Finite Group Databasebeta

← #49170577

Subgroups — #49170577

Order 37504.

460 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34036941C211yesmaximal, characteristic, complemented
2#34036939C211yesmaximal, characteristic, complemented
3#34036938C211yesmaximal, characteristic, complemented
4#34036935C211yesmaximal, characteristic, complemented
5#34036500C211yesmaximal, characteristic, complemented
6#34036395C211yesmaximal, characteristic, complemented
7#34036393C211yesmaximal, characteristic, complemented
8#34036391C211yesmaximal, characteristic, complemented
9#34036314C211yesmaximal, characteristic, complemented
10#34036310C211yesmaximal, characteristic, complemented
11#34036299C211yesmaximal, characteristic, complemented
12#34036295C211yesmaximal, characteristic, complemented
13#34036245C211yesmaximal, characteristic, complemented
14#34036244C211yesmaximal, characteristic, complemented
15#471×C293C211yesmaximal, characteristic, complemented
16#22627672C2211yescharacteristic
17#22627671C2211yescharacteristic
18#22627670C2211yescharacteristic, complemented
19#22627669C2211yescharacteristic, complemented
20#22627657—21no—
21#22627657C2211yescharacteristic, complemented
22#22627657C2211yescharacteristic, complemented
23#22627656—21no—
24#22627655C2211yescharacteristic, complemented
25#22627653—21no—
26#22627653—21no—
27#22627652—21no—
28#22627652C2211yescharacteristic
29#22627652C2211yescharacteristic
30#22627651—21no—
31#22627651—21no—
32#22627650C2211yescharacteristic
33#22627648—21no—
34#22627647—21no—
35#22627646—21no—
36#22627646C2211yescharacteristic
37#22627645—21no—
38#22627644C2211yescharacteristic
39#22627643—21no—
40C2×#17989791—21no—
41C2×#17989791C2211yescharacteristic, complemented
42C2×#17989790C2211yescharacteristic, complemented
43C2×#17989789C2211yescharacteristic, complemented
44SD16×D293—21no—
45SD16×D293—21no—
46SD16×D293—21no—
47#22627632C2211yescharacteristic, complemented
48#22627632C2211yescharacteristic, complemented
49#22627630C2211yescharacteristic, complemented
50#22627622C2211yescharacteristic, complemented
51#22627620C2211yescharacteristic
52#22627559C2211yescharacteristic, complemented
53#22627551C2211yescharacteristic, complemented
54#22627550C2211yescharacteristic, complemented
55#22627550C2211yescharacteristic, complemented
56#22627550C2211yescharacteristic, complemented
57#22627542C2211yescharacteristic, complemented
58#22627540C2211yescharacteristic, complemented
59C8×Dic293C2211yescharacteristic, complemented
60#137×C293C2211yescharacteristic, complemented
61AGL(1,ℤ/8)×C293C2211yescharacteristic
62C2×SD16×C293C2211yescharacteristic, complemented
63#131×C293C2211yescharacteristic, complemented
64#108×C293C2211yescharacteristic, complemented
65#101×C293C2211yescharacteristic, complemented
66#100×C293C2211yescharacteristic, complemented
67#17989803—21no—
68#17989803—21no—
69#17989803—21no—
70#17989802—21no—
71#17989802—21no—
72#17989802—41no—
73#17989802C2311yescharacteristic
74#17989802C2311yescharacteristic
75#17989802C2311yescharacteristic
76#17989801—21no—
77#17989801—21no—
78#17989801—21no—
79#17989801—41no—
80C2×Dic586—21no—
81C2×Dic586C2311yescharacteristic
82Q8×D293—21no—
83Q8×D293—21no—
84Q8×D293—21no—
85C2×D1172—21no—
86C2×D1172C2311yescharacteristic
87C2×C22⋊D293—41no—
88C2×C22⋊D293—41no—
89D4×D293—21no—
90D4×D293—21no—
91D4×D293—21no—
92D4×D293—41no—
93Dic1172—21no—
94Dic1172—21no—
95Dic1172D411yescharacteristic, complemented
96#17989792—21no—
97#17989792D411yescharacteristic, complemented
98#17989791—21no—
99#17989791—21no—
100#17989791—21no—
101#17989791—21no—
102#17989791—21no—
103#17989791—21no—
104#17989791D411yescharacteristic, complemented
105#17989791D411yescharacteristic, complemented
106#17989790—21no—
107#17989790—21no—
108#17989790—21no—
109#17989790D411yescharacteristic, complemented
110#17989789—21no—
111#17989789—21no—
112#17989789—21no—
113#17989789D411yescharacteristic, complemented
114D2344—21no—
115D2344—21no—
116D2344D411yescharacteristic, complemented
117#17989787—21no—
118#17989787D411yescharacteristic, complemented
119#17989786C2311yescharacteristic
120#17989785—21no—
121#17989785—21no—
122#17989785—21no—
123C2×#14556873C2311yescharacteristic
124C8×D293—21no—
125C8×D293—21no—
126C8×D293—21no—
127#17989781—41no—
128C4×Dic293C2311yescharacteristic
129#41×C293C2311yescharacteristic
130C2×Q8×C293C2311yescharacteristic
131C2×D4×C293C2311yescharacteristic
132Q16×C293—21no—
133SD16×C293—21no—
134SD16×C293—21no—
135SD16×C293—21no—
136SD16×C293—21no—
137D8×C293—21no—
138M16×C293—21no—
139M16×C293C2311yescharacteristic
140M16×C293C2311yescharacteristic
141M16×C293C2311yescharacteristic
142C2×C8×C293—21no—
143C2×C8×C293C2311yescharacteristic
144C22×D293—21no—
145C22×D293—41no—
146C22×D293—41no—
147Dic586—21no—
148Dic586—21no—
149Dic586—41no—
150Dic586—41no—
151Dic586C2×D411yescharacteristic
152Dic586C2×D411yescharacteristic
153Dic586C2×D411yescharacteristic
154D1172—21no—
155D1172—21no—
156D1172—41no—
157D1172—41no—
158D1172C2×D411yescharacteristic
159D1172C2×D411yescharacteristic
160D1172C2×D411yescharacteristic
161C22⋊D293—21no—
162C22⋊D293—21no—
163C22⋊D293—21no—
164C22⋊D293—41no—
165C22⋊D293—41no—
166C22⋊D293—41no—
167C22⋊D293—41no—
168C22⋊D293—82no—
169C2×Dic293—21no—
170C2×Dic293—21no—
171C2×Dic293—41no—
172C2×Dic293—41no—
173C4×D293—21no—
174C4×D293—21no—
175C4×D293—21no—
176C4×D293—41no—
177C4×D293—41no—
178C4×D293—41no—
179C4×D293—41no—
180#14556873—21no—
181#14556873C2×D411yescharacteristic
182#14556873C2×D411yescharacteristic
183C23×C293—21no—
184Q8×C293—21no—
185Q8×C293—21no—
186Q8×C293C2×D411yescharacteristic
187D4×C293—21no—
188D4×C293—21no—
189D4×C293—21no—
190D4×C293C2×D411yescharacteristic
191C2×C4×C293—21no—
192C2×C4×C293—21no—
193C2×C4×C293C2411yescharacteristic
194C8×C293—21no—
195C8×C293—21no—
196C8×C293—21no—
197C8×C293C2×D411yescharacteristic
198C8×C293C2×D411yescharacteristic
199C22×C293—21no—
200C22×C293—41no—
201C22×C293—81no—
202C22×C293H(5,𝔽2)11yescharacteristic
203C2×D293—21no—
204C2×D293—21no—
205C2×D293—21no—
206C2×D293—41no—
207C2×D293—81no—
208C2×D293—161no—
209C2×D293—161no—
210C4×C293—21no—
211C4×C293—41no—
212C4×C293C22×D411yescharacteristic
213C4×C293C22×D411yescharacteristic
214Dic293—22no—
215Dic293—21no—
216Dic293—21no—
217Dic293—21no—
218Dic293—41no—
219C2×C293—21no—
220C2×C293—41no—
221C2×C293—81no—
222C2×C293D4211yescharacteristic
223D293—41no—
224D293—41no—
225D293—41no—
226D293—81no—
227C293#329111yescharacteristic, complemented, Sylow
228#3291—2931nomaximal, Sylow
229#577—2931no—
230#577—2931no—
231#577—2931no—
232#577—2931no—
233#492—2931no—
234#471—2931no—
235#471—2931no—
236#471—2931no—
237#471D29311yescharacteristic, complemented
238#456—2931no—
239#456—2931no—
240#453—2931no—
241#453—2931no—
242#443—2931no—
243#443—2931no—
2442−1+4—2931no—
2452−1+4—2931no—
246H(5,𝔽2)—2931no—
247H(5,𝔽2)—2931no—
248#137—2931no—
249#137—2931no—
250#137—2931no—
251#137—5861no—
252#137—5861no—
253#137—5861no—
254#137—5861no—
255#137C2×D29311yescharacteristic, complemented
256AGL(1,ℤ/8)—2931no—
257AGL(1,ℤ/8)—2931no—
258AGL(1,ℤ/8)—2931no—
259AGL(1,ℤ/8)—5861no—
260AGL(1,ℤ/8)—5861no—
261AGL(1,ℤ/8)—5861no—
262AGL(1,ℤ/8)—5861no—
263AGL(1,ℤ/8)C2×D29311yescharacteristic
264#135—2931no—
265#135—2931no—
266#135—5861no—
267#135—5861no—
268#135—5861no—
269#135—5861no—
270C2×SD16—2931no—
271C2×SD16—2931no—
272C2×SD16—2931no—
273C2×SD16—5861no—
274C2×SD16—5861no—
275C2×SD16—5861no—
276C2×SD16—5861no—
277C2×SD16C2×D29311yescharacteristic, complemented
278#131—2931no—
279#131—2931no—
280#131—2931no—
281#131C2×D29311yescharacteristic, complemented
282C4⋊Q8—2931no—
283#127—2931no—
284#108—2931no—
285#108C2×D29311yescharacteristic, complemented
286#104—2931no—
287#104—2931no—
288#104—2931no—
289#104—2931no—
290#101—2931no—
291#101C2×D29311yescharacteristic, complemented
292#100—2931no—
293#100C2×D29311yescharacteristic, complemented
294C4×C8—2931no—
295#41—2931no—
296#41—2931no—
297#41—2931no—
298#41—5861no—
299#41—5861no—
300#41—5861no—
301#41—5861no—
302#41—5861no—
303#41—5861no—
304#41—5861no—
305#41—5861no—
306#41—11721no—
307#41—11721no—
308#41C22×D29311yescharacteristic
309C2×Q8—2931no—
310C2×Q8—5861no—
311C2×Q8—5861no—
312C2×Q8—5861no—
313C2×Q8—5861no—
314C2×Q8C22×D29311yescharacteristic
315C2×D4—2931no—
316C2×D4—5861no—
317C2×D4—5861no—
318C2×D4—5861no—
319C2×D4—5861no—
320C2×D4—11721no—
321C2×D4—11721no—
322C2×D4—11721no—
323C2×D4C22×D29311yescharacteristic
324Q16—21no—
325Q16—2931no—
326Q16—2931no—
327Q16—5861no—
328Q16—5861no—
329Q16—5861no—
330SD16—21no—
331SD16—21no—
332SD16—21no—
333SD16—21no—
334SD16—2931no—
335SD16—2931no—
336SD16—2931no—
337SD16—2931no—
338SD16—5861no—
339SD16—5861no—
340SD16—5861no—
341SD16—5861no—
342SD16—5861no—
343SD16—5861no—
344SD16—5861no—
345SD16—5861no—
346SD16—5861no—
347SD16—5861no—
348SD16—5861no—
349SD16—5861no—
350D8—21no—
351D8—2931no—
352D8—2931no—
353D8—5861no—
354D8—5861no—
355D8—5861no—
356M16—21no—
357M16—2931no—
358M16—5861no—
359M16—5861no—
360M16—5861no—
361M16C22×D29311yescharacteristic
362M16C22×D29311yescharacteristic
363M16C22×D29311yescharacteristic
364C2×C8—21no—
365C2×C8—2931no—
366C2×C8—5861no—
367C2×C8—5861no—
368C2×C8—5861no—
369C2×C8C22×D29311yescharacteristic
370C4⋊C4—11721no—
371C42—2931no—
372C23—21no—
373C23—5861no—
374C23—11721no—
375C23—11721no—
376Q8—21no—
377Q8—21no—
378Q8—2931no—
379Q8—2931no—
380Q8—2931no—
381Q8—5861no—
382Q8—5861no—
383Q8—11721no—
384Q8—11721no—
385Q8D4×D29311yescharacteristic
386D4—21no—
387D4—21no—
388D4—21no—
389D4—2931no—
390D4—2931no—
391D4—2931no—
392D4—23442no—
393D4—5861no—
394D4—5861no—
395D4—5861no—
396D4—5861no—
397D4—5861no—
398D4—11721no—
399D4—11721no—
400D4—11721no—
401D4—11721no—
402D4—11721no—
403D4—11721no—
404D4D4×D29311yescharacteristic
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—5861no—
408C2×C4—5861no—
409C2×C4—5861no—
410C2×C4—5861no—
411C2×C4—5861no—
412C2×C4—11721no—
413C2×C4—11721no—
414C2×C4—11721no—
415C2×C4—11721no—
416C2×C4—11721no—
417C2×C4—11721no—
418C2×C4C23×D29311yescharacteristic
419C8—21no—
420C8—21no—
421C8—21no—
422C8—2931no—
423C8—2931no—
424C8—5861no—
425C8D4×D29311yescharacteristic
426C8D4×D29311yescharacteristic
427C22—21no—
428C22—41no—
429C22—81no—
430C22—23441no—
431C22—5861no—
432C22—5861no—
433C22—5861no—
434C22—46881no—
435C22—46881no—
436C22—11721no—
437C22#2262767011yescharacteristic
438C4—21no—
439C4—41no—
440C4—5862no—
441C4—5861no—
442C4—5861no—
443C4—5861no—
444C4—11721no—
445C4C2×D4×D29311yescharacteristic
446C4C2×D4×D29311yescharacteristic
447C2—21no—
448C2—41no—
449C2—81no—
450C2—23441no—
451C2—11721no—
452C2—11721no—
453C2—11721no—
454C2#3403676311yescharacteristic
subgroupG/H
centerC2#34036763
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#471×C293C2
socleC2×C293D42
hypercenter#471D293
hypocenterC293#3291
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C2 ◁ C2×C4 ◁ #471

Complements

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

NH ≅ G/Ntype
1. #34036941C2semidirect
2. #34036939C2semidirect
3. #34036938C2semidirect
4. #34036935C2semidirect
5. #34036500C2semidirect
6. #34036395C2semidirect
7. #34036393C2semidirect
8. #34036391C2semidirect
9. #34036314C2semidirect
10. #34036310C2semidirect
11. #34036299C2semidirect
12. #34036295C2semidirect
13. #34036245C2semidirect
14. #34036244C2semidirect
15. #471×C293C2semidirect
18. #22627670C22semidirect
19. #22627669C22semidirect
21. #22627657C22semidirect
22. #22627657C22semidirect
24. #22627655C22semidirect
41. C2×#17989791C22semidirect
42. C2×#17989790C22semidirect
43. C2×#17989789C22semidirect
47. #22627632C22semidirect
48. #22627632C22semidirect
49. #22627630C22semidirect
50. #22627622C22semidirect
52. #22627559C22semidirect
53. #22627551C22semidirect
54. #22627550C22semidirect
55. #22627550C22semidirect
56. #22627550C22semidirect
57. #22627542C22semidirect
58. #22627540C22semidirect
59. C8×Dic293C22semidirect
60. #137×C293C22semidirect
62. C2×SD16×C293C22semidirect
63. #131×C293C22semidirect
64. #108×C293C22semidirect
65. #101×C293C22semidirect
66. #100×C293C22semidirect
95. Dic1172D4semidirect
97. #17989792D4semidirect
104. #17989791D4semidirect
105. #17989791D4semidirect
109. #17989790D4semidirect
113. #17989789D4semidirect
116. D2344D4semidirect
118. #17989787D4semidirect
227. C293#3291semidirect
237. #471D293semidirect
255. #137C2×D293semidirect
277. C2×SD16C2×D293semidirect
281. #131C2×D293semidirect
285. #108C2×D293semidirect
291. #101C2×D293semidirect
293. #100C2×D293semidirect

Maximal subgroups

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

subgroupindex
#340369412
#340369392
#340369382
#340369352
#340365002
#340363952
#340363932
#340363912
#340363142
#340363102
#340362992
#340362952
#340362452
#340362442
#471×C2932
#3291293