Finite Group Databasebeta

← #46822729

Subgroups — #46822729

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#31771251C211yesmaximal, characteristic, complemented
2#31770962C211yesmaximal, characteristic, complemented
3#31770940C211yesmaximal, characteristic, complemented
4#31770929C211yesmaximal, characteristic, complemented
5#31770902C211yesmaximal, characteristic, complemented
6#31770839C211yesmaximal, characteristic, complemented
7#31770832C211yesmaximal, characteristic, complemented
8#31770820C211yesmaximal, characteristic, complemented
9#31770748C211yesmaximal, characteristic, complemented
10#31770738C211yesmaximal, characteristic, complemented
11C4×#16864387C211yesmaximal, characteristic, complemented
12D4×#14524130C211yesmaximal, characteristic, complemented
13#31770599C211yesmaximal, characteristic, complemented
14C2×#22575373C211yesmaximal, characteristic, complemented
15#553×C269C211yesmaximal, characteristic, complemented
16C2×#16864402C2211yescharacteristic
17C22×D1076C2211yescharacteristic
18C2×#16864390C2211yescharacteristic, complemented
19C2×#16864387C2211yescharacteristic
20C2×#16864387C2211yescharacteristic
21C22×#14524130C2211yescharacteristic
22C22×#14524130C2211yescharacteristic, complemented
23#22575441C2211yescharacteristic
24#22575438—21no—
25#22575432C2211yescharacteristic
26#22575427C2211yescharacteristic
27#22575426—21no—
28#22575421—21no—
29C4×D1076C2211yescharacteristic
30C22⋊C4×D269—21no—
31#22575386C2211yescharacteristic, complemented
32#22575386C2211yescharacteristic, complemented
33#22575384C2211yescharacteristic, complemented
34C4⋊#14524130C2211yescharacteristic, complemented
35#22575376C2211yescharacteristic, complemented
36#22575376C2211yescharacteristic, complemented
37#22575376C2211yescharacteristic, complemented
38#22575374C2211yescharacteristic, complemented
39#22575373C2211yescharacteristic
40#22575373C2211yescharacteristic
41#22575373C2211yescharacteristic
42#22575373C2211yescharacteristic
43#22575373C2211yescharacteristic
44#22575373C2211yescharacteristic
45C22⋊#14524130C2211yescharacteristic
46C22⋊#14524130C2211yescharacteristic, complemented
47C4×#14524130C2211yescharacteristic, complemented
48#126×C269—21no—
49#125×C269C2211yescharacteristic, complemented
50#123×C269—21no—
51#121×C269—21no—
52#121×C269C2211yescharacteristic
53#121×C269C2211yescharacteristic, complemented
54C4×D4×C269—21no—
55C4×D4×C269C2211yescharacteristic
56C4×D4×C269C2211yescharacteristic, complemented
57#117×C269C2211yescharacteristic, complemented
58C2×C4⋊C4×C269C2211yescharacteristic
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×C22⋊D269—21no—
69C2×C22⋊D269—21no—
70C2×C4×D269—21no—
71C2×C4×D269—21no—
72#16864390—41no—
73#16864387—22no—
74#16864387—41no—
75C2×#14524130—41no—
76C2×#14524130—41no—
77C2×#14524130C2311yescharacteristic
78C2×#14524130C2311yescharacteristic
79C2×#14524130C2311yescharacteristic
80C2×#14524130C2311yescharacteristic
81C4⋊Dic269C2311yescharacteristic
82#16864381—21no—
83C22⋊Dic269—21no—
84#16864379—21no—
85#16864379—21no—
86#16864379—21no—
87#16864379—21no—
88C2×D4×C269—21no—
89C2×D4×C269—21no—
90C2×D4×C269D411yescharacteristic
91C2×D4×C269C2311yescharacteristic
92C22×C4×C269—21no—
93C22×C4×C269—21no—
94C22×C4×C269C2311yescharacteristic
95C22×C4×C269C2311yescharacteristic
96C4⋊C4×C269—21no—
97C4⋊C4×C269—21no—
98C4⋊C4×C269—21no—
99C4⋊C4×C269D411yescharacteristic
100C4⋊C4×C269C2311yescharacteristic
101C4⋊C4×C269C2311yescharacteristic
102C4⋊C4×C269C2311yescharacteristic
103C22⋊C4×C269—21no—
104C22⋊C4×C269—21no—
105C22⋊C4×C269—21no—
106C22⋊C4×C269—21no—
107C22⋊C4×C269D411yescharacteristic
108C22⋊C4×C269D411yescharacteristic
109C42×C269—21no—
110C42×C269C2311yescharacteristic
111C22×D269—21no—
112C22×D269—21no—
113C22×D269—42no—
114C22×D269—42no—
115C22×D269—81no—
116Dic538—21no—
117Dic538—21no—
118D1076—21no—
119D1076—21no—
120D1076—21no—
121D1076—42no—
122D1076#4112yes—
123C22⋊D269—42no—
124C22⋊D269—81no—
125C2×Dic269—21no—
126C2×Dic269—21no—
127C4×D269—42no—
128C4×D269—81no—
129#14524130—21no—
130#14524130—21no—
131#14524130—21no—
132#14524130—21no—
133C23×C269—21no—
134C23×C269C2×D411yescharacteristic
135C23×C269C2×D411yescharacteristic
136D4×C269—22no—
137D4×C269—21no—
138D4×C269—21no—
139D4×C269—42no—
140D4×C269—81no—
141C2×C4×C269—21no—
142C2×C4×C269—21no—
143C2×C4×C269—21no—
144C2×C4×C269—21no—
145C2×C4×C269—21no—
146C2×C4×C269—21no—
147C2×C4×C269—21no—
148C2×C4×C269—81no—
149C2×C4×C269—81no—
150C2×C4×C269C2×D411yescharacteristic
151C2×C4×C269C2×D411yescharacteristic
152C2×C4×C269C2×D411yescharacteristic
153C2×C4×C269C2×D411yescharacteristic
154C2×C4×C269C2411yescharacteristic
155C22×C269—21no—
156C22×C269—21no—
157C22×C269—21no—
158C22×C269—21no—
159C22×C269—21no—
160C22×C269—41no—
161C22×C269—41no—
162C22×C269—41no—
163C22×C269#13512yes—
164C22×C269C22×D411yescharacteristic
165C2×D269—22no—
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×C269—41no—
177C4×C269—41no—
178C4×C269—41no—
179C4×C269C2×#4111yescharacteristic
180C4×C269H(5,𝔽2)11yescharacteristic
181Dic269—41no—
182Dic269—41no—
183C2×C269—22no—
184C2×C269—41no—
185C2×C269—81no—
186C2×C269#54611yescharacteristic
187C2×C269C2×#13511yescharacteristic
188C2×C269#57611yescharacteristic
189D269—42no—
190D269—81no—
191C269#329711yescharacteristic, complemented, Sylow
192#3297—2691nomaximal, Sylow
193#553D26911yescharacteristic, complemented
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×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—21no—
216#125C2×D26911yescharacteristic, complemented
217#124—2691no—
218#123—21no—
219#123—5381no—
220C22⋊Q8—2691no—
221#121—21no—
222#121—2691no—
223#121—5381no—
224#121C2×D26911yescharacteristic
225#121C2×D26911yescharacteristic, complemented
226#120—5381no—
227C4×D4—21no—
228C4×D4—2691no—
229C4×D4C2×D26911yescharacteristic
230C4×D4C2×D26911yescharacteristic, complemented
231#117C2×D26911yescharacteristic, complemented
232C2×C4⋊C4C2×D26911yescharacteristic
233C2×C22⋊C4—5381no—
234#107—2691no—
235#107—2691no—
236#106—2691no—
237C4⋊C8—2691no—
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⋊C8—2691no—
249C22⋊C8—2691no—
250C4×C8—2691no—
251C24—5381no—
252#41—5382no—
253#41—10761no—
254C2×Q8—2691no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—2691no—
258C2×D4—2691no—
259C2×D4—5382no—
260C2×D4—5381no—
261C2×D4—5381no—
262C2×D4—5381no—
263C2×D4—10761no—
264C2×D4C22⋊D26911yescharacteristic
265C2×D4C22×D26911yescharacteristic
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—5381no—
269C22×C4—5381no—
270C22×C4C22×D26911yescharacteristic
271C22×C4C22×D26911yescharacteristic
272SD16—10761no—
273D8—5382no—
274D8—10761no—
275C2×C8—2691no—
276C2×C8—2691no—
277C2×C8—2691no—
278C2×C8—2691no—
279C2×C8—10761no—
280C2×C8—10761no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—2691no—
285C4⋊C4—5381no—
286C4⋊C4C22⋊D26911yescharacteristic
287C4⋊C4C22×D26911yescharacteristic
288C4⋊C4C22×D26911yescharacteristic
289C4⋊C4C22×D26911yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—5381no—
295C22⋊C4—5381no—
296C22⋊C4—5381no—
297C22⋊C4—5381no—
298C22⋊C4—5381no—
299C22⋊C4C22⋊D26911yescharacteristic
300C22⋊C4C22⋊D26911yescharacteristic
301C42—21no—
302C42C22×D26911yescharacteristic
303C23—21no—
304C23—5381no—
305C23—5381no—
306C23—10762no—
307C23—10762no—
308C23—21521no—
309C23C2×C22⋊D26911yescharacteristic
310C23C2×C22⋊D26911yescharacteristic
311Q8—5381no—
312Q8—5381no—
313D4—22no—
314D4—21no—
315D4—21no—
316D4—42no—
317D4—81no—
318D4—2692no—
319D4—5381no—
320D4—5381no—
321D4—5381no—
322D4—10762no—
323D4—10762no—
324D4—21521no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—5381no—
335C2×C4—5381no—
336C2×C4—10762no—
337C2×C4—21521no—
338C2×C4C2×C22⋊D26911yescharacteristic
339C2×C4C2×C22⋊D26911yescharacteristic
340C2×C4C2×C22⋊D26911yescharacteristic
341C2×C4C2×C22⋊D26911yescharacteristic
342C2×C4C23×D26911yescharacteristic
343C8—5381no—
344C8—5381no—
345C8—5381no—
346C8—5381no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—5382no—
356C22—10761no—
357C22—10761no—
358C22—10761no—
359C22—10761no—
360C22—10761no—
361C22—21524no—
362C22#2257547312yes—
363C22C22×C22⋊D26911yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4—10761no—
372C4—10761no—
373C4#41×D26911yescharacteristic
374C4#2257549911yescharacteristic
375C2—22no—
376C2—41no—
377C2—81no—
378C2—10762no—
379C2—21521no—
380C2#3177124311yescharacteristic
381C2C2×#2257547311yescharacteristic
382C2#3177140811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#553×C269C2
socleC22×C269C22×D4
hypercenter#553D269
hypocenterC269#3297
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C4×D4 ◁ #553

Complements

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

NH ≅ G/Ntype
1. #31771251C2semidirect
2. #31770962C2semidirect
3. #31770940C2semidirect
4. #31770929C2semidirect
5. #31770902C2semidirect
6. #31770839C2semidirect
7. #31770832C2semidirect
8. #31770820C2semidirect
9. #31770748C2semidirect
10. #31770738C2semidirect
11. C4×#16864387C2semidirect
12. D4×#14524130C2semidirect
13. #31770599C2semidirect
14. C2×#22575373C2semidirect
15. #553×C269C2semidirect
18. C2×#16864390C22semidirect
22. C22×#14524130C22semidirect
31. #22575386C22semidirect
32. #22575386C22semidirect
33. #22575384C22semidirect
34. C4⋊#14524130C22semidirect
35. #22575376C22semidirect
36. #22575376C22semidirect
37. #22575376C22semidirect
38. #22575374C22semidirect
46. C22⋊#14524130C22semidirect
47. C4×#14524130C22semidirect
49. #125×C269C22semidirect
53. #121×C269C22semidirect
56. C4×D4×C269C22semidirect
57. #117×C269C22semidirect
191. C269#3297semidirect
193. #553D269semidirect
216. #125C2×D269semidirect
225. #121C2×D269semidirect
230. C4×D4C2×D269semidirect
231. #117C2×D269semidirect

Maximal subgroups

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

subgroupindex
#317712512
#317709622
#317709402
#317709292
#317709022
#317708392
#317708322
#317708202
#317707482
#317707382
C4×#168643872
D4×#145241302
#317705992
C2×#225753732
#553×C2692
#3297269