Finite Group Databasebeta

← #46822841

Subgroups — #46822841

Order 34432.

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#31771255C211yesmaximal, characteristic, complemented
2#31771252C211yesmaximal, characteristic, complemented
3#31770963C211yesmaximal, characteristic, complemented
4#31770937C211yesmaximal, characteristic, complemented
5#31770921C211yesmaximal, characteristic, complemented
6#31770894C211yesmaximal, characteristic, complemented
7#31770833C211yesmaximal, characteristic, complemented
8#31770818C211yesmaximal, characteristic, complemented
9#31770748C211yesmaximal, characteristic, complemented
10#31770737C211yesmaximal, characteristic, complemented
11#31770714C211yesmaximal, characteristic, complemented
12#31770674C211yesmaximal, characteristic, complemented
13#31770607C211yesmaximal, characteristic, complemented
14#102×D269C211yesmaximal, characteristic, complemented
15#469×C269C211yesmaximal, characteristic, complemented
16C2×#16864402C2211yescharacteristic
17C2×#16864401—21no—
18C2×D4×D269C2211yescharacteristic
19C2×#16864390C2211yescharacteristic, complemented
20C2×#16864387C2211yescharacteristic
21C2×C8×D269C2211yescharacteristic
22#22575452C2211yescharacteristic, complemented
23#22575440C2211yescharacteristic
24#22575435—21no—
25#22575435—21no—
26#22575434—21no—
27#22575433C2211yescharacteristic
28#22575430—21no—
29#22575424—21no—
30#22575424C2211yescharacteristic
31#22575421—21no—
32D4×Dic269C2211yescharacteristic, complemented
33#22575415C2211yescharacteristic
34C4×C22⋊D269C2211yescharacteristic
35C2×C4⋊Dic269C2211yescharacteristic, complemented
36C4⋊C4×D269C2211yescharacteristic
37C22⋊C4×D269—21no—
38#22575384C2211yescharacteristic, complemented
39#22575384C2211yescharacteristic, complemented
40#22575382C2211yescharacteristic, complemented
41#22575378C2211yescharacteristic, complemented
42#22575378C2211yescharacteristic, complemented
43#22575376C2211yescharacteristic, complemented
44#22575375C2211yescharacteristic
45#22575375C2211yescharacteristic
46#22575374C2211yescharacteristic
47#22575374C2211yescharacteristic
48#22575373C2211yescharacteristic
49C22⋊#14524130C2211yescharacteristic, complemented
50#22575365C2211yescharacteristic
51#22575363C2211yescharacteristic, complemented
52C2×D8×C269C2211yescharacteristic
53C2×M16×C269C2211yescharacteristic, complemented
54#121×C269C2211yescharacteristic
55#121×C269C2211yescharacteristic, complemented
56#106×C269C2211yescharacteristic, complemented
57#102×C269C2211yescharacteristic
58#102×C269C2211yescharacteristic, complemented
59C23×D269—21no—
60#16864402—22no—
61#16864402—41no—
62#16864401—44no—
63C2×Dic538—21no—
64C2×Dic538C2311yescharacteristic
65C2×D1076C2311yescharacteristic
66C2×C22⋊D269—21no—
67C2×C22⋊D269—21no—
68C2×C22⋊D269—21no—
69C2×C22⋊D269D411yescharacteristic
70D4×D269—22no—
71D4×D269—41no—
72C22×Dic269—21no—
73C22×Dic269—21no—
74C2×C4×D269—21no—
75C2×C4×D269—21no—
76C2×C4×D269—21no—
77C2×C4×D269C2311yescharacteristic
78#16864390—41no—
79#16864387—22no—
80C2×#14524130C2311yescharacteristic
81C2×#14524130C2311yescharacteristic
82C8×D269—41no—
83C4⋊Dic269—21no—
84C4⋊Dic269—21no—
85C4⋊Dic269C2311yescharacteristic
86C4⋊Dic269C2311yescharacteristic
87#16864381—21no—
88#16864381—21no—
89#16864381D411yescharacteristic
90C22⋊Dic269—21no—
91C22⋊Dic269—21no—
92C22⋊Dic269—21no—
93C22⋊Dic269—21no—
94C22⋊Dic269D411yescharacteristic
95#16864379—21no—
96#16864379—21no—
97#16864379D411yescharacteristic
98C4×Dic269C2311yescharacteristic
99C2×D4×C269—21no—
100C2×D4×C269—21no—
101C2×D4×C269C2311yescharacteristic
102C2×D4×C269C2311yescharacteristic
103C22×C4×C269C2311yescharacteristic
104D8×C269—41no—
105M16×C269—41no—
106C2×C8×C269C2311yescharacteristic
107C2×C8×C269C2311yescharacteristic
108C4⋊C4×C269C2311yescharacteristic
109C4⋊C4×C269C2311yescharacteristic
110C22⋊C4×C269—21no—
111C22⋊C4×C269—21no—
112C22×D269—21no—
113C22×D269—42no—
114C22×D269—42no—
115C22×D269—81no—
116C22×D269C2×D411yescharacteristic
117Dic538—21no—
118Dic538—21no—
119Dic538—42no—
120D1076—21no—
121D1076#4112yescomplemented
122C22⋊D269—22no—
123C22⋊D269—42no—
124C22⋊D269—42no—
125C22⋊D269—42no—
126C2×Dic269—21no—
127C2×Dic269—21no—
128C2×Dic269—21no—
129C2×Dic269—42no—
130C2×Dic269—42no—
131C2×Dic269C2×D411yescharacteristic
132C2×Dic269C2×D411yescharacteristic
133C2×Dic269C2×D411yescharacteristic
134C4×D269—21no—
135C4×D269—21no—
136C4×D269—42no—
137C4×D269—42no—
138C4×D269—81no—
139#14524130—21no—
140#14524130—21no—
141C23×C269—21no—
142C23×C269—21no—
143C23×C269C2×D411yescharacteristic
144D4×C269—21no—
145D4×C269—21no—
146D4×C269—21no—
147D4×C269—21no—
148D4×C269—42no—
149D4×C269—81no—
150C2×C4×C269—21no—
151C2×C4×C269—21no—
152C2×C4×C269—21no—
153C2×C4×C269—21no—
154C2×C4×C269C2×D411yescharacteristic
155C2×C4×C269C2411yescharacteristic
156C8×C269—21no—
157C8×C269—21no—
158C22×C269—21no—
159C22×C269—21no—
160C22×C269—21no—
161C22×C269—41no—
162C22×C269—41no—
163C22×C269—41no—
164C22×C269—41no—
165C22×C269—41no—
166C22×C269—41no—
167C22×C269C22×D411yescharacteristic
168C2×D269—21no—
169C2×D269—21no—
170C2×D269—22no—
171C2×D269—41no—
172C2×D269—41no—
173C2×D269—84no—
174C2×D269AGL(1,ℤ/8)12yes—
175C4×C269—21no—
176C4×C269—41no—
177C4×C269—41no—
178C4×C269C2×#4111yescharacteristic
179C4×C269H(5,𝔽2)11yescharacteristic
180Dic269—21no—
181Dic269—21no—
182Dic269—21no—
183Dic269—41no—
184Dic269—41no—
185Dic269—41no—
186C2×C269—41no—
187C2×C269—81no—
188C2×C269—81no—
189C2×C269#54611yescharacteristic
190C2×C269C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C269#57711yescharacteristic
192D269—22no—
193D269—42no—
194C269#330611yescharacteristic, complemented, Sylow
195#3306—2691nomaximal, Sylow
196#547—2691no—
197#546—2691no—
198#489—2691no—
199#483—2691no—
200#481—2691no—
201#476—2691no—
202#469D26911yescharacteristic, complemented
203#465—2691no—
204#463—2691no—
205#449—2691no—
206#447—2691no—
207#442—2691no—
208#435—2691no—
209#417—2691no—
210C2×#102—2691no—
211C2×#41—2691no—
212C2×#41—5381no—
213C22×D4—2691no—
214C2×SD16—2691no—
215C2×D8—2691no—
216C2×D8C2×D26911yescharacteristic
217C2×M16C2×D26911yescharacteristic, complemented
218C22×C8—2691no—
219C4⋊Q8—2691no—
220#124—2691no—
221#123—5381no—
222#123—5381no—
223#123—5381no—
224C22⋊Q8—2691no—
225C22⋊Q8—5381no—
226#121—2691no—
227#121—5381no—
228#121C2×D26911yescharacteristic
229#121C2×D26911yescharacteristic, complemented
230#120—5381no—
231C4×D4—2691no—
232C4×D4—2691no—
233C4×D4—2691no—
234C2×C4⋊C4—2691no—
235C2×C4⋊C4—2691no—
236C2×C22⋊C4—5381no—
237#106—2691no—
238#106—2691no—
239#106C2×D26911yescharacteristic, complemented
240C4⋊C8—2691no—
241#103—2691no—
242#103—2691no—
243#103—2691no—
244#102—2691no—
245#102—2691no—
246#102—2691no—
247#102—2691no—
248#102—2691no—
249#102C2×D26911yescharacteristic
250#102C2×D26911yescharacteristic, complemented
251C22⋊C8—2691no—
252C22⋊C8—2691no—
253#97—2691no—
254C24—5381no—
255#41—5382no—
256#41—10761no—
257#41—10764no—
258C2×Q8—2691no—
259C2×Q8—5381no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—2691no—
263C2×D4—2691no—
264C2×D4—5381no—
265C2×D4—5382no—
266C2×D4—5381no—
267C2×D4—5381no—
268C2×D4—10761no—
269C2×D4C22×D26911yescharacteristic
270C2×D4C22×D26911yescharacteristic
271C22×C4—2691no—
272C22×C4—5381no—
273C22×C4—5381no—
274C22×C4—5381no—
275C22×C4—5381no—
276C22×C4—5381no—
277C22×C4C22×D26911yescharacteristic
278SD16—10761no—
279D8—41no—
280D8—5382no—
281M16—41no—
282C2×C8—2691no—
283C2×C8—2691no—
284C2×C8—10761no—
285C2×C8C22×D26911yescharacteristic
286C2×C8C22×D26911yescharacteristic
287C4⋊C4—2691no—
288C4⋊C4—2691no—
289C4⋊C4—2691no—
290C4⋊C4—5381no—
291C4⋊C4—5381no—
292C4⋊C4—5381no—
293C4⋊C4—5381no—
294C4⋊C4C22×D26911yescharacteristic
295C4⋊C4C22×D26911yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—2691no—
299C22⋊C4—2691no—
300C22⋊C4—5381no—
301C22⋊C4—5381no—
302C22⋊C4—5381no—
303C22⋊C4—5381no—
304C22⋊C4—5381no—
305C22⋊C4—5381no—
306C42—2691no—
307C23—21no—
308C23—21no—
309C23—2691no—
310C23—5381no—
311C23—10762no—
312C23—10762no—
313C23—21521no—
314C23D4×D26911yescharacteristic
315Q8—5381no—
316Q8—5381no—
317Q8—10762no—
318D4—21no—
319D4—21no—
320D4—21no—
321D4—21no—
322D4—42no—
323D4—81no—
324D4—2692no—
325D4—5382no—
326D4—5381no—
327D4—10762no—
328D4—10762no—
329D4—10762no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—2691no—
335C2×C4—2691no—
336C2×C4—2691no—
337C2×C4—5381no—
338C2×C4—5381no—
339C2×C4—5381no—
340C2×C4—5381no—
341C2×C4—5381no—
342C2×C4—10762no—
343C2×C4—10762no—
344C2×C4—10762no—
345C2×C4—10762no—
346C2×C4—21521no—
347C2×C4D4×D26911yescharacteristic
348C2×C4C23×D26911yescharacteristic
349C8—21no—
350C8—21no—
351C8—5381no—
352C8—5381no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—2692no—
363C22—5381no—
364C22—5381no—
365C22—5382no—
366C22—10761no—
367C22—10761no—
368C22—21524no—
369C22C2×D4×D26911yescharacteristic
370C4—21no—
371C4—41no—
372C4—41no—
373C4—5381no—
374C4—5381no—
375C4—5381no—
376C4—10761no—
377C4—10761no—
378C4—10761no—
379C4#41×D26911yescharacteristic
380C4#2257549811yescharacteristic
381C2—41no—
382C2—81no—
383C2—81no—
384C2—5382no—
385C2—10762no—
386C2#3177124411yescharacteristic
387C2AGL(1,ℤ/8)×D26911yescharacteristic
388C2#3177141511yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#469×C269C2
socleC22×C269C22×D4
hypercenter#469D269
hypocenterC269#3306
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. #31771255C2semidirect
2. #31771252C2semidirect
3. #31770963C2semidirect
4. #31770937C2semidirect
5. #31770921C2semidirect
6. #31770894C2semidirect
7. #31770833C2semidirect
8. #31770818C2semidirect
9. #31770748C2semidirect
10. #31770737C2semidirect
11. #31770714C2semidirect
12. #31770674C2semidirect
13. #31770607C2semidirect
14. #102×D269C2semidirect
15. #469×C269C2semidirect
19. C2×#16864390C22semidirect
22. #22575452C22semidirect
32. D4×Dic269C22semidirect
35. C2×C4⋊Dic269C22semidirect
38. #22575384C22semidirect
39. #22575384C22semidirect
40. #22575382C22semidirect
41. #22575378C22semidirect
42. #22575378C22semidirect
43. #22575376C22semidirect
49. C22⋊#14524130C22semidirect
51. #22575363C22semidirect
53. C2×M16×C269C22semidirect
55. #121×C269C22semidirect
56. #106×C269C22semidirect
58. #102×C269C22semidirect
121. D1076#41semidirect
194. C269#3306semidirect
202. #469D269semidirect
217. C2×M16C2×D269semidirect
229. #121C2×D269semidirect
239. #106C2×D269semidirect
250. #102C2×D269semidirect

Maximal subgroups

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

subgroupindex
#317712552
#317712522
#317709632
#317709372
#317709212
#317708942
#317708332
#317708182
#317707482
#317707372
#317707142
#317706742
#317706072
#102×D2692
#469×C2692
#3306269