Finite Group Databasebeta

← #49170277

Subgroups — #49170277

Order 37504.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22627648C211yesmaximal, characteristic, complemented
2C22×#17989791C211yesmaximal, characteristic, complemented
3#34036785C211yesmaximal, characteristic, complemented
4#34036763C211yesmaximal, characteristic, complemented
5#34036517C211yesmaximal, characteristic, complemented
6#34036376C211yesmaximal, characteristic, complemented
7#34036343C211yesmaximal, characteristic, complemented
8#34036328C211yesmaximal, characteristic, complemented
9#34036281C211yesmaximal, characteristic, complemented
10#34036275C211yesmaximal, characteristic, complemented
11#34036267C211yesmaximal, characteristic, complemented
12#34036262C211yesmaximal, characteristic, complemented
13#34036228C211yesmaximal, characteristic, complemented
14#34036200C211yesmaximal, characteristic, complemented
15#466×C293C211yesmaximal, characteristic, complemented
16C2×#17989801C2211yescharacteristic, complemented
17C22×Dic586C2211yescharacteristic, complemented
18C22×D1172C2211yescharacteristic
19C2×D4×D293—21no—
20C2×D4×D293C2211yescharacteristic
21#22627648—24no—
22C2×#17989791—24no—
23C2×#17989791C2211yescharacteristic
24C2×#17989791C2211yescharacteristic
25C2×#17989791C2211yescharacteristic, complemented
26C2×#17989791C2211yescharacteristic, complemented
27C2×#17989790C2211yescharacteristic, complemented
28C2×#17989790C2211yescharacteristic, complemented
29C2×#17989787C2211yescharacteristic
30C2×#17989787C2211yescharacteristic, complemented
31C2×#17989785C2211yescharacteristic, complemented
32#22627620C2211yescharacteristic
33#22627611C2211yescharacteristic, complemented
34#22627610—21no—
35#22627604C2211yescharacteristic, complemented
36#22627601—21no—
37#22627599C2211yescharacteristic
38#22627597—21no—
39#22627596—21no—
40#22627594—21no—
41#22627592—21no—
42#22627589C2211yescharacteristic, complemented
43D4×Dic293—21no—
44#22627586C2211yescharacteristic
45C4×C22⋊D293C2211yescharacteristic
46#22627553C2211yescharacteristic, complemented
47#22627547C2211yescharacteristic, complemented
48#22627547C2211yescharacteristic, complemented
49#22627544C2211yescharacteristic
50#22627544C2211yescharacteristic, complemented
51C22⋊#14556873C2211yescharacteristic, complemented
52#22627536C2211yescharacteristic, complemented
53#22627534C2211yescharacteristic, complemented
54C2×D8×C293C2211yescharacteristic, complemented
55C22×C8×C293C2211yescharacteristic, complemented
56#121×C293C2211yescharacteristic
57#121×C293C2211yescharacteristic, complemented
58#107×C293C2211yescharacteristic, complemented
59#102×C293C2211yescharacteristic, complemented
60#102×C293C2211yescharacteristic, complemented
61C23×D293—21no—
62C23×D293—21no—
63#17989801—22no—
64#17989801—41no—
65C2×Dic586—22no—
66C2×Dic586—21no—
67C2×Dic586—41no—
68C2×Dic586C2311yescharacteristic
69C2×Dic586C2311yescharacteristic
70C2×D1172—22no—
71C2×D1172—21no—
72C2×D1172—21no—
73C2×D1172—41no—
74C2×D1172C2311yescharacteristic
75C2×D1172C2311yescharacteristic
76C2×C22⋊D293—21no—
77C2×C22⋊D293—21no—
78C2×C22⋊D293—21no—
79C2×C22⋊D293—21no—
80C2×C22⋊D293D411yescharacteristic
81D4×D293—22no—
82D4×D293—41no—
83D4×D293—44no—
84C22×Dic293—21no—
85C22×Dic293—21no—
86C2×C4×D293—21no—
87C2×C4×D293C2311yescharacteristic
88#17989791—22no—
89#17989791—22no—
90#17989791—22no—
91#17989791D414yescomplemented
92#17989790—22no—
93#17989790—41no—
94#17989787—22no—
95#17989787—41no—
96#17989785—22no—
97C2×#14556873C2311yescharacteristic
98C2×#14556873C2311yescharacteristic
99C4⋊Dic293—21no—
100#17989781—21no—
101#17989781D411yescharacteristic
102C22⋊Dic293—21no—
103C22⋊Dic293—21no—
104C22⋊Dic293D411yescharacteristic
105#17989779—21no—
106#17989779—21no—
107#17989779—21no—
108#17989779D411yescharacteristic
109C4×Dic293—21no—
110C4×Dic293C2311yescharacteristic
111C2×D4×C293—21no—
112C2×D4×C293—21no—
113C2×D4×C293C2311yescharacteristic
114C2×D4×C293C2311yescharacteristic
115C22×C4×C293C2311yescharacteristic
116D8×C293—22no—
117C2×C8×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—42no—
127C22×D293—42no—
128C22×D293—42no—
129C22×D293—42no—
130C22×D293—81no—
131C22×D293—81no—
132C22×D293C2×D411yescharacteristic
133Dic586—21no—
134Dic586—21no—
135Dic586—21no—
136Dic586—42no—
137Dic586C2×D412yescomplemented
138D1172—21no—
139D1172—21no—
140D1172—21no—
141D1172—42no—
142D1172—42no—
143D1172C2×D412yes—
144C22⋊D293—22no—
145C22⋊D293—42no—
146C22⋊D293—42no—
147C22⋊D293—42no—
148C22⋊D293—81no—
149C2×Dic293—21no—
150C2×Dic293—21no—
151C2×Dic293—42no—
152C2×Dic293—42no—
153C2×Dic293C2×D411yescharacteristic
154C2×Dic293C2×D411yescharacteristic
155C2×Dic293C2×D411yescharacteristic
156C4×D293—21no—
157C4×D293—21no—
158C4×D293—42no—
159#14556873—21no—
160#14556873—21no—
161C23×C293—21no—
162C23×C293—21no—
163C23×C293C2×D411yescharacteristic
164D4×C293—21no—
165D4×C293—21no—
166D4×C293—21no—
167D4×C293—21no—
168D4×C293—42no—
169D4×C293—42no—
170C2×C4×C293—21no—
171C2×C4×C293—21no—
172C2×C4×C293—21no—
173C2×C4×C293—21no—
174C2×C4×C293C2×D411yescharacteristic
175C2×C4×C293C2411yescharacteristic
176C8×C293—21no—
177C8×C293C2×D412yescomplemented
178C22×C293—21no—
179C22×C293—21no—
180C22×C293—41no—
181C22×C293—41no—
182C22×C293—41no—
183C22×C293—41no—
184C22×C293—41no—
185C22×C293—41no—
186C22×C293AGL(1,ℤ/8)12yescomplemented
187C22×C293C22×D411yescharacteristic
188C2×D293—21no—
189C2×D293—22no—
190C2×D293—21no—
191C2×D293—21no—
192C2×D293—41no—
193C2×D293—41no—
194C2×D293—41no—
195C2×D293—41no—
196C2×D293—41no—
197C2×D293—84no—
198C2×D293—84no—
199C4×C293—21no—
200C4×C293—41no—
201C4×C293—41no—
202C4×C293C22×D411yescharacteristic
203C4×C293H(5,𝔽2)11yescharacteristic
204Dic293—21no—
205Dic293—22no—
206Dic293—21no—
207Dic293—21no—
208Dic293—41no—
209C2×C293—22no—
210C2×C293—81no—
211C2×C293—81no—
212C2×C293D4211yescharacteristic
213C2×C293C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C293#57711yescharacteristic
215D293—41no—
216D293—42no—
217D293—81no—
218C293#326511yescharacteristic, complemented, Sylow
219#3265—2931nomaximal, Sylow
220C2×AGL(1,ℤ/8)—2931no—
221C22×SD16—2931no—
222#548—2931no—
223D42—2931no—
224#496—2931no—
225#468—2931no—
226#466D29311yescharacteristic, complemented
227#463—2931no—
228#460—2931no—
229#450—2931no—
230#449—2931no—
231#448—2931no—
232#447—2931no—
233#440—2931no—
234#435—2931no—
235C2×#41—2931no—
236C22×Q8—2931no—
237C22×D4—2931no—
238C22×D4—2931no—
239C22×D4—5861no—
240AGL(1,ℤ/8)—5864no—
241C2×SD16—2931no—
242C2×SD16—2931no—
243C2×SD16—2931no—
244C2×SD16—2931no—
245C2×SD16—2931no—
246C2×SD16—2931no—
247C2×SD16—5864no—
248C2×D8—2931no—
249C2×D8—2931no—
250C2×D8C2×D29311yescharacteristic, complemented
251C2×M16—2931no—
252C22×C8C2×D29311yescharacteristic, complemented
253#127—2931no—
254#124—2931no—
255#124—5861no—
256C22⋊Q8—2931no—
257C22⋊Q8—5861no—
258#121—2931no—
259#121—5861no—
260#121—5861no—
261#121C2×D29311yescharacteristic
262#121C2×D29311yescharacteristic, complemented
263#120—5861no—
264#120—5861no—
265C4×Q8—2931no—
266C4×D4—2931no—
267C4×D4—2931no—
268C4×D4—5861no—
269#107C2×D29311yescharacteristic, complemented
270C4⋊C8—2931no—
271#103—2931no—
272#103—2931no—
273#102—2931no—
274#102—2931no—
275#102C2×D29311yescharacteristic, complemented
276#102C2×D29311yescharacteristic, complemented
277C22⋊C8—2931no—
278C22⋊C8—2931no—
279#97—2931no—
280C24—5861no—
281C24—5861no—
282#41—5862no—
283#41—11721no—
284C2×Q8—2931no—
285C2×Q8—2931no—
286C2×Q8—5862no—
287C2×Q8—5861no—
288C2×Q8—11721no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—2931no—
292C2×D4—2931no—
293C2×D4—2931no—
294C2×D4—5862no—
295C2×D4—5861no—
296C2×D4—5861no—
297C2×D4—5861no—
298C2×D4—5861no—
299C2×D4—5862no—
300C2×D4—5861no—
301C2×D4—5861no—
302C2×D4—11721no—
303C2×D4—11721no—
304C2×D4—11724no—
305C2×D4C22×D29311yescharacteristic
306C2×D4C22×D29311yescharacteristic
307C22×C4—2931no—
308C22×C4—5861no—
309C22×C4—5861no—
310C22×C4—5861no—
311C22×C4C22×D29311yescharacteristic
312SD16—2934no—
313SD16—5862no—
314SD16—5862no—
315SD16—5862no—
316SD16—5862no—
317SD16—11721no—
318D8—22no—
319D8—5862no—
320D8—11721no—
321M16—5862no—
322C2×C8—22no—
323C2×C8—2931no—
324C2×C8—2931no—
325C2×C8C22×D29311yescharacteristic
326C2×C8C22×D29311yescharacteristic
327C4⋊C4—2931no—
328C4⋊C4—5861no—
329C4⋊C4—5861no—
330C4⋊C4C22×D29311yescharacteristic
331C4⋊C4C22×D29311yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—2931no—
335C22⋊C4—2931no—
336C22⋊C4—5861no—
337C22⋊C4—5861no—
338C22⋊C4—5861no—
339C22⋊C4—5861no—
340C22⋊C4—5861no—
341C42—2931no—
342C42—5861no—
343C23—21no—
344C23—21no—
345C23—2931no—
346C23—23441no—
347C23—23441no—
348C23—5861no—
349C23—5861no—
350C23—11722no—
351C23—11722no—
352C23—11722no—
353C23—11722no—
354C23D4×D29311yescharacteristic
355Q8—2932no—
356Q8—5861no—
357Q8—5861no—
358Q8—5861no—
359Q8—11722no—
360D4—21no—
361D4—21no—
362D4—21no—
363D4—21no—
364D4—42no—
365D4—42no—
366D4—2932no—
367D4—23441no—
368D4—5862no—
369D4—5861no—
370D4—5861no—
371D4—5861no—
372D4—11722no—
373D4—11722no—
374D4—11722no—
375D4—11722no—
376D4—11722no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—2931no—
382C2×C4—2931no—
383C2×C4—2931no—
384C2×C4—5861no—
385C2×C4—5861no—
386C2×C4—5861no—
387C2×C4—5861no—
388C2×C4—11722no—
389C2×C4—11722no—
390C2×C4—11722no—
391C2×C4D4×D29311yescharacteristic
392C2×C4C23×D29311yescharacteristic
393C8—21no—
394C8—5861no—
395C8—5861no—
396C8D4×D29312yescomplemented
397C22—21no—
398C22—21no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—23444no—
406C22—23444no—
407C22—5861no—
408C22—5862no—
409C22—5861no—
410C22—5861no—
411C22—11721no—
412C22—11721no—
413C22—11721no—
414C22—11721no—
415C22—11721no—
416C22#2262764812yescomplemented
417C22C2×D4×D29311yescharacteristic
418C4—21no—
419C4—41no—
420C4—41no—
421C4—5861no—
422C4—5862no—
423C4—5861no—
424C4—5861no—
425C4—11721no—
426C4C2×D4×D29311yescharacteristic
427C4#2262767011yescharacteristic
428C2—22no—
429C2—81no—
430C2—81no—
431C2—23441no—
432C2—11721no—
433C2—11722no—
434C2#3403676311yescharacteristic
435C2C2×#2262764811yescharacteristic
436C2#3403693611yescharacteristic
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#466×C293C2
socleC22×C293C22×D4
hypercenter#466D293
hypocenterC293#3265
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. C2×#22627648C2semidirect
2. C22×#17989791C2semidirect
3. #34036785C2semidirect
4. #34036763C2semidirect
5. #34036517C2semidirect
6. #34036376C2semidirect
7. #34036343C2semidirect
8. #34036328C2semidirect
9. #34036281C2semidirect
10. #34036275C2semidirect
11. #34036267C2semidirect
12. #34036262C2semidirect
13. #34036228C2semidirect
14. #34036200C2semidirect
15. #466×C293C2semidirect
16. C2×#17989801C22semidirect
17. C22×Dic586C22semidirect
25. C2×#17989791C22semidirect
26. C2×#17989791C22semidirect
27. C2×#17989790C22semidirect
28. C2×#17989790C22semidirect
30. C2×#17989787C22semidirect
31. C2×#17989785C22semidirect
33. #22627611C22semidirect
35. #22627604C22semidirect
42. #22627589C22semidirect
46. #22627553C22semidirect
47. #22627547C22semidirect
48. #22627547C22semidirect
50. #22627544C22semidirect
51. C22⋊#14556873C22semidirect
52. #22627536C22semidirect
53. #22627534C22semidirect
54. C2×D8×C293C22semidirect
55. C22×C8×C293C22semidirect
57. #121×C293C22semidirect
58. #107×C293C22semidirect
59. #102×C293C22semidirect
60. #102×C293C22semidirect
91. #17989791D4semidirect
137. Dic586C2×D4semidirect
177. C8×C293C2×D4semidirect
186. C22×C293AGL(1,ℤ/8)semidirect
218. C293#3265semidirect
226. #466D293semidirect
250. C2×D8C2×D293semidirect
252. C22×C8C2×D293semidirect
262. #121C2×D293semidirect
269. #107C2×D293semidirect
275. #102C2×D293semidirect
276. #102C2×D293semidirect
396. C8D4×D293semidirect
416. C22#22627648semidirect

Maximal subgroups

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

subgroupindex
C2×#226276482
C22×#179897912
#340367852
#340367632
#340365172
#340363762
#340363432
#340363282
#340362812
#340362752
#340362672
#340362622
#340362282
#340362002
#466×C2932
#3265293