Finite Group Databasebeta

← #49170279

Subgroups — #49170279

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
2C2×SD16×D293C211yesmaximal, characteristic, complemented
3#34036787C211yesmaximal, characteristic, complemented
4#34036764C211yesmaximal, characteristic, complemented
5#34036372C211yesmaximal, characteristic, complemented
6#34036362C211yesmaximal, characteristic, complemented
7#34036341C211yesmaximal, characteristic, complemented
8#34036326C211yesmaximal, characteristic, complemented
9#34036278C211yesmaximal, characteristic, complemented
10#34036271C211yesmaximal, characteristic, complemented
11#34036266C211yesmaximal, characteristic, complemented
12#34036261C211yesmaximal, characteristic, complemented
13#34036231C211yesmaximal, characteristic, complemented
14#34036202C211yesmaximal, characteristic, complemented
15#496×C293C211yesmaximal, characteristic, complemented
16C2×#17989801C2211yescharacteristic, complemented
17C2×Q8×D293C2211yescharacteristic, complemented
18C2×D4×D293—21no—
19C2×D4×D293C2211yescharacteristic
20C2×D4×D293C2211yescharacteristic
21#22627648—24no—
22C2×#17989791C2211yescharacteristic
23C2×#17989790C2211yescharacteristic, complemented
24C2×#17989790C2211yescharacteristic, complemented
25C2×#17989789C2211yescharacteristic
26SD16×D293—24no—
27C2×#17989787C2211yescharacteristic
28C2×#17989785C2211yescharacteristic, complemented
29C2×C8×D293C2211yescharacteristic, complemented
30#22627610—21no—
31#22627601—21no—
32#22627600C2211yescharacteristic, complemented
33#22627597—21no—
34#22627595—21no—
35#22627595C2211yescharacteristic
36#22627595C2211yescharacteristic
37#22627595C2211yescharacteristic, complemented
38#22627592—21no—
39#22627592—21no—
40C4×Dic586C2211yescharacteristic, complemented
41C4×D1172C2211yescharacteristic
42C4×D1172C2211yescharacteristic
43#22627584—21no—
44#22627558C2211yescharacteristic, complemented
45C4⋊#14556873C2211yescharacteristic, complemented
46#22627549C2211yescharacteristic, complemented
47#22627548C2211yescharacteristic, complemented
48#22627546C2211yescharacteristic
49#22627545C2211yescharacteristic, complemented
50#22627545C2211yescharacteristic, complemented
51#22627545C2211yescharacteristic, complemented
52#22627536C2211yescharacteristic, complemented
53#22627536C2211yescharacteristic, complemented
54C2×SD16×C293C2211yescharacteristic, complemented
55C2×SD16×C293C2211yescharacteristic, complemented
56C2×D8×C293C2211yescharacteristic, complemented
57C2×D8×C293C2211yescharacteristic, complemented
58#127×C293C2211yescharacteristic
59#124×C293C2211yescharacteristic, complemented
60#97×C293C2211yescharacteristic, complemented
61C23×D293—21no—
62C23×D293—21no—
63#17989801—22no—
64#17989801—41no—
65C2×Dic586—21no—
66C2×Dic586C2311yescharacteristic
67Q8×D293—22no—
68Q8×D293—41no—
69C2×D1172D411yescharacteristic
70C2×D1172C2311yescharacteristic
71C2×C22⋊D293—21no—
72C2×C22⋊D293—21no—
73C2×C22⋊D293—21no—
74C2×C22⋊D293—21no—
75C2×C22⋊D293—21no—
76C2×C22⋊D293—21no—
77D4×D293—22no—
78D4×D293—22no—
79D4×D293—41no—
80D4×D293—41no—
81D4×D293—44no—
82C22×Dic293—21no—
83C2×C4×D293—21no—
84C2×C4×D293—21no—
85C2×C4×D293C2311yescharacteristic
86C2×C4×D293C2311yescharacteristic
87#17989791D414yescomplemented
88#17989790—22no—
89#17989790—22no—
90#17989789—22no—
91#17989787—22no—
92#17989785—22no—
93C2×#14556873C2311yescharacteristic
94C2×#14556873C2311yescharacteristic
95C8×D293—22no—
96C4⋊Dic293D411yescharacteristic
97C4⋊Dic293C2311yescharacteristic
98C4⋊Dic293C2311yescharacteristic
99#17989781—21no—
100#17989781—21no—
101C22⋊Dic293—21no—
102C22⋊Dic293—21no—
103C22⋊Dic293—21no—
104#17989779—21no—
105#17989779—21no—
106#17989779D411yescharacteristic
107#17989779D411yescharacteristic
108C4×Dic293—21no—
109C2×Q8×C293C2311yescharacteristic
110C2×D4×C293—21no—
111C2×D4×C293—21no—
112C2×D4×C293C2311yescharacteristic
113C2×D4×C293C2311yescharacteristic
114C2×D4×C293C2311yescharacteristic
115SD16×C293—22no—
116SD16×C293—41no—
117D8×C293—22no—
118D8×C293—41no—
119C2×C8×C293C2311yescharacteristic
120C2×C8×C293C2311yescharacteristic
121C22⋊C4×C293—21no—
122C22⋊C4×C293—21no—
123C42×C293C2311yescharacteristic
124C22×D293—21no—
125C22×D293—42no—
126C22×D293—42no—
127C22×D293—42no—
128C22×D293—42no—
129C22×D293—81no—
130C22×D293—81no—
131C22×D293C2×D411yescharacteristic
132C22×D293C2×D411yescharacteristic
133Dic586—21no—
134Dic586—42no—
135Dic586C2×D412yescomplemented
136D1172—22no—
137D1172—21no—
138D1172C2×D412yes—
139C22⋊D293—42no—
140C22⋊D293—42no—
141C22⋊D293—42no—
142C22⋊D293—42no—
143C22⋊D293—42no—
144C22⋊D293—42no—
145C2×Dic293—21no—
146C2×Dic293—21no—
147C2×Dic293—21no—
148C2×Dic293—42no—
149C2×Dic293C2×D411yescharacteristic
150C2×Dic293C2×D411yescharacteristic
151C4×D293—21no—
152C4×D293—21no—
153C4×D293—21no—
154C4×D293—21no—
155C4×D293—42no—
156C4×D293—42no—
157#14556873—21no—
158#14556873—21no—
159C23×C293—21no—
160C23×C293—21no—
161C23×C293—21no—
162Q8×C293—21no—
163Q8×C293—21no—
164D4×C293—21no—
165D4×C293—21no—
166D4×C293—21no—
167D4×C293—21no—
168D4×C293—21no—
169D4×C293—21no—
170D4×C293—42no—
171D4×C293—81no—
172C2×C4×C293—21no—
173C2×C4×C293C2×D411yescharacteristic
174C2×C4×C293C2×D411yescharacteristic
175C2×C4×C293C2411yescharacteristic
176C8×C293—21no—
177C8×C293C2×D412yescomplemented
178C22×C293—41no—
179C22×C293—41no—
180C22×C293—41no—
181C22×C293—41no—
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—84no—
197C2×D293—84no—
198C2×D293AGL(1,ℤ/8)12yescomplemented
199C4×C293—21no—
200C4×C293—21no—
201C4×C293—41no—
202C4×C293C22×D411yescharacteristic
203C4×C293H(5,𝔽2)11yescharacteristic
204Dic293—21no—
205Dic293—21no—
206Dic293—22no—
207Dic293—41no—
208Dic293—41no—
209C2×C293—81no—
210C2×C293—81no—
211C2×C293—81no—
212C2×C293D4211yescharacteristic
213C2×C293C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C293#57711yescharacteristic
215D293—22no—
216D293—41no—
217D293—42no—
218C293#326511yescharacteristic, complemented, Sylow
219#3265—2931nomaximal, Sylow
220C2×AGL(1,ℤ/8)—2931no—
221C22×SD16—2931no—
222#548—2931no—
223D42—2931no—
224#496D29311yescharacteristic, complemented
225#468—2931no—
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—5864no—
246C2×SD16C2×D29311yescharacteristic, complemented
247C2×SD16C2×D29311yescharacteristic, complemented
248C2×D8—2931no—
249C2×D8C2×D29311yescharacteristic, complemented
250C2×D8C2×D29311yescharacteristic, complemented
251C2×M16—2931no—
252C22×C8—2931no—
253#127C2×D29311yescharacteristic
254#124—5861no—
255#124C2×D29311yescharacteristic, complemented
256C22⋊Q8—2931no—
257C22⋊Q8—5861no—
258#121—2931no—
259#121—2931no—
260#121—2931no—
261#121—5861no—
262#121—5861no—
263#120—5861no—
264#120—5861no—
265C4×Q8—2931no—
266C4×D4—2931no—
267C4×D4—2931no—
268C4×D4—5861no—
269#107—2931no—
270C4⋊C8—2931no—
271#103—2931no—
272#103—2931no—
273#102—2931no—
274#102—2931no—
275#102—2931no—
276#102—2931no—
277C22⋊C8—2931no—
278C22⋊C8—2931no—
279#97C2×D29311yescharacteristic, complemented
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—21no—
291C2×D4—2931no—
292C2×D4—2931no—
293C2×D4—5861no—
294C2×D4—5861no—
295C2×D4—5862no—
296C2×D4—5862no—
297C2×D4—5861no—
298C2×D4—5861no—
299C2×D4—5861no—
300C2×D4—5861no—
301C2×D4—11721no—
302C2×D4—11721no—
303C2×D4—11724no—
304C2×D4C22×D29311yescharacteristic
305C2×D4C22×D29311yescharacteristic
306C2×D4C22×D29311yescharacteristic
307C22×C4—2931no—
308C22×C4—2931no—
309C22×C4—5861no—
310C22×C4—5861no—
311C22×C4—5861no—
312SD16—22no—
313SD16—41no—
314SD16—2934no—
315SD16—5862no—
316SD16—5862no—
317SD16—5862no—
318D8—22no—
319D8—41no—
320D8—5862no—
321M16—5862no—
322C2×C8—2931no—
323C2×C8—2931no—
324C2×C8—5862no—
325C2×C8C22×D29311yescharacteristic
326C2×C8C22×D29311yescharacteristic
327C4⋊C4—2931no—
328C4⋊C4—2931no—
329C4⋊C4—2931no—
330C4⋊C4—5861no—
331C4⋊C4—5861no—
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—5861no—
342C42C22×D29311yescharacteristic
343C23—21no—
344C23—21no—
345C23—21no—
346C23—2931no—
347C23—2931no—
348C23—23441no—
349C23—23441no—
350C23—5861no—
351C23—11722no—
352C23—11722no—
353C23—11722no—
354C23—11722no—
355Q8—21no—
356Q8—21no—
357Q8—2932no—
358Q8—5861no—
359Q8—11722no—
360D4—21no—
361D4—21no—
362D4—21no—
363D4—21no—
364D4—21no—
365D4—21no—
366D4—42no—
367D4—81no—
368D4—2932no—
369D4—5862no—
370D4—5861no—
371D4—11722no—
372D4—11722no—
373D4—11722no—
374D4—11722no—
375D4—11722no—
376D4—11722no—
377C2×C4—21no—
378C2×C4—2931no—
379C2×C4—2931no—
380C2×C4—5861no—
381C2×C4—5861no—
382C2×C4—5861no—
383C2×C4—5861no—
384C2×C4—5861no—
385C2×C4—5861no—
386C2×C4—5861no—
387C2×C4—11722no—
388C2×C4—11722no—
389C2×C4—11722no—
390C2×C4D4×D29311yescharacteristic
391C2×C4D4×D29311yescharacteristic
392C2×C4C23×D29311yescharacteristic
393C8—21no—
394C8—5861no—
395C8—5861no—
396C8D4×D29312yescomplemented
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—2932no—
407C22—23444no—
408C22—23444no—
409C22—5861no—
410C22—5861no—
411C22—5861no—
412C22—5862no—
413C22—5861no—
414C22—5861no—
415C22—11721no—
416C22—11721no—
417C22C2×D4×D29311yescharacteristic
418C4—21no—
419C4—21no—
420C4—41no—
421C4—5861no—
422C4—5861no—
423C4—5862no—
424C4—11721no—
425C4—11721no—
426C4C2×D4×D29311yescharacteristic
427C4#2262766911yescharacteristic
428C2—81no—
429C2—81no—
430C2—81no—
431C2—5862no—
432C2—11721no—
433C2—11722no—
434C2#3403676411yescharacteristic
435C2AGL(1,ℤ/8)×D29311yescharacteristic
436C2#3403693911yescharacteristic
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#496×C293C2
socleC22×C293C22×D4
hypercenter#496D293
hypocenterC293#3265
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#22627648C2semidirect
2. C2×SD16×D293C2semidirect
3. #34036787C2semidirect
4. #34036764C2semidirect
5. #34036372C2semidirect
6. #34036362C2semidirect
7. #34036341C2semidirect
8. #34036326C2semidirect
9. #34036278C2semidirect
10. #34036271C2semidirect
11. #34036266C2semidirect
12. #34036261C2semidirect
13. #34036231C2semidirect
14. #34036202C2semidirect
15. #496×C293C2semidirect
16. C2×#17989801C22semidirect
17. C2×Q8×D293C22semidirect
23. C2×#17989790C22semidirect
24. C2×#17989790C22semidirect
28. C2×#17989785C22semidirect
29. C2×C8×D293C22semidirect
32. #22627600C22semidirect
37. #22627595C22semidirect
40. C4×Dic586C22semidirect
44. #22627558C22semidirect
45. C4⋊#14556873C22semidirect
46. #22627549C22semidirect
47. #22627548C22semidirect
49. #22627545C22semidirect
50. #22627545C22semidirect
51. #22627545C22semidirect
52. #22627536C22semidirect
53. #22627536C22semidirect
54. C2×SD16×C293C22semidirect
55. C2×SD16×C293C22semidirect
56. C2×D8×C293C22semidirect
57. C2×D8×C293C22semidirect
59. #124×C293C22semidirect
60. #97×C293C22semidirect
87. #17989791D4semidirect
135. Dic586C2×D4semidirect
177. C8×C293C2×D4semidirect
198. C2×D293AGL(1,ℤ/8)semidirect
218. C293#3265semidirect
224. #496D293semidirect
246. C2×SD16C2×D293semidirect
247. C2×SD16C2×D293semidirect
249. C2×D8C2×D293semidirect
250. C2×D8C2×D293semidirect
255. #124C2×D293semidirect
279. #97C2×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×#226276482
C2×SD16×D2932
#340367872
#340367642
#340363722
#340363622
#340363412
#340363262
#340362782
#340362712
#340362662
#340362612
#340362312
#340362022
#496×C2932
#3265293