Finite Group Databasebeta

← #35315009

Subgroups — #35315009

Order 21248.

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

normal subgroups only

This group has 460 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
1C4×#14581707C211yesmaximal, characteristic
2#24905801C211yesmaximal, characteristic
3#24905487C212yesmaximal, complemented
4#24905484C212yesmaximal, complemented
5C83×#2580C211yesmaximal, characteristic
6C2×#14581707—21no—
7#19127665—21no—
8C4×C22⋊Dic83—21no—
9#19127468C2212yes—
10#19127427—21no—
11#19127425C2212yes—
12C2×C4×Q8×C83C411yescharacteristic, complemented
13C2×C4×D4×C83C2211yescharacteristic
14#399×C83C411yescharacteristic, complemented
15#399×C83C2211yescharacteristic
16#393×C83C411yescharacteristic, complemented
17#393×C83C2211yescharacteristic
18#390×C83—21no—
19#389×C83—21no—
20#385×C83—21no—
21#385×C83—21no—
22#382×C83C411yescharacteristic, complemented
23C2×#13050574—41no—
24C2×C22⋊Dic83—22no—
25C2×C4×Dic83—41no—
26#14581707—24no—
27C22⋊(C83⋊C8)—22no—
28C22⋊(C83⋊C8)—41no—
29C83⋊#95—41no—
30C22×Q8×C83C2×C411yescharacteristic
31C22×D4×C83D411yescharacteristic
32C23×C4×C83D411yescharacteristic
33C23×C4×C83D411yescharacteristic
34C4×Q8×C83—41no—
35C4×Q8×C83—41no—
36C4×D4×C83—41no—
37C4×D4×C83—41no—
38C2×C4⋊C4×C83—21no—
39C2×C4⋊C4×C83—21no—
40C2×C4⋊C4×C83—21no—
41C2×C4⋊C4×C83—21no—
42C2×C4⋊C4×C83C2×C411yescharacteristic
43C2×C4⋊C4×C83D411yescharacteristic
44C2×C22⋊C4×C83—21no—
45C2×C22⋊C4×C83—21no—
46C2×C22⋊C4×C83—21no—
47C2×C22⋊C4×C83—21no—
48C2×C22⋊C4×C83C2×C411yescharacteristic
49C2×C22⋊C4×C83C2311yescharacteristic
50C2×C42×C83—21no—
51C2×C42×C83C2×C411yescharacteristic
52#95×C83—21no—
53#95×C83—21no—
54#95×C83—21no—
55#95×C83C2×C411yescharacteristic
56#95×C83C2×C411yescharacteristic
57C22×Dic83—42no—
58#13050574—82no—
59C2×C83⋊C8—42no—
60C22⋊Dic83—24no—
61C22⋊Dic83—42no—
62C4×Dic83—82no—
63C24×C83C22⋊C411yescharacteristic
64C24×C83C2×D411yescharacteristic
65C2×Q8×C83—42no—
66C2×Q8×C83—41no—
67C2×D4×C83—22no—
68C2×D4×C83—22no—
69C2×D4×C83—41no—
70C22×C4×C83—21no—
71C22×C4×C83—21no—
72C22×C4×C83—21no—
73C22×C4×C83—21no—
74C22×C4×C83—21no—
75C22×C4×C83—21no—
76C22×C4×C83—21no—
77C22×C4×C83—21no—
78C22×C4×C83—81no—
79C22×C4×C83—81no—
80C22×C4×C83C22⋊C411yescharacteristic
81C22×C4×C83C22⋊C411yescharacteristic
82C22×C4×C83C22⋊C411yescharacteristic
83C22×C4×C83C22×C411yescharacteristic
84C22×C4×C83C2×D411yescharacteristic
85C4⋊C4×C83—41no—
86C4⋊C4×C83—41no—
87C4⋊C4×C83—41no—
88C4⋊C4×C83—41no—
89C4⋊C4×C83—42no—
90C4⋊C4×C83—81no—
91C4⋊C4×C83—81no—
92C22⋊C4×C83—41no—
93C22⋊C4×C83—41no—
94C22⋊C4×C83—42no—
95C22⋊C4×C83—42no—
96C22⋊C4×C83—81no—
97C22⋊C4×C83—81no—
98C42×C83—21no—
99C42×C83—21no—
100C42×C83—42no—
101C23×C83—21no—
102C23×C83—21no—
103C23×C83—21no—
104C23×C83—21no—
105C23×C83—21no—
106C23×C83—41no—
107C23×C83—41no—
108C23×C83—41no—
109C23×C83—41no—
110C23×C83C2×C22⋊C411yescharacteristic
111C23×C83#13712yes—
112Q8×C83—42no—
113Q8×C83—42no—
114D4×C83—44no—
115C2×C4×C83—21no—
116C2×C4×C83—21no—
117C2×C4×C83—22no—
118C2×C4×C83—21no—
119C2×C4×C83—21no—
120C2×C4×C83—21no—
121C2×C4×C83—41no—
122C2×C4×C83—42no—
123C2×C4×C83—42no—
124C2×C4×C83—41no—
125C2×C4×C83—41no—
126C2×C4×C83—41no—
127C2×C4×C83—41no—
128C2×C4×C83—41no—
129C2×C4×C83—41no—
130C2×C4×C83—41no—
131C2×C4×C83—41no—
132C2×C4×C83—41no—
133C2×C4×C83—41no—
134C2×C4×C83—41no—
135C2×C4×C83—41no—
136C2×C4×C83—41no—
137C2×C4×C83—41no—
138C2×C4×C83—81no—
139C2×C4×C83—81no—
140C2×C4×C83#9911yescharacteristic
141C2×C4×C83#9911yescharacteristic
142C2×C4×C83#10412yes—
143C2×Dic83—44no—
144C2×Dic83—82no—
145C2×Dic83—82no—
146C83⋊C8—82no—
147C22×C83—21no—
148C22×C83—21no—
149C22×C83—22no—
150C22×C83—42no—
151C22×C83—42no—
152C22×C83—41no—
153C22×C83—41no—
154C22×C83—41no—
155C22×C83—41no—
156C22×C83—41no—
157C22×C83—81no—
158C22×C83—81no—
159C22×C83C2×#9911yescharacteristic
160C22×C83#41911yescharacteristic
161C22×C83C2×#10411yescharacteristic
162C4×C83—21no—
163C4×C83—21no—
164C4×C83—22no—
165C4×C83—42no—
166C4×C83—41no—
167C4×C83—41no—
168C4×C83—41no—
169C4×C83—81no—
170C4×C83—81no—
171C4×C83—81no—
172Dic83—84no—
173#11784—831nomaximal, Sylow
174C2×C83—21no—
175C2×C83—21no—
176C2×C83—42no—
177C2×C83—81no—
178C2×C83#150811yescharacteristic
179C2×C83#212011yescharacteristic
180C2×C83#212711yescharacteristic
181#2580D8311yescharacteristic
182C4×#99—831no—
183#1465—831no—
184#1352—832no—
185#1351—832no—
186C83#1178411yescharacteristic, complemented, Sylow
187C2×C4×Q8Dic8311yescharacteristic, complemented
188C2×C4×D4C2×D8311yescharacteristic
189C2×#99—1661no—
190#407—1661no—
191#399Dic8311yescharacteristic, complemented
192#399C2×D8311yescharacteristic
193#393Dic8311yescharacteristic, complemented
194#393C2×D8311yescharacteristic
195#390—21no—
196#389—21no—
197#385—21no—
198#385—21no—
199#382Dic8311yescharacteristic, complemented
200C4×C22⋊C4—1661no—
201#342—832no—
202#324—1661no—
203#323—832no—
204C22×Q8C2×Dic8311yescharacteristic
205C22×D4C22⋊D8311yescharacteristic
206C23×C4C22⋊D8311yescharacteristic
207C23×C4C22⋊D8311yescharacteristic
208C2×M16—3321no—
209C4×Q8—41no—
210C4×Q8—41no—
211C4×D4—41no—
212C4×D4—41no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4C2×Dic8311yescharacteristic
218C2×C4⋊C4C22⋊D8311yescharacteristic
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—1662no—
224C2×C22⋊C4C2×Dic8311yescharacteristic
225C2×C22⋊C4C22×D8311yescharacteristic
226C2×C42—21no—
227C2×C42—3321no—
228C2×C42C2×Dic8311yescharacteristic
229#99—1664no—
230C22⋊C8—3321no—
231C22⋊C8—1662no—
232#95—21no—
233#95—21no—
234#95—21no—
235#95—3321no—
236#95C2×Dic8311yescharacteristic
237#95C2×Dic8311yescharacteristic
238C24C22⋊Dic8311yescharacteristic
239C24C2×C22⋊D8311yescharacteristic
240C2×Q8—42no—
241C2×Q8—41no—
242C2×D4—22no—
243C2×D4—22no—
244C2×D4—41no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—81no—
254C22×C4—81no—
255C22×C4—3322no—
256C22×C4C22⋊Dic8311yescharacteristic
257C22×C4C22⋊Dic8311yescharacteristic
258C22×C4C22⋊Dic8311yescharacteristic
259C22×C4C22×Dic8311yescharacteristic
260C22×C4C2×C22⋊D8311yescharacteristic
261M16—6642no—
262C2×C8—3322no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—42no—
268C4⋊C4—81no—
269C4⋊C4—81no—
270C22⋊C4—41no—
271C22⋊C4—41no—
272C22⋊C4—42no—
273C22⋊C4—42no—
274C22⋊C4—81no—
275C22⋊C4—81no—
276C22⋊C4—3322no—
277C22⋊C4—1664no—
278C42—21no—
279C42—21no—
280C42—42no—
281C42—6642no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23C2×C22⋊Dic8311yescharacteristic
292C23#1458182312yes—
293Q8—42no—
294Q8—42no—
295D4—44no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—3324no—
322C2×C4—6642no—
323C2×C4—6642no—
324C2×C4#1458170711yescharacteristic
325C2×C4#1458170711yescharacteristic
326C2×C4#1458171912yes—
327C8—6642no—
328C22—21no—
329C22—21no—
330C22—22no—
331C22—42no—
332C22—42no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—81no—
339C22—81no—
340C22C2×#1458170711yescharacteristic
341C22#1912771011yescharacteristic
342C22C2×#1458171911yescharacteristic
343C4—21no—
344C4—21no—
345C4—22no—
346C4—42no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—81no—
351C4—81no—
352C4—81no—
353C4—6644no—
354C2—21no—
355C2—21no—
356C2—42no—
357C2—81no—
358C2#2490598311yescharacteristic
359C2#2490910411yescharacteristic
360C2#2490913211yescharacteristic
subgroupG/H
centerC22C2×#14581707
commutatorC22×C4×C83C22×C4
FrattiniC2×C22⋊C4C22×D83
FittingC83×#2580C2
socleC22×C83C2×#99
prime radicalC22×D4×C83D4
cosocleC2×C22⋊C4×C83C23
hypercenter#2580D83
hypocenterC83#11784
LCSG ▷ C22×C4×C83 ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2580

Complements

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

NH ≅ G/Ntype
3. #24905487C2semidirect
4. #24905484C2semidirect
12. C2×C4×Q8×C83C4semidirect
14. #399×C83C4semidirect
16. #393×C83C4semidirect
22. #382×C83C4semidirect
186. C83#11784semidirect
187. C2×C4×Q8Dic83semidirect
191. #399Dic83semidirect
193. #393Dic83semidirect
199. #382Dic83semidirect

Maximal subgroups

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

subgroupindex
C4×#145817072
#249058012
#249054872
#249054842
C83×#25802
#1178483