Finite Group Databasebeta

← #46822915

Subgroups — #46822915

Order 34432.

402 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31771251C211yesmaximal, characteristic
2#31770967C211yesmaximal, characteristic, complemented
3#31770946C211yesmaximal, characteristic, complemented
4#31770929C211yesmaximal, characteristic, complemented
5#31770892C211yesmaximal, characteristic, complemented
6#31770861C211yesmaximal, characteristic, complemented
7#31770836C211yesmaximal, characteristic, complemented
8#31770827C211yesmaximal, characteristic, complemented
9#31770765C211yesmaximal, characteristic, complemented
10#31770755C211yesmaximal, characteristic, complemented
11#31770703C211yesmaximal, characteristic, complemented
12#31770677C211yesmaximal, characteristic, complemented
13#31770609C211yesmaximal, characteristic, complemented
14#31770599C211yesmaximal, characteristic, complemented
15#556×C269C211yesmaximal, characteristic, complemented
16C2×#16864402C2211yescharacteristic
17C22×D1076C2211yescharacteristic
18C2×#16864392C2211yescharacteristic, complemented
19C2×#16864389C2211yescharacteristic
20C2×#16864389C2211yescharacteristic
21C2×#16864386C2211yescharacteristic
22C22×#14524130C2211yescharacteristic, complemented
23#22575441C2211yescharacteristic
24#22575438—21no—
25#22575432C2211yescharacteristic
26#22575427C2211yescharacteristic
27#22575426—21no—
28#22575421—21no—
29C4×D1076C2211yescharacteristic
30C22⋊C4×D269—21no—
31#22575386C2211yescharacteristic, complemented
32#22575384C2211yescharacteristic, complemented
33#22575384C2211yescharacteristic, complemented
34C4⋊#14524130C2211yescharacteristic, complemented
35#22575377C2211yescharacteristic, complemented
36#22575376C2211yescharacteristic, complemented
37#22575376C2211yescharacteristic, complemented
38#22575376C2211yescharacteristic, complemented
39#22575373C2211yescharacteristic
40#22575373C2211yescharacteristic
41#22575373C2211yescharacteristic
42#22575373C2211yescharacteristic
43#22575373C2211yescharacteristic
44#22575373C2211yescharacteristic
45C22⋊#14524130C2211yescharacteristic
46C22⋊#14524130C2211yescharacteristic, complemented
47#22575364C2211yescharacteristic, complemented
48C4⋊Q8×C269C2211yescharacteristic, complemented
49#126×C269—21no—
50#126×C269—21no—
51#124×C269—21no—
52C22⋊Q8×C269C2211yescharacteristic
53C22⋊Q8×C269C2211yescharacteristic, complemented
54C4×Q8×C269—21no—
55C4×Q8×C269C2211yescharacteristic, complemented
56C4×D4×C269C2211yescharacteristic
57#117×C269C2211yescharacteristic
58#117×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#16864392—41no—
73#16864389—22no—
74#16864389—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—
88C2×Q8×C269—21no—
89C2×Q8×C269C2311yescharacteristic
90C2×D4×C269D411yescharacteristic
91C22×C4×C269C2311yescharacteristic
92C22×C4×C269C2311yescharacteristic
93C4⋊C4×C269—21no—
94C4⋊C4×C269—21no—
95C4⋊C4×C269—21no—
96C4⋊C4×C269—21no—
97C4⋊C4×C269D411yescharacteristic
98C4⋊C4×C269C2311yescharacteristic
99C4⋊C4×C269C2311yescharacteristic
100C4⋊C4×C269C2311yescharacteristic
101C22⋊C4×C269—21no—
102C22⋊C4×C269—21no—
103C22⋊C4×C269—21no—
104C22⋊C4×C269—21no—
105C22⋊C4×C269D411yescharacteristic
106C22⋊C4×C269D411yescharacteristic
107C42×C269—21no—
108C42×C269—21no—
109C42×C269—21no—
110C42×C269C2311yescharacteristic
111C22×D269—21no—
112C22×D269—21no—
113C22×D269—42no—
114C22×D269—42no—
115C22×D269—81no—
116Dic538—21no—
117Dic538—21no—
118D1076—21no—
119D1076—21no—
120D1076—21no—
121D1076—42no—
122D1076#4112yes—
123C22⋊D269—42no—
124C22⋊D269—81no—
125C2×Dic269—21no—
126C2×Dic269—21no—
127C4×D269—42no—
128C4×D269—81no—
129#14524130—21no—
130#14524130—21no—
131#14524130—21no—
132#14524130—21no—
133C23×C269C2×D411yescharacteristic
134C23×C269C2×D411yescharacteristic
135Q8×C269—21no—
136Q8×C269—21no—
137Q8×C269—42no—
138D4×C269—41no—
139C2×C4×C269—21no—
140C2×C4×C269—21no—
141C2×C4×C269—21no—
142C2×C4×C269—21no—
143C2×C4×C269—21no—
144C2×C4×C269—21no—
145C2×C4×C269—21no—
146C2×C4×C269—21no—
147C2×C4×C269C2×D411yescharacteristic
148C2×C4×C269C2×D411yescharacteristic
149C2×C4×C269C2×D411yescharacteristic
150C2×C4×C269C2×D411yescharacteristic
151C2×C4×C269C2411yescharacteristic
152C22×C269—21no—
153C22×C269—21no—
154C22×C269—21no—
155C22×C269—21no—
156C22×C269—21no—
157C22×C269—21no—
158C22×C269C22×D411yescharacteristic
159C2×D269—22no—
160C2×D269—41no—
161C2×D269—41no—
162C2×D269—41no—
163C2×D269—41no—
164C2×D269—41no—
165C2×D269—84no—
166C4×C269—21no—
167C4×C269—21no—
168C4×C269—21no—
169C4×C269—21no—
170C4×C269—41no—
171C4×C269—41no—
172C4×C269—41no—
173C4×C269—41no—
174C4×C269C2×#4111yescharacteristic
175C4×C269H(5,𝔽2)11yescharacteristic
176Dic269—41no—
177Dic269—41no—
178C2×C269—41no—
179C2×C269—41no—
180C2×C269#54611yescharacteristic
181C2×C269#57511yescharacteristic
182C2×C269#57711yescharacteristic
183D269—42no—
184D269—81no—
185C269#331111yescharacteristic, complemented, Sylow
186#3311—2691nomaximal, Sylow
187#556D26911yescharacteristic, complemented
188#546—2691no—
189#489—2691no—
190#485—2691no—
191#482—2691no—
192#476—2691no—
193#470—2691no—
194#465—2691no—
195#464—2691no—
196#452—2691no—
197#450—2691no—
198#440—2691no—
199#435—2691no—
200#418—2691no—
201#416—2691no—
202C2×#41—2691no—
203C22×D4—2691no—
204C2×Q16—2691no—
205C2×SD16—2691no—
206C2×SD16—2691no—
207C2×M16—2691no—
208C22×C8—2691no—
209C4⋊Q8C2×D26911yescharacteristic, complemented
210#126—21no—
211#126—21no—
212#124—21no—
213#124—2691no—
214#123—5381no—
215C22⋊Q8—2691no—
216C22⋊Q8C2×D26911yescharacteristic
217C22⋊Q8C2×D26911yescharacteristic, complemented
218#121—2691no—
219#121—5381no—
220#120—5381no—
221C4×Q8—21no—
222C4×Q8C2×D26911yescharacteristic, complemented
223C4×D4—2691no—
224C4×D4C2×D26911yescharacteristic
225#117C2×D26911yescharacteristic
226#117C2×D26911yescharacteristic, complemented
227C2×C22⋊C4—5381no—
228#107—2691no—
229#106—2691no—
230#106—2691no—
231C4⋊C8—2691no—
232#103—2691no—
233#103—2691no—
234#103—2691no—
235#103—2691no—
236#102—2691no—
237#102—2691no—
238#102—2691no—
239#102—2691no—
240#102—2691no—
241#102—2691no—
242C22⋊C8—2691no—
243C22⋊C8—2691no—
244#97—2691no—
245C24—5381no—
246#41—5382no—
247#41—10761no—
248C2×Q8—21no—
249C2×Q8—2691no—
250C2×Q8C22×D26911yescharacteristic
251C2×D4—2691no—
252C2×D4—2691no—
253C2×D4—5382no—
254C2×D4—5381no—
255C2×D4—5381no—
256C2×D4—5381no—
257C2×D4—10761no—
258C2×D4C22⋊D26911yescharacteristic
259C22×C4—5381no—
260C22×C4—5381no—
261C22×C4C22×D26911yescharacteristic
262C22×C4C22×D26911yescharacteristic
263Q16—10761no—
264SD16—5382no—
265SD16—10761no—
266M16—10761no—
267C2×C8—2691no—
268C2×C8—2691no—
269C2×C8—2691no—
270C2×C8—2691no—
271C2×C8—10761no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—2691no—
277C4⋊C4—5381no—
278C4⋊C4C22⋊D26911yescharacteristic
279C4⋊C4C22×D26911yescharacteristic
280C4⋊C4C22×D26911yescharacteristic
281C4⋊C4C22×D26911yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—5381no—
287C22⋊C4—5381no—
288C22⋊C4—5381no—
289C22⋊C4—5381no—
290C22⋊C4—5381no—
291C22⋊C4C22⋊D26911yescharacteristic
292C22⋊C4C22⋊D26911yescharacteristic
293C42—21no—
294C42—21no—
295C42—21no—
296C42C22×D26911yescharacteristic
297C23—5381no—
298C23—5381no—
299C23—10762no—
300C23—10762no—
301C23—21521no—
302C23C2×C22⋊D26911yescharacteristic
303C23C2×C22⋊D26911yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—42no—
307Q8—5381no—
308Q8—5381no—
309D4—41no—
310D4—2692no—
311D4—5381no—
312D4—5381no—
313D4—5381no—
314D4—10762no—
315D4—10762no—
316D4—21521no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—5381no—
326C2×C4—5381no—
327C2×C4—10762no—
328C2×C4—21521no—
329C2×C4C2×C22⋊D26911yescharacteristic
330C2×C4C2×C22⋊D26911yescharacteristic
331C2×C4C2×C22⋊D26911yescharacteristic
332C2×C4C2×C22⋊D26911yescharacteristic
333C2×C4C23×D26911yescharacteristic
334C8—5381no—
335C8—5381no—
336C8—5381no—
337C8—5381no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—5382no—
345C22—10761no—
346C22—10761no—
347C22—10761no—
348C22—10761no—
349C22—10761no—
350C22—21524no—
351C22C22×C22⋊D26911yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—10761no—
361C4—10761no—
362C4#41×D26911yescharacteristic
363C4#2257549911yescharacteristic
364C2—41no—
365C2—41no—
366C2—10762no—
367C2—21521no—
368C2#3177124311yescharacteristic
369C2#3177140311yescharacteristic
370C2#3177141311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#556×C269C2
socleC22×C269C22×D4
prime radical#31771251C2
hypercenter#556D269
hypocenterC269#3311
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C4×D4 ◁ #556

Complements

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

NH ≅ G/Ntype
2. #31770967C2semidirect
3. #31770946C2semidirect
4. #31770929C2semidirect
5. #31770892C2semidirect
6. #31770861C2semidirect
7. #31770836C2semidirect
8. #31770827C2semidirect
9. #31770765C2semidirect
10. #31770755C2semidirect
11. #31770703C2semidirect
12. #31770677C2semidirect
13. #31770609C2semidirect
14. #31770599C2semidirect
15. #556×C269C2semidirect
18. C2×#16864392C22semidirect
22. C22×#14524130C22semidirect
31. #22575386C22semidirect
32. #22575384C22semidirect
33. #22575384C22semidirect
34. C4⋊#14524130C22semidirect
35. #22575377C22semidirect
36. #22575376C22semidirect
37. #22575376C22semidirect
38. #22575376C22semidirect
46. C22⋊#14524130C22semidirect
47. #22575364C22semidirect
48. C4⋊Q8×C269C22semidirect
53. C22⋊Q8×C269C22semidirect
55. C4×Q8×C269C22semidirect
58. #117×C269C22semidirect
185. C269#3311semidirect
187. #556D269semidirect
209. C4⋊Q8C2×D269semidirect
217. C22⋊Q8C2×D269semidirect
222. C4×Q8C2×D269semidirect
226. #117C2×D269semidirect

Maximal subgroups

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

subgroupindex
#317712512
#317709672
#317709462
#317709292
#317708922
#317708612
#317708362
#317708272
#317707652
#317707552
#317707032
#317706772
#317706092
#317705992
#556×C2692
#3311269