Finite Group Databasebeta

← #49130804

Subgroups — #49130804

Order 36224.

504 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 141 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×#22583630C211yesmaximal, characteristic, complemented
2#31827032C211yesmaximal, characteristic, complemented
3#31826948C211yesmaximal, characteristic, complemented
4#31826685C212yesmaximal, complemented
5#31826678C212yesmaximal, complemented
6#31826663C212yesmaximal, complemented
7#31826658C212yesmaximal, complemented
8#31826493C211yesmaximal, characteristic, complemented
9#31826467C211yesmaximal, characteristic, complemented
10C2×#22583531C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C283C211yesmaximal, characteristic, complemented
12C2×#17985392C2211yescharacteristic
13#22583630—24no—
14C2×#17985382C2211yescharacteristic, complemented
15C2×#17985380C2211yescharacteristic, complemented
16C2×#17985379C2211yescharacteristic, complemented
17C2×#17985377C2211yescharacteristic, complemented
18C22×C283⋊C8C2211yescharacteristic
19#22583606C2212yes—
20#22583597C2212yes—
21#22583581—21no—
22#22583581—21no—
23#22583579—22no—
24Q8×Dic283C2212yescomplemented
25D4×Dic283—22no—
26D4×Dic283C2212yescomplemented
27#22583569—21no—
28#22583569C2211yescharacteristic
29C2×C4⋊Dic283—21no—
30C2×C4⋊Dic283C2211yescharacteristic
31Dic283⋊C8C2212yescomplemented
32Dic283⋊C8C2212yescomplemented
33#22583535C2212yescomplemented
34#22583532C2212yescomplemented
35#22583531C2212yes—
36#22583531C2212yes—
37C2×#41×C283C2211yescharacteristic
38C22×D4×C283C2211yescharacteristic
39AGL(1,ℤ/8)×C283—24no—
40C2×SD16×C283C2212yes—
41C2×D8×C283C2212yes—
42C2×M16×C283C2211yescharacteristic
43#17985392—22no—
44#17985392—22no—
45C2×Dic566C2311yescharacteristic
46C2×D1132C2311yescharacteristic
47C2×C22⋊D283—22no—
48C2×C22⋊D283—22no—
49C2×C22⋊D283—21no—
50C22×Dic283—22no—
51C22×Dic283—21no—
52C2×C4×D283—21no—
53#17985382—22no—
54#17985380—22no—
55#17985379—22no—
56#17985377—22no—
57C2×C283⋊C8—22no—
58C2×C283⋊C8C2311yescharacteristic
59C2×C283⋊C8C2311yescharacteristic
60C4⋊Dic283—22no—
61C4⋊Dic283—21no—
62C4⋊Dic283D411yescharacteristic
63C4⋊Dic283D411yescharacteristic
64C4⋊Dic283C2312yes—
65C22⋊Dic283—22no—
66C22⋊Dic283—22no—
67C22⋊Dic283—22no—
68C22⋊Dic283D412yes—
69C4×Dic283—22no—
70C4×Dic283C2312yes—
71C24×C283—21no—
72#41×C283—22no—
73#41×C283D414yescomplemented
74C2×Q8×C283C2311yescharacteristic
75C2×D4×C283—21no—
76C2×D4×C283—22no—
77C2×D4×C283—21no—
78C2×D4×C283—22no—
79C2×D4×C283C2312yes—
80C2×D4×C283C2311yescharacteristic
81C22×C4×C283—21no—
82C22×C4×C283C2311yescharacteristic
83SD16×C283—24no—
84D8×C283—24no—
85M16×C283—22no—
86C2×C8×C283C2312yes—
87C22×D283—21no—
88Dic566—21no—
89Dic566—21no—
90D1132—21no—
91D1132—21no—
92C22⋊D283—42no—
93C22⋊D283—82no—
94C22⋊D283—82no—
95C2×Dic283—22no—
96C2×Dic283—21no—
97C2×Dic283—82no—
98C2×Dic283—81no—
99C2×Dic283C2×D412yes—
100C2×Dic283C2×D412yes—
101C4×D283—42no—
102C283⋊C8—21no—
103C283⋊C8—21no—
104C23×C283—21no—
105C23×C283—22no—
106C23×C283—42no—
107C23×C283—81no—
108C23×C283—81no—
109C23×C283C2×D411yescharacteristic
110Q8×C283—21no—
111Q8×C283C2×D412yes—
112D4×C283—24no—
113D4×C283—22no—
114D4×C283—21no—
115D4×C283—22no—
116D4×C283—42no—
117D4×C283C2×D412yes—
118C2×C4×C283—21no—
119C2×C4×C283—24no—
120C2×C4×C283C2×D411yescharacteristic
121C2×C4×C283C2×D412yes—
122C2×C4×C283#4112yes—
123C2×C4×C283C2411yescharacteristic
124C8×C283—22no—
125C22×C283—21no—
126C22×C283—21no—
127C22×C283—22no—
128C22×C283—41no—
129C22×C283—41no—
130C22×C283—42no—
131C22×C283—42no—
132C22×C283—42no—
133C22×C283—84no—
134C22×C283#12112yes—
135C22×C283C22×D411yescharacteristic
136C2×D283—41no—
137C2×D283—41no—
138C2×D283—41no—
139C4×C283—22no—
140C4×C283#12112yes—
141C4×C283C22×D411yescharacteristic
142C4×C283C2×#4111yescharacteristic
143Dic283—22no—
144Dic283—22no—
145Dic283—41no—
146Dic283—42no—
147C2×C283—22no—
148C2×C283—42no—
149C2×C283—82no—
150C2×C283C2×#12111yescharacteristic
151C2×C283#57611yescharacteristic
152C2×C283#57711yescharacteristic
153D283—81no—
154C283#304511yescharacteristic, complemented, Sylow
155#3045—2831nomaximal, Sylow
156C2×AGL(1,ℤ/8)D28311yescharacteristic, complemented
157C2×#135—2831no—
158#535—2831no—
159#520—2831no—
160#464—2832no—
161#463—2832no—
162#460—2832no—
163#459—2832no—
164#424—2831no—
165#417—2831no—
166C2×#102—2831no—
167C2×#41—2831no—
168C2×#41C2×D28311yescharacteristic
169C22×D4C2×D28311yescharacteristic
170AGL(1,ℤ/8)—24no—
171#135—5664no—
172C2×Q16—2831no—
173C2×SD16—2831no—
174C2×SD16—2831no—
175C2×SD16C2×D28312yes—
176C2×D8—2831no—
177C2×D8C2×D28312yes—
178C2×M16C2×D28311yescharacteristic
179C22×C8—2831no—
180#127—2832no—
181#124—2832no—
182#121—5661no—
183#121—5661no—
184#120—5662no—
185C4×Q8—2832no—
186C4×D4—2832no—
187C4×D4—5662no—
188#117—2831no—
189#117—5661no—
190C2×C4⋊C4—2831no—
191C2×C4⋊C4—5661no—
192C4⋊C8—2832no—
193C4⋊C8—2832no—
194#103—2832no—
195#102—2832no—
196#102—2832no—
197#102—2832no—
198C24—21no—
199#41—22no—
200#41—5662no—
201#41—5662no—
202#41C22⋊D28314yescomplemented
203C2×Q8—2831no—
204C2×Q8C22×D28311yescharacteristic
205C2×D4—21no—
206C2×D4—21no—
207C2×D4—22no—
208C2×D4—22no—
209C2×D4—2831no—
210C2×D4—5662no—
211C2×D4—5662no—
212C2×D4—5661no—
213C2×D4C22×D28312yes—
214C2×D4C22×D28311yescharacteristic
215C22×C4—21no—
216C22×C4—5662no—
217C22×C4—5661no—
218C22×C4—5661no—
219C22×C4C22×D28311yescharacteristic
220Q16—5662no—
221SD16—24no—
222SD16—5662no—
223SD16—5662no—
224D8—24no—
225D8—5662no—
226M16—22no—
227C2×C8—2831no—
228C2×C8—2831no—
229C2×C8—5662no—
230C2×C8C22×D28312yes—
231C4⋊C4—2831no—
232C4⋊C4—2831no—
233C4⋊C4—2832no—
234C4⋊C4—5662no—
235C4⋊C4—5661no—
236C22⋊C4—2832no—
237C22⋊C4—5662no—
238C22⋊C4—5662no—
239C22⋊C4—5662no—
240C42—2832no—
241C42—5662no—
242C23—21no—
243C23—22no—
244C23—42no—
245C23—81no—
246C23—81no—
247C23—5661no—
248C23D4×D28311yescharacteristic
249Q8—21no—
250Q8—5661no—
251Q8—5661no—
252Q8C2×C22⋊D28312yes—
253D4—24no—
254D4—22no—
255D4—21no—
256D4—22no—
257D4—42no—
258D4—5661no—
259D4—5661no—
260D4—11322no—
261D4—22642no—
262D4—22642no—
263D4C2×C22⋊D28312yes—
264C2×C4—21no—
265C2×C4—24no—
266C2×C4—2832no—
267C2×C4—2832no—
268C2×C4—5662no—
269C2×C4—5661no—
270C2×C4—11322no—
271C2×C4—22642no—
272C2×C4—22641no—
273C2×C4D4×D28311yescharacteristic
274C2×C4C2×C22⋊D28312yes—
275C2×C4#1798539112yes—
276C2×C4C23×D28311yescharacteristic
277C8—22no—
278C8—5661no—
279C8—5661no—
280C22—21no—
281C22—21no—
282C22—22no—
283C22—41no—
284C22—41no—
285C22—42no—
286C22—42no—
287C22—42no—
288C22—84no—
289C22—11321no—
290C22—11321no—
291C22—11321no—
292C22#2258358212yes—
293C22C2×D4×D28311yescharacteristic
294C4—22no—
295C4—5662no—
296C4—5662no—
297C4—11321no—
298C4—11322no—
299C4#2258358212yes—
300C4C22×C22⋊D28311yescharacteristic
301C4C2×#1798539111yescharacteristic
302C2—22no—
303C2—42no—
304C2—82no—
305C2—22641no—
306C2C2×#2258358211yescharacteristic
307C2#3182726911yescharacteristic
308C2#3182727511yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
FittingC2×AGL(1,ℤ/8)×C283C2
socleC22×C283C22×D4
hypercenterC2×AGL(1,ℤ/8)D283
hypocenterC283#3045
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#22583630C2semidirect
2. #31827032C2semidirect
3. #31826948C2semidirect
4. #31826685C2semidirect
5. #31826678C2semidirect
6. #31826663C2semidirect
7. #31826658C2semidirect
8. #31826493C2semidirect
9. #31826467C2semidirect
10. C2×#22583531C2semidirect
11. C2×AGL(1,ℤ/8)×C283C2semidirect
14. C2×#17985382C22semidirect
15. C2×#17985380C22semidirect
16. C2×#17985379C22semidirect
17. C2×#17985377C22semidirect
24. Q8×Dic283C22semidirect
26. D4×Dic283C22semidirect
31. Dic283⋊C8C22semidirect
32. Dic283⋊C8C22semidirect
33. #22583535C22semidirect
34. #22583532C22semidirect
73. #41×C283D4semidirect
154. C283#3045semidirect
156. C2×AGL(1,ℤ/8)D283semidirect
202. #41C22⋊D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836302
#318270322
#318269482
#318266852
#318266782
#318266632
#318266582
#318264932
#318264672
C2×#225835312
C2×AGL(1,ℤ/8)×C2832
#3045283