Finite Group Databasebeta

← #49170272

Subgroups — #49170272

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×#22627652C211yesmaximal, characteristic, complemented
2C2×D8×D293C211yesmaximal, characteristic, complemented
3#34036776C211yesmaximal, characteristic, complemented
4#34036763C211yesmaximal, characteristic, complemented
5#34036517C211yesmaximal, characteristic, complemented
6#34036359C211yesmaximal, characteristic, complemented
7#34036343C211yesmaximal, characteristic, complemented
8#34036323C211yesmaximal, characteristic, complemented
9#34036281C211yesmaximal, characteristic, complemented
10#34036272C211yesmaximal, characteristic, complemented
11#34036264C211yesmaximal, characteristic, complemented
12#34036262C211yesmaximal, characteristic, complemented
13#34036240C211yesmaximal, characteristic, complemented
14#34036198C211yesmaximal, characteristic, complemented
15#469×C293C211yesmaximal, characteristic, complemented
16C2×#17989802C2211yescharacteristic, complemented
17C22×D1172C2211yescharacteristic, complemented
18C2×D4×D293—21no—
19C2×D4×D293C2211yescharacteristic
20C2×D4×D293C2211yescharacteristic, complemented
21#22627652—24no—
22C2×#17989791C2211yescharacteristic, complemented
23C2×#17989791C2211yescharacteristic, complemented
24C2×#17989790C2211yescharacteristic, complemented
25C2×D2344C2211yescharacteristic
26C2×D2344C2211yescharacteristic, complemented
27C2×#17989787C2211yescharacteristic
28C2×#17989787C2211yescharacteristic, complemented
29C2×#17989787C2211yescharacteristic, complemented
30D8×D293—24no—
31C2×C8×D293C2211yescharacteristic, complemented
32#22627620C2211yescharacteristic, complemented
33#22627611C2211yescharacteristic, complemented
34#22627606—21no—
35#22627604C2211yescharacteristic, complemented
36#22627599C2211yescharacteristic
37#22627597—21no—
38#22627595—21no—
39#22627594—21no—
40#22627592—21no—
41#22627592—21no—
42#22627586C2211yescharacteristic, complemented
43#22627586C2211yescharacteristic, complemented
44C4×C22⋊D293C2211yescharacteristic
45C22⋊C4×D293—21no—
46#22627553C2211yescharacteristic, complemented
47#22627547C2211yescharacteristic, complemented
48#22627544C2211yescharacteristic, complemented
49#22627544C2211yescharacteristic, complemented
50#22627544C2211yescharacteristic, complemented
51C22⋊#14556873C2211yescharacteristic, complemented
52#22627536C2211yescharacteristic, complemented
53#22627534C2211yescharacteristic, complemented
54C2×D8×C293C2211yescharacteristic, complemented
55C2×M16×C293C2211yescharacteristic, complemented
56#121×C293C2211yescharacteristic, complemented
57#121×C293C2211yescharacteristic, complemented
58#106×C293C2211yescharacteristic, complemented
59#102×C293C2211yescharacteristic, complemented
60#102×C293C2211yescharacteristic, complemented
61C23×D293—21no—
62C23×D293—21no—
63C23×D293—21no—
64#17989802—22no—
65#17989802—41no—
66C2×Dic586C2311yescharacteristic
67C2×D1172—22no—
68C2×D1172—21no—
69C2×D1172—21no—
70C2×D1172—41no—
71C2×D1172C2311yescharacteristic
72C2×D1172C2311yescharacteristic
73C2×D1172C2311yescharacteristic
74C2×C22⋊D293—21no—
75C2×C22⋊D293—21no—
76C2×C22⋊D293—21no—
77C2×C22⋊D293—21no—
78C2×C22⋊D293—21no—
79C2×C22⋊D293D411yescharacteristic
80D4×D293—22no—
81D4×D293—22no—
82D4×D293—41no—
83D4×D293—41no—
84D4×D293—44no—
85C2×C4×D293—21no—
86C2×C4×D293—21no—
87C2×C4×D293—21no—
88C2×C4×D293C2311yescharacteristic
89#17989791—22no—
90#17989791—22no—
91#17989790—41no—
92D2344—22no—
93D2344D414yescomplemented
94#17989787—22no—
95#17989787—22no—
96#17989787—41no—
97C2×#14556873C2311yescharacteristic
98C2×#14556873C2311yescharacteristic
99C8×D293—22no—
100C4⋊Dic293—21no—
101#17989781D411yescharacteristic
102C22⋊Dic293—21no—
103C22⋊Dic293—21no—
104C22⋊Dic293D411yescharacteristic
105#17989779—21no—
106#17989779—21no—
107#17989779—21no—
108#17989779—21no—
109#17989779D411yescharacteristic
110C4×Dic293C2311yescharacteristic
111C2×D4×C293—21no—
112C2×D4×C293—21no—
113C2×D4×C293C2311yescharacteristic
114C2×D4×C293C2311yescharacteristic
115C22×C4×C293C2311yescharacteristic
116D8×C293—22no—
117M16×C293—22no—
118C2×C8×C293C2311yescharacteristic
119C2×C8×C293C2311yescharacteristic
120C4⋊C4×C293C2311yescharacteristic
121C4⋊C4×C293C2311yescharacteristic
122C22⋊C4×C293—21no—
123C22⋊C4×C293—21no—
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
137Dic586—21no—
138Dic586—21no—
139D1172—21no—
140D1172—21no—
141D1172—21no—
142D1172—21no—
143D1172—42no—
144D1172—42no—
145D1172C2×D412yes—
146D1172C2×D412yescomplemented
147C22⋊D293—22no—
148C22⋊D293—42no—
149C22⋊D293—42no—
150C22⋊D293—42no—
151C22⋊D293—42no—
152C22⋊D293—81no—
153C2×Dic293—21no—
154C2×Dic293C2×D411yescharacteristic
155C2×Dic293C2×D411yescharacteristic
156C2×Dic293C2×D411yescharacteristic
157C4×D293—21no—
158C4×D293—21no—
159C4×D293—42no—
160C4×D293—42no—
161C4×D293—81no—
162#14556873—21no—
163#14556873—21no—
164C23×C293—21no—
165C23×C293—21no—
166C23×C293C2×D411yescharacteristic
167D4×C293—21no—
168D4×C293—21no—
169D4×C293—21no—
170D4×C293—21no—
171D4×C293—42no—
172D4×C293—81no—
173C2×C4×C293—21no—
174C2×C4×C293—21no—
175C2×C4×C293—21no—
176C2×C4×C293—21no—
177C2×C4×C293C2×D411yescharacteristic
178C2×C4×C293C2411yescharacteristic
179C8×C293—21no—
180C8×C293C2×D412yescomplemented
181C22×C293—21no—
182C22×C293—21no—
183C22×C293—21no—
184C22×C293—41no—
185C22×C293—41no—
186C22×C293—41no—
187C22×C293—41no—
188C22×C293—41no—
189C22×C293—41no—
190C22×C293C22×D411yescharacteristic
191C2×D293—21no—
192C2×D293—21no—
193C2×D293—22no—
194C2×D293—22no—
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—41no—
208C4×C293—41no—
209C4×C293C22×D411yescharacteristic
210C4×C293H(5,𝔽2)11yescharacteristic
211Dic293—21no—
212Dic293—21no—
213Dic293—21no—
214Dic293—41no—
215C2×C293—41no—
216C2×C293—81no—
217C2×C293—81no—
218C2×C293D4211yescharacteristic
219C2×C293C2×AGL(1,ℤ/8)11yescharacteristic
220C2×C293#57611yescharacteristic
221D293—22no—
222D293—42no—
223D293—42no—
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#469D29311yescharacteristic, complemented
233#465—2931no—
234#463—2931no—
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×SD16—2931no—
251C2×D8—2931no—
252C2×D8—2931no—
253C2×D8—2931no—
254C2×D8—2931no—
255C2×D8—2931no—
256C2×D8—5864no—
257C2×D8C2×D29311yescharacteristic, complemented
258C2×M16C2×D29311yescharacteristic, complemented
259C22×C8—2931no—
260#127—2931no—
261#124—2931no—
262#123—5861no—
263C22⋊Q8—2931no—
264#121—2931no—
265#121—5861no—
266#121—5861no—
267#121C2×D29311yescharacteristic, complemented
268#121C2×D29311yescharacteristic, complemented
269#120—5861no—
270#120—5861no—
271#120—5861no—
272C4×D4—2931no—
273C4×D4—2931no—
274C4×D4—2931no—
275C2×C22⋊C4—5861no—
276#106C2×D29311yescharacteristic, complemented
277C4⋊C8—2931no—
278#103—2931no—
279#102—2931no—
280#102—2931no—
281#102—2931no—
282#102C2×D29311yescharacteristic, complemented
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×Q8—2931no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—2931no—
296C2×D4—2931no—
297C2×D4—2931no—
298C2×D4—2931no—
299C2×D4—5862no—
300C2×D4—5861no—
301C2×D4—5861no—
302C2×D4—5861no—
303C2×D4—5861no—
304C2×D4—5861no—
305C2×D4—5862no—
306C2×D4—5862no—
307C2×D4—5861no—
308C2×D4—5861no—
309C2×D4—11721no—
310C2×D4—11721no—
311C2×D4—11721no—
312C2×D4—11724no—
313C2×D4C22×D29311yescharacteristic
314C2×D4C22×D29311yescharacteristic
315C22×C4—2931no—
316C22×C4—5861no—
317C22×C4—5861no—
318C22×C4—5861no—
319C22×C4C22×D29311yescharacteristic
320SD16—5862no—
321SD16—5862no—
322SD16—11721no—
323D8—22no—
324D8—2934no—
325D8—5862no—
326D8—5862no—
327D8—5862no—
328D8—11721no—
329M16—22no—
330C2×C8—2931no—
331C2×C8—2931no—
332C2×C8—5862no—
333C2×C8C22×D29311yescharacteristic
334C2×C8C22×D29311yescharacteristic
335C4⋊C4—2931no—
336C4⋊C4—5861no—
337C4⋊C4C22×D29311yescharacteristic
338C4⋊C4C22×D29311yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—2931no—
342C22⋊C4—2931no—
343C22⋊C4—5861no—
344C22⋊C4—5861no—
345C22⋊C4—5861no—
346C22⋊C4—5861no—
347C22⋊C4—5861no—
348C22⋊C4—5861no—
349C42—2931no—
350C23—21no—
351C23—21no—
352C23—2931no—
353C23—23441no—
354C23—23441no—
355C23—23441no—
356C23—5861no—
357C23—5861no—
358C23—5861no—
359C23—11722no—
360C23—11722no—
361C23—11722no—
362C23—11722no—
363C23—11722no—
364C23—11722no—
365C23D4×D29311yescharacteristic
366Q8—5861no—
367Q8—5861no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—42no—
373D4—81no—
374D4—2932no—
375D4—2932no—
376D4—23441no—
377D4—5862no—
378D4—5861no—
379D4—5861no—
380D4—5861no—
381D4—5861no—
382D4—11722no—
383D4—11722no—
384D4—11722no—
385D4—11722no—
386D4—11722no—
387D4—11722no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—2931no—
393C2×C4—2931no—
394C2×C4—2931no—
395C2×C4—23441no—
396C2×C4—5861no—
397C2×C4—5861no—
398C2×C4—5861no—
399C2×C4—11722no—
400C2×C4—11722no—
401C2×C4D4×D29311yescharacteristic
402C2×C4C23×D29311yescharacteristic
403C8—21no—
404C8—5861no—
405C8—5861no—
406C8D4×D29312yescomplemented
407C22—21no—
408C22—21no—
409C22—21no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—2932no—
417C22—23444no—
418C22—23444no—
419C22—23444no—
420C22—5861no—
421C22—5861no—
422C22—5862no—
423C22—5862no—
424C22—11721no—
425C22—11721no—
426C22—11721no—
427C22—11721no—
428C22—11721no—
429C22—11721no—
430C22—11721no—
431C22C2×D4×D29311yescharacteristic
432C4—21no—
433C4—41no—
434C4—41no—
435C4—5861no—
436C4—5861no—
437C4—5861no—
438C4—11721no—
439C4C2×D4×D29311yescharacteristic
440C4#2262767011yescharacteristic
441C2—41no—
442C2—81no—
443C2—81no—
444C2—23441no—
445C2—5862no—
446C2—11722no—
447C2—11722no—
448C2#3403676311yescharacteristic
449C2AGL(1,ℤ/8)×D29311yescharacteristic
450C2#3403693311yescharacteristic
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#469×C293C2
socleC22×C293C22×D4
hypercenter#469D293
hypocenterC293#3264
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. C2×#22627652C2semidirect
2. C2×D8×D293C2semidirect
3. #34036776C2semidirect
4. #34036763C2semidirect
5. #34036517C2semidirect
6. #34036359C2semidirect
7. #34036343C2semidirect
8. #34036323C2semidirect
9. #34036281C2semidirect
10. #34036272C2semidirect
11. #34036264C2semidirect
12. #34036262C2semidirect
13. #34036240C2semidirect
14. #34036198C2semidirect
15. #469×C293C2semidirect
16. C2×#17989802C22semidirect
17. C22×D1172C22semidirect
20. C2×D4×D293C22semidirect
22. C2×#17989791C22semidirect
23. C2×#17989791C22semidirect
24. C2×#17989790C22semidirect
26. C2×D2344C22semidirect
28. C2×#17989787C22semidirect
29. C2×#17989787C22semidirect
31. C2×C8×D293C22semidirect
32. #22627620C22semidirect
33. #22627611C22semidirect
35. #22627604C22semidirect
42. #22627586C22semidirect
43. #22627586C22semidirect
46. #22627553C22semidirect
47. #22627547C22semidirect
48. #22627544C22semidirect
49. #22627544C22semidirect
50. #22627544C22semidirect
51. C22⋊#14556873C22semidirect
52. #22627536C22semidirect
53. #22627534C22semidirect
54. C2×D8×C293C22semidirect
55. C2×M16×C293C22semidirect
56. #121×C293C22semidirect
57. #121×C293C22semidirect
58. #106×C293C22semidirect
59. #102×C293C22semidirect
60. #102×C293C22semidirect
93. D2344D4semidirect
146. D1172C2×D4semidirect
180. C8×C293C2×D4semidirect
205. C2×D293AGL(1,ℤ/8)semidirect
225. C293#3264semidirect
232. #469D293semidirect
257. C2×D8C2×D293semidirect
258. C2×M16C2×D293semidirect
267. #121C2×D293semidirect
268. #121C2×D293semidirect
276. #106C2×D293semidirect
282. #102C2×D293semidirect
283. #102C2×D293semidirect
406. C8D4×D293semidirect

Maximal subgroups

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

subgroupindex
C2×#226276522
C2×D8×D2932
#340367762
#340367632
#340365172
#340363592
#340363432
#340363232
#340362812
#340362722
#340362642
#340362622
#340362402
#340361982
#469×C2932
#3264293