Finite Group Databasebeta

← #35310613

Subgroups — #35310613

Order 21248.

974 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24910075C211yesmaximal, characteristic
2C2×#19127468C211yesmaximal, characteristic
3C2×#19127467C212yesmaximal
4C2×#19127467C212yesmaximal
5C83×#2273C211yesmaximal, characteristic
6C22×C22⋊Dic83C2212yes—
7C22×C22⋊Dic83C2211yescharacteristic
8C22×C22⋊Dic83C2211yescharacteristic
9#19127574—22no—
10#19127573C414yescomplemented
11#19127559—22no—
12#19127468—22no—
13#19127467—24no—
14#19127467C418yescomplemented
15C22×C22⋊C4×C83C2211yescharacteristic
16C22×C22⋊C4×C83C2212yes—
17C4×C22⋊C4×C83—22no—
18C4×C22⋊C4×C83—24no—
19C23×Dic83—21no—
20C23×Dic83—22no—
21C23×Dic83C2×C412yes—
22C2×Dic83⋊C4—24no—
23C2×C22⋊Dic83—21no—
24C2×C22⋊Dic83—22no—
25C2×C22⋊Dic83—22no—
26C2×C22⋊Dic83—21no—
27C2×C22⋊Dic83—21no—
28C2×C22⋊Dic83—22no—
29C2×C22⋊Dic83—22no—
30C2×C22⋊Dic83—22no—
31C2×C22⋊Dic83—22no—
32C2×C22⋊Dic83—22no—
33C2×C22⋊Dic83—22no—
34C2×C22⋊Dic83C2×C414yes—
35C2×C22⋊Dic83C2×C412yes—
36C2×C22⋊Dic83C2×C414yes—
37C83⋊#95—24no—
38C25×C83C2311yescharacteristic
39C23×C4×C83—22no—
40C23×C4×C83D412yes—
41C2×C22⋊C4×C83—22no—
42C2×C22⋊C4×C83—22no—
43C2×C22⋊C4×C83—22no—
44C2×C22⋊C4×C83—22no—
45C2×C22⋊C4×C83—22no—
46C2×C22⋊C4×C83C2×C412yes—
47C2×C22⋊C4×C83C2×C414yes—
48C2×C22⋊C4×C83D412yes—
49C2×C22⋊C4×C83D414yes—
50C2×C22⋊C4×C83Q814yes—
51C2×C42×C83—24no—
52#95×C83—24no—
53C22×Dic83—22no—
54C22×Dic83—24no—
55C22×Dic83—22no—
56C22×Dic83—24no—
57C22×Dic83—24no—
58C22×Dic83—42no—
59C22×Dic83—42no—
60C22×Dic83—82no—
61C22×Dic83—82no—
62C22×Dic83—84no—
63Dic83⋊C4—84no—
64C22⋊Dic83—22no—
65C22⋊Dic83—24no—
66C22⋊Dic83—24no—
67C22⋊Dic83—24no—
68C22⋊Dic83—41no—
69C22⋊Dic83—42no—
70C22⋊Dic83—41no—
71C22⋊Dic83—42no—
72C22⋊Dic83—42no—
73C22⋊Dic83C4214yescomplemented
74C24×C83—82no—
75C24×C83C22×C411yescharacteristic
76C24×C83C22×C412yes—
77C24×C83C2×D411yescharacteristic
78C24×C83C2×D412yes—
79C24×C83C2×D411yescharacteristic
80C24×C83C2×Q812yes—
81C24×C83#4112yes—
82C24×C83#4112yes—
83C24×C83#4112yes—
84C22×C4×C83—22no—
85C22×C4×C83—24no—
86C22×C4×C83—24no—
87C22×C4×C83—24no—
88C22×C4×C83—22no—
89C22×C4×C83—82no—
90C22×C4×C83—84no—
91C22×C4×C83C22⋊C414yes—
92C22⋊C4×C83—24no—
93C22⋊C4×C83—24no—
94C22⋊C4×C83—24no—
95C22⋊C4×C83—24no—
96C22⋊C4×C83—24no—
97C22⋊C4×C83C4⋊C418yescomplemented
98C42×C83—84no—
99C23×C83—42no—
100C23×C83—44no—
101C23×C83—42no—
102C23×C83—42no—
103C23×C83—44no—
104C23×C83—82no—
105C23×C83—82no—
106C23×C83—82no—
107C23×C83—82no—
108C23×C83C2×C4211yescharacteristic
109C23×C83C2×C22⋊C411yescharacteristic
110C23×C83C2×C4⋊C412yes—
111C23×C83#11711yescharacteristic
112C23×C83#11712yes—
113C23×C83C4×D412yes—
114C23×C83C4×D412yes—
115C23×C83C4×D412yes—
116C23×C83C4×D412yes—
117C23×C83C4×Q812yes—
118C23×C83C4×Q812yes—
119C23×C83#12011yescharacteristic
120C23×C83#12112yes—
121C23×C83C22⋊Q812yes—
122C23×C83C22⋊Q812yes—
123C23×C83#12311yescharacteristic
124C23×C83#12312yes—
125C23×C83#12411yescharacteristic
126C23×C83#12512yes—
127C23×C83#12612yes—
128C23×C83C4⋊Q811yescharacteristic
129C2×C4×C83—28no—
130C2×C4×C83—24no—
131C2×C4×C83—48no—
132C2×C4×C83—44no—
133C2×C4×C83—44no—
134C2×C4×C83—82no—
135C2×C4×C83—84no—
136C2×C4×C83—84no—
137C2×Dic83—24no—
138C2×Dic83—44no—
139C2×Dic83—44no—
140C2×Dic83—42no—
141C2×Dic83—44no—
142C2×Dic83—84no—
143C2×Dic83—84no—
144C2×Dic83—84no—
145C2×Dic83—84no—
146C2×Dic83—84no—
147C22×C83—22no—
148C22×C83—24no—
149C22×C83—24no—
150C22×C83—22no—
151C22×C83—22no—
152C22×C83—42no—
153C22×C83—42no—
154C22×C83—42no—
155C22×C83—42no—
156C22×C83—42no—
157C22×C83—41no—
158C22×C83—42no—
159C22×C83—41no—
160C22×C83—42no—
161C22×C83—42no—
162C22×C83—42no—
163C22×C83—42no—
164C22×C83—42no—
165C22×C83—41no—
166C22×C83—42no—
167C22×C83—41no—
168C22×C83C4×C22⋊C411yescharacteristic
169C22×C83C4×C4⋊C412yes—
170C22×C83#38512yes—
171C22×C83#38611yescharacteristic
172C22×C83#38712yes—
173C22×C83#38712yes—
174C22×C83#38812yes—
175C22×C83#38912yes—
176C22×C83#39111yescharacteristic
177C4×C83—48no—
178C4×C83—44no—
179Dic83—44no—
180Dic83—84no—
181Dic83—82no—
182#10915—831nomaximal, Sylow
183C2×C83—21no—
184C2×C83—22no—
185C2×C83—22no—
186C2×C83—22no—
187C2×C83—22no—
188C2×C83—22no—
189C2×C83—22no—
190C2×C83—21no—
191C2×C83#143611yescharacteristic
192C2×C83#191312yes—
193#2315—831no—
194#2273D8311yescharacteristic
195C2×#342—832no—
196C2×#342—832no—
197C2×#342—831no—
198C83#1091511yescharacteristic, complemented, Sylow
199C22×C22⋊C4—831no—
200C22×C22⋊C4—832no—
201C22×C22⋊C4—831no—
202C22×C22⋊C4C2×D8311yescharacteristic
203C22×C22⋊C4C2×D8312yes—
204#385—834no—
205#385—1662no—
206#380—1662no—
207C4×C22⋊C4—22no—
208C4×C22⋊C4—24no—
209#342—838no—
210#342—1662no—
211#342—1664no—
212C25C22×D8311yescharacteristic
213C23×C4—22no—
214C23×C4—832no—
215C23×C4—1662no—
216C23×C4—1661no—
217C23×C4C22⋊D8312yes—
218C2×C4⋊C4—1664no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—834no—
225C2×C22⋊C4—832no—
226C2×C22⋊C4—834no—
227C2×C22⋊C4—1662no—
228C2×C22⋊C4—1661no—
229C2×C22⋊C4—1661no—
230C2×C22⋊C4—1662no—
231C2×C22⋊C4—1662no—
232C2×C22⋊C4—1662no—
233C2×C22⋊C4—1662no—
234C2×C22⋊C4—1662no—
235C2×C22⋊C4—1662no—
236C2×C22⋊C4—1661no—
237C2×C22⋊C4—1662no—
238C2×C22⋊C4Dic16614yes—
239C2×C22⋊C4C4×D8312yes—
240C2×C22⋊C4C4×D8314yes—
241C2×C22⋊C4D33212yes—
242C2×C22⋊C4C22⋊D8314yes—
243C2×C42—24no—
244#95—24no—
245#95—1664no—
246C24—82no—
247C24C2×C4×D8311yescharacteristic
248C24C2×C4×D8312yes—
249C24C2×C22⋊D8312yes—
250C24C2×C22⋊D8311yescharacteristic
251C24C2×D33211yescharacteristic
252C24C2×Dic16612yes—
253C24#1305059012yes—
254C24#1305059012yes—
255C24#1305059012yes—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—82no—
262C22×C4—84no—
263C22×C4—3322no—
264C22×C4—3322no—
265C22×C4—6642no—
266C22×C4—6642no—
267C22×C4—6644no—
268C22×C4—1662no—
269C22×C4—1662no—
270C22×C4—1664no—
271C22×C4—1664no—
272C22×C4—1664no—
273C22×C4(C2×D83)⋊C414yes—
274C4⋊C4—6644no—
275C22⋊C4—24no—
276C22⋊C4—24no—
277C22⋊C4—24no—
278C22⋊C4—24no—
279C22⋊C4—24no—
280C22⋊C4—3321no—
281C22⋊C4—3322no—
282C22⋊C4—3321no—
283C22⋊C4—3322no—
284C22⋊C4—3322no—
285C22⋊C4—834no—
286C22⋊C4—1664no—
287C22⋊C4—1664no—
288C22⋊C4—1662no—
289C22⋊C4—1664no—
290C22⋊C4Dic83⋊C418yescomplemented
291C42—84no—
292C23—42no—
293C23—44no—
294C23—42no—
295C23—42no—
296C23—44no—
297C23—82no—
298C23—82no—
299C23—82no—
300C23—82no—
301C23C42×D8311yescharacteristic
302C23C2×(C2×D83)⋊C411yescharacteristic
303C23C2×Dic83⋊C412yes—
304C23#1458174912yes—
305C23#1458174911yescharacteristic
306C23C4×C22⋊D8312yes—
307C23C4×C22⋊D8312yes—
308C23C4×C22⋊D8312yes—
309C23C4×D33212yes—
310C23C4×Dic16612yes—
311C23C4×Dic16612yes—
312C23#1458176111yescharacteristic
313C23#1458176612yes—
314C23#1458177112yes—
315C23#1458177112yes—
316C23#1458177511yescharacteristic
317C23#1458177512yes—
318C23#1458178011yescharacteristic
319C23#1458178412yes—
320C23#1458178612yes—
321C23#1458179211yescharacteristic
322C2×C4—28no—
323C2×C4—24no—
324C2×C4—48no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—82no—
328C2×C4—84no—
329C2×C4—84no—
330C2×C4—3324no—
331C2×C4—3324no—
332C2×C4—3322no—
333C2×C4—3324no—
334C2×C4—6644no—
335C2×C4—6644no—
336C2×C4—6644no—
337C2×C4—6644no—
338C2×C4—6644no—
339C2×C4—1664no—
340C22—22no—
341C22—22no—
342C22—22no—
343C22—24no—
344C22—24no—
345C22—42no—
346C22—42no—
347C22—42no—
348C22—42no—
349C22—42no—
350C22—42no—
351C22—42no—
352C22—41no—
353C22—42no—
354C22—41no—
355C22—41no—
356C22—42no—
357C22—41no—
358C22—42no—
359C22—42no—
360C22—42no—
361C22C4×(C2×D83)⋊C411yescharacteristic
362C22C4×Dic83⋊C412yes—
363C22#1912757412yes—
364C22#1912758011yescharacteristic
365C22#1912758512yes—
366C22#1912758512yes—
367C22#1912759212yes—
368C22#1912759912yes—
369C22#1912760711yescharacteristic
370C4—48no—
371C4—44no—
372C4—3324no—
373C4—6644no—
374C4—6642no—
375C2—22no—
376C2—21no—
377C2—22no—
378C2—22no—
379C2—22no—
380C2—22no—
381C2—22no—
382C2—21no—
383C2#2490569611yescharacteristic
384C2#2490800512yes—
subgroupG/H
centerC22#19127580
commutatorC23×C83C2×C42
FrattiniC25C22×D83
FittingC83×#2273C2
socleC22×C83#386
prime radical, cosocleC25×C83C23
hypercenter#2273D83
hypocenterC83#10915
LCSG ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C25 ◁ #2273

Complements

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

NH ≅ G/Ntype
10. #19127573C4semidirect
14. #19127467C4semidirect
73. C22⋊Dic83C42semidirect
97. C22⋊C4×C83C4⋊C4semidirect
198. C83#10915semidirect
290. C22⋊C4Dic83⋊C4semidirect

Maximal subgroups

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

subgroupindex
#249100752
C2×#191274682
C2×#191274672
C83×#22732
#1091583