Finite Group Databasebeta

← #49132792

Subgroups — #49132792

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×#22583637C211yesmaximal, characteristic, complemented
2C2×#22583631C211yesmaximal, characteristic, complemented
3#31827118C211yesmaximal, characteristic, complemented
4#31827110C211yesmaximal, characteristic, complemented
5#31826857C211yesmaximal, characteristic, complemented
6#31826722C211yesmaximal, characteristic, complemented
7#31826697C211yesmaximal, characteristic, complemented
8#31826681C211yesmaximal, characteristic, complemented
9#31826630C211yesmaximal, characteristic, complemented
10#31826624C211yesmaximal, characteristic, complemented
11#31826614C211yesmaximal, characteristic, complemented
12#31826613C211yesmaximal, characteristic, complemented
13#31826575C211yesmaximal, characteristic, complemented
14#31826535C211yesmaximal, characteristic, complemented
15#461×C283C211yesmaximal, characteristic, complemented
16C2×#17985393C2211yescharacteristic
17C2×#17985393C2211yescharacteristic, complemented
18C2×#17985391C2211yescharacteristic
19C2×D4×D283C2211yescharacteristic
20#22583637—24no—
21#22583631—24no—
22C2×#17985382C2211yescharacteristic, complemented
23C2×#17985382C2211yescharacteristic, complemented
24C2×#17985381C2211yescharacteristic, complemented
25C2×#17985379C2211yescharacteristic
26C2×#17985379C2211yescharacteristic
27C2×D2264C2211yescharacteristic
28C2×#17985377C2211yescharacteristic
29C2×#17985375C2211yescharacteristic
30C2×C8×D283C2211yescharacteristic, complemented
31#22583598C2211yescharacteristic
32#22583595—21no—
33#22583595—21no—
34#22583594—21no—
35#22583590C2211yescharacteristic
36#22583586C2211yescharacteristic
37#22583586C2211yescharacteristic, complemented
38#22583585—21no—
39#22583585C2211yescharacteristic
40#22583584—21no—
41#22583580—21no—
42C4×D1132C2211yescharacteristic
43C4×D1132C2211yescharacteristic, complemented
44C4⋊C4×D283—21no—
45C2×(C2×D283)⋊C4—21no—
46#22583541C2211yescharacteristic, complemented
47#22583535C2211yescharacteristic, complemented
48#22583532C2211yescharacteristic
49#22583531C2211yescharacteristic
50#22583530C2211yescharacteristic, complemented
51(C2×D283)⋊C8C2211yescharacteristic
52(C2×D283)⋊C8C2211yescharacteristic, complemented
53#22583521C2211yescharacteristic, complemented
54C2×SD16×C283C2211yescharacteristic
55C2×SD16×C283C2211yescharacteristic
56C4⋊Q8×C283C2211yescharacteristic, complemented
57C4×D4×C283C2211yescharacteristic
58C4⋊C8×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#17985391—22no—
67#17985391—41no—
68C2×Dic566C2311yescharacteristic
69C2×D1132—21no—
70C2×D1132—21no—
71C2×D1132—21no—
72C2×D1132D411yescharacteristic
73C2×D1132C2311yescharacteristic
74C2×D1132C2311yescharacteristic
75C2×C22⋊D283—21no—
76C2×C22⋊D283—21no—
77D4×D283—22no—
78D4×D283—41no—
79C22×Dic283—21no—
80C2×C4×D283—21no—
81C2×C4×D283—21no—
82C2×C4×D283—21no—
83C2×C4×D283C2311yescharacteristic
84C2×C4×D283C2311yescharacteristic
85#17985382—22no—
86#17985382—41no—
87#17985381—22no—
88#17985379—22no—
89#17985379—41no—
90D2264—22no—
91#17985377D414yescomplemented
92#17985375—22no—
93C2×C283⋊C8C2311yescharacteristic
94C2×C283⋊C8C2311yescharacteristic
95C8×D283—22no—
96C4⋊Dic283—21no—
97C4⋊Dic283D411yescharacteristic
98C4⋊Dic283C2311yescharacteristic
99Dic283⋊C4—21no—
100Dic283⋊C4—21no—
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)⋊C4—21no—
107(C2×D283)⋊C4—21no—
108(C2×D283)⋊C4D411yescharacteristic
109(C2×D283)⋊C4D411yescharacteristic
110C2×Q8×C283C2311yescharacteristic
111C2×Q8×C283C2311yescharacteristic
112C2×D4×C283C2311yescharacteristic
113C22×C4×C283—21no—
114SD16×C283—22no—
115SD16×C283—22no—
116C2×C8×C283C2311yescharacteristic
117C2×C8×C283C2311yescharacteristic
118C4⋊C4×C283—21no—
119C4⋊C4×C283—21no—
120C4⋊C4×C283C2311yescharacteristic
121C22⋊C4×C283—21no—
122C42×C283C2311yescharacteristic
123C22×D283—21no—
124C22×D283—21no—
125C22×D283—42no—
126C22×D283—42no—
127C22×D283—81no—
128C22×D283C2×D411yescharacteristic
129C22×D283C2×D411yescharacteristic
130Dic566—21no—
131Dic566—21no—
132D1132—21no—
133D1132—21no—
134D1132—21no—
135D1132—41no—
136D1132—42no—
137D1132—42no—
138D1132—81no—
139D1132C2×D412yes—
140C22⋊D283—42no—
141C22⋊D283—42no—
142C2×Dic283—21no—
143C2×Dic283—21no—
144C2×Dic283—42no—
145C2×Dic283C2×D411yescharacteristic
146C2×Dic283C2×D411yescharacteristic
147C4×D283—21no—
148C4×D283—21no—
149C4×D283—21no—
150C4×D283—21no—
151C4×D283—42no—
152C4×D283—42no—
153C4×D283—42no—
154C283⋊C8—21no—
155C283⋊C8C2×D412yes—
156C23×C283—21no—
157Q8×C283—21no—
158Q8×C283—21no—
159Q8×C283—21no—
160Q8×C283—21no—
161D4×C283—21no—
162D4×C283C2×D412yes—
163C2×C4×C283—21no—
164C2×C4×C283—21no—
165C2×C4×C283—21no—
166C2×C4×C283—81no—
167C2×C4×C283C2×D411yescharacteristic
168C2×C4×C283C2×D411yescharacteristic
169C2×C4×C283C2411yescharacteristic
170C8×C283—21no—
171C8×C283—21no—
172C22×C283—22no—
173C22×C283—41no—
174C22×C283—41no—
175C22×C283C22×D411yescharacteristic
176C2×D283—21no—
177C2×D283—21no—
178C2×D283—22no—
179C2×D283—21no—
180C2×D283—21no—
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—21no—
191C4×C283—41no—
192C4×C283—41no—
193C4×C283—41no—
194C4×C283C22×D411yescharacteristic
195C4×C283H(5,𝔽2)11yescharacteristic
196Dic283—21no—
197Dic283—21no—
198Dic283—41no—
199Dic283—41no—
200C2×C283—42no—
201C2×C283D4211yescharacteristic
202C2×C283#57511yescharacteristic
203C2×C283#57711yescharacteristic
204D283—41no—
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#461D28311yescharacteristic, complemented
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×Q16—2831no—
233C2×SD16—2831no—
234C2×SD16—2831no—
235C2×SD16—2831no—
236C2×SD16C2×D28311yescharacteristic
237C2×SD16C2×D28311yescharacteristic
238C2×D8—2831no—
239C2×D8—2831no—
240C2×M16—2831no—
241C22×C8—2831no—
242C4⋊Q8C2×D28311yescharacteristic, complemented
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×D4C2×D28311yescharacteristic
257C2×C4⋊C4—5661no—
258C2×C22⋊C4—5661no—
259#106—2831no—
260C4⋊C8C2×D28311yescharacteristic, complemented
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—5661no—
284C2×D4—5661no—
285C2×D4—5661no—
286C2×D4—5661no—
287C2×D4—5662no—
288C2×D4—5661no—
289C2×D4—11321no—
290C2×D4C22×D28311yescharacteristic
291C22×C4—21no—
292C22×C4—2831no—
293C22×C4—2831no—
294C22×C4—5661no—
295C22×C4—5661no—
296C22×C4—5661no—
297C22×C4—5661no—
298Q16—5662no—
299Q16—11321no—
300SD16—22no—
301SD16—22no—
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—21no—
315C4⋊C4—2831no—
316C4⋊C4—2831no—
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—
331C42C22×D28311yescharacteristic
332C23—21no—
333C23—2831no—
334C23—2831no—
335C23—5661no—
336C23—5661no—
337C23—11322no—
338C23—11322no—
339C23—22641no—
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—5661no—
345Q8—5661no—
346D4—21no—
347D4—2832no—
348D4—5661no—
349D4—5661no—
350D4—5661no—
351D4—11321no—
352D4—11322no—
353D4—11322no—
354D4—11322no—
355D4—11322no—
356D4—22641no—
357D4D4×D28312yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—81no—
362C2×C4—2831no—
363C2×C4—2831no—
364C2×C4—5661no—
365C2×C4—5661no—
366C2×C4—5661no—
367C2×C4—5661no—
368C2×C4—5661no—
369C2×C4—5661no—
370C2×C4—11322no—
371C2×C4—11322no—
372C2×C4—11322no—
373C2×C4—11322no—
374C2×C4D4×D28311yescharacteristic
375C2×C4D4×D28311yescharacteristic
376C2×C4C23×D28311yescharacteristic
377C8—21no—
378C8—21no—
379C8—2832no—
380C8—5661no—
381C22—22no—
382C22—41no—
383C22—41no—
384C22—5661no—
385C22—5661no—
386C22—5661no—
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—21no—
400C4—41no—
401C4—41no—
402C4—41no—
403C4—5661no—
404C4—5661no—
405C4—11321no—
406C4—11321no—
407C4C2×D4×D28311yescharacteristic
408C4#2258365611yescharacteristic
409C2—42no—
410C2—11321no—
411C2—11321no—
412C2—11322no—
413C2—22641no—
414C2#3182709911yescharacteristic
415C2#3182726411yescharacteristic
416C2#3182727111yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#461×C283C2
socleC22×C283C22×D4
hypercenter#461D283
hypocenterC283#3267
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. C2×#22583637C2semidirect
2. C2×#22583631C2semidirect
3. #31827118C2semidirect
4. #31827110C2semidirect
5. #31826857C2semidirect
6. #31826722C2semidirect
7. #31826697C2semidirect
8. #31826681C2semidirect
9. #31826630C2semidirect
10. #31826624C2semidirect
11. #31826614C2semidirect
12. #31826613C2semidirect
13. #31826575C2semidirect
14. #31826535C2semidirect
15. #461×C283C2semidirect
17. C2×#17985393C22semidirect
22. C2×#17985382C22semidirect
23. C2×#17985382C22semidirect
24. C2×#17985381C22semidirect
30. C2×C8×D283C22semidirect
37. #22583586C22semidirect
43. C4×D1132C22semidirect
46. #22583541C22semidirect
47. #22583535C22semidirect
50. #22583530C22semidirect
52. (C2×D283)⋊C8C22semidirect
53. #22583521C22semidirect
56. C4⋊Q8×C283C22semidirect
58. C4⋊C8×C283C22semidirect
59. #103×C283C22semidirect
60. #103×C283C22semidirect
91. #17985377D4semidirect
208. C283#3267semidirect
218. #461D283semidirect
242. C4⋊Q8C2×D283semidirect
260. C4⋊C8C2×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×#225836372
C2×#225836312
#318271182
#318271102
#318268572
#318267222
#318266972
#318266812
#318266302
#318266242
#318266142
#318266132
#318265752
#318265352
#461×C2832
#3267283