Finite Group Databasebeta

← #35557749

Subgroups — #35557749

Order 21248.

696 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
1C2×#19128136C211yesmaximal, characteristic
2#24916339C211yesmaximal, characteristic, complemented
3#24916314C211yesmaximal, characteristic, complemented
4#24909336C212yesmaximal, complemented
5#24909330C212yesmaximal, complemented
6#24909312C212yesmaximal, complemented
7#24909307C212yesmaximal, complemented
8#24909252C211yesmaximal, characteristic, complemented
9C2×#19127503C211yesmaximal, characteristic, complemented
10#24909059C211yesmaximal, characteristic, complemented
11C83×#2138C211yesmaximal, characteristic, complemented
12C22×D664C2211yescharacteristic
13#19128184—21no—
14#19128175—21no—
15#19128136C418yescomplemented
16C2×#14581735C2212yes—
17C2×#14581733C2212yes—
18C2×#14581732C2211yescharacteristic
19#19127812—22no—
20#19127802—22no—
21Q16×Dic83C2212yescomplemented
22D8×Dic83C2212yescomplemented
23#19127783C2212yes—
24#19127783C2212yes—
25C2×#14581726C2211yescharacteristic
26#19127705—21no—
27#19127702—21no—
28#19127656C2211yescharacteristic
29#19127528C2212yescomplemented
30#19127526C2212yescomplemented
31#19127503C2212yes—
32#19127503C2212yes—
33#19127476C2212yescomplemented
34C16×Dic83C2212yescomplemented
35C2×#135×C83C2211yescharacteristic
36C22×C16×C83C2211yescharacteristic
37#427×C83C2211yescharacteristic
38#358×C83C2211yescharacteristic, complemented
39#358×C83C2211yescharacteristic, complemented
40#357×C83C2211yescharacteristic, complemented
41#357×C83C2211yescharacteristic, complemented
42C22×D332—21no—
43C2×D664—21no—
44C2×D664—22no—
45C2×D664C2×C414yescomplemented
46C2×D664C2312yes—
47C2×#13050574D411yescharacteristic
48Q8×Dic83—22no—
49C4×D332—41no—
50C4×D332—41no—
51D4×Dic83—22no—
52D4×Dic83—42no—
53#14581754—22no—
54#14581754—22no—
55#14581751—41no—
56#14581751D411yescharacteristic
57C2×C4⋊Dic83—21no—
58C2×C4⋊Dic83—41no—
59C22⋊C4×D83—42no—
60#14581735C2×C418yescomplemented
61#14581733C2×C418yescomplemented
62#14581732C2×C414yescomplemented
63C2×C83⋊C16C2312yes—
64#14581726—21no—
65#14581726C2312yes—
66#14581716—22no—
67#14581713—22no—
68#14581712—22no—
69#14581712—22no—
70#14581702D412yes—
71C8×Dic83C2312yes—
72C2×#41×C83—21no—
73#135×C83—22no—
74#135×C83C2×C414yescomplemented
75C2×Q16×C83C2311yescharacteristic
76C2×SD16×C83—21no—
77C2×D8×C83C2311yescharacteristic
78C22×C8×C83C2311yescharacteristic
79#117×C83—21no—
80C2×C16×C83—22no—
81C2×C16×C83C2311yescharacteristic
82C2×C16×C83C2311yescharacteristic
83#107×C83C2311yescharacteristic
84#107×C83C2311yescharacteristic
85#106×C83—21no—
86C23×D83—41no—
87C2×D332—24no—
88C2×D332—22no—
89C2×D332—41no—
90C2×D332—42no—
91C22×Dic83—41no—
92C22×Dic83—42no—
93C2×C4×D83—42no—
94C2×C4×D83—42no—
95D664—24no—
96D664—22no—
97D664C22×C414yes—
98#13050574—22no—
99C2×C83⋊C8C2×D412yes—
100C4⋊Dic83—22no—
101C4⋊Dic83—41no—
102C4⋊Dic83—42no—
103C4⋊Dic83D811yescharacteristic
104C4⋊Dic83D811yescharacteristic
105C22⋊Dic83—42no—
106C22⋊Dic83D812yes—
107(C2×D83)⋊C4—42no—
108(C2×D83)⋊C4—42no—
109C4×Dic83—42no—
110C4×Dic83C2×D412yes—
111C83⋊C16C22×C414yes—
112#41×C83—24no—
113#41×C83—42no—
114C2×Q8×C83—21no—
115C2×D4×C83—21no—
116C2×D4×C83—41no—
117C22×C4×C83—41no—
118C22×C4×C83C2×D411yescharacteristic
119Q16×C83—21no—
120Q16×C83C22×C412yes—
121SD16×C83—24no—
122D8×C83—21no—
123D8×C83C22×C412yes—
124C2×C8×C83C22×C412yes—
125C2×C8×C83C2×D411yescharacteristic
126C2×C8×C83#4112yes—
127C2×C8×C83C2411yescharacteristic
128C4⋊C4×C83—21no—
129C4⋊C4×C83—21no—
130C22⋊C4×C83—41no—
131C42×C83—41no—
132C16×C83—21no—
133C16×C83—21no—
134C23×C83—41no—
135C23×C83C2×D811yescharacteristic
136C22×D83—42no—
137C22×D83—44no—
138C22×D83—161no—
139C22×D83—161no—
140Q8×C83—22no—
141Q8×C83—41no—
142D4×C83—22no—
143D4×C83—44no—
144D4×C83—41no—
145C2×C4×C83—41no—
146C2×C4×C83—44no—
147C2×C4×C83—41no—
148C2×C4×C83—41no—
149C2×C4×C83C4×D412yes—
150C2×C4×C83C2×D811yescharacteristic
151C2×C4×C83#13512yes—
152C2×C4×C83C22×D411yescharacteristic
153C2×Dic83—42no—
154C2×Dic83—162no—
155C2×Dic83—161no—
156C2×Dic83C2×D812yes—
157C2×Dic83C2×D812yes—
158D332—24no—
159D332—42no—
160D332—44no—
161C4×D83—162no—
162C4×D83—162no—
163C8×C83C4×D412yes—
164C8×C83C23×C411yescharacteristic
165C8×C83C2×#4111yescharacteristic
166C83⋊C8—22no—
167C22×C83—21no—
168C22×C83—21no—
169C22×C83—42no—
170C22×C83—81no—
171C22×C83—81no—
172C22×C83C4×D812yes—
173C22×C83C22×D811yescharacteristic
174C2×D83—44no—
175C2×D83—82no—
176C2×D83—82no—
177C2×D83—164no—
178C4×C83—42no—
179C4×C83—81no—
180C4×C83—81no—
181C4×C83C4×D812yes—
182C4×C83C2×C4×D411yescharacteristic
183C4×C83C2×#13511yescharacteristic
184Dic83—22no—
185Dic83—22no—
186Dic83—82no—
187#33454—831nomaximal, Sylow
188C2×C83—22no—
189C2×C83—82no—
190C2×C83C2×C4×D811yescharacteristic
191C2×C83#341511yescharacteristic
192C2×C83#341611yescharacteristic
193D83—84no—
194C2×#509—831no—
195#2951—831no—
196#2948—831no—
197#2177—832no—
198#2175—832no—
199C4×SD32—832no—
200C4×D16—832no—
201#2159—831no—
202#2138D8311yescharacteristic, complemented
203C2×#357—831no—
204#2108—831no—
205C83#3345411yescharacteristic, complemented, Sylow
206C2×#135C2×D8311yescharacteristic
207C22×D8—831no—
208#520—1661no—
209#518—1661no—
210#509—838no—
211C2×SD32—832no—
212C2×D16—832no—
213C2×M32—831no—
214C22×C16C2×D8311yescharacteristic
215#442—1662no—
216#440—1662no—
217C4×Q16—832no—
218C4×D8—832no—
219C4×D8—832no—
220C4×D8—832no—
221#427C2×D8311yescharacteristic
222C2×#107—831no—
223#418—1661no—
224#417—1661no—
225#405—831no—
226#367—832no—
227#366—832no—
228#358C2×D8311yescharacteristic, complemented
229#358C2×D8311yescharacteristic, complemented
230#357—832no—
231#357—832no—
232#357C2×D8311yescharacteristic, complemented
233#357C2×D8311yescharacteristic, complemented
234#346—832no—
235C4×C16—832no—
236C2×#41—21no—
237C22×D4—1661no—
238#135—22no—
239#135C4×D8314yescomplemented
240C2×Q16C22×D8311yescharacteristic
241C2×SD16—21no—
242C2×D8—834no—
243C2×D8—832no—
244C2×D8—1661no—
245C2×D8—1662no—
246C2×D8C22×D8311yescharacteristic
247C2×M16—831no—
248C22×C8C22×D8311yescharacteristic
249C4×Q8—1662no—
250C4×D4—3321no—
251C4×D4—3321no—
252C4×D4—3322no—
253C4×D4—1662no—
254C4×D4—1662no—
255C4×D4—1662no—
256#117—21no—
257#117—3321no—
258#117—831no—
259C2×C4⋊C4—3321no—
260C2×C4⋊C4—1661no—
261C2×C22⋊C4—3322no—
262SD32—838no—
263D16—838no—
264M32—834no—
265C2×C16—22no—
266C2×C16—832no—
267C2×C16C22×D8311yescharacteristic
268C2×C16C22×D8311yescharacteristic
269#107—832no—
270#107—1661no—
271#107C22×D8311yescharacteristic
272#107C22×D8311yescharacteristic
273#106—21no—
274#103—1662no—
275#102—1662no—
276#102—1662no—
277#102—1662no—
278#97—832no—
279C4×C8—832no—
280C24—3321no—
281#41—24no—
282#41—42no—
283C2×Q8—21no—
284C2×D4—21no—
285C2×D4—41no—
286C2×D4—3321no—
287C2×D4—3322no—
288C2×D4—1662no—
289C2×D4—1664no—
290C22×C4—41no—
291C22×C4—3322no—
292C22×C4—3322no—
293C22×C4—3321no—
294C22×C4—3322no—
295C22×C4D4×D8311yescharacteristic
296Q16—21no—
297Q16C2×C4×D8312yes—
298SD16—24no—
299D8—21no—
300D8—834no—
301D8—1662no—
302D8—1664no—
303D8C2×C4×D8312yes—
304M16—1662no—
305C2×C8—832no—
306C2×C8C2×C4×D8312yes—
307C2×C8D4×D8311yescharacteristic
308C2×C8#1305058912yes—
309C2×C8C23×D8311yescharacteristic
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—3322no—
313C4⋊C4—3321no—
314C4⋊C4—831no—
315C4⋊C4—831no—
316C4⋊C4—1662no—
317C22⋊C4—41no—
318C22⋊C4—3322no—
319C22⋊C4—3322no—
320C22⋊C4—3322no—
321C22⋊C4—832no—
322C42—41no—
323C42—3322no—
324C42—832no—
325C16—21no—
326C16—21no—
327C16—834no—
328C23—41no—
329C23—13281no—
330C23—13281no—
331C23—3324no—
332C23—3322no—
333C23D8×D8311yescharacteristic
334Q8—22no—
335Q8—41no—
336D4—22no—
337D4—44no—
338D4—41no—
339D4—3322no—
340D4—3324no—
341D4—1664no—
342C2×C4—41no—
343C2×C4—44no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—13282no—
347C2×C4—13282no—
348C2×C4—13281no—
349C2×C4—13282no—
350C2×C4—3322no—
351C2×C4—832no—
352C2×C4—832no—
353C2×C4(C22⋊D83)⋊C412yes—
354C2×C4D8×D8311yescharacteristic
355C2×C4#1458181312yes—
356C2×C4C2×D4×D8311yescharacteristic
357C8—1662no—
358C8(C22⋊D83)⋊C412yes—
359C8C22×C4×D8311yescharacteristic
360C8C2×#1305058911yescharacteristic
361C22—21no—
362C22—21no—
363C22—42no—
364C22—81no—
365C22—81no—
366C22—13284no—
367C22—3324no—
368C22—6642no—
369C22—6642no—
370C22#1912778112yes—
371C22C2×D8×D8311yescharacteristic
372C4—42no—
373C4—81no—
374C4—81no—
375C4—6642no—
376C4—1662no—
377C4—1662no—
378C4#1912778112yes—
379C4C2×(C22⋊D83)⋊C411yescharacteristic
380C4C2×#1458181311yescharacteristic
381C2—22no—
382C2—82no—
383C2—6644no—
384C2C2×#1912778111yescharacteristic
385C2#2492060111yescharacteristic
386C2#2492060611yescharacteristic
subgroupG/H
centerC22C2×D8×D83
commutatorC8×C83C23×C4
FrattiniC2×C8C23×D83
FittingC83×#2138C2
socleC22×C83C22×D8
prime radicalC2×#19128136C2
cosocleC2×C8×C83C24
hypercenter#2138D83
hypocenterC83#33454
LCSG ▷ C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2138

Complements

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

NH ≅ G/Ntype
2. #24916339C2semidirect
3. #24916314C2semidirect
4. #24909336C2semidirect
5. #24909330C2semidirect
6. #24909312C2semidirect
7. #24909307C2semidirect
8. #24909252C2semidirect
9. C2×#19127503C2semidirect
10. #24909059C2semidirect
11. C83×#2138C2semidirect
15. #19128136C4semidirect
21. Q16×Dic83C22semidirect
22. D8×Dic83C22semidirect
29. #19127528C22semidirect
30. #19127526C22semidirect
33. #19127476C22semidirect
34. C16×Dic83C22semidirect
38. #358×C83C22semidirect
39. #358×C83C22semidirect
40. #357×C83C22semidirect
41. #357×C83C22semidirect
45. C2×D664C2×C4semidirect
60. #14581735C2×C4semidirect
61. #14581733C2×C4semidirect
62. #14581732C2×C4semidirect
74. #135×C83C2×C4semidirect
202. #2138D83semidirect
205. C83#33454semidirect
228. #358C2×D83semidirect
229. #358C2×D83semidirect
232. #357C2×D83semidirect
233. #357C2×D83semidirect
239. #135C4×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#191281362
#249163392
#249163142
#249093362
#249093302
#249093122
#249093072
#249092522
C2×#191275032
#249090592
C83×#21382
#3345483