Finite Group Databasebeta

← #35314690

Subgroups — #35314690

Order 21248.

592 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24913268C211yesmaximal, characteristic
2C2×#19127468C211yesmaximal, characteristic
3#24905488C212yesmaximal, complemented
4#24905485C212yesmaximal, complemented
5C2×#323×C83C211yesmaximal, characteristic
6C2×#14581771C411yescharacteristic, complemented
7C22×C22⋊Dic83—21no—
8#19127637C411yescharacteristic, complemented
9#19127637C2211yescharacteristic
10#19127624—21no—
11#19127624—21no—
12#19127613C411yescharacteristic, complemented
13#19127613C2211yescharacteristic
14#19127574—21no—
15#19127561C411yescharacteristic, complemented
16#19127557—21no—
17#19127468—21no—
18#19127468C2212yes—
19C22×C22⋊C4×C83C2211yescharacteristic
20C2×C22⋊C8×C83—21no—
21#323×C83—21no—
22#323×C83C2212yes—
23C22×Dic166C2×C411yescharacteristic, complemented
24C23×Dic83—41no—
25#14581771—41no—
26#14581771—41no—
27C2×C4⋊Dic83—21no—
28C2×C4⋊Dic83C2×C411yescharacteristic, complemented
29C2×Dic83⋊C4—21no—
30C2×Dic83⋊C4—21no—
31C2×Dic83⋊C4—21no—
32C2×C22⋊Dic83—24no—
33C2×C22⋊Dic83—21no—
34C2×C22⋊Dic83—21no—
35C2×C22⋊Dic83—21no—
36C2×C22⋊Dic83—21no—
37C2×C22⋊Dic83—22no—
38C2×C22⋊Dic83—21no—
39C2×C22⋊Dic83—21no—
40C2×C22⋊Dic83—41no—
41C2×C22⋊Dic83—42no—
42C83⋊#95—21no—
43C83⋊#95—21no—
44C83⋊#95C2×C411yescharacteristic, complemented
45C83⋊#95C2×C411yescharacteristic, complemented
46C25×C83D411yescharacteristic
47C23×C4×C83C2×C411yescharacteristic
48C23×C4×C83D411yescharacteristic
49C22×C8×C83—41no—
50C2×C22⋊C4×C83—41no—
51C2×C22⋊C4×C83—41no—
52C2×C22⋊C4×C83C2×C411yescharacteristic
53C2×C22⋊C4×C83D411yescharacteristic
54C2×C22⋊C4×C83D411yescharacteristic
55C2×C22⋊C4×C83C2311yescharacteristic
56C22⋊C8×C83—22no—
57C22⋊C8×C83—41no—
58C2×Dic166—41no—
59C2×Dic166—81no—
60C22×Dic83—21no—
61C22×Dic83—21no—
62C22×Dic83—21no—
63C22×Dic83—21no—
64C22×Dic83—44no—
65C22×Dic83—42no—
66C22×Dic83—82no—
67C22×Dic83—82no—
68C4⋊Dic83—41no—
69C4⋊Dic83—81no—
70Dic83⋊C4—42no—
71Dic83⋊C4—81no—
72Dic83⋊C4—81no—
73C22⋊Dic83—24no—
74C22⋊Dic83—44no—
75C22⋊Dic83—41no—
76C22⋊Dic83—41no—
77C22⋊Dic83—42no—
78C22⋊Dic83—42no—
79C22⋊Dic83—81no—
80C22⋊Dic83—81no—
81C22⋊Dic83—81no—
82C22⋊Dic83—81no—
83C24×C83—21no—
84C24×C83—21no—
85C24×C83—22no—
86C24×C83—81no—
87C24×C83—81no—
88C24×C83C22⋊C411yescharacteristic
89C24×C83C22⋊C411yescharacteristic
90C24×C83SD1612yes—
91C24×C83Q1612yes—
92C24×C83C2×D411yescharacteristic
93C22×C4×C83—21no—
94C22×C4×C83—21no—
95C22×C4×C83—21no—
96C22×C4×C83—21no—
97C22×C4×C83—81no—
98C22×C4×C83—81no—
99C22×C4×C83C22⋊C411yescharacteristic
100C22×C4×C83C22⋊C411yescharacteristic
101C22×C4×C83C22×C411yescharacteristic
102C22×C4×C83C2×D411yescharacteristic
103C2×C8×C83—42no—
104C2×C8×C83—82no—
105C22⋊C4×C83—41no—
106C22⋊C4×C83—41no—
107C22⋊C4×C83—41no—
108C22⋊C4×C83—41no—
109C23×C83—22no—
110C23×C83—22no—
111C23×C83—21no—
112C23×C83—21no—
113C23×C83—22no—
114C23×C83—22no—
115C23×C83—21no—
116C23×C83—21no—
117C23×C83—41no—
118C23×C83—41no—
119C23×C83—42no—
120C23×C83—42no—
121C23×C83—41no—
122C23×C83—41no—
123C23×C83—41no—
124C23×C83—41no—
125C23×C83—81no—
126C23×C83—81no—
127C23×C83—82no—
128C23×C83—82no—
129C23×C83—84no—
130C23×C83#9911yescharacteristic
131C23×C83#9911yescharacteristic
132C23×C83#10312yes—
133C23×C83#10312yes—
134C23×C83#10412yes—
135C23×C83C2×C22⋊C411yescharacteristic
136C23×C83C2×SD1611yescharacteristic
137C23×C83C2×Q1611yescharacteristic
138C2×C4×C83—21no—
139C2×C4×C83—21no—
140C2×C4×C83—21no—
141C2×C4×C83—21no—
142C2×C4×C83—42no—
143C2×C4×C83—41no—
144C2×C4×C83—41no—
145C2×C4×C83—41no—
146C2×C4×C83—41no—
147C2×C4×C83—81no—
148C2×C4×C83—81no—
149C2×C4×C83—81no—
150C2×Dic83—44no—
151C2×Dic83—41no—
152C2×Dic83—41no—
153C2×Dic83—41no—
154C2×Dic83—41no—
155C2×Dic83—41no—
156C2×Dic83—41no—
157C2×Dic83—41no—
158C2×Dic83—41no—
159C2×Dic83—41no—
160C2×Dic83—41no—
161C2×Dic83—41no—
162C2×Dic83—41no—
163C2×Dic83—88no—
164C2×Dic83—82no—
165C2×Dic83—82no—
166Dic166—82no—
167C8×C83—82no—
168C22×C83—21no—
169C22×C83—21no—
170C22×C83—21no—
171C22×C83—21no—
172C22×C83—21no—
173C22×C83—21no—
174C22×C83—21no—
175C22×C83—22no—
176C22×C83—21no—
177C22×C83—42no—
178C22×C83—42no—
179C22×C83—42no—
180C22×C83—41no—
181C22×C83—41no—
182C22×C83—44no—
183C22×C83—44no—
184C22×C83—41no—
185C22×C83—41no—
186C22×C83—41no—
187C22×C83—41no—
188C22×C83—81no—
189C22×C83—81no—
190C22×C83—82no—
191C22×C83—82no—
192C22×C83#32812yes—
193C22×C83#32812yes—
194C22×C83C2×#9911yescharacteristic
195C22×C83C2×#10311yescharacteristic
196C22×C83C2×#10411yescharacteristic
197C4×C83—41no—
198C4×C83—41no—
199C4×C83—41no—
200C4×C83—41no—
201Dic83—81no—
202Dic83—81no—
203Dic83—84no—
204Dic83—81no—
205Dic83—81no—
206#11716—831nomaximal, Sylow
207C2×C83—21no—
208C2×C83—22no—
209C2×C83—21no—
210C2×C83—22no—
211C2×C83—41no—
212C2×C83—42no—
213C2×C83—41no—
214C2×C83C2×#32811yescharacteristic
215C2×C83#212111yescharacteristic
216C2×C83#212711yescharacteristic
217#2626—831no—
218C2×#342—831no—
219C2×#323D8311yescharacteristic
220#1352—832no—
221#1351—832no—
222C83#1171611yescharacteristic, complemented, Sylow
223C2×C22⋊Q8—831no—
224C22×C22⋊C4—1661no—
225C22×C22⋊C4C2×D8311yescharacteristic
226C2×C22⋊C8—21no—
227#399—831no—
228#399—831no—
229#396—1661no—
230#396—1661no—
231#393—831no—
232#393—831no—
233#385—1661no—
234#380—831no—
235#379—1661no—
236#342—832no—
237#342—1661no—
238#323—21no—
239#323C2×D8312yes—
240C25D33211yescharacteristic
241C22×Q8—831no—
242C23×C4—3321no—
243C23×C4C4×D8311yescharacteristic
244C23×C4C22⋊D8311yescharacteristic
245C22×C8—41no—
246C22⋊Q8—3321no—
247C22⋊Q8—3321no—
248C2×C4⋊C4—831no—
249C2×C4⋊C4—1661no—
250C2×C4⋊C4—1661no—
251C2×C4⋊C4—1661no—
252C2×C4⋊C4—1661no—
253C2×C22⋊C4—41no—
254C2×C22⋊C4—41no—
255C2×C22⋊C4—3321no—
256C2×C22⋊C4—3322no—
257C2×C22⋊C4—1661no—
258C2×C22⋊C4—1661no—
259C2×C22⋊C4—1664no—
260C2×C22⋊C4—1662no—
261C2×C22⋊C4—1661no—
262C2×C22⋊C4—1661no—
263C2×C22⋊C4—1661no—
264C2×C22⋊C4—1661no—
265C2×C22⋊C4C4×D8311yescharacteristic
266C2×C22⋊C4D33211yescharacteristic
267C2×C22⋊C4C22⋊D8311yescharacteristic
268C2×C22⋊C4C22×D8311yescharacteristic
269C22⋊C8—22no—
270C22⋊C8—41no—
271#95—831no—
272#95—831no—
273#95—1661no—
274#95—1661no—
275C24—21no—
276C24—21no—
277C24—22no—
278C24—81no—
279C24—81no—
280C24(C2×D83)⋊C411yescharacteristic
281C24(C2×D83)⋊C411yescharacteristic
282C24#1305057912yes—
283C24Dic33212yes—
284C24C2×D33211yescharacteristic
285C2×Q8—3321no—
286C2×Q8—6641no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—81no—
292C22×C4—81no—
293C22×C4—3322no—
294C22×C4—3324no—
295C22×C4—6642no—
296C22×C4—6642no—
297C22×C4—1661no—
298C22×C4—1661no—
299C22×C4—1661no—
300C22×C4—1661no—
301C22×C4(C2×D83)⋊C411yescharacteristic
302C22×C4(C2×D83)⋊C411yescharacteristic
303C22×C4C2×C4×D8311yescharacteristic
304C22×C4C2×C22⋊D8311yescharacteristic
305C2×C8—42no—
306C2×C8—82no—
307C4⋊C4—3321no—
308C4⋊C4—3322no—
309C4⋊C4—6641no—
310C4⋊C4—6641no—
311C4⋊C4—6641no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—3324no—
317C22⋊C4—3322no—
318C22⋊C4—3321no—
319C22⋊C4—3321no—
320C22⋊C4—3322no—
321C22⋊C4—6641no—
322C22⋊C4—6641no—
323C22⋊C4—6641no—
324C22⋊C4—6641no—
325C22⋊C4—1664no—
326C23—21no—
327C23—21no—
328C23—22no—
329C23—22no—
330C23—22no—
331C23—22no—
332C23—21no—
333C23—21no—
334C23—41no—
335C23—41no—
336C23—42no—
337C23—42no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—81no—
343C23—81no—
344C23—82no—
345C23—82no—
346C23—84no—
347C23#1458170611yescharacteristic
348C23#1458170611yescharacteristic
349C23#1458171712yes—
350C23#1458171712yes—
351C23#1458171812yes—
352C23C2×(C2×D83)⋊C411yescharacteristic
353C23C2×#1305057911yescharacteristic
354C23C2×Dic33211yescharacteristic
355Q8—6642no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—3324no—
369C2×C4—3321no—
370C2×C4—3321no—
371C2×C4—3321no—
372C2×C4—3321no—
373C2×C4—3321no—
374C2×C4—3321no—
375C2×C4—3321no—
376C2×C4—3321no—
377C2×C4—3321no—
378C2×C4—3321no—
379C2×C4—3321no—
380C2×C4—3321no—
381C2×C4—6648no—
382C2×C4—6642no—
383C2×C4—6642no—
384C8—82no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—22no—
394C22—42no—
395C22—42no—
396C22—41no—
397C22—41no—
398C22—42no—
399C22—44no—
400C22—44no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—81no—
406C22—81no—
407C22—82no—
408C22—82no—
409C22#1912743912yes—
410C22#1912743912yes—
411C22C2×#1458170611yescharacteristic
412C22C2×#1458171711yescharacteristic
413C22C2×#1458171811yescharacteristic
414C4—41no—
415C4—41no—
416C4—41no—
417C4—41no—
418C4—6641no—
419C4—6641no—
420C4—6644no—
421C4—6641no—
422C4—6641no—
423C2—21no—
424C2—21no—
425C2—22no—
426C2—22no—
427C2—41no—
428C2—41no—
429C2—42no—
430C2C2×#1912743911yescharacteristic
431C2#2490911011yescharacteristic
432C2#2490913011yescharacteristic
subgroupG/H
centerC22C2×#14581706
commutatorC22×C4×C83C22×C4
FrattiniC2×C22⋊C4C22×D83
FittingC2×#323×C83C2
socleC22×C83C2×#99
prime radicalC25×C83D4
cosocleC2×C22⋊C4×C83C23
hypercenterC2×#323D83
hypocenterC83#11716
LCSG ▷ C22×C4×C83 ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ C2×#323

Complements

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

NH ≅ G/Ntype
3. #24905488C2semidirect
4. #24905485C2semidirect
6. C2×#14581771C4semidirect
8. #19127637C4semidirect
12. #19127613C4semidirect
15. #19127561C4semidirect
23. C22×Dic166C2×C4semidirect
28. C2×C4⋊Dic83C2×C4semidirect
44. C83⋊#95C2×C4semidirect
45. C83⋊#95C2×C4semidirect
222. C83#11716semidirect

Maximal subgroups

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

subgroupindex
#249132682
C2×#191274682
#249054882
#249054852
C2×#323×C832
#1171683