Finite Group Databasebeta

← #49133223

Subgroups — #49133223

Order 36224.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31827116C211yesmaximal, characteristic, complemented
2#31827103C211yesmaximal, characteristic, complemented
3#31826826C211yesmaximal, characteristic, complemented
4#31826794C211yesmaximal, characteristic, complemented
5#31826783C211yesmaximal, characteristic, complemented
6#31826753C211yesmaximal, characteristic, complemented
7#31826718C211yesmaximal, characteristic, complemented
8#31826697C211yesmaximal, characteristic, complemented
9#31826683C211yesmaximal, characteristic, complemented
10#31826610C211yesmaximal, characteristic, complemented
11#31826600C211yesmaximal, characteristic, complemented
12#31826574C211yesmaximal, characteristic, complemented
13#31826534C211yesmaximal, characteristic, complemented
14#102×D283C211yesmaximal, characteristic, complemented
15#417×C283C211yesmaximal, characteristic, complemented
16C2×#17985392—21no—
17C2×D4×D283C2211yescharacteristic
18C2×#17985379C2211yescharacteristic, complemented
19C2×D2264C2211yescharacteristic
20C2×#17985377C2211yescharacteristic
21C2×C8×D283C2211yescharacteristic
22#22583609C2211yescharacteristic, complemented
23#22583599C2211yescharacteristic
24#22583595—21no—
25#22583593—21no—
26#22583593—21no—
27#22583588—21no—
28#22583586C2211yescharacteristic
29#22583585C2211yescharacteristic
30#22583584C2211yescharacteristic
31#22583584C2211yescharacteristic, complemented
32#22583582—21no—
33#22583578—21no—
34D4×Dic283C2211yescharacteristic
35#22583572C2211yescharacteristic, complemented
36C4×C22⋊D283C2211yescharacteristic
37C4⋊C4×D283C2211yescharacteristic
38C2×(C2×D283)⋊C4—21no—
39#22583542C2211yescharacteristic, complemented
40#22583541C2211yescharacteristic, complemented
41#22583541C2211yescharacteristic, complemented
42Dic283⋊C8C2211yescharacteristic, complemented
43#22583534C2211yescharacteristic, complemented
44#22583532C2211yescharacteristic
45#22583532C2211yescharacteristic, complemented
46#22583531C2211yescharacteristic
47#22583530C2211yescharacteristic
48#22583530C2211yescharacteristic
49C22⋊(C283⋊C8)C2211yescharacteristic, complemented
50(C2×D283)⋊C8C2211yescharacteristic
51#22583520C2211yescharacteristic, complemented
52C2×#41×C283C2211yescharacteristic
53C2×M16×C283C2211yescharacteristic, complemented
54C2×C4⋊C4×C283C2211yescharacteristic, complemented
55#103×C283C2211yescharacteristic, complemented
56#103×C283C2211yescharacteristic, complemented
57#102×C283C2211yescharacteristic
58#102×C283C2211yescharacteristic
59C23×D283—21no—
60#17985392—44no—
61C2×Dic566—21no—
62C2×D1132—21no—
63C2×D1132C2311yescharacteristic
64C2×D1132C2311yescharacteristic
65C2×C22⋊D283—21no—
66C2×C22⋊D283—21no—
67C2×C22⋊D283—21no—
68C2×C22⋊D283D411yescharacteristic
69D4×D283—22no—
70D4×D283—41no—
71C22×Dic283—21no—
72C2×C4×D283—21no—
73C2×C4×D283—21no—
74C2×C4×D283C2311yescharacteristic
75#17985379—41no—
76D2264—41no—
77#17985377—22no—
78C2×C283⋊C8C2311yescharacteristic
79C2×C283⋊C8C2311yescharacteristic
80C8×D283—41no—
81C4⋊Dic283C2311yescharacteristic
82C4⋊Dic283C2311yescharacteristic
83Dic283⋊C4—21no—
84Dic283⋊C4—21no—
85Dic283⋊C4—21no—
86Dic283⋊C4D411yescharacteristic
87C22⋊Dic283—21no—
88C22⋊Dic283D411yescharacteristic
89(C2×D283)⋊C4—21no—
90(C2×D283)⋊C4—21no—
91(C2×D283)⋊C4—21no—
92(C2×D283)⋊C4—21no—
93(C2×D283)⋊C4—21no—
94(C2×D283)⋊C4—21no—
95(C2×D283)⋊C4—21no—
96(C2×D283)⋊C4D411yescharacteristic
97C4×Dic283C2311yescharacteristic
98#41×C283—22no—
99#41×C283—41no—
100C2×Q8×C283C2311yescharacteristic
101C2×D4×C283—21no—
102C2×D4×C283C2311yescharacteristic
103C22×C4×C283—21no—
104C22×C4×C283—21no—
105C22×C4×C283C2311yescharacteristic
106M16×C283—41no—
107C2×C8×C283C2311yescharacteristic
108C2×C8×C283C2311yescharacteristic
109C4⋊C4×C283—21no—
110C4⋊C4×C283C2311yescharacteristic
111C4⋊C4×C283C2311yescharacteristic
112C22×D283—21no—
113C22×D283—21no—
114C22×D283—42no—
115C22×D283—42no—
116C22×D283—81no—
117C22×D283C2×D411yescharacteristic
118Dic566—42no—
119D1132—21no—
120D1132—21no—
121D1132—21no—
122D1132—21no—
123D1132—42no—
124C22⋊D283—22no—
125C22⋊D283—42no—
126C22⋊D283—42no—
127C22⋊D283—81no—
128C2×Dic283—21no—
129C2×Dic283—21no—
130C2×Dic283—81no—
131C2×Dic283C2×D411yescharacteristic
132C2×Dic283C2×D411yescharacteristic
133C2×Dic283C2×D411yescharacteristic
134C4×D283—21no—
135C4×D283—21no—
136C4×D283—42no—
137C4×D283—42no—
138C283⋊C8—21no—
139C283⋊C8—21no—
140C23×C283—21no—
141C23×C283C2×D411yescharacteristic
142Q8×C283—21no—
143Q8×C283—21no—
144D4×C283—21no—
145D4×C283—42no—
146D4×C283#4112yescomplemented
147C2×C4×C283—21no—
148C2×C4×C283—21no—
149C2×C4×C283—21no—
150C2×C4×C283—21no—
151C2×C4×C283—21no—
152C2×C4×C283—42no—
153C2×C4×C283—42no—
154C2×C4×C283C2×D411yescharacteristic
155C2×C4×C283C2411yescharacteristic
156C8×C283—21no—
157C8×C283—21no—
158C22×C283—21no—
159C22×C283—22no—
160C22×C283—21no—
161C22×C283—21no—
162C22×C283—41no—
163C22×C283—41no—
164C22×C283C22×D411yescharacteristic
165C2×D283—21no—
166C2×D283—21no—
167C2×D283—41no—
168C2×D283—41no—
169C2×D283—41no—
170C2×D283—41no—
171C2×D283—41no—
172C2×D283—41no—
173C2×D283—84no—
174C2×D283AGL(1,ℤ/8)12yes—
175C4×C283—21no—
176C4×C283—41no—
177C4×C283—41no—
178C4×C283—41no—
179C4×C283C2×#4111yescharacteristic
180C4×C283H(5,𝔽2)11yescharacteristic
181Dic283—21no—
182Dic283—21no—
183Dic283—21no—
184Dic283—41no—
185Dic283—41no—
186C2×C283—42no—
187C2×C283—41no—
188C2×C283#54611yescharacteristic
189C2×C283C2×AGL(1,ℤ/8)11yescharacteristic
190C2×C283#57711yescharacteristic
191D283—22no—
192D283—81no—
193D283—81no—
194C283#330611yescharacteristic, complemented, Sylow
195#3306—2831nomaximal, Sylow
196#547—2831no—
197#546—2831no—
198#489—2831no—
199#483—2831no—
200#481—2831no—
201#476—2831no—
202#469—2831no—
203#465—2831no—
204#463—2831no—
205#449—2831no—
206#447—2831no—
207#442—2831no—
208#435—2831no—
209#417D28311yescharacteristic, complemented
210C2×#102—2831no—
211C2×#41—5661no—
212C2×#41C2×D28311yescharacteristic
213C22×D4—2831no—
214C2×SD16—2831no—
215C2×D8—2831no—
216C2×D8—2831no—
217C2×M16C2×D28311yescharacteristic, complemented
218C22×C8—2831no—
219C4⋊Q8—2831no—
220#124—2831no—
221#123—5661no—
222#123—5661no—
223#123—5661no—
224C22⋊Q8—2831no—
225C22⋊Q8—5661no—
226#121—2831no—
227#121—2831no—
228#121—2831no—
229#121—5661no—
230#120—5661no—
231C4×D4—2831no—
232C4×D4—2831no—
233C4×D4—2831no—
234C2×C4⋊C4—2831no—
235C2×C4⋊C4C2×D28311yescharacteristic, complemented
236C2×C22⋊C4—5661no—
237#106—2831no—
238#106—2831no—
239#106—2831no—
240C4⋊C8—2831no—
241#103—2831no—
242#103C2×D28311yescharacteristic, complemented
243#103C2×D28311yescharacteristic, complemented
244#102—2831no—
245#102—2831no—
246#102—2831no—
247#102—2831no—
248#102—2831no—
249#102C2×D28311yescharacteristic
250#102C2×D28311yescharacteristic
251C22⋊C8—2831no—
252C22⋊C8—2831no—
253#97—2831no—
254C24—5661no—
255#41—22no—
256#41—41no—
257#41—11324no—
258C2×Q8—5661no—
259C2×Q8C22×D28311yescharacteristic
260C2×D4—21no—
261C2×D4—2831no—
262C2×D4—2831no—
263C2×D4—2831no—
264C2×D4—5661no—
265C2×D4—5661no—
266C2×D4—5662no—
267C2×D4—5661no—
268C2×D4—5661no—
269C2×D4—11321no—
270C2×D4C22×D28311yescharacteristic
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—2831no—
274C22×C4—5661no—
275C22×C4—5661no—
276C22×C4—5661no—
277C22×C4C22×D28311yescharacteristic
278SD16—11321no—
279D8—5662no—
280D8—11321no—
281M16—41no—
282C2×C8—2831no—
283C2×C8—2831no—
284C2×C8—11321no—
285C2×C8C22×D28311yescharacteristic
286C2×C8C22×D28311yescharacteristic
287C4⋊C4—21no—
288C4⋊C4—2831no—
289C4⋊C4—2831no—
290C4⋊C4—2831no—
291C4⋊C4—5661no—
292C4⋊C4—5661no—
293C4⋊C4—5661no—
294C4⋊C4C22×D28311yescharacteristic
295C4⋊C4C22×D28311yescharacteristic
296C22⋊C4—2831no—
297C22⋊C4—2831no—
298C22⋊C4—5661no—
299C22⋊C4—5661no—
300C22⋊C4—5661no—
301C22⋊C4—5661no—
302C22⋊C4—5661no—
303C22⋊C4—5661no—
304C22⋊C4—5661no—
305C22⋊C4—5661no—
306C42—2831no—
307C23—21no—
308C23—2831no—
309C23—5661no—
310C23—5661no—
311C23—11322no—
312C23—11322no—
313C23—22641no—
314C23D4×D28311yescharacteristic
315Q8—21no—
316Q8—21no—
317Q8—11322no—
318D4—21no—
319D4—42no—
320D4—5662no—
321D4—5661no—
322D4—5661no—
323D4—5661no—
324D4—5661no—
325D4—11322no—
326D4—11322no—
327D4—11322no—
328D4—22641no—
329D4#1798539212yescomplemented
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—2831no—
338C2×C4—2831no—
339C2×C4—2831no—
340C2×C4—5661no—
341C2×C4—5661no—
342C2×C4—5661no—
343C2×C4—5661no—
344C2×C4—11322no—
345C2×C4—11322no—
346C2×C4—22641no—
347C2×C4D4×D28311yescharacteristic
348C2×C4C23×D28311yescharacteristic
349C8—21no—
350C8—21no—
351C8—5661no—
352C8—5661no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—22no—
357C22—41no—
358C22—41no—
359C22—2832no—
360C22—5661no—
361C22—5661no—
362C22—11321no—
363C22—11321no—
364C22—11321no—
365C22—11321no—
366C22—11321no—
367C22—11321no—
368C22—22644no—
369C22C2×D4×D28311yescharacteristic
370C4—21no—
371C4—41no—
372C4—41no—
373C4—41no—
374C4—5661no—
375C4—5661no—
376C4—5661no—
377C4—11321no—
378C4—11321no—
379C4C2×#1798539211yescharacteristic
380C4#2258365511yescharacteristic
381C2—41no—
382C2—42no—
383C2—5662no—
384C2—22641no—
385C2—22641no—
386C2#3182710511yescharacteristic
387C2AGL(1,ℤ/8)×D28311yescharacteristic
388C2#3182727411yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#417×C283C2
socleC22×C283C22×D4
hypercenter#417D283
hypocenterC283#3306
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. #31827116C2semidirect
2. #31827103C2semidirect
3. #31826826C2semidirect
4. #31826794C2semidirect
5. #31826783C2semidirect
6. #31826753C2semidirect
7. #31826718C2semidirect
8. #31826697C2semidirect
9. #31826683C2semidirect
10. #31826610C2semidirect
11. #31826600C2semidirect
12. #31826574C2semidirect
13. #31826534C2semidirect
14. #102×D283C2semidirect
15. #417×C283C2semidirect
18. C2×#17985379C22semidirect
22. #22583609C22semidirect
31. #22583584C22semidirect
35. #22583572C22semidirect
39. #22583542C22semidirect
40. #22583541C22semidirect
41. #22583541C22semidirect
42. Dic283⋊C8C22semidirect
43. #22583534C22semidirect
45. #22583532C22semidirect
49. C22⋊(C283⋊C8)C22semidirect
51. #22583520C22semidirect
53. C2×M16×C283C22semidirect
54. C2×C4⋊C4×C283C22semidirect
55. #103×C283C22semidirect
56. #103×C283C22semidirect
146. D4×C283#41semidirect
194. C283#3306semidirect
209. #417D283semidirect
217. C2×M16C2×D283semidirect
235. C2×C4⋊C4C2×D283semidirect
242. #103C2×D283semidirect
243. #103C2×D283semidirect
329. D4#17985392semidirect

Maximal subgroups

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

subgroupindex
#318271162
#318271032
#318268262
#318267942
#318267832
#318267532
#318267182
#318266972
#318266832
#318266102
#318266002
#318265742
#318265342
#102×D2832
#417×C2832
#3306283