Finite Group Databasebeta

← #35610505

Subgroups — #35610505

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×#19128248C216yesmaximal, complemented
2#24915860C211yesmaximal, characteristic, complemented
3#24915857C211yesmaximal, characteristic, complemented
4#24915817C216yesmaximal, complemented
5#24915809C211yesmaximal, characteristic, complemented
6#24915807C216yesmaximal, complemented
7#24915805C211yesmaximal, characteristic, complemented
8C4×#14581789C211yesmaximal, characteristic, complemented
9C4×#14581788C216yesmaximal, complemented
10C43×D83C211yesmaximal, characteristic, complemented
11C4×#127×C83C211yesmaximal, characteristic, complemented
12C22×#13050590C2214yes—
13#19128248—248no—
14C2×#14581791C2216yes—
15C2×#14581789C2211yescharacteristic
16C2×#14581788C2216yes—
17C2×#14581788C2216yes—
18C2×#14581778C22112yes—
19C2×#14581767C2216yes—
20C2×#14581765C22112yes—
21C2×#14581763C2216yes—
22C2×C4×D332C2216yes—
23C2×C4×C22⋊D83C22112yes—
24C2×C42×D83C2211yescharacteristic
25C2×C42×D83C2213yes—
26C2×C42×D83C2213yes—
27#19127604C2216yes—
28#19127603C2216yes—
29#19127602C2216yes—
30#19127601C22124yes—
31#19127571C2211yescharacteristic
32#19127570C2216yes—
33C42×Dic83C2213yes—
34C42×Dic83C2211yescharacteristic
35C42×Dic83C2213yes—
36C2×#127×C83C2211yescharacteristic
37C2×C4×D4×C83C2216yes—
38#390×C83C2216yes—
39#384×C83C2211yescharacteristic
40C43×C83C2211yescharacteristic
41C2×#13050590—248no—
42C22×Dic166C2314yes—
43C22×D332C2316yes—
44C22×D332C2314yes—
45C22×C22⋊D83C23112yes—
46C22×C22⋊D83C2314yes—
47C22×C22⋊D83C2314yes—
48C22×C4×D83C2314yes—
49C22×C4×D83C2313yes—
50C22×C4×D83C2313yes—
51C22×C4×D83C2311yescharacteristic
52#14581791—224no—
53#14581789—24no—
54#14581788—224no—
55#14581788—224no—
56#14581778—248no—
57#14581767—224no—
58#14581765—248no—
59#14581763—224no—
60C4×D332—224no—
61C4×C22⋊D83—248no—
62C2×C4⋊Dic83C2316yes—
63C2×Dic83⋊C4C23112yes—
64C2×C22⋊Dic83C23112yes—
65C2×(C2×D83)⋊C4C23112yes—
66C2×(C2×D83)⋊C4C23112yes—
67C2×C4×Dic83C2316yes—
68C2×C4×Dic83C2313yes—
69C2×C4×Dic83C2313yes—
70C2×C4×Dic83C2313yes—
71C2×C4×Dic83C2313yes—
72C2×C4×Dic83C2313yes—
73C42×D83—212no—
74C42×D83—24no—
75C42×D83D4124yescomplemented
76C22×D4×C83C2316yes—
77C23×C4×C83C2314yes—
78#127×C83—24no—
79C4×D4×C83—224no—
80C2×C4⋊C4×C83C2316yes—
81C2×C22⋊C4×C83C23112yes—
82C2×C42×C83C2313yes—
83C2×C42×C83C2313yes—
84C2×C42×C83C2311yescharacteristic
85C23×D83C2414yes—
86C23×D83C2411yescharacteristic
87#13050590—464no—
88C2×Dic166—224no—
89C2×D332—224no—
90C2×D332—212no—
91C2×D332—212no—
92C2×D332—212no—
93C2×C22⋊D83—224no—
94C2×C22⋊D83—224no—
95C2×C22⋊D83—224no—
96C2×C22⋊D83—248no—
97C22×Dic83C2414yes—
98C22×Dic83C2413yes—
99C22×Dic83C2411yescharacteristic
100C22×Dic83C2413yes—
101C2×C4×D83—26no—
102C2×C4×D83—26no—
103C2×C4×D83—212no—
104C2×C4×D83—224no—
105C2×C4×D83C2×D4112yes—
106C2×C4×D83C2×D4112yes—
107C2×C4×D83C2×D4112yes—
108C4⋊Dic83—212no—
109Dic83⋊C4—224no—
110C22⋊Dic83—224no—
111(C2×D83)⋊C4—224no—
112(C2×D83)⋊C4—224no—
113C4×Dic83—26no—
114C4×Dic83—212no—
115C4×Dic83C2×D4112yes—
116C4×Dic83C2×D4112yes—
117C4×Dic83C2×D4112yes—
118C4×Dic83#41112yescomplemented
119C24×C83C2414yes—
120C2×D4×C83—212no—
121C2×D4×C83—212no—
122C2×D4×C83—212no—
123C22×C4×C83—224no—
124C22×C4×C83C2414yes—
125C22×C4×C83C2413yes—
126C22×C4×C83C2413yes—
127C22×C4×C83C2411yescharacteristic
128C4⋊C4×C83—212no—
129C22⋊C4×C83—224no—
130C42×C83—26no—
131C42×C83C2×D4112yes—
132C42×C83#4114yescomplemented
133C23×C83—212no—
134C23×C83—24no—
135C23×C83—212no—
136C23×C83C2511yescharacteristic
137C22×D83—23no—
138C22×D83—21no—
139C22×D83—212no—
140C22×D83—212no—
141C22×D83—24no—
142C22×D83C22×D416yes—
143D4×C83—424no—
144C2×C4×C83—26no—
145C2×C4×C83—212no—
146C2×C4×C83—416no—
147C2×C4×C83C22×D416yes—
148C2×C4×C83C22×D416yes—
149C2×C4×C83C22×D416yes—
150C2×C4×C83C2×#4116yes—
151C2×C4×C83C2×#4116yes—
152C22⋊D83—416no—
153C22⋊D83—416no—
154C22⋊D83—448no—
155C2×Dic83—23no—
156C2×Dic83—23no—
157C2×Dic83—212no—
158C2×Dic83C22×D416yes—
159C2×Dic83C22×D416yes—
160C2×Dic83C22×D416yes—
161C2×Dic83C2×#41112yes—
162D332—416no—
163D332—424no—
164C4×D83—224no—
165C4×D83—224no—
166C4×D83—416no—
167C4×D83#127116yescomplemented
168Dic166—416no—
169C22×C83—44no—
170C22×C83—412no—
171C22×C83—412no—
172C22×C83C23×D413yes—
173C22×C83C22×#4113yes—
174C22×C83C22×#4111yescharacteristic
175C2×D83—26no—
176C2×D83—26no—
177C2×D83—44no—
178C2×D83—412no—
179C2×D83—412no—
180C2×D83C2×#12714yes—
181C4×C83—26no—
182C4×C83—44no—
183C4×C83C2×#12714yes—
184C4×C83#532112yescomplemented
185Dic83—26no—
186Dic83—44no—
187Dic83C2×#12714yes—
188Dic83#532112yescomplemented
189#36644—831nomaximal, Sylow
190C2×C83—84no—
191C2×C83C22×#12711yescharacteristic
192C2×C83C2×#53213yes—
193C2×C83C2×#53213yes—
194D83—24no—
195D83—84no—
196C2×#532—836no—
197#2868—831no—
198#2867—831no—
199#2858—836no—
200#2856—836no—
201#2856—831no—
202#2856—831no—
203C4×#127—831no—
204C4×#127—836no—
205C4×#127D8311yescharacteristic, complemented
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—836no—
217C2×#121—836no—
218C2×#121—8312no—
219C2×C4×D4—8312no—
220C2×C4×D4—836no—
221C2×C4×D4C2×D8316yes—
222C22×C42—833no—
223C22×C42—833no—
224C22×C42—831no—
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—833no—
235C43—831no—
236C43C2×D8311yescharacteristic
237C2×#41—16648no—
238C22×Q8—834no—
239C22×D4—8312no—
240C22×D4—834no—
241C22×D4—834no—
242C22×D4—836no—
243C22×D4—834no—
244C22×D4C22×D8316yes—
245C23×C4—831no—
246C23×C4—833no—
247C23×C4—833no—
248C23×C4—834no—
249C23×C4C22×D8314yes—
250C4⋊Q8—16624no—
251#127—24no—
252#127—16624no—
253#127—16624no—
254#127—1664no—
255#124—16648no—
256#121—16648no—
257#121—16624no—
258#121—16624no—
259C4×D4—224no—
260C4×D4—16648no—
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—833no—
270C2×C42—833no—
271C2×C42—833no—
272C2×C42—833no—
273C2×C42—8324no—
274C2×C42—836no—
275C2×C42—833no—
276C2×C42—16612no—
277C2×C42—1664no—
278C2×C42C22×D8313yes—
279C2×C42C22×D8313yes—
280C2×C42C22×D8311yescharacteristic
281C24—831no—
282C24—834no—
283C24C23×D8314yes—
284#41—33264no—
285C2×Q8—16624no—
286C2×D4—212no—
287C2×D4—212no—
288C2×D4—212no—
289C2×D4—16624no—
290C2×D4—16624no—
291C2×D4—16624no—
292C2×D4—16648no—
293C2×D4—16624no—
294C2×D4—16612no—
295C2×D4—16612no—
296C2×D4—16612no—
297C22×C4—224no—
298C22×C4—833no—
299C22×C4—831no—
300C22×C4—8312no—
301C22×C4—833no—
302C22×C4—8312no—
303C22×C4—8312no—
304C22×C4—834no—
305C22×C4—16612no—
306C22×C4—16624no—
307C22×C4—1666no—
308C22×C4—1666no—
309C22×C4C23×D8314yes—
310C22×C4C23×D8313yes—
311C22×C4C23×D8313yes—
312C22×C4C23×D8311yescharacteristic
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—16612no—
326C42—1666no—
327C42D4×D83112yes—
328C42#1305059014yescomplemented
329C23—212no—
330C23—212no—
331C23—24no—
332C23—836no—
333C23—16612no—
334C23—16612no—
335C23—1664no—
336C23—1663no—
337C23—1661no—
338C23C24×D8311yescharacteristic
339Q8—33216no—
340D4—424no—
341D4—33216no—
342D4—33216no—
343D4—33216no—
344D4—33248no—
345D4—33224no—
346C2×C4—26no—
347C2×C4—212no—
348C2×C4—416no—
349C2×C4—33216no—
350C2×C4—836no—
351C2×C4—836no—
352C2×C4—8312no—
353C2×C4—8316no—
354C2×C4—836no—
355C2×C4—16624no—
356C2×C4—16612no—
357C2×C4—1663no—
358C2×C4—1663no—
359C2×C4—16624no—
360C2×C4C2×D4×D8316yes—
361C2×C4C2×D4×D8316yes—
362C2×C4C2×D4×D8316yes—
363C2×C4#41×D8316yes—
364C2×C4C2×#1305059016yes—
365C22—44no—
366C22—412no—
367C22—412no—
368C22—3324no—
369C22—33212no—
370C22—33212no—
371C22—834no—
372C22—1666no—
373C22—1666no—
374C22C22×D4×D8313yes—
375C22C2×#41×D8313yes—
376C22C22×#1305059011yescharacteristic
377C4—26no—
378C4—44no—
379C4—3324no—
380C4—834no—
381C4—8312no—
382C4—1666no—
383C4#127×D8314yes—
384C4#19128248112yescomplemented
385C2—84no—
386C2—6644no—
387C2—1664no—
388C2C2×#127×D8311yescharacteristic
389C2#532×D8313yes—
390C2C2×#1912824813yes—
subgroupG/H
centerC22×C4C23×D83
commutator, socle, cosocleC23×C83C25
FrattiniC23C24×D83
FittingC4×#127×C83C2
hypercenterC4×#127D83
hypocenterC83#36644
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C22×C4 ◁ C4×#127

Complements

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

NH ≅ G/Ntype
1. C2×#19128248C2semidirect
2. #24915860C2semidirect
3. #24915857C2semidirect
4. #24915817C2semidirect
5. #24915809C2semidirect
6. #24915807C2semidirect
7. #24915805C2semidirect
8. C4×#14581789C2semidirect
9. C4×#14581788C2semidirect
10. C43×D83C2semidirect
11. C4×#127×C83C2semidirect
75. C42×D83D4semidirect
118. C4×Dic83#41semidirect
132. C42×C83#41semidirect
167. C4×D83#127semidirect
184. C4×C83#532semidirect
188. Dic83#532semidirect
205. C4×#127D83semidirect
207. C83#36644semidirect
328. C42#13050590semidirect
384. C4#19128248semidirect

Maximal subgroups

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

subgroupindex
C2×#191282482
#249158602
#249158572
#249158172
#249158092
#249158072
#249158052
C4×#145817892
C4×#145817882
C43×D832
C4×#127×C832
#3664483