Finite Group Databasebeta

← #35295293

Subgroups — #35295293

Order 21248.

628 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24907225C211yesmaximal, characteristic
2C2×#19127438C212yesmaximal
3#24905690C211yesmaximal, characteristic
4#24905341C212yesmaximal
5C4×C22⋊Q8×C83C211yesmaximal, characteristic
6C2×C22⋊(C83⋊C8)C2212yes—
7C2×#14581703—21no—
8C2×C83⋊#95—21no—
9C2×C83⋊#95—21no—
10C2×C83⋊#95C2212yes—
11#19127456—21no—
12#19127438C418yescomplemented
13C2×C22⋊Q8×C83C2211yescharacteristic
14C2×C4×Q8×C83C411yescharacteristic, complemented
15C22×C42×C83C2211yescharacteristic
16#391×C83C411yescharacteristic, complemented
17#389×C83C411yescharacteristic, complemented
18#387×C83—21no—
19#385×C83—21no—
20#380×C83C2211yescharacteristic
21C4×C4⋊C4×C83—21no—
22C4×C4⋊C4×C83C411yescharacteristic, complemented
23C4×C22⋊C4×C83—21no—
24C23×Dic83—22no—
25C22×C83⋊C8—22no—
26C22⋊(C83⋊C8)—22no—
27C22⋊(C83⋊C8)C2×C414yes—
28#14581703—42no—
29C83⋊#95—22no—
30C83⋊#95—42no—
31C83⋊#95—42no—
32C83⋊#95C2×C414yes—
33C22×Q8×C83C2×C411yescharacteristic
34C23×C4×C83D411yescharacteristic
35C23×C4×C83D411yescharacteristic
36C23×C4×C83C2311yescharacteristic
37C22⋊Q8×C83—22no—
38C22⋊Q8×C83C2×C414yes—
39C4×Q8×C83—42no—
40C2×C4⋊C4×C83—21no—
41C2×C4⋊C4×C83—21no—
42C2×C4⋊C4×C83—21no—
43C2×C4⋊C4×C83C2×C411yescharacteristic
44C2×C4⋊C4×C83C2×C411yescharacteristic
45C2×C4⋊C4×C83C2×C411yescharacteristic
46C2×C22⋊C4×C83—21no—
47C2×C22⋊C4×C83—21no—
48C2×C42×C83—21no—
49C2×C42×C83—21no—
50C2×C42×C83—42no—
51C2×C42×C83C2×C411yescharacteristic
52C2×C42×C83C2×C411yescharacteristic
53C2×C42×C83D411yescharacteristic
54C2×C42×C83D411yescharacteristic
55#95×C83—21no—
56#95×C83—21no—
57#95×C83—21no—
58#95×C83—21no—
59#95×C83—21no—
60C22×Dic83—24no—
61C22×Dic83—42no—
62C22×Dic83—42no—
63C22×Dic83—42no—
64C22×Dic83—42no—
65C22×Dic83—42no—
66C22×Dic83—42no—
67C2×C83⋊C8—24no—
68C2×C83⋊C8—42no—
69C2×C83⋊C8—42no—
70C24×C83C2×D411yescharacteristic
71C2×Q8×C83—21no—
72C2×Q8×C83—41no—
73C2×Q8×C83—41no—
74C2×Q8×C83C4212yes—
75C22×C4×C83—21no—
76C22×C4×C83—21no—
77C22×C4×C83—21no—
78C22×C4×C83—21no—
79C22×C4×C83—22no—
80C22×C4×C83—22no—
81C22×C4×C83—21no—
82C22×C4×C83—21no—
83C22×C4×C83—41no—
84C22×C4×C83—41no—
85C22×C4×C83—41no—
86C22×C4×C83—41no—
87C22×C4×C83—41no—
88C22×C4×C83C22⋊C411yescharacteristic
89C22×C4×C83C22⋊C411yescharacteristic
90C22×C4×C83C22⋊C411yescharacteristic
91C22×C4×C83C22⋊C411yescharacteristic
92C22×C4×C83C22×C412yes—
93C22×C4×C83C22×C411yescharacteristic
94C22×C4×C83C2×D411yescharacteristic
95C22×C4×C83#4112yes—
96C4⋊C4×C83—21no—
97C4⋊C4×C83—24no—
98C4⋊C4×C83—41no—
99C4⋊C4×C83—41no—
100C4⋊C4×C83—42no—
101C4⋊C4×C83—42no—
102C4⋊C4×C83C4212yes—
103C22⋊C4×C83—24no—
104C22⋊C4×C83—41no—
105C22⋊C4×C83—41no—
106C42×C83—22no—
107C42×C83—22no—
108C42×C83—41no—
109C42×C83—41no—
110C42×C83—42no—
111C42×C83—42no—
112C23×C83—21no—
113C23×C83—21no—
114C23×C83—21no—
115C23×C83—21no—
116C23×C83C2×C22⋊C411yescharacteristic
117C23×C83C4×D412yes—
118C23×C83AGL(1,ℤ/8)11yescharacteristic
119C23×C83#13711yescharacteristic
120C23×C83#13712yes—
121Q8×C83—42no—
122Q8×C83—42no—
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—22no—
130C2×C4×C83—22no—
131C2×C4×C83—21no—
132C2×C4×C83—21no—
133C2×C4×C83—21no—
134C2×C4×C83—21no—
135C2×C4×C83—21no—
136C2×C4×C83—21no—
137C2×C4×C83—21no—
138C2×C4×C83—21no—
139C2×C4×C83—21no—
140C2×C4×C83—42no—
141C2×C4×C83—42no—
142C2×C4×C83—44no—
143C2×C4×C83—42no—
144C2×C4×C83—42no—
145C2×C4×C83—41no—
146C2×C4×C83—41no—
147C2×C4×C83—41no—
148C2×C4×C83—41no—
149C2×C4×C83—41no—
150C2×C4×C83—41no—
151C2×C4×C83—41no—
152C2×C4×C83—41no—
153C2×C4×C83—41no—
154C2×C4×C83—41no—
155C2×C4×C83#9912yes—
156C2×C4×C83C2×C4211yescharacteristic
157C2×C4×C83#11711yescharacteristic
158C2×C4×C83C4×D412yes—
159C2×Dic83—44no—
160C2×Dic83—44no—
161C2×Dic83—44no—
162C2×Dic83—82no—
163C2×Dic83—82no—
164C2×Dic83—82no—
165C2×Dic83—82no—
166C83⋊C8—44no—
167C22×C83—22no—
168C22×C83—22no—
169C22×C83—22no—
170C22×C83—21no—
171C22×C83—21no—
172C22×C83—21no—
173C22×C83—21no—
174C22×C83—22no—
175C22×C83C4×C22⋊C411yescharacteristic
176C22×C83C2×#9911yescharacteristic
177C22×C83#41011yescharacteristic
178C22×C83#41711yescharacteristic
179C22×C83#41911yescharacteristic
180C22×C83#42111yescharacteristic
181C22×C83#42111yescharacteristic
182C22×C83#44012yes—
183C22×C83#44112yes—
184C4×C83—22no—
185C4×C83—21no—
186C4×C83—21no—
187C4×C83—22no—
188C4×C83—42no—
189C4×C83—42no—
190C4×C83—41no—
191C4×C83—41no—
192C4×C83—41no—
193C4×C83—81no—
194C4×C83—81no—
195Dic83—84no—
196#8217—831nomaximal, Sylow
197C2×C83—22no—
198C2×C83—22no—
199C2×C83#151111yescharacteristic
200C2×C83#151211yescharacteristic
201C2×C83#151311yescharacteristic
202C2×C83#151511yescharacteristic
203C2×C83C4×#9911yescharacteristic
204C2×C83#175911yescharacteristic
205C2×C83#176311yescharacteristic
206C4×C22⋊Q8D8311yescharacteristic
207#1749—831no—
208C2×#328—832no—
209#1434—831no—
210#1294—832no—
211C83#821711yescharacteristic, complemented, Sylow
212C2×C22⋊Q8C2×D8311yescharacteristic
213C2×C4×Q8Dic8311yescharacteristic, complemented
214C22×C42C2×D8311yescharacteristic
215C2×C22⋊C8—832no—
216C2×#97—1661no—
217#391Dic8311yescharacteristic, complemented
218#389Dic8311yescharacteristic, complemented
219#387—21no—
220#385—21no—
221#380C2×D8311yescharacteristic
222C4×C4⋊C4—21no—
223C4×C4⋊C4Dic8311yescharacteristic, complemented
224C4×C22⋊C4—21no—
225C2×#95—832no—
226C2×#95—1661no—
227C2×#95—1661no—
228#336—1661no—
229#328—838no—
230C22×Q8C2×Dic8311yescharacteristic
231C23×C4—1662no—
232C23×C4C22⋊D8311yescharacteristic
233C23×C4C22⋊D8311yescharacteristic
234C23×C4C22×D8311yescharacteristic
235C22×C8—1662no—
236C22⋊Q8—22no—
237C22⋊Q8C4×D8314yes—
238C4×Q8—42no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4C2×Dic8311yescharacteristic
243C2×C4⋊C4C2×Dic8311yescharacteristic
244C2×C4⋊C4C2×Dic8311yescharacteristic
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—42no—
250C2×C42C2×Dic8311yescharacteristic
251C2×C42C2×Dic8311yescharacteristic
252C2×C42C22⋊D8311yescharacteristic
253C2×C42C22⋊D8311yescharacteristic
254C22⋊C8—834no—
255C22⋊C8—1662no—
256#97—3322no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—3322no—
263#95—3322no—
264#95—834no—
265#95—1662no—
266C24C2×C22⋊D8311yescharacteristic
267C2×Q8—21no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8C4×Dic8312yes—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—3322no—
285C22×C4—3322no—
286C22×C4—3322no—
287C22×C4—3322no—
288C22×C4—3322no—
289C22×C4—3322no—
290C22×C4—1664no—
291C22×C4C22⋊Dic8311yescharacteristic
292C22×C4C22⋊Dic8311yescharacteristic
293C22×C4C22⋊Dic8311yescharacteristic
294C22×C4C22⋊Dic8311yescharacteristic
295C22×C4C2×C4×D8312yes—
296C22×C4C22×Dic8311yescharacteristic
297C22×C4C2×C22⋊D8311yescharacteristic
298C22×C4#1305059012yes—
299C2×C8—3322no—
300C2×C8—3322no—
301C2×C8—1664no—
302C4⋊C4—21no—
303C4⋊C4—24no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4C4×Dic8312yes—
309C22⋊C4—24no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C42—22no—
313C42—22no—
314C42—41no—
315C42—41no—
316C42—42no—
317C42—42no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23C2×C22⋊Dic8311yescharacteristic
323C23C4×C22⋊D8312yes—
324C23#1458181711yescharacteristic
325C23#1458182211yescharacteristic
326C23#1458182312yes—
327Q8—42no—
328Q8—42no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—44no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—3324no—
362C2×C4—3324no—
363C2×C4—3324no—
364C2×C4—6642no—
365C2×C4—6642no—
366C2×C4—6642no—
367C2×C4—6642no—
368C2×C4#1458170712yes—
369C2×C4C2×C4×Dic8311yescharacteristic
370C2×C4#1458175111yescharacteristic
371C2×C4C4×C22⋊D8312yes—
372C8—3324no—
373C22—22no—
374C22—22no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—22no—
380C22—22no—
381C22C4×C22⋊Dic8311yescharacteristic
382C22C2×#1458170711yescharacteristic
383C22#1912767511yescharacteristic
384C22#1912770211yescharacteristic
385C22#1912771011yescharacteristic
386C22#1912771811yescharacteristic
387C22#1912771811yescharacteristic
388C22#1912779912yes—
389C22#1912780612yes—
390C4—22no—
391C4—22no—
392C4—21no—
393C4—21no—
394C4—42no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—42no—
399C4—81no—
400C4—81no—
401C4—6644no—
402C2—22no—
403C2—22no—
404C2#2490599611yescharacteristic
405C2#2490600111yescharacteristic
406C2#2490600611yescharacteristic
407C2#2490601411yescharacteristic
408C2C4×#1458170711yescharacteristic
409C2#2490727311yescharacteristic
410C2#2490729311yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic83
commutatorC2×C4×C83C2×C42
FrattiniC23×C4C22×D83
FittingC4×C22⋊Q8×C83C2
socleC23×C83C2×C22⋊C4
prime radicalC24×C83C2×D4
cosocleC23×C4×C83C23
hypercenterC4×C22⋊Q8D83
hypocenterC83#8217
LCSG ▷ C2×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
12. #19127438C4semidirect
14. C2×C4×Q8×C83C4semidirect
16. #391×C83C4semidirect
17. #389×C83C4semidirect
22. C4×C4⋊C4×C83C4semidirect
211. C83#8217semidirect
213. C2×C4×Q8Dic83semidirect
217. #391Dic83semidirect
218. #389Dic83semidirect
223. C4×C4⋊C4Dic83semidirect

Maximal subgroups

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

subgroupindex
#249072252
C2×#191274382
#249056902
#249053412
C4×C22⋊Q8×C832
#821783