Finite Group Databasebeta

← #49170274

Subgroups — #49170274

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×SD16×D293C211yesmaximal, characteristic, complemented
2#34036786C211yesmaximal, characteristic, complemented
3D4×C22⋊D293C211yesmaximal, characteristic, complemented
4#34036517C211yesmaximal, characteristic, complemented
5#34036375C211yesmaximal, characteristic, complemented
6#34036362C211yesmaximal, characteristic, complemented
7#34036342C211yesmaximal, characteristic, complemented
8#34036327C211yesmaximal, characteristic, complemented
9#34036280C211yesmaximal, characteristic, complemented
10#34036274C211yesmaximal, characteristic, complemented
11#34036266C211yesmaximal, characteristic, complemented
12#34036261C211yesmaximal, characteristic, complemented
13#34036230C211yesmaximal, characteristic, complemented
14#34036198C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C293C211yesmaximal, characteristic, complemented
16C2×Q8×D293C2211yescharacteristic, complemented
17C22×C22⋊D293—21no—
18C2×D4×D293C2211yescharacteristic
19C2×#17989791C2211yescharacteristic, complemented
20C2×#17989790C2211yescharacteristic
21C2×#17989790C2211yescharacteristic, complemented
22C2×#17989789C2211yescharacteristic, complemented
23SD16×D293—24no—
24C2×#17989787C2211yescharacteristic, complemented
25C2×C8×D293C2211yescharacteristic, complemented
26#22627620C2211yescharacteristic
27#22627611—21no—
28#22627611C2211yescharacteristic, complemented
29#22627603—21no—
30#22627603C2211yescharacteristic, complemented
31#22627598C2211yescharacteristic
32#22627596—21no—
33#22627595—21no—
34#22627595C2211yescharacteristic
35#22627595C2211yescharacteristic, complemented
36#22627593—21no—
37#22627592—21no—
38Q8×Dic293C2211yescharacteristic, complemented
39D4×Dic293C2211yescharacteristic
40C4×C22⋊D293—21no—
41C4×C22⋊D293C2211yescharacteristic
42#22627558C2211yescharacteristic, complemented
43#22627553C2211yescharacteristic, complemented
44#22627549C2211yescharacteristic, complemented
45#22627548C2211yescharacteristic, complemented
46#22627546C2211yescharacteristic, complemented
47#22627545C2211yescharacteristic
48#22627545C2211yescharacteristic, complemented
49#22627545C2211yescharacteristic, complemented
50C22⋊#14556873C2211yescharacteristic, complemented
51#22627536C2211yescharacteristic, complemented
52#22627534C2211yescharacteristic, complemented
53C2×#41×C293C2211yescharacteristic, complemented
54C22×D4×C293C2211yescharacteristic
55AGL(1,ℤ/8)×C293—24no—
56C2×SD16×C293C2211yescharacteristic
57C2×SD16×C293C2211yescharacteristic, complemented
58C2×D8×C293C2211yescharacteristic
59C2×D8×C293C2211yescharacteristic, complemented
60C2×M16×C293C2211yescharacteristic, complemented
61C23×D293—21no—
62C2×Dic586—21no—
63C2×Dic586C2311yescharacteristic
64Q8×D293—22no—
65Q8×D293—41no—
66C2×D1172C2311yescharacteristic
67C2×C22⋊D293—21no—
68C2×C22⋊D293—21no—
69C2×C22⋊D293—21no—
70C2×C22⋊D293—21no—
71C2×C22⋊D293—21no—
72C2×C22⋊D293—21no—
73C2×C22⋊D293—44no—
74C2×C22⋊D293D411yescharacteristic
75D4×D293—22no—
76D4×D293—41no—
77C22×Dic293—21no—
78C2×C4×D293—21no—
79C2×C4×D293C2311yescharacteristic
80#17989791—22no—
81#17989790—22no—
82#17989790—41no—
83#17989789—22no—
84#17989787—41no—
85C2×#14556873C2311yescharacteristic
86C2×#14556873C2311yescharacteristic
87C8×D293—22no—
88C4⋊Dic293—21no—
89C4⋊Dic293C2311yescharacteristic
90C4⋊Dic293C2311yescharacteristic
91#17989781—21no—
92#17989781D411yescharacteristic
93C22⋊Dic293—21no—
94C22⋊Dic293—21no—
95C22⋊Dic293—21no—
96C22⋊Dic293—21no—
97C22⋊Dic293—21no—
98C22⋊Dic293—21no—
99C22⋊Dic293D411yescharacteristic
100#17989779—21no—
101#17989779D411yescharacteristic
102C4×Dic293—21no—
103C4×Dic293C2311yescharacteristic
104C24×C293—21no—
105#41×C293—22no—
106#41×C293—41no—
107C2×Q8×C293C2311yescharacteristic
108C2×D4×C293—22no—
109C2×D4×C293—21no—
110C2×D4×C293—21no—
111C2×D4×C293—41no—
112C2×D4×C293C2311yescharacteristic
113C2×D4×C293C2311yescharacteristic
114C2×D4×C293C2311yescharacteristic
115C22×C4×C293—21no—
116C22×C4×C293C2311yescharacteristic
117SD16×C293—22no—
118SD16×C293D414yescomplemented
119D8×C293—22no—
120D8×C293—22no—
121M16×C293—22no—
122C2×C8×C293C2311yescharacteristic
123C2×C8×C293C2311yescharacteristic
124C22×D293—21no—
125C22×D293—42no—
126C22×D293—42no—
127C22×D293—81no—
128C22×D293C2×D411yescharacteristic
129Dic586—21no—
130Dic586—21no—
131Dic586—42no—
132D1172—21no—
133D1172—21no—
134C22⋊D293—22no—
135C22⋊D293—42no—
136C22⋊D293—42no—
137C22⋊D293—42no—
138C22⋊D293—42no—
139C22⋊D293—42no—
140C22⋊D293—81no—
141C2×Dic293—21no—
142C2×Dic293—21no—
143C2×Dic293—21no—
144C2×Dic293—42no—
145C2×Dic293C2×D411yescharacteristic
146C2×Dic293C2×D411yescharacteristic
147C2×Dic293C2×D411yescharacteristic
148C4×D293—21no—
149C4×D293—21no—
150C4×D293—42no—
151#14556873—21no—
152#14556873—21no—
153C23×C293—21no—
154C23×C293—21no—
155C23×C293—21no—
156C23×C293—42no—
157C23×C293—42no—
158C23×C293—81no—
159C23×C293C2×D411yescharacteristic
160Q8×C293—21no—
161Q8×C293C2×D412yescomplemented
162D4×C293—21no—
163D4×C293—21no—
164D4×C293—21no—
165D4×C293—21no—
166D4×C293—21no—
167D4×C293—42no—
168D4×C293—42no—
169D4×C293C2×D412yes—
170C2×C4×C293—21no—
171C2×C4×C293—21no—
172C2×C4×C293—21no—
173C2×C4×C293—42no—
174C2×C4×C293C2×D411yescharacteristic
175C2×C4×C293C2411yescharacteristic
176C8×C293—21no—
177C8×C293C2×D412yescomplemented
178C22×C293—21no—
179C22×C293—22no—
180C22×C293—21no—
181C22×C293—21no—
182C22×C293—41no—
183C22×C293—41no—
184C22×C293—41no—
185C22×C293—41no—
186C22×C293—41no—
187C22×C293—41no—
188C22×C293—41no—
189C22×C293—41no—
190C22×C293—84no—
191C22×C293C22×D411yescharacteristic
192C2×D293—21no—
193C2×D293—21no—
194C2×D293—41no—
195C2×D293—41no—
196C2×D293—41no—
197C2×D293—84no—
198C2×D293AGL(1,ℤ/8)12yescomplemented
199C4×C293—21no—
200C4×C293—22no—
201C4×C293C22×D411yescharacteristic
202C4×C293H(5,𝔽2)11yescharacteristic
203Dic293—21no—
204Dic293—21no—
205Dic293—21no—
206Dic293—41no—
207Dic293—41no—
208Dic293—41no—
209C2×C293—42no—
210C2×C293—41no—
211C2×C293—81no—
212C2×C293—81no—
213C2×C293D4211yescharacteristic
214C2×C293C2×AGL(1,ℤ/8)11yescharacteristic
215C2×C293#57711yescharacteristic
216D293—22no—
217D293—81no—
218C293#326511yescharacteristic, complemented, Sylow
219#3265—2931nomaximal, Sylow
220C2×AGL(1,ℤ/8)D29311yescharacteristic, complemented
221C22×SD16—2931no—
222#548—2931no—
223D42—2931no—
224#496—2931no—
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×#41C2×D29311yescharacteristic, complemented
236C22×Q8—2931no—
237C22×D4—2931no—
238C22×D4—5861no—
239C22×D4C2×D29311yescharacteristic
240AGL(1,ℤ/8)—24no—
241C2×SD16—2931no—
242C2×SD16—2931no—
243C2×SD16—2931no—
244C2×SD16—2931no—
245C2×SD16—5864no—
246C2×SD16C2×D29311yescharacteristic
247C2×SD16C2×D29311yescharacteristic, complemented
248C2×D8—2931no—
249C2×D8C2×D29311yescharacteristic
250C2×D8C2×D29311yescharacteristic, complemented
251C2×M16C2×D29311yescharacteristic, complemented
252C22×C8—2931no—
253#127—2931no—
254#124—2931no—
255#124—5861no—
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#97—2931no—
280C24—21no—
281C24—5861no—
282#41—22no—
283#41—41no—
284C2×Q8—2931no—
285C2×Q8—5861no—
286C2×Q8—5862no—
287C2×Q8—11721no—
288C2×Q8C22×D29311yescharacteristic
289C2×D4—22no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—41no—
293C2×D4—2931no—
294C2×D4—2931no—
295C2×D4—5861no—
296C2×D4—5861no—
297C2×D4—5861no—
298C2×D4—5862no—
299C2×D4—5861no—
300C2×D4—5861no—
301C2×D4—5861no—
302C2×D4—11721no—
303C2×D4—11724no—
304C2×D4C22×D29311yescharacteristic
305C2×D4C22×D29311yescharacteristic
306C2×D4C22×D29311yescharacteristic
307C22×C4—21no—
308C22×C4—2931no—
309C22×C4—5861no—
310C22×C4—5861no—
311C22×C4C22×D29311yescharacteristic
312SD16—22no—
313SD16—5862no—
314SD16—5862no—
315SD16—5862no—
316SD16—11721no—
317SD16C22⋊D29314yescomplemented
318D8—22no—
319D8—22no—
320D8—11721no—
321M16—22no—
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—2931no—
333C22⋊C4—2931no—
334C22⋊C4—5861no—
335C22⋊C4—5861no—
336C22⋊C4—5861no—
337C22⋊C4—5861no—
338C22⋊C4—5861no—
339C22⋊C4—5861no—
340C22⋊C4—5861no—
341C42—2931no—
342C42—5861no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—42no—
347C23—42no—
348C23—81no—
349C23—2931no—
350C23—23441no—
351C23—5861no—
352C23—11722no—
353C23—11722no—
354C23D4×D29311yescharacteristic
355Q8—21no—
356Q8—5861no—
357Q8—5861no—
358Q8—11722no—
359Q8C2×C22⋊D29312yescomplemented
360D4—21no—
361D4—21no—
362D4—21no—
363D4—21no—
364D4—21no—
365D4—42no—
366D4—42no—
367D4—23441no—
368D4—5862no—
369D4—5861no—
370D4—5861no—
371D4—11722no—
372D4—11722no—
373D4—11722no—
374D4—11722no—
375D4—11722no—
376D4C2×C22⋊D29312yes—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—42no—
381C2×C4—2931no—
382C2×C4—2931no—
383C2×C4—2931no—
384C2×C4—5861no—
385C2×C4—5861no—
386C2×C4—5861no—
387C2×C4—5861no—
388C2×C4—5861no—
389C2×C4—11722no—
390C2×C4—11722no—
391C2×C4D4×D29311yescharacteristic
392C2×C4C23×D29311yescharacteristic
393C8—21no—
394C8—5861no—
395C8—5861no—
396C8C2×C22⋊D29312yescomplemented
397C22—21no—
398C22—22no—
399C22—21no—
400C22—21no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—84no—
410C22—2932no—
411C22—23444no—
412C22—5861no—
413C22—5861no—
414C22—11721no—
415C22—11721no—
416C22—11721no—
417C22C2×D4×D29311yescharacteristic
418C4—21no—
419C4—22no—
420C4—5861no—
421C4—5861no—
422C4—5861no—
423C4—11721no—
424C4—11721no—
425C4—11721no—
426C4C22×C22⋊D29311yescharacteristic
427C4#2262766911yescharacteristic
428C2—41no—
429C2—42no—
430C2—81no—
431C2—81no—
432C2—23441no—
433C2—5862no—
434C2D4×C22⋊D29311yescharacteristic
435C2AGL(1,ℤ/8)×D29311yescharacteristic
436C2#3403693911yescharacteristic
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
FittingC2×AGL(1,ℤ/8)×C293C2
socleC22×C293C22×D4
hypercenterC2×AGL(1,ℤ/8)D293
hypocenterC293#3265
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D293C2semidirect
2. #34036786C2semidirect
3. D4×C22⋊D293C2semidirect
4. #34036517C2semidirect
5. #34036375C2semidirect
6. #34036362C2semidirect
7. #34036342C2semidirect
8. #34036327C2semidirect
9. #34036280C2semidirect
10. #34036274C2semidirect
11. #34036266C2semidirect
12. #34036261C2semidirect
13. #34036230C2semidirect
14. #34036198C2semidirect
15. C2×AGL(1,ℤ/8)×C293C2semidirect
16. C2×Q8×D293C22semidirect
19. C2×#17989791C22semidirect
21. C2×#17989790C22semidirect
22. C2×#17989789C22semidirect
24. C2×#17989787C22semidirect
25. C2×C8×D293C22semidirect
28. #22627611C22semidirect
30. #22627603C22semidirect
35. #22627595C22semidirect
38. Q8×Dic293C22semidirect
42. #22627558C22semidirect
43. #22627553C22semidirect
44. #22627549C22semidirect
45. #22627548C22semidirect
46. #22627546C22semidirect
48. #22627545C22semidirect
49. #22627545C22semidirect
50. C22⋊#14556873C22semidirect
51. #22627536C22semidirect
52. #22627534C22semidirect
53. C2×#41×C293C22semidirect
57. C2×SD16×C293C22semidirect
59. C2×D8×C293C22semidirect
60. C2×M16×C293C22semidirect
118. SD16×C293D4semidirect
161. Q8×C293C2×D4semidirect
177. C8×C293C2×D4semidirect
198. C2×D293AGL(1,ℤ/8)semidirect
218. C293#3265semidirect
220. C2×AGL(1,ℤ/8)D293semidirect
235. C2×#41C2×D293semidirect
247. C2×SD16C2×D293semidirect
250. C2×D8C2×D293semidirect
251. C2×M16C2×D293semidirect
317. SD16C22⋊D293semidirect
359. Q8C2×C22⋊D293semidirect
396. C8C2×C22⋊D293semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D2932
#340367862
D4×C22⋊D2932
#340365172
#340363752
#340363622
#340363422
#340363272
#340362802
#340362742
#340362662
#340362612
#340362302
#340361982
C2×AGL(1,ℤ/8)×C2932
#3265293