Finite Group Databasebeta

← #46822417

Subgroups — #46822417

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×#22575481C211yesmaximal, characteristic, complemented
2C2×#22575476C211yesmaximal, characteristic, complemented
3#31771257C211yesmaximal, characteristic, complemented
4#31771246C211yesmaximal, characteristic, complemented
5#31770998C211yesmaximal, characteristic, complemented
6#31770835C211yesmaximal, characteristic, complemented
7#31770823C211yesmaximal, characteristic, complemented
8#31770809C211yesmaximal, characteristic, complemented
9#31770770C211yesmaximal, characteristic, complemented
10#31770765C211yesmaximal, characteristic, complemented
11#31770755C211yesmaximal, characteristic, complemented
12#31770753C211yesmaximal, characteristic, complemented
13#31770716C211yesmaximal, characteristic, complemented
14#31770674C211yesmaximal, characteristic, complemented
15#470×C269C211yesmaximal, characteristic, complemented
16C2×#16864403C2211yescharacteristic
17C2×#16864403C2211yescharacteristic, complemented
18C2×#16864402C2211yescharacteristic
19C22×D1076C2211yescharacteristic
20#22575481—24no—
21#22575476—24no—
22C2×#16864392C2211yescharacteristic, complemented
23C2×#16864391C2211yescharacteristic
24C2×#16864391C2211yescharacteristic
25C2×#16864389C2211yescharacteristic
26C2×#16864389C2211yescharacteristic
27C2×#16864389C2211yescharacteristic, complemented
28C2×D2152C2211yescharacteristic
29C2×D2152C2211yescharacteristic
30C2×C8×D269C2211yescharacteristic, complemented
31#22575451C2211yescharacteristic, complemented
32#22575442C2211yescharacteristic
33#22575438—21no—
34#22575438—21no—
35#22575435—21no—
36#22575433C2211yescharacteristic, complemented
37#22575428—21no—
38#22575428C2211yescharacteristic
39#22575426—21no—
40#22575423—21no—
41#22575415C2211yescharacteristic
42#22575415C2211yescharacteristic, complemented
43C4×C22⋊D269C2211yescharacteristic
44C4⋊C4×D269—21no—
45C22⋊C4×D269—21no—
46#22575382C2211yescharacteristic, complemented
47#22575376C2211yescharacteristic, complemented
48#22575373C2211yescharacteristic
49#22575373C2211yescharacteristic
50#22575373C2211yescharacteristic, complemented
51C22⋊#14524130C2211yescharacteristic
52#22575365C2211yescharacteristic, complemented
53#22575363C2211yescharacteristic, complemented
54C2×Q16×C269C2211yescharacteristic, complemented
55C2×M16×C269C2211yescharacteristic
56C22⋊Q8×C269C2211yescharacteristic
57C22⋊Q8×C269C2211yescharacteristic
58#106×C269C2211yescharacteristic, complemented
59#103×C269C2211yescharacteristic, complemented
60#103×C269C2211yescharacteristic, complemented
61C23×D269—21no—
62#16864403—22no—
63#16864403—22no—
64#16864403—41no—
65#16864403—41no—
66#16864402—22no—
67#16864402—41no—
68C2×Dic538C2311yescharacteristic
69C2×D1076—22no—
70C2×D1076—21no—
71C2×D1076—21no—
72C2×D1076—21no—
73C2×D1076—21no—
74C2×D1076—41no—
75C2×D1076C2311yescharacteristic
76C2×D1076C2311yescharacteristic
77C2×D1076C2311yescharacteristic
78C2×C22⋊D269—21no—
79C2×C22⋊D269D411yescharacteristic
80C2×C4×D269—21no—
81C2×C4×D269—21no—
82C2×C4×D269—21no—
83C2×C4×D269—21no—
84C2×C4×D269—21no—
85C2×C4×D269C2311yescharacteristic
86#16864392—41no—
87#16864391—22no—
88#16864391—22no—
89#16864389—22no—
90#16864389—22no—
91#16864389—41no—
92D2152—22no—
93D2152D414yescomplemented
94C2×#14524130C2311yescharacteristic
95C2×#14524130C2311yescharacteristic
96C8×D269—22no—
97C4⋊Dic269—21no—
98#16864381—21no—
99#16864381—21no—
100#16864381D411yescharacteristic
101C22⋊Dic269D411yescharacteristic
102#16864379—21no—
103#16864379—21no—
104#16864379—21no—
105#16864379—21no—
106#16864379—21no—
107#16864379—21no—
108#16864379D411yescharacteristic
109C4×Dic269C2311yescharacteristic
110C2×Q8×C269C2311yescharacteristic
111C2×Q8×C269C2311yescharacteristic
112C22×C4×C269C2311yescharacteristic
113Q16×C269—22no—
114M16×C269—22no—
115C2×C8×C269C2311yescharacteristic
116C2×C8×C269C2311yescharacteristic
117C4⋊C4×C269—21no—
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—21no—
126C22×D269—42no—
127C22×D269—42no—
128C22×D269—81no—
129C22×D269C2×D411yescharacteristic
130Dic538—21no—
131Dic538—21no—
132D1076—21no—
133D1076—21no—
134D1076—21no—
135D1076—21no—
136D1076—42no—
137D1076—42no—
138D1076—42no—
139D1076—81no—
140D1076C2×D412yes—
141D1076C2×D412yes—
142C22⋊D269—41no—
143C22⋊D269—42no—
144C2×Dic269—21no—
145C2×Dic269C2×D411yescharacteristic
146C2×Dic269C2×D411yescharacteristic
147C2×Dic269C2×D411yescharacteristic
148C4×D269—21no—
149C4×D269—21no—
150C4×D269—42no—
151C4×D269—42no—
152C4×D269—42no—
153C4×D269—42no—
154C4×D269—81no—
155#14524130—21no—
156#14524130—21no—
157C23×C269C2×D411yescharacteristic
158Q8×C269—21no—
159Q8×C269—21no—
160Q8×C269—21no—
161Q8×C269—21no—
162C2×C4×C269—21no—
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×C269C22×D411yescharacteristic
176C2×D269—21no—
177C2×D269—22no—
178C2×D269—22no—
179C2×D269—21no—
180C2×D269—21no—
181C2×D269—41no—
182C2×D269—41no—
183C2×D269—41no—
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×C269—41no—
194C4×C269C22×D411yescharacteristic
195C4×C269H(5,𝔽2)11yescharacteristic
196Dic269—21no—
197Dic269—21no—
198Dic269—21no—
199Dic269—41no—
200C2×C269—41no—
201C2×C269D4211yescharacteristic
202C2×C269#57511yescharacteristic
203C2×C269#57711yescharacteristic
204D269—41no—
205D269—42no—
206D269—42no—
207D269—81no—
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#470D26911yescharacteristic, complemented
216#465—2691no—
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×Q16C2×D26911yescharacteristic, complemented
233C2×SD16—2691no—
234C2×SD16—2691no—
235C2×SD16—2691no—
236C2×SD16—2691no—
237C2×SD16—2691no—
238C2×D8—2691no—
239C2×D8—2691no—
240C2×M16C2×D26911yescharacteristic
241C22×C8—2691no—
242C4⋊Q8—2691no—
243#124—2691no—
244#123—5381no—
245#123—5381no—
246#123—5381no—
247C22⋊Q8—2691no—
248C22⋊Q8C2×D26911yescharacteristic
249C22⋊Q8C2×D26911yescharacteristic
250#121—2691no—
251#121—5381no—
252#121—5381no—
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#103C2×D26911yescharacteristic, complemented
263#103C2×D26911yescharacteristic, complemented
264#102—2691no—
265#102—2691no—
266#102—2691no—
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×Q8C22×D26911yescharacteristic
279C2×Q8C22×D26911yescharacteristic
280C2×D4—2691no—
281C2×D4—2691no—
282C2×D4—2691no—
283C2×D4—2691no—
284C2×D4—5382no—
285C2×D4—5381no—
286C2×D4—5381no—
287C2×D4—5381no—
288C2×D4—5381no—
289C2×D4—5381no—
290C2×D4—10761no—
291C22×C4—2691no—
292C22×C4—5381no—
293C22×C4—5381no—
294C22×C4—5381no—
295C22×C4—5381no—
296C22×C4—5381no—
297C22×C4C22×D26911yescharacteristic
298Q16—22no—
299Q16—10761no—
300SD16—5382no—
301SD16—5382no—
302SD16—5382no—
303SD16—5382no—
304SD16—10761no—
305D8—2694no—
306D8—5382no—
307M16—22no—
308C2×C8—2691no—
309C2×C8—2691no—
310C2×C8—5382no—
311C2×C8C22×D26911yescharacteristic
312C2×C8C22×D26911yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—2691no—
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—2691no—
333C23—5381no—
334C23—5381no—
335C23—5381no—
336C23—10762no—
337C23—10762no—
338C23—21521no—
339C23D4×D26911yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—5381no—
345Q8—5381no—
346D4—2692no—
347D4—2692no—
348D4—5381no—
349D4—5381no—
350D4—5381no—
351D4—5381no—
352D4—10762no—
353D4—10762no—
354D4—10762no—
355D4—10761no—
356D4—10762no—
357D4—21521no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—2691no—
365C2×C4—2691no—
366C2×C4—2691no—
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—5381no—
385C22—5381no—
386C22—5381no—
387C22—5382no—
388C22—5382no—
389C22—10761no—
390C22—10761no—
391C22—10761no—
392C22—10761no—
393C22—10761no—
394C22—10761no—
395C22—10761no—
396C22—21524no—
397C22C2×D4×D26911yescharacteristic
398C4—21no—
399C4—41no—
400C4—41no—
401C4—41no—
402C4—41no—
403C4—5381no—
404C4—5381no—
405C4—5381no—
406C4—10761no—
407C4C2×D4×D26911yescharacteristic
408C4#2257549911yescharacteristic
409C2—41no—
410C2—10761no—
411C2—10762no—
412C2—10762no—
413C2—21521no—
414C2#3177123911yescharacteristic
415C2#3177140411yescharacteristic
416C2#3177141411yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#470×C269C2
socleC22×C269C22×D4
hypercenter#470D269
hypocenterC269#3267
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. C2×#22575481C2semidirect
2. C2×#22575476C2semidirect
3. #31771257C2semidirect
4. #31771246C2semidirect
5. #31770998C2semidirect
6. #31770835C2semidirect
7. #31770823C2semidirect
8. #31770809C2semidirect
9. #31770770C2semidirect
10. #31770765C2semidirect
11. #31770755C2semidirect
12. #31770753C2semidirect
13. #31770716C2semidirect
14. #31770674C2semidirect
15. #470×C269C2semidirect
17. C2×#16864403C22semidirect
22. C2×#16864392C22semidirect
27. C2×#16864389C22semidirect
30. C2×C8×D269C22semidirect
31. #22575451C22semidirect
36. #22575433C22semidirect
42. #22575415C22semidirect
46. #22575382C22semidirect
47. #22575376C22semidirect
50. #22575373C22semidirect
52. #22575365C22semidirect
53. #22575363C22semidirect
54. C2×Q16×C269C22semidirect
58. #106×C269C22semidirect
59. #103×C269C22semidirect
60. #103×C269C22semidirect
93. D2152D4semidirect
208. C269#3267semidirect
215. #470D269semidirect
232. C2×Q16C2×D269semidirect
259. #106C2×D269semidirect
262. #103C2×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×#225754812
C2×#225754762
#317712572
#317712462
#317709982
#317708352
#317708232
#317708092
#317707702
#317707652
#317707552
#317707532
#317707162
#317706742
#470×C2692
#3267269