Finite Group Databasebeta

← #35314738

Subgroups — #35314738

Order 21248.

456 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 95 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#24905784C211yesmaximal, characteristic
3#24905487C212yesmaximal, complemented
4#24905463C212yesmaximal, complemented
5C83×#2395C211yesmaximal, characteristic
6#19127754—21no—
7C2×#14581707—21no—
8C4×C22⋊Dic83—21no—
9#19127468C2212yes—
10#19127425—21no—
11#19127425C2212yes—
12C2×C4×D4×C83C2211yescharacteristic
13C2×#117×C83C411yescharacteristic, complemented
14#400×C83C411yescharacteristic, complemented
15#399×C83C2211yescharacteristic
16#398×C83C411yescharacteristic, complemented
17#397×C83—21no—
18#397×C83C2211yescharacteristic
19#394×C83—21no—
20C4×C4⋊C4×C83—21no—
21C4×C22⋊C4×C83—21no—
22C4×C22⋊C4×C83C411yescharacteristic, complemented
23C2×C22⋊Dic83—22no—
24C2×C4×Dic83—41no—
25C4⋊(C83⋊C8)—41no—
26#14581707—24no—
27C22⋊(C83⋊C8)—22no—
28C4×C83⋊C8—41no—
29C83⋊#95—41no—
30C22×D4×C83D411yescharacteristic
31C23×C4×C83C2×C411yescharacteristic
32C23×C4×C83D411yescharacteristic
33C4×D4×C83—41no—
34C4×D4×C83—41no—
35#117×C83—41no—
36#117×C83—41no—
37C2×C4⋊C4×C83—21no—
38C2×C4⋊C4×C83—21no—
39C2×C4⋊C4×C83C2×C411yescharacteristic
40C2×C4⋊C4×C83C2×C411yescharacteristic
41C2×C4⋊C4×C83C2×C411yescharacteristic
42C2×C22⋊C4×C83—21no—
43C2×C22⋊C4×C83—21no—
44C2×C22⋊C4×C83—21no—
45C2×C22⋊C4×C83—21no—
46C2×C22⋊C4×C83D411yescharacteristic
47C2×C22⋊C4×C83C2311yescharacteristic
48C2×C42×C83—21no—
49C2×C42×C83—21no—
50C2×C42×C83D411yescharacteristic
51#95×C83—21no—
52#95×C83—21no—
53#95×C83—21no—
54#95×C83—21no—
55#95×C83C2×C411yescharacteristic
56#95×C83C2×C411yescharacteristic
57C22×Dic83—42no—
58C2×C83⋊C8—42no—
59C22⋊Dic83—24no—
60C22⋊Dic83—42no—
61C4×Dic83—82no—
62C24×C83C22⋊C411yescharacteristic
63C24×C83C2×D411yescharacteristic
64C2×D4×C83—22no—
65C2×D4×C83—22no—
66C2×D4×C83—41no—
67C22×C4×C83—21no—
68C22×C4×C83—21no—
69C22×C4×C83—21no—
70C22×C4×C83—21no—
71C22×C4×C83—21no—
72C22×C4×C83—21no—
73C22×C4×C83—21no—
74C22×C4×C83—21no—
75C22×C4×C83—42no—
76C22×C4×C83—42no—
77C22×C4×C83C22⋊C411yescharacteristic
78C22×C4×C83C22⋊C411yescharacteristic
79C22×C4×C83C22⋊C411yescharacteristic
80C22×C4×C83C22×C411yescharacteristic
81C22×C4×C83C2×D411yescharacteristic
82C4⋊C4×C83—41no—
83C4⋊C4×C83—41no—
84C4⋊C4×C83—41no—
85C4⋊C4×C83—42no—
86C4⋊C4×C83—42no—
87C22⋊C4×C83—41no—
88C22⋊C4×C83—41no—
89C22⋊C4×C83—42no—
90C22⋊C4×C83—42no—
91C22⋊C4×C83—42no—
92C22⋊C4×C83—42no—
93C42×C83—21no—
94C42×C83—21no—
95C42×C83—42no—
96C42×C83—41no—
97C42×C83—41no—
98C23×C83—21no—
99C23×C83—21no—
100C23×C83—21no—
101C23×C83—21no—
102C23×C83—21no—
103C23×C83—41no—
104C23×C83—41no—
105C23×C83—41no—
106C23×C83—41no—
107C23×C83C2×C22⋊C411yescharacteristic
108C23×C83#13512yes—
109D4×C83—44no—
110C2×C4×C83—21no—
111C2×C4×C83—21no—
112C2×C4×C83—22no—
113C2×C4×C83—22no—
114C2×C4×C83—21no—
115C2×C4×C83—21no—
116C2×C4×C83—21no—
117C2×C4×C83—42no—
118C2×C4×C83—41no—
119C2×C4×C83—41no—
120C2×C4×C83—42no—
121C2×C4×C83—42no—
122C2×C4×C83—41no—
123C2×C4×C83—41no—
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—81no—
135C2×C4×C83#9911yescharacteristic
136C2×C4×C83#9911yescharacteristic
137C2×C4×C83#10412yes—
138C2×Dic83—44no—
139C2×Dic83—82no—
140C2×Dic83—82no—
141C83⋊C8—82no—
142C22×C83—21no—
143C22×C83—21no—
144C22×C83—22no—
145C22×C83—42no—
146C22×C83—41no—
147C22×C83—41no—
148C22×C83—41no—
149C22×C83—41no—
150C22×C83—41no—
151C22×C83—81no—
152C22×C83—81no—
153C22×C83—81no—
154C22×C83C2×#9911yescharacteristic
155C22×C83#41611yescharacteristic
156C22×C83C2×#10411yescharacteristic
157C4×C83—21no—
158C4×C83—22no—
159C4×C83—42no—
160C4×C83—41no—
161C4×C83—41no—
162C4×C83—41no—
163C4×C83—81no—
164C4×C83—81no—
165C4×C83—81no—
166C4×C83#35212yes—
167Dic83—84no—
168#11727—831nomaximal, Sylow
169C2×C83—21no—
170C2×C83—21no—
171C2×C83—42no—
172C2×C83—81no—
173C2×C83#150111yescharacteristic
174C2×C83C2×#35211yescharacteristic
175C2×C83#212611yescharacteristic
176#2395D8311yescharacteristic
177C4×#99—831no—
178#1460—831no—
179#1352—832no—
180#1344—832no—
181C83#1172711yescharacteristic, complemented, Sylow
182C2×C4×D4C2×D8311yescharacteristic
183C2×#117Dic8311yescharacteristic, complemented
184#431—1661no—
185C2×#99—1661no—
186#400Dic8311yescharacteristic, complemented
187#399C2×D8311yescharacteristic
188#398Dic8311yescharacteristic, complemented
189#397—21no—
190#397C2×D8311yescharacteristic
191#394—21no—
192C4×C4⋊C4—21no—
193C4×C22⋊C4—21no—
194C4×C22⋊C4—1661no—
195C4×C22⋊C4Dic8311yescharacteristic, complemented
196#342—832no—
197#323—832no—
198#323—1661no—
199C22×D4C22⋊D8311yescharacteristic
200C23×C4C2×Dic8311yescharacteristic
201C23×C4C22⋊D8311yescharacteristic
202C4×D4—41no—
203C4×D4—41no—
204#117—41no—
205#117—41no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4C2×Dic8311yescharacteristic
209C2×C4⋊C4C2×Dic8311yescharacteristic
210C2×C4⋊C4C2×Dic8311yescharacteristic
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—1662no—
216C2×C22⋊C4C22⋊D8311yescharacteristic
217C2×C22⋊C4C22×D8311yescharacteristic
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—3321no—
221C2×C42C22⋊D8311yescharacteristic
222C4⋊C8—3321no—
223#99—1664no—
224C22⋊C8—1662no—
225C4×C8—3321no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—3321no—
231#95C2×Dic8311yescharacteristic
232#95C2×Dic8311yescharacteristic
233C24C22⋊Dic8311yescharacteristic
234C24C2×C22⋊D8311yescharacteristic
235C2×D4—22no—
236C2×D4—22no—
237C2×D4—41no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—3322no—
249C22×C4C22⋊Dic8311yescharacteristic
250C22×C4C22⋊Dic8311yescharacteristic
251C22×C4C22⋊Dic8311yescharacteristic
252C22×C4C22×Dic8311yescharacteristic
253C22×C4C2×C22⋊D8311yescharacteristic
254C2×C8—3322no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C22⋊C4—42no—
265C22⋊C4—42no—
266C22⋊C4—3322no—
267C22⋊C4—1664no—
268C42—21no—
269C42—21no—
270C42—42no—
271C42—41no—
272C42—41no—
273C42—6642no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23C2×C22⋊Dic8311yescharacteristic
284C23#1458181212yes—
285D4—44no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—81no—
311C2×C4—3324no—
312C2×C4—6642no—
313C2×C4—6642no—
314C2×C4#1458170711yescharacteristic
315C2×C4#1458170711yescharacteristic
316C2×C4#1458171912yes—
317C8—6642no—
318C22—21no—
319C22—21no—
320C22—22no—
321C22—42no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22C2×#1458170711yescharacteristic
331C22#1912769711yescharacteristic
332C22C2×#1458171911yescharacteristic
333C4—21no—
334C4—22no—
335C4—42no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—81no—
340C4—81no—
341C4—81no—
342C4—6644no—
343C4#1912749112yes—
344C2—21no—
345C2—21no—
346C2—42no—
347C2—81no—
348C2#2490595411yescharacteristic
349C2C2×#1912749111yescharacteristic
350C2#2490912811yescharacteristic
subgroupG/H
centerC2×C4#14581707
commutatorC22×C4×C83C22×C4
FrattiniC2×C22⋊C4C22×D83
FittingC83×#2395C2
socleC22×C83C2×#99
prime radicalC22×D4×C83D4
cosocleC2×C22⋊C4×C83C23
hypercenter#2395D83
hypocenterC83#11727
LCSG ▷ C22×C4×C83 ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2395

Complements

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

NH ≅ G/Ntype
3. #24905487C2semidirect
4. #24905463C2semidirect
13. C2×#117×C83C4semidirect
14. #400×C83C4semidirect
16. #398×C83C4semidirect
22. C4×C22⋊C4×C83C4semidirect
181. C83#11727semidirect
183. C2×#117Dic83semidirect
186. #400Dic83semidirect
188. #398Dic83semidirect
195. C4×C22⋊C4Dic83semidirect

Maximal subgroups

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

subgroupindex
C4×#145817072
#249057842
#249054872
#249054632
C83×#23952
#1172783