Finite Group Databasebeta

← #49130722

Subgroups — #49130722

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×#22583636C211yesmaximal, characteristic, complemented
2#31827011C211yesmaximal, characteristic, complemented
3C2×D4×Dic283C211yesmaximal, characteristic, complemented
4#31826682C212yesmaximal, complemented
5#31826678C212yesmaximal, complemented
6#31826668C212yesmaximal, complemented
7#31826658C212yesmaximal, complemented
8#31826488C211yesmaximal, characteristic, complemented
9#31826467C211yesmaximal, characteristic, complemented
10#31826466C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C283C211yesmaximal, characteristic, complemented
12C2×#17985392C2211yescharacteristic
13C23×Dic283—21no—
14#22583636—24no—
15C2×#17985380C2212yes—
16C2×#17985377C2212yes—
17C2×#17985376C2211yescharacteristic
18#22583608C2211yescharacteristic, complemented
19#22583606C2211yescharacteristic, complemented
20#22583599C2211yescharacteristic, complemented
21#22583597C2211yescharacteristic, complemented
22#22583582—21no—
23#22583582—21no—
24D4×Dic283—21no—
25D4×Dic283—21no—
26D4×Dic283C2212yes—
27D4×Dic283C2212yes—
28C4×C22⋊D283—21no—
29C4×C22⋊D283—21no—
30C2×C4⋊Dic283C2211yescharacteristic
31C2×C22⋊Dic283—21no—
32C2×C4×Dic283C2211yescharacteristic
33Dic283⋊C8C2212yescomplemented
34Dic283⋊C8C2212yescomplemented
35#22583535C2212yescomplemented
36#22583534C2212yescomplemented
37#22583532C2212yescomplemented
38#22583531C2212yescomplemented
39C2×#41×C283C2211yescharacteristic
40C22×D4×C283C2211yescharacteristic
41AGL(1,ℤ/8)×C283—24no—
42C2×SD16×C283C2212yes—
43C2×D8×C283C2212yes—
44C2×M16×C283C2211yescharacteristic
45#17985392—22no—
46#17985392—22no—
47C2×Dic566C2311yescharacteristic
48C2×D1132C2311yescharacteristic
49C2×C22⋊D283—21no—
50C2×C22⋊D283—21no—
51C2×C22⋊D283—21no—
52C22×Dic283—22no—
53C22×Dic283—22no—
54C22×Dic283—21no—
55C22×Dic283—44no—
56C22×Dic283D411yescharacteristic
57C22×Dic283D411yescharacteristic
58C2×C4×D283—21no—
59#17985380—24no—
60#17985377—24no—
61#17985376—22no—
62C2×C283⋊C8C2312yes—
63C4⋊Dic283—21no—
64C4⋊Dic283C2312yes—
65Dic283⋊C4—21no—
66Dic283⋊C4—21no—
67C22⋊Dic283—22no—
68C22⋊Dic283—22no—
69C22⋊Dic283—21no—
70C22⋊Dic283—21no—
71(C2×D283)⋊C4—21no—
72(C2×D283)⋊C4—21no—
73C4×Dic283D411yescharacteristic
74C4×Dic283D411yescharacteristic
75C4×Dic283C2311yescharacteristic
76C4×Dic283C2311yescharacteristic
77C24×C283—21no—
78#41×C283—22no—
79#41×C283—22no—
80C2×Q8×C283C2311yescharacteristic
81C2×D4×C283—21no—
82C2×D4×C283—21no—
83C2×D4×C283—22no—
84C2×D4×C283D414yescomplemented
85C2×D4×C283C2311yescharacteristic
86C2×D4×C283C2312yes—
87C22×C4×C283—21no—
88C22×C4×C283C2311yescharacteristic
89SD16×C283—24no—
90D8×C283—24no—
91M16×C283—22no—
92C2×C8×C283C2312yes—
93C22×D283—21no—
94Dic566—21no—
95Dic566—21no—
96D1132—21no—
97D1132—21no—
98C22⋊D283—42no—
99C22⋊D283—42no—
100C22⋊D283—42no—
101C2×Dic283—22no—
102C2×Dic283—21no—
103C2×Dic283—22no—
104C2×Dic283—41no—
105C2×Dic283—44no—
106C2×Dic283—42no—
107C2×Dic283—44no—
108C2×Dic283C2×D411yescharacteristic
109C2×Dic283C2×D411yescharacteristic
110C2×Dic283C2×D411yescharacteristic
111C2×Dic283C2×D411yescharacteristic
112C4×D283—42no—
113C283⋊C8—22no—
114C23×C283—24no—
115C23×C283—22no—
116C23×C283—21no—
117C23×C283—42no—
118C23×C283—42no—
119C23×C283C2×D411yescharacteristic
120Q8×C283—21no—
121Q8×C283—21no—
122D4×C283—24no—
123D4×C283—21no—
124D4×C283—22no—
125D4×C283—21no—
126D4×C283—42no—
127D4×C283C2×D414yes—
128C2×C4×C283—21no—
129C2×C4×C283—42no—
130C2×C4×C283C2×D411yescharacteristic
131C2×C4×C283C2×D412yes—
132C2×C4×C283#4112yes—
133C2×C4×C283C2411yescharacteristic
134C8×C283—22no—
135C22×C283—21no—
136C22×C283—21no—
137C22×C283—24no—
138C22×C283—48no—
139C22×C283—42no—
140C22×C283—42no—
141C22×C283—41no—
142C22×C283—41no—
143C22×C283—41no—
144C22×C283#12112yes—
145C22×C283C22×D411yescharacteristic
146C2×D283—41no—
147C2×D283—41no—
148C2×D283—41no—
149C4×C283—41no—
150C4×C283#12112yes—
151C4×C283C22×D411yescharacteristic
152C4×C283C2×#4111yescharacteristic
153Dic283—21no—
154Dic283—21no—
155Dic283—21no—
156Dic283—41no—
157Dic283—42no—
158Dic283AGL(1,ℤ/8)12yescomplemented
159C2×C283—22no—
160C2×C283—44no—
161C2×C283—81no—
162C2×C283C2×#12111yescharacteristic
163C2×C283C2×AGL(1,ℤ/8)11yescharacteristic
164C2×C283#57511yescharacteristic
165D283—81no—
166C283#303611yescharacteristic, complemented, Sylow
167#3036—2831nomaximal, Sylow
168C2×AGL(1,ℤ/8)—2831no—
169C2×AGL(1,ℤ/8)D28311yescharacteristic, complemented
170#532—2831no—
171C2×C4×D4—2831no—
172#463—2832no—
173#463—2832no—
174#461—2832no—
175#459—2832no—
176#423—2831no—
177#417—2831no—
178#417—2831no—
179C2×#41—2831no—
180C2×#41C2×D28311yescharacteristic
181C22×D4C2×D28311yescharacteristic
182C23×C4—5661no—
183AGL(1,ℤ/8)—24no—
184AGL(1,ℤ/8)—5664no—
185C2×SD16—2832no—
186C2×SD16C2×D28312yes—
187C2×D8—2832no—
188C2×D8C2×D28312yes—
189C2×M16—2831no—
190C2×M16C2×D28311yescharacteristic
191C4⋊Q8—2831no—
192#127—2831no—
193#124—2831no—
194#124—2831no—
195#121—5661no—
196#121—5661no—
197C4×D4—2832no—
198C4×D4—2832no—
199C4×D4—5661no—
200C4×D4—5661no—
201C4×D4—5661no—
202C4×D4—5661no—
203C2×C4⋊C4—2831no—
204C2×C22⋊C4—5661no—
205C2×C42—2831no—
206C4⋊C8—2832no—
207C4⋊C8—2832no—
208#103—2832no—
209#103—2832no—
210#102—2832no—
211#102—2832no—
212C24—21no—
213#41—22no—
214#41—22no—
215#41—5662no—
216#41—5662no—
217C2×Q8—2831no—
218C2×Q8C22×D28311yescharacteristic
219C2×D4—21no—
220C2×D4—21no—
221C2×D4—22no—
222C2×D4—2831no—
223C2×D4—5661no—
224C2×D4—5661no—
225C2×D4—5661no—
226C2×D4C22⋊D28314yescomplemented
227C2×D4C22×D28311yescharacteristic
228C2×D4C22×D28312yes—
229C22×C4—21no—
230C22×C4—2831no—
231C22×C4—2831no—
232C22×C4—5662no—
233C22×C4—5662no—
234C22×C4—5661no—
235C22×C4—5661no—
236C22×C4—11324no—
237C22×C4C22×D28311yescharacteristic
238SD16—24no—
239SD16—5664no—
240D8—24no—
241D8—5664no—
242M16—22no—
243M16—5662no—
244C2×C8—2832no—
245C2×C8C22×D28312yes—
246C4⋊C4—2832no—
247C4⋊C4—5661no—
248C4⋊C4—5661no—
249C4⋊C4—5661no—
250C22⋊C4—5661no—
251C22⋊C4—5661no—
252C22⋊C4—5662no—
253C22⋊C4—5662no—
254C22⋊C4—5661no—
255C22⋊C4—5661no—
256C42—2831no—
257C42—2831no—
258C42—2831no—
259C42—2831no—
260C23—24no—
261C23—22no—
262C23—21no—
263C23—42no—
264C23—42no—
265C23—5661no—
266C23D4×D28311yescharacteristic
267Q8—21no—
268Q8—21no—
269Q8—5661no—
270Q8—5661no—
271D4—24no—
272D4—21no—
273D4—22no—
274D4—21no—
275D4—42no—
276D4—5661no—
277D4—5661no—
278D4—11322no—
279D4—11322no—
280D4—11322no—
281D4C2×C22⋊D28314yes—
282C2×C4—21no—
283C2×C4—42no—
284C2×C4—2831no—
285C2×C4—2831no—
286C2×C4—2831no—
287C2×C4—2831no—
288C2×C4—5662no—
289C2×C4—5662no—
290C2×C4—5661no—
291C2×C4—11324no—
292C2×C4—11321no—
293C2×C4—11324no—
294C2×C4—11322no—
295C2×C4—11322no—
296C2×C4D4×D28311yescharacteristic
297C2×C4C2×C22⋊D28312yes—
298C2×C4#1798539112yes—
299C2×C4C23×D28311yescharacteristic
300C8—22no—
301C8—5662no—
302C22—21no—
303C22—21no—
304C22—24no—
305C22—48no—
306C22—42no—
307C22—42no—
308C22—41no—
309C22—41no—
310C22—41no—
311C22—11321no—
312C22—11321no—
313C22—11321no—
314C22#2258358212yes—
315C22C2×D4×D28311yescharacteristic
316C4—41no—
317C4—2832no—
318C4—5661no—
319C4—5661no—
320C4—5661no—
321C4—11321no—
322C4—11322no—
323C4#2258358212yes—
324C4C22×C22⋊D28311yescharacteristic
325C4C2×#1798539111yescharacteristic
326C2—22no—
327C2—44no—
328C2—81no—
329C2—22641no—
330C2C2×#2258358211yescharacteristic
331C2AGL(1,ℤ/8)×D28311yescharacteristic
332C2#3182726111yescharacteristic
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#3036
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×#22583636C2semidirect
2. #31827011C2semidirect
3. C2×D4×Dic283C2semidirect
4. #31826682C2semidirect
5. #31826678C2semidirect
6. #31826668C2semidirect
7. #31826658C2semidirect
8. #31826488C2semidirect
9. #31826467C2semidirect
10. #31826466C2semidirect
11. C2×AGL(1,ℤ/8)×C283C2semidirect
18. #22583608C22semidirect
19. #22583606C22semidirect
20. #22583599C22semidirect
21. #22583597C22semidirect
33. Dic283⋊C8C22semidirect
34. Dic283⋊C8C22semidirect
35. #22583535C22semidirect
36. #22583534C22semidirect
37. #22583532C22semidirect
38. #22583531C22semidirect
84. C2×D4×C283D4semidirect
158. Dic283AGL(1,ℤ/8)semidirect
166. C283#3036semidirect
169. C2×AGL(1,ℤ/8)D283semidirect
226. C2×D4C22⋊D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836362
#318270112
C2×D4×Dic2832
#318266822
#318266782
#318266682
#318266582
#318264882
#318264672
#318264662
C2×AGL(1,ℤ/8)×C2832
#3036283