Finite Group Databasebeta

← #44560621

Subgroups — #44560621

Order 32128.

384 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30632167C211yesmaximal, characteristic, complemented
2#30632020C211yesmaximal, characteristic, complemented
3#30631998C211yesmaximal, characteristic, complemented
4#30631979C211yesmaximal, characteristic, complemented
5#30631964C211yesmaximal, characteristic, complemented
6#30631906C211yesmaximal, characteristic, complemented
7#30631899C211yesmaximal, characteristic, complemented
8#30631847C211yesmaximal, characteristic, complemented
9#30631830C211yesmaximal, characteristic
10#30631787C211yesmaximal, characteristic, complemented
11#30631785C211yesmaximal, characteristic, complemented
12C4×#16860800C211yesmaximal, characteristic, complemented
13C4×#16860795C211yesmaximal, characteristic, complemented
14#30631748C211yesmaximal, characteristic, complemented
15#542×C251C211yesmaximal, characteristic, complemented
16C2×#16860810C2211yescharacteristic
17C2×#16860800C2211yescharacteristic
18C2×#16860798C2211yescharacteristic
19C2×#16860797C2211yescharacteristic
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#22548648C2211yescharacteristic
33#22548648C2211yescharacteristic
34C4⋊(C251⋊C8)C2211yescharacteristic
35C4⋊(C251⋊C8)C2211yescharacteristic
36#22548639C2211yescharacteristic
37#22548639C2211yescharacteristic
38#22548638C2211yescharacteristic
39#22548638C2211yescharacteristic
40#22548636C2211yescharacteristic
41#22548636C2211yescharacteristic
42#22548635C2211yescharacteristic
43#22548635C2211yescharacteristic
44C22⋊(C251⋊C8)C2211yescharacteristic
45C22⋊(C251⋊C8)C2211yescharacteristic
46#22548626C2211yescharacteristic
47C4×C251⋊C8C2211yescharacteristic
48C4⋊Q8×C251C2211yescharacteristic
49#126×C251—21no—
50#124×C251—21no—
51#124×C251C2211yescharacteristic
52#123×C251—21no—
53C22⋊Q8×C251—21no—
54C22⋊Q8×C251C2211yescharacteristic
55#121×C251C2211yescharacteristic
56C4×Q8×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#16860800—22no—
69#16860798—22no—
70#16860797—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×Q8×C251—21no—
86C2×Q8×C251C2311yescharacteristic
87C2×D4×C251—21no—
88C2×D4×C251C2311yescharacteristic
89C22×C4×C251—21no—
90C22×C4×C251C2311yescharacteristic
91C4⋊C4×C251—21no—
92C4⋊C4×C251—21no—
93C4⋊C4×C251—21no—
94C4⋊C4×C251D411yescharacteristic
95C4⋊C4×C251D411yescharacteristic
96C4⋊C4×C251C2311yescharacteristic
97C4⋊C4×C251C2311yescharacteristic
98C22⋊C4×C251—21no—
99C22⋊C4×C251—21no—
100C22⋊C4×C251—21no—
101C22⋊C4×C251—21no—
102C22⋊C4×C251—21no—
103C22⋊C4×C251D411yescharacteristic
104C22⋊C4×C251D411yescharacteristic
105C42×C251—21no—
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×C251C2×D411yescharacteristic
127Q8×C251—21no—
128Q8×C251—21no—
129Q8×C251—81no—
130D4×C251—21no—
131D4×C251—21no—
132D4×C251—81no—
133C2×C4×C251—21no—
134C2×C4×C251—21no—
135C2×C4×C251—21no—
136C2×C4×C251—21no—
137C2×C4×C251—21no—
138C2×C4×C251—81no—
139C2×C4×C251C2×D411yescharacteristic
140C2×C4×C251C2×D411yescharacteristic
141C2×C4×C251C2×D411yescharacteristic
142C2×C4×C251C2×D411yescharacteristic
143C2×C4×C251C2×D411yescharacteristic
144C2×C4×C251C2411yescharacteristic
145C22×C251—21no—
146C22×C251—21no—
147C22×C251—21no—
148C22×C251—41no—
149C22×C251—41no—
150C22×C251—41no—
151C22×C251C22×D411yescharacteristic
152C2×D251—22no—
153C2×D251—41no—
154C2×D251—41no—
155C4×C251—21no—
156C4×C251—21no—
157C4×C251—21no—
158C4×C251—21no—
159C4×C251—21no—
160C4×C251—41no—
161C4×C251—41no—
162C4×C251—41no—
163C4×C251C2×#4111yescharacteristic
164C4×C251C2×#4111yescharacteristic
165Dic251—22no—
166Dic251—41no—
167Dic251—41no—
168C2×C251—41no—
169C2×C251—81no—
170C2×C251#52511yescharacteristic
171C2×C251#57611yescharacteristic
172C2×C251#57811yescharacteristic
173D251—42no—
174C251#312411yescharacteristic, complemented, Sylow
175#3124—2511nomaximal, Sylow
176#542D25111yescharacteristic, complemented
177#520—2511no—
178#484—2511no—
179#480—2511no—
180#477—2511no—
181#474—2511no—
182#464—2511no—
183#463—2511no—
184#452—2511no—
185#449—2511no—
186#440—2511no—
187#440—2511no—
188C4×Q16—2511no—
189C4×D8—2511no—
190#433—2511no—
191C2×#41—2511no—
192C2×Q16—2511no—
193C2×SD16—2511no—
194C2×SD16—2511no—
195C2×D8—2511no—
196C4⋊Q8C2×D25111yescharacteristic
197#126—21no—
198#124—21no—
199#124C2×D25111yescharacteristic
200#123—21no—
201C22⋊Q8—21no—
202C22⋊Q8C2×D25111yescharacteristic
203#121C2×D25111yescharacteristic
204C4×Q8—2511no—
205C4×Q8—2511no—
206C4×Q8C2×D25111yescharacteristic
207C4×D4—5021no—
208C4×D4—5021no—
209C4×D4—2511no—
210C4×D4—2511no—
211C4×D4C2×D25111yescharacteristic
212#117—5021no—
213#117C2×D25111yescharacteristic
214C2×C4⋊C4—5021no—
215C2×C4⋊C4—2511no—
216#107—2511no—
217#107—2511no—
218#107—2511no—
219#107—2511no—
220C4⋊C8—2511no—
221C4⋊C8—2511no—
222#103—2511no—
223#103—2511no—
224#103—2511no—
225#103—2511no—
226#102—2511no—
227#102—2511no—
228#102—2511no—
229#102—2511no—
230C22⋊C8—2511no—
231C22⋊C8—2511no—
232#97—2511no—
233C4×C8—2511no—
234#41—5022no—
235#41—5022no—
236C2×Q8—21no—
237C2×Q8—2511no—
238C2×Q8C22×D25111yescharacteristic
239C2×D4—21no—
240C2×D4—5021no—
241C2×D4—2511no—
242C2×D4C22×D25111yescharacteristic
243C22×C4—21no—
244C22×C4—5021no—
245C22×C4—5021no—
246C22×C4—5021no—
247C22×C4—5021no—
248C22×C4C22×D25111yescharacteristic
249Q16—5022no—
250SD16—5022no—
251SD16—5022no—
252D8—5022no—
253C2×C8—2511no—
254C2×C8—2511no—
255C2×C8—2511no—
256C2×C8—2511no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—5021no—
261C4⋊C4—5021no—
262C4⋊C4—5021no—
263C4⋊C4—2511no—
264C4⋊C4—2511no—
265C4⋊C4C22⋊D25111yescharacteristic
266C4⋊C4C22⋊D25111yescharacteristic
267C4⋊C4C22×D25111yescharacteristic
268C4⋊C4C22×D25111yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—21no—
272C22⋊C4—21no—
273C22⋊C4—21no—
274C22⋊C4—5021no—
275C22⋊C4—5021no—
276C22⋊C4C22⋊D25111yescharacteristic
277C22⋊C4C22⋊D25111yescharacteristic
278C42—21no—
279C42—5021no—
280C42—5021no—
281C42C22×D25111yescharacteristic
282C42C22×D25111yescharacteristic
283C23—21no—
284C23—5021no—
285C23C2×C22⋊D25111yescharacteristic
286Q8—21no—
287Q8—21no—
288Q8—81no—
289Q8—5021no—
290Q8—2512no—
291D4—21no—
292D4—21no—
293D4—81no—
294D4—5024no—
295D4—5021no—
296D4—2512no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—81no—
303C2×C4—20081no—
304C2×C4—20081no—
305C2×C4—20081no—
306C2×C4—5024no—
307C2×C4—5021no—
308C2×C4—5021no—
309C2×C4—5021no—
310C2×C4C2×C22⋊D25111yescharacteristic
311C2×C4C2×C22⋊D25111yescharacteristic
312C2×C4C2×C22⋊D25111yescharacteristic
313C2×C4C2×C22⋊D25111yescharacteristic
314C2×C4C2×C22⋊D25111yescharacteristic
315C2×C4C23×D25111yescharacteristic
316C8—5021no—
317C8—5021no—
318C8—5021no—
319C8—5021no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—10041no—
327C22—10041no—
328C22—5022no—
329C22C22×C22⋊D25111yescharacteristic
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—10041no—
339C4—10041no—
340C4—5022no—
341C4#41×D25111yescharacteristic
342C4#41×D25111yescharacteristic
343C2—41no—
344C2—81no—
345C2—10042no—
346C2#3063219411yescharacteristic
347C2#3063249011yescharacteristic
348C2#3063250211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#542×C251C2
socleC22×C251C22×D4
prime radical#30631830C2
hypercenter#542D251
hypocenterC251#3124
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ #117 ◁ #542

Complements

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

NH ≅ G/Ntype
1. #30632167C2semidirect
2. #30632020C2semidirect
3. #30631998C2semidirect
4. #30631979C2semidirect
5. #30631964C2semidirect
6. #30631906C2semidirect
7. #30631899C2semidirect
8. #30631847C2semidirect
10. #30631787C2semidirect
11. #30631785C2semidirect
12. C4×#16860800C2semidirect
13. C4×#16860795C2semidirect
14. #30631748C2semidirect
15. #542×C251C2semidirect
174. C251#3124semidirect
176. #542D251semidirect

Maximal subgroups

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

subgroupindex
#306321672
#306320202
#306319982
#306319792
#306319642
#306319062
#306318992
#306318472
#306318302
#306317872
#306317852
C4×#168608002
C4×#168607952
#306317482
#542×C2512
#3124251