Finite Group Databasebeta

← #35610503

Subgroups — #35610503

Order 21248.

4086 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128246C216yesmaximal, complemented
2#24915860C211yesmaximal, characteristic, complemented
3#24915857C211yesmaximal, characteristic, complemented
4#24915817C216yesmaximal, complemented
5#24915808C211yesmaximal, characteristic, complemented
6#24915806C216yesmaximal, complemented
7#127×Dic83C211yesmaximal, characteristic, complemented
8#24910027C216yesmaximal, complemented
9#24910025C211yesmaximal, characteristic, complemented
10C2×C42×Dic83C211yesmaximal, characteristic, complemented
11C83×#2856C211yesmaximal, characteristic, complemented
12C22×#13050589C2214yes—
13#19128246—248no—
14C2×#14581791C2216yes—
15C2×#14581789C2211yescharacteristic
16C2×#14581788C2216yes—
17C2×#14581788C2216yes—
18C2×#14581778C22112yes—
19C2×#14581766C2216yes—
20C2×#14581764C22112yes—
21C2×D4×Dic83C2216yes—
22C2×#14581754C2216yes—
23C2×(C22⋊D83)⋊C4C22112yes—
24C22×C4×Dic83C2213yes—
25C22×C4×Dic83C2213yes—
26#19127604C2216yes—
27#19127603C2216yes—
28#19127602C2216yes—
29#19127601C22124yes—
30#19127571C2211yescharacteristic
31#19127570C2216yes—
32C42×Dic83C2213yes—
33C42×Dic83C2211yescharacteristic
34C42×Dic83C2211yescharacteristic
35C42×Dic83C2213yes—
36C2×#127×C83C2211yescharacteristic
37C2×#121×C83C2216yes—
38C22×C42×C83C2211yescharacteristic
39#390×C83C2216yes—
40#384×C83C2211yescharacteristic
41C2×#13050589—248no—
42C22×Dic166C2314yes—
43C22×D332C2316yes—
44C22×C22⋊D83C23112yes—
45C22×C22⋊D83C2314yes—
46C22×C22⋊D83C2314yes—
47C23×Dic83C2314yes—
48C23×Dic83C2313yes—
49C23×Dic83C2311yescharacteristic
50C22×C4×D83C2314yes—
51#14581791—224no—
52#14581789—24no—
53#14581788—224no—
54#14581788—224no—
55#14581778—248no—
56#14581766—224no—
57#14581764—248no—
58D4×Dic83—224no—
59#14581754—224no—
60(C22⋊D83)⋊C4—248no—
61C2×C4⋊Dic83C2316yes—
62C2×Dic83⋊C4C23112yes—
63C2×C22⋊Dic83C23112yes—
64C2×(C2×D83)⋊C4C23112yes—
65C2×(C2×D83)⋊C4C23112yes—
66C2×C4×Dic83—212no—
67C2×C4×Dic83D4124yescomplemented
68C2×C4×Dic83C2316yes—
69C2×C4×Dic83C2313yes—
70C2×C4×Dic83C2313yes—
71C2×C4×Dic83C2313yes—
72C2×C4×Dic83C2313yes—
73C2×C4×Dic83C2313yes—
74C2×C4×Dic83C2313yes—
75C22×D4×C83C2316yes—
76C22×D4×C83C2314yes—
77C23×C4×C83C2313yes—
78#127×C83—24no—
79#121×C83—224no—
80C2×C4⋊C4×C83C2316yes—
81C2×C22⋊C4×C83C23112yes—
82C2×C42×C83—24no—
83C2×C42×C83C2313yes—
84C2×C42×C83C2311yescharacteristic
85C23×D83C2414yes—
86#13050589—464no—
87C2×Dic166—224no—
88C2×D332—212no—
89C2×D332—212no—
90C2×D332—212no—
91C2×C22⋊D83—224no—
92C2×C22⋊D83—224no—
93C2×C22⋊D83—224no—
94C2×C22⋊D83—248no—
95C22×Dic83—212no—
96C22×Dic83—224no—
97C22×Dic83C2×D4112yes—
98C22×Dic83C2×D4112yes—
99C22×Dic83C2414yes—
100C22×Dic83C2413yes—
101C22×Dic83C2411yescharacteristic
102C22×Dic83C2411yescharacteristic
103C22×Dic83C2413yes—
104C2×C4×D83—224no—
105C4⋊Dic83—212no—
106Dic83⋊C4—224no—
107C22⋊Dic83—224no—
108(C2×D83)⋊C4—224no—
109(C2×D83)⋊C4—224no—
110C4×Dic83—26no—
111C4×Dic83—212no—
112C4×Dic83C2×D4112yes—
113C4×Dic83C2×D4112yes—
114C4×Dic83C2×D4112yes—
115C4×Dic83C2×D4112yes—
116C4×Dic83#41112yescomplemented
117C24×C83C2414yes—
118C24×C83C2411yescharacteristic
119C2×D4×C83—224no—
120C2×D4×C83—212no—
121C2×D4×C83—212no—
122C2×D4×C83—212no—
123C22×C4×C83—26no—
124C22×C4×C83—26no—
125C22×C4×C83C2×D4112yes—
126C22×C4×C83C2414yes—
127C22×C4×C83C2413yes—
128C22×C4×C83C2413yes—
129C4⋊C4×C83—212no—
130C22⋊C4×C83—224no—
131C42×C83—26no—
132C42×C83#4114yescomplemented
133C23×C83—21no—
134C23×C83—23no—
135C23×C83—212no—
136C23×C83—24no—
137C23×C83—212no—
138C23×C83C22×D416yes—
139C23×C83C2511yescharacteristic
140C22×D83—212no—
141C22×D83—212no—
142C22×D83—24no—
143D4×C83—416no—
144D4×C83—424no—
145C2×C4×C83—23no—
146C2×C4×C83—23no—
147C2×C4×C83—224no—
148C2×C4×C83—212no—
149C2×C4×C83C22×D416yes—
150C2×C4×C83C22×D416yes—
151C2×C4×C83C2×#4116yes—
152C2×C4×C83C2×#4116yes—
153C22⋊D83—416no—
154C22⋊D83—416no—
155C22⋊D83—448no—
156C2×Dic83—26no—
157C2×Dic83—224no—
158C2×Dic83—212no—
159C2×Dic83—416no—
160C2×Dic83#127116yescomplemented
161C2×Dic83C22×D416yes—
162C2×Dic83C22×D416yes—
163C2×Dic83C22×D416yes—
164C2×Dic83C22×D416yes—
165C2×Dic83C2×#41112yes—
166D332—424no—
167C4×D83—416no—
168Dic166—416no—
169C22×C83—26no—
170C22×C83—26no—
171C22×C83—44no—
172C22×C83—412no—
173C22×C83—412no—
174C22×C83C2×#12714yes—
175C22×C83C23×D413yes—
176C22×C83C22×#4113yes—
177C22×C83C22×#4111yescharacteristic
178C2×D83—44no—
179C2×D83—412no—
180C2×D83—412no—
181C4×C83—26no—
182C4×C83—44no—
183C4×C83#532112yescomplemented
184Dic83—26no—
185Dic83—44no—
186Dic83C2×#12714yes—
187Dic83C2×#12714yes—
188Dic83#532112yescomplemented
189#36644—831nomaximal, Sylow
190C2×C83—24no—
191C2×C83—84no—
192C2×C83C22×#12711yescharacteristic
193C2×C83C2×#53213yes—
194C2×C83C2×#53213yes—
195D83—84no—
196C2×#532—836no—
197#2868—831no—
198#2867—831no—
199#2858—836no—
200#2856—831no—
201#2856—836no—
202#2856D8311yescharacteristic, complemented
203C4×#127—836no—
204C4×#127—831no—
205C4×#127—831no—
206C2×C43—831no—
207C83#3664411yescharacteristic, complemented, Sylow
208C22×#41—834no—
209#532—16648no—
210C2×C4⋊Q8—836no—
211C2×#127—836no—
212C2×#127—836no—
213C2×#127—831no—
214C2×#127C2×D8311yescharacteristic
215C2×#124—8312no—
216C2×#121—8312no—
217C2×#121—836no—
218C2×#121C2×D8316yes—
219C2×C4×D4—836no—
220C2×C4×D4—836no—
221C2×C4×D4—8312no—
222C22×C42—833no—
223C22×C42—833no—
224C22×C42C2×D8311yescharacteristic
225#390—836no—
226#390—836no—
227#390—8324no—
228#390—836no—
229#390C2×D8316yes—
230#384—836no—
231#384—831no—
232#384C2×D8311yescharacteristic
233C43—833no—
234C43—831no—
235C43—833no—
236C43—831no—
237C2×#41—16648no—
238C22×Q8—834no—
239C22×D4—8312no—
240C22×D4—834no—
241C22×D4—834no—
242C22×D4—836no—
243C22×D4C22×D8316yes—
244C22×D4C22×D8314yes—
245C23×C4—831no—
246C23×C4—833no—
247C23×C4—834no—
248C23×C4—834no—
249C23×C4C22×D8313yes—
250C4⋊Q8—16624no—
251#127—24no—
252#127—16624no—
253#127—16624no—
254#127—1664no—
255#124—16648no—
256#121—224no—
257#121—16648no—
258#121—16624no—
259C4×D4—16648no—
260C4×D4—16624no—
261C4×D4—16624no—
262C2×C4⋊C4—8312no—
263C2×C4⋊C4—836no—
264C2×C4⋊C4C22×D8316yes—
265C2×C22⋊C4—8312no—
266C2×C22⋊C4—8312no—
267C2×C22⋊C4—8312no—
268C2×C22⋊C4C22×D83112yes—
269C2×C42—24no—
270C2×C42—8324no—
271C2×C42—833no—
272C2×C42—833no—
273C2×C42—833no—
274C2×C42—833no—
275C2×C42—833no—
276C2×C42—836no—
277C2×C42—833no—
278C2×C42—16612no—
279C2×C42C22×D8313yes—
280C2×C42C22×D8311yescharacteristic
281C24—834no—
282C24C23×D8314yes—
283C24C23×D8311yescharacteristic
284#41—33264no—
285C2×Q8—16624no—
286C2×D4—224no—
287C2×D4—212no—
288C2×D4—212no—
289C2×D4—212no—
290C2×D4—16624no—
291C2×D4—16624no—
292C2×D4—16648no—
293C2×D4—16624no—
294C2×D4—16612no—
295C2×D4—16612no—
296C2×D4—16612no—
297C22×C4—26no—
298C22×C4—26no—
299C22×C4—8312no—
300C22×C4—833no—
301C22×C4—831no—
302C22×C4—831no—
303C22×C4—8312no—
304C22×C4—833no—
305C22×C4—834no—
306C22×C4—16624no—
307C22×C4—16624no—
308C22×C4—16612no—
309C22×C4D4×D83112yes—
310C22×C4C23×D8314yes—
311C22×C4C23×D8313yes—
312C22×C4C23×D8313yes—
313C4⋊C4—212no—
314C4⋊C4—16624no—
315C4⋊C4—16612no—
316C22⋊C4—224no—
317C22⋊C4—16624no—
318C22⋊C4—16624no—
319C22⋊C4—16624no—
320C42—26no—
321C42—8312no—
322C42—8312no—
323C42—8312no—
324C42—8312no—
325C42—8312no—
326C42—16612no—
327C42—1666no—
328C42#1305058914yescomplemented
329C23—23no—
330C23—21no—
331C23—212no—
332C23—212no—
333C23—24no—
334C23—16612no—
335C23—16612no—
336C23—1664no—
337C23C2×D4×D8316yes—
338C23C24×D8311yescharacteristic
339Q8—33216no—
340D4—416no—
341D4—424no—
342D4—33216no—
343D4—33216no—
344D4—33248no—
345D4—33224no—
346C2×C4—23no—
347C2×C4—23no—
348C2×C4—224no—
349C2×C4—212no—
350C2×C4—33216no—
351C2×C4—33216no—
352C2×C4—8316no—
353C2×C4—836no—
354C2×C4—836no—
355C2×C4—836no—
356C2×C4—8312no—
357C2×C4—836no—
358C2×C4—16612no—
359C2×C4—16624no—
360C2×C4—1666no—
361C2×C4C2×D4×D8316yes—
362C2×C4C2×D4×D8316yes—
363C2×C4#41×D8316yes—
364C2×C4C2×#1305058916yes—
365C22—26no—
366C22—26no—
367C22—44no—
368C22—412no—
369C22—412no—
370C22—3324no—
371C22—33212no—
372C22—33212no—
373C22#127×D8314yes—
374C22C22×D4×D8313yes—
375C22C2×#41×D8313yes—
376C22C22×#1305058911yescharacteristic
377C4—26no—
378C4—44no—
379C4—3324no—
380C4—834no—
381C4—834no—
382C4—8312no—
383C4—1666no—
384C4#19128246112yescomplemented
385C2—24no—
386C2—84no—
387C2—6644no—
388C2C2×#127×D8311yescharacteristic
389C2#532×D8313yes—
390C2C2×#1912824613yes—
subgroupG/H
center, FrattiniC23C24×D83
commutator, socle, cosocleC23×C83C25
FittingC83×#2856C2
hypercenter#2856D83
hypocenterC83#36644
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #2856

Complements

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

NH ≅ G/Ntype
1. C2×#19128246C2semidirect
2. #24915860C2semidirect
3. #24915857C2semidirect
4. #24915817C2semidirect
5. #24915808C2semidirect
6. #24915806C2semidirect
7. #127×Dic83C2semidirect
8. #24910027C2semidirect
9. #24910025C2semidirect
10. C2×C42×Dic83C2semidirect
11. C83×#2856C2semidirect
67. C2×C4×Dic83D4semidirect
116. C4×Dic83#41semidirect
132. C42×C83#41semidirect
160. C2×Dic83#127semidirect
183. C4×C83#532semidirect
188. Dic83#532semidirect
202. #2856D83semidirect
207. C83#36644semidirect
328. C42#13050589semidirect
384. C4#19128246semidirect

Maximal subgroups

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

subgroupindex
C2×#191282462
#249158602
#249158572
#249158172
#249158082
#249158062
#127×Dic832
#249100272
#249100252
C2×C42×Dic832
C83×#28562
#3664483