Finite Group Databasebeta

← #46822407

Subgroups — #46822407

Order 34432.

500 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22575480C211yesmaximal, characteristic, complemented
2C2×#22575475C211yesmaximal, characteristic, complemented
3#31771257C211yesmaximal, characteristic, complemented
4#31771252C211yesmaximal, characteristic, complemented
5#31770996C211yesmaximal, characteristic, complemented
6#31770864C211yesmaximal, characteristic, complemented
7#31770819C211yesmaximal, characteristic, complemented
8#31770809C211yesmaximal, characteristic, complemented
9#31770770C211yesmaximal, characteristic, complemented
10#31770765C211yesmaximal, characteristic, complemented
11#31770757C211yesmaximal, characteristic, complemented
12#31770748C211yesmaximal, characteristic, complemented
13#31770716C211yesmaximal, characteristic, complemented
14#31770676C211yesmaximal, characteristic, complemented
15#465×C269C211yesmaximal, characteristic, complemented
16C2×#16864403C2211yescharacteristic
17C2×#16864402C2211yescharacteristic
18C2×#16864402C2211yescharacteristic, complemented
19C2×D4×D269C2211yescharacteristic
20#22575480—24no—
21#22575475—24no—
22C2×Dic1076C2211yescharacteristic, complemented
23C2×#16864392C2211yescharacteristic, complemented
24C2×#16864391C2211yescharacteristic
25C2×#16864391C2211yescharacteristic, complemented
26C2×#16864390C2211yescharacteristic
27C2×#16864389C2211yescharacteristic
28C2×D2152C2211yescharacteristic
29C2×#16864387C2211yescharacteristic
30C2×#16864385C2211yescharacteristic
31#22575451C2211yescharacteristic, complemented
32#22575440C2211yescharacteristic
33#22575438—21no—
34#22575435—21no—
35#22575435—21no—
36#22575433C2211yescharacteristic
37#22575433C2211yescharacteristic
38#22575426—21no—
39#22575424—21no—
40#22575421—21no—
41#22575415C2211yescharacteristic
42#22575415C2211yescharacteristic, complemented
43C4×C22⋊D269C2211yescharacteristic
44C4⋊C4×D269—21no—
45C22⋊C4×D269—21no—
46#22575382C2211yescharacteristic, complemented
47#22575376C2211yescharacteristic, complemented
48#22575376C2211yescharacteristic, complemented
49#22575373C2211yescharacteristic
50#22575373C2211yescharacteristic, complemented
51C22⋊#14524130C2211yescharacteristic, complemented
52#22575365C2211yescharacteristic
53#22575363C2211yescharacteristic, complemented
54C2×SD16×C269C2211yescharacteristic
55C22×C8×C269C2211yescharacteristic, complemented
56C22⋊Q8×C269C2211yescharacteristic, complemented
57#121×C269C2211yescharacteristic
58#106×C269C2211yescharacteristic, complemented
59#103×C269C2211yescharacteristic, complemented
60#102×C269C2211yescharacteristic
61C23×D269—21no—
62#16864403—22no—
63#16864403—41no—
64#16864402—22no—
65#16864402—22no—
66#16864402—41no—
67#16864402—41no—
68C2×Dic538C2311yescharacteristic
69C2×Dic538C2311yescharacteristic
70C2×D1076—21no—
71C2×D1076C2311yescharacteristic
72C2×D1076C2311yescharacteristic
73C2×C22⋊D269—21no—
74C2×C22⋊D269—21no—
75C2×C22⋊D269—21no—
76C2×C22⋊D269D411yescharacteristic
77D4×D269—22no—
78D4×D269—41no—
79C2×C4×D269—21no—
80C2×C4×D269—21no—
81C2×C4×D269—21no—
82C2×C4×D269—21no—
83C2×C4×D269—21no—
84C2×C4×D269C2311yescharacteristic
85Dic1076—22no—
86#16864392—41no—
87#16864391—22no—
88#16864391—22no—
89#16864390—41no—
90#16864389—22no—
91D2152D414yescomplemented
92#16864387—22no—
93#16864385—22no—
94C2×#14524130C2311yescharacteristic
95C2×#14524130C2311yescharacteristic
96C4⋊Dic269—21no—
97C4⋊Dic269—21no—
98#16864381—21no—
99#16864381—21no—
100#16864381D411yescharacteristic
101C22⋊Dic269—21no—
102C22⋊Dic269—21no—
103C22⋊Dic269D411yescharacteristic
104#16864379—21no—
105#16864379—21no—
106#16864379—21no—
107#16864379—21no—
108#16864379D411yescharacteristic
109C4×Dic269C2311yescharacteristic
110C2×Q8×C269C2311yescharacteristic
111C2×D4×C269—21no—
112C2×D4×C269C2311yescharacteristic
113C22×C4×C269C2311yescharacteristic
114SD16×C269—22no—
115C2×C8×C269—22no—
116C2×C8×C269C2311yescharacteristic
117C2×C8×C269C2311yescharacteristic
118C4⋊C4×C269—21no—
119C4⋊C4×C269C2311yescharacteristic
120C4⋊C4×C269C2311yescharacteristic
121C22⋊C4×C269—21no—
122C22⋊C4×C269—21no—
123C22×D269—21no—
124C22×D269—21no—
125C22×D269—42no—
126C22×D269—42no—
127C22×D269—81no—
128C22×D269C2×D411yescharacteristic
129Dic538—21no—
130Dic538—21no—
131Dic538—21no—
132Dic538—21no—
133D1076—21no—
134D1076—21no—
135D1076—42no—
136D1076C2×D412yes—
137D1076C2×D412yes—
138C22⋊D269—41no—
139C22⋊D269—42no—
140C22⋊D269—42no—
141C22⋊D269—42no—
142C2×Dic269—21no—
143C2×Dic269—21no—
144C2×Dic269C2×D411yescharacteristic
145C2×Dic269C2×D411yescharacteristic
146C2×Dic269C2×D411yescharacteristic
147C4×D269—21no—
148C4×D269—21no—
149C4×D269—42no—
150C4×D269—42no—
151C4×D269—42no—
152C4×D269—42no—
153C4×D269—81no—
154#14524130—21no—
155#14524130—21no—
156C23×C269—21no—
157C23×C269C2×D411yescharacteristic
158Q8×C269—21no—
159Q8×C269—21no—
160D4×C269—21no—
161D4×C269—21no—
162D4×C269—81no—
163C2×C4×C269—21no—
164C2×C4×C269—21no—
165C2×C4×C269—21no—
166C2×C4×C269—21no—
167C2×C4×C269—21no—
168C2×C4×C269C2×D411yescharacteristic
169C2×C4×C269C2411yescharacteristic
170C8×C269—21no—
171C8×C269C2×D412yes—
172C22×C269—21no—
173C22×C269—21no—
174C22×C269—21no—
175C22×C269—41no—
176C22×C269—41no—
177C22×C269—41no—
178C22×C269C22×D411yescharacteristic
179C2×D269—21no—
180C2×D269—22no—
181C2×D269—22no—
182C2×D269—21no—
183C2×D269—21no—
184C2×D269—41no—
185C2×D269—41no—
186C2×D269—41no—
187C2×D269—41no—
188C2×D269—84no—
189C4×C269—21no—
190C4×C269—41no—
191C4×C269—41no—
192C4×C269—41no—
193C4×C269C22×D411yescharacteristic
194C4×C269H(5,𝔽2)11yescharacteristic
195Dic269—21no—
196Dic269—21no—
197Dic269—21no—
198Dic269—41no—
199Dic269—41no—
200C2×C269—41no—
201C2×C269—81no—
202C2×C269D4211yescharacteristic
203C2×C269#57511yescharacteristic
204C2×C269#57711yescharacteristic
205D269—41no—
206D269—42no—
207D269—42no—
208C269#326711yescharacteristic, complemented, Sylow
209#3267—2691nomaximal, Sylow
210C2×AGL(1,ℤ/8)—2691no—
211C2×#135—2691no—
212#547—2691no—
213#546—2691no—
214#497—2691no—
215#470—2691no—
216#465D26911yescharacteristic, complemented
217#463—2691no—
218#461—2691no—
219#452—2691no—
220#452—2691no—
221#450—2691no—
222#449—2691no—
223#442—2691no—
224#435—2691no—
225C2×#41—2691no—
226C2×#41—2691no—
227C2×#41—2691no—
228C22×D4—2691no—
229AGL(1,ℤ/8)—5384no—
230#135—5384no—
231C2×Q16—2691no—
232C2×Q16—2691no—
233C2×SD16—2691no—
234C2×SD16—2691no—
235C2×SD16—2691no—
236C2×SD16—2691no—
237C2×SD16C2×D26911yescharacteristic
238C2×D8—2691no—
239C2×D8—2691no—
240C2×M16—2691no—
241C22×C8C2×D26911yescharacteristic, complemented
242C4⋊Q8—2691no—
243#124—2691no—
244#123—5381no—
245#123—5381no—
246#123—5381no—
247C22⋊Q8—2691no—
248C22⋊Q8—2691no—
249C22⋊Q8C2×D26911yescharacteristic, complemented
250#121—5381no—
251#121—5381no—
252#121C2×D26911yescharacteristic
253#120—5381no—
254C4×D4—2691no—
255C4×D4—2691no—
256C4×D4—2691no—
257C2×C4⋊C4—5381no—
258C2×C22⋊C4—5381no—
259#106C2×D26911yescharacteristic, complemented
260C4⋊C8—2691no—
261#103—2691no—
262#103—2691no—
263#103C2×D26911yescharacteristic, complemented
264#102—2691no—
265#102—2691no—
266#102C2×D26911yescharacteristic
267C22⋊C8—2691no—
268C22⋊C8—2691no—
269#97—2691no—
270C24—5381no—
271#41—5382no—
272#41—5382no—
273#41—5382no—
274#41—10761no—
275#41—10761no—
276#41—10761no—
277C2×Q8—2691no—
278C2×Q8—2691no—
279C2×Q8C22×D26911yescharacteristic
280C2×D4—21no—
281C2×D4—2691no—
282C2×D4—2691no—
283C2×D4—2691no—
284C2×D4—5381no—
285C2×D4—5381no—
286C2×D4—5382no—
287C2×D4—5381no—
288C2×D4—5381no—
289C2×D4—10761no—
290C2×D4C22×D26911yescharacteristic
291C22×C4—2691no—
292C22×C4—5381no—
293C22×C4—5381no—
294C22×C4—5381no—
295C22×C4—5381no—
296C22×C4—5381no—
297C22×C4C22×D26911yescharacteristic
298Q16—5382no—
299Q16—10761no—
300SD16—22no—
301SD16—5382no—
302SD16—5382no—
303SD16—5382no—
304SD16—10761no—
305D8—2694no—
306D8—5382no—
307M16—5382no—
308C2×C8—22no—
309C2×C8—2691no—
310C2×C8—2691no—
311C2×C8C22×D26911yescharacteristic
312C2×C8C22×D26911yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—2691no—
315C4⋊C4—5381no—
316C4⋊C4—5381no—
317C4⋊C4—5381no—
318C4⋊C4—5381no—
319C4⋊C4C22×D26911yescharacteristic
320C4⋊C4C22×D26911yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—21no—
323C22⋊C4—2691no—
324C22⋊C4—2691no—
325C22⋊C4—5381no—
326C22⋊C4—5381no—
327C22⋊C4—5381no—
328C22⋊C4—5381no—
329C22⋊C4—5381no—
330C22⋊C4—5381no—
331C42—2691no—
332C23—21no—
333C23—2691no—
334C23—5381no—
335C23—5381no—
336C23—10762no—
337C23—10762no—
338C23—21521no—
339C23D4×D26911yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—5381no—
343Q8—5381no—
344Q8—5381no—
345Q8—5381no—
346D4—21no—
347D4—21no—
348D4—81no—
349D4—2692no—
350D4—2692no—
351D4—5381no—
352D4—5381no—
353D4—10762no—
354D4—10761no—
355D4—10762no—
356D4—10762no—
357D4—10762no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—2691no—
364C2×C4—2691no—
365C2×C4—2691no—
366C2×C4—5381no—
367C2×C4—5381no—
368C2×C4—5381no—
369C2×C4—5381no—
370C2×C4—10762no—
371C2×C4—10762no—
372C2×C4—10762no—
373C2×C4—10762no—
374C2×C4—21521no—
375C2×C4D4×D26911yescharacteristic
376C2×C4C23×D26911yescharacteristic
377C8—21no—
378C8—5381no—
379C8—5381no—
380C8D4×D26912yes—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—5381no—
388C22—5381no—
389C22—5381no—
390C22—5382no—
391C22—5382no—
392C22—10761no—
393C22—10761no—
394C22—10761no—
395C22—10761no—
396C22—21524no—
397C22C2×D4×D26911yescharacteristic
398C4—21no—
399C4—41no—
400C4—41no—
401C4—41no—
402C4—5381no—
403C4—5381no—
404C4—5381no—
405C4—10761no—
406C4—10761no—
407C4C2×D4×D26911yescharacteristic
408C4#2257549911yescharacteristic
409C2—41no—
410C2—81no—
411C2—10761no—
412C2—10762no—
413C2—10762no—
414C2#3177123911yescharacteristic
415C2#3177140211yescharacteristic
416C2#3177141111yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#465×C269C2
socleC22×C269C22×D4
hypercenter#465D269
hypocenterC269#3267
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×#22575480C2semidirect
2. C2×#22575475C2semidirect
3. #31771257C2semidirect
4. #31771252C2semidirect
5. #31770996C2semidirect
6. #31770864C2semidirect
7. #31770819C2semidirect
8. #31770809C2semidirect
9. #31770770C2semidirect
10. #31770765C2semidirect
11. #31770757C2semidirect
12. #31770748C2semidirect
13. #31770716C2semidirect
14. #31770676C2semidirect
15. #465×C269C2semidirect
18. C2×#16864402C22semidirect
22. C2×Dic1076C22semidirect
23. C2×#16864392C22semidirect
25. C2×#16864391C22semidirect
31. #22575451C22semidirect
42. #22575415C22semidirect
46. #22575382C22semidirect
47. #22575376C22semidirect
48. #22575376C22semidirect
50. #22575373C22semidirect
51. C22⋊#14524130C22semidirect
53. #22575363C22semidirect
55. C22×C8×C269C22semidirect
56. C22⋊Q8×C269C22semidirect
58. #106×C269C22semidirect
59. #103×C269C22semidirect
91. D2152D4semidirect
208. C269#3267semidirect
216. #465D269semidirect
241. C22×C8C2×D269semidirect
249. C22⋊Q8C2×D269semidirect
259. #106C2×D269semidirect
263. #103C2×D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754802
C2×#225754752
#317712572
#317712522
#317709962
#317708642
#317708192
#317708092
#317707702
#317707652
#317707572
#317707482
#317707162
#317706762
#465×C2692
#3267269