Finite Group Databasebeta

← #24919568

Subgroups — #24919568

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#19128360C211yesmaximal, characteristic
2#19128063C211yesmaximal, characteristic, complemented
3#19128031C211yesmaximal, characteristic, complemented
4#19128019C211yesmaximal, characteristic, complemented
5#19127983C211yesmaximal, characteristic, complemented
6#19127957C211yesmaximal, characteristic, complemented
7#19127932C211yesmaximal, characteristic, complemented
8#19127923C211yesmaximal, characteristic, complemented
9#19127861C211yesmaximal, characteristic, complemented
10#19127856C211yesmaximal, characteristic, complemented
11#19127806C211yesmaximal, characteristic, complemented
12#19127773C211yesmaximal, characteristic, complemented
13#19127700C211yesmaximal, characteristic, complemented
14C2×#14581715C211yesmaximal, characteristic, complemented
15#554×C83C211yesmaximal, characteristic, complemented
16C2×#13050590C2211yescharacteristic
17C22×Dic166C2211yescharacteristic
18C2×#13050580C2211yescharacteristic
19C2×#13050580C2211yescharacteristic
20C2×#13050577C2211yescharacteristic
21C2×#13050574C2211yescharacteristic
22C22×C83⋊C8C2211yescharacteristic
23#14581780C2211yescharacteristic
24#14581778—21no—
25#14581771C2211yescharacteristic
26#14581770—21no—
27#14581766C2211yescharacteristic
28#14581764—21no—
29C4×Dic166C2211yescharacteristic
30(C22⋊D83)⋊C4—21no—
31#14581723C2211yescharacteristic
32#14581723C2211yescharacteristic
33#14581723C2211yescharacteristic
34C4⋊(C83⋊C8)C2211yescharacteristic
35#14581716C2211yescharacteristic
36#14581715C2211yescharacteristic
37#14581715C2211yescharacteristic
38#14581715C2211yescharacteristic
39#14581715C2211yescharacteristic
40#14581715C2211yescharacteristic
41#14581715C2211yescharacteristic
42#14581712C2211yescharacteristic
43#14581712C2211yescharacteristic
44#14581712C2211yescharacteristic
45C22⋊(C83⋊C8)C2211yescharacteristic
46C22⋊(C83⋊C8)C2211yescharacteristic
47#14581703C2211yescharacteristic
48C4⋊Q8×C83C2211yescharacteristic
49#125×C83—21no—
50C22⋊Q8×C83—21no—
51C22⋊Q8×C83—21no—
52C22⋊Q8×C83C2211yescharacteristic
53C22⋊Q8×C83C2211yescharacteristic
54C4×Q8×C83C2211yescharacteristic
55C4×D4×C83—21no—
56C4×D4×C83C2211yescharacteristic
57C2×C4⋊C4×C83C2211yescharacteristic
58C2×C4⋊C4×C83C2211yescharacteristic
59#13050590—22no—
60#13050590—41no—
61C2×Dic166—22no—
62C2×Dic166—21no—
63C2×Dic166—41no—
64C2×Dic166C2311yescharacteristic
65C2×Dic166C2311yescharacteristic
66C2×D332C2311yescharacteristic
67C2×C22⋊D83—21no—
68C2×C22⋊D83—21no—
69C22×Dic83—21no—
70C2×C4×D83—21no—
71#13050580—22no—
72#13050580—41no—
73#13050577—41no—
74#13050574—41no—
75C2×C83⋊C8—41no—
76C2×C83⋊C8C2311yescharacteristic
77C2×C83⋊C8C2311yescharacteristic
78C2×C83⋊C8C2311yescharacteristic
79C2×C83⋊C8C2311yescharacteristic
80C4⋊Dic83C2311yescharacteristic
81Dic83⋊C4—21no—
82Dic83⋊C4—21no—
83C22⋊Dic83—21no—
84(C2×D83)⋊C4—21no—
85(C2×D83)⋊C4—21no—
86(C2×D83)⋊C4—21no—
87C4×Dic83—21no—
88C2×Q8×C83—21no—
89C2×Q8×C83C2311yescharacteristic
90C2×D4×C83D411yescharacteristic
91C22×C4×C83—21no—
92C22×C4×C83—21no—
93C22×C4×C83C2311yescharacteristic
94C22×C4×C83C2311yescharacteristic
95C4⋊C4×C83—21no—
96C4⋊C4×C83—21no—
97C4⋊C4×C83—21no—
98C4⋊C4×C83—21no—
99C4⋊C4×C83—21no—
100C4⋊C4×C83—21no—
101C4⋊C4×C83D411yescharacteristic
102C4⋊C4×C83C2311yescharacteristic
103C4⋊C4×C83C2311yescharacteristic
104C4⋊C4×C83C2311yescharacteristic
105C22⋊C4×C83—21no—
106C22⋊C4×C83—21no—
107C22⋊C4×C83D411yescharacteristic
108C22⋊C4×C83D411yescharacteristic
109C42×C83—21no—
110C42×C83C2311yescharacteristic
111C23×C83C2×D411yescharacteristic
112C23×C83C2×D411yescharacteristic
113C22×D83—21no—
114Q8×C83—21no—
115Q8×C83—21no—
116Q8×C83—81no—
117D4×C83—22no—
118C2×C4×C83—21no—
119C2×C4×C83—21no—
120C2×C4×C83—21no—
121C2×C4×C83—21no—
122C2×C4×C83—21no—
123C2×C4×C83—21no—
124C2×C4×C83—21no—
125C2×C4×C83—21no—
126C2×C4×C83—81no—
127C2×C4×C83—81no—
128C2×C4×C83C2×D411yescharacteristic
129C2×C4×C83C2×D411yescharacteristic
130C2×C4×C83C2×D411yescharacteristic
131C2×C4×C83C2×D411yescharacteristic
132C2×C4×C83C2411yescharacteristic
133C22⋊D83—42no—
134C22⋊D83—42no—
135C2×Dic83—21no—
136C2×Dic83—21no—
137C2×Dic83—21no—
138C2×Dic83—42no—
139D332—21no—
140D332—21no—
141C4×D83—42no—
142Dic166—21no—
143Dic166—21no—
144Dic166—21no—
145Dic166—42no—
146Dic166#4112yes—
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#13712yes—
157C22×C83C22×D411yescharacteristic
158C2×D83—41no—
159C2×D83—41no—
160C2×D83—41no—
161C4×C83—21no—
162C4×C83—21no—
163C4×C83—21no—
164C4×C83—21no—
165C4×C83—41no—
166C4×C83—41no—
167C4×C83—41no—
168C4×C83—41no—
169C4×C83C2×#4111yescharacteristic
170C4×C83H(5,𝔽2)11yescharacteristic
171Dic83—22no—
172Dic83—41no—
173Dic83—41no—
174C2×C83—22no—
175C2×C83—41no—
176C2×C83#54611yescharacteristic
177C2×C83C2×#13711yescharacteristic
178C2×C83#57711yescharacteristic
179D83—81no—
180#3316—831nomaximal, Sylow
181C83#331611yescharacteristic, complemented, Sylow
182#554D8311yescharacteristic, complemented
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#451—831no—
193#441—831no—
194#435—831no—
195#417—831no—
196C2×#103—831no—
197C2×#41—831no—
198C22×Q8—831no—
199C2×Q16—831no—
200C2×Q16—831no—
201C2×SD16—831no—
202C2×M16—831no—
203C22×C8—831no—
204C4⋊Q8C2×D8311yescharacteristic
205#125—21no—
206#124—831no—
207#124—1661no—
208C22⋊Q8—21no—
209C22⋊Q8—21no—
210C22⋊Q8—831no—
211C22⋊Q8—1661no—
212C22⋊Q8C2×D8311yescharacteristic
213C22⋊Q8C2×D8311yescharacteristic
214#121—831no—
215#121—1661no—
216C4×Q8—831no—
217C4×Q8C2×D8311yescharacteristic
218C4×D4—21no—
219C4×D4—1661no—
220C4×D4C2×D8311yescharacteristic
221C2×C4⋊C4C2×D8311yescharacteristic
222C2×C4⋊C4C2×D8311yescharacteristic
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—3321no—
241#41—1662no—
242C2×Q8—21no—
243C2×Q8—3321no—
244C2×Q8—831no—
245C2×Q8—831no—
246C2×Q8—1662no—
247C2×Q8—1661no—
248C2×Q8C22×D8311yescharacteristic
249C2×D4—831no—
250C2×D4—1661no—
251C2×D4—1661no—
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—21no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—831no—
275C4⋊C4—1661no—
276C4⋊C4—1661no—
277C4⋊C4C22⋊D8311yescharacteristic
278C4⋊C4C22×D8311yescharacteristic
279C4⋊C4C22×D8311yescharacteristic
280C4⋊C4C22×D8311yescharacteristic
281C22⋊C4—21no—
282C22⋊C4—21no—
283C22⋊C4—1661no—
284C22⋊C4—1661no—
285C22⋊C4—1661no—
286C22⋊C4—1661no—
287C22⋊C4C22⋊D8311yescharacteristic
288C22⋊C4C22⋊D8311yescharacteristic
289C42—21no—
290C42—1661no—
291C42C22×D8311yescharacteristic
292C23—1661no—
293C23C2×C22⋊D8311yescharacteristic
294C23C2×C22⋊D8311yescharacteristic
295Q8—21no—
296Q8—21no—
297Q8—81no—
298Q8—3322no—
299Q8—832no—
300Q8—1661no—
301Q8—1661no—
302Q8—1661no—
303D4—22no—
304D4—3322no—
305D4—3322no—
306D4—1661no—
307D4—1661no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—3322no—
319C2×C4—3322no—
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—3321no—
338C22—3321no—
339C22—3321no—
340C22#1458182312yes—
341C22C22×C22⋊D8311yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—3321no—
351C4—3321no—
352C4—1662no—
353C4#41×D8311yescharacteristic
354C4#1458183811yescharacteristic
355C2—22no—
356C2—41no—
357C2—6641no—
358C2#1912833911yescharacteristic
359C2C2×#1458182311yescharacteristic
360C2#1912851211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#554×C83C2
socleC22×C83C22×D4
prime radical#19128360C2
hypercenter#554D83
hypocenterC83#3316
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C4×D4 ◁ #554

Complements

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

NH ≅ G/Ntype
2. #19128063C2semidirect
3. #19128031C2semidirect
4. #19128019C2semidirect
5. #19127983C2semidirect
6. #19127957C2semidirect
7. #19127932C2semidirect
8. #19127923C2semidirect
9. #19127861C2semidirect
10. #19127856C2semidirect
11. #19127806C2semidirect
12. #19127773C2semidirect
13. #19127700C2semidirect
14. C2×#14581715C2semidirect
15. #554×C83C2semidirect
181. C83#3316semidirect
182. #554D83semidirect

Maximal subgroups

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

subgroupindex
#191283602
#191280632
#191280312
#191280192
#191279832
#191279572
#191279322
#191279232
#191278612
#191278562
#191278062
#191277732
#191277002
C2×#145817152
#554×C832
#331683