Finite Group Databasebeta

← #49130883

Subgroups — #49130883

Order 36224.

548 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 135 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22583635C211yesmaximal, characteristic, complemented
2C2×M16×D283C211yesmaximal, characteristic, complemented
3#31826972C211yesmaximal, characteristic, complemented
4C2×#22583585C211yesmaximal, characteristic, complemented
5#31826717C212yesmaximal, complemented
6#31826714C212yesmaximal, complemented
7#31826700C212yesmaximal, complemented
8#31826697C212yesmaximal, complemented
9#31826505C211yesmaximal, characteristic, complemented
10#31826471C211yesmaximal, characteristic, complemented
11#417×C283C211yesmaximal, characteristic, complemented
12C22×D1132—21no—
13C22×D1132C2211yescharacteristic
14C22×C4×D283C2211yescharacteristic
15#22583635—24no—
16C2×#17985379C2212yescomplemented
17C2×#17985377C2212yescomplemented
18C2×#17985376C2211yescharacteristic
19C2×#17985375C2212yescomplemented
20M16×D283—24no—
21C2×C8×D283C2212yescomplemented
22#22583591—21no—
23#22583586C2211yescharacteristic, complemented
24#22583586C2211yescharacteristic, complemented
25#22583585—21no—
26#22583585—21no—
27#22583585C2212yes—
28#22583585C2212yes—
29#22583581C2211yescharacteristic, complemented
30#22583581C2211yescharacteristic, complemented
31#22583578—21no—
32C4×C22⋊D283—21no—
33C4×C22⋊D283—21no—
34#22583569C2211yescharacteristic
35C2×(C2×D283)⋊C4—21no—
36#22583543C2212yescomplemented
37#22583541C2212yescomplemented
38#22583532C2212yescomplemented
39#22583532C2212yescomplemented
40C2×#41×C283C2211yescharacteristic
41C2×M16×C283C2211yescharacteristic
42C2×C4⋊C4×C283C2211yescharacteristic
43#103×C283C2212yescomplemented
44#102×C283C2212yescomplemented
45C23×D283—21no—
46C23×D283D411yescharacteristic
47C2×D1132—22no—
48C2×D1132—22no—
49C2×D1132—22no—
50C2×D1132—21no—
51C2×D1132—22no—
52C2×D1132—44no—
53C2×D1132C2312yes—
54C2×C22⋊D283—21no—
55C2×C22⋊D283—21no—
56C22×Dic283D411yescharacteristic
57C2×C4×D283—22no—
58C2×C4×D283—22no—
59C2×C4×D283D411yescharacteristic
60C2×C4×D283D411yescharacteristic
61C2×C4×D283C2311yescharacteristic
62C2×C4×D283C2311yescharacteristic
63#17985379—24no—
64#17985377—24no—
65#17985376D414yescomplemented
66#17985375—24no—
67C2×C283⋊C8C2312yes—
68C8×D283—24no—
69C4⋊Dic283C2311yescharacteristic
70C4⋊Dic283C2311yescharacteristic
71Dic283⋊C4—21no—
72Dic283⋊C4—21no—
73C22⋊Dic283—21no—
74C22⋊Dic283—21no—
75C22⋊Dic283—21no—
76(C2×D283)⋊C4—21no—
77(C2×D283)⋊C4—21no—
78(C2×D283)⋊C4—22no—
79(C2×D283)⋊C4—22no—
80C4×Dic283—21no—
81#41×C283—22no—
82#41×C283—22no—
83C2×Q8×C283C2311yescharacteristic
84C2×D4×C283—21no—
85C2×D4×C283C2311yescharacteristic
86C22×C4×C283—21no—
87C22×C4×C283—21no—
88C22×C4×C283C2311yescharacteristic
89M16×C283—22no—
90C2×C8×C283C2312yes—
91C4⋊C4×C283—21no—
92C4⋊C4×C283C2312yes—
93C22×D283—22no—
94C22×D283—22no—
95C22×D283—22no—
96C22×D283—41no—
97C22×D283—42no—
98C22×D283—42no—
99C22×D283—42no—
100C22×D283C2×D411yescharacteristic
101C22×D283C2×D411yescharacteristic
102D1132—22no—
103D1132—22no—
104D1132—44no—
105D1132—44no—
106D1132—42no—
107C22⋊D283—42no—
108C22⋊D283—42no—
109C2×Dic283—22no—
110C2×Dic283—21no—
111C2×Dic283—21no—
112C2×Dic283C2×D411yescharacteristic
113C2×Dic283C2×D411yescharacteristic
114C4×D283—22no—
115C4×D283—22no—
116C4×D283—21no—
117C4×D283—21no—
118C4×D283—21no—
119C4×D283—21no—
120C283⋊C8C2×D414yescomplemented
121C23×C283—21no—
122C23×C283C2×D411yescharacteristic
123Q8×C283—21no—
124Q8×C283—21no—
125D4×C283—21no—
126D4×C283—21no—
127D4×C283—42no—
128C2×C4×C283—22no—
129C2×C4×C283—21no—
130C2×C4×C283—42no—
131C2×C4×C283—42no—
132C2×C4×C283C2×D411yescharacteristic
133C2×C4×C283C2×D412yes—
134C2×C4×C283#4112yes—
135C2×C4×C283C2411yescharacteristic
136C8×C283—22no—
137C22×C283—21no—
138C22×C283—21no—
139C22×C283—41no—
140C22×C283—41no—
141C22×C283—41no—
142C22×C283#12112yes—
143C22×C283C22×D411yescharacteristic
144C2×D283—21no—
145C2×D283—21no—
146C2×D283—21no—
147C2×D283—21no—
148C2×D283—21no—
149C2×D283—44no—
150C2×D283—42no—
151C2×D283—42no—
152C2×D283—42no—
153C2×D283—84no—
154C2×D283AGL(1,ℤ/8)12yescomplemented
155C4×C283—41no—
156C4×C283—42no—
157C4×C283#12112yes—
158C4×C283C22×D411yescharacteristic
159C4×C283C2×#4111yescharacteristic
160Dic283—21no—
161Dic283—21no—
162Dic283—41no—
163Dic283—41no—
164C2×C283—22no—
165C2×C283—81no—
166C2×C283C2×#12111yescharacteristic
167C2×C283C2×AGL(1,ℤ/8)11yescharacteristic
168C2×C283#57511yescharacteristic
169D283—22no—
170D283—41no—
171D283—82no—
172C283#305511yescharacteristic, complemented, Sylow
173#3055—2831nomaximal, Sylow
174C2×AGL(1,ℤ/8)—2831no—
175C22×M16—2831no—
176#525—2831no—
177C2×#121—2831no—
178#469—2832no—
179#468—2832no—
180#466—2832no—
181#465—2832no—
182#428—2831no—
183#418—2831no—
184#417D28311yescharacteristic, complemented
185C2×#41C2×D28311yescharacteristic
186C22×D4—2831no—
187C22×D4—5661no—
188C23×C4—2831no—
189AGL(1,ℤ/8)—5664no—
190C2×SD16—2832no—
191C2×D8—2832no—
192C2×M16—2831no—
193C2×M16—2832no—
194C2×M16—5664no—
195C2×M16C2×D28311yescharacteristic
196C22×C8—2832no—
197#123—5661no—
198C22⋊Q8—2831no—
199C22⋊Q8—2831no—
200#121—2831no—
201#121—2831no—
202#121—2832no—
203#121—2832no—
204#121—5661no—
205#121—5661no—
206#120—5661no—
207C4×D4—5661no—
208C4×D4—5661no—
209#117—2831no—
210C2×C4⋊C4C2×D28311yescharacteristic
211C2×C22⋊C4—5661no—
212#107—2832no—
213#106—2832no—
214#103C2×D28312yescomplemented
215#102—2832no—
216#102—2832no—
217#102C2×D28312yescomplemented
218C24—2831no—
219C24—5661no—
220#41—22no—
221#41—22no—
222C2×Q8C22×D28311yescharacteristic
223C2×D4—21no—
224C2×D4—2832no—
225C2×D4—5662no—
226C2×D4—5662no—
227C2×D4—5661no—
228C2×D4—5661no—
229C2×D4—5662no—
230C2×D4—5661no—
231C2×D4—5662no—
232C2×D4—11324no—
233C2×D4C22×D28311yescharacteristic
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—2831no—
237C22×C4—2831no—
238C22×C4—2831no—
239C22×C4—2831no—
240C22×C4—2831no—
241C22×C4—5662no—
242C22×C4—5662no—
243C22×C4C22×D28311yescharacteristic
244SD16—5664no—
245D8—5664no—
246M16—22no—
247M16—2834no—
248M16—5664no—
249C2×C8—2832no—
250C2×C8—5664no—
251C2×C8C22×D28312yes—
252C4⋊C4—21no—
253C4⋊C4—2831no—
254C4⋊C4—2831no—
255C4⋊C4—5661no—
256C4⋊C4—5661no—
257C4⋊C4C22×D28312yes—
258C22⋊C4—5662no—
259C22⋊C4—5662no—
260C22⋊C4—5661no—
261C22⋊C4—5661no—
262C22⋊C4—5661no—
263C22⋊C4—5661no—
264C22⋊C4—5661no—
265C42—5661no—
266C23—21no—
267C23—2831no—
268C23—2831no—
269C23—5662no—
270C23—5662no—
271C23—5662no—
272C23—11321no—
273C23—11322no—
274C23—11322no—
275C23—11322no—
276C23D4×D28311yescharacteristic
277Q8—21no—
278Q8—21no—
279D4—21no—
280D4—21no—
281D4—42no—
282D4—5662no—
283D4—5662no—
284D4—11324no—
285D4—11324no—
286D4—11322no—
287D4—11322no—
288D4—11322no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—2831no—
294C2×C4—2831no—
295C2×C4—5662no—
296C2×C4—5662no—
297C2×C4—5662no—
298C2×C4—5661no—
299C2×C4—5661no—
300C2×C4—5661no—
301C2×C4—5661no—
302C2×C4—5661no—
303C2×C4—5661no—
304C2×C4D4×D28311yescharacteristic
305C2×C4D4×D28312yes—
306C2×C4#1798539212yes—
307C2×C4C23×D28311yescharacteristic
308C8—22no—
309C8—2834no—
310C22—21no—
311C22—21no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—2832no—
316C22—5661no—
317C22—5661no—
318C22—5661no—
319C22—5661no—
320C22—5661no—
321C22—11324no—
322C22—11322no—
323C22—11322no—
324C22—11322no—
325C22—22644no—
326C22#2258358312yes—
327C22C2×D4×D28311yescharacteristic
328C4—41no—
329C4—42no—
330C4—5661no—
331C4—5661no—
332C4—11321no—
333C4—11321no—
334C4#2258358312yes—
335C4C2×D4×D28311yescharacteristic
336C4C2×#1798539211yescharacteristic
337C2—22no—
338C2—81no—
339C2—5662no—
340C2—11321no—
341C2—22642no—
342C2C2×#2258358311yescharacteristic
343C2AGL(1,ℤ/8)×D28311yescharacteristic
344C2#3182726411yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#417×C283C2
socleC22×C283C22×D4
hypercenter#417D283
hypocenterC283#3055
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. C2×#22583635C2semidirect
2. C2×M16×D283C2semidirect
3. #31826972C2semidirect
4. C2×#22583585C2semidirect
5. #31826717C2semidirect
6. #31826714C2semidirect
7. #31826700C2semidirect
8. #31826697C2semidirect
9. #31826505C2semidirect
10. #31826471C2semidirect
11. #417×C283C2semidirect
16. C2×#17985379C22semidirect
17. C2×#17985377C22semidirect
19. C2×#17985375C22semidirect
21. C2×C8×D283C22semidirect
23. #22583586C22semidirect
24. #22583586C22semidirect
29. #22583581C22semidirect
30. #22583581C22semidirect
36. #22583543C22semidirect
37. #22583541C22semidirect
38. #22583532C22semidirect
39. #22583532C22semidirect
43. #103×C283C22semidirect
44. #102×C283C22semidirect
65. #17985376D4semidirect
120. C283⋊C8C2×D4semidirect
154. C2×D283AGL(1,ℤ/8)semidirect
172. C283#3055semidirect
184. #417D283semidirect
214. #103C2×D283semidirect
217. #102C2×D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836352
C2×M16×D2832
#318269722
C2×#225835852
#318267172
#318267142
#318267002
#318266972
#318265052
#318264712
#417×C2832
#3055283