Finite Group Databasebeta

← #46822737

Subgroups — #46822737

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#31771301C211yesmaximal, characteristic, complemented
2#31771251C211yesmaximal, characteristic, complemented
3#31770960C211yesmaximal, characteristic, complemented
4#31770941C211yesmaximal, characteristic, complemented
5#31770930C211yesmaximal, characteristic, complemented
6#31770906C211yesmaximal, characteristic, complemented
7#31770841C211yesmaximal, characteristic, complemented
8#31770834C211yesmaximal, characteristic, complemented
9#31770821C211yesmaximal, characteristic, complemented
10#31770752C211yesmaximal, characteristic, complemented
11#31770741C211yesmaximal, characteristic, complemented
12C4×D2152C211yesmaximal, characteristic, complemented
13#31770602C211yesmaximal, characteristic, complemented
14C2×#22575375C211yesmaximal, characteristic, complemented
15C8×D4×C269C211yesmaximal, characteristic, complemented
16C2×#16864402C2211yescharacteristic
17C22×D1076C2211yescharacteristic
18C2×#16864391C2211yescharacteristic, complemented
19C2×D2152C2211yescharacteristic
20C2×D2152C2211yescharacteristic
21#22575448—21no—
22#22575445C2211yescharacteristic, complemented
23#22575441C2211yescharacteristic
24#22575438—21no—
25#22575435—21no—
26#22575432C2211yescharacteristic
27#22575427C2211yescharacteristic
28#22575427C2211yescharacteristic
29#22575427C2211yescharacteristic, complemented
30#22575426—21no—
31#22575425—21no—
32#22575421—21no—
33C4×D1076C2211yescharacteristic
34C4×D1076C2211yescharacteristic, complemented
35#22575413—21no—
36#22575412C2211yescharacteristic, complemented
37C2×C4⋊Dic269C2211yescharacteristic
38C22⋊C4×D269—21no—
39#22575387C2211yescharacteristic, complemented
40#22575387C2211yescharacteristic, complemented
41#22575385C2211yescharacteristic, complemented
42#22575378C2211yescharacteristic, complemented
43#22575378C2211yescharacteristic, complemented
44#22575378C2211yescharacteristic, complemented
45#22575375C2211yescharacteristic
46#22575375C2211yescharacteristic
47#22575375C2211yescharacteristic
48#22575375C2211yescharacteristic
49#22575375C2211yescharacteristic
50#22575375C2211yescharacteristic
51#22575375C2211yescharacteristic, complemented
52C22×C8×C269C2211yescharacteristic
53C22×C8×C269C2211yescharacteristic, complemented
54C4×D4×C269C2211yescharacteristic
55C4⋊C8×C269C2211yescharacteristic, complemented
56C22⋊C8×C269C2211yescharacteristic
57C22⋊C8×C269C2211yescharacteristic, complemented
58C4×C8×C269C2211yescharacteristic, complemented
59C23×D269—21no—
60#16864402—22no—
61#16864402—41no—
62C2×Dic538C2311yescharacteristic
63C2×D1076—22no—
64C2×D1076—21no—
65C2×D1076—41no—
66C2×D1076C2311yescharacteristic
67C2×D1076C2311yescharacteristic
68C2×D1076C2311yescharacteristic
69C2×C22⋊D269—21no—
70C2×C22⋊D269—21no—
71C2×C22⋊D269—21no—
72C2×C22⋊D269—21no—
73C22×Dic269—21no—
74C2×C4×D269—21no—
75C2×C4×D269—21no—
76C2×C4×D269—21no—
77#16864391—41no—
78D2152—22no—
79D2152—41no—
80C4⋊Dic269—21no—
81C4⋊Dic269C2311yescharacteristic
82C4⋊Dic269C2311yescharacteristic
83C4⋊Dic269C2311yescharacteristic
84C4⋊Dic269C2311yescharacteristic
85#16864381—21no—
86#16864381—21no—
87#16864381—21no—
88C22⋊Dic269—21no—
89C22⋊Dic269—21no—
90#16864379—21no—
91#16864379—21no—
92#16864379—21no—
93#16864379—21no—
94#16864379—21no—
95#16864379—21no—
96#16864379—21no—
97C4×Dic269—21no—
98C2×D4×C269D411yescharacteristic
99C22×C4×C269C2311yescharacteristic
100C22×C4×C269C2311yescharacteristic
101C2×C8×C269—41no—
102C2×C8×C269—41no—
103C2×C8×C269C2311yescharacteristic
104C2×C8×C269C2311yescharacteristic
105C2×C8×C269C2311yescharacteristic
106C2×C8×C269C2311yescharacteristic
107C4⋊C4×C269D411yescharacteristic
108C22⋊C4×C269D411yescharacteristic
109C22⋊C4×C269D411yescharacteristic
110C42×C269C2311yescharacteristic
111C22×D269—21no—
112C22×D269—21no—
113C22×D269—21no—
114C22×D269—42no—
115C22×D269—42no—
116C22×D269—81no—
117Dic538—21no—
118Dic538—21no—
119D1076—21no—
120D1076—21no—
121D1076—21no—
122D1076—21no—
123D1076—21no—
124D1076—42no—
125D1076#4112yes—
126C22⋊D269—42no—
127C22⋊D269—42no—
128C22⋊D269—81no—
129C22⋊D269—81no—
130C2×Dic269—21no—
131C2×Dic269—21no—
132C2×Dic269—21no—
133C2×Dic269—21no—
134C2×Dic269—21no—
135C2×Dic269—81no—
136C4×D269—42no—
137C4×D269—81no—
138C4×D269—81no—
139C23×C269C2×D411yescharacteristic
140C23×C269C2×D411yescharacteristic
141D4×C269—22no—
142C2×C4×C269—21no—
143C2×C4×C269—21no—
144C2×C4×C269—21no—
145C2×C4×C269—21no—
146C2×C4×C269C2×D411yescharacteristic
147C2×C4×C269C2×D411yescharacteristic
148C2×C4×C269C2×D411yescharacteristic
149C2×C4×C269C2×D411yescharacteristic
150C2×C4×C269C2411yescharacteristic
151C8×C269—21no—
152C8×C269—21no—
153C8×C269—21no—
154C8×C269—21no—
155C22×C269—21no—
156C22×C269—21no—
157C22×C269—21no—
158C22×C269—21no—
159C22×C269—21no—
160C22×C269#13512yes—
161C22×C269C22×D411yescharacteristic
162C2×D269—22no—
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—21no—
174C4×C269—21no—
175C4×C269—21no—
176C4×C269C2×#4111yescharacteristic
177C4×C269H(5,𝔽2)11yescharacteristic
178Dic269—41no—
179Dic269—41no—
180Dic269—41no—
181Dic269—41no—
182Dic269—41no—
183C2×C269—22no—
184C2×C269—41no—
185C2×C269#54611yescharacteristic
186C2×C269C2×#13511yescharacteristic
187C2×C269#57611yescharacteristic
188D269—42no—
189D269—81no—
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#449—2691no—
203#447—2691no—
204C4×D8—2691no—
205C8×D4D26911yescharacteristic, complemented
206#416—2691no—
207C2×#102—2691no—
208C2×#41—2691no—
209C22×D4—2691no—
210C2×SD16—2691no—
211C2×D8—2691no—
212C2×D8—2691no—
213C22×C8C2×D26911yescharacteristic
214C22×C8C2×D26911yescharacteristic, complemented
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—2691no—
224#121—5381no—
225#121—5381no—
226#120—5381no—
227C4×D4—2691no—
228C4×D4—2691no—
229C4×D4—5381no—
230C4×D4C2×D26911yescharacteristic
231#117—2691no—
232C2×C4⋊C4—2691no—
233C2×C22⋊C4—5381no—
234#107—2691no—
235#107—2691no—
236#106—2691no—
237C4⋊C8C2×D26911yescharacteristic, complemented
238#103—2691no—
239#103—2691no—
240#103—2691no—
241#102—2691no—
242#102—2691no—
243#102—2691no—
244#102—2691no—
245#102—2691no—
246#102—2691no—
247#102—2691no—
248C22⋊C8C2×D26911yescharacteristic
249C22⋊C8C2×D26911yescharacteristic, complemented
250C4×C8C2×D26911yescharacteristic, complemented
251C24—5381no—
252#41—5382no—
253#41—10761no—
254C2×Q8—2691no—
255C2×D4—2691no—
256C2×D4—2691no—
257C2×D4—2691no—
258C2×D4—5382no—
259C2×D4—5381no—
260C2×D4—5381no—
261C2×D4—5381no—
262C2×D4—5381no—
263C2×D4—5381no—
264C2×D4—10761no—
265C2×D4D107611yescharacteristic
266C22×C4—5381no—
267C22×C4—5381no—
268C22×C4—5381no—
269C22×C4—5381no—
270C22×C4C22×D26911yescharacteristic
271C22×C4C22×D26911yescharacteristic
272SD16—10761no—
273D8—5382no—
274D8—10761no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8C22×D26911yescharacteristic
278C2×C8C22×D26911yescharacteristic
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⋊C4D107611yescharacteristic
290C22⋊C4—5381no—
291C22⋊C4—5381no—
292C22⋊C4—5381no—
293C22⋊C4—5381no—
294C22⋊C4—5381no—
295C22⋊C4—5381no—
296C22⋊C4—5381no—
297C22⋊C4—5381no—
298C22⋊C4—5381no—
299C22⋊C4D107611yescharacteristic
300C22⋊C4D107611yescharacteristic
301C42—5381no—
302C42C22×D26911yescharacteristic
303C23—5381no—
304C23—5381no—
305C23—5381no—
306C23—10762no—
307C23—10762no—
308C23—21521no—
309C23C2×D107611yescharacteristic
310C23C2×D107611yescharacteristic
311Q8—5381no—
312Q8—5381no—
313D4—22no—
314D4—2692no—
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—5381no—
330C2×C4—5381no—
331C2×C4—5381no—
332C2×C4—5381no—
333C2×C4—5381no—
334C2×C4—10762no—
335C2×C4—21521no—
336C2×C4—21521no—
337C2×C4—21521no—
338C2×C4C2×D107611yescharacteristic
339C2×C4C2×D107611yescharacteristic
340C2×C4C2×D107611yescharacteristic
341C2×C4C2×D107611yescharacteristic
342C2×C4C23×D26911yescharacteristic
343C8—21no—
344C8—21no—
345C8—21no—
346C8—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—5382no—
353C22—10761no—
354C22—10761no—
355C22—10761no—
356C22—10761no—
357C22—10761no—
358C22—10761no—
359C22—10761no—
360C22—10761no—
361C22—21524no—
362C22#2257547512yes—
363C22C22×D107611yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—10761no—
369C4—10761no—
370C4—10761no—
371C4—10761no—
372C4—10761no—
373C4#41×D26911yescharacteristic
374C4#2257549811yescharacteristic
375C2—22no—
376C2—41no—
377C2—10762no—
378C2—21521no—
379C2—21521no—
380C2#3177125011yescharacteristic
381C2C2×#2257547511yescharacteristic
382C2#3177141011yescharacteristic
subgroupG/H
centerC22C22×D1076
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
FittingC8×D4×C269C2
socleC22×C269C22×D4
hypercenterC8×D4D269
hypocenterC269#3297
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #31771301C2semidirect
2. #31771251C2semidirect
3. #31770960C2semidirect
4. #31770941C2semidirect
5. #31770930C2semidirect
6. #31770906C2semidirect
7. #31770841C2semidirect
8. #31770834C2semidirect
9. #31770821C2semidirect
10. #31770752C2semidirect
11. #31770741C2semidirect
12. C4×D2152C2semidirect
13. #31770602C2semidirect
14. C2×#22575375C2semidirect
15. C8×D4×C269C2semidirect
18. C2×#16864391C22semidirect
22. #22575445C22semidirect
29. #22575427C22semidirect
34. C4×D1076C22semidirect
36. #22575412C22semidirect
39. #22575387C22semidirect
40. #22575387C22semidirect
41. #22575385C22semidirect
42. #22575378C22semidirect
43. #22575378C22semidirect
44. #22575378C22semidirect
51. #22575375C22semidirect
53. C22×C8×C269C22semidirect
55. C4⋊C8×C269C22semidirect
57. C22⋊C8×C269C22semidirect
58. C4×C8×C269C22semidirect
191. C269#3297semidirect
205. C8×D4D269semidirect
214. C22×C8C2×D269semidirect
237. C4⋊C8C2×D269semidirect
249. C22⋊C8C2×D269semidirect
250. C4×C8C2×D269semidirect

Maximal subgroups

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

subgroupindex
#317713012
#317712512
#317709602
#317709412
#317709302
#317709062
#317708412
#317708342
#317708212
#317707522
#317707412
C4×D21522
#317706022
C2×#225753752
C8×D4×C2692
#3297269