Finite Group Databasebeta

← #46822900

Subgroups — #46822900

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
1D4×D1076C211yesmaximal, characteristic
2#31770971C211yesmaximal, characteristic, complemented
3#31770929C211yesmaximal, characteristic, complemented
4#31770918C211yesmaximal, characteristic, complemented
5#31770902C211yesmaximal, characteristic, complemented
6#31770853C211yesmaximal, characteristic, complemented
7#31770836C211yesmaximal, characteristic, complemented
8#31770805C211yesmaximal, characteristic, complemented
9#31770755C211yesmaximal, characteristic, complemented
10#31770755C211yesmaximal, characteristic, complemented
11#31770703C211yesmaximal, characteristic, complemented
12#31770677C211yesmaximal, characteristic, complemented
13#31770609C211yesmaximal, characteristic, complemented
14C2×#22575373C211yesmaximal, characteristic, complemented
15#552×C269C211yesmaximal, characteristic, complemented
16C22×D1076C2211yescharacteristic
17C22×D1076C2211yescharacteristic
18C2×D4×D269—21no—
19C2×#16864389C2211yescharacteristic
20C2×#16864389C2211yescharacteristic, complemented
21C2×#16864389C2211yescharacteristic, complemented
22C2×#16864386C2211yescharacteristic, complemented
23C22×#14524130C2211yescharacteristic, complemented
24#22575450C2211yescharacteristic
25#22575428—21no—
26#22575427C2211yescharacteristic
27#22575427C2211yescharacteristic
28#22575423—21no—
29#22575423—21no—
30C4×D1076C2211yescharacteristic
31#22575386C2211yescharacteristic, complemented
32#22575386C2211yescharacteristic, complemented
33#22575384C2211yescharacteristic, complemented
34C4⋊#14524130C2211yescharacteristic, complemented
35#22575377C2211yescharacteristic, complemented
36#22575373C2211yescharacteristic
37#22575373C2211yescharacteristic
38#22575373C2211yescharacteristic
39#22575373C2211yescharacteristic, complemented
40#22575373C2211yescharacteristic, complemented
41#22575373C2211yescharacteristic, complemented
42#22575373C2211yescharacteristic, complemented
43#22575373C2211yescharacteristic, complemented
44#22575373C2211yescharacteristic, complemented
45C22⋊#14524130C2211yescharacteristic, complemented
46C22⋊#14524130C2211yescharacteristic, complemented
47#22575364C2211yescharacteristic, complemented
48#126×C269—21no—
49#125×C269—21no—
50#125×C269C2211yescharacteristic, complemented
51#123×C269—21no—
52C22⋊Q8×C269C2211yescharacteristic, complemented
53C22⋊Q8×C269C2211yescharacteristic, complemented
54C4×Q8×C269C2211yescharacteristic, complemented
55C4×D4×C269C2211yescharacteristic
56#117×C269—21no—
57#117×C269C2211yescharacteristic, complemented
58C2×C4⋊C4×C269C2211yescharacteristic, complemented
59C23×D269—21no—
60C23×D269—21no—
61C2×D1076—22no—
62C2×D1076—22no—
63C2×D1076—21no—
64C2×D1076—21no—
65C2×D1076—21no—
66C2×D1076—21no—
67C2×D1076—41no—
68C2×D1076—41no—
69C2×D1076C2311yescharacteristic
70C2×D1076C2311yescharacteristic
71C2×D1076C2311yescharacteristic
72C2×C22⋊D269—21no—
73C2×C22⋊D269—21no—
74D4×D269—44no—
75C2×C4×D269—21no—
76#16864389—22no—
77#16864389—41no—
78#16864389—41no—
79#16864386—41no—
80C2×#14524130—41no—
81C2×#14524130C2311yescharacteristic
82C2×#14524130C2311yescharacteristic
83C2×#14524130C2311yescharacteristic
84C2×#14524130C2311yescharacteristic
85C4⋊Dic269C2311yescharacteristic
86#16864379—21no—
87#16864379—21no—
88#16864379—21no—
89C2×Q8×C269C2311yescharacteristic
90C2×D4×C269D411yescharacteristic
91C22×C4×C269—21no—
92C22×C4×C269C2311yescharacteristic
93C22×C4×C269C2311yescharacteristic
94C4⋊C4×C269—21no—
95C4⋊C4×C269—21no—
96C4⋊C4×C269—21no—
97C4⋊C4×C269—21no—
98C4⋊C4×C269—21no—
99C4⋊C4×C269—21no—
100C4⋊C4×C269D411yescharacteristic
101C4⋊C4×C269C2311yescharacteristic
102C4⋊C4×C269C2311yescharacteristic
103C4⋊C4×C269C2311yescharacteristic
104C22⋊C4×C269—21no—
105C22⋊C4×C269—21no—
106C22⋊C4×C269—21no—
107C22⋊C4×C269D411yescharacteristic
108C22⋊C4×C269D411yescharacteristic
109C42×C269—21no—
110C42×C269—21no—
111C42×C269C2311yescharacteristic
112C22×D269—21no—
113C22×D269—21no—
114C22×D269—21no—
115C22×D269—42no—
116C22×D269—42no—
117C22×D269—42no—
118C22×D269—42no—
119C22×D269—81no—
120C22×D269—81no—
121D1076—21no—
122D1076—21no—
123D1076—21no—
124D1076—21no—
125D1076—21no—
126D1076—42no—
127D1076—42no—
128D1076—42no—
129D1076—81no—
130D1076#4112yes—
131C22⋊D269—42no—
132C22⋊D269—42no—
133C2×Dic269—21no—
134C4×D269—42no—
135#14524130—21no—
136#14524130—21no—
137#14524130—21no—
138#14524130—21no—
139C23×C269C2×D411yescharacteristic
140C23×C269C2×D411yescharacteristic
141Q8×C269—21no—
142Q8×C269—21no—
143D4×C269—22no—
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—21no—
151C2×C4×C269—21no—
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×C269AGL(1,ℤ/8)12yes—
164C22×C269C22×D411yescharacteristic
165C2×D269—22no—
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—
175C2×D269—84no—
176C4×C269—21no—
177C4×C269—21no—
178C4×C269—21no—
179C4×C269—21no—
180C4×C269—41no—
181C4×C269—41no—
182C4×C269—41no—
183C4×C269—41no—
184C4×C269C2×#4111yescharacteristic
185C4×C269H(5,𝔽2)11yescharacteristic
186Dic269—41no—
187C2×C269—22no—
188C2×C269—41no—
189C2×C269#54611yescharacteristic
190C2×C269C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C269#57711yescharacteristic
192D269—42no—
193D269—81no—
194D269—81no—
195C269#331011yescharacteristic, complemented, Sylow
196#3310—2691nomaximal, Sylow
197#552D26911yescharacteristic, complemented
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×#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#127—2691no—
221#126—21no—
222#125—21no—
223#125C2×D26911yescharacteristic, complemented
224#123—21no—
225C22⋊Q8C2×D26911yescharacteristic, complemented
226C22⋊Q8C2×D26911yescharacteristic, complemented
227#121—2691no—
228#121—2691no—
229#121—5381no—
230#120—5381no—
231#120—5381no—
232C4×Q8C2×D26911yescharacteristic, complemented
233C4×D4—2691no—
234C4×D4C2×D26911yescharacteristic
235#117—21no—
236#117C2×D26911yescharacteristic, complemented
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#102—2691no—
249#102—2691no—
250#102—2691no—
251#102—2691no—
252C22⋊C8—2691no—
253C22⋊C8—2691no—
254#97—2691no—
255C24—5381no—
256C24—5381no—
257C2×Q8C22×D26911yescharacteristic
258C2×D4—2691no—
259C2×D4—2691no—
260C2×D4—2691no—
261C2×D4—5382no—
262C2×D4—5382no—
263C2×D4—5381no—
264C2×D4—5381no—
265C2×D4—5381no—
266C2×D4—5381no—
267C2×D4—5381no—
268C2×D4—5381no—
269C2×D4—10761no—
270C2×D4—10761no—
271C2×D4—10764no—
272C2×D4C22⋊D26911yescharacteristic
273C22×C4—21no—
274C22×C4—5381no—
275C22×C4C22×D26911yescharacteristic
276C22×C4C22×D26911yescharacteristic
277SD16—5382no—
278SD16—10761no—
279SD16—10761no—
280M16—10761no—
281C2×C8—2691no—
282C2×C8—2691no—
283C2×C8—2691no—
284C2×C8—2691no—
285C2×C8—10761no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—2691no—
293C4⋊C4C22⋊D26911yescharacteristic
294C4⋊C4C22×D26911yescharacteristic
295C4⋊C4C22×D26911yescharacteristic
296C4⋊C4C22×D26911yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—5381no—
301C22⋊C4—5381no—
302C22⋊C4—5381no—
303C22⋊C4C22⋊D26911yescharacteristic
304C22⋊C4C22⋊D26911yescharacteristic
305C42—21no—
306C42—21no—
307C42C22×D26911yescharacteristic
308C23—5381no—
309C23—5381no—
310C23—5381no—
311C23—10762no—
312C23—10762no—
313C23—10762no—
314C23—10762no—
315C23—21521no—
316C23—21521no—
317C23C2×C22⋊D26911yescharacteristic
318C23C2×C22⋊D26911yescharacteristic
319Q8—21no—
320Q8—21no—
321D4—22no—
322D4—2692no—
323D4—5381no—
324D4—5381no—
325D4—5381no—
326D4—5381no—
327D4—5381no—
328D4—10762no—
329D4—10762no—
330D4—10762no—
331D4—10762no—
332D4—10762no—
333D4—21521no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—5381no—
344C2×C4—10762no—
345C2×C4C2×C22⋊D26911yescharacteristic
346C2×C4C2×C22⋊D26911yescharacteristic
347C2×C4C2×C22⋊D26911yescharacteristic
348C2×C4C2×C22⋊D26911yescharacteristic
349C2×C4C23×D26911yescharacteristic
350C8—5381no—
351C8—5381no—
352C8—5381no—
353C8—5381no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—5382no—
360C22—10761no—
361C22—10761no—
362C22—10761no—
363C22—10761no—
364C22—10761no—
365C22—10761no—
366C22—10761no—
367C22—10761no—
368C22—21524no—
369C22—21524no—
370C22#2257547812yes—
371C22C22×C22⋊D26911yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—10761no—
381C4#41×D26911yescharacteristic
382C4#2257549911yescharacteristic
383C2—22no—
384C2—41no—
385C2—10762no—
386C2—21521no—
387C2—21521no—
388C2#3177124311yescharacteristic
389C2C2×#2257547811yescharacteristic
390C2#3177141311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#552×C269C2
socleC22×C269C22×D4
prime radicalD4×D1076C2
hypercenter#552D269
hypocenterC269#3310
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C4×D4 ◁ #552

Complements

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

NH ≅ G/Ntype
2. #31770971C2semidirect
3. #31770929C2semidirect
4. #31770918C2semidirect
5. #31770902C2semidirect
6. #31770853C2semidirect
7. #31770836C2semidirect
8. #31770805C2semidirect
9. #31770755C2semidirect
10. #31770755C2semidirect
11. #31770703C2semidirect
12. #31770677C2semidirect
13. #31770609C2semidirect
14. C2×#22575373C2semidirect
15. #552×C269C2semidirect
20. C2×#16864389C22semidirect
21. C2×#16864389C22semidirect
22. C2×#16864386C22semidirect
23. C22×#14524130C22semidirect
31. #22575386C22semidirect
32. #22575386C22semidirect
33. #22575384C22semidirect
34. C4⋊#14524130C22semidirect
35. #22575377C22semidirect
39. #22575373C22semidirect
40. #22575373C22semidirect
41. #22575373C22semidirect
42. #22575373C22semidirect
43. #22575373C22semidirect
44. #22575373C22semidirect
45. C22⋊#14524130C22semidirect
46. C22⋊#14524130C22semidirect
47. #22575364C22semidirect
50. #125×C269C22semidirect
52. C22⋊Q8×C269C22semidirect
53. C22⋊Q8×C269C22semidirect
54. C4×Q8×C269C22semidirect
57. #117×C269C22semidirect
58. C2×C4⋊C4×C269C22semidirect
195. C269#3310semidirect
197. #552D269semidirect
223. #125C2×D269semidirect
225. C22⋊Q8C2×D269semidirect
226. C22⋊Q8C2×D269semidirect
232. C4×Q8C2×D269semidirect
236. #117C2×D269semidirect
237. C2×C4⋊C4C2×D269semidirect

Maximal subgroups

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

subgroupindex
D4×D10762
#317709712
#317709292
#317709182
#317709022
#317708532
#317708362
#317708052
#317707552
#317707032
#317706772
#317706092
C2×#225753732
#552×C2692
#3310269