Finite Group Databasebeta

← #46822728

Subgroups — #46822728

Order 34432.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31771302C211yesmaximal, characteristic, complemented
2#31771243C211yesmaximal, characteristic, complemented
3#31770961C211yesmaximal, characteristic, complemented
4#31770939C211yesmaximal, characteristic, complemented
5#31770930C211yesmaximal, characteristic, complemented
6#31770907C211yesmaximal, characteristic, complemented
7#31770839C211yesmaximal, characteristic, complemented
8#31770831C211yesmaximal, characteristic, complemented
9#31770822C211yesmaximal, characteristic, complemented
10#31770737C211yesmaximal, characteristic, complemented
11D8×Dic269C211yesmaximal, characteristic, complemented
12#31770668C211yesmaximal, characteristic, complemented
13#31770601C211yesmaximal, characteristic, complemented
14#102×D269C211yesmaximal, characteristic, complemented
15#449×C269C211yesmaximal, characteristic, complemented
16C2×D4×D269C2211yescharacteristic
17C2×#16864387C2211yescharacteristic
18C22×#14524130C2211yescharacteristic, complemented
19C2×C8×D269C2211yescharacteristic
20#22575446—21no—
21#22575444C2211yescharacteristic, complemented
22#22575442C2211yescharacteristic
23#22575436—21no—
24#22575434—21no—
25#22575429C2211yescharacteristic
26#22575427C2211yescharacteristic
27#22575425—21no—
28#22575424—21no—
29#22575424C2211yescharacteristic
30#22575421—21no—
31D4×Dic269C2211yescharacteristic
32D4×Dic269C2211yescharacteristic, complemented
33C4×C22⋊D269C2211yescharacteristic
34#22575413—21no—
35#22575412C2211yescharacteristic, complemented
36C4⋊C4×D269C2211yescharacteristic
37C2×#16864379—21no—
38#22575387C2211yescharacteristic, complemented
39#22575386C2211yescharacteristic, complemented
40#22575385C2211yescharacteristic, complemented
41#22575382C2211yescharacteristic, complemented
42#22575377C2211yescharacteristic, complemented
43#22575377C2211yescharacteristic, complemented
44#22575375C2211yescharacteristic
45#22575375C2211yescharacteristic
46#22575374C2211yescharacteristic
47#22575374C2211yescharacteristic
48#22575374C2211yescharacteristic, complemented
49#22575373C2211yescharacteristic
50#22575365C2211yescharacteristic
51C8×Dic269C2211yescharacteristic, complemented
52C2×#41×C269C2211yescharacteristic
53C2×SD16×C269C2211yescharacteristic, complemented
54C2×D8×C269C2211yescharacteristic
55#121×C269C2211yescharacteristic, complemented
56#103×C269C2211yescharacteristic, complemented
57#102×C269C2211yescharacteristic
58C22⋊C8×C269C2211yescharacteristic, complemented
59C23×D269—21no—
60C2×D1076C2311yescharacteristic
61C2×C22⋊D269—21no—
62C2×C22⋊D269—21no—
63C2×C22⋊D269—21no—
64C2×C22⋊D269D411yescharacteristic
65D4×D269—22no—
66D4×D269—41no—
67C22×Dic269—21no—
68C22×Dic269—21no—
69C2×C4×D269—21no—
70C2×C4×D269C2311yescharacteristic
71#16864387—41no—
72C2×#14524130—41no—
73C2×#14524130C2311yescharacteristic
74C2×#14524130C2311yescharacteristic
75C8×D269—41no—
76C4⋊Dic269—21no—
77C4⋊Dic269C2311yescharacteristic
78C4⋊Dic269C2311yescharacteristic
79C4⋊Dic269C2311yescharacteristic
80#16864381—21no—
81#16864381—21no—
82#16864381—21no—
83#16864381D411yescharacteristic
84C22⋊Dic269—21no—
85C22⋊Dic269—21no—
86C22⋊Dic269—21no—
87C22⋊Dic269—21no—
88C22⋊Dic269D411yescharacteristic
89#16864379—21no—
90#16864379—21no—
91#16864379—21no—
92#16864379—21no—
93#16864379D411yescharacteristic
94C4×Dic269—21no—
95C4×Dic269C2311yescharacteristic
96#41×C269—22no—
97#41×C269—41no—
98C2×Q8×C269C2311yescharacteristic
99C2×D4×C269—21no—
100C2×D4×C269—21no—
101C2×D4×C269C2311yescharacteristic
102C2×D4×C269C2311yescharacteristic
103C22×C4×C269—21no—
104C22×C4×C269C2311yescharacteristic
105SD16×C269—41no—
106D8×C269—22no—
107C2×C8×C269C2311yescharacteristic
108C2×C8×C269C2311yescharacteristic
109C4⋊C4×C269C2311yescharacteristic
110C22⋊C4×C269—21no—
111C22×D269—21no—
112C22×D269—42no—
113C22×D269—42no—
114C22×D269—81no—
115C22×D269C2×D411yescharacteristic
116D1076—21no—
117D1076—21no—
118C22⋊D269—22no—
119C22⋊D269—42no—
120C22⋊D269—42no—
121C22⋊D269—81no—
122C2×Dic269—21no—
123C2×Dic269—21no—
124C2×Dic269—21no—
125C2×Dic269—81no—
126C2×Dic269—81no—
127C2×Dic269C2×D411yescharacteristic
128C2×Dic269C2×D411yescharacteristic
129C2×Dic269C2×D411yescharacteristic
130C4×D269—21no—
131C4×D269—21no—
132C4×D269—81no—
133#14524130—21no—
134#14524130—21no—
135C23×C269—21no—
136C23×C269—21no—
137C23×C269C2×D411yescharacteristic
138Q8×C269—21no—
139Q8×C269—21no—
140D4×C269—21no—
141D4×C269—21no—
142D4×C269—21no—
143D4×C269—42no—
144D4×C269—81no—
145D4×C269#4112yes—
146C2×C4×C269—21no—
147C2×C4×C269—21no—
148C2×C4×C269—21no—
149C2×C4×C269—21no—
150C2×C4×C269—42no—
151C2×C4×C269C2×D411yescharacteristic
152C2×C4×C269C2411yescharacteristic
153C8×C269—21no—
154C8×C269—21no—
155C22×C269—21no—
156C22×C269—22no—
157C22×C269—21no—
158C22×C269—21no—
159C22×C269—41no—
160C22×C269—41no—
161C22×C269—41no—
162C22×C269—41no—
163C22×C269—41no—
164C22×C269C22×D411yescharacteristic
165C2×D269—21no—
166C2×D269—21no—
167C2×D269—41no—
168C2×D269—41no—
169C2×D269—41no—
170C2×D269—84no—
171C2×D269#13512yes—
172C4×C269—21no—
173C4×C269—41no—
174C4×C269—41no—
175C4×C269C2×#4111yescharacteristic
176C4×C269H(5,𝔽2)11yescharacteristic
177Dic269—21no—
178Dic269—21no—
179Dic269—21no—
180Dic269—41no—
181Dic269—41no—
182Dic269—41no—
183C2×C269—42no—
184C2×C269—41no—
185C2×C269—81no—
186C2×C269#54611yescharacteristic
187C2×C269C2×#13511yescharacteristic
188C2×C269#57611yescharacteristic
189D269—22no—
190D269—81no—
191C269#329711yescharacteristic, complemented, Sylow
192#3297—2691nomaximal, Sylow
193#553—2691no—
194#546—2691no—
195#488—2691no—
196#484—2691no—
197#482—2691no—
198#478—2691no—
199#466—2691no—
200#465—2691no—
201#463—2691no—
202#449D26911yescharacteristic, complemented
203#447—2691no—
204C4×D8—2691no—
205C8×D4—2691no—
206#416—2691no—
207C2×#102—2691no—
208C2×#41C2×D26911yescharacteristic
209C22×D4—2691no—
210C2×SD16C2×D26911yescharacteristic, complemented
211C2×D8—2691no—
212C2×D8C2×D26911yescharacteristic
213C22×C8—2691no—
214C22×C8—2691no—
215#126—5381no—
216#125—2691no—
217#124—2691no—
218#123—5381no—
219#123—5381no—
220C22⋊Q8—2691no—
221#121—2691no—
222#121—2691no—
223#121—5381no—
224#121—5381no—
225#121C2×D26911yescharacteristic, complemented
226#120—5381no—
227C4×D4—2691no—
228C4×D4—2691no—
229C4×D4—2691no—
230C4×D4—5381no—
231#117—2691no—
232C2×C4⋊C4—2691no—
233C2×C22⋊C4—5381no—
234#107—2691no—
235#107—2691no—
236#106—2691no—
237C4⋊C8—2691no—
238#103—2691no—
239#103—2691no—
240#103C2×D26911yescharacteristic, complemented
241#102—2691no—
242#102—2691no—
243#102—2691no—
244#102—2691no—
245#102—2691no—
246#102—2691no—
247#102C2×D26911yescharacteristic
248C22⋊C8—2691no—
249C22⋊C8C2×D26911yescharacteristic, complemented
250C4×C8—2691no—
251C24—5381no—
252#41—22no—
253#41—41no—
254C2×Q8C22×D26911yescharacteristic
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—2691no—
258C2×D4—2691no—
259C2×D4—5381no—
260C2×D4—5382no—
261C2×D4—5381no—
262C2×D4—5381no—
263C2×D4—10761no—
264C2×D4C22×D26911yescharacteristic
265C2×D4C22×D26911yescharacteristic
266C22×C4—21no—
267C22×C4—2691no—
268C22×C4—5381no—
269C22×C4—5381no—
270C22×C4—5381no—
271C22×C4C22×D26911yescharacteristic
272SD16—41no—
273D8—22no—
274D8—10761no—
275C2×C8—2691no—
276C2×C8—2691no—
277C2×C8—10761no—
278C2×C8—10761no—
279C2×C8C22×D26911yescharacteristic
280C2×C8C22×D26911yescharacteristic
281C4⋊C4—2691no—
282C4⋊C4—2691no—
283C4⋊C4—2691no—
284C4⋊C4—2691no—
285C4⋊C4—5381no—
286C4⋊C4—5381no—
287C4⋊C4—5381no—
288C4⋊C4—5381no—
289C4⋊C4C22×D26911yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—2691no—
292C22⋊C4—2691no—
293C22⋊C4—5381no—
294C22⋊C4—5381no—
295C22⋊C4—5381no—
296C22⋊C4—5381no—
297C22⋊C4—5381no—
298C22⋊C4—5381no—
299C22⋊C4—5381no—
300C22⋊C4—5381no—
301C42—2691no—
302C42—5381no—
303C23—21no—
304C23—21no—
305C23—2691no—
306C23—5381no—
307C23—10762no—
308C23—10762no—
309C23—21521no—
310C23D4×D26911yescharacteristic
311Q8—21no—
312Q8—21no—
313D4—21no—
314D4—21no—
315D4—21no—
316D4—42no—
317D4—81no—
318D4—5382no—
319D4—5381no—
320D4—5381no—
321D4—10762no—
322D4—10762no—
323D4—21521no—
324D4#1686440112yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—42no—
330C2×C4—2691no—
331C2×C4—2691no—
332C2×C4—2691no—
333C2×C4—5381no—
334C2×C4—5381no—
335C2×C4—5381no—
336C2×C4—5381no—
337C2×C4—5381no—
338C2×C4—21521no—
339C2×C4—21521no—
340C2×C4—21521no—
341C2×C4D4×D26911yescharacteristic
342C2×C4C23×D26911yescharacteristic
343C8—21no—
344C8—21no—
345C8—5381no—
346C8—5381no—
347C22—21no—
348C22—21no—
349C22—22no—
350C22—21no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—2692no—
357C22—5381no—
358C22—5381no—
359C22—10761no—
360C22—10761no—
361C22—10761no—
362C22—21524no—
363C22C2×D4×D26911yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—5381no—
368C4—5381no—
369C4—5381no—
370C4—10761no—
371C4—10761no—
372C4—10761no—
373C4C2×#1686440111yescharacteristic
374C4#2257549811yescharacteristic
375C2—41no—
376C2—42no—
377C2—81no—
378C2—5382no—
379C2—21521no—
380C2#3177124211yescharacteristic
381C2#135×D26911yescharacteristic
382C2#3177140911yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#449×C269C2
socleC22×C269C22×D4
hypercenter#449D269
hypocenterC269#3297
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #31771302C2semidirect
2. #31771243C2semidirect
3. #31770961C2semidirect
4. #31770939C2semidirect
5. #31770930C2semidirect
6. #31770907C2semidirect
7. #31770839C2semidirect
8. #31770831C2semidirect
9. #31770822C2semidirect
10. #31770737C2semidirect
11. D8×Dic269C2semidirect
12. #31770668C2semidirect
13. #31770601C2semidirect
14. #102×D269C2semidirect
15. #449×C269C2semidirect
18. C22×#14524130C22semidirect
21. #22575444C22semidirect
32. D4×Dic269C22semidirect
35. #22575412C22semidirect
38. #22575387C22semidirect
39. #22575386C22semidirect
40. #22575385C22semidirect
41. #22575382C22semidirect
42. #22575377C22semidirect
43. #22575377C22semidirect
48. #22575374C22semidirect
51. C8×Dic269C22semidirect
53. C2×SD16×C269C22semidirect
55. #121×C269C22semidirect
56. #103×C269C22semidirect
58. C22⋊C8×C269C22semidirect
191. C269#3297semidirect
202. #449D269semidirect
210. C2×SD16C2×D269semidirect
225. #121C2×D269semidirect
240. #103C2×D269semidirect
249. C22⋊C8C2×D269semidirect

Maximal subgroups

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

subgroupindex
#317713022
#317712432
#317709612
#317709392
#317709302
#317709072
#317708392
#317708312
#317708222
#317707372
D8×Dic2692
#317706682
#317706012
#102×D2692
#449×C2692
#3297269