Finite Group Databasebeta

← #35308632

Subgroups — #35308632

Order 21248.

506 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128049C211yesmaximal, characteristic
2#24908389C211yesmaximal, characteristic
3#24908387C211yesmaximal, characteristic, complemented
4#24908179C211yesmaximal, characteristic, complemented
5C2×#19127444C211yesmaximal, characteristic
6#24905297C211yesmaximal, characteristic, complemented
7C2×#332×C83C211yesmaximal, characteristic, complemented
8C2×#14581789C2211yescharacteristic
9#19128049—22no—
10#19128049—22no—
11C2×C4⋊(C83⋊C8)C2211yescharacteristic
12C2×Dic83⋊C8—21no—
13C2×#14581716—21no—
14C2×#14581716—21no—
15C2×#14581714—21no—
16C2×#14581714—21no—
17C2×#14581712—21no—
18C2×#14581712—21no—
19C2×C4×C83⋊C8C2211yescharacteristic
20C4×C4⋊Dic83C2211yescharacteristic
21#19127455—21no—
22#19127444—22no—
23C2×C4⋊Q8×C83C2211yescharacteristic
24C2×C4⋊C8×C83C2211yescharacteristic
25C2×C4⋊C8×C83C2211yescharacteristic
26#332×C83—22no—
27C22×D332—21no—
28C22×D332—21no—
29C22×D332—21no—
30C22×C83⋊C8—21no—
31C22×C83⋊C8D411yescharacteristic
32C22×C83⋊C8D411yescharacteristic
33#14581789—21no—
34#14581789—21no—
35#14581789—21no—
36#14581789D412yes—
37C2×C4⋊Dic83D411yescharacteristic
38C2×C4⋊Dic83D411yescharacteristic
39C2×C4×Dic83—21no—
40C4⋊(C83⋊C8)—21no—
41C4⋊(C83⋊C8)—21no—
42Dic83⋊C8—42no—
43#14581716—42no—
44#14581716—42no—
45#14581714—42no—
46#14581714—42no—
47#14581712—24no—
48#14581712—24no—
49C4×C83⋊C8—21no—
50C4×C83⋊C8—21no—
51C83⋊#95—21no—
52C22×Q8×C83—21no—
53C22×C8×C83—21no—
54C22×C8×C83—21no—
55C4⋊Q8×C83—21no—
56C4⋊Q8×C83—21no—
57C4⋊Q8×C83—21no—
58C4⋊Q8×C83D412yes—
59C2×C4⋊C4×C83—21no—
60C2×C4⋊C4×C83—21no—
61C2×C42×C83C2311yescharacteristic
62C4⋊C8×C83—21no—
63C4⋊C8×C83—21no—
64C4⋊C8×C83—21no—
65C4⋊C8×C83—21no—
66C23×D83—41no—
67C2×D332—22no—
68C2×D332—22no—
69C2×D332—22no—
70C2×D332—41no—
71C2×D332—41no—
72C2×D332—41no—
73C2×D332—41no—
74C2×D332—41no—
75C2×D332—41no—
76C2×D332—41no—
77C2×D332—41no—
78C2×D332—41no—
79C2×D332—41no—
80C2×D332—41no—
81C2×D332—81no—
82C2×D332—81no—
83C22×Dic83—21no—
84C22×Dic83—21no—
85C2×C83⋊C8—21no—
86C2×C83⋊C8—21no—
87C2×C83⋊C8—21no—
88C2×C83⋊C8—21no—
89C2×C83⋊C8—21no—
90C2×C83⋊C8—21no—
91C2×C83⋊C8—41no—
92C2×C83⋊C8—41no—
93C2×C83⋊C8—41no—
94C4⋊Dic83—21no—
95C4⋊Dic83—21no—
96C4⋊Dic83—21no—
97C4⋊Dic83D812yes—
98C4×Dic83—41no—
99C4×Dic83—41no—
100C2×Q8×C83—22no—
101C2×Q8×C83—41no—
102C2×Q8×C83—41no—
103C2×Q8×C83—41no—
104C2×Q8×C83—41no—
105C2×Q8×C83—41no—
106C22×C4×C83—41no—
107C22×C4×C83C2×D411yescharacteristic
108C22×C4×C83C2×D411yescharacteristic
109C22×C4×C83#4111yescharacteristic
110C2×C8×C83—41no—
111C2×C8×C83—41no—
112C2×C8×C83—41no—
113C2×C8×C83—41no—
114C2×C8×C83—41no—
115C2×C8×C83—41no—
116C4⋊C4×C83—22no—
117C4⋊C4×C83—22no—
118C4⋊C4×C83—41no—
119C4⋊C4×C83—41no—
120C42×C83C2×D411yescharacteristic
121C42×C83C2×D411yescharacteristic
122C42×C83#4111yescharacteristic
123C42×C83#4111yescharacteristic
124C23×C83#12111yescharacteristic
125C22×D83—42no—
126C22×D83—81no—
127C22×D83—81no—
128C22×D83—81no—
129C22×D83—81no—
130C22×D83—81no—
131C22×D83—81no—
132Q8×C83—42no—
133Q8×C83—42no—
134Q8×C83—81no—
135Q8×C83—81no—
136C2×C4×C83—21no—
137C2×C4×C83—21no—
138C2×C4×C83—42no—
139C2×C4×C83—81no—
140C2×C4×C83—81no—
141C2×C4×C83#12011yescharacteristic
142C2×C4×C83#12111yescharacteristic
143C2×C4×C83#12111yescharacteristic
144C2×C4×C83#12311yescharacteristic
145C2×C4×C83#12311yescharacteristic
146C2×C4×C83#12411yescharacteristic
147C2×C4×C83C2×D811yescharacteristic
148C2×C4×C83#13511yescharacteristic
149C2×C4×C83#13511yescharacteristic
150C2×C4×C83#13511yescharacteristic
151C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
152C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
153C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
154C2×C4×C83#13711yescharacteristic
155C2×Dic83—22no—
156C2×Dic83—41no—
157C2×Dic83—41no—
158C2×Dic83—41no—
159C2×Dic83—41no—
160C2×Dic83—41no—
161D332—42no—
162D332—42no—
163D332—42no—
164D332—42no—
165D332—82no—
166D332—82no—
167D332—81no—
168D332—81no—
169D332—81no—
170D332—81no—
171C8×C83—81no—
172C8×C83—81no—
173C83⋊C8—41no—
174C83⋊C8—41no—
175C83⋊C8—81no—
176C22×C83#39411yescharacteristic
177C22×C83#46311yescharacteristic
178C22×C83#46411yescharacteristic
179C22×C83#46611yescharacteristic
180C22×C83#46911yescharacteristic
181C22×C83#47111yescharacteristic
182C22×C83#47111yescharacteristic
183C2×D83—82no—
184C2×D83—82no—
185C2×D83—82no—
186C2×D83—161no—
187C2×D83—161no—
188C2×D83—161no—
189C2×D83—161no—
190C4×C83—21no—
191C4×C83—21no—
192C4×C83—82no—
193C4×C83#44911yescharacteristic
194C4×C83#45211yescharacteristic
195C4×C83#45911yescharacteristic
196C4×C83#45911yescharacteristic
197C4×C83#46311yescharacteristic
198C4×C83#46311yescharacteristic
199C4×C83#48811yescharacteristic
200C4×C83#48911yescharacteristic
201Dic83—42no—
202Dic83—81no—
203#10540—831nomaximal, Sylow
204C2×C83#168111yescharacteristic
205C2×C83#168211yescharacteristic
206C2×C83#170111yescharacteristic
207C2×C83#170211yescharacteristic
208C2×C83#204211yescharacteristic
209C2×C83#204611yescharacteristic
210C2×C83#205411yescharacteristic
211D83—162no—
212C2×#486—831no—
213#1982—831no—
214#1981—831no—
215#1939—831no—
216C2×#332D8311yescharacteristic, complemented
217C2×#331—831no—
218#1277—831no—
219C83#1054011yescharacteristic, complemented, Sylow
220C2×C4⋊Q8C2×D8311yescharacteristic
221C2×#127—831no—
222#486—1662no—
223#486—1662no—
224C2×C4⋊C8—831no—
225C2×C4⋊C8—1661no—
226C2×C4⋊C8C2×D8311yescharacteristic
227C2×C4⋊C8C2×D8311yescharacteristic
228C2×#103—1661no—
229C2×#103—1661no—
230C2×#102—1661no—
231C2×#102—1661no—
232C2×#102—1661no—
233C2×#102—1661no—
234C2×C4×C8—831no—
235C4×C4⋊C4—831no—
236#336—1661no—
237#332—22no—
238#331—1662no—
239C22×Q8—21no—
240C22×D4—1661no—
241C22×D4—1661no—
242C22×D4—1661no—
243C22×C8—21no—
244C22×C8—21no—
245C22×C8—831no—
246C22×C8—831no—
247C22×C8—1661no—
248C4⋊Q8—21no—
249C4⋊Q8—21no—
250C4⋊Q8—21no—
251C4⋊Q8C22⋊D8312yes—
252#127—832no—
253#127—1661no—
254#127—1661no—
255#127—1661no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—831no—
259C2×C4⋊C4—831no—
260C2×C42—1661no—
261C2×C42C22×D8311yescharacteristic
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—3322no—
267C4⋊C8—1661no—
268C4⋊C8—1661no—
269#103—3322no—
270#103—3322no—
271#102—3322no—
272#102—3322no—
273#102—1664no—
274#102—1664no—
275C4×C8—1661no—
276C4×C8—1661no—
277#95—1661no—
278C24—3321no—
279C2×Q8—22no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×D4—3321no—
286C2×D4—3321no—
287C2×D4—3321no—
288C2×D4—3321no—
289C2×D4—3321no—
290C2×D4—3321no—
291C2×D4—3321no—
292C2×D4—3321no—
293C2×D4—3321no—
294C2×D4—3321no—
295C2×D4—3321no—
296C2×D4—6641no—
297C2×D4—6641no—
298C2×D4—1662no—
299C2×D4—1662no—
300C2×D4—1662no—
301C22×C4—41no—
302C22×C4—1661no—
303C22×C4—1661no—
304C22×C4D4×D8311yescharacteristic
305C22×C4D4×D8311yescharacteristic
306C22×C4#1305059011yescharacteristic
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—3321no—
314C2×C8—3321no—
315C2×C8—3321no—
316C2×C8—1661no—
317C2×C8—1661no—
318C2×C8—1661no—
319C2×C8—1661no—
320C2×C8—1661no—
321C2×C8—1661no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—832no—
327C4⋊C4—1661no—
328C4⋊C4—1661no—
329C4⋊C4—1661no—
330C42—3321no—
331C42—3321no—
332C42D4×D8311yescharacteristic
333C42C2×C22⋊D8311yescharacteristic
334C42#1305058911yescharacteristic
335C42#1305059011yescharacteristic
336C23—3322no—
337C23—6641no—
338C23—6641no—
339C23—6641no—
340C23—6641no—
341C23—6641no—
342C23—6641no—
343C23#1458176511yescharacteristic
344Q8—42no—
345Q8—42no—
346Q8—81no—
347Q8—81no—
348D4—3322no—
349D4—3322no—
350D4—3322no—
351D4—3322no—
352D4—6642no—
353D4—6642no—
354D4—6641no—
355D4—6641no—
356D4—6641no—
357D4—6641no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—42no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—3321no—
364C2×C4—3321no—
365C2×C4—3321no—
366C2×C4—3321no—
367C2×C4—3321no—
368C2×C4—1662no—
369C2×C4#1458176011yescharacteristic
370C2×C4#1458176411yescharacteristic
371C2×C4#1458176511yescharacteristic
372C2×C4#1458177411yescharacteristic
373C2×C4#1458177511yescharacteristic
374C2×C4#1458177811yescharacteristic
375C2×C4D8×D8311yescharacteristic
376C2×C4#1458181311yescharacteristic
377C2×C4#1458181311yescharacteristic
378C2×C4#1458181511yescharacteristic
379C2×C4#1458181611yescharacteristic
380C2×C4#1458181911yescharacteristic
381C2×C4#1458181911yescharacteristic
382C2×C4#1458182411yescharacteristic
383C8—81no—
384C8—81no—
385C8—3321no—
386C8—3321no—
387C8—6641no—
388C22—13281no—
389C22—13281no—
390C22—13281no—
391C22—13281no—
392C22—6642no—
393C22—6642no—
394C22—6642no—
395C22#1912761611yescharacteristic
396C22#1912791911yescharacteristic
397C22#1912792611yescharacteristic
398C22#1912793611yescharacteristic
399C22#1912795311yescharacteristic
400C22#1912796711yescharacteristic
401C22#1912796711yescharacteristic
402C4—21no—
403C4—21no—
404C4—82no—
405C4—3322no—
406C4—6641no—
407C4#1912784511yescharacteristic
408C4#1912786211yescharacteristic
409C4#1912789411yescharacteristic
410C4#1912789511yescharacteristic
411C4#1912791511yescharacteristic
412C4#1912791811yescharacteristic
413C4#1912805711yescharacteristic
414C4#1912806211yescharacteristic
415C2—13282no—
416C2#2490690311yescharacteristic
417C2#2490691011yescharacteristic
418C2#2490702711yescharacteristic
419C2#2490703411yescharacteristic
420C2#2490870011yescharacteristic
421C2#2490872211yescharacteristic
422C2#2490877211yescharacteristic
subgroupG/H
centerC23#14581765
commutator, cosocleC2×C42×C83C23
FrattiniC2×C42C22×D83
FittingC2×#332×C83C2
socleC23×C83#121
prime radicalC2×#14581789C22
hypercenterC2×#332D83
hypocenterC83#10540
LCSG ▷ C2×C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#332

Complements

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

NH ≅ G/Ntype
3. #24908387C2semidirect
4. #24908179C2semidirect
6. #24905297C2semidirect
7. C2×#332×C83C2semidirect
216. C2×#332D83semidirect
219. C83#10540semidirect

Maximal subgroups

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

subgroupindex
C2×#191280492
#249083892
#249083872
#249081792
C2×#191274442
#249052972
C2×#332×C832
#1054083