Finite Group Databasebeta

← #49170287

Subgroups — #49170287

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×#22627652C211yesmaximal, characteristic, complemented
2C2×SD16×D293C211yesmaximal, characteristic, complemented
3#34036789C211yesmaximal, characteristic, complemented
4#34036763C211yesmaximal, characteristic, complemented
5#34036515C211yesmaximal, characteristic, complemented
6#34036366C211yesmaximal, characteristic, complemented
7#34036347C211yesmaximal, characteristic, complemented
8#34036328C211yesmaximal, characteristic, complemented
9#34036279C211yesmaximal, characteristic, complemented
10#34036272C211yesmaximal, characteristic, complemented
11#34036269C211yesmaximal, characteristic, complemented
12#34036264C211yesmaximal, characteristic, complemented
13#34036228C211yesmaximal, characteristic, complemented
14#34036198C211yesmaximal, characteristic, complemented
15#468×C293C211yesmaximal, characteristic, complemented
16C2×#17989802C2211yescharacteristic, complemented
17C2×Q8×D293C2211yescharacteristic, complemented
18C22×D1172C2211yescharacteristic
19C2×D4×D293—21no—
20C2×D4×D293C2211yescharacteristic
21#22627652—24no—
22C2×#17989791C2211yescharacteristic
23C2×#17989791C2211yescharacteristic, complemented
24C2×#17989790C2211yescharacteristic, complemented
25C2×#17989789C2211yescharacteristic
26C2×#17989789C2211yescharacteristic, complemented
27SD16×D293—24no—
28C2×D2344C2211yescharacteristic
29C2×D2344C2211yescharacteristic, complemented
30C2×#17989787C2211yescharacteristic, complemented
31C2×C8×D293C2211yescharacteristic, complemented
32#22627620C2211yescharacteristic
33#22627613C2211yescharacteristic, complemented
34#22627610—21no—
35#22627602—21no—
36#22627599C2211yescharacteristic
37#22627599C2211yescharacteristic, complemented
38#22627597—21no—
39#22627597—21no—
40#22627594—21no—
41#22627592—21no—
42#22627589C2211yescharacteristic, complemented
43#22627586—21no—
44#22627586C2211yescharacteristic
45C4×C22⋊D293C2211yescharacteristic
46#22627553C2211yescharacteristic, complemented
47#22627547C2211yescharacteristic, complemented
48#22627544C2211yescharacteristic
49#22627544C2211yescharacteristic, complemented
50#22627544C2211yescharacteristic, complemented
51C22⋊#14556873C2211yescharacteristic, complemented
52#22627536C2211yescharacteristic, complemented
53#22627534C2211yescharacteristic, complemented
54C2×SD16×C293C2211yescharacteristic, complemented
55C2×M16×C293C2211yescharacteristic, complemented
56C22⋊Q8×C293C2211yescharacteristic, complemented
57#121×C293C2211yescharacteristic
58#107×C293C2211yescharacteristic, complemented
59#103×C293C2211yescharacteristic, complemented
60#102×C293C2211yescharacteristic, complemented
61C23×D293—21no—
62C23×D293—21no—
63#17989802—22no—
64#17989802—41no—
65C2×Dic586—21no—
66C2×Dic586C2311yescharacteristic
67Q8×D293—22no—
68Q8×D293—41no—
69C2×D1172—22no—
70C2×D1172—21no—
71C2×D1172—21no—
72C2×D1172—21no—
73C2×D1172—41no—
74C2×D1172C2311yescharacteristic
75C2×D1172C2311yescharacteristic
76C2×D1172C2311yescharacteristic
77C2×C22⋊D293—21no—
78C2×C22⋊D293—21no—
79C2×C22⋊D293—21no—
80C2×C22⋊D293—21no—
81C2×C22⋊D293D411yescharacteristic
82D4×D293—22no—
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#17989791D414yescomplemented
91#17989790—22no—
92#17989789—22no—
93#17989789—41no—
94D2344—22no—
95D2344—22no—
96#17989787—41no—
97C2×#14556873C2311yescharacteristic
98C2×#14556873C2311yescharacteristic
99C8×D293—22no—
100#17989781—21no—
101#17989781D411yescharacteristic
102C22⋊Dic293D411yescharacteristic
103#17989779—21no—
104#17989779—21no—
105#17989779—21no—
106#17989779—21no—
107#17989779—21no—
108#17989779D411yescharacteristic
109C4×Dic293—21no—
110C4×Dic293C2311yescharacteristic
111C2×Q8×C293C2311yescharacteristic
112C2×D4×C293—21no—
113C2×D4×C293C2311yescharacteristic
114C22×C4×C293C2311yescharacteristic
115SD16×C293—22no—
116M16×C293—22no—
117C2×C8×C293C2311yescharacteristic
118C2×C8×C293C2311yescharacteristic
119C4⋊C4×C293—21no—
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—81no—
132C22×D293—81no—
133C22×D293C2×D411yescharacteristic
134Dic586—21no—
135Dic586—42no—
136Dic586C2×D412yescomplemented
137D1172—21no—
138D1172—21no—
139D1172—21no—
140D1172—21no—
141D1172—21no—
142D1172—42no—
143D1172—42no—
144D1172—42no—
145D1172C2×D412yes—
146C22⋊D293—22no—
147C22⋊D293—42no—
148C22⋊D293—42no—
149C22⋊D293—42no—
150C22⋊D293—81no—
151C2×Dic293—21no—
152C2×Dic293C2×D411yescharacteristic
153C2×Dic293C2×D411yescharacteristic
154C2×Dic293C2×D411yescharacteristic
155C4×D293—21no—
156C4×D293—21no—
157C4×D293—42no—
158C4×D293—42no—
159C4×D293—42no—
160#14556873—21no—
161#14556873—21no—
162C23×C293—21no—
163C23×C293C2×D411yescharacteristic
164Q8×C293—21no—
165Q8×C293—21no—
166D4×C293—21no—
167D4×C293—21no—
168D4×C293—42no—
169C2×C4×C293—21no—
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—21no—
181C22×C293—41no—
182C22×C293—41no—
183C22×C293—41no—
184C22×C293C22×D411yescharacteristic
185C2×D293—21no—
186C2×D293—21no—
187C2×D293—22no—
188C2×D293—41no—
189C2×D293—41no—
190C2×D293—41no—
191C2×D293—41no—
192C2×D293—41no—
193C2×D293—41no—
194C2×D293—41no—
195C2×D293—41no—
196C2×D293—84no—
197C2×D293—84no—
198C2×D293AGL(1,ℤ/8)12yescomplemented
199C4×C293—21no—
200C4×C293—41no—
201C4×C293—41no—
202C4×C293—41no—
203C4×C293C22×D411yescharacteristic
204C4×C293H(5,𝔽2)11yescharacteristic
205Dic293—21no—
206Dic293—22no—
207Dic293—21no—
208Dic293—21no—
209C2×C293—41no—
210C2×C293—81no—
211C2×C293D4211yescharacteristic
212C2×C293C2×AGL(1,ℤ/8)11yescharacteristic
213C2×C293#57711yescharacteristic
214D293—22no—
215D293—42no—
216D293—81no—
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#468D29311yescharacteristic, complemented
226#466—2931no—
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—5864no—
247C2×SD16C2×D29311yescharacteristic, complemented
248C2×D8—2931no—
249C2×D8—2931no—
250C2×D8—2931no—
251C2×M16C2×D29311yescharacteristic, complemented
252C22×C8—2931no—
253#127—2931no—
254#124—2931no—
255#124—5861no—
256C22⋊Q8—5861no—
257C22⋊Q8C2×D29311yescharacteristic, complemented
258#121—2931no—
259#121—2931no—
260#121—5861no—
261#121—5861no—
262#121C2×D29311yescharacteristic
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#103C2×D29311yescharacteristic, complemented
273#102—2931no—
274#102—2931no—
275#102—2931no—
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—5861no—
286C2×Q8—5862no—
287C2×Q8—11721no—
288C2×Q8C22×D29311yescharacteristic
289C2×D4—21no—
290C2×D4—2931no—
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—5861no—
300C2×D4—5862no—
301C2×D4—5861no—
302C2×D4—5861no—
303C2×D4—11721no—
304C2×D4—11721no—
305C2×D4—11724no—
306C2×D4C22×D29311yescharacteristic
307C22×C4—2931no—
308C22×C4—5861no—
309C22×C4—5861no—
310C22×C4—5861no—
311C22×C4C22×D29311yescharacteristic
312SD16—22no—
313SD16—2934no—
314SD16—5862no—
315SD16—5862no—
316SD16—5862no—
317SD16—11721no—
318D8—5862no—
319D8—5862no—
320D8—11721no—
321M16—22no—
322C2×C8—2931no—
323C2×C8—2931no—
324C2×C8—5862no—
325C2×C8C22×D29311yescharacteristic
326C2×C8C22×D29311yescharacteristic
327C4⋊C4—21no—
328C4⋊C4—2931no—
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—2931no—
345C23—23441no—
346C23—23441no—
347C23—5861no—
348C23—5861no—
349C23—5861no—
350C23—11722no—
351C23—11722no—
352C23—11722no—
353C23—11722no—
354C23D4×D29311yescharacteristic
355Q8—21no—
356Q8—21no—
357Q8—2932no—
358Q8—5861no—
359Q8—11722no—
360D4—21no—
361D4—21no—
362D4—42no—
363D4—2932no—
364D4—23441no—
365D4—5862no—
366D4—5861no—
367D4—5861no—
368D4—5861no—
369D4—5861no—
370D4—5861no—
371D4—11722no—
372D4—11722no—
373D4—11722no—
374D4—11722no—
375D4—11722no—
376D4—11722no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—2931no—
383C2×C4—2931no—
384C2×C4—2931no—
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—21no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—2932no—
404C22—23444no—
405C22—23444no—
406C22—5861no—
407C22—5861no—
408C22—5862no—
409C22—11721no—
410C22—11721no—
411C22—11721no—
412C22—11721no—
413C22—11721no—
414C22—11721no—
415C22—11721no—
416C22—11721no—
417C22C2×D4×D29311yescharacteristic
418C4—21no—
419C4—41no—
420C4—41no—
421C4—41no—
422C4—5861no—
423C4—5862no—
424C4—5861no—
425C4—5861no—
426C4C2×D4×D29311yescharacteristic
427C4#2262767011yescharacteristic
428C2—41no—
429C2—81no—
430C2—23441no—
431C2—23441no—
432C2—5862no—
433C2—11722no—
434C2#3403676311yescharacteristic
435C2AGL(1,ℤ/8)×D29311yescharacteristic
436C2#3403693811yescharacteristic
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#468×C293C2
socleC22×C293C22×D4
hypercenter#468D293
hypocenterC293#3265
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×#22627652C2semidirect
2. C2×SD16×D293C2semidirect
3. #34036789C2semidirect
4. #34036763C2semidirect
5. #34036515C2semidirect
6. #34036366C2semidirect
7. #34036347C2semidirect
8. #34036328C2semidirect
9. #34036279C2semidirect
10. #34036272C2semidirect
11. #34036269C2semidirect
12. #34036264C2semidirect
13. #34036228C2semidirect
14. #34036198C2semidirect
15. #468×C293C2semidirect
16. C2×#17989802C22semidirect
17. C2×Q8×D293C22semidirect
23. C2×#17989791C22semidirect
24. C2×#17989790C22semidirect
26. C2×#17989789C22semidirect
29. C2×D2344C22semidirect
30. C2×#17989787C22semidirect
31. C2×C8×D293C22semidirect
33. #22627613C22semidirect
37. #22627599C22semidirect
42. #22627589C22semidirect
46. #22627553C22semidirect
47. #22627547C22semidirect
49. #22627544C22semidirect
50. #22627544C22semidirect
51. C22⋊#14556873C22semidirect
52. #22627536C22semidirect
53. #22627534C22semidirect
54. C2×SD16×C293C22semidirect
55. C2×M16×C293C22semidirect
56. C22⋊Q8×C293C22semidirect
58. #107×C293C22semidirect
59. #103×C293C22semidirect
60. #102×C293C22semidirect
90. #17989791D4semidirect
136. Dic586C2×D4semidirect
177. C8×C293C2×D4semidirect
198. C2×D293AGL(1,ℤ/8)semidirect
218. C293#3265semidirect
225. #468D293semidirect
247. C2×SD16C2×D293semidirect
251. C2×M16C2×D293semidirect
257. C22⋊Q8C2×D293semidirect
269. #107C2×D293semidirect
272. #103C2×D293semidirect
276. #102C2×D293semidirect
396. 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×SD16×D2932
#340367892
#340367632
#340365152
#340363662
#340363472
#340363282
#340362792
#340362722
#340362692
#340362642
#340362282
#340361982
#468×C2932
#3265293