Finite Group Databasebeta

← #18004057

Subgroups — #18004057

Order 4736.

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×#11924562C211yesmaximal, characteristic, complemented
2#14558079C211yesmaximal, characteristic, complemented
3C2×D4×Dic37C211yesmaximal, characteristic, complemented
4#14557750C212yesmaximal, complemented
5#14557746C212yesmaximal, complemented
6#14557736C212yesmaximal, complemented
7#14557726C212yesmaximal, complemented
8#14557556C211yesmaximal, characteristic, complemented
9#14557535C211yesmaximal, characteristic, complemented
10#14557534C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C37C211yesmaximal, characteristic, complemented
12C2×#10596962C2211yescharacteristic
13C23×Dic37—21no—
14#11924562—24no—
15C2×#10596940C2212yes—
16C2×#10596939C2212yes—
17C2×#10596934C2211yescharacteristic
18#11924534C2211yescharacteristic, complemented
19#11924532C2211yescharacteristic, complemented
20#11924525C2211yescharacteristic, complemented
21#11924523C2211yescharacteristic, complemented
22#11924508—21no—
23#11924508—21no—
24D4×Dic37—21no—
25D4×Dic37—21no—
26D4×Dic37C2212yes—
27D4×Dic37C2212yes—
28C4×C22⋊D37—21no—
29C4×C22⋊D37—21no—
30C2×C4⋊Dic37C2211yescharacteristic
31C2×C22⋊Dic37—21no—
32C2×C4×Dic37C2211yescharacteristic
33#11924465C2212yescomplemented
34#11924465C2212yescomplemented
35#11924461C2212yescomplemented
36#11924460C2212yescomplemented
37#11924458C2212yescomplemented
38#11924457C2212yescomplemented
39C2×#41×C37C2211yescharacteristic
40C22×D4×C37C2211yescharacteristic
41AGL(1,ℤ/8)×C37—24no—
42C2×SD16×C37C2212yes—
43C2×D8×C37C2212yes—
44C2×M16×C37C2211yescharacteristic
45C24×C37—21no—
46#41×C37—22no—
47#41×C37—22no—
48C2×Q8×C37C2311yescharacteristic
49C2×D4×C37—21no—
50C2×D4×C37—21no—
51C2×D4×C37—22no—
52C2×D4×C37D414yescomplemented
53C2×D4×C37C2311yescharacteristic
54C2×D4×C37C2312yes—
55C22×C4×C37—21no—
56C22×C4×C37C2311yescharacteristic
57C2×C22⋊D37—21no—
58C2×C22⋊D37—21no—
59C2×C22⋊D37—21no—
60C22×Dic37—21no—
61C22×Dic37—22no—
62C22×Dic37—22no—
63C22×Dic37—44no—
64C22×Dic37D411yescharacteristic
65C22×Dic37D411yescharacteristic
66#10596962—22no—
67#10596962—22no—
68C2×D148C2311yescharacteristic
69C2×C4×D37—21no—
70C2×Dic74C2311yescharacteristic
71SD16×C37—24no—
72D8×C37—24no—
73M16×C37—22no—
74C2×C8×C37C2312yes—
75C22⋊Dic37—22no—
76C22⋊Dic37—22no—
77C22⋊Dic37—21no—
78C22⋊Dic37—21no—
79#10596940—24no—
80#10596939—24no—
81#10596938—21no—
82#10596938—21no—
83C4⋊Dic37—21no—
84C4⋊Dic37C2312yes—
85#10596936—21no—
86#10596936—21no—
87C4×Dic37D411yescharacteristic
88C4×Dic37D411yescharacteristic
89C4×Dic37C2311yescharacteristic
90C4×Dic37C2311yescharacteristic
91#10596934—22no—
92C2×#64436C2312yes—
93C23×C37—24no—
94C23×C37—22no—
95C23×C37—21no—
96C23×C37—42no—
97C23×C37—42no—
98C23×C37C2×D411yescharacteristic
99C22×D37—21no—
100Q8×C37—21no—
101Q8×C37—21no—
102D4×C37—24no—
103D4×C37—21no—
104D4×C37—22no—
105D4×C37—21no—
106D4×C37—42no—
107D4×C37C2×D414yes—
108C2×C4×C37—21no—
109C2×C4×C37—42no—
110C2×C4×C37C2×D411yescharacteristic
111C2×C4×C37C2×D412yes—
112C2×C4×C37#4112yes—
113C2×C4×C37C2411yescharacteristic
114C22⋊D37—42no—
115C22⋊D37—42no—
116C22⋊D37—42no—
117C2×Dic37—22no—
118C2×Dic37—21no—
119C2×Dic37—22no—
120C2×Dic37—41no—
121C2×Dic37—44no—
122C2×Dic37—42no—
123C2×Dic37—44no—
124C2×Dic37C2×D411yescharacteristic
125C2×Dic37C2×D411yescharacteristic
126C2×Dic37C2×D411yescharacteristic
127C2×Dic37C2×D411yescharacteristic
128D148—21no—
129D148—21no—
130C4×D37—42no—
131Dic74—21no—
132Dic74—21no—
133C8×C37—22no—
134#64436—22no—
135C22×C37—21no—
136C22×C37—21no—
137C22×C37—24no—
138C22×C37—48no—
139C22×C37—42no—
140C22×C37—42no—
141C22×C37—41no—
142C22×C37—41no—
143C22×C37—41no—
144C22×C37#12112yes—
145C22×C37C22×D411yescharacteristic
146C2×D37—41no—
147C2×D37—41no—
148C2×D37—41no—
149C4×C37—41no—
150C4×C37#12112yes—
151C4×C37C22×D411yescharacteristic
152C4×C37C2×#4111yescharacteristic
153Dic37—21no—
154Dic37—21no—
155Dic37—21no—
156Dic37—41no—
157Dic37—42no—
158Dic37AGL(1,ℤ/8)12yescomplemented
159#3036—371nomaximal, Sylow
160C2×C37—22no—
161C2×C37—44no—
162C2×C37—81no—
163C2×C37C2×#12111yescharacteristic
164C2×C37C2×AGL(1,ℤ/8)11yescharacteristic
165C2×C37#57511yescharacteristic
166D37—81no—
167C2×AGL(1,ℤ/8)—371no—
168C2×AGL(1,ℤ/8)D3711yescharacteristic, complemented
169#532—371no—
170C2×C4×D4—371no—
171#463—372no—
172#463—372no—
173#461—372no—
174#459—372no—
175#423—371no—
176#417—371no—
177#417—371no—
178C37#303611yescharacteristic, complemented, Sylow
179C2×#41—371no—
180C2×#41C2×D3711yescharacteristic
181C22×D4C2×D3711yescharacteristic
182C23×C4—741no—
183AGL(1,ℤ/8)—24no—
184AGL(1,ℤ/8)—744no—
185C2×SD16—372no—
186C2×SD16C2×D3712yes—
187C2×D8—372no—
188C2×D8C2×D3712yes—
189C2×M16—371no—
190C2×M16C2×D3711yescharacteristic
191C4⋊Q8—371no—
192#127—371no—
193#124—371no—
194#124—371no—
195#121—741no—
196#121—741no—
197C4×D4—372no—
198C4×D4—372no—
199C4×D4—741no—
200C4×D4—741no—
201C4×D4—741no—
202C4×D4—741no—
203C2×C4⋊C4—371no—
204C2×C22⋊C4—741no—
205C2×C42—371no—
206C4⋊C8—372no—
207C4⋊C8—372no—
208#103—372no—
209#103—372no—
210#102—372no—
211#102—372no—
212C24—21no—
213#41—22no—
214#41—22no—
215#41—742no—
216#41—742no—
217C2×Q8—371no—
218C2×Q8C22×D3711yescharacteristic
219C2×D4—21no—
220C2×D4—21no—
221C2×D4—22no—
222C2×D4—371no—
223C2×D4—741no—
224C2×D4—741no—
225C2×D4—741no—
226C2×D4C22⋊D3714yescomplemented
227C2×D4C22×D3711yescharacteristic
228C2×D4C22×D3712yes—
229C22×C4—21no—
230C22×C4—371no—
231C22×C4—371no—
232C22×C4—742no—
233C22×C4—742no—
234C22×C4—741no—
235C22×C4—741no—
236C22×C4—1484no—
237C22×C4C22×D3711yescharacteristic
238SD16—24no—
239SD16—744no—
240D8—24no—
241D8—744no—
242M16—22no—
243M16—742no—
244C2×C8—372no—
245C2×C8C22×D3712yes—
246C4⋊C4—372no—
247C4⋊C4—741no—
248C4⋊C4—741no—
249C4⋊C4—741no—
250C22⋊C4—741no—
251C22⋊C4—741no—
252C22⋊C4—742no—
253C22⋊C4—742no—
254C22⋊C4—741no—
255C22⋊C4—741no—
256C42—371no—
257C42—371no—
258C42—371no—
259C42—371no—
260C23—24no—
261C23—22no—
262C23—21no—
263C23—42no—
264C23—42no—
265C23—741no—
266C23D4×D3711yescharacteristic
267Q8—21no—
268Q8—21no—
269Q8—741no—
270Q8—741no—
271D4—24no—
272D4—21no—
273D4—22no—
274D4—21no—
275D4—42no—
276D4—741no—
277D4—741no—
278D4—1482no—
279D4—1482no—
280D4—1482no—
281D4C2×C22⋊D3714yes—
282C2×C4—21no—
283C2×C4—42no—
284C2×C4—371no—
285C2×C4—371no—
286C2×C4—371no—
287C2×C4—371no—
288C2×C4—742no—
289C2×C4—742no—
290C2×C4—741no—
291C2×C4—1484no—
292C2×C4—1481no—
293C2×C4—1484no—
294C2×C4—1482no—
295C2×C4—1482no—
296C2×C4D4×D3711yescharacteristic
297C2×C4#1059696412yes—
298C2×C4C2×C22⋊D3712yes—
299C2×C4C23×D3711yescharacteristic
300C8—22no—
301C8—742no—
302C22—21no—
303C22—21no—
304C22—24no—
305C22—48no—
306C22—42no—
307C22—42no—
308C22—41no—
309C22—41no—
310C22—41no—
311C22—1481no—
312C22—1481no—
313C22—1481no—
314C22#1192450812yes—
315C22C2×D4×D3711yescharacteristic
316C4—41no—
317C4—372no—
318C4—741no—
319C4—741no—
320C4—741no—
321C4—1481no—
322C4—1482no—
323C4#1192450812yes—
324C4C22×C22⋊D3711yescharacteristic
325C4C2×#1059696411yescharacteristic
326C2—22no—
327C2—44no—
328C2—81no—
329C2—2961no—
330C2C2×#1192450811yescharacteristic
331C2AGL(1,ℤ/8)×D3711yescharacteristic
332C2#1455832911yescharacteristic
subgroupG/H
centerC22C2×D4×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC2×AGL(1,ℤ/8)×C37C2
socleC22×C37C22×D4
hypercenterC2×AGL(1,ℤ/8)D37
hypocenterC37#3036
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
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×#11924562C2semidirect
2. #14558079C2semidirect
3. C2×D4×Dic37C2semidirect
4. #14557750C2semidirect
5. #14557746C2semidirect
6. #14557736C2semidirect
7. #14557726C2semidirect
8. #14557556C2semidirect
9. #14557535C2semidirect
10. #14557534C2semidirect
11. C2×AGL(1,ℤ/8)×C37C2semidirect
18. #11924534C22semidirect
19. #11924532C22semidirect
20. #11924525C22semidirect
21. #11924523C22semidirect
33. #11924465C22semidirect
34. #11924465C22semidirect
35. #11924461C22semidirect
36. #11924460C22semidirect
37. #11924458C22semidirect
38. #11924457C22semidirect
52. C2×D4×C37D4semidirect
158. Dic37AGL(1,ℤ/8)semidirect
168. C2×AGL(1,ℤ/8)D37semidirect
178. C37#3036semidirect
226. C2×D4C22⋊D37semidirect

Maximal subgroups

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

subgroupindex
C2×#119245622
#145580792
C2×D4×Dic372
#145577502
#145577462
#145577362
#145577262
#145575562
#145575352
#145575342
C2×AGL(1,ℤ/8)×C372
#303637