Finite Group Databasebeta

← #49132802

Subgroups — #49132802

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×#22583635C211yesmaximal, characteristic, complemented
2C2×#22583630C211yesmaximal, characteristic, complemented
3#31827111C211yesmaximal, characteristic, complemented
4#31826852C211yesmaximal, characteristic, complemented
5#31826709C211yesmaximal, characteristic, complemented
6#31826699C211yesmaximal, characteristic, complemented
7#31826687C211yesmaximal, characteristic, complemented
8#31826657C211yesmaximal, characteristic, complemented
9#31826627C211yesmaximal, characteristic, complemented
10#31826610C211yesmaximal, characteristic, complemented
11#31826608C211yesmaximal, characteristic, complemented
12#31826598C211yesmaximal, characteristic, complemented
13#31826563C211yesmaximal, characteristic, complemented
14#31826537C211yesmaximal, characteristic, complemented
15#550×C283C211yesmaximal, characteristic, complemented
16C2×#17985392C2211yescharacteristic
17C22×D1132C2211yescharacteristic
18#22583635—24no—
19#22583630—24no—
20C2×#17985382C2211yescharacteristic, complemented
21C2×#17985380C2211yescharacteristic, complemented
22C2×#17985380C2211yescharacteristic, complemented
23C2×#17985379C2211yescharacteristic
24C2×#17985379C2211yescharacteristic
25C2×#17985377C2211yescharacteristic
26C2×#17985377C2211yescharacteristic
27C2×#17985377C2211yescharacteristic
28C2×#17985377C2211yescharacteristic
29C2×#17985376C2211yescharacteristic
30C22×C283⋊C8C2211yescharacteristic, complemented
31#22583598C2211yescharacteristic
32#22583595—21no—
33#22583589C2211yescharacteristic
34#22583584C2211yescharacteristic
35#22583583—21no—
36#22583578—21no—
37C4×D1132C2211yescharacteristic
38C22⋊C4×D283—21no—
39#22583543C2211yescharacteristic, complemented
40C4⋊(C283⋊C8)C2211yescharacteristic, complemented
41#22583534C2211yescharacteristic, complemented
42#22583533C2211yescharacteristic, complemented
43#22583531C2211yescharacteristic, complemented
44#22583531C2211yescharacteristic, complemented
45#22583530C2211yescharacteristic
46#22583530C2211yescharacteristic
47C22⋊(C283⋊C8)C2211yescharacteristic
48C22⋊(C283⋊C8)C2211yescharacteristic, complemented
49#22583521C2211yescharacteristic, complemented
50C2×#41×C283C2211yescharacteristic
51C2×#41×C283C2211yescharacteristic, complemented
52#127×C283—21no—
53#127×C283C2211yescharacteristic, complemented
54#121×C283—21no—
55#121×C283—21no—
56#121×C283C2211yescharacteristic
57#121×C283C2211yescharacteristic, complemented
58C4×Q8×C283C2211yescharacteristic, complemented
59C4×D4×C283—21no—
60C4×D4×C283C2211yescharacteristic
61C23×D283—21no—
62#17985392—22no—
63#17985392—41no—
64C2×Dic566C2311yescharacteristic
65C2×D1132—22no—
66C2×D1132—21no—
67C2×D1132—41no—
68C2×D1132C2311yescharacteristic
69C2×D1132C2311yescharacteristic
70C2×C22⋊D283—21no—
71C2×C22⋊D283—21no—
72C2×C4×D283—21no—
73C2×C4×D283—21no—
74#17985382—22no—
75#17985380—22no—
76#17985380—41no—
77#17985379—22no—
78#17985379D414yescomplemented
79#17985377—22no—
80#17985377—22no—
81#17985377—22no—
82#17985377—41no—
83#17985376—22no—
84C2×C283⋊C8—22no—
85C2×C283⋊C8C2311yescharacteristic
86C2×C283⋊C8C2311yescharacteristic
87C2×C283⋊C8C2311yescharacteristic
88C2×C283⋊C8C2311yescharacteristic
89C4⋊Dic283C2311yescharacteristic
90Dic283⋊C4—21no—
91C22⋊Dic283—21no—
92(C2×D283)⋊C4—21no—
93(C2×D283)⋊C4—21no—
94(C2×D283)⋊C4—21no—
95(C2×D283)⋊C4—21no—
96#41×C283—22no—
97#41×C283—22no—
98#41×C283—41no—
99#41×C283—41no—
100C2×Q8×C283C2311yescharacteristic
101C2×D4×C283—21no—
102C2×D4×C283—21no—
103C2×D4×C283—21no—
104C2×D4×C283—21no—
105C2×D4×C283—21no—
106C2×D4×C283—21no—
107C2×D4×C283D411yescharacteristic
108C2×D4×C283C2311yescharacteristic
109C2×D4×C283C2311yescharacteristic
110C22×C4×C283—21no—
111C22×C4×C283—21no—
112C22×C4×C283C2311yescharacteristic
113C22×C4×C283C2311yescharacteristic
114C4⋊C4×C283—21no—
115C4⋊C4×C283D411yescharacteristic
116C4⋊C4×C283C2311yescharacteristic
117C22⋊C4×C283—21no—
118C22⋊C4×C283—21no—
119C22⋊C4×C283D411yescharacteristic
120C22⋊C4×C283D411yescharacteristic
121C42×C283—21no—
122C42×C283C2311yescharacteristic
123C22×D283—21no—
124C22×D283—21no—
125C22×D283—42no—
126C22×D283—42no—
127C22×D283—81no—
128Dic566—21no—
129Dic566—21no—
130D1132—21no—
131D1132—21no—
132D1132—21no—
133D1132—42no—
134D1132C2×D412yes—
135C22⋊D283—42no—
136C22⋊D283—81no—
137C2×Dic283—21no—
138C2×Dic283—21no—
139C4×D283—42no—
140C4×D283—81no—
141C283⋊C8—21no—
142C283⋊C8—21no—
143C283⋊C8—21no—
144C283⋊C8C2×D412yes—
145C23×C283—21no—
146C23×C283—21no—
147C23×C283C2×D411yescharacteristic
148C23×C283C2×D411yescharacteristic
149Q8×C283—21no—
150Q8×C283C2×D412yes—
151D4×C283—21no—
152D4×C283—21no—
153D4×C283—21no—
154D4×C283—21no—
155D4×C283—41no—
156D4×C283—42no—
157D4×C283—42no—
158D4×C283—42no—
159D4×C283—81no—
160D4×C283—81no—
161D4×C283—81no—
162C2×C4×C283—21no—
163C2×C4×C283—21no—
164C2×C4×C283—21no—
165C2×C4×C283—21no—
166C2×C4×C283—21no—
167C2×C4×C283—21no—
168C2×C4×C283—42no—
169C2×C4×C283—42no—
170C2×C4×C283C2×D411yescharacteristic
171C2×C4×C283C2×D411yescharacteristic
172C2×C4×C283C2×D411yescharacteristic
173C2×C4×C283C2×D411yescharacteristic
174C2×C4×C283C2411yescharacteristic
175C22×C283—21no—
176C22×C283—21no—
177C22×C283—21no—
178C22×C283—21no—
179C22×C283—21no—
180C22×C283—21no—
181C22×C283—41no—
182C22×C283—41no—
183C22×C283—41no—
184C22×C283—41no—
185C22×C283—41no—
186C22×C283—41no—
187C22×C283C22×D411yescharacteristic
188C2×D283—22no—
189C2×D283—41no—
190C2×D283—41no—
191C2×D283—41no—
192C2×D283—41no—
193C2×D283—41no—
194C2×D283—84no—
195C4×C283—21no—
196C4×C283—21no—
197C4×C283—22no—
198C4×C283—21no—
199C4×C283—21no—
200C4×C283—41no—
201C4×C283C22×D411yescharacteristic
202C4×C283H(5,𝔽2)11yescharacteristic
203Dic283—41no—
204Dic283—41no—
205C2×C283—41no—
206C2×C283—41no—
207C2×C283—81no—
208C2×C283—81no—
209C2×C283D4211yescharacteristic
210C2×C283#57511yescharacteristic
211C2×C283#57611yescharacteristic
212D283—42no—
213D283—81no—
214C283#326811yescharacteristic, complemented, Sylow
215#3268—2831nomaximal, Sylow
216C2×AGL(1,ℤ/8)—2831no—
217C2×#135—2831no—
218#550D28311yescharacteristic, complemented
219#546—2831no—
220#496—2831no—
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×#41C2×D28311yescharacteristic
233C2×#41C2×D28311yescharacteristic, complemented
234C22×D4—2831no—
235AGL(1,ℤ/8)—5664no—
236#135—5664no—
237C2×Q16—2831no—
238C2×SD16—2831no—
239C2×SD16—2831no—
240C2×SD16—2831no—
241C2×SD16—2831no—
242C2×D8—2831no—
243C2×D8—2831no—
244C2×D8—2831no—
245C2×D8—2831no—
246C2×M16—2831no—
247C22×C8—2831no—
248#127—21no—
249#127C2×D28311yescharacteristic, complemented
250#124—2831no—
251#123—5661no—
252C22⋊Q8—2831no—
253#121—21no—
254#121—21no—
255#121—2831no—
256#121—5661no—
257#121C2×D28311yescharacteristic
258#121C2×D28311yescharacteristic, complemented
259#120—5661no—
260C4×Q8C2×D28311yescharacteristic, complemented
261C4×D4—21no—
262C4×D4—2831no—
263C4×D4C2×D28311yescharacteristic
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#97—2831no—
276C24—5661no—
277#41—22no—
278#41—22no—
279#41—41no—
280#41—41no—
281#41—5662no—
282#41—11321no—
283C2×Q8—2831no—
284C2×Q8C22×D28311yescharacteristic
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—2831no—
292C2×D4—2831no—
293C2×D4—5662no—
294C2×D4—5661no—
295C2×D4—5661no—
296C2×D4—5661no—
297C2×D4—11321no—
298C2×D4C22⋊D28311yescharacteristic
299C2×D4C22×D28311yescharacteristic
300C2×D4C22×D28311yescharacteristic
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—5661no—
304C22×C4—5661no—
305C22×C4C22×D28311yescharacteristic
306C22×C4C22×D28311yescharacteristic
307Q16—5662no—
308SD16—2834no—
309SD16—5662no—
310SD16—5662no—
311SD16—11321no—
312D8—5662no—
313D8—5662no—
314D8—5662no—
315D8—11321no—
316M16—5662no—
317C2×C8—2831no—
318C2×C8—2831no—
319C2×C8—2831no—
320C2×C8—2831no—
321C2×C8—5662no—
322C4⋊C4—21no—
323C4⋊C4—2831no—
324C4⋊C4—5661no—
325C4⋊C4C22⋊D28311yescharacteristic
326C4⋊C4C22×D28311yescharacteristic
327C22⋊C4—21no—
328C22⋊C4—21no—
329C22⋊C4—5661no—
330C22⋊C4—5661no—
331C22⋊C4—5661no—
332C22⋊C4—5661no—
333C22⋊C4—5661no—
334C22⋊C4C22⋊D28311yescharacteristic
335C22⋊C4C22⋊D28311yescharacteristic
336C42—21no—
337C42C22×D28311yescharacteristic
338C23—21no—
339C23—21no—
340C23—5661no—
341C23—5661no—
342C23—11322no—
343C23—11322no—
344C23—22641no—
345C23C2×C22⋊D28311yescharacteristic
346C23C2×C22⋊D28311yescharacteristic
347Q8—21no—
348Q8—5661no—
349Q8—5661no—
350Q8D4×D28312yes—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—21no—
355D4—41no—
356D4—42no—
357D4—42no—
358D4—42no—
359D4—81no—
360D4—81no—
361D4—81no—
362D4—2832no—
363D4—5661no—
364D4—5661no—
365D4—5661no—
366D4—11322no—
367D4—11322no—
368D4—22641no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—5661no—
378C2×C4—5661no—
379C2×C4—11322no—
380C2×C4—22641no—
381C2×C4C2×C22⋊D28311yescharacteristic
382C2×C4C2×C22⋊D28311yescharacteristic
383C2×C4C2×C22⋊D28311yescharacteristic
384C2×C4C2×C22⋊D28311yescharacteristic
385C2×C4C23×D28311yescharacteristic
386C8—2832no—
387C8—5661no—
388C8—5661no—
389C8—5661no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—5662no—
403C22—11321no—
404C22—11321no—
405C22—11321no—
406C22—11321no—
407C22—11321no—
408C22—22644no—
409C22C22×C22⋊D28311yescharacteristic
410C4—21no—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—22no—
415C4—41no—
416C4—11321no—
417C4—11321no—
418C4C2×D4×D28311yescharacteristic
419C4#2258365511yescharacteristic
420C2—41no—
421C2—41no—
422C2—81no—
423C2—81no—
424C2—11322no—
425C2—22641no—
426C2D4×C22⋊D28311yescharacteristic
427C2#3182726311yescharacteristic
428C2#3182726811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#550×C283C2
socleC22×C283C22×D4
hypercenter#550D283
hypocenterC283#3268
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C4×D4 ◁ #550

Complements

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

NH ≅ G/Ntype
1. C2×#22583635C2semidirect
2. C2×#22583630C2semidirect
3. #31827111C2semidirect
4. #31826852C2semidirect
5. #31826709C2semidirect
6. #31826699C2semidirect
7. #31826687C2semidirect
8. #31826657C2semidirect
9. #31826627C2semidirect
10. #31826610C2semidirect
11. #31826608C2semidirect
12. #31826598C2semidirect
13. #31826563C2semidirect
14. #31826537C2semidirect
15. #550×C283C2semidirect
20. C2×#17985382C22semidirect
21. C2×#17985380C22semidirect
22. C2×#17985380C22semidirect
30. C22×C283⋊C8C22semidirect
39. #22583543C22semidirect
40. C4⋊(C283⋊C8)C22semidirect
41. #22583534C22semidirect
42. #22583533C22semidirect
43. #22583531C22semidirect
44. #22583531C22semidirect
48. C22⋊(C283⋊C8)C22semidirect
49. #22583521C22semidirect
51. C2×#41×C283C22semidirect
53. #127×C283C22semidirect
57. #121×C283C22semidirect
58. C4×Q8×C283C22semidirect
78. #17985379D4semidirect
214. C283#3268semidirect
218. #550D283semidirect
233. C2×#41C2×D283semidirect
249. #127C2×D283semidirect
258. #121C2×D283semidirect
260. C4×Q8C2×D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836352
C2×#225836302
#318271112
#318268522
#318267092
#318266992
#318266872
#318266572
#318266272
#318266102
#318266082
#318265982
#318265632
#318265372
#550×C2832
#3268283