Finite Group Databasebeta

← #46822780

Subgroups — #46822780

Order 34432.

398 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31771302C211yesmaximal, characteristic, complemented
2#31771262C211yesmaximal, characteristic, complemented
3#31770958C211yesmaximal, characteristic, complemented
4#31770947C211yesmaximal, characteristic, complemented
5#31770922C211yesmaximal, characteristic, complemented
6#31770914C211yesmaximal, characteristic, complemented
7#31770838C211yesmaximal, characteristic, complemented
8#31770832C211yesmaximal, characteristic, complemented
9#31770818C211yesmaximal, characteristic, complemented
10#31770742C211yesmaximal, characteristic, complemented
11SD16×Dic269C211yesmaximal, characteristic, complemented
12#31770668C211yesmaximal, characteristic, complemented
13#31770601C211yesmaximal, characteristic, complemented
14#103×D269C211yesmaximal, characteristic, complemented
15#449×C269C211yesmaximal, characteristic, complemented
16C2×Q8×D269C2211yescharacteristic
17C2×#16864390C2211yescharacteristic
18C22×#14524130C2211yescharacteristic
19C2×C8×D269C2211yescharacteristic
20#22575446—21no—
21#22575444C2211yescharacteristic
22#22575440—21no—
23#22575440C2211yescharacteristic
24#22575434—21no—
25#22575432—21no—
26#22575432C2211yescharacteristic
27#22575424—21no—
28#22575424—21no—
29#22575424C2211yescharacteristic
30#22575424C2211yescharacteristic
31Q8×Dic269C2211yescharacteristic
32D4×Dic269C2211yescharacteristic
33C4×C22⋊D269—21no—
34C4×C22⋊D269C2211yescharacteristic
35#22575413—21no—
36#22575412C2211yescharacteristic
37C4⋊C4×D269C2211yescharacteristic
38#22575387C2211yescharacteristic
39#22575385C2211yescharacteristic
40#22575384C2211yescharacteristic
41#22575382C2211yescharacteristic
42#22575378C2211yescharacteristic
43#22575378C2211yescharacteristic
44#22575377C2211yescharacteristic
45#22575377C2211yescharacteristic
46#22575376C2211yescharacteristic
47#22575374C2211yescharacteristic
48#22575374C2211yescharacteristic
49#22575374C2211yescharacteristic
50#22575365C2211yescharacteristic
51C8×Dic269C2211yescharacteristic
52C2×#41×C269C2211yescharacteristic
53C2×SD16×C269C2211yescharacteristic
54C2×D8×C269C2211yescharacteristic
55#121×C269C2211yescharacteristic
56#103×C269C2211yescharacteristic
57#102×C269C2211yescharacteristic
58C22⋊C8×C269C2211yescharacteristic
59C2×Dic538—21no—
60C2×Dic538C2311yescharacteristic
61Q8×D269—22no—
62Q8×D269—41no—
63C2×C22⋊D269—21no—
64C2×C22⋊D269—21no—
65C2×C22⋊D269D411yescharacteristic
66C22×Dic269—21no—
67C2×C4×D269—21no—
68C2×C4×D269—21no—
69C2×C4×D269C2311yescharacteristic
70#16864390—41no—
71C2×#14524130—41no—
72C2×#14524130C2311yescharacteristic
73C2×#14524130C2311yescharacteristic
74C8×D269—41no—
75C4⋊Dic269—21no—
76C4⋊Dic269—21no—
77C4⋊Dic269C2311yescharacteristic
78C4⋊Dic269C2311yescharacteristic
79C4⋊Dic269C2311yescharacteristic
80#16864381—21no—
81#16864381—21no—
82#16864381—21no—
83#16864381D411yescharacteristic
84C22⋊Dic269—21no—
85C22⋊Dic269—21no—
86C22⋊Dic269—21no—
87C22⋊Dic269—21no—
88C22⋊Dic269—21no—
89C22⋊Dic269—21no—
90C22⋊Dic269—21no—
91C22⋊Dic269D411yescharacteristic
92#16864379D411yescharacteristic
93C4×Dic269—21no—
94C4×Dic269—21no—
95C4×Dic269C2311yescharacteristic
96#41×C269—22no—
97#41×C269—41no—
98C2×Q8×C269C2311yescharacteristic
99C2×D4×C269—21no—
100C2×D4×C269—21no—
101C2×D4×C269C2311yescharacteristic
102C2×D4×C269C2311yescharacteristic
103C22×C4×C269—21no—
104C22×C4×C269C2311yescharacteristic
105SD16×C269—22no—
106D8×C269—41no—
107C2×C8×C269C2311yescharacteristic
108C2×C8×C269C2311yescharacteristic
109C4⋊C4×C269C2311yescharacteristic
110C22⋊C4×C269—21no—
111C22×D269C2×D411yescharacteristic
112Dic538—21no—
113Dic538—21no—
114Dic538—42no—
115C22⋊D269—22no—
116C22⋊D269—42no—
117C22⋊D269—42no—
118C2×Dic269—21no—
119C2×Dic269—21no—
120C2×Dic269—21no—
121C2×Dic269—21no—
122C2×Dic269—81no—
123C2×Dic269C2×D411yescharacteristic
124C2×Dic269C2×D411yescharacteristic
125C2×Dic269C2×D411yescharacteristic
126C4×D269—21no—
127C4×D269—21no—
128C4×D269—42no—
129C4×D269—81no—
130#14524130—21no—
131#14524130—21no—
132C23×C269—21no—
133C23×C269—21no—
134C23×C269C2×D411yescharacteristic
135Q8×C269—21no—
136Q8×C269#4112yes—
137D4×C269—21no—
138D4×C269—21no—
139D4×C269—21no—
140D4×C269—21no—
141D4×C269—42no—
142D4×C269—81no—
143C2×C4×C269—21no—
144C2×C4×C269—21no—
145C2×C4×C269—21no—
146C2×C4×C269—21no—
147C2×C4×C269—42no—
148C2×C4×C269C2×D411yescharacteristic
149C2×C4×C269C2411yescharacteristic
150C8×C269—21no—
151C8×C269—21no—
152C22×C269—21no—
153C22×C269—21no—
154C22×C269—21no—
155C22×C269—41no—
156C22×C269—41no—
157C22×C269—41no—
158C22×C269—41no—
159C22×C269—41no—
160C22×C269—41no—
161C22×C269C22×D411yescharacteristic
162C2×D269—21no—
163C2×D269—21no—
164C2×D269#13512yes—
165C4×C269—21no—
166C4×C269—22no—
167C4×C269—41no—
168C4×C269C2×#4111yescharacteristic
169C4×C269H(5,𝔽2)11yescharacteristic
170Dic269—21no—
171Dic269—21no—
172Dic269—21no—
173Dic269—41no—
174Dic269—41no—
175Dic269—41no—
176Dic269—41no—
177C2×C269—41no—
178C2×C269—81no—
179C2×C269—81no—
180C2×C269#54611yescharacteristic
181C2×C269C2×#13511yescharacteristic
182C2×C269#57711yescharacteristic
183D269—22no—
184C269#330111yescharacteristic, complemented, Sylow
185#3301—2691nomaximal, Sylow
186#553—2691no—
187#548—2691no—
188#488—2691no—
189#485—2691no—
190#481—2691no—
191#479—2691no—
192#466—2691no—
193#465—2691no—
194#463—2691no—
195#449D26911yescharacteristic, complemented
196#448—2691no—
197C4×SD16—2691no—
198C8×D4—2691no—
199#416—2691no—
200C2×#103—2691no—
201C2×#41C2×D26911yescharacteristic
202C22×Q8—2691no—
203C2×SD16—2691no—
204C2×SD16C2×D26911yescharacteristic
205C2×D8C2×D26911yescharacteristic
206C22×C8—2691no—
207C22×C8—2691no—
208#126—5381no—
209#125—2691no—
210#124—2691no—
211#124—5381no—
212#123—5381no—
213C22⋊Q8—2691no—
214C22⋊Q8—5381no—
215#121—2691no—
216#121—2691no—
217#121—5381no—
218#121—5381no—
219#121C2×D26911yescharacteristic
220C4×Q8—2691no—
221C4×D4—2691no—
222C4×D4—2691no—
223C4×D4—5381no—
224C4×D4—5381no—
225#117—2691no—
226C2×C4⋊C4—2691no—
227#107—2691no—
228#106—2691no—
229#106—2691no—
230C4⋊C8—2691no—
231#103—2691no—
232#103—2691no—
233#103—2691no—
234#103—2691no—
235#103—2691no—
236#103C2×D26911yescharacteristic
237#102—2691no—
238#102—2691no—
239#102—2691no—
240#102C2×D26911yescharacteristic
241C22⋊C8—2691no—
242C22⋊C8C2×D26911yescharacteristic
243C4×C8—2691no—
244#41—22no—
245#41—41no—
246C2×Q8—2691no—
247C2×Q8—5381no—
248C2×Q8—5382no—
249C2×Q8—10761no—
250C2×Q8C22×D26911yescharacteristic
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—2691no—
254C2×D4—5381no—
255C2×D4—5381no—
256C2×D4C22×D26911yescharacteristic
257C2×D4C22×D26911yescharacteristic
258C22×C4—21no—
259C22×C4—2691no—
260C22×C4—5381no—
261C22×C4—5381no—
262C22×C4—5381no—
263C22×C4C22×D26911yescharacteristic
264SD16—22no—
265SD16—10761no—
266D8—41no—
267C2×C8—2691no—
268C2×C8—2691no—
269C2×C8—10761no—
270C2×C8—10761no—
271C2×C8C22×D26911yescharacteristic
272C2×C8C22×D26911yescharacteristic
273C4⋊C4—2691no—
274C4⋊C4—2691no—
275C4⋊C4—2691no—
276C4⋊C4—2691no—
277C4⋊C4—5381no—
278C4⋊C4—5381no—
279C4⋊C4—5381no—
280C4⋊C4—5381no—
281C4⋊C4—5381no—
282C4⋊C4C22×D26911yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—2691no—
285C22⋊C4—2691no—
286C22⋊C4—5381no—
287C22⋊C4—5381no—
288C22⋊C4—5381no—
289C22⋊C4—5381no—
290C22⋊C4—5381no—
291C22⋊C4—5381no—
292C22⋊C4—5381no—
293C42—2691no—
294C42—5381no—
295C42—5381no—
296C23—21no—
297C23—21no—
298C23—2691no—
299C23D4×D26911yescharacteristic
300Q8—21no—
301Q8—5381no—
302Q8—5381no—
303Q8—10762no—
304Q8#1686440112yes—
305D4—21no—
306D4—21no—
307D4—21no—
308D4—21no—
309D4—42no—
310D4—81no—
311D4—5382no—
312D4—10762no—
313D4—10762no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—42no—
319C2×C4—2691no—
320C2×C4—2691no—
321C2×C4—2691no—
322C2×C4—5381no—
323C2×C4—5381no—
324C2×C4—5381no—
325C2×C4—5381no—
326C2×C4—5381no—
327C2×C4—5381no—
328C2×C4—10762no—
329C2×C4—21521no—
330C2×C4—21521no—
331C2×C4D4×D26911yescharacteristic
332C2×C4C23×D26911yescharacteristic
333C8—21no—
334C8—21no—
335C8—5381no—
336C8—5381no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—2692no—
347C22—5381no—
348C22—5381no—
349C22C2×D4×D26911yescharacteristic
350C4—21no—
351C4—22no—
352C4—41no—
353C4—5381no—
354C4—5381no—
355C4—5381no—
356C4—10761no—
357C4—10761no—
358C4—10761no—
359C4—10761no—
360C4C2×#1686440111yescharacteristic
361C4#2257549811yescharacteristic
362C2—41no—
363C2—81no—
364C2—81no—
365C2—5382no—
366C2#3177124211yescharacteristic
367C2#135×D26911yescharacteristic
368C2#3177141511yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#449×C269C2
socleC22×C269C22×D4
hypercenter#449D269
hypocenterC269#3301
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #31771302C2semidirect
2. #31771262C2semidirect
3. #31770958C2semidirect
4. #31770947C2semidirect
5. #31770922C2semidirect
6. #31770914C2semidirect
7. #31770838C2semidirect
8. #31770832C2semidirect
9. #31770818C2semidirect
10. #31770742C2semidirect
11. SD16×Dic269C2semidirect
12. #31770668C2semidirect
13. #31770601C2semidirect
14. #103×D269C2semidirect
15. #449×C269C2semidirect
184. C269#3301semidirect
195. #449D269semidirect

Maximal subgroups

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

subgroupindex
#317713022
#317712622
#317709582
#317709472
#317709222
#317709142
#317708382
#317708322
#317708182
#317707422
SD16×Dic2692
#317706682
#317706012
#103×D2692
#449×C2692
#3301269