Finite Group Databasebeta

← #24919716

Subgroups — #24919716

Order 10624.

392 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19128397C211yesmaximal, characteristic, complemented
2#19128055C211yesmaximal, characteristic, complemented
3#19128029C211yesmaximal, characteristic, complemented
4#19128022C211yesmaximal, characteristic, complemented
5#19128008C211yesmaximal, characteristic, complemented
6#19127952C211yesmaximal, characteristic, complemented
7#19127949C211yesmaximal, characteristic, complemented
8#19127913C211yesmaximal, characteristic, complemented
9#19127846C211yesmaximal, characteristic, complemented
10#19127840C211yesmaximal, characteristic, complemented
11C4×#13050577C211yesmaximal, characteristic, complemented
12#19127779C211yesmaximal, characteristic, complemented
13#19127710C211yesmaximal, characteristic, complemented
14#19127702C211yesmaximal, characteristic, complemented
15#548×C83C211yesmaximal, characteristic, complemented
16C2×#13050577C2211yescharacteristic
17C2×#13050577C2211yescharacteristic
18C2×#13050575C2211yescharacteristic
19C2×#13050574C2211yescharacteristic
20C2×#13050574C2211yescharacteristic
21#14581787—21no—
22#14581784C2211yescharacteristic
23#14581774—21no—
24#14581766C2211yescharacteristic
25#14581766C2211yescharacteristic
26#14581764—21no—
27C4×D332C2211yescharacteristic
28(C22⋊D83)⋊C4—21no—
29#14581751C2211yescharacteristic
30C2×C4⋊Dic83C2211yescharacteristic
31#14581725C2211yescharacteristic
32#14581723C2211yescharacteristic
33#14581723C2211yescharacteristic
34C4⋊(C83⋊C8)C2211yescharacteristic
35#14581716C2211yescharacteristic
36#14581716C2211yescharacteristic
37#14581716C2211yescharacteristic
38#14581716C2211yescharacteristic
39#14581716C2211yescharacteristic
40#14581716C2211yescharacteristic
41#14581713C2211yescharacteristic
42#14581713C2211yescharacteristic
43#14581713C2211yescharacteristic
44#14581712C2211yescharacteristic
45C22⋊(C83⋊C8)C2211yescharacteristic
46C22⋊(C83⋊C8)C2211yescharacteristic
47C4×C83⋊C8C2211yescharacteristic
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×D332C2311yescharacteristic
60C2×C22⋊D83—21no—
61C2×C22⋊D83—21no—
62C22×Dic83—21no—
63C2×C4×D83—21no—
64#13050577—22no—
65#13050577—41no—
66#13050575—41no—
67#13050574—41no—
68#13050574—41no—
69C2×C83⋊C8C2311yescharacteristic
70C2×C83⋊C8C2311yescharacteristic
71C2×C83⋊C8C2311yescharacteristic
72C2×C83⋊C8C2311yescharacteristic
73C4⋊Dic83—21no—
74C4⋊Dic83C2311yescharacteristic
75C4⋊Dic83C2311yescharacteristic
76C4⋊Dic83C2311yescharacteristic
77Dic83⋊C4—21no—
78Dic83⋊C4—21no—
79C22⋊Dic83—21no—
80(C2×D83)⋊C4—21no—
81(C2×D83)⋊C4—21no—
82(C2×D83)⋊C4—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
114C22×D83—21no—
115Q8×C83—21no—
116Q8×C83—21no—
117Q8×C83—21no—
118Q8×C83—42no—
119Q8×C83#4112yes—
120D4×C83—21no—
121D4×C83—21no—
122D4×C83—41no—
123D4×C83—42no—
124D4×C83—42no—
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—21no—
132C2×C4×C83—42no—
133C2×C4×C83—42no—
134C2×C4×C83C2×D411yescharacteristic
135C2×C4×C83C2×D411yescharacteristic
136C2×C4×C83C2×D411yescharacteristic
137C2×C4×C83C2×D411yescharacteristic
138C2×C4×C83C2411yescharacteristic
139C22⋊D83—42no—
140C22⋊D83—81no—
141C2×Dic83—21no—
142C2×Dic83—21no—
143C2×Dic83—21no—
144C2×Dic83—81no—
145D332—21no—
146D332—21no—
147C4×D83—81no—
148C83⋊C8—21no—
149C83⋊C8—21no—
150C83⋊C8—21no—
151C83⋊C8—21no—
152C22×C83—21no—
153C22×C83—21no—
154C22×C83—21no—
155C22×C83—21no—
156C22×C83—21no—
157C22×C83—21no—
158C22×C83—41no—
159C22×C83—41no—
160C22×C83—41no—
161C22×C83C22×D411yescharacteristic
162C2×D83—41no—
163C2×D83—41no—
164C2×D83—41no—
165C4×C83—21no—
166C4×C83—21no—
167C4×C83—22no—
168C4×C83—21no—
169C4×C83—21no—
170C4×C83—41no—
171C4×C83—41no—
172C4×C83C2×#4111yescharacteristic
173C4×C83H(5,𝔽2)11yescharacteristic
174Dic83—41no—
175Dic83—41no—
176Dic83—41no—
177C2×C83—41no—
178C2×C83—41no—
179C2×C83—81no—
180C2×C83#54611yescharacteristic
181C2×C83#57511yescharacteristic
182C2×C83#57711yescharacteristic
183D83—81no—
184#3326—831nomaximal, Sylow
185C83#332611yescharacteristic, complemented, Sylow
186#553—831no—
187#548D8311yescharacteristic, complemented
188#488—831no—
189#483—831no—
190#482—831no—
191#479—831no—
192#469—831no—
193#468—831no—
194#463—831no—
195#449—831no—
196#448—831no—
197C4×SD16—831no—
198#436—831no—
199#419—831no—
200#417—831no—
201C2×#41C2×D8311yescharacteristic
202C22×Q8C2×D8311yescharacteristic
203C2×SD16—831no—
204C2×SD16—831no—
205C2×D8—831no—
206C2×M16—831no—
207C2×M16—831no—
208#126—1661no—
209#125—831no—
210#124—21no—
211#124C2×D8311yescharacteristic
212#123—1661no—
213C22⋊Q8—21no—
214C22⋊Q8C2×D8311yescharacteristic
215#121—21no—
216#121—831no—
217#121—831no—
218#121—1661no—
219#121C2×D8311yescharacteristic
220C4×Q8C2×D8311yescharacteristic
221C4×D4—21no—
222C4×D4—831no—
223C4×D4—1661no—
224C4×D4C2×D8311yescharacteristic
225#117—831no—
226C2×C4⋊C4—831no—
227#107—831no—
228#106—831no—
229#106—831no—
230C4⋊C8—831no—
231#103—831no—
232#103—831no—
233#103—831no—
234#103—831no—
235#103—831no—
236#103—831no—
237#102—831no—
238#102—831no—
239#102—831no—
240#102—831no—
241C22⋊C8—831no—
242C22⋊C8—831no—
243C4×C8—831no—
244#41—22no—
245#41—41no—
246C2×Q8—21no—
247C2×Q8—22no—
248C2×Q8—41no—
249C2×Q8C22×D8311yescharacteristic
250C2×Q8C22×D8311yescharacteristic
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—831no—
254C2×D4—1661no—
255C2×D4—1661no—
256C2×D4C22⋊D8311yescharacteristic
257C2×D4C22×D8311yescharacteristic
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—1661no—
261C22×C4—1661no—
262C22×C4C22×D8311yescharacteristic
263C22×C4C22×D8311yescharacteristic
264SD16—3321no—
265SD16—1662no—
266D8—3321no—
267M16—3321no—
268M16—3321no—
269C2×C8—831no—
270C2×C8—831no—
271C2×C8—831no—
272C2×C8—831no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—831no—
276C4⋊C4—831no—
277C4⋊C4—831no—
278C4⋊C4—1661no—
279C4⋊C4—1661no—
280C4⋊C4—1661no—
281C4⋊C4C22⋊D8311yescharacteristic
282C4⋊C4C22×D8311yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—1661no—
288C22⋊C4—1661no—
289C22⋊C4—1661no—
290C22⋊C4—1661no—
291C22⋊C4C22⋊D8311yescharacteristic
292C22⋊C4C22⋊D8311yescharacteristic
293C42—21no—
294C42—1661no—
295C42C22×D8311yescharacteristic
296C23—21no—
297C23—1661no—
298C23C2×C22⋊D8311yescharacteristic
299C23C2×C22⋊D8311yescharacteristic
300Q8—21no—
301Q8—21no—
302Q8—21no—
303Q8—42no—
304Q8#1305058912yes—
305D4—21no—
306D4—21no—
307D4—41no—
308D4—42no—
309D4—42no—
310D4—3322no—
311D4—6641no—
312D4—1661no—
313D4—1661no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—6641no—
324C2×C4—6641no—
325C2×C4—1661no—
326C2×C4—1661no—
327C2×C4—1661no—
328C2×C4C2×C22⋊D8311yescharacteristic
329C2×C4C2×C22⋊D8311yescharacteristic
330C2×C4C2×C22⋊D8311yescharacteristic
331C2×C4C2×C22⋊D8311yescharacteristic
332C2×C4C23×D8311yescharacteristic
333C8—1661no—
334C8—1661no—
335C8—1661no—
336C8—1661no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—3321no—
347C22—3321no—
348C22—3321no—
349C22C22×C22⋊D8311yescharacteristic
350C4—21no—
351C4—21no—
352C4—22no—
353C4—21no—
354C4—21no—
355C4—41no—
356C4—41no—
357C4—3321no—
358C4—3321no—
359C4—3321no—
360C4C2×#1305058911yescharacteristic
361C4#1458183711yescharacteristic
362C2—41no—
363C2—41no—
364C2—81no—
365C2—6641no—
366C2#1912834511yescharacteristic
367C2#1912849911yescharacteristic
368C2#1912850911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#548×C83C2
socleC22×C83C22×D4
hypercenter#548D83
hypocenterC83#3326
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. #19128397C2semidirect
2. #19128055C2semidirect
3. #19128029C2semidirect
4. #19128022C2semidirect
5. #19128008C2semidirect
6. #19127952C2semidirect
7. #19127949C2semidirect
8. #19127913C2semidirect
9. #19127846C2semidirect
10. #19127840C2semidirect
11. C4×#13050577C2semidirect
12. #19127779C2semidirect
13. #19127710C2semidirect
14. #19127702C2semidirect
15. #548×C83C2semidirect
185. C83#3326semidirect
187. #548D83semidirect

Maximal subgroups

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

subgroupindex
#191283972
#191280552
#191280292
#191280222
#191280082
#191279522
#191279492
#191279132
#191278462
#191278402
C4×#130505772
#191277792
#191277102
#191277022
#548×C832
#332683