Finite Group Databasebeta

← #49130892

Subgroups — #49130892

Order 36224.

496 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 143 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×#22583643C211yesmaximal, characteristic, complemented
2C2×M16×D283C211yesmaximal, characteristic, complemented
3#31826977C211yesmaximal, characteristic
4C2×#22583590C211yesmaximal, characteristic, complemented
5#31826724C212yesmaximal, complemented
6#31826712C212yesmaximal, complemented
7#31826707C212yesmaximal, complemented
8#31826695C212yesmaximal, complemented
9#31826473C211yesmaximal, characteristic, complemented
10#31826464C211yesmaximal, characteristic, complemented
11#428×C283C211yesmaximal, characteristic, complemented
12C2×#17985392C2211yescharacteristic
13C22×Dic566C2211yescharacteristic
14C22×C4×D283C2211yescharacteristic
15#22583643—24no—
16C2×Dic1132C2212yescomplemented
17C2×#17985381C2212yescomplemented
18C2×#17985376C2211yescharacteristic
19C2×#17985375C2212yes—
20M16×D283—24no—
21C2×C8×D283C2212yes—
22#22583594—21no—
23#22583590—21no—
24#22583590—21no—
25#22583590C2212yes—
26#22583590C2211yescharacteristic
27#22583590C2212yes—
28#22583590C2211yescharacteristic
29#22583585C2211yescharacteristic
30#22583585C2211yescharacteristic
31#22583580—21no—
32C4×D1132—21no—
33C4×D1132—21no—
34C2×C4⋊Dic283—21no—
35C2×(C2×D283)⋊C4—21no—
36#22583533C2212yescomplemented
37#22583533C2212yescomplemented
38#22583533C2212yescomplemented
39#22583530C2212yescomplemented
40C2×M16×C283C2211yescharacteristic
41#117×C283C2211yescharacteristic
42C2×C4⋊C4×C283C2211yescharacteristic
43#107×C283C2212yescomplemented
44#106×C283C2212yescomplemented
45C23×D283D411yescharacteristic
46#17985392—22no—
47#17985392—22no—
48C2×Dic566—21no—
49C2×Dic566—22no—
50C2×Dic566—22no—
51C2×Dic566C2312yes—
52C2×Dic566C2311yescharacteristic
53C2×D1132—21no—
54C2×D1132—21no—
55C2×D1132C2311yescharacteristic
56C2×C22⋊D283—21no—
57C22×Dic283—21no—
58C22×Dic283D411yescharacteristic
59C2×C4×D283—22no—
60C2×C4×D283—21no—
61C2×C4×D283—22no—
62C2×C4×D283D411yescharacteristic
63C2×C4×D283D411yescharacteristic
64C2×C4×D283C2311yescharacteristic
65C2×C4×D283C2311yescharacteristic
66Dic1132—24no—
67#17985381—24no—
68#17985376—22no—
69#17985375—24no—
70C2×C283⋊C8C2312yes—
71C8×D283—24no—
72C4⋊Dic283—22no—
73C4⋊Dic283—22no—
74C4⋊Dic283—21no—
75C4⋊Dic283—21no—
76(C2×D283)⋊C4—22no—
77(C2×D283)⋊C4—22no—
78(C2×D283)⋊C4—21no—
79(C2×D283)⋊C4—21no—
80(C2×D283)⋊C4—21no—
81(C2×D283)⋊C4—21no—
82C22×C4×C283—21no—
83C22×C4×C283C2311yescharacteristic
84M16×C283D414yescomplemented
85C2×C8×C283C2312yes—
86C4⋊C4×C283—21no—
87C4⋊C4×C283C2311yescharacteristic
88C4⋊C4×C283C2311yescharacteristic
89C4⋊C4×C283C2312yes—
90C22⋊C4×C283—21no—
91C42×C283—21no—
92C22×D283—22no—
93C22×D283—22no—
94C22×D283—21no—
95C22×D283—41no—
96C22×D283C2×D411yescharacteristic
97C22×D283C2×D411yescharacteristic
98Dic566—22no—
99Dic566—21no—
100Dic566—22no—
101Dic566—21no—
102Dic566—42no—
103D1132—21no—
104D1132—21no—
105D1132—42no—
106D1132—42no—
107C22⋊D283—42no—
108C2×Dic283—22no—
109C2×Dic283—21no—
110C2×Dic283—22no—
111C2×Dic283—42no—
112C2×Dic283C2×D411yescharacteristic
113C2×Dic283C2×D411yescharacteristic
114C4×D283—22no—
115C4×D283—22no—
116C4×D283—21no—
117C4×D283—21no—
118C4×D283—21no—
119C4×D283—21no—
120C4×D283—42no—
121C283⋊C8—22no—
122C23×C283C2×D411yescharacteristic
123C2×C4×C283—21no—
124C2×C4×C283—21no—
125C2×C4×C283—22no—
126C2×C4×C283—42no—
127C2×C4×C283C2×D411yescharacteristic
128C2×C4×C283C2×D412yes—
129C2×C4×C283#4112yes—
130C2×C4×C283C2411yescharacteristic
131C8×C283C2×D414yes—
132C22×C283—21no—
133C22×C283—21no—
134C22×C283#12112yes—
135C22×C283C22×D411yescharacteristic
136C2×D283—21no—
137C2×D283—21no—
138C2×D283—21no—
139C2×D283—21no—
140C2×D283—21no—
141C2×D283—44no—
142C2×D283—41no—
143C2×D283—41no—
144C2×D283—41no—
145C2×D283#13712yescomplemented
146C4×C283—41no—
147C4×C283—42no—
148C4×C283—41no—
149C4×C283#12112yes—
150C4×C283C22×D411yescharacteristic
151C4×C283C2×#4111yescharacteristic
152Dic283—21no—
153Dic283—21no—
154Dic283—41no—
155Dic283—42no—
156C2×C283—22no—
157C2×C283C2×#12111yescharacteristic
158C2×C283C2×#13711yescharacteristic
159C2×C283#57511yescharacteristic
160D283—22no—
161D283—41no—
162D283—81no—
163C283#305611yescharacteristic, complemented, Sylow
164#3056—2831nomaximal, Sylow
165C2×#137—2831no—
166C22×M16—2831no—
167#525—2831no—
168C2×C22⋊Q8—2831no—
169#470—2832no—
170#468—2832no—
171#467—2832no—
172#465—2832no—
173#428D28311yescharacteristic, complemented
174#419—2831no—
175#417—2831no—
176C2×#41—2831no—
177C22×Q8—2831no—
178C23×C4—2831no—
179#137—5664no—
180C2×Q16—2832no—
181C2×SD16—2832no—
182C2×M16—2831no—
183C2×M16—2832no—
184C2×M16—5664no—
185C2×M16C2×D28311yescharacteristic
186C22×C8—2832no—
187#123—5661no—
188C22⋊Q8—2832no—
189C22⋊Q8—2832no—
190C22⋊Q8—2831no—
191C22⋊Q8—2831no—
192C22⋊Q8—5661no—
193C22⋊Q8—5661no—
194#121—2831no—
195#121—2831no—
196#120—5661no—
197C4×D4—5661no—
198C4×D4—5661no—
199#117C2×D28311yescharacteristic
200C2×C4⋊C4—5661no—
201C2×C4⋊C4C2×D28311yescharacteristic
202C2×C22⋊C4—5661no—
203#107C2×D28312yescomplemented
204#106C2×D28312yescomplemented
205#103—2832no—
206#103—2832no—
207#103—2832no—
208#102—2832no—
209C24—2831no—
210#41—5662no—
211#41—5662no—
212C2×Q8—2832no—
213C2×Q8—2831no—
214C2×Q8—5661no—
215C2×Q8—5662no—
216C2×Q8—5662no—
217C2×D4—2831no—
218C2×D4—5661no—
219C2×D4—5661no—
220C2×D4—5661no—
221C22×C4—21no—
222C22×C4—2831no—
223C22×C4—2831no—
224C22×C4—2831no—
225C22×C4—2831no—
226C22×C4—2831no—
227C22×C4—5662no—
228C22×C4—5662no—
229C22×C4—5661no—
230C22×C4—5661no—
231C22×C4C22×D28311yescharacteristic
232Q16—5664no—
233SD16—5664no—
234M16—5662no—
235M16—5664no—
236M16D113214yescomplemented
237C2×C8—2832no—
238C2×C8—5664no—
239C2×C8C22×D28312yes—
240C4⋊C4—21no—
241C4⋊C4—5662no—
242C4⋊C4—5662no—
243C4⋊C4—5661no—
244C4⋊C4—5661no—
245C4⋊C4C22×D28312yes—
246C4⋊C4C22×D28311yescharacteristic
247C4⋊C4C22×D28311yescharacteristic
248C22⋊C4—21no—
249C22⋊C4—5661no—
250C22⋊C4—5661no—
251C22⋊C4—5662no—
252C22⋊C4—5662no—
253C22⋊C4—5661no—
254C22⋊C4—5661no—
255C42—21no—
256C23—2831no—
257C23—2831no—
258C23—5662no—
259C23—5662no—
260C23—5661no—
261C23—11321no—
262C23D4×D28311yescharacteristic
263Q8—5662no—
264Q8—5661no—
265Q8—5662no—
266Q8—5661no—
267Q8—11322no—
268D4—5661no—
269D4—5661no—
270D4—11322no—
271D4—11322no—
272D4—11322no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—42no—
277C2×C4—2831no—
278C2×C4—2831no—
279C2×C4—5662no—
280C2×C4—5662no—
281C2×C4—5662no—
282C2×C4—5661no—
283C2×C4—5661no—
284C2×C4—5661no—
285C2×C4—5661no—
286C2×C4—5661no—
287C2×C4—5662no—
288C2×C4—11322no—
289C2×C4—11322no—
290C2×C4D4×D28311yescharacteristic
291C2×C4C2×D113212yes—
292C2×C4#1798539312yes—
293C2×C4C23×D28311yescharacteristic
294C8—5662no—
295C8C2×D113214yes—
296C22—21no—
297C22—21no—
298C22—2832no—
299C22—5661no—
300C22—5661no—
301C22—5661no—
302C22—5661no—
303C22—5661no—
304C22—11324no—
305C22—11321no—
306C22—11321no—
307C22—11321no—
308C22#2258358512yes—
309C22C2×D4×D28311yescharacteristic
310C4—41no—
311C4—42no—
312C4—41no—
313C4—5661no—
314C4—5661no—
315C4—11321no—
316C4—11322no—
317C4#2258358512yes—
318C4C22×D113211yescharacteristic
319C4C2×#1798539311yescharacteristic
320C2—22no—
321C2—5662no—
322C2—11321no—
323C2—22641no—
324C2C2×#2258358511yescharacteristic
325C2#137×D28311yescharacteristic
326C2#3182726111yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#428×C283C2
socleC22×C283C22×D4
prime radical#31826977C2
hypercenter#428D283
hypocenterC283#3056
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. C2×#22583643C2semidirect
2. C2×M16×D283C2semidirect
4. C2×#22583590C2semidirect
5. #31826724C2semidirect
6. #31826712C2semidirect
7. #31826707C2semidirect
8. #31826695C2semidirect
9. #31826473C2semidirect
10. #31826464C2semidirect
11. #428×C283C2semidirect
16. C2×Dic1132C22semidirect
17. C2×#17985381C22semidirect
36. #22583533C22semidirect
37. #22583533C22semidirect
38. #22583533C22semidirect
39. #22583530C22semidirect
43. #107×C283C22semidirect
44. #106×C283C22semidirect
84. M16×C283D4semidirect
145. C2×D283#137semidirect
163. C283#3056semidirect
173. #428D283semidirect
203. #107C2×D283semidirect
204. #106C2×D283semidirect
236. M16D1132semidirect

Maximal subgroups

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

subgroupindex
C2×#225836432
C2×M16×D2832
#318269772
C2×#225835902
#318267242
#318267122
#318267072
#318266952
#318264732
#318264642
#428×C2832
#3056283