Finite Group Databasebeta

← #49170278

Subgroups — #49170278

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×#22627650C211yesmaximal, characteristic, complemented
2C2×SD16×D293C211yesmaximal, characteristic, complemented
3#34036789C211yesmaximal, characteristic, complemented
4D4×C22⋊D293C211yesmaximal, characteristic, complemented
5#34036515C211yesmaximal, characteristic, complemented
6#34036377C211yesmaximal, characteristic, complemented
7#34036364C211yesmaximal, characteristic, complemented
8#34036346C211yesmaximal, characteristic, complemented
9#34036328C211yesmaximal, characteristic, complemented
10#34036276C211yesmaximal, characteristic, complemented
11#34036266C211yesmaximal, characteristic, complemented
12#34036264C211yesmaximal, characteristic, complemented
13#34036226C211yesmaximal, characteristic, complemented
14#34036203C211yesmaximal, characteristic, complemented
15#450×C293C211yesmaximal, characteristic, complemented
16C2×#17989802C2211yescharacteristic, complemented
17C2×Q8×D293C2211yescharacteristic, complemented
18C22×C22⋊D293—21no—
19C2×D4×D293C2211yescharacteristic
20#22627650—24no—
21C2×#17989791C2211yescharacteristic, complemented
22C2×#17989790C2211yescharacteristic
23C2×#17989790C2211yescharacteristic, complemented
24C2×#17989789C2211yescharacteristic, complemented
25SD16×D293—24no—
26C2×D2344C2211yescharacteristic, complemented
27C2×#17989787C2211yescharacteristic
28C2×#17989787C2211yescharacteristic, complemented
29C2×#17989786C2211yescharacteristic, complemented
30C2×C8×D293C2211yescharacteristic, complemented
31#22627620C2211yescharacteristic
32#22627613C2211yescharacteristic, complemented
33#22627610—21no—
34#22627602—21no—
35#22627599C2211yescharacteristic, complemented
36#22627598C2211yescharacteristic
37#22627597—21no—
38#22627596—21no—
39#22627595C2211yescharacteristic
40#22627593—21no—
41#22627592—21no—
42#22627589C2211yescharacteristic, complemented
43D4×Dic293C2211yescharacteristic
44#22627586—21no—
45C4×C22⋊D293C2211yescharacteristic
46#22627557C2211yescharacteristic, complemented
47#22627553C2211yescharacteristic, complemented
48#22627549C2211yescharacteristic, complemented
49#22627548C2211yescharacteristic, complemented
50#22627546C2211yescharacteristic, complemented
51#22627544C2211yescharacteristic, complemented
52#22627536C2211yescharacteristic, complemented
53#22627534C2211yescharacteristic, complemented
54C22×D4×C293C2211yescharacteristic
55C2×SD16×C293C2211yescharacteristic
56C2×SD16×C293C2211yescharacteristic, complemented
57C22⋊Q8×C293C2211yescharacteristic, complemented
58#102×C293C2211yescharacteristic
59#102×C293C2211yescharacteristic, complemented
60C22⋊C8×C293C2211yescharacteristic, complemented
61C23×D293—21no—
62#17989802—22no—
63#17989802—41no—
64C2×Dic586—21no—
65C2×Dic586C2311yescharacteristic
66Q8×D293—22no—
67Q8×D293—41no—
68C2×D1172—21no—
69C2×D1172C2311yescharacteristic
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—
77C2×C22⋊D293—44no—
78C2×C22⋊D293D411yescharacteristic
79D4×D293—22no—
80D4×D293—41no—
81C22×Dic293—21no—
82C2×C4×D293—21no—
83C2×C4×D293—21no—
84C2×C4×D293C2311yescharacteristic
85#17989791—22no—
86#17989790—22no—
87#17989790D414yescomplemented
88#17989789—22no—
89D2344—41no—
90#17989787—22no—
91#17989787—22no—
92#17989786—22no—
93C2×#14556873C2311yescharacteristic
94C2×#14556873C2311yescharacteristic
95C8×D293—22no—
96C4⋊Dic293C2311yescharacteristic
97#17989781—21no—
98#17989781D411yescharacteristic
99C22⋊Dic293—21no—
100C22⋊Dic293—21no—
101C22⋊Dic293D411yescharacteristic
102#17989779—21no—
103#17989779—21no—
104#17989779—21no—
105#17989779—21no—
106#17989779D411yescharacteristic
107C4×Dic293—21no—
108C4×Dic293C2311yescharacteristic
109C24×C293—21no—
110C2×Q8×C293C2311yescharacteristic
111C2×D4×C293—22no—
112C2×D4×C293—21no—
113C2×D4×C293—41no—
114C2×D4×C293C2311yescharacteristic
115C2×D4×C293C2311yescharacteristic
116C22×C4×C293C2311yescharacteristic
117SD16×C293—22no—
118SD16×C293—41no—
119C2×C8×C293C2311yescharacteristic
120C2×C8×C293C2311yescharacteristic
121C4⋊C4×C293—21no—
122C4⋊C4×C293C2311yescharacteristic
123C22⋊C4×C293—21no—
124C22×D293—21no—
125C22×D293—21no—
126C22×D293—42no—
127C22×D293—42no—
128C22×D293—81no—
129C22×D293C2×D411yescharacteristic
130Dic586—21no—
131Dic586—42no—
132Dic586C2×D412yescomplemented
133D1172—21no—
134D1172—21no—
135D1172—21no—
136D1172—21no—
137D1172—42no—
138C22⋊D293—22no—
139C22⋊D293—42no—
140C22⋊D293—42no—
141C22⋊D293—42no—
142C22⋊D293—42no—
143C22⋊D293—42no—
144C22⋊D293—81no—
145C2×Dic293—21no—
146C2×Dic293—21no—
147C2×Dic293—42no—
148C2×Dic293C2×D411yescharacteristic
149C2×Dic293C2×D411yescharacteristic
150C2×Dic293C2×D411yescharacteristic
151C4×D293—21no—
152C4×D293—21no—
153C4×D293—42no—
154C4×D293—42no—
155#14556873—21no—
156#14556873C2×D412yescomplemented
157C23×C293—21no—
158C23×C293—21no—
159C23×C293—42no—
160C23×C293—42no—
161C23×C293—81no—
162C23×C293C2×D411yescharacteristic
163Q8×C293—21no—
164Q8×C293—21no—
165D4×C293—21no—
166D4×C293—21no—
167D4×C293—21no—
168D4×C293—42no—
169D4×C293C2×D412yes—
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×C293—21no—
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—84no—
188C22×C293C22×D411yescharacteristic
189C2×D293—21no—
190C2×D293—21no—
191C2×D293—41no—
192C2×D293—41no—
193C2×D293—41no—
194C2×D293—41no—
195C2×D293—41no—
196C2×D293—41no—
197C2×D293—84no—
198C2×D293AGL(1,ℤ/8)12yescomplemented
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—42no—
210C2×C293—41no—
211C2×C293—81no—
212C2×C293D4211yescharacteristic
213C2×C293C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C293#57711yescharacteristic
215D293—22no—
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#468—2931no—
226#466—2931no—
227#463—2931no—
228#460—2931no—
229#450D29311yescharacteristic, complemented
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—5861no—
239C22×D4C2×D29311yescharacteristic
240AGL(1,ℤ/8)—5864no—
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×D8—2931no—
250C2×D8—2931no—
251C2×M16—2931no—
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—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#102C2×D29311yescharacteristic
276#102C2×D29311yescharacteristic, complemented
277C22⋊C8—2931no—
278C22⋊C8C2×D29311yescharacteristic, complemented
279#97—2931no—
280C24—21no—
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—22no—
290C2×D4—21no—
291C2×D4—41no—
292C2×D4—2931no—
293C2×D4—2931no—
294C2×D4—2931no—
295C2×D4—5861no—
296C2×D4—5861no—
297C2×D4—5861no—
298C2×D4—5861no—
299C2×D4—5862no—
300C2×D4—5861no—
301C2×D4—5861no—
302C2×D4—5861no—
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—22no—
313SD16—41no—
314SD16—2934no—
315SD16—5862no—
316SD16—5862no—
317SD16—5862no—
318D8—5862no—
319D8—5862no—
320D8—11721no—
321M16—5862no—
322C2×C8—2931no—
323C2×C8—2931no—
324C2×C8—5862no—
325C2×C8C22×D29311yescharacteristic
326C2×C8C22×D29311yescharacteristic
327C4⋊C4—21no—
328C4⋊C4—2931no—
329C4⋊C4—2931no—
330C4⋊C4—5861no—
331C4⋊C4C22×D29311yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—2931no—
334C22⋊C4—2931no—
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—42no—
346C23—42no—
347C23—81no—
348C23—2931no—
349C23—23441no—
350C23—5861no—
351C23—5861no—
352C23—11722no—
353C23—11722no—
354C23D4×D29311yescharacteristic
355Q8—21no—
356Q8—21no—
357Q8—2932no—
358Q8—5861no—
359Q8—11722no—
360D4—21no—
361D4—21no—
362D4—21no—
363D4—42no—
364D4—23441no—
365D4—5862no—
366D4—5861no—
367D4—5861no—
368D4—5861no—
369D4—5861no—
370D4—11722no—
371D4—11722no—
372D4—11722no—
373D4—11722no—
374D4—11722no—
375D4—11722no—
376D4D4×D29312yes—
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—21no—
395C8—2932no—
396C8—5861no—
397C22—21no—
398C22—22no—
399C22—21no—
400C22—21no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—84no—
407C22—2932no—
408C22—23444no—
409C22—5861no—
410C22—5861no—
411C22—11721no—
412C22—11721no—
413C22—11721no—
414C22—11721no—
415C22—11721no—
416C22—11721no—
417C22C2×D4×D29311yescharacteristic
418C4—21no—
419C4—41no—
420C4—41no—
421C4—5861no—
422C4—5862no—
423C4—5861no—
424C4—5861no—
425C4—11721no—
426C4C2×D4×D29311yescharacteristic
427C4#2262766911yescharacteristic
428C2—41no—
429C2—42no—
430C2—81no—
431C2—23441no—
432C2—23441no—
433C2—5862no—
434C2#3403676111yescharacteristic
435C2AGL(1,ℤ/8)×D29311yescharacteristic
436C2#3403693511yescharacteristic
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#450×C293C2
socleC22×C293C22×D4
hypercenter#450D293
hypocenterC293#3265
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C2×#22627650C2semidirect
2. C2×SD16×D293C2semidirect
3. #34036789C2semidirect
4. D4×C22⋊D293C2semidirect
5. #34036515C2semidirect
6. #34036377C2semidirect
7. #34036364C2semidirect
8. #34036346C2semidirect
9. #34036328C2semidirect
10. #34036276C2semidirect
11. #34036266C2semidirect
12. #34036264C2semidirect
13. #34036226C2semidirect
14. #34036203C2semidirect
15. #450×C293C2semidirect
16. C2×#17989802C22semidirect
17. C2×Q8×D293C22semidirect
21. C2×#17989791C22semidirect
23. C2×#17989790C22semidirect
24. C2×#17989789C22semidirect
26. C2×D2344C22semidirect
28. C2×#17989787C22semidirect
29. C2×#17989786C22semidirect
30. C2×C8×D293C22semidirect
32. #22627613C22semidirect
35. #22627599C22semidirect
42. #22627589C22semidirect
46. #22627557C22semidirect
47. #22627553C22semidirect
48. #22627549C22semidirect
49. #22627548C22semidirect
50. #22627546C22semidirect
51. #22627544C22semidirect
52. #22627536C22semidirect
53. #22627534C22semidirect
56. C2×SD16×C293C22semidirect
57. C22⋊Q8×C293C22semidirect
59. #102×C293C22semidirect
60. C22⋊C8×C293C22semidirect
87. #17989790D4semidirect
132. Dic586C2×D4semidirect
156. #14556873C2×D4semidirect
198. C2×D293AGL(1,ℤ/8)semidirect
218. C293#3265semidirect
229. #450D293semidirect
247. C2×SD16C2×D293semidirect
257. C22⋊Q8C2×D293semidirect
276. #102C2×D293semidirect
278. C22⋊C8C2×D293semidirect

Maximal subgroups

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

subgroupindex
C2×#226276502
C2×SD16×D2932
#340367892
D4×C22⋊D2932
#340365152
#340363772
#340363642
#340363462
#340363282
#340362762
#340362662
#340362642
#340362262
#340362032
#450×C2932
#3265293