Finite Group Databasebeta

← #35314885

Subgroups — #35314885

Order 21248.

486 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 91 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#24907196C211yesmaximal, characteristic, complemented
2#24905797C211yesmaximal, characteristic, complemented
3#24905484C212yesmaximal, complemented
4#24905460C212yesmaximal
5C83×#2610C211yesmaximal, characteristic
6C2×#14581751—21no—
7#19127758—21no—
8#19127468—21no—
9#19127468C2212yes—
10#19127427—21no—
11#19127425C2212yes—
12C2×#124×C83C411yescharacteristic, complemented
13C2×#117×C83C2211yescharacteristic
14#395×C83C411yescharacteristic, complemented
15#394×C83C411yescharacteristic, complemented
16#393×C83C2211yescharacteristic
17#392×C83C2211yescharacteristic
18#390×C83—21no—
19#388×C83—21no—
20#388×C83—21no—
21#386×C83—21no—
22#376×C83C411yescharacteristic, complemented
23#14581751—24no—
24#14581751—42no—
25C2×C4⋊Dic83—41no—
26C2×C22⋊Dic83—22no—
27C2×C4×Dic83—41no—
28C4⋊(C83⋊C8)—41no—
29C22⋊(C83⋊C8)—22no—
30#14581703—41no—
31C22×Q8×C83C2×C411yescharacteristic
32C22×D4×C83—21no—
33C22×D4×C83C2×C411yescharacteristic
34C23×C4×C83D411yescharacteristic
35#124×C83—41no—
36#124×C83—41no—
37#117×C83—41no—
38#117×C83—41no—
39C2×C4⋊C4×C83—21no—
40C2×C4⋊C4×C83C2×C411yescharacteristic
41C2×C4⋊C4×C83D411yescharacteristic
42C2×C22⋊C4×C83—21no—
43C2×C22⋊C4×C83—21no—
44C2×C22⋊C4×C83—21no—
45C2×C22⋊C4×C83—21no—
46C2×C22⋊C4×C83—21no—
47C2×C22⋊C4×C83—21no—
48C2×C22⋊C4×C83D411yescharacteristic
49C2×C22⋊C4×C83C2311yescharacteristic
50C2×C42×C83—21no—
51C2×C42×C83C2×C411yescharacteristic
52C2×C42×C83D411yescharacteristic
53#95×C83—21no—
54#95×C83—21no—
55#95×C83—21no—
56#95×C83C2×C411yescharacteristic
57#95×C83C2×C411yescharacteristic
58C22×Dic83—42no—
59C2×C83⋊C8—42no—
60C4⋊Dic83—44no—
61C22⋊Dic83—24no—
62C22⋊Dic83—42no—
63C4×Dic83—44no—
64C24×C83—21no—
65C24×C83C2×D411yescharacteristic
66C2×Q8×C83—41no—
67C2×Q8×C83—81no—
68C2×D4×C83—41no—
69C2×D4×C83—42no—
70C2×D4×C83—41no—
71C2×D4×C83—41no—
72C2×D4×C83—81no—
73C2×D4×C83—81no—
74C22×C4×C83—21no—
75C22×C4×C83—21no—
76C22×C4×C83—21no—
77C22×C4×C83—21no—
78C22×C4×C83—22no—
79C22×C4×C83—21no—
80C22×C4×C83—21no—
81C22×C4×C83—21no—
82C22×C4×C83C22⋊C411yescharacteristic
83C22×C4×C83C22⋊C411yescharacteristic
84C22×C4×C83C22⋊C411yescharacteristic
85C22×C4×C83C22⋊C411yescharacteristic
86C22×C4×C83C22×C411yescharacteristic
87C22×C4×C83C2×D411yescharacteristic
88C4⋊C4×C83—41no—
89C4⋊C4×C83—41no—
90C4⋊C4×C83—81no—
91C22⋊C4×C83—41no—
92C22⋊C4×C83—41no—
93C22⋊C4×C83—42no—
94C22⋊C4×C83—42no—
95C22⋊C4×C83—41no—
96C22⋊C4×C83—41no—
97C22⋊C4×C83—42no—
98C22⋊C4×C83—81no—
99C22⋊C4×C83—81no—
100C42×C83—21no—
101C42×C83—21no—
102C42×C83—21no—
103C42×C83—21no—
104C42×C83—42no—
105C23×C83—21no—
106C23×C83—21no—
107C23×C83—41no—
108C23×C83—41no—
109C23×C83—41no—
110C23×C83—41no—
111C23×C83—41no—
112C23×C83—41no—
113C23×C83—41no—
114C23×C83—41no—
115C23×C83—41no—
116C23×C83C2×C22⋊C411yescharacteristic
117C23×C83AGL(1,ℤ/8)11yescharacteristic
118C23×C83#13711yescharacteristic
119Q8×C83—82no—
120D4×C83—82no—
121D4×C83—82no—
122D4×C83—82no—
123C2×C4×C83—22no—
124C2×C4×C83—22no—
125C2×C4×C83—22no—
126C2×C4×C83—22no—
127C2×C4×C83—21no—
128C2×C4×C83—21no—
129C2×C4×C83—21no—
130C2×C4×C83—21no—
131C2×C4×C83—22no—
132C2×C4×C83—21no—
133C2×C4×C83—21no—
134C2×C4×C83—21no—
135C2×C4×C83—21no—
136C2×C4×C83—41no—
137C2×C4×C83—41no—
138C2×C4×C83—41no—
139C2×C4×C83—41no—
140C2×C4×C83—41no—
141C2×C4×C83—41no—
142C2×C4×C83—41no—
143C2×C4×C83—41no—
144C2×C4×C83—41no—
145C2×C4×C83—41no—
146C2×C4×C83—41no—
147C2×C4×C83—41no—
148C2×C4×C83#9911yescharacteristic
149C2×C4×C83#9911yescharacteristic
150C2×C4×C83#10412yes—
151C2×Dic83—44no—
152C2×Dic83—82no—
153C2×Dic83—82no—
154C83⋊C8—82no—
155C22×C83—21no—
156C22×C83—21no—
157C22×C83—21no—
158C22×C83—21no—
159C22×C83—41no—
160C22×C83—41no—
161C22×C83—41no—
162C22×C83—41no—
163C22×C83—81no—
164C22×C83—81no—
165C22×C83—81no—
166C22×C83—81no—
167C22×C83—81no—
168C22×C83—81no—
169C22×C83—81no—
170C22×C83—81no—
171C22×C83C2×#9911yescharacteristic
172C22×C83#41711yescharacteristic
173C22×C83C2×#10411yescharacteristic
174C4×C83—21no—
175C4×C83—21no—
176C4×C83—22no—
177C4×C83—42no—
178C4×C83—41no—
179C4×C83—41no—
180C4×C83—41no—
181C4×C83—41no—
182C4×C83—81no—
183C4×C83—81no—
184Dic83—84no—
185#11758—831nomaximal, Sylow
186C2×C83—21no—
187C2×C83—21no—
188C2×C83—41no—
189C2×C83—41no—
190C2×C83—161no—
191C2×C83#150411yescharacteristic
192C2×C83#212211yescharacteristic
193C2×C83#212611yescharacteristic
194#2610D8311yescharacteristic
195#1740—831no—
196#1464—831no—
197#1351—832no—
198#1343—832no—
199C83#1175811yescharacteristic, complemented, Sylow
200C2×#124Dic8311yescharacteristic, complemented
201C2×#117—1661no—
202C2×#117C2×D8311yescharacteristic
203#432—1661no—
204#395Dic8311yescharacteristic, complemented
205#394Dic8311yescharacteristic, complemented
206#393C2×D8311yescharacteristic
207#392C2×D8311yescharacteristic
208#390—21no—
209#388—21no—
210#388—21no—
211#386—21no—
212#376Dic8311yescharacteristic, complemented
213#342—832no—
214#342—1661no—
215#324—1661no—
216#323—832no—
217C22×Q8C2×Dic8311yescharacteristic
218C22×D4—21no—
219C22×D4C2×Dic8311yescharacteristic
220C23×C4C22⋊D8311yescharacteristic
221#124—41no—
222#124—41no—
223#117—41no—
224#117—41no—
225#117—3322no—
226#117—1664no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—3321no—
229C2×C4⋊C4C2×Dic8311yescharacteristic
230C2×C4⋊C4C22⋊D8311yescharacteristic
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—1662no—
238C2×C22⋊C4C22⋊D8311yescharacteristic
239C2×C22⋊C4C22×D8311yescharacteristic
240C2×C42—21no—
241C2×C42—3321no—
242C2×C42C2×Dic8311yescharacteristic
243C2×C42C22⋊D8311yescharacteristic
244C4⋊C8—3321no—
245C22⋊C8—1662no—
246#97—3321no—
247#95—21no—
248#95—21no—
249#95—21no—
250#95C2×Dic8311yescharacteristic
251#95C2×Dic8311yescharacteristic
252C24—21no—
253C24C2×C22⋊D8311yescharacteristic
254C2×Q8—41no—
255C2×Q8—81no—
256C2×D4—41no—
257C2×D4—42no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—81no—
261C2×D4—81no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—3322no—
271C22×C4C22⋊Dic8311yescharacteristic
272C22×C4C22⋊Dic8311yescharacteristic
273C22×C4C22⋊Dic8311yescharacteristic
274C22×C4C22⋊Dic8311yescharacteristic
275C22×C4C22×Dic8311yescharacteristic
276C22×C4C2×C22⋊D8311yescharacteristic
277C2×C8—3322no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—81no—
281C4⋊C4—3324no—
282C22⋊C4—41no—
283C22⋊C4—41no—
284C22⋊C4—42no—
285C22⋊C4—42no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—42no—
289C22⋊C4—81no—
290C22⋊C4—81no—
291C22⋊C4—3322no—
292C22⋊C4—1664no—
293C42—21no—
294C42—21no—
295C42—21no—
296C42—21no—
297C42—42no—
298C42—3324no—
299C23—21no—
300C23—21no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23C2×C22⋊Dic8311yescharacteristic
311C23#1458181711yescharacteristic
312C23#1458182211yescharacteristic
313Q8—82no—
314D4—82no—
315D4—82no—
316D4—82no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—3324no—
343C2×C4—6642no—
344C2×C4—6642no—
345C2×C4#1458170711yescharacteristic
346C2×C4#1458170711yescharacteristic
347C2×C4#1458171912yes—
348C8—6642no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22C2×#1458170711yescharacteristic
366C22#1912770211yescharacteristic
367C22C2×#1458171911yescharacteristic
368C4—21no—
369C4—21no—
370C4—22no—
371C4—42no—
372C4—41no—
373C4—41no—
374C4—41no—
375C4—41no—
376C4—81no—
377C4—81no—
378C4—6644no—
379C2—21no—
380C2—21no—
381C2—41no—
382C2—41no—
383C2—161no—
384C2#2490596711yescharacteristic
385C2#2490911311yescharacteristic
386C2#2490912811yescharacteristic
subgroupG/H
centerC22C2×#14581707
commutatorC22×C4×C83C22×C4
FrattiniC2×C22⋊C4C22×D83
FittingC83×#2610C2
socleC22×C83C2×#99
prime radical#392×C83C22
cosocleC2×C22⋊C4×C83C23
hypercenter#2610D83
hypocenterC83#11758
LCSG ▷ C22×C4×C83 ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2610

Complements

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

NH ≅ G/Ntype
1. #24907196C2semidirect
2. #24905797C2semidirect
3. #24905484C2semidirect
12. C2×#124×C83C4semidirect
14. #395×C83C4semidirect
15. #394×C83C4semidirect
22. #376×C83C4semidirect
199. C83#11758semidirect
200. C2×#124Dic83semidirect
204. #395Dic83semidirect
205. #394Dic83semidirect
212. #376Dic83semidirect

Maximal subgroups

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

subgroupindex
#249071962
#249057972
#249054842
#249054602
C83×#26102
#1175883