Finite Group Databasebeta

← #35304596

Subgroups — #35304596

Order 21248.

548 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24907443C211yesmaximal, characteristic
2#24905690C211yesmaximal, characteristic
3#24905341C212yesmaximal
4#24905341C212yesmaximal
5C83×#2851C211yesmaximal, characteristic
6C2×C4⋊(C83⋊C8)—21no—
7C2×C22⋊(C83⋊C8)C2212yes—
8C2×C83⋊#95—21no—
9C2×C83⋊#95—21no—
10C2×C83⋊#95C2212yes—
11#19127456—21no—
12C2×#125×C83C411yescharacteristic, complemented
13C22×C42×C83C2211yescharacteristic
14#389×C83C411yescharacteristic, complemented
15#389×C83C411yescharacteristic, complemented
16#387×C83—21no—
17#387×C83—21no—
18#385×C83—21no—
19#385×C83—21no—
20#382×C83C411yescharacteristic, complemented
21#380×C83C2211yescharacteristic
22#380×C83C2211yescharacteristic
23C23×Dic83—22no—
24C22×C83⋊C8—22no—
25C4⋊(C83⋊C8)—42no—
26C22⋊(C83⋊C8)—24no—
27C83⋊#95—22no—
28C83⋊#95—42no—
29C83⋊#95—42no—
30C83⋊#95Q814yes—
31C23×C4×C83D411yescharacteristic
32C23×C4×C83D411yescharacteristic
33C23×C4×C83C2311yescharacteristic
34#125×C83—42no—
35C2×C4⋊C4×C83—21no—
36C2×C4⋊C4×C83—21no—
37C2×C4⋊C4×C83—21no—
38C2×C4⋊C4×C83—21no—
39C2×C4⋊C4×C83C2×C411yescharacteristic
40C2×C4⋊C4×C83C2×C411yescharacteristic
41C2×C4⋊C4×C83C2×C411yescharacteristic
42C2×C4⋊C4×C83C2×C411yescharacteristic
43C2×C22⋊C4×C83—21no—
44C2×C22⋊C4×C83—21no—
45C2×C42×C83—42no—
46C2×C42×C83C2×C411yescharacteristic
47C2×C42×C83C2×C411yescharacteristic
48C2×C42×C83D411yescharacteristic
49C2×C42×C83D411yescharacteristic
50#95×C83—21no—
51#95×C83—21no—
52#95×C83—21no—
53#95×C83—21no—
54#95×C83—21no—
55#95×C83—21no—
56C22×Dic83—24no—
57C22×Dic83—42no—
58C22×Dic83—42no—
59C22×Dic83—42no—
60C22×Dic83—42no—
61C22×Dic83—42no—
62C22×Dic83—42no—
63C2×C83⋊C8—42no—
64C2×C83⋊C8—42no—
65C2×C83⋊C8—42no—
66C24×C83C2×D411yescharacteristic
67C22×C4×C83—22no—
68C22×C4×C83—21no—
69C22×C4×C83—21no—
70C22×C4×C83—22no—
71C22×C4×C83—21no—
72C22×C4×C83—21no—
73C22×C4×C83—21no—
74C22×C4×C83—21no—
75C22×C4×C83—41no—
76C22×C4×C83—41no—
77C22×C4×C83—41no—
78C22×C4×C83—41no—
79C22×C4×C83—41no—
80C22×C4×C83C22⋊C411yescharacteristic
81C22×C4×C83C22⋊C411yescharacteristic
82C22×C4×C83C22⋊C411yescharacteristic
83C22×C4×C83C22⋊C411yescharacteristic
84C22×C4×C83C22×C411yescharacteristic
85C22×C4×C83C2×D411yescharacteristic
86C22×C4×C83C2×Q812yes—
87C22×C4×C83#4112yes—
88C4⋊C4×C83—41no—
89C4⋊C4×C83—41no—
90C4⋊C4×C83—41no—
91C4⋊C4×C83—41no—
92C4⋊C4×C83—42no—
93C4⋊C4×C83—42no—
94C4⋊C4×C83—42no—
95C4⋊C4×C83—42no—
96C22⋊C4×C83—42no—
97C22⋊C4×C83—41no—
98C22⋊C4×C83—41no—
99C42×C83—22no—
100C42×C83—22no—
101C42×C83—41no—
102C42×C83—41no—
103C23×C83—21no—
104C23×C83—21no—
105C23×C83—21no—
106C23×C83—21no—
107C23×C83C2×C22⋊C411yescharacteristic
108C23×C83C22⋊Q812yes—
109C23×C83#13512yes—
110C23×C83AGL(1,ℤ/8)11yescharacteristic
111C23×C83#13711yescharacteristic
112C2×C4×C83—21no—
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—42no—
128C2×C4×C83—42no—
129C2×C4×C83—44no—
130C2×C4×C83—42no—
131C2×C4×C83—42no—
132C2×C4×C83—41no—
133C2×C4×C83—41no—
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#9912yes—
145C2×C4×C83C4×Q812yes—
146C2×C4×C83#12411yescharacteristic
147C2×C4×C83C4⋊Q811yescharacteristic
148C2×Dic83—44no—
149C2×Dic83—44no—
150C2×Dic83—44no—
151C2×Dic83—82no—
152C2×Dic83—82no—
153C2×Dic83—82no—
154C2×Dic83—82no—
155C83⋊C8—82no—
156C22×C83—22no—
157C22×C83—22no—
158C22×C83—21no—
159C22×C83—21no—
160C22×C83—21no—
161C22×C83—21no—
162C22×C83—22no—
163C22×C83—22no—
164C22×C83#39111yescharacteristic
165C22×C83C2×#9911yescharacteristic
166C22×C83#41011yescharacteristic
167C22×C83#41611yescharacteristic
168C22×C83#41711yescharacteristic
169C22×C83#42111yescharacteristic
170C22×C83#42111yescharacteristic
171C22×C83#47812yes—
172C22×C83#47912yes—
173C4×C83—21no—
174C4×C83—22no—
175C4×C83—21no—
176C4×C83—22no—
177C4×C83—41no—
178C4×C83—41no—
179C4×C83—41no—
180C4×C83—81no—
181C4×C83—81no—
182C4×C83—81no—
183C4×C83—81no—
184Dic83—84no—
185#9913—831nomaximal, Sylow
186C2×C83—22no—
187C2×C83—22no—
188C2×C83#151211yescharacteristic
189C2×C83#151311yescharacteristic
190C2×C83#151611yescharacteristic
191C2×C83#152111yescharacteristic
192C2×C83#198911yescharacteristic
193C2×C83#199511yescharacteristic
194C2×C83#199611yescharacteristic
195#2851D8311yescharacteristic
196#1797—831no—
197#1434—831no—
198#1294—832no—
199#1294—832no—
200C83#991311yescharacteristic, complemented, Sylow
201C2×#125Dic8311yescharacteristic, complemented
202C22×C42C2×D8311yescharacteristic
203C2×C4⋊C8—1661no—
204C2×C22⋊C8—832no—
205#389Dic8311yescharacteristic, complemented
206#389Dic8311yescharacteristic, complemented
207#387—21no—
208#387—21no—
209#385—21no—
210#385—21no—
211#382Dic8311yescharacteristic, complemented
212#380C2×D8311yescharacteristic
213#380C2×D8311yescharacteristic
214C2×#95—832no—
215C2×#95—1661no—
216C2×#95—1661no—
217#336—1661no—
218C23×C4—1662no—
219C23×C4C22⋊D8311yescharacteristic
220C23×C4C22⋊D8311yescharacteristic
221C23×C4C22×D8311yescharacteristic
222C22×C8—1662no—
223#125—42no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4C2×Dic8311yescharacteristic
229C2×C4⋊C4C2×Dic8311yescharacteristic
230C2×C4⋊C4C2×Dic8311yescharacteristic
231C2×C4⋊C4C2×Dic8311yescharacteristic
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C42—42no—
235C2×C42C2×Dic8311yescharacteristic
236C2×C42C2×Dic8311yescharacteristic
237C2×C42C22⋊D8311yescharacteristic
238C2×C42C22⋊D8311yescharacteristic
239C4⋊C8—3322no—
240C22⋊C8—1664no—
241#95—21no—
242#95—21no—
243#95—21no—
244#95—21no—
245#95—21no—
246#95—21no—
247#95—3322no—
248#95—3322no—
249#95—834no—
250#95—1662no—
251C24C2×C22⋊D8311yescharacteristic
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—22no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—3322no—
266C22×C4—3322no—
267C22×C4—3322no—
268C22×C4—3322no—
269C22×C4—3322no—
270C22×C4—3322no—
271C22×C4—1664no—
272C22×C4C22⋊Dic8311yescharacteristic
273C22×C4C22⋊Dic8311yescharacteristic
274C22×C4C22⋊Dic8311yescharacteristic
275C22×C4C22⋊Dic8311yescharacteristic
276C22×C4C22×Dic8311yescharacteristic
277C22×C4C2×C22⋊D8311yescharacteristic
278C22×C4Q8×D8312yes—
279C22×C4#1305059112yes—
280C2×C8—3322no—
281C2×C8—3322no—
282C2×C8—3322no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C22⋊C4—42no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C42—22no—
295C42—22no—
296C42—41no—
297C42—41no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23C2×C22⋊Dic8311yescharacteristic
303C23#1458176812yes—
304C23#1458181212yes—
305C23#1458181711yescharacteristic
306C23#1458182211yescharacteristic
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—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—44no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—3324no—
340C2×C4—3324no—
341C2×C4—3324no—
342C2×C4—6642no—
343C2×C4—6642no—
344C2×C4—6642no—
345C2×C4—6642no—
346C2×C4#1458170712yes—
347C2×C4Q8×Dic8312yes—
348C2×C4#1458178111yescharacteristic
349C2×C4#1458179011yescharacteristic
350C8—6642no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—22no—
356C22—22no—
357C22—22no—
358C22—22no—
359C22#1912760811yescharacteristic
360C22C2×#1458170711yescharacteristic
361C22#1912767511yescharacteristic
362C22#1912769711yescharacteristic
363C22#1912770211yescharacteristic
364C22#1912771811yescharacteristic
365C22#1912771811yescharacteristic
366C22#1912799812yes—
367C22#1912800512yes—
368C4—22no—
369C4—21no—
370C4—21no—
371C4—22no—
372C4—41no—
373C4—41no—
374C4—41no—
375C4—81no—
376C4—81no—
377C4—81no—
378C4—81no—
379C4—6644no—
380C2—22no—
381C2—22no—
382C2#2490600111yescharacteristic
383C2#2490600611yescharacteristic
384C2#2490601911yescharacteristic
385C2#2490604011yescharacteristic
386C2#2490842111yescharacteristic
387C2#2490844511yescharacteristic
388C2#2490844911yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic83
commutatorC22×C4×C83C22×C4
FrattiniC23×C4C22×D83
FittingC83×#2851C2
socleC23×C83C2×C22⋊C4
prime radicalC24×C83C2×D4
cosocleC23×C4×C83C23
hypercenter#2851D83
hypocenterC83#9913
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
12. C2×#125×C83C4semidirect
14. #389×C83C4semidirect
15. #389×C83C4semidirect
20. #382×C83C4semidirect
200. C83#9913semidirect
201. C2×#125Dic83semidirect
205. #389Dic83semidirect
206. #389Dic83semidirect
211. #382Dic83semidirect

Maximal subgroups

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

subgroupindex
#249074432
#249056902
#249053412
C83×#28512
#991383