Finite Group Databasebeta

← #49133256

Subgroups — #49133256

Order 36224.

420 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#31827126C211yesmaximal, characteristic, complemented
2#31827115C211yesmaximal, characteristic, complemented
3#31826826C211yesmaximal, characteristic, complemented
4#31826799C211yesmaximal, characteristic, complemented
5#31826779C211yesmaximal, characteristic, complemented
6#31826758C211yesmaximal, characteristic, complemented
7#31826708C211yesmaximal, characteristic, complemented
8#31826699C211yesmaximal, characteristic, complemented
9#31826682C211yesmaximal, characteristic, complemented
10#31826609C211yesmaximal, characteristic, complemented
11#31826566C211yesmaximal, characteristic, complemented
12#31826533C211yesmaximal, characteristic, complemented
13#31826465C211yesmaximal, characteristic, complemented
14C2×#22583534C211yesmaximal, characteristic, complemented
15#448×C283C211yesmaximal, characteristic, complemented
16C2×#17985393C2211yescharacteristic
17C2×#17985391—21no—
18C2×#17985377C2211yescharacteristic
19C2×#17985375C2211yescharacteristic
20C22×C283⋊C8C2211yescharacteristic
21#22583609C2211yescharacteristic
22#22583599—21no—
23#22583599C2211yescharacteristic
24#22583592—21no—
25#22583591—21no—
26#22583587—21no—
27#22583586—21no—
28#22583586C2211yescharacteristic
29#22583584—21no—
30#22583584C2211yescharacteristic
31#22583581C2211yescharacteristic
32Q8×Dic283C2211yescharacteristic
33D4×Dic283C2211yescharacteristic
34C4×C22⋊D283—21no—
35C4×C22⋊D283C2211yescharacteristic
36C2×C4⋊Dic283C2211yescharacteristic
37C4⋊C4×D283C2211yescharacteristic
38#22583544C2211yescharacteristic
39#22583544C2211yescharacteristic
40#22583543C2211yescharacteristic
41Dic283⋊C8C2211yescharacteristic
42#22583534C2211yescharacteristic
43#22583534C2211yescharacteristic
44#22583534C2211yescharacteristic
45#22583534C2211yescharacteristic
46#22583532C2211yescharacteristic
47#22583532C2211yescharacteristic
48#22583531C2211yescharacteristic
49#22583530C2211yescharacteristic
50(C2×D283)⋊C8C2211yescharacteristic
51#22583520C2211yescharacteristic
52C22×Q8×C283C2211yescharacteristic
53C2×SD16×C283C2211yescharacteristic
54C2×SD16×C283C2211yescharacteristic
55#121×C283C2211yescharacteristic
56#103×C283C2211yescharacteristic
57#103×C283C2211yescharacteristic
58C22⋊C8×C283C2211yescharacteristic
59#17985393—22no—
60#17985393—41no—
61#17985391—44no—
62C2×Dic566—21no—
63C2×D1132—21no—
64C2×D1132C2311yescharacteristic
65C2×C22⋊D283—21no—
66C2×C22⋊D283—21no—
67C2×C22⋊D283D411yescharacteristic
68C22×Dic283—21no—
69C22×Dic283—21no—
70C2×C4×D283—21no—
71C2×C4×D283—21no—
72C2×C4×D283C2311yescharacteristic
73#17985377—41no—
74#17985375—41no—
75C2×C283⋊C8—41no—
76C2×C283⋊C8C2311yescharacteristic
77C2×C283⋊C8C2311yescharacteristic
78C4⋊Dic283—21no—
79C4⋊Dic283—21no—
80C4⋊Dic283C2311yescharacteristic
81C4⋊Dic283C2311yescharacteristic
82C4⋊Dic283C2311yescharacteristic
83Dic283⋊C4—21no—
84Dic283⋊C4—21no—
85Dic283⋊C4—21no—
86Dic283⋊C4D411yescharacteristic
87C22⋊Dic283—21no—
88C22⋊Dic283—21no—
89C22⋊Dic283D411yescharacteristic
90(C2×D283)⋊C4—21no—
91(C2×D283)⋊C4—21no—
92(C2×D283)⋊C4—21no—
93(C2×D283)⋊C4—21no—
94(C2×D283)⋊C4D411yescharacteristic
95C4×Dic283—21no—
96C4×Dic283C2311yescharacteristic
97C2×Q8×C283—21no—
98C2×Q8×C283—22no—
99C2×Q8×C283—41no—
100C2×Q8×C283C2311yescharacteristic
101C2×Q8×C283C2311yescharacteristic
102C2×D4×C283—21no—
103C2×D4×C283C2311yescharacteristic
104C22×C4×C283—21no—
105C22×C4×C283C2311yescharacteristic
106SD16×C283—22no—
107SD16×C283—41no—
108C2×C8×C283C2311yescharacteristic
109C2×C8×C283C2311yescharacteristic
110C4⋊C4×C283C2311yescharacteristic
111C22⋊C4×C283—21no—
112C22×D283—21no—
113C22×D283C2×D411yescharacteristic
114Dic566—42no—
115D1132—21no—
116D1132—21no—
117D1132—42no—
118C22⋊D283—22no—
119C22⋊D283—42no—
120C22⋊D283—42no—
121C2×Dic283—21no—
122C2×Dic283—21no—
123C2×Dic283—21no—
124C2×Dic283—42no—
125C2×Dic283—42no—
126C2×Dic283C2×D411yescharacteristic
127C2×Dic283C2×D411yescharacteristic
128C2×Dic283C2×D411yescharacteristic
129C4×D283—21no—
130C4×D283—21no—
131C4×D283—42no—
132C4×D283—42no—
133C283⋊C8—21no—
134C283⋊C8—21no—
135C23×C283—21no—
136C23×C283C2×D411yescharacteristic
137Q8×C283—21no—
138Q8×C283—21no—
139Q8×C283—21no—
140Q8×C283—42no—
141Q8×C283#4112yescomplemented
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×C283—42no—
151C2×C4×C283C2×D411yescharacteristic
152C2×C4×C283C2411yescharacteristic
153C8×C283—21no—
154C8×C283—21no—
155C22×C283—21no—
156C22×C283—21no—
157C22×C283—41no—
158C22×C283—41no—
159C22×C283—41no—
160C22×C283AGL(1,ℤ/8)12yes—
161C22×C283C22×D411yescharacteristic
162C2×D283—21no—
163C2×D283—21no—
164C2×D283—21no—
165C2×D283—41no—
166C2×D283—41no—
167C2×D283—41no—
168C4×C283—21no—
169C4×C283—22no—
170C4×C283—41no—
171C4×C283—41no—
172C4×C283C2×#4111yescharacteristic
173C4×C283H(5,𝔽2)11yescharacteristic
174Dic283—21no—
175Dic283—21no—
176Dic283—21no—
177Dic283—41no—
178Dic283—41no—
179Dic283—41no—
180C2×C283—22no—
181C2×C283—81no—
182C2×C283#54611yescharacteristic
183C2×C283C2×AGL(1,ℤ/8)11yescharacteristic
184C2×C283#57811yescharacteristic
185D283—41no—
186D283—81no—
187C283#330811yescharacteristic, complemented, Sylow
188#3308—2831nomaximal, Sylow
189#548—2831no—
190#547—2831no—
191#489—2831no—
192#484—2831no—
193#480—2831no—
194#477—2831no—
195#468—2831no—
196#466—2831no—
197#463—2831no—
198#449—2831no—
199#448D28311yescharacteristic, complemented
200#440—2831no—
201#435—2831no—
202#417—2831no—
203C2×#103—2831no—
204C2×#41—2831no—
205C2×#41—5661no—
206C22×Q8C2×D28311yescharacteristic
207C2×SD16C2×D28311yescharacteristic
208C2×SD16C2×D28311yescharacteristic
209C2×D8—2831no—
210C2×M16—2831no—
211C22×C8—2831no—
212C4⋊Q8—2831no—
213#124—2831no—
214#124—5661no—
215#123—5661no—
216#123—5661no—
217C22⋊Q8—2831no—
218C22⋊Q8—5661no—
219C22⋊Q8—5661no—
220#121—2831no—
221#121—2831no—
222#121—5661no—
223#121C2×D28311yescharacteristic
224C4×Q8—2831no—
225C4×D4—2831no—
226C4×D4—2831no—
227C4×D4—5661no—
228C2×C4⋊C4—2831no—
229C2×C4⋊C4—2831no—
230#107—2831no—
231#107—2831no—
232#107—2831no—
233C4⋊C8—2831no—
234#103—2831no—
235#103—2831no—
236#103—2831no—
237#103—2831no—
238#103C2×D28311yescharacteristic
239#103C2×D28311yescharacteristic
240#102—2831no—
241#102—2831no—
242#102—2831no—
243#102—2831no—
244C22⋊C8—2831no—
245C22⋊C8C2×D28311yescharacteristic
246#97—2831no—
247#41—5662no—
248#41—11321no—
249#41—11324no—
250C2×Q8—21no—
251C2×Q8—22no—
252C2×Q8—41no—
253C2×Q8—5661no—
254C2×Q8C22×D28311yescharacteristic
255C2×Q8C22×D28311yescharacteristic
256C2×D4—21no—
257C2×D4—2831no—
258C2×D4—2831no—
259C2×D4—5661no—
260C2×D4—5661no—
261C2×D4—5661no—
262C2×D4C22×D28311yescharacteristic
263C22×C4—21no—
264C22×C4—2831no—
265C22×C4—5661no—
266C22×C4—5661no—
267C22×C4—5661no—
268C22×C4—5661no—
269C22×C4C22×D28311yescharacteristic
270SD16—22no—
271SD16—41no—
272D8—11321no—
273M16—11321no—
274C2×C8—2831no—
275C2×C8—2831no—
276C2×C8—11321no—
277C2×C8C22×D28311yescharacteristic
278C2×C8C22×D28311yescharacteristic
279C4⋊C4—2831no—
280C4⋊C4—2831no—
281C4⋊C4—2831no—
282C4⋊C4—2831no—
283C4⋊C4—5661no—
284C4⋊C4—5661no—
285C4⋊C4—5661no—
286C4⋊C4—5661no—
287C4⋊C4—5661no—
288C4⋊C4C22×D28311yescharacteristic
289C22⋊C4—21no—
290C22⋊C4—2831no—
291C22⋊C4—2831no—
292C22⋊C4—5661no—
293C22⋊C4—5661no—
294C22⋊C4—5661no—
295C22⋊C4—5661no—
296C22⋊C4—5661no—
297C22⋊C4—5661no—
298C42—2831no—
299C42—5661no—
300C23—21no—
301C23—2831no—
302C23—5661no—
303C23D4×D28311yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—42no—
308Q8—11322no—
309Q8#1798539112yescomplemented
310D4—21no—
311D4—21no—
312D4—42no—
313D4—5662no—
314D4—5661no—
315D4—5661no—
316D4—11322no—
317D4—11322no—
318D4—11322no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—42no—
325C2×C4—2831no—
326C2×C4—2831no—
327C2×C4—2831no—
328C2×C4—5661no—
329C2×C4—5661no—
330C2×C4—5661no—
331C2×C4—5661no—
332C2×C4—5661no—
333C2×C4—11322no—
334C2×C4—11322no—
335C2×C4—11322no—
336C2×C4—11322no—
337C2×C4D4×D28311yescharacteristic
338C2×C4C23×D28311yescharacteristic
339C8—21no—
340C8—21no—
341C8—5661no—
342C8—5661no—
343C22—21no—
344C22—21no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—5661no—
349C22—5661no—
350C22—5661no—
351C22—11321no—
352C22—11321no—
353C22—11321no—
354C22#2258363712yes—
355C22C2×D4×D28311yescharacteristic
356C4—21no—
357C4—22no—
358C4—41no—
359C4—41no—
360C4—5661no—
361C4—5661no—
362C4—5661no—
363C4—11321no—
364C4—11321no—
365C4—11321no—
366C4C2×#1798539111yescharacteristic
367C4#2258365511yescharacteristic
368C2—22no—
369C2—81no—
370C2—11321no—
371C2—22641no—
372C2#3182710211yescharacteristic
373C2C2×#2258363711yescharacteristic
374C2#3182728111yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#448×C283C2
socleC22×C283C22×D4
hypercenter#448D283
hypocenterC283#3308
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. #31827126C2semidirect
2. #31827115C2semidirect
3. #31826826C2semidirect
4. #31826799C2semidirect
5. #31826779C2semidirect
6. #31826758C2semidirect
7. #31826708C2semidirect
8. #31826699C2semidirect
9. #31826682C2semidirect
10. #31826609C2semidirect
11. #31826566C2semidirect
12. #31826533C2semidirect
13. #31826465C2semidirect
14. C2×#22583534C2semidirect
15. #448×C283C2semidirect
141. Q8×C283#41semidirect
187. C283#3308semidirect
199. #448D283semidirect
309. Q8#17985391semidirect

Maximal subgroups

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

subgroupindex
#318271262
#318271152
#318268262
#318267992
#318267792
#318267582
#318267082
#318266992
#318266822
#318266092
#318265662
#318265332
#318264652
C2×#225835342
#448×C2832
#3308283