Finite Group Databasebeta

← #35561228

Subgroups — #35561228

Order 21248.

654 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24919811C211yesmaximal, characteristic, complemented
2D4×D664C211yesmaximal, characteristic, complemented
3#24909687C211yesmaximal, characteristic, complemented
4#24909644C212yesmaximal, complemented
5#24909601C211yesmaximal, characteristic, complemented
6#24909549C212yesmaximal, complemented
7#24909499C211yesmaximal, characteristic, complemented
8#24909377C212yesmaximal, complemented
9C4×D1328C211yesmaximal, characteristic, complemented
10C2×#19127503C212yesmaximal, complemented
11D4×C16×C83C211yesmaximal, characteristic, complemented
12C22×D664C2212yes—
13#19128355—21no—
14D4×D332—21no—
15C2×D1328C2211yescharacteristic
16C2×D1328C2212yescomplemented
17#19128106C2211yescharacteristic, complemented
18#19128078C2211yescharacteristic
19#19128015—22no—
20#19127982—21no—
21#19127937C2212yes—
22#19127937C2212yescomplemented
23#19127897—21no—
24#19127837—22no—
25C4×D664C2211yescharacteristic
26C4×D664C2211yescharacteristic, complemented
27C2×#14581726C2212yescomplemented
28#19127527C2212yescomplemented
29#19127527C2211yescharacteristic, complemented
30#19127503C2212yes—
31#19127503C2211yescharacteristic
32#19127503C2212yescomplemented
33#19127503C2212yescomplemented
34#19127503C2212yescomplemented
35#19127503C2211yescharacteristic, complemented
36C22×C16×C83C2212yescomplemented
37C8×D4×C83C2211yescharacteristic
38C4⋊C16×C83C2211yescharacteristic, complemented
39C22⋊C16×C83C2212yescomplemented
40C4×C16×C83C2211yescharacteristic, complemented
41C2×#13050589—41no—
42C22×D332—22no—
43C2×D4×D83—41no—
44C2×D664—24no—
45C2×D664—22no—
46C2×D664—21no—
47C2×D664—42no—
48C2×D664C2312yes—
49C2×D664C2311yescharacteristic
50C2×D664C2311yescharacteristic
51#14581792—21no—
52#14581789—21no—
53#14581776—42no—
54#14581772—41no—
55#14581767—41no—
56#14581766—22no—
57#14581766—22no—
58#14581762—42no—
59C4×D332—21no—
60C4×D332—21no—
61C2×C4⋊Dic83—22no—
62D1328—22no—
63D1328—42no—
64#14581726—21no—
65#14581726—22no—
66#14581726C2312yes—
67#14581726C2311yescharacteristic
68#14581726C2311yescharacteristic
69#14581714—21no—
70#14581714—22no—
71#14581714—22no—
72#14581714—21no—
73C22×C8×C83C2312yes—
74C4×D4×C83D411yescharacteristic
75C2×C16×C83—42no—
76C2×C16×C83C2311yescharacteristic
77C2×C16×C83C2312yes—
78C2×C16×C83C2311yescharacteristic
79C4⋊C8×C83D411yescharacteristic
80C22⋊C8×C83D412yes—
81C4×C8×C83C2311yescharacteristic
82C23×D83—42no—
83#13050589—84no—
84C2×Dic166—41no—
85C2×D332—21no—
86C2×D332—21no—
87C2×D332—22no—
88C2×D332—44no—
89C2×D332—42no—
90C2×D332—41no—
91C2×D332—41no—
92C2×D332—82no—
93C2×C22⋊D83—42no—
94C2×C22⋊D83—42no—
95D4×D83—84no—
96C22×Dic83—42no—
97C2×C4×D83—41no—
98C2×C4×D83—41no—
99D664—22no—
100D664—21no—
101D664—21no—
102D664—22no—
103D664—21no—
104D664—44no—
105D664—81no—
106D664#4112yescomplemented
107C4⋊Dic83—21no—
108C4⋊Dic83—22no—
109C4⋊Dic83—21no—
110C4⋊Dic83—42no—
111C4⋊Dic83—41no—
112(C2×D83)⋊C4—41no—
113(C2×D83)⋊C4—42no—
114(C2×D83)⋊C4—42no—
115(C2×D83)⋊C4—41no—
116C2×D4×C83D811yescharacteristic
117C22×C4×C83C2×D412yes—
118C2×C8×C83—22no—
119C2×C8×C83—22no—
120C2×C8×C83C2×D411yescharacteristic
121C2×C8×C83C2×D412yes—
122C2×C8×C83C2411yescharacteristic
123C4⋊C4×C83D811yescharacteristic
124C22⋊C4×C83D812yes—
125C42×C83C2×D411yescharacteristic
126C16×C83—21no—
127C16×C83—21no—
128C16×C83—22no—
129C23×C83C2×D812yes—
130C22×D83—41no—
131C22×D83—41no—
132C22×D83—42no—
133C22×D83—84no—
134C22×D83—84no—
135C22×D83—162no—
136D4×C83D1614yescomplemented
137C2×C4×C83—22no—
138C2×C4×C83—22no—
139C2×C4×C83C2×D811yescharacteristic
140C2×C4×C83C2×D812yes—
141C2×C4×C83C2×D811yescharacteristic
142C2×C4×C83C22×D411yescharacteristic
143C22⋊D83—84no—
144C22⋊D83—84no—
145C2×Dic83—41no—
146C2×Dic83—41no—
147C2×Dic83—42no—
148C2×Dic83—84no—
149D332—22no—
150D332—42no—
151D332—41no—
152D332—41no—
153D332—42no—
154D332—41no—
155D332—84no—
156D332—82no—
157D332—161no—
158C4×D83—82no—
159C4×D83—82no—
160Dic166—82no—
161C8×C83—22no—
162C8×C83—21no—
163C8×C83C2×#4111yescharacteristic
164C8×C83H(5,𝔽2)11yescharacteristic
165C22×C83—22no—
166C22×C83—22no—
167C22×C83C2×D1614yes—
168C22×C83C22×D811yescharacteristic
169C2×D83—42no—
170C2×D83—81no—
171C2×D83—81no—
172C2×D83—81no—
173C2×D83—82no—
174C2×D83—82no—
175C2×D83—81no—
176C2×D83—82no—
177C2×D83—81no—
178C2×D83—168no—
179C4×C83—21no—
180C4×C83—22no—
181C4×C83C2×D1612yes—
182C4×C83#54611yescharacteristic
183C4×C83#57711yescharacteristic
184Dic83—82no—
185Dic83—81no—
186Dic83—81no—
187#33861—831nomaximal, Sylow
188C2×C83—24no—
189C2×C83#329411yescharacteristic
190C2×C83C22×D1611yescharacteristic
191C2×C83#341611yescharacteristic
192D83—82no—
193D83—161no—
194D83—162no—
195#3334—831no—
196D4×D8—831no—
197#2240—831no—
198#2232—832no—
199#2224—831no—
200#2215—832no—
201#2206—831no—
202#2184—832no—
203C4×D16—831no—
204D4×C16D8311yescharacteristic, complemented
205C2×#357—832no—
206C83#3386111yescharacteristic, complemented, Sylow
207C22×D8—832no—
208#547—1661no—
209D42—1661no—
210C2×D16—832no—
211C2×D16—831no—
212C22×C16C2×D8312yescomplemented
213#500—831no—
214#493—831no—
215#480—1662no—
216#474—1661no—
217#466—832no—
218#466—832no—
219#459—1661no—
220#447—1662no—
221C4×D8—831no—
222C4×D8—831no—
223C8×D4C2×D8311yescharacteristic
224C2×#107—832no—
225#366—831no—
226#366—832no—
227C4⋊C16C2×D8311yescharacteristic, complemented
228#357—831no—
229#357—832no—
230#357—832no—
231#357—832no—
232#357—831no—
233#357—832no—
234C22⋊C16C2×D8312yescomplemented
235C4×C16C2×D8311yescharacteristic, complemented
236C2×#41—3321no—
237C22×D4—3321no—
238C22×D4—1662no—
239C2×D8—3322no—
240C2×D8—831no—
241C2×D8—832no—
242C2×D8—831no—
243C2×D8—1664no—
244C2×D8—1662no—
245C2×D8—1661no—
246C22×C8C22×D8312yes—
247C4⋊Q8—1661no—
248#127—1661no—
249#123—3322no—
250C22⋊Q8—3321no—
251#121—3321no—
252#121—1662no—
253#121—1662no—
254#120—3322no—
255C4×D4—1661no—
256C4×D4—1661no—
257C4×D4D33211yescharacteristic
258C2×C4⋊C4—1662no—
259D16—3322no—
260D16—1662no—
261C2×C16—42no—
262C2×C16C22×D8311yescharacteristic
263C2×C16C22×D8311yescharacteristic
264C2×C16C22×D8312yes—
265#107—831no—
266#107—832no—
267#107—831no—
268#107—1662no—
269#107—1661no—
270C4⋊C8D33211yescharacteristic
271#102—1661no—
272#102—1661no—
273#102—1662no—
274#102—1662no—
275C22⋊C8D33212yes—
276C4×C8C22×D8311yescharacteristic
277C24—3322no—
278#41—6644no—
279C2×Q8—3321no—
280C2×D4—3324no—
281C2×D4—3322no—
282C2×D4—3322no—
283C2×D4—3321no—
284C2×D4—3322no—
285C2×D4—3321no—
286C2×D4—6642no—
287C2×D4—6644no—
288C2×D4—1661no—
289C2×D4—1662no—
290C2×D4—1661no—
291C2×D4D66411yescharacteristic
292C22×C4—3322no—
293C22×C4—3321no—
294C22×C4—3321no—
295C22×C4C2×D33212yes—
296D8—3324no—
297D8—832no—
298D8—6641no—
299D8—1661no—
300D8—1662no—
301D8—1661no—
302D8—1662no—
303D8—1661no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8C2×D33211yescharacteristic
307C2×C8C2×D33212yes—
308C2×C8C23×D8311yescharacteristic
309C4⋊C4—3322no—
310C4⋊C4—3321no—
311C4⋊C4—1661no—
312C4⋊C4—1661no—
313C4⋊C4—1662no—
314C4⋊C4D66411yescharacteristic
315C22⋊C4—3321no—
316C22⋊C4—3321no—
317C22⋊C4—3322no—
318C22⋊C4—3322no—
319C22⋊C4D66412yes—
320C42C2×D33211yescharacteristic
321C16—21no—
322C16—21no—
323C16—22no—
324C23—13282no—
325C23—3321no—
326C23—3321no—
327C23—3322no—
328C23—6644no—
329C23—6644no—
330C23C2×D66412yes—
331Q8—6642no—
332D4—13281no—
333D4—3322no—
334D4—3321no—
335D4—3321no—
336D4—3322no—
337D4—3321no—
338D4—6644no—
339D4—6644no—
340D4—6644no—
341D4—6642no—
342D4—1662no—
343D4D132814yescomplemented
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—3321no—
347C2×C4—3321no—
348C2×C4—3322no—
349C2×C4—6644no—
350C2×C4—6642no—
351C2×C4—6642no—
352C2×C4C2×D66411yescharacteristic
353C2×C4C2×D66412yes—
354C2×C4C2×D66411yescharacteristic
355C2×C4C22×D33211yescharacteristic
356C8—22no—
357C8—21no—
358C8#41×D8311yescharacteristic
359C8#1458183711yescharacteristic
360C22—22no—
361C22—22no—
362C22—13288no—
363C22—3322no—
364C22—6642no—
365C22—6641no—
366C22—6642no—
367C22—6641no—
368C22—6641no—
369C22—6642no—
370C22—6641no—
371C22—6641no—
372C22C2×D132814yes—
373C22C22×D66411yescharacteristic
374C4—21no—
375C4—22no—
376C4—6641no—
377C4—6641no—
378C4—6642no—
379C4C2×D132812yes—
380C4#1912834611yescharacteristic
381C4#1912851311yescharacteristic
382C2—24no—
383C2—13282no—
384C2—13281no—
385C2—6642no—
386C2#2491927711yescharacteristic
387C2C22×D132811yescharacteristic
388C2#2492061111yescharacteristic
subgroupG/H
centerC22C22×D664
commutator, cosocleC2×C8×C83C24
FrattiniC2×C8C23×D83
FittingD4×C16×C83C2
socleC22×C83C22×D8
hypercenterD4×C16D83
hypocenterC83#33861
LCSG ▷ C2×C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ D4×C16

Complements

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

NH ≅ G/Ntype
1. #24919811C2semidirect
2. D4×D664C2semidirect
3. #24909687C2semidirect
4. #24909644C2semidirect
5. #24909601C2semidirect
6. #24909549C2semidirect
7. #24909499C2semidirect
8. #24909377C2semidirect
9. C4×D1328C2semidirect
10. C2×#19127503C2semidirect
11. D4×C16×C83C2semidirect
16. C2×D1328C22semidirect
17. #19128106C22semidirect
22. #19127937C22semidirect
26. C4×D664C22semidirect
27. C2×#14581726C22semidirect
28. #19127527C22semidirect
29. #19127527C22semidirect
32. #19127503C22semidirect
33. #19127503C22semidirect
34. #19127503C22semidirect
35. #19127503C22semidirect
36. C22×C16×C83C22semidirect
38. C4⋊C16×C83C22semidirect
39. C22⋊C16×C83C22semidirect
40. C4×C16×C83C22semidirect
106. D664#41semidirect
136. D4×C83D16semidirect
204. D4×C16D83semidirect
206. C83#33861semidirect
212. C22×C16C2×D83semidirect
227. C4⋊C16C2×D83semidirect
234. C22⋊C16C2×D83semidirect
235. C4×C16C2×D83semidirect
343. D4D1328semidirect

Maximal subgroups

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

subgroupindex
#249198112
D4×D6642
#249096872
#249096442
#249096012
#249095492
#249094992
#249093772
C4×D13282
C2×#191275032
D4×C16×C832
#3386183