Finite Group Databasebeta

← #35297965

Subgroups — #35297965

Order 21248.

472 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 115 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#24907451C211yesmaximal, characteristic
2#24905781C211yesmaximal, characteristic
3#24905342C212yesmaximal, complemented
4#24905330C212yesmaximal, complemented
5C83×#2353C211yesmaximal, characteristic
6C2×#14581707—21no—
7C2×C22⋊(C83⋊C8)C2212yes—
8#19127561C2212yes—
9#19127468—21no—
10#19127427—21no—
11#19127425—21no—
12C2×C4×D4×C83C2211yescharacteristic
13#388×C83—21no—
14#387×C83—21no—
15#386×C83—21no—
16#383×C83C411yescharacteristic, complemented
17#381×C83C411yescharacteristic, complemented
18#380×C83C2211yescharacteristic
19C4×C22⋊C4×C83—21no—
20C4×C22⋊C4×C83C411yescharacteristic, complemented
21#376×C83C411yescharacteristic, complemented
22C2×#95×C83C2211yescharacteristic
23C22×C83⋊C8—22no—
24C2×C4⋊Dic83—22no—
25C2×C22⋊Dic83—22no—
26#14581707—42no—
27C22⋊(C83⋊C8)—22no—
28C22⋊(C83⋊C8)—22no—
29C83⋊#95—22no—
30C22×D4×C83D411yescharacteristic
31C23×C4×C83—21no—
32C23×C4×C83C2×C411yescharacteristic
33C23×C4×C83C2311yescharacteristic
34C4×D4×C83—22no—
35C4×D4×C83—22no—
36C2×C4⋊C4×C83—21no—
37C2×C4⋊C4×C83D411yescharacteristic
38C2×C22⋊C4×C83—21no—
39C2×C22⋊C4×C83—21no—
40C2×C22⋊C4×C83—21no—
41C2×C22⋊C4×C83D411yescharacteristic
42C2×C22⋊C4×C83D411yescharacteristic
43C2×C42×C83—21no—
44C2×C42×C83—21no—
45C2×C42×C83C2×C411yescharacteristic
46#95×C83—21no—
47#95×C83—21no—
48#95×C83—21no—
49#95×C83—21no—
50#95×C83—21no—
51#95×C83C2×C411yescharacteristic
52#95×C83C2×C411yescharacteristic
53#95×C83C2×C411yescharacteristic
54#95×C83C2×C411yescharacteristic
55C22×Dic83—42no—
56C2×C83⋊C8—42no—
57C2×C83⋊C8—82no—
58C4⋊Dic83—82no—
59C22⋊Dic83—42no—
60C22⋊Dic83—42no—
61C24×C83C22⋊C411yescharacteristic
62C24×C83C2×D411yescharacteristic
63C2×D4×C83—22no—
64C2×D4×C83—22no—
65C2×D4×C83—21no—
66C2×D4×C83—21no—
67C22×C4×C83—21no—
68C22×C4×C83—21no—
69C22×C4×C83—21no—
70C22×C4×C83—21no—
71C22×C4×C83—21no—
72C22×C4×C83—21no—
73C22×C4×C83—42no—
74C22×C4×C83—42no—
75C22×C4×C83—81no—
76C22×C4×C83—81no—
77C22×C4×C83—81no—
78C22×C4×C83C22⋊C411yescharacteristic
79C22×C4×C83C22⋊C411yescharacteristic
80C22×C4×C83C22⋊C411yescharacteristic
81C22×C4×C83C22×C411yescharacteristic
82C22×C4×C83C2×D411yescharacteristic
83C22×C4×C83#4112yes—
84C22×C4×C83#4112yes—
85C4⋊C4×C83—22no—
86C4⋊C4×C83—81no—
87C22⋊C4×C83—22no—
88C22⋊C4×C83—22no—
89C22⋊C4×C83—42no—
90C22⋊C4×C83—41no—
91C22⋊C4×C83—41no—
92C22⋊C4×C83—81no—
93C42×C83—21no—
94C42×C83—21no—
95C42×C83—42no—
96C42×C83—81no—
97C23×C83—21no—
98C23×C83—21no—
99C23×C83—21no—
100C23×C83—41no—
101C23×C83—41no—
102C23×C83—41no—
103C23×C83—41no—
104C23×C83C2×C22⋊C411yescharacteristic
105C23×C83#12312yes—
106C23×C83#13512yes—
107C23×C83AGL(1,ℤ/8)11yescharacteristic
108C23×C83#13711yescharacteristic
109D4×C83—42no—
110D4×C83—42no—
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—41no—
118C2×C4×C83—41no—
119C2×C4×C83—42no—
120C2×C4×C83—42no—
121C2×C4×C83—41no—
122C2×C4×C83—41no—
123C2×C4×C83—41no—
124C2×C4×C83—41no—
125C2×C4×C83—41no—
126C2×C4×C83—41no—
127C2×C4×C83—41no—
128C2×C4×C83—41no—
129C2×C4×C83—41no—
130C2×C4×C83—41no—
131C2×C4×C83—41no—
132C2×C4×C83—41no—
133C2×C4×C83—41no—
134C2×C4×C83—81no—
135C2×C4×C83—81no—
136C2×C4×C83—84no—
137C2×C4×C83#11711yescharacteristic
138C2×C4×C83#11711yescharacteristic
139C2×C4×C83#12312yes—
140C2×Dic83—82no—
141C2×Dic83—82no—
142C2×Dic83—82no—
143C83⋊C8—82no—
144C22×C83—22no—
145C22×C83—21no—
146C22×C83—22no—
147C22×C83—21no—
148C22×C83—42no—
149C22×C83—41no—
150C22×C83—41no—
151C22×C83—41no—
152C22×C83—41no—
153C22×C83—41no—
154C22×C83—81no—
155C22×C83—81no—
156C22×C83#38111yescharacteristic
157C22×C83#41611yescharacteristic
158C22×C83#41711yescharacteristic
159C22×C83#48212yes—
160C22×C83#48512yes—
161C4×C83—22no—
162C4×C83—21no—
163C4×C83—21no—
164C4×C83—42no—
165C4×C83—41no—
166C4×C83—41no—
167C4×C83—41no—
168C4×C83—81no—
169C4×C83—81no—
170C4×C83—81no—
171Dic83—162no—
172#8763—831nomaximal, Sylow
173C2×C83—22no—
174C2×C83—21no—
175C2×C83—21no—
176C2×C83—22no—
177C2×C83—81no—
178C2×C83#182511yescharacteristic
179C2×C83#182711yescharacteristic
180C2×C83#182811yescharacteristic
181#2353D8311yescharacteristic
182#1799—831no—
183#1459—831no—
184#1294—832no—
185#1289—832no—
186C83#876311yescharacteristic, complemented, Sylow
187C2×C4×D4C2×D8311yescharacteristic
188C2×#99—1661no—
189C2×C22⋊C8—832no—
190#388—21no—
191#387—21no—
192#386—21no—
193#383Dic8311yescharacteristic, complemented
194#381Dic8311yescharacteristic, complemented
195#380—832no—
196#380C2×D8311yescharacteristic
197C4×C22⋊C4—21no—
198C4×C22⋊C4Dic8311yescharacteristic, complemented
199#376Dic8311yescharacteristic, complemented
200C2×#95C2×D8311yescharacteristic
201#342—1661no—
202#324—1661no—
203#323—1661no—
204C22×D4C22⋊D8311yescharacteristic
205C23×C4—21no—
206C23×C4C2×Dic8311yescharacteristic
207C23×C4C22×D8311yescharacteristic
208C22×C8—1662no—
209C4×D4—22no—
210C4×D4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—1662no—
213C2×C4⋊C4C22⋊D8311yescharacteristic
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—1662no—
218C2×C22⋊C4C22⋊D8311yescharacteristic
219C2×C22⋊C4C22⋊D8311yescharacteristic
220C2×C42—21no—
221C2×C42—21no—
222C2×C42C2×Dic8311yescharacteristic
223#99—3322no—
224C22⋊C8—1662no—
225C22⋊C8—1662no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—1662no—
232#95C2×Dic8311yescharacteristic
233#95C2×Dic8311yescharacteristic
234#95C2×Dic8311yescharacteristic
235#95C2×Dic8311yescharacteristic
236C24C22⋊Dic8311yescharacteristic
237C24C2×C22⋊D8311yescharacteristic
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—21no—
241C2×D4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—81no—
251C22×C4—81no—
252C22×C4—81no—
253C22×C4—3322no—
254C22×C4C22⋊Dic8311yescharacteristic
255C22×C4C22⋊Dic8311yescharacteristic
256C22×C4C22⋊Dic8311yescharacteristic
257C22×C4C22×Dic8311yescharacteristic
258C22×C4C2×C22⋊D8311yescharacteristic
259C22×C4#1305059012yes—
260C22×C4#1305059012yes—
261C2×C8—3322no—
262C2×C8—6642no—
263C4⋊C4—22no—
264C4⋊C4—81no—
265C4⋊C4—6642no—
266C22⋊C4—22no—
267C22⋊C4—22no—
268C22⋊C4—42no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C22⋊C4—81no—
272C22⋊C4—3322no—
273C22⋊C4—3322no—
274C42—21no—
275C42—21no—
276C42—42no—
277C42—81no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23C2×C22⋊Dic8311yescharacteristic
286C23#1458177512yes—
287C23#1458181212yes—
288C23#1458181711yescharacteristic
289C23#1458182211yescharacteristic
290D4—42no—
291D4—42no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—84no—
318C2×C4—6642no—
319C2×C4—6642no—
320C2×C4—6642no—
321C2×C4#1458175111yescharacteristic
322C2×C4#1458175111yescharacteristic
323C2×C4#1458177512yes—
324C8—6642no—
325C22—21no—
326C22—22no—
327C22—21no—
328C22—22no—
329C22—42no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—81no—
336C22—81no—
337C22#1912756411yescharacteristic
338C22#1912769711yescharacteristic
339C22#1912770211yescharacteristic
340C22#1912802512yes—
341C22#1912804212yes—
342C4—22no—
343C4—21no—
344C4—21no—
345C4—42no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—81no—
350C4—81no—
351C4—81no—
352C4—13282no—
353C2—21no—
354C2—21no—
355C2—22no—
356C2—22no—
357C2—81no—
358C2#2490755811yescharacteristic
359C2#2490756711yescharacteristic
360C2#2490757111yescharacteristic
subgroupG/H
centerC22#19127564
commutatorC22×C4×C83C22×C4
FrattiniC23×C4C22×D83
FittingC83×#2353C2
socleC22×C83#381
prime radicalC22×D4×C83D4
cosocleC23×C4×C83C23
hypercenter#2353D83
hypocenterC83#8763
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2353

Complements

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

NH ≅ G/Ntype
3. #24905342C2semidirect
4. #24905330C2semidirect
16. #383×C83C4semidirect
17. #381×C83C4semidirect
20. C4×C22⋊C4×C83C4semidirect
21. #376×C83C4semidirect
186. C83#8763semidirect
193. #383Dic83semidirect
194. #381Dic83semidirect
198. C4×C22⋊C4Dic83semidirect
199. #376Dic83semidirect

Maximal subgroups

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

subgroupindex
#249074512
#249057812
#249053422
#249053302
C83×#23532
#876383