Finite Group Databasebeta

← #35297386

Subgroups — #35297386

Order 21248.

456 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24907501C211yesmaximal, characteristic, complemented
2#24907492C211yesmaximal, characteristic, complemented
3#24906006C212yesmaximal
4#24905368C212yesmaximal, complemented
5C83×#2929C211yesmaximal, characteristic
6C2×#14581728—21no—
7C2×#14581726—21no—
8#19127665C2212yes—
9#19127561C2212yes—
10#19127466—21no—
11#19127464—21no—
12#19127438—22no—
13#19127435—22no—
14C2×#131×C83C411yescharacteristic, complemented
15C23×C8×C83C2211yescharacteristic
16#525×C83C2211yescharacteristic
17#435×C83—21no—
18#435×C83—21no—
19C8×D4×C83—21no—
20C8×D4×C83—21no—
21#424×C83C411yescharacteristic, complemented
22#408×C83C411yescharacteristic, complemented
23#407×C83C2211yescharacteristic
24#405×C83C411yescharacteristic, complemented
25C2×#13050574—22no—
26C2×C4⋊Dic83—22no—
27C2×C22⋊Dic83—22no—
28#14581728—42no—
29#14581726—42no—
30C22⋊(C83⋊C8)—22no—
31C22⋊(C83⋊C8)—22no—
32C83⋊#95—22no—
33C2×#41×C83C2×C411yescharacteristic
34C23×C4×C83C2311yescharacteristic
35#131×C83—42no—
36C2×M16×C83—21no—
37C2×M16×C83C2×C411yescharacteristic
38C2×M16×C83C2×C411yescharacteristic
39C22×C8×C83—21no—
40C22×C8×C83—42no—
41C22×C8×C83C2×C411yescharacteristic
42C22×C8×C83C2×C411yescharacteristic
43C22×C8×C83D411yescharacteristic
44C22×C8×C83D411yescharacteristic
45C22×C8×C83D411yescharacteristic
46C22×C8×C83D411yescharacteristic
47#123×C83—21no—
48C22⋊Q8×C83D411yescharacteristic
49C22⋊Q8×C83Q811yescharacteristic
50#121×C83D411yescharacteristic
51#121×C83Q811yescharacteristic
52#120×C83—21no—
53C4×D4×C83—21no—
54C4×D4×C83—21no—
55#117×C83C2×C411yescharacteristic
56C4⋊C8×C83—21no—
57C4⋊C8×C83—21no—
58C22⋊C8×C83—21no—
59C22⋊C8×C83—21no—
60C22⋊C8×C83—21no—
61C22⋊C8×C83—21no—
62#97×C83—21no—
63C4×C8×C83—21no—
64C22×Dic83—42no—
65#13050574—42no—
66#13050574—42no—
67C2×C83⋊C8—42no—
68C4⋊Dic83—42no—
69C4⋊Dic83—42no—
70C22⋊Dic83—82no—
71C24×C83#4111yescharacteristic
72#41×C83—41no—
73#41×C83—41no—
74C2×Q8×C83C4⋊C411yescharacteristic
75C2×D4×C83—21no—
76C2×D4×C83—21no—
77C2×D4×C83—21no—
78C2×D4×C83C4⋊C411yescharacteristic
79C22×C4×C83—21no—
80C22×C4×C83—41no—
81C22×C4×C83—41no—
82C22×C4×C83C22×C411yescharacteristic
83C22×C4×C83C2×D411yescharacteristic
84C22×C4×C83C2×D411yescharacteristic
85C22×C4×C83C2×D411yescharacteristic
86C22×C4×C83C2×Q811yescharacteristic
87C22×C4×C83#4111yescharacteristic
88M16×C83—42no—
89M16×C83—41no—
90M16×C83—41no—
91C2×C8×C83—22no—
92C2×C8×C83—22no—
93C2×C8×C83—22no—
94C2×C8×C83—21no—
95C2×C8×C83—21no—
96C2×C8×C83—41no—
97C2×C8×C83—41no—
98C2×C8×C83—41no—
99C2×C8×C83—42no—
100C2×C8×C83C22⋊C411yescharacteristic
101C2×C8×C83C22⋊C411yescharacteristic
102C2×C8×C83C22⋊C411yescharacteristic
103C2×C8×C83C22⋊C411yescharacteristic
104C4⋊C4×C83—21no—
105C4⋊C4×C83—21no—
106C4⋊C4×C83C4⋊C411yescharacteristic
107C4⋊C4×C83C4⋊C411yescharacteristic
108C22⋊C4×C83—21no—
109C22⋊C4×C83—21no—
110C22⋊C4×C83—21no—
111C22⋊C4×C83—21no—
112C22⋊C4×C83—21no—
113C42×C83—21no—
114C23×C83—21no—
115C23×C83—21no—
116C23×C83—21no—
117C23×C83—41no—
118C23×C83—41no—
119C23×C83#11711yescharacteristic
120C23×C83#12111yescharacteristic
121C23×C83C22⋊Q811yescharacteristic
122Q8×C83—41no—
123D4×C83—41no—
124D4×C83—42no—
125D4×C83—42no—
126D4×C83—41no—
127D4×C83—41no—
128C2×C4×C83—21no—
129C2×C4×C83—21no—
130C2×C4×C83—21no—
131C2×C4×C83—21no—
132C2×C4×C83—21no—
133C2×C4×C83—21no—
134C2×C4×C83—21no—
135C2×C4×C83—21no—
136C2×C4×C83—21no—
137C2×C4×C83—41no—
138C2×C4×C83—41no—
139C2×C4×C83—41no—
140C2×C4×C83—41no—
141C2×C4×C83—41no—
142C2×C4×C83C2×C22⋊C411yescharacteristic
143C2×C4×C83C2×C4⋊C411yescharacteristic
144C2×C4×C83#12111yescharacteristic
145C2×C4×C83C22⋊Q811yescharacteristic
146C2×Dic83—82no—
147C2×Dic83—82no—
148C2×Dic83—82no—
149C8×C83—21no—
150C8×C83—21no—
151C8×C83—21no—
152C8×C83—41no—
153C8×C83—41no—
154C8×C83#9912yes—
155C83⋊C8—82no—
156C22×C83—21no—
157C22×C83—21no—
158C22×C83—21no—
159C22×C83—21no—
160C22×C83—21no—
161C22×C83—21no—
162C22×C83—21no—
163C22×C83—21no—
164C22×C83—42no—
165C22×C83—42no—
166C22×C83—41no—
167C22×C83—41no—
168C22×C83—41no—
169C22×C83#38011yescharacteristic
170C22×C83#47211yescharacteristic
171C22×C83#47311yescharacteristic
172C4×C83—21no—
173C4×C83—21no—
174C4×C83—21no—
175C4×C83—41no—
176C4×C83—41no—
177C4×C83—41no—
178C4×C83C2×#9911yescharacteristic
179C4×C83#41011yescharacteristic
180Dic83—162no—
181#8628—831nomaximal, Sylow
182C2×C83—21no—
183C2×C83—21no—
184C2×C83—41no—
185C2×C83—41no—
186C2×C83—81no—
187C2×C83#179911yescharacteristic
188C2×C83#181411yescharacteristic
189C2×C83#181611yescharacteristic
190#2929D8311yescharacteristic
191#1810—831no—
192#1808—831no—
193#1513—832no—
194#1305—832no—
195C83#862811yescharacteristic, complemented, Sylow
196C2×#131Dic8311yescharacteristic, complemented
197C23×C8C2×D8311yescharacteristic
198#525C2×D8311yescharacteristic
199#435—21no—
200#435—21no—
201C8×D4—21no—
202C8×D4—21no—
203C2×#108—1661no—
204C2×#107—1661no—
205#424Dic8311yescharacteristic, complemented
206#408Dic8311yescharacteristic, complemented
207#407—832no—
208#407C2×D8311yescharacteristic
209#405Dic8311yescharacteristic, complemented
210#380—832no—
211#341—1661no—
212#340—1661no—
213#328—1662no—
214#327—1662no—
215C2×#41C2×Dic8311yescharacteristic
216C23×C4C22×D8311yescharacteristic
217#131—42no—
218C2×M16—21no—
219C2×M16—1662no—
220C2×M16C2×Dic8311yescharacteristic
221C2×M16C2×Dic8311yescharacteristic
222C22×C8—21no—
223C22×C8—42no—
224C22×C8C2×Dic8311yescharacteristic
225C22×C8C2×Dic8311yescharacteristic
226C22×C8C22⋊D8311yescharacteristic
227C22×C8C22⋊D8311yescharacteristic
228C22×C8C22⋊D8311yescharacteristic
229C22×C8C22⋊D8311yescharacteristic
230#123—21no—
231C22⋊Q8Dic16611yescharacteristic
232C22⋊Q8D33211yescharacteristic
233#121Dic16611yescharacteristic
234#121D33211yescharacteristic
235#120—21no—
236C4×D4—21no—
237C4×D4—21no—
238#117C2×Dic8311yescharacteristic
239C2×C4⋊C4—1662no—
240C2×C22⋊C4—1662no—
241#108—3322no—
242#107—3322no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C22⋊C8—21no—
246C22⋊C8—21no—
247C22⋊C8—21no—
248C22⋊C8—21no—
249C22⋊C8—1662no—
250C22⋊C8—1662no—
251#97—21no—
252C4×C8—21no—
253#95—1662no—
254C24#1305059011yescharacteristic
255#41—41no—
256#41—41no—
257C2×Q8C4⋊Dic8311yescharacteristic
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4C4⋊Dic8311yescharacteristic
262C22×C4—21no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—3322no—
266C22×C4C22×Dic8311yescharacteristic
267C22×C4C2×C22⋊D8311yescharacteristic
268C22×C4C2×C22⋊D8311yescharacteristic
269C22×C4C2×D33211yescharacteristic
270C22×C4C2×Dic16611yescharacteristic
271C22×C4#1305059011yescharacteristic
272M16—42no—
273M16—41no—
274M16—41no—
275M16—3322no—
276M16—3322no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—42no—
286C2×C8—3322no—
287C2×C8C22⋊Dic8311yescharacteristic
288C2×C8C22⋊Dic8311yescharacteristic
289C2×C8C22⋊Dic8311yescharacteristic
290C2×C8C22⋊Dic8311yescharacteristic
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—3322no—
294C4⋊C4—3322no—
295C4⋊C4C4⋊Dic8311yescharacteristic
296C4⋊C4C4⋊Dic8311yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—6642no—
303C42—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—41no—
308C23—41no—
309C23#1458175111yescharacteristic
310C23#1458176611yescharacteristic
311C23#1458177111yescharacteristic
312Q8—41no—
313D4—41no—
314D4—42no—
315D4—42no—
316D4—41no—
317D4—41no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—6642no—
333C2×C4—6642no—
334C2×C4—6642no—
335C2×C4C2×C22⋊Dic8311yescharacteristic
336C2×C4C2×C4⋊Dic8311yescharacteristic
337C2×C4#1458176611yescharacteristic
338C2×C4#1458177111yescharacteristic
339C8—21no—
340C8—21no—
341C8—21no—
342C8—41no—
343C8—41no—
344C8—6642no—
345C8#1458170712yes—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—42no—
355C22—42no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22#1912756111yescharacteristic
360C22#1912797111yescharacteristic
361C22#1912797611yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—13282no—
369C4C2×#1458170711yescharacteristic
370C4#1912767511yescharacteristic
371C2—21no—
372C2—21no—
373C2—41no—
374C2—41no—
375C2—81no—
376C2#2490745111yescharacteristic
377C2#2490751711yescharacteristic
378C2#2490752711yescharacteristic
subgroupG/H
centerC22#19127561
commutatorC22×C4×C83C22×C4
FrattiniC23×C4C22×D83
FittingC83×#2929C2
socleC22×C83#380
prime radical#525×C83C22
cosocleC23×C4×C83C23
hypercenter#2929D83
hypocenterC83#8628
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2929

Complements

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

NH ≅ G/Ntype
1. #24907501C2semidirect
2. #24907492C2semidirect
4. #24905368C2semidirect
14. C2×#131×C83C4semidirect
21. #424×C83C4semidirect
22. #408×C83C4semidirect
24. #405×C83C4semidirect
195. C83#8628semidirect
196. C2×#131Dic83semidirect
205. #424Dic83semidirect
206. #408Dic83semidirect
209. #405Dic83semidirect

Maximal subgroups

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

subgroupindex
#249075012
#249074922
#249060062
#249053682
C83×#29292
#862883