Finite Group Databasebeta

← #35300190

Subgroups — #35300190

Order 21248.

648 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 135 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
1C2×#19127569C211yesmaximal, characteristic, complemented
2#24907531C211yesmaximal, characteristic, complemented
3C2×#19127435C212yesmaximal
4#24905341C212yesmaximal, complemented
5C83×#2854C211yesmaximal, characteristic
6C2×C22⋊(C83⋊C8)C2212yes—
7#19127569—24no—
8C2×C83⋊#95—21no—
9C2×C83⋊#95—21no—
10C2×C83⋊#95C2212yes—
11#19127456—22no—
12#19127435—24no—
13C2×#123×C83—21no—
14C2×#121×C83C2211yescharacteristic
15C22×C42×C83C2211yescharacteristic
16#390×C83C412yescomplemented
17#389×C83C412yescomplemented
18#388×C83—22no—
19#386×C83—21no—
20#380×C83C2211yescharacteristic
21C23×Dic83—22no—
22C22×C83⋊C8—22no—
23C22⋊(C83⋊C8)—22no—
24C22⋊(C83⋊C8)—22no—
25C83⋊#95—24no—
26C83⋊#95—24no—
27C83⋊#95—22no—
28C83⋊#95D414yes—
29C22×D4×C83—21no—
30C22×D4×C83C2×C411yescharacteristic
31C23×C4×C83D411yescharacteristic
32C23×C4×C83D411yescharacteristic
33C23×C4×C83C2311yescharacteristic
34#123×C83—44no—
35#121×C83—22no—
36#121×C83—22no—
37C2×C4⋊C4×C83—22no—
38C2×C4⋊C4×C83C2×C411yescharacteristic
39C2×C4⋊C4×C83C2×C412yes—
40C2×C22⋊C4×C83—21no—
41C2×C22⋊C4×C83—22no—
42C2×C22⋊C4×C83—22no—
43C2×C22⋊C4×C83—21no—
44C2×C42×C83—22no—
45C2×C42×C83—22no—
46C2×C42×C83C2×C412yes—
47C2×C42×C83D412yes—
48#95×C83—22no—
49#95×C83—21no—
50C22×Dic83—24no—
51C22×Dic83—44no—
52C22×Dic83—44no—
53C22×Dic83—42no—
54C22×Dic83—42no—
55C2×C83⋊C8—44no—
56C2×C83⋊C8—42no—
57C24×C83—21no—
58C24×C83C2×D411yescharacteristic
59C2×D4×C83—22no—
60C2×D4×C83—42no—
61C2×D4×C83—41no—
62C2×D4×C83—41no—
63C2×D4×C83—42no—
64C2×D4×C83—42no—
65C22×C4×C83—21no—
66C22×C4×C83—21no—
67C22×C4×C83—21no—
68C22×C4×C83—21no—
69C22×C4×C83—22no—
70C22×C4×C83—22no—
71C22×C4×C83—22no—
72C22×C4×C83—21no—
73C22×C4×C83—42no—
74C22×C4×C83—42no—
75C22×C4×C83C22⋊C412yes—
76C22×C4×C83C22⋊C412yes—
77C22×C4×C83C22×C411yescharacteristic
78C22×C4×C83C2×D412yes—
79C22×C4×C83C2×D411yescharacteristic
80C22×C4×C83#4112yes—
81C4⋊C4×C83—22no—
82C4⋊C4×C83—42no—
83C4⋊C4×C83—44no—
84C22⋊C4×C83—44no—
85C22⋊C4×C83—44no—
86C22⋊C4×C83—42no—
87C22⋊C4×C83—42no—
88C42×C83—24no—
89C42×C83—24no—
90C23×C83—22no—
91C23×C83—22no—
92C23×C83—41no—
93C23×C83—42no—
94C23×C83—41no—
95C23×C83—41no—
96C23×C83—42no—
97C23×C83C2×C22⋊C411yescharacteristic
98C23×C83#12011yescharacteristic
99C23×C83#12311yescharacteristic
100C23×C83#13512yes—
101C23×C83AGL(1,ℤ/8)12yes—
102D4×C83—82no—
103D4×C83—82no—
104D4×C83—82no—
105C2×C4×C83—22no—
106C2×C4×C83—22no—
107C2×C4×C83—22no—
108C2×C4×C83—22no—
109C2×C4×C83—22no—
110C2×C4×C83—22no—
111C2×C4×C83—22no—
112C2×C4×C83—22no—
113C2×C4×C83—42no—
114C2×C4×C83—42no—
115C2×C4×C83—42no—
116C2×C4×C83—42no—
117C2×C4×C83—42no—
118C2×C4×C83—42no—
119C2×C4×C83—41no—
120C2×C4×C83—42no—
121C2×C4×C83—41no—
122C2×C4×C83—42no—
123C2×C4×C83—41no—
124C2×C4×C83—42no—
125C2×C4×C83C4×D412yes—
126C2×C4×C83#12112yes—
127C2×Dic83—44no—
128C2×Dic83—44no—
129C2×Dic83—44no—
130C2×Dic83—84no—
131C2×Dic83—84no—
132C83⋊C8—82no—
133C22×C83—21no—
134C22×C83—21no—
135C22×C83—21no—
136C22×C83—21no—
137C22×C83—24no—
138C22×C83—24no—
139C22×C83—81no—
140C22×C83—81no—
141C22×C83—81no—
142C22×C83—82no—
143C22×C83—82no—
144C22×C83#38611yescharacteristic
145C22×C83#41012yes—
146C22×C83#41611yescharacteristic
147C22×C83#41811yescharacteristic
148C22×C83#42112yes—
149C22×C83#44912yes—
150C22×C83#48212yes—
151C4×C83—22no—
152C4×C83—22no—
153C4×C83—42no—
154C4×C83—42no—
155C4×C83—81no—
156C4×C83—82no—
157Dic83—84no—
158#9157—831nomaximal, Sylow
159C2×C83—22no—
160C2×C83—22no—
161C2×C83—161no—
162C2×C83#151412yes—
163C2×C83#152112yes—
164C2×C83#187511yescharacteristic
165C2×C83#189711yescharacteristic
166C2×C83#189911yescharacteristic
167#2854D8311yescharacteristic
168C2×#383—831no—
169#1817—831no—
170C2×#327—832no—
171#1294—832no—
172C83#915711yescharacteristic, complemented, Sylow
173C2×#123—21no—
174C2×#121C2×D8311yescharacteristic
175C22×C42C2×D8311yescharacteristic
176C2×C22⋊C8—832no—
177#390Dic8312yescomplemented
178#389Dic8312yescomplemented
179#388—22no—
180#386—21no—
181#383—1664no—
182#380C2×D8311yescharacteristic
183C2×#95—832no—
184C2×#95—1661no—
185C2×#95—1661no—
186#336—1662no—
187#327—1664no—
188C22×D4—21no—
189C22×D4C2×Dic8311yescharacteristic
190C23×C4—1662no—
191C23×C4C22⋊D8311yescharacteristic
192C23×C4C22⋊D8311yescharacteristic
193C23×C4C22×D8311yescharacteristic
194C22×C8—1662no—
195#123—44no—
196#121—22no—
197#121—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4C2×Dic8311yescharacteristic
200C2×C4⋊C4C2×Dic8312yes—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—21no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42C2×Dic8312yes—
208C2×C42C22⋊D8312yes—
209C22⋊C8—1662no—
210C22⋊C8—1662no—
211#95—22no—
212#95—21no—
213#95—834no—
214#95—1664no—
215#95—1664no—
216#95—1662no—
217C24—21no—
218C24C2×C22⋊D8311yescharacteristic
219C2×D4—22no—
220C2×D4—42no—
221C2×D4—41no—
222C2×D4—41no—
223C2×D4—42no—
224C2×D4—42no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—3324no—
236C22×C4—3324no—
237C22×C4—3322no—
238C22×C4—3322no—
239C22×C4—1664no—
240C22×C4C22⋊Dic8312yes—
241C22×C4C22⋊Dic8312yes—
242C22×C4C22×Dic8311yescharacteristic
243C22×C4D4×D8312yes—
244C22×C4C2×C22⋊D8311yescharacteristic
245C22×C4#1305058912yes—
246C2×C8—3324no—
247C2×C8—3322no—
248C4⋊C4—22no—
249C4⋊C4—42no—
250C4⋊C4—44no—
251C22⋊C4—44no—
252C22⋊C4—44no—
253C22⋊C4—42no—
254C22⋊C4—42no—
255C42—24no—
256C42—24no—
257C23—22no—
258C23—22no—
259C23—41no—
260C23—42no—
261C23—41no—
262C23—41no—
263C23—42no—
264C23C2×C22⋊Dic8311yescharacteristic
265C23#1458176011yescharacteristic
266C23#1458177311yescharacteristic
267C23#1458181212yes—
268C23#1458181812yes—
269D4—82no—
270D4—82no—
271D4—82no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—3324no—
293C2×C4—3324no—
294C2×C4—3324no—
295C2×C4—6644no—
296C2×C4—6644no—
297C2×C4D4×Dic8312yes—
298C2×C4#1458176412yes—
299C8—6642no—
300C22—24no—
301C22—24no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—81no—
307C22—82no—
308C22—82no—
309C22—81no—
310C22—81no—
311C22#1912758111yescharacteristic
312C22#1912767512yes—
313C22#1912769711yescharacteristic
314C22#1912770611yescharacteristic
315C22#1912771812yes—
316C22#1912784412yes—
317C22#1912802212yes—
318C4—22no—
319C4—22no—
320C4—42no—
321C4—42no—
322C4—81no—
323C4—82no—
324C4—6644no—
325C2—22no—
326C2—22no—
327C2—161no—
328C2#2490601012yes—
329C2#2490604012yes—
330C2#2490778611yescharacteristic
331C2#2490791211yescharacteristic
332C2#2490792211yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic83
commutatorC22×C4×C83C22×C4
FrattiniC23×C4C22×D83
FittingC83×#2854C2
socleC23×C83C2×C22⋊C4
prime radicalC2×#121×C83C22
cosocleC23×C4×C83C23
hypercenter#2854D83
hypocenterC83#9157
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2854

Complements

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

NH ≅ G/Ntype
1. C2×#19127569C2semidirect
2. #24907531C2semidirect
4. #24905341C2semidirect
16. #390×C83C4semidirect
17. #389×C83C4semidirect
172. C83#9157semidirect
177. #390Dic83semidirect
178. #389Dic83semidirect

Maximal subgroups

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

subgroupindex
C2×#191275692
#249075312
C2×#191274352
#249053412
C83×#28542
#915783