Finite Group Databasebeta

← #46820376

Subgroups — #46820376

Order 34432.

500 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22575485C211yesmaximal, characteristic, complemented
2C2×#22575480C211yesmaximal, characteristic
3#124×D269C211yesmaximal, characteristic, complemented
4C4×#16864403C211yesmaximal, characteristic, complemented
5#31770829C211yesmaximal, characteristic, complemented
6#31770827C211yesmaximal, characteristic, complemented
7#31770825C211yesmaximal, characteristic, complemented
8#31770823C211yesmaximal, characteristic, complemented
9#31770821C211yesmaximal, characteristic, complemented
10#31770819C211yesmaximal, characteristic, complemented
11#31770817C211yesmaximal, characteristic, complemented
12#31770627C211yesmaximal, characteristic, complemented
13#31770612C211yesmaximal, characteristic, complemented
14#31770608C211yesmaximal, characteristic, complemented
15#464×C269C211yesmaximal, characteristic, complemented
16C2×#16864403C2211yescharacteristic
17C2×Q8×D269C2211yescharacteristic
18C2×D4×D269C2211yescharacteristic
19#22575485—24no—
20#22575480—24no—
21C2×#16864392C2211yescharacteristic, complemented
22C2×#16864391C2211yescharacteristic, complemented
23C2×#16864389C2211yescharacteristic
24C2×#16864389C2211yescharacteristic, complemented
25C2×D2152C2211yescharacteristic
26C2×#16864387C2211yescharacteristic
27C2×#16864385C2211yescharacteristic
28C2×#16864385C2211yescharacteristic
29#22575442C2211yescharacteristic
30#22575441C2211yescharacteristic
31#22575440C2211yescharacteristic
32#22575439—21no—
33#22575439—21no—
34Q8×Dic269C2211yescharacteristic, complemented
35C4×D1076—21no—
36C4×D1076C2211yescharacteristic, complemented
37#22575415—21no—
38#22575415C2211yescharacteristic, complemented
39#22575411—21no—
40#22575411C2211yescharacteristic
41C22⋊C4×D269—21no—
42C22⋊C4×D269—21no—
43C42×D269—21no—
44C42×D269C2211yescharacteristic
45C4⋊#14524130C2211yescharacteristic, complemented
46#22575382C2211yescharacteristic, complemented
47#22575382C2211yescharacteristic, complemented
48#22575378C2211yescharacteristic, complemented
49#22575377C2211yescharacteristic, complemented
50#22575376C2211yescharacteristic, complemented
51#22575375C2211yescharacteristic
52#22575374C2211yescharacteristic
53#22575373C2211yescharacteristic
54C2×Q16×C269C2211yescharacteristic, complemented
55C2×SD16×C269C2211yescharacteristic
56#124×C269C2211yescharacteristic
57C4×Q8×C269C2211yescharacteristic, complemented
58C4⋊C8×C269C2211yescharacteristic, complemented
59#103×C269C2211yescharacteristic, complemented
60#102×C269C2211yescharacteristic
61C23×D269—21no—
62#16864403—22no—
63#16864403D414yes—
64C2×Dic538—21no—
65C2×Dic538C2311yescharacteristic
66Q8×D269—22no—
67Q8×D269—22no—
68C2×D1076—21no—
69C2×D1076C2311yescharacteristic
70C2×D1076C2311yescharacteristic
71C2×C22⋊D269—21no—
72D4×D269—22no—
73D4×D269—22no—
74C2×C4×D269—21no—
75C2×C4×D269—21no—
76C2×C4×D269—21no—
77C2×C4×D269—21no—
78C2×C4×D269D411yescharacteristic
79C2×C4×D269D411yescharacteristic
80C2×C4×D269C2311yescharacteristic
81#16864392—22no—
82#16864391—22no—
83#16864389—22no—
84#16864389—22no—
85D2152—22no—
86#16864387—22no—
87#16864385—22no—
88#16864385—22no—
89C2×#14524130C2311yescharacteristic
90C2×#14524130C2311yescharacteristic
91C4⋊Dic269—21no—
92C4⋊Dic269C2311yescharacteristic
93C22⋊Dic269—21no—
94C22⋊Dic269—21no—
95#16864379—21no—
96#16864379—21no—
97#16864379—21no—
98#16864379—21no—
99#16864379—21no—
100#16864379—21no—
101#16864379—21no—
102C4×Dic269—21no—
103C4×Dic269—21no—
104C4×Dic269D411yescharacteristic
105C4×Dic269D411yescharacteristic
106C4×Dic269C2311yescharacteristic
107C2×Q8×C269C2311yescharacteristic
108C2×Q8×C269C2311yescharacteristic
109C2×D4×C269C2311yescharacteristic
110Q16×C269—22no—
111SD16×C269—22no—
112C2×C8×C269C2311yescharacteristic
113C2×C8×C269C2311yescharacteristic
114C4⋊C4×C269—21no—
115C4⋊C4×C269C2311yescharacteristic
116C22⋊C4×C269—21no—
117C22⋊C4×C269—21no—
118C42×C269—21no—
119C42×C269C2311yescharacteristic
120C22×D269—21no—
121C22×D269—21no—
122C22×D269—42no—
123C22×D269—42no—
124C22×D269—42no—
125C22×D269C2×D411yescharacteristic
126Dic538—21no—
127Dic538—21no—
128Dic538—42no—
129D1076—24no—
130D1076—21no—
131D1076—21no—
132D1076—21no—
133D1076C2×D412yes—
134C22⋊D269—42no—
135C2×Dic269—21no—
136C2×Dic269—21no—
137C2×Dic269C2×D411yescharacteristic
138C2×Dic269C2×D411yescharacteristic
139C2×Dic269C2×D411yescharacteristic
140C4×D269—24no—
141C4×D269—41no—
142C4×D269—41no—
143C4×D269—42no—
144C4×D269—42no—
145C4×D269—42no—
146C4×D269C2×D412yes—
147C4×D269#4112yes—
148#14524130—21no—
149#14524130—21no—
150C23×C269—21no—
151Q8×C269—21no—
152Q8×C269—21no—
153Q8×C269—21no—
154Q8×C269C2×D412yes—
155D4×C269—21no—
156D4×C269—21no—
157C2×C4×C269—21no—
158C2×C4×C269—21no—
159C2×C4×C269—21no—
160C2×C4×C269C2×D411yescharacteristic
161C2×C4×C269C2×D411yescharacteristic
162C2×C4×C269C2411yescharacteristic
163C8×C269—21no—
164C8×C269—21no—
165C22×C269—41no—
166C22×C269—41no—
167C22×C269—41no—
168C22×C269C22×D411yescharacteristic
169C2×D269—21no—
170C2×D269—21no—
171C2×D269—22no—
172C2×D269—41no—
173C2×D269—41no—
174C2×D269—41no—
175C2×D269—41no—
176C2×D269—41no—
177C2×D269—84no—
178C2×D269#12112yes—
179C4×C269—22no—
180C4×C269—21no—
181C4×C269—21no—
182C4×C269—41no—
183C4×C269—41no—
184C4×C269C22×D411yescharacteristic
185C4×C269C2×#4111yescharacteristic
186Dic269—21no—
187Dic269—21no—
188Dic269—41no—
189Dic269—41no—
190Dic269#12112yes—
191C2×C269—81no—
192C2×C269C2×#12111yescharacteristic
193C2×C269#57511yescharacteristic
194C2×C269#57511yescharacteristic
195D269—22no—
196D269—42no—
197D269—81no—
198C269#304011yescharacteristic, complemented, Sylow
199#3040—2691nomaximal, Sylow
200C2×#137—2691no—
201C2×AGL(1,ℤ/8)—2691no—
202C2×#124—2691no—
203C4×#41—2691no—
204#464—2691no—
205#464—2691no—
206#464—2691no—
207#464D26911yescharacteristic, complemented
208#463—2691no—
209#463—2691no—
210#463—2691no—
211#463—2691no—
212#423—2691no—
213#419—2691no—
214#418—2691no—
215C2×#41—2691no—
216C22×Q8—2691no—
217C22×D4—2691no—
218#137—5384no—
219AGL(1,ℤ/8)—5384no—
220C2×Q16—2691no—
221C2×Q16C2×D26911yescharacteristic, complemented
222C2×SD16—2691no—
223C2×SD16—2691no—
224C2×SD16—2691no—
225C2×SD16C2×D26911yescharacteristic
226C2×D8—2691no—
227C2×D8—2691no—
228C2×M16—2691no—
229C2×M16—2691no—
230#124—2691no—
231#124—2691no—
232#124—2691no—
233#124—5381no—
234#124—5381no—
235#124C2×D26911yescharacteristic
236C4×Q8—2691no—
237C4×Q8C2×D26911yescharacteristic, complemented
238C4×D4—2691no—
239C4×D4—2691no—
240C4×D4—5381no—
241C4×D4—5381no—
242#117—2691no—
243#117—5381no—
244C2×C22⋊C4—5381no—
245C2×C22⋊C4—5381no—
246C2×C42—2691no—
247C2×C42—5381no—
248C4⋊C8—2691no—
249C4⋊C8—2691no—
250C4⋊C8—2691no—
251C4⋊C8C2×D26911yescharacteristic, complemented
252#103—2691no—
253#103—2691no—
254#103—2691no—
255#103C2×D26911yescharacteristic, complemented
256#102—2691no—
257#102—2691no—
258#102—2691no—
259#102C2×D26911yescharacteristic
260C24—5381no—
261#41—2694no—
262#41—5382no—
263C2×Q8—2691no—
264C2×Q8—5381no—
265C2×Q8—5382no—
266C2×Q8—5382no—
267C2×Q8C22×D26911yescharacteristic
268C2×Q8C22×D26911yescharacteristic
269C2×D4—2691no—
270C2×D4—2691no—
271C2×D4—5381no—
272C2×D4—5382no—
273C2×D4—5382no—
274C2×D4—5381no—
275C2×D4C22×D26911yescharacteristic
276C22×C4—2691no—
277C22×C4—2691no—
278C22×C4—2691no—
279C22×C4—5381no—
280C22×C4—5381no—
281C22×C4—5381no—
282C22×C4—5381no—
283Q16—22no—
284Q16—5382no—
285SD16—22no—
286SD16—5382no—
287SD16—5382no—
288SD16—5382no—
289D8—5382no—
290D8—5382no—
291M16—5382no—
292M16—5382no—
293C2×C8—2691no—
294C2×C8—2691no—
295C2×C8C22×D26911yescharacteristic
296C2×C8C22×D26911yescharacteristic
297C4⋊C4—21no—
298C4⋊C4—2691no—
299C4⋊C4—5381no—
300C4⋊C4C22×D26911yescharacteristic
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—5381no—
304C22⋊C4—5381no—
305C22⋊C4—5381no—
306C22⋊C4—5381no—
307C22⋊C4—5381no—
308C22⋊C4—5381no—
309C22⋊C4—5381no—
310C22⋊C4—5381no—
311C22⋊C4—5381no—
312C42—21no—
313C42—2691no—
314C42—2691no—
315C42—2691no—
316C42—5381no—
317C42—5381no—
318C42C22×D26911yescharacteristic
319C23—21no—
320C23—2691no—
321C23—5381no—
322C23—5381no—
323C23—10762no—
324C23—10762no—
325C23—10762no—
326Q8—21no—
327Q8—21no—
328Q8—21no—
329Q8—5381no—
330Q8—5381no—
331Q8—10762no—
332Q8D4×D26912yes—
333D4—21no—
334D4—21no—
335D4—2692no—
336D4—5384no—
337D4—5381no—
338D4—5381no—
339D4—5381no—
340D4—10762no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—2691no—
345C2×C4—2692no—
346C2×C4—2692no—
347C2×C4—2691no—
348C2×C4—2691no—
349C2×C4—5384no—
350C2×C4—5381no—
351C2×C4—5381no—
352C2×C4—10761no—
353C2×C4—10761no—
354C2×C4—10762no—
355C2×C4—10762no—
356C2×C4—10762no—
357C2×C4D4×D26911yescharacteristic
358C2×C4D4×D26911yescharacteristic
359C2×C4C23×D26911yescharacteristic
360C8—21no—
361C8—21no—
362C8—5381no—
363C8—5381no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—2692no—
368C22—5381no—
369C22—5381no—
370C22—5382no—
371C22—10761no—
372C22—10761no—
373C22—10761no—
374C22—10761no—
375C22—10761no—
376C22—21524no—
377C22C2×D4×D26911yescharacteristic
378C4—21no—
379C4—21no—
380C4—22no—
381C4—41no—
382C4—41no—
383C4—2692no—
384C4—5381no—
385C4—5381no—
386C4—10761no—
387C4—10761no—
388C4C2×D4×D26911yescharacteristic
389C4#41×D26911yescharacteristic
390C2—81no—
391C2—5382no—
392C2—10762no—
393C2—21521no—
394C2#121×D26911yescharacteristic
395C2#3177140211yescharacteristic
396C2#3177140411yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#464×C269C2
socleC22×C269C22×D4
prime radicalC2×#22575480C2
hypercenter#464D269
hypocenterC269#3040
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. C2×#22575485C2semidirect
3. #124×D269C2semidirect
4. C4×#16864403C2semidirect
5. #31770829C2semidirect
6. #31770827C2semidirect
7. #31770825C2semidirect
8. #31770823C2semidirect
9. #31770821C2semidirect
10. #31770819C2semidirect
11. #31770817C2semidirect
12. #31770627C2semidirect
13. #31770612C2semidirect
14. #31770608C2semidirect
15. #464×C269C2semidirect
21. C2×#16864392C22semidirect
22. C2×#16864391C22semidirect
24. C2×#16864389C22semidirect
34. Q8×Dic269C22semidirect
36. C4×D1076C22semidirect
38. #22575415C22semidirect
45. C4⋊#14524130C22semidirect
46. #22575382C22semidirect
47. #22575382C22semidirect
48. #22575378C22semidirect
49. #22575377C22semidirect
50. #22575376C22semidirect
54. C2×Q16×C269C22semidirect
57. C4×Q8×C269C22semidirect
58. C4⋊C8×C269C22semidirect
59. #103×C269C22semidirect
198. C269#3040semidirect
207. #464D269semidirect
221. C2×Q16C2×D269semidirect
237. C4×Q8C2×D269semidirect
251. C4⋊C8C2×D269semidirect
255. #103C2×D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754852
C2×#225754802
#124×D2692
C4×#168644032
#317708292
#317708272
#317708252
#317708232
#317708212
#317708192
#317708172
#317706272
#317706122
#317706082
#464×C2692
#3040269