Finite Group Databasebeta

← #35323563

Subgroups — #35323563

Order 21248.

498 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127528C211yesmaximal, characteristic, complemented
2C2×#19127502C211yesmaximal, characteristic, complemented
3C2×#19127502C211yesmaximal, characteristic, complemented
4C2×C22⋊(C83⋊C16)C211yesmaximal, characteristic, complemented
5#24907492C211yesmaximal, characteristic, complemented
6#24905554C211yesmaximal, characteristic, complemented
7C2×#466×C83C211yesmaximal, characteristic
8C22×C83⋊C16C2211yescharacteristic
9C22×C83⋊C16C2211yescharacteristic
10C2×#14581726C2211yescharacteristic
11C2×#14581726C2211yescharacteristic
12#19127561—21no—
13#19127528—22no—
14#19127502—22no—
15#19127502—22no—
16C22⋊(C83⋊C16)—22no—
17#19127464—21no—
18C22×D8×C83C2211yescharacteristic
19C23×C8×C83C2211yescharacteristic
20C2×#121×C83—21no—
21#466×C83—22no—
22#466×C83—22no—
23C2×#107×C83C2211yescharacteristic
24C2×#102×C83—21no—
25C2×C4⋊Dic83—21no—
26C2×C4⋊Dic83—21no—
27C2×C22⋊Dic83—41no—
28C2×C83⋊C16—21no—
29C2×C83⋊C16—21no—
30C2×C83⋊C16—21no—
31C2×C83⋊C16—21no—
32C2×C83⋊C16—21no—
33C2×C83⋊C16D412yescomplemented
34#14581726—21no—
35#14581726—21no—
36#14581726—21no—
37#14581726—21no—
38C83⋊#95—41no—
39C22×D4×C83—21no—
40C22×D4×C83—41no—
41C23×C4×C83D411yescharacteristic
42C2×D8×C83—21no—
43C2×D8×C83—21no—
44C2×D8×C83—21no—
45C2×D8×C83—21no—
46C2×D8×C83—21no—
47C2×D8×C83D412yes—
48C22×C8×C83—21no—
49C22×C8×C83—21no—
50C22×C8×C83—21no—
51C22×C8×C83—21no—
52C22×C8×C83—21no—
53C22×C8×C83—21no—
54C22×C8×C83D411yescharacteristic
55C22×C8×C83C2311yescharacteristic
56#121×C83—24no—
57#121×C83—42no—
58C2×C4⋊C4×C83—21no—
59C2×C22⋊C4×C83—41no—
60#107×C83—21no—
61#107×C83D412yes—
62#102×C83—24no—
63C22×Dic83—41no—
64C22×Dic83—41no—
65C4⋊Dic83—41no—
66C4⋊Dic83—41no—
67C4⋊Dic83—41no—
68C4⋊Dic83—41no—
69C22⋊Dic83—82no—
70C83⋊C16—22no—
71C83⋊C16—41no—
72C24×C83—41no—
73C24×C83D811yescharacteristic
74C2×D4×C83—22no—
75C2×D4×C83—44no—
76C2×D4×C83—41no—
77C2×D4×C83—41no—
78C2×D4×C83—41no—
79C2×D4×C83—41no—
80C2×D4×C83—41no—
81C2×D4×C83—82no—
82C2×D4×C83—82no—
83C22×C4×C83—21no—
84C22×C4×C83—21no—
85C22×C4×C83—21no—
86C22×C4×C83—21no—
87C22×C4×C83—21no—
88C22×C4×C83—21no—
89C22×C4×C83—41no—
90C22×C4×C83D811yescharacteristic
91C22×C4×C83C2×D411yescharacteristic
92D8×C83—22no—
93D8×C83—22no—
94D8×C83—41no—
95D8×C83—41no—
96C2×C8×C83—21no—
97C2×C8×C83—21no—
98C2×C8×C83—21no—
99C2×C8×C83—41no—
100C2×C8×C83—41no—
101C2×C8×C83—41no—
102C2×C8×C83—41no—
103C2×C8×C83C2×D411yescharacteristic
104C2×C8×C83C2×D411yescharacteristic
105C2×C8×C83C2×D411yescharacteristic
106C2×C8×C83#4111yescharacteristic
107C2×C8×C83#4111yescharacteristic
108C2×C8×C83#4111yescharacteristic
109C4⋊C4×C83—22no—
110C4⋊C4×C83—41no—
111C22⋊C4×C83—44no—
112C23×C83—21no—
113C23×C83—21no—
114C23×C83—21no—
115C23×C83—21no—
116C23×C83—21no—
117C23×C83—21no—
118C23×C83—42no—
119C23×C83—81no—
120C23×C83—81no—
121C23×C83—81no—
122C23×C83—81no—
123C23×C83—81no—
124C23×C83—81no—
125C23×C83SD3212yes—
126C23×C83C2×D811yescharacteristic
127D4×C83—42no—
128D4×C83—42no—
129D4×C83—88no—
130D4×C83—81no—
131D4×C83—81no—
132C2×C4×C83—21no—
133C2×C4×C83—21no—
134C2×C4×C83—21no—
135C2×C4×C83—41no—
136C2×C4×C83—41no—
137C2×C4×C83—41no—
138C2×C4×C83—41no—
139C2×C4×C83—42no—
140C2×C4×C83—81no—
141C2×C4×C83—81no—
142C2×C4×C83#12011yescharacteristic
143C2×C4×C83#12111yescharacteristic
144C2×C4×C83#12311yescharacteristic
145C2×C4×C83#13511yescharacteristic
146C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C83#13711yescharacteristic
148C2×Dic83—81no—
149C2×Dic83—81no—
150C2×Dic83—81no—
151C2×Dic83—81no—
152C2×Dic83—81no—
153C2×Dic83—81no—
154C8×C83—41no—
155C8×C83#12111yescharacteristic
156C8×C83#12111yescharacteristic
157C8×C83#12311yescharacteristic
158C8×C83#12411yescharacteristic
159C22×C83—22no—
160C22×C83—22no—
161C22×C83—22no—
162C22×C83—41no—
163C22×C83—41no—
164C22×C83—41no—
165C22×C83—41no—
166C22×C83—82no—
167C22×C83—82no—
168C22×C83—82no—
169C22×C83—161no—
170C22×C83—161no—
171C22×C83—161no—
172C22×C83—161no—
173C22×C83#44711yescharacteristic
174C22×C83#46611yescharacteristic
175C22×C83#48011yescharacteristic
176C22×C83C2×SD3211yescharacteristic
177C22×C83#50811yescharacteristic
178C22×C83#50911yescharacteristic
179C22×C83#50911yescharacteristic
180C4×C83—41no—
181C4×C83—82no—
182C4×C83#39411yescharacteristic
183C4×C83#45211yescharacteristic
184C4×C83#46811yescharacteristic
185C4×C83#48211yescharacteristic
186Dic83—161no—
187Dic83—161no—
188#13381—831nomaximal, Sylow
189C2×C83—42no—
190C2×C83—162no—
191C2×C83#203311yescharacteristic
192C2×C83#218611yescharacteristic
193C2×C83#218711yescharacteristic
194C2×C83#221711yescharacteristic
195C2×C83#221811yescharacteristic
196C2×C83#223311yescharacteristic
197C2×C83#223411yescharacteristic
198C2×#466D8311yescharacteristic
199C2×#367—831no—
200C2×#357—831no—
201C2×#357—831no—
202C2×C22⋊C16—831no—
203#1808—831no—
204#1380—831no—
205C83#1338111yescharacteristic, complemented, Sylow
206C22×D8C2×D8311yescharacteristic
207C23×C8C2×D8311yescharacteristic
208C2×#121—21no—
209C22×C16—831no—
210C22×C16—831no—
211#466—22no—
212#466—22no—
213C2×#107—831no—
214C2×#107—831no—
215C2×#107C2×D8311yescharacteristic
216C2×#102—21no—
217#380—1661no—
218#367—1662no—
219#357—1662no—
220#357—1662no—
221C22⋊C16—1662no—
222#340—1661no—
223C22×D4—21no—
224C22×D4—41no—
225C23×C4C22⋊D8311yescharacteristic
226C2×D8—21no—
227C2×D8—21no—
228C2×D8—21no—
229C2×D8—21no—
230C2×D8—21no—
231C2×D8C22⋊D8312yes—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8C22⋊D8311yescharacteristic
239C22×C8C22×D8311yescharacteristic
240#121—24no—
241#121—42no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—1661no—
244C2×C4⋊C4—1661no—
245C2×C22⋊C4—41no—
246C2×C22⋊C4—3321no—
247C2×C16—832no—
248C2×C16—1661no—
249C2×C16—1661no—
250C2×C16—1661no—
251C2×C16—1661no—
252C2×C16—1661no—
253#107—21no—
254#107—1661no—
255#107—1661no—
256#107—1661no—
257#107—1661no—
258#107C22⋊D8312yes—
259#102—24no—
260#95—3321no—
261C24—41no—
262C24#1305057511yescharacteristic
263C2×D4—22no—
264C2×D4—44no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—82no—
271C2×D4—82no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—41no—
279C22×C4—3321no—
280C22×C4—3321no—
281C22×C4#1305057511yescharacteristic
282C22×C4C2×C22⋊D8311yescharacteristic
283D8—22no—
284D8—22no—
285D8—41no—
286D8—41no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8D4×D8311yescharacteristic
295C2×C8C2×C22⋊D8311yescharacteristic
296C2×C8C2×C22⋊D8311yescharacteristic
297C2×C8#1305058911yescharacteristic
298C2×C8#1305058911yescharacteristic
299C2×C8#1305058911yescharacteristic
300C4⋊C4—22no—
301C4⋊C4—41no—
302C4⋊C4—3321no—
303C4⋊C4—3321no—
304C4⋊C4—3321no—
305C4⋊C4—3321no—
306C22⋊C4—44no—
307C22⋊C4—6642no—
308C16—3321no—
309C16—1662no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—42no—
317C23—81no—
318C23—81no—
319C23—81no—
320C23—81no—
321C23—81no—
322C23—81no—
323C23#1458173612yes—
324C23C2×#1305057511yescharacteristic
325D4—42no—
326D4—42no—
327D4—88no—
328D4—81no—
329D4—81no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—42no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—6641no—
341C2×C4—6641no—
342C2×C4—6641no—
343C2×C4—6641no—
344C2×C4—6641no—
345C2×C4—6641no—
346C2×C4#1458176111yescharacteristic
347C2×C4#1458176411yescharacteristic
348C2×C4#1458177311yescharacteristic
349C2×C4#1458181211yescharacteristic
350C2×C4#1458181811yescharacteristic
351C2×C4#1458182211yescharacteristic
352C8—41no—
353C8#1458176311yescharacteristic
354C8#1458176411yescharacteristic
355C8#1458177311yescharacteristic
356C8#1458177911yescharacteristic
357C22—22no—
358C22—22no—
359C22—22no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—82no—
365C22—82no—
366C22—82no—
367C22—161no—
368C22—161no—
369C22—161no—
370C22—161no—
371C22#1912783511yescharacteristic
372C22#1912793511yescharacteristic
373C22#1912801211yescharacteristic
374C22C2×#1458173611yescharacteristic
375C22#1912813111yescharacteristic
376C22#1912813611yescharacteristic
377C22#1912813711yescharacteristic
378C4—41no—
379C4—82no—
380C4—13281no—
381C4—13281no—
382C4#1912761511yescharacteristic
383C4#1912786311yescharacteristic
384C4#1912794511yescharacteristic
385C4#1912802211yescharacteristic
386C2—42no—
387C2—162no—
388C2#2490864211yescharacteristic
389C2#2490938611yescharacteristic
390C2#2490939211yescharacteristic
391C2#2490955711yescharacteristic
392C2#2490956411yescharacteristic
393C2#2490964611yescharacteristic
394C2#2490965111yescharacteristic
subgroupG/H
centerC23C2×#13050575
commutator, cosocleC22×C8×C83C23
FrattiniC22×C8C22×D83
Fitting, prime radicalC2×#466×C83C2
socleC23×C83C2×D8
hypercenterC2×#466D83
hypocenterC83#13381
LCSG ▷ C22×C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ C2×#466

Complements

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

NH ≅ G/Ntype
1. C2×#19127528C2semidirect
2. C2×#19127502C2semidirect
3. C2×#19127502C2semidirect
4. C2×C22⋊(C83⋊C16)C2semidirect
5. #24907492C2semidirect
6. #24905554C2semidirect
33. C2×C83⋊C16D4semidirect
205. C83#13381semidirect

Maximal subgroups

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

subgroupindex
C2×#191275282
C2×#191275022
C2×C22⋊(C83⋊C16)2
#249074922
#249055542
C2×#466×C832
#1338183