Finite Group Databasebeta

← #49132780

Subgroups — #49132780

Order 36224.

500 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22583629C211yesmaximal, characteristic, complemented
2#31827114C211yesmaximal, characteristic, complemented
3#31827109C211yesmaximal, characteristic, complemented
4#31826856C211yesmaximal, characteristic, complemented
5#31826723C211yesmaximal, characteristic, complemented
6#31826691C211yesmaximal, characteristic, complemented
7#31826678C211yesmaximal, characteristic, complemented
8#31826668C211yesmaximal, characteristic, complemented
9#31826627C211yesmaximal, characteristic, complemented
10#31826624C211yesmaximal, characteristic, complemented
11#31826616C211yesmaximal, characteristic, complemented
12#31826607C211yesmaximal, characteristic, complemented
13#31826577C211yesmaximal, characteristic, complemented
14#31826534C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C283C211yesmaximal, characteristic, complemented
16C2×#17985391C2211yescharacteristic
17C2×#17985391C2211yescharacteristic, complemented
18#22583629—24no—
19C2×Dic1132C2211yescharacteristic, complemented
20C2×#17985382C2211yescharacteristic, complemented
21C2×#17985380C2211yescharacteristic
22C2×#17985380C2211yescharacteristic
23C2×#17985380C2211yescharacteristic, complemented
24C2×C8×D283C2211yescharacteristic, complemented
25#22583608C2211yescharacteristic, complemented
26#22583597C2211yescharacteristic
27#22583593—21no—
28#22583591—21no—
29#22583591—21no—
30#22583589C2211yescharacteristic
31#22583589C2211yescharacteristic
32#22583586C2211yescharacteristic, complemented
33#22583582—21no—
34#22583582—21no—
35#22583581C2211yescharacteristic
36#22583579—21no—
37D4×Dic283C2211yescharacteristic
38D4×Dic283C2211yescharacteristic, complemented
39C4×C22⋊D283C2211yescharacteristic
40C2×Dic283⋊C4—21no—
41C2×C22⋊Dic283—21no—
42#22583542C2211yescharacteristic, complemented
43Dic283⋊C8C2211yescharacteristic, complemented
44#22583535C2211yescharacteristic, complemented
45#22583535C2211yescharacteristic, complemented
46#22583534C2211yescharacteristic, complemented
47#22583531C2211yescharacteristic
48#22583531C2211yescharacteristic
49#22583531C2211yescharacteristic, complemented
50C22⋊(C283⋊C8)C2211yescharacteristic
51(C2×D283)⋊C8C2211yescharacteristic, complemented
52#22583520C2211yescharacteristic, complemented
53C2×#41×C283C2211yescharacteristic
54C22×D4×C283C2211yescharacteristic
55AGL(1,ℤ/8)×C283—24no—
56C2×SD16×C283C2211yescharacteristic
57C2×SD16×C283C2211yescharacteristic
58C2×D8×C283C2211yescharacteristic
59C2×D8×C283C2211yescharacteristic
60C2×M16×C283C2211yescharacteristic
61#17985391—22no—
62#17985391—22no—
63#17985391—41no—
64#17985391—41no—
65C2×Dic566C2311yescharacteristic
66C2×Dic566C2311yescharacteristic
67C2×C22⋊D283—21no—
68C2×C22⋊D283—21no—
69C2×C22⋊D283—21no—
70C2×C22⋊D283D411yescharacteristic
71C22×Dic283—21no—
72C22×Dic283—21no—
73C22×Dic283—21no—
74C22×Dic283—21no—
75C2×C4×D283C2311yescharacteristic
76Dic1132—22no—
77#17985382—41no—
78#17985380—22no—
79#17985380—22no—
80#17985380—41no—
81C2×C283⋊C8C2311yescharacteristic
82C2×C283⋊C8C2311yescharacteristic
83C8×D283—22no—
84C4⋊Dic283C2311yescharacteristic
85C4⋊Dic283C2311yescharacteristic
86Dic283⋊C4—21no—
87Dic283⋊C4—21no—
88Dic283⋊C4—21no—
89Dic283⋊C4—21no—
90Dic283⋊C4—21no—
91Dic283⋊C4D411yescharacteristic
92C22⋊Dic283—21no—
93C22⋊Dic283—21no—
94C22⋊Dic283—21no—
95C22⋊Dic283—21no—
96C22⋊Dic283—21no—
97C22⋊Dic283—21no—
98C22⋊Dic283—21no—
99C22⋊Dic283D411yescharacteristic
100(C2×D283)⋊C4—21no—
101(C2×D283)⋊C4D411yescharacteristic
102C4×Dic283C2311yescharacteristic
103C24×C283—21no—
104#41×C283—22no—
105#41×C283—41no—
106C2×Q8×C283C2311yescharacteristic
107C2×D4×C283—22no—
108C2×D4×C283—21no—
109C2×D4×C283—21no—
110C2×D4×C283—41no—
111C2×D4×C283C2311yescharacteristic
112C2×D4×C283C2311yescharacteristic
113C2×D4×C283C2311yescharacteristic
114C22×C4×C283—21no—
115C22×C4×C283C2311yescharacteristic
116SD16×C283—22no—
117SD16×C283—22no—
118D8×C283—22no—
119D8×C283D414yescomplemented
120M16×C283—22no—
121C2×C8×C283C2311yescharacteristic
122C2×C8×C283C2311yescharacteristic
123C22×D283C2×D411yescharacteristic
124Dic566—21no—
125Dic566—21no—
126Dic566—21no—
127Dic566—21no—
128C22⋊D283—41no—
129C22⋊D283—42no—
130C22⋊D283—42no—
131C22⋊D283—81no—
132C2×Dic283—21no—
133C2×Dic283—21no—
134C2×Dic283—21no—
135C2×Dic283—21no—
136C2×Dic283—42no—
137C2×Dic283—42no—
138C2×Dic283—42no—
139C2×Dic283—81no—
140C2×Dic283C2×D411yescharacteristic
141C2×Dic283C2×D411yescharacteristic
142C2×Dic283C2×D411yescharacteristic
143C4×D283—21no—
144C4×D283—21no—
145C283⋊C8—21no—
146C283⋊C8—21no—
147C23×C283—21no—
148C23×C283—21no—
149C23×C283—21no—
150C23×C283—42no—
151C23×C283—42no—
152C23×C283—81no—
153C23×C283C2×D411yescharacteristic
154Q8×C283—21no—
155Q8×C283—21no—
156D4×C283—21no—
157D4×C283—21no—
158D4×C283—21no—
159D4×C283—21no—
160D4×C283—42no—
161D4×C283—42no—
162D4×C283C2×D412yes—
163D4×C283C2×D412yes—
164C2×C4×C283—21no—
165C2×C4×C283—21no—
166C2×C4×C283—21no—
167C2×C4×C283—42no—
168C2×C4×C283C2×D411yescharacteristic
169C2×C4×C283C2411yescharacteristic
170C8×C283—21no—
171C8×C283C2×D412yes—
172C22×C283—21no—
173C22×C283—22no—
174C22×C283—22no—
175C22×C283—21no—
176C22×C283—21no—
177C22×C283—41no—
178C22×C283—41no—
179C22×C283—41no—
180C22×C283—41no—
181C22×C283—41no—
182C22×C283—41no—
183C22×C283—41no—
184C22×C283—84no—
185C22×C283C22×D411yescharacteristic
186C2×D283—21no—
187C2×D283—21no—
188C2×D283—21no—
189C4×C283—21no—
190C4×C283—41no—
191C4×C283C22×D411yescharacteristic
192C4×C283H(5,𝔽2)11yescharacteristic
193Dic283—21no—
194Dic283—21no—
195Dic283—21no—
196Dic283—41no—
197Dic283—41no—
198Dic283—41no—
199Dic283—41no—
200C2×C283—41no—
201C2×C283—42no—
202C2×C283—42no—
203C2×C283—81no—
204C2×C283D4211yescharacteristic
205C2×C283#57511yescharacteristic
206C2×C283#57711yescharacteristic
207D283—41no—
208C283#326711yescharacteristic, complemented, Sylow
209#3267—2831nomaximal, Sylow
210C2×AGL(1,ℤ/8)D28311yescharacteristic, complemented
211C2×#135—2831no—
212#547—2831no—
213#546—2831no—
214#497—2831no—
215#470—2831no—
216#465—2831no—
217#463—2831no—
218#461—2831no—
219#452—2831no—
220#452—2831no—
221#450—2831no—
222#449—2831no—
223#442—2831no—
224#435—2831no—
225C2×#41—2831no—
226C2×#41—2831no—
227C2×#41C2×D28311yescharacteristic
228C22×D4C2×D28311yescharacteristic
229AGL(1,ℤ/8)—24no—
230#135—5664no—
231C2×Q16—2831no—
232C2×Q16—2831no—
233C2×SD16—2831no—
234C2×SD16—2831no—
235C2×SD16—2831no—
236C2×SD16C2×D28311yescharacteristic
237C2×SD16C2×D28311yescharacteristic
238C2×D8C2×D28311yescharacteristic
239C2×D8C2×D28311yescharacteristic
240C2×M16C2×D28311yescharacteristic
241C22×C8—2831no—
242C4⋊Q8—2831no—
243#124—2831no—
244#123—5661no—
245#123—5661no—
246#123—5661no—
247C22⋊Q8—2831no—
248C22⋊Q8—2831no—
249C22⋊Q8—2831no—
250#121—2831no—
251#121—5661no—
252#121—5661no—
253#120—5661no—
254C4×D4—2831no—
255C4×D4—2831no—
256C4×D4—2831no—
257C2×C4⋊C4—5661no—
258C2×C22⋊C4—5661no—
259#106—2831no—
260C4⋊C8—2831no—
261#103—2831no—
262#103—2831no—
263#103—2831no—
264#102—2831no—
265#102—2831no—
266#102—2831no—
267C22⋊C8—2831no—
268C22⋊C8—2831no—
269#97—2831no—
270C24—21no—
271#41—22no—
272#41—41no—
273#41—5662no—
274#41—5662no—
275#41—11321no—
276#41—11321no—
277C2×Q8—2831no—
278C2×Q8—2831no—
279C2×Q8C22×D28311yescharacteristic
280C2×D4—22no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—41no—
284C2×D4—2831no—
285C2×D4—5661no—
286C2×D4—5661no—
287C2×D4—5661no—
288C2×D4C22×D28311yescharacteristic
289C2×D4C22×D28311yescharacteristic
290C2×D4C22×D28311yescharacteristic
291C22×C4—21no—
292C22×C4—2831no—
293C22×C4—5661no—
294C22×C4—5661no—
295C22×C4—5661no—
296C22×C4—5661no—
297C22×C4C22×D28311yescharacteristic
298Q16—5662no—
299Q16—11321no—
300SD16—22no—
301SD16—22no—
302SD16—5662no—
303SD16—5662no—
304SD16—11321no—
305D8—22no—
306D8C22⋊D28314yescomplemented
307M16—22no—
308C2×C8—2831no—
309C2×C8—2831no—
310C2×C8—5662no—
311C2×C8C22×D28311yescharacteristic
312C2×C8C22×D28311yescharacteristic
313C4⋊C4—2831no—
314C4⋊C4—2831no—
315C4⋊C4—2831no—
316C4⋊C4—5661no—
317C4⋊C4—5661no—
318C4⋊C4—5661no—
319C4⋊C4—5661no—
320C4⋊C4—5661no—
321C22⋊C4—2831no—
322C22⋊C4—2831no—
323C22⋊C4—5661no—
324C22⋊C4—5661no—
325C22⋊C4—5661no—
326C22⋊C4—5661no—
327C22⋊C4—5661no—
328C22⋊C4—5661no—
329C22⋊C4—5661no—
330C22⋊C4—5661no—
331C42—2831no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—42no—
336C23—42no—
337C23—81no—
338C23—2831no—
339C23D4×D28311yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—5661no—
343Q8—5661no—
344Q8—5661no—
345Q8—5661no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—21no—
350D4—42no—
351D4—42no—
352D4—11321no—
353D4—11322no—
354D4—11322no—
355D4—22641no—
356D4C2×C22⋊D28312yes—
357D4C2×C22⋊D28312yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—42no—
362C2×C4—2831no—
363C2×C4—2831no—
364C2×C4—2831no—
365C2×C4—5661no—
366C2×C4—5661no—
367C2×C4—5661no—
368C2×C4—5661no—
369C2×C4—5661no—
370C2×C4—5661no—
371C2×C4—11322no—
372C2×C4—11322no—
373C2×C4—11322no—
374C2×C4—22641no—
375C2×C4D4×D28311yescharacteristic
376C2×C4C23×D28311yescharacteristic
377C8—21no—
378C8—5661no—
379C8—5661no—
380C8C2×C22⋊D28312yes—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—22no—
385C22—22no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—84no—
394C22—5661no—
395C22—5661no—
396C22—5661no—
397C22C2×D4×D28311yescharacteristic
398C4—21no—
399C4—41no—
400C4—5661no—
401C4—5661no—
402C4—5661no—
403C4—11321no—
404C4—11321no—
405C4—11321no—
406C4—11321no—
407C4C22×C22⋊D28311yescharacteristic
408C4#2258365511yescharacteristic
409C2—41no—
410C2—42no—
411C2—42no—
412C2—81no—
413C2—11321no—
414C2D4×C22⋊D28311yescharacteristic
415C2#3182726111yescharacteristic
416C2#3182727511yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
FittingC2×AGL(1,ℤ/8)×C283C2
socleC22×C283C22×D4
hypercenterC2×AGL(1,ℤ/8)D283
hypocenterC283#3267
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
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×#22583629C2semidirect
2. #31827114C2semidirect
3. #31827109C2semidirect
4. #31826856C2semidirect
5. #31826723C2semidirect
6. #31826691C2semidirect
7. #31826678C2semidirect
8. #31826668C2semidirect
9. #31826627C2semidirect
10. #31826624C2semidirect
11. #31826616C2semidirect
12. #31826607C2semidirect
13. #31826577C2semidirect
14. #31826534C2semidirect
15. C2×AGL(1,ℤ/8)×C283C2semidirect
17. C2×#17985391C22semidirect
19. C2×Dic1132C22semidirect
20. C2×#17985382C22semidirect
23. C2×#17985380C22semidirect
24. C2×C8×D283C22semidirect
25. #22583608C22semidirect
32. #22583586C22semidirect
38. D4×Dic283C22semidirect
42. #22583542C22semidirect
43. Dic283⋊C8C22semidirect
44. #22583535C22semidirect
45. #22583535C22semidirect
46. #22583534C22semidirect
49. #22583531C22semidirect
51. (C2×D283)⋊C8C22semidirect
52. #22583520C22semidirect
119. D8×C283D4semidirect
208. C283#3267semidirect
210. C2×AGL(1,ℤ/8)D283semidirect
306. D8C22⋊D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836292
#318271142
#318271092
#318268562
#318267232
#318266912
#318266782
#318266682
#318266272
#318266242
#318266162
#318266072
#318265772
#318265342
C2×AGL(1,ℤ/8)×C2832
#3267283