Finite Group Databasebeta

← #49170391

Subgroups — #49170391

Order 37504.

520 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22627651C211yesmaximal, characteristic, complemented
2C2×#22627648C211yesmaximal, characteristic, complemented
3#34036774C211yesmaximal, characteristic, complemented
4#34036772C211yesmaximal, characteristic, complemented
5#34036511C211yesmaximal, characteristic, complemented
6#34036378C211yesmaximal, characteristic, complemented
7#34036374C211yesmaximal, characteristic, complemented
8#34036345C211yesmaximal, characteristic, complemented
9#34036294C211yesmaximal, characteristic, complemented
10#34036288C211yesmaximal, characteristic, complemented
11#34036264C211yesmaximal, characteristic, complemented
12#34036261C211yesmaximal, characteristic, complemented
13C4×#17989787C211yesmaximal, characteristic, complemented
14#34036206C211yesmaximal, characteristic, complemented
15#463×C293C211yesmaximal, characteristic, complemented
16C2×#17989803C2211yescharacteristic
17C2×#17989801C2211yescharacteristic
18C2×D4×D293C2211yescharacteristic, complemented
19C2×D4×D293C2211yescharacteristic, complemented
20#22627651—24no—
21#22627648—24no—
22C2×#17989792C2211yescharacteristic, complemented
23C2×#17989791C2211yescharacteristic, complemented
24C2×#17989790C2211yescharacteristic, complemented
25C2×#17989789C2211yescharacteristic, complemented
26C2×D2344C2211yescharacteristic
27C2×#17989787C2211yescharacteristic
28C2×#17989787C2211yescharacteristic
29C2×#17989785C2211yescharacteristic, complemented
30C2×#17989785C2211yescharacteristic, complemented
31#22627612C2211yescharacteristic, complemented
32#22627608—21no—
33#22627606—21no—
34#22627604C2211yescharacteristic, complemented
35#22627600C2211yescharacteristic, complemented
36#22627599C2211yescharacteristic
37#22627598—21no—
38#22627597—21no—
39#22627595C2211yescharacteristic
40#22627594—21no—
41#22627594—21no—
42C4×D1172C2211yescharacteristic
43C4×D1172C2211yescharacteristic, complemented
44C2×#17989779—21no—
45C22⋊C4×D293—21no—
46#22627557C2211yescharacteristic, complemented
47#22627549C2211yescharacteristic, complemented
48#22627546C2211yescharacteristic, complemented
49#22627545C2211yescharacteristic, complemented
50#22627544C2211yescharacteristic, complemented
51#22627536C2211yescharacteristic, complemented
52#22627536C2211yescharacteristic, complemented
53C4×#14556873C2211yescharacteristic, complemented
54C2×SD16×C293C2211yescharacteristic, complemented
55C2×D8×C293C2211yescharacteristic
56#124×C293C2211yescharacteristic, complemented
57C4×D4×C293C2211yescharacteristic, complemented
58C4⋊C8×C293C2211yescharacteristic, complemented
59#103×C293C2211yescharacteristic, complemented
60#102×C293C2211yescharacteristic, complemented
61C23×D293—21no—
62C23×D293—21no—
63#17989803—22no—
64#17989803—41no—
65#17989801—22no—
66#17989801—41no—
67C2×Dic586C2311yescharacteristic
68C2×D1172—21no—
69C2×D1172—21no—
70C2×D1172D411yescharacteristic
71C2×D1172C2311yescharacteristic
72C2×D1172C2311yescharacteristic
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—
81C22×Dic293—21no—
82C2×C4×D293—21no—
83C2×C4×D293—21no—
84C2×C4×D293C2311yescharacteristic
85C2×C4×D293C2311yescharacteristic
86#17989792—41no—
87#17989791—22no—
88#17989790—22no—
89#17989789—22no—
90D2344—22no—
91#17989787—41no—
92#17989787D414yescomplemented
93#17989785—22no—
94#17989785—22no—
95C2×#14556873C2311yescharacteristic
96C2×#14556873C2311yescharacteristic
97C4⋊Dic293—21no—
98C4⋊Dic293D411yescharacteristic
99C4⋊Dic293C2311yescharacteristic
100#17989781—21no—
101C22⋊Dic293—21no—
102#17989779—21no—
103#17989779—21no—
104#17989779—21no—
105#17989779—21no—
106#17989779—21no—
107#17989779—21no—
108#17989779D411yescharacteristic
109#17989779D411yescharacteristic
110C2×Q8×C293C2311yescharacteristic
111C2×D4×C293C2311yescharacteristic
112C2×D4×C293C2311yescharacteristic
113C22×C4×C293—21no—
114SD16×C293—22no—
115D8×C293—22no—
116C2×C8×C293C2311yescharacteristic
117C2×C8×C293C2311yescharacteristic
118C4⋊C4×C293C2311yescharacteristic
119C22⋊C4×C293—21no—
120C22⋊C4×C293—21no—
121C22⋊C4×C293—21no—
122C42×C293C2311yescharacteristic
123C22×D293—21no—
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—21no—
135D1172—21no—
136D1172—21no—
137D1172—21no—
138D1172—41no—
139D1172—42no—
140D1172—81no—
141D1172C2×D412yes—
142C22⋊D293—42no—
143C22⋊D293—42no—
144C22⋊D293—42no—
145C22⋊D293—81no—
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—81no—
157#14556873—21no—
158#14556873C2×D412yes—
159C23×C293—21no—
160C23×C293—21no—
161Q8×C293—21no—
162Q8×C293—21no—
163D4×C293—21no—
164D4×C293—21no—
165D4×C293—21no—
166D4×C293C2×D412yes—
167C2×C4×C293—21no—
168C2×C4×C293—21no—
169C2×C4×C293—81no—
170C2×C4×C293C2×D411yescharacteristic
171C2×C4×C293C2×D411yescharacteristic
172C2×C4×C293C2411yescharacteristic
173C8×C293—21no—
174C8×C293—21no—
175C22×C293—22no—
176C22×C293—41no—
177C22×C293—41no—
178C22×C293—41no—
179C22×C293—41no—
180C22×C293—41no—
181C22×C293C22×D411yescharacteristic
182C2×D293—21no—
183C2×D293—21no—
184C2×D293—22no—
185C2×D293—21no—
186C2×D293—21no—
187C2×D293—21no—
188C2×D293—21no—
189C2×D293—41no—
190C2×D293—41no—
191C2×D293—41no—
192C2×D293—41no—
193C2×D293—41no—
194C2×D293—84no—
195C2×D293—84no—
196C4×C293—21no—
197C4×C293—21no—
198C4×C293—41no—
199C4×C293—41no—
200C4×C293C22×D411yescharacteristic
201C4×C293H(5,𝔽2)11yescharacteristic
202Dic293—21no—
203Dic293—21no—
204Dic293—41no—
205Dic293—41no—
206C2×C293—42no—
207C2×C293—81no—
208C2×C293D4211yescharacteristic
209C2×C293#57511yescharacteristic
210C2×C293#57611yescharacteristic
211D293—41no—
212D293—41no—
213D293—42no—
214D293—81no—
215C293#327311yescharacteristic, complemented, Sylow
216#3273—2931nomaximal, Sylow
217C2×AGL(1,ℤ/8)—2931no—
218C2×AGL(1,ℤ/8)—2931no—
219#546—2931no—
220#546—2931no—
221#495—2931no—
222#468—2931no—
223#468—2931no—
224#463—2931no—
225#463D29311yescharacteristic, complemented
226#452—2931no—
227#452—2931no—
228#447—2931no—
229#447—2931no—
230C4×D8—2931no—
231#436—2931no—
232C2×#41—2931no—
233C2×#41—2931no—
234C22×D4—2931no—
235C22×D4—2931no—
236AGL(1,ℤ/8)—5864no—
237AGL(1,ℤ/8)—5864no—
238C2×Q16—2931no—
239C2×SD16—2931no—
240C2×SD16—2931no—
241C2×SD16—2931no—
242C2×SD16C2×D29311yescharacteristic, complemented
243C2×D8—2931no—
244C2×D8—2931no—
245C2×D8—2931no—
246C2×D8C2×D29311yescharacteristic
247C2×M16—2931no—
248C2×M16—2931no—
249#124—2931no—
250#124C2×D29311yescharacteristic, complemented
251#123—5861no—
252#123—5861no—
253C22⋊Q8—2931no—
254C22⋊Q8—2931no—
255#121—2931no—
256#121—2931no—
257#121—5861no—
258#121—5861no—
259#120—5861no—
260#120—5861no—
261C4×D4—2931no—
262C4×D4—2931no—
263C4×D4C2×D29311yescharacteristic, complemented
264C2×C22⋊C4—5861no—
265C2×C22⋊C4—5861no—
266#107—2931no—
267C4⋊C8C2×D29311yescharacteristic, complemented
268#103—2931no—
269#103C2×D29311yescharacteristic, complemented
270#102—2931no—
271#102—2931no—
272#102—2931no—
273#102C2×D29311yescharacteristic, complemented
274C22⋊C8—2931no—
275C22⋊C8—2931no—
276C4×C8—2931no—
277C24—5861no—
278C24—5861no—
279#41—5862no—
280#41—5862no—
281#41—11721no—
282#41—11721no—
283C2×Q8—2931no—
284C2×Q8C22×D29311yescharacteristic
285C2×D4—2931no—
286C2×D4—2931no—
287C2×D4—2931no—
288C2×D4—5861no—
289C2×D4—5861no—
290C2×D4—5861no—
291C2×D4—5861no—
292C2×D4—5862no—
293C2×D4—5861no—
294C2×D4—5862no—
295C2×D4—5861no—
296C2×D4—11721no—
297C2×D4—11721no—
298C2×D4C22×D29311yescharacteristic
299C2×D4C22×D29311yescharacteristic
300C22×C4—21no—
301C22×C4—2931no—
302C22×C4—2931no—
303C22×C4—5861no—
304C22×C4—5861no—
305C22×C4—5861no—
306Q16—11721no—
307SD16—22no—
308SD16—5862no—
309SD16—5862no—
310SD16—5862no—
311D8—22no—
312D8—2934no—
313D8—5862no—
314D8—11721no—
315M16—5862no—
316M16—5862no—
317C2×C8—2931no—
318C2×C8—2931no—
319C2×C8C22×D29311yescharacteristic
320C2×C8C22×D29311yescharacteristic
321C4⋊C4—2931no—
322C4⋊C4—2931no—
323C4⋊C4—5861no—
324C4⋊C4—5861no—
325C4⋊C4C22×D29311yescharacteristic
326C22⋊C4—21no—
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—2931no—
330C22⋊C4—2931no—
331C22⋊C4—5861no—
332C22⋊C4—5861no—
333C22⋊C4—5861no—
334C22⋊C4—5861no—
335C22⋊C4—5861no—
336C22⋊C4—5861no—
337C22⋊C4—5861no—
338C42C22×D29311yescharacteristic
339C23—21no—
340C23—21no—
341C23—2931no—
342C23—2931no—
343C23—23441no—
344C23—23441no—
345C23—5861no—
346C23—5861no—
347C23—11722no—
348C23—11722no—
349C23—11722no—
350C23—11722no—
351Q8—21no—
352Q8—21no—
353Q8—5861no—
354Q8—5861no—
355D4—21no—
356D4—21no—
357D4—21no—
358D4—2932no—
359D4—23441no—
360D4—23441no—
361D4—5861no—
362D4—5861no—
363D4—5861no—
364D4—11721no—
365D4—11722no—
366D4—11722no—
367D4—11722no—
368D4—11722no—
369D4D4×D29312yes—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—81no—
373C2×C4—2931no—
374C2×C4—2931no—
375C2×C4—23441no—
376C2×C4—5861no—
377C2×C4—5861no—
378C2×C4—5861no—
379C2×C4—5861no—
380C2×C4—5861no—
381C2×C4—5861no—
382C2×C4—11722no—
383C2×C4—11722no—
384C2×C4D4×D29311yescharacteristic
385C2×C4D4×D29311yescharacteristic
386C2×C4C23×D29311yescharacteristic
387C8—21no—
388C8—21no—
389C8—2932no—
390C8—5861no—
391C22—22no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—23444no—
398C22—23444no—
399C22—5861no—
400C22—5861no—
401C22—5862no—
402C22—5861no—
403C22—5861no—
404C22—5861no—
405C22—5861no—
406C22—11721no—
407C22—11721no—
408C22—11721no—
409C22—11721no—
410C22—11721no—
411C22C2×D4×D29311yescharacteristic
412C4—21no—
413C4—21no—
414C4—41no—
415C4—41no—
416C4—5861no—
417C4—5861no—
418C4—11721no—
419C4—11721no—
420C4C2×D4×D29311yescharacteristic
421C4#2262767011yescharacteristic
422C2—42no—
423C2—81no—
424C2—23441no—
425C2—11721no—
426C2—11721no—
427C2—11722no—
428C2#3403676311yescharacteristic
429C2#3403692611yescharacteristic
430C2#3403693311yescharacteristic
subgroupG/H
centerC22C2×D4×D293
commutator, cosocleC2×C4×C293C24
FrattiniC2×C4C23×D293
Fitting#463×C293C2
socleC22×C293C22×D4
hypercenter#463D293
hypocenterC293#3273
LCSG ▷ C2×C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#22627651C2semidirect
2. C2×#22627648C2semidirect
3. #34036774C2semidirect
4. #34036772C2semidirect
5. #34036511C2semidirect
6. #34036378C2semidirect
7. #34036374C2semidirect
8. #34036345C2semidirect
9. #34036294C2semidirect
10. #34036288C2semidirect
11. #34036264C2semidirect
12. #34036261C2semidirect
13. C4×#17989787C2semidirect
14. #34036206C2semidirect
15. #463×C293C2semidirect
18. C2×D4×D293C22semidirect
19. C2×D4×D293C22semidirect
22. C2×#17989792C22semidirect
23. C2×#17989791C22semidirect
24. C2×#17989790C22semidirect
25. C2×#17989789C22semidirect
29. C2×#17989785C22semidirect
30. C2×#17989785C22semidirect
31. #22627612C22semidirect
34. #22627604C22semidirect
35. #22627600C22semidirect
43. C4×D1172C22semidirect
46. #22627557C22semidirect
47. #22627549C22semidirect
48. #22627546C22semidirect
49. #22627545C22semidirect
50. #22627544C22semidirect
51. #22627536C22semidirect
52. #22627536C22semidirect
53. C4×#14556873C22semidirect
54. C2×SD16×C293C22semidirect
56. #124×C293C22semidirect
57. C4×D4×C293C22semidirect
58. C4⋊C8×C293C22semidirect
59. #103×C293C22semidirect
60. #102×C293C22semidirect
92. #17989787D4semidirect
215. C293#3273semidirect
225. #463D293semidirect
242. C2×SD16C2×D293semidirect
250. #124C2×D293semidirect
263. C4×D4C2×D293semidirect
267. C4⋊C8C2×D293semidirect
269. #103C2×D293semidirect
273. #102C2×D293semidirect

Maximal subgroups

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

subgroupindex
C2×#226276512
C2×#226276482
#340367742
#340367722
#340365112
#340363782
#340363742
#340363452
#340362942
#340362882
#340362642
#340362612
C4×#179897872
#340362062
#463×C2932
#3273293