Finite Group Databasebeta

← #44560624

Subgroups — #44560624

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#30632306C211yesmaximal, characteristic, complemented
2#30632166C211yesmaximal, characteristic, complemented
3#30632021C211yesmaximal, characteristic, complemented
4#30631999C211yesmaximal, characteristic, complemented
5#30631983C211yesmaximal, characteristic, complemented
6#30631967C211yesmaximal, characteristic, complemented
7#30631906C211yesmaximal, characteristic, complemented
8#30631846C211yesmaximal, characteristic, complemented
9#30631829C211yesmaximal, characteristic
10#30631788C211yesmaximal, characteristic, complemented
11#30631784C211yesmaximal, characteristic, complemented
12#30631779C211yesmaximal, characteristic, complemented
13D8×Dic251C211yesmaximal, characteristic, complemented
14#30631747C211yesmaximal, characteristic, complemented
15#463×C251C211yesmaximal, characteristic, complemented
16C2×#16860809C2211yescharacteristic
17C2×#16860800C2211yescharacteristic
18C2×#16860798C2211yescharacteristic
19#22548715C2211yescharacteristic
20#22548708—21no—
21#22548701—21no—
22#22548699—21no—
23#22548692—21no—
24#22548691C2211yescharacteristic
25#22548686C2211yescharacteristic
26Q8×Dic251C2211yescharacteristic
27C4×Dic502C2211yescharacteristic
28#22548680C2211yescharacteristic
29D4×Dic251C2211yescharacteristic
30D4×Dic251C2211yescharacteristic
31C4×C22⋊D251—21no—
32(C22⋊D251)⋊C4—21no—
33#22548672C2211yescharacteristic
34#22548671—21no—
35C2×Dic251⋊C4—21no—
36C4⋊C4×D251C2211yescharacteristic
37#22548649C2211yescharacteristic
38#22548649C2211yescharacteristic
39#22548648C2211yescharacteristic
40#22548648C2211yescharacteristic
41C4⋊(C251⋊C8)C2211yescharacteristic
42#22548640C2211yescharacteristic
43#22548640C2211yescharacteristic
44#22548639C2211yescharacteristic
45#22548636C2211yescharacteristic
46#22548636C2211yescharacteristic
47#22548636C2211yescharacteristic
48(C2×D251)⋊C8C2211yescharacteristic
49(C2×D251)⋊C8C2211yescharacteristic
50#22548625C2211yescharacteristic
51C8×Dic251C2211yescharacteristic
52C2×SD16×C251C2211yescharacteristic
53C2×D8×C251C2211yescharacteristic
54#124×C251C2211yescharacteristic
55C4×D4×C251C2211yescharacteristic
56C4⋊C8×C251C2211yescharacteristic
57#103×C251C2211yescharacteristic
58#102×C251C2211yescharacteristic
59#16860809—22no—
60#16860809—22no—
61C2×Dic502—21no—
62C2×Dic502C2311yescharacteristic
63C2×C22⋊D251—21no—
64C2×C22⋊D251—21no—
65C22×Dic251—21no—
66C22×Dic251—21no—
67C22×Dic251—21no—
68C2×C4×D251—21no—
69C2×C4×D251C2311yescharacteristic
70#16860800—22no—
71#16860798—22no—
72C2×C251⋊C8C2311yescharacteristic
73C2×C251⋊C8C2311yescharacteristic
74C4⋊Dic251—21no—
75C4⋊Dic251—21no—
76C4⋊Dic251C2311yescharacteristic
77C4⋊Dic251C2311yescharacteristic
78C4⋊Dic251C2311yescharacteristic
79Dic251⋊C4—21no—
80Dic251⋊C4—21no—
81Dic251⋊C4—21no—
82Dic251⋊C4—21no—
83Dic251⋊C4D411yescharacteristic
84Dic251⋊C4D411yescharacteristic
85C22⋊Dic251—21no—
86C22⋊Dic251—21no—
87C22⋊Dic251—21no—
88(C2×D251)⋊C4—21no—
89(C2×D251)⋊C4D411yescharacteristic
90(C2×D251)⋊C4D411yescharacteristic
91C4×Dic251—21no—
92C4×Dic251—21no—
93C4×Dic251—21no—
94C4×Dic251C2311yescharacteristic
95C2×Q8×C251C2311yescharacteristic
96C2×D4×C251C2311yescharacteristic
97C2×D4×C251C2311yescharacteristic
98C22×C4×C251—21no—
99SD16×C251—22no—
100D8×C251—22no—
101C2×C8×C251C2311yescharacteristic
102C2×C8×C251C2311yescharacteristic
103C4⋊C4×C251C2311yescharacteristic
104C22⋊C4×C251—21no—
105C22⋊C4×C251—21no—
106C22⋊C4×C251—21no—
107C42×C251C2311yescharacteristic
108C22×D251C2×D411yescharacteristic
109Dic502—21no—
110Dic502—81no—
111Dic502#4112yes—
112C22⋊D251—24no—
113C22⋊D251—81no—
114C2×Dic251—24no—
115C2×Dic251—21no—
116C2×Dic251—21no—
117C2×Dic251—21no—
118C2×Dic251—21no—
119C2×Dic251—81no—
120C2×Dic251—81no—
121C2×Dic251C2×D411yescharacteristic
122C2×Dic251C2×D411yescharacteristic
123C2×Dic251C2×D411yescharacteristic
124C4×D251—21no—
125C4×D251—21no—
126C4×D251—81no—
127C251⋊C8—21no—
128C251⋊C8—21no—
129C23×C251—21no—
130C23×C251—21no—
131Q8×C251—21no—
132Q8×C251—21no—
133D4×C251—21no—
134D4×C251—21no—
135D4×C251—21no—
136D4×C251#4112yes—
137C2×C4×C251—21no—
138C2×C4×C251—21no—
139C2×C4×C251—81no—
140C2×C4×C251C2×D411yescharacteristic
141C2×C4×C251C2×D411yescharacteristic
142C2×C4×C251C2411yescharacteristic
143C8×C251—21no—
144C8×C251—21no—
145C22×C251—22no—
146C22×C251—41no—
147C22×C251—41no—
148C22×C251—41no—
149C22×C251—41no—
150C22×C251—41no—
151C22×C251C22×D411yescharacteristic
152C2×D251—21no—
153C2×D251—21no—
154C2×D251—21no—
155C4×C251—21no—
156C4×C251—21no—
157C4×C251—41no—
158C4×C251—41no—
159C4×C251C2×#4111yescharacteristic
160C4×C251C2×#4111yescharacteristic
161Dic251—21no—
162Dic251—22no—
163Dic251—21no—
164Dic251—21no—
165Dic251—41no—
166Dic251—41no—
167Dic251—41no—
168C2×C251—42no—
169C2×C251—81no—
170C2×C251#52511yescharacteristic
171C2×C251#57611yescharacteristic
172C2×C251#57811yescharacteristic
173D251—41no—
174C251#312411yescharacteristic, complemented, Sylow
175#3124—2511nomaximal, Sylow
176#542—2511no—
177#520—2511no—
178#484—2511no—
179#480—2511no—
180#477—2511no—
181#474—2511no—
182#464—2511no—
183#463D25111yescharacteristic, complemented
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×SD16C2×D25111yescharacteristic
195C2×D8C2×D25111yescharacteristic
196C4⋊Q8—2511no—
197#126—5021no—
198#124—5021no—
199#124C2×D25111yescharacteristic
200#123—5021no—
201C22⋊Q8—5021no—
202C22⋊Q8—2511no—
203#121—2511no—
204C4×Q8—2511no—
205C4×Q8—2511no—
206C4×Q8—2511no—
207C4×D4—5021no—
208C4×D4—5021no—
209C4×D4—2511no—
210C4×D4—2511no—
211C4×D4C2×D25111yescharacteristic
212#117—5021no—
213#117—2511no—
214C2×C4⋊C4—5021no—
215C2×C4⋊C4—2511no—
216#107—2511no—
217#107—2511no—
218#107—2511no—
219#107—2511no—
220C4⋊C8—2511no—
221C4⋊C8C2×D25111yescharacteristic
222#103—2511no—
223#103—2511no—
224#103—2511no—
225#103C2×D25111yescharacteristic
226#102—2511no—
227#102—2511no—
228#102—2511no—
229#102C2×D25111yescharacteristic
230C22⋊C8—2511no—
231C22⋊C8—2511no—
232#97—2511no—
233C4×C8—2511no—
234#41—5022no—
235#41—5022no—
236C2×Q8—5021no—
237C2×Q8—2511no—
238C2×Q8C22×D25111yescharacteristic
239C2×D4—5021no—
240C2×D4—5021no—
241C2×D4C22×D25111yescharacteristic
242C2×D4C22×D25111yescharacteristic
243C22×C4—21no—
244C22×C4—5021no—
245C22×C4—5021no—
246C22×C4—5021no—
247C22×C4—5021no—
248C22×C4—2511no—
249Q16—5022no—
250SD16—22no—
251SD16—5022no—
252D8—22no—
253C2×C8—2511no—
254C2×C8—2511no—
255C2×C8C22×D25111yescharacteristic
256C2×C8C22×D25111yescharacteristic
257C4⋊C4—5021no—
258C4⋊C4—5021no—
259C4⋊C4—5021no—
260C4⋊C4—5021no—
261C4⋊C4—5021no—
262C4⋊C4—5021no—
263C4⋊C4—2511no—
264C4⋊C4—2511no—
265C4⋊C4—2511no—
266C4⋊C4—2511no—
267C4⋊C4—2511no—
268C4⋊C4C22×D25111yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—21no—
272C22⋊C4—5021no—
273C22⋊C4—5021no—
274C22⋊C4—5021no—
275C22⋊C4—5021no—
276C22⋊C4—2511no—
277C22⋊C4—2511no—
278C42—5021no—
279C42—5021no—
280C42—5021no—
281C42—2511no—
282C42C22×D25111yescharacteristic
283C23—21no—
284C23—21no—
285C23—2511no—
286Q8—21no—
287Q8—21no—
288Q8—20081no—
289Q8—5021no—
290Q8—2512no—
291D4—21no—
292D4—21no—
293D4—21no—
294D4—20081no—
295D4—5024no—
296D4#1686080912yes—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—81no—
300C2×C4—20081no—
301C2×C4—20081no—
302C2×C4—20081no—
303C2×C4—5021no—
304C2×C4—5021no—
305C2×C4—5021no—
306C2×C4—5024no—
307C2×C4—5021no—
308C2×C4—5021no—
309C2×C4—5021no—
310C2×C4—2511no—
311C2×C4—2511no—
312C2×C4—2511no—
313C2×C4D4×D25111yescharacteristic
314C2×C4D4×D25111yescharacteristic
315C2×C4C23×D25111yescharacteristic
316C8—21no—
317C8—21no—
318C8—5021no—
319C8—5021no—
320C22—22no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—5021no—
327C22—5021no—
328C22—5021no—
329C22C2×D4×D25111yescharacteristic
330C4—21no—
331C4—21no—
332C4—41no—
333C4—41no—
334C4—10041no—
335C4—10041no—
336C4—10041no—
337C4—5021no—
338C4—5022no—
339C4—5021no—
340C4—5021no—
341C4#41×D25111yescharacteristic
342C4C2×#1686080911yescharacteristic
343C2—42no—
344C2—81no—
345C2—10041no—
346C2#3063219711yescharacteristic
347C2#3063249111yescharacteristic
348C2#3063250111yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#463×C251C2
socleC22×C251C22×D4
prime radical#30631829C2
hypercenter#463D251
hypocenterC251#3124
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. #30632306C2semidirect
2. #30632166C2semidirect
3. #30632021C2semidirect
4. #30631999C2semidirect
5. #30631983C2semidirect
6. #30631967C2semidirect
7. #30631906C2semidirect
8. #30631846C2semidirect
10. #30631788C2semidirect
11. #30631784C2semidirect
12. #30631779C2semidirect
13. D8×Dic251C2semidirect
14. #30631747C2semidirect
15. #463×C251C2semidirect
174. C251#3124semidirect
183. #463D251semidirect

Maximal subgroups

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

subgroupindex
#306323062
#306321662
#306320212
#306319992
#306319832
#306319672
#306319062
#306318462
#306318292
#306317882
#306317842
#306317792
D8×Dic2512
#306317472
#463×C2512
#3124251