Finite Group Databasebeta

← #35389680

Subgroups — #35389680

Order 21248.

686 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24918406C211yesmaximal, characteristic
2#24917268C211yesmaximal, characteristic, complemented
3C2×#19127901C211yesmaximal, characteristic
4#24906879C214yesmaximal, complemented
5#24906811C214yesmaximal, complemented
6#24906234C211yesmaximal, characteristic, complemented
7#24906130C211yesmaximal, characteristic
8#24906118C211yesmaximal, characteristic, complemented
9C83×#2982C211yesmaximal, characteristic, complemented
10C22×#13050577—21no—
11#19128258C2211yescharacteristic
12C2×#14581789C2211yescharacteristic
13#19128074C2212yes—
14#19128048C2212yes—
15#19128008—22no—
16#19127996C2212yescomplemented
17#19127986C2212yescomplemented
18#19127902—22no—
19#19127901—22no—
20#19127901C2214yes—
21#19127854—22no—
22#19127741—21no—
23#19127725C2211yescharacteristic
24C2×C4⋊(C83⋊C8)C2211yescharacteristic
25#19127710—21no—
26C2×#14581712—21no—
27#19127452C2214yescomplemented
28#19127449C2212yescomplemented
29#19127448C2212yescomplemented
30#19127444C2214yes—
31C2×C4×Q8×C83C2211yescharacteristic
32C8×Q8×C83—22no—
33C8×Q8×C83C2214yescomplemented
34#431×C83—22no—
35#431×C83C2211yescharacteristic
36#431×C83C2211yescharacteristic
37C22×D332—21no—
38C22×D332—21no—
39C22×D332—21no—
40C2×#13050579—24no—
41C2×#13050577—24no—
42C2×#13050577—44no—
43C2×#13050574—21no—
44C22×C83⋊C8—21no—
45#14581792C2311yescharacteristic
46#14581792C2311yescharacteristic
47#14581789—21no—
48#14581789—21no—
49#14581789—21no—
50#14581789C2312yes—
51#14581784—21no—
52#14581771—21no—
53#14581771—21no—
54C4×Dic166—21no—
55C4×Dic166—21no—
56#14581751—21no—
57#14581725—22no—
58#14581723—22no—
59C4⋊(C83⋊C8)—21no—
60C4⋊(C83⋊C8)—21no—
61C4⋊(C83⋊C8)C2312yes—
62C4⋊(C83⋊C8)C2312yes—
63#14581716—22no—
64#14581716—22no—
65#14581714—24no—
66#14581712—24no—
67C22×Q8×C83D411yescharacteristic
68C4×Q8×C83—21no—
69C4×Q8×C83—21no—
70C4×Q8×C83D412yes—
71C4×Q8×C83C2312yes—
72C2×C4⋊C4×C83—21no—
73C2×C4⋊C4×C83D411yescharacteristic
74C2×C42×C83—21no—
75C2×C42×C83C2311yescharacteristic
76C4⋊C8×C83—24no—
77C4⋊C8×C83D412yes—
78C4⋊C8×C83C2312yes—
79C22⋊C8×C83—21no—
80C22⋊C8×C83D412yes—
81C4×C8×C83—24no—
82C4×C8×C83—21no—
83C4×C8×C83C2312yes—
84C23×D83—41no—
85C2×Dic166—21no—
86C2×Dic166—21no—
87C2×D332—22no—
88C2×D332—22no—
89C2×D332—22no—
90C2×D332—42no—
91C2×D332—42no—
92C2×D332—41no—
93C2×D332—41no—
94C2×D332—41no—
95C2×D332—82no—
96C2×D332—82no—
97#13050579—84no—
98#13050577—48no—
99#13050574—42no—
100C2×C83⋊C8—22no—
101C2×C83⋊C8—22no—
102C2×C83⋊C8—42no—
103C4⋊Dic83—21no—
104C4⋊Dic83—21no—
105C4⋊Dic83—21no—
106C4⋊Dic83—21no—
107Dic83⋊C4—41no—
108Dic83⋊C4—41no—
109C22⋊Dic83—41no—
110C4×Dic83—41no—
111C2×Q8×C83—21no—
112C2×Q8×C83—22no—
113C2×Q8×C83SD1614yescomplemented
114C2×Q8×C83C2×D412yes—
115C22×C4×C83—21no—
116C22×C4×C83—21no—
117C22×C4×C83C2×D411yescharacteristic
118C22×C4×C83C2×D411yescharacteristic
119C22×C4×C83#4111yescharacteristic
120C2×C8×C83—22no—
121C2×C8×C83C2×D414yes—
122C4⋊C4×C83—22no—
123C4⋊C4×C83—21no—
124C4⋊C4×C83—22no—
125C4⋊C4×C83C2×D412yes—
126C42×C83—22no—
127C42×C83—22no—
128C42×C83C2×D411yescharacteristic
129C42×C83C2×D411yescharacteristic
130C42×C83#4111yescharacteristic
131C42×C83C2411yescharacteristic
132C23×C83#12111yescharacteristic
133C22×D83—42no—
134C22×D83—82no—
135C22×D83—82no—
136C22×D83—82no—
137Q8×C83—22no—
138Q8×C83—24no—
139Q8×C83C2×SD1614yes—
140C2×C4×C83—22no—
141C2×C4×C83—24no—
142C2×C4×C83—22no—
143C2×C4×C83—22no—
144C2×C4×C83—41no—
145C2×C4×C83—81no—
146C2×C4×C83#12111yescharacteristic
147C2×C4×C83#12111yescharacteristic
148C2×C4×C83#12111yescharacteristic
149C2×C4×C83C2×SD1612yes—
150C2×C4×C83#13512yes—
151C2×C4×C83C22×D411yescharacteristic
152C2×C4×C83C22×D411yescharacteristic
153C2×C4×C83C2×#4111yescharacteristic
154C2×Dic83—41no—
155C2×Dic83—41no—
156D332—42no—
157D332—42no—
158D332—44no—
159D332—82no—
160D332—82no—
161D332—82no—
162D332—82no—
163Dic166—42no—
164Dic166—81no—
165C8×C83—24no—
166C8×C83—42no—
167C83⋊C8—42no—
168C83⋊C8—42no—
169C22×C83—21no—
170C22×C83—21no—
171C22×C83#46512yes—
172C22×C83C2×#12111yescharacteristic
173C2×D83—82no—
174C2×D83—82no—
175C2×D83—82no—
176C2×D83—164no—
177C4×C83—21no—
178C4×C83—24no—
179C4×C83—21no—
180C4×C83—21no—
181C4×C83—42no—
182C4×C83#46512yes—
183C4×C83C22×SD1611yescharacteristic
184C4×C83C2×#13511yescharacteristic
185C4×C83#57712yes—
186Dic83—81no—
187Dic83—81no—
188#19396—831nomaximal, Sylow
189C2×C83—22no—
190C2×C83#304311yescharacteristic
191C2×C83C2×#46511yescharacteristic
192C2×C83#306811yescharacteristic
193D83—162no—
194#3208—831no—
195#3078—831no—
196C2×#460—831no—
197#2982D8311yescharacteristic, complemented
198#1678—834no—
199#1666—834no—
200#1570—831no—
201#1545—831no—
202#1542—831no—
203C83#1939611yescharacteristic, complemented, Sylow
204C22×SD16—1661no—
205#533—831no—
206C2×#127—831no—
207C2×C4×Q8C2×D8311yescharacteristic
208#492—832no—
209#486—832no—
210#479—1662no—
211#477—832no—
212#475—832no—
213#460—834no—
214#460—1662no—
215#460—1662no—
216#450—1662no—
217C8×Q8—22no—
218C8×Q8C2×D8314yescomplemented
219#431—22no—
220#431C2×D8311yescharacteristic
221#431C2×D8311yescharacteristic
222#428—1661no—
223#423—831no—
224C2×C4⋊C8—831no—
225#419—1661no—
226C2×#102—1661no—
227#334—834no—
228#333—832no—
229#332—832no—
230#331—834no—
231C22×Q8D33211yescharacteristic
232C22×D4—1661no—
233C22×D4—1661no—
234C22×D4—1661no—
235C2×SD16—3324no—
236C2×SD16—1664no—
237C2×SD16—1664no—
238C2×M16—1661no—
239C22×C8—1661no—
240C4⋊Q8—831no—
241C4⋊Q8—831no—
242#127—832no—
243#127—1661no—
244#127—1661no—
245#127—1661no—
246#125—1661no—
247C22⋊Q8—1661no—
248C22⋊Q8—1661no—
249C4×Q8—21no—
250C4×Q8—21no—
251C4×Q8—1661no—
252C4×Q8—1661no—
253C4×Q8D33212yes—
254C4×Q8C22×D8312yes—
255#117—1661no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4D33211yescharacteristic
258C2×C42—21no—
259C2×C42C22×D8311yescharacteristic
260#107—1662no—
261#106—1662no—
262C4⋊C8—24no—
263C4⋊C8—832no—
264C4⋊C8—832no—
265C4⋊C8—1661no—
266C4⋊C8—1661no—
267C4⋊C8C22⋊D8312yes—
268C4⋊C8C22×D8312yes—
269#103—1662no—
270#103—1662no—
271#102—1664no—
272#102—1664no—
273C22⋊C8—21no—
274C22⋊C8C22⋊D8312yes—
275C4×C8—24no—
276C4×C8—21no—
277C4×C8C22×D8312yes—
278C24—3321no—
279C2×Q8—21no—
280C2×Q8—22no—
281C2×Q8—1661no—
282C2×Q8—1661no—
283C2×Q8#1305057914yescomplemented
284C2×Q8C2×D33212yes—
285C2×D4—3321no—
286C2×D4—3321no—
287C2×D4—3321no—
288C2×D4—3322no—
289C2×D4—3322no—
290C2×D4—6642no—
291C2×D4—6642no—
292C2×D4—1662no—
293C2×D4—1662no—
294C2×D4—1662no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4C2×C22⋊D8311yescharacteristic
298C22×C4C2×D33211yescharacteristic
299C22×C4#1305059011yescharacteristic
300SD16—3328no—
301SD16—6644no—
302M16—3322no—
303C2×C8—22no—
304C2×C8—3322no—
305C2×C8—1662no—
306C2×C8—1662no—
307C2×C8C2×C22⋊D8314yes—
308C4⋊C4—22no—
309C4⋊C4—21no—
310C4⋊C4—22no—
311C4⋊C4—3321no—
312C4⋊C4—3321no—
313C4⋊C4—1661no—
314C4⋊C4—1661no—
315C4⋊C4—1661no—
316C4⋊C4—1661no—
317C4⋊C4C2×D33212yes—
318C22⋊C4—3321no—
319C42—22no—
320C42—22no—
321C42—3321no—
322C42C2×C22⋊D8311yescharacteristic
323C42C2×D33211yescharacteristic
324C42#1305059011yescharacteristic
325C42C23×D8311yescharacteristic
326C23—3322no—
327C23—6642no—
328C23—6642no—
329C23—6642no—
330C23#1458176611yescharacteristic
331Q8—24no—
332Q8—22no—
333Q8—3322no—
334Q8—6641no—
335Q8C2×#1305057914yes—
336D4—3322no—
337D4—3322no—
338D4—3324no—
339D4—6642no—
340D4—6642no—
341D4—6642no—
342D4—6642no—
343C2×C4—24no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—41no—
348C2×C4—81no—
349C2×C4—3321no—
350C2×C4—3321no—
351C2×C4#1458176611yescharacteristic
352C2×C4#1458176611yescharacteristic
353C2×C4#1458176611yescharacteristic
354C2×C4C2×#1305057912yes—
355C2×C4#1458181412yes—
356C2×C4C22×C22⋊D8311yescharacteristic
357C2×C4C22×D33211yescharacteristic
358C2×C4C2×#1305059011yescharacteristic
359C8—24no—
360C8—42no—
361C8—3322no—
362C8—3322no—
363C22—21no—
364C22—21no—
365C22—13284no—
366C22—6642no—
367C22—6642no—
368C22—6642no—
369C22#1912793012yes—
370C22C2×#1458176611yescharacteristic
371C4—24no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—42no—
376C4—6641no—
377C4—6641no—
378C4#1912793012yes—
379C4C22×#1305057911yescharacteristic
380C4C2×#1458181411yescharacteristic
381C4#1912850912yes—
382C2—22no—
383C2—13282no—
384C2#2491698411yescharacteristic
385C2C2×#1912793011yescharacteristic
386C2#2491719111yescharacteristic
subgroupG/H
centerC22C2×#14581766
commutator, cosocleC42×C83C24
FrattiniC42C23×D83
FittingC83×#2982C2
socleC22×C83C2×#121
prime radicalC2×#14581789C22
hypercenter#2982D83
hypocenterC83#19396
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2982

Complements

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

NH ≅ G/Ntype
2. #24917268C2semidirect
4. #24906879C2semidirect
5. #24906811C2semidirect
6. #24906234C2semidirect
8. #24906118C2semidirect
9. C83×#2982C2semidirect
16. #19127996C22semidirect
17. #19127986C22semidirect
27. #19127452C22semidirect
28. #19127449C22semidirect
29. #19127448C22semidirect
33. C8×Q8×C83C22semidirect
113. C2×Q8×C83SD16semidirect
197. #2982D83semidirect
203. C83#19396semidirect
218. C8×Q8C2×D83semidirect
283. C2×Q8#13050579semidirect

Maximal subgroups

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

subgroupindex
#249184062
#249172682
C2×#191279012
#249068792
#249068112
#249062342
#249061302
#249061182
C83×#29822
#1939683