Finite Group Databasebeta

← #49132793

Subgroups — #49132793

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×#22583638C211yesmaximal, characteristic, complemented
2C2×#22583633C211yesmaximal, characteristic, complemented
3#31827117C211yesmaximal, characteristic, complemented
4#31827106C211yesmaximal, characteristic, complemented
5#31826858C211yesmaximal, characteristic, complemented
6#31826695C211yesmaximal, characteristic, complemented
7#31826683C211yesmaximal, characteristic, complemented
8#31826669C211yesmaximal, characteristic, complemented
9#31826630C211yesmaximal, characteristic, complemented
10#31826625C211yesmaximal, characteristic, complemented
11#31826615C211yesmaximal, characteristic, complemented
12#31826613C211yesmaximal, characteristic, complemented
13#31826576C211yesmaximal, characteristic, complemented
14#31826534C211yesmaximal, characteristic, complemented
15#470×C283C211yesmaximal, characteristic, complemented
16C2×#17985393C2211yescharacteristic
17C2×#17985393C2211yescharacteristic, complemented
18C2×#17985392C2211yescharacteristic
19C22×D1132C2211yescharacteristic
20#22583638—24no—
21#22583633—24no—
22C2×#17985382C2211yescharacteristic, complemented
23C2×#17985381C2211yescharacteristic
24C2×#17985381C2211yescharacteristic
25C2×#17985379C2211yescharacteristic
26C2×#17985379C2211yescharacteristic
27C2×#17985379C2211yescharacteristic, complemented
28C2×D2264C2211yescharacteristic
29C2×D2264C2211yescharacteristic
30C2×C8×D283C2211yescharacteristic, complemented
31#22583608C2211yescharacteristic, complemented
32#22583599C2211yescharacteristic
33#22583595—21no—
34#22583595—21no—
35#22583592—21no—
36#22583590C2211yescharacteristic, complemented
37#22583585—21no—
38#22583585C2211yescharacteristic
39#22583583—21no—
40#22583580—21no—
41#22583572C2211yescharacteristic
42#22583572C2211yescharacteristic, complemented
43C4×C22⋊D283C2211yescharacteristic
44C4⋊C4×D283—21no—
45C22⋊C4×D283—21no—
46Dic283⋊C8C2211yescharacteristic, complemented
47#22583533C2211yescharacteristic, complemented
48#22583530C2211yescharacteristic
49#22583530C2211yescharacteristic
50#22583530C2211yescharacteristic, complemented
51C22⋊(C283⋊C8)C2211yescharacteristic
52(C2×D283)⋊C8C2211yescharacteristic, complemented
53#22583520C2211yescharacteristic, complemented
54C2×Q16×C283C2211yescharacteristic, complemented
55C2×M16×C283C2211yescharacteristic
56C22⋊Q8×C283C2211yescharacteristic
57C22⋊Q8×C283C2211yescharacteristic
58#106×C283C2211yescharacteristic, complemented
59#103×C283C2211yescharacteristic, complemented
60#103×C283C2211yescharacteristic, complemented
61C23×D283—21no—
62#17985393—22no—
63#17985393—22no—
64#17985393—41no—
65#17985393—41no—
66#17985392—22no—
67#17985392—41no—
68C2×Dic566C2311yescharacteristic
69C2×D1132—22no—
70C2×D1132—21no—
71C2×D1132—21no—
72C2×D1132—21no—
73C2×D1132—21no—
74C2×D1132—41no—
75C2×D1132C2311yescharacteristic
76C2×D1132C2311yescharacteristic
77C2×D1132C2311yescharacteristic
78C2×C22⋊D283—21no—
79C2×C22⋊D283D411yescharacteristic
80C2×C4×D283—21no—
81C2×C4×D283—21no—
82C2×C4×D283—21no—
83C2×C4×D283—21no—
84C2×C4×D283—21no—
85C2×C4×D283C2311yescharacteristic
86#17985382—41no—
87#17985381—22no—
88#17985381—22no—
89#17985379—22no—
90#17985379—22no—
91#17985379—41no—
92D2264—22no—
93D2264D414yescomplemented
94C2×C283⋊C8C2311yescharacteristic
95C2×C283⋊C8C2311yescharacteristic
96C8×D283—22no—
97C4⋊Dic283—21no—
98Dic283⋊C4—21no—
99Dic283⋊C4—21no—
100Dic283⋊C4D411yescharacteristic
101C22⋊Dic283D411yescharacteristic
102(C2×D283)⋊C4—21no—
103(C2×D283)⋊C4—21no—
104(C2×D283)⋊C4—21no—
105(C2×D283)⋊C4—21no—
106(C2×D283)⋊C4—21no—
107(C2×D283)⋊C4—21no—
108(C2×D283)⋊C4D411yescharacteristic
109C4×Dic283C2311yescharacteristic
110C2×Q8×C283C2311yescharacteristic
111C2×Q8×C283C2311yescharacteristic
112C22×C4×C283C2311yescharacteristic
113Q16×C283—22no—
114M16×C283—22no—
115C2×C8×C283C2311yescharacteristic
116C2×C8×C283C2311yescharacteristic
117C4⋊C4×C283—21no—
118C4⋊C4×C283—21no—
119C4⋊C4×C283C2311yescharacteristic
120C4⋊C4×C283C2311yescharacteristic
121C22⋊C4×C283—21no—
122C22⋊C4×C283—21no—
123C22×D283—21no—
124C22×D283—21no—
125C22×D283—21no—
126C22×D283—42no—
127C22×D283—42no—
128C22×D283—81no—
129C22×D283C2×D411yescharacteristic
130Dic566—21no—
131Dic566—21no—
132D1132—21no—
133D1132—21no—
134D1132—21no—
135D1132—21no—
136D1132—42no—
137D1132—42no—
138D1132—42no—
139D1132—81no—
140D1132C2×D412yes—
141D1132C2×D412yes—
142C22⋊D283—41no—
143C22⋊D283—42no—
144C2×Dic283—21no—
145C2×Dic283C2×D411yescharacteristic
146C2×Dic283C2×D411yescharacteristic
147C2×Dic283C2×D411yescharacteristic
148C4×D283—21no—
149C4×D283—21no—
150C4×D283—42no—
151C4×D283—42no—
152C4×D283—42no—
153C4×D283—42no—
154C4×D283—81no—
155C283⋊C8—21no—
156C283⋊C8—21no—
157C23×C283C2×D411yescharacteristic
158Q8×C283—21no—
159Q8×C283—21no—
160Q8×C283—21no—
161Q8×C283—21no—
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×C283C2×D411yescharacteristic
169C2×C4×C283C2411yescharacteristic
170C8×C283—21no—
171C8×C283C2×D412yes—
172C22×C283—21no—
173C22×C283—21no—
174C22×C283—21no—
175C22×C283C22×D411yescharacteristic
176C2×D283—21no—
177C2×D283—22no—
178C2×D283—22no—
179C2×D283—21no—
180C2×D283—21no—
181C2×D283—41no—
182C2×D283—41no—
183C2×D283—41no—
184C2×D283—41no—
185C2×D283—41no—
186C2×D283—41no—
187C2×D283—41no—
188C2×D283—84no—
189C4×C283—21no—
190C4×C283—41no—
191C4×C283—41no—
192C4×C283—41no—
193C4×C283—41no—
194C4×C283C22×D411yescharacteristic
195C4×C283H(5,𝔽2)11yescharacteristic
196Dic283—21no—
197Dic283—21no—
198Dic283—21no—
199Dic283—41no—
200C2×C283—41no—
201C2×C283D4211yescharacteristic
202C2×C283#57511yescharacteristic
203C2×C283#57711yescharacteristic
204D283—41no—
205D283—42no—
206D283—42no—
207D283—81no—
208C283#326711yescharacteristic, complemented, Sylow
209#3267—2831nomaximal, Sylow
210C2×AGL(1,ℤ/8)—2831no—
211C2×#135—2831no—
212#547—2831no—
213#546—2831no—
214#497—2831no—
215#470D28311yescharacteristic, complemented
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×#41—2831no—
228C22×D4—2831no—
229AGL(1,ℤ/8)—5664no—
230#135—5664no—
231C2×Q16—2831no—
232C2×Q16C2×D28311yescharacteristic, complemented
233C2×SD16—2831no—
234C2×SD16—2831no—
235C2×SD16—2831no—
236C2×SD16—2831no—
237C2×SD16—2831no—
238C2×D8—2831no—
239C2×D8—2831no—
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⋊Q8C2×D28311yescharacteristic
249C22⋊Q8C2×D28311yescharacteristic
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#106C2×D28311yescharacteristic, complemented
260C4⋊C8—2831no—
261#103—2831no—
262#103C2×D28311yescharacteristic, complemented
263#103C2×D28311yescharacteristic, complemented
264#102—2831no—
265#102—2831no—
266#102—2831no—
267C22⋊C8—2831no—
268C22⋊C8—2831no—
269#97—2831no—
270C24—5661no—
271#41—5662no—
272#41—5662no—
273#41—5662no—
274#41—11321no—
275#41—11321no—
276#41—11321no—
277C2×Q8—2831no—
278C2×Q8C22×D28311yescharacteristic
279C2×Q8C22×D28311yescharacteristic
280C2×D4—2831no—
281C2×D4—2831no—
282C2×D4—2831no—
283C2×D4—2831no—
284C2×D4—5662no—
285C2×D4—5661no—
286C2×D4—5661no—
287C2×D4—5661no—
288C2×D4—5661no—
289C2×D4—5661no—
290C2×D4—11321no—
291C22×C4—2831no—
292C22×C4—5661no—
293C22×C4—5661no—
294C22×C4—5661no—
295C22×C4—5661no—
296C22×C4—5661no—
297C22×C4C22×D28311yescharacteristic
298Q16—22no—
299Q16—11321no—
300SD16—5662no—
301SD16—5662no—
302SD16—5662no—
303SD16—5662no—
304SD16—11321no—
305D8—2834no—
306D8—5662no—
307M16—22no—
308C2×C8—2831no—
309C2×C8—2831no—
310C2×C8—5662no—
311C2×C8C22×D28311yescharacteristic
312C2×C8C22×D28311yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—2831no—
316C4⋊C4—5661no—
317C4⋊C4—5661no—
318C4⋊C4—5661no—
319C4⋊C4C22×D28311yescharacteristic
320C4⋊C4C22×D28311yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—2831no—
324C22⋊C4—2831no—
325C22⋊C4—5661no—
326C22⋊C4—5661no—
327C22⋊C4—5661no—
328C22⋊C4—5661no—
329C22⋊C4—5661no—
330C22⋊C4—5661no—
331C42—2831no—
332C23—2831no—
333C23—5661no—
334C23—5661no—
335C23—5661no—
336C23—11322no—
337C23—11322no—
338C23—22641no—
339C23D4×D28311yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—5661no—
345Q8—5661no—
346D4—2832no—
347D4—2832no—
348D4—5661no—
349D4—5661no—
350D4—5661no—
351D4—5661no—
352D4—11322no—
353D4—11322no—
354D4—11322no—
355D4—11321no—
356D4—11322no—
357D4—22641no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—2831no—
365C2×C4—2831no—
366C2×C4—2831no—
367C2×C4—5661no—
368C2×C4—5661no—
369C2×C4—5661no—
370C2×C4—11322no—
371C2×C4—11322no—
372C2×C4—11322no—
373C2×C4—11322no—
374C2×C4—22641no—
375C2×C4D4×D28311yescharacteristic
376C2×C4C23×D28311yescharacteristic
377C8—21no—
378C8—5661no—
379C8—5661no—
380C8D4×D28312yes—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—5661no—
385C22—5661no—
386C22—5661no—
387C22—5662no—
388C22—5662no—
389C22—11321no—
390C22—11321no—
391C22—11321no—
392C22—11321no—
393C22—11321no—
394C22—11321no—
395C22—11321no—
396C22—22644no—
397C22C2×D4×D28311yescharacteristic
398C4—21no—
399C4—41no—
400C4—41no—
401C4—41no—
402C4—41no—
403C4—5661no—
404C4—5661no—
405C4—5661no—
406C4—11321no—
407C4C2×D4×D28311yescharacteristic
408C4#2258365611yescharacteristic
409C2—41no—
410C2—11321no—
411C2—11322no—
412C2—11322no—
413C2—22641no—
414C2#3182709911yescharacteristic
415C2#3182726411yescharacteristic
416C2#3182727411yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#470×C283C2
socleC22×C283C22×D4
hypercenter#470D283
hypocenterC283#3267
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. C2×#22583638C2semidirect
2. C2×#22583633C2semidirect
3. #31827117C2semidirect
4. #31827106C2semidirect
5. #31826858C2semidirect
6. #31826695C2semidirect
7. #31826683C2semidirect
8. #31826669C2semidirect
9. #31826630C2semidirect
10. #31826625C2semidirect
11. #31826615C2semidirect
12. #31826613C2semidirect
13. #31826576C2semidirect
14. #31826534C2semidirect
15. #470×C283C2semidirect
17. C2×#17985393C22semidirect
22. C2×#17985382C22semidirect
27. C2×#17985379C22semidirect
30. C2×C8×D283C22semidirect
31. #22583608C22semidirect
36. #22583590C22semidirect
42. #22583572C22semidirect
46. Dic283⋊C8C22semidirect
47. #22583533C22semidirect
50. #22583530C22semidirect
52. (C2×D283)⋊C8C22semidirect
53. #22583520C22semidirect
54. C2×Q16×C283C22semidirect
58. #106×C283C22semidirect
59. #103×C283C22semidirect
60. #103×C283C22semidirect
93. D2264D4semidirect
208. C283#3267semidirect
215. #470D283semidirect
232. C2×Q16C2×D283semidirect
259. #106C2×D283semidirect
262. #103C2×D283semidirect
263. #103C2×D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836382
C2×#225836332
#318271172
#318271062
#318268582
#318266952
#318266832
#318266692
#318266302
#318266252
#318266152
#318266132
#318265762
#318265342
#470×C2832
#3267283