Finite Group Databasebeta

← #35302199

Subgroups — #35302199

Order 21248.

576 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 123 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#24916492C211yesmaximal, characteristic
2#24907549C212yesmaximal, complemented
3#24907451C211yesmaximal, characteristic
4#24905330C212yesmaximal, complemented
5C83×#2486C211yesmaximal, characteristic
6#19127773—24no—
7#19127758—21no—
8C2×#14581707—21no—
9#19127665—21no—
10C2×C22⋊(C83⋊C8)C2212yes—
11#19127561C2212yes—
12#19127468—21no—
13#19127463—24no—
14C2×#123×C83—21no—
15C2×C4×D4×C83C2211yescharacteristic
16C22×C4⋊C4×C83C2211yescharacteristic
17#398×C83C412yescomplemented
18#397×C83C412yescomplemented
19#389×C83—21no—
20#381×C83—21no—
21#380×C83—21no—
22#380×C83C2211yescharacteristic
23C2×#13050574—22no—
24C22×C83⋊C8D414yes—
25C2×C4⋊Dic83—22no—
26C2×C22⋊Dic83—22no—
27C4⋊(C83⋊C8)—22no—
28#14581707—42no—
29C22⋊(C83⋊C8)—22no—
30C22⋊(C83⋊C8)—22no—
31C22⋊(C83⋊C8)—22no—
32#14581703—22no—
33C83⋊#95—22no—
34C22×D4×C83D411yescharacteristic
35C23×C4×C83—21no—
36C23×C4×C83D411yescharacteristic
37C23×C4×C83C2311yescharacteristic
38#123×C83—44no—
39C4×D4×C83—22no—
40C4×D4×C83—22no—
41C2×C4⋊C4×C83—22no—
42C2×C4⋊C4×C83—22no—
43C2×C4⋊C4×C83—22no—
44C2×C4⋊C4×C83—21no—
45C2×C4⋊C4×C83—21no—
46C2×C4⋊C4×C83C2×C412yes—
47C2×C4⋊C4×C83C2×C411yescharacteristic
48C2×C22⋊C4×C83—22no—
49C2×C22⋊C4×C83—21no—
50C2×C22⋊C4×C83C2×C411yescharacteristic
51C2×C22⋊C4×C83D411yescharacteristic
52C2×C42×C83D411yescharacteristic
53#95×C83—22no—
54#95×C83—22no—
55#95×C83C2×C412yes—
56C22×Dic83—42no—
57#13050574—82no—
58C2×C83⋊C8—24no—
59C2×C83⋊C8—44no—
60C4⋊Dic83—44no—
61C22⋊Dic83—42no—
62C22⋊Dic83—42no—
63C24×C83C22⋊C411yescharacteristic
64C24×C83C2×D411yescharacteristic
65C2×D4×C83—22no—
66C2×D4×C83—22no—
67C2×D4×C83—21no—
68C2×D4×C83—21no—
69C22×C4×C83—21no—
70C22×C4×C83—21no—
71C22×C4×C83—22no—
72C22×C4×C83—22no—
73C22×C4×C83—42no—
74C22×C4×C83—42no—
75C22×C4×C83—42no—
76C22×C4×C83—42no—
77C22×C4×C83—41no—
78C22×C4×C83—41no—
79C22×C4×C83C22⋊C411yescharacteristic
80C22×C4×C83C22⋊C411yescharacteristic
81C22×C4×C83C22⋊C411yescharacteristic
82C22×C4×C83C22×C411yescharacteristic
83C22×C4×C83C2×D412yes—
84C22×C4×C83C2×D411yescharacteristic
85C22×C4×C83#4112yes—
86C4⋊C4×C83—22no—
87C4⋊C4×C83—24no—
88C4⋊C4×C83—42no—
89C4⋊C4×C83—44no—
90C4⋊C4×C83—41no—
91C4⋊C4×C83—41no—
92C22⋊C4×C83—22no—
93C22⋊C4×C83—22no—
94C22⋊C4×C83—44no—
95C22⋊C4×C83—42no—
96C42×C83—21no—
97C42×C83—21no—
98C23×C83—21no—
99C23×C83—21no—
100C23×C83—21no—
101C23×C83—41no—
102C23×C83—41no—
103C23×C83—41no—
104C23×C83—41no—
105C23×C83C2×C22⋊C411yescharacteristic
106C23×C83#12112yes—
107C23×C83AGL(1,ℤ/8)11yescharacteristic
108C23×C83AGL(1,ℤ/8)12yes—
109C23×C83#13711yescharacteristic
110D4×C83—42no—
111D4×C83—42no—
112C2×C4×C83—22no—
113C2×C4×C83—24no—
114C2×C4×C83—21no—
115C2×C4×C83—21no—
116C2×C4×C83—21no—
117C2×C4×C83—21no—
118C2×C4×C83—21no—
119C2×C4×C83—41no—
120C2×C4×C83—42no—
121C2×C4×C83—42no—
122C2×C4×C83—41no—
123C2×C4×C83—41no—
124C2×C4×C83—42no—
125C2×C4×C83—42no—
126C2×C4×C83—42no—
127C2×C4×C83—42no—
128C2×C4×C83—42no—
129C2×C4×C83—42no—
130C2×C4×C83—81no—
131C2×C4×C83—82no—
132C2×C4×C83—82no—
133C2×C4×C83C4×D412yes—
134C2×C4×C83#12011yescharacteristic
135C2×C4×C83#12311yescharacteristic
136C2×Dic83—82no—
137C2×Dic83—82no—
138C2×Dic83—82no—
139C83⋊C8—44no—
140C22×C83—22no—
141C22×C83—21no—
142C22×C83—22no—
143C22×C83—21no—
144C22×C83—41no—
145C22×C83—42no—
146C22×C83—41no—
147C22×C83—42no—
148C22×C83—41no—
149C22×C83—41no—
150C22×C83—41no—
151C22×C83—81no—
152C22×C83#38611yescharacteristic
153C22×C83#41711yescharacteristic
154C22×C83#41811yescharacteristic
155C22×C83#46812yes—
156C22×C83#46912yes—
157C4×C83—21no—
158C4×C83—22no—
159C4×C83—41no—
160C4×C83—41no—
161C4×C83—41no—
162C4×C83—82no—
163C4×C83—82no—
164C4×C83#45812yes—
165Dic83—162no—
166#9474—831nomaximal, Sylow
167C2×C83—22no—
168C2×C83—21no—
169C2×C83—21no—
170C2×C83—22no—
171C2×C83—81no—
172C2×C83#189011yescharacteristic
173C2×C83#189511yescharacteristic
174C2×C83#190911yescharacteristic
175#2973—831no—
176#2486D8311yescharacteristic
177#1823—832no—
178#1799—831no—
179#1289—832no—
180C83#947411yescharacteristic, complemented, Sylow
181C2×#123—21no—
182C2×C4×D4C2×D8311yescharacteristic
183C22×C4⋊C4C2×D8311yescharacteristic
184#435—1664no—
185#432—1661no—
186C2×#99—1661no—
187#407—1661no—
188C2×C22⋊C8—832no—
189#398Dic8312yescomplemented
190#397Dic8312yescomplemented
191#389—21no—
192#381—21no—
193#380—21no—
194#380—832no—
195#380C2×D8311yescharacteristic
196#342—1661no—
197#340—1664no—
198C22×D4C22⋊D8311yescharacteristic
199C23×C4—21no—
200C23×C4C22⋊D8311yescharacteristic
201C23×C4C22×D8311yescharacteristic
202C2×M16—1662no—
203C22×C8—834no—
204#123—44no—
205C4×D4—22no—
206C4×D4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—1662no—
213C2×C4⋊C4C2×Dic8312yes—
214C2×C4⋊C4C2×Dic8311yescharacteristic
215C2×C22⋊C4—22no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—1662no—
218C2×C22⋊C4C2×Dic8311yescharacteristic
219C2×C22⋊C4C22⋊D8311yescharacteristic
220C2×C42C22⋊D8311yescharacteristic
221C4⋊C8—1662no—
222#99—3322no—
223C22⋊C8—1662no—
224C22⋊C8—1662no—
225C22⋊C8—1662no—
226#97—1662no—
227#95—22no—
228#95—22no—
229#95—1662no—
230#95C2×Dic8312yes—
231C24C22⋊Dic8311yescharacteristic
232C24C2×C22⋊D8311yescharacteristic
233C2×D4—22no—
234C2×D4—21no—
235C2×D4—22no—
236C2×D4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—41no—
245C22×C4—42no—
246C22×C4—41no—
247C22×C4—3322no—
248C22×C4C22⋊Dic8311yescharacteristic
249C22×C4C22⋊Dic8311yescharacteristic
250C22×C4C22⋊Dic8311yescharacteristic
251C22×C4C22×Dic8311yescharacteristic
252C22×C4D4×D8312yes—
253C22×C4C2×C22⋊D8311yescharacteristic
254C22×C4#1305058912yes—
255M16—6642no—
256C2×C8—3324no—
257C2×C8—1664no—
258C4⋊C4—22no—
259C4⋊C4—24no—
260C4⋊C4—44no—
261C4⋊C4—42no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—3324no—
265C22⋊C4—22no—
266C22⋊C4—22no—
267C22⋊C4—44no—
268C22⋊C4—42no—
269C22⋊C4—3322no—
270C22⋊C4—3322no—
271C42—21no—
272C42—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23C2×C22⋊Dic8311yescharacteristic
281C23#1458176412yes—
282C23#1458181711yescharacteristic
283C23#1458181812yes—
284C23#1458182211yescharacteristic
285D4—42no—
286D4—42no—
287C2×C4—22no—
288C2×C4—24no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—81no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—6642no—
309C2×C4—6642no—
310C2×C4—6642no—
311C2×C4D4×Dic8312yes—
312C2×C4#1458176011yescharacteristic
313C2×C4#1458177311yescharacteristic
314C8—3324no—
315C22—21no—
316C22—22no—
317C22—21no—
318C22—22no—
319C22—41no—
320C22—42no—
321C22—41no—
322C22—42no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—81no—
327C22#1912758111yescharacteristic
328C22#1912770211yescharacteristic
329C22#1912770611yescharacteristic
330C22#1912794512yes—
331C22#1912795212yes—
332C4—21no—
333C4—22no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—82no—
338C4—82no—
339C4—13282no—
340C4#1912789012yes—
341C2—21no—
342C2—22no—
343C2—21no—
344C2—22no—
345C2—81no—
346C2#2490787711yescharacteristic
347C2#2490790211yescharacteristic
348C2#2490798611yescharacteristic
subgroupG/H
centerC22#19127581
commutatorC22×C4×C83C22×C4
FrattiniC23×C4C22×D83
FittingC83×#2486C2
socleC22×C83#386
prime radicalC22×D4×C83D4
cosocleC23×C4×C83C23
hypercenter#2486D83
hypocenterC83#9474
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2486

Complements

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

NH ≅ G/Ntype
2. #24907549C2semidirect
4. #24905330C2semidirect
17. #398×C83C4semidirect
18. #397×C83C4semidirect
180. C83#9474semidirect
189. #398Dic83semidirect
190. #397Dic83semidirect

Maximal subgroups

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

subgroupindex
#249164922
#249075492
#249074512
#249053302
C83×#24862
#947483