Finite Group Databasebeta

← #35560665

Subgroups — #35560665

Order 21248.

454 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24918102C211yesmaximal, characteristic
2#24909750C211yesmaximal, characteristic, complemented
3#24909736C211yesmaximal, characteristic, complemented
4#24909702C211yesmaximal, characteristic, complemented
5#24909697C211yesmaximal, characteristic, complemented
6#24909671C211yesmaximal, characteristic, complemented
7#24909665C211yesmaximal, characteristic, complemented
8#24909654C211yesmaximal, characteristic
9#24909643C211yesmaximal, characteristic
10#24909624C211yesmaximal, characteristic, complemented
11#24909617C211yesmaximal, characteristic, complemented
12#24909612C211yesmaximal, characteristic, complemented
13#24909597C211yesmaximal, characteristic, complemented
14#24909270C211yesmaximal, characteristic, complemented
15C83×#3164C211yesmaximal, characteristic, complemented
16#19128322—21no—
17#19128106C2211yescharacteristic
18#19128085C2211yescharacteristic
19#19128061—21no—
20#19127947—21no—
21#19127942C2211yescharacteristic
22#19127937C2211yescharacteristic
23#19127803—21no—
24C4×Dic332C2211yescharacteristic
25C4×D664C2211yescharacteristic
26#19127528C2211yescharacteristic
27#19127528C2211yescharacteristic
28#19127526C2211yescharacteristic
29#19127526C2211yescharacteristic
30#19127526C2211yescharacteristic
31#19127526C2211yescharacteristic
32#19127526C2211yescharacteristic
33#19127526C2211yescharacteristic
34C4⋊(C83⋊C16)C2211yescharacteristic
35C4⋊(C83⋊C16)C2211yescharacteristic
36#19127504C2211yescharacteristic
37#19127504C2211yescharacteristic
38#19127504C2211yescharacteristic
39#19127504C2211yescharacteristic
40#19127501C2211yescharacteristic
41#19127501C2211yescharacteristic
42#19127501C2211yescharacteristic
43#19127501C2211yescharacteristic
44C22⋊(C83⋊C16)C2211yescharacteristic
45C22⋊(C83⋊C16)C2211yescharacteristic
46#19127477C2211yescharacteristic
47C4×C83⋊C16C2211yescharacteristic
48#544×C83—21no—
49#501×C83—21no—
50#501×C83—21no—
51#500×C83C2211yescharacteristic
52#500×C83C2211yescharacteristic
53#499×C83C2211yescharacteristic
54#499×C83C2211yescharacteristic
55#428×C83—21no—
56#427×C83C2211yescharacteristic
57C2×#107×C83C2211yescharacteristic
58#405×C83C2211yescharacteristic
59C2×Dic332C2311yescharacteristic
60C2×#13050579—21no—
61C2×D664C2311yescharacteristic
62#14581792—21no—
63#14581787—41no—
64#14581780—21no—
65#14581778—41no—
66#14581774—41no—
67#14581771—21no—
68#14581770—41no—
69#14581766—21no—
70C4×Dic166—21no—
71C4×D332—21no—
72C2×C83⋊C16C2311yescharacteristic
73C2×C83⋊C16C2311yescharacteristic
74C2×C83⋊C16C2311yescharacteristic
75C2×C83⋊C16C2311yescharacteristic
76#14581726—21no—
77#14581726C2311yescharacteristic
78#14581726C2311yescharacteristic
79#14581717—21no—
80#14581717—21no—
81#14581714—21no—
82#14581714—21no—
83C2×M16×C83D411yescharacteristic
84C22×C8×C83C2311yescharacteristic
85C4⋊Q8×C83—21no—
86C4⋊Q8×C83—21no—
87C4⋊Q8×C83—41no—
88#125×C83—21no—
89#125×C83—21no—
90#125×C83—41no—
91C22⋊Q8×C83—41no—
92C22⋊Q8×C83—41no—
93#117×C83—21no—
94#117×C83D411yescharacteristic
95C2×C4⋊C4×C83—21no—
96#107×C83—21no—
97#107×C83—21no—
98#107×C83—21no—
99#107×C83C2311yescharacteristic
100#107×C83C2311yescharacteristic
101#107×C83C2311yescharacteristic
102#107×C83C2311yescharacteristic
103#106×C83—21no—
104#106×C83—21no—
105#106×C83—21no—
106#97×C83D411yescharacteristic
107#97×C83D411yescharacteristic
108C4×C8×C83C2311yescharacteristic
109C4×C8×C83C2311yescharacteristic
110C2×Dic166—21no—
111C2×Dic166—41no—
112C2×D332—21no—
113C2×C22⋊D83—41no—
114C2×C4×D83—41no—
115Dic332—21no—
116Dic332—21no—
117#13050579—42no—
118D664—21no—
119D664—21no—
120C4⋊Dic83—21no—
121C4⋊Dic83—21no—
122C4⋊Dic83—41no—
123Dic83⋊C4—41no—
124Dic83⋊C4—41no—
125C22⋊Dic83—41no—
126(C2×D83)⋊C4—41no—
127(C2×D83)⋊C4—41no—
128(C2×D83)⋊C4—41no—
129(C2×D83)⋊C4—41no—
130C4×Dic83—41no—
131C83⋊C16—21no—
132C83⋊C16—21no—
133C83⋊C16—21no—
134C83⋊C16—21no—
135C2×Q8×C83—41no—
136C2×Q8×C83—41no—
137C22×C4×C83—41no—
138C22×C4×C83C2×D411yescharacteristic
139M16×C83—22no—
140C2×C8×C83—21no—
141C2×C8×C83—21no—
142C2×C8×C83C2×D411yescharacteristic
143C2×C8×C83C2×D411yescharacteristic
144C2×C8×C83C2×D411yescharacteristic
145C2×C8×C83C2411yescharacteristic
146C4⋊C4×C83—21no—
147C4⋊C4×C83—21no—
148C4⋊C4×C83—21no—
149C4⋊C4×C83—21no—
150C4⋊C4×C83—41no—
151C4⋊C4×C83—41no—
152C4⋊C4×C83—41no—
153C4⋊C4×C83—41no—
154C4⋊C4×C83—41no—
155C4⋊C4×C83—41no—
156C4⋊C4×C83—41no—
157C4⋊C4×C83D811yescharacteristic
158C4⋊C4×C83D811yescharacteristic
159C22⋊C4×C83—41no—
160C22⋊C4×C83D811yescharacteristic
161C22⋊C4×C83D811yescharacteristic
162C42×C83—41no—
163C42×C83C2×D411yescharacteristic
164C42×C83C2×D411yescharacteristic
165C23×C83C2×D811yescharacteristic
166C22×D83—41no—
167Q8×C83—161no—
168Q8×C83—161no—
169C2×C4×C83—21no—
170C2×C4×C83—21no—
171C2×C4×C83—41no—
172C2×C4×C83—41no—
173C2×C4×C83—41no—
174C2×C4×C83—41no—
175C2×C4×C83—161no—
176C2×C4×C83C2×D811yescharacteristic
177C2×C4×C83C2×D811yescharacteristic
178C2×C4×C83C2×D811yescharacteristic
179C2×C4×C83C2×D811yescharacteristic
180C2×C4×C83C2×D811yescharacteristic
181C2×C4×C83C22×D411yescharacteristic
182C22⋊D83—161no—
183C2×Dic83—41no—
184C2×Dic83—41no—
185C2×Dic83—41no—
186D332—41no—
187D332—41no—
188C4×D83—161no—
189Dic166—41no—
190Dic166—41no—
191Dic166—161no—
192C8×C83—21no—
193C8×C83—21no—
194C8×C83#13712yes—
195C8×C832−1+411yescharacteristic
196C8×C832−1+411yescharacteristic
197C22×C83—21no—
198C22×C83—21no—
199C22×C83—21no—
200C22×C83C22×D811yescharacteristic
201C2×D83—81no—
202C2×D83—81no—
203C2×D83—81no—
204C4×C83—21no—
205C4×C83—21no—
206C4×C83—21no—
207C4×C83—21no—
208C4×C83—21no—
209C4×C83—81no—
210C4×C83—81no—
211C4×C83—81no—
212C4×C83—81no—
213C4×C83#53611yescharacteristic
214C4×C83C2×#13711yescharacteristic
215Dic83—81no—
216Dic83—81no—
217Dic83—81no—
218#33809—831nomaximal, Sylow
219C2×C83—41no—
220C2×C83#322611yescharacteristic
221C2×C83#341511yescharacteristic
222C2×C83#341711yescharacteristic
223D83—161no—
224#3175—831no—
225#3164D8311yescharacteristic, complemented
226#2255—831no—
227#2252—831no—
228#2243—831no—
229#2242—831no—
230#2237—831no—
231#2236—831no—
232#2234—831no—
233#2232—831no—
234#2229—831no—
235#2228—831no—
236#2227—831no—
237#2224—831no—
238#2164—831no—
239C83#3380911yescharacteristic, complemented, Sylow
240#544—21no—
241#542—1661no—
242#501—21no—
243#501—21no—
244#500—831no—
245#500C2×D8311yescharacteristic
246#500C2×D8311yescharacteristic
247#499C2×D8311yescharacteristic
248#499C2×D8311yescharacteristic
249#495—831no—
250#489—1661no—
251#468—1661no—
252#467—831no—
253#466—831no—
254#440—1661no—
255C4×Q16—831no—
256C4×D8—831no—
257#428—21no—
258#427C2×D8311yescharacteristic
259C2×#107C2×D8311yescharacteristic
260#405C2×D8311yescharacteristic
261#367—831no—
262#367—831no—
263#366—831no—
264#366—831no—
265#366—831no—
266#366—831no—
267#366—831no—
268#366—831no—
269C4⋊C16—831no—
270C4⋊C16—831no—
271#358—831no—
272#358—831no—
273#358—831no—
274#358—831no—
275#357—831no—
276#357—831no—
277#357—831no—
278#357—831no—
279C22⋊C16—831no—
280C22⋊C16—831no—
281#346—831no—
282C4×C16—831no—
283C2×Q16—831no—
284C2×SD16—1661no—
285C2×D8—831no—
286C2×M16C22⋊D8311yescharacteristic
287C22×C8C22×D8311yescharacteristic
288C4⋊Q8—21no—
289C4⋊Q8—21no—
290C4⋊Q8—41no—
291C4⋊Q8—1661no—
292#126—3321no—
293#125—21no—
294#125—21no—
295#125—41no—
296#124—3321no—
297#124—1661no—
298#123—3321no—
299C22⋊Q8—41no—
300C22⋊Q8—41no—
301C22⋊Q8—3321no—
302C22⋊Q8—1661no—
303#121—1661no—
304C4×Q8—1661no—
305C4×D4—1661no—
306#117—21no—
307#117C22⋊D8311yescharacteristic
308C2×C4⋊C4—21no—
309C2×C16—831no—
310C2×C16—831no—
311C2×C16—831no—
312C2×C16—831no—
313#107—21no—
314#107—21no—
315#107—21no—
316#107—831no—
317#107—831no—
318#107—1661no—
319#107C22×D8311yescharacteristic
320#107C22×D8311yescharacteristic
321#107C22×D8311yescharacteristic
322#107C22×D8311yescharacteristic
323#106—21no—
324#106—21no—
325#106—21no—
326#103—1661no—
327#103—1661no—
328#102—1661no—
329#102—1661no—
330#97C22⋊D8311yescharacteristic
331#97C22⋊D8311yescharacteristic
332C4×C8C22×D8311yescharacteristic
333C4×C8C22×D8311yescharacteristic
334C2×Q8—41no—
335C2×Q8—41no—
336C2×Q8—3321no—
337C2×Q8—1661no—
338C2×D4—3321no—
339C2×D4—1661no—
340C22×C4—41no—
341C22×C4—3321no—
342C22×C4C2×C22⋊D8311yescharacteristic
343Q16—1661no—
344Q16—1661no—
345SD16—3322no—
346D8—1661no—
347D8—1661no—
348M16—22no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8C2×C22⋊D8311yescharacteristic
352C2×C8C2×C22⋊D8311yescharacteristic
353C2×C8C2×C22⋊D8311yescharacteristic
354C2×C8C23×D8311yescharacteristic
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—3321no—
367C4⋊C4—3321no—
368C4⋊C4—3321no—
369C4⋊C4—1661no—
370C4⋊C4—1661no—
371C4⋊C4#1305057511yescharacteristic
372C4⋊C4#1305057511yescharacteristic
373C22⋊C4—41no—
374C22⋊C4—3321no—
375C22⋊C4—3321no—
376C22⋊C4—3321no—
377C22⋊C4—3321no—
378C22⋊C4—3321no—
379C22⋊C4#1305057511yescharacteristic
380C22⋊C4#1305057511yescharacteristic
381C42—41no—
382C42—3321no—
383C42C2×C22⋊D8311yescharacteristic
384C42C2×C22⋊D8311yescharacteristic
385C16—1661no—
386C16—1661no—
387C16—1661no—
388C16—1661no—
389C23—3321no—
390C23C2×#1305057511yescharacteristic
391Q8—161no—
392Q8—161no—
393Q8—13281no—
394Q8—3321no—
395Q8—3321no—
396D4—13281no—
397D4—3321no—
398D4—3321no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—161no—
406C2×C4—13281no—
407C2×C4—3321no—
408C2×C4—3321no—
409C2×C4—3321no—
410C2×C4C2×#1305057511yescharacteristic
411C2×C4C2×#1305057511yescharacteristic
412C2×C4C2×#1305057511yescharacteristic
413C2×C4C2×#1305057511yescharacteristic
414C2×C4C2×#1305057511yescharacteristic
415C2×C4C22×C22⋊D8311yescharacteristic
416C8—21no—
417C8—21no—
418C8#1458182312yes—
419C8#1458184011yescharacteristic
420C8#1458184011yescharacteristic
421C22—21no—
422C22—21no—
423C22—21no—
424C22—6641no—
425C22—6641no—
426C22—6641no—
427C22C22×#1305057511yescharacteristic
428C4—21no—
429C4—21no—
430C4—21no—
431C4—21no—
432C4—21no—
433C4—81no—
434C4—81no—
435C4—81no—
436C4—81no—
437C4—6641no—
438C4—6641no—
439C4—6641no—
440C4#1912827511yescharacteristic
441C4C2×#1458182311yescharacteristic
442C2—41no—
443C2—13281no—
444C2#2491858411yescharacteristic
445C2#2492060211yescharacteristic
446C2#2492061411yescharacteristic
subgroupG/H
centerC22C22×#13050575
commutator, cosocleC2×C8×C83C24
FrattiniC2×C8C23×D83
FittingC83×#3164C2
socleC22×C83C22×D8
prime radical#19127937C22
hypercenter#3164D83
hypocenterC83#33809
LCSG ▷ C2×C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #3164

Complements

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

NH ≅ G/Ntype
2. #24909750C2semidirect
3. #24909736C2semidirect
4. #24909702C2semidirect
5. #24909697C2semidirect
6. #24909671C2semidirect
7. #24909665C2semidirect
10. #24909624C2semidirect
11. #24909617C2semidirect
12. #24909612C2semidirect
13. #24909597C2semidirect
14. #24909270C2semidirect
15. C83×#3164C2semidirect
225. #3164D83semidirect
239. C83#33809semidirect

Maximal subgroups

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

subgroupindex
#249181022
#249097502
#249097362
#249097022
#249096972
#249096712
#249096652
#249096542
#249096432
#249096242
#249096172
#249096122
#249095972
#249092702
C83×#31642
#3380983