

Subgroups — #49133105

Order 36224.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31827111C211yesmaximal, characteristic, complemented
2#31826822C211yesmaximal, characteristic, complemented
3#31826800C211yesmaximal, characteristic, complemented
4#31826789C211yesmaximal, characteristic, complemented
5#31826762C211yesmaximal, characteristic, complemented
6#31826699C211yesmaximal, characteristic, complemented
7#31826692C211yesmaximal, characteristic, complemented
8#31826680C211yesmaximal, characteristic, complemented
9#31826608C211yesmaximal, characteristic, complemented
10#31826598C211yesmaximal, characteristic, complemented
11C4×#17985377C211yesmaximal, characteristic, complemented
12D4×C283⋊C8C211yesmaximal, characteristic, complemented
13#31826459C211yesmaximal, characteristic, complemented
14C2×#22583530C211yesmaximal, characteristic, complemented
15#553×C283C211yesmaximal, characteristic, complemented
16C2×#17985392C2211yescharacteristic
17C22×D1132C2211yescharacteristic
18C2×#17985380C2211yescharacteristic, complemented
19C2×#17985377C2211yescharacteristic
20C2×#17985377C2211yescharacteristic
21C22×C283⋊C8C2211yescharacteristic
22C22×C283⋊C8C2211yescharacteristic, complemented
23#22583598C2211yescharacteristic
24#22583595—21no—
25#22583589C2211yescharacteristic
26#22583584C2211yescharacteristic
27#22583583—21no—
28#22583578—21no—
29C4×D1132C2211yescharacteristic
30C22⋊C4×D283—21no—
31#22583543C2211yescharacteristic, complemented
32#22583543C2211yescharacteristic, complemented
33#22583541C2211yescharacteristic, complemented
34C4⋊(C283⋊C8)C2211yescharacteristic, complemented
35#22583533C2211yescharacteristic, complemented
36#22583533C2211yescharacteristic, complemented
37#22583533C2211yescharacteristic, complemented
38#22583531C2211yescharacteristic, complemented
39#22583530C2211yescharacteristic
40#22583530C2211yescharacteristic
41#22583530C2211yescharacteristic
42#22583530C2211yescharacteristic
43#22583530C2211yescharacteristic
44#22583530C2211yescharacteristic
45C22⋊(C283⋊C8)C2211yescharacteristic
46C22⋊(C283⋊C8)C2211yescharacteristic, complemented
47C4×C283⋊C8C2211yescharacteristic, complemented
48#126×C283—21no—
49#125×C283C2211yescharacteristic, complemented
50#123×C283—21no—
51#121×C283—21no—
52#121×C283C2211yescharacteristic
53#121×C283C2211yescharacteristic, complemented
54C4×D4×C283—21no—
55C4×D4×C283C2211yescharacteristic
56C4×D4×C283C2211yescharacteristic, complemented
57#117×C283C2211yescharacteristic, complemented
58C2×C4⋊C4×C283C2211yescharacteristic
59C23×D283—21no—
60#17985392—22no—
61#17985392—41no—
62C2×Dic566C2311yescharacteristic
63C2×D1132—22no—
64C2×D1132—21no—
65C2×D1132—41no—
66C2×D1132C2311yescharacteristic
67C2×D1132C2311yescharacteristic
68C2×C22⋊D283—21no—
69C2×C22⋊D283—21no—
70C2×C4×D283—21no—
71C2×C4×D283—21no—
72#17985380—41no—
73#17985377—22no—
74#17985377—41no—
75C2×C283⋊C8—41no—
76C2×C283⋊C8—41no—
77C2×C283⋊C8C2311yescharacteristic
78C2×C283⋊C8C2311yescharacteristic
79C2×C283⋊C8C2311yescharacteristic
80C2×C283⋊C8C2311yescharacteristic
81C4⋊Dic283C2311yescharacteristic
82Dic283⋊C4—21no—
83C22⋊Dic283—21no—
84(C2×D283)⋊C4—21no—
85(C2×D283)⋊C4—21no—
86(C2×D283)⋊C4—21no—
87(C2×D283)⋊C4—21no—
88C2×D4×C283—21no—
89C2×D4×C283—21no—
90C2×D4×C283D411yescharacteristic
91C2×D4×C283C2311yescharacteristic
92C22×C4×C283—21no—
93C22×C4×C283—21no—
94C22×C4×C283C2311yescharacteristic
95C22×C4×C283C2311yescharacteristic
96C4⋊C4×C283—21no—
97C4⋊C4×C283—21no—
98C4⋊C4×C283—21no—
99C4⋊C4×C283D411yescharacteristic
100C4⋊C4×C283C2311yescharacteristic
101C4⋊C4×C283C2311yescharacteristic
102C4⋊C4×C283C2311yescharacteristic
103C22⋊C4×C283—21no—
104C22⋊C4×C283—21no—
105C22⋊C4×C283—21no—
106C22⋊C4×C283—21no—
107C22⋊C4×C283D411yescharacteristic
108C22⋊C4×C283D411yescharacteristic
109C42×C283—21no—
110C42×C283C2311yescharacteristic
111C22×D283—21no—
112C22×D283—21no—
113C22×D283—42no—
114C22×D283—42no—
115C22×D283—81no—
116Dic566—21no—
117Dic566—21no—
118D1132—21no—
119D1132—21no—
120D1132—21no—
121D1132—42no—
122D1132#4112yes—
123C22⋊D283—42no—
124C22⋊D283—81no—
125C2×Dic283—21no—
126C2×Dic283—21no—
127C4×D283—42no—
128C4×D283—81no—
129C283⋊C8—21no—
130C283⋊C8—21no—
131C283⋊C8—21no—
132C283⋊C8—21no—
133C23×C283—21no—
134C23×C283C2×D411yescharacteristic
135C23×C283C2×D411yescharacteristic
136D4×C283—22no—
137D4×C283—21no—
138D4×C283—21no—
139D4×C283—42no—
140D4×C283—81no—
141C2×C4×C283—21no—
142C2×C4×C283—21no—
143C2×C4×C283—21no—
144C2×C4×C283—21no—
145C2×C4×C283—21no—
146C2×C4×C283—21no—
147C2×C4×C283—21no—
148C2×C4×C283—81no—
149C2×C4×C283—81no—
150C2×C4×C283C2×D411yescharacteristic
151C2×C4×C283C2×D411yescharacteristic
152C2×C4×C283C2×D411yescharacteristic
153C2×C4×C283C2×D411yescharacteristic
154C2×C4×C283C2411yescharacteristic
155C22×C283—21no—
156C22×C283—21no—
157C22×C283—21no—
158C22×C283—21no—
159C22×C283—21no—
160C22×C283—41no—
161C22×C283—41no—
162C22×C283—41no—
163C22×C283#13512yes—
164C22×C283C22×D411yescharacteristic
165C2×D283—22no—
166C2×D283—41no—
167C2×D283—41no—
168C2×D283—41no—
169C2×D283—41no—
170C2×D283—41no—
171C2×D283—84no—
172C4×C283—21no—
173C4×C283—21no—
174C4×C283—21no—
175C4×C283—21no—
176C4×C283—41no—
177C4×C283—41no—
178C4×C283—41no—
179C4×C283C2×#4111yescharacteristic
180C4×C283H(5,𝔽2)11yescharacteristic
181Dic283—41no—
182Dic283—41no—
183C2×C283—22no—
184C2×C283—41no—
185C2×C283—81no—
186C2×C283#54611yescharacteristic
187C2×C283C2×#13511yescharacteristic
188C2×C283#57611yescharacteristic
189D283—42no—
190D283—81no—
191C283#329711yescharacteristic, complemented, Sylow
192#3297—2831nomaximal, Sylow
193#553D28311yescharacteristic, complemented
194#546—2831no—
195#488—2831no—
196#484—2831no—
197#482—2831no—
198#478—2831no—
199#466—2831no—
200#465—2831no—
201#463—2831no—
202#449—2831no—
203#447—2831no—
204C4×D8—2831no—
205C8×D4—2831no—
206#416—2831no—
207C2×#102—2831no—
208C2×#41—2831no—
209C22×D4—2831no—
210C2×SD16—2831no—
211C2×D8—2831no—
212C2×D8—2831no—
213C22×C8—2831no—
214C22×C8—2831no—
215#126—21no—
216#125C2×D28311yescharacteristic, complemented
217#124—2831no—
218#123—21no—
219#123—5661no—
220C22⋊Q8—2831no—
221#121—21no—
222#121—2831no—
223#121—5661no—
224#121C2×D28311yescharacteristic
225#121C2×D28311yescharacteristic, complemented
226#120—5661no—
227C4×D4—21no—
228C4×D4—2831no—
229C4×D4C2×D28311yescharacteristic
230C4×D4C2×D28311yescharacteristic, complemented
231#117C2×D28311yescharacteristic, complemented
232C2×C4⋊C4C2×D28311yescharacteristic
233C2×C22⋊C4—5661no—
234#107—2831no—
235#107—2831no—
236#106—2831no—
237C4⋊C8—2831no—
238#103—2831no—
239#103—2831no—
240#103—2831no—
241#102—2831no—
242#102—2831no—
243#102—2831no—
244#102—2831no—
245#102—2831no—
246#102—2831no—
247#102—2831no—
248C22⋊C8—2831no—
249C22⋊C8—2831no—
250C4×C8—2831no—
251C24—5661no—
252#41—5662no—
253#41—11321no—
254C2×Q8—2831no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—2831no—
258C2×D4—2831no—
259C2×D4—5662no—
260C2×D4—5661no—
261C2×D4—5661no—
262C2×D4—5661no—
263C2×D4—11321no—
264C2×D4C22⋊D28311yescharacteristic
265C2×D4C22×D28311yescharacteristic
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—5661no—
269C22×C4—5661no—
270C22×C4C22×D28311yescharacteristic
271C22×C4C22×D28311yescharacteristic
272SD16—11321no—
273D8—5662no—
274D8—11321no—
275C2×C8—2831no—
276C2×C8—2831no—
277C2×C8—2831no—
278C2×C8—2831no—
279C2×C8—11321no—
280C2×C8—11321no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—2831no—
285C4⋊C4—5661no—
286C4⋊C4C22⋊D28311yescharacteristic
287C4⋊C4C22×D28311yescharacteristic
288C4⋊C4C22×D28311yescharacteristic
289C4⋊C4C22×D28311yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—5661no—
295C22⋊C4—5661no—
296C22⋊C4—5661no—
297C22⋊C4—5661no—
298C22⋊C4—5661no—
299C22⋊C4C22⋊D28311yescharacteristic
300C22⋊C4C22⋊D28311yescharacteristic
301C42—21no—
302C42C22×D28311yescharacteristic
303C23—21no—
304C23—5661no—
305C23—5661no—
306C23—11322no—
307C23—11322no—
308C23—22641no—
309C23C2×C22⋊D28311yescharacteristic
310C23C2×C22⋊D28311yescharacteristic
311Q8—5661no—
312Q8—5661no—
313D4—22no—
314D4—21no—
315D4—21no—
316D4—42no—
317D4—81no—
318D4—2832no—
319D4—5661no—
320D4—5661no—
321D4—5661no—
322D4—11322no—
323D4—11322no—
324D4—22641no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—5661no—
335C2×C4—5661no—
336C2×C4—11322no—
337C2×C4—22641no—
338C2×C4C2×C22⋊D28311yescharacteristic
339C2×C4C2×C22⋊D28311yescharacteristic
340C2×C4C2×C22⋊D28311yescharacteristic
341C2×C4C2×C22⋊D28311yescharacteristic
342C2×C4C23×D28311yescharacteristic
343C8—5661no—
344C8—5661no—
345C8—5661no—
346C8—5661no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—5662no—
356C22—11321no—
357C22—11321no—
358C22—11321no—
359C22—11321no—
360C22—11321no—
361C22—22644no—
362C22#2258363012yes—
363C22C22×C22⋊D28311yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4—11321no—
372C4—11321no—
373C4#41×D28311yescharacteristic
374C4#2258365611yescharacteristic
375C2—22no—
376C2—41no—
377C2—81no—
378C2—11322no—
379C2—22641no—
380C2#3182710311yescharacteristic
381C2C2×#2258363011yescharacteristic
382C2#3182726811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#553×C283C2
socleC22×C283C22×D4
hypercenter#553D283
hypocenterC283#3297
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C4×D4 ◁ #553

Complements

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

NH ≅ G/Ntype
1. #31827111C2semidirect
2. #31826822C2semidirect
3. #31826800C2semidirect
4. #31826789C2semidirect
5. #31826762C2semidirect
6. #31826699C2semidirect
7. #31826692C2semidirect
8. #31826680C2semidirect
9. #31826608C2semidirect
10. #31826598C2semidirect
11. C4×#17985377C2semidirect
12. D4×C283⋊C8C2semidirect
13. #31826459C2semidirect
14. C2×#22583530C2semidirect
15. #553×C283C2semidirect
18. C2×#17985380C22semidirect
22. C22×C283⋊C8C22semidirect
31. #22583543C22semidirect
32. #22583543C22semidirect
33. #22583541C22semidirect
34. C4⋊(C283⋊C8)C22semidirect
35. #22583533C22semidirect
36. #22583533C22semidirect
37. #22583533C22semidirect
38. #22583531C22semidirect
46. C22⋊(C283⋊C8)C22semidirect
47. C4×C283⋊C8C22semidirect
49. #125×C283C22semidirect
53. #121×C283C22semidirect
56. C4×D4×C283C22semidirect
57. #117×C283C22semidirect
191. C283#3297semidirect
193. #553D283semidirect
216. #125C2×D283semidirect
225. #121C2×D283semidirect
230. C4×D4C2×D283semidirect
231. #117C2×D283semidirect

Maximal subgroups

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

subgroupindex
#318271112
#318268222
#318268002
#318267892
#318267622
#318266992
#318266922
#318266802
#318266082
#318265982
C4×#179853772
D4×C283⋊C82
#318264592
C2×#225835302
#553×C2832
#3297283