Finite Group Databasebeta

← #49133259

Subgroups — #49133259

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#31827123C211yesmaximal, characteristic, complemented
2#31827120C211yesmaximal, characteristic, complemented
3#31826824C211yesmaximal, characteristic, complemented
4#31826799C211yesmaximal, characteristic, complemented
5#31826777C211yesmaximal, characteristic, complemented
6#31826761C211yesmaximal, characteristic, complemented
7#31826708C211yesmaximal, characteristic, complemented
8#31826698C211yesmaximal, characteristic, complemented
9#31826607C211yesmaximal, characteristic, complemented
10#31826602C211yesmaximal, characteristic, complemented
11#31826565C211yesmaximal, characteristic, complemented
12#31826535C211yesmaximal, characteristic, complemented
13#31826463C211yesmaximal, characteristic, complemented
14#103×D283C211yesmaximal, characteristic, complemented
15#463×C283C211yesmaximal, characteristic, complemented
16C2×#17985392—21no—
17C2×#17985391C2211yescharacteristic
18C2×Q8×D283C2211yescharacteristic
19C2×#17985380C2211yescharacteristic
20C2×#17985375C2211yescharacteristic
21C2×C8×D283C2211yescharacteristic
22#22583610C2211yescharacteristic
23#22583596—21no—
24#22583592—21no—
25#22583592—21no—
26#22583589—21no—
27#22583587—21no—
28#22583586C2211yescharacteristic
29#22583583—21no—
30#22583581C2211yescharacteristic
31#22583581C2211yescharacteristic
32#22583581C2211yescharacteristic
33C4×Dic566C2211yescharacteristic
34C4×D1132C2211yescharacteristic
35(C22⋊D283)⋊C4—21no—
36C4⋊C4×D283C2211yescharacteristic
37C4⋊C4×D283C2211yescharacteristic
38#22583544C2211yescharacteristic
39#22583544C2211yescharacteristic
40#22583543C2211yescharacteristic
41#22583535C2211yescharacteristic
42#22583535C2211yescharacteristic
43#22583534C2211yescharacteristic
44#22583534C2211yescharacteristic
45#22583533C2211yescharacteristic
46#22583531C2211yescharacteristic
47#22583531C2211yescharacteristic
48#22583531C2211yescharacteristic
49(C2×D283)⋊C8C2211yescharacteristic
50(C2×D283)⋊C8C2211yescharacteristic
51#22583521C2211yescharacteristic
52C2×SD16×C283C2211yescharacteristic
53C2×D8×C283C2211yescharacteristic
54#124×C283C2211yescharacteristic
55C4×D4×C283C2211yescharacteristic
56C4⋊C8×C283C2211yescharacteristic
57#103×C283C2211yescharacteristic
58#102×C283C2211yescharacteristic
59#17985392—44no—
60#17985391—22no—
61#17985391—41no—
62C2×Dic566—21no—
63C2×Dic566—21no—
64C2×Dic566C2311yescharacteristic
65Q8×D283—22no—
66Q8×D283—41no—
67C2×D1132D411yescharacteristic
68C2×C22⋊D283—21no—
69C2×C22⋊D283—21no—
70C2×C22⋊D283—21no—
71C2×C22⋊D283—21no—
72C22×Dic283—21no—
73C2×C4×D283—21no—
74C2×C4×D283—21no—
75C2×C4×D283—21no—
76C2×C4×D283C2311yescharacteristic
77C2×C4×D283C2311yescharacteristic
78#17985380—22no—
79#17985375—41no—
80C2×C283⋊C8C2311yescharacteristic
81C2×C283⋊C8C2311yescharacteristic
82C8×D283—41no—
83C4⋊Dic283—21no—
84C4⋊Dic283D411yescharacteristic
85C4⋊Dic283C2311yescharacteristic
86C4⋊Dic283C2311yescharacteristic
87C4⋊Dic283C2311yescharacteristic
88Dic283⋊C4—21no—
89Dic283⋊C4—21no—
90Dic283⋊C4—21no—
91Dic283⋊C4—21no—
92C22⋊Dic283—21no—
93C22⋊Dic283—21no—
94C22⋊Dic283—21no—
95(C2×D283)⋊C4—21no—
96(C2×D283)⋊C4D411yescharacteristic
97(C2×D283)⋊C4D411yescharacteristic
98C4×Dic283—21no—
99C2×Q8×C283C2311yescharacteristic
100C2×D4×C283C2311yescharacteristic
101C2×D4×C283C2311yescharacteristic
102C22×C4×C283—21no—
103SD16×C283—41no—
104D8×C283—41no—
105C2×C8×C283C2311yescharacteristic
106C2×C8×C283C2311yescharacteristic
107C4⋊C4×C283C2311yescharacteristic
108C22⋊C4×C283—21no—
109C22⋊C4×C283—21no—
110C22⋊C4×C283—21no—
111C42×C283C2311yescharacteristic
112C22×D283C2×D411yescharacteristic
113C22×D283C2×D411yescharacteristic
114Dic566—21no—
115Dic566—42no—
116Dic566—42no—
117Dic566#4112yescomplemented
118D1132—22no—
119C22⋊D283—42no—
120C22⋊D283—42no—
121C22⋊D283—42no—
122C22⋊D283—42no—
123C2×Dic283—21no—
124C2×Dic283—21no—
125C2×Dic283—21no—
126C2×Dic283—21no—
127C2×Dic283—42no—
128C2×Dic283C2×D411yescharacteristic
129C2×Dic283C2×D411yescharacteristic
130C4×D283—21no—
131C4×D283—21no—
132C4×D283—21no—
133C4×D283—21no—
134C4×D283—42no—
135C4×D283—42no—
136C4×D283—42no—
137C283⋊C8—21no—
138C283⋊C8—21no—
139C23×C283—21no—
140C23×C283—21no—
141Q8×C283—21no—
142Q8×C283—21no—
143D4×C283—21no—
144D4×C283—21no—
145D4×C283—21no—
146D4×C283—21no—
147C2×C4×C283—21no—
148C2×C4×C283—21no—
149C2×C4×C283—42no—
150C2×C4×C283C2×D411yescharacteristic
151C2×C4×C283C2×D411yescharacteristic
152C2×C4×C283C2411yescharacteristic
153C8×C283—21no—
154C8×C283—21no—
155C22×C283—41no—
156C22×C283—41no—
157C22×C283—41no—
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—21no—
167C2×D283AGL(1,ℤ/8)12yes—
168C4×C283—21no—
169C4×C283—21no—
170C4×C283—41no—
171C4×C283—41no—
172C4×C283C2×#4111yescharacteristic
173C4×C283H(5,𝔽2)11yescharacteristic
174Dic283—21no—
175Dic283—21no—
176Dic283—22no—
177Dic283—41no—
178Dic283—41no—
179Dic283—41no—
180C2×C283—81no—
181C2×C283—81no—
182C2×C283#54611yescharacteristic
183C2×C283C2×AGL(1,ℤ/8)11yescharacteristic
184C2×C283#57811yescharacteristic
185D283—22no—
186D283—41no—
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#463D28311yescharacteristic, complemented
198#449—2831no—
199#448—2831no—
200#440—2831no—
201#435—2831no—
202#417—2831no—
203C2×#103—2831no—
204C2×#41—2831no—
205C2×#41—5661no—
206C22×Q8—2831no—
207C2×SD16—2831no—
208C2×SD16C2×D28311yescharacteristic
209C2×D8C2×D28311yescharacteristic
210C2×M16—2831no—
211C22×C8—2831no—
212C4⋊Q8—2831no—
213#124—5661no—
214#124C2×D28311yescharacteristic
215#123—5661no—
216#123—5661no—
217C22⋊Q8—2831no—
218C22⋊Q8—5661no—
219C22⋊Q8—5661no—
220#121—2831no—
221#121—2831no—
222#121—2831no—
223#121—5661no—
224C4×Q8—2831no—
225C4×D4—2831no—
226C4×D4—5661no—
227C4×D4C2×D28311yescharacteristic
228C2×C4⋊C4—2831no—
229C2×C4⋊C4—2831no—
230#107—2831no—
231#107—2831no—
232#107—2831no—
233C4⋊C8C2×D28311yescharacteristic
234#103—2831no—
235#103—2831no—
236#103—2831no—
237#103—2831no—
238#103—2831no—
239#103C2×D28311yescharacteristic
240#102—2831no—
241#102—2831no—
242#102—2831no—
243#102C2×D28311yescharacteristic
244C22⋊C8—2831no—
245C22⋊C8—2831no—
246#97—2831no—
247#41—5662no—
248#41—11324no—
249#41—11321no—
250C2×Q8—2831no—
251C2×Q8—5661no—
252C2×Q8—5661no—
253C2×Q8—5662no—
254C2×Q8—11321no—
255C2×Q8C22×D28311yescharacteristic
256C2×D4—2831no—
257C2×D4—5661no—
258C2×D4—5661no—
259C2×D4—5661no—
260C2×D4—5661no—
261C2×D4C22×D28311yescharacteristic
262C2×D4C22×D28311yescharacteristic
263C22×C4—21no—
264C22×C4—2831no—
265C22×C4—2831no—
266C22×C4—5661no—
267C22×C4—5661no—
268C22×C4—5661no—
269C22×C4—5661no—
270SD16—41no—
271SD16—5662no—
272D8—41no—
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—21no—
291C22⋊C4—21no—
292C22⋊C4—2831no—
293C22⋊C4—2831no—
294C22⋊C4—5661no—
295C22⋊C4—5661no—
296C22⋊C4—5661no—
297C22⋊C4—5661no—
298C42—5661no—
299C42C22×D28311yescharacteristic
300C23—21no—
301C23—21no—
302C23—2831no—
303C23—2831no—
304Q8—21no—
305Q8—21no—
306Q8—2832no—
307Q8—5661no—
308Q8—11322no—
309Q8—11322no—
310D4—21no—
311D4—21no—
312D4—21no—
313D4—21no—
314D4—5662no—
315D4—11322no—
316D4—11322no—
317D4—11322no—
318D4—11322no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
322C2×C4—2831no—
323C2×C4—2831no—
324C2×C4—5661no—
325C2×C4—5661no—
326C2×C4—5661no—
327C2×C4—5661no—
328C2×C4—5661no—
329C2×C4—5661no—
330C2×C4—5661no—
331C2×C4—5661no—
332C2×C4—11322no—
333C2×C4—11322no—
334C2×C4—11322no—
335C2×C4—11322no—
336C2×C4D4×D28311yescharacteristic
337C2×C4D4×D28311yescharacteristic
338C2×C4C23×D28311yescharacteristic
339C8—21no—
340C8—21no—
341C8—5661no—
342C8—5661no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—2832no—
350C22—5661no—
351C22—5661no—
352C22—5661no—
353C22—5661no—
354C22—5661no—
355C22C2×D4×D28311yescharacteristic
356C4—21no—
357C4—21no—
358C4—41no—
359C4—41no—
360C4—5661no—
361C4—5661no—
362C4—5662no—
363C4—11321no—
364C4—11321no—
365C4—11321no—
366C4#41×D28311yescharacteristic
367C4#2258365511yescharacteristic
368C2—81no—
369C2—81no—
370C2—5662no—
371C2—11321no—
372C2#3182711211yescharacteristic
373C2AGL(1,ℤ/8)×D28311yescharacteristic
374C2#3182727911yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#463×C283C2
socleC22×C283C22×D4
hypercenter#463D283
hypocenterC283#3308
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. #31827123C2semidirect
2. #31827120C2semidirect
3. #31826824C2semidirect
4. #31826799C2semidirect
5. #31826777C2semidirect
6. #31826761C2semidirect
7. #31826708C2semidirect
8. #31826698C2semidirect
9. #31826607C2semidirect
10. #31826602C2semidirect
11. #31826565C2semidirect
12. #31826535C2semidirect
13. #31826463C2semidirect
14. #103×D283C2semidirect
15. #463×C283C2semidirect
117. Dic566#41semidirect
187. C283#3308semidirect
197. #463D283semidirect

Maximal subgroups

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

subgroupindex
#318271232
#318271202
#318268242
#318267992
#318267772
#318267612
#318267082
#318266982
#318266072
#318266022
#318265652
#318265352
#318264632
#103×D2832
#463×C2832
#3308283