Finite Group Databasebeta

← #24920529

Subgroups — #24920529

Order 10624.

368 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19128436C211yesmaximal, characteristic, complemented
2#19128093C211yesmaximal, characteristic, complemented
3#19128091C211yesmaximal, characteristic
4#19128083C211yesmaximal, characteristic, complemented
5#19127923C211yesmaximal, characteristic, complemented
6#19127920C211yesmaximal, characteristic, complemented
7#19127916C211yesmaximal, characteristic, complemented
8#19127913C211yesmaximal, characteristic, complemented
9#19127903C211yesmaximal, characteristic, complemented
10#19127901C211yesmaximal, characteristic, complemented
11#19127832C211yesmaximal, characteristic, complemented
12#19127811C211yesmaximal, characteristic, complemented
13#19127806C211yesmaximal, characteristic, complemented
14C4×#13050577C211yesmaximal, characteristic, complemented
15#556×C83C211yesmaximal, characteristic, complemented
16C2×#13050580C2211yescharacteristic
17C2×#13050580C2211yescharacteristic
18C2×#13050578C2211yescharacteristic
19C2×#13050577C2211yescharacteristic
20C2×#13050577C2211yescharacteristic
21C2×#13050577C2211yescharacteristic
22C2×#13050577C2211yescharacteristic
23C2×#13050575C2211yescharacteristic
24C2×#13050575C2211yescharacteristic
25#14581789C2211yescharacteristic
26#14581781—21no—
27#14581780C2211yescharacteristic
28#14581780C2211yescharacteristic
29#14581777—21no—
30#14581777—21no—
31C4×Dic166C2211yescharacteristic
32C4×D332C2211yescharacteristic
33C4×D332C2211yescharacteristic
34#14581754—21no—
35#14581723C2211yescharacteristic
36C4⋊(C83⋊C8)C2211yescharacteristic
37C4⋊(C83⋊C8)C2211yescharacteristic
38C4⋊(C83⋊C8)C2211yescharacteristic
39#14581716C2211yescharacteristic
40#14581716C2211yescharacteristic
41#14581715C2211yescharacteristic
42#14581713C2211yescharacteristic
43#14581712C2211yescharacteristic
44#14581712C2211yescharacteristic
45#14581703C2211yescharacteristic
46#14581703C2211yescharacteristic
47C4×C83⋊C8C2211yescharacteristic
48C4⋊Q8×C83C2211yescharacteristic
49#126×C83—21no—
50#126×C83—21no—
51#124×C83C2211yescharacteristic
52#124×C83C2211yescharacteristic
53C22⋊Q8×C83—21no—
54C4×Q8×C83C2211yescharacteristic
55C4×Q8×C83C2211yescharacteristic
56C4×Q8×C83C2211yescharacteristic
57C4×D4×C83C2211yescharacteristic
58#117×C83—21no—
59C2×Dic166C2311yescharacteristic
60C2×D332—21no—
61C2×D332—21no—
62C2×D332C2311yescharacteristic
63C2×D332C2311yescharacteristic
64C2×C4×D83—21no—
65C2×C4×D83—21no—
66#13050580—22no—
67#13050580—41no—
68#13050578—41no—
69#13050577—22no—
70#13050577—41no—
71#13050577—41no—
72#13050577—41no—
73#13050575—22no—
74#13050575—41no—
75C2×C83⋊C8C2311yescharacteristic
76C2×C83⋊C8C2311yescharacteristic
77C2×C83⋊C8C2311yescharacteristic
78C2×C83⋊C8C2311yescharacteristic
79C4⋊Dic83C2311yescharacteristic
80Dic83⋊C4—21no—
81(C2×D83)⋊C4—21no—
82(C2×D83)⋊C4—21no—
83(C2×D83)⋊C4—21no—
84(C2×D83)⋊C4—21no—
85(C2×D83)⋊C4—21no—
86(C2×D83)⋊C4—21no—
87C4×Dic83—21no—
88C2×Q8×C83D411yescharacteristic
89C2×Q8×C83C2311yescharacteristic
90C2×Q8×C83C2311yescharacteristic
91C2×D4×C83C2311yescharacteristic
92C22×C4×C83—21no—
93C4⋊C4×C83—21no—
94C4⋊C4×C83—21no—
95C4⋊C4×C83—21no—
96C4⋊C4×C83—21no—
97C4⋊C4×C83D411yescharacteristic
98C4⋊C4×C83D411yescharacteristic
99C4⋊C4×C83D411yescharacteristic
100C4⋊C4×C83C2311yescharacteristic
101C22⋊C4×C83—21no—
102C22⋊C4×C83—21no—
103C22⋊C4×C83—21no—
104C22⋊C4×C83—21no—
105C22⋊C4×C83—21no—
106C42×C83—21no—
107C42×C83—21no—
108C42×C83C2311yescharacteristic
109C42×C83C2311yescharacteristic
110C42×C83C2311yescharacteristic
111C23×C83—21no—
112C22×D83—21no—
113C22×D83—21no—
114Q8×C83—21no—
115Q8×C83—21no—
116Q8×C83—21no—
117Q8×C83—21no—
118Q8×C83—41no—
119D4×C83—21no—
120D4×C83—21no—
121C2×C4×C83—21no—
122C2×C4×C83—21no—
123C2×C4×C83—21no—
124C2×C4×C83—42no—
125C2×C4×C83C2×D411yescharacteristic
126C2×C4×C83C2×D411yescharacteristic
127C2×C4×C83C2×D411yescharacteristic
128C2×C4×C83C2×D411yescharacteristic
129C2×C4×C83C2×D411yescharacteristic
130C2×C4×C83C2×D411yescharacteristic
131C2×C4×C83C2411yescharacteristic
132C2×Dic83—21no—
133C2×Dic83—21no—
134D332—21no—
135D332—21no—
136D332—21no—
137D332—21no—
138D332—42no—
139D332—81no—
140C4×D83—81no—
141C4×D83—81no—
142Dic166—21no—
143Dic166—21no—
144C83⋊C8—21no—
145C83⋊C8—21no—
146C83⋊C8—21no—
147C83⋊C8#4112yes—
148C22×C83—41no—
149C22×C83—41no—
150C22×C83—41no—
151C22×C83C22×D411yescharacteristic
152C2×D83—41no—
153C2×D83—41no—
154C2×D83—41no—
155C2×D83—41no—
156C2×D83—41no—
157C2×D83—41no—
158C4×C83—21no—
159C4×C83—21no—
160C4×C83—21no—
161C4×C83—21no—
162C4×C83—21no—
163C4×C83—21no—
164C4×C83—41no—
165C4×C83—41no—
166C4×C83—41no—
167C4×C83C2×#4111yescharacteristic
168C4×C83H(5,𝔽2)11yescharacteristic
169Dic83—41no—
170Dic83—41no—
171C2×C83—81no—
172C2×C83#55011yescharacteristic
173C2×C83#57511yescharacteristic
174C2×C83#57711yescharacteristic
175D83—81no—
176D83—81no—
177#3401—831nomaximal, Sylow
178C83#340111yescharacteristic, complemented, Sylow
179#559—831no—
180#556D8311yescharacteristic, complemented
181#497—831no—
182#496—831no—
183#495—831no—
184#464—831no—
185#464—831no—
186#463—831no—
187#463—831no—
188#461—831no—
189#460—831no—
190#446—831no—
191#442—831no—
192#441—831no—
193C4×SD16—831no—
194C2×Q16—831no—
195C2×Q16—831no—
196C2×SD16—831no—
197C2×SD16—831no—
198C2×SD16—831no—
199C2×SD16—831no—
200C2×SD16—831no—
201C2×D8—831no—
202C2×D8—831no—
203C4⋊Q8C2×D8311yescharacteristic
204#127—831no—
205#126—21no—
206#126—21no—
207#124—831no—
208#124—831no—
209#124—1661no—
210#124C2×D8311yescharacteristic
211#124C2×D8311yescharacteristic
212#123—1661no—
213#123—1661no—
214C22⋊Q8—21no—
215C4×Q8—831no—
216C4×Q8C2×D8311yescharacteristic
217C4×Q8C2×D8311yescharacteristic
218C4×Q8C2×D8311yescharacteristic
219C4×D4—831no—
220C4×D4—831no—
221C4×D4—1661no—
222C4×D4C2×D8311yescharacteristic
223#117—21no—
224#106—831no—
225C4⋊C8—831no—
226C4⋊C8—831no—
227C4⋊C8—831no—
228#103—831no—
229#103—831no—
230#103—831no—
231#102—831no—
232#102—831no—
233#102—831no—
234#97—831no—
235#97—831no—
236C4×C8—831no—
237C2×Q8—831no—
238C2×Q8C22⋊D8311yescharacteristic
239C2×Q8C22×D8311yescharacteristic
240C2×Q8C22×D8311yescharacteristic
241C2×D4—831no—
242C2×D4—831no—
243C2×D4—1661no—
244C2×D4—1661no—
245C2×D4C22×D8311yescharacteristic
246C22×C4—21no—
247C22×C4—1661no—
248C22×C4—1661no—
249Q16—3321no—
250Q16—1662no—
251SD16—3321no—
252SD16—3321no—
253SD16—3321no—
254SD16—3321no—
255SD16—1662no—
256D8—3321no—
257D8—1662no—
258C2×C8—831no—
259C2×C8—831no—
260C2×C8—831no—
261C2×C8—831no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—831no—
267C4⋊C4—1661no—
268C4⋊C4C22⋊D8311yescharacteristic
269C4⋊C4C22⋊D8311yescharacteristic
270C4⋊C4C22⋊D8311yescharacteristic
271C4⋊C4C22×D8311yescharacteristic
272C22⋊C4—21no—
273C22⋊C4—21no—
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—1661no—
278C22⋊C4—1661no—
279C22⋊C4—1661no—
280C22⋊C4—1661no—
281C22⋊C4—1661no—
282C22⋊C4—1661no—
283C42—21no—
284C42—21no—
285C42—1661no—
286C42C22×D8311yescharacteristic
287C42C22×D8311yescharacteristic
288C42C22×D8311yescharacteristic
289C23—21no—
290C23—1661no—
291C23—1661no—
292Q8—21no—
293Q8—21no—
294Q8—21no—
295Q8—21no—
296Q8—41no—
297Q8—1661no—
298Q8—1661no—
299D4—21no—
300D4—21no—
301D4—3322no—
302D4—6641no—
303D4—1661no—
304D4—1661no—
305D4—1661no—
306D4—1661no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—42no—
311C2×C4—6641no—
312C2×C4—6641no—
313C2×C4—1661no—
314C2×C4—1661no—
315C2×C4C2×C22⋊D8311yescharacteristic
316C2×C4C2×C22⋊D8311yescharacteristic
317C2×C4C2×C22⋊D8311yescharacteristic
318C2×C4C2×C22⋊D8311yescharacteristic
319C2×C4C2×C22⋊D8311yescharacteristic
320C2×C4C2×C22⋊D8311yescharacteristic
321C2×C4C23×D8311yescharacteristic
322C8—832no—
323C8—1661no—
324C8—1661no—
325C8—1661no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—3321no—
330C22—3321no—
331C22—3321no—
332C22—3321no—
333C22—3321no—
334C22—3321no—
335C22C22×C22⋊D8311yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—3321no—
346C4—3321no—
347C4#41×D8311yescharacteristic
348C4#1458183811yescharacteristic
349C2—81no—
350C2—6641no—
351C2—6641no—
352C2#1912837011yescharacteristic
353C2#1912849911yescharacteristic
354C2#1912850911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#556×C83C2
socleC22×C83C22×D4
prime radical#19128091C2
hypercenter#556D83
hypocenterC83#3401
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #556

Complements

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

NH ≅ G/Ntype
1. #19128436C2semidirect
2. #19128093C2semidirect
4. #19128083C2semidirect
5. #19127923C2semidirect
6. #19127920C2semidirect
7. #19127916C2semidirect
8. #19127913C2semidirect
9. #19127903C2semidirect
10. #19127901C2semidirect
11. #19127832C2semidirect
12. #19127811C2semidirect
13. #19127806C2semidirect
14. C4×#13050577C2semidirect
15. #556×C83C2semidirect
178. C83#3401semidirect
180. #556D83semidirect

Maximal subgroups

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

subgroupindex
#191284362
#191280932
#191280912
#191280832
#191279232
#191279202
#191279162
#191279132
#191279032
#191279012
#191278322
#191278112
#191278062
C4×#130505772
#556×C832
#340183