Finite Group Databasebeta

← #18003966

Subgroups — #18003966

Order 4736.

736 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 133 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
1#14558370C211yesmaximal, characteristic, complemented
2AGL(1,ℤ/8)×D37C212yesmaximal, complemented
3#14558101C211yesmaximal, characteristic, complemented
4#14557708C211yesmaximal, characteristic, complemented
5#14557706C211yesmaximal, characteristic, complemented
6#14557705C211yesmaximal, characteristic, complemented
7#14557702C212yesmaximal, complemented
8#14557699C212yesmaximal, complemented
9#14557548C212yesmaximal, complemented
10#14557516C211yesmaximal, characteristic, complemented
11C37×#454C211yesmaximal, characteristic, complemented
12#11924584—22no—
13#11924584C2211yescharacteristic, complemented
14#11924582—22no—
15#11924582—22no—
16#11924582C2212yescomplemented
17C2×#10596966—22no—
18C2×#10596966C2211yescharacteristic
19#41×D37—22no—
20#41×D37—22no—
21#41×D37C2212yes—
22C2×D4×D37C2211yescharacteristic, complemented
23#11924564C2212yescomplemented
24#11924563—22no—
25#11924562C2212yescomplemented
26#11924561C2212yescomplemented
27#11924560—22no—
28SD16×D37—22no—
29D8×D37—22no—
30M16×D37C2212yescomplemented
31#11924533C2211yescharacteristic, complemented
32#11924532C2211yescharacteristic, complemented
33#11924525C2211yescharacteristic, complemented
34#11924509—22no—
35#11924506—22no—
36#11924493C2211yescharacteristic, complemented
37#11924464C2212yescomplemented
38#11924463C2212yescomplemented
39#11924462C2212yescomplemented
40#11924455C2211yescharacteristic, complemented
41#11924452C2212yescomplemented
422−1+4×C37C2211yescharacteristic, complemented
43AGL(1,ℤ/8)×C37C2212yescomplemented
44#124×C37C2211yescharacteristic, complemented
45#104×C37C2212yescomplemented
46#101×C37C2211yescharacteristic, complemented
47C23×D37—21no—
48#41×C37—22no—
49#41×C37—22no—
50#41×C37C2312yes—
51C2×Q8×C37—22no—
52C2×Q8×C37C2311yescharacteristic
53C2×D4×C37C2311yescharacteristic
54C2×C22⋊D37—22no—
55C2×C22⋊D37—21no—
56#10596966—22no—
57#10596966—24no—
58#10596966—22no—
59#10596966—22no—
60#10596966—22no—
61#10596966—22no—
62#10596966D412yescomplemented
63Q8×D37—22no—
64Q8×D37—22no—
65Q8×D37D412yescomplemented
66#10596964—22no—
67#10596964—22no—
68#10596964D412yescomplemented
69D4×D37—24no—
70D4×D37—22no—
71D4×D37—24no—
72D4×D37—22no—
73D4×D37—22no—
74D4×D37—22no—
75D4×D37—22no—
76D4×D37D412yescomplemented
77#10596962—24no—
78#10596962—24no—
79#10596962—22no—
80#10596962—22no—
81#10596962—22no—
82#10596962C2312yes—
83C2×D148—24no—
84C2×D148—22no—
85C2×D148—22no—
86C2×D148D412yes—
87C2×D148C2311yescharacteristic
88C2×D148C2311yescharacteristic
89C2×C4×D37—24no—
90C2×C4×D37—22no—
91C2×C4×D37D412yes—
92C2×C4×D37C2311yescharacteristic
93SD16×C37—22no—
94D8×C37—22no—
95M16×C37C2312yes—
96C22⋊C4×C37—22no—
97C42×C37C2311yescharacteristic
98#10596941—22no—
99#10596940—22no—
100#10596939—22no—
101#10596939—22no—
102#10596938—21no—
103#10596938—22no—
104#10596936—21no—
105C4×Dic37C2311yescharacteristic
106#10596934C2312yes—
107D296—22no—
108#10596930—22no—
109#10596929—22no—
110C8×D37—22no—
111C23×C37—21no—
112C22×D37—24no—
113C22×D37—22no—
114C22×D37—21no—
115C22×D37—42no—
116C22×D37—82no—
117C22×D37C2×D412yes—
118C22×D37C2×D411yescharacteristic
119Q8×C37—22no—
120Q8×C37—22no—
121Q8×C37C2×D412yes—
122D4×C37—22no—
123D4×C37—22no—
124D4×C37—22no—
125D4×C37C2×D412yes—
126C2×C4×C37—24no—
127C2×C4×C37—22no—
128C2×C4×C37—21no—
129C2×C4×C37C2×D412yes—
130C2×C4×C37C2411yescharacteristic
131C22⋊D37—24no—
132C22⋊D37—22no—
133C22⋊D37—22no—
134C22⋊D37—22no—
135C22⋊D37—41no—
136C22⋊D37—41no—
137C22⋊D37—82no—
138C2×Dic37—22no—
139C2×Dic37—21no—
140C2×Dic37C2×D411yescharacteristic
141D148—24no—
142D148—22no—
143D148—22no—
144D148—24no—
145D148—22no—
146D148—22no—
147D148—82no—
148D148C2×D412yes—
149C4×D37—24no—
150C4×D37—24no—
151C4×D37—22no—
152C4×D37—22no—
153C4×D37—22no—
154C4×D37—22no—
155C4×D37—22no—
156C4×D37C2×D412yes—
157C4×D37C2×D412yes—
158Dic74—22no—
159Dic74—22no—
160Dic74C2×D412yes—
161C8×C37—22no—
162#64436—22no—
163C22×C37—22no—
164C22×C37—42no—
165C22×C37—41no—
166C22×C37C22×D411yescharacteristic
167C2×D37—22no—
168C2×D37—22no—
169C2×D37—41no—
170C2×D37—42no—
171C2×D37—42no—
172C2×D37—41no—
173C2×D37—82no—
174C2×D37—84no—
175C2×D37—84no—
176C2×D37#12012yes—
177C4×C37—22no—
178C4×C37—22no—
179C4×C37—41no—
180C4×C37C22×D412yes—
181Dic37—22no—
182Dic37—41no—
183Dic37#12011yescharacteristic
184Dic37#12011yescharacteristic
185#3021—371nomaximal, Sylow
186C2×C37—21no—
187C2×C37—42no—
188C2×C37—81no—
189C2×C37C2×#12011yescharacteristic
190D37—22no—
191D37—42no—
192D37—42no—
193D37—81no—
194#585—371no—
195C2×AGL(1,ℤ/8)—372no—
196#535—371no—
197#454—371no—
198#454—371no—
199#454—371no—
200#454D3711yescharacteristic, complemented
201#453—372no—
202#453—372no—
203#421—372no—
204#413—371no—
205C37#302111yescharacteristic, complemented, Sylow
2062−1+4—371no—
2072−1+4—742no—
2082−1+4C2×D3711yescharacteristic, complemented
209H(5,𝔽2)—372no—
210H(5,𝔽2)—742no—
211H(5,𝔽2)—742no—
212C2×#41—371no—
213C2×#41—372no—
214C2×#41—742no—
215C2×#41—742no—
216C2×#41—742no—
217C22×D4—371no—
218AGL(1,ℤ/8)—372no—
219AGL(1,ℤ/8)—372no—
220AGL(1,ℤ/8)—372no—
221AGL(1,ℤ/8)—742no—
222AGL(1,ℤ/8)—742no—
223AGL(1,ℤ/8)C2×D3712yescomplemented
224C2×SD16—742no—
225C2×D8—742no—
226C2×M16—372no—
227#127—371no—
228#127—371no—
229#124—371no—
230#124C2×D3711yescharacteristic, complemented
231#121—742no—
232#120—742no—
233#117—371no—
234#104—372no—
235#104—372no—
236#104—372no—
237#104C2×D3712yescomplemented
238#101—371no—
239#101C2×D3711yescharacteristic, complemented
240#100—372no—
241C24—741no—
242#41—22no—
243#41—22no—
244#41—372no—
245#41—372no—
246#41—372no—
247#41—742no—
248#41—742no—
249#41—744no—
250#41—744no—
251#41—742no—
252#41—742no—
253#41—742no—
254#41—742no—
255#41—742no—
256#41—742no—
257#41—744no—
258#41—742no—
259#41—742no—
260#41C22×D3712yes—
261C2×Q8—22no—
262C2×Q8—372no—
263C2×Q8—742no—
264C2×Q8—742no—
265C2×Q8C22×D3711yescharacteristic
266C2×D4—372no—
267C2×D4—372no—
268C2×D4—371no—
269C2×D4—371no—
270C2×D4—742no—
271C2×D4—742no—
272C2×D4—744no—
273C2×D4—742no—
274C2×D4—742no—
275C2×D4—741no—
276C2×D4—742no—
277C2×D4—744no—
278C2×D4—742no—
279C2×D4—744no—
280C2×D4—742no—
281C2×D4—742no—
282C2×D4C22×D3711yescharacteristic
283C22×C4—371no—
284C22×C4—372no—
285C22×C4—742no—
286C22×C4—744no—
287SD16—22no—
288SD16—742no—
289SD16—742no—
290SD16—742no—
291D8—22no—
292D8—742no—
293D8—742no—
294D8—742no—
295M16—372no—
296M16—742no—
297M16C22×D3712yes—
298C2×C8—742no—
299C4⋊C4—741no—
300C22⋊C4—22no—
301C22⋊C4—741no—
302C22⋊C4—742no—
303C42—371no—
304C42C22×D3711yescharacteristic
305C23—21no—
306C23—371no—
307C23—372no—
308C23—2962no—
309C23—742no—
310C23—744no—
311C23—741no—
312C23—1482no—
313Q8—22no—
314Q8—22no—
315Q8—372no—
316Q8—742no—
317Q8—742no—
318Q8D4×D3712yes—
319D4—22no—
320D4—22no—
321D4—22no—
322D4—372no—
323D4—2962no—
324D4—2962no—
325D4—744no—
326D4—742no—
327D4—744no—
328D4—742no—
329D4—742no—
330D4—742no—
331D4—744no—
332D4—742no—
333D4—742no—
334D4—742no—
335D4—1481no—
336D4—1481no—
337D4D4×D3712yes—
338C2×C4—24no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—371no—
342C2×C4—372no—
343C2×C4—372no—
344C2×C4—742no—
345C2×C4—742no—
346C2×C4—742no—
347C2×C4—742no—
348C2×C4—744no—
349C2×C4—744no—
350C2×C4—742no—
351C2×C4—742no—
352C2×C4—741no—
353C2×C4D4×D3712yes—
354C2×C4C23×D3711yescharacteristic
355C8—22no—
356C8—742no—
357C22—22no—
358C22—42no—
359C22—41no—
360C22—372no—
361C22—2962no—
362C22—2964no—
363C22—2964no—
364C22—742no—
365C22—742no—
366C22—1482no—
367C22—1481no—
368C22—1482no—
369C22—1481no—
370C22C2×D4×D3711yescharacteristic
371C4—22no—
372C4—22no—
373C4—41no—
374C4—371no—
375C4—371no—
376C4—742no—
377C4—1481no—
378C4C2×D4×D3712yes—
379C2—21no—
380C2—42no—
381C2—81no—
382C2—2961no—
383C2—742no—
384C2—1482no—
385C2—1482no—
386C2#120×D3711yescharacteristic
subgroupG/H
centerC2#120×D37
commutator, cosocleC2×C4×C37C24
FrattiniC2×C4C23×D37
FittingC37×#454C2
socleC2×C37C2×#120
hypercenter#454D37
hypocenterC37#3021
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C2 ◁ C2×C4 ◁ #454

Complements

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

NH ≅ G/Ntype
1. #14558370C2semidirect
2. AGL(1,ℤ/8)×D37C2semidirect
3. #14558101C2semidirect
4. #14557708C2semidirect
5. #14557706C2semidirect
6. #14557705C2semidirect
7. #14557702C2semidirect
8. #14557699C2semidirect
9. #14557548C2semidirect
10. #14557516C2semidirect
11. C37×#454C2semidirect
13. #11924584C22semidirect
16. #11924582C22semidirect
22. C2×D4×D37C22semidirect
23. #11924564C22semidirect
25. #11924562C22semidirect
26. #11924561C22semidirect
30. M16×D37C22semidirect
31. #11924533C22semidirect
32. #11924532C22semidirect
33. #11924525C22semidirect
36. #11924493C22semidirect
37. #11924464C22semidirect
38. #11924463C22semidirect
39. #11924462C22semidirect
40. #11924455C22semidirect
41. #11924452C22semidirect
42. 2−1+4×C37C22semidirect
43. AGL(1,ℤ/8)×C37C22semidirect
44. #124×C37C22semidirect
45. #104×C37C22semidirect
46. #101×C37C22semidirect
62. #10596966D4semidirect
65. Q8×D37D4semidirect
68. #10596964D4semidirect
76. D4×D37D4semidirect
200. #454D37semidirect
205. C37#3021semidirect
208. 2−1+4C2×D37semidirect
223. AGL(1,ℤ/8)C2×D37semidirect
230. #124C2×D37semidirect
237. #104C2×D37semidirect
239. #101C2×D37semidirect

Maximal subgroups

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

subgroupindex
#145583702
AGL(1,ℤ/8)×D372
#145581012
#145577082
#145577062
#145577052
#145577022
#145576992
#145575482
#145575162
C37×#4542
#302137