Finite Group Databasebeta

← #44560558

Subgroups — #44560558

Order 32128.

396 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30632167C211yesmaximal, characteristic, complemented
2#30632003C211yesmaximal, characteristic, complemented
3#30631998C211yesmaximal, characteristic, complemented
4#30631991C211yesmaximal, characteristic, complemented
5#30631984C211yesmaximal, characteristic, complemented
6#30631884C211yesmaximal, characteristic, complemented
7#30631879C211yesmaximal, characteristic, complemented
8#30631830C211yesmaximal, characteristic, complemented
9#30631830C211yesmaximal, characteristic, complemented
10#30631797C211yesmaximal, characteristic, complemented
11#30631787C211yesmaximal, characteristic, complemented
12C4×#16860798C211yesmaximal, characteristic, complemented
13C4×#16860795C211yesmaximal, characteristic, complemented
14#30631748C211yesmaximal, characteristic, complemented
15#540×C251C211yesmaximal, characteristic, complemented
16C2×#16860810C2211yescharacteristic
17C2×#16860798C2211yescharacteristic
18C2×#16860798C2211yescharacteristic
19C2×#16860795C2211yescharacteristic
20C2×#16860795C2211yescharacteristic
21C4×Dic502C2211yescharacteristic
22C4×Dic502C2211yescharacteristic
23C4×D1004C2211yescharacteristic
24C4×D1004C2211yescharacteristic
25(C22⋊D251)⋊C4—21no—
26(C22⋊D251)⋊C4—21no—
27#22548673—21no—
28C2×C4⋊Dic251C2211yescharacteristic
29C4⋊C4×D251—21no—
30#22548648C2211yescharacteristic
31#22548648C2211yescharacteristic
32#22548646C2211yescharacteristic
33#22548646C2211yescharacteristic
34C4⋊(C251⋊C8)C2211yescharacteristic
35C4⋊(C251⋊C8)C2211yescharacteristic
36#22548638C2211yescharacteristic
37#22548638C2211yescharacteristic
38#22548636C2211yescharacteristic
39#22548636C2211yescharacteristic
40#22548636C2211yescharacteristic
41#22548636C2211yescharacteristic
42#22548635C2211yescharacteristic
43#22548635C2211yescharacteristic
44C22⋊(C251⋊C8)C2211yescharacteristic
45C22⋊(C251⋊C8)C2211yescharacteristic
46#22548626C2211yescharacteristic
47C4×C251⋊C8C2211yescharacteristic
48#127×C251C2211yescharacteristic
49#125×C251C2211yescharacteristic
50#123×C251—21no—
51#123×C251—21no—
52#121×C251—21no—
53#121×C251—21no—
54#121×C251C2211yescharacteristic
55#121×C251C2211yescharacteristic
56C4×D4×C251C2211yescharacteristic
57C4×D4×C251C2211yescharacteristic
58#117×C251C2211yescharacteristic
59#16860810—22no—
60#16860810—22no—
61C2×Dic502C2311yescharacteristic
62C2×D1004C2311yescharacteristic
63C2×C22⋊D251—21no—
64C22×Dic251—21no—
65C2×C4×D251—21no—
66C2×C4×D251—21no—
67C2×C4×D251—21no—
68#16860798—22no—
69#16860798—22no—
70#16860795—22no—
71#16860795—22no—
72C2×C251⋊C8C2311yescharacteristic
73C2×C251⋊C8C2311yescharacteristic
74C2×C251⋊C8C2311yescharacteristic
75C2×C251⋊C8C2311yescharacteristic
76C4⋊Dic251—21no—
77C4⋊Dic251C2311yescharacteristic
78C4⋊Dic251C2311yescharacteristic
79Dic251⋊C4—21no—
80Dic251⋊C4—21no—
81(C2×D251)⋊C4—21no—
82(C2×D251)⋊C4—21no—
83C4×Dic251—21no—
84C4×Dic251—21no—
85C2×D4×C251—21no—
86C2×D4×C251—21no—
87C2×D4×C251—21no—
88C2×D4×C251—21no—
89C2×D4×C251C2311yescharacteristic
90C2×D4×C251C2311yescharacteristic
91C22×C4×C251—21no—
92C22×C4×C251—21no—
93C22×C4×C251C2311yescharacteristic
94C4⋊C4×C251—21no—
95C4⋊C4×C251—21no—
96C4⋊C4×C251D411yescharacteristic
97C4⋊C4×C251D411yescharacteristic
98C4⋊C4×C251C2311yescharacteristic
99C4⋊C4×C251C2311yescharacteristic
100C22⋊C4×C251—21no—
101C22⋊C4×C251—21no—
102C22⋊C4×C251—21no—
103C22⋊C4×C251—21no—
104C22⋊C4×C251D411yescharacteristic
105C22⋊C4×C251D411yescharacteristic
106C42×C251C2311yescharacteristic
107C42×C251C2311yescharacteristic
108C22×D251—21no—
109Dic502—21no—
110Dic502#4112yes—
111D1004—21no—
112D1004#4112yes—
113C22⋊D251—24no—
114C2×Dic251—21no—
115C2×Dic251—21no—
116C2×Dic251—21no—
117C2×Dic251—81no—
118C4×D251—24no—
119C4×D251—81no—
120C4×D251—81no—
121C251⋊C8—21no—
122C251⋊C8—21no—
123C251⋊C8—21no—
124C251⋊C8—21no—
125C23×C251—21no—
126C23×C251—21no—
127C23×C251C2×D411yescharacteristic
128D4×C251—21no—
129D4×C251—21no—
130D4×C251—21no—
131D4×C251—21no—
132D4×C251—81no—
133D4×C251—81no—
134D4×C251—81no—
135D4×C251—81no—
136C2×C4×C251—21no—
137C2×C4×C251—21no—
138C2×C4×C251—21no—
139C2×C4×C251—21no—
140C2×C4×C251—81no—
141C2×C4×C251—81no—
142C2×C4×C251C2×D411yescharacteristic
143C2×C4×C251C2×D411yescharacteristic
144C2×C4×C251C2×D411yescharacteristic
145C2×C4×C251C2×D411yescharacteristic
146C2×C4×C251C2×D411yescharacteristic
147C2×C4×C251C2411yescharacteristic
148C22×C251—21no—
149C22×C251—21no—
150C22×C251—21no—
151C22×C251—41no—
152C22×C251—41no—
153C22×C251—41no—
154C22×C251—41no—
155C22×C251—41no—
156C22×C251—41no—
157C22×C251C22×D411yescharacteristic
158C2×D251—22no—
159C2×D251—41no—
160C2×D251—41no—
161C4×C251—21no—
162C4×C251—21no—
163C4×C251—21no—
164C4×C251—21no—
165C4×C251—21no—
166C4×C251—41no—
167C4×C251—41no—
168C4×C251C2×#4111yescharacteristic
169C4×C251C2×#4111yescharacteristic
170Dic251—22no—
171Dic251—41no—
172Dic251—41no—
173C2×C251—41no—
174C2×C251—81no—
175C2×C251—81no—
176C2×C251#52511yescharacteristic
177C2×C251#57611yescharacteristic
178C2×C251#57711yescharacteristic
179D251—42no—
180C251#311911yescharacteristic, complemented, Sylow
181#3119—2511nomaximal, Sylow
182#540D25111yescharacteristic, complemented
183#520—2511no—
184#481—2511no—
185#480—2511no—
186#479—2511no—
187#478—2511no—
188#460—2511no—
189#459—2511no—
190#449—2511no—
191#449—2511no—
192#442—2511no—
193#440—2511no—
194C4×SD16—2511no—
195C4×D8—2511no—
196#433—2511no—
197C2×#41—2511no—
198C2×SD16—2511no—
199C2×SD16—2511no—
200C2×D8—2511no—
201C2×D8—2511no—
202#127C2×D25111yescharacteristic
203#125C2×D25111yescharacteristic
204#123—21no—
205#123—21no—
206#121—21no—
207#121—21no—
208#121C2×D25111yescharacteristic
209#121C2×D25111yescharacteristic
210C4×Q8—2511no—
211C4×Q8—2511no—
212C4×D4—5021no—
213C4×D4—5021no—
214C4×D4—2511no—
215C4×D4—2511no—
216C4×D4C2×D25111yescharacteristic
217C4×D4C2×D25111yescharacteristic
218#117—5021no—
219#117C2×D25111yescharacteristic
220C2×C4⋊C4—5021no—
221C2×C4⋊C4—2511no—
222#107—2511no—
223#107—2511no—
224#106—2511no—
225#106—2511no—
226C4⋊C8—2511no—
227C4⋊C8—2511no—
228#103—2511no—
229#103—2511no—
230#102—2511no—
231#102—2511no—
232#102—2511no—
233#102—2511no—
234#102—2511no—
235#102—2511no—
236C22⋊C8—2511no—
237C22⋊C8—2511no—
238#97—2511no—
239C4×C8—2511no—
240#41—5022no—
241#41—5022no—
242C2×Q8—2511no—
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4—5021no—
248C2×D4—2511no—
249C2×D4C22×D25111yescharacteristic
250C2×D4C22×D25111yescharacteristic
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—5021no—
254C22×C4—5021no—
255C22×C4—5021no—
256C22×C4—5021no—
257C22×C4C22×D25111yescharacteristic
258SD16—5022no—
259SD16—5022no—
260D8—5022no—
261D8—5022no—
262C2×C8—2511no—
263C2×C8—2511no—
264C2×C8—2511no—
265C2×C8—2511no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—5021no—
269C4⋊C4—5021no—
270C4⋊C4—5021no—
271C4⋊C4—2511no—
272C4⋊C4—2511no—
273C4⋊C4C22⋊D25111yescharacteristic
274C4⋊C4C22⋊D25111yescharacteristic
275C4⋊C4C22×D25111yescharacteristic
276C4⋊C4C22×D25111yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—21no—
280C22⋊C4—21no—
281C22⋊C4—5021no—
282C22⋊C4—5021no—
283C22⋊C4C22⋊D25111yescharacteristic
284C22⋊C4C22⋊D25111yescharacteristic
285C42—5021no—
286C42—5021no—
287C42C22×D25111yescharacteristic
288C42C22×D25111yescharacteristic
289C23—21no—
290C23—21no—
291C23—5021no—
292C23C2×C22⋊D25111yescharacteristic
293Q8—5021no—
294Q8—2512no—
295D4—21no—
296D4—21no—
297D4—21no—
298D4—21no—
299D4—81no—
300D4—81no—
301D4—81no—
302D4—81no—
303D4—5024no—
304D4—5021no—
305D4—2512no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—20081no—
313C2×C4—20081no—
314C2×C4—20081no—
315C2×C4—5024no—
316C2×C4—5021no—
317C2×C4—5021no—
318C2×C4—5021no—
319C2×C4C2×C22⋊D25111yescharacteristic
320C2×C4C2×C22⋊D25111yescharacteristic
321C2×C4C2×C22⋊D25111yescharacteristic
322C2×C4C2×C22⋊D25111yescharacteristic
323C2×C4C2×C22⋊D25111yescharacteristic
324C2×C4C23×D25111yescharacteristic
325C8—5021no—
326C8—5021no—
327C8—5021no—
328C8—5021no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—10041no—
339C22—10041no—
340C22—5022no—
341C22C22×C22⋊D25111yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—41no—
348C4—41no—
349C4—10041no—
350C4—10041no—
351C4—5022no—
352C4#41×D25111yescharacteristic
353C4#41×D25111yescharacteristic
354C2—41no—
355C2—81no—
356C2—81no—
357C2—10042no—
358C2#3063219411yescharacteristic
359C2#3063249011yescharacteristic
360C2#3063249811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#540×C251C2
socleC22×C251C22×D4
hypercenter#540D251
hypocenterC251#3119
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ #117 ◁ #540

Complements

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

NH ≅ G/Ntype
1. #30632167C2semidirect
2. #30632003C2semidirect
3. #30631998C2semidirect
4. #30631991C2semidirect
5. #30631984C2semidirect
6. #30631884C2semidirect
7. #30631879C2semidirect
8. #30631830C2semidirect
9. #30631830C2semidirect
10. #30631797C2semidirect
11. #30631787C2semidirect
12. C4×#16860798C2semidirect
13. C4×#16860795C2semidirect
14. #30631748C2semidirect
15. #540×C251C2semidirect
180. C251#3119semidirect
182. #540D251semidirect

Maximal subgroups

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

subgroupindex
#306321672
#306320032
#306319982
#306319912
#306319842
#306318842
#306318792
#306318302
#306317972
#306317872
C4×#168607982
C4×#168607952
#306317482
#540×C2512
#3119251