Finite Group Databasebeta

← #46822902

Subgroups — #46822902

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#31771290C211yesmaximal, characteristic, complemented
2D4×C22⋊D269C211yesmaximal, characteristic
3#31770968C211yesmaximal, characteristic, complemented
4#31770928C211yesmaximal, characteristic, complemented
5#31770918C211yesmaximal, characteristic, complemented
6#31770907C211yesmaximal, characteristic, complemented
7#31770850C211yesmaximal, characteristic, complemented
8#31770834C211yesmaximal, characteristic, complemented
9#31770804C211yesmaximal, characteristic, complemented
10#31770757C211yesmaximal, characteristic, complemented
11#31770756C211yesmaximal, characteristic, complemented
12#31770702C211yesmaximal, characteristic, complemented
13#31770676C211yesmaximal, characteristic, complemented
14#31770608C211yesmaximal, characteristic, complemented
15C2×#102×C269C211yesmaximal, characteristic, complemented
16C22×C22⋊D269—21no—
17C2×D4×D269C2211yescharacteristic
18C2×#16864391C2211yescharacteristic, complemented
19C2×#16864390C2211yescharacteristic
20C2×#16864390C2211yescharacteristic, complemented
21C2×#16864385C2211yescharacteristic, complemented
22#22575449C2211yescharacteristic
23#22575448—21no—
24#22575444C2211yescharacteristic, complemented
25#22575443—21no—
26#22575436—21no—
27#22575433C2211yescharacteristic, complemented
28#22575432C2211yescharacteristic, complemented
29#22575427C2211yescharacteristic
30#22575425—21no—
31#22575424C2211yescharacteristic
32#22575422—21no—
33#22575421—21no—
34#22575418C2211yescharacteristic, complemented
35D4×Dic269C2211yescharacteristic
36C4×C22⋊D269C2211yescharacteristic
37#22575412—21no—
38#22575411C2211yescharacteristic, complemented
39#22575386C2211yescharacteristic, complemented
40#22575386C2211yescharacteristic, complemented
41#22575385C2211yescharacteristic, complemented
42#22575382C2211yescharacteristic, complemented
43#22575378C2211yescharacteristic, complemented
44#22575375C2211yescharacteristic, complemented
45#22575374C2211yescharacteristic
46#22575374C2211yescharacteristic, complemented
47#22575373C2211yescharacteristic, complemented
48#22575373C2211yescharacteristic, complemented
49C22⋊#14524130C2211yescharacteristic, complemented
50#22575365C2211yescharacteristic, complemented
51#22575363C2211yescharacteristic, complemented
52C22×D4×C269C2211yescharacteristic
53C22×C8×C269C2211yescharacteristic, complemented
54C2×C4⋊C4×C269C2211yescharacteristic, complemented
55#102×C269C2211yescharacteristic
56#102×C269C2211yescharacteristic
57#102×C269C2211yescharacteristic, complemented
58#102×C269C2211yescharacteristic, complemented
59C23×D269—21no—
60C2×Dic538C2311yescharacteristic
61C2×D1076C2311yescharacteristic
62C2×C22⋊D269—21no—
63C2×C22⋊D269—21no—
64C2×C22⋊D269—21no—
65C2×C22⋊D269—21no—
66C2×C22⋊D269—21no—
67C2×C22⋊D269—44no—
68C2×C22⋊D269D411yescharacteristic
69D4×D269—22no—
70D4×D269—41no—
71C22×Dic269—21no—
72C2×C4×D269C2311yescharacteristic
73#16864391—41no—
74#16864390—22no—
75#16864390—41no—
76#16864385—41no—
77C2×#14524130C2311yescharacteristic
78C2×#14524130C2311yescharacteristic
79C4⋊Dic269—21no—
80C4⋊Dic269—21no—
81C4⋊Dic269C2311yescharacteristic
82C4⋊Dic269C2311yescharacteristic
83#16864381—21no—
84#16864381—21no—
85#16864381—21no—
86#16864381D411yescharacteristic
87C22⋊Dic269—21no—
88C22⋊Dic269—21no—
89C22⋊Dic269—21no—
90C22⋊Dic269D411yescharacteristic
91#16864379—21no—
92#16864379—21no—
93#16864379—21no—
94#16864379D411yescharacteristic
95C4×Dic269—21no—
96C4×Dic269—21no—
97C4×Dic269C2311yescharacteristic
98C24×C269—21no—
99C2×D4×C269—22no—
100C2×D4×C269—21no—
101C2×D4×C269—41no—
102C2×D4×C269C2311yescharacteristic
103C2×D4×C269C2311yescharacteristic
104C22×C4×C269—21no—
105C22×C4×C269C2311yescharacteristic
106C2×C8×C269—41no—
107C2×C8×C269C2311yescharacteristic
108C2×C8×C269C2311yescharacteristic
109C4⋊C4×C269—21no—
110C4⋊C4×C269C2311yescharacteristic
111C4⋊C4×C269C2311yescharacteristic
112C22×D269—21no—
113C22×D269—42no—
114C22×D269—42no—
115C22×D269—81no—
116C22×D269C2×D411yescharacteristic
117Dic538—21no—
118Dic538—21no—
119D1076—21no—
120D1076—21no—
121C22⋊D269—22no—
122C22⋊D269—42no—
123C22⋊D269—42no—
124C22⋊D269—42no—
125C22⋊D269—42no—
126C22⋊D269—81no—
127C2×Dic269—21no—
128C2×Dic269—21no—
129C2×Dic269—21no—
130C2×Dic269—42no—
131C2×Dic269C2×D411yescharacteristic
132C2×Dic269C2×D411yescharacteristic
133C2×Dic269C2×D411yescharacteristic
134C4×D269—21no—
135C4×D269—21no—
136#14524130—21no—
137#14524130—21no—
138C23×C269—21no—
139C23×C269—21no—
140C23×C269—42no—
141C23×C269—42no—
142C23×C269—81no—
143C23×C269C2×D411yescharacteristic
144D4×C269—21no—
145D4×C269—21no—
146D4×C269—21no—
147D4×C269—42no—
148D4×C269#4112yes—
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×C269—22no—
161C22×C269—41no—
162C22×C269—41no—
163C22×C269—41no—
164C22×C269—41no—
165C22×C269—41no—
166C22×C269—84no—
167C22×C269AGL(1,ℤ/8)12yes—
168C22×C269C22×D411yescharacteristic
169C2×D269—21no—
170C2×D269—21no—
171C2×D269—21no—
172C2×D269—41no—
173C2×D269—41no—
174C2×D269—41no—
175C2×D269—84no—
176C4×C269—21no—
177C4×C269—41no—
178C4×C269—41no—
179C4×C269C2×#4111yescharacteristic
180C4×C269H(5,𝔽2)11yescharacteristic
181Dic269—21no—
182Dic269—21no—
183Dic269—21no—
184Dic269—41no—
185Dic269—41no—
186Dic269—41no—
187C2×C269—22no—
188C2×C269—42no—
189C2×C269—81no—
190C2×C269#54611yescharacteristic
191C2×C269C2×AGL(1,ℤ/8)11yescharacteristic
192C2×C269#57711yescharacteristic
193D269—41no—
194D269—81no—
195C269#331011yescharacteristic, complemented, Sylow
196#3310—2691nomaximal, Sylow
197#552—2691no—
198D42—2691no—
199#490—2691no—
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×#102D26911yescharacteristic, complemented
212C22×D4—2691no—
213C22×D4—5381no—
214C22×D4C2×D26911yescharacteristic
215C2×SD16—2691no—
216C2×SD16—2691no—
217C2×SD16—2691no—
218C2×M16—2691no—
219C22×C8C2×D26911yescharacteristic, complemented
220#127—2691no—
221#126—5381no—
222#125—2691no—
223#125—5381no—
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⋊C4C2×D26911yescharacteristic, complemented
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
249#102C2×D26911yescharacteristic
250#102C2×D26911yescharacteristic, complemented
251#102C2×D26911yescharacteristic, complemented
252C22⋊C8—2691no—
253C22⋊C8—2691no—
254#97—2691no—
255C24—21no—
256C24—5381no—
257C2×Q8—2691no—
258C2×D4—22no—
259C2×D4—21no—
260C2×D4—41no—
261C2×D4—2691no—
262C2×D4—2691no—
263C2×D4—5381no—
264C2×D4—5381no—
265C2×D4—5381no—
266C2×D4—5382no—
267C2×D4—5381no—
268C2×D4—5381no—
269C2×D4—10761no—
270C2×D4—10764no—
271C2×D4C22×D26911yescharacteristic
272C2×D4C22×D26911yescharacteristic
273C22×C4—21no—
274C22×C4—2691no—
275C22×C4—5381no—
276C22×C4C22×D26911yescharacteristic
277SD16—5382no—
278SD16—10761no—
279SD16—10761no—
280M16—10761no—
281C2×C8—41no—
282C2×C8—2691no—
283C2×C8—2691no—
284C2×C8C22×D26911yescharacteristic
285C2×C8C22×D26911yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—2691no—
288C4⋊C4—2691no—
289C4⋊C4—2691no—
290C4⋊C4—5381no—
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—2691no—
306C42—5381no—
307C42—5381no—
308C23—21no—
309C23—21no—
310C23—42no—
311C23—42no—
312C23—81no—
313C23—2691no—
314C23—5381no—
315C23—10762no—
316C23—10762no—
317C23—21521no—
318C23D4×D26911yescharacteristic
319Q8—5381no—
320Q8—5381no—
321D4—21no—
322D4—21no—
323D4—21no—
324D4—42no—
325D4—5382no—
326D4—5381no—
327D4—5381no—
328D4—10762no—
329D4—10762no—
330D4—10762no—
331D4—10762no—
332D4—21521no—
333D4#1686440212yes—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—42no—
339C2×C4—2691no—
340C2×C4—2691no—
341C2×C4—2691no—
342C2×C4—5381no—
343C2×C4—5381no—
344C2×C4—5381no—
345C2×C4—5381no—
346C2×C4—5381no—
347C2×C4—10762no—
348C2×C4D4×D26911yescharacteristic
349C2×C4C23×D26911yescharacteristic
350C8—21no—
351C8—21no—
352C8—5381no—
353C8—5381no—
354C22—21no—
355C22—21no—
356C22—22no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—84no—
363C22—5381no—
364C22—5381no—
365C22—5381no—
366C22—10761no—
367C22—10761no—
368C22—10761no—
369C22—21524no—
370C22#2257547712yes—
371C22C2×D4×D26911yescharacteristic
372C4—21no—
373C4—41no—
374C4—41no—
375C4—5381no—
376C4—5381no—
377C4—5381no—
378C4—10761no—
379C4—10761no—
380C4—10761no—
381C4C2×#1686440211yescharacteristic
382C4#2257549811yescharacteristic
383C2—22no—
384C2—42no—
385C2—81no—
386C2—10761no—
387C2—21521no—
388C2#3177124511yescharacteristic
389C2C2×#2257547711yescharacteristic
390C2#3177141111yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
FittingC2×#102×C269C2
socleC22×C269C22×D4
prime radicalD4×C22⋊D269C2
hypercenterC2×#102D269
hypocenterC269#3310
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. #31771290C2semidirect
3. #31770968C2semidirect
4. #31770928C2semidirect
5. #31770918C2semidirect
6. #31770907C2semidirect
7. #31770850C2semidirect
8. #31770834C2semidirect
9. #31770804C2semidirect
10. #31770757C2semidirect
11. #31770756C2semidirect
12. #31770702C2semidirect
13. #31770676C2semidirect
14. #31770608C2semidirect
15. C2×#102×C269C2semidirect
18. C2×#16864391C22semidirect
20. C2×#16864390C22semidirect
21. C2×#16864385C22semidirect
24. #22575444C22semidirect
27. #22575433C22semidirect
28. #22575432C22semidirect
34. #22575418C22semidirect
38. #22575411C22semidirect
39. #22575386C22semidirect
40. #22575386C22semidirect
41. #22575385C22semidirect
42. #22575382C22semidirect
43. #22575378C22semidirect
44. #22575375C22semidirect
46. #22575374C22semidirect
47. #22575373C22semidirect
48. #22575373C22semidirect
49. C22⋊#14524130C22semidirect
50. #22575365C22semidirect
51. #22575363C22semidirect
53. C22×C8×C269C22semidirect
54. C2×C4⋊C4×C269C22semidirect
57. #102×C269C22semidirect
58. #102×C269C22semidirect
195. C269#3310semidirect
211. C2×#102D269semidirect
219. C22×C8C2×D269semidirect
237. C2×C4⋊C4C2×D269semidirect
250. #102C2×D269semidirect
251. #102C2×D269semidirect

Maximal subgroups

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

subgroupindex
#317712902
D4×C22⋊D2692
#317709682
#317709282
#317709182
#317709072
#317708502
#317708342
#317708042
#317707572
#317707562
#317707022
#317706762
#317706082
C2×#102×C2692
#3310269