Finite Group Databasebeta

← #44560316

Subgroups — #44560316

Order 32128.

418 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
1#30632253C211yesmaximal, characteristic
2#30632168C211yesmaximal, characteristic, complemented
3#30632014C211yesmaximal, characteristic, complemented
4#30632012C211yesmaximal, characteristic, complemented
5#30632009C211yesmaximal, characteristic, complemented
6#30632008C211yesmaximal, characteristic, complemented
7#30632005C211yesmaximal, characteristic, complemented
8#30632004C211yesmaximal, characteristic, complemented
9#30632002C211yesmaximal, characteristic, complemented
10#30631728C211yesmaximal, characteristic, complemented
11#30631724C211yesmaximal, characteristic, complemented
12#30631690C211yesmaximal, characteristic, complemented
13#102×D251C211yesmaximal, characteristic, complemented
14#30631652C211yesmaximal, characteristic, complemented
15#480×C251C211yesmaximal, characteristic, complemented
16C2×#16860810C2211yescharacteristic
17C2×D4×D251C2211yescharacteristic
18C2×#16860793C2211yescharacteristic
19C2×C8×D251C2211yescharacteristic
20#22548711—21no—
21#22548701—21no—
22#22548690C2211yescharacteristic
23#22548689C2211yescharacteristic
24#22548686C2211yescharacteristic
25#22548685—21no—
26#22548683—21no—
27#22548680—21no—
28#22548677—21no—
29C4×C22⋊D251—21no—
30C4×C22⋊D251—21no—
31#22548674C2211yescharacteristic, complemented
32#22548673C2211yescharacteristic
33#22548673C2211yescharacteristic
34#22548673C2211yescharacteristic, complemented
35C4⋊C4×D251C2211yescharacteristic
36C4⋊C4×D251C2211yescharacteristic, complemented
37#22548649C2211yescharacteristic, complemented
38#22548648C2211yescharacteristic, complemented
39#22548647C2211yescharacteristic, complemented
40#22548646C2211yescharacteristic, complemented
41#22548646C2211yescharacteristic, complemented
42#22548646C2211yescharacteristic, complemented
43#22548637C2211yescharacteristic
44#22548637C2211yescharacteristic
45#22548636C2211yescharacteristic
46#22548636C2211yescharacteristic
47#22548635C2211yescharacteristic
48#22548635C2211yescharacteristic
49C22⋊(C251⋊C8)C2211yescharacteristic, complemented
50(C2×D251)⋊C8C2211yescharacteristic, complemented
51(C2×D251)⋊C8C2211yescharacteristic, complemented
52#121×C251C2211yescharacteristic
53C2×C4⋊C4×C251C2211yescharacteristic, complemented
54#107×C251C2211yescharacteristic, complemented
55#107×C251C2211yescharacteristic, complemented
56#102×C251C2211yescharacteristic
57#102×C251C2211yescharacteristic
58C22⋊C8×C251C2211yescharacteristic, complemented
59C23×D251—21no—
60#16860810—22no—
61#16860810—22no—
62C2×Dic502D411yescharacteristic
63C2×D1004—21no—
64C2×D1004D411yescharacteristic
65C2×D1004C2311yescharacteristic
66C2×C22⋊D251—21no—
67C2×C22⋊D251—21no—
68C2×C22⋊D251—21no—
69C2×C22⋊D251D411yescharacteristic
70C2×C22⋊D251D411yescharacteristic
71D4×D251—22no—
72D4×D251—22no—
73C2×C4×D251—21no—
74C2×C4×D251—21no—
75C2×C4×D251C2311yescharacteristic
76C2×C4×D251C2311yescharacteristic
77#16860793—22no—
78C2×C251⋊C8C2311yescharacteristic
79C2×C251⋊C8C2311yescharacteristic
80C8×D251—22no—
81C4⋊Dic251C2311yescharacteristic
82C4⋊Dic251C2311yescharacteristic
83C4⋊Dic251C2311yescharacteristic
84Dic251⋊C4—21no—
85Dic251⋊C4—21no—
86C22⋊Dic251—21no—
87C22⋊Dic251—21no—
88(C2×D251)⋊C4—21no—
89(C2×D251)⋊C4—21no—
90(C2×D251)⋊C4—21no—
91(C2×D251)⋊C4—21no—
92(C2×D251)⋊C4—21no—
93C4×Dic251—21no—
94C4×Dic251—21no—
95C4×Dic251—21no—
96C4×Dic251—21no—
97C2×D4×C251—21no—
98C2×D4×C251C2311yescharacteristic
99C22×C4×C251—21no—
100C22×C4×C251C2311yescharacteristic
101C2×C8×C251C2311yescharacteristic
102C2×C8×C251C2311yescharacteristic
103C4⋊C4×C251—21no—
104C4⋊C4×C251C2311yescharacteristic
105C4⋊C4×C251C2311yescharacteristic
106C4⋊C4×C251C2311yescharacteristic
107C22⋊C4×C251—21no—
108C22×D251—21no—
109C22×D251—42no—
110C22×D251—81no—
111C22×D251—81no—
112C22×D251C2×D411yescharacteristic
113C22×D251C2×D411yescharacteristic
114Dic502—22no—
115D1004—22no—
116D1004—21no—
117D1004—21no—
118D1004—81no—
119C22⋊D251—22no—
120C22⋊D251—22no—
121C22⋊D251—42no—
122C22⋊D251—81no—
123C22⋊D251—81no—
124C2×Dic251—21no—
125C2×Dic251—21no—
126C2×Dic251—21no—
127C2×Dic251C2×D411yescharacteristic
128C2×Dic251C2×D411yescharacteristic
129C4×D251—21no—
130C4×D251—21no—
131C4×D251—81no—
132C4×D251—81no—
133C4×D251#4112yes—
134C4×D251#4112yes—
135C251⋊C8—21no—
136C251⋊C8—21no—
137C23×C251—21no—
138C23×C251C2×D411yescharacteristic
139D4×C251—21no—
140D4×C251—21no—
141D4×C251—81no—
142C2×C4×C251—21no—
143C2×C4×C251—21no—
144C2×C4×C251—21no—
145C2×C4×C251—21no—
146C2×C4×C251—21no—
147C2×C4×C251—81no—
148C2×C4×C251C2×D411yescharacteristic
149C2×C4×C251C2411yescharacteristic
150C8×C251—21no—
151C8×C251—21no—
152C22×C251—21no—
153C22×C251—21no—
154C22×C251—21no—
155C22×C251—41no—
156C22×C251—41no—
157C22×C251—41no—
158C22×C251C22×D411yescharacteristic
159C2×D251—21no—
160C2×D251—21no—
161C2×D251—21no—
162C2×D251—21no—
163C2×D251—21no—
164C2×D251—41no—
165C2×D251—41no—
166C2×D251—41no—
167C2×D251—84no—
168C2×D251#12312yes—
169C4×C251—21no—
170C4×C251—41no—
171C4×C251—41no—
172C4×C251—41no—
173C4×C251C2×#4111yescharacteristic
174C4×C251C2×#4111yescharacteristic
175Dic251—21no—
176Dic251—41no—
177Dic251—41no—
178Dic251—41no—
179Dic251#12312yes—
180C2×C251—41no—
181C2×C251—81no—
182C2×C251C2×#12311yescharacteristic
183C2×C251#57611yescharacteristic
184C2×C251#57711yescharacteristic
185D251—22no—
186D251—41no—
187D251—81no—
188C251#309611yescharacteristic, complemented, Sylow
189#3096—2511nomaximal, Sylow
190#535—2511no—
191#520—2511no—
192#482—2511no—
193#482—2511no—
194#482—2511no—
195#482—2511no—
196#481—2511no—
197#481—2511no—
198#480—2511no—
199#480D25111yescharacteristic, complemented
200#428—2511no—
201#427—2511no—
202#418—2511no—
203C2×#102—2511no—
204#408—2511no—
205C2×#41—2511no—
206C22×D4—2511no—
207C2×M16—2511no—
208C22×C8—2511no—
209#127—5021no—
210#124—5021no—
211#121—2511no—
212#121—2511no—
213#121—2511no—
214#121C2×D25111yescharacteristic
215#120—5021no—
216#120—5021no—
217C4×Q8—5021no—
218C4×D4—5021no—
219C4×D4—5021no—
220C4×D4—5021no—
221#117—2511no—
222#117—2511no—
223#117—2511no—
224#117—2511no—
225C2×C4⋊C4—2511no—
226C2×C4⋊C4—2511no—
227C2×C4⋊C4C2×D25111yescharacteristic, complemented
228#107—2511no—
229#107—2511no—
230#107C2×D25111yescharacteristic, complemented
231#107C2×D25111yescharacteristic, complemented
232#106—2511no—
233#106—2511no—
234#106—2511no—
235#106—2511no—
236#102—2511no—
237#102—2511no—
238#102—2511no—
239#102—2511no—
240#102—2511no—
241#102—2511no—
242#102C2×D25111yescharacteristic
243#102C2×D25111yescharacteristic
244C22⋊C8—2511no—
245C22⋊C8—2511no—
246C22⋊C8—2511no—
247C22⋊C8C2×D25111yescharacteristic, complemented
248C24—5021no—
249#41—5022no—
250#41—5022no—
251C2×Q8—2511no—
252C2×D4—21no—
253C2×D4—5021no—
254C2×D4—5021no—
255C2×D4—5021no—
256C2×D4—5022no—
257C2×D4—5022no—
258C2×D4—5021no—
259C2×D4—2511no—
260C2×D4—2511no—
261C2×D4—2511no—
262C2×D4—2511no—
263C2×D4C22×D25111yescharacteristic
264C22×C4—21no—
265C22×C4—5021no—
266C22×C4—5021no—
267C22×C4—2511no—
268C22×C4—2511no—
269C22×C4C22×D25111yescharacteristic
270M16—5022no—
271C2×C8—5022no—
272C2×C8—2511no—
273C2×C8—2511no—
274C2×C8C22×D25111yescharacteristic
275C2×C8C22×D25111yescharacteristic
276C4⋊C4—21no—
277C4⋊C4—5021no—
278C4⋊C4—5021no—
279C4⋊C4—2511no—
280C4⋊C4—2511no—
281C4⋊C4—2511no—
282C4⋊C4C22×D25111yescharacteristic
283C4⋊C4C22×D25111yescharacteristic
284C4⋊C4C22×D25111yescharacteristic
285C22⋊C4—21no—
286C22⋊C4—5021no—
287C22⋊C4—5021no—
288C22⋊C4—5021no—
289C22⋊C4—5021no—
290C22⋊C4—5021no—
291C22⋊C4—5021no—
292C22⋊C4—5021no—
293C42—5021no—
294C42—5021no—
295C42—5021no—
296C42—5021no—
297C23—21no—
298C23—20081no—
299C23—20081no—
300C23—10042no—
301C23—5021no—
302C23—2511no—
303C23—2511no—
304C23D4×D25111yescharacteristic
305Q8—5022no—
306D4—21no—
307D4—21no—
308D4—81no—
309D4—20081no—
310D4—20081no—
311D4—20081no—
312D4—10042no—
313D4—5022no—
314D4—5022no—
315D4—5022no—
316D4—5021no—
317D4—5021no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—81no—
324C2×C4—20081no—
325C2×C4—20081no—
326C2×C4—5021no—
327C2×C4—5021no—
328C2×C4—5021no—
329C2×C4—5021no—
330C2×C4—5021no—
331C2×C4—2511no—
332C2×C4—2511no—
333C2×C4—2512no—
334C2×C4—2512no—
335C2×C4D4×D25111yescharacteristic
336C2×C4C23×D25111yescharacteristic
337C8—21no—
338C8—21no—
339C8—5021no—
340C8—5021no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—20084no—
348C22—10041no—
349C22—10041no—
350C22—10041no—
351C22—5021no—
352C22—5021no—
353C22—5021no—
354C22—5021no—
355C22—5021no—
356C22—2512no—
357C22C2×D4×D25111yescharacteristic
358C4—21no—
359C4—41no—
360C4—41no—
361C4—41no—
362C4—10041no—
363C4—10041no—
364C4—10041no—
365C4—5021no—
366C4—2512no—
367C4#41×D25111yescharacteristic
368C4#41×D25111yescharacteristic
369C2—41no—
370C2—81no—
371C2—20081no—
372C2—10041no—
373C2—5022no—
374C2#123×D25111yescharacteristic
375C2#3063248811yescharacteristic
376C2#3063249611yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#480×C251C2
socleC22×C251C22×D4
prime radical#30632253C2
hypercenter#480D251
hypocenterC251#3096
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
2. #30632168C2semidirect
3. #30632014C2semidirect
4. #30632012C2semidirect
5. #30632009C2semidirect
6. #30632008C2semidirect
7. #30632005C2semidirect
8. #30632004C2semidirect
9. #30632002C2semidirect
10. #30631728C2semidirect
11. #30631724C2semidirect
12. #30631690C2semidirect
13. #102×D251C2semidirect
14. #30631652C2semidirect
15. #480×C251C2semidirect
31. #22548674C22semidirect
34. #22548673C22semidirect
36. C4⋊C4×D251C22semidirect
37. #22548649C22semidirect
38. #22548648C22semidirect
39. #22548647C22semidirect
40. #22548646C22semidirect
41. #22548646C22semidirect
42. #22548646C22semidirect
49. C22⋊(C251⋊C8)C22semidirect
50. (C2×D251)⋊C8C22semidirect
51. (C2×D251)⋊C8C22semidirect
53. C2×C4⋊C4×C251C22semidirect
54. #107×C251C22semidirect
55. #107×C251C22semidirect
58. C22⋊C8×C251C22semidirect
188. C251#3096semidirect
199. #480D251semidirect
227. C2×C4⋊C4C2×D251semidirect
230. #107C2×D251semidirect
231. #107C2×D251semidirect
247. C22⋊C8C2×D251semidirect

Maximal subgroups

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

subgroupindex
#306322532
#306321682
#306320142
#306320122
#306320092
#306320082
#306320052
#306320042
#306320022
#306317282
#306317242
#306316902
#102×D2512
#306316522
#480×C2512
#3096251