Finite Group Databasebeta

← #35300010

Subgroups — #35300010

Order 21248.

626 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14581703C211yesmaximal, characteristic
2#24907225C211yesmaximal, characteristic
3C2×#19127443C212yesmaximal, complemented
4#24905344C212yesmaximal, complemented
5C83×#2851C211yesmaximal, characteristic
6C23×C83⋊C8—21no—
7C2×C22⋊(C83⋊C8)—21no—
8C2×#14581703—24no—
9C2×#14581703—21no—
10C2×#14581703C2212yes—
11C2×#14581703C2212yes—
12#19127456—21no—
13#19127443—24no—
14C2×#125×C83C2211yescharacteristic
15C22×C42×C83C2211yescharacteristic
16#389×C83C411yescharacteristic
17#389×C83C411yescharacteristic
18#387×C83—21no—
19#387×C83—21no—
20#385×C83—21no—
21#385×C83—21no—
22#382×C83C2211yescharacteristic
23#380×C83C411yescharacteristic
24#380×C83C411yescharacteristic
25C22×C83⋊C8—24no—
26C22×C83⋊C8—24no—
27C22×C83⋊C8—24no—
28C22×C83⋊C8—22no—
29C22×C83⋊C8—22no—
30C22⋊(C83⋊C8)—24no—
31#14581703—24no—
32#14581703—22no—
33#14581703—22no—
34#14581703—22no—
35#14581703D414yes—
36C23×C4×C83C2×C411yescharacteristic
37C23×C4×C83D411yescharacteristic
38C23×C4×C83D411yescharacteristic
39#125×C83—22no—
40#125×C83—22no—
41C2×C4⋊C4×C83—21no—
42C2×C4⋊C4×C83—21no—
43C2×C4⋊C4×C83—21no—
44C2×C4⋊C4×C83—21no—
45C2×C4⋊C4×C83C2×C411yescharacteristic
46C2×C4⋊C4×C83C2×C411yescharacteristic
47C2×C4⋊C4×C83C2×C411yescharacteristic
48C2×C4⋊C4×C83C2×C411yescharacteristic
49C2×C22⋊C4×C83—21no—
50C2×C22⋊C4×C83—21no—
51C2×C42×C83—22no—
52C2×C42×C83—22no—
53C2×C42×C83C2×C411yescharacteristic
54C2×C42×C83D411yescharacteristic
55C2×C42×C83D411yescharacteristic
56C2×C42×C83C2311yescharacteristic
57#95×C83—21no—
58#95×C83—21no—
59#95×C83—21no—
60#95×C83—21no—
61#95×C83—21no—
62#95×C83—21no—
63C2×C83⋊C8—216no—
64C2×C83⋊C8—24no—
65C2×C83⋊C8—24no—
66C2×C83⋊C8—24no—
67C2×C83⋊C8—24no—
68C2×C83⋊C8—42no—
69C2×C83⋊C8—42no—
70C24×C83C22⋊C411yescharacteristic
71C22×C4×C83—22no—
72C22×C4×C83—22no—
73C22×C4×C83—21no—
74C22×C4×C83—21no—
75C22×C4×C83—22no—
76C22×C4×C83—22no—
77C22×C4×C83—22no—
78C22×C4×C83—22no—
79C22×C4×C83—21no—
80C22×C4×C83—21no—
81C22×C4×C83—21no—
82C22×C4×C83—21no—
83C22×C4×C83—41no—
84C22×C4×C83—41no—
85C22×C4×C83C22⋊C411yescharacteristic
86C22×C4×C83C22⋊C411yescharacteristic
87C22×C4×C83C22⋊C411yescharacteristic
88C22×C4×C83C22×C411yescharacteristic
89C22×C4×C83C2×D411yescharacteristic
90C22×C4×C83C2×D411yescharacteristic
91C4⋊C4×C83—22no—
92C4⋊C4×C83—22no—
93C4⋊C4×C83—41no—
94C4⋊C4×C83—41no—
95C4⋊C4×C83—42no—
96C4⋊C4×C83—42no—
97C4⋊C4×C83—41no—
98C4⋊C4×C83—42no—
99C4⋊C4×C83—41no—
100C22⋊C4×C83—42no—
101C22⋊C4×C83—42no—
102C42×C83—22no—
103C42×C83—21no—
104C42×C83—22no—
105C42×C83—21no—
106C42×C83C2×D412yes—
107C42×C83#4112yes—
108C23×C83—21no—
109C23×C83—21no—
110C23×C83—21no—
111C23×C83—21no—
112C23×C83#10112yes—
113C23×C83#10412yes—
114C23×C83#10412yes—
115C23×C83C2×C22⋊C411yescharacteristic
116C2×C4×C83—24no—
117C2×C4×C83—24no—
118C2×C4×C83—21no—
119C2×C4×C83—21no—
120C2×C4×C83—21no—
121C2×C4×C83—21no—
122C2×C4×C83—21no—
123C2×C4×C83—21no—
124C2×C4×C83—21no—
125C2×C4×C83—21no—
126C2×C4×C83—21no—
127C2×C4×C83—21no—
128C2×C4×C83—21no—
129C2×C4×C83—21no—
130C2×C4×C83—21no—
131C2×C4×C83—21no—
132C2×C4×C83—44no—
133C2×C4×C83—44no—
134C2×C4×C83—41no—
135C2×C4×C83—41no—
136C2×C4×C83—41no—
137C2×C4×C83—41no—
138C2×C4×C83—41no—
139C2×C4×C83—41no—
140C2×C4×C83—41no—
141C2×C4×C83—41no—
142C2×C4×C83—41no—
143C2×C4×C83—41no—
144C2×C4×C83—41no—
145C2×C4×C83—41no—
146C2×C4×C83#10412yes—
147C2×C4×C83C4×D412yes—
148C2×C4×C83#12011yescharacteristic
149C2×C4×C83#12112yes—
150C2×C4×C83#12311yescharacteristic
151C83⋊C8—28no—
152C83⋊C8—44no—
153C22×C83—22no—
154C22×C83—22no—
155C22×C83—22no—
156C22×C83—22no—
157C22×C83—22no—
158C22×C83—22no—
159C22×C83#33014yescomplemented
160C22×C83#38611yescharacteristic
161C22×C83C2×#10111yescharacteristic
162C22×C83#41311yescharacteristic
163C22×C83C2×#10411yescharacteristic
164C22×C83C2×#10411yescharacteristic
165C22×C83C2×#10411yescharacteristic
166C22×C83#42111yescharacteristic
167C4×C83—22no—
168C4×C83—21no—
169C4×C83—22no—
170C4×C83—21no—
171C4×C83—22no—
172C4×C83—41no—
173C4×C83—41no—
174C4×C83—81no—
175C4×C83—81no—
176C4×C83—81no—
177C4×C83—81no—
178#9121—831nomaximal, Sylow
179C2×C83—24no—
180C2×C83C2×#33011yescharacteristic
181C2×C83#152411yescharacteristic
182C2×C83#153111yescharacteristic
183C2×C83#153211yescharacteristic
184C2×C83#187111yescharacteristic
185C2×C83#189611yescharacteristic
186C2×C83#190111yescharacteristic
187C22×#97—831no—
188#2851D8311yescharacteristic
189#1749—831no—
190C2×#330—832no—
191#1295—832no—
192C83#912111yescharacteristic, complemented, Sylow
193C23×C8—1661no—
194C2×#125C2×D8311yescharacteristic
195C22×C42C2×D8311yescharacteristic
196C2×C22⋊C8—1661no—
197C2×#97—832no—
198C2×#97—832no—
199C2×#97—1664no—
200C2×#97—1661no—
201#389Dic8311yescharacteristic
202#389Dic8311yescharacteristic
203#387—21no—
204#387—21no—
205#385—21no—
206#385—21no—
207#382C2×D8311yescharacteristic
208#380Dic8311yescharacteristic
209#380Dic8311yescharacteristic
210#336—1661no—
211#330—1664no—
212C23×C4C2×Dic8311yescharacteristic
213C23×C4C22⋊D8311yescharacteristic
214C23×C4C22⋊D8311yescharacteristic
215C22×C8—1662no—
216C22×C8—1664no—
217C22×C8—1664no—
218C22×C8—1664no—
219C22×C8—1662no—
220#125—22no—
221#125—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4C2×Dic8311yescharacteristic
227C2×C4⋊C4C2×Dic8311yescharacteristic
228C2×C4⋊C4C2×Dic8311yescharacteristic
229C2×C4⋊C4C2×Dic8311yescharacteristic
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42C2×Dic8311yescharacteristic
235C2×C42C22⋊D8311yescharacteristic
236C2×C42C22⋊D8311yescharacteristic
237C2×C42C22×D8311yescharacteristic
238C22⋊C8—1664no—
239#97—834no—
240#97—1664no—
241#97—1662no—
242#97—1662no—
243#97—1662no—
244#95—21no—
245#95—21no—
246#95—21no—
247#95—21no—
248#95—21no—
249#95—21no—
250C24C22⋊Dic8311yescharacteristic
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4C22⋊Dic8311yescharacteristic
266C22×C4C22⋊Dic8311yescharacteristic
267C22×C4C22⋊Dic8311yescharacteristic
268C22×C4C22×Dic8311yescharacteristic
269C22×C4C2×C22⋊D8311yescharacteristic
270C22×C4C2×C22⋊D8311yescharacteristic
271C2×C8—3322no—
272C2×C8—3322no—
273C2×C8—1664no—
274C2×C8—16616no—
275C2×C8—1664no—
276C2×C8—1664no—
277C2×C8—1664no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—42no—
283C4⋊C4—41no—
284C4⋊C4—42no—
285C4⋊C4—41no—
286C4⋊C4—42no—
287C22⋊C4—42no—
288C22⋊C4—42no—
289C42—21no—
290C42—22no—
291C42—21no—
292C42—22no—
293C42D4×D8312yes—
294C42#1305058912yes—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23#1458171112yes—
300C23#1458171912yes—
301C23#1458171912yes—
302C23C2×C22⋊Dic8311yescharacteristic
303C2×C4—24no—
304C2×C4—24no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4#1458171912yes—
334C2×C4D4×Dic8312yes—
335C2×C4#1458176011yescharacteristic
336C2×C4#1458176412yes—
337C2×C4#1458177311yescharacteristic
338C8—3324no—
339C8—1668no—
340C22—22no—
341C22—22no—
342C22—22no—
343C22—22no—
344C22—22no—
345C22—22no—
346C22#1912744314yescomplemented
347C22#1912758111yescharacteristic
348C22C2×#1458171111yescharacteristic
349C22#1912768511yescharacteristic
350C22C2×#1458171911yescharacteristic
351C22C2×#1458171911yescharacteristic
352C22C2×#1458171911yescharacteristic
353C22#1912771811yescharacteristic
354C4—21no—
355C4—21no—
356C4—22no—
357C4—22no—
358C4—22no—
359C4—41no—
360C4—41no—
361C4—81no—
362C4—81no—
363C4—81no—
364C4—81no—
365C2—24no—
366C2C2×#1912744311yescharacteristic
367C2#2490605111yescharacteristic
368C2#2490607811yescharacteristic
369C2#2490608311yescharacteristic
370C2#2490776611yescharacteristic
371C2#2490790711yescharacteristic
372C2#2490793211yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic83
commutatorC22×C4×C83C22×C4
FrattiniC2×C42C22×D83
FittingC83×#2851C2
socleC23×C83C2×C22⋊C4
prime radicalC24×C83C22⋊C4
cosocleC2×C42×C83C23
hypercenter#2851D83
hypocenterC83#9121
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2851

Complements

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

NH ≅ G/Ntype
3. C2×#19127443C2semidirect
4. #24905344C2semidirect
159. C22×C83#330semidirect
192. C83#9121semidirect
346. C22#19127443semidirect

Maximal subgroups

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

subgroupindex
C22×#145817032
#249072252
C2×#191274432
#249053442
C83×#28512
#912183