Finite Group Databasebeta

← #35314716

Subgroups — #35314716

Order 21248.

452 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 107 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#24913276C211yesmaximal, characteristic
2C2×#19127470C211yesmaximal, characteristic
3#24905494C212yesmaximal, complemented
4#24905491C212yesmaximal, complemented
5C2×#324×C83C211yesmaximal, characteristic
6C22×#13050574—21no—
7C2×#14581771C411yescharacteristic
8#19127640C411yescharacteristic
9#19127640C2211yescharacteristic
10#19127634—21no—
11#19127634—21no—
12#19127621C411yescharacteristic
13#19127621C2211yescharacteristic
14#19127574—21no—
15#19127564—21no—
16#19127561C411yescharacteristic
17#19127470—21no—
18#19127470C2212yes—
19C22×C4⋊C4×C83C2211yescharacteristic
20C2×C22⋊C8×C83—21no—
21#324×C83—21no—
22#324×C83C2212yes—
23C22×Dic166C2×C411yescharacteristic
24C2×#13050574—24no—
25C2×#13050574—22no—
26C2×#13050574—41no—
27C2×#13050574—42no—
28C22×C83⋊C8—41no—
29#14581771—41no—
30#14581771—41no—
31C2×C4⋊Dic83—21no—
32C2×C4⋊Dic83C2×C411yescharacteristic
33C2×Dic83⋊C4—21no—
34C2×Dic83⋊C4—21no—
35C2×Dic83⋊C4—21no—
36C2×C22⋊Dic83—21no—
37C2×C22⋊Dic83—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
49C22×C8×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—
58C2×Dic166—41no—
59C2×Dic166—81no—
60C22×Dic83—21no—
61C22×Dic83—21no—
62C22×Dic83—21no—
63C22×Dic83—21no—
64#13050574—24no—
65#13050574—44no—
66#13050574—42no—
67C2×C83⋊C8—44no—
68C2×C83⋊C8—42no—
69C4⋊Dic83—41no—
70C4⋊Dic83—81no—
71Dic83⋊C4—42no—
72Dic83⋊C4—81no—
73Dic83⋊C4—81no—
74C22⋊Dic83—41no—
75C22⋊Dic83—41no—
76C22⋊Dic83—42no—
77C24×C83C22⋊C411yescharacteristic
78C22×C4×C83—21no—
79C22×C4×C83—21no—
80C22×C4×C83—22no—
81C22×C4×C83—21no—
82C22×C4×C83—21no—
83C22×C4×C83—21no—
84C22×C4×C83—21no—
85C22×C4×C83—81no—
86C22×C4×C83—81no—
87C22×C4×C83C22⋊C411yescharacteristic
88C22×C4×C83C22⋊C411yescharacteristic
89C22×C4×C83C22⋊C411yescharacteristic
90C22×C4×C83SD1612yes—
91C22×C4×C83Q1612yes—
92C22×C4×C83C22×C411yescharacteristic
93C22×C4×C83C2×D411yescharacteristic
94C22×C4×C83C2×D411yescharacteristic
95C2×C8×C83—42no—
96C2×C8×C83—82no—
97C4⋊C4×C83—41no—
98C4⋊C4×C83—41no—
99C4⋊C4×C83—41no—
100C4⋊C4×C83—41no—
101C23×C83—41no—
102C23×C83—41no—
103C23×C83#9911yescharacteristic
104C23×C83#9911yescharacteristic
105C23×C83#10312yes—
106C23×C83#10412yes—
107C23×C83C2×C22⋊C411yescharacteristic
108C2×C4×C83—22no—
109C2×C4×C83—22no—
110C2×C4×C83—22no—
111C2×C4×C83—22no—
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—21no—
120C2×C4×C83—42no—
121C2×C4×C83—41no—
122C2×C4×C83—41no—
123C2×C4×C83—41no—
124C2×C4×C83—41no—
125C2×C4×C83—81no—
126C2×C4×C83—81no—
127C2×C4×C83—81no—
128C2×C4×C83#10312yes—
129C2×C4×C83C2×SD1611yescharacteristic
130C2×C4×C83C2×Q1611yescharacteristic
131C2×Dic83—41no—
132C2×Dic83—41no—
133C2×Dic83—41no—
134C2×Dic83—41no—
135C2×Dic83—41no—
136C2×Dic83—41no—
137C2×Dic83—42no—
138C2×Dic83—41no—
139C2×Dic83—41no—
140C2×Dic83—41no—
141C2×Dic83—41no—
142Dic166—82no—
143C8×C83—82no—
144C83⋊C8—44no—
145C22×C83—21no—
146C22×C83—21no—
147C22×C83—21no—
148C22×C83—21no—
149C22×C83—21no—
150C22×C83—21no—
151C22×C83—42no—
152C22×C83—42no—
153C22×C83#32812yes—
154C22×C83#32812yes—
155C22×C83C2×#9911yescharacteristic
156C22×C83C2×#10311yescharacteristic
157C22×C83C2×#10411yescharacteristic
158C4×C83—21no—
159C4×C83—22no—
160C4×C83—21no—
161C4×C83—41no—
162C4×C83—41no—
163C4×C83—41no—
164C4×C83—41no—
165Dic83—81no—
166Dic83—81no—
167Dic83—81no—
168Dic83—81no—
169#11722—831nomaximal, Sylow
170C2×C83—21no—
171C2×C83—22no—
172C2×C83—22no—
173C2×C83—21no—
174C2×C83C2×#32811yescharacteristic
175C2×C83#212111yescharacteristic
176C2×C83#213311yescharacteristic
177#2627—831no—
178C2×#343—831no—
179C2×#324D8311yescharacteristic
180#1354—832no—
181#1353—832no—
182C83#1172211yescharacteristic, complemented, Sylow
183C22×M16—1661no—
184C2×C22⋊Q8—831no—
185C22×C4⋊C4C2×D8311yescharacteristic
186C2×C22⋊C8—21no—
187#400—831no—
188#400—831no—
189#398—1661no—
190#398—1661no—
191#395—831no—
192#395—831no—
193#385—1661no—
194#381—1661no—
195#380—831no—
196#343—832no—
197#343—1661no—
198#324—21no—
199#324C2×D8312yes—
200C22×Q8—831no—
201C23×C4C4×D8311yescharacteristic
202C23×C4D33211yescharacteristic
203C23×C4C22⋊D8311yescharacteristic
204C2×M16—3321no—
205C2×M16—3322no—
206C2×M16—1664no—
207C2×M16—1662no—
208C22×C8—41no—
209C22×C8—3321no—
210C22⋊Q8—3321no—
211C22⋊Q8—3321no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—831no—
215C2×C4⋊C4—1661no—
216C2×C4⋊C4—1661no—
217C2×C4⋊C4—1661no—
218C2×C4⋊C4—1661no—
219C2×C4⋊C4C4×D8311yescharacteristic
220C2×C4⋊C4D33211yescharacteristic
221C2×C4⋊C4C22⋊D8311yescharacteristic
222C2×C4⋊C4C22×D8311yescharacteristic
223C2×C22⋊C4—1661no—
224C2×C22⋊C4—1661no—
225C22⋊C8—22no—
226C22⋊C8—41no—
227#95—831no—
228#95—831no—
229#95—1661no—
230#95—1661no—
231#95—1661no—
232#95—1661no—
233#95—1661no—
234#95—1661no—
235C24(C2×D83)⋊C411yescharacteristic
236C2×Q8—3321no—
237C2×Q8—6641no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—81no—
246C22×C4—81no—
247C22×C4—1661no—
248C22×C4—1661no—
249C22×C4—1661no—
250C22×C4—1661no—
251C22×C4(C2×D83)⋊C411yescharacteristic
252C22×C4(C2×D83)⋊C411yescharacteristic
253C22×C4(C2×D83)⋊C411yescharacteristic
254C22×C4#1305057912yes—
255C22×C4Dic33212yes—
256C22×C4C2×C4×D8311yescharacteristic
257C22×C4C2×C22⋊D8311yescharacteristic
258C22×C4C2×D33211yescharacteristic
259M16—3324no—
260M16—3322no—
261M16—1664no—
262C2×C8—42no—
263C2×C8—82no—
264C2×C8—3322no—
265C2×C8—3324no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—3321no—
271C4⋊C4—3322no—
272C4⋊C4—6641no—
273C4⋊C4—6641no—
274C4⋊C4—6641no—
275C22⋊C4—3321no—
276C22⋊C4—3321no—
277C22⋊C4—3322no—
278C23—41no—
279C23—41no—
280C23#1458170611yescharacteristic
281C23#1458170611yescharacteristic
282C23#1458171712yes—
283C23#1458171812yes—
284C23C2×(C2×D83)⋊C411yescharacteristic
285Q8—6642no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—3321no—
307C2×C4—3321no—
308C2×C4—3321no—
309C2×C4—3321no—
310C2×C4—3321no—
311C2×C4—3321no—
312C2×C4—3322no—
313C2×C4—3321no—
314C2×C4—3321no—
315C2×C4—3321no—
316C2×C4—3321no—
317C2×C4#1458171712yes—
318C2×C4C2×#1305057911yescharacteristic
319C2×C4C2×Dic33211yescharacteristic
320C8—82no—
321C8—3324no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—42no—
329C22—42no—
330C22#1912743912yes—
331C22#1912743912yes—
332C22C2×#1458170611yescharacteristic
333C22C2×#1458171711yescharacteristic
334C22C2×#1458171811yescharacteristic
335C4—21no—
336C4—21no—
337C4—22no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—6641no—
343C4—6641no—
344C4—6641no—
345C4—6641no—
346C2—21no—
347C2—21no—
348C2—22no—
349C2—22no—
350C2C2×#1912743911yescharacteristic
351C2#2490910711yescharacteristic
352C2#2490915011yescharacteristic
subgroupG/H
centerC22C2×#14581706
commutatorC22×C4×C83C22×C4
FrattiniC2×C4⋊C4C22×D83
FittingC2×#324×C83C2
socleC22×C83C2×#99
prime radicalC24×C83C22⋊C4
cosocleC2×C4⋊C4×C83C23
hypercenterC2×#324D83
hypocenterC83#11722
LCSG ▷ C22×C4×C83 ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
3. #24905494C2semidirect
4. #24905491C2semidirect
182. C83#11722semidirect

Maximal subgroups

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

subgroupindex
#249132762
C2×#191274702
#249054942
#249054912
C2×#324×C832
#1172283