Finite Group Databasebeta

← #49170552

Subgroups — #49170552

Order 37504.

464 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34036941C211yesmaximal, characteristic, complemented
2#34036936C211yesmaximal, characteristic, complemented
3#34036934C211yesmaximal, characteristic, complemented
4#34036931C211yesmaximal, characteristic, complemented
5#34036518C211yesmaximal, characteristic, complemented
6#34036400C211yesmaximal, characteristic, complemented
7#34036399C211yesmaximal, characteristic, complemented
8#34036393C211yesmaximal, characteristic, complemented
9#34036392C211yesmaximal, characteristic, complemented
10#34036308C211yesmaximal, characteristic, complemented
11#34036303C211yesmaximal, characteristic, complemented
12#34036298C211yesmaximal, characteristic, complemented
13#34036298C211yesmaximal, characteristic, complemented
14#34036252C211yesmaximal, characteristic, complemented
15#444×C293C211yesmaximal, characteristic, complemented
16#22627671C2211yescharacteristic, complemented
17#22627670C2211yescharacteristic
18#22627670C2211yescharacteristic, complemented
19#22627670C2211yescharacteristic, complemented
20#22627657—21no—
21#22627657C2211yescharacteristic, complemented
22#22627656—21no—
23#22627654C2211yescharacteristic, complemented
24#22627652—21no—
25#22627652—21no—
26#22627652—21no—
27#22627652C2211yescharacteristic
28#22627652C2211yescharacteristic, complemented
29#22627652C2211yescharacteristic, complemented
30#22627651—21no—
31#22627651—21no—
32#22627649C2211yescharacteristic
33#22627649C2211yescharacteristic, complemented
34#22627649C2211yescharacteristic, complemented
35#22627648—21no—
36#22627647—21no—
37#22627646—21no—
38#22627646—21no—
39#22627646C2211yescharacteristic
40#22627645—21no—
41C2×#17989791—21no—
42C2×#17989791C2211yescharacteristic, complemented
43C2×#17989791C2211yescharacteristic, complemented
44SD16×D293—21no—
45C2×D2344—21no—
46C2×D2344C2211yescharacteristic, complemented
47C2×D2344C2211yescharacteristic, complemented
48D8×D293—21no—
49#22627630C2211yescharacteristic, complemented
50#22627630C2211yescharacteristic, complemented
51#22627621C2211yescharacteristic, complemented
52#22627612C2211yescharacteristic, complemented
53#22627560C2211yescharacteristic, complemented
54#22627552C2211yescharacteristic, complemented
55#22627552C2211yescharacteristic, complemented
56#22627542C2211yescharacteristic, complemented
57#22627540C2211yescharacteristic, complemented
58#22627540C2211yescharacteristic, complemented
59#22627540C2211yescharacteristic, complemented
60#135×C293C2211yescharacteristic
61#131×C293C2211yescharacteristic, complemented
62#131×C293C2211yescharacteristic, complemented
63#108×C293C2211yescharacteristic, complemented
64#104×C293C2211yescharacteristic, complemented
65#104×C293C2211yescharacteristic, complemented
66#97×C293C2211yescharacteristic, complemented
67#17989803—21no—
68#17989803—21no—
69#17989803—21no—
70#17989802—21no—
71#17989802—21no—
72#17989802—21no—
73#17989802—21no—
74#17989802C2311yescharacteristic
75#17989802C2311yescharacteristic
76#17989801—21no—
77#17989801—41no—
78C2×Dic586—21no—
79C2×Dic586C2311yescharacteristic
80Q8×D293—21no—
81C2×D1172—21no—
82C2×D1172—21no—
83C2×D1172—21no—
84C2×D1172—41no—
85C2×D1172C2311yescharacteristic
86C2×D1172C2311yescharacteristic
87C2×D1172C2311yescharacteristic
88D4×D293—21no—
89D4×D293—21no—
90D4×D293—21no—
91D4×D293—41no—
92D4×D293—41no—
93D4×D293—41no—
94Dic1172—21no—
95Dic1172D411yescharacteristic, complemented
96#17989792—21no—
97#17989791—21no—
98#17989791—21no—
99#17989791—21no—
100#17989791—21no—
101#17989791—21no—
102#17989791—21no—
103#17989791—21no—
104#17989791—21no—
105#17989791D411yescharacteristic, complemented
106#17989791D411yescharacteristic, complemented
107#17989790—21no—
108#17989789—21no—
109#17989789—21no—
110#17989789—21no—
111D2344—21no—
112D2344—21no—
113D2344—21no—
114D2344—21no—
115D2344—21no—
116D2344—21no—
117D2344—21no—
118D2344D411yescharacteristic, complemented
119#17989787—21no—
120#17989787—21no—
121#17989787—21no—
122#17989786C2311yescharacteristic
123#17989786C2311yescharacteristic
124#17989785—21no—
125#17989785—21no—
126C8×D293—21no—
127C8×D293—21no—
128#17989779—41no—
129#41×C293C2311yescharacteristic
130#41×C293C2311yescharacteristic
131Q16×C293D411yescharacteristic, complemented
132SD16×C293D411yescharacteristic, complemented
133SD16×C293D411yescharacteristic, complemented
134D8×C293D411yescharacteristic, complemented
135M16×C293—21no—
136M16×C293—21no—
137M16×C293C2311yescharacteristic
138M16×C293C2311yescharacteristic
139C2×C8×C293—21no—
140C2×C8×C293—21no—
141C2×C8×C293C2311yescharacteristic
142C2×C8×C293C2311yescharacteristic
143C42×C293C2311yescharacteristic
144C22×D293—21no—
145C22×D293—21no—
146C22×D293—21no—
147C22×D293—41no—
148C22×D293—41no—
149C22×D293—41no—
150Dic586—21no—
151Dic586—21no—
152Dic586—41no—
153Dic586C2×D411yescharacteristic
154Dic586C2×D411yescharacteristic
155D1172—21no—
156D1172—21no—
157D1172—21no—
158D1172—21no—
159D1172—21no—
160D1172—21no—
161D1172—41no—
162D1172—41no—
163D1172—41no—
164D1172—82no—
165D1172C2×D411yescharacteristic
166D1172C2×D411yescharacteristic
167C22⋊D293—21no—
168C22⋊D293—21no—
169C22⋊D293—41no—
170C22⋊D293—41no—
171C22⋊D293—41no—
172C22⋊D293—41no—
173C2×Dic293—21no—
174C2×Dic293—41no—
175C4×D293—21no—
176C4×D293—21no—
177C4×D293—41no—
178C4×D293—41no—
179C4×D293—41no—
180C4×D293—41no—
181#14556873—21no—
182#14556873—21no—
183Q8×C293C2×D411yescharacteristic
184Q8×C293C2×D411yescharacteristic
185D4×C293—21no—
186D4×C293—21no—
187D4×C293C2×D411yescharacteristic
188D4×C293C2×D411yescharacteristic
189C2×C4×C293—21no—
190C2×C4×C293—21no—
191C2×C4×C293—21no—
192C2×C4×C293C2411yescharacteristic
193C8×C293—21no—
194C8×C293—21no—
195C8×C293C2×D411yescharacteristic
196C8×C293C2×D411yescharacteristic
197C8×C293C2×D411yescharacteristic
198C8×C293C2×D411yescharacteristic
199C22×C293—21no—
200C22×C293—21no—
201C22×C293H(5,𝔽2)11yescharacteristic
202C2×D293—21no—
203C2×D293—21no—
204C2×D293—41no—
205C2×D293—41no—
206C2×D293—41no—
207C2×D293—81no—
208C2×D293—81no—
209C2×D293—81no—
210C2×D293—161no—
211C2×D293—161no—
212C2×D293—161no—
213C4×C293—21no—
214C4×C293—21no—
215C4×C293—41no—
216C4×C293C22×D411yescharacteristic
217C4×C293C22×D411yescharacteristic
218Dic293—21no—
219Dic293—21no—
220Dic293—41no—
221C2×C293—21no—
222C2×C293—41no—
223C2×C293—41no—
224C2×C293D4211yescharacteristic
225D293—41no—
226D293—41no—
227D293—81no—
228D293—81no—
229D293—81no—
230C293#328811yescharacteristic, complemented, Sylow
231#3288—2931nomaximal, Sylow
232#577—2931no—
233#577—2931no—
234#576—2931no—
235#576—2931no—
236#496—2931no—
237#472—2931no—
238#472—2931no—
239#471—2931no—
240#471—2931no—
241#455—2931no—
242#454—2931no—
243#453—2931no—
244#453—2931no—
245#444—2931no—
246#444D29311yescharacteristic, complemented
2472−1+4—2931no—
248H(5,𝔽2)—2931no—
249H(5,𝔽2)—2931no—
250H(5,𝔽2)—2931no—
251#137—2931no—
252#137—2931no—
253#137—5861no—
254#137—5861no—
255AGL(1,ℤ/8)—2931no—
256AGL(1,ℤ/8)—2931no—
257AGL(1,ℤ/8)—2931no—
258AGL(1,ℤ/8)—2931no—
259AGL(1,ℤ/8)—2931no—
260AGL(1,ℤ/8)—2931no—
261AGL(1,ℤ/8)—5861no—
262AGL(1,ℤ/8)—5861no—
263AGL(1,ℤ/8)—5861no—
264AGL(1,ℤ/8)—5861no—
265AGL(1,ℤ/8)—5861no—
266AGL(1,ℤ/8)—5861no—
267#135—2931no—
268#135—5861no—
269#135—5861no—
270#135—5861no—
271#135—5861no—
272#135C2×D29311yescharacteristic
273C2×SD16—2931no—
274C2×SD16—2931no—
275C2×SD16—5861no—
276C2×SD16—5861no—
277C2×D8—2931no—
278C2×D8—2931no—
279C2×D8—5861no—
280C2×D8—5861no—
281#131—2931no—
282#131—2931no—
283#131C2×D29311yescharacteristic, complemented
284#131C2×D29311yescharacteristic, complemented
285#127—2931no—
286#124—2931no—
287#108—2931no—
288#108C2×D29311yescharacteristic, complemented
289#104—2931no—
290#104—2931no—
291#104C2×D29311yescharacteristic, complemented
292#104C2×D29311yescharacteristic, complemented
293#101—2931no—
294#100—2931no—
295#100—2931no—
296#100—2931no—
297#97C2×D29311yescharacteristic, complemented
298#41—2931no—
299#41—2931no—
300#41—5861no—
301#41—5861no—
302#41—5861no—
303#41—5861no—
304#41—5861no—
305#41—5861no—
306#41—5861no—
307#41—5861no—
308#41—11721no—
309#41C22×D29311yescharacteristic
310#41C22×D29311yescharacteristic
311C2×Q8—2931no—
312C2×Q8—5861no—
313C2×Q8—5861no—
314C2×D4—2931no—
315C2×D4—2931no—
316C2×D4—2931no—
317C2×D4—5861no—
318C2×D4—5861no—
319C2×D4—5861no—
320C2×D4—5861no—
321C2×D4—5861no—
322C2×D4—5861no—
323C2×D4—11721no—
324C2×D4—11721no—
325C2×D4—11721no—
326C2×D4—11721no—
327Q16—2931no—
328Q16—5861no—
329Q16—5861no—
330Q16D117211yescharacteristic, complemented
331SD16—2931no—
332SD16—2931no—
333SD16—5861no—
334SD16—5861no—
335SD16—5861no—
336SD16—5861no—
337SD16—5861no—
338SD16—5861no—
339SD16—5861no—
340SD16—5861no—
341SD16—5861no—
342SD16—5861no—
343SD16—5861no—
344SD16—5861no—
345SD16D117211yescharacteristic, complemented
346SD16D117211yescharacteristic, complemented
347D8—2931no—
348D8—5861no—
349D8—5861no—
350D8—5861no—
351D8—5861no—
352D8—5861no—
353D8—5861no—
354D8—5861no—
355D8—5861no—
356D8—5861no—
357D8—5861no—
358D8D117211yescharacteristic, complemented
359M16—21no—
360M16—21no—
361M16—2931no—
362M16—2931no—
363M16—5861no—
364M16—5861no—
365M16C22×D29311yescharacteristic
366M16C22×D29311yescharacteristic
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—5861no—
370C2×C8—5861no—
371C2×C8C22×D29311yescharacteristic
372C2×C8C22×D29311yescharacteristic
373C22⋊C4—11721no—
374C42C22×D29311yescharacteristic
375C23—5861no—
376C23—5861no—
377C23—5861no—
378C23—11721no—
379C23—11721no—
380C23—11721no—
381Q8—2931no—
382Q8—2931no—
383Q8—5861no—
384Q8—5861no—
385Q8—11721no—
386Q8C2×D117211yescharacteristic
387Q8C2×D117211yescharacteristic
388D4—21no—
389D4—21no—
390D4—2931no—
391D4—2931no—
392D4—23442no—
393D4—5861no—
394D4—5861no—
395D4—5861no—
396D4—5861no—
397D4—5861no—
398D4—5861no—
399D4—5861no—
400D4—5861no—
401D4—11721no—
402D4—11721no—
403D4—11721no—
404D4—11721no—
405D4—11721no—
406D4—11721no—
407D4—11721no—
408D4C2×D117211yescharacteristic
409D4C2×D117211yescharacteristic
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—5861no—
414C2×C4—5861no—
415C2×C4—5861no—
416C2×C4—11721no—
417C2×C4—11721no—
418C2×C4—11721no—
419C2×C4—11721no—
420C2×C4—11721no—
421C2×C4C23×D29311yescharacteristic
422C8—21no—
423C8—21no—
424C8—5861no—
425C8—5861no—
426C8D4×D29311yescharacteristic
427C8D4×D29311yescharacteristic
428C8C2×D117211yescharacteristic
429C8C2×D117211yescharacteristic
430C22—21no—
431C22—21no—
432C22—23441no—
433C22—23441no—
434C22—23441no—
435C22—5861no—
436C22—5861no—
437C22—46881no—
438C22—46881no—
439C22—46881no—
440C22—11721no—
441C22—11721no—
442C22—11721no—
443C22#2262767011yescharacteristic
444C4—21no—
445C4—21no—
446C4—41no—
447C4—5861no—
448C4—5861no—
449C4—11721no—
450C4C2×D4×D29311yescharacteristic
451C4C22×D117211yescharacteristic
452C2—21no—
453C2—41no—
454C2—41no—
455C2—23441no—
456C2—23441no—
457C2—23441no—
458C2—11721no—
459C2—11721no—
460C2D4×D117211yescharacteristic
subgroupG/H
centerC2D4×D1172
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#444×C293C2
socleC2×C293D42
hypercenter#444D293
hypocenterC293#3288
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C2 ◁ C2×C4 ◁ #444

Complements

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

NH ≅ G/Ntype
1. #34036941C2semidirect
2. #34036936C2semidirect
3. #34036934C2semidirect
4. #34036931C2semidirect
5. #34036518C2semidirect
6. #34036400C2semidirect
7. #34036399C2semidirect
8. #34036393C2semidirect
9. #34036392C2semidirect
10. #34036308C2semidirect
11. #34036303C2semidirect
12. #34036298C2semidirect
13. #34036298C2semidirect
14. #34036252C2semidirect
15. #444×C293C2semidirect
16. #22627671C22semidirect
18. #22627670C22semidirect
19. #22627670C22semidirect
21. #22627657C22semidirect
23. #22627654C22semidirect
28. #22627652C22semidirect
29. #22627652C22semidirect
33. #22627649C22semidirect
34. #22627649C22semidirect
42. C2×#17989791C22semidirect
43. C2×#17989791C22semidirect
46. C2×D2344C22semidirect
47. C2×D2344C22semidirect
49. #22627630C22semidirect
50. #22627630C22semidirect
51. #22627621C22semidirect
52. #22627612C22semidirect
53. #22627560C22semidirect
54. #22627552C22semidirect
55. #22627552C22semidirect
56. #22627542C22semidirect
57. #22627540C22semidirect
58. #22627540C22semidirect
59. #22627540C22semidirect
61. #131×C293C22semidirect
62. #131×C293C22semidirect
63. #108×C293C22semidirect
64. #104×C293C22semidirect
65. #104×C293C22semidirect
66. #97×C293C22semidirect
95. Dic1172D4semidirect
105. #17989791D4semidirect
106. #17989791D4semidirect
118. D2344D4semidirect
131. Q16×C293D4semidirect
132. SD16×C293D4semidirect
133. SD16×C293D4semidirect
134. D8×C293D4semidirect
230. C293#3288semidirect
246. #444D293semidirect
283. #131C2×D293semidirect
284. #131C2×D293semidirect
288. #108C2×D293semidirect
291. #104C2×D293semidirect
292. #104C2×D293semidirect
297. #97C2×D293semidirect
330. Q16D1172semidirect
345. SD16D1172semidirect
346. SD16D1172semidirect
358. D8D1172semidirect

Maximal subgroups

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

subgroupindex
#340369412
#340369362
#340369342
#340369312
#340365182
#340364002
#340363992
#340363932
#340363922
#340363082
#340363032
#340362982
#340362522
#444×C2932
#3288293