Finite Group Databasebeta

← #49132787

Subgroups — #49132787

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×#22583630C211yesmaximal, characteristic, complemented
3#31827111C211yesmaximal, characteristic, complemented
4#31826857C211yesmaximal, characteristic, complemented
5#31826721C211yesmaximal, characteristic, complemented
6#31826696C211yesmaximal, characteristic, complemented
7#31826680C211yesmaximal, characteristic, complemented
8#31826667C211yesmaximal, characteristic, complemented
9#31826627C211yesmaximal, characteristic, complemented
10#31826625C211yesmaximal, characteristic, complemented
11#31826615C211yesmaximal, characteristic, complemented
12#31826610C211yesmaximal, characteristic, complemented
13#31826575C211yesmaximal, characteristic, complemented
14#31826537C211yesmaximal, characteristic, complemented
15#547×C283C211yesmaximal, characteristic, complemented
16C2×#17985392C2211yescharacteristic
17C22×D1132C2211yescharacteristic
18#22583635—24no—
19#22583630—24no—
20C2×#17985382C2211yescharacteristic, complemented
21C2×#17985382C2211yescharacteristic, complemented
22C2×#17985380C2211yescharacteristic, complemented
23C2×#17985379C2211yescharacteristic
24C2×#17985379C2211yescharacteristic
25C2×#17985379C2211yescharacteristic
26C2×#17985379C2211yescharacteristic
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#22583541C2211yescharacteristic, complemented
40C4⋊(C283⋊C8)C2211yescharacteristic, complemented
41#22583534C2211yescharacteristic, complemented
42#22583534C2211yescharacteristic, complemented
43#22583533C2211yescharacteristic, 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
52C4⋊Q8×C283C2211yescharacteristic, complemented
53#123×C283—21no—
54#123×C283—21no—
55C22⋊Q8×C283C2211yescharacteristic
56C22⋊Q8×C283C2211yescharacteristic, complemented
57#121×C283—21no—
58C4×D4×C283C2211yescharacteristic
59C4×D4×C283C2211yescharacteristic, complemented
60C2×C4⋊C4×C283—21no—
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#17985382—41no—
76#17985380—22no—
77#17985379—22no—
78#17985379—22no—
79#17985379—22no—
80#17985379—41no—
81#17985377—22no—
82#17985377D414yescomplemented
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×Q8×C283C2311yescharacteristic
102C2×D4×C283—21no—
103C2×D4×C283—21no—
104C2×D4×C283D411yescharacteristic
105C2×D4×C283C2311yescharacteristic
106C22×C4×C283—21no—
107C22×C4×C283—21no—
108C22×C4×C283—21no—
109C22×C4×C283C2311yescharacteristic
110C22×C4×C283C2311yescharacteristic
111C4⋊C4×C283—21no—
112C4⋊C4×C283—21no—
113C4⋊C4×C283—21no—
114C4⋊C4×C283—21no—
115C4⋊C4×C283D411yescharacteristic
116C4⋊C4×C283C2311yescharacteristic
117C22⋊C4×C283—21no—
118C22⋊C4×C283—21no—
119C22⋊C4×C283—21no—
120C22⋊C4×C283D411yescharacteristic
121C22⋊C4×C283D411yescharacteristic
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×C283C2×D411yescharacteristic
147C23×C283C2×D411yescharacteristic
148Q8×C283—21no—
149Q8×C283—21no—
150Q8×C283—21no—
151Q8×C283—21no—
152D4×C283—21no—
153D4×C283—41no—
154D4×C283—42no—
155D4×C283—42no—
156D4×C283C2×D412yes—
157C2×C4×C283—21no—
158C2×C4×C283—21no—
159C2×C4×C283—21no—
160C2×C4×C283—21no—
161C2×C4×C283—21no—
162C2×C4×C283—21no—
163C2×C4×C283—21no—
164C2×C4×C283—42no—
165C2×C4×C283—42no—
166C2×C4×C283—42no—
167C2×C4×C283C2×D411yescharacteristic
168C2×C4×C283C2×D411yescharacteristic
169C2×C4×C283C2×D411yescharacteristic
170C2×C4×C283C2×D411yescharacteristic
171C2×C4×C283C2411yescharacteristic
172C22×C283—21no—
173C22×C283—21no—
174C22×C283—22no—
175C22×C283—21no—
176C22×C283—21no—
177C22×C283—21no—
178C22×C283—21no—
179C22×C283—41no—
180C22×C283—41no—
181C22×C283C22×D411yescharacteristic
182C2×D283—22no—
183C2×D283—41no—
184C2×D283—41no—
185C2×D283—41no—
186C2×D283—41no—
187C2×D283—41no—
188C2×D283—84no—
189C4×C283—21no—
190C4×C283—21no—
191C4×C283—21no—
192C4×C283—21no—
193C4×C283—41no—
194C4×C283—41no—
195C4×C283—41no—
196C4×C283C22×D411yescharacteristic
197C4×C283H(5,𝔽2)11yescharacteristic
198Dic283—41no—
199Dic283—41no—
200C2×C283—41no—
201C2×C283—41no—
202C2×C283—42no—
203C2×C283D4211yescharacteristic
204C2×C283#57511yescharacteristic
205C2×C283#57711yescharacteristic
206D283—42no—
207D283—81no—
208C283#326711yescharacteristic, complemented, Sylow
209#3267—2831nomaximal, Sylow
210C2×AGL(1,ℤ/8)—2831no—
211C2×#135—2831no—
212#547D28311yescharacteristic, complemented
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×#41C2×D28311yescharacteristic
227C2×#41C2×D28311yescharacteristic, complemented
228C22×D4—2831no—
229AGL(1,ℤ/8)—5664no—
230#135—5664no—
231C2×Q16—2831no—
232C2×Q16—2831no—
233C2×SD16—2831no—
234C2×SD16—2831no—
235C2×SD16—2831no—
236C2×SD16—2831no—
237C2×SD16—2831no—
238C2×D8—2831no—
239C2×D8—2831no—
240C2×M16—2831no—
241C22×C8—2831no—
242C4⋊Q8C2×D28311yescharacteristic, complemented
243#124—2831no—
244#123—21no—
245#123—21no—
246#123—5661no—
247C22⋊Q8—2831no—
248C22⋊Q8C2×D28311yescharacteristic
249C22⋊Q8C2×D28311yescharacteristic, complemented
250#121—21no—
251#121—2831no—
252#121—5661no—
253#120—5661no—
254C4×D4—2831no—
255C4×D4C2×D28311yescharacteristic
256C4×D4C2×D28311yescharacteristic, complemented
257C2×C4⋊C4—21no—
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—5661no—
271#41—22no—
272#41—22no—
273#41—41no—
274#41—41no—
275#41—5662no—
276#41—11321no—
277C2×Q8—2831no—
278C2×Q8C22×D28311yescharacteristic
279C2×Q8C22×D28311yescharacteristic
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—2831no—
283C2×D4—2831no—
284C2×D4—5662no—
285C2×D4—5661no—
286C2×D4—5661no—
287C2×D4—5661no—
288C2×D4—11321no—
289C2×D4C22⋊D28311yescharacteristic
290C2×D4C22×D28311yescharacteristic
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—5661no—
295C22×C4—5661no—
296C22×C4C22×D28311yescharacteristic
297C22×C4C22×D28311yescharacteristic
298Q16—5662no—
299Q16—11321no—
300SD16—5662no—
301SD16—5662no—
302SD16—5662no—
303SD16—5662no—
304SD16—11321no—
305D8—2834no—
306D8—5662no—
307M16—5662no—
308C2×C8—2831no—
309C2×C8—2831no—
310C2×C8—2831no—
311C2×C8—2831no—
312C2×C8—5662no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—2831no—
318C4⋊C4—5661no—
319C4⋊C4C22⋊D28311yescharacteristic
320C4⋊C4C22×D28311yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—21no—
324C22⋊C4—5661no—
325C22⋊C4—5661no—
326C22⋊C4—5661no—
327C22⋊C4—5661no—
328C22⋊C4—5661no—
329C22⋊C4C22⋊D28311yescharacteristic
330C22⋊C4C22⋊D28311yescharacteristic
331C42C22×D28311yescharacteristic
332C23—21no—
333C23—5661no—
334C23—5661no—
335C23—11322no—
336C23—11322no—
337C23—22641no—
338C23C2×C22⋊D28311yescharacteristic
339C23C2×C22⋊D28311yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—5661no—
345Q8—5661no—
346D4—21no—
347D4—41no—
348D4—42no—
349D4—42no—
350D4—2832no—
351D4—5661no—
352D4—5661no—
353D4—5661no—
354D4—11322no—
355D4—11322no—
356D4—22641no—
357D4D4×D28312yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—5661no—
369C2×C4—5661no—
370C2×C4—11322no—
371C2×C4—22641no—
372C2×C4C2×C22⋊D28311yescharacteristic
373C2×C4C2×C22⋊D28311yescharacteristic
374C2×C4C2×C22⋊D28311yescharacteristic
375C2×C4C2×C22⋊D28311yescharacteristic
376C2×C4C23×D28311yescharacteristic
377C8—2832no—
378C8—5661no—
379C8—5661no—
380C8—5661no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—22no—
388C22—41no—
389C22—41no—
390C22—5662no—
391C22—11321no—
392C22—11321no—
393C22—11321no—
394C22—11321no—
395C22—11321no—
396C22—22644no—
397C22C22×C22⋊D28311yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—41no—
403C4—41no—
404C4—41no—
405C4—11321no—
406C4—11321no—
407C4C2×D4×D28311yescharacteristic
408C4#2258365511yescharacteristic
409C2—41no—
410C2—41no—
411C2—42no—
412C2—11322no—
413C2—22641no—
414C2D4×C22⋊D28311yescharacteristic
415C2#3182726311yescharacteristic
416C2#3182727311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#547×C283C2
socleC22×C283C22×D4
hypercenter#547D283
hypocenterC283#3267
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. C2×#22583635C2semidirect
2. C2×#22583630C2semidirect
3. #31827111C2semidirect
4. #31826857C2semidirect
5. #31826721C2semidirect
6. #31826696C2semidirect
7. #31826680C2semidirect
8. #31826667C2semidirect
9. #31826627C2semidirect
10. #31826625C2semidirect
11. #31826615C2semidirect
12. #31826610C2semidirect
13. #31826575C2semidirect
14. #31826537C2semidirect
15. #547×C283C2semidirect
20. C2×#17985382C22semidirect
21. C2×#17985382C22semidirect
22. C2×#17985380C22semidirect
30. C22×C283⋊C8C22semidirect
39. #22583541C22semidirect
40. C4⋊(C283⋊C8)C22semidirect
41. #22583534C22semidirect
42. #22583534C22semidirect
43. #22583533C22semidirect
44. #22583531C22semidirect
48. C22⋊(C283⋊C8)C22semidirect
49. #22583521C22semidirect
51. C2×#41×C283C22semidirect
52. C4⋊Q8×C283C22semidirect
56. C22⋊Q8×C283C22semidirect
59. C4×D4×C283C22semidirect
82. #17985377D4semidirect
208. C283#3267semidirect
212. #547D283semidirect
227. C2×#41C2×D283semidirect
242. C4⋊Q8C2×D283semidirect
249. C22⋊Q8C2×D283semidirect
256. C4×D4C2×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
#318268572
#318267212
#318266962
#318266802
#318266672
#318266272
#318266252
#318266152
#318266102
#318265752
#318265372
#547×C2832
#3267283