Finite Group Databasebeta

← #35315211

Subgroups — #35315211

Order 21248.

416 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 83 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#24910746C211yesmaximal, characteristic
2#24907459C211yesmaximal, characteristic, complemented
3#24905491C212yesmaximal, complemented
4#24905467C212yesmaximal
5C83×#1467C211yesmaximal, characteristic, complemented
6C2×C4×D332C411yescharacteristic
7#19127725—21no—
8C2×#14581711—21no—
9#19127640C411yescharacteristic
10#19127640C2211yescharacteristic
11#19127627—21no—
12#19127618C411yescharacteristic
13#19127618C2211yescharacteristic
14#19127612—21no—
15C4×(C2×D83)⋊C4—21no—
16C4×(C2×D83)⋊C4—21no—
17#19127547C411yescharacteristic
18#19127470C2212yes—
19C2×C4×Q8×C83C2211yescharacteristic
20#432×C83—21no—
21#324×C83—21no—
22#324×C83C2212yes—
23C22×D332C2×C411yescharacteristic
24C22×C4×D83—21no—
25C2×#13050574—22no—
26C4×D332—41no—
27C4×D332—41no—
28C2×C4⋊Dic83C2×C411yescharacteristic
29C2×(C2×D83)⋊C4—21no—
30C2×(C2×D83)⋊C4—21no—
31C2×(C2×D83)⋊C4—21no—
32C2×(C2×D83)⋊C4—21no—
33C2×(C2×D83)⋊C4—21no—
34C2×C4×Dic83—21no—
35C4⋊(C83⋊C8)—41no—
36C4⋊(C83⋊C8)—41no—
37#14581711—42no—
38C83⋊#95—21no—
39C83⋊#95—21no—
40C83⋊#95—21no—
41C83⋊#95—21no—
42C83⋊#95—21no—
43C83⋊#95C2×C411yescharacteristic
44C83⋊#95C2×C411yescharacteristic
45C22×Q8×C83D411yescharacteristic
46C4×Q8×C83—41no—
47C4×Q8×C83—41no—
48C2×C4⋊C4×C83C2×C411yescharacteristic
49C2×C4⋊C4×C83D411yescharacteristic
50C2×C4⋊C4×C83C2311yescharacteristic
51C2×C42×C83C2×C411yescharacteristic
52C2×C42×C83D411yescharacteristic
53C2×C42×C83D411yescharacteristic
54C4⋊C8×C83—41no—
55C22⋊C8×C83—22no—
56#97×C83—41no—
57C23×D83—21no—
58C2×D332—41no—
59C2×D332—42no—
60C22×Dic83—21no—
61C22×Dic83—21no—
62C22×Dic83—21no—
63C2×C4×D83—42no—
64C2×C4×D83—42no—
65C2×C4×D83—41no—
66C2×C4×D83—41no—
67#13050574—42no—
68#13050574—42no—
69C2×C83⋊C8—42no—
70C4⋊Dic83—41no—
71(C2×D83)⋊C4—42no—
72(C2×D83)⋊C4—42no—
73(C2×D83)⋊C4—42no—
74(C2×D83)⋊C4—42no—
75(C2×D83)⋊C4—41no—
76(C2×D83)⋊C4—41no—
77C4×Dic83—42no—
78C2×Q8×C83—21no—
79C2×Q8×C83—21no—
80C2×Q8×C83—41no—
81C2×Q8×C83—41no—
82C22×C4×C83C22⋊C411yescharacteristic
83C22×C4×C83C22⋊C411yescharacteristic
84C22×C4×C83C22⋊C411yescharacteristic
85C22×C4×C83C22⋊C411yescharacteristic
86C22×C4×C83C22×C411yescharacteristic
87C22×C4×C83C2×D411yescharacteristic
88C22×C4×C83C2×D411yescharacteristic
89C2×C8×C83—42no—
90C4⋊C4×C83—41no—
91C4⋊C4×C83—41no—
92C4⋊C4×C83—41no—
93C42×C83—21no—
94C42×C83—21no—
95C42×C83—21no—
96C42×C83—21no—
97C42×C83—21no—
98C42×C83—21no—
99C23×C83C2×C22⋊C411yescharacteristic
100C22×D83—22no—
101C22×D83—41no—
102C22×D83—41no—
103C22×D83—41no—
104C22×D83—41no—
105C22×D83—41no—
106C22×D83—41no—
107Q8×C83—42no—
108Q8×C83—42no—
109C2×C4×C83—21no—
110C2×C4×C83—21no—
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—21no—
121C2×C4×C83—41no—
122C2×C4×C83—41no—
123C2×C4×C83—41no—
124C2×C4×C83#9911yescharacteristic
125C2×C4×C83#9911yescharacteristic
126C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
127C2×C4×C83#13711yescharacteristic
128C2×Dic83—41no—
129C2×Dic83—41no—
130C2×Dic83—41no—
131C2×Dic83—41no—
132C2×Dic83—41no—
133C2×Dic83—41no—
134C2×Dic83—41no—
135C2×Dic83—41no—
136C2×Dic83—41no—
137D332—42no—
138D332—42no—
139C4×D83—82no—
140C4×D83—82no—
141C8×C83—82no—
142C83⋊C8—82no—
143C22×C83—21no—
144C22×C83—21no—
145C22×C83C2×#9911yescharacteristic
146C22×C83#41711yescharacteristic
147C22×C83#42111yescharacteristic
148C2×D83—42no—
149C2×D83—42no—
150C2×D83—42no—
151C2×D83—81no—
152C2×D83—81no—
153C2×D83—81no—
154C2×D83—81no—
155C4×C83—21no—
156C4×C83—21no—
157C4×C83—21no—
158C4×C83—41no—
159C4×C83—41no—
160C4×C83—41no—
161C4×C83—41no—
162C4×C83—41no—
163C4×C83#35512yes—
164Dic83—81no—
165Dic83—81no—
166Dic83—81no—
167#11829—831nomaximal, Sylow
168C2×C83—21no—
169C2×C83—21no—
170C2×C83#151311yescharacteristic
171C2×C83#212011yescharacteristic
172C2×C83C2×#35511yescharacteristic
173D83—82no—
174#2398—831no—
175#1801—831no—
176#1467D8311yescharacteristic, complemented
177#1353—832no—
178#1345—832no—
179C83#1182911yescharacteristic, complemented, Sylow
180C2×C4×Q8C2×D8311yescharacteristic
181C2×C4×D4—831no—
182#432—21no—
183#423—1661no—
184C2×#101—1661no—
185#400—831no—
186#400—831no—
187#397—1661no—
188#394—831no—
189#394—831no—
190#393—1661no—
191C4×C22⋊C4—1661no—
192C4×C22⋊C4—1661no—
193#376—831no—
194#343—832no—
195#324—21no—
196#324C2×D8312yes—
197C22×Q8D33211yescharacteristic
198C22×D4—831no—
199C23×C4—1661no—
200C2×M16—1662no—
201C4×Q8—41no—
202C4×Q8—41no—
203C4×D4—3321no—
204C4×D4—3321no—
205C2×C4⋊C4—831no—
206C2×C4⋊C4C4×D8311yescharacteristic
207C2×C4⋊C4C22⋊D8311yescharacteristic
208C2×C4⋊C4C22×D8311yescharacteristic
209C2×C22⋊C4—1661no—
210C2×C22⋊C4—1661no—
211C2×C22⋊C4—1661no—
212C2×C22⋊C4—1661no—
213C2×C22⋊C4—1661no—
214C2×C42—1661no—
215C2×C42C4×D8311yescharacteristic
216C2×C42D33211yescharacteristic
217C2×C42C22⋊D8311yescharacteristic
218C4⋊C8—41no—
219C4⋊C8—3321no—
220C4⋊C8—3321no—
221#101—3322no—
222C22⋊C8—22no—
223#97—41no—
224#95—831no—
225#95—831no—
226#95—1661no—
227#95—1661no—
228#95—1661no—
229#95—1661no—
230#95—1661no—
231C24—1661no—
232C2×Q8—21no—
233C2×Q8—21no—
234C2×Q8—41no—
235C2×Q8—41no—
236C2×D4—3321no—
237C2×D4—3322no—
238C22×C4—3322no—
239C22×C4—3322no—
240C22×C4—3321no—
241C22×C4—3321no—
242C22×C4—1661no—
243C22×C4—1661no—
244C22×C4—1661no—
245C22×C4(C2×D83)⋊C411yescharacteristic
246C22×C4(C2×D83)⋊C411yescharacteristic
247C22×C4(C2×D83)⋊C411yescharacteristic
248C22×C4(C2×D83)⋊C411yescharacteristic
249C22×C4C2×C4×D8311yescharacteristic
250C22×C4C2×C22⋊D8311yescharacteristic
251C22×C4C2×D33211yescharacteristic
252M16—3322no—
253M16—3322no—
254C2×C8—42no—
255C2×C8—3322no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—3321no—
260C22⋊C4—3322no—
261C22⋊C4—3322no—
262C22⋊C4—3322no—
263C22⋊C4—3322no—
264C22⋊C4—3321no—
265C22⋊C4—3321no—
266C42—21no—
267C42—21no—
268C42—21no—
269C42—21no—
270C42—21no—
271C42—21no—
272C42—3322no—
273C23—3321no—
274C23—3321no—
275C23—3321no—
276C23—3321no—
277C23—3321no—
278C23—3321no—
279C23—1662no—
280C23C2×(C2×D83)⋊C411yescharacteristic
281Q8—42no—
282Q8—42no—
283D4—3322no—
284D4—3322no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—3321no—
301C2×C4—3321no—
302C2×C4—3321no—
303C2×C4—3321no—
304C2×C4—3321no—
305C2×C4—3321no—
306C2×C4—3321no—
307C2×C4—3321no—
308C2×C4—3321no—
309C2×C4—6642no—
310C2×C4—6642no—
311C2×C4#1458170611yescharacteristic
312C2×C4#1458170611yescharacteristic
313C2×C4#1458182011yescharacteristic
314C2×C4#1458182511yescharacteristic
315C8—82no—
316C8—6642no—
317C22—21no—
318C22—21no—
319C22—3322no—
320C22—3322no—
321C22—3322no—
322C22—6641no—
323C22—6641no—
324C22—6641no—
325C22—6641no—
326C22C2×#1458170611yescharacteristic
327C22#1912770311yescharacteristic
328C22#1912771711yescharacteristic
329C4—21no—
330C4—21no—
331C4—21no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—6641no—
338C4—6641no—
339C4—6641no—
340C4#1912749712yes—
341C2—21no—
342C2—21no—
343C2—6642no—
344C2#2490600711yescharacteristic
345C2#2490910211yescharacteristic
346C2C2×#1912749711yescharacteristic
subgroupG/H
centerC22C2×#14581706
commutatorC22×C4×C83C22×C4
FrattiniC2×C4⋊C4C22×D83
FittingC83×#1467C2
socleC22×C83C2×#99
prime radicalC22×D332C2×C4
cosocleC2×C4⋊C4×C83C23
hypercenter#1467D83
hypocenterC83#11829
LCSG ▷ C22×C4×C83 ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1467

Complements

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

NH ≅ G/Ntype
2. #24907459C2semidirect
3. #24905491C2semidirect
5. C83×#1467C2semidirect
176. #1467D83semidirect
179. C83#11829semidirect

Maximal subgroups

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

subgroupindex
#249107462
#249074592
#249054912
#249054672
C83×#14672
#1182983