Finite Group Databasebeta

← #35303647

Subgroups — #35303647

Order 21248.

654 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128077C211yesmaximal, characteristic
2#24908391C211yesmaximal, characteristic, complemented
3#24908389C211yesmaximal, characteristic, complemented
4#24907328C211yesmaximal, characteristic, complemented
5C2×#19127453C211yesmaximal, characteristic, complemented
6C2×#19127444C211yesmaximal, characteristic, complemented
7C2×#331×C83C211yesmaximal, characteristic, complemented
8C22×#13050575—21no—
9C22×#13050575—21no—
10C2×#14581789C2211yescharacteristic
11#19128077—22no—
12#19128077C414yescomplemented
13C2×C4⋊(C83⋊C8)C2211yescharacteristic
14C2×#14581714—22no—
15C2×#14581713—22no—
16C2×C4×C83⋊C8C2211yescharacteristic
17C2×C8×Dic83—21no—
18C4×C4⋊Dic83C2211yescharacteristic
19#19127455—21no—
20#19127453—22no—
21#19127444—22no—
22C2×#127×C83C2211yescharacteristic
23C2×C4⋊C8×C83C2211yescharacteristic
24C2×C4⋊C8×C83C2211yescharacteristic
25#331×C83—22no—
26C22×D332—21no—
27C22×D332—21no—
28C22×D332—21no—
29C2×#13050575—24no—
30C2×#13050575—24no—
31C2×#13050575—42no—
32C2×#13050575—42no—
33C2×#13050575—42no—
34C2×#13050575—42no—
35C22×C83⋊C8—21no—
36C22×C83⋊C8D411yescharacteristic
37C22×C83⋊C8D411yescharacteristic
38#14581789—21no—
39#14581789—21no—
40#14581789—21no—
41#14581789C2×C412yes—
42C2×C4⋊Dic83D412yes—
43C2×C4×Dic83—21no—
44C4⋊(C83⋊C8)—21no—
45C4⋊(C83⋊C8)—21no—
46#14581714—44no—
47#14581713—44no—
48C4×C83⋊C8C2×C412yes—
49C4×C83⋊C8D412yes—
50C8×Dic83—42no—
51C83⋊#95—21no—
52C22×D4×C83—21no—
53C22×D4×C83—21no—
54C22×D4×C83—21no—
55C22×C8×C83—21no—
56C22×C8×C83—21no—
57#127×C83—21no—
58#127×C83—21no—
59#127×C83—21no—
60#127×C83C2×C412yes—
61C2×C42×C83C2311yescharacteristic
62C4⋊C8×C83—21no—
63C4⋊C8×C83—21no—
64C4⋊C8×C83—21no—
65C4⋊C8×C83D412yes—
66C23×D83—41no—
67C2×D332—22no—
68C2×D332—22no—
69C2×D332—22no—
70C2×D332—41no—
71C2×D332—42no—
72C2×D332—41no—
73C2×D332—42no—
74C2×D332—41no—
75C2×D332—41no—
76C2×D332—41no—
77C2×D332—41no—
78C2×D332—41no—
79C2×D332—82no—
80C22×Dic83—22no—
81#13050575—48no—
82#13050575—48no—
83C2×C83⋊C8—21no—
84C2×C83⋊C8—21no—
85C2×C83⋊C8—41no—
86C2×C83⋊C8—42no—
87C2×C83⋊C8C22⋊C412yes—
88C2×C83⋊C8C22⋊C412yes—
89C2×C83⋊C8D812yes—
90C2×C83⋊C8SD1612yes—
91C4⋊Dic83—22no—
92C4⋊Dic83D814yescomplemented
93C4×Dic83—41no—
94C4×Dic83—41no—
95C24×C83—41no—
96C2×D4×C83—22no—
97C2×D4×C83—22no—
98C2×D4×C83—22no—
99C2×D4×C83—41no—
100C2×D4×C83—41no—
101C2×D4×C83—41no—
102C2×D4×C83—41no—
103C2×D4×C83—42no—
104C2×D4×C83—41no—
105C2×D4×C83—42no—
106C2×D4×C83—41no—
107C2×D4×C83—41no—
108C2×D4×C83—82no—
109C22×C4×C83C2×D411yescharacteristic
110C22×C4×C83C2×D411yescharacteristic
111C22×C4×C83#4111yescharacteristic
112C2×C8×C83—22no—
113C2×C8×C83—41no—
114C2×C8×C83—41no—
115C2×C8×C83—41no—
116C2×C8×C83—42no—
117C42×C83C22×C411yescharacteristic
118C42×C83C2×D411yescharacteristic
119C42×C83C2×D411yescharacteristic
120C42×C83#4111yescharacteristic
121C23×C83—42no—
122C23×C83—81no—
123C23×C83—81no—
124C23×C83—82no—
125C23×C83—82no—
126C23×C83#12111yescharacteristic
127C22×D83—42no—
128C22×D83—81no—
129C22×D83—81no—
130C22×D83—82no—
131C22×D83—82no—
132D4×C83—42no—
133D4×C83—42no—
134D4×C83—44no—
135D4×C83—82no—
136D4×C83—82no—
137D4×C83—81no—
138D4×C83—81no—
139D4×C83—81no—
140D4×C83—81no—
141C2×C4×C83—22no—
142C2×C4×C83C2×C22⋊C411yescharacteristic
143C2×C4×C83C4×D411yescharacteristic
144C2×C4×C83C4×D411yescharacteristic
145C2×C4×C83#12111yescharacteristic
146C2×C4×C83#12411yescharacteristic
147C2×C4×C83#12711yescharacteristic
148C2×C4×C83C2×D811yescharacteristic
149C2×C4×C83C2×D812yes—
150C2×C4×C83C2×SD1611yescharacteristic
151C2×C4×C83#13512yes—
152C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
153C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
154C2×Dic83—24no—
155C2×Dic83—42no—
156C2×Dic83—42no—
157D332—42no—
158D332—42no—
159D332—44no—
160D332—82no—
161D332—82no—
162D332—81no—
163D332—81no—
164D332—81no—
165D332—81no—
166C8×C83—42no—
167C8×C83—81no—
168C83⋊C8—22no—
169C83⋊C8—81no—
170C83⋊C8#10214yescomplemented
171C22×C83—82no—
172C22×C83—82no—
173C22×C83—82no—
174C22×C83—162no—
175C22×C83—162no—
176C22×C83#39011yescharacteristic
177C22×C83#45911yescharacteristic
178C22×C83#46011yescharacteristic
179C22×C83#46612yes—
180C22×C83#47212yes—
181C2×D83—82no—
182C2×D83—82no—
183C2×D83—82no—
184C2×D83—162no—
185C2×D83—162no—
186C4×C83—21no—
187C4×C83—21no—
188C4×C83C2×#10211yescharacteristic
189C4×C83#41811yescharacteristic
190C4×C83C4×D812yes—
191C4×C83#45911yescharacteristic
192C4×C83#46011yescharacteristic
193C4×C83#49311yescharacteristic
194C4×C83#49511yescharacteristic
195Dic83—44no—
196#9734—831nomaximal, Sylow
197C2×C83—162no—
198C2×C83#166712yes—
199C2×C83#166812yes—
200C2×C83#197011yescharacteristic
201C2×C83#197811yescharacteristic
202C2×C83#198211yescharacteristic
203D83—162no—
204C2×#493—831no—
205#1982—831no—
206#1982—831no—
207C8×C4⋊C4—831no—
208C2×#335—831no—
209C2×#331—831no—
210C2×#331D8311yescharacteristic, complemented
211C83#973411yescharacteristic, complemented, Sylow
212C22×D8—1661no—
213C22×D8—1661no—
214C2×#127—831no—
215C2×#127C2×D8311yescharacteristic
216#493—834no—
217#493—1662no—
218C2×C4⋊C8—831no—
219C2×C4⋊C8C2×D8311yescharacteristic
220C2×C4⋊C8C2×D8311yescharacteristic
221C2×#102—1662no—
222C2×#102—1662no—
223C2×C4×C8—831no—
224C2×C4×C8—1661no—
225C4×C4⋊C4—831no—
226#336—1661no—
227#335—1662no—
228#331—22no—
229#331—1662no—
230C22×D4—21no—
231C22×D4—21no—
232C22×D4—21no—
233C22×D4—1661no—
234C22×D4—1661no—
235C22×D4—1661no—
236C2×D8—3322no—
237C2×D8—3322no—
238C2×D8—3322no—
239C2×D8—3322no—
240C2×D8—1664no—
241C2×D8—1664no—
242C22×C8—21no—
243C22×C8—21no—
244C22×C8—831no—
245C22×C8—831no—
246C22×C8—1661no—
247#127—21no—
248#127—21no—
249#127—21no—
250#127—832no—
251#127—1661no—
252#127—1661no—
253#127—1661no—
254#127C4×D8312yes—
255C2×C4⋊C4—832no—
256C2×C42—1661no—
257C2×C42C22×D8311yescharacteristic
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—1661no—
262C4⋊C8—1661no—
263C4⋊C8C22⋊D8312yes—
264#102—3324no—
265#102—3324no—
266C4×C8—3322no—
267C4×C8—832no—
268C4×C8—832no—
269#95—1661no—
270C24—41no—
271C24—3321no—
272C2×D4—22no—
273C2×D4—22no—
274C2×D4—22no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—42no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—42no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—82no—
285C2×D4—3321no—
286C2×D4—3321no—
287C2×D4—3321no—
288C2×D4—3322no—
289C2×D4—3321no—
290C2×D4—3321no—
291C2×D4—3322no—
292C2×D4—3321no—
293C2×D4—3321no—
294C2×D4—6642no—
295C2×D4—1662no—
296C2×D4—1662no—
297C2×D4—1662no—
298C22×C4—1662no—
299C22×C4D4×D8311yescharacteristic
300C22×C4D4×D8311yescharacteristic
301C22×C4#1305059011yescharacteristic
302D8—3328no—
303D8—3328no—
304C2×C8—22no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—42no—
309C2×C8—3321no—
310C2×C8—3322no—
311C2×C8—832no—
312C2×C8—832no—
313C2×C8—832no—
314C2×C8—832no—
315C2×C8—1661no—
316C2×C8—1661no—
317C4⋊C4—834no—
318C4⋊C4—1662no—
319C42—3321no—
320C42—3321no—
321C42C2×C4×D8311yescharacteristic
322C42D4×D8311yescharacteristic
323C42C2×C22⋊D8311yescharacteristic
324C42#1305058911yescharacteristic
325C23—42no—
326C23—81no—
327C23—81no—
328C23—82no—
329C23—82no—
330C23—3322no—
331C23—6641no—
332C23—6641no—
333C23—6642no—
334C23—6642no—
335C23#1458176511yescharacteristic
336D4—42no—
337D4—42no—
338D4—44no—
339D4—82no—
340D4—82no—
341D4—81no—
342D4—81no—
343D4—81no—
344D4—81no—
345D4—3322no—
346D4—3322no—
347D4—3324no—
348D4—6642no—
349D4—6642no—
350D4—6641no—
351D4—6641no—
352D4—6641no—
353D4—6641no—
354C2×C4—22no—
355C2×C4—3322no—
356C2×C4—3322no—
357C2×C4—1664no—
358C2×C4C22⋊C4×D8311yescharacteristic
359C2×C4(C22⋊D83)⋊C411yescharacteristic
360C2×C4C4×C22⋊D8311yescharacteristic
361C2×C4#1458176411yescharacteristic
362C2×C4#1458177811yescharacteristic
363C2×C4#1458178811yescharacteristic
364C2×C4D8×D8312yes—
365C2×C4D8×D8311yescharacteristic
366C2×C4SD16×D8311yescharacteristic
367C2×C4#1458181312yes—
368C2×C4#1458181611yescharacteristic
369C2×C4#1458181911yescharacteristic
370C8—42no—
371C8—81no—
372C8—834no—
373C8—6641no—
374C8—1662no—
375C22—82no—
376C22—82no—
377C22—82no—
378C22—162no—
379C22—162no—
380C22—13282no—
381C22—13282no—
382C22—6642no—
383C22—6642no—
384C22—6642no—
385C22#1912760111yescharacteristic
386C22#1912789511yescharacteristic
387C22#1912790011yescharacteristic
388C22#1912793612yes—
389C22#1912797012yes—
390C4—21no—
391C4—21no—
392C4—3324no—
393C4#102×D8311yescharacteristic
394C4#1912770411yescharacteristic
395C4#1912778112yes—
396C4#1912789411yescharacteristic
397C4#1912789911yescharacteristic
398C4#1912807611yescharacteristic
399C4#1912808211yescharacteristic
400C2—162no—
401C2—13282no—
402C2#2490681712yes—
403C2#2490682212yes—
404C2#2490832611yescharacteristic
405C2#2490837011yescharacteristic
406C2#2490838811yescharacteristic
subgroupG/H
centerC23#14581765
commutatorC42×C83C22×C4
FrattiniC2×C42C22×D83
FittingC2×#331×C83C2
socleC23×C83#121
prime radicalC2×#19128077C2
cosocleC2×C42×C83C23
hypercenterC2×#331D83
hypocenterC83#9734
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#331

Complements

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

NH ≅ G/Ntype
2. #24908391C2semidirect
3. #24908389C2semidirect
4. #24907328C2semidirect
5. C2×#19127453C2semidirect
6. C2×#19127444C2semidirect
7. C2×#331×C83C2semidirect
12. #19128077C4semidirect
92. C4⋊Dic83D8semidirect
170. C83⋊C8#102semidirect
210. C2×#331D83semidirect
211. C83#9734semidirect

Maximal subgroups

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

subgroupindex
C2×#191280772
#249083912
#249083892
#249073282
C2×#191274532
C2×#191274442
C2×#331×C832
#973483