Finite Group Databasebeta

← #24919927

Subgroups — #24919927

Order 10624.

388 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19128355C211yesmaximal, characteristic, complemented
2#19128111C211yesmaximal, characteristic, complemented
3#19128025C211yesmaximal, characteristic, complemented
4#19128019C211yesmaximal, characteristic, complemented
5#19128017C211yesmaximal, characteristic, complemented
6#19128012C211yesmaximal, characteristic, complemented
7#19127998C211yesmaximal, characteristic, complemented
8#19127991C211yesmaximal, characteristic, complemented
9#19127952C211yesmaximal, characteristic, complemented
10#19127935C211yesmaximal, characteristic, complemented
11#19127811C211yesmaximal, characteristic, complemented
12#19127773C211yesmaximal, characteristic, complemented
13#19127744C211yesmaximal, characteristic, complemented
14C2×#14581723C211yesmaximal, characteristic, complemented
15#553×C83C211yesmaximal, characteristic, complemented
16C2×#13050589—21no—
17C2×#13050575C2211yescharacteristic
18C2×#13050574C2211yescharacteristic
19C22×C83⋊C8C2211yescharacteristic
20#14581792C2211yescharacteristic
21#14581776—21no—
22#14581776—21no—
23#14581772—21no—
24#14581766C2211yescharacteristic
25#14581766C2211yescharacteristic
26C4×D332C2211yescharacteristic
27C2×C4⋊Dic83C2211yescharacteristic
28C2×C4⋊Dic83C2211yescharacteristic
29#14581725C2211yescharacteristic
30#14581725C2211yescharacteristic
31#14581725C2211yescharacteristic
32#14581723C2211yescharacteristic
33#14581723C2211yescharacteristic
34#14581723C2211yescharacteristic
35#14581723C2211yescharacteristic
36#14581723C2211yescharacteristic
37#14581723C2211yescharacteristic
38C4⋊(C83⋊C8)C2211yescharacteristic
39#14581713C2211yescharacteristic
40#14581713C2211yescharacteristic
41#14581713C2211yescharacteristic
42#14581712C2211yescharacteristic
43#14581712C2211yescharacteristic
44#14581712C2211yescharacteristic
45C22⋊(C83⋊C8)C2211yescharacteristic
46C22⋊(C83⋊C8)C2211yescharacteristic
47#14581703C2211yescharacteristic
48#126×C83—21no—
49#125×C83C2211yescharacteristic
50#123×C83—21no—
51#121×C83—21no—
52#121×C83C2211yescharacteristic
53#121×C83C2211yescharacteristic
54C4×D4×C83—21no—
55C4×D4×C83C2211yescharacteristic
56C4×D4×C83C2211yescharacteristic
57#117×C83C2211yescharacteristic
58C2×C4⋊C4×C83C2211yescharacteristic
59#13050589—44no—
60C2×Dic166—21no—
61C2×D332C2311yescharacteristic
62C2×C22⋊D83—21no—
63C2×C22⋊D83—21no—
64C22×Dic83—21no—
65C22×Dic83—21no—
66C2×C4×D83—21no—
67#13050575—22no—
68#13050574—22no—
69C2×C83⋊C8—22no—
70C2×C83⋊C8C2311yescharacteristic
71C2×C83⋊C8C2311yescharacteristic
72C2×C83⋊C8C2311yescharacteristic
73C2×C83⋊C8C2311yescharacteristic
74C4⋊Dic83—21no—
75C4⋊Dic83—21no—
76C4⋊Dic83—21no—
77C4⋊Dic83C2311yescharacteristic
78C4⋊Dic83C2311yescharacteristic
79C4⋊Dic83C2311yescharacteristic
80(C2×D83)⋊C4—21no—
81(C2×D83)⋊C4—21no—
82(C2×D83)⋊C4—21no—
83C2×D4×C83—21no—
84C2×D4×C83—21no—
85C2×D4×C83D411yescharacteristic
86C2×D4×C83C2311yescharacteristic
87C22×C4×C83—21no—
88C22×C4×C83—21no—
89C22×C4×C83C2311yescharacteristic
90C22×C4×C83C2311yescharacteristic
91C4⋊C4×C83—21no—
92C4⋊C4×C83—21no—
93C4⋊C4×C83—21no—
94C4⋊C4×C83D411yescharacteristic
95C4⋊C4×C83C2311yescharacteristic
96C4⋊C4×C83C2311yescharacteristic
97C4⋊C4×C83C2311yescharacteristic
98C22⋊C4×C83—21no—
99C22⋊C4×C83—21no—
100C22⋊C4×C83—21no—
101C22⋊C4×C83—21no—
102C22⋊C4×C83D411yescharacteristic
103C22⋊C4×C83D411yescharacteristic
104C42×C83—21no—
105C42×C83C2311yescharacteristic
106C23×C83—21no—
107C23×C83C2×D411yescharacteristic
108C23×C83C2×D411yescharacteristic
109C22×D83—21no—
110D4×C83—22no—
111D4×C83—21no—
112D4×C83—21no—
113D4×C83—42no—
114D4×C83—81no—
115C2×C4×C83—21no—
116C2×C4×C83—21no—
117C2×C4×C83—21no—
118C2×C4×C83—21no—
119C2×C4×C83—21no—
120C2×C4×C83—21no—
121C2×C4×C83—21no—
122C2×C4×C83—81no—
123C2×C4×C83—81no—
124C2×C4×C83C2×D411yescharacteristic
125C2×C4×C83C2×D411yescharacteristic
126C2×C4×C83C2×D411yescharacteristic
127C2×C4×C83C2×D411yescharacteristic
128C2×C4×C83C2411yescharacteristic
129C22⋊D83—42no—
130C22⋊D83—42no—
131C2×Dic83—21no—
132C2×Dic83—21no—
133C2×Dic83—21no—
134C2×Dic83—42no—
135C2×Dic83—42no—
136D332—21no—
137D332—21no—
138C4×D83—42no—
139Dic166—42no—
140C83⋊C8—21no—
141C83⋊C8—21no—
142C83⋊C8—21no—
143C83⋊C8#4112yescomplemented
144C22×C83—21no—
145C22×C83—21no—
146C22×C83—21no—
147C22×C83—21no—
148C22×C83—21no—
149C22×C83—41no—
150C22×C83—41no—
151C22×C83—41no—
152C22×C83AGL(1,ℤ/8)12yes—
153C22×C83C22×D411yescharacteristic
154C2×D83—41no—
155C2×D83—41no—
156C2×D83—41no—
157C4×C83—21no—
158C4×C83—21no—
159C4×C83—21no—
160C4×C83—21no—
161C4×C83—41no—
162C4×C83—41no—
163C4×C83—41no—
164C4×C83C2×#4111yescharacteristic
165C4×C832−1+411yescharacteristic
166Dic83—41no—
167Dic83—41no—
168Dic83—41no—
169C2×C83—22no—
170C2×C83—41no—
171C2×C83—81no—
172C2×C83#54711yescharacteristic
173C2×C83C2×AGL(1,ℤ/8)11yescharacteristic
174C2×C83#57711yescharacteristic
175D83—81no—
176#3345—831nomaximal, Sylow
177C83#334511yescharacteristic, complemented, Sylow
178#553D8311yescharacteristic, complemented
179#547—831no—
180#501—831no—
181#482—831no—
182#481—831no—
183#481—831no—
184#480—831no—
185#478—831no—
186#476—831no—
187#469—831no—
188#466—831no—
189#442—831no—
190#435—831no—
191#428—831no—
192C2×#106—831no—
193C2×#41—1661no—
194C2×D8—831no—
195C2×M16—831no—
196C22×C8—831no—
197C4⋊Q8—831no—
198#126—21no—
199#125C2×D8311yescharacteristic
200#123—21no—
201#123—1661no—
202#123—1661no—
203C22⋊Q8—1661no—
204#121—21no—
205#121—831no—
206#121—831no—
207#121C2×D8311yescharacteristic
208#121C2×D8311yescharacteristic
209C4×D4—21no—
210C4×D4—831no—
211C4×D4C2×D8311yescharacteristic
212C4×D4C2×D8311yescharacteristic
213#117C2×D8311yescharacteristic
214C2×C4⋊C4—831no—
215C2×C4⋊C4—831no—
216C2×C4⋊C4C2×D8311yescharacteristic
217#107—831no—
218#107—831no—
219#107—831no—
220#106—831no—
221#106—831no—
222#106—831no—
223#106—831no—
224#106—831no—
225#106—831no—
226C4⋊C8—831no—
227#102—831no—
228#102—831no—
229#102—831no—
230#102—831no—
231#102—831no—
232#102—831no—
233C22⋊C8—831no—
234C22⋊C8—831no—
235#97—831no—
236#41—3324no—
237C2×Q8—1661no—
238C2×D4—21no—
239C2×D4—21no—
240C2×D4—831no—
241C2×D4—1661no—
242C2×D4—1661no—
243C2×D4C22⋊D8311yescharacteristic
244C2×D4C22×D8311yescharacteristic
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—1661no—
248C22×C4—1661no—
249C22×C4—1661no—
250C22×C4C22×D8311yescharacteristic
251C22×C4C22×D8311yescharacteristic
252D8—1662no—
253M16—1662no—
254C2×C8—831no—
255C2×C8—831no—
256C2×C8—831no—
257C2×C8—831no—
258C2×C8—1662no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—831no—
263C4⋊C4—831no—
264C4⋊C4—831no—
265C4⋊C4—1661no—
266C4⋊C4—1661no—
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—1661no—
277C22⋊C4—1661no—
278C22⋊C4—1661no—
279C22⋊C4C22⋊D8311yescharacteristic
280C22⋊C4C22⋊D8311yescharacteristic
281C42—21no—
282C42C22×D8311yescharacteristic
283C23—21no—
284C23—1661no—
285C23C2×C22⋊D8311yescharacteristic
286C23C2×C22⋊D8311yescharacteristic
287Q8—3322no—
288D4—22no—
289D4—21no—
290D4—21no—
291D4—42no—
292D4—81no—
293D4—3322no—
294D4—3322no—
295D4—1661no—
296D4—1661no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—3322no—
307C2×C4—3322no—
308C2×C4—3322no—
309C2×C4—1661no—
310C2×C4—1661no—
311C2×C4—1661no—
312C2×C4C2×C22⋊D8311yescharacteristic
313C2×C4C2×C22⋊D8311yescharacteristic
314C2×C4C2×C22⋊D8311yescharacteristic
315C2×C4C2×C22⋊D8311yescharacteristic
316C2×C4C23×D8311yescharacteristic
317C8—832no—
318C8—1661no—
319C8—1661no—
320C8—1661no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—3321no—
330C22—3321no—
331C22—3321no—
332C22#1458181712yes—
333C22C22×C22⋊D8311yescharacteristic
334C4—21no—
335C4—21no—
336C4—21no—
337C4—21no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—3321no—
342C4—3321no—
343C4—3321no—
344C4#41×D8311yescharacteristic
345C4#1458183911yescharacteristic
346C2—22no—
347C2—41no—
348C2—81no—
349C2—6641no—
350C2#1912835011yescharacteristic
351C2C2×#1458181711yescharacteristic
352C2#1912851211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#553×C83C2
socleC22×C83C22×D4
hypercenter#553D83
hypocenterC83#3345
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C4×D4 ◁ #553

Complements

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

NH ≅ G/Ntype
1. #19128355C2semidirect
2. #19128111C2semidirect
3. #19128025C2semidirect
4. #19128019C2semidirect
5. #19128017C2semidirect
6. #19128012C2semidirect
7. #19127998C2semidirect
8. #19127991C2semidirect
9. #19127952C2semidirect
10. #19127935C2semidirect
11. #19127811C2semidirect
12. #19127773C2semidirect
13. #19127744C2semidirect
14. C2×#14581723C2semidirect
15. #553×C83C2semidirect
143. C83⋊C8#41semidirect
177. C83#3345semidirect
178. #553D83semidirect

Maximal subgroups

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

subgroupindex
#191283552
#191281112
#191280252
#191280192
#191280172
#191280122
#191279982
#191279912
#191279522
#191279352
#191278112
#191277732
#191277442
C2×#145817232
#553×C832
#334583