Finite Group Databasebeta

← #35387161

Subgroups — #35387161

Order 21248.

598 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24918049C211yesmaximal, characteristic, complemented
2#24917534C212yesmaximal
3#24907017C212yesmaximal, complemented
4#24907010C212yesmaximal, complemented
5#24906981C211yesmaximal, characteristic, complemented
6#24906976C211yesmaximal, characteristic, complemented
7#24906957C211yesmaximal, characteristic, complemented
8#24906952C211yesmaximal, characteristic, complemented
9#24906211C211yesmaximal, characteristic, complemented
10#24906127C212yesmaximal, complemented
11C83×#2965C211yesmaximal, characteristic
12#19128258C2212yes—
13#19128106C2211yescharacteristic
14#19128081C2211yescharacteristic
15#19128052C2212yes—
16#19128042—22no—
17#19127986C2212yes—
18#19127981C2212yes—
19#19127942—21no—
20#19127942—21no—
21#19127911C2212yes—
22#19127906C2212yes—
23#19127863—22no—
24C4×Dic332—21no—
25C4×Dic332—21no—
26C4×#13050580—22no—
27C4×#13050578—22no—
28#19127754C2212yes—
29#19127454C2212yes—
30#19127452C2211yescharacteristic
31#19127452C2211yescharacteristic
32#19127448C2212yes—
33#19127448C2212yes—
34#19127446C2212yes—
35#19127446C2212yes—
36C4×#41×C83C2211yescharacteristic
37#446×C83C2211yescharacteristic
38#446×C83C2211yescharacteristic
39#436×C83C2211yescharacteristic
40#436×C83C2211yescharacteristic
41#435×C83—21no—
42#435×C83—21no—
43#431×C83—21no—
44#431×C83C2211yescharacteristic
45C4×M16×C83—21no—
46C2×C4×C8×C83C2211yescharacteristic
47C2×Dic332—21no—
48C2×Dic332—21no—
49C2×#13050580—22no—
50C2×#13050578—22no—
51#14581792C2312yes—
52#14581792C2312yes—
53#14581784—22no—
54#14581771—22no—
55#14581771—22no—
56C4×Dic166—22no—
57C4×Dic166—22no—
58#14581751—22no—
59#14581726—21no—
60#14581726—21no—
61#14581725—22no—
62#14581723—22no—
63C4⋊(C83⋊C8)C2312yes—
64C4⋊(C83⋊C8)C2312yes—
65#14581717—22no—
66#14581717—22no—
67#14581716—22no—
68#14581715—22no—
69#14581715—22no—
70#14581713—22no—
71C22⋊(C83⋊C8)—22no—
72C4×C83⋊C8—22no—
73C2×#41×C83D411yescharacteristic
74C2×M16×C83—21no—
75C2×M16×C83—21no—
76C22×C8×C83D411yescharacteristic
77C22×C8×C83D411yescharacteristic
78C4×Q8×C83D411yescharacteristic
79C4×Q8×C83C2311yescharacteristic
80C4×D4×C83—21no—
81C4×D4×C83—21no—
82C4×D4×C83D411yescharacteristic
83C4×D4×C83C2311yescharacteristic
84#117×C83—21no—
85#117×C83D411yescharacteristic
86C2×C42×C83—21no—
87C2×C42×C83C2311yescharacteristic
88C4⋊C8×C83—21no—
89C4⋊C8×C83—21no—
90C4⋊C8×C83C2311yescharacteristic
91C4⋊C8×C83C2311yescharacteristic
92C22⋊C8×C83—21no—
93C22⋊C8×C83—21no—
94C22⋊C8×C83—21no—
95#97×C83—21no—
96#97×C83—21no—
97C4×C8×C83—21no—
98C4×C8×C83D411yescharacteristic
99C4×C8×C83D411yescharacteristic
100C4×C8×C83C2311yescharacteristic
101C4×C8×C83C2311yescharacteristic
102C2×Dic166—22no—
103C2×Dic166—22no—
104Dic332—42no—
105Dic332—42no—
106#13050580—44no—
107#13050578—44no—
108C2×C83⋊C8—22no—
109C2×C83⋊C8—22no—
110C4⋊Dic83—22no—
111C4⋊Dic83—22no—
112C4⋊Dic83—22no—
113C4⋊Dic83—22no—
114Dic83⋊C4—42no—
115Dic83⋊C4—42no—
116C22⋊Dic83—42no—
117C4×Dic83—42no—
118#41×C83—42no—
119C2×Q8×C83C2×D411yescharacteristic
120C2×D4×C83—21no—
121C2×D4×C83C2×D411yescharacteristic
122C22×C4×C83—21no—
123C22×C4×C83—21no—
124C22×C4×C83—21no—
125C22×C4×C83C2×D411yescharacteristic
126C22×C4×C83C2×D411yescharacteristic
127C22×C4×C83#4111yescharacteristic
128M16×C83—42no—
129M16×C83—42no—
130C2×C8×C83—22no—
131C2×C8×C83—21no—
132C2×C8×C83—21no—
133C2×C8×C83—41no—
134C2×C8×C83C2×D411yescharacteristic
135C2×C8×C83C2×D411yescharacteristic
136C2×C8×C83C2×D411yescharacteristic
137C2×C8×C83C2×D411yescharacteristic
138C4⋊C4×C83—21no—
139C4⋊C4×C83—21no—
140C4⋊C4×C83C2×D411yescharacteristic
141C4⋊C4×C83C2×D411yescharacteristic
142C22⋊C4×C83—21no—
143C22⋊C4×C83—21no—
144C22⋊C4×C83—21no—
145C42×C83—21no—
146C42×C83—21no—
147C42×C83—21no—
148C42×C83C2×D411yescharacteristic
149C42×C83C2×D411yescharacteristic
150C42×C83#4111yescharacteristic
151C42×C83C2411yescharacteristic
152C23×C83—21no—
153C23×C83#12111yescharacteristic
154Q8×C83—22no—
155D4×C83—22no—
156D4×C83—42no—
157C2×C4×C83—21no—
158C2×C4×C83—21no—
159C2×C4×C83—22no—
160C2×C4×C83—22no—
161C2×C4×C83—21no—
162C2×C4×C83—21no—
163C2×C4×C83—21no—
164C2×C4×C83—42no—
165C2×C4×C83—41no—
166C2×C4×C83—41no—
167C2×C4×C83—42no—
168C2×C4×C83#12111yescharacteristic
169C2×C4×C83#12111yescharacteristic
170C2×C4×C83#12111yescharacteristic
171C2×C4×C83C22×D411yescharacteristic
172C2×C4×C83C22×D411yescharacteristic
173C2×C4×C83C2×#4111yescharacteristic
174C2×Dic83—42no—
175C2×Dic83—42no—
176Dic166—42no—
177Dic166—42no—
178Dic166—42no—
179Dic166—42no—
180C8×C83—21no—
181C8×C83—21no—
182C8×C83—21no—
183C8×C83—41no—
184C8×C83—41no—
185C8×C83#13512yes—
186C83⋊C8—42no—
187C83⋊C8—42no—
188C22×C83—21no—
189C22×C83—21no—
190C22×C83—21no—
191C22×C83—41no—
192C22×C83—41no—
193C22×C83—41no—
194C22×C83C2×#12111yescharacteristic
195C4×C83—21no—
196C4×C83—21no—
197C4×C83—21no—
198C4×C83—41no—
199C4×C83—41no—
200C4×C83—41no—
201C4×C83#47312yes—
202C4×C83C2×#13511yescharacteristic
203C4×C83#57511yescharacteristic
204C4×C83#57512yes—
205Dic83—82no—
206Dic83—82no—
207#19161—831nomaximal, Sylow
208C2×C83—41no—
209C2×C83—81no—
210C2×C83#303511yescharacteristic
211C2×C83#305411yescharacteristic
212C2×C83C2×#47311yescharacteristic
213#3170—831no—
214#3107—832no—
215#2965D8311yescharacteristic
216#1700—832no—
217#1699—832no—
218#1694—831no—
219#1693—831no—
220#1690—831no—
221#1689—831no—
222#1564—831no—
223#1544—832no—
224C83#1916111yescharacteristic, complemented, Sylow
225#533—832no—
226C4×#41C2×D8311yescharacteristic
227#500—831no—
228#494—831no—
229#487—832no—
230#485—1662no—
231#475—832no—
232#474—832no—
233#467—1661no—
234#467—1661no—
235#462—832no—
236#461—832no—
237#452—1662no—
238#446C2×D8311yescharacteristic
239#446C2×D8311yescharacteristic
240C4×Q16—1662no—
241C4×Q16—1661no—
242C4×Q16—1661no—
243C4×SD16—1662no—
244#436C2×D8311yescharacteristic
245#436C2×D8311yescharacteristic
246#435—21no—
247#435—21no—
248#431—21no—
249#431—832no—
250#431C2×D8311yescharacteristic
251C4×M16—21no—
252C2×C4×C8C2×D8311yescharacteristic
253#335—832no—
254#334—831no—
255#334—831no—
256#332—832no—
257#332—832no—
258#332—832no—
259#332—832no—
260C2×#41D33211yescharacteristic
261C2×Q16—1662no—
262C2×Q16—1661no—
263C2×Q16—1661no—
264C2×SD16—1662no—
265C2×M16—21no—
266C2×M16—21no—
267C22×C8C22⋊D8311yescharacteristic
268C22×C8C22⋊D8311yescharacteristic
269C4⋊Q8—832no—
270C4⋊Q8—832no—
271#125—1662no—
272C22⋊Q8—1662no—
273C22⋊Q8—1662no—
274C4×Q8—1662no—
275C4×Q8—1662no—
276C4×Q8D33211yescharacteristic
277C4×Q8C22×D8311yescharacteristic
278C4×D4—21no—
279C4×D4—21no—
280C4×D4D33211yescharacteristic
281C4×D4C22×D8311yescharacteristic
282#117—21no—
283#117—1662no—
284#117D33211yescharacteristic
285C2×C42—21no—
286C2×C42C22×D8311yescharacteristic
287#107—1661no—
288#107—1661no—
289#107—1662no—
290#106—1662no—
291C4⋊C8—21no—
292C4⋊C8—21no—
293C4⋊C8—832no—
294C4⋊C8—832no—
295C4⋊C8C22×D8311yescharacteristic
296C4⋊C8C22×D8311yescharacteristic
297#103—1662no—
298#103—1662no—
299#103—1662no—
300#103—1662no—
301#103—1662no—
302#102—1662no—
303C22⋊C8—21no—
304C22⋊C8—21no—
305C22⋊C8—21no—
306C22⋊C8—1662no—
307#97—21no—
308#97—21no—
309C4×C8—21no—
310C4×C8—1662no—
311C4×C8C22⋊D8311yescharacteristic
312C4×C8C22⋊D8311yescharacteristic
313C4×C8C22×D8311yescharacteristic
314C4×C8C22×D8311yescharacteristic
315#41—42no—
316C2×Q8—1662no—
317C2×Q8—1662no—
318C2×Q8C2×D33211yescharacteristic
319C2×D4—21no—
320C2×D4C2×D33211yescharacteristic
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4C2×C22⋊D8311yescharacteristic
325C22×C4C2×D33211yescharacteristic
326C22×C4#1305059011yescharacteristic
327Q16—3322no—
328Q16—3324no—
329Q16—3322no—
330SD16—3324no—
331M16—42no—
332M16—42no—
333C2×C8—22no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—41no—
337C2×C8—1662no—
338C2×C8—1662no—
339C2×C8C2×C22⋊D8311yescharacteristic
340C2×C8C2×C22⋊D8311yescharacteristic
341C2×C8C2×C22⋊D8311yescharacteristic
342C2×C8C2×C22⋊D8311yescharacteristic
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—3322no—
346C4⋊C4—3322no—
347C4⋊C4—1662no—
348C4⋊C4—1662no—
349C4⋊C4—1662no—
350C4⋊C4—1662no—
351C4⋊C4C2×D33211yescharacteristic
352C4⋊C4C2×D33211yescharacteristic
353C22⋊C4—21no—
354C22⋊C4—21no—
355C22⋊C4—21no—
356C22⋊C4—3322no—
357C42—21no—
358C42—21no—
359C42—21no—
360C42—3322no—
361C42C2×C22⋊D8311yescharacteristic
362C42C2×D33211yescharacteristic
363C42#1305059011yescharacteristic
364C42C23×D8311yescharacteristic
365C23—21no—
366C23#1458176611yescharacteristic
367Q8—22no—
368Q8—3322no—
369Q8—3322no—
370Q8—3322no—
371Q8—3322no—
372D4—22no—
373D4—42no—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—21no—
377C2×C4—22no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—3322no—
386C2×C4—3322no—
387C2×C4#1458176611yescharacteristic
388C2×C4#1458176611yescharacteristic
389C2×C4#1458176611yescharacteristic
390C2×C4C22×C22⋊D8311yescharacteristic
391C2×C4C22×D33211yescharacteristic
392C2×C4C2×#1305059011yescharacteristic
393C8—21no—
394C8—21no—
395C8—21no—
396C8—41no—
397C8—41no—
398C8—3322no—
399C8—3322no—
400C8#1458181212yes—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22C2×#1458176611yescharacteristic
408C4—21no—
409C4—21no—
410C4—21no—
411C4—41no—
412C4—41no—
413C4—41no—
414C4—6642no—
415C4—6642no—
416C4#1912797612yes—
417C4C2×#1458181211yescharacteristic
418C4#1912849911yescharacteristic
419C4#1912850112yes—
420C2—41no—
421C2—81no—
422C2#2491691811yescharacteristic
423C2#2491707211yescharacteristic
424C2C2×#1912797611yescharacteristic
subgroupG/H
centerC2×C4#14581766
commutator, cosocleC42×C83C24
FrattiniC42C23×D83
FittingC83×#2965C2
socleC22×C83C2×#121
prime radicalC2×#41×C83D4
hypercenter#2965D83
hypocenterC83#19161
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2965

Complements

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

NH ≅ G/Ntype
1. #24918049C2semidirect
3. #24907017C2semidirect
4. #24907010C2semidirect
5. #24906981C2semidirect
6. #24906976C2semidirect
7. #24906957C2semidirect
8. #24906952C2semidirect
9. #24906211C2semidirect
10. #24906127C2semidirect
224. C83#19161semidirect

Maximal subgroups

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

subgroupindex
#249180492
#249175342
#249070172
#249070102
#249069812
#249069762
#249069572
#249069522
#249062112
#249061272
C83×#29652
#1916183