Finite Group Databasebeta

← #49132800

Subgroups — #49132800

Order 36224.

510 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22583634C211yesmaximal, characteristic, complemented
2C2×#22583631C211yesmaximal, characteristic, complemented
3#31827135C211yesmaximal, characteristic, complemented
4#31827108C211yesmaximal, characteristic, complemented
5#31826708C211yesmaximal, characteristic, complemented
6#31826698C211yesmaximal, characteristic, complemented
7#31826684C211yesmaximal, characteristic, complemented
8#31826657C211yesmaximal, characteristic, complemented
9#31826624C211yesmaximal, characteristic, complemented
10#31826609C211yesmaximal, characteristic, complemented
11#31826607C211yesmaximal, characteristic, complemented
12#31826597C211yesmaximal, characteristic, complemented
13#31826567C211yesmaximal, characteristic, complemented
14#31826538C211yesmaximal, characteristic, complemented
15#496×C283C211yesmaximal, characteristic, complemented
16C2×#17985393C2211yescharacteristic
17C2×#17985391C2211yescharacteristic
18C2×#17985391C2211yescharacteristic, complemented
19C2×D4×D283C2211yescharacteristic
20#22583634—24no—
21#22583631—24no—
22C2×#17985382C2211yescharacteristic, complemented
23C2×#17985381C2211yescharacteristic
24C2×#17985380C2211yescharacteristic
25C2×#17985377C2211yescharacteristic
26C2×#17985377C2211yescharacteristic
27C2×#17985375C2211yescharacteristic
28C2×C8×D283C2211yescharacteristic, complemented
29#22583606—21no—
30#22583592—21no—
31#22583586C2211yescharacteristic
32#22583583—21no—
33#22583582—21no—
34#22583582—21no—
35#22583581C2211yescharacteristic
36#22583581C2211yescharacteristic
37#22583581C2211yescharacteristic, complemented
38#22583580—21no—
39C4×Dic566C2211yescharacteristic, complemented
40C4×D1132C2211yescharacteristic
41C4×D1132C2211yescharacteristic
42D4×Dic283—21no—
43C22⋊C4×D283—21no—
44#22583544C2211yescharacteristic, complemented
45C4⋊(C283⋊C8)C2211yescharacteristic, complemented
46#22583535C2211yescharacteristic, complemented
47#22583534C2211yescharacteristic, complemented
48#22583532C2211yescharacteristic
49#22583531C2211yescharacteristic
50#22583531C2211yescharacteristic, complemented
51#22583531C2211yescharacteristic, complemented
52(C2×D283)⋊C8C2211yescharacteristic
53(C2×D283)⋊C8C2211yescharacteristic, complemented
54C2×SD16×C283C2211yescharacteristic, complemented
55C2×SD16×C283C2211yescharacteristic, complemented
56C2×D8×C283C2211yescharacteristic
57C2×D8×C283C2211yescharacteristic
58#127×C283C2211yescharacteristic, complemented
59#124×C283C2211yescharacteristic
60#97×C283C2211yescharacteristic, complemented
61C23×D283—21no—
62#17985393—22no—
63#17985393—41no—
64#17985391—22no—
65#17985391—22no—
66#17985391—41no—
67#17985391—41no—
68C2×Dic566C2311yescharacteristic
69C2×D1132—21no—
70C2×D1132D411yescharacteristic
71C2×D1132C2311yescharacteristic
72C2×C22⋊D283—21no—
73C2×C22⋊D283—21no—
74C2×C22⋊D283—21no—
75C2×C22⋊D283—21no—
76C2×C22⋊D283—21no—
77C2×C22⋊D283—21no—
78D4×D283—22no—
79D4×D283—41no—
80C22×Dic283—21no—
81C22×Dic283—21no—
82C2×C4×D283—21no—
83C2×C4×D283—21no—
84C2×C4×D283C2311yescharacteristic
85C2×C4×D283C2311yescharacteristic
86#17985382—22no—
87#17985381D414yescomplemented
88#17985380—22no—
89#17985377—22no—
90#17985377—22no—
91#17985375—22no—
92C2×C283⋊C8C2311yescharacteristic
93C2×C283⋊C8C2311yescharacteristic
94C8×D283—22no—
95C4⋊Dic283D411yescharacteristic
96C4⋊Dic283C2311yescharacteristic
97C4⋊Dic283C2311yescharacteristic
98Dic283⋊C4—21no—
99Dic283⋊C4—21no—
100C22⋊Dic283—21no—
101C22⋊Dic283—21no—
102C22⋊Dic283—21no—
103(C2×D283)⋊C4—21no—
104(C2×D283)⋊C4—21no—
105(C2×D283)⋊C4D411yescharacteristic
106(C2×D283)⋊C4D411yescharacteristic
107C4×Dic283—21no—
108C2×Q8×C283C2311yescharacteristic
109C2×D4×C283—21no—
110C2×D4×C283—21no—
111C2×D4×C283C2311yescharacteristic
112C2×D4×C283C2311yescharacteristic
113C2×D4×C283C2311yescharacteristic
114SD16×C283—22no—
115SD16×C283—41no—
116D8×C283—22no—
117D8×C283—41no—
118C2×C8×C283C2311yescharacteristic
119C2×C8×C283C2311yescharacteristic
120C22⋊C4×C283—21no—
121C22⋊C4×C283—21no—
122C42×C283C2311yescharacteristic
123C22×D283—21no—
124C22×D283—42no—
125C22×D283—42no—
126C22×D283—81no—
127C22×D283C2×D411yescharacteristic
128C22×D283C2×D411yescharacteristic
129Dic566—21no—
130Dic566C2×D412yes—
131D1132—21no—
132D1132—41no—
133D1132—42no—
134D1132C2×D412yes—
135C22⋊D283—42no—
136C22⋊D283—42no—
137C22⋊D283—42no—
138C22⋊D283—81no—
139C22⋊D283—81no—
140C22⋊D283—81no—
141C2×Dic283—21no—
142C2×Dic283—21no—
143C2×Dic283—21no—
144C2×Dic283—42no—
145C2×Dic283—42no—
146C2×Dic283C2×D411yescharacteristic
147C2×Dic283C2×D411yescharacteristic
148C4×D283—21no—
149C4×D283—21no—
150C4×D283—21no—
151C4×D283—21no—
152C4×D283—42no—
153C4×D283—81no—
154C283⋊C8—21no—
155C283⋊C8—21no—
156C23×C283—21no—
157C23×C283—21no—
158C23×C283—21no—
159Q8×C283—21no—
160Q8×C283—21no—
161D4×C283—21no—
162D4×C283—21no—
163D4×C283—21no—
164D4×C283—21no—
165D4×C283—21no—
166D4×C283—21no—
167D4×C283—42no—
168D4×C283—81no—
169C2×C4×C283—21no—
170C2×C4×C283C2×D411yescharacteristic
171C2×C4×C283C2×D411yescharacteristic
172C2×C4×C283C2411yescharacteristic
173C8×C283—21no—
174C8×C283C2×D412yes—
175C22×C283—41no—
176C22×C283—41no—
177C22×C283—41no—
178C22×C283—41no—
179C22×C283—41no—
180C22×C283—41no—
181C22×C283—41no—
182C22×C283—41no—
183C22×C283—41no—
184C22×C283C22×D411yescharacteristic
185C2×D283—21no—
186C2×D283—21no—
187C2×D283—22no—
188C2×D283—21no—
189C2×D283—21no—
190C2×D283—21no—
191C2×D283—21no—
192C2×D283—41no—
193C2×D283—41no—
194C2×D283—84no—
195C4×C283—21no—
196C4×C283—21no—
197C4×C283—41no—
198C4×C283C22×D411yescharacteristic
199C4×C283H(5,𝔽2)11yescharacteristic
200Dic283—21no—
201Dic283—21no—
202Dic283—22no—
203Dic283—41no—
204Dic283—41no—
205C2×C283—81no—
206C2×C283—81no—
207C2×C283—81no—
208C2×C283D4211yescharacteristic
209C2×C283#57511yescharacteristic
210C2×C283#57611yescharacteristic
211D283—41no—
212D283—41no—
213D283—42no—
214C283#326811yescharacteristic, complemented, Sylow
215#3268—2831nomaximal, Sylow
216C2×AGL(1,ℤ/8)—2831no—
217C2×#135—2831no—
218#550—2831no—
219#546—2831no—
220#496D28311yescharacteristic, complemented
221#468—2831no—
222#466—2831no—
223#464—2831no—
224#459—2831no—
225#452—2831no—
226#449—2831no—
227#449—2831no—
228#447—2831no—
229#440—2831no—
230#435—2831no—
231C2×#41—2831no—
232C2×#41—2831no—
233C2×#41—2831no—
234C22×D4—2831no—
235AGL(1,ℤ/8)—5664no—
236#135—5664no—
237C2×Q16—2831no—
238C2×SD16—2831no—
239C2×SD16—2831no—
240C2×SD16C2×D28311yescharacteristic, complemented
241C2×SD16C2×D28311yescharacteristic, complemented
242C2×D8—2831no—
243C2×D8—2831no—
244C2×D8C2×D28311yescharacteristic
245C2×D8C2×D28311yescharacteristic
246C2×M16—2831no—
247C22×C8—2831no—
248#127—5661no—
249#127C2×D28311yescharacteristic, complemented
250#124C2×D28311yescharacteristic
251#123—5661no—
252C22⋊Q8—2831no—
253#121—2831no—
254#121—2831no—
255#121—2831no—
256#121—5661no—
257#121—5661no—
258#121—5661no—
259#120—5661no—
260C4×Q8—2831no—
261C4×D4—2831no—
262C4×D4—2831no—
263C4×D4—5661no—
264C2×C22⋊C4—5661no—
265#107—2831no—
266C4⋊C8—2831no—
267#103—2831no—
268#103—2831no—
269#102—2831no—
270#102—2831no—
271#102—2831no—
272#102—2831no—
273C22⋊C8—2831no—
274C22⋊C8—2831no—
275#97C2×D28311yescharacteristic, complemented
276C24—5661no—
277#41—5662no—
278#41—5662no—
279#41—5662no—
280#41—11321no—
281#41—11321no—
282#41—11321no—
283C2×Q8—2831no—
284C2×Q8C22×D28311yescharacteristic
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—2831no—
288C2×D4—2831no—
289C2×D4—5661no—
290C2×D4—5661no—
291C2×D4—5662no—
292C2×D4—5661no—
293C2×D4—5661no—
294C2×D4—5661no—
295C2×D4—5661no—
296C2×D4—5661no—
297C2×D4—11321no—
298C2×D4C22×D28311yescharacteristic
299C2×D4C22×D28311yescharacteristic
300C2×D4C22×D28311yescharacteristic
301C22×C4—2831no—
302C22×C4—2831no—
303C22×C4—5661no—
304C22×C4—5661no—
305C22×C4—5661no—
306C22×C4—5661no—
307Q16—5662no—
308SD16—22no—
309SD16—41no—
310SD16—2834no—
311SD16—5662no—
312D8—22no—
313D8—41no—
314D8—5662no—
315D8—5662no—
316M16—5662no—
317C2×C8—2831no—
318C2×C8—2831no—
319C2×C8—5662no—
320C2×C8C22×D28311yescharacteristic
321C2×C8C22×D28311yescharacteristic
322C4⋊C4—2831no—
323C4⋊C4—2831no—
324C4⋊C4—2831no—
325C4⋊C4—5661no—
326C4⋊C4—5661no—
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—2831no—
330C22⋊C4—2831no—
331C22⋊C4—5661no—
332C22⋊C4—5661no—
333C22⋊C4—5661no—
334C22⋊C4—5661no—
335C22⋊C4—5661no—
336C42—5661no—
337C42C22×D28311yescharacteristic
338C23—21no—
339C23—21no—
340C23—21no—
341C23—2831no—
342C23—2831no—
343C23—5661no—
344C23—11322no—
345C23—11322no—
346C23—22641no—
347Q8—21no—
348Q8—21no—
349Q8—2832no—
350Q8—5661no—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—21no—
355D4—21no—
356D4—21no—
357D4—42no—
358D4—81no—
359D4—2832no—
360D4—5661no—
361D4—11321no—
362D4—11322no—
363D4—11322no—
364D4—11322no—
365D4—11322no—
366D4—22641no—
367D4—22641no—
368D4—22641no—
369C2×C4—21no—
370C2×C4—2831no—
371C2×C4—2831no—
372C2×C4—5661no—
373C2×C4—5661no—
374C2×C4—5661no—
375C2×C4—5661no—
376C2×C4—5661no—
377C2×C4—5661no—
378C2×C4—5661no—
379C2×C4—11322no—
380C2×C4—11322no—
381C2×C4—11322no—
382C2×C4—22641no—
383C2×C4D4×D28311yescharacteristic
384C2×C4D4×D28311yescharacteristic
385C2×C4C23×D28311yescharacteristic
386C8—21no—
387C8—5661no—
388C8—5661no—
389C8D4×D28312yes—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—5661no—
400C22—5661no—
401C22—5661no—
402C22—5661no—
403C22—5661no—
404C22—5661no—
405C22—5662no—
406C22—11321no—
407C22—11321no—
408C22—22644no—
409C22C2×D4×D28311yescharacteristic
410C4—21no—
411C4—21no—
412C4—41no—
413C4—5661no—
414C4—5661no—
415C4—5662no—
416C4—11321no—
417C4—11321no—
418C4C2×D4×D28311yescharacteristic
419C4#2258365511yescharacteristic
420C2—81no—
421C2—81no—
422C2—81no—
423C2—11321no—
424C2—11321no—
425C2—11322no—
426C2#3182710011yescharacteristic
427C2#3182726111yescharacteristic
428C2#3182726911yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#496×C283C2
socleC22×C283C22×D4
hypercenter#496D283
hypocenterC283#3268
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#22583634C2semidirect
2. C2×#22583631C2semidirect
3. #31827135C2semidirect
4. #31827108C2semidirect
5. #31826708C2semidirect
6. #31826698C2semidirect
7. #31826684C2semidirect
8. #31826657C2semidirect
9. #31826624C2semidirect
10. #31826609C2semidirect
11. #31826607C2semidirect
12. #31826597C2semidirect
13. #31826567C2semidirect
14. #31826538C2semidirect
15. #496×C283C2semidirect
18. C2×#17985391C22semidirect
22. C2×#17985382C22semidirect
28. C2×C8×D283C22semidirect
37. #22583581C22semidirect
39. C4×Dic566C22semidirect
44. #22583544C22semidirect
45. C4⋊(C283⋊C8)C22semidirect
46. #22583535C22semidirect
47. #22583534C22semidirect
50. #22583531C22semidirect
51. #22583531C22semidirect
53. (C2×D283)⋊C8C22semidirect
54. C2×SD16×C283C22semidirect
55. C2×SD16×C283C22semidirect
58. #127×C283C22semidirect
60. #97×C283C22semidirect
87. #17985381D4semidirect
214. C283#3268semidirect
220. #496D283semidirect
240. C2×SD16C2×D283semidirect
241. C2×SD16C2×D283semidirect
249. #127C2×D283semidirect
275. #97C2×D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836342
C2×#225836312
#318271352
#318271082
#318267082
#318266982
#318266842
#318266572
#318266242
#318266092
#318266072
#318265972
#318265672
#318265382
#496×C2832
#3268283