Finite Group Databasebeta

← #46823620

Subgroups — #46823620

Order 34432.

338 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×Dic538C211yesmaximal, characteristic, complemented
2#31771324C211yesmaximal, characteristic
3#31770964C211yesmaximal, characteristic
4#31770963C211yesmaximal, characteristic
5#31770954C211yesmaximal, characteristic, complemented
6#31770893C211yesmaximal, characteristic
7#31770891C211yesmaximal, characteristic, complemented
8#31770888C211yesmaximal, characteristic, complemented
9#31770825C211yesmaximal, characteristic
10#31770813C211yesmaximal, characteristic, complemented
11#31770734C211yesmaximal, characteristic, complemented
12#31770712C211yesmaximal, characteristic, complemented
13#31770710C211yesmaximal, characteristic, complemented
14SD16×Dic269C211yesmaximal, characteristic
15#464×C269C211yesmaximal, characteristic
16C2×#16864392C2211yescharacteristic
17#22575453—21no—
18#22575453C2211yescharacteristic
19#22575452—21no—
20#22575452—21no—
21#22575452C2211yescharacteristic
22#22575452C2211yescharacteristic
23#22575451C2211yescharacteristic
24#22575446—21no—
25#22575446—21no—
26#22575440C2211yescharacteristic
27#22575430—21no—
28Q8×Dic269C2211yescharacteristic
29Q8×Dic269C2211yescharacteristic
30Q8×Dic269C2211yescharacteristic
31C4×Dic538—21no—
32C4×Dic538C2211yescharacteristic
33C4×Dic538C2211yescharacteristic
34D4×Dic269C2211yescharacteristic
35#22575409—21no—
36#22575385C2211yescharacteristic
37#22575385C2211yescharacteristic
38#22575384C2211yescharacteristic
39#22575382C2211yescharacteristic
40#22575382C2211yescharacteristic
41#22575378C2211yescharacteristic
42#22575378C2211yescharacteristic
43#22575377C2211yescharacteristic
44#22575377C2211yescharacteristic
45#22575377C2211yescharacteristic
46#22575376C2211yescharacteristic
47#22575374C2211yescharacteristic
48#22575374C2211yescharacteristic
49#22575364C2211yescharacteristic
50#22575363C2211yescharacteristic
51C8×Dic269C2211yescharacteristic
52C2×Q16×C269C2211yescharacteristic
53C2×SD16×C269C2211yescharacteristic
54#124×C269C2211yescharacteristic
55C4×Q8×C269C2211yescharacteristic
56C4⋊C8×C269C2211yescharacteristic
57#103×C269C2211yescharacteristic
58#102×C269C2211yescharacteristic
59C2×Dic538—21no—
60C2×Dic538—21no—
61C2×Dic538—21no—
62C2×Dic538D411yescharacteristic
63C2×Dic538C2311yescharacteristic
64C22×Dic269—21no—
65#16864392—22no—
66C2×#14524130C2311yescharacteristic
67C2×#14524130C2311yescharacteristic
68C4⋊Dic269—21no—
69C4⋊Dic269—21no—
70C4⋊Dic269—21no—
71C4⋊Dic269—21no—
72C4⋊Dic269D411yescharacteristic
73C4⋊Dic269C2311yescharacteristic
74C4⋊Dic269C2311yescharacteristic
75C4⋊Dic269C2311yescharacteristic
76#16864381—21no—
77#16864381—21no—
78#16864381—21no—
79#16864381—21no—
80#16864381—21no—
81#16864381D411yescharacteristic
82#16864381D411yescharacteristic
83C22⋊Dic269—21no—
84C22⋊Dic269—21no—
85C22⋊Dic269—21no—
86C4×Dic269—21no—
87C4×Dic269—21no—
88C4×Dic269—21no—
89C4×Dic269—21no—
90C4×Dic269C2311yescharacteristic
91C4×Dic269C2311yescharacteristic
92C2×Q8×C269C2311yescharacteristic
93C2×Q8×C269C2311yescharacteristic
94C2×D4×C269C2311yescharacteristic
95Q16×C269—22no—
96SD16×C269—22no—
97C2×C8×C269C2311yescharacteristic
98C2×C8×C269C2311yescharacteristic
99C4⋊C4×C269—21no—
100C4⋊C4×C269C2311yescharacteristic
101C22⋊C4×C269—21no—
102C22⋊C4×C269—21no—
103C42×C269—21no—
104C42×C269C2311yescharacteristic
105Dic538—22no—
106Dic538—21no—
107Dic538—21no—
108Dic538—81no—
109Dic538—81no—
110Dic538—81no—
111C2×Dic269—21no—
112C2×Dic269—21no—
113C2×Dic269—21no—
114C2×Dic269—21no—
115C2×Dic269—42no—
116C2×Dic269C2×D411yescharacteristic
117C2×Dic269C2×D411yescharacteristic
118C2×Dic269C2×D411yescharacteristic
119C2×Dic269C2×D411yescharacteristic
120#14524130—21no—
121#14524130—21no—
122C23×C269—21no—
123Q8×C269—21no—
124Q8×C269—21no—
125Q8×C269—21no—
126Q8×C269#4112yes—
127D4×C269—21no—
128D4×C269—21no—
129C2×C4×C269—21no—
130C2×C4×C269—21no—
131C2×C4×C269—21no—
132C2×C4×C269C2×D411yescharacteristic
133C2×C4×C269C2×D411yescharacteristic
134C2×C4×C269C2411yescharacteristic
135C8×C269—21no—
136C8×C269—21no—
137C22×C269—41no—
138C22×C269—41no—
139C22×C269—41no—
140C22×C269C22×D411yescharacteristic
141C4×C269—22no—
142C4×C269—21no—
143C4×C269—21no—
144C4×C269—41no—
145C4×C269—41no—
146C4×C269C2×#4111yescharacteristic
147C4×C2692−1+411yescharacteristic
148Dic269—21no—
149Dic269—21no—
150Dic269—21no—
151Dic269—41no—
152Dic269—41no—
153Dic269—41no—
154Dic269—41no—
155Dic269#13712yes—
156C2×C269—81no—
157C2×C269#54811yescharacteristic
158C2×C269C2×#13711yescharacteristic
159C2×C269#57711yescharacteristic
160C269#336911yescharacteristic, complemented, Sylow
161#3369—2691nomaximal, Sylow
162Q82—2691no—
163#556—2691no—
164#489—2691no—
165#489—2691no—
166#487—2691no—
167#476—2691no—
168#475—2691no—
169#475—2691no—
170#464—2691no—
171#464D26911yescharacteristic
172#462—2691no—
173#446—2691no—
174#441—2691no—
175#441—2691no—
176C4×SD16—2691no—
177C2×Q16—2691no—
178C2×Q16C2×D26911yescharacteristic
179C2×SD16C2×D26911yescharacteristic
180C4⋊Q8—2691no—
181C4⋊Q8—2691no—
182C4⋊Q8—2691no—
183C4⋊Q8—2691no—
184C4⋊Q8—5381no—
185C4⋊Q8—5381no—
186C4⋊Q8—5381no—
187#126—5381no—
188#126—5381no—
189#124—2691no—
190#124C2×D26911yescharacteristic
191C22⋊Q8—5381no—
192C4×Q8—2691no—
193C4×Q8—2691no—
194C4×Q8—2691no—
195C4×Q8—2691no—
196C4×Q8—2691no—
197C4×Q8—5381no—
198C4×Q8C2×D26911yescharacteristic
199C4×D4—2691no—
200#117—5381no—
201#106—2691no—
202#106—2691no—
203#106—2691no—
204C4⋊C8—2691no—
205C4⋊C8—2691no—
206C4⋊C8C2×D26911yescharacteristic
207#103—2691no—
208#103—2691no—
209#103—2691no—
210#103—2691no—
211#103—2691no—
212#103—2691no—
213#103C2×D26911yescharacteristic
214#102—2691no—
215#102—2691no—
216#102C2×D26911yescharacteristic
217#97—2691no—
218#97—2691no—
219C4×C8—2691no—
220C2×Q8—2691no—
221C2×Q8—2691no—
222C2×Q8—5381no—
223C2×Q8—5381no—
224C2×Q8—5381no—
225C2×Q8C22×D26911yescharacteristic
226C2×Q8C22×D26911yescharacteristic
227C2×D4C22×D26911yescharacteristic
228C22×C4—5381no—
229Q16—22no—
230Q16—5382no—
231SD16—22no—
232C2×C8—2691no—
233C2×C8—2691no—
234C2×C8C22×D26911yescharacteristic
235C2×C8C22×D26911yescharacteristic
236C4⋊C4—21no—
237C4⋊C4—2691no—
238C4⋊C4—2691no—
239C4⋊C4—2691no—
240C4⋊C4—2691no—
241C4⋊C4—2691no—
242C4⋊C4—2691no—
243C4⋊C4—5381no—
244C4⋊C4—5381no—
245C4⋊C4—5381no—
246C4⋊C4—5381no—
247C4⋊C4—5381no—
248C4⋊C4—5381no—
249C4⋊C4—5381no—
250C4⋊C4—5381no—
251C4⋊C4—5381no—
252C4⋊C4C22×D26911yescharacteristic
253C22⋊C4—21no—
254C22⋊C4—21no—
255C22⋊C4—5381no—
256C22⋊C4—5381no—
257C22⋊C4—5381no—
258C42—21no—
259C42—2691no—
260C42—2691no—
261C42—5381no—
262C42—5381no—
263C42—5381no—
264C42—5381no—
265C42C22×D26911yescharacteristic
266C23—21no—
267Q8—21no—
268Q8—21no—
269Q8—21no—
270Q8—5382no—
271Q8—5381no—
272Q8—5381no—
273Q8—21521no—
274Q8—21521no—
275Q8—21521no—
276Q8#1686440112yes—
277D4—21no—
278D4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—2691no—
283C2×C4—2691no—
284C2×C4—2691no—
285C2×C4—2691no—
286C2×C4—5381no—
287C2×C4—5381no—
288C2×C4—5381no—
289C2×C4—5381no—
290C2×C4—10762no—
291C2×C4D4×D26911yescharacteristic
292C2×C4D4×D26911yescharacteristic
293C2×C4C23×D26911yescharacteristic
294C8—21no—
295C8—21no—
296C8—5381no—
297C8—5381no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22C2×D4×D26911yescharacteristic
302C4—21no—
303C4—21no—
304C4—22no—
305C4—41no—
306C4—41no—
307C4—2692no—
308C4—5381no—
309C4—5381no—
310C4—5381no—
311C4—10761no—
312C4—10761no—
313C4—10761no—
314C4—10761no—
315C4C2×#1686440111yescharacteristic
316C4#2257550011yescharacteristic
317C2—81no—
318C2#3177126111yescharacteristic
319C2#137×D26911yescharacteristic
320C2#3177141211yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#464×C269C2
socleC22×C269C22×D4
prime radicalC2×D4×C269C23
hypercenter#464D269
hypocenterC269#3369
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
1. Q8×Dic538C2semidirect
5. #31770954C2semidirect
7. #31770891C2semidirect
8. #31770888C2semidirect
10. #31770813C2semidirect
11. #31770734C2semidirect
12. #31770712C2semidirect
13. #31770710C2semidirect
160. C269#3369semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic5382
#317713242
#317709642
#317709632
#317709542
#317708932
#317708912
#317708882
#317708252
#317708132
#317707342
#317707122
#317707102
SD16×Dic2692
#464×C2692
#3369269