Finite Group Databasebeta

← #35314736

Subgroups — #35314736

Order 21248.

456 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 87 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#24910722C211yesmaximal, characteristic
2#24905784C211yesmaximal, characteristic
3#24905486C212yesmaximal, complemented
4#24905462C212yesmaximal, complemented
5C4×#99×C83C211yesmaximal, characteristic
6C2×#14581751C411yescharacteristic, complemented
7#19127754—21no—
8#19127640C411yescharacteristic, complemented
9#19127637C2211yescharacteristic
10#19127631C411yescharacteristic, complemented
11#19127629—21no—
12#19127626C2211yescharacteristic
13#19127615—21no—
14C4×Dic83⋊C4—21no—
15C4×C22⋊Dic83—21no—
16C4×C22⋊Dic83C411yescharacteristic, complemented
17#19127425—21no—
18#19127425C2212yes—
19C2×C4×D4×C83C2211yescharacteristic
20C2×#99×C83—21no—
21C4×C22⋊C4×C83—21no—
22#342×C83C2212yes—
23#14581751—41no—
24#14581751—41no—
25C2×C4⋊Dic83C2×C411yescharacteristic
26C2×C4⋊Dic83C2×C411yescharacteristic
27C2×Dic83⋊C4—21no—
28C2×Dic83⋊C4—21no—
29C2×C22⋊Dic83—21no—
30C2×C22⋊Dic83—21no—
31C2×C22⋊Dic83—21no—
32C2×C22⋊Dic83—21no—
33C2×C4×Dic83—21no—
34C2×C4×Dic83—21no—
35C4⋊(C83⋊C8)—41no—
36C22⋊(C83⋊C8)—22no—
37C4×C83⋊C8—41no—
38C83⋊#95—21no—
39C83⋊#95—21no—
40C83⋊#95—21no—
41C83⋊#95—21no—
42C83⋊#95C2×C411yescharacteristic
43C83⋊#95C2×C411yescharacteristic
44C22×D4×C83D411yescharacteristic
45C23×C4×C83C2×C411yescharacteristic
46C23×C4×C83D411yescharacteristic
47C4×D4×C83—41no—
48C4×D4×C83—41no—
49C2×C4⋊C4×C83C2×C411yescharacteristic
50C2×C22⋊C4×C83—22no—
51C2×C22⋊C4×C83D411yescharacteristic
52C2×C22⋊C4×C83C2311yescharacteristic
53C2×C42×C83—41no—
54C2×C42×C83D411yescharacteristic
55#99×C83—24no—
56#95×C83—41no—
57C22×Dic83—21no—
58C22×Dic83—21no—
59C22×Dic83—21no—
60C22×Dic83—21no—
61C2×C83⋊C8—42no—
62C4⋊Dic83—41no—
63C4⋊Dic83—41no—
64Dic83⋊C4—42no—
65Dic83⋊C4—42no—
66C22⋊Dic83—42no—
67C22⋊Dic83—42no—
68C22⋊Dic83—42no—
69C22⋊Dic83—42no—
70C4×Dic83—42no—
71C4×Dic83—41no—
72C4×Dic83—41no—
73C24×C83C22⋊C411yescharacteristic
74C24×C83C2×D411yescharacteristic
75C2×D4×C83—22no—
76C2×D4×C83—22no—
77C2×D4×C83—41no—
78C22×C4×C83—21no—
79C22×C4×C83—21no—
80C22×C4×C83—21no—
81C22×C4×C83—21no—
82C22×C4×C83—42no—
83C22×C4×C83—42no—
84C22×C4×C83—42no—
85C22×C4×C83C22⋊C411yescharacteristic
86C22×C4×C83C22⋊C411yescharacteristic
87C22×C4×C83C22⋊C411yescharacteristic
88C22×C4×C83C22×C411yescharacteristic
89C22×C4×C83C2×D411yescharacteristic
90C4⋊C4×C83—41no—
91C22⋊C4×C83—24no—
92C22⋊C4×C83—41no—
93C22⋊C4×C83—41no—
94C22⋊C4×C83—42no—
95C42×C83—21no—
96C42×C83—21no—
97C42×C83—82no—
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—21no—
114C2×C4×C83—21no—
115C2×C4×C83—21no—
116C2×C4×C83—42no—
117C2×C4×C83—42no—
118C2×C4×C83—42no—
119C2×C4×C83—41no—
120C2×C4×C83—41no—
121C2×C4×C83—41no—
122C2×C4×C83—44no—
123C2×C4×C83—81no—
124C2×C4×C83—82no—
125C2×C4×C83—82no—
126C2×C4×C83#9911yescharacteristic
127C2×C4×C83#9911yescharacteristic
128C2×C4×C83#10412yes—
129C2×Dic83—22no—
130C2×Dic83—41no—
131C2×Dic83—41no—
132C2×Dic83—41no—
133C2×Dic83—41no—
134C2×Dic83—41no—
135C2×Dic83—41no—
136C2×Dic83—41no—
137C2×Dic83—41no—
138C2×Dic83—41no—
139C2×Dic83—41no—
140C2×Dic83—41no—
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—41no—
160C4×C83—41no—
161C4×C83—41no—
162C4×C83—84no—
163C4×C83#35212yes—
164Dic83—42no—
165Dic83—81no—
166Dic83—81no—
167Dic83—81no—
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#2395—831no—
177C4×#99D8311yescharacteristic
178#1460—831no—
179#1352—832no—
180#1344—832no—
181C83#1172711yescharacteristic, complemented, Sylow
182C2×C4×D4C2×D8311yescharacteristic
183C2×#117—831no—
184#431—1661no—
185C2×#99—21no—
186#400—831no—
187#399—831no—
188#398—831no—
189#397—831no—
190#397—1661no—
191#394—1661no—
192C4×C4⋊C4—1661no—
193C4×C22⋊C4—21no—
194C4×C22⋊C4—831no—
195C4×C22⋊C4—1661no—
196#342C2×D8312yes—
197#323—832no—
198#323—1661no—
199C22×D4C22⋊D8311yescharacteristic
200C23×C4C4×D8311yescharacteristic
201C23×C4C22⋊D8311yescharacteristic
202C4×D4—41no—
203C4×D4—41no—
204#117—3321no—
205#117—3321no—
206C2×C4⋊C4—831no—
207C2×C4⋊C4—831no—
208C2×C4⋊C4—1661no—
209C2×C4⋊C4—1661no—
210C2×C4⋊C4C4×D8311yescharacteristic
211C2×C22⋊C4—22no—
212C2×C22⋊C4—1661no—
213C2×C22⋊C4—1661no—
214C2×C22⋊C4—1661no—
215C2×C22⋊C4—1661no—
216C2×C22⋊C4D33211yescharacteristic
217C2×C22⋊C4C22×D8311yescharacteristic
218C2×C42—41no—
219C2×C42—1661no—
220C2×C42—1661no—
221C2×C42D33211yescharacteristic
222C4⋊C8—3321no—
223#99—24no—
224C22⋊C8—1662no—
225C4×C8—3321no—
226#95—41no—
227#95—831no—
228#95—831no—
229#95—1661no—
230#95—1661no—
231#95—1661no—
232#95—1661no—
233C24(C2×D83)⋊C411yescharacteristic
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—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—1661no—
246C22×C4—1661no—
247C22×C4—1661no—
248C22×C4—1661no—
249C22×C4(C2×D83)⋊C411yescharacteristic
250C22×C4(C2×D83)⋊C411yescharacteristic
251C22×C4(C2×D83)⋊C411yescharacteristic
252C22×C4C2×C4×D8311yescharacteristic
253C22×C4C2×D33211yescharacteristic
254C2×C8—3322no—
255C4⋊C4—41no—
256C4⋊C4—3321no—
257C4⋊C4—3321no—
258C4⋊C4—3322no—
259C4⋊C4—3322no—
260C22⋊C4—24no—
261C22⋊C4—41no—
262C22⋊C4—41no—
263C22⋊C4—42no—
264C22⋊C4—3322no—
265C22⋊C4—3322no—
266C22⋊C4—3322no—
267C22⋊C4—3322no—
268C42—21no—
269C42—21no—
270C42—82no—
271C42—3322no—
272C42—3321no—
273C42—3321no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23C2×(C2×D83)⋊C411yescharacteristic
284C23#1458181212yes—
285D4—44no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—44no—
299C2×C4—81no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—3321no—
303C2×C4—3321no—
304C2×C4—3321no—
305C2×C4—3321no—
306C2×C4—3321no—
307C2×C4—3321no—
308C2×C4—3321no—
309C2×C4—3321no—
310C2×C4—3321no—
311C2×C4—3321no—
312C2×C4—3321no—
313C2×C4—1662no—
314C2×C4#1458170611yescharacteristic
315C2×C4#1458170611yescharacteristic
316C2×C4#1458172012yes—
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×#1458170611yescharacteristic
331C22#1912769511yescharacteristic
332C22C2×#1458172011yescharacteristic
333C4—21no—
334C4—22no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—84no—
339C4—3322no—
340C4—6641no—
341C4—6641no—
342C4—6641no—
343C4#1912749012yes—
344C2—21no—
345C2—21no—
346C2—42no—
347C2—81no—
348C2#2490595211yescharacteristic
349C2C2×#1912749011yescharacteristic
350C2#2490912611yescharacteristic
subgroupG/H
centerC2×C4#14581706
commutatorC22×C4×C83C22×C4
FrattiniC2×C22⋊C4C22×D83
FittingC4×#99×C83C2
socleC22×C83C2×#99
prime radicalC22×D4×C83D4
cosocleC2×C22⋊C4×C83C23
hypercenterC4×#99D83
hypocenterC83#11727
LCSG ▷ C22×C4×C83 ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
3. #24905486C2semidirect
4. #24905462C2semidirect
6. C2×#14581751C4semidirect
8. #19127640C4semidirect
10. #19127631C4semidirect
16. C4×C22⋊Dic83C4semidirect
181. C83#11727semidirect

Maximal subgroups

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

subgroupindex
#249107222
#249057842
#249054862
#249054622
C4×#99×C832
#1172783