Finite Group Databasebeta

← #46822905

Subgroups — #46822905

Order 34432.

450 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31771293C211yesmaximal, characteristic, complemented
2#31771240C211yesmaximal, characteristic
3#31770927C211yesmaximal, characteristic, complemented
4#31770917C211yesmaximal, characteristic, complemented
5#31770902C211yesmaximal, characteristic, complemented
6#31770850C211yesmaximal, characteristic, complemented
7#31770833C211yesmaximal, characteristic, complemented
8#31770802C211yesmaximal, characteristic, complemented
9#31770757C211yesmaximal, characteristic, complemented
10#31770757C211yesmaximal, characteristic, complemented
11#31770707C211yesmaximal, characteristic, complemented
12#31770678C211yesmaximal, characteristic, complemented
13#31770608C211yesmaximal, characteristic, complemented
14#102×D269C211yesmaximal, characteristic, complemented
15#490×C269C211yesmaximal, characteristic, complemented
16C2×D4×D269—21no—
17C2×D4×D269C2211yescharacteristic
18C2×D4×D269C2211yescharacteristic
19C2×#16864391C2211yescharacteristic
20C2×#16864390C2211yescharacteristic, complemented
21C2×#16864390C2211yescharacteristic, complemented
22C2×#16864385C2211yescharacteristic, complemented
23C2×C8×D269C2211yescharacteristic, complemented
24#22575448—21no—
25#22575444—21no—
26#22575438—21no—
27#22575433C2211yescharacteristic, complemented
28#22575433C2211yescharacteristic, complemented
29#22575424—21no—
30#22575424C2211yescharacteristic
31#22575424C2211yescharacteristic
32#22575421—21no—
33#22575421—21no—
34C4×Dic538C2211yescharacteristic, complemented
35C4×D1076C2211yescharacteristic
36C4×D1076C2211yescharacteristic
37#22575411—21no—
38#22575411C2211yescharacteristic, complemented
39C4⋊C4×D269C2211yescharacteristic, complemented
40#22575387C2211yescharacteristic, complemented
41#22575386C2211yescharacteristic, complemented
42#22575384C2211yescharacteristic, complemented
43C4⋊#14524130C2211yescharacteristic, complemented
44#22575378C2211yescharacteristic, complemented
45#22575375C2211yescharacteristic
46#22575374C2211yescharacteristic, complemented
47#22575374C2211yescharacteristic, complemented
48#22575373C2211yescharacteristic
49#22575373C2211yescharacteristic
50#22575365C2211yescharacteristic, complemented
51#22575365C2211yescharacteristic, complemented
52#127×C269C2211yescharacteristic
53#125×C269C2211yescharacteristic, complemented
54#102×C269C2211yescharacteristic, complemented
55#102×C269C2211yescharacteristic, complemented
56#102×C269C2211yescharacteristic, complemented
57#102×C269C2211yescharacteristic, complemented
58#97×C269C2211yescharacteristic, complemented
59C23×D269—21no—
60C23×D269—21no—
61C2×Dic538C2311yescharacteristic
62C2×D1076D411yescharacteristic
63C2×D1076C2311yescharacteristic
64C2×C22⋊D269—21no—
65C2×C22⋊D269—21no—
66C2×C22⋊D269—21no—
67C2×C22⋊D269—21no—
68D4×D269—22no—
69D4×D269—22no—
70D4×D269—41no—
71D4×D269—41no—
72D4×D269—44no—
73C2×C4×D269—21no—
74C2×C4×D269—21no—
75C2×C4×D269C2311yescharacteristic
76C2×C4×D269C2311yescharacteristic
77#16864391—22no—
78#16864390—41no—
79#16864390—41no—
80#16864385—41no—
81C2×#14524130C2311yescharacteristic
82C2×#14524130C2311yescharacteristic
83C8×D269—41no—
84C4⋊Dic269—21no—
85C4⋊Dic269—21no—
86C4⋊Dic269D411yescharacteristic
87C4⋊Dic269C2311yescharacteristic
88C4⋊Dic269C2311yescharacteristic
89#16864381—21no—
90#16864381—21no—
91C22⋊Dic269—21no—
92C22⋊Dic269—21no—
93#16864379—21no—
94#16864379—21no—
95#16864379—21no—
96#16864379—21no—
97#16864379D411yescharacteristic
98#16864379D411yescharacteristic
99C4×Dic269—21no—
100C4×Dic269—21no—
101C2×D4×C269—21no—
102C2×D4×C269—21no—
103C2×D4×C269C2311yescharacteristic
104C2×D4×C269C2311yescharacteristic
105C2×C8×C269C2311yescharacteristic
106C2×C8×C269C2311yescharacteristic
107C4⋊C4×C269—21no—
108C4⋊C4×C269—21no—
109C4⋊C4×C269C2311yescharacteristic
110C4⋊C4×C269C2311yescharacteristic
111C42×C269C2311yescharacteristic
112C22×D269—21no—
113C22×D269—42no—
114C22×D269—42no—
115C22×D269—42no—
116C22×D269—42no—
117C22×D269—81no—
118C22×D269—81no—
119C22×D269C2×D411yescharacteristic
120C22×D269C2×D411yescharacteristic
121Dic538—21no—
122Dic538—21no—
123D1076—22no—
124D1076—21no—
125D1076#4112yes—
126C22⋊D269—42no—
127C22⋊D269—42no—
128C22⋊D269—42no—
129C22⋊D269—42no—
130C2×Dic269—21no—
131C2×Dic269—21no—
132C2×Dic269—21no—
133C2×Dic269C2×D411yescharacteristic
134C2×Dic269C2×D411yescharacteristic
135C4×D269—21no—
136C4×D269—21no—
137C4×D269—21no—
138C4×D269—21no—
139C4×D269—42no—
140C4×D269—42no—
141#14524130—21no—
142#14524130—21no—
143C23×C269—21no—
144C23×C269—21no—
145D4×C269—21no—
146D4×C269—21no—
147D4×C269—21no—
148D4×C269—21no—
149D4×C269—42no—
150D4×C269—81no—
151C2×C4×C269—21no—
152C2×C4×C269—21no—
153C2×C4×C269C2×D411yescharacteristic
154C2×C4×C269C2×D411yescharacteristic
155C2×C4×C269C2411yescharacteristic
156C8×C269—21no—
157C8×C269—21no—
158C22×C269—41no—
159C22×C269—41no—
160C22×C269—41no—
161C22×C269—41no—
162C22×C269—41no—
163C22×C269—41no—
164C22×C269C22×D411yescharacteristic
165C2×D269—21no—
166C2×D269—21no—
167C2×D269—21no—
168C2×D269—22no—
169C2×D269—21no—
170C2×D269—21no—
171C2×D269—41no—
172C2×D269—41no—
173C2×D269—84no—
174C2×D269—84no—
175C2×D269AGL(1,ℤ/8)12yes—
176C4×C269—21no—
177C4×C269—21no—
178C4×C269—41no—
179C4×C269—41no—
180C4×C269C2×#4111yescharacteristic
181C4×C269H(5,𝔽2)11yescharacteristic
182Dic269—21no—
183Dic269—21no—
184Dic269—41no—
185Dic269—41no—
186Dic269—41no—
187C2×C269—81no—
188C2×C269—81no—
189C2×C269#54611yescharacteristic
190C2×C269C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C269#57711yescharacteristic
192D269—22no—
193D269—41no—
194D269—42no—
195C269#331011yescharacteristic, complemented, Sylow
196#3310—2691nomaximal, Sylow
197#552—2691no—
198D42—2691no—
199#490D26911yescharacteristic, complemented
200#482—2691no—
201#480—2691no—
202#478—2691no—
203#468—2691no—
204#465—2691no—
205#460—2691no—
206#450—2691no—
207#450—2691no—
208#440—2691no—
209#435—2691no—
210#418—2691no—
211C2×#102—2691no—
212C22×D4—2691no—
213C22×D4—2691no—
214C22×D4—5381no—
215C2×SD16—2691no—
216C2×SD16—2691no—
217C2×SD16—2691no—
218C2×M16—2691no—
219C22×C8—2691no—
220#127C2×D26911yescharacteristic
221#126—5381no—
222#125—5381no—
223#125C2×D26911yescharacteristic, complemented
224#123—5381no—
225C22⋊Q8—2691no—
226C22⋊Q8—2691no—
227#121—2691no—
228#121—2691no—
229#121—5381no—
230#120—5381no—
231#120—5381no—
232C4×Q8—2691no—
233C4×D4—2691no—
234C4×D4—2691no—
235#117—2691no—
236#117—5381no—
237C2×C4⋊C4—2691no—
238#107—2691no—
239#107—2691no—
240#106—2691no—
241C4⋊C8—2691no—
242#103—2691no—
243#102—2691no—
244#102—2691no—
245#102—2691no—
246#102—2691no—
247#102—2691no—
248#102C2×D26911yescharacteristic, complemented
249#102C2×D26911yescharacteristic, complemented
250#102C2×D26911yescharacteristic, complemented
251#102C2×D26911yescharacteristic, complemented
252C22⋊C8—2691no—
253C22⋊C8—2691no—
254#97C2×D26911yescharacteristic, complemented
255C24—5381no—
256C24—5381no—
257C2×Q8—2691no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—2691no—
261C2×D4—2691no—
262C2×D4—5381no—
263C2×D4—5381no—
264C2×D4—5382no—
265C2×D4—5382no—
266C2×D4—5381no—
267C2×D4—5381no—
268C2×D4—10761no—
269C2×D4—10761no—
270C2×D4—10764no—
271C2×D4C22×D26911yescharacteristic
272C2×D4C22×D26911yescharacteristic
273C22×C4—2691no—
274C22×C4—2691no—
275C22×C4—5381no—
276C22×C4—5381no—
277SD16—5382no—
278SD16—10761no—
279SD16—10761no—
280M16—10761no—
281C2×C8—2691no—
282C2×C8—2691no—
283C2×C8—10761no—
284C2×C8C22×D26911yescharacteristic
285C2×C8C22×D26911yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—2691no—
289C4⋊C4—2691no—
290C4⋊C4—2691no—
291C4⋊C4—5381no—
292C4⋊C4—5381no—
293C4⋊C4—5381no—
294C4⋊C4—5381no—
295C4⋊C4C22×D26911yescharacteristic
296C4⋊C4C22×D26911yescharacteristic
297C22⋊C4—2691no—
298C22⋊C4—2691no—
299C22⋊C4—5381no—
300C22⋊C4—5381no—
301C22⋊C4—5381no—
302C22⋊C4—5381no—
303C22⋊C4—5381no—
304C22⋊C4—5381no—
305C42—5381no—
306C42—5381no—
307C42C22×D26911yescharacteristic
308C23—21no—
309C23—21no—
310C23—2691no—
311C23—2691no—
312C23—5381no—
313C23—10762no—
314C23—10762no—
315C23—10762no—
316C23—10762no—
317C23—21521no—
318C23—21521no—
319Q8—5381no—
320Q8—5381no—
321D4—21no—
322D4—21no—
323D4—21no—
324D4—21no—
325D4—42no—
326D4—81no—
327D4—2692no—
328D4—5382no—
329D4—5381no—
330D4—10762no—
331D4—10762no—
332D4—10762no—
333D4—10762no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—2691no—
337C2×C4—2691no—
338C2×C4—5381no—
339C2×C4—5381no—
340C2×C4—5381no—
341C2×C4—5381no—
342C2×C4—5381no—
343C2×C4—5381no—
344C2×C4—5381no—
345C2×C4—10762no—
346C2×C4—10762no—
347C2×C4D4×D26911yescharacteristic
348C2×C4D4×D26911yescharacteristic
349C2×C4C23×D26911yescharacteristic
350C8—21no—
351C8—21no—
352C8—5381no—
353C8—5381no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—2692no—
361C22—5381no—
362C22—5381no—
363C22—5381no—
364C22—5381no—
365C22—5382no—
366C22—5381no—
367C22—10761no—
368C22—10761no—
369C22—21524no—
370C22—21524no—
371C22C2×D4×D26911yescharacteristic
372C4—21no—
373C4—21no—
374C4—41no—
375C4—41no—
376C4—5381no—
377C4—5381no—
378C4—10761no—
379C4—10761no—
380C4—10761no—
381C4#41×D26911yescharacteristic
382C4#2257549911yescharacteristic
383C2—81no—
384C2—81no—
385C2—5382no—
386C2—10761no—
387C2—10762no—
388C2#3177124811yescharacteristic
389C2AGL(1,ℤ/8)×D26911yescharacteristic
390C2#3177141511yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#490×C269C2
socleC22×C269C22×D4
prime radical#31771240C2
hypercenter#490D269
hypocenterC269#3310
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
1. #31771293C2semidirect
3. #31770927C2semidirect
4. #31770917C2semidirect
5. #31770902C2semidirect
6. #31770850C2semidirect
7. #31770833C2semidirect
8. #31770802C2semidirect
9. #31770757C2semidirect
10. #31770757C2semidirect
11. #31770707C2semidirect
12. #31770678C2semidirect
13. #31770608C2semidirect
14. #102×D269C2semidirect
15. #490×C269C2semidirect
20. C2×#16864390C22semidirect
21. C2×#16864390C22semidirect
22. C2×#16864385C22semidirect
23. C2×C8×D269C22semidirect
27. #22575433C22semidirect
28. #22575433C22semidirect
34. C4×Dic538C22semidirect
38. #22575411C22semidirect
39. C4⋊C4×D269C22semidirect
40. #22575387C22semidirect
41. #22575386C22semidirect
42. #22575384C22semidirect
43. C4⋊#14524130C22semidirect
44. #22575378C22semidirect
46. #22575374C22semidirect
47. #22575374C22semidirect
50. #22575365C22semidirect
51. #22575365C22semidirect
53. #125×C269C22semidirect
54. #102×C269C22semidirect
55. #102×C269C22semidirect
56. #102×C269C22semidirect
57. #102×C269C22semidirect
58. #97×C269C22semidirect
195. C269#3310semidirect
199. #490D269semidirect
223. #125C2×D269semidirect
248. #102C2×D269semidirect
249. #102C2×D269semidirect
250. #102C2×D269semidirect
251. #102C2×D269semidirect
254. #97C2×D269semidirect

Maximal subgroups

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

subgroupindex
#317712932
#317712402
#317709272
#317709172
#317709022
#317708502
#317708332
#317708022
#317707572
#317707072
#317706782
#317706082
#102×D2692
#490×C2692
#3310269