Finite Group Databasebeta

← #49133121

Subgroups — #49133121

Order 36224.

420 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31827128C211yesmaximal, characteristic, complemented
2#31827115C211yesmaximal, characteristic
3#31826814C211yesmaximal, characteristic, complemented
4#31826797C211yesmaximal, characteristic, complemented
5#31826793C211yesmaximal, characteristic, complemented
6#31826749C211yesmaximal, characteristic, complemented
7#31826693C211yesmaximal, characteristic, complemented
8#31826668C211yesmaximal, characteristic, complemented
9#31826603C211yesmaximal, characteristic, complemented
10#31826602C211yesmaximal, characteristic, complemented
11#31826556C211yesmaximal, characteristic, complemented
12C8×C22⋊D283C211yesmaximal, characteristic, complemented
13C2×#22583535C211yesmaximal, characteristic, complemented
14#103×D283C211yesmaximal, characteristic, complemented
15#465×C283C211yesmaximal, characteristic, complemented
16C2×#17985391—21no—
17C22×Dic566C2211yescharacteristic
18C2×Q8×D283C2211yescharacteristic
19C2×#17985380C2211yescharacteristic
20C2×#17985380C2211yescharacteristic
21C2×C8×D283C2211yescharacteristic
22#22583609—21no—
23#22583609C2211yescharacteristic
24#22583608C2211yescharacteristic
25#22583592—21no—
26#22583591—21no—
27#22583590C2211yescharacteristic
28#22583588—21no—
29#22583587—21no—
30#22583587—21no—
31#22583581C2211yescharacteristic
32#22583575C2211yescharacteristic
33D4×Dic283C2211yescharacteristic
34C4×C22⋊D283C2211yescharacteristic
35(C22⋊D283)⋊C4—21no—
36C2×C4⋊Dic283C2211yescharacteristic
37C4⋊C4×D283C2211yescharacteristic
38#22583541C2211yescharacteristic
39#22583541C2211yescharacteristic
40Dic283⋊C8C2211yescharacteristic
41#22583535C2211yescharacteristic
42#22583535C2211yescharacteristic
43#22583535C2211yescharacteristic
44#22583535C2211yescharacteristic
45#22583534C2211yescharacteristic
46#22583534C2211yescharacteristic
47#22583533C2211yescharacteristic
48#22583533C2211yescharacteristic
49C22⋊(C283⋊C8)C2211yescharacteristic
50(C2×D283)⋊C8C2211yescharacteristic
51C8×Dic283C2211yescharacteristic
52C2×SD16×C283C2211yescharacteristic
53C22×C8×C283C2211yescharacteristic
54C22⋊Q8×C283C2211yescharacteristic
55#121×C283C2211yescharacteristic
56#106×C283C2211yescharacteristic
57#103×C283C2211yescharacteristic
58#102×C283C2211yescharacteristic
59#17985391—44no—
60C2×Dic566—21no—
61C2×Dic566—21no—
62C2×Dic566—22no—
63C2×Dic566—21no—
64C2×Dic566—41no—
65C2×Dic566C2311yescharacteristic
66C2×Dic566C2311yescharacteristic
67Q8×D283—22no—
68Q8×D283—41no—
69C2×C22⋊D283—21no—
70C2×C22⋊D283D411yescharacteristic
71C22×Dic283—21no—
72C22×Dic283—21no—
73C22×Dic283—21no—
74C2×C4×D283—21no—
75C2×C4×D283—21no—
76C2×C4×D283C2311yescharacteristic
77#17985380—22no—
78#17985380—41no—
79C2×C283⋊C8C2311yescharacteristic
80C2×C283⋊C8C2311yescharacteristic
81C8×D283—41no—
82C4⋊Dic283—21no—
83C4⋊Dic283—21no—
84C4⋊Dic283C2311yescharacteristic
85C4⋊Dic283C2311yescharacteristic
86Dic283⋊C4—21no—
87Dic283⋊C4—21no—
88Dic283⋊C4—21no—
89Dic283⋊C4—21no—
90Dic283⋊C4—21no—
91Dic283⋊C4D411yescharacteristic
92C22⋊Dic283—21no—
93C22⋊Dic283—21no—
94C22⋊Dic283D411yescharacteristic
95(C2×D283)⋊C4—21no—
96(C2×D283)⋊C4D411yescharacteristic
97C4×Dic283—21no—
98C4×Dic283C2311yescharacteristic
99C2×Q8×C283C2311yescharacteristic
100C2×D4×C283—21no—
101C2×D4×C283C2311yescharacteristic
102C22×C4×C283C2311yescharacteristic
103SD16×C283—41no—
104C2×C8×C283—41no—
105C2×C8×C283C2311yescharacteristic
106C2×C8×C283C2311yescharacteristic
107C4⋊C4×C283—21no—
108C4⋊C4×C283C2311yescharacteristic
109C4⋊C4×C283C2311yescharacteristic
110C22⋊C4×C283—21no—
111C22⋊C4×C283—21no—
112C22×D283C2×D411yescharacteristic
113Dic566—21no—
114Dic566—21no—
115Dic566—21no—
116Dic566—42no—
117Dic566—42no—
118Dic566—42no—
119Dic566#4112yescomplemented
120C22⋊D283—42no—
121C22⋊D283SD1614yescomplemented
122C2×Dic283—21no—
123C2×Dic283—21no—
124C2×Dic283—21no—
125C2×Dic283—21no—
126C2×Dic283—42no—
127C2×Dic283—42no—
128C2×Dic283—42no—
129C2×Dic283C2×D411yescharacteristic
130C2×Dic283C2×D411yescharacteristic
131C2×Dic283C2×D411yescharacteristic
132C4×D283—21no—
133C4×D283—21no—
134C4×D283—42no—
135C4×D283—42no—
136C283⋊C8—21no—
137C283⋊C8—21no—
138C23×C283—21no—
139C23×C283C2×D411yescharacteristic
140Q8×C283—21no—
141Q8×C283—21no—
142D4×C283—21no—
143D4×C283—21no—
144D4×C283—42no—
145C2×C4×C283—21no—
146C2×C4×C283—21no—
147C2×C4×C283—21no—
148C2×C4×C283—21no—
149C2×C4×C283—21no—
150C2×C4×C283C2×D411yescharacteristic
151C2×C4×C283C2411yescharacteristic
152C8×C283—21no—
153C8×C283—21no—
154C22×C283—21no—
155C22×C283—21no—
156C22×C283—41no—
157C22×C283—41no—
158C22×C283—41no—
159C22×C283C2×SD1612yes—
160C22×C283C22×D411yescharacteristic
161C2×D283—21no—
162C2×D283—21no—
163C2×D283C2×SD1612yes—
164C4×C283—21no—
165C4×C283—41no—
166C4×C283—41no—
167C4×C283—41no—
168C4×C283C2×#4111yescharacteristic
169C4×C283H(5,𝔽2)11yescharacteristic
170Dic283—21no—
171Dic283—22no—
172Dic283—21no—
173Dic283—41no—
174Dic283—41no—
175Dic283—41no—
176Dic283C2×SD1612yes—
177C2×C283—22no—
178C2×C283—81no—
179C2×C283#54611yescharacteristic
180C2×C283C22×SD1611yescharacteristic
181C2×C283#57811yescharacteristic
182D283—22no—
183C283#329811yescharacteristic, complemented, Sylow
184#3298—2831nomaximal, Sylow
185D4×Q8—2831no—
186#547—2831no—
187#487—2831no—
188#483—2831no—
189#483—2831no—
190#475—2831no—
191#465—2831no—
192#465D28311yescharacteristic, complemented
193#461—2831no—
194#448—2831no—
195#448—2831no—
196C4×SD16—2831no—
197C8×D4—2831no—
198C2×#103—2831no—
199C2×#103—2831no—
200C2×#41—5661no—
201C22×Q8—2831no—
202C22×Q8—2831no—
203C2×SD16—2831no—
204C2×SD16—2831no—
205C2×SD16C2×D28311yescharacteristic
206C22×C8—2831no—
207C22×C8C2×D28311yescharacteristic
208C4⋊Q8—2831no—
209C4⋊Q8—2831no—
210C4⋊Q8—5661no—
211#123—5661no—
212#123—5661no—
213C22⋊Q8—2831no—
214C22⋊Q8—5661no—
215C22⋊Q8—5661no—
216C22⋊Q8—5661no—
217C22⋊Q8C2×D28311yescharacteristic
218#121—2831no—
219#121C2×D28311yescharacteristic
220C4×Q8—2831no—
221C4×D4—2831no—
222C4×D4—2831no—
223C4×D4—5661no—
224C2×C4⋊C4—2831no—
225C2×C4⋊C4—2831no—
226#106—2831no—
227#106—2831no—
228#106C2×D28311yescharacteristic
229C4⋊C8—2831no—
230#103—2831no—
231#103—2831no—
232#103—2831no—
233#103—2831no—
234#103—2831no—
235#103—2831no—
236#103—2831no—
237#103—2831no—
238#103C2×D28311yescharacteristic
239#102C2×D28311yescharacteristic
240C22⋊C8—2831no—
241C22⋊C8—2831no—
242C4×C8—2831no—
243#41—11324no—
244C2×Q8—2831no—
245C2×Q8—2831no—
246C2×Q8—5661no—
247C2×Q8—5661no—
248C2×Q8—5662no—
249C2×Q8—5661no—
250C2×Q8—5662no—
251C2×Q8—11321no—
252C2×Q8—11321no—
253C2×Q8C22×D28311yescharacteristic
254C2×D4—21no—
255C2×D4—2831no—
256C2×D4—5661no—
257C2×D4C22×D28311yescharacteristic
258C22×C4—2831no—
259C22×C4—5661no—
260C22×C4—5661no—
261C22×C4—5661no—
262C22×C4—5661no—
263C22×C4—5661no—
264C22×C4C22×D28311yescharacteristic
265SD16—41no—
266SD16—5662no—
267SD16—11321no—
268C2×C8—41no—
269C2×C8—2831no—
270C2×C8—2831no—
271C2×C8—11321no—
272C2×C8C22×D28311yescharacteristic
273C2×C8C22×D28311yescharacteristic
274C4⋊C4—21no—
275C4⋊C4—2831no—
276C4⋊C4—2831no—
277C4⋊C4—2831no—
278C4⋊C4—5661no—
279C4⋊C4—5661no—
280C4⋊C4—5661no—
281C4⋊C4—5661no—
282C4⋊C4—5661no—
283C4⋊C4—5661no—
284C4⋊C4—5661no—
285C4⋊C4C22×D28311yescharacteristic
286C4⋊C4C22×D28311yescharacteristic
287C22⋊C4—21no—
288C22⋊C4—21no—
289C22⋊C4—2831no—
290C22⋊C4—2831no—
291C22⋊C4—5661no—
292C22⋊C4—5661no—
293C22⋊C4—5661no—
294C42—2831no—
295C42—5661no—
296C23—21no—
297C23—2831no—
298C23D4×D28311yescharacteristic
299Q8—21no—
300Q8—21no—
301Q8—2832no—
302Q8—5661no—
303Q8—5661no—
304Q8—5661no—
305Q8—11322no—
306Q8—11322no—
307Q8—11322no—
308D4—21no—
309D4—21no—
310D4—42no—
311D4—2834no—
312D4—11322no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—2831no—
319C2×C4—2831no—
320C2×C4—2831no—
321C2×C4—5661no—
322C2×C4—5661no—
323C2×C4—5661no—
324C2×C4—5661no—
325C2×C4—5661no—
326C2×C4—5661no—
327C2×C4—11322no—
328C2×C4—11322no—
329C2×C4—11322no—
330C2×C4—11322no—
331C2×C4—11322no—
332C2×C4D4×D28311yescharacteristic
333C2×C4C23×D28311yescharacteristic
334C8—21no—
335C8—21no—
336C8—5661no—
337C8—5661no—
338C22—21no—
339C22—21no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—2832no—
344C22—5661no—
345C22—5661no—
346C22SD16×D28312yes—
347C22C2×D4×D28311yescharacteristic
348C4—21no—
349C4—41no—
350C4—41no—
351C4—41no—
352C4—2832no—
353C4—5661no—
354C4—5661no—
355C4—5662no—
356C4—11321no—
357C4—11321no—
358C4—11321no—
359C4#41×D28311yescharacteristic
360C4#2258365511yescharacteristic
361C2—22no—
362C2—81no—
363C2—5662no—
364C2#3182710411yescharacteristic
365C2C2×SD16×D28311yescharacteristic
366C2#3182727911yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#465×C283C2
socleC22×C283C22×D4
prime radical#31827115C2
hypercenter#465D283
hypocenterC283#3298
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #31827128C2semidirect
3. #31826814C2semidirect
4. #31826797C2semidirect
5. #31826793C2semidirect
6. #31826749C2semidirect
7. #31826693C2semidirect
8. #31826668C2semidirect
9. #31826603C2semidirect
10. #31826602C2semidirect
11. #31826556C2semidirect
12. C8×C22⋊D283C2semidirect
13. C2×#22583535C2semidirect
14. #103×D283C2semidirect
15. #465×C283C2semidirect
119. Dic566#41semidirect
121. C22⋊D283SD16semidirect
183. C283#3298semidirect
192. #465D283semidirect

Maximal subgroups

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

subgroupindex
#318271282
#318271152
#318268142
#318267972
#318267932
#318267492
#318266932
#318266682
#318266032
#318266022
#318265562
C8×C22⋊D2832
C2×#225835352
#103×D2832
#465×C2832
#3298283