Finite Group Databasebeta

← #35314974

Subgroups — #35314974

Order 21248.

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

normal subgroups only

This group has 652 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
1C4×#14581707C211yesmaximal, characteristic, complemented
2#24905801C211yesmaximal, characteristic, complemented
3#24905460C214yesmaximal, complemented
4C83×#2541C211yesmaximal, characteristic
5C2×#14581707—21no—
6#19127665—21no—
7C4×C22⋊Dic83—21no—
8#19127468C2212yes—
9#19127427—21no—
10#19127425C2212yes—
11C2×#127×C83—21no—
12C2×#127×C83C411yescharacteristic, complemented
13C2×#121×C83—21no—
14C2×#121×C83—21no—
15C2×C4×D4×C83C2211yescharacteristic
16#392×C83C412yescomplemented
17#392×C83C2212yes—
18#390×C83—21no—
19C4×C4⋊C4×C83C411yescharacteristic, complemented
20C2×#13050574—41no—
21C2×C22⋊Dic83—22no—
22C2×C4×Dic83—41no—
23#14581707—24no—
24C22⋊(C83⋊C8)—22no—
25C22⋊(C83⋊C8)—41no—
26C83⋊#95—41no—
27C22×D4×C83—22no—
28C22×D4×C83—22no—
29C22×D4×C83—21no—
30C22×D4×C83—21no—
31C22×D4×C83C2×C412yes—
32C22×D4×C83D411yescharacteristic
33C23×C4×C83D411yescharacteristic
34C23×C4×C83D411yescharacteristic
35#127×C83—41no—
36#127×C83—41no—
37#127×C83—44no—
38#121×C83—44no—
39#121×C83—44no—
40C4×D4×C83—41no—
41C4×D4×C83—41no—
42C2×C4⋊C4×C83—21no—
43C2×C4⋊C4×C83D411yescharacteristic
44C2×C22⋊C4×C83—22no—
45C2×C22⋊C4×C83—22no—
46C2×C22⋊C4×C83C2×C411yescharacteristic
47C2×C22⋊C4×C83C2311yescharacteristic
48C2×C42×C83—21no—
49C2×C42×C83C2×C411yescharacteristic
50#95×C83C2×C412yes—
51C22×Dic83—42no—
52#13050574—82no—
53C2×C83⋊C8—42no—
54C22⋊Dic83—24no—
55C22⋊Dic83—42no—
56C4×Dic83—82no—
57C24×C83—22no—
58C24×C83C22⋊C411yescharacteristic
59C24×C83C2×D411yescharacteristic
60C2×D4×C83—22no—
61C2×D4×C83—22no—
62C2×D4×C83—41no—
63C2×D4×C83—42no—
64C2×D4×C83—42no—
65C2×D4×C83—42no—
66C2×D4×C83—44no—
67C2×D4×C83—44no—
68C2×D4×C83—42no—
69C2×D4×C83—42no—
70C2×D4×C83—44no—
71C2×D4×C83—44no—
72C2×D4×C83—41no—
73C2×D4×C83—41no—
74C2×D4×C83—42no—
75C2×D4×C83—82no—
76C2×D4×C83—82no—
77C2×D4×C83—82no—
78C22×C4×C83—22no—
79C22×C4×C83—21no—
80C22×C4×C83—21no—
81C22×C4×C83—21no—
82C22×C4×C83—21no—
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—24no—
91C4⋊C4×C83—41no—
92C22⋊C4×C83—41no—
93C22⋊C4×C83—41no—
94C22⋊C4×C83—44no—
95C22⋊C4×C83—44no—
96C42×C83—21no—
97C42×C83—21no—
98C42×C83—42no—
99C23×C83—21no—
100C23×C83—21no—
101C23×C83—21no—
102C23×C83—21no—
103C23×C83—21no—
104C23×C83—41no—
105C23×C83—41no—
106C23×C83—41no—
107C23×C83—41no—
108C23×C83—42no—
109C23×C83—42no—
110C23×C83—42no—
111C23×C83—42no—
112C23×C83—42no—
113C23×C83—42no—
114C23×C83—42no—
115C23×C83C2×C22⋊C411yescharacteristic
116C23×C83AGL(1,ℤ/8)12yes—
117D4×C83—44no—
118D4×C83—82no—
119D4×C83—84no—
120D4×C83—88no—
121D4×C83—88no—
122D4×C83—84no—
123D4×C83—82no—
124C2×C4×C83—21no—
125C2×C4×C83—21no—
126C2×C4×C83—22no—
127C2×C4×C83—24no—
128C2×C4×C83—21no—
129C2×C4×C83—21no—
130C2×C4×C83—21no—
131C2×C4×C83—42no—
132C2×C4×C83—42no—
133C2×C4×C83—42no—
134C2×C4×C83—42no—
135C2×C4×C83—41no—
136C2×C4×C83—41no—
137C2×C4×C83—41no—
138C2×C4×C83—81no—
139C2×C4×C83—81no—
140C2×C4×C83#9911yescharacteristic
141C2×C4×C83#9911yescharacteristic
142C2×C4×C83#10412yes—
143C2×Dic83—44no—
144C2×Dic83—82no—
145C2×Dic83—82no—
146C83⋊C8—82no—
147C22×C83—21no—
148C22×C83—21no—
149C22×C83—22no—
150C22×C83—42no—
151C22×C83—42no—
152C22×C83—41no—
153C22×C83—41no—
154C22×C83—41no—
155C22×C83—41no—
156C22×C83—41no—
157C22×C83—81no—
158C22×C83—81no—
159C22×C83—82no—
160C22×C83—82no—
161C22×C83—82no—
162C22×C83—82no—
163C22×C83—82no—
164C22×C83—82no—
165C22×C83—82no—
166C22×C83C2×#9911yescharacteristic
167C22×C83#41811yescharacteristic
168C22×C83C2×#10411yescharacteristic
169C4×C83—21no—
170C4×C83—22no—
171C4×C83—44no—
172C4×C83—41no—
173C4×C83—41no—
174C4×C83—41no—
175C4×C83#35312yes—
176Dic83—84no—
177#11776—831nomaximal, Sylow
178C2×C83—21no—
179C2×C83—21no—
180C2×C83—42no—
181C2×C83—81no—
182C2×C83—162no—
183C2×C83#150711yescharacteristic
184C2×C83#212211yescharacteristic
185C2×C83C2×#35311yescharacteristic
186#2541D8311yescharacteristic
187C4×#99—831no—
188#1465—831no—
189#1343—834no—
190C83#1177611yescharacteristic, complemented, Sylow
191C2×#127—21no—
192C2×#127Dic8311yescharacteristic, complemented
193C2×#121—21no—
194C2×#121—21no—
195C2×C4×D4C2×D8311yescharacteristic
196C2×#99—1661no—
197#407—1661no—
198#392Dic8312yescomplemented
199#392C2×D8312yes—
200#390—21no—
201C4×C4⋊C4Dic8311yescharacteristic, complemented
202C4×C22⋊C4—1661no—
203#342—832no—
204#324—1661no—
205#323—832no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4—21no—
209C22×D4—21no—
210C22×D4C2×Dic8312yes—
211C22×D4C22⋊D8311yescharacteristic
212C23×C4C22⋊D8311yescharacteristic
213C23×C4C22⋊D8311yescharacteristic
214C2×M16—3321no—
215#127—41no—
216#127—41no—
217#127—44no—
218#121—44no—
219#121—44no—
220C4×D4—41no—
221C4×D4—41no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4C22⋊D8311yescharacteristic
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—1662no—
227C2×C22⋊C4C2×Dic8311yescharacteristic
228C2×C22⋊C4C22×D8311yescharacteristic
229C2×C42—21no—
230C2×C42—3321no—
231C2×C42C2×Dic8311yescharacteristic
232#99—1664no—
233C22⋊C8—3321no—
234C22⋊C8—1662no—
235#95—3321no—
236#95C2×Dic8312yes—
237C24—22no—
238C24C22⋊Dic8311yescharacteristic
239C24C2×C22⋊D8311yescharacteristic
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—41no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—42no—
246C2×D4—44no—
247C2×D4—44no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—44no—
251C2×D4—44no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—42no—
255C2×D4—82no—
256C2×D4—82no—
257C2×D4—82no—
258C22×C4—22no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—3322no—
266C22×C4C22⋊Dic8311yescharacteristic
267C22×C4C22⋊Dic8311yescharacteristic
268C22×C4C22⋊Dic8311yescharacteristic
269C22×C4C22×Dic8311yescharacteristic
270C22×C4C2×C22⋊D8311yescharacteristic
271M16—6642no—
272C2×C8—3322no—
273C4⋊C4—24no—
274C4⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4—44no—
278C22⋊C4—44no—
279C22⋊C4—3322no—
280C22⋊C4—1664no—
281C42—21no—
282C42—21no—
283C42—42no—
284C42—6642no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—42no—
295C23—42no—
296C23—42no—
297C23—42no—
298C23—42no—
299C23—42no—
300C23—42no—
301C23C2×C22⋊Dic8311yescharacteristic
302C23#1458181812yes—
303D4—44no—
304D4—82no—
305D4—84no—
306D4—88no—
307D4—88no—
308D4—84no—
309D4—82no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—24no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—3324no—
327C2×C4—6642no—
328C2×C4—6642no—
329C2×C4#1458170711yescharacteristic
330C2×C4#1458170711yescharacteristic
331C2×C4#1458171912yes—
332C8—6642no—
333C22—21no—
334C22—21no—
335C22—22no—
336C22—42no—
337C22—42no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—81no—
344C22—81no—
345C22—82no—
346C22—82no—
347C22—82no—
348C22—82no—
349C22—82no—
350C22—82no—
351C22—82no—
352C22C2×#1458170711yescharacteristic
353C22#1912770611yescharacteristic
354C22C2×#1458171911yescharacteristic
355C4—21no—
356C4—22no—
357C4—44no—
358C4—41no—
359C4—41no—
360C4—41no—
361C4—6644no—
362C4#1912749412yes—
363C2—21no—
364C2—21no—
365C2—42no—
366C2—81no—
367C2—162no—
368C2#2490597911yescharacteristic
369C2#2490911311yescharacteristic
370C2C2×#1912749411yescharacteristic
subgroupG/H
centerC22C2×#14581707
commutatorC22×C4×C83C22×C4
FrattiniC2×C22⋊C4C22×D83
Fitting, prime radicalC83×#2541C2
socleC22×C83C2×#99
cosocleC2×C22⋊C4×C83C23
hypercenter#2541D83
hypocenterC83#11776
LCSG ▷ C22×C4×C83 ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2541

Complements

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

NH ≅ G/Ntype
1. C4×#14581707C2semidirect
2. #24905801C2semidirect
3. #24905460C2semidirect
12. C2×#127×C83C4semidirect
16. #392×C83C4semidirect
19. C4×C4⋊C4×C83C4semidirect
190. C83#11776semidirect
192. C2×#127Dic83semidirect
198. #392Dic83semidirect
201. C4×C4⋊C4Dic83semidirect

Maximal subgroups

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

subgroupindex
C4×#145817072
#249058012
#249054602
C83×#25412
#1177683