Finite Group Databasebeta

← #35409108

Subgroups — #35409108

Order 21248.

646 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24909937C211yesmaximal, characteristic
2#24908403C211yesmaximal, characteristic
3#24908402C211yesmaximal, characteristic
4#24908187C211yesmaximal, characteristic
5#24908030C212yesmaximal
6#24908024C212yesmaximal
7#24907322C212yesmaximal
8#24907318C211yesmaximal, characteristic
9#24907213C212yesmaximal
10#97×Dic83C211yesmaximal, characteristic
11C83×#1940C211yesmaximal, characteristic
12C2×C4×Dic166C2211yescharacteristic
13C2×C4⋊(C83⋊C8)C2211yescharacteristic
14C2×C4⋊(C83⋊C8)C2212yes—
15C2×#14581703C2211yescharacteristic
16C2×#14581702C2212yes—
17C2×#14581702C2211yescharacteristic
18C2×#14581702C2211yescharacteristic
19C2×C8×Dic83C2211yescharacteristic
20C2×C8×Dic83C2211yescharacteristic
21#19127606C412yes—
22#19127597C412yes—
23#19127583C414yes—
24C4×Dic83⋊C4C2212yes—
25#19127547C2212yes—
26C42×Dic83C2211yescharacteristic
27#19127456C2212yes—
28#19127455C2212yes—
29#19127455C2212yes—
30#19127455C2212yes—
31#19127455C2212yes—
32#19127455C2212yes—
33C2×C4⋊C8×C83C2211yescharacteristic
34C2×C4⋊C8×C83C2212yes—
35C2×#97×C83C2211yescharacteristic
36C4×C4⋊C4×C83C2211yescharacteristic
37#336×C83C2212yes—
38C22×Dic166C2×C411yescharacteristic
39C22×C83⋊C8C2312yes—
40C22×C83⋊C8C2311yescharacteristic
41C22×C83⋊C8C2311yescharacteristic
42C4×Dic166—22no—
43C4×Dic166—22no—
44C2×C4⋊Dic83C2×C411yescharacteristic
45C2×Dic83⋊C4C2×C412yes—
46C2×Dic83⋊C4C2×C414yes—
47C2×C4×Dic83C2×C412yes—
48C2×C4×Dic83C2×C412yes—
49C2×C4×Dic83C2311yescharacteristic
50C2×C4×Dic83C2312yes—
51C2×C4×Dic83C2311yescharacteristic
52C4⋊(C83⋊C8)—24no—
53C4⋊(C83⋊C8)—22no—
54#14581703—22no—
55#14581702—22no—
56#14581702—24no—
57#14581702—22no—
58C8×Dic83—22no—
59C8×Dic83Q814yescomplemented
60C83⋊#95C2×C414yes—
61C83⋊#95C2×C414yes—
62C83⋊#95C2×C414yes—
63C22×C8×C83C2312yes—
64C22×C8×C83C2311yescharacteristic
65C22×C8×C83C2311yescharacteristic
66C2×C4⋊C4×C83C2×C412yes—
67C2×C42×C83C2312yes—
68C2×C42×C83C2311yescharacteristic
69C4⋊C8×C83—24no—
70C4⋊C8×C83—22no—
71#97×C83—22no—
72#95×C83C2×C412yes—
73C2×Dic166—22no—
74C2×Dic166—42no—
75C22×Dic83C22×C412yes—
76C22×Dic83C22×C412yes—
77C22×Dic83C22×C412yes—
78C22×Dic83C22×C412yes—
79C2×C83⋊C8—21no—
80C2×C83⋊C8—21no—
81C2×C83⋊C8—21no—
82C2×C83⋊C8—21no—
83C2×C83⋊C8—22no—
84C2×C83⋊C8—22no—
85C2×C83⋊C8—22no—
86C2×C83⋊C8C2×Q812yes—
87C2×C83⋊C8#4112yes—
88C4⋊Dic83—22no—
89Dic83⋊C4—24no—
90Dic83⋊C4—44no—
91C4×Dic83—21no—
92C4×Dic83—21no—
93C4×Dic83—22no—
94C4×Dic83—22no—
95C4×Dic83—42no—
96C4×Dic83—42no—
97C4×Dic83C2×Q812yes—
98C4×Dic83#4112yes—
99C22×C4×C83C22×C414yes—
100C22×C4×C83C22×C411yescharacteristic
101C22×C4×C83C22×C411yescharacteristic
102C22×C4×C83C2411yescharacteristic
103C2×C8×C83—21no—
104C2×C8×C83—21no—
105C2×C8×C83—21no—
106C2×C8×C83—21no—
107C2×C8×C83—22no—
108C2×C8×C83—22no—
109C2×C8×C83—22no—
110C2×C8×C83C2×Q812yes—
111C2×C8×C83#4112yes—
112C4⋊C4×C83—42no—
113C42×C83—22no—
114C42×C83—22no—
115C42×C83#4112yes—
116C42×C83#4112yes—
117C23×C83C23×C411yescharacteristic
118C2×C4×C83—22no—
119C2×C4×C83—22no—
120C2×C4×C83—24no—
121C2×C4×C83—24no—
122C2×C4×C83—24no—
123C2×C4×C83#11712yes—
124C2×C4×C83#11712yes—
125C2×C4×C83C22×Q811yescharacteristic
126C2×C4×C83C2×#4111yescharacteristic
127C2×C4×C83C2×#4111yescharacteristic
128C2×C4×C83C2×#4111yescharacteristic
129C2×C4×C83C2×#4111yescharacteristic
130C2×C4×C83C2×#4111yescharacteristic
131C2×Dic83—22no—
132C2×Dic83—22no—
133C2×Dic83—22no—
134C2×Dic83—22no—
135C2×Dic83—22no—
136C2×Dic83—22no—
137C2×Dic83—22no—
138C2×Dic83—24no—
139C2×Dic83—24no—
140C2×Dic83C4×Q814yes—
141Dic166—44no—
142C8×C83—22no—
143C8×C83—22no—
144C8×C83—42no—
145C83⋊C8—22no—
146C83⋊C8—22no—
147C83⋊C8—42no—
148C22×C83C2×#11711yescharacteristic
149C22×C83C2×C4×Q811yescharacteristic
150C22×C83C4×#4111yescharacteristic
151C22×C83#56812yes—
152C22×C83#56812yes—
153C4×C83—22no—
154C4×C83—22no—
155C4×C83—44no—
156C4×C83#52911yescharacteristic
157C4×C83#52911yescharacteristic
158C4×C83#53311yescharacteristic
159C4×C83#53311yescharacteristic
160Dic83—24no—
161Dic83—42no—
162Dic83—42no—
163Dic83—42no—
164#21342—831nomaximal, Sylow
165C2×C83#229511yescharacteristic
166C2×C83#292111yescharacteristic
167C2×C83#292311yescharacteristic
168C2×C83#296112yes—
169C2×C83#296712yes—
170#2295—831no—
171#1985—831no—
172#1985—831no—
173#1940—831no—
174#1940D8311yescharacteristic
175#1917—832no—
176#1917—832no—
177#1768—832no—
178#1767—831no—
179#1746—832no—
180C4×#97—831no—
181C83#2134211yescharacteristic, complemented, Sylow
182C2×C4×Q8—831no—
183C2×C4⋊C8—832no—
184C2×C4⋊C8—831no—
185C2×C4⋊C8C2×D8312yes—
186C2×C4⋊C8C2×D8311yescharacteristic
187C2×#97—831no—
188C2×#97—831no—
189C2×#97—832no—
190C2×#97—831no—
191C2×#97C2×D8311yescharacteristic
192C2×C4×C8—831no—
193C2×C4×C8—831no—
194#391—832no—
195#389—832no—
196#387—834no—
197C4×C4⋊C4—832no—
198C4×C4⋊C4C2×D8311yescharacteristic
199#376—832no—
200C43—831no—
201#336—832no—
202#336—832no—
203#336—832no—
204#336—832no—
205#336—832no—
206#336—832no—
207#336C2×D8312yes—
208C22×Q8—831no—
209C22×C8—831no—
210C22×C8—831no—
211C22×C8—832no—
212C22×C8C22×D8312yes—
213C22×C8C22×D8311yescharacteristic
214C22×C8C22×D8311yescharacteristic
215C4×Q8—1662no—
216C4×Q8—1662no—
217C2×C4⋊C4—834no—
218C2×C4⋊C4—832no—
219C2×C4⋊C4—831no—
220C2×C4⋊C4C4×D8312yes—
221C2×C42—832no—
222C2×C42—832no—
223C2×C42—831no—
224C2×C42—832no—
225C2×C42—831no—
226C2×C42C22×D8312yes—
227C2×C42C22×D8311yescharacteristic
228C4⋊C8—24no—
229C4⋊C8—22no—
230C4⋊C8—1664no—
231C4⋊C8—1662no—
232#97—22no—
233#97—1664no—
234#97—1662no—
235#97—1662no—
236#97—1662no—
237C4×C8—834no—
238C4×C8—1662no—
239#95—834no—
240#95—834no—
241#95—834no—
242#95C4×D8312yes—
243C2×Q8—3322no—
244C2×Q8—1662no—
245C22×C4—832no—
246C22×C4—832no—
247C22×C4—832no—
248C22×C4—832no—
249C22×C4C2×C4×D8314yes—
250C22×C4C2×C4×D8311yescharacteristic
251C22×C4C2×C4×D8311yescharacteristic
252C22×C4C23×D8311yescharacteristic
253C2×C8—21no—
254C2×C8—21no—
255C2×C8—21no—
256C2×C8—21no—
257C2×C8—22no—
258C2×C8—22no—
259C2×C8—22no—
260C2×C8—832no—
261C2×C8—832no—
262C2×C8—1662no—
263C2×C8—1662no—
264C2×C8—1662no—
265C2×C8—1661no—
266C2×C8—1661no—
267C2×C8—1661no—
268C2×C8—1661no—
269C2×C8Q8×D8312yes—
270C2×C8#1305058912yes—
271C4⋊C4—42no—
272C4⋊C4—3324no—
273C4⋊C4—1664no—
274C4⋊C4—1662no—
275C42—22no—
276C42—22no—
277C42—3322no—
278C42—3322no—
279C42—832no—
280C42—832no—
281C42—1661no—
282C42—1661no—
283C42—1662no—
284C42—1662no—
285C42#1305058912yes—
286C42#1305059112yes—
287C23C22×C4×D8311yescharacteristic
288Q8—3324no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—24no—
292C2×C4—24no—
293C2×C4—24no—
294C2×C4—834no—
295C2×C4—1662no—
296C2×C4—1662no—
297C2×C4—1662no—
298C2×C4—1662no—
299C2×C4—1664no—
300C2×C4—1662no—
301C2×C4—1664no—
302C2×C4—1662no—
303C2×C4—1662no—
304C2×C4#1458175012yes—
305C2×C4#1458175012yes—
306C2×C4C2×Q8×D8311yescharacteristic
307C2×C4#41×D8311yescharacteristic
308C2×C4#41×D8311yescharacteristic
309C2×C4C2×#1305058911yescharacteristic
310C2×C4C2×#1305058911yescharacteristic
311C2×C4C2×#1305059111yescharacteristic
312C8—22no—
313C8—22no—
314C8—42no—
315C8—3322no—
316C8—1662no—
317C8—1662no—
318C22C2×#1458175011yescharacteristic
319C22C4×Q8×D8311yescharacteristic
320C22C4×#1305058911yescharacteristic
321C22#1912846612yes—
322C22#1912846712yes—
323C4—22no—
324C4—22no—
325C4—44no—
326C4—3322no—
327C4—3322no—
328C4—3322no—
329C4—1664no—
330C4#1912823211yescharacteristic
331C4#1912823611yescharacteristic
332C4#1912825311yescharacteristic
333C4#1912825611yescharacteristic
334C2#2490993411yescharacteristic
335C2#2491610011yescharacteristic
336C2#2491611311yescharacteristic
337C2#2491639912yes—
338C2#2491644512yes—
subgroupG/H
center, FrattiniC22×C4C23×D83
commutator, socle, prime radicalC23×C83C23×C4
FittingC83×#1940C2
cosocleC22×C4×C83C24
hypercenter#1940D83
hypocenterC83#21342
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C22×C4 ◁ #1940

Complements

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

NH ≅ G/Ntype
59. C8×Dic83Q8semidirect
181. C83#21342semidirect

Maximal subgroups

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

subgroupindex
#249099372
#249084032
#249084022
#249081872
#249080302
#249080242
#249073222
#249073182
#249072132
#97×Dic832
C83×#19402
#2134283