Finite Group Databasebeta

← #35314916

Subgroups — #35314916

Order 21248.

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

normal subgroups only

This group has 446 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#24911750C211yesmaximal, characteristic
2C2×#19127470C211yesmaximal, characteristic
3#24905491C212yesmaximal, complemented
4#24905470C212yesmaximal, complemented
5C83×#1463C211yesmaximal, characteristic
6C22×#13050574—21no—
7C2×#14581751C411yescharacteristic
8#19127640C411yescharacteristic
9#19127640C2211yescharacteristic
10#19127631C411yescharacteristic
11#19127631C2211yescharacteristic
12#19127582—21no—
13#19127582—21no—
14#19127574—21no—
15#19127564—21no—
16#19127564C411yescharacteristic
17#19127470—21no—
18#19127470C2212yes—
19C22×C4⋊C4×C83C2211yescharacteristic
20#407×C83—21no—
21#324×C83—21no—
22#324×C83C2212yes—
23C2×#13050574—24no—
24C2×#13050574—22no—
25C2×#13050574—41no—
26C2×#13050574—42no—
27C22×C83⋊C8—41no—
28#14581751—41no—
29#14581751—41no—
30C2×C4⋊Dic83C2×C411yescharacteristic
31C2×C4⋊Dic83C2×C411yescharacteristic
32C2×Dic83⋊C4—21no—
33C2×Dic83⋊C4—21no—
34C2×C22⋊Dic83—21no—
35C2×C22⋊Dic83—21no—
36C2×C4×Dic83—21no—
37C83⋊#95—21no—
38C83⋊#95—21no—
39C83⋊#95—21no—
40C83⋊#95—21no—
41C83⋊#95—21no—
42C83⋊#95—21no—
43C83⋊#95—21no—
44C83⋊#95C2×C411yescharacteristic
45C83⋊#95C2×C411yescharacteristic
46C23×C4×C83C2×C411yescharacteristic
47C23×C4×C83D411yescharacteristic
48C23×C4×C83D411yescharacteristic
49C2×M16×C83—41no—
50C2×C4⋊C4×C83—41no—
51C2×C4⋊C4×C83—41no—
52C2×C4⋊C4×C83C2×C411yescharacteristic
53C2×C4⋊C4×C83D411yescharacteristic
54C2×C4⋊C4×C83D411yescharacteristic
55C2×C4⋊C4×C83C2311yescharacteristic
56C22⋊C8×C83—22no—
57C22⋊C8×C83—41no—
58C22×Dic83—21no—
59C22×Dic83—21no—
60C22×Dic83—21no—
61C22×Dic83—21no—
62#13050574—24no—
63#13050574—44no—
64#13050574—42no—
65C2×C83⋊C8—44no—
66C2×C83⋊C8—42no—
67C4⋊Dic83—41no—
68C4⋊Dic83—41no—
69Dic83⋊C4—42no—
70Dic83⋊C4—41no—
71Dic83⋊C4—41no—
72C22⋊Dic83—42no—
73C22⋊Dic83—41no—
74C22⋊Dic83—41no—
75C4×Dic83—42no—
76C24×C83C22⋊C411yescharacteristic
77C22×C4×C83—21no—
78C22×C4×C83—21no—
79C22×C4×C83—22no—
80C22×C4×C83—21no—
81C22×C4×C83—21no—
82C22×C4×C83—21no—
83C22×C4×C83—21no—
84C22×C4×C83—21no—
85C22×C4×C83—21no—
86C22×C4×C83—42no—
87C22×C4×C83—42no—
88C22×C4×C83C22⋊C411yescharacteristic
89C22×C4×C83C22⋊C411yescharacteristic
90C22×C4×C83C22⋊C411yescharacteristic
91C22×C4×C83C22×C411yescharacteristic
92C22×C4×C83C2×D411yescharacteristic
93C22×C4×C83C2×D411yescharacteristic
94M16×C83—82no—
95C2×C8×C83—42no—
96C4⋊C4×C83—41no—
97C4⋊C4×C83—41no—
98C4⋊C4×C83—41no—
99C4⋊C4×C83—41no—
100C23×C83—22no—
101C23×C83—21no—
102C23×C83—21no—
103C23×C83—41no—
104C23×C83#9911yescharacteristic
105C23×C83#9911yescharacteristic
106C23×C83C2×C22⋊C411yescharacteristic
107C2×C4×C83—22no—
108C2×C4×C83—22no—
109C2×C4×C83—22no—
110C2×C4×C83—22no—
111C2×C4×C83—21no—
112C2×C4×C83—21no—
113C2×C4×C83—21no—
114C2×C4×C83—21no—
115C2×C4×C83—21no—
116C2×C4×C83—21no—
117C2×C4×C83—21no—
118C2×C4×C83—21no—
119C2×C4×C83—42no—
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—81no—
127C2×C4×C83#10412yes—
128C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
129C2×C4×C83#13711yescharacteristic
130C2×Dic83—41no—
131C2×Dic83—41no—
132C2×Dic83—41no—
133C2×Dic83—41no—
134C2×Dic83—42no—
135C2×Dic83—41no—
136C2×Dic83—41no—
137C2×Dic83—41no—
138C2×Dic83—41no—
139C2×Dic83—41no—
140C2×Dic83—41no—
141C8×C83—82no—
142C83⋊C8—44no—
143C22×C83—21no—
144C22×C83—21no—
145C22×C83—21no—
146C22×C83—21no—
147C22×C83—21no—
148C22×C83—21no—
149C22×C83—21no—
150C22×C83—42no—
151C22×C83—42no—
152C22×C83#35212yes—
153C22×C83C2×#9911yescharacteristic
154C22×C83#41711yescharacteristic
155C22×C83C2×#10411yescharacteristic
156C4×C83—21no—
157C4×C83—22no—
158C4×C83—21no—
159C4×C83—41no—
160C4×C83—41no—
161C4×C83—41no—
162C4×C83—41no—
163Dic83—81no—
164Dic83—81no—
165Dic83—81no—
166Dic83—81no—
167#11764—831nomaximal, Sylow
168C2×C83—21no—
169C2×C83—21no—
170C2×C83—22no—
171C2×C83—41no—
172C2×C83#150411yescharacteristic
173C2×C83C2×#35211yescharacteristic
174C2×C83#213311yescharacteristic
175#2484—831no—
176C2×#343—831no—
177#1463D8311yescharacteristic
178#1353—832no—
179#1346—832no—
180C83#1176411yescharacteristic, complemented, Sylow
181C22×M16—1661no—
182C2×#117—831no—
183C22×C4⋊C4C2×D8311yescharacteristic
184#407—21no—
185#400—831no—
186#400—831no—
187#398—831no—
188#398—831no—
189#387—1661no—
190#387—1661no—
191#385—1661no—
192#381—831no—
193#381—1661no—
194#343—832no—
195#343—1661no—
196#324—21no—
197#324C2×D8312yes—
198C23×C4C4×D8311yescharacteristic
199C23×C4D33211yescharacteristic
200C23×C4C22⋊D8311yescharacteristic
201C2×M16—41no—
202C2×M16—3321no—
203C2×M16—3322no—
204C2×M16—1664no—
205C2×M16—1662no—
206C22×C8—3321no—
207#117—3321no—
208#117—3321no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—831no—
212C2×C4⋊C4—831no—
213C2×C4⋊C4—1661no—
214C2×C4⋊C4—1661no—
215C2×C4⋊C4C4×D8311yescharacteristic
216C2×C4⋊C4D33211yescharacteristic
217C2×C4⋊C4C22⋊D8311yescharacteristic
218C2×C4⋊C4C22×D8311yescharacteristic
219C2×C22⋊C4—1661no—
220C2×C22⋊C4—1661no—
221C2×C42—1661no—
222C22⋊C8—22no—
223C22⋊C8—41no—
224#95—831no—
225#95—831no—
226#95—1661no—
227#95—1661no—
228#95—1661no—
229#95—1661no—
230#95—1661no—
231#95—1661no—
232#95—1661no—
233C24(C2×D83)⋊C411yescharacteristic
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
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×C22⋊D8311yescharacteristic
254C22×C4C2×D33211yescharacteristic
255M16—82no—
256M16—3324no—
257M16—3322no—
258M16—1664no—
259C2×C8—42no—
260C2×C8—3324no—
261C2×C8—3322no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—3321no—
267C4⋊C4—3321no—
268C4⋊C4—3321no—
269C4⋊C4—3321no—
270C4⋊C4—3322no—
271C22⋊C4—3322no—
272C22⋊C4—3321no—
273C22⋊C4—3321no—
274C42—3322no—
275C23—21no—
276C23—21no—
277C23—22no—
278C23—41no—
279C23#1458170611yescharacteristic
280C23#1458170611yescharacteristic
281C23C2×(C2×D83)⋊C411yescharacteristic
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—81no—
302C2×C4—3321no—
303C2×C4—3321no—
304C2×C4—3321no—
305C2×C4—3321no—
306C2×C4—3322no—
307C2×C4—3321no—
308C2×C4—3321no—
309C2×C4—3321no—
310C2×C4—3321no—
311C2×C4—3321no—
312C2×C4—3321no—
313C2×C4#1458171812yes—
314C2×C4#1458182011yescharacteristic
315C2×C4#1458182511yescharacteristic
316C8—82no—
317C8—3324no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—42no—
326C22—42no—
327C22#1912749012yes—
328C22C2×#1458170611yescharacteristic
329C22#1912770311yescharacteristic
330C22C2×#1458171811yescharacteristic
331C4—21no—
332C4—21no—
333C4—22no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—6641no—
339C4—6641no—
340C4—6641no—
341C4—6641no—
342C2—21no—
343C2—21no—
344C2—22no—
345C2—41no—
346C2#2490596811yescharacteristic
347C2C2×#1912749011yescharacteristic
348C2#2490915011yescharacteristic
subgroupG/H
centerC22C2×#14581706
commutatorC22×C4×C83C22×C4
FrattiniC2×C4⋊C4C22×D83
FittingC83×#1463C2
socleC22×C83C2×#99
prime radicalC24×C83C22⋊C4
cosocleC2×C4⋊C4×C83C23
hypercenter#1463D83
hypocenterC83#11764
LCSG ▷ C22×C4×C83 ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
3. #24905491C2semidirect
4. #24905470C2semidirect
180. C83#11764semidirect

Maximal subgroups

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

subgroupindex
#249117502
C2×#191274702
#249054912
#249054702
C83×#14632
#1176483