Finite Group Databasebeta

← #46821131

Subgroups — #46821131

Order 34432.

394 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31771215C211yesmaximal, characteristic, complemented
2#31771080C211yesmaximal, characteristic
3#31770947C211yesmaximal, characteristic, complemented
4#31770943C211yesmaximal, characteristic, complemented
5#31770907C211yesmaximal, characteristic, complemented
6#31770903C211yesmaximal, characteristic, complemented
7#31770808C211yesmaximal, characteristic, complemented
8#31770808C211yesmaximal, characteristic, complemented
9#31770756C211yesmaximal, characteristic
10#31770706C211yesmaximal, characteristic, complemented
11#31770702C211yesmaximal, characteristic, complemented
12#31770696C211yesmaximal, characteristic, complemented
13SD16×Dic269C211yesmaximal, characteristic, complemented
14#31770663C211yesmaximal, characteristic, complemented
15#450×C269C211yesmaximal, characteristic
16C2×#16864390C2211yescharacteristic
17C2×#16864390C2211yescharacteristic
18#22575451C2211yescharacteristic, complemented
19#22575446—21no—
20#22575446—21no—
21#22575444—21no—
22#22575444C2211yescharacteristic, complemented
23#22575443—21no—
24#22575432C2211yescharacteristic
25Q8×Dic269C2211yescharacteristic, complemented
26#22575418C2211yescharacteristic, complemented
27D4×Dic269—21no—
28D4×Dic269—21no—
29D4×Dic269C2211yescharacteristic
30D4×Dic269C2211yescharacteristic
31D4×Dic269C2211yescharacteristic
32D4×Dic269C2211yescharacteristic
33#22575412C2211yescharacteristic
34#22575412C2211yescharacteristic
35C2×C22⋊Dic269—21no—
36C2×C22⋊Dic269—21no—
37#22575387C2211yescharacteristic, complemented
38#22575386C2211yescharacteristic, complemented
39#22575385C2211yescharacteristic, complemented
40#22575384C2211yescharacteristic, complemented
41#22575382C2211yescharacteristic, complemented
42#22575382C2211yescharacteristic, complemented
43#22575378C2211yescharacteristic, complemented
44#22575378C2211yescharacteristic, complemented
45#22575377C2211yescharacteristic, complemented
46#22575377C2211yescharacteristic, complemented
47#22575374C2211yescharacteristic
48#22575374C2211yescharacteristic
49C22⋊#14524130C2211yescharacteristic
50#22575363C2211yescharacteristic, complemented
51C8×Dic269C2211yescharacteristic, complemented
52C22×D4×C269C2211yescharacteristic
53C2×SD16×C269C2211yescharacteristic
54C2×SD16×C269C2211yescharacteristic
55C22⋊Q8×C269C2211yescharacteristic
56#102×C269C2211yescharacteristic
57#102×C269C2211yescharacteristic
58C22⋊C8×C269C2211yescharacteristic
59C2×Dic538C2311yescharacteristic
60C22×Dic269—21no—
61C22×Dic269—21no—
62C22×Dic269—21no—
63C22×Dic269—21no—
64#16864390—22no—
65#16864390—22no—
66C2×#14524130C2311yescharacteristic
67C2×#14524130C2311yescharacteristic
68C4⋊Dic269—21no—
69C4⋊Dic269—21no—
70C4⋊Dic269D411yescharacteristic
71C4⋊Dic269D411yescharacteristic
72C4⋊Dic269C2311yescharacteristic
73C4⋊Dic269C2311yescharacteristic
74C4⋊Dic269C2311yescharacteristic
75#16864381—21no—
76#16864381—21no—
77#16864381—21no—
78#16864381—21no—
79#16864381—21no—
80C22⋊Dic269—21no—
81C22⋊Dic269—21no—
82C22⋊Dic269—21no—
83C22⋊Dic269—21no—
84C22⋊Dic269—21no—
85C22⋊Dic269—21no—
86C22⋊Dic269D411yescharacteristic
87C22⋊Dic269D411yescharacteristic
88C4×Dic269—21no—
89C4×Dic269—21no—
90C4×Dic269—21no—
91C4×Dic269C2311yescharacteristic
92C4×Dic269C2311yescharacteristic
93C24×C269—21no—
94C2×Q8×C269C2311yescharacteristic
95C2×D4×C269—22no—
96C2×D4×C269—22no—
97C2×D4×C269—21no—
98C2×D4×C269C2311yescharacteristic
99C2×D4×C269C2311yescharacteristic
100C22×C4×C269C2311yescharacteristic
101SD16×C269—22no—
102SD16×C269—22no—
103C2×C8×C269C2311yescharacteristic
104C2×C8×C269C2311yescharacteristic
105C4⋊C4×C269—21no—
106C4⋊C4×C269C2311yescharacteristic
107C22⋊C4×C269—21no—
108Dic538—21no—
109Dic538—21no—
110C2×Dic269—21no—
111C2×Dic269—21no—
112C2×Dic269—21no—
113C2×Dic269—21no—
114C2×Dic269—81no—
115C2×Dic269—81no—
116C2×Dic269—81no—
117C2×Dic269—81no—
118C2×Dic269C2×D411yescharacteristic
119C2×Dic269C2×D411yescharacteristic
120C2×Dic269C2×D411yescharacteristic
121C2×Dic269C2×D411yescharacteristic
122#14524130—21no—
123#14524130—21no—
124C23×C269—24no—
125C23×C269—21no—
126C23×C269—21no—
127C23×C269—81no—
128C23×C269—81no—
129C23×C269C2×D411yescharacteristic
130Q8×C269—21no—
131Q8×C269—21no—
132D4×C269—24no—
133D4×C269—21no—
134D4×C269—21no—
135D4×C269#4112yes—
136D4×C269#4112yes—
137C2×C4×C269—21no—
138C2×C4×C269—21no—
139C2×C4×C269—21no—
140C2×C4×C269—21no—
141C2×C4×C269C2×D411yescharacteristic
142C2×C4×C269C2411yescharacteristic
143C8×C269—21no—
144C8×C269—21no—
145C22×C269—21no—
146C22×C269—22no—
147C22×C269—22no—
148C22×C269—21no—
149C22×C269—21no—
150C22×C269—41no—
151C22×C269—41no—
152C22×C269—41no—
153C22×C269—41no—
154C22×C269—84no—
155C22×C269C22×D411yescharacteristic
156C4×C269—21no—
157C4×C269—41no—
158C4×C269—41no—
159C4×C269C2×#4111yescharacteristic
160C4×C269C2×#4111yescharacteristic
161Dic269—21no—
162Dic269—21no—
163Dic269—21no—
164Dic269—21no—
165Dic269—41no—
166Dic269—41no—
167Dic269—41no—
168Dic269—41no—
169C2×C269—41no—
170C2×C269—42no—
171C2×C269—42no—
172C2×C269#52511yescharacteristic
173C2×C269#57711yescharacteristic
174C2×C269#57711yescharacteristic
175C269#312311yescharacteristic, complemented, Sylow
176#3123—2691nomaximal, Sylow
177#541—2691no—
178#518—2691no—
179#485—2691no—
180#485—2691no—
181#478—2691no—
182#478—2691no—
183#461—2691no—
184#461—2691no—
185#450—2691no—
186#450D26911yescharacteristic
187#440—2691no—
188#440—2691no—
189C4×SD16—2691no—
190C4×SD16—2691no—
191#433—2691no—
192C22×D4C2×D26911yescharacteristic
193C2×SD16—2691no—
194C2×SD16—2691no—
195C2×SD16C2×D26911yescharacteristic
196C2×SD16C2×D26911yescharacteristic
197C4⋊Q8—2691no—
198#126—5381no—
199#126—5381no—
200#125—2691no—
201#125—5381no—
202#125—5381no—
203C22⋊Q8—2691no—
204C22⋊Q8C2×D26911yescharacteristic
205C4×Q8—2691no—
206C4×Q8—2691no—
207C4×D4—2691no—
208C4×D4—2691no—
209C4×D4—2691no—
210C4×D4—2691no—
211C4×D4—5381no—
212C4×D4—5381no—
213#117—2691no—
214#117—2691no—
215C2×C22⋊C4—5381no—
216C2×C22⋊C4—5381no—
217#107—2691no—
218#107—2691no—
219#106—2691no—
220#106—2691no—
221C4⋊C8—2691no—
222C4⋊C8—2691no—
223#103—2691no—
224#103—2691no—
225#103—2691no—
226#103—2691no—
227#102—2691no—
228#102—2691no—
229#102C2×D26911yescharacteristic
230#102C2×D26911yescharacteristic
231C22⋊C8—2691no—
232C22⋊C8C2×D26911yescharacteristic
233#97—2691no—
234C4×C8—2691no—
235C24—21no—
236C2×Q8—2691no—
237C2×Q8C22×D26911yescharacteristic
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—21no—
241C2×D4C22×D26911yescharacteristic
242C2×D4C22×D26911yescharacteristic
243C22×C4—5381no—
244C22×C4—5381no—
245C22×C4—5381no—
246C22×C4—5381no—
247C22×C4C22×D26911yescharacteristic
248SD16—22no—
249SD16—22no—
250SD16—5382no—
251SD16—5382no—
252C2×C8—2691no—
253C2×C8—2691no—
254C2×C8C22×D26911yescharacteristic
255C2×C8C22×D26911yescharacteristic
256C4⋊C4—21no—
257C4⋊C4—2691no—
258C4⋊C4—2691no—
259C4⋊C4—2691no—
260C4⋊C4—2691no—
261C4⋊C4—2691no—
262C4⋊C4—5381no—
263C4⋊C4—5381no—
264C4⋊C4—5381no—
265C4⋊C4—5381no—
266C4⋊C4—5381no—
267C4⋊C4—5381no—
268C4⋊C4—5381no—
269C4⋊C4C22×D26911yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—2691no—
272C22⋊C4—2691no—
273C22⋊C4—5381no—
274C22⋊C4—5381no—
275C22⋊C4—5381no—
276C22⋊C4—5381no—
277C22⋊C4—5381no—
278C22⋊C4—5381no—
279C42—2691no—
280C42—2691no—
281C42—5381no—
282C42—5381no—
283C42—5381no—
284C23—24no—
285C23—21no—
286C23—21no—
287C23—81no—
288C23—81no—
289C23D4×D26911yescharacteristic
290Q8—21no—
291Q8—21no—
292Q8—5381no—
293Q8—5381no—
294D4—24no—
295D4—21no—
296D4—21no—
297D4#1686440112yes—
298D4#1686440112yes—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—2691no—
304C2×C4—2691no—
305C2×C4—2691no—
306C2×C4—2691no—
307C2×C4—5381no—
308C2×C4—5381no—
309C2×C4—5381no—
310C2×C4—5381no—
311C2×C4—21521no—
312C2×C4—21521no—
313C2×C4—21521no—
314C2×C4—21521no—
315C2×C4D4×D26911yescharacteristic
316C2×C4C23×D26911yescharacteristic
317C8—21no—
318C8—21no—
319C8—5381no—
320C8—5381no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—22no—
325C22—22no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—84no—
331C22C2×D4×D26911yescharacteristic
332C4—21no—
333C4—41no—
334C4—41no—
335C4—5381no—
336C4—5381no—
337C4—5381no—
338C4—5381no—
339C4—10761no—
340C4—10761no—
341C4—10761no—
342C4—10761no—
343C4C2×#1686440111yescharacteristic
344C4C2×#1686440111yescharacteristic
345C2—41no—
346C2—42no—
347C2—42no—
348C2#3177111111yescharacteristic
349C2#3177141111yescharacteristic
350C2#3177141511yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#450×C269C2
socleC22×C269C22×D4
prime radicalC22×D4×C269C22
hypercenter#450D269
hypocenterC269#3123
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #31771215C2semidirect
3. #31770947C2semidirect
4. #31770943C2semidirect
5. #31770907C2semidirect
6. #31770903C2semidirect
7. #31770808C2semidirect
8. #31770808C2semidirect
10. #31770706C2semidirect
11. #31770702C2semidirect
12. #31770696C2semidirect
13. SD16×Dic269C2semidirect
14. #31770663C2semidirect
18. #22575451C22semidirect
22. #22575444C22semidirect
25. Q8×Dic269C22semidirect
26. #22575418C22semidirect
37. #22575387C22semidirect
38. #22575386C22semidirect
39. #22575385C22semidirect
40. #22575384C22semidirect
41. #22575382C22semidirect
42. #22575382C22semidirect
43. #22575378C22semidirect
44. #22575378C22semidirect
45. #22575377C22semidirect
46. #22575377C22semidirect
50. #22575363C22semidirect
51. C8×Dic269C22semidirect
175. C269#3123semidirect

Maximal subgroups

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

subgroupindex
#317712152
#317710802
#317709472
#317709432
#317709072
#317709032
#317708082
#317707562
#317707062
#317707022
#317706962
SD16×Dic2692
#317706632
#450×C2692
#3123269