Finite Group Databasebeta

← #46823475

Subgroups — #46823475

Order 34432.

386 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31771319C211yesmaximal, characteristic, complemented
2#31771255C211yesmaximal, characteristic, complemented
3#31771002C211yesmaximal, characteristic, complemented
4#31770927C211yesmaximal, characteristic, complemented
5#31770926C211yesmaximal, characteristic, complemented
6#31770922C211yesmaximal, characteristic, complemented
7#31770921C211yesmaximal, characteristic, complemented
8#31770893C211yesmaximal, characteristic, complemented
9#31770883C211yesmaximal, characteristic, complemented
10#31770855C211yesmaximal, characteristic, complemented
11D8×Dic269C211yesmaximal, characteristic, complemented
12#31770681C211yesmaximal, characteristic, complemented
13#31770647C211yesmaximal, characteristic, complemented
14#31770644C211yesmaximal, characteristic, complemented
15#469×C269C211yesmaximal, characteristic, complemented
16C2×#16864401—21no—
17C2×#16864385C2211yescharacteristic
18#22575452C2211yescharacteristic
19#22575452C2211yescharacteristic
20#22575440—21no—
21#22575439—21no—
22#22575435—21no—
23#22575434—21no—
24#22575430—21no—
25#22575425—21no—
26#22575424C2211yescharacteristic
27#22575424C2211yescharacteristic
28D4×Dic269C2211yescharacteristic
29D4×Dic269C2211yescharacteristic
30C4×C22⋊D269C2211yescharacteristic
31#22575412C2211yescharacteristic
32#22575411C2211yescharacteristic
33#22575409—21no—
34C2×C4⋊Dic269C2211yescharacteristic
35C4⋊C4×D269C2211yescharacteristic
36#22575387C2211yescharacteristic
37#22575387C2211yescharacteristic
38#22575386C2211yescharacteristic
39#22575385C2211yescharacteristic
40#22575385C2211yescharacteristic
41#22575384C2211yescharacteristic
42#22575384C2211yescharacteristic
43#22575384C2211yescharacteristic
44#22575382C2211yescharacteristic
45#22575374C2211yescharacteristic
46#22575374C2211yescharacteristic
47#22575374C2211yescharacteristic
48#22575374C2211yescharacteristic
49C22⋊#14524130C2211yescharacteristic
50#22575365C2211yescharacteristic
51C8×Dic269C2211yescharacteristic
52C2×D8×C269C2211yescharacteristic
53C2×M16×C269C2211yescharacteristic
54#121×C269C2211yescharacteristic
55#121×C269C2211yescharacteristic
56#106×C269C2211yescharacteristic
57#102×C269C2211yescharacteristic
58#102×C269C2211yescharacteristic
59#16864401—44no—
60C2×Dic538—21no—
61C2×Dic538—21no—
62C2×C22⋊D269—21no—
63C2×C22⋊D269—21no—
64C2×C22⋊D269D411yescharacteristic
65C22×Dic269—21no—
66C22×Dic269—21no—
67C22×Dic269—21no—
68C2×C4×D269—21no—
69C2×C4×D269C2311yescharacteristic
70#16864385—22no—
71C2×#14524130C2311yescharacteristic
72C2×#14524130C2311yescharacteristic
73C4⋊Dic269—21no—
74C4⋊Dic269C2311yescharacteristic
75C4⋊Dic269C2311yescharacteristic
76C4⋊Dic269C2311yescharacteristic
77C4⋊Dic269C2311yescharacteristic
78#16864381—21no—
79#16864381—21no—
80#16864381—21no—
81#16864381D411yescharacteristic
82C22⋊Dic269—21no—
83C22⋊Dic269—21no—
84C22⋊Dic269—21no—
85C22⋊Dic269—21no—
86C22⋊Dic269—21no—
87C22⋊Dic269D411yescharacteristic
88#16864379—21no—
89#16864379D411yescharacteristic
90C4×Dic269—21no—
91C4×Dic269—21no—
92C4×Dic269C2311yescharacteristic
93C2×D4×C269—21no—
94C2×D4×C269—21no—
95C2×D4×C269C2311yescharacteristic
96C2×D4×C269C2311yescharacteristic
97C22×C4×C269C2311yescharacteristic
98D8×C269—22no—
99M16×C269—22no—
100C2×C8×C269C2311yescharacteristic
101C2×C8×C269C2311yescharacteristic
102C4⋊C4×C269C2311yescharacteristic
103C4⋊C4×C269C2311yescharacteristic
104C22⋊C4×C269—21no—
105C22⋊C4×C269—21no—
106C22×D269C2×D411yescharacteristic
107Dic538—42no—
108Dic538—81no—
109C22⋊D269—22no—
110C22⋊D269—42no—
111C22⋊D269—81no—
112C2×Dic269—21no—
113C2×Dic269—21no—
114C2×Dic269—21no—
115C2×Dic269—21no—
116C2×Dic269—42no—
117C2×Dic269—42no—
118C2×Dic269—42no—
119C2×Dic269C2×D411yescharacteristic
120C2×Dic269C2×D411yescharacteristic
121C2×Dic269C2×D411yescharacteristic
122C4×D269—21no—
123C4×D269—21no—
124C4×D269—42no—
125#14524130—21no—
126#14524130—21no—
127C23×C269—21no—
128C23×C269—21no—
129C23×C269C2×D411yescharacteristic
130D4×C269—21no—
131D4×C269—21no—
132D4×C269—21no—
133D4×C269—21no—
134D4×C269—42no—
135D4×C269—81no—
136C2×C4×C269—21no—
137C2×C4×C269—21no—
138C2×C4×C269—21no—
139C2×C4×C269—21no—
140C2×C4×C269C2×D411yescharacteristic
141C2×C4×C269C2411yescharacteristic
142C8×C269—21no—
143C8×C269#4112yescomplemented
144C22×C269—21no—
145C22×C269—21no—
146C22×C269—21no—
147C22×C269—41no—
148C22×C269—41no—
149C22×C269—41no—
150C22×C269—41no—
151C22×C269—41no—
152C22×C269—41no—
153C22×C269C22×D411yescharacteristic
154C2×D269—21no—
155C2×D269—21no—
156C2×D269—21no—
157C4×C269—21no—
158C4×C269—41no—
159C4×C269—41no—
160C4×C269C2×#4111yescharacteristic
161C4×C2692−1+411yescharacteristic
162Dic269—21no—
163Dic269—21no—
164Dic269—41no—
165Dic269—41no—
166Dic269—41no—
167Dic269—41no—
168Dic269AGL(1,ℤ/8)12yes—
169C2×C269—41no—
170C2×C269—81no—
171C2×C269—81no—
172C2×C269#54711yescharacteristic
173C2×C269C2×AGL(1,ℤ/8)11yescharacteristic
174C2×C269#57711yescharacteristic
175D269—41no—
176C269#335411yescharacteristic, complemented, Sylow
177#3354—2691nomaximal, Sylow
178#555—2691no—
179#547—2691no—
180#498—2691no—
181#482—2691no—
182#482—2691no—
183#481—2691no—
184#481—2691no—
185#476—2691no—
186#474—2691no—
187#469—2691no—
188#469D26911yescharacteristic, complemented
189C4×D8—2691no—
190#436—2691no—
191#428—2691no—
192#428—2691no—
193C2×#41—5381no—
194C2×D8C2×D26911yescharacteristic
195C2×M16—2691no—
196C2×M16C2×D26911yescharacteristic
197C4⋊Q8—2691no—
198C4⋊Q8—2691no—
199#124—5381no—
200#124—5381no—
201#123—5381no—
202#123—5381no—
203C22⋊Q8—5381no—
204#121—2691no—
205#121—2691no—
206#121—5381no—
207#121C2×D26911yescharacteristic
208#121C2×D26911yescharacteristic
209C4×D4—2691no—
210C4×D4—2691no—
211C4×D4—2691no—
212#117—2691no—
213#117—2691no—
214#117—5381no—
215C2×C4⋊C4—2691no—
216C2×C4⋊C4—2691no—
217#107—2691no—
218#107—2691no—
219#107—2691no—
220#106—2691no—
221#106—2691no—
222#106—2691no—
223#106—2691no—
224#106—2691no—
225#106C2×D26911yescharacteristic
226C4⋊C8—2691no—
227#102—2691no—
228#102—2691no—
229#102—2691no—
230#102—2691no—
231#102C2×D26911yescharacteristic
232#102C2×D26911yescharacteristic
233C22⋊C8—2691no—
234C22⋊C8—2691no—
235C4×C8—2691no—
236#41—10764no—
237C2×Q8—5381no—
238C2×Q8—5381no—
239C2×D4—21no—
240C2×D4—21no—
241C2×D4—2691no—
242C2×D4—5381no—
243C2×D4—5381no—
244C2×D4C22×D26911yescharacteristic
245C2×D4C22×D26911yescharacteristic
246C22×C4—2691no—
247C22×C4—5381no—
248C22×C4—5381no—
249C22×C4—5381no—
250C22×C4—5381no—
251C22×C4C22×D26911yescharacteristic
252D8—22no—
253M16—22no—
254M16—5382no—
255C2×C8—2691no—
256C2×C8—2691no—
257C2×C8C22×D26911yescharacteristic
258C2×C8C22×D26911yescharacteristic
259C4⋊C4—2691no—
260C4⋊C4—2691no—
261C4⋊C4—2691no—
262C4⋊C4—2691no—
263C4⋊C4—2691no—
264C4⋊C4—5381no—
265C4⋊C4—5381no—
266C4⋊C4—5381no—
267C4⋊C4—5381no—
268C4⋊C4C22×D26911yescharacteristic
269C4⋊C4C22×D26911yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—21no—
272C22⋊C4—2691no—
273C22⋊C4—2691no—
274C22⋊C4—5381no—
275C22⋊C4—5381no—
276C22⋊C4—5381no—
277C22⋊C4—5381no—
278C22⋊C4—5381no—
279C22⋊C4—5381no—
280C42—2691no—
281C42—5381no—
282C42—5381no—
283C23—21no—
284C23—21no—
285C23—2691no—
286C23D4×D26911yescharacteristic
287Q8—10762no—
288Q8—21521no—
289D4—21no—
290D4—21no—
291D4—21no—
292D4—21no—
293D4—42no—
294D4—81no—
295D4—5382no—
296D4—10762no—
297D4—21521no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—2691no—
303C2×C4—2691no—
304C2×C4—2691no—
305C2×C4—5381no—
306C2×C4—5381no—
307C2×C4—5381no—
308C2×C4—5381no—
309C2×C4—5381no—
310C2×C4—5381no—
311C2×C4—10762no—
312C2×C4—10762no—
313C2×C4—10762no—
314C2×C4—10762no—
315C2×C4D4×D26911yescharacteristic
316C2×C4C23×D26911yescharacteristic
317C8—21no—
318C8—5381no—
319C8—5381no—
320C8#1686440112yescomplemented
321C22—21no—
322C22—21no—
323C22—21no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—5381no—
331C22—5381no—
332C22—5381no—
333C22C2×D4×D26911yescharacteristic
334C4—21no—
335C4—41no—
336C4—41no—
337C4—2692no—
338C4—5381no—
339C4—5381no—
340C4—10761no—
341C4—10761no—
342C4—10761no—
343C4—10761no—
344C4C2×#1686440111yescharacteristic
345C4#2257550011yescharacteristic
346C2—41no—
347C2—81no—
348C2—81no—
349C2—10761no—
350C2#3177125511yescharacteristic
351C2AGL(1,ℤ/8)×D26911yescharacteristic
352C2#3177141511yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#469×C269C2
socleC22×C269C22×D4
hypercenter#469D269
hypocenterC269#3354
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. #31771319C2semidirect
2. #31771255C2semidirect
3. #31771002C2semidirect
4. #31770927C2semidirect
5. #31770926C2semidirect
6. #31770922C2semidirect
7. #31770921C2semidirect
8. #31770893C2semidirect
9. #31770883C2semidirect
10. #31770855C2semidirect
11. D8×Dic269C2semidirect
12. #31770681C2semidirect
13. #31770647C2semidirect
14. #31770644C2semidirect
15. #469×C269C2semidirect
143. C8×C269#41semidirect
176. C269#3354semidirect
188. #469D269semidirect
320. C8#16864401semidirect

Maximal subgroups

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

subgroupindex
#317713192
#317712552
#317710022
#317709272
#317709262
#317709222
#317709212
#317708932
#317708832
#317708552
D8×Dic2692
#317706812
#317706472
#317706442
#469×C2692
#3354269