Finite Group Databasebeta

← #35300175

Subgroups — #35300175

Order 21248.

902 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 902 conjugacy classes of subgroups — too many to draw. Only its 143 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×#19127564C211yesmaximal, characteristic
2#24907373C211yesmaximal, characteristic
3C2×#19127438C212yesmaximal, complemented
4#24905341C212yesmaximal, complemented
5C83×#2848C211yesmaximal, characteristic
6C22×C22⋊Dic83—21no—
7C2×C22⋊(C83⋊C8)—22no—
8C2×C22⋊(C83⋊C8)C2212yes—
9#19127564—24no—
10C2×C83⋊#95—21no—
11C2×C83⋊#95C2212yes—
12#19127438—24no—
13C24×C4×C83C2211yescharacteristic
14C2×C22⋊Q8×C83C412yescomplemented
15C2×C22⋊Q8×C83C2211yescharacteristic
16#385×C83—21no—
17#385×C83—22no—
18#380×C83C412yescomplemented
19#380×C83C2211yescharacteristic
20#379×C83—21no—
21C23×Dic83—22no—
22C22×C83⋊C8—22no—
23C2×C22⋊Dic83—24no—
24C2×C22⋊Dic83—44no—
25C22⋊(C83⋊C8)—22no—
26C22⋊(C83⋊C8)—22no—
27C22⋊(C83⋊C8)—44no—
28C83⋊#95—24no—
29C83⋊#95—22no—
30C83⋊#95D414yes—
31C25×C83D411yescharacteristic
32C22×Q8×C83C2×C411yescharacteristic
33C23×C4×C83—22no—
34C23×C4×C83—22no—
35C23×C4×C83—44no—
36C23×C4×C83C2×C412yes—
37C23×C4×C83D412yes—
38C23×C4×C83D411yescharacteristic
39C23×C4×C83C2311yescharacteristic
40C22⋊Q8×C83—22no—
41C22⋊Q8×C83—22no—
42C22⋊Q8×C83—44no—
43C2×C4⋊C4×C83—22no—
44C2×C4⋊C4×C83—21no—
45C2×C4⋊C4×C83C2×C411yescharacteristic
46C2×C4⋊C4×C83C2×C412yes—
47C2×C22⋊C4×C83—22no—
48C2×C22⋊C4×C83—22no—
49C2×C22⋊C4×C83—21no—
50C2×C22⋊C4×C83—21no—
51#95×C83—22no—
52#95×C83—21no—
53C22×Dic83—24no—
54C22×Dic83—44no—
55C22×Dic83—44no—
56C22×Dic83—42no—
57C22×Dic83—42no—
58C2×C83⋊C8—44no—
59C2×C83⋊C8—42no—
60C22⋊Dic83—48no—
61C24×C83—24no—
62C24×C83—21no—
63C24×C83—21no—
64C24×C83—22no—
65C24×C83—22no—
66C24×C83—42no—
67C24×C83C22⋊C412yes—
68C24×C83C2×D411yescharacteristic
69C2×Q8×C83—22no—
70C2×Q8×C83—42no—
71C22×C4×C83—24no—
72C22×C4×C83—24no—
73C22×C4×C83—21no—
74C22×C4×C83—21no—
75C22×C4×C83—24no—
76C22×C4×C83—24no—
77C22×C4×C83—21no—
78C22×C4×C83—22no—
79C22×C4×C83—21no—
80C22×C4×C83—42no—
81C22×C4×C83—42no—
82C22×C4×C83—42no—
83C22×C4×C83—42no—
84C22×C4×C83—48no—
85C22×C4×C83—48no—
86C22×C4×C83C22⋊C412yes—
87C22×C4×C83C22×C411yescharacteristic
88C22×C4×C83C2×D412yes—
89C22×C4×C83C2×D411yescharacteristic
90C22×C4×C83#4112yes—
91C4⋊C4×C83—22no—
92C4⋊C4×C83—42no—
93C4⋊C4×C83—44no—
94C4⋊C4×C83—42no—
95C22⋊C4×C83—44no—
96C22⋊C4×C83—44no—
97C22⋊C4×C83—41no—
98C22⋊C4×C83—41no—
99C22⋊C4×C83—42no—
100C23×C83—22no—
101C23×C83—22no—
102C23×C83—24no—
103C23×C83—24no—
104C23×C83—22no—
105C23×C83—22no—
106C23×C83—22no—
107C23×C83—22no—
108C23×C83—22no—
109C23×C83—22no—
110C23×C83—48no—
111C23×C83—48no—
112C23×C83—42no—
113C23×C83—42no—
114C23×C83—42no—
115C23×C83—42no—
116C23×C83#9914yes—
117C23×C83C2×C22⋊C411yescharacteristic
118C23×C83#12011yescharacteristic
119C23×C83#12311yescharacteristic
120C23×C83#13512yes—
121C23×C83#13712yes—
122Q8×C83—82no—
123C2×C4×C83—22no—
124C2×C4×C83—22no—
125C2×C4×C83—22no—
126C2×C4×C83—22no—
127C2×C4×C83—22no—
128C2×C4×C83—42no—
129C2×C4×C83—42no—
130C2×C4×C83—48no—
131C2×C4×C83—48no—
132C2×C4×C83—42no—
133C2×C4×C83—42no—
134C2×C4×C83—41no—
135C2×C4×C83—41no—
136C2×C4×C83—42no—
137C2×C4×C83—42no—
138C2×C4×C83—42no—
139C2×C4×C83—42no—
140C2×C4×C83—42no—
141C2×C4×C83C4×D412yes—
142C2×C4×C83#12112yes—
143C2×Dic83—44no—
144C2×Dic83—44no—
145C2×Dic83—44no—
146C2×Dic83—84no—
147C2×Dic83—84no—
148C83⋊C8—82no—
149C22×C83—21no—
150C22×C83—21no—
151C22×C83—21no—
152C22×C83—21no—
153C22×C83—24no—
154C22×C83—24no—
155C22×C83—24no—
156C22×C83—24no—
157C22×C83—24no—
158C22×C83—48no—
159C22×C83—48no—
160C22×C83—42no—
161C22×C83—42no—
162C22×C83—42no—
163C22×C83—42no—
164C22×C83#38611yescharacteristic
165C22×C83C2×#9912yes—
166C22×C83#41611yescharacteristic
167C22×C83#41911yescharacteristic
168C22×C83#42112yes—
169C22×C83#45212yes—
170C22×C83#48512yes—
171C4×C83—22no—
172C4×C83—22no—
173C4×C83—42no—
174C4×C83—81no—
175C4×C83—81no—
176C4×C83—82no—
177Dic83—84no—
178#9154—831nomaximal, Sylow
179C2×C83—22no—
180C2×C83—22no—
181C2×C83—44no—
182C2×C83#151112yes—
183C2×C83#151612yes—
184C2×C83#187311yescharacteristic
185C2×C83#189711yescharacteristic
186C2×C83#189811yescharacteristic
187#2848D8311yescharacteristic
188C2×#381—831no—
189#1779—831no—
190C2×#328—832no—
191#1294—832no—
192C83#915411yescharacteristic, complemented, Sylow
193C24×C4C2×D8311yescharacteristic
194C2×C22⋊Q8Dic8312yescomplemented
195C2×C22⋊Q8C2×D8311yescharacteristic
196C22×C22⋊C4—1661no—
197C2×C22⋊C8—832no—
198C2×C22⋊C8—1662no—
199#385—21no—
200#385—22no—
201#381—1664no—
202#380Dic8312yescomplemented
203#380C2×D8311yescharacteristic
204#379—21no—
205C2×#95—832no—
206C2×#95—1661no—
207#328—1664no—
208C25C22⋊D8311yescharacteristic
209C22×Q8C2×Dic8311yescharacteristic
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—44no—
213C23×C4—1662no—
214C23×C4C2×Dic8312yes—
215C23×C4C22⋊D8312yes—
216C23×C4C22⋊D8311yescharacteristic
217C23×C4C22×D8311yescharacteristic
218C22×C8—1662no—
219C22⋊Q8—22no—
220C22⋊Q8—22no—
221C22⋊Q8—44no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4C2×Dic8311yescharacteristic
225C2×C4⋊C4C2×Dic8312yes—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—3324no—
231C2×C22⋊C4—1664no—
232C22⋊C8—3324no—
233C22⋊C8—1662no—
234C22⋊C8—1662no—
235#95—22no—
236#95—21no—
237#95—834no—
238#95—1664no—
239#95—1662no—
240C24—21no—
241C24—21no—
242C24—24no—
243C24—22no—
244C24—22no—
245C24—42no—
246C24C22⋊Dic8312yes—
247C24C2×C22⋊D8311yescharacteristic
248C2×Q8—22no—
249C2×Q8—42no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—48no—
264C22×C4—48no—
265C22×C4—3324no—
266C22×C4—3324no—
267C22×C4—3322no—
268C22×C4—3322no—
269C22×C4—1664no—
270C22×C4C22⋊Dic8312yes—
271C22×C4C22×Dic8311yescharacteristic
272C22×C4D4×D8312yes—
273C22×C4C2×C22⋊D8311yescharacteristic
274C22×C4#1305058912yes—
275C2×C8—3324no—
276C2×C8—3322no—
277C4⋊C4—22no—
278C4⋊C4—42no—
279C4⋊C4—44no—
280C4⋊C4—42no—
281C22⋊C4—44no—
282C22⋊C4—41no—
283C22⋊C4—41no—
284C22⋊C4—42no—
285C22⋊C4—44no—
286C22⋊C4—3328no—
287C23—22no—
288C23—22no—
289C23—24no—
290C23—24no—
291C23—22no—
292C23—22no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—22no—
297C23—48no—
298C23—48no—
299C23—42no—
300C23—42no—
301C23—42no—
302C23—42no—
303C23#1458170714yes—
304C23C2×C22⋊Dic8311yescharacteristic
305C23#1458176011yescharacteristic
306C23#1458177311yescharacteristic
307C23#1458181212yes—
308C23#1458182312yes—
309Q8—82no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—48no—
318C2×C4—48no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—3324no—
329C2×C4—3324no—
330C2×C4—3324no—
331C2×C4—6644no—
332C2×C4—6644no—
333C2×C4D4×Dic8312yes—
334C2×C4#1458176412yes—
335C8—6642no—
336C22—24no—
337C22—24no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—24no—
343C22—24no—
344C22—24no—
345C22—48no—
346C22—48no—
347C22—42no—
348C22—42no—
349C22—42no—
350C22—42no—
351C22#1912758111yescharacteristic
352C22C2×#1458170712yes—
353C22#1912769711yescharacteristic
354C22#1912771011yescharacteristic
355C22#1912771812yes—
356C22#1912786112yes—
357C22#1912803912yes—
358C4—22no—
359C4—22no—
360C4—42no—
361C4—82no—
362C4—81no—
363C4—81no—
364C4—6644no—
365C2—22no—
366C2—22no—
367C2—44no—
368C2#2490599612yes—
369C2#2490601912yes—
370C2#2490777611yescharacteristic
371C2#2490791211yescharacteristic
372C2#2490791711yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic83
commutatorC22×C4×C83C22×C4
FrattiniC23×C4C22×D83
FittingC83×#2848C2
socleC23×C83C2×C22⋊C4
prime radicalC25×C83D4
cosocleC23×C4×C83C23
hypercenter#2848D83
hypocenterC83#9154
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C24×C4 ◁ #2848

Complements

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

NH ≅ G/Ntype
3. C2×#19127438C2semidirect
4. #24905341C2semidirect
14. C2×C22⋊Q8×C83C4semidirect
18. #380×C83C4semidirect
192. C83#9154semidirect
194. C2×C22⋊Q8Dic83semidirect
202. #380Dic83semidirect

Maximal subgroups

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

subgroupindex
C2×#191275642
#249073732
C2×#191274382
#249053412
C83×#28482
#915483