Finite Group Databasebeta

← #35322869

Subgroups — #35322869

Order 21248.

598 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128076C211yesmaximal, characteristic
2C2×Dic83⋊C16C211yesmaximal, characteristic, complemented
3C2×#19127503C211yesmaximal, characteristic, complemented
4C2×#19127502C211yesmaximal, characteristic, complemented
5C2×#19127501C211yesmaximal, characteristic, complemented
6#107×Dic83C211yesmaximal, characteristic, complemented
7C2×#357×C83C211yesmaximal, characteristic, complemented
8C22×D664C2211yescharacteristic
9C22×#13050575—21no—
10C2×#14581788—21no—
11C22×C83⋊C16C2211yescharacteristic
12#19128076—22no—
13#19128076C414yescomplemented
14C2×#14581726C2211yescharacteristic
15C2×C8×Dic83C2211yescharacteristic
16C4⋊C4×Dic83—21no—
17Dic83⋊C16—22no—
18#19127503—22no—
19#19127502—22no—
20#19127501—22no—
21#19127463—21no—
22C22×D8×C83C2211yescharacteristic
23C22×C16×C83C2211yescharacteristic
24C2×#107×C83C2211yescharacteristic
25#357×C83—22no—
26C22×D332—21no—
27C22×C22⋊D83—41no—
28C22×C22⋊D83—41no—
29C2×D664—21no—
30C2×D664—21no—
31C2×D664—21no—
32C2×D664—21no—
33C2×D664—21no—
34C2×D664C2×C412yes—
35C2×#13050575—24no—
36C2×#13050575—42no—
37C2×#13050575—42no—
38C22×C83⋊C8D411yescharacteristic
39#14581788—24no—
40#14581788—42no—
41C2×C4⋊Dic83—21no—
42C2×C4×Dic83—41no—
43C2×C4×Dic83D411yescharacteristic
44C2×C83⋊C16—21no—
45C2×C83⋊C16—21no—
46C2×C83⋊C16—21no—
47#14581726—21no—
48#14581726D412yescomplemented
49C8×Dic83C2×C412yes—
50C8×Dic83D412yes—
51C83⋊#95—41no—
52C22×D4×C83—21no—
53C2×D8×C83—21no—
54C2×D8×C83—21no—
55C2×D8×C83—21no—
56C2×D8×C83—21no—
57C2×D8×C83—21no—
58C2×D8×C83C2×C412yes—
59C22×C8×C83C2311yescharacteristic
60C2×C4⋊C4×C83—21no—
61C2×C16×C83—21no—
62C2×C16×C83—21no—
63C2×C16×C83—21no—
64#107×C83—21no—
65#107×C83D412yescomplemented
66C23×D83—41no—
67C2×D332—22no—
68C2×D332—41no—
69C2×D332—41no—
70C2×D332—41no—
71C2×D332—41no—
72C2×D332—41no—
73C2×C22⋊D83—44no—
74C2×C22⋊D83—44no—
75C2×C22⋊D83—82no—
76C2×C22⋊D83—82no—
77C2×C22⋊D83—82no—
78C2×C22⋊D83—82no—
79C22×Dic83—41no—
80C22×Dic83D811yescharacteristic
81C22×Dic83D811yescharacteristic
82D664—22no—
83D664—22no—
84D664—41no—
85D664—41no—
86#13050575—48no—
87C2×C83⋊C8—21no—
88C2×C83⋊C8—21no—
89C2×C83⋊C8C22⋊C412yes—
90C4⋊Dic83—22no—
91C4⋊Dic83—41no—
92C4×Dic83—82no—
93C4×Dic83C22⋊C412yes—
94C4×Dic83SD1612yes—
95C83⋊C16—41no—
96C24×C83—41no—
97C2×D4×C83—22no—
98C2×D4×C83—41no—
99C2×D4×C83—41no—
100C2×D4×C83—41no—
101C2×D4×C83—41no—
102C2×D4×C83—41no—
103C22×C4×C83—41no—
104C22×C4×C83C2×D411yescharacteristic
105D8×C83—22no—
106D8×C83—22no—
107D8×C83—41no—
108D8×C83—41no—
109C2×C8×C83C22×C411yescharacteristic
110C2×C8×C83C2×D411yescharacteristic
111C2×C8×C83C2×D411yescharacteristic
112C2×C8×C83C2×D411yescharacteristic
113C2×C8×C83#4111yescharacteristic
114C2×C8×C83#4111yescharacteristic
115C4⋊C4×C83—22no—
116C4⋊C4×C83—41no—
117C16×C83—41no—
118C23×C83—42no—
119C23×C83—81no—
120C23×C83—81no—
121C23×C83—81no—
122C23×C83—81no—
123C23×C83—81no—
124C23×C83—81no—
125C23×C83C2×D811yescharacteristic
126C22×D83—42no—
127C22×D83—81no—
128C22×D83—81no—
129C22×D83—81no—
130C22×D83—81no—
131C22×D83—81no—
132C22×D83—81no—
133D4×C83—42no—
134D4×C83—42no—
135D4×C83—81no—
136D4×C83—81no—
137C2×C4×C83—42no—
138C2×C4×C83—81no—
139C2×C4×C83—81no—
140C2×C4×C83C2×C22⋊C411yescharacteristic
141C2×C4×C83#12111yescharacteristic
142C2×C4×C83#12111yescharacteristic
143C2×C4×C83C2×SD1611yescharacteristic
144C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
145C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
146C22⋊D83—88no—
147C22⋊D83—88no—
148C2×Dic83—21no—
149C2×Dic83—21no—
150C2×Dic83—42no—
151C2×Dic83—81no—
152C2×Dic83—81no—
153C2×Dic83#10212yes—
154C2×Dic83#10212yes—
155C2×Dic83D1612yes—
156C2×Dic83SD3212yes—
157D332—42no—
158D332—42no—
159D332—81no—
160D332—81no—
161C8×C83C4×D411yescharacteristic
162C8×C83C4×D411yescharacteristic
163C8×C83#12411yescharacteristic
164C8×C83#12711yescharacteristic
165C83⋊C8—22no—
166C22×C83—82no—
167C22×C83—82no—
168C22×C83—82no—
169C22×C83—161no—
170C22×C83—161no—
171C22×C83—161no—
172C22×C83—161no—
173C22×C83C2×#10211yescharacteristic
174C22×C83#45911yescharacteristic
175C22×C83#45911yescharacteristic
176C22×C83C2×D1611yescharacteristic
177C22×C83C2×SD3211yescharacteristic
178C22×C83#50911yescharacteristic
179C22×C83#50911yescharacteristic
180C2×D83—82no—
181C2×D83—82no—
182C2×D83—82no—
183C2×D83—161no—
184C2×D83—161no—
185C2×D83—161no—
186C2×D83—161no—
187C4×C83—82no—
188C4×C83#39011yescharacteristic
189C4×C83#41811yescharacteristic
190C4×C83#46011yescharacteristic
191C4×C83#46011yescharacteristic
192Dic83—22no—
193Dic83—82no—
194Dic83#35714yescomplemented
195#13252—831nomaximal, Sylow
196C2×C83—162no—
197C2×C83#198211yescharacteristic
198C2×C83C2×#35711yescharacteristic
199C2×C83#214011yescharacteristic
200C2×C83#220611yescharacteristic
201C2×C83#220611yescharacteristic
202C2×C83#220711yescharacteristic
203C2×C83#220711yescharacteristic
204D83—162no—
205C2×#493—831no—
206C2×C4⋊C16—831no—
207C2×#357—831no—
208C2×#357—831no—
209C2×#357—831no—
210C2×#357D8311yescharacteristic, complemented
211C4×#107—831no—
212C83#1325211yescharacteristic, complemented, Sylow
213C22×D8—831no—
214C22×D8—1661no—
215C22×D8C2×D8311yescharacteristic
216C2×#127—1661no—
217C22×C16—831no—
218C22×C16C2×D8311yescharacteristic
219#493—834no—
220#493—1662no—
221C2×#107—831no—
222C2×#107C2×D8311yescharacteristic
223C2×C4×C8—831no—
224C4×C4⋊C4—1661no—
225C4⋊C16—1662no—
226#357—22no—
227#357—1662no—
228#357—1662no—
229#357—1662no—
230#340—1661no—
231C22×D4—21no—
232C22×D4—3321no—
233C22×D4—3321no—
234C22×D4—1661no—
235C2×D8—21no—
236C2×D8—21no—
237C2×D8—21no—
238C2×D8—21no—
239C2×D8—21no—
240C2×D8—3322no—
241C2×D8—3322no—
242C2×D8—832no—
243C2×D8—1664no—
244C2×D8—1661no—
245C2×D8—1661no—
246C2×D8—1661no—
247C2×D8—1661no—
248C2×D8—1661no—
249C2×D8C4×D8312yes—
250C22×C8—831no—
251C22×C8C22×D8311yescharacteristic
252#127—3322no—
253#127—1664no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—1661no—
256C2×C42—3321no—
257C2×C42—831no—
258C2×C16—21no—
259C2×C16—21no—
260C2×C16—21no—
261C2×C16—1661no—
262C2×C16—1661no—
263C2×C16—1661no—
264#107—21no—
265#107—832no—
266#107—1661no—
267#107C22⋊D8312yescomplemented
268C4×C8—832no—
269C4×C8—832no—
270#95—3321no—
271C24—41no—
272C24—3321no—
273C2×D4—22no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—3324no—
280C2×D4—3324no—
281C2×D4—3321no—
282C2×D4—3321no—
283C2×D4—3321no—
284C2×D4—3321no—
285C2×D4—3321no—
286C2×D4—6642no—
287C2×D4—6642no—
288C2×D4—6642no—
289C2×D4—6642no—
290C2×D4—1662no—
291C22×C4—41no—
292C22×C4—3321no—
293C22×C4—831no—
294C22×C4—831no—
295C22×C4D4×D8311yescharacteristic
296D8—22no—
297D8—22no—
298D8—41no—
299D8—41no—
300D8—3328no—
301D8—3321no—
302D8—3321no—
303D8—1662no—
304D8—1662no—
305C2×C8—832no—
306C2×C8—1661no—
307C2×C8—1661no—
308C2×C8C2×C4×D8311yescharacteristic
309C2×C8D4×D8311yescharacteristic
310C2×C8D4×D8311yescharacteristic
311C2×C8C2×C22⋊D8311yescharacteristic
312C2×C8#1305058911yescharacteristic
313C2×C8#1305059011yescharacteristic
314C4⋊C4—22no—
315C4⋊C4—41no—
316C4⋊C4—3321no—
317C4⋊C4—1662no—
318C42—832no—
319C42—832no—
320C42—6642no—
321C16—41no—
322C16—3321no—
323C23—42no—
324C23—81no—
325C23—81no—
326C23—81no—
327C23—81no—
328C23—81no—
329C23—81no—
330C23—3322no—
331C23—6641no—
332C23—6641no—
333C23—6641no—
334C23—6641no—
335C23—6641no—
336C23—6641no—
337C23D8×D8311yescharacteristic
338D4—42no—
339D4—42no—
340D4—81no—
341D4—81no—
342D4—3322no—
343D4—3322no—
344D4—6648no—
345D4—6648no—
346D4—6641no—
347D4—6641no—
348C2×C4—42no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—3322no—
352C2×C4—832no—
353C2×C4—832no—
354C2×C4—832no—
355C2×C4—832no—
356C2×C4—6641no—
357C2×C4—6641no—
358C2×C4—1661no—
359C2×C4—1661no—
360C2×C4C22⋊C4×D8311yescharacteristic
361C2×C4#1458176411yescharacteristic
362C2×C4#1458176511yescharacteristic
363C2×C4SD16×D8311yescharacteristic
364C2×C4#1458181611yescharacteristic
365C2×C4#1458181911yescharacteristic
366C8—1662no—
367C8(C22⋊D83)⋊C411yescharacteristic
368C8C4×C22⋊D8311yescharacteristic
369C8#1458177811yescharacteristic
370C8#1458178811yescharacteristic
371C22—82no—
372C22—82no—
373C22—82no—
374C22—161no—
375C22—161no—
376C22—161no—
377C22—161no—
378C22—13281no—
379C22—13281no—
380C22—13281no—
381C22—13281no—
382C22—6642no—
383C22—6642no—
384C22—6642no—
385C22#102×D8311yescharacteristic
386C22#1912789411yescharacteristic
387C22#1912789511yescharacteristic
388C22D16×D8311yescharacteristic
389C22SD32×D8311yescharacteristic
390C22#1912813511yescharacteristic
391C22#1912813811yescharacteristic
392C4—82no—
393C4—834no—
394C4—6642no—
395C4—1662no—
396C4#1912760111yescharacteristic
397C4#1912770411yescharacteristic
398C4#1912789911yescharacteristic
399C4#1912790011yescharacteristic
400C2—162no—
401C2—13282no—
402C2#2490838811yescharacteristic
403C2#357×D8311yescharacteristic
404C2#2490917711yescharacteristic
405C2#2490949611yescharacteristic
406C2#2490949711yescharacteristic
407C2#2490950111yescharacteristic
408C2#2490950211yescharacteristic
subgroupG/H
centerC23D8×D83
commutatorC2×C8×C83C22×C4
FrattiniC22×C8C22×D83
FittingC2×#357×C83C2
socleC23×C83C2×D8
prime radicalC2×#19128076C2
cosocleC22×C8×C83C23
hypercenterC2×#357D83
hypocenterC83#13252
LCSG ▷ C2×C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#357

Complements

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

NH ≅ G/Ntype
2. C2×Dic83⋊C16C2semidirect
3. C2×#19127503C2semidirect
4. C2×#19127502C2semidirect
5. C2×#19127501C2semidirect
6. #107×Dic83C2semidirect
7. C2×#357×C83C2semidirect
13. #19128076C4semidirect
48. #14581726D4semidirect
65. #107×C83D4semidirect
194. Dic83#357semidirect
210. C2×#357D83semidirect
212. C83#13252semidirect
267. #107C22⋊D83semidirect

Maximal subgroups

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

subgroupindex
C2×#191280762
C2×Dic83⋊C162
C2×#191275032
C2×#191275022
C2×#191275012
#107×Dic832
C2×#357×C832
#1325283