Finite Group Databasebeta

← #49130881

Subgroups — #49130881

Order 36224.

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

normal subgroups only

This group has 548 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×#22583638C211yesmaximal, characteristic, complemented
2C2×M16×D283C211yesmaximal, characteristic, complemented
3#31826977C211yesmaximal, characteristic, complemented
4C2×#22583585C211yesmaximal, characteristic, complemented
5#31826719C212yesmaximal, complemented
6#31826712C212yesmaximal, complemented
7#31826702C212yesmaximal, complemented
8#31826695C212yesmaximal, complemented
9#31826469C211yesmaximal, characteristic, complemented
10#31826464C211yesmaximal, characteristic, complemented
11#428×C283C211yesmaximal, characteristic, complemented
12C2×#17985392C2211yescharacteristic
13C22×D1132—21no—
14C22×D1132C2211yescharacteristic
15C22×C4×D283C2211yescharacteristic
16#22583638—24no—
17C2×#17985381C2212yescomplemented
18C2×D2264C2212yescomplemented
19C2×#17985376C2211yescharacteristic
20C2×#17985375C2212yescomplemented
21M16×D283—24no—
22C2×C8×D283C2212yescomplemented
23#22583594—21no—
24#22583590C2211yescharacteristic, complemented
25#22583590C2211yescharacteristic, complemented
26#22583585—21no—
27#22583585—21no—
28#22583585C2212yes—
29#22583585C2212yes—
30#22583585C2211yescharacteristic, complemented
31#22583585C2211yescharacteristic, complemented
32#22583580—21no—
33C4×D1132—21no—
34C4×D1132—21no—
35C2×(C2×D283)⋊C4—21no—
36#22583533C2212yescomplemented
37#22583530C2212yescomplemented
38#22583530C2212yescomplemented
39#22583530C2212yescomplemented
40C2×M16×C283C2211yescharacteristic
41#117×C283C2211yescharacteristic
42C2×C4⋊C4×C283C2211yescharacteristic
43#107×C283C2212yescomplemented
44#106×C283C2212yescomplemented
45C23×D283—21no—
46C23×D283D411yescharacteristic
47#17985392—22no—
48#17985392—22no—
49C2×Dic566C2311yescharacteristic
50C2×D1132—21no—
51C2×D1132—21no—
52C2×D1132—22no—
53C2×D1132—22no—
54C2×D1132—22no—
55C2×D1132—21no—
56C2×D1132—22no—
57C2×D1132—44no—
58C2×D1132C2312yes—
59C2×D1132C2311yescharacteristic
60C2×C22⋊D283—21no—
61C22×Dic283D411yescharacteristic
62C2×C4×D283—22no—
63C2×C4×D283—21no—
64C2×C4×D283—22no—
65C2×C4×D283D411yescharacteristic
66C2×C4×D283D411yescharacteristic
67C2×C4×D283C2311yescharacteristic
68C2×C4×D283C2311yescharacteristic
69#17985381—24no—
70D2264—24no—
71#17985376—22no—
72#17985375—24no—
73C2×C283⋊C8C2312yes—
74C8×D283—24no—
75C4⋊Dic283—21no—
76C4⋊Dic283—21no—
77(C2×D283)⋊C4—21no—
78(C2×D283)⋊C4—21no—
79(C2×D283)⋊C4—22no—
80(C2×D283)⋊C4—22no—
81(C2×D283)⋊C4—21no—
82(C2×D283)⋊C4—21no—
83C22×C4×C283—21no—
84C22×C4×C283C2311yescharacteristic
85M16×C283D414yescomplemented
86C2×C8×C283C2312yes—
87C4⋊C4×C283—21no—
88C4⋊C4×C283C2312yes—
89C4⋊C4×C283C2311yescharacteristic
90C4⋊C4×C283C2311yescharacteristic
91C22⋊C4×C283—21no—
92C42×C283—21no—
93C22×D283—22no—
94C22×D283—22no—
95C22×D283—21no—
96C22×D283—22no—
97C22×D283—41no—
98C22×D283—42no—
99C22×D283—42no—
100C22×D283—42no—
101C22×D283C2×D411yescharacteristic
102C22×D283C2×D411yescharacteristic
103Dic566—21no—
104Dic566—21no—
105D1132—21no—
106D1132—22no—
107D1132—22no—
108D1132—21no—
109D1132—42no—
110D1132—44no—
111D1132—42no—
112D1132—44no—
113D1132—42no—
114C22⋊D283—42no—
115C2×Dic283—22no—
116C2×Dic283—21no—
117C2×Dic283C2×D411yescharacteristic
118C2×Dic283C2×D411yescharacteristic
119C4×D283—22no—
120C4×D283—22no—
121C4×D283—21no—
122C4×D283—21no—
123C4×D283—21no—
124C4×D283—21no—
125C4×D283—42no—
126C283⋊C8—22no—
127C23×C283C2×D411yescharacteristic
128C2×C4×C283—21no—
129C2×C4×C283—21no—
130C2×C4×C283—22no—
131C2×C4×C283—42no—
132C2×C4×C283C2×D411yescharacteristic
133C2×C4×C283C2×D412yes—
134C2×C4×C283#4112yes—
135C2×C4×C283C2411yescharacteristic
136C8×C283C2×D414yescomplemented
137C22×C283—21no—
138C22×C283—21no—
139C22×C283#12112yes—
140C22×C283C22×D411yescharacteristic
141C2×D283—21no—
142C2×D283—21no—
143C2×D283—21no—
144C2×D283—21no—
145C2×D283—21no—
146C2×D283—44no—
147C2×D283—41no—
148C2×D283—42no—
149C2×D283—42no—
150C2×D283—41no—
151C2×D283—41no—
152C2×D283—42no—
153C2×D283—84no—
154C2×D283AGL(1,ℤ/8)12yescomplemented
155C4×C283—42no—
156C4×C283—41no—
157C4×C283—41no—
158C4×C283#12112yes—
159C4×C283C22×D411yescharacteristic
160C4×C283C2×#4111yescharacteristic
161Dic283—21no—
162Dic283—21no—
163Dic283—41no—
164C2×C283—22no—
165C2×C283C2×#12111yescharacteristic
166C2×C283C2×AGL(1,ℤ/8)11yescharacteristic
167C2×C283#57511yescharacteristic
168D283—22no—
169D283—41no—
170D283—81no—
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#428D28311yescharacteristic, complemented
183#418—2831no—
184#417—2831no—
185C2×#41—2831no—
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—2832no—
201#121—2832no—
202#121—2831no—
203#121—2831no—
204#121—5661no—
205#121—5661no—
206#120—5661no—
207C4×D4—5661no—
208C4×D4—5661no—
209#117C2×D28311yescharacteristic
210C2×C4⋊C4C2×D28311yescharacteristic
211C2×C22⋊C4—5661no—
212#107C2×D28312yescomplemented
213#106C2×D28312yescomplemented
214#103—2832no—
215#102—2832no—
216#102—2832no—
217#102—2832no—
218C24—2831no—
219C24—5661no—
220#41—5662no—
221#41—5662no—
222C2×Q8—2831no—
223C2×D4—2831no—
224C2×D4—2832no—
225C2×D4—5661no—
226C2×D4—5661no—
227C2×D4—5662no—
228C2×D4—5662no—
229C2×D4—5661no—
230C2×D4—5661no—
231C2×D4—5662no—
232C2×D4—5662no—
233C2×D4—11324no—
234C22×C4—21no—
235C22×C4—2831no—
236C22×C4—2831no—
237C22×C4—2831no—
238C22×C4—2831no—
239C22×C4—2831no—
240C22×C4—5662no—
241C22×C4—5662no—
242C22×C4—5661no—
243C22×C4C22×D28311yescharacteristic
244SD16—5664no—
245D8—5664no—
246M16—5662no—
247M16—5664no—
248M16D113214yescomplemented
249C2×C8—2832no—
250C2×C8—5664no—
251C2×C8C22×D28312yes—
252C4⋊C4—21no—
253C4⋊C4—5661no—
254C4⋊C4—5661no—
255C4⋊C4C22×D28311yescharacteristic
256C4⋊C4C22×D28311yescharacteristic
257C4⋊C4C22×D28312yes—
258C22⋊C4—21no—
259C22⋊C4—5662no—
260C22⋊C4—5662no—
261C22⋊C4—5661no—
262C22⋊C4—5661no—
263C22⋊C4—5661no—
264C22⋊C4—5661no—
265C42—21no—
266C23—2831no—
267C23—2831no—
268C23—5662no—
269C23—5662no—
270C23—5661no—
271C23—5662no—
272C23—11321no—
273C23—11322no—
274C23—11322no—
275C23—11322no—
276C23D4×D28311yescharacteristic
277Q8—5661no—
278Q8—5661no—
279D4—5661no—
280D4—5662no—
281D4—5662no—
282D4—5661no—
283D4—11324no—
284D4—11322no—
285D4—11324no—
286D4—11322no—
287D4—11322no—
288D4—11322no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—22no—
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—11322no—
304C2×C4D4×D28311yescharacteristic
305C2×C4C2×D113212yes—
306C2×C4#1798539312yes—
307C2×C4C23×D28311yescharacteristic
308C8—5662no—
309C8C2×D113214yescomplemented
310C22—21no—
311C22—21no—
312C22—2832no—
313C22—5661no—
314C22—5661no—
315C22—5661no—
316C22—5661no—
317C22—5661no—
318C22—11324no—
319C22—11321no—
320C22—11322no—
321C22—11322no—
322C22—11321no—
323C22—11321no—
324C22—11322no—
325C22—22644no—
326C22#2258358512yes—
327C22C2×D4×D28311yescharacteristic
328C4—42no—
329C4—41no—
330C4—41no—
331C4—5661no—
332C4—5661no—
333C4—11321no—
334C4#2258358512yes—
335C4C22×D113211yescharacteristic
336C4C2×#1798539311yescharacteristic
337C2—22no—
338C2—5662no—
339C2—11321no—
340C2—22642no—
341C2—22641no—
342C2C2×#2258358511yescharacteristic
343C2AGL(1,ℤ/8)×D28311yescharacteristic
344C2#3182726411yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#428×C283C2
socleC22×C283C22×D4
hypercenter#428D283
hypocenterC283#3055
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×#22583638C2semidirect
2. C2×M16×D283C2semidirect
3. #31826977C2semidirect
4. C2×#22583585C2semidirect
5. #31826719C2semidirect
6. #31826712C2semidirect
7. #31826702C2semidirect
8. #31826695C2semidirect
9. #31826469C2semidirect
10. #31826464C2semidirect
11. #428×C283C2semidirect
17. C2×#17985381C22semidirect
18. C2×D2264C22semidirect
20. C2×#17985375C22semidirect
22. C2×C8×D283C22semidirect
24. #22583590C22semidirect
25. #22583590C22semidirect
30. #22583585C22semidirect
31. #22583585C22semidirect
36. #22583533C22semidirect
37. #22583530C22semidirect
38. #22583530C22semidirect
39. #22583530C22semidirect
43. #107×C283C22semidirect
44. #106×C283C22semidirect
85. M16×C283D4semidirect
136. C8×C283C2×D4semidirect
154. C2×D283AGL(1,ℤ/8)semidirect
172. C283#3055semidirect
182. #428D283semidirect
212. #107C2×D283semidirect
213. #106C2×D283semidirect
248. M16D1132semidirect
309. C8C2×D1132semidirect

Maximal subgroups

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

subgroupindex
C2×#225836382
C2×M16×D2832
#318269772
C2×#225835852
#318267192
#318267122
#318267022
#318266952
#318264692
#318264642
#428×C2832
#3055283