Finite Group Databasebeta

← #35315447

Subgroups — #35315447

Order 21248.

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

normal subgroups only

This group has 440 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
1#24912368C211yesmaximal, characteristic
2#24907459C211yesmaximal, characteristic, complemented
3#24905470C212yesmaximal, complemented
4#24905467C212yesmaximal
5C83×#1461C211yesmaximal, characteristic, complemented
6C2×#14581768—21no—
7C2×C4×D332C411yescharacteristic
8#19127725—21no—
9C2×#14581711—21no—
10#19127631C411yescharacteristic
11#19127631C2211yescharacteristic
12#19127618C411yescharacteristic
13#19127618C2211yescharacteristic
14#19127592—21no—
15#19127592—21no—
16#19127567C411yescharacteristic
17#19127558—21no—
18#19127470C2212yes—
19C2×C4×Q8×C83C2211yescharacteristic
20#431×C83—21no—
21#324×C83—21no—
22#324×C83C2212yes—
23C22×D332C2×C411yescharacteristic
24C22×C4×D83—21no—
25C2×#13050574—22no—
26#14581768—42no—
27#14581768—42no—
28C4×D332—41no—
29C4×D332—41no—
30C2×C4⋊Dic83—21no—
31C2×C4⋊Dic83C2×C411yescharacteristic
32C2×Dic83⋊C4—21no—
33C2×Dic83⋊C4—21no—
34C2×(C2×D83)⋊C4—21no—
35C2×(C2×D83)⋊C4—21no—
36C2×(C2×D83)⋊C4—21no—
37C2×(C2×D83)⋊C4—21no—
38C2×(C2×D83)⋊C4—21no—
39C4⋊(C83⋊C8)—41no—
40C4⋊(C83⋊C8)—41no—
41#14581711—42no—
42C83⋊#95—21no—
43C83⋊#95—21no—
44C83⋊#95—21no—
45C83⋊#95C2×C411yescharacteristic
46C83⋊#95C2×C411yescharacteristic
47C22×Q8×C83D411yescharacteristic
48C4×Q8×C83—41no—
49C4×Q8×C83—41no—
50C2×C4⋊C4×C83C2×C411yescharacteristic
51C2×C4⋊C4×C83D411yescharacteristic
52C2×C4⋊C4×C83C2311yescharacteristic
53C2×C42×C83C2×C411yescharacteristic
54C2×C42×C83D411yescharacteristic
55C2×C42×C83D411yescharacteristic
56C4⋊C8×C83—41no—
57C22⋊C8×C83—22no—
58C4×C8×C83—41no—
59C23×D83—21no—
60C2×D332—41no—
61C2×D332—42no—
62C22×Dic83—21no—
63C22×Dic83—21no—
64C22×Dic83—21no—
65C2×C4×D83—42no—
66C2×C4×D83—42no—
67C2×C4×D83—42no—
68#13050574—42no—
69#13050574—42no—
70C2×C83⋊C8—42no—
71C4⋊Dic83—42no—
72C4⋊Dic83—41no—
73Dic83⋊C4—42no—
74Dic83⋊C4—41no—
75Dic83⋊C4—41no—
76(C2×D83)⋊C4—42no—
77(C2×D83)⋊C4—42no—
78(C2×D83)⋊C4—41no—
79(C2×D83)⋊C4—41no—
80(C2×D83)⋊C4—42no—
81(C2×D83)⋊C4—41no—
82(C2×D83)⋊C4—41no—
83C2×Q8×C83—41no—
84C2×Q8×C83—41no—
85C2×Q8×C83D812yes—
86C2×Q8×C83SD1612yes—
87C22×C4×C83C22⋊C411yescharacteristic
88C22×C4×C83C22⋊C411yescharacteristic
89C22×C4×C83C22⋊C411yescharacteristic
90C22×C4×C83C22⋊C411yescharacteristic
91C22×C4×C83C22×C411yescharacteristic
92C22×C4×C83C2×D411yescharacteristic
93C22×C4×C83C2×D411yescharacteristic
94C2×C8×C83—42no—
95C4⋊C4×C83—41no—
96C4⋊C4×C83—41no—
97C4⋊C4×C83—41no—
98C42×C83—21no—
99C42×C83—21no—
100C42×C83—21no—
101C42×C83—21no—
102C42×C83—21no—
103C42×C83—21no—
104C23×C83C2×C22⋊C411yescharacteristic
105C22×D83—22no—
106C22×D83—41no—
107C22×D83—41no—
108C22×D83—41no—
109C22×D83—41no—
110C22×D83—41no—
111C22×D83—41no—
112Q8×C83—42no—
113Q8×C83—42no—
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—21no—
121C2×C4×C83—21no—
122C2×C4×C83—21no—
123C2×C4×C83—41no—
124C2×C4×C83—41no—
125C2×C4×C83—41no—
126C2×C4×C83—41no—
127C2×C4×C83#9911yescharacteristic
128C2×C4×C83#9911yescharacteristic
129C2×C4×C83#10212yes—
130C2×C4×C83#10212yes—
131C2×C4×C83C2×D811yescharacteristic
132C2×C4×C83C2×SD1611yescharacteristic
133C2×Dic83—41no—
134C2×Dic83—41no—
135C2×Dic83—41no—
136C2×Dic83—41no—
137C2×Dic83—41no—
138C2×Dic83—41no—
139C2×Dic83—41no—
140C2×Dic83—41no—
141C2×Dic83—41no—
142D332—42no—
143D332—42no—
144C4×D83—84no—
145C8×C83—82no—
146C83⋊C8—82no—
147C22×C83—21no—
148C22×C83—21no—
149C22×C83C2×#9911yescharacteristic
150C22×C83C2×#10211yescharacteristic
151C22×C83#42111yescharacteristic
152C2×D83—42no—
153C2×D83—42no—
154C2×D83—42no—
155C2×D83—81no—
156C2×D83—81no—
157C2×D83—81no—
158C2×D83—81no—
159C4×C83—21no—
160C4×C83—21no—
161C4×C83—21no—
162C4×C83—22no—
163C4×C83—22no—
164C4×C83—21no—
165C4×C83—41no—
166C4×C83—41no—
167C4×C83—41no—
168Dic83—81no—
169Dic83—81no—
170Dic83—81no—
171#11879—831nomaximal, Sylow
172C2×C83—21no—
173C2×C83—21no—
174C2×C83#151911yescharacteristic
175C2×C83#212011yescharacteristic
176C2×C83#213311yescharacteristic
177D83—82no—
178#2536—831no—
179#1801—831no—
180#1461D8311yescharacteristic, complemented
181#1346—832no—
182#1345—832no—
183C83#1187911yescharacteristic, complemented, Sylow
184C2×C22⋊Q8—1661no—
185C2×C4×Q8C2×D8311yescharacteristic
186C2×C4×D4—831no—
187#431—21no—
188#423—1661no—
189C2×#101—1661no—
190#398—831no—
191#398—831no—
192#394—831no—
193#394—831no—
194#388—1661no—
195#388—1661no—
196#382—831no—
197#380—1661no—
198#343—832no—
199#324—21no—
200#324C2×D8312yes—
201C22×Q8D33211yescharacteristic
202C22×D4—831no—
203C23×C4—1661no—
204C2×M16—1662no—
205C22⋊Q8—3322no—
206C22⋊Q8—3322no—
207C4×Q8—41no—
208C4×Q8—41no—
209C4×D4—3321no—
210C4×D4—3321no—
211C2×C4⋊C4—831no—
212C2×C4⋊C4—1661no—
213C2×C4⋊C4—1661no—
214C2×C4⋊C4—1661no—
215C2×C4⋊C4C4×D8311yescharacteristic
216C2×C4⋊C4C22⋊D8311yescharacteristic
217C2×C4⋊C4C22×D8311yescharacteristic
218C2×C22⋊C4—1661no—
219C2×C22⋊C4—1661no—
220C2×C22⋊C4—1661no—
221C2×C22⋊C4—1661no—
222C2×C22⋊C4—1661no—
223C2×C42C4×D8311yescharacteristic
224C2×C42D33211yescharacteristic
225C2×C42C22⋊D8311yescharacteristic
226C4⋊C8—41no—
227C4⋊C8—3321no—
228C4⋊C8—3321no—
229#101—3322no—
230C22⋊C8—22no—
231C4×C8—41no—
232#95—831no—
233#95—831no—
234#95—1661no—
235#95—1661no—
236#95—1661no—
237C24—1661no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8D66412yes—
241C2×Q8#1305057912yes—
242C2×D4—3321no—
243C2×D4—3322no—
244C22×C4—3322no—
245C22×C4—3322no—
246C22×C4—3322no—
247C22×C4—1661no—
248C22×C4—1661no—
249C22×C4—1661no—
250C22×C4(C2×D83)⋊C411yescharacteristic
251C22×C4(C2×D83)⋊C411yescharacteristic
252C22×C4(C2×D83)⋊C411yescharacteristic
253C22×C4(C2×D83)⋊C411yescharacteristic
254C22×C4C2×C4×D8311yescharacteristic
255C22×C4C2×C22⋊D8311yescharacteristic
256C22×C4C2×D33211yescharacteristic
257M16—3322no—
258M16—3322no—
259C2×C8—42no—
260C2×C8—3322no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—3321no—
265C4⋊C4—3321no—
266C4⋊C4—3321no—
267C4⋊C4—3322no—
268C4⋊C4—3322no—
269C22⋊C4—3322no—
270C22⋊C4—3322no—
271C22⋊C4—3321no—
272C22⋊C4—3321no—
273C22⋊C4—3321no—
274C22⋊C4—3321no—
275C22⋊C4—3322no—
276C42—21no—
277C42—21no—
278C42—21no—
279C42—21no—
280C42—21no—
281C42—21no—
282C23—3321no—
283C23—3321no—
284C23—3321no—
285C23—3321no—
286C23—3321no—
287C23—3321no—
288C23—1662no—
289C23C2×(C2×D83)⋊C411yescharacteristic
290Q8—42no—
291Q8—42no—
292D4—3322no—
293D4—3322no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—3321no—
308C2×C4—3321no—
309C2×C4—3321no—
310C2×C4—3321no—
311C2×C4—3321no—
312C2×C4—3321no—
313C2×C4—3321no—
314C2×C4—3321no—
315C2×C4—3321no—
316C2×C4—6644no—
317C2×C4#1458170611yescharacteristic
318C2×C4#1458170611yescharacteristic
319C2×C4#1458171412yes—
320C2×C4#1458171412yes—
321C2×C4C2×D66411yescharacteristic
322C2×C4C2×#1305057911yescharacteristic
323C8—82no—
324C8—6642no—
325C22—21no—
326C22—21no—
327C22—3322no—
328C22—3322no—
329C22—3322no—
330C22—6641no—
331C22—6641no—
332C22—6641no—
333C22—6641no—
334C22C2×#1458170611yescharacteristic
335C22C2×#1458171411yescharacteristic
336C22#1912771711yescharacteristic
337C4—21no—
338C4—22no—
339C4—21no—
340C4—22no—
341C4—21no—
342C4—21no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—6641no—
347C4—6641no—
348C4—6641no—
349C2—21no—
350C2—21no—
351C2—6642no—
352C2#2490603211yescharacteristic
353C2#2490910211yescharacteristic
354C2#2490915011yescharacteristic
subgroupG/H
centerC22C2×#14581706
commutatorC22×C4×C83C22×C4
FrattiniC2×C4⋊C4C22×D83
FittingC83×#1461C2
socleC22×C83C2×#99
prime radicalC22×D332C2×C4
cosocleC2×C4⋊C4×C83C23
hypercenter#1461D83
hypocenterC83#11879
LCSG ▷ C22×C4×C83 ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1461

Complements

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

NH ≅ G/Ntype
2. #24907459C2semidirect
3. #24905470C2semidirect
5. C83×#1461C2semidirect
180. #1461D83semidirect
183. C83#11879semidirect

Maximal subgroups

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

subgroupindex
#249123682
#249074592
#249054702
#249054672
C83×#14612
#1187983