Finite Group Databasebeta

← #49132789

Subgroups — #49132789

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×#22583635C211yesmaximal, characteristic, complemented
2C2×#22583631C211yesmaximal, characteristic, complemented
3#31827114C211yesmaximal, characteristic, complemented
4#31827106C211yesmaximal, characteristic, complemented
5#31826858C211yesmaximal, characteristic, complemented
6#31826721C211yesmaximal, characteristic, complemented
7#31826697C211yesmaximal, characteristic, complemented
8#31826682C211yesmaximal, characteristic, complemented
9#31826669C211yesmaximal, characteristic, complemented
10#31826626C211yesmaximal, characteristic, complemented
11#31826618C211yesmaximal, characteristic, complemented
12#31826611C211yesmaximal, characteristic, complemented
13#31826574C211yesmaximal, characteristic, complemented
14#31826539C211yesmaximal, characteristic, complemented
15#452×C283C211yesmaximal, characteristic, complemented
16C2×#17985393C2211yescharacteristic
17C2×#17985391C2211yescharacteristic, complemented
18C22×D1132C2211yescharacteristic
19#22583635—24no—
20#22583631—24no—
21C2×#17985382C2211yescharacteristic, complemented
22C2×#17985381C2211yescharacteristic
23C2×#17985381C2211yescharacteristic
24C2×#17985379C2211yescharacteristic
25C2×#17985379C2211yescharacteristic
26C2×#17985377C2211yescharacteristic
27C2×#17985377C2211yescharacteristic
28C2×#17985376C2211yescharacteristic
29C2×C8×D283C2211yescharacteristic, complemented
30#22583608C2211yescharacteristic, complemented
31#22583599C2211yescharacteristic
32#22583595—21no—
33#22583593—21no—
34#22583591—21no—
35#22583589C2211yescharacteristic
36#22583586C2211yescharacteristic, complemented
37#22583585—21no—
38#22583585C2211yescharacteristic
39#22583582—21no—
40#22583580—21no—
41D4×Dic283C2211yescharacteristic, complemented
42#22583572C2211yescharacteristic
43C4×C22⋊D283C2211yescharacteristic
44C2×Dic283⋊C4—21no—
45C22⋊C4×D283—21no—
46#22583541C2211yescharacteristic, complemented
47Dic283⋊C8C2211yescharacteristic, complemented
48#22583534C2211yescharacteristic, complemented
49#22583533C2211yescharacteristic, complemented
50#22583532C2211yescharacteristic
51#22583532C2211yescharacteristic
52(C2×D283)⋊C8C2211yescharacteristic, complemented
53#22583520C2211yescharacteristic, complemented
54C2×#41×C283C2211yescharacteristic
55C2×Q16×C283C2211yescharacteristic, complemented
56C2×SD16×C283C2211yescharacteristic, complemented
57C22⋊Q8×C283C2211yescharacteristic
58#103×C283C2211yescharacteristic, complemented
59#102×C283C2211yescharacteristic, complemented
60C22⋊C8×C283C2211yescharacteristic
61C23×D283—21no—
62#17985393—22no—
63#17985393—41no—
64#17985391—22no—
65#17985391—41no—
66C2×Dic566C2311yescharacteristic
67C2×D1132—22no—
68C2×D1132—21no—
69C2×D1132—21no—
70C2×D1132—21no—
71C2×D1132—41no—
72C2×D1132C2311yescharacteristic
73C2×D1132C2311yescharacteristic
74C2×C22⋊D283—21no—
75C2×C22⋊D283D411yescharacteristic
76C22×Dic283—21no—
77C22×Dic283—21no—
78C2×C4×D283—21no—
79C2×C4×D283—21no—
80C2×C4×D283C2311yescharacteristic
81#17985382—22no—
82#17985381—22no—
83#17985381—41no—
84#17985379—22no—
85#17985379—22no—
86#17985377—22no—
87#17985377D414yescomplemented
88#17985376—22no—
89C2×C283⋊C8C2311yescharacteristic
90C2×C283⋊C8C2311yescharacteristic
91C8×D283—22no—
92C4⋊Dic283C2311yescharacteristic
93Dic283⋊C4—21no—
94Dic283⋊C4—21no—
95Dic283⋊C4—21no—
96Dic283⋊C4—21no—
97Dic283⋊C4D411yescharacteristic
98C22⋊Dic283—21no—
99C22⋊Dic283—21no—
100C22⋊Dic283D411yescharacteristic
101(C2×D283)⋊C4—21no—
102(C2×D283)⋊C4—21no—
103(C2×D283)⋊C4—21no—
104(C2×D283)⋊C4—21no—
105(C2×D283)⋊C4—21no—
106(C2×D283)⋊C4D411yescharacteristic
107C4×Dic283C2311yescharacteristic
108#41×C283—22no—
109#41×C283—41no—
110C2×Q8×C283C2311yescharacteristic
111C2×Q8×C283C2311yescharacteristic
112C2×D4×C283—21no—
113C2×D4×C283C2311yescharacteristic
114C22×C4×C283—21no—
115C22×C4×C283C2311yescharacteristic
116Q16×C283—41no—
117SD16×C283—22no—
118C2×C8×C283C2311yescharacteristic
119C2×C8×C283C2311yescharacteristic
120C4⋊C4×C283—21no—
121C4⋊C4×C283C2311yescharacteristic
122C22⋊C4×C283—21no—
123C22×D283—21no—
124C22×D283—21no—
125C22×D283—42no—
126C22×D283—42no—
127C22×D283—81no—
128C22×D283C2×D411yescharacteristic
129Dic566—21no—
130Dic566—21no—
131D1132—21no—
132D1132—21no—
133D1132—21no—
134D1132—42no—
135D1132—42no—
136D1132—81no—
137D1132C2×D412yes—
138C22⋊D283—41no—
139C22⋊D283—42no—
140C2×Dic283—21no—
141C2×Dic283—21no—
142C2×Dic283—42no—
143C2×Dic283—42no—
144C2×Dic283C2×D411yescharacteristic
145C2×Dic283C2×D411yescharacteristic
146C2×Dic283C2×D411yescharacteristic
147C4×D283—21no—
148C4×D283—21no—
149C4×D283—42no—
150C4×D283—81no—
151C283⋊C8—21no—
152C283⋊C8C2×D412yes—
153C23×C283—21no—
154C23×C283C2×D411yescharacteristic
155Q8×C283—21no—
156Q8×C283—21no—
157Q8×C283—21no—
158Q8×C283—21no—
159D4×C283—21no—
160D4×C283—42no—
161D4×C283C2×D412yes—
162C2×C4×C283—21no—
163C2×C4×C283—21no—
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×C283—21no—
172C22×C283—21no—
173C22×C283—22no—
174C22×C283—21no—
175C22×C283—21no—
176C22×C283—41no—
177C22×C283—41no—
178C22×C283C22×D411yescharacteristic
179C2×D283—21no—
180C2×D283—22no—
181C2×D283—21no—
182C2×D283—21no—
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×C283C22×D411yescharacteristic
194C4×C283H(5,𝔽2)11yescharacteristic
195Dic283—21no—
196Dic283—21no—
197Dic283—21no—
198Dic283—41no—
199Dic283—41no—
200C2×C283—41no—
201C2×C283—42no—
202C2×C283D4211yescharacteristic
203C2×C283#57511yescharacteristic
204C2×C283#57711yescharacteristic
205D283—41no—
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#470—2831no—
216#465—2831no—
217#463—2831no—
218#461—2831no—
219#452—2831no—
220#452D28311yescharacteristic, complemented
221#450—2831no—
222#449—2831no—
223#442—2831no—
224#435—2831no—
225C2×#41—2831no—
226C2×#41—2831no—
227C2×#41C2×D28311yescharacteristic
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×SD16C2×D28311yescharacteristic, complemented
238C2×D8—2831no—
239C2×D8—2831no—
240C2×M16—2831no—
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⋊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#106—2831no—
260C4⋊C8—2831no—
261#103—2831no—
262#103—2831no—
263#103C2×D28311yescharacteristic, complemented
264#102—2831no—
265#102—2831no—
266#102C2×D28311yescharacteristic, complemented
267C22⋊C8—2831no—
268C22⋊C8C2×D28311yescharacteristic
269#97—2831no—
270C24—5661no—
271#41—22no—
272#41—41no—
273#41—5662no—
274#41—5662no—
275#41—11321no—
276#41—11321no—
277C2×Q8—2831no—
278C2×Q8C22×D28311yescharacteristic
279C2×Q8C22×D28311yescharacteristic
280C2×D4—21no—
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—11321no—
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—41no—
299Q16—5662no—
300SD16—22no—
301SD16—5662no—
302SD16—5662no—
303SD16—5662no—
304SD16—11321no—
305D8—2834no—
306D8—5662no—
307M16—5662no—
308C2×C8—2831no—
309C2×C8—2831no—
310C2×C8—5662no—
311C2×C8C22×D28311yescharacteristic
312C2×C8C22×D28311yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—2831no—
315C4⋊C4—2831no—
316C4⋊C4—5661no—
317C4⋊C4—5661no—
318C4⋊C4—5661no—
319C4⋊C4—5661no—
320C4⋊C4C22×D28311yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—2831no—
323C22⋊C4—2831no—
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—2831no—
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—21no—
347D4—42no—
348D4—2832no—
349D4—5661no—
350D4—5661no—
351D4—5661no—
352D4—11322no—
353D4—11322no—
354D4—11321no—
355D4—11322no—
356D4—22641no—
357D4D4×D28312yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—42no—
364C2×C4—2831no—
365C2×C4—2831no—
366C2×C4—2831no—
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—21no—
379C8—2832no—
380C8—5661no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—22no—
385C22—41no—
386C22—41no—
387C22—5661no—
388C22—5661no—
389C22—5661no—
390C22—5662no—
391C22—11321no—
392C22—11321no—
393C22—11321no—
394C22—11321no—
395C22—11321no—
396C22—22644no—
397C22C2×D4×D28311yescharacteristic
398C4—21no—
399C4—41no—
400C4—41no—
401C4—41no—
402C4—5661no—
403C4—5661no—
404C4—5661no—
405C4—11321no—
406C4—11321no—
407C4C2×D4×D28311yescharacteristic
408C4#2258365511yescharacteristic
409C2—41no—
410C2—42no—
411C2—11321no—
412C2—11322no—
413C2—22641no—
414C2#3182709711yescharacteristic
415C2#3182726411yescharacteristic
416C2#3182727211yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#452×C283C2
socleC22×C283C22×D4
hypercenter#452D283
hypocenterC283#3267
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. C2×#22583635C2semidirect
2. C2×#22583631C2semidirect
3. #31827114C2semidirect
4. #31827106C2semidirect
5. #31826858C2semidirect
6. #31826721C2semidirect
7. #31826697C2semidirect
8. #31826682C2semidirect
9. #31826669C2semidirect
10. #31826626C2semidirect
11. #31826618C2semidirect
12. #31826611C2semidirect
13. #31826574C2semidirect
14. #31826539C2semidirect
15. #452×C283C2semidirect
17. C2×#17985391C22semidirect
21. C2×#17985382C22semidirect
29. C2×C8×D283C22semidirect
30. #22583608C22semidirect
36. #22583586C22semidirect
41. D4×Dic283C22semidirect
46. #22583541C22semidirect
47. Dic283⋊C8C22semidirect
48. #22583534C22semidirect
49. #22583533C22semidirect
52. (C2×D283)⋊C8C22semidirect
53. #22583520C22semidirect
55. C2×Q16×C283C22semidirect
56. C2×SD16×C283C22semidirect
58. #103×C283C22semidirect
59. #102×C283C22semidirect
87. #17985377D4semidirect
208. C283#3267semidirect
220. #452D283semidirect
232. C2×Q16C2×D283semidirect
237. C2×SD16C2×D283semidirect
263. #103C2×D283semidirect
266. #102C2×D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836352
C2×#225836312
#318271142
#318271062
#318268582
#318267212
#318266972
#318266822
#318266692
#318266262
#318266182
#318266112
#318265742
#318265392
#452×C2832
#3267283