Finite Group Databasebeta

← #49130894

Subgroups — #49130894

Order 36224.

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

normal subgroups only

This group has 496 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×#22583640C211yesmaximal, characteristic, complemented
2C2×M16×D283C211yesmaximal, characteristic, complemented
3#31826972C211yesmaximal, characteristic
4C2×#22583590C211yesmaximal, characteristic, complemented
5#31826722C212yesmaximal, complemented
6#31826710C212yesmaximal, complemented
7#31826705C212yesmaximal, complemented
8#31826693C212yesmaximal, complemented
9#31826505C211yesmaximal, characteristic, complemented
10#31826475C211yesmaximal, characteristic, complemented
11#417×C283C211yesmaximal, characteristic, complemented
12C22×Dic566C2211yescharacteristic
13C22×C4×D283C2211yescharacteristic
14#22583640—24no—
15C2×#17985382C2212yescomplemented
16C2×#17985380C2212yescomplemented
17C2×#17985376C2211yescharacteristic
18C2×#17985375C2212yes—
19M16×D283—24no—
20C2×C8×D283C2212yes—
21#22583591—21no—
22#22583590—21no—
23#22583590—21no—
24#22583590C2212yes—
25#22583590C2212yes—
26#22583586C2211yescharacteristic
27#22583586C2211yescharacteristic
28#22583581C2211yescharacteristic
29#22583581C2211yescharacteristic
30#22583578—21no—
31C4×C22⋊D283—21no—
32C4×C22⋊D283—21no—
33#22583569C2211yescharacteristic
34C2×C4⋊Dic283—21no—
35C2×(C2×D283)⋊C4—21no—
36#22583543C2212yescomplemented
37#22583541C2212yescomplemented
38#22583535C2212yescomplemented
39#22583535C2212yescomplemented
40C2×#41×C283C2211yescharacteristic
41C2×M16×C283C2211yescharacteristic
42C2×C4⋊C4×C283C2211yescharacteristic
43#103×C283C2212yescomplemented
44#102×C283C2212yescomplemented
45C23×D283D411yescharacteristic
46C2×Dic566—21no—
47C2×Dic566—22no—
48C2×Dic566—22no—
49C2×Dic566C2312yes—
50C2×C22⋊D283—21no—
51C2×C22⋊D283—21no—
52C22×Dic283—21no—
53C22×Dic283D411yescharacteristic
54C2×C4×D283—22no—
55C2×C4×D283—22no—
56C2×C4×D283D411yescharacteristic
57C2×C4×D283D411yescharacteristic
58C2×C4×D283C2311yescharacteristic
59C2×C4×D283C2311yescharacteristic
60#17985382—24no—
61#17985380—24no—
62#17985376D414yescomplemented
63#17985375—24no—
64C2×C283⋊C8C2312yes—
65C8×D283—24no—
66C4⋊Dic283—22no—
67C4⋊Dic283—22no—
68C4⋊Dic283C2311yescharacteristic
69C4⋊Dic283C2311yescharacteristic
70Dic283⋊C4—21no—
71Dic283⋊C4—21no—
72C22⋊Dic283—21no—
73C22⋊Dic283—21no—
74C22⋊Dic283—21no—
75(C2×D283)⋊C4—22no—
76(C2×D283)⋊C4—22no—
77(C2×D283)⋊C4—21no—
78(C2×D283)⋊C4—21no—
79C4×Dic283—21no—
80#41×C283—22no—
81#41×C283—22no—
82C2×Q8×C283C2311yescharacteristic
83C2×D4×C283—21no—
84C2×D4×C283C2311yescharacteristic
85C22×C4×C283—21no—
86C22×C4×C283—21no—
87C22×C4×C283C2311yescharacteristic
88M16×C283—22no—
89C2×C8×C283C2312yes—
90C4⋊C4×C283—21no—
91C4⋊C4×C283C2312yes—
92C22×D283—22no—
93C22×D283—22no—
94C22×D283—41no—
95C22×D283C2×D411yescharacteristic
96C22×D283C2×D411yescharacteristic
97Dic566—22no—
98Dic566—22no—
99Dic566—42no—
100C22⋊D283—42no—
101C22⋊D283—42no—
102C2×Dic283—22no—
103C2×Dic283—21no—
104C2×Dic283—21no—
105C2×Dic283—22no—
106C2×Dic283—42no—
107C2×Dic283C2×D411yescharacteristic
108C2×Dic283C2×D411yescharacteristic
109C4×D283—22no—
110C4×D283—22no—
111C4×D283—21no—
112C4×D283—21no—
113C4×D283—21no—
114C4×D283—21no—
115C283⋊C8C2×D414yes—
116C23×C283—21no—
117C23×C283C2×D411yescharacteristic
118Q8×C283—21no—
119Q8×C283—21no—
120D4×C283—21no—
121D4×C283—21no—
122D4×C283—42no—
123C2×C4×C283—22no—
124C2×C4×C283—21no—
125C2×C4×C283—42no—
126C2×C4×C283—42no—
127C2×C4×C283C2×D411yescharacteristic
128C2×C4×C283C2×D412yes—
129C2×C4×C283#4112yes—
130C2×C4×C283C2411yescharacteristic
131C8×C283—22no—
132C22×C283—21no—
133C22×C283—21no—
134C22×C283—41no—
135C22×C283—41no—
136C22×C283—41no—
137C22×C283#12112yes—
138C22×C283C22×D411yescharacteristic
139C2×D283—21no—
140C2×D283—21no—
141C2×D283—21no—
142C2×D283—21no—
143C2×D283—21no—
144C2×D283—44no—
145C2×D283#13712yescomplemented
146C4×C283—41no—
147C4×C283—42no—
148C4×C283#12112yes—
149C4×C283C22×D411yescharacteristic
150C4×C283C2×#4111yescharacteristic
151Dic283—21no—
152Dic283—21no—
153Dic283—42no—
154Dic283—41no—
155Dic283—41no—
156C2×C283—22no—
157C2×C283—81no—
158C2×C283C2×#12111yescharacteristic
159C2×C283C2×#13711yescharacteristic
160C2×C283#57511yescharacteristic
161D283—22no—
162D283—41no—
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#428—2831no—
174#419—2831no—
175#417D28311yescharacteristic, complemented
176C2×#41C2×D28311yescharacteristic
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—2831no—
189C22⋊Q8—2831no—
190C22⋊Q8—2832no—
191C22⋊Q8—2832no—
192C22⋊Q8—5661no—
193C22⋊Q8—5661no—
194#121—2831no—
195#121—2831no—
196#120—5661no—
197C4×D4—5661no—
198C4×D4—5661no—
199#117—2831no—
200C2×C4⋊C4—5661no—
201C2×C4⋊C4C2×D28311yescharacteristic
202C2×C22⋊C4—5661no—
203#107—2832no—
204#106—2832no—
205#103—2832no—
206#103—2832no—
207#103C2×D28312yescomplemented
208#102C2×D28312yescomplemented
209C24—2831no—
210#41—22no—
211#41—22no—
212C2×Q8—2832no—
213C2×Q8—5661no—
214C2×Q8—5662no—
215C2×Q8—5662no—
216C2×Q8C22×D28311yescharacteristic
217C2×D4—21no—
218C2×D4—5661no—
219C2×D4—5661no—
220C2×D4C22×D28311yescharacteristic
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—2831no—
224C22×C4—2831no—
225C22×C4—2831no—
226C22×C4—2831no—
227C22×C4—2831no—
228C22×C4—5662no—
229C22×C4—5662no—
230C22×C4—5661no—
231C22×C4C22×D28311yescharacteristic
232Q16—5664no—
233SD16—5664no—
234M16—22no—
235M16—2834no—
236M16—5664no—
237C2×C8—2832no—
238C2×C8—5664no—
239C2×C8C22×D28312yes—
240C4⋊C4—21no—
241C4⋊C4—2831no—
242C4⋊C4—2831no—
243C4⋊C4—5661no—
244C4⋊C4—5661no—
245C4⋊C4—5662no—
246C4⋊C4—5662no—
247C4⋊C4C22×D28312yes—
248C22⋊C4—5662no—
249C22⋊C4—5662no—
250C22⋊C4—5661no—
251C22⋊C4—5661no—
252C22⋊C4—5661no—
253C22⋊C4—5661no—
254C22⋊C4—5661no—
255C42—5661no—
256C23—21no—
257C23—2831no—
258C23—2831no—
259C23—5662no—
260C23—5662no—
261C23—11321no—
262C23D4×D28311yescharacteristic
263Q8—21no—
264Q8—21no—
265Q8—5662no—
266Q8—5662no—
267Q8—11322no—
268D4—21no—
269D4—21no—
270D4—42no—
271D4—11322no—
272D4—11322no—
273C2×C4—22no—
274C2×C4—21no—
275C2×C4—42no—
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—5661no—
288C2×C4—5662no—
289C2×C4—11322no—
290C2×C4D4×D28311yescharacteristic
291C2×C4D4×D28312yes—
292C2×C4#1798539212yes—
293C2×C4C23×D28311yescharacteristic
294C8—22no—
295C8—2834no—
296C22—21no—
297C22—21no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—2832no—
302C22—5661no—
303C22—5661no—
304C22—5661no—
305C22—5661no—
306C22—5661no—
307C22—11324no—
308C22#2258358312yes—
309C22C2×D4×D28311yescharacteristic
310C4—41no—
311C4—42no—
312C4—5661no—
313C4—5661no—
314C4—11322no—
315C4—11321no—
316C4—11321no—
317C4#2258358312yes—
318C4C2×D4×D28311yescharacteristic
319C4C2×#1798539211yescharacteristic
320C2—22no—
321C2—81no—
322C2—5662no—
323C2—11321no—
324C2C2×#2258358311yescharacteristic
325C2#137×D28311yescharacteristic
326C2#3182726111yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#417×C283C2
socleC22×C283C22×D4
prime radical#31826972C2
hypercenter#417D283
hypocenterC283#3056
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×#22583640C2semidirect
2. C2×M16×D283C2semidirect
4. C2×#22583590C2semidirect
5. #31826722C2semidirect
6. #31826710C2semidirect
7. #31826705C2semidirect
8. #31826693C2semidirect
9. #31826505C2semidirect
10. #31826475C2semidirect
11. #417×C283C2semidirect
15. C2×#17985382C22semidirect
16. C2×#17985380C22semidirect
36. #22583543C22semidirect
37. #22583541C22semidirect
38. #22583535C22semidirect
39. #22583535C22semidirect
43. #103×C283C22semidirect
44. #102×C283C22semidirect
62. #17985376D4semidirect
145. C2×D283#137semidirect
163. C283#3056semidirect
175. #417D283semidirect
207. #103C2×D283semidirect
208. #102C2×D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836402
C2×M16×D2832
#318269722
C2×#225835902
#318267222
#318267102
#318267052
#318266932
#318265052
#318264752
#417×C2832
#3056283