Finite Group Databasebeta

← #35323575

Subgroups — #35323575

Order 21248.

446 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
1C2×#19127943C211yesmaximal, characteristic
2C2×#19127506C211yesmaximal, characteristic, complemented
3C2×#19127504C211yesmaximal, characteristic, complemented
4C2×(C2×D83)⋊C16C211yesmaximal, characteristic
5#24907489C211yesmaximal, characteristic
6#24905554C211yesmaximal, characteristic, complemented
7C2×#367×C83C211yesmaximal, characteristic, complemented
8C22×Dic332C2211yescharacteristic
9C22×C8×D83C2211yescharacteristic
10C2×#14581772—21no—
11C22×C83⋊C16C2211yescharacteristic
12#19127943—22no—
13#19127943—22no—
14C2×#14581726C2211yescharacteristic
15C2×#14581715—21no—
16#19127558—21no—
17#19127506—22no—
18#19127504—22no—
19(C2×D83)⋊C16—22no—
20#19127463—21no—
21C22×C16×C83C2211yescharacteristic
22C2×#107×C83C2211yescharacteristic
23C2×#107×C83C2211yescharacteristic
24#367×C83—22no—
25C22×Dic166—21no—
26C22×C4×D83D411yescharacteristic
27C2×Dic332—21no—
28C2×Dic332—21no—
29C2×Dic332—21no—
30C2×Dic332—21no—
31C2×Dic332—21no—
32C2×Dic332D412yes—
33C22×C83⋊C8D411yescharacteristic
34C2×C8×D83—21no—
35C2×C8×D83—21no—
36C2×C8×D83—21no—
37C2×C8×D83—21no—
38C2×C8×D83—21no—
39C2×C8×D83—21no—
40#14581772—24no—
41#14581772—42no—
42C2×C4⋊Dic83—21no—
43C2×C4⋊Dic83—41no—
44C2×(C2×D83)⋊C4—41no—
45C2×(C2×D83)⋊C4—41no—
46C2×C83⋊C16—21no—
47C2×C83⋊C16—21no—
48C2×C83⋊C16—21no—
49#14581726—21no—
50#14581726—21no—
51#14581715—24no—
52C83⋊#95—41no—
53C22×C8×C83C2311yescharacteristic
54C2×C4⋊C4×C83—21no—
55C2×C4⋊C4×C83—21no—
56C2×C16×C83—21no—
57C2×C16×C83—21no—
58C2×C16×C83D412yes—
59#107×C83—21no—
60#107×C83—21no—
61#107×C83—21no—
62#107×C83D412yes—
63C23×D83D811yescharacteristic
64C2×Dic166—22no—
65C2×Dic166—41no—
66C2×Dic166—41no—
67C2×Dic166—41no—
68C2×Dic166—41no—
69C2×Dic166—41no—
70C22×Dic83—41no—
71C22×Dic83—41no—
72C22×Dic83D811yescharacteristic
73C2×C4×D83—21no—
74C2×C4×D83—21no—
75C2×C4×D83—21no—
76C2×C4×D83—21no—
77C2×C4×D83—21no—
78C2×C4×D83—21no—
79Dic332—22no—
80Dic332—22no—
81Dic332—41no—
82Dic332—41no—
83C2×C83⋊C8—21no—
84C2×C83⋊C8—21no—
85C2×C83⋊C8—21no—
86C8×D83—41no—
87C8×D83—41no—
88C8×D83—41no—
89C8×D83—41no—
90C4⋊Dic83—44no—
91C4⋊Dic83—41no—
92C4⋊Dic83—41no—
93(C2×D83)⋊C4—44no—
94(C2×D83)⋊C4—82no—
95C83⋊C16—41no—
96C22×C4×C83—41no—
97C22×C4×C83—41no—
98C22×C4×C83C2×D411yescharacteristic
99C2×C8×C83C2×D411yescharacteristic
100C2×C8×C83C2×D411yescharacteristic
101C2×C8×C83C2×D411yescharacteristic
102C2×C8×C83#4111yescharacteristic
103C2×C8×C83#4111yescharacteristic
104C2×C8×C83#4111yescharacteristic
105C4⋊C4×C83—22no—
106C4⋊C4×C83—41no—
107C4⋊C4×C83—41no—
108C4⋊C4×C83—41no—
109C16×C83—22no—
110C23×C83C2×D811yescharacteristic
111C22×D83—21no—
112C22×D83—21no—
113C22×D83—21no—
114C22×D83—21no—
115C22×D83—21no—
116C22×D83—21no—
117C22×D83SD3212yes—
118C2×C4×C83—42no—
119C2×C4×C83—81no—
120C2×C4×C83—81no—
121C2×C4×C83—81no—
122C2×C4×C83—81no—
123C2×C4×C83—81no—
124C2×C4×C83#12011yescharacteristic
125C2×C4×C83#12111yescharacteristic
126C2×C4×C83#12311yescharacteristic
127C2×C4×C83#13511yescharacteristic
128C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
129C2×C4×C83#13711yescharacteristic
130C2×Dic83—21no—
131C2×Dic83—21no—
132C2×Dic83—21no—
133C2×Dic83—42no—
134C2×Dic83—81no—
135C2×Dic83—81no—
136C2×Dic83—81no—
137C2×Dic83—81no—
138C2×Dic83—81no—
139C4×D83—41no—
140C4×D83—41no—
141C4×D83—41no—
142C4×D83—41no—
143Dic166—42no—
144Dic166—42no—
145Dic166—81no—
146Dic166—81no—
147C8×C83#12111yescharacteristic
148C8×C83#12111yescharacteristic
149C8×C83#12311yescharacteristic
150C8×C83#12411yescharacteristic
151C83⋊C8—41no—
152C22×C83#44711yescharacteristic
153C22×C83#46611yescharacteristic
154C22×C83#48011yescharacteristic
155C22×C83C2×SD3211yescharacteristic
156C22×C83#50811yescharacteristic
157C22×C83#51011yescharacteristic
158C22×C83#51011yescharacteristic
159C2×D83—22no—
160C2×D83—22no—
161C2×D83—22no—
162C2×D83—41no—
163C2×D83—41no—
164C2×D83—41no—
165C2×D83—41no—
166C4×C83—82no—
167C4×C83—161no—
168C4×C83#39411yescharacteristic
169C4×C83#45211yescharacteristic
170C4×C83#46811yescharacteristic
171C4×C83#48211yescharacteristic
172Dic83—41no—
173Dic83—82no—
174Dic83—161no—
175#13382—831nomaximal, Sylow
176C2×C83#203311yescharacteristic
177C2×C83#218511yescharacteristic
178C2×C83#218911yescharacteristic
179C2×C83#221711yescharacteristic
180C2×C83#221911yescharacteristic
181C2×C83#223511yescharacteristic
182C2×C83#223711yescharacteristic
183D83—42no—
184C2×#467—831no—
185C2×#367D8311yescharacteristic, complemented
186C2×#358—831no—
187C2×#358—831no—
188C2×C22⋊C16—831no—
189#1808—831no—
190#1380—831no—
191C83#1338211yescharacteristic, complemented, Sylow
192C22×Q16—831no—
193C23×C8—831no—
194C2×C22⋊Q8—1661no—
195C22×C16—831no—
196C22×C16C2×D8311yescharacteristic
197#467—1662no—
198#467—1662no—
199C2×#107—831no—
200C2×#107C2×D8311yescharacteristic
201C2×#107C2×D8311yescharacteristic
202C2×#103—1661no—
203#380—1661no—
204#367—22no—
205#358—1662no—
206#358—1662no—
207C22⋊C16—1662no—
208#340—1661no—
209C22×Q8—1661no—
210C23×C4—831no—
211C2×Q16—832no—
212C2×Q16—1661no—
213C2×Q16—1661no—
214C2×Q16—1661no—
215C2×Q16—1661no—
216C2×Q16—1661no—
217C22×C8—831no—
218C22×C8—1661no—
219C22×C8—1661no—
220C22×C8—1661no—
221C22×C8—1661no—
222C22×C8—1661no—
223C22×C8—1661no—
224C22×C8C22×D8311yescharacteristic
225C22⋊Q8—3322no—
226C22⋊Q8—1664no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—3321no—
230C2×C4⋊C4—1661no—
231C2×C22⋊C4—3321no—
232C2×C22⋊C4—3321no—
233C2×C16—21no—
234C2×C16—21no—
235C2×C16—1661no—
236C2×C16—1661no—
237C2×C16—1661no—
238C2×C16D33212yes—
239#107—21no—
240#107—21no—
241#107—21no—
242#107—1661no—
243#107—1661no—
244#107C22⋊D8312yes—
245#103—1664no—
246#95—3321no—
247C24—831no—
248C2×Q8—3321no—
249C2×Q8—3321no—
250C2×Q8—3321no—
251C2×Q8—3321no—
252C2×Q8—3321no—
253C2×Q8—1662no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—3321no—
257C22×C4—3321no—
258C22×C4—831no—
259C22×C4—1661no—
260C22×C4—1661no—
261C22×C4—1661no—
262C22×C4—1661no—
263C22×C4—1661no—
264C22×C4—1661no—
265C22×C4D4×D8311yescharacteristic
266Q16—3321no—
267Q16—3321no—
268Q16—1662no—
269Q16—1662no—
270C2×C8—3321no—
271C2×C8—3321no—
272C2×C8—3321no—
273C2×C8—3321no—
274C2×C8—1661no—
275C2×C8—1661no—
276C2×C8—1661no—
277C2×C8D4×D8311yescharacteristic
278C2×C8C2×C22⋊D8311yescharacteristic
279C2×C8C2×D33211yescharacteristic
280C2×C8#1305059011yescharacteristic
281C2×C8#1305059111yescharacteristic
282C2×C8#1305059111yescharacteristic
283C4⋊C4—22no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—3324no—
288C4⋊C4—3321no—
289C4⋊C4—3321no—
290C22⋊C4—3324no—
291C22⋊C4—6642no—
292C16—22no—
293C16—3321no—
294C23—832no—
295C23—1661no—
296C23—1661no—
297C23—1661no—
298C23—1661no—
299C23—1661no—
300C23—1661no—
301C23D8×D8311yescharacteristic
302Q8—3322no—
303Q8—3322no—
304Q8—6641no—
305Q8—6641no—
306C2×C4—42no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—3321no—
313C2×C4—3321no—
314C2×C4—3321no—
315C2×C4—3321no—
316C2×C4—3322no—
317C2×C4—6641no—
318C2×C4—6641no—
319C2×C4—6641no—
320C2×C4—6641no—
321C2×C4—6641no—
322C2×C4—1661no—
323C2×C4—1661no—
324C2×C4—1661no—
325C2×C4#1458176011yescharacteristic
326C2×C4#1458176711yescharacteristic
327C2×C4#1458177711yescharacteristic
328C2×C4#1458181511yescharacteristic
329C2×C4#1458181611yescharacteristic
330C2×C4#1458182411yescharacteristic
331C8—3321no—
332C8#1458176611yescharacteristic
333C8#1458176711yescharacteristic
334C8#1458177711yescharacteristic
335C8#1458178111yescharacteristic
336C22—3321no—
337C22—3321no—
338C22—3321no—
339C22—3321no—
340C22—1662no—
341C22—1662no—
342C22—1662no—
343C22#1912783311yescharacteristic
344C22#1912793811yescharacteristic
345C22#1912801611yescharacteristic
346C22SD32×D8311yescharacteristic
347C22#1912813211yescharacteristic
348C22#1912814011yescharacteristic
349C22#1912814011yescharacteristic
350C4—82no—
351C4—161no—
352C4—13281no—
353C4—3321no—
354C4—6642no—
355C4#1912761811yescharacteristic
356C4#1912786211yescharacteristic
357C4#1912794811yescharacteristic
358C4#1912802811yescharacteristic
359C2—3322no—
360C2#2490864711yescharacteristic
361C2#2490937811yescharacteristic
362C2#2490940011yescharacteristic
363C2#2490956011yescharacteristic
364C2#2490957211yescharacteristic
365C2#2490966211yescharacteristic
366C2#2490967411yescharacteristic
subgroupG/H
centerC23D8×D83
commutator, cosocleC22×C8×C83C23
FrattiniC22×C8C22×D83
FittingC2×#367×C83C2
socleC23×C83C2×D8
prime radicalC23×D83D8
hypercenterC2×#367D83
hypocenterC83#13382
LCSG ▷ C22×C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#367

Complements

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

NH ≅ G/Ntype
2. C2×#19127506C2semidirect
3. C2×#19127504C2semidirect
6. #24905554C2semidirect
7. C2×#367×C83C2semidirect
185. C2×#367D83semidirect
191. C83#13382semidirect

Maximal subgroups

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

subgroupindex
C2×#191279432
C2×#191275062
C2×#191275042
C2×(C2×D83)⋊C162
#249074892
#249055542
C2×#367×C832
#1338283