Finite Group Databasebeta

← #49132899

Subgroups — #49132899

Order 36224.

498 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×#22583643C211yesmaximal, characteristic, complemented
2C2×#22583634C211yesmaximal, characteristic, complemented
3#31827121C211yesmaximal, characteristic, complemented
4#31827112C211yesmaximal, characteristic, complemented
5#31826848C211yesmaximal, characteristic, complemented
6#31826724C211yesmaximal, characteristic, complemented
7#31826686C211yesmaximal, characteristic, complemented
8#31826679C211yesmaximal, characteristic, complemented
9#31826628C211yesmaximal, characteristic, complemented
10#31826617C211yesmaximal, characteristic, complemented
11#31826608C211yesmaximal, characteristic, complemented
12#31826603C211yesmaximal, characteristic, complemented
13#31826552C211yesmaximal, characteristic, complemented
14#31826541C211yesmaximal, characteristic, complemented
15#469×C283C211yesmaximal, characteristic, complemented
16C2×#17985392C2211yescharacteristic
17C2×#17985391C2211yescharacteristic
18C22×Dic566C2211yescharacteristic, complemented
19C2×D4×D283C2211yescharacteristic
20#22583643—24no—
21#22583634—24no—
22C2×Dic1132C2211yescharacteristic, complemented
23C2×Dic1132C2211yescharacteristic, complemented
24C2×#17985381C2211yescharacteristic
25C2×#17985381C2211yescharacteristic
26C2×#17985380C2211yescharacteristic
27C2×#17985380C2211yescharacteristic
28C2×#17985380C2211yescharacteristic, complemented
29C2×#17985377C2211yescharacteristic
30C2×#17985375C2211yescharacteristic
31#22583597C2211yescharacteristic
32#22583597C2211yescharacteristic, complemented
33#22583596—21no—
34#22583592—21no—
35#22583590C2211yescharacteristic
36#22583590C2211yescharacteristic
37#22583587—21no—
38#22583582—21no—
39#22583581—21no—
40#22583578—21no—
41#22583575C2211yescharacteristic, complemented
42D4×Dic283—21no—
43#22583572C2211yescharacteristic
44C4×C22⋊D283C2211yescharacteristic
45C22⋊C4×D283—21no—
46Dic283⋊C8C2211yescharacteristic, complemented
47#22583533C2211yescharacteristic, complemented
48#22583533C2211yescharacteristic, complemented
49#22583533C2211yescharacteristic, complemented
50#22583530C2211yescharacteristic
51C22⋊(C283⋊C8)C2211yescharacteristic, complemented
52(C2×D283)⋊C8C2211yescharacteristic
53C8×Dic283C2211yescharacteristic, complemented
54C2×D8×C283C2211yescharacteristic
55C2×M16×C283C2211yescharacteristic, complemented
56#121×C283C2211yescharacteristic
57#121×C283C2211yescharacteristic, complemented
58#106×C283C2211yescharacteristic, complemented
59#102×C283C2211yescharacteristic
60#102×C283C2211yescharacteristic, complemented
61C23×D283—21no—
62#17985392—22no—
63#17985392—41no—
64#17985391—22no—
65#17985391—41no—
66C2×Dic566—22no—
67C2×Dic566—21no—
68C2×Dic566—41no—
69C2×Dic566C2311yescharacteristic
70C2×Dic566C2311yescharacteristic
71C2×Dic566C2311yescharacteristic
72C2×D1132C2311yescharacteristic
73C2×C22⋊D283—21no—
74C2×C22⋊D283—21no—
75C2×C22⋊D283—21no—
76C2×C22⋊D283D411yescharacteristic
77D4×D283—22no—
78D4×D283—41no—
79C22×Dic283—21no—
80C22×Dic283—21no—
81C2×C4×D283—21no—
82C2×C4×D283—21no—
83C2×C4×D283C2311yescharacteristic
84Dic1132—22no—
85Dic1132—22no—
86#17985381—22no—
87#17985381D414yescomplemented
88#17985380—22no—
89#17985380—41no—
90#17985380—41no—
91#17985377—22no—
92#17985375—22no—
93C2×C283⋊C8C2311yescharacteristic
94C2×C283⋊C8C2311yescharacteristic
95C4⋊Dic283—21no—
96C4⋊Dic283—21no—
97Dic283⋊C4—21no—
98Dic283⋊C4D411yescharacteristic
99C22⋊Dic283—21no—
100C22⋊Dic283—21no—
101C22⋊Dic283—21no—
102C22⋊Dic283—21no—
103C22⋊Dic283D411yescharacteristic
104(C2×D283)⋊C4—21no—
105(C2×D283)⋊C4—21no—
106(C2×D283)⋊C4—21no—
107(C2×D283)⋊C4D411yescharacteristic
108C4×Dic283—21no—
109C4×Dic283C2311yescharacteristic
110C2×D4×C283—21no—
111C2×D4×C283—21no—
112C2×D4×C283C2311yescharacteristic
113C2×D4×C283C2311yescharacteristic
114C22×C4×C283C2311yescharacteristic
115D8×C283—22no—
116M16×C283—22no—
117C2×C8×C283C2311yescharacteristic
118C2×C8×C283C2311yescharacteristic
119C4⋊C4×C283C2311yescharacteristic
120C4⋊C4×C283C2311yescharacteristic
121C22⋊C4×C283—21no—
122C22⋊C4×C283—21no—
123C22×D283—21no—
124C22×D283—42no—
125C22×D283—42no—
126C22×D283—81no—
127C22×D283C2×D411yescharacteristic
128Dic566—21no—
129Dic566—21no—
130Dic566—21no—
131Dic566—21no—
132Dic566—21no—
133Dic566—42no—
134Dic566C2×D412yes—
135D1132—21no—
136D1132C2×D412yes—
137C22⋊D283—41no—
138C22⋊D283—42no—
139C22⋊D283—42no—
140C22⋊D283—42no—
141C2×Dic283—21no—
142C2×Dic283—21no—
143C2×Dic283—21no—
144C2×Dic283—42no—
145C2×Dic283—42no—
146C2×Dic283C2×D411yescharacteristic
147C2×Dic283C2×D411yescharacteristic
148C2×Dic283C2×D411yescharacteristic
149C4×D283—21no—
150C4×D283—21no—
151C4×D283—42no—
152C4×D283—81no—
153C283⋊C8—21no—
154C283⋊C8—21no—
155C23×C283—21no—
156C23×C283—21no—
157C23×C283C2×D411yescharacteristic
158D4×C283—21no—
159D4×C283—21no—
160D4×C283—21no—
161D4×C283—21no—
162D4×C283—42no—
163D4×C283—81no—
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×C283—41no—
176C22×C283—41no—
177C22×C283—41no—
178C22×C283—41no—
179C22×C283—41no—
180C22×C283—41no—
181C22×C283C22×D411yescharacteristic
182C2×D283—21no—
183C2×D283—22no—
184C2×D283—21no—
185C2×D283—21no—
186C2×D283—41no—
187C2×D283—41no—
188C2×D283—84no—
189C4×C283—21no—
190C4×C283—41no—
191C4×C283—41no—
192C4×C283C22×D411yescharacteristic
193C4×C283H(5,𝔽2)11yescharacteristic
194Dic283—21no—
195Dic283—22no—
196Dic283—21no—
197Dic283—21no—
198Dic283—41no—
199Dic283—41no—
200C2×C283—41no—
201C2×C283—81no—
202C2×C283—81no—
203C2×C283D4211yescharacteristic
204C2×C283#57511yescharacteristic
205C2×C283#57711yescharacteristic
206D283—41no—
207D283—42no—
208C283#327511yescharacteristic, complemented, Sylow
209#3275—2831nomaximal, Sylow
210C2×#137—2831no—
211C2×AGL(1,ℤ/8)—2831no—
212#548—2831no—
213#546—2831no—
214#495—2831no—
215#470—2831no—
216#469D28311yescharacteristic, complemented
217#464—2831no—
218#463—2831no—
219#452—2831no—
220#450—2831no—
221#449—2831no—
222#448—2831no—
223C4×SD16—2831no—
224#436—2831no—
225C2×#41—2831no—
226C2×#41—2831no—
227C22×Q8—2831no—
228C22×D4—2831no—
229#137—5664no—
230AGL(1,ℤ/8)—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×D8C2×D28311yescharacteristic
240C2×M16—2831no—
241C2×M16C2×D28311yescharacteristic, complemented
242#124—2831no—
243#124—2831no—
244#124—5661no—
245#123—5661no—
246C22⋊Q8—2831no—
247C22⋊Q8—2831no—
248C22⋊Q8—5661no—
249#121—5661no—
250#121—5661no—
251#121C2×D28311yescharacteristic
252#121C2×D28311yescharacteristic, complemented
253#120—5661no—
254C4×Q8—2831no—
255C4×D4—2831no—
256C4×D4—2831no—
257C4×D4—5661no—
258C2×C22⋊C4—5661no—
259#106C2×D28311yescharacteristic, complemented
260C4⋊C8—2831no—
261#103—2831no—
262#103—2831no—
263#103—2831no—
264#102—2831no—
265#102C2×D28311yescharacteristic
266#102C2×D28311yescharacteristic, complemented
267C22⋊C8—2831no—
268C22⋊C8—2831no—
269C4×C8—2831no—
270C24—5661no—
271#41—5662no—
272#41—5662no—
273#41—11321no—
274#41—11321no—
275C2×Q8—2831no—
276C2×Q8—2831no—
277C2×Q8—2831no—
278C2×Q8—5662no—
279C2×Q8—5661no—
280C2×Q8—11321no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—2831no—
284C2×D4—2831no—
285C2×D4—5661no—
286C2×D4—5662no—
287C2×D4—5661no—
288C2×D4—5661no—
289C2×D4—11321no—
290C2×D4C22×D28311yescharacteristic
291C2×D4C22×D28311yescharacteristic
292C22×C4—2831no—
293C22×C4—5661no—
294C22×C4—5661no—
295C22×C4—5661no—
296C22×C4—5661no—
297C22×C4C22×D28311yescharacteristic
298Q16—5662no—
299Q16—5662no—
300SD16—2834no—
301SD16—5662no—
302SD16—5662no—
303SD16—11321no—
304SD16—11321no—
305D8—22no—
306D8—5662no—
307M16—22no—
308M16—5662no—
309C2×C8—2831no—
310C2×C8—2831no—
311C2×C8C22×D28311yescharacteristic
312C2×C8C22×D28311yescharacteristic
313C4⋊C4—2831no—
314C4⋊C4—5661no—
315C4⋊C4—5661no—
316C4⋊C4—5661no—
317C4⋊C4C22×D28311yescharacteristic
318C4⋊C4C22×D28311yescharacteristic
319C22⋊C4—21no—
320C22⋊C4—21no—
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—
330C42—2831no—
331C42—5661no—
332C23—21no—
333C23—21no—
334C23—2831no—
335C23—5661no—
336C23—11322no—
337C23—11322no—
338C23—22641no—
339C23D4×D28311yescharacteristic
340Q8—2832no—
341Q8—5661no—
342Q8—5661no—
343Q8—5661no—
344Q8—5661no—
345Q8—5661no—
346Q8—11322no—
347D4—21no—
348D4—21no—
349D4—21no—
350D4—21no—
351D4—42no—
352D4—81no—
353D4—2832no—
354D4—5661no—
355D4—11321no—
356D4—11322no—
357D4—11322no—
358D4—11322no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—2831no—
364C2×C4—2831no—
365C2×C4—2831no—
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—
380C8D4×D28312yes—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—5661no—
391C22—5661no—
392C22—5661no—
393C22—5662no—
394C22—11321no—
395C22—11321no—
396C22—22644no—
397C22C2×D4×D28311yescharacteristic
398C4—21no—
399C4—41no—
400C4—41no—
401C4—5661no—
402C4—5661no—
403C4—5661no—
404C4—5662no—
405C4—11321no—
406C4—11321no—
407C4C2×D4×D28311yescharacteristic
408C4#2258365611yescharacteristic
409C2—41no—
410C2—81no—
411C2—81no—
412C2—11321no—
413C2—11322no—
414C2#3182709911yescharacteristic
415C2#3182726111yescharacteristic
416C2#3182727511yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#469×C283C2
socleC22×C283C22×D4
hypercenter#469D283
hypocenterC283#3275
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. C2×#22583643C2semidirect
2. C2×#22583634C2semidirect
3. #31827121C2semidirect
4. #31827112C2semidirect
5. #31826848C2semidirect
6. #31826724C2semidirect
7. #31826686C2semidirect
8. #31826679C2semidirect
9. #31826628C2semidirect
10. #31826617C2semidirect
11. #31826608C2semidirect
12. #31826603C2semidirect
13. #31826552C2semidirect
14. #31826541C2semidirect
15. #469×C283C2semidirect
18. C22×Dic566C22semidirect
22. C2×Dic1132C22semidirect
23. C2×Dic1132C22semidirect
28. C2×#17985380C22semidirect
32. #22583597C22semidirect
41. #22583575C22semidirect
46. Dic283⋊C8C22semidirect
47. #22583533C22semidirect
48. #22583533C22semidirect
49. #22583533C22semidirect
51. C22⋊(C283⋊C8)C22semidirect
53. C8×Dic283C22semidirect
55. C2×M16×C283C22semidirect
57. #121×C283C22semidirect
58. #106×C283C22semidirect
60. #102×C283C22semidirect
87. #17985381D4semidirect
208. C283#3275semidirect
216. #469D283semidirect
241. C2×M16C2×D283semidirect
252. #121C2×D283semidirect
259. #106C2×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×#225836432
C2×#225836342
#318271212
#318271122
#318268482
#318267242
#318266862
#318266792
#318266282
#318266172
#318266082
#318266032
#318265522
#318265412
#469×C2832
#3275283