Finite Group Databasebeta

← #46822844

Subgroups — #46822844

Order 34432.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31771256C211yesmaximal, characteristic, complemented
2#31771246C211yesmaximal, characteristic, complemented
3#31770966C211yesmaximal, characteristic, complemented
4#31770925C211yesmaximal, characteristic, complemented
5#31770894C211yesmaximal, characteristic, complemented
6#31770859C211yesmaximal, characteristic, complemented
7#31770836C211yesmaximal, characteristic, complemented
8#31770823C211yesmaximal, characteristic, complemented
9#31770753C211yesmaximal, characteristic, complemented
10#31770741C211yesmaximal, characteristic, complemented
11#31770716C211yesmaximal, characteristic, complemented
12#31770673C211yesmaximal, characteristic, complemented
13#31770605C211yesmaximal, characteristic, complemented
14C2×#22575373C211yesmaximal, characteristic, complemented
15#483×C269C211yesmaximal, characteristic, complemented
16C2×#16864403C2211yescharacteristic
17C2×#16864402—21no—
18C22×D1076C2211yescharacteristic
19C2×#16864389C2211yescharacteristic, complemented
20C2×D2152C2211yescharacteristic
21C2×D2152C2211yescharacteristic
22C2×#16864385C2211yescharacteristic, complemented
23C22×#14524130C2211yescharacteristic
24#22575452C2211yescharacteristic, complemented
25#22575442C2211yescharacteristic
26#22575438—21no—
27#22575438—21no—
28#22575436—21no—
29#22575431—21no—
30#22575428—21no—
31#22575428C2211yescharacteristic
32#22575428C2211yescharacteristic, complemented
33#22575427C2211yescharacteristic
34#22575423—21no—
35#22575415C2211yescharacteristic
36#22575415C2211yescharacteristic, complemented
37C4×C22⋊D269C2211yescharacteristic
38C4⋊C4×D269C2211yescharacteristic, complemented
39C22⋊C4×D269—21no—
40#22575384C2211yescharacteristic, complemented
41#22575382C2211yescharacteristic, complemented
42#22575377C2211yescharacteristic, complemented
43#22575375C2211yescharacteristic
44#22575375C2211yescharacteristic
45#22575373C2211yescharacteristic
46#22575373C2211yescharacteristic
47#22575373C2211yescharacteristic
48#22575373C2211yescharacteristic
49#22575373C2211yescharacteristic, complemented
50#22575365C2211yescharacteristic, complemented
51#22575363C2211yescharacteristic, complemented
52C22⋊Q8×C269C2211yescharacteristic
53C2×C4⋊C4×C269C2211yescharacteristic
54#106×C269C2211yescharacteristic, complemented
55#106×C269C2211yescharacteristic, complemented
56#103×C269C2211yescharacteristic, complemented
57#103×C269C2211yescharacteristic, complemented
58C22⋊C8×C269C2211yescharacteristic
59C23×D269—21no—
60#16864403—22no—
61#16864403—41no—
62#16864402—44no—
63C2×Dic538—21no—
64C2×D1076—22no—
65C2×D1076—21no—
66C2×D1076—21no—
67C2×D1076—21no—
68C2×D1076—21no—
69C2×D1076—41no—
70C2×D1076C2311yescharacteristic
71C2×D1076C2311yescharacteristic
72C2×D1076C2311yescharacteristic
73C2×C22⋊D269—21no—
74C2×C22⋊D269D411yescharacteristic
75C2×C4×D269—21no—
76C2×C4×D269—21no—
77C2×C4×D269—21no—
78C2×C4×D269—21no—
79C2×C4×D269C2311yescharacteristic
80#16864389—41no—
81D2152—22no—
82D2152—41no—
83#16864385—41no—
84C2×#14524130—41no—
85C2×#14524130C2311yescharacteristic
86C2×#14524130C2311yescharacteristic
87C4⋊Dic269C2311yescharacteristic
88#16864381—21no—
89#16864381—21no—
90#16864381D411yescharacteristic
91C22⋊Dic269D411yescharacteristic
92#16864379—21no—
93#16864379—21no—
94#16864379—21no—
95#16864379—21no—
96#16864379—21no—
97#16864379—21no—
98#16864379—21no—
99#16864379D411yescharacteristic
100C4×Dic269C2311yescharacteristic
101C2×Q8×C269C2311yescharacteristic
102C22×C4×C269—21no—
103C22×C4×C269C2311yescharacteristic
104C2×C8×C269C2311yescharacteristic
105C2×C8×C269C2311yescharacteristic
106C4⋊C4×C269—21no—
107C4⋊C4×C269—21no—
108C4⋊C4×C269C2311yescharacteristic
109C4⋊C4×C269C2311yescharacteristic
110C4⋊C4×C269C2311yescharacteristic
111C22⋊C4×C269—21no—
112C22×D269—21no—
113C22×D269—21no—
114C22×D269—21no—
115C22×D269—42no—
116C22×D269—42no—
117C22×D269—81no—
118C22×D269C2×D411yescharacteristic
119Dic538—42no—
120D1076—21no—
121D1076—21no—
122D1076—21no—
123D1076—21no—
124D1076—21no—
125D1076—42no—
126D1076—42no—
127D1076—42no—
128D1076—81no—
129D1076#4112yescomplemented
130C22⋊D269—22no—
131C22⋊D269—42no—
132C2×Dic269—21no—
133C2×Dic269C2×D411yescharacteristic
134C2×Dic269C2×D411yescharacteristic
135C2×Dic269C2×D411yescharacteristic
136C4×D269—21no—
137C4×D269—21no—
138C4×D269—42no—
139C4×D269—42no—
140C4×D269—42no—
141C4×D269—81no—
142#14524130—21no—
143#14524130—21no—
144C23×C269C2×D411yescharacteristic
145Q8×C269—21no—
146Q8×C269—21no—
147C2×C4×C269—21no—
148C2×C4×C269—21no—
149C2×C4×C269—21no—
150C2×C4×C269—21no—
151C2×C4×C269—21no—
152C2×C4×C269—21no—
153C2×C4×C269—42no—
154C2×C4×C269C2×D411yescharacteristic
155C2×C4×C269C2411yescharacteristic
156C8×C269—21no—
157C8×C269—21no—
158C22×C269—21no—
159C22×C269—21no—
160C22×C269AGL(1,ℤ/8)12yes—
161C22×C269C22×D411yescharacteristic
162C2×D269—21no—
163C2×D269—22no—
164C2×D269—21no—
165C2×D269—21no—
166C2×D269—41no—
167C2×D269—41no—
168C2×D269—41no—
169C2×D269—41no—
170C2×D269—41no—
171C2×D269—41no—
172C2×D269—41no—
173C2×D269—41no—
174C2×D269—84no—
175C4×C269—21no—
176C4×C269—41no—
177C4×C269—41no—
178C4×C269—41no—
179C4×C269—41no—
180C4×C269C2×#4111yescharacteristic
181C4×C269H(5,𝔽2)11yescharacteristic
182Dic269—21no—
183Dic269—21no—
184Dic269—21no—
185Dic269—41no—
186C2×C269—22no—
187C2×C269#54611yescharacteristic
188C2×C269C2×AGL(1,ℤ/8)11yescharacteristic
189C2×C269#57711yescharacteristic
190D269—41no—
191D269—42no—
192D269—81no—
193D269—81no—
194C269#330611yescharacteristic, complemented, Sylow
195#3306—2691nomaximal, Sylow
196#547—2691no—
197#546—2691no—
198#489—2691no—
199#483D26911yescharacteristic, complemented
200#481—2691no—
201#476—2691no—
202#469—2691no—
203#465—2691no—
204#463—2691no—
205#449—2691no—
206#447—2691no—
207#442—2691no—
208#435—2691no—
209#417—2691no—
210C2×#102—2691no—
211C2×#41—2691no—
212C2×#41—5381no—
213C22×D4—2691no—
214C2×SD16—2691no—
215C2×D8—2691no—
216C2×D8—2691no—
217C2×M16—2691no—
218C22×C8—2691no—
219C4⋊Q8—2691no—
220#124—2691no—
221#123—5381no—
222#123—5381no—
223#123—5381no—
224C22⋊Q8—5381no—
225C22⋊Q8C2×D26911yescharacteristic
226#121—2691no—
227#121—2691no—
228#121—2691no—
229#121—5381no—
230#120—5381no—
231C4×D4—2691no—
232C4×D4—2691no—
233C4×D4—2691no—
234C2×C4⋊C4—2691no—
235C2×C4⋊C4C2×D26911yescharacteristic
236C2×C22⋊C4—5381no—
237#106—2691no—
238#106C2×D26911yescharacteristic, complemented
239#106C2×D26911yescharacteristic, complemented
240C4⋊C8—2691no—
241#103—2691no—
242#103C2×D26911yescharacteristic, complemented
243#103C2×D26911yescharacteristic, complemented
244#102—2691no—
245#102—2691no—
246#102—2691no—
247#102—2691no—
248#102—2691no—
249#102—2691no—
250#102—2691no—
251C22⋊C8—2691no—
252C22⋊C8C2×D26911yescharacteristic
253#97—2691no—
254C24—5381no—
255#41—5382no—
256#41—10764no—
257#41—10761no—
258C2×Q8—5381no—
259C2×Q8C22×D26911yescharacteristic
260C2×D4—2691no—
261C2×D4—2691no—
262C2×D4—2691no—
263C2×D4—2691no—
264C2×D4—5382no—
265C2×D4—5381no—
266C2×D4—5381no—
267C2×D4—5381no—
268C2×D4—5381no—
269C2×D4—5381no—
270C2×D4—10761no—
271C22×C4—21no—
272C22×C4—2691no—
273C22×C4—5381no—
274C22×C4—5381no—
275C22×C4—5381no—
276C22×C4—5381no—
277C22×C4C22×D26911yescharacteristic
278SD16—10761no—
279D8—5382no—
280D8—10761no—
281M16—10761no—
282C2×C8—2691no—
283C2×C8—2691no—
284C2×C8—10761no—
285C2×C8C22×D26911yescharacteristic
286C2×C8C22×D26911yescharacteristic
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—2691no—
290C4⋊C4—2691no—
291C4⋊C4—5381no—
292C4⋊C4—5381no—
293C4⋊C4C22×D26911yescharacteristic
294C4⋊C4C22×D26911yescharacteristic
295C4⋊C4C22×D26911yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—2691no—
298C22⋊C4—2691no—
299C22⋊C4—5381no—
300C22⋊C4—5381no—
301C22⋊C4—5381no—
302C22⋊C4—5381no—
303C22⋊C4—5381no—
304C22⋊C4—5381no—
305C22⋊C4—5381no—
306C42—2691no—
307C23—2691no—
308C23—5381no—
309C23—5381no—
310C23—5381no—
311C23—10762no—
312C23—10762no—
313C23—21521no—
314C23D4×D26911yescharacteristic
315Q8—21no—
316Q8—21no—
317Q8—10762no—
318D4—2692no—
319D4—5382no—
320D4—5381no—
321D4—5381no—
322D4—5381no—
323D4—5381no—
324D4—5381no—
325D4—10762no—
326D4—10762no—
327D4—10762no—
328D4—10762no—
329D4—21521no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—42no—
337C2×C4—2691no—
338C2×C4—2691no—
339C2×C4—2691no—
340C2×C4—5381no—
341C2×C4—5381no—
342C2×C4—5381no—
343C2×C4—10762no—
344C2×C4—10762no—
345C2×C4—10762no—
346C2×C4—21521no—
347C2×C4D4×D26911yescharacteristic
348C2×C4C23×D26911yescharacteristic
349C8—21no—
350C8—21no—
351C8—5381no—
352C8—5381no—
353C22—21no—
354C22—21no—
355C22—5381no—
356C22—5381no—
357C22—5381no—
358C22—5382no—
359C22—10761no—
360C22—10761no—
361C22—10761no—
362C22—10761no—
363C22—10761no—
364C22—10761no—
365C22—10761no—
366C22—10761no—
367C22—21524no—
368C22#2257548012yes—
369C22C2×D4×D26911yescharacteristic
370C4—21no—
371C4—41no—
372C4—41no—
373C4—41no—
374C4—41no—
375C4—5381no—
376C4—5381no—
377C4—5381no—
378C4—10761no—
379C4#41×D26911yescharacteristic
380C4#2257549911yescharacteristic
381C2—22no—
382C2—10761no—
383C2—10762no—
384C2—21521no—
385C2—21521no—
386C2#3177124711yescharacteristic
387C2C2×#2257548011yescharacteristic
388C2#3177141411yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#483×C269C2
socleC22×C269C22×D4
hypercenter#483D269
hypocenterC269#3306
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
1. #31771256C2semidirect
2. #31771246C2semidirect
3. #31770966C2semidirect
4. #31770925C2semidirect
5. #31770894C2semidirect
6. #31770859C2semidirect
7. #31770836C2semidirect
8. #31770823C2semidirect
9. #31770753C2semidirect
10. #31770741C2semidirect
11. #31770716C2semidirect
12. #31770673C2semidirect
13. #31770605C2semidirect
14. C2×#22575373C2semidirect
15. #483×C269C2semidirect
19. C2×#16864389C22semidirect
22. C2×#16864385C22semidirect
24. #22575452C22semidirect
32. #22575428C22semidirect
36. #22575415C22semidirect
38. C4⋊C4×D269C22semidirect
40. #22575384C22semidirect
41. #22575382C22semidirect
42. #22575377C22semidirect
49. #22575373C22semidirect
50. #22575365C22semidirect
51. #22575363C22semidirect
54. #106×C269C22semidirect
55. #106×C269C22semidirect
56. #103×C269C22semidirect
57. #103×C269C22semidirect
129. D1076#41semidirect
194. C269#3306semidirect
199. #483D269semidirect
238. #106C2×D269semidirect
239. #106C2×D269semidirect
242. #103C2×D269semidirect
243. #103C2×D269semidirect

Maximal subgroups

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

subgroupindex
#317712562
#317712462
#317709662
#317709252
#317708942
#317708592
#317708362
#317708232
#317707532
#317707412
#317707162
#317706732
#317706052
C2×#225753732
#483×C2692
#3306269