Finite Group Databasebeta

← #44559192

Subgroups — #44559192

Order 32128.

506 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22548742C211yesmaximal, characteristic
2#30632169C211yesmaximal, characteristic, complemented
3#30632159C211yesmaximal, characteristic, complemented
4#30631798C211yesmaximal, characteristic, complemented
5#30631797C211yesmaximal, characteristic, complemented
6#30631788C211yesmaximal, characteristic, complemented
7#30631785C211yesmaximal, characteristic, complemented
8#30631772C211yesmaximal, characteristic, complemented
9C4×D2008C211yesmaximal, characteristic, complemented
10#30631767C211yesmaximal, characteristic, complemented
11#30631726C211yesmaximal, characteristic, complemented
12#30631682C211yesmaximal, characteristic, complemented
13#102×D251C211yesmaximal, characteristic, complemented
14#30631641C211yesmaximal, characteristic, complemented
15C4×SD16×C251C211yesmaximal, characteristic, complemented
16C2×#16860811C2211yescharacteristic
17C2×D4×D251C2211yescharacteristic
18#22548742C418yescomplemented
19C2×#16860797C2211yescharacteristic
20C2×D2008C2211yescharacteristic
21C2×#16860795C2211yescharacteristic
22C2×#16860793C2211yescharacteristic
23C2×C8×D251C2211yescharacteristic
24Q8×Dic251C2211yescharacteristic, complemented
25C4×D1004—21no—
26C4×D1004C2211yescharacteristic
27C4×D1004C2211yescharacteristic, complemented
28D4×Dic251C2211yescharacteristic
29#22548677—21no—
30#22548677C2211yescharacteristic
31#22548677C2211yescharacteristic, complemented
32C4×C22⋊D251—21no—
33(C22⋊D251)⋊C4—21no—
34#22548673C2211yescharacteristic
35#22548672—21no—
36#22548672C2211yescharacteristic
37C4⋊C4×D251—21no—
38C4⋊C4×D251C2211yescharacteristic
39C2×(C2×D251)⋊C4—21no—
40C22⋊C4×D251—21no—
41#22548649C2211yescharacteristic, complemented
42#22548648C2211yescharacteristic, complemented
43#22548646C2211yescharacteristic, complemented
44#22548639C2211yescharacteristic, complemented
45#22548637C2211yescharacteristic
46#22548637C2211yescharacteristic, complemented
47#22548636C2211yescharacteristic
48#22548635C2211yescharacteristic
49#22548635C2211yescharacteristic, complemented
50#22548626C2211yescharacteristic, complemented
51#22548625C2211yescharacteristic, complemented
52C8×Dic251C2211yescharacteristic, complemented
53C2×SD16×C251C2211yescharacteristic
54C4×Q8×C251C2211yescharacteristic, complemented
55C4×D4×C251C2211yescharacteristic
56#106×C251C2211yescharacteristic, complemented
57#103×C251C2211yescharacteristic, complemented
58#102×C251C2211yescharacteristic
59C4×C8×C251C2211yescharacteristic, complemented
60C23×D251—21no—
61#16860811—22no—
62#16860811C2×C414yescomplemented
63C2×D1004—21no—
64C2×D1004C2311yescharacteristic
65C2×D1004C2311yescharacteristic
66C2×C22⋊D251—21no—
67D4×D251—22no—
68D4×D251C2×C414yescomplemented
69C22×Dic251—21no—
70C2×C4×D251—21no—
71C2×C4×D251—21no—
72C2×C4×D251—21no—
73C2×C4×D251—21no—
74C2×C4×D251—21no—
75C2×C4×D251—21no—
76C2×C4×D251C2311yescharacteristic
77#16860797C2×C414yescomplemented
78D2008C2×C414yescomplemented
79#16860795C2×C414yescomplemented
80#16860793C2×C414yescomplemented
81C2×C251⋊C8C2311yescharacteristic
82C2×C251⋊C8C2311yescharacteristic
83C8×D251—22no—
84C4⋊Dic251—21no—
85C4⋊Dic251C2311yescharacteristic
86C4⋊Dic251C2311yescharacteristic
87Dic251⋊C4—21no—
88Dic251⋊C4D411yescharacteristic
89Dic251⋊C4D411yescharacteristic
90C22⋊Dic251—21no—
91(C2×D251)⋊C4—21no—
92(C2×D251)⋊C4—21no—
93(C2×D251)⋊C4—21no—
94(C2×D251)⋊C4—21no—
95(C2×D251)⋊C4—21no—
96(C2×D251)⋊C4D411yescharacteristic
97(C2×D251)⋊C4D411yescharacteristic
98C4×Dic251—21no—
99C4×Dic251—21no—
100C4×Dic251C2311yescharacteristic
101C2×Q8×C251C2311yescharacteristic
102C2×D4×C251C2311yescharacteristic
103C22×C4×C251—21no—
104SD16×C251C2×C414yescomplemented
105C2×C8×C251C2311yescharacteristic
106C2×C8×C251C2311yescharacteristic
107C4⋊C4×C251—21no—
108C4⋊C4×C251C2311yescharacteristic
109C4⋊C4×C251C2311yescharacteristic
110C22⋊C4×C251—21no—
111C42×C251—21no—
112C42×C251C2311yescharacteristic
113C22×D251—21no—
114C22×D251—21no—
115C22×D251—24no—
116C22×D251—81no—
117C22×D251—81no—
118C22×D251C2×D411yescharacteristic
119D1004—24no—
120D1004—21no—
121D1004—21no—
122D1004C22×C412yes—
123D1004C22×C412yes—
124C22⋊D251—24no—
125C2×Dic251—21no—
126C2×Dic251—21no—
127C2×Dic251—81no—
128C2×Dic251C2×D411yescharacteristic
129C2×Dic251C2×D411yescharacteristic
130C2×Dic251C2×D411yescharacteristic
131C4×D251—24no—
132C4×D251—81no—
133C4×D251—81no—
134C4×D251—81no—
135C4×D251—81no—
136C4×D251—81no—
137C4×D251C22×C412yes—
138C4×D251#4112yes—
139C251⋊C8—21no—
140C251⋊C8C22×C412yes—
141C23×C251—21no—
142Q8×C251—21no—
143Q8×C251C22×C412yes—
144D4×C251—21no—
145D4×C251C22×C412yes—
146C2×C4×C251—21no—
147C2×C4×C251—21no—
148C2×C4×C251—21no—
149C2×C4×C251—81no—
150C2×C4×C251C2×D411yescharacteristic
151C2×C4×C251C2×D411yescharacteristic
152C2×C4×C251C2411yescharacteristic
153C8×C251—21no—
154C8×C251C22×C412yes—
155C22×C251—22no—
156C22×C251—41no—
157C22×C251—41no—
158C22×C251C22×D411yescharacteristic
159C2×D251—21no—
160C2×D251—21no—
161C2×D251—22no—
162C2×D251—22no—
163C2×D251—41no—
164C2×D251—41no—
165C2×D251—41no—
166C2×D251—41no—
167C2×D251—84no—
168C2×D251C4×D412yes—
169C4×C251—22no—
170C4×C251—21no—
171C4×C251—21no—
172C4×C251—41no—
173C4×C251—41no—
174C4×C251C23×C411yescharacteristic
175C4×C251C2×#4111yescharacteristic
176Dic251—21no—
177Dic251—21no—
178Dic251—41no—
179Dic251—41no—
180Dic251C4×D412yes—
181C2×C251—42no—
182C2×C251C2×C4×D411yescharacteristic
183C2×C251#57611yescharacteristic
184C2×C251#57711yescharacteristic
185D251—22no—
186D251—42no—
187D251—42no—
188C251#294611yescharacteristic, complemented, Sylow
189#2946—2511nomaximal, Sylow
190C2×AGL(1,ℤ/8)—2511no—
191#520—2511no—
192#518—2511no—
193#442—2511no—
194#442—2511no—
195#440—2511no—
196#440—2511no—
197C4×SD16—2511no—
198C4×SD16D25111yescharacteristic, complemented
199C4×D8—2511no—
200C4×D8—2511no—
201#428—2511no—
202#416—2511no—
203C2×#102—2511no—
204#405—2511no—
205C2×#41—2511no—
206C22×D4—2511no—
207AGL(1,ℤ/8)—2518no—
208C2×SD16—2511no—
209C2×SD16C2×D25111yescharacteristic
210C2×D8—2511no—
211C2×D8—2511no—
212C2×M16—2511no—
213C22×C8—2511no—
214C4×Q8—2511no—
215C4×Q8C2×D25111yescharacteristic, complemented
216C4×D4—5021no—
217C4×D4—5021no—
218C4×D4—5021no—
219C4×D4—5021no—
220C4×D4—2511no—
221C4×D4—2511no—
222C4×D4—2511no—
223C4×D4—2511no—
224C4×D4—2511no—
225C4×D4C2×D25111yescharacteristic
226#117—5021no—
227#117—2511no—
228#117—2511no—
229C2×C4⋊C4—5021no—
230C2×C4⋊C4—2511no—
231C2×C22⋊C4—5021no—
232C2×C22⋊C4—5021no—
233#107—2511no—
234#107—2511no—
235#106—2511no—
236#106C2×D25111yescharacteristic, complemented
237#103—2511no—
238#103C2×D25111yescharacteristic, complemented
239#102—2511no—
240#102—2511no—
241#102—2511no—
242#102—2511no—
243#102—2511no—
244#102C2×D25111yescharacteristic
245#97—2511no—
246#97—2511no—
247C4×C8—2511no—
248C4×C8C2×D25111yescharacteristic, complemented
249C24—5021no—
250#41—5022no—
251#41—2514no—
252C2×Q8C22×D25111yescharacteristic
253C2×D4—5021no—
254C2×D4—5022no—
255C2×D4—5021no—
256C2×D4—2514no—
257C2×D4—2511no—
258C2×D4—2511no—
259C2×D4C22×D25111yescharacteristic
260C22×C4—21no—
261C22×C4—5021no—
262C22×C4—5021no—
263C22×C4—5021no—
264C22×C4—5021no—
265C22×C4—5021no—
266C22×C4—5021no—
267C22×C4—5021no—
268C22×C4—2511no—
269SD16—2514no—
270SD16C4×D25114yescomplemented
271D8—2514no—
272D8—2514no—
273M16—2514no—
274C2×C8—5022no—
275C2×C8—2511no—
276C2×C8—2511no—
277C2×C8C22×D25111yescharacteristic
278C2×C8C22×D25111yescharacteristic
279C4⋊C4—21no—
280C4⋊C4—5021no—
281C4⋊C4—5021no—
282C4⋊C4—2511no—
283C4⋊C4—2511no—
284C4⋊C4—2511no—
285C4⋊C4—2511no—
286C4⋊C4C22×D25111yescharacteristic
287C4⋊C4C22×D25111yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—5021no—
290C22⋊C4—5021no—
291C22⋊C4—5021no—
292C22⋊C4—5021no—
293C22⋊C4—5021no—
294C22⋊C4—5021no—
295C22⋊C4—2511no—
296C22⋊C4—2511no—
297C42—21no—
298C42—5021no—
299C42—5021no—
300C42—2511no—
301C42C22×D25111yescharacteristic
302C23—21no—
303C23—20081no—
304C23—20081no—
305C23—5024no—
306C23—5021no—
307C23—5021no—
308C23—2511no—
309Q8—21no—
310Q8C2×C4×D25112yes—
311D4—21no—
312D4—5024no—
313D4—5024no—
314D4—5021no—
315D4—5021no—
316D4—2512no—
317D4—2512no—
318D4C2×C4×D25112yes—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—81no—
323C2×C4—20081no—
324C2×C4—20081no—
325C2×C4—20081no—
326C2×C4—20081no—
327C2×C4—20081no—
328C2×C4—20081no—
329C2×C4—5024no—
330C2×C4—5021no—
331C2×C4—5021no—
332C2×C4—2511no—
333C2×C4—2511no—
334C2×C4—2511no—
335C2×C4—2512no—
336C2×C4—2512no—
337C2×C4D4×D25111yescharacteristic
338C2×C4D4×D25111yescharacteristic
339C2×C4C23×D25111yescharacteristic
340C8—21no—
341C8—5021no—
342C8—2512no—
343C8C2×C4×D25112yes—
344C22—22no—
345C22—41no—
346C22—41no—
347C22—20084no—
348C22—10041no—
349C22—10041no—
350C22—10041no—
351C22—10041no—
352C22—5022no—
353C22—5022no—
354C22—5021no—
355C22—5021no—
356C22—2512no—
357C22C2×D4×D25111yescharacteristic
358C4—21no—
359C4—21no—
360C4—22no—
361C4—41no—
362C4—41no—
363C4—10041no—
364C4—10041no—
365C4—5021no—
366C4—5021no—
367C4—2512no—
368C4C22×C4×D25111yescharacteristic
369C4#41×D25111yescharacteristic
370C2—42no—
371C2—10042no—
372C2—10042no—
373C2—5022no—
374C2C4×D4×D25111yescharacteristic
375C2#3063248911yescharacteristic
376C2#3063249311yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutatorC4×C251C23×C4
FrattiniC2×C4C23×D251
FittingC4×SD16×C251C2
socleC22×C251C22×D4
prime radicalC2×#22548742C2
cosocleC2×C4×C251C24
hypercenterC4×SD16D251
hypocenterC251#2946
LCSG ▷ C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
2. #30632169C2semidirect
3. #30632159C2semidirect
4. #30631798C2semidirect
5. #30631797C2semidirect
6. #30631788C2semidirect
7. #30631785C2semidirect
8. #30631772C2semidirect
9. C4×D2008C2semidirect
10. #30631767C2semidirect
11. #30631726C2semidirect
12. #30631682C2semidirect
13. #102×D251C2semidirect
14. #30631641C2semidirect
15. C4×SD16×C251C2semidirect
18. #22548742C4semidirect
24. Q8×Dic251C22semidirect
27. C4×D1004C22semidirect
31. #22548677C22semidirect
41. #22548649C22semidirect
42. #22548648C22semidirect
43. #22548646C22semidirect
44. #22548639C22semidirect
46. #22548637C22semidirect
49. #22548635C22semidirect
50. #22548626C22semidirect
51. #22548625C22semidirect
52. C8×Dic251C22semidirect
54. C4×Q8×C251C22semidirect
56. #106×C251C22semidirect
57. #103×C251C22semidirect
59. C4×C8×C251C22semidirect
62. #16860811C2×C4semidirect
68. D4×D251C2×C4semidirect
77. #16860797C2×C4semidirect
78. D2008C2×C4semidirect
79. #16860795C2×C4semidirect
80. #16860793C2×C4semidirect
104. SD16×C251C2×C4semidirect
188. C251#2946semidirect
198. C4×SD16D251semidirect
215. C4×Q8C2×D251semidirect
236. #106C2×D251semidirect
238. #103C2×D251semidirect
248. C4×C8C2×D251semidirect
270. SD16C4×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487422
#306321692
#306321592
#306317982
#306317972
#306317882
#306317852
#306317722
C4×D20082
#306317672
#306317262
#306316822
#102×D2512
#306316412
C4×SD16×C2512
#2946251