Finite Group Databasebeta

← #35557746

Subgroups — #35557746

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×#19128137C211yesmaximal, characteristic
2#24916340C211yesmaximal, characteristic, complemented
3#24909337C212yesmaximal, complemented
4#24909327C212yesmaximal, complemented
5C4×#14581736C212yesmaximal, complemented
6C4×#14581733C212yesmaximal, complemented
7#24909252C211yesmaximal, characteristic, complemented
8#24909168C211yesmaximal, characteristic, complemented
9C2×#19127502C211yesmaximal, characteristic, complemented
10#24909060C211yesmaximal, characteristic, complemented
11C83×#2948C211yesmaximal, characteristic, complemented
12C2×#14581814C2211yescharacteristic
13#19128181—21no—
14#19128137C418yescomplemented
15C2×#14581736C2212yes—
16C2×#14581733C2212yes—
17C2×#14581732C2211yescharacteristic
18C22×C83⋊C16C2211yescharacteristic
19#19127803—22no—
20C4×Dic332C2212yescomplemented
21C4×D664C2212yescomplemented
22C2×#14581726C2211yescharacteristic
23#19127703—21no—
24#19127528C2212yescomplemented
25#19127526C2212yescomplemented
26#19127504C2211yescharacteristic, complemented
27#19127504C2211yescharacteristic, complemented
28#19127502C2212yes—
29#19127502C2212yes—
30#19127501C2211yescharacteristic, complemented
31#19127501C2211yescharacteristic, complemented
32#19127477C2212yescomplemented
33C4×C83⋊C16C2212yescomplemented
34C22×D8×C83C2211yescharacteristic
35#518×C83—21no—
36#442×C83—22no—
37C4×D8×C83C2212yes—
38C4×D8×C83C2212yes—
39#427×C83C2211yescharacteristic
40#418×C83—21no—
41#405×C83C2211yescharacteristic
42C2×#13050590—21no—
43#14581814—22no—
44#14581814C2×C414yescomplemented
45C2×Dic332C2311yescharacteristic
46C2×#13050579—21no—
47C2×D664C2311yescharacteristic
48C4×Dic166—22no—
49C4×D332—22no—
50(C22⋊D83)⋊C4—42no—
51#14581750—41no—
52C2×C4⋊Dic83—21no—
53C4⋊C4×D83—41no—
54#14581736C2×C418yescomplemented
55#14581733C2×C418yescomplemented
56#14581732C2×C414yescomplemented
57C2×C83⋊C16—22no—
58C2×C83⋊C16C2311yescharacteristic
59C2×C83⋊C16C2311yescharacteristic
60C2×C83⋊C16C2312yes—
61#14581726—21no—
62#14581726C2312yes—
63#14581717—22no—
64#14581714—22no—
65C22×D4×C83—21no—
66C2×D8×C83—21no—
67C2×D8×C83—22no—
68C2×D8×C83C2×C414yescomplemented
69C2×D8×C83C2312yes—
70C2×M16×C83D411yescharacteristic
71C22×C8×C83C2311yescharacteristic
72C4×D4×C83—22no—
73C4×D4×C83—22no—
74C4×D4×C83—41no—
75C4×D4×C83—41no—
76#117×C83—21no—
77#117×C83D411yescharacteristic
78C2×C22⋊C4×C83—42no—
79#107×C83C2311yescharacteristic
80#107×C83C2311yescharacteristic
81#106×C83—21no—
82#102×C83—22no—
83#102×C83—22no—
84#97×C83D412yes—
85C4×C8×C83C2312yes—
86#13050590—24no—
87#13050590—42no—
88C2×Dic166—21no—
89C2×D332—21no—
90C2×C22⋊D83—41no—
91C22×Dic83—41no—
92C2×C4×D83—41no—
93C2×C4×D83—42no—
94Dic332—21no—
95Dic332C22×C412yes—
96#13050579—24no—
97D664—21no—
98D664C22×C412yes—
99C4⋊Dic83—22no—
100C4⋊Dic83—41no—
101Dic83⋊C4—42no—
102(C2×D83)⋊C4—42no—
103C4×Dic83—42no—
104C83⋊C16—21no—
105C83⋊C16—21no—
106C83⋊C16C22×C414yes—
107C24×C83—41no—
108C2×D4×C83—24no—
109C2×D4×C83—22no—
110C2×D4×C83—41no—
111C2×D4×C83—42no—
112C22×C4×C83—42no—
113C22×C4×C83—42no—
114C22×C4×C83C2×D411yescharacteristic
115D8×C83—24no—
116D8×C83—22no—
117D8×C83C22×C414yes—
118M16×C83—22no—
119C2×C8×C83C22×C412yes—
120C2×C8×C83C2×D412yes—
121C2×C8×C83C2×D411yescharacteristic
122C2×C8×C83#4112yes—
123C2×C8×C83C2411yescharacteristic
124C4⋊C4×C83—21no—
125C4⋊C4×C83—21no—
126C4⋊C4×C83D811yescharacteristic
127C4⋊C4×C83D811yescharacteristic
128C22⋊C4×C83—41no—
129C22⋊C4×C83—42no—
130C22⋊C4×C83—42no—
131C22⋊C4×C83D812yes—
132C42×C83—41no—
133C42×C83C2×D412yes—
134C23×C83—42no—
135C23×C83—44no—
136C23×C83—161no—
137C23×C83—161no—
138C23×C83C2×D811yescharacteristic
139C22×D83—41no—
140D4×C83—24no—
141D4×C83—42no—
142D4×C83—44no—
143C2×C4×C83—41no—
144C2×C4×C83—41no—
145C2×C4×C83—162no—
146C2×C4×C83—162no—
147C2×C4×C83C4×D412yes—
148C2×C4×C83C2×D812yes—
149C2×C4×C83C2×D812yes—
150C2×C4×C83C2×D811yescharacteristic
151C2×C4×C83#13512yes—
152C2×C4×C83C22×D411yescharacteristic
153C22⋊D83—44no—
154C2×Dic83—41no—
155C2×Dic83—42no—
156C2×Dic83—161no—
157D332—22no—
158D332—41no—
159C4×D83—44no—
160C4×D83—162no—
161Dic166—22no—
162Dic166—41no—
163C8×C83—22no—
164C8×C83C4×D412yes—
165C8×C83C23×C411yescharacteristic
166C8×C83C2×#4111yescharacteristic
167C22×C83—21no—
168C22×C83—21no—
169C22×C83—44no—
170C22×C83—82no—
171C22×C83—82no—
172C22×C83—164no—
173C22×C83C4×D812yes—
174C22×C83C22×D811yescharacteristic
175C2×D83—42no—
176C2×D83—81no—
177C2×D83—81no—
178C4×C83—22no—
179C4×C83—22no—
180C4×C83—81no—
181C4×C83—81no—
182C4×C83C4×D812yes—
183C4×C83C2×C4×D411yescharacteristic
184C4×C83C2×#13511yescharacteristic
185Dic83—42no—
186Dic83—82no—
187#33454—831nomaximal, Sylow
188C2×C83—22no—
189C2×C83—84no—
190C2×C83C2×C4×D811yescharacteristic
191C2×C83#341511yescharacteristic
192C2×C83#341611yescharacteristic
193D83—82no—
194C2×#509—831no—
195#2951—831no—
196#2948D8311yescharacteristic, complemented
197#2177—832no—
198#2175—832no—
199C4×SD32—832no—
200C4×D16—832no—
201#2159—831no—
202#2138—831no—
203C2×#357—831no—
204#2108—831no—
205C83#3345411yescharacteristic, complemented, Sylow
206C2×#135—831no—
207C22×D8C2×D8311yescharacteristic
208#520—1661no—
209#518—21no—
210#509—838no—
211C2×SD32—832no—
212C2×D16—832no—
213C2×M32—831no—
214C22×C16—831no—
215#442—22no—
216#440—1662no—
217C4×Q16—832no—
218C4×D8—832no—
219C4×D8C2×D8312yes—
220C4×D8C2×D8312yes—
221#427C2×D8311yescharacteristic
222C2×#107—831no—
223#418—21no—
224#417—1661no—
225#405C2×D8311yescharacteristic
226#367—832no—
227#366—832no—
228#358—831no—
229#358—831no—
230#357—832no—
231#357—832no—
232#357—831no—
233#357—831no—
234#346—832no—
235C4×C16—832no—
236C2×#41—1661no—
237C22×D4—21no—
238#135—834no—
239#135—1662no—
240C2×Q16—831no—
241C2×SD16—1661no—
242C2×D8—21no—
243C2×D8—22no—
244C2×D8—831no—
245C2×D8C4×D8314yescomplemented
246C2×D8C22×D8312yes—
247C2×M16C22⋊D8311yescharacteristic
248C22×C8C22×D8311yescharacteristic
249C4×Q8—1662no—
250C4×D4—22no—
251C4×D4—22no—
252C4×D4—41no—
253C4×D4—41no—
254C4×D4—3322no—
255C4×D4—1662no—
256#117—21no—
257#117—3321no—
258#117C22⋊D8311yescharacteristic
259C2×C4⋊C4—3321no—
260C2×C4⋊C4—1661no—
261C2×C22⋊C4—42no—
262SD32—838no—
263D16—838no—
264M32—834no—
265C2×C16—832no—
266C2×C16—831no—
267C2×C16—831no—
268C2×C16—1662no—
269#107—832no—
270#107—1661no—
271#107C22×D8311yescharacteristic
272#107C22×D8311yescharacteristic
273#106—21no—
274#103—1662no—
275#102—22no—
276#102—22no—
277#102—1662no—
278#97C22⋊D8312yes—
279C4×C8C22×D8312yes—
280C24—41no—
281#41—3322no—
282#41—1664no—
283C2×Q8—1661no—
284C2×D4—22no—
285C2×D4—24no—
286C2×D4—41no—
287C2×D4—42no—
288C2×D4—3321no—
289C2×D4—1661no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—3321no—
293C22×C4—3322no—
294C22×C4—3321no—
295C22×C4C2×C22⋊D8311yescharacteristic
296Q16—832no—
297Q16—1661no—
298SD16—1664no—
299D8—22no—
300D8—24no—
301D8—832no—
302D8—1661no—
303D8C2×C4×D8314yes—
304M16—22no—
305C2×C8C2×C4×D8312yes—
306C2×C8C2×C22⋊D8312yes—
307C2×C8C2×C22⋊D8311yescharacteristic
308C2×C8#1305059012yes—
309C2×C8C23×D8311yescharacteristic
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—3321no—
313C4⋊C4—3322no—
314C4⋊C4—1662no—
315C4⋊C4#1305057511yescharacteristic
316C4⋊C4#1305057511yescharacteristic
317C22⋊C4—41no—
318C22⋊C4—42no—
319C22⋊C4—42no—
320C22⋊C4—3322no—
321C22⋊C4#1305057512yes—
322C42—41no—
323C42—3322no—
324C42C2×C22⋊D8312yes—
325C16—834no—
326C16—1661no—
327C16—1661no—
328C23—42no—
329C23—44no—
330C23—161no—
331C23—161no—
332C23—3321no—
333C23C2×#1305057511yescharacteristic
334Q8—3321no—
335Q8—1662no—
336D4—24no—
337D4—42no—
338D4—44no—
339D4—3324no—
340D4—3321no—
341D4—1662no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—162no—
345C2×C4—162no—
346C2×C4—13282no—
347C2×C4—13281no—
348C2×C4—3321no—
349C2×C4—3324no—
350C2×C4—3322no—
351C2×C4C4×C22⋊D8312yes—
352C2×C4C2×#1305057512yes—
353C2×C4C2×#1305057512yes—
354C2×C4C2×#1305057511yescharacteristic
355C2×C4#1458181212yes—
356C2×C4C22×C22⋊D8311yescharacteristic
357C8—22no—
358C8C4×C22⋊D8312yes—
359C8C22×C4×D8311yescharacteristic
360C8C2×#1305059011yescharacteristic
361C22—21no—
362C22—21no—
363C22—44no—
364C22—82no—
365C22—82no—
366C22—164no—
367C22—3322no—
368C22—6641no—
369C22—6641no—
370C22C4×#1305057512yes—
371C22C22×#1305057511yescharacteristic
372C4—22no—
373C4—22no—
374C4—81no—
375C4—81no—
376C4—3322no—
377C4—6642no—
378C4C4×#1305057512yes—
379C4C2×C4×C22⋊D8311yescharacteristic
380C4C2×#1458181211yescharacteristic
381C2—22no—
382C2—84no—
383C2—6642no—
384C2C2×C4×#1305057511yescharacteristic
385C2#2492060211yescharacteristic
386C2#2492061011yescharacteristic
subgroupG/H
centerC22C22×#13050575
commutatorC8×C83C23×C4
FrattiniC2×C8C23×D83
FittingC83×#2948C2
socleC22×C83C22×D8
prime radicalC2×#19128137C2
cosocleC2×C8×C83C24
hypercenter#2948D83
hypocenterC83#33454
LCSG ▷ C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #2948

Complements

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

NH ≅ G/Ntype
2. #24916340C2semidirect
3. #24909337C2semidirect
4. #24909327C2semidirect
5. C4×#14581736C2semidirect
6. C4×#14581733C2semidirect
7. #24909252C2semidirect
8. #24909168C2semidirect
9. C2×#19127502C2semidirect
10. #24909060C2semidirect
11. C83×#2948C2semidirect
14. #19128137C4semidirect
20. C4×Dic332C22semidirect
21. C4×D664C22semidirect
24. #19127528C22semidirect
25. #19127526C22semidirect
26. #19127504C22semidirect
27. #19127504C22semidirect
30. #19127501C22semidirect
31. #19127501C22semidirect
32. #19127477C22semidirect
33. C4×C83⋊C16C22semidirect
44. #14581814C2×C4semidirect
54. #14581736C2×C4semidirect
55. #14581733C2×C4semidirect
56. #14581732C2×C4semidirect
68. C2×D8×C83C2×C4semidirect
196. #2948D83semidirect
205. C83#33454semidirect
245. C2×D8C4×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#191281372
#249163402
#249093372
#249093272
C4×#145817362
C4×#145817332
#249092522
#249091682
C2×#191275022
#249090602
C83×#29482
#3345483