Finite Group Databasebeta

← #49170271

Subgroups — #49170271

Order 37504.

568 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22627651C211yesmaximal, characteristic, complemented
2C2×D8×D293C211yesmaximal, characteristic, complemented
3#34036772C211yesmaximal, characteristic, complemented
4D4×D1172C211yesmaximal, characteristic, complemented
5#34036519C211yesmaximal, characteristic, complemented
6#34036381C211yesmaximal, characteristic, complemented
7#34036361C211yesmaximal, characteristic, complemented
8#34036325C211yesmaximal, characteristic, complemented
9#34036278C211yesmaximal, characteristic, complemented
10#34036277C211yesmaximal, characteristic, complemented
11#34036264C211yesmaximal, characteristic, complemented
12#34036261C211yesmaximal, characteristic, complemented
13#34036239C211yesmaximal, characteristic, complemented
14#34036199C211yesmaximal, characteristic, complemented
15#463×C293C211yesmaximal, characteristic, complemented
16C2×#17989803C2211yescharacteristic, complemented
17C22×D1172—21no—
18C2×D4×D293C2211yescharacteristic
19C2×D4×D293C2211yescharacteristic, complemented
20C2×D4×D293C2211yescharacteristic, complemented
21#22627651—24no—
22C2×#17989789C2211yescharacteristic, complemented
23C2×#17989789C2211yescharacteristic, complemented
24C2×D2344C2211yescharacteristic, complemented
25C2×D2344C2211yescharacteristic, complemented
26C2×#17989787C2211yescharacteristic
27C2×#17989787C2211yescharacteristic, complemented
28C2×#17989787C2211yescharacteristic, complemented
29D8×D293—24no—
30C2×#17989785C2211yescharacteristic, complemented
31C2×C8×D293C2211yescharacteristic, complemented
32#22627621C2211yescharacteristic, complemented
33#22627606—21no—
34#22627600C2211yescharacteristic, complemented
35#22627599C2211yescharacteristic
36#22627599C2211yescharacteristic, complemented
37#22627598—21no—
38#22627597—21no—
39#22627595C2211yescharacteristic, complemented
40#22627594—21no—
41#22627594—21no—
42#22627594—21no—
43C4×D1172C2211yescharacteristic
44C4×D1172C2211yescharacteristic, complemented
45C22⋊C4×D293—21no—
46#22627555C2211yescharacteristic, complemented
47#22627546C2211yescharacteristic, complemented
48#22627546C2211yescharacteristic, complemented
49#22627545C2211yescharacteristic, complemented
50#22627544C2211yescharacteristic, complemented
51#22627536C2211yescharacteristic, complemented
52#22627536C2211yescharacteristic, complemented
53#22627535C2211yescharacteristic, complemented
54C2×SD16×C293C2211yescharacteristic, complemented
55C2×D8×C293C2211yescharacteristic
56#124×C293C2211yescharacteristic, complemented
57C4×D4×C293C2211yescharacteristic, complemented
58C4⋊C8×C293C2211yescharacteristic, complemented
59#103×C293C2211yescharacteristic, complemented
60#102×C293C2211yescharacteristic, complemented
61C23×D293—21no—
62C23×D293—21no—
63C23×D293—21no—
64#17989803—22no—
65#17989803—41no—
66C2×D1172—21no—
67C2×D1172—21no—
68C2×D1172—21no—
69C2×D1172—21no—
70C2×D1172—21no—
71C2×D1172—44no—
72C2×D1172D411yescharacteristic
73C2×D1172C2311yescharacteristic
74C2×D1172C2311yescharacteristic
75C2×D1172C2311yescharacteristic
76C2×C22⋊D293—21no—
77C2×C22⋊D293—21no—
78C2×C22⋊D293—21no—
79C2×C22⋊D293—21no—
80D4×D293—22no—
81D4×D293—22no—
82D4×D293—22no—
83D4×D293—41no—
84D4×D293—41no—
85D4×D293—41no—
86C2×C4×D293—21no—
87C2×C4×D293—21no—
88C2×C4×D293C2311yescharacteristic
89C2×C4×D293C2311yescharacteristic
90#17989789—22no—
91#17989789—41no—
92D2344—22no—
93D2344—22no—
94#17989787—22no—
95#17989787—41no—
96#17989787D414yescomplemented
97#17989785—22no—
98C2×#14556873C2311yescharacteristic
99C2×#14556873C2311yescharacteristic
100C8×D293—22no—
101C4⋊Dic293D411yescharacteristic
102C4⋊Dic293C2311yescharacteristic
103#17989781—21no—
104C22⋊Dic293—21no—
105#17989779—21no—
106#17989779—21no—
107#17989779—21no—
108#17989779—21no—
109#17989779D411yescharacteristic
110#17989779D411yescharacteristic
111C2×Q8×C293C2311yescharacteristic
112C2×D4×C293C2311yescharacteristic
113C2×D4×C293C2311yescharacteristic
114C22×C4×C293—21no—
115SD16×C293—22no—
116D8×C293—22no—
117C2×C8×C293C2311yescharacteristic
118C2×C8×C293C2311yescharacteristic
119C4⋊C4×C293C2311yescharacteristic
120C22⋊C4×C293—21no—
121C22⋊C4×C293—21no—
122C22⋊C4×C293—21no—
123C42×C293C2311yescharacteristic
124C22×D293—21no—
125C22×D293—21no—
126C22×D293—21no—
127C22×D293—42no—
128C22×D293—42no—
129C22×D293—42no—
130C22×D293—42no—
131C22×D293—42no—
132C22×D293—42no—
133C22×D293—81no—
134C22×D293—81no—
135C22×D293—81no—
136C22×D293C2×D411yescharacteristic
137C22×D293C2×D411yescharacteristic
138D1172—22no—
139D1172—21no—
140D1172—21no—
141D1172—21no—
142D1172—21no—
143D1172—21no—
144D1172—42no—
145D1172—42no—
146D1172—42no—
147D1172—42no—
148D1172—81no—
149D1172C2×D412yescomplemented
150C22⋊D293—42no—
151C22⋊D293—42no—
152C22⋊D293—42no—
153C22⋊D293—81no—
154C2×Dic293—21no—
155C2×Dic293C2×D411yescharacteristic
156C2×Dic293C2×D411yescharacteristic
157C4×D293—21no—
158C4×D293—21no—
159C4×D293—21no—
160C4×D293—21no—
161C4×D293—42no—
162C4×D293—81no—
163#14556873—21no—
164#14556873C2×D412yescomplemented
165C23×C293—21no—
166C23×C293—21no—
167Q8×C293—21no—
168Q8×C293—21no—
169D4×C293—21no—
170D4×C293—21no—
171D4×C293—21no—
172D4×C293C2×D412yes—
173C2×C4×C293—21no—
174C2×C4×C293—21no—
175C2×C4×C293—42no—
176C2×C4×C293C2×D411yescharacteristic
177C2×C4×C293C2×D411yescharacteristic
178C2×C4×C293C2411yescharacteristic
179C8×C293—21no—
180C8×C293—21no—
181C22×C293—22no—
182C22×C293—41no—
183C22×C293—41no—
184C22×C293—41no—
185C22×C293—41no—
186C22×C293—41no—
187C22×C293C22×D411yescharacteristic
188C2×D293—21no—
189C2×D293—21no—
190C2×D293—21no—
191C2×D293—22no—
192C2×D293—21no—
193C2×D293—21no—
194C2×D293—41no—
195C2×D293—41no—
196C2×D293—41no—
197C2×D293—41no—
198C2×D293—41no—
199C2×D293—41no—
200C2×D293—41no—
201C2×D293—41no—
202C2×D293—84no—
203C2×D293—84no—
204C2×D293—84no—
205C2×D293AGL(1,ℤ/8)12yescomplemented
206C4×C293—21no—
207C4×C293—21no—
208C4×C293—41no—
209C4×C293—41no—
210C4×C293C22×D411yescharacteristic
211C4×C293H(5,𝔽2)11yescharacteristic
212Dic293—21no—
213Dic293—21no—
214Dic293—41no—
215C2×C293—42no—
216C2×C293—81no—
217C2×C293D4211yescharacteristic
218C2×C293C2×AGL(1,ℤ/8)11yescharacteristic
219C2×C293#57611yescharacteristic
220D293—22no—
221D293—41no—
222D293—42no—
223D293—81no—
224D293—81no—
225C293#326411yescharacteristic, complemented, Sylow
226#3264—2931nomaximal, Sylow
227C2×AGL(1,ℤ/8)—2931no—
228C22×D8—2931no—
229#546—2931no—
230D42—2931no—
231#496—2931no—
232#469—2931no—
233#465—2931no—
234#463D29311yescharacteristic, complemented
235#459—2931no—
236#450—2931no—
237#449—2931no—
238#447—2931no—
239#447—2931no—
240#442—2931no—
241#435—2931no—
242C2×#41—2931no—
243C22×D4—2931no—
244C22×D4—2931no—
245C22×D4—2931no—
246C22×D4—5861no—
247AGL(1,ℤ/8)—5864no—
248C2×SD16—2931no—
249C2×SD16—2931no—
250C2×SD16C2×D29311yescharacteristic, complemented
251C2×D8—2931no—
252C2×D8—2931no—
253C2×D8—2931no—
254C2×D8—2931no—
255C2×D8—2931no—
256C2×D8—5864no—
257C2×D8C2×D29311yescharacteristic
258C2×M16—2931no—
259C22×C8—2931no—
260#127—2931no—
261#124C2×D29311yescharacteristic, complemented
262#123—5861no—
263C22⋊Q8—2931no—
264#121—2931no—
265#121—2931no—
266#121—2931no—
267#121—5861no—
268#121—5861no—
269#120—5861no—
270#120—5861no—
271#120—5861no—
272C4×D4—2931no—
273C4×D4—2931no—
274C4×D4C2×D29311yescharacteristic, complemented
275C2×C22⋊C4—5861no—
276#106—2931no—
277C4⋊C8C2×D29311yescharacteristic, complemented
278#103C2×D29311yescharacteristic, complemented
279#102—2931no—
280#102—2931no—
281#102—2931no—
282#102—2931no—
283#102C2×D29311yescharacteristic, complemented
284C22⋊C8—2931no—
285C22⋊C8—2931no—
286#97—2931no—
287C24—5861no—
288C24—5861no—
289C24—5861no—
290#41—5862no—
291#41—11721no—
292C2×Q8C22×D29311yescharacteristic
293C2×D4—2931no—
294C2×D4—2931no—
295C2×D4—2931no—
296C2×D4—2931no—
297C2×D4—5861no—
298C2×D4—5861no—
299C2×D4—5861no—
300C2×D4—5861no—
301C2×D4—5861no—
302C2×D4—5861no—
303C2×D4—5862no—
304C2×D4—5861no—
305C2×D4—5862no—
306C2×D4—5862no—
307C2×D4—5861no—
308C2×D4—5861no—
309C2×D4—11724no—
310C2×D4—11721no—
311C2×D4—11721no—
312C2×D4—11721no—
313C2×D4C22×D29311yescharacteristic
314C2×D4C22×D29311yescharacteristic
315C22×C4—21no—
316C22×C4—2931no—
317C22×C4—2931no—
318C22×C4—5861no—
319C22×C4—5861no—
320SD16—22no—
321SD16—5862no—
322SD16—11721no—
323D8—22no—
324D8—2934no—
325D8—5862no—
326D8—5862no—
327D8—5862no—
328D8—11721no—
329M16—5862no—
330C2×C8—2931no—
331C2×C8—2931no—
332C2×C8—5862no—
333C2×C8C22×D29311yescharacteristic
334C2×C8C22×D29311yescharacteristic
335C4⋊C4—2931no—
336C4⋊C4—2931no—
337C4⋊C4—5861no—
338C4⋊C4C22×D29311yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—21no—
342C22⋊C4—2931no—
343C22⋊C4—2931no—
344C22⋊C4—5861no—
345C22⋊C4—5861no—
346C22⋊C4—5861no—
347C22⋊C4—5861no—
348C22⋊C4—5861no—
349C42C22×D29311yescharacteristic
350C23—21no—
351C23—21no—
352C23—2931no—
353C23—2931no—
354C23—23441no—
355C23—23441no—
356C23—23441no—
357C23—5861no—
358C23—5861no—
359C23—5861no—
360C23—11722no—
361C23—11722no—
362C23—11722no—
363C23—11722no—
364C23—11722no—
365C23—11722no—
366Q8—21no—
367Q8—21no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—2932no—
372D4—23441no—
373D4—23441no—
374D4—5862no—
375D4—5861no—
376D4—5861no—
377D4—5861no—
378D4—5861no—
379D4—5861no—
380D4—11722no—
381D4—11722no—
382D4—11722no—
383D4—11722no—
384D4—11722no—
385D4—11722no—
386D4—11722no—
387D4D4×D29312yes—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—42no—
391C2×C4—2931no—
392C2×C4—2931no—
393C2×C4—23441no—
394C2×C4—5861no—
395C2×C4—5861no—
396C2×C4—5861no—
397C2×C4—5861no—
398C2×C4—5861no—
399C2×C4—11722no—
400C2×C4D4×D29311yescharacteristic
401C2×C4D4×D29311yescharacteristic
402C2×C4C23×D29311yescharacteristic
403C8—21no—
404C8—21no—
405C8—2932no—
406C8—5861no—
407C22—22no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—2932no—
414C22—23444no—
415C22—23444no—
416C22—23444no—
417C22—5861no—
418C22—5861no—
419C22—5861no—
420C22—5862no—
421C22—5861no—
422C22—5861no—
423C22—11721no—
424C22—11721no—
425C22—11721no—
426C22—11721no—
427C22—11721no—
428C22—11721no—
429C22—11721no—
430C22—11721no—
431C22C2×D4×D29311yescharacteristic
432C4—21no—
433C4—21no—
434C4—41no—
435C4—41no—
436C4—5861no—
437C4—5861no—
438C4—11721no—
439C4C2×D4×D29311yescharacteristic
440C4#2262767011yescharacteristic
441C2—42no—
442C2—81no—
443C2—23441no—
444C2—23441no—
445C2—5862no—
446C2—11721no—
447C2—11722no—
448C2#3403676311yescharacteristic
449C2AGL(1,ℤ/8)×D29311yescharacteristic
450C2#3403693111yescharacteristic
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#463×C293C2
socleC22×C293C22×D4
hypercenter#463D293
hypocenterC293#3264
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#22627651C2semidirect
2. C2×D8×D293C2semidirect
3. #34036772C2semidirect
4. D4×D1172C2semidirect
5. #34036519C2semidirect
6. #34036381C2semidirect
7. #34036361C2semidirect
8. #34036325C2semidirect
9. #34036278C2semidirect
10. #34036277C2semidirect
11. #34036264C2semidirect
12. #34036261C2semidirect
13. #34036239C2semidirect
14. #34036199C2semidirect
15. #463×C293C2semidirect
16. C2×#17989803C22semidirect
19. C2×D4×D293C22semidirect
20. C2×D4×D293C22semidirect
22. C2×#17989789C22semidirect
23. C2×#17989789C22semidirect
24. C2×D2344C22semidirect
25. C2×D2344C22semidirect
27. C2×#17989787C22semidirect
28. C2×#17989787C22semidirect
30. C2×#17989785C22semidirect
31. C2×C8×D293C22semidirect
32. #22627621C22semidirect
34. #22627600C22semidirect
36. #22627599C22semidirect
39. #22627595C22semidirect
44. C4×D1172C22semidirect
46. #22627555C22semidirect
47. #22627546C22semidirect
48. #22627546C22semidirect
49. #22627545C22semidirect
50. #22627544C22semidirect
51. #22627536C22semidirect
52. #22627536C22semidirect
53. #22627535C22semidirect
54. C2×SD16×C293C22semidirect
56. #124×C293C22semidirect
57. C4×D4×C293C22semidirect
58. C4⋊C8×C293C22semidirect
59. #103×C293C22semidirect
60. #102×C293C22semidirect
96. #17989787D4semidirect
149. D1172C2×D4semidirect
164. #14556873C2×D4semidirect
205. C2×D293AGL(1,ℤ/8)semidirect
225. C293#3264semidirect
234. #463D293semidirect
250. C2×SD16C2×D293semidirect
261. #124C2×D293semidirect
274. C4×D4C2×D293semidirect
277. C4⋊C8C2×D293semidirect
278. #103C2×D293semidirect
283. #102C2×D293semidirect

Maximal subgroups

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

subgroupindex
C2×#226276512
C2×D8×D2932
#340367722
D4×D11722
#340365192
#340363812
#340363612
#340363252
#340362782
#340362772
#340362642
#340362612
#340362392
#340361992
#463×C2932
#3264293