Finite Group Databasebeta

← #24919574

Subgroups — #24919574

Order 10624.

388 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#19128408C211yesmaximal, characteristic, complemented
2#19128060C211yesmaximal, characteristic, complemented
3#19128029C211yesmaximal, characteristic, complemented
4#19128017C211yesmaximal, characteristic, complemented
5#19127986C211yesmaximal, characteristic, complemented
6#19127957C211yesmaximal, characteristic, complemented
7#19127928C211yesmaximal, characteristic, complemented
8#19127920C211yesmaximal, characteristic, complemented
9#19127863C211yesmaximal, characteristic, complemented
10#19127857C211yesmaximal, characteristic, complemented
11#19127806C211yesmaximal, characteristic, complemented
12#19127773C211yesmaximal, characteristic, complemented
13#19127702C211yesmaximal, characteristic, complemented
14C2×#14581716C211yesmaximal, characteristic, complemented
15#548×C83C211yesmaximal, characteristic
16C2×#13050580C2211yescharacteristic
17C2×#13050580C2211yescharacteristic
18C2×#13050578C2211yescharacteristic
19C2×#13050574C2211yescharacteristic
20C22×C83⋊C8C2211yescharacteristic
21#14581792C2211yescharacteristic
22#14581783—21no—
23#14581771C2211yescharacteristic
24#14581771C2211yescharacteristic
25#14581769—21no—
26#14581769—21no—
27C4×Dic166C2211yescharacteristic
28D4×Dic83—21no—
29C2×C4⋊Dic83C2211yescharacteristic
30C2×C4⋊Dic83C2211yescharacteristic
31#14581723C2211yescharacteristic
32#14581723C2211yescharacteristic
33#14581723C2211yescharacteristic
34C4⋊(C83⋊C8)C2211yescharacteristic
35#14581716C2211yescharacteristic
36#14581716C2211yescharacteristic
37#14581716C2211yescharacteristic
38#14581716C2211yescharacteristic
39#14581716C2211yescharacteristic
40#14581716C2211yescharacteristic
41#14581715C2211yescharacteristic
42#14581713C2211yescharacteristic
43#14581713C2211yescharacteristic
44#14581713C2211yescharacteristic
45C22⋊(C83⋊C8)C2211yescharacteristic
46C22⋊(C83⋊C8)C2211yescharacteristic
47#14581703C2211yescharacteristic
48C2×#41×C83C2211yescharacteristic
49C22×Q8×C83C2211yescharacteristic
50#124×C83—21no—
51#124×C83C2211yescharacteristic
52C22⋊Q8×C83—21no—
53C22⋊Q8×C83C2211yescharacteristic
54#121×C83—21no—
55#121×C83C2211yescharacteristic
56C4×Q8×C83C2211yescharacteristic
57C4×D4×C83—21no—
58C4×D4×C83C2211yescharacteristic
59C2×Dic166—21no—
60C2×Dic166C2311yescharacteristic
61C22×Dic83—21no—
62C22×Dic83—21no—
63#13050580—22no—
64#13050580—41no—
65#13050578—41no—
66#13050574—41no—
67C2×C83⋊C8—41no—
68C2×C83⋊C8C2311yescharacteristic
69C2×C83⋊C8C2311yescharacteristic
70C2×C83⋊C8C2311yescharacteristic
71C2×C83⋊C8C2311yescharacteristic
72C4⋊Dic83—21no—
73C4⋊Dic83—21no—
74C4⋊Dic83—21no—
75C4⋊Dic83C2311yescharacteristic
76C4⋊Dic83C2311yescharacteristic
77C4⋊Dic83C2311yescharacteristic
78Dic83⋊C4—21no—
79Dic83⋊C4—21no—
80Dic83⋊C4—21no—
81C22⋊Dic83—21no—
82C22⋊Dic83—21no—
83C4×Dic83—21no—
84#41×C83—22no—
85#41×C83—41no—
86C2×Q8×C83—21no—
87C2×Q8×C83—22no—
88C2×Q8×C83—41no—
89C2×Q8×C83C2311yescharacteristic
90C2×Q8×C83C2311yescharacteristic
91C2×D4×C83—21no—
92C2×D4×C83—21no—
93C2×D4×C83D411yescharacteristic
94C2×D4×C83C2311yescharacteristic
95C22×C4×C83—21no—
96C22×C4×C83—21no—
97C22×C4×C83C2311yescharacteristic
98C22×C4×C83C2311yescharacteristic
99C4⋊C4×C83—21no—
100C4⋊C4×C83—21no—
101C4⋊C4×C83D411yescharacteristic
102C4⋊C4×C83C2311yescharacteristic
103C22⋊C4×C83—21no—
104C22⋊C4×C83—21no—
105C22⋊C4×C83—21no—
106C22⋊C4×C83—21no—
107C22⋊C4×C83D411yescharacteristic
108C22⋊C4×C83D411yescharacteristic
109C42×C83—21no—
110C42×C83C2311yescharacteristic
111C23×C83—21no—
112C23×C83C2×D411yescharacteristic
113C23×C83C2×D411yescharacteristic
114Q8×C83—21no—
115Q8×C83—21no—
116Q8×C83—21no—
117Q8×C83—42no—
118Q8×C83#4112yes—
119D4×C83—22no—
120D4×C83—21no—
121D4×C83—21no—
122D4×C83—42no—
123D4×C83—42no—
124C2×C4×C83—21no—
125C2×C4×C83—21no—
126C2×C4×C83—21no—
127C2×C4×C83—21no—
128C2×C4×C83—21no—
129C2×C4×C83—21no—
130C2×C4×C83—21no—
131C2×C4×C83—42no—
132C2×C4×C83—42no—
133C2×C4×C83C2×D411yescharacteristic
134C2×C4×C83C2×D411yescharacteristic
135C2×C4×C83C2×D411yescharacteristic
136C2×C4×C83C2×D411yescharacteristic
137C2×C4×C83C2411yescharacteristic
138C2×Dic83—21no—
139C2×Dic83—21no—
140C2×Dic83—21no—
141C2×Dic83—21no—
142C2×Dic83—81no—
143C2×Dic83—81no—
144Dic166—21no—
145Dic166—21no—
146Dic166—81no—
147C83⋊C8—21no—
148C83⋊C8—21no—
149C83⋊C8—21no—
150C83⋊C8—21no—
151C22×C83—21no—
152C22×C83—21no—
153C22×C83—21no—
154C22×C83—21no—
155C22×C83—21no—
156C22×C83—41no—
157C22×C83—41no—
158C22×C83—41no—
159C22×C83#13712yes—
160C22×C83C22×D411yescharacteristic
161C4×C83—21no—
162C4×C83—21no—
163C4×C83—22no—
164C4×C83—21no—
165C4×C83—21no—
166C4×C83—41no—
167C4×C83—41no—
168C4×C83C2×#4111yescharacteristic
169C4×C83H(5,𝔽2)11yescharacteristic
170Dic83—41no—
171Dic83—41no—
172Dic83—41no—
173Dic83—41no—
174C2×C83—22no—
175C2×C83—41no—
176C2×C83—81no—
177C2×C83#54611yescharacteristic
178C2×C83C2×#13711yescharacteristic
179C2×C83#57711yescharacteristic
180#3316—831nomaximal, Sylow
181C83#331611yescharacteristic, complemented, Sylow
182#554—831no—
183#548D8311yescharacteristic
184#489—831no—
185#483—831no—
186#481—831no—
187#475—831no—
188#470—831no—
189#465—831no—
190#464—831no—
191#452—831no—
192#451—831no—
193#441—831no—
194#435—831no—
195#417—831no—
196C2×#103—831no—
197C2×#41C2×D8311yescharacteristic
198C22×Q8C2×D8311yescharacteristic
199C2×Q16—831no—
200C2×Q16—831no—
201C2×SD16—831no—
202C2×M16—831no—
203C22×C8—831no—
204C4⋊Q8—831no—
205#125—1661no—
206#124—21no—
207#124C2×D8311yescharacteristic
208C22⋊Q8—21no—
209C22⋊Q8—831no—
210C22⋊Q8—831no—
211C22⋊Q8—1661no—
212C22⋊Q8—1661no—
213C22⋊Q8C2×D8311yescharacteristic
214#121—21no—
215#121C2×D8311yescharacteristic
216C4×Q8—831no—
217C4×Q8C2×D8311yescharacteristic
218C4×D4—21no—
219C4×D4—1661no—
220C4×D4C2×D8311yescharacteristic
221C2×C4⋊C4—831no—
222C2×C4⋊C4—831no—
223#106—831no—
224#106—831no—
225#106—831no—
226C4⋊C8—831no—
227#103—831no—
228#103—831no—
229#103—831no—
230#103—831no—
231#103—831no—
232#103—831no—
233#103—831no—
234#102—831no—
235#102—831no—
236#102—831no—
237C22⋊C8—831no—
238C22⋊C8—831no—
239#97—831no—
240#41—22no—
241#41—41no—
242C2×Q8—21no—
243C2×Q8—22no—
244C2×Q8—41no—
245C2×Q8—831no—
246C2×Q8—1661no—
247C2×Q8C22×D8311yescharacteristic
248C2×Q8C22×D8311yescharacteristic
249C2×D4—21no—
250C2×D4—21no—
251C2×D4C22⋊D8311yescharacteristic
252C2×D4C22×D8311yescharacteristic
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—1661no—
256C22×C4—1661no—
257C22×C4C22×D8311yescharacteristic
258C22×C4C22×D8311yescharacteristic
259Q16—3321no—
260Q16—1662no—
261SD16—3321no—
262M16—3321no—
263C2×C8—3321no—
264C2×C8—831no—
265C2×C8—831no—
266C2×C8—831no—
267C2×C8—831no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—831no—
271C4⋊C4—831no—
272C4⋊C4—831no—
273C4⋊C4—1661no—
274C4⋊C4—1661no—
275C4⋊C4—1661no—
276C4⋊C4—1661no—
277C4⋊C4—1661no—
278C4⋊C4—1661no—
279C4⋊C4C22⋊D8311yescharacteristic
280C4⋊C4C22×D8311yescharacteristic
281C22⋊C4—21no—
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—1661no—
286C22⋊C4—1661no—
287C22⋊C4C22⋊D8311yescharacteristic
288C22⋊C4C22⋊D8311yescharacteristic
289C42—21no—
290C42—1661no—
291C42C22×D8311yescharacteristic
292C23—21no—
293C23C2×C22⋊D8311yescharacteristic
294C23C2×C22⋊D8311yescharacteristic
295Q8—21no—
296Q8—21no—
297Q8—21no—
298Q8—42no—
299Q8—6641no—
300Q8—1661no—
301Q8—1661no—
302Q8#1305058912yes—
303D4—22no—
304D4—21no—
305D4—21no—
306D4—42no—
307D4—42no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—6641no—
318C2×C4—6641no—
319C2×C4—1661no—
320C2×C4—1661no—
321C2×C4—1661no—
322C2×C4—1661no—
323C2×C4C2×C22⋊D8311yescharacteristic
324C2×C4C2×C22⋊D8311yescharacteristic
325C2×C4C2×C22⋊D8311yescharacteristic
326C2×C4C2×C22⋊D8311yescharacteristic
327C2×C4C23×D8311yescharacteristic
328C8—1661no—
329C8—1661no—
330C8—1661no—
331C8—1661no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22#1458182212yes—
341C22C22×C22⋊D8311yescharacteristic
342C4—21no—
343C4—21no—
344C4—22no—
345C4—21no—
346C4—21no—
347C4—41no—
348C4—41no—
349C4—3321no—
350C4—3321no—
351C4—3321no—
352C4—3321no—
353C4C2×#1305058911yescharacteristic
354C4#1458183711yescharacteristic
355C2—22no—
356C2—41no—
357C2—81no—
358C2#1912834511yescharacteristic
359C2C2×#1458182211yescharacteristic
360C2#1912850911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting, prime radical#548×C83C2
socleC22×C83C22×D4
hypercenter#548D83
hypocenterC83#3316
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #19128408C2semidirect
2. #19128060C2semidirect
3. #19128029C2semidirect
4. #19128017C2semidirect
5. #19127986C2semidirect
6. #19127957C2semidirect
7. #19127928C2semidirect
8. #19127920C2semidirect
9. #19127863C2semidirect
10. #19127857C2semidirect
11. #19127806C2semidirect
12. #19127773C2semidirect
13. #19127702C2semidirect
14. C2×#14581716C2semidirect
181. C83#3316semidirect

Maximal subgroups

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

subgroupindex
#191284082
#191280602
#191280292
#191280172
#191279862
#191279572
#191279282
#191279202
#191278632
#191278572
#191278062
#191277732
#191277022
C2×#145817162
#548×C832
#331683