Finite Group Databasebeta

← #46820508

Subgroups — #46820508

Order 34432.

496 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22575482C211yesmaximal, characteristic, complemented
2C22×#16864385C211yesmaximal, characteristic, complemented
3#31771115C211yesmaximal, characteristic
4C22⋊Q8×D269C211yesmaximal, characteristic, complemented
5#31770864C211yesmaximal, characteristic, complemented
6#31770862C211yesmaximal, characteristic, complemented
7#31770850C211yesmaximal, characteristic, complemented
8#31770848C211yesmaximal, characteristic, complemented
9#31770846C211yesmaximal, characteristic, complemented
10#31770844C211yesmaximal, characteristic, complemented
11#31770832C211yesmaximal, characteristic, complemented
12#31770644C211yesmaximal, characteristic, complemented
13#31770612C211yesmaximal, characteristic, complemented
14#31770603C211yesmaximal, characteristic, complemented
15#465×C269C211yesmaximal, characteristic, complemented
16C2×#16864401C2211yescharacteristic
17C2×Q8×D269C2211yescharacteristic
18C22×C4×D269C2211yescharacteristic
19#22575482—24no—
20C2×Dic1076C2211yescharacteristic, complemented
21C2×#16864392C2211yescharacteristic, complemented
22C2×#16864390C2211yescharacteristic, complemented
23C2×#16864385—24no—
24C2×#16864385C2211yescharacteristic
25C2×#16864385C2211yescharacteristic
26C2×#16864385C2211yescharacteristic
27C2×#16864385C2211yescharacteristic
28C22×#14524130C2211yescharacteristic
29#22575435—21no—
30#22575433C2211yescharacteristic
31#22575433C2211yescharacteristic
32#22575432C2211yescharacteristic
33#22575432C2211yescharacteristic
34#22575431—21no—
35#22575430—21no—
36#22575429C2211yescharacteristic
37#22575424C2211yescharacteristic
38#22575421—21no—
39D4×Dic269—21no—
40#22575413—21no—
41#22575411C2211yescharacteristic
42C4⋊C4×D269—21no—
43C4⋊C4×D269C2211yescharacteristic
44C22⋊C4×D269—21no—
45#22575387C2211yescharacteristic, complemented
46#22575386C2211yescharacteristic, complemented
47#22575384C2211yescharacteristic, complemented
48#22575378C2211yescharacteristic, complemented
49#22575378C2211yescharacteristic, complemented
50#22575377C2211yescharacteristic, complemented
51#22575376C2211yescharacteristic, complemented
52#22575376C2211yescharacteristic, complemented
53#22575374C2211yescharacteristic, complemented
54C2×SD16×C269C2211yescharacteristic, complemented
55C22×C8×C269C2211yescharacteristic
56C22⋊Q8×C269C2211yescharacteristic
57#121×C269C2211yescharacteristic
58#106×C269C2211yescharacteristic, complemented
59#103×C269C2211yescharacteristic, complemented
60#102×C269C2211yescharacteristic, complemented
61C23×D269D411yescharacteristic
62#16864401—22no—
63#16864401—22no—
64C2×Dic538—21no—
65C2×Dic538C2311yescharacteristic
66C2×Dic538C2311yescharacteristic
67Q8×D269—22no—
68Q8×D269—22no—
69C2×C22⋊D269—21no—
70C2×C22⋊D269—21no—
71C22×Dic269—21no—
72C22×Dic269D411yescharacteristic
73C2×C4×D269—22no—
74C2×C4×D269—22no—
75C2×C4×D269—21no—
76C2×C4×D269—21no—
77C2×C4×D269D411yescharacteristic
78C2×C4×D269D411yescharacteristic
79C2×C4×D269C2311yescharacteristic
80C2×C4×D269C2311yescharacteristic
81Dic1076—22no—
82#16864392—22no—
83#16864390—22no—
84#16864385—22no—
85#16864385—22no—
86#16864385—22no—
87#16864385D414yescomplemented
88C2×#14524130—22no—
89C2×#14524130C2311yescharacteristic
90C2×#14524130C2311yescharacteristic
91C4⋊Dic269—21no—
92C4⋊Dic269—21no—
93C4⋊Dic269C2311yescharacteristic
94C4⋊Dic269C2311yescharacteristic
95#16864381—21no—
96#16864381—21no—
97#16864381—21no—
98#16864381—21no—
99C22⋊Dic269—21no—
100C22⋊Dic269—21no—
101C22⋊Dic269—21no—
102#16864379—21no—
103#16864379—21no—
104#16864379—21no—
105#16864379—21no—
106C4×Dic269—21no—
107C2×Q8×C269C2311yescharacteristic
108C2×D4×C269—21no—
109C2×D4×C269C2311yescharacteristic
110C22×C4×C269C2311yescharacteristic
111SD16×C269—22no—
112C2×C8×C269—22no—
113C2×C8×C269C2311yescharacteristic
114C2×C8×C269C2311yescharacteristic
115C4⋊C4×C269—21no—
116C4⋊C4×C269C2311yescharacteristic
117C4⋊C4×C269C2311yescharacteristic
118C22⋊C4×C269—21no—
119C22⋊C4×C269—21no—
120C22×D269—22no—
121C22×D269—22no—
122C22×D269—41no—
123C22×D269C2×D411yescharacteristic
124C22×D269C2×D411yescharacteristic
125Dic538—21no—
126Dic538—21no—
127Dic538—21no—
128Dic538—21no—
129Dic538—42no—
130C22⋊D269—42no—
131C22⋊D269—42no—
132C2×Dic269—22no—
133C2×Dic269—21no—
134C2×Dic269—21no—
135C2×Dic269—21no—
136C2×Dic269—21no—
137C2×Dic269—42no—
138C2×Dic269C2×D411yescharacteristic
139C2×Dic269C2×D411yescharacteristic
140C4×D269—22no—
141C4×D269—21no—
142C4×D269—21no—
143C4×D269—22no—
144C4×D269—42no—
145C4×D269—42no—
146C4×D269C2×D412yes—
147C4×D269#4112yes—
148#14524130—21no—
149#14524130C2×D412yes—
150C23×C269—21no—
151C23×C269C2×D411yescharacteristic
152Q8×C269—21no—
153Q8×C269—21no—
154D4×C269—21no—
155D4×C269—21no—
156D4×C269—42no—
157C2×C4×C269—21no—
158C2×C4×C269—21no—
159C2×C4×C269—21no—
160C2×C4×C269—21no—
161C2×C4×C269—21no—
162C2×C4×C269C2×D411yescharacteristic
163C2×C4×C269C2411yescharacteristic
164C8×C269—21no—
165C8×C269C2×D412yes—
166C22×C269—21no—
167C22×C269—21no—
168C22×C269—41no—
169C22×C269—41no—
170C22×C269—41no—
171C22×C269#13712yescomplemented
172C22×C269C22×D411yescharacteristic
173C2×D269—21no—
174C2×D269—21no—
175C2×D269—21no—
176C2×D269—21no—
177C2×D269—21no—
178C2×D269—44no—
179C2×D269#12112yes—
180C4×C269—21no—
181C4×C269—41no—
182C4×C269—41no—
183C4×C269—41no—
184C4×C269C22×D411yescharacteristic
185C4×C269C2×#4111yescharacteristic
186Dic269—21no—
187Dic269—41no—
188Dic269—41no—
189Dic269—41no—
190Dic269—41no—
191Dic269#12112yes—
192C2×C269—22no—
193C2×C269—81no—
194C2×C269C2×#12111yescharacteristic
195C2×C269C2×#13711yescharacteristic
196C2×C269#57511yescharacteristic
197D269—22no—
198D269—41no—
199C269#305611yescharacteristic, complemented, Sylow
200#3056—2691nomaximal, Sylow
201C2×#137—2691no—
202C22×M16—2691no—
203#525—2691no—
204C2×C22⋊Q8—2691no—
205#470—2691no—
206#470—2691no—
207#468—2691no—
208#468—2691no—
209#467—2691no—
210#467—2691no—
211#465—2691no—
212#465D26911yescharacteristic, complemented
213#428—2691no—
214#419—2691no—
215#417—2691no—
216C2×#41—2691no—
217C22×Q8—2691no—
218C23×C4—2691no—
219#137—5384no—
220C2×Q16—2691no—
221C2×Q16—2691no—
222C2×SD16—2691no—
223C2×SD16C2×D26911yescharacteristic, complemented
224C2×M16—2691no—
225C2×M16—2691no—
226C2×M16—2691no—
227C2×M16—2691no—
228C2×M16—5384no—
229C22×C8—2691no—
230C22×C8C2×D26911yescharacteristic
231#123—5381no—
232C22⋊Q8—2691no—
233C22⋊Q8—2691no—
234C22⋊Q8—2691no—
235C22⋊Q8—2691no—
236C22⋊Q8—2691no—
237C22⋊Q8—5381no—
238C22⋊Q8—5381no—
239C22⋊Q8C2×D26911yescharacteristic
240#121—2691no—
241#121C2×D26911yescharacteristic
242#120—5381no—
243C4×D4—5381no—
244C4×D4—5381no—
245#117—2691no—
246C2×C4⋊C4—2691no—
247C2×C4⋊C4—5381no—
248C2×C22⋊C4—5381no—
249#107—2691no—
250#107—2691no—
251#106—2691no—
252#106C2×D26911yescharacteristic, complemented
253#103—2691no—
254#103—2691no—
255#103—2691no—
256#103—2691no—
257#103—2691no—
258#103C2×D26911yescharacteristic, complemented
259#102—2691no—
260#102C2×D26911yescharacteristic, complemented
261C24—2691no—
262#41—5382no—
263#41—5382no—
264C2×Q8—2691no—
265C2×Q8—2691no—
266C2×Q8—5381no—
267C2×Q8—5382no—
268C2×Q8—5382no—
269C2×Q8C22×D26911yescharacteristic
270C2×D4—21no—
271C2×D4—5381no—
272C2×D4—5381no—
273C2×D4C22×D26911yescharacteristic
274C22×C4—2691no—
275C22×C4—2691no—
276C22×C4—2691no—
277C22×C4—2691no—
278C22×C4—2691no—
279C22×C4—5382no—
280C22×C4—5382no—
281C22×C4—5381no—
282C22×C4—5381no—
283C22×C4—5381no—
284C22×C4C22×D26911yescharacteristic
285Q16—5382no—
286Q16—5382no—
287SD16—22no—
288SD16—5382no—
289M16—2694no—
290M16—5382no—
291M16—5382no—
292M16—5382no—
293C2×C8—22no—
294C2×C8—2691no—
295C2×C8—2691no—
296C2×C8—5382no—
297C2×C8C22×D26911yescharacteristic
298C2×C8C22×D26911yescharacteristic
299C4⋊C4—21no—
300C4⋊C4—2691no—
301C4⋊C4—2691no—
302C4⋊C4—5381no—
303C4⋊C4—5381no—
304C4⋊C4—5381no—
305C4⋊C4—5381no—
306C4⋊C4—5381no—
307C4⋊C4—5381no—
308C4⋊C4C22×D26911yescharacteristic
309C4⋊C4C22×D26911yescharacteristic
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—5381no—
313C22⋊C4—5381no—
314C22⋊C4—5381no—
315C22⋊C4—5381no—
316C22⋊C4—5381no—
317C22⋊C4—5381no—
318C22⋊C4—5381no—
319C42—5381no—
320C23—21no—
321C23—2691no—
322C23—2691no—
323C23—5382no—
324C23—5382no—
325C23—10761no—
326C23D4×D26911yescharacteristic
327Q8—21no—
328Q8—21no—
329Q8—5381no—
330Q8—5381no—
331Q8—5381no—
332Q8—5381no—
333Q8—10762no—
334D4—21no—
335D4—21no—
336D4—42no—
337D4—10762no—
338D4—10762no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—2691no—
345C2×C4—2692no—
346C2×C4—2692no—
347C2×C4—2691no—
348C2×C4—5382no—
349C2×C4—5382no—
350C2×C4—5382no—
351C2×C4—5381no—
352C2×C4—5381no—
353C2×C4—5381no—
354C2×C4—5381no—
355C2×C4—5381no—
356C2×C4—5381no—
357C2×C4—10762no—
358C2×C4—10762no—
359C2×C4—10762no—
360C2×C4D4×D26911yescharacteristic
361C2×C4C23×D26911yescharacteristic
362C8—21no—
363C8—2692no—
364C8—5381no—
365C8D4×D26912yes—
366C22—21no—
367C22—21no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—2692no—
372C22—5381no—
373C22—5381no—
374C22—5381no—
375C22—5381no—
376C22—5381no—
377C22—10764no—
378C22#2257548212yescomplemented
379C22C2×D4×D26911yescharacteristic
380C4—21no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—2692no—
385C4—5381no—
386C4—10761no—
387C4—10761no—
388C4—10761no—
389C4—10761no—
390C4C2×D4×D26911yescharacteristic
391C4#41×D26911yescharacteristic
392C2—22no—
393C2—81no—
394C2—5382no—
395C2—10761no—
396C2#121×D26911yescharacteristic
397C2C2×#2257548211yescharacteristic
398C2#3177140211yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#465×C269C2
socleC22×C269C22×D4
prime radical#31771115C2
hypercenter#465D269
hypocenterC269#3056
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. C2×#22575482C2semidirect
2. C22×#16864385C2semidirect
4. C22⋊Q8×D269C2semidirect
5. #31770864C2semidirect
6. #31770862C2semidirect
7. #31770850C2semidirect
8. #31770848C2semidirect
9. #31770846C2semidirect
10. #31770844C2semidirect
11. #31770832C2semidirect
12. #31770644C2semidirect
13. #31770612C2semidirect
14. #31770603C2semidirect
15. #465×C269C2semidirect
20. C2×Dic1076C22semidirect
21. C2×#16864392C22semidirect
22. C2×#16864390C22semidirect
45. #22575387C22semidirect
46. #22575386C22semidirect
47. #22575384C22semidirect
48. #22575378C22semidirect
49. #22575378C22semidirect
50. #22575377C22semidirect
51. #22575376C22semidirect
52. #22575376C22semidirect
53. #22575374C22semidirect
54. C2×SD16×C269C22semidirect
58. #106×C269C22semidirect
59. #103×C269C22semidirect
60. #102×C269C22semidirect
87. #16864385D4semidirect
171. C22×C269#137semidirect
199. C269#3056semidirect
212. #465D269semidirect
223. C2×SD16C2×D269semidirect
252. #106C2×D269semidirect
258. #103C2×D269semidirect
260. #102C2×D269semidirect
378. C22#22575482semidirect

Maximal subgroups

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

subgroupindex
C2×#225754822
C22×#168643852
#317711152
C22⋊Q8×D2692
#317708642
#317708622
#317708502
#317708482
#317708462
#317708442
#317708322
#317706442
#317706122
#317706032
#465×C2692
#3056269