Finite Group Databasebeta

← #44560485

Subgroups — #44560485

Order 32128.

424 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30632234C211yesmaximal, characteristic
2C4×Q8×D251C211yesmaximal, characteristic, complemented
3#30632002C211yesmaximal, characteristic, complemented
4#30631999C211yesmaximal, characteristic, complemented
5#30631981C211yesmaximal, characteristic, complemented
6#30631980C211yesmaximal, characteristic, complemented
7#30631886C211yesmaximal, characteristic, complemented
8#30631885C211yesmaximal, characteristic, complemented
9#30631827C211yesmaximal, characteristic, complemented
10#30631824C211yesmaximal, characteristic, complemented
11#30631789C211yesmaximal, characteristic, complemented
12#30631787C211yesmaximal, characteristic, complemented
13#30631785C211yesmaximal, characteristic, complemented
14#30631742C211yesmaximal, characteristic, complemented
15#440×C251C211yesmaximal, characteristic, complemented
16C2×#16860810—21no—
17C2×Q8×D251C2211yescharacteristic
18C2×#16860799C2211yescharacteristic
19C2×#16860798C2211yescharacteristic
20C2×#16860797C2211yescharacteristic
21#22548714C2211yescharacteristic
22#22548711C2211yescharacteristic
23#22548701—21no—
24#22548690C2211yescharacteristic
25#22548688—21no—
26#22548686C2211yescharacteristic
27Q8×Dic251C2211yescharacteristic
28C4×Dic502—21no—
29C4×Dic502C2211yescharacteristic
30#22548680—21no—
31#22548680C2211yescharacteristic
32C4×D1004C2211yescharacteristic
33C4×C22⋊D251—21no—
34C4⋊C4×D251—21no—
35C4⋊C4×D251C2211yescharacteristic
36C42×D251—21no—
37C42×D251C2211yescharacteristic
38#22548649C2211yescharacteristic
39#22548648C2211yescharacteristic
40#22548648C2211yescharacteristic
41Dic251⋊C8C2211yescharacteristic
42Dic251⋊C8C2211yescharacteristic
43#22548640C2211yescharacteristic
44#22548639C2211yescharacteristic
45#22548638C2211yescharacteristic
46#22548637C2211yescharacteristic
47#22548636C2211yescharacteristic
48#22548635C2211yescharacteristic
49(C2×D251)⋊C8C2211yescharacteristic
50(C2×D251)⋊C8C2211yescharacteristic
51#22548626C2211yescharacteristic
52C2×SD16×C251C2211yescharacteristic
53C4×Q8×C251C2211yescharacteristic
54C4×D4×C251C2211yescharacteristic
55#107×C251C2211yescharacteristic
56#103×C251C2211yescharacteristic
57#102×C251C2211yescharacteristic
58#97×C251C2211yescharacteristic
59#16860810—44no—
60C2×Dic502—21no—
61C2×Dic502—21no—
62C2×Dic502C2311yescharacteristic
63Q8×D251—22no—
64Q8×D251—22no—
65C2×D1004—21no—
66C2×D1004C2311yescharacteristic
67C2×C22⋊D251—21no—
68C2×C22⋊D251—21no—
69C2×C22⋊D251—21no—
70C2×C4×D251—21no—
71C2×C4×D251—21no—
72C2×C4×D251—21no—
73C2×C4×D251D411yescharacteristic
74C2×C4×D251D411yescharacteristic
75C2×C4×D251C2311yescharacteristic
76#16860799—22no—
77#16860798—22no—
78#16860797—22no—
79C2×C251⋊C8C2311yescharacteristic
80C2×C251⋊C8C2311yescharacteristic
81C4⋊Dic251—21no—
82C4⋊Dic251C2311yescharacteristic
83C4⋊Dic251C2311yescharacteristic
84Dic251⋊C4—21no—
85Dic251⋊C4—21no—
86Dic251⋊C4—21no—
87Dic251⋊C4—21no—
88C22⋊Dic251—21no—
89(C2×D251)⋊C4—21no—
90(C2×D251)⋊C4—21no—
91C4×Dic251—21no—
92C4×Dic251—21no—
93C4×Dic251—21no—
94C4×Dic251D411yescharacteristic
95C4×Dic251D411yescharacteristic
96C4×Dic251C2311yescharacteristic
97C2×Q8×C251C2311yescharacteristic
98C2×D4×C251C2311yescharacteristic
99C22×C4×C251—21no—
100SD16×C251—22no—
101C2×C8×C251C2311yescharacteristic
102C2×C8×C251C2311yescharacteristic
103C4⋊C4×C251—21no—
104C4⋊C4×C251C2311yescharacteristic
105C4⋊C4×C251C2311yescharacteristic
106C22⋊C4×C251—21no—
107C42×C251—21no—
108C42×C251C2311yescharacteristic
109C22×D251—21no—
110C22×D251C2×D411yescharacteristic
111Dic502—24no—
112Dic502—21no—
113Dic502—42no—
114Dic502#4112yescomplemented
115D1004—21no—
116D1004—21no—
117D1004—42no—
118C22⋊D251—42no—
119C22⋊D251—42no—
120C22⋊D251—42no—
121C2×Dic251—21no—
122C2×Dic251—21no—
123C2×Dic251—21no—
124C2×Dic251C2×D411yescharacteristic
125C2×Dic251C2×D411yescharacteristic
126C2×Dic251C2×D411yescharacteristic
127C4×D251—24no—
128C4×D251—22no—
129C4×D251—21no—
130C4×D251—21no—
131C4×D251—22no—
132C4×D251—42no—
133C4×D251—42no—
134C251⋊C8—21no—
135C251⋊C8—21no—
136C23×C251—21no—
137Q8×C251—21no—
138Q8×C251#4112yescomplemented
139D4×C251—21no—
140D4×C251—21no—
141C2×C4×C251—21no—
142C2×C4×C251—21no—
143C2×C4×C251—21no—
144C2×C4×C251—42no—
145C2×C4×C251C2×D411yescharacteristic
146C2×C4×C251C2×D411yescharacteristic
147C2×C4×C251C2411yescharacteristic
148C8×C251—21no—
149C8×C251—21no—
150C22×C251—41no—
151C22×C251—41no—
152C22×C251—41no—
153C22×C251C22×D411yescharacteristic
154C2×D251—21no—
155C2×D251—21no—
156C2×D251—41no—
157C2×D251—41no—
158C2×D251—41no—
159C2×D251AGL(1,ℤ/8)12yes—
160C4×C251—22no—
161C4×C251—21no—
162C4×C251—21no—
163C4×C251—41no—
164C4×C251—41no—
165C4×C251C2×#4111yescharacteristic
166C4×C251C2×#4111yescharacteristic
167Dic251—22no—
168Dic251—21no—
169Dic251—21no—
170Dic251—41no—
171Dic251—41no—
172Dic251#13712yes—
173C2×C251—81no—
174C2×C251#52511yescharacteristic
175C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
176C2×C251C2×#13711yescharacteristic
177D251—22no—
178D251—81no—
179C251#311211yescharacteristic, complemented, Sylow
180#3112—2511nomaximal, Sylow
181#532—2511no—
182C2×C4×Q8—2511no—
183#480—2511no—
184#480—2511no—
185#477—2511no—
186#477—2511no—
187#460—2511no—
188#460—2511no—
189#448—2511no—
190#448—2511no—
191#440—2511no—
192#440—2511no—
193#440—2511no—
194#440D25111yescharacteristic, complemented
195#432—2511no—
196C2×#41—5021no—
197C22×Q8—2511no—
198C2×SD16—2511no—
199C2×SD16—2511no—
200C2×SD16—2511no—
201C2×SD16C2×D25111yescharacteristic
202C4⋊Q8—2511no—
203#127—2511no—
204#124—5021no—
205#121—5021no—
206#121—2511no—
207#121—2511no—
208C4×Q8—5021no—
209C4×Q8—5021no—
210C4×Q8—2511no—
211C4×Q8—2511no—
212C4×Q8—2511no—
213C4×Q8C2×D25111yescharacteristic
214C4×D4—5021no—
215C4×D4—2511no—
216C4×D4C2×D25111yescharacteristic
217C2×C4⋊C4—5021no—
218C2×C4⋊C4—2511no—
219C2×C42—5021no—
220C2×C42—2511no—
221#107—2511no—
222#107—2511no—
223#107—2511no—
224#107C2×D25111yescharacteristic
225C4⋊C8—2511no—
226C4⋊C8—2511no—
227#103—2511no—
228#103—2511no—
229#103—2511no—
230#103C2×D25111yescharacteristic
231#102—2511no—
232#102—2511no—
233#102—2511no—
234#102C2×D25111yescharacteristic
235C22⋊C8—2511no—
236C22⋊C8—2511no—
237#97—2511no—
238#97C2×D25111yescharacteristic
239#41—10044no—
240C2×Q8—5021no—
241C2×Q8—5021no—
242C2×Q8—5022no—
243C2×Q8—5022no—
244C2×Q8—2511no—
245C2×Q8C22×D25111yescharacteristic
246C2×D4—5021no—
247C2×D4—5021no—
248C2×D4—5021no—
249C2×D4—5021no—
250C2×D4—2511no—
251C2×D4C22×D25111yescharacteristic
252C22×C4—21no—
253C22×C4—5021no—
254C22×C4—5021no—
255C22×C4—5021no—
256C22×C4—2511no—
257C22×C4—2511no—
258C22×C4—2511no—
259SD16—22no—
260SD16—5022no—
261SD16—5022no—
262SD16—5022no—
263C2×C8—2511no—
264C2×C8—2511no—
265C2×C8C22×D25111yescharacteristic
266C2×C8C22×D25111yescharacteristic
267C4⋊C4—21no—
268C4⋊C4—5021no—
269C4⋊C4—5021no—
270C4⋊C4—5021no—
271C4⋊C4—5021no—
272C4⋊C4—5021no—
273C4⋊C4—2511no—
274C4⋊C4—2511no—
275C4⋊C4C22×D25111yescharacteristic
276C4⋊C4C22×D25111yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—5021no—
279C22⋊C4—5021no—
280C22⋊C4—5021no—
281C42—21no—
282C42—5021no—
283C42—5021no—
284C42—5021no—
285C42—2511no—
286C42—2511no—
287C42—2511no—
288C42C22×D25111yescharacteristic
289C23—21no—
290C23—5021no—
291C23—2511no—
292Q8—21no—
293Q8—10042no—
294Q8—5024no—
295Q8—5021no—
296Q8—2512no—
297Q8#1686081012yescomplemented
298D4—21no—
299D4—21no—
300D4—10042no—
301D4—10042no—
302D4—10042no—
303D4—10042no—
304D4—5021no—
305D4—5021no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—42no—
310C2×C4—10042no—
311C2×C4—10042no—
312C2×C4—5024no—
313C2×C4—5022no—
314C2×C4—5022no—
315C2×C4—5021no—
316C2×C4—5021no—
317C2×C4—5021no—
318C2×C4—5021no—
319C2×C4—5021no—
320C2×C4—2511no—
321C2×C4—2511no—
322C2×C4—2511no—
323C2×C4D4×D25111yescharacteristic
324C2×C4D4×D25111yescharacteristic
325C2×C4C23×D25111yescharacteristic
326C8—21no—
327C8—21no—
328C8—5021no—
329C8—5021no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—10041no—
334C22—10041no—
335C22—10041no—
336C22—5021no—
337C22—5021no—
338C22—2512no—
339C22C2×D4×D25111yescharacteristic
340C4—22no—
341C4—21no—
342C4—21no—
343C4—41no—
344C4—41no—
345C4—10041no—
346C4—10041no—
347C4—5022no—
348C4—5021no—
349C4—5021no—
350C4—2512no—
351C4#41×D25111yescharacteristic
352C4C2×#1686081011yescharacteristic
353C2—81no—
354C2—20081no—
355C2—5022no—
356C2#3063219611yescharacteristic
357C2AGL(1,ℤ/8)×D25111yescharacteristic
358C2#137×D25111yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#440×C251C2
socleC22×C251C22×D4
prime radical#30632234C2
hypercenter#440D251
hypocenterC251#3112
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
2. C4×Q8×D251C2semidirect
3. #30632002C2semidirect
4. #30631999C2semidirect
5. #30631981C2semidirect
6. #30631980C2semidirect
7. #30631886C2semidirect
8. #30631885C2semidirect
9. #30631827C2semidirect
10. #30631824C2semidirect
11. #30631789C2semidirect
12. #30631787C2semidirect
13. #30631785C2semidirect
14. #30631742C2semidirect
15. #440×C251C2semidirect
114. Dic502#41semidirect
138. Q8×C251#41semidirect
179. C251#3112semidirect
194. #440D251semidirect
297. Q8#16860810semidirect

Maximal subgroups

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

subgroupindex
#306322342
C4×Q8×D2512
#306320022
#306319992
#306319812
#306319802
#306318862
#306318852
#306318272
#306318242
#306317892
#306317872
#306317852
#306317422
#440×C2512
#3112251