Finite Group Databasebeta

← #46822840

Subgroups — #46822840

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#31771251C211yesmaximal, characteristic, complemented
2#31770967C211yesmaximal, characteristic, complemented
3#31770935C211yesmaximal, characteristic, complemented
4#31770923C211yesmaximal, characteristic, complemented
5#31770892C211yesmaximal, characteristic, complemented
6#31770856C211yesmaximal, characteristic, complemented
7#31770832C211yesmaximal, characteristic, complemented
8#31770820C211yesmaximal, characteristic, complemented
9#31770748C211yesmaximal, characteristic, complemented
10#31770738C211yesmaximal, characteristic, complemented
11#31770715C211yesmaximal, characteristic, complemented
12#31770677C211yesmaximal, characteristic, complemented
13#31770604C211yesmaximal, characteristic, complemented
14C2×#22575373C211yesmaximal, characteristic, complemented
15#547×C269C211yesmaximal, characteristic, complemented
16C2×#16864402C2211yescharacteristic
17C22×D1076C2211yescharacteristic
18C2×#16864390C2211yescharacteristic, complemented
19C2×#16864387C2211yescharacteristic
20C2×#16864387C2211yescharacteristic
21C2×#16864386C2211yescharacteristic, complemented
22C22×#14524130C2211yescharacteristic
23#22575441C2211yescharacteristic
24#22575438—21no—
25#22575432C2211yescharacteristic
26#22575427C2211yescharacteristic
27#22575426—21no—
28#22575421—21no—
29C4×D1076C2211yescharacteristic
30C22⋊C4×D269—21no—
31#22575384C2211yescharacteristic, complemented
32#22575384C2211yescharacteristic, complemented
33#22575384C2211yescharacteristic, complemented
34C4⋊#14524130C2211yescharacteristic, complemented
35#22575376C2211yescharacteristic, complemented
36#22575376C2211yescharacteristic, complemented
37#22575376C2211yescharacteristic, complemented
38#22575374C2211yescharacteristic, complemented
39#22575373C2211yescharacteristic
40#22575373C2211yescharacteristic
41#22575373C2211yescharacteristic
42#22575373C2211yescharacteristic
43#22575373C2211yescharacteristic
44#22575373C2211yescharacteristic
45C22⋊#14524130C2211yescharacteristic
46C22⋊#14524130C2211yescharacteristic, complemented
47#22575364C2211yescharacteristic, complemented
48C2×#41×C269—21no—
49C4⋊Q8×C269C2211yescharacteristic, complemented
50#123×C269—21no—
51#123×C269—21no—
52C22⋊Q8×C269—21no—
53#121×C269C2211yescharacteristic
54#121×C269C2211yescharacteristic, complemented
55C4×D4×C269C2211yescharacteristic
56C4×D4×C269C2211yescharacteristic, complemented
57C2×C4⋊C4×C269C2211yescharacteristic
58C2×C4⋊C4×C269C2211yescharacteristic, complemented
59C23×D269—21no—
60#16864402—22no—
61#16864402—41no—
62C2×Dic538C2311yescharacteristic
63C2×D1076—22no—
64C2×D1076—21no—
65C2×D1076—41no—
66C2×D1076C2311yescharacteristic
67C2×D1076C2311yescharacteristic
68C2×C22⋊D269—21no—
69C2×C22⋊D269—21no—
70C2×C4×D269—21no—
71C2×C4×D269—21no—
72#16864390—41no—
73#16864387—22no—
74#16864387—41no—
75#16864386—41no—
76C2×#14524130—41no—
77C2×#14524130C2311yescharacteristic
78C2×#14524130C2311yescharacteristic
79C2×#14524130C2311yescharacteristic
80C2×#14524130C2311yescharacteristic
81C4⋊Dic269C2311yescharacteristic
82#16864381—21no—
83C22⋊Dic269—21no—
84#16864379—21no—
85#16864379—21no—
86#16864379—21no—
87#16864379—21no—
88#41×C269—44no—
89C2×Q8×C269—21no—
90C2×D4×C269—21no—
91C2×D4×C269—21no—
92C2×D4×C269D411yescharacteristic
93C2×D4×C269C2311yescharacteristic
94C22×C4×C269—21no—
95C22×C4×C269—21no—
96C22×C4×C269—21no—
97C22×C4×C269C2311yescharacteristic
98C22×C4×C269C2311yescharacteristic
99C4⋊C4×C269—21no—
100C4⋊C4×C269—21no—
101C4⋊C4×C269—21no—
102C4⋊C4×C269D411yescharacteristic
103C4⋊C4×C269C2311yescharacteristic
104C4⋊C4×C269C2311yescharacteristic
105C4⋊C4×C269C2311yescharacteristic
106C22⋊C4×C269—21no—
107C22⋊C4×C269—21no—
108C22⋊C4×C269—21no—
109C22⋊C4×C269D411yescharacteristic
110C22⋊C4×C269D411yescharacteristic
111C42×C269C2311yescharacteristic
112C22×D269—21no—
113C22×D269—21no—
114C22×D269—42no—
115C22×D269—42no—
116C22×D269—81no—
117Dic538—21no—
118Dic538—21no—
119D1076—21no—
120D1076—21no—
121D1076—21no—
122D1076—42no—
123D1076#4112yescomplemented
124C22⋊D269—42no—
125C22⋊D269—81no—
126C2×Dic269—21no—
127C2×Dic269—21no—
128C4×D269—42no—
129C4×D269—81no—
130#14524130—21no—
131#14524130—21no—
132#14524130—21no—
133#14524130—21no—
134C23×C269—21no—
135C23×C269C2×D411yescharacteristic
136C23×C269C2×D411yescharacteristic
137Q8×C269—42no—
138D4×C269—22no—
139D4×C269—21no—
140D4×C269—21no—
141D4×C269—42no—
142D4×C269—42no—
143C2×C4×C269—21no—
144C2×C4×C269—21no—
145C2×C4×C269—21no—
146C2×C4×C269—21no—
147C2×C4×C269—21no—
148C2×C4×C269—21no—
149C2×C4×C269—21no—
150C2×C4×C269—42no—
151C2×C4×C269—42no—
152C2×C4×C269—42no—
153C2×C4×C269C2×D411yescharacteristic
154C2×C4×C269C2×D411yescharacteristic
155C2×C4×C269C2×D411yescharacteristic
156C2×C4×C269C2×D411yescharacteristic
157C2×C4×C269C2411yescharacteristic
158C22×C269—21no—
159C22×C269—21no—
160C22×C269—21no—
161C22×C269—21no—
162C22×C269—21no—
163C22×C269—41no—
164C22×C269—41no—
165C22×C269—41no—
166C22×C269AGL(1,ℤ/8)12yes—
167C22×C269C22×D411yescharacteristic
168C2×D269—22no—
169C2×D269—41no—
170C2×D269—41no—
171C2×D269—41no—
172C2×D269—41no—
173C2×D269—41no—
174C2×D269—84no—
175C4×C269—21no—
176C4×C269—21no—
177C4×C269—21no—
178C4×C269—21no—
179C4×C269—41no—
180C4×C269—41no—
181C4×C269—41no—
182C4×C269C2×#4111yescharacteristic
183C4×C269H(5,𝔽2)11yescharacteristic
184Dic269—41no—
185Dic269—41no—
186C2×C269—22no—
187C2×C269—41no—
188C2×C269—81no—
189C2×C269#54611yescharacteristic
190C2×C269C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C269#57711yescharacteristic
192D269—42no—
193D269—81no—
194C269#330611yescharacteristic, complemented, Sylow
195#3306—2691nomaximal, Sylow
196#547D26911yescharacteristic, complemented
197#546—2691no—
198#489—2691no—
199#483—2691no—
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—21no—
212C2×#41—2691no—
213C22×D4—2691no—
214C2×SD16—2691no—
215C2×D8—2691no—
216C2×D8—2691no—
217C2×M16—2691no—
218C22×C8—2691no—
219C4⋊Q8C2×D26911yescharacteristic, complemented
220#124—2691no—
221#123—21no—
222#123—21no—
223#123—5381no—
224C22⋊Q8—21no—
225C22⋊Q8—2691no—
226#121—2691no—
227#121—5381no—
228#121C2×D26911yescharacteristic
229#121C2×D26911yescharacteristic, complemented
230#120—5381no—
231C4×D4—2691no—
232C4×D4C2×D26911yescharacteristic
233C4×D4C2×D26911yescharacteristic, complemented
234C2×C4⋊C4C2×D26911yescharacteristic
235C2×C4⋊C4C2×D26911yescharacteristic, complemented
236C2×C22⋊C4—5381no—
237#106—2691no—
238#106—2691no—
239#106—2691no—
240C4⋊C8—2691no—
241#103—2691no—
242#103—2691no—
243#103—2691no—
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⋊C8—2691no—
253#97—2691no—
254C24—5381no—
255#41—44no—
256#41—5382no—
257#41—10761no—
258C2×Q8—21no—
259C2×Q8—2691no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—2691no—
263C2×D4—2691no—
264C2×D4—5382no—
265C2×D4—5381no—
266C2×D4—5381no—
267C2×D4—5381no—
268C2×D4—10761no—
269C2×D4C22⋊D26911yescharacteristic
270C2×D4C22×D26911yescharacteristic
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—5381no—
275C22×C4—5381no—
276C22×C4C22×D26911yescharacteristic
277C22×C4C22×D26911yescharacteristic
278SD16—10761no—
279D8—5382no—
280D8—10761no—
281M16—10761no—
282C2×C8—2691no—
283C2×C8—2691no—
284C2×C8—2691no—
285C2×C8—2691no—
286C2×C8—10761no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—2691no—
291C4⋊C4—5381no—
292C4⋊C4C22⋊D26911yescharacteristic
293C4⋊C4C22×D26911yescharacteristic
294C4⋊C4C22×D26911yescharacteristic
295C4⋊C4C22×D26911yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—5381no—
300C22⋊C4—5381no—
301C22⋊C4—5381no—
302C22⋊C4—5381no—
303C22⋊C4—5381no—
304C22⋊C4C22⋊D26911yescharacteristic
305C22⋊C4C22⋊D26911yescharacteristic
306C42C22×D26911yescharacteristic
307C23—21no—
308C23—5381no—
309C23—5381no—
310C23—10762no—
311C23—10762no—
312C23—21521no—
313C23C2×C22⋊D26911yescharacteristic
314C23C2×C22⋊D26911yescharacteristic
315Q8—42no—
316Q8—5381no—
317Q8—5381no—
318D4—22no—
319D4—21no—
320D4—21no—
321D4—42no—
322D4—42no—
323D4—2692no—
324D4—5381no—
325D4—5381no—
326D4—5381no—
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—21no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—5381no—
341C2×C4—5381no—
342C2×C4—10762no—
343C2×C4—21521no—
344C2×C4C2×C22⋊D26911yescharacteristic
345C2×C4C2×C22⋊D26911yescharacteristic
346C2×C4C2×C22⋊D26911yescharacteristic
347C2×C4C2×C22⋊D26911yescharacteristic
348C2×C4C23×D26911yescharacteristic
349C8—5381no—
350C8—5381no—
351C8—5381no—
352C8—5381no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—5382no—
362C22—10761no—
363C22—10761no—
364C22—10761no—
365C22—10761no—
366C22—10761no—
367C22—21524no—
368C22#2257547912yes—
369C22C22×C22⋊D26911yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—41no—
375C4—41no—
376C4—41no—
377C4—10761no—
378C4—10761no—
379C4#41×D26911yescharacteristic
380C4#2257549911yescharacteristic
381C2—22no—
382C2—41no—
383C2—81no—
384C2—10762no—
385C2—21521no—
386C2#3177124311yescharacteristic
387C2C2×#2257547911yescharacteristic
388C2#3177141311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#547×C269C2
socleC22×C269C22×D4
hypercenter#547D269
hypocenterC269#3306
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. #31771251C2semidirect
2. #31770967C2semidirect
3. #31770935C2semidirect
4. #31770923C2semidirect
5. #31770892C2semidirect
6. #31770856C2semidirect
7. #31770832C2semidirect
8. #31770820C2semidirect
9. #31770748C2semidirect
10. #31770738C2semidirect
11. #31770715C2semidirect
12. #31770677C2semidirect
13. #31770604C2semidirect
14. C2×#22575373C2semidirect
15. #547×C269C2semidirect
18. C2×#16864390C22semidirect
21. C2×#16864386C22semidirect
31. #22575384C22semidirect
32. #22575384C22semidirect
33. #22575384C22semidirect
34. C4⋊#14524130C22semidirect
35. #22575376C22semidirect
36. #22575376C22semidirect
37. #22575376C22semidirect
38. #22575374C22semidirect
46. C22⋊#14524130C22semidirect
47. #22575364C22semidirect
49. C4⋊Q8×C269C22semidirect
54. #121×C269C22semidirect
56. C4×D4×C269C22semidirect
58. C2×C4⋊C4×C269C22semidirect
123. D1076#41semidirect
194. C269#3306semidirect
196. #547D269semidirect
219. C4⋊Q8C2×D269semidirect
229. #121C2×D269semidirect
233. C4×D4C2×D269semidirect
235. C2×C4⋊C4C2×D269semidirect

Maximal subgroups

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

subgroupindex
#317712512
#317709672
#317709352
#317709232
#317708922
#317708562
#317708322
#317708202
#317707482
#317707382
#317707152
#317706772
#317706042
C2×#225753732
#547×C2692
#3306269