Finite Group Databasebeta

← #35300034

Subgroups — #35300034

Order 21248.

530 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24915773C211yesmaximal, characteristic
2#24907225C211yesmaximal, characteristic
3C2×#19127442C212yesmaximal, complemented
4#24905345C212yesmaximal, complemented
5C2×#432×C83C211yesmaximal, characteristic
6C2×#14581784C2211yescharacteristic
7C2×C22⋊(C83⋊C8)—21no—
8C2×#14581703C2212yes—
9#19127599C411yescharacteristic
10#19127599C411yescharacteristic
11#19127585—21no—
12#19127585—21no—
13#19127574—21no—
14#19127574—21no—
15#19127567C2211yescharacteristic
16#19127561C411yescharacteristic
17#19127561C411yescharacteristic
18#19127456—21no—
19#19127442—24no—
20C22×C42×C83C2211yescharacteristic
21#432×C83—24no—
22C2×C4⋊C8×C83—21no—
23C2×C22⋊C8×C83—21no—
24C2×#97×C83C2212yes—
25C22×C83⋊C8—22no—
26#14581784—22no—
27#14581784—22no—
28C2×C4⋊Dic83C2×C411yescharacteristic
29C2×C4⋊Dic83C2×C411yescharacteristic
30C2×C4⋊Dic83C2×C411yescharacteristic
31C2×C4⋊Dic83C2×C411yescharacteristic
32C2×Dic83⋊C4—21no—
33C2×Dic83⋊C4—21no—
34C2×Dic83⋊C4—21no—
35C2×Dic83⋊C4—21no—
36C2×C22⋊Dic83—21no—
37C2×C22⋊Dic83—21no—
38C22⋊(C83⋊C8)—24no—
39#14581703—22no—
40#14581703—22no—
41C83⋊#95—21no—
42C83⋊#95—21no—
43C83⋊#95—21no—
44C83⋊#95—21no—
45C83⋊#95—21no—
46C83⋊#95—21no—
47C23×C4×C83C2×C411yescharacteristic
48C23×C4×C83D411yescharacteristic
49C23×C4×C83D411yescharacteristic
50C22×C8×C83—22no—
51C2×C42×C83—22no—
52C2×C42×C83—22no—
53C2×C42×C83C2×C411yescharacteristic
54C2×C42×C83D411yescharacteristic
55C2×C42×C83D411yescharacteristic
56C2×C42×C83C2311yescharacteristic
57C4⋊C8×C83—24no—
58C22⋊C8×C83—24no—
59#97×C83—22no—
60#97×C83D414yes—
61C22×Dic83—21no—
62C22×Dic83—21no—
63C22×Dic83—21no—
64C22×Dic83—21no—
65C2×C83⋊C8—24no—
66C2×C83⋊C8—42no—
67C2×C83⋊C8—42no—
68C4⋊Dic83—22no—
69C4⋊Dic83—22no—
70C4⋊Dic83—41no—
71C4⋊Dic83—41no—
72Dic83⋊C4—42no—
73Dic83⋊C4—42no—
74Dic83⋊C4—41no—
75Dic83⋊C4—42no—
76Dic83⋊C4—41no—
77C22⋊Dic83—42no—
78C22⋊Dic83—42no—
79C24×C83C22⋊C411yescharacteristic
80C22×C4×C83—22no—
81C22×C4×C83—22no—
82C22×C4×C83—22no—
83C22×C4×C83—21no—
84C22×C4×C83—22no—
85C22×C4×C83—21no—
86C22×C4×C83—41no—
87C22×C4×C83—41no—
88C22×C4×C83—41no—
89C22×C4×C83—41no—
90C22×C4×C83C22⋊C411yescharacteristic
91C22×C4×C83C22⋊C411yescharacteristic
92C22×C4×C83C22⋊C411yescharacteristic
93C22×C4×C83C22×C411yescharacteristic
94C22×C4×C83C2×D411yescharacteristic
95C22×C4×C83C2×D411yescharacteristic
96C2×C8×C83—24no—
97C2×C8×C83—42no—
98C2×C8×C83—42no—
99C42×C83—22no—
100C42×C83—21no—
101C42×C83—22no—
102C42×C83—21no—
103C42×C83C2×D412yes—
104C42×C83#4112yes—
105C23×C83—21no—
106C23×C83—21no—
107C23×C83—21no—
108C23×C83—21no—
109C23×C83—21no—
110C23×C83—21no—
111C23×C83#10112yes—
112C23×C83C2×C22⋊C411yescharacteristic
113C2×C4×C83—21no—
114C2×C4×C83—21no—
115C2×C4×C83—21no—
116C2×C4×C83—21no—
117C2×C4×C83—21no—
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—44no—
128C2×C4×C83—44no—
129C2×C4×C83—42no—
130C2×C4×C83—42no—
131C2×C4×C83#10412yes—
132C2×C4×C83C4×D412yes—
133C2×C4×C83#12011yescharacteristic
134C2×C4×C83#12112yes—
135C2×C4×C83#12311yescharacteristic
136C2×Dic83—41no—
137C2×Dic83—41no—
138C2×Dic83—41no—
139C2×Dic83—41no—
140C2×Dic83—41no—
141C2×Dic83—41no—
142C2×Dic83—41no—
143C2×Dic83—41no—
144C2×Dic83—41no—
145C2×Dic83—41no—
146C2×Dic83—41no—
147C2×Dic83—41no—
148C8×C83—44no—
149C83⋊C8—44no—
150C22×C83—22no—
151C22×C83—22no—
152C22×C83—22no—
153C22×C83—41no—
154C22×C83—41no—
155C22×C83—41no—
156C22×C83—41no—
157C22×C83#38611yescharacteristic
158C22×C83C2×#10111yescharacteristic
159C22×C83#41311yescharacteristic
160C22×C83C2×#10411yescharacteristic
161C22×C83#42111yescharacteristic
162C22×C83#42111yescharacteristic
163C22×C83#42111yescharacteristic
164C4×C83—22no—
165C4×C83—22no—
166C4×C83—21no—
167C4×C83—21no—
168C4×C83—22no—
169C4×C83—41no—
170C4×C83—41no—
171Dic83—81no—
172Dic83—81no—
173Dic83—81no—
174Dic83—81no—
175#9126—831nomaximal, Sylow
176C2×C83—42no—
177C2×C83#152911yescharacteristic
178C2×C83#153111yescharacteristic
179C2×C83#153411yescharacteristic
180C2×C83#153511yescharacteristic
181C2×C83#187111yescharacteristic
182C2×C83#189711yescharacteristic
183C2×C83#190111yescharacteristic
184C2×#432D8311yescharacteristic
185#2851—831no—
186#1749—831no—
187C2×#330—832no—
188#1295—832no—
189C83#912611yescharacteristic, complemented, Sylow
190C2×#125—831no—
191C22×C42C2×D8311yescharacteristic
192#432—24no—
193C2×C4⋊C8—21no—
194C2×C22⋊C8—21no—
195C2×C22⋊C8—1661no—
196C2×#97—832no—
197C2×#97C2×D8312yes—
198#389—831no—
199#389—831no—
200#387—1661no—
201#387—1661no—
202#385—1661no—
203#385—1661no—
204#382—831no—
205#380—831no—
206#380—831no—
207#336—1661no—
208#330—1664no—
209C23×C4C4×D8311yescharacteristic
210C23×C4D33211yescharacteristic
211C23×C4C22⋊D8311yescharacteristic
212C22×C8—22no—
213C22×C8—1662no—
214#125—1662no—
215#125—1662no—
216C2×C4⋊C4—831no—
217C2×C4⋊C4—831no—
218C2×C4⋊C4—831no—
219C2×C4⋊C4—831no—
220C2×C4⋊C4—1661no—
221C2×C4⋊C4—1661no—
222C2×C4⋊C4—1661no—
223C2×C4⋊C4—1661no—
224C2×C22⋊C4—1661no—
225C2×C22⋊C4—1661no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42C4×D8311yescharacteristic
229C2×C42D33211yescharacteristic
230C2×C42C22⋊D8311yescharacteristic
231C2×C42C22×D8311yescharacteristic
232C4⋊C8—24no—
233C22⋊C8—24no—
234C22⋊C8—1664no—
235#97—22no—
236#97—1662no—
237#97—1662no—
238#97C22⋊D8314yes—
239#95—1661no—
240#95—1661no—
241#95—1661no—
242#95—1661no—
243#95—1661no—
244#95—1661no—
245C24(C2×D83)⋊C411yescharacteristic
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—1661no—
257C22×C4—1661no—
258C22×C4—1661no—
259C22×C4—1661no—
260C22×C4(C2×D83)⋊C411yescharacteristic
261C22×C4(C2×D83)⋊C411yescharacteristic
262C22×C4(C2×D83)⋊C411yescharacteristic
263C22×C4C2×C4×D8311yescharacteristic
264C22×C4C2×C22⋊D8311yescharacteristic
265C22×C4C2×D33211yescharacteristic
266C2×C8—24no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—3322no—
270C2×C8—3322no—
271C2×C8—1664no—
272C4⋊C4—3321no—
273C4⋊C4—3321no—
274C4⋊C4—3322no—
275C4⋊C4—3321no—
276C4⋊C4—3322no—
277C4⋊C4—3321no—
278C4⋊C4—3322no—
279C4⋊C4—1662no—
280C4⋊C4—1662no—
281C22⋊C4—3322no—
282C22⋊C4—3322no—
283C42—21no—
284C42—22no—
285C42—21no—
286C42—22no—
287C42C2×C22⋊D8312yes—
288C42#1305059012yes—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23#1458171012yes—
296C23C2×(C2×D83)⋊C411yescharacteristic
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—3321no—
316C2×C4—3321no—
317C2×C4—3321no—
318C2×C4—3321no—
319C2×C4—3321no—
320C2×C4—3321no—
321C2×C4—3321no—
322C2×C4—3321no—
323C2×C4—3321no—
324C2×C4—3321no—
325C2×C4—3321no—
326C2×C4—3321no—
327C2×C4#1458171812yes—
328C2×C4C4×C22⋊D8312yes—
329C2×C4#1458176111yescharacteristic
330C2×C4#1458176612yes—
331C2×C4#1458177511yescharacteristic
332C8—44no—
333C8—3324no—
334C22—22no—
335C22—22no—
336C22—22no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22#1912758011yescharacteristic
342C22C2×#1458171011yescharacteristic
343C22#1912768311yescharacteristic
344C22C2×#1458171811yescharacteristic
345C22#1912771711yescharacteristic
346C22#1912771911yescharacteristic
347C22#1912771911yescharacteristic
348C4—21no—
349C4—21no—
350C4—22no—
351C4—22no—
352C4—22no—
353C4—41no—
354C4—41no—
355C4—6641no—
356C4—6641no—
357C4—6641no—
358C4—6641no—
359C2—42no—
360C2#2490607111yescharacteristic
361C2#2490607911yescharacteristic
362C2#2490608911yescharacteristic
363C2#2490609211yescharacteristic
364C2#2490776511yescharacteristic
365C2#2490791111yescharacteristic
366C2#2490793111yescharacteristic
subgroupG/H
centerC23C2×(C2×D83)⋊C4
commutatorC22×C4×C83C22×C4
FrattiniC2×C42C22×D83
FittingC2×#432×C83C2
socleC23×C83C2×C22⋊C4
prime radicalC24×C83C22⋊C4
cosocleC2×C42×C83C23
hypercenterC2×#432D83
hypocenterC83#9126
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#432

Complements

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

NH ≅ G/Ntype
3. C2×#19127442C2semidirect
4. #24905345C2semidirect
189. C83#9126semidirect

Maximal subgroups

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

subgroupindex
#249157732
#249072252
C2×#191274422
#249053452
C2×#432×C832
#912683