Finite Group Databasebeta

← #46823456

Subgroups — #46823456

Order 34432.

336 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31771294C211yesmaximal, characteristic
2#31771290C211yesmaximal, characteristic
3#31771005C211yesmaximal, characteristic, complemented
4#31770949C211yesmaximal, characteristic, complemented
5#31770946C211yesmaximal, characteristic, complemented
6#31770942C211yesmaximal, characteristic, complemented
7#31770940C211yesmaximal, characteristic, complemented
8#31770915C211yesmaximal, characteristic, complemented
9#31770911C211yesmaximal, characteristic, complemented
10#31770864C211yesmaximal, characteristic, complemented
11#31770863C211yesmaximal, characteristic, complemented
12#31770699C211yesmaximal, characteristic, complemented
13#31770681C211yesmaximal, characteristic
14#31770646C211yesmaximal, characteristic, complemented
15#428×C269C211yesmaximal, characteristic, complemented
16C2×Dic1076C2211yescharacteristic
17C2×#16864385C2211yescharacteristic
18#22575448—21no—
19#22575446—21no—
20#22575445—21no—
21#22575444C2211yescharacteristic
22#22575444C2211yescharacteristic
23#22575443—21no—
24#22575436—21no—
25#22575435—21no—
26#22575433C2211yescharacteristic
27#22575433C2211yescharacteristic
28#22575432C2211yescharacteristic
29#22575432C2211yescharacteristic
30#22575418C2211yescharacteristic
31#22575418C2211yescharacteristic
32C4×C22⋊D269C2211yescharacteristic
33#22575412—21no—
34#22575411C2211yescharacteristic
35#22575410—21no—
36C4⋊C4×D269C2211yescharacteristic
37#22575386C2211yescharacteristic
38#22575386C2211yescharacteristic
39#22575386C2211yescharacteristic
40#22575385C2211yescharacteristic
41#22575384C2211yescharacteristic
42#22575382C2211yescharacteristic
43#22575378C2211yescharacteristic
44#22575378C2211yescharacteristic
45#22575376C2211yescharacteristic
46#22575376C2211yescharacteristic
47#22575376C2211yescharacteristic
48#22575376C2211yescharacteristic
49C22⋊#14524130C2211yescharacteristic
50#22575365C2211yescharacteristic
51C8×Dic269C2211yescharacteristic
52C2×M16×C269C2211yescharacteristic
53#117×C269C2211yescharacteristic
54C2×C4⋊C4×C269C2211yescharacteristic
55#107×C269C2211yescharacteristic
56#107×C269C2211yescharacteristic
57#106×C269C2211yescharacteristic
58#106×C269C2211yescharacteristic
59C2×Dic538C2311yescharacteristic
60C2×Dic538C2311yescharacteristic
61C2×C22⋊D269D411yescharacteristic
62C2×C4×D269—21no—
63C2×C4×D269C2311yescharacteristic
64Dic1076—22no—
65#16864385—22no—
66C2×#14524130C2311yescharacteristic
67C2×#14524130C2311yescharacteristic
68C4⋊Dic269—21no—
69C4⋊Dic269—21no—
70C4⋊Dic269—21no—
71C4⋊Dic269—21no—
72C4⋊Dic269C2311yescharacteristic
73C4⋊Dic269C2311yescharacteristic
74#16864381—21no—
75#16864381—21no—
76#16864381—21no—
77#16864381—21no—
78#16864381—21no—
79#16864381—21no—
80#16864381—21no—
81#16864381D411yescharacteristic
82C22⋊Dic269—21no—
83C22⋊Dic269—21no—
84C22⋊Dic269D411yescharacteristic
85#16864379—21no—
86#16864379—21no—
87#16864379—21no—
88#16864379D411yescharacteristic
89C4×Dic269—21no—
90C4×Dic269—21no—
91C4×Dic269—21no—
92C4×Dic269C2311yescharacteristic
93C22×C4×C269—21no—
94C22×C4×C269C2311yescharacteristic
95M16×C269—22no—
96C2×C8×C269C2311yescharacteristic
97C2×C8×C269C2311yescharacteristic
98C4⋊C4×C269—21no—
99C4⋊C4×C269C2311yescharacteristic
100C4⋊C4×C269C2311yescharacteristic
101C4⋊C4×C269C2311yescharacteristic
102C4⋊C4×C269C2311yescharacteristic
103C22⋊C4×C269—21no—
104C42×C269—21no—
105C22×D269C2×D411yescharacteristic
106Dic538—21no—
107Dic538—21no—
108Dic538—21no—
109Dic538—21no—
110C22⋊D269—41no—
111C2×Dic269—21no—
112C2×Dic269—21no—
113C2×Dic269—21no—
114C2×Dic269—21no—
115C2×Dic269C2×D411yescharacteristic
116C2×Dic269C2×D411yescharacteristic
117C2×Dic269C2×D411yescharacteristic
118C4×D269—21no—
119C4×D269—21no—
120C4×D269—42no—
121#14524130—21no—
122#14524130—21no—
123C23×C269C2×D411yescharacteristic
124C2×C4×C269—21no—
125C2×C4×C269—21no—
126C2×C4×C269—21no—
127C2×C4×C269—21no—
128C2×C4×C269—21no—
129C2×C4×C269—21no—
130C2×C4×C269—42no—
131C2×C4×C269C2×D411yescharacteristic
132C2×C4×C269C2411yescharacteristic
133C8×C269—21no—
134C8×C269#4112yes—
135C22×C269—21no—
136C22×C269—21no—
137C22×C269—21no—
138C22×C269C22×D411yescharacteristic
139C2×D269—21no—
140C2×D269—21no—
141C2×D269—21no—
142C4×C269—21no—
143C4×C269—41no—
144C4×C269—41no—
145C4×C269—41no—
146C4×C269—41no—
147C4×C269C2×#4111yescharacteristic
148C4×C2692−1+411yescharacteristic
149Dic269—21no—
150Dic269—21no—
151Dic269—21no—
152Dic269—41no—
153Dic269—41no—
154Dic269—41no—
155Dic269—41no—
156C2×C269—41no—
157C2×C269#54711yescharacteristic
158C2×C269#57511yescharacteristic
159C2×C269#57811yescharacteristic
160D269—41no—
161C269#335211yescharacteristic, complemented, Sylow
162#3352—2691nomaximal, Sylow
163#552—2691no—
164#552—2691no—
165#499—2691no—
166#485—2691no—
167#485—2691no—
168#484—2691no—
169#484—2691no—
170#479—2691no—
171#479—2691no—
172#470—2691no—
173#470—2691no—
174C4×Q16—2691no—
175#436—2691no—
176#428—2691no—
177#428D26911yescharacteristic, complemented
178C2×Q16—2691no—
179C2×M16—2691no—
180C2×M16C2×D26911yescharacteristic
181#126—5381no—
182#126—5381no—
183#125—2691no—
184#125—2691no—
185#125—5381no—
186#125—5381no—
187#123—5381no—
188#123—5381no—
189C22⋊Q8—2691no—
190C22⋊Q8—2691no—
191C22⋊Q8—2691no—
192C22⋊Q8—2691no—
193C4×Q8—2691no—
194C4×Q8—2691no—
195C4×D4—2691no—
196#117—2691no—
197#117—5381no—
198#117—5381no—
199#117C2×D26911yescharacteristic
200C2×C4⋊C4—2691no—
201C2×C4⋊C4C2×D26911yescharacteristic
202#107—2691no—
203#107—2691no—
204#107—2691no—
205#107C2×D26911yescharacteristic
206#107C2×D26911yescharacteristic
207#106—2691no—
208#106—2691no—
209#106C2×D26911yescharacteristic
210#106C2×D26911yescharacteristic
211C4⋊C8—2691no—
212#103—2691no—
213#103—2691no—
214#103—2691no—
215#103—2691no—
216#103—2691no—
217#103—2691no—
218C22⋊C8—2691no—
219C22⋊C8—2691no—
220C4×C8—2691no—
221C2×Q8—2691no—
222C2×Q8—2691no—
223C2×D4—2691no—
224C22×C4—21no—
225C22×C4—2691no—
226C22×C4—5381no—
227C22×C4C22×D26911yescharacteristic
228Q16—5382no—
229M16—22no—
230M16—5382no—
231C2×C8—2691no—
232C2×C8—2691no—
233C2×C8C22×D26911yescharacteristic
234C2×C8C22×D26911yescharacteristic
235C4⋊C4—21no—
236C4⋊C4—2691no—
237C4⋊C4—2691no—
238C4⋊C4—2691no—
239C4⋊C4—5381no—
240C4⋊C4—5381no—
241C4⋊C4—5381no—
242C4⋊C4—5381no—
243C4⋊C4—5381no—
244C4⋊C4—5381no—
245C4⋊C4—5381no—
246C4⋊C4—5381no—
247C4⋊C4—5381no—
248C4⋊C4—5381no—
249C4⋊C4—5381no—
250C4⋊C4C22×D26911yescharacteristic
251C4⋊C4C22×D26911yescharacteristic
252C4⋊C4C22×D26911yescharacteristic
253C4⋊C4C22×D26911yescharacteristic
254C22⋊C4—21no—
255C22⋊C4—2691no—
256C22⋊C4—2691no—
257C22⋊C4—5381no—
258C22⋊C4—5381no—
259C22⋊C4—5381no—
260C22⋊C4—5381no—
261C22⋊C4—5381no—
262C42—21no—
263C42—2691no—
264C42—5381no—
265C42—5381no—
266C42—5381no—
267C23—2691no—
268C23D4×D26911yescharacteristic
269Q8—5381no—
270Q8—5381no—
271Q8—5381no—
272Q8—5381no—
273D4—10761no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—42no—
281C2×C4—2691no—
282C2×C4—2691no—
283C2×C4—2691no—
284C2×C4—5381no—
285C2×C4—5381no—
286C2×C4—5381no—
287C2×C4—5381no—
288C2×C4—5381no—
289C2×C4—5381no—
290C2×C4—10762no—
291C2×C4D4×D26911yescharacteristic
292C2×C4C23×D26911yescharacteristic
293C8—21no—
294C8—5381no—
295C8—5381no—
296C8#1686440212yes—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—5381no—
301C22—5381no—
302C22—5381no—
303C22C2×D4×D26911yescharacteristic
304C4—21no—
305C4—41no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—5381no—
310C4—5381no—
311C4—5381no—
312C4—10761no—
313C4—10761no—
314C4—10761no—
315C4—10761no—
316C4C2×#1686440211yescharacteristic
317C4#2257550111yescharacteristic
318C2—41no—
319C2—10761no—
320C2#3177125611yescharacteristic
321C2#3177140111yescharacteristic
322C2#3177142111yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#428×C269C2
socleC22×C269C22×D4
prime radicalC2×C22⋊D269D4
hypercenter#428D269
hypocenterC269#3352
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
3. #31771005C2semidirect
4. #31770949C2semidirect
5. #31770946C2semidirect
6. #31770942C2semidirect
7. #31770940C2semidirect
8. #31770915C2semidirect
9. #31770911C2semidirect
10. #31770864C2semidirect
11. #31770863C2semidirect
12. #31770699C2semidirect
14. #31770646C2semidirect
15. #428×C269C2semidirect
161. C269#3352semidirect
177. #428D269semidirect

Maximal subgroups

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

subgroupindex
#317712942
#317712902
#317710052
#317709492
#317709462
#317709422
#317709402
#317709152
#317709112
#317708642
#317708632
#317706992
#317706812
#317706462
#428×C2692
#3352269