Finite Group Databasebeta

← #44562350

Subgroups — #44562350

Order 32128.

420 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30632346C211yesmaximal, characteristic, complemented
2#30632049C211yesmaximal, characteristic, complemented
3#30632022C211yesmaximal, characteristic, complemented
4#30632000C211yesmaximal, characteristic, complemented
5#30631979C211yesmaximal, characteristic, complemented
6#30631931C211yesmaximal, characteristic, complemented
7#30631921C211yesmaximal, characteristic, complemented
8#30631902C211yesmaximal, characteristic, complemented
9#30631830C211yesmaximal, characteristic, complemented
10#30631825C211yesmaximal, characteristic, complemented
11#30631787C211yesmaximal, characteristic, complemented
12#30631759C211yesmaximal, characteristic, complemented
13#30631686C211yesmaximal, characteristic, complemented
14C2×#22548638C211yesmaximal, characteristic, complemented
15#547×C251C211yesmaximal, characteristic, complemented
16C2×#16860810C2211yescharacteristic
17C22×Dic502C2211yescharacteristic
18C2×#16860798C2211yescharacteristic
19C2×#16860798C2211yescharacteristic
20C2×#16860795C2211yescharacteristic
21C2×#16860794C2211yescharacteristic
22C22×C251⋊C8C2211yescharacteristic
23#22548703C2211yescharacteristic
24#22548701—21no—
25#22548694C2211yescharacteristic
26#22548693—21no—
27#22548689C2211yescharacteristic
28#22548687—21no—
29C4×Dic502C2211yescharacteristic
30(C22⋊D251)⋊C4—21no—
31#22548648C2211yescharacteristic
32#22548648C2211yescharacteristic
33#22548648C2211yescharacteristic
34C4⋊(C251⋊C8)C2211yescharacteristic
35#22548638C2211yescharacteristic
36#22548638C2211yescharacteristic
37#22548638C2211yescharacteristic
38#22548638C2211yescharacteristic
39#22548638C2211yescharacteristic
40#22548638C2211yescharacteristic
41#22548636C2211yescharacteristic
42#22548635C2211yescharacteristic
43#22548635C2211yescharacteristic
44#22548635C2211yescharacteristic
45C22⋊(C251⋊C8)C2211yescharacteristic
46C22⋊(C251⋊C8)C2211yescharacteristic
47#22548626C2211yescharacteristic
48C2×#41×C251—21no—
49C4⋊Q8×C251C2211yescharacteristic
50#123×C251—21no—
51#123×C251—21no—
52C22⋊Q8×C251—21no—
53#121×C251C2211yescharacteristic
54#121×C251C2211yescharacteristic
55C4×D4×C251C2211yescharacteristic
56C4×D4×C251C2211yescharacteristic
57C2×C4⋊C4×C251C2211yescharacteristic
58C2×C4⋊C4×C251C2211yescharacteristic
59#16860810—22no—
60#16860810—41no—
61C2×Dic502—22no—
62C2×Dic502—21no—
63C2×Dic502—41no—
64C2×Dic502C2311yescharacteristic
65C2×Dic502C2311yescharacteristic
66C2×D1004C2311yescharacteristic
67C2×C22⋊D251—21no—
68C2×C22⋊D251—21no—
69C22×Dic251—21no—
70C2×C4×D251—21no—
71#16860798—22no—
72#16860798—41no—
73#16860795—41no—
74#16860794—41no—
75C2×C251⋊C8—41no—
76C2×C251⋊C8C2311yescharacteristic
77C2×C251⋊C8C2311yescharacteristic
78C2×C251⋊C8C2311yescharacteristic
79C2×C251⋊C8C2311yescharacteristic
80C4⋊Dic251C2311yescharacteristic
81Dic251⋊C4—21no—
82Dic251⋊C4—21no—
83C22⋊Dic251—21no—
84(C2×D251)⋊C4—21no—
85(C2×D251)⋊C4—21no—
86(C2×D251)⋊C4—21no—
87C4×Dic251—21no—
88#41×C251—44no—
89C2×Q8×C251—21no—
90C2×D4×C251—21no—
91C2×D4×C251—21no—
92C2×D4×C251D411yescharacteristic
93C2×D4×C251C2311yescharacteristic
94C22×C4×C251—21no—
95C22×C4×C251—21no—
96C22×C4×C251—21no—
97C22×C4×C251C2311yescharacteristic
98C22×C4×C251C2311yescharacteristic
99C4⋊C4×C251—21no—
100C4⋊C4×C251—21no—
101C4⋊C4×C251—21no—
102C4⋊C4×C251D411yescharacteristic
103C4⋊C4×C251C2311yescharacteristic
104C4⋊C4×C251C2311yescharacteristic
105C4⋊C4×C251C2311yescharacteristic
106C22⋊C4×C251—21no—
107C22⋊C4×C251—21no—
108C22⋊C4×C251—21no—
109C22⋊C4×C251D411yescharacteristic
110C22⋊C4×C251D411yescharacteristic
111C42×C251C2311yescharacteristic
112C22×D251—21no—
113Dic502—21no—
114Dic502—21no—
115Dic502—21no—
116Dic502—42no—
117Dic502#4112yescomplemented
118D1004—21no—
119D1004—21no—
120C22⋊D251—42no—
121C22⋊D251—42no—
122C2×Dic251—21no—
123C2×Dic251—21no—
124C2×Dic251—21no—
125C2×Dic251—42no—
126C4×D251—42no—
127C251⋊C8—21no—
128C251⋊C8—21no—
129C251⋊C8—21no—
130C251⋊C8—21no—
131C23×C251—21no—
132C23×C251C2×D411yescharacteristic
133C23×C251C2×D411yescharacteristic
134Q8×C251—42no—
135D4×C251—22no—
136D4×C251—21no—
137D4×C251—21no—
138D4×C251—42no—
139D4×C251—42no—
140C2×C4×C251—21no—
141C2×C4×C251—21no—
142C2×C4×C251—21no—
143C2×C4×C251—21no—
144C2×C4×C251—21no—
145C2×C4×C251—21no—
146C2×C4×C251—21no—
147C2×C4×C251—42no—
148C2×C4×C251—42no—
149C2×C4×C251—42no—
150C2×C4×C251C2×D411yescharacteristic
151C2×C4×C251C2×D411yescharacteristic
152C2×C4×C251C2×D411yescharacteristic
153C2×C4×C251C2×D411yescharacteristic
154C2×C4×C251C2411yescharacteristic
155C22×C251—21no—
156C22×C251—21no—
157C22×C251—21no—
158C22×C251—21no—
159C22×C251—21no—
160C22×C251—41no—
161C22×C251—41no—
162C22×C251—41no—
163C22×C251AGL(1,ℤ/8)12yes—
164C22×C251C22×D411yescharacteristic
165C2×D251—41no—
166C2×D251—41no—
167C2×D251—41no—
168C4×C251—21no—
169C4×C251—21no—
170C4×C251—21no—
171C4×C251—21no—
172C4×C251—41no—
173C4×C251—41no—
174C4×C251—41no—
175C4×C251C2×#4111yescharacteristic
176C4×C251H(5,𝔽2)11yescharacteristic
177Dic251—22no—
178Dic251—41no—
179Dic251—41no—
180C2×C251—22no—
181C2×C251—41no—
182C2×C251—81no—
183C2×C251#54611yescharacteristic
184C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
185C2×C251#57811yescharacteristic
186D251—81no—
187C251#330811yescharacteristic, complemented, Sylow
188#3308—2511nomaximal, Sylow
189#548—2511no—
190#547D25111yescharacteristic, complemented
191#489—2511no—
192#484—2511no—
193#480—2511no—
194#477—2511no—
195#468—2511no—
196#466—2511no—
197#463—2511no—
198#449—2511no—
199#448—2511no—
200#440—2511no—
201#435—2511no—
202#417—2511no—
203C2×#103—2511no—
204C2×#41—21no—
205C2×#41—2511no—
206C22×Q8—2511no—
207C2×SD16—2511no—
208C2×SD16—2511no—
209C2×D8—2511no—
210C2×M16—2511no—
211C22×C8—2511no—
212C4⋊Q8C2×D25111yescharacteristic
213#124—5021no—
214#124—2511no—
215#123—21no—
216#123—21no—
217C22⋊Q8—21no—
218C22⋊Q8—5021no—
219C22⋊Q8—2511no—
220#121—5021no—
221#121—2511no—
222#121C2×D25111yescharacteristic
223#121C2×D25111yescharacteristic
224C4×Q8—2511no—
225C4×D4—5021no—
226C4×D4C2×D25111yescharacteristic
227C4×D4C2×D25111yescharacteristic
228C2×C4⋊C4C2×D25111yescharacteristic
229C2×C4⋊C4C2×D25111yescharacteristic
230#107—2511no—
231#107—2511no—
232#107—2511no—
233C4⋊C8—2511no—
234#103—2511no—
235#103—2511no—
236#103—2511no—
237#103—2511no—
238#103—2511no—
239#103—2511no—
240#102—2511no—
241#102—2511no—
242#102—2511no—
243#102—2511no—
244C22⋊C8—2511no—
245C22⋊C8—2511no—
246#97—2511no—
247#41—44no—
248#41—10041no—
249#41—5022no—
250C2×Q8—21no—
251C2×Q8—10041no—
252C2×Q8—5022no—
253C2×Q8—5021no—
254C2×Q8—2511no—
255C2×Q8—2511no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—5021no—
259C2×D4—5021no—
260C2×D4—2511no—
261C2×D4C22⋊D25111yescharacteristic
262C2×D4C22×D25111yescharacteristic
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—5021no—
267C22×C4—5021no—
268C22×C4C22×D25111yescharacteristic
269C22×C4C22×D25111yescharacteristic
270SD16—10041no—
271SD16—5022no—
272D8—10041no—
273M16—10041no—
274C2×C8—10041no—
275C2×C8—2511no—
276C2×C8—2511no—
277C2×C8—2511no—
278C2×C8—2511no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—5021no—
283C4⋊C4—5021no—
284C4⋊C4—2511no—
285C4⋊C4C22⋊D25111yescharacteristic
286C4⋊C4C22×D25111yescharacteristic
287C4⋊C4C22×D25111yescharacteristic
288C4⋊C4C22×D25111yescharacteristic
289C22⋊C4—21no—
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—5021no—
293C22⋊C4—5021no—
294C22⋊C4—5021no—
295C22⋊C4—5021no—
296C22⋊C4C22⋊D25111yescharacteristic
297C22⋊C4C22⋊D25111yescharacteristic
298C42—5021no—
299C42C22×D25111yescharacteristic
300C23—21no—
301C23—5021no—
302C23C2×C22⋊D25111yescharacteristic
303C23C2×C22⋊D25111yescharacteristic
304Q8—42no—
305Q8—10042no—
306Q8—5021no—
307Q8—5021no—
308Q8—5021no—
309Q8—2512no—
310D4—22no—
311D4—21no—
312D4—21no—
313D4—42no—
314D4—42no—
315D4—10042no—
316D4—10042no—
317D4—5021no—
318D4—5021no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—10042no—
330C2×C4—10042no—
331C2×C4—5021no—
332C2×C4—5021no—
333C2×C4—5021no—
334C2×C4C2×C22⋊D25111yescharacteristic
335C2×C4C2×C22⋊D25111yescharacteristic
336C2×C4C2×C22⋊D25111yescharacteristic
337C2×C4C2×C22⋊D25111yescharacteristic
338C2×C4C23×D25111yescharacteristic
339C8—5021no—
340C8—5021no—
341C8—5021no—
342C8—5021no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—10041no—
352C22—10041no—
353C22—10041no—
354C22#2254874112yes—
355C22C22×C22⋊D25111yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—41no—
361C4—41no—
362C4—41no—
363C4—10041no—
364C4—10041no—
365C4—5022no—
366C4#41×D25111yescharacteristic
367C4#2254876111yescharacteristic
368C2—22no—
369C2—41no—
370C2—81no—
371C2—20081no—
372C2#3063232511yescharacteristic
373C2C2×#2254874111yescharacteristic
374C2#3063250211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#547×C251C2
socleC22×C251C22×D4
hypercenter#547D251
hypocenterC251#3308
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. #30632346C2semidirect
2. #30632049C2semidirect
3. #30632022C2semidirect
4. #30632000C2semidirect
5. #30631979C2semidirect
6. #30631931C2semidirect
7. #30631921C2semidirect
8. #30631902C2semidirect
9. #30631830C2semidirect
10. #30631825C2semidirect
11. #30631787C2semidirect
12. #30631759C2semidirect
13. #30631686C2semidirect
14. C2×#22548638C2semidirect
15. #547×C251C2semidirect
117. Dic502#41semidirect
187. C251#3308semidirect
190. #547D251semidirect

Maximal subgroups

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

subgroupindex
#306323462
#306320492
#306320222
#306320002
#306319792
#306319312
#306319212
#306319022
#306318302
#306318252
#306317872
#306317592
#306316862
C2×#225486382
#547×C2512
#3308251