Finite Group Databasebeta

← #46822733

Subgroups — #46822733

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#31771303C211yesmaximal, characteristic, complemented
2#31771246C211yesmaximal, characteristic, complemented
3#31770961C211yesmaximal, characteristic, complemented
4#31770932C211yesmaximal, characteristic, complemented
5#31770908C211yesmaximal, characteristic, complemented
6#31770842C211yesmaximal, characteristic, complemented
7#31770836C211yesmaximal, characteristic, complemented
8#31770823C211yesmaximal, characteristic, complemented
9#31770753C211yesmaximal, characteristic, complemented
10#31770741C211yesmaximal, characteristic, complemented
11#31770687C211yesmaximal, characteristic, complemented
12#31770668C211yesmaximal, characteristic, complemented
13#31770600C211yesmaximal, characteristic, complemented
14C2×#22575373C211yesmaximal, characteristic, complemented
15#484×C269C211yesmaximal, characteristic, complemented
16C2×#16864403C2211yescharacteristic
17C22×D1076C2211yescharacteristic
18C2×#16864389C2211yescharacteristic, complemented
19C2×D2152C2211yescharacteristic
20C2×D2152C2211yescharacteristic
21C22×#14524130C2211yescharacteristic
22C2×C8×D269C2211yescharacteristic, complemented
23#22575448—21no—
24#22575444C2211yescharacteristic, complemented
25#22575442C2211yescharacteristic
26#22575438—21no—
27#22575438—21no—
28#22575428—21no—
29#22575428C2211yescharacteristic
30#22575428C2211yescharacteristic, complemented
31#22575427—21no—
32#22575427C2211yescharacteristic
33#22575423—21no—
34#22575415C2211yescharacteristic
35#22575415C2211yescharacteristic, complemented
36C4×C22⋊D269—21no—
37C4×C22⋊D269C2211yescharacteristic
38#22575411C2211yescharacteristic, complemented
39C22⋊C4×D269—21no—
40#22575384C2211yescharacteristic, complemented
41#22575382C2211yescharacteristic, complemented
42#22575377C2211yescharacteristic, complemented
43#22575375C2211yescharacteristic
44#22575375C2211yescharacteristic
45#22575373C2211yescharacteristic
46#22575373C2211yescharacteristic
47#22575373C2211yescharacteristic
48#22575373C2211yescharacteristic
49#22575373C2211yescharacteristic, complemented
50#22575365C2211yescharacteristic, complemented
51C8×Dic269C2211yescharacteristic, complemented
52C22⋊Q8×C269C2211yescharacteristic
53C2×C4⋊C4×C269C2211yescharacteristic
54#107×C269C2211yescharacteristic, complemented
55#107×C269C2211yescharacteristic, complemented
56#103×C269C2211yescharacteristic, complemented
57#103×C269C2211yescharacteristic, complemented
58C22⋊C8×C269C2211yescharacteristic
59C23×D269—21no—
60#16864403—22no—
61#16864403—41no—
62C2×D1076—22no—
63C2×D1076—21no—
64C2×D1076—21no—
65C2×D1076—21no—
66C2×D1076—21no—
67C2×D1076—41no—
68C2×D1076C2311yescharacteristic
69C2×D1076C2311yescharacteristic
70C2×D1076C2311yescharacteristic
71C2×C22⋊D269—21no—
72C2×C22⋊D269D411yescharacteristic
73C2×C4×D269—21no—
74C2×C4×D269—21no—
75C2×C4×D269—21no—
76C2×C4×D269C2311yescharacteristic
77#16864389—41no—
78D2152—22no—
79D2152—41no—
80C2×#14524130—41no—
81C2×#14524130C2311yescharacteristic
82C2×#14524130C2311yescharacteristic
83C8×D269—41no—
84C4⋊Dic269—21no—
85C4⋊Dic269C2311yescharacteristic
86#16864381—21no—
87#16864381D411yescharacteristic
88C22⋊Dic269D411yescharacteristic
89#16864379—21no—
90#16864379—21no—
91#16864379—21no—
92#16864379—21no—
93#16864379—21no—
94#16864379—21no—
95#16864379—21no—
96#16864379—21no—
97#16864379D411yescharacteristic
98C4×Dic269—21no—
99C4×Dic269C2311yescharacteristic
100C2×Q8×C269C2311yescharacteristic
101C22×C4×C269—21no—
102C22×C4×C269C2311yescharacteristic
103C2×C8×C269C2311yescharacteristic
104C2×C8×C269C2311yescharacteristic
105C4⋊C4×C269—21no—
106C4⋊C4×C269—21no—
107C4⋊C4×C269C2311yescharacteristic
108C4⋊C4×C269C2311yescharacteristic
109C4⋊C4×C269C2311yescharacteristic
110C22⋊C4×C269—21no—
111C22×D269—21no—
112C22×D269—21no—
113C22×D269—21no—
114C22×D269—42no—
115C22×D269—42no—
116C22×D269—81no—
117C22×D269C2×D411yescharacteristic
118D1076—21no—
119D1076—21no—
120D1076—21no—
121D1076—21no—
122D1076—21no—
123D1076—42no—
124D1076—42no—
125D1076—81no—
126D1076—81no—
127D1076#4112yes—
128C22⋊D269—22no—
129C22⋊D269—42no—
130C2×Dic269—21no—
131C2×Dic269C2×D411yescharacteristic
132C2×Dic269C2×D411yescharacteristic
133C2×Dic269C2×D411yescharacteristic
134C4×D269—21no—
135C4×D269—21no—
136C4×D269—42no—
137C4×D269—81no—
138C4×D269—81no—
139#14524130—21no—
140#14524130—21no—
141C23×C269C2×D411yescharacteristic
142Q8×C269—21no—
143Q8×C269—21no—
144C2×C4×C269—21no—
145C2×C4×C269—21no—
146C2×C4×C269—21no—
147C2×C4×C269—21no—
148C2×C4×C269—21no—
149C2×C4×C269—21no—
150C2×C4×C269—81no—
151C2×C4×C269C2×D411yescharacteristic
152C2×C4×C269C2411yescharacteristic
153C8×C269—21no—
154C8×C269—21no—
155C22×C269—21no—
156C22×C269—21no—
157C22×C269#13512yes—
158C22×C269C22×D411yescharacteristic
159C2×D269—21no—
160C2×D269—22no—
161C2×D269—21no—
162C2×D269—21no—
163C2×D269—41no—
164C2×D269—41no—
165C2×D269—41no—
166C2×D269—41no—
167C2×D269—41no—
168C2×D269—41no—
169C2×D269—41no—
170C2×D269—41no—
171C2×D269—84no—
172C4×C269—21no—
173C4×C269—41no—
174C4×C269—41no—
175C4×C269—41no—
176C4×C269—41no—
177C4×C269C2×#4111yescharacteristic
178C4×C269H(5,𝔽2)11yescharacteristic
179Dic269—21no—
180Dic269—21no—
181Dic269—21no—
182Dic269—41no—
183C2×C269—22no—
184C2×C269#54611yescharacteristic
185C2×C269C2×#13511yescharacteristic
186C2×C269#57611yescharacteristic
187D269—41no—
188D269—42no—
189D269—81no—
190D269—81no—
191C269#329711yescharacteristic, complemented, Sylow
192#3297—2691nomaximal, Sylow
193#553—2691no—
194#546—2691no—
195#488—2691no—
196#484D26911yescharacteristic, complemented
197#482—2691no—
198#478—2691no—
199#466—2691no—
200#465—2691no—
201#463—2691no—
202#449—2691no—
203#447—2691no—
204C4×D8—2691no—
205C8×D4—2691no—
206#416—2691no—
207C2×#102—2691no—
208C2×#41—2691no—
209C22×D4—2691no—
210C2×SD16—2691no—
211C2×D8—2691no—
212C2×D8—2691no—
213C22×C8—2691no—
214C22×C8—2691no—
215#126—5381no—
216#125—2691no—
217#124—2691no—
218#123—5381no—
219#123—5381no—
220C22⋊Q8C2×D26911yescharacteristic
221#121—2691no—
222#121—2691no—
223#121—2691no—
224#121—5381no—
225#121—5381no—
226#120—5381no—
227C4×D4—2691no—
228C4×D4—2691no—
229C4×D4—2691no—
230C4×D4—5381no—
231#117—2691no—
232C2×C4⋊C4C2×D26911yescharacteristic
233C2×C22⋊C4—5381no—
234#107C2×D26911yescharacteristic, complemented
235#107C2×D26911yescharacteristic, complemented
236#106—2691no—
237C4⋊C8—2691no—
238#103—2691no—
239#103C2×D26911yescharacteristic, complemented
240#103C2×D26911yescharacteristic, complemented
241#102—2691no—
242#102—2691no—
243#102—2691no—
244#102—2691no—
245#102—2691no—
246#102—2691no—
247#102—2691no—
248C22⋊C8—2691no—
249C22⋊C8C2×D26911yescharacteristic
250C4×C8—2691no—
251C24—5381no—
252#41—5382no—
253#41—10761no—
254C2×Q8C22×D26911yescharacteristic
255C2×D4—2691no—
256C2×D4—2691no—
257C2×D4—2691no—
258C2×D4—2691no—
259C2×D4—5382no—
260C2×D4—5381no—
261C2×D4—5381no—
262C2×D4—5381no—
263C2×D4—5381no—
264C2×D4—5381no—
265C2×D4—10761no—
266C22×C4—21no—
267C22×C4—2691no—
268C22×C4—5381no—
269C22×C4—5381no—
270C22×C4—5381no—
271C22×C4C22×D26911yescharacteristic
272SD16—10761no—
273D8—5382no—
274D8—10761no—
275C2×C8—2691no—
276C2×C8—2691no—
277C2×C8—10761no—
278C2×C8—10761no—
279C2×C8C22×D26911yescharacteristic
280C2×C8C22×D26911yescharacteristic
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—2691no—
284C4⋊C4—2691no—
285C4⋊C4—5381no—
286C4⋊C4—5381no—
287C4⋊C4C22×D26911yescharacteristic
288C4⋊C4C22×D26911yescharacteristic
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—2691no—
304C23—5381no—
305C23—5381no—
306C23—5381no—
307C23—10762no—
308C23—10762no—
309C23—21521no—
310C23D4×D26911yescharacteristic
311Q8—21no—
312Q8—21no—
313D4—2692no—
314D4—5382no—
315D4—5381no—
316D4—5381no—
317D4—5381no—
318D4—5381no—
319D4—5381no—
320D4—10762no—
321D4—10762no—
322D4—10762no—
323D4—21521no—
324D4—21521no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—81no—
332C2×C4—2691no—
333C2×C4—2691no—
334C2×C4—2691no—
335C2×C4—5381no—
336C2×C4—5381no—
337C2×C4—5381no—
338C2×C4—10762no—
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—5381no—
350C22—5381no—
351C22—5382no—
352C22—5381no—
353C22—10761no—
354C22—10761no—
355C22—10761no—
356C22—10761no—
357C22—10761no—
358C22—10761no—
359C22—10761no—
360C22—10761no—
361C22—21524no—
362C22#2257547612yes—
363C22C2×D4×D26911yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—5381no—
370C4—5381no—
371C4—5381no—
372C4—10761no—
373C4#41×D26911yescharacteristic
374C4#2257549911yescharacteristic
375C2—22no—
376C2—10761no—
377C2—10762no—
378C2—21521no—
379C2—21521no—
380C2#3177124711yescharacteristic
381C2C2×#2257547611yescharacteristic
382C2#3177140911yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#484×C269C2
socleC22×C269C22×D4
hypercenter#484D269
hypocenterC269#3297
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
1. #31771303C2semidirect
2. #31771246C2semidirect
3. #31770961C2semidirect
4. #31770932C2semidirect
5. #31770908C2semidirect
6. #31770842C2semidirect
7. #31770836C2semidirect
8. #31770823C2semidirect
9. #31770753C2semidirect
10. #31770741C2semidirect
11. #31770687C2semidirect
12. #31770668C2semidirect
13. #31770600C2semidirect
14. C2×#22575373C2semidirect
15. #484×C269C2semidirect
18. C2×#16864389C22semidirect
22. C2×C8×D269C22semidirect
24. #22575444C22semidirect
30. #22575428C22semidirect
35. #22575415C22semidirect
38. #22575411C22semidirect
40. #22575384C22semidirect
41. #22575382C22semidirect
42. #22575377C22semidirect
49. #22575373C22semidirect
50. #22575365C22semidirect
51. C8×Dic269C22semidirect
54. #107×C269C22semidirect
55. #107×C269C22semidirect
56. #103×C269C22semidirect
57. #103×C269C22semidirect
191. C269#3297semidirect
196. #484D269semidirect
234. #107C2×D269semidirect
235. #107C2×D269semidirect
239. #103C2×D269semidirect
240. #103C2×D269semidirect

Maximal subgroups

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

subgroupindex
#317713032
#317712462
#317709612
#317709322
#317709082
#317708422
#317708362
#317708232
#317707532
#317707412
#317706872
#317706682
#317706002
C2×#225753732
#484×C2692
#3297269