Finite Group Databasebeta

← #24919578

Subgroups — #24919578

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#19128409C211yesmaximal, characteristic, complemented
2#19128362C211yesmaximal, characteristic
3#19128062C211yesmaximal, characteristic, complemented
4#19128030C211yesmaximal, characteristic, complemented
5#19128020C211yesmaximal, characteristic, complemented
6#19127984C211yesmaximal, characteristic, complemented
7#19127956C211yesmaximal, characteristic, complemented
8#19127932C211yesmaximal, characteristic, complemented
9#19127925C211yesmaximal, characteristic, complemented
10#19127862C211yesmaximal, characteristic, complemented
11#19127808C211yesmaximal, characteristic, complemented
12#19127769C211yesmaximal, characteristic, complemented
13#19127701C211yesmaximal, characteristic, complemented
14C2×#14581716C211yesmaximal, characteristic, complemented
15#451×C83C211yesmaximal, characteristic, complemented
16C2×#13050591C2211yescharacteristic
17C2×#13050577C2211yescharacteristic
18C2×#13050573C2211yescharacteristic
19C22×C83⋊C8C2211yescharacteristic
20#14581791C2211yescharacteristic
21#14581782—21no—
22#14581781—21no—
23#14581781C2211yescharacteristic
24#14581772—21no—
25#14581769—21no—
26#14581768—21no—
27#14581768C2211yescharacteristic
28#14581768C2211yescharacteristic
29#14581766—21no—
30#14581766C2211yescharacteristic
31Q8×Dic83C2211yescharacteristic
32Q8×Dic83C2211yescharacteristic
33C4×C22⋊D83—21no—
34C4×C22⋊D83C2211yescharacteristic
35(C22⋊D83)⋊C4—21no—
36C2×C4⋊Dic83C2211yescharacteristic
37C4⋊C4×D83C2211yescharacteristic
38#14581724C2211yescharacteristic
39#14581724C2211yescharacteristic
40#14581723C2211yescharacteristic
41Dic83⋊C8C2211yescharacteristic
42#14581716C2211yescharacteristic
43#14581716C2211yescharacteristic
44#14581716C2211yescharacteristic
45#14581716C2211yescharacteristic
46#14581716C2211yescharacteristic
47#14581714C2211yescharacteristic
48#14581714C2211yescharacteristic
49#14581712C2211yescharacteristic
50(C2×D83)⋊C8C2211yescharacteristic
51#14581702C2211yescharacteristic
52C22×Q8×C83C2211yescharacteristic
53C2×Q16×C83C2211yescharacteristic
54C2×Q16×C83C2211yescharacteristic
55C22⋊Q8×C83C2211yescharacteristic
56#103×C83C2211yescharacteristic
57#103×C83C2211yescharacteristic
58C22⋊C8×C83C2211yescharacteristic
59#13050591—22no—
60#13050591—41no—
61C2×Dic166—21no—
62C2×D332—21no—
63C2×D332C2311yescharacteristic
64C2×C22⋊D83—21no—
65C2×C22⋊D83D411yescharacteristic
66C22×Dic83—21no—
67C2×C4×D83—21no—
68C2×C4×D83—21no—
69C2×C4×D83C2311yescharacteristic
70#13050577—41no—
71#13050573—41no—
72C2×C83⋊C8—41no—
73C2×C83⋊C8C2311yescharacteristic
74C2×C83⋊C8C2311yescharacteristic
75C4⋊Dic83—21no—
76C4⋊Dic83—21no—
77C4⋊Dic83—21no—
78C4⋊Dic83C2311yescharacteristic
79C4⋊Dic83C2311yescharacteristic
80C4⋊Dic83C2311yescharacteristic
81Dic83⋊C4—21no—
82Dic83⋊C4—21no—
83Dic83⋊C4—21no—
84Dic83⋊C4—21no—
85Dic83⋊C4D411yescharacteristic
86C22⋊Dic83D411yescharacteristic
87(C2×D83)⋊C4—21no—
88(C2×D83)⋊C4—21no—
89(C2×D83)⋊C4—21no—
90(C2×D83)⋊C4—21no—
91(C2×D83)⋊C4—21no—
92(C2×D83)⋊C4D411yescharacteristic
93C4×Dic83—21no—
94C4×Dic83—21no—
95C4×Dic83C2311yescharacteristic
96C2×Q8×C83—21no—
97C2×Q8×C83—22no—
98C2×Q8×C83—41no—
99C2×Q8×C83C2311yescharacteristic
100C2×Q8×C83C2311yescharacteristic
101C2×Q8×C83C2311yescharacteristic
102C22×C4×C83—21no—
103C22×C4×C83C2311yescharacteristic
104Q16×C83—22no—
105Q16×C83—41no—
106C2×C8×C83C2311yescharacteristic
107C2×C8×C83C2311yescharacteristic
108C4⋊C4×C83—21no—
109C4⋊C4×C83C2311yescharacteristic
110C22⋊C4×C83—21no—
111C23×C83C2×D411yescharacteristic
112C22×D83—21no—
113C22×D83C2×D411yescharacteristic
114Q8×C83—21no—
115Q8×C83—21no—
116Q8×C83—21no—
117Q8×C83—21no—
118Q8×C83—21no—
119Q8×C83—42no—
120Q8×C83#4112yes—
121C2×C4×C83—21no—
122C2×C4×C83—21no—
123C2×C4×C83—21no—
124C2×C4×C83—21no—
125C2×C4×C83—21no—
126C2×C4×C83—21no—
127C2×C4×C83—42no—
128C2×C4×C83C2×D411yescharacteristic
129C2×C4×C83C2411yescharacteristic
130C22⋊D83—22no—
131C22⋊D83—42no—
132C2×Dic83—21no—
133C2×Dic83—21no—
134C2×Dic83—21no—
135C2×Dic83—81no—
136C2×Dic83C2×D411yescharacteristic
137C2×Dic83C2×D411yescharacteristic
138C2×Dic83C2×D411yescharacteristic
139D332—21no—
140D332—21no—
141D332—42no—
142C4×D83—21no—
143C4×D83—21no—
144C4×D83—42no—
145C4×D83—81no—
146Dic166—81no—
147C8×C83—21no—
148C8×C83—21no—
149C83⋊C8—21no—
150C83⋊C8—21no—
151C22×C83—21no—
152C22×C83—21no—
153C22×C83#13712yes—
154C22×C83C22×D411yescharacteristic
155C2×D83—21no—
156C2×D83—21no—
157C2×D83—21no—
158C2×D83—41no—
159C2×D83—41no—
160C2×D83—41no—
161C4×C83—22no—
162C4×C83—21no—
163C4×C83—41no—
164C4×C83—41no—
165C4×C83—41no—
166C4×C83C2×#4111yescharacteristic
167C4×C83H(5,𝔽2)11yescharacteristic
168Dic83—21no—
169Dic83—21no—
170Dic83—21no—
171Dic83—41no—
172Dic83—41no—
173Dic83—41no—
174C2×C83—22no—
175C2×C83#54611yescharacteristic
176C2×C83C2×#13711yescharacteristic
177C2×C83#57711yescharacteristic
178D83—41no—
179D83—81no—
180#3316—831nomaximal, Sylow
181C83#331611yescharacteristic, complemented, Sylow
182#554—831no—
183#548—831no—
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#451D8311yescharacteristic, complemented
193#441—831no—
194#435—831no—
195#417—831no—
196C2×#103—831no—
197C2×#41—831no—
198C22×Q8C2×D8311yescharacteristic
199C2×Q16C2×D8311yescharacteristic
200C2×Q16C2×D8311yescharacteristic
201C2×SD16—831no—
202C2×M16—831no—
203C22×C8—831no—
204C4⋊Q8—831no—
205#125—1661no—
206#124—831no—
207#124—1661no—
208C22⋊Q8—831no—
209C22⋊Q8—831no—
210C22⋊Q8—1661no—
211C22⋊Q8—1661no—
212C22⋊Q8—1661no—
213C22⋊Q8C2×D8311yescharacteristic
214#121—831no—
215#121—1661no—
216C4×Q8—831no—
217C4×Q8—831no—
218C4×D4—831no—
219C4×D4—1661no—
220C4×D4—1661no—
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#103C2×D8311yescharacteristic
233#103C2×D8311yescharacteristic
234#102—831no—
235#102—831no—
236#102—831no—
237C22⋊C8—831no—
238C22⋊C8C2×D8311yescharacteristic
239#97—831no—
240#41—3321no—
241#41—1662no—
242C2×Q8—21no—
243C2×Q8—22no—
244C2×Q8—41no—
245C2×Q8—1661no—
246C2×Q8C22×D8311yescharacteristic
247C2×Q8C22×D8311yescharacteristic
248C2×Q8C22×D8311yescharacteristic
249C2×D4—831no—
250C2×D4—831no—
251C2×D4—1661no—
252C2×D4—1661no—
253C22×C4—21no—
254C22×C4—831no—
255C22×C4—1661no—
256C22×C4—1661no—
257C22×C4—1661no—
258C22×C4C22×D8311yescharacteristic
259Q16—22no—
260Q16—41no—
261SD16—3321no—
262M16—3321no—
263C2×C8—3321no—
264C2×C8—831no—
265C2×C8—831no—
266C2×C8C22×D8311yescharacteristic
267C2×C8C22×D8311yescharacteristic
268C4⋊C4—21no—
269C4⋊C4—831no—
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⋊C4—1661no—
280C4⋊C4C22×D8311yescharacteristic
281C22⋊C4—21no—
282C22⋊C4—831no—
283C22⋊C4—831no—
284C22⋊C4—1661no—
285C22⋊C4—1661no—
286C22⋊C4—1661no—
287C22⋊C4—1661no—
288C22⋊C4—1661no—
289C42—831no—
290C42—1661no—
291C42—1661no—
292C23—831no—
293C23—1661no—
294C23D4×D8311yescharacteristic
295Q8—21no—
296Q8—21no—
297Q8—21no—
298Q8—21no—
299Q8—21no—
300Q8—42no—
301Q8—6641no—
302Q8#1305058912yes—
303D4—3322no—
304D4—3322no—
305D4—1662no—
306D4—1661no—
307D4—1661no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—42no—
315C2×C4—3322no—
316C2×C4—831no—
317C2×C4—831no—
318C2×C4—831no—
319C2×C4—6641no—
320C2×C4—6641no—
321C2×C4—1661no—
322C2×C4—1661no—
323C2×C4—1661no—
324C2×C4—1661no—
325C2×C4—1661no—
326C2×C4D4×D8311yescharacteristic
327C2×C4C23×D8311yescharacteristic
328C8—21no—
329C8—21no—
330C8—1661no—
331C8—1661no—
332C22—21no—
333C22—21no—
334C22—3321no—
335C22—3321no—
336C22—3321no—
337C22—1661no—
338C22—1661no—
339C22—1661no—
340C22#1458182412yes—
341C22C2×D4×D8311yescharacteristic
342C4—21no—
343C4—22no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—3321no—
348C4—3321no—
349C4—3321no—
350C4—1661no—
351C4—1661no—
352C4—1661no—
353C4C2×#1305058911yescharacteristic
354C4#1458183711yescharacteristic
355C2—22no—
356C2—3321no—
357C2—6641no—
358C2#1912833811yescharacteristic
359C2C2×#1458182411yescharacteristic
360C2#1912851011yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#451×C83C2
socleC22×C83C22×D4
prime radical#19128362C2
hypercenter#451D83
hypocenterC83#3316
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. #19128409C2semidirect
3. #19128062C2semidirect
4. #19128030C2semidirect
5. #19128020C2semidirect
6. #19127984C2semidirect
7. #19127956C2semidirect
8. #19127932C2semidirect
9. #19127925C2semidirect
10. #19127862C2semidirect
11. #19127808C2semidirect
12. #19127769C2semidirect
13. #19127701C2semidirect
14. C2×#14581716C2semidirect
15. #451×C83C2semidirect
181. C83#3316semidirect
192. #451D83semidirect

Maximal subgroups

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

subgroupindex
#191284092
#191283622
#191280622
#191280302
#191280202
#191279842
#191279562
#191279322
#191279252
#191278622
#191278082
#191277692
#191277012
C2×#145817162
#451×C832
#331683