Finite Group Databasebeta

← #35308255

Subgroups — #35308255

Order 21248.

474 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127988C211yesmaximal, characteristic
2#24908425C211yesmaximal, characteristic, complemented
3#24908256C211yesmaximal, characteristic
4#24907464C211yesmaximal, characteristic
5C2×#19127453C211yesmaximal, characteristic, complemented
6#24905347C211yesmaximal, characteristic, complemented
7C2×#333×C83C211yesmaximal, characteristic, complemented
8C2×C4×D332C2211yescharacteristic
9#19127988—22no—
10#19127988—22no—
11C2×#14581723—21no—
12C2×C4⋊(C83⋊C8)C2211yescharacteristic
13C2×#14581714—21no—
14C2×#14581712—21no—
15C2×(C2×D83)⋊C8—21no—
16C2×(C2×D83)⋊C8—21no—
17C2×C4×C83⋊C8C2211yescharacteristic
18#19127571C2211yescharacteristic
19#19127464—21no—
20#19127463—21no—
21#19127463—21no—
22#19127453—22no—
23C2×C4⋊Q8×C83C2211yescharacteristic
24C2×C4⋊C8×C83C2211yescharacteristic
25C2×C4⋊C8×C83C2211yescharacteristic
26#333×C83—22no—
27C22×D332D411yescharacteristic
28C22×C4×D83—21no—
29C22×C83⋊C8—21no—
30C22×C83⋊C8D411yescharacteristic
31C22×C83⋊C8D411yescharacteristic
32C4×D332—21no—
33C4×D332—21no—
34C4×D332—21no—
35C4×D332—21no—
36C2×C4⋊Dic83—21no—
37C2×C4⋊Dic83—21no—
38C2×C4⋊Dic83—21no—
39C2×C4⋊Dic83D411yescharacteristic
40C2×(C2×D83)⋊C4—21no—
41#14581723—24no—
42C4⋊(C83⋊C8)—21no—
43C4⋊(C83⋊C8)D412yes—
44#14581714—42no—
45#14581712—24no—
46(C2×D83)⋊C8—42no—
47(C2×D83)⋊C8—42no—
48C4×C83⋊C8—21no—
49C4×C83⋊C8—21no—
50C22×Q8×C83—21no—
51C22×C8×C83—21no—
52C22×C8×C83—21no—
53C4⋊Q8×C83—21no—
54C4⋊Q8×C83—21no—
55C4⋊Q8×C83—21no—
56C4⋊Q8×C83D412yes—
57C2×C4⋊C4×C83—21no—
58C2×C4⋊C4×C83—21no—
59C2×C42×C83C2311yescharacteristic
60C4⋊C8×C83—21no—
61C4⋊C8×C83—21no—
62C4⋊C8×C83—21no—
63C4⋊C8×C83—21no—
64C23×D83—21no—
65C2×D332—21no—
66C2×D332—21no—
67C2×D332—21no—
68C2×D332—21no—
69C2×D332—21no—
70C2×D332Q1612yes—
71C22×Dic83—21no—
72C22×Dic83—41no—
73C2×C4×D83—41no—
74C2×C4×D83—41no—
75C2×C4×D83—41no—
76C2×C4×D83—41no—
77C2×C4×D83—41no—
78C2×C4×D83—41no—
79C2×C83⋊C8—21no—
80C2×C83⋊C8—21no—
81C2×C83⋊C8—22no—
82C2×C83⋊C8—21no—
83C2×C83⋊C8—21no—
84C2×C83⋊C8—21no—
85C2×C83⋊C8—41no—
86C2×C83⋊C8—41no—
87C2×C83⋊C8SD1612yes—
88C4⋊Dic83—21no—
89C4⋊Dic83—21no—
90C4⋊Dic83—41no—
91C4⋊Dic83—41no—
92C4⋊Dic83—41no—
93C4⋊Dic83—41no—
94C4⋊Dic83—81no—
95(C2×D83)⋊C4—41no—
96(C2×D83)⋊C4—41no—
97C2×Q8×C83—22no—
98C2×Q8×C83—41no—
99C2×Q8×C83—81no—
100C2×Q8×C83—81no—
101C22×C4×C83—41no—
102C22×C4×C83C2×D411yescharacteristic
103C22×C4×C83C2×D411yescharacteristic
104C22×C4×C83#4111yescharacteristic
105C2×C8×C83—41no—
106C2×C8×C83—41no—
107C2×C8×C83—41no—
108C2×C8×C83—41no—
109C2×C8×C83—41no—
110C2×C8×C83—41no—
111C4⋊C4×C83—22no—
112C4⋊C4×C83—22no—
113C4⋊C4×C83—41no—
114C4⋊C4×C83—41no—
115C42×C83C2×D411yescharacteristic
116C42×C83C2×D411yescharacteristic
117C42×C83#4111yescharacteristic
118C42×C83#4111yescharacteristic
119C23×C83#12111yescharacteristic
120C22×D83—22no—
121C22×D83—41no—
122C22×D83—41no—
123C22×D83—41no—
124C22×D83—41no—
125C22×D83—41no—
126C22×D83—41no—
127Q8×C83—82no—
128Q8×C83—82no—
129C2×C4×C83—21no—
130C2×C4×C83—21no—
131C2×C4×C83—42no—
132C2×C4×C83—81no—
133C2×C4×C83—81no—
134C2×C4×C83#12011yescharacteristic
135C2×C4×C83#12111yescharacteristic
136C2×C4×C83#12111yescharacteristic
137C2×C4×C83#12311yescharacteristic
138C2×C4×C83#12311yescharacteristic
139C2×C4×C83#12411yescharacteristic
140C2×C4×C83C2×SD1611yescharacteristic
141C2×C4×C83C2×Q1611yescharacteristic
142C2×C4×C83#13511yescharacteristic
143C2×C4×C83#13511yescharacteristic
144C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
145C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
146C2×C4×C83#13711yescharacteristic
147C2×C4×C83#13711yescharacteristic
148C2×Dic83—41no—
149C2×Dic83—41no—
150C2×Dic83—41no—
151C2×Dic83—81no—
152C2×Dic83—81no—
153C2×Dic83—81no—
154D332—22no—
155D332—22no—
156D332—41no—
157D332—41no—
158C4×D83—81no—
159C4×D83—81no—
160C4×D83—81no—
161C4×D83—81no—
162C8×C83—81no—
163C8×C83—81no—
164C83⋊C8—22no—
165C83⋊C8—42no—
166C83⋊C8—41no—
167C22×C83#39411yescharacteristic
168C22×C83#46111yescharacteristic
169C22×C83#46311yescharacteristic
170C22×C83#46711yescharacteristic
171C22×C83#46811yescharacteristic
172C22×C83#47111yescharacteristic
173C22×C83#47211yescharacteristic
174C2×D83—42no—
175C2×D83—42no—
176C2×D83—42no—
177C2×D83—81no—
178C2×D83—81no—
179C2×D83—81no—
180C2×D83—81no—
181C4×C83—21no—
182C4×C83—21no—
183C4×C83—82no—
184C4×C83#44911yescharacteristic
185C4×C83#45111yescharacteristic
186C4×C83#46011yescharacteristic
187C4×C83#46411yescharacteristic
188C4×C83#46511yescharacteristic
189C4×C83#46811yescharacteristic
190C4×C83#48411yescharacteristic
191C4×C83#48511yescharacteristic
192Dic83—81no—
193Dic83—161no—
194#10481—831nomaximal, Sylow
195C2×C83#167511yescharacteristic
196C2×C83#168311yescharacteristic
197C2×C83#169111yescharacteristic
198C2×C83#169711yescharacteristic
199C2×C83#203611yescharacteristic
200C2×C83#205111yescharacteristic
201C2×C83#205511yescharacteristic
202D83—82no—
203C2×#476—831no—
204#1990—831no—
205#1954—831no—
206#1802—831no—
207C2×#335—831no—
208C2×#333D8311yescharacteristic, complemented
209#1296—831no—
210C83#1048111yescharacteristic, complemented, Sylow
211C2×C4⋊Q8C2×D8311yescharacteristic
212C2×C4×D4—831no—
213#476—1662no—
214#476—1662no—
215C2×#106—1661no—
216C2×C4⋊C8—831no—
217C2×C4⋊C8C2×D8311yescharacteristic
218C2×C4⋊C8C2×D8311yescharacteristic
219C2×#102—1661no—
220C2×#102—1661no—
221C2×C22⋊C8—1661no—
222C2×C22⋊C8—1661no—
223C2×C4×C8—831no—
224#384—831no—
225#340—1661no—
226#340—1661no—
227#340—1661no—
228#335—1662no—
229#333—22no—
230C22×Q8—21no—
231C22×D4—831no—
232C23×C4—1661no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—831no—
236C22×C8—831no—
237C22×C8—1661no—
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8—21no—
241C4⋊Q8C22⋊D8312yes—
242C4×D4—1661no—
243C4×D4—1661no—
244C4×D4—1661no—
245C4×D4—1661no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—831no—
249C2×C4⋊C4—1661no—
250C2×C4⋊C4—1661no—
251C2×C4⋊C4—1661no—
252C2×C22⋊C4—1661no—
253C2×C42C22×D8311yescharacteristic
254#106—1664no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—832no—
260C4⋊C8—1661no—
261#102—3322no—
262#102—1664no—
263C22⋊C8—3322no—
264C22⋊C8—3322no—
265C4×C8—1661no—
266C4×C8—1661no—
267C24—1661no—
268C2×Q8—22no—
269C2×Q8—41no—
270C2×Q8—81no—
271C2×Q8—81no—
272C2×D4—832no—
273C2×D4—1661no—
274C2×D4—1661no—
275C2×D4—1661no—
276C2×D4—1661no—
277C2×D4—1661no—
278C22×C4—41no—
279C22×C4—3321no—
280C22×C4—3321no—
281C22×C4—3321no—
282C22×C4—3321no—
283C22×C4—3321no—
284C22×C4—3321no—
285C22×C4—3321no—
286C22×C4—1661no—
287C22×C4D4×D8311yescharacteristic
288C22×C4D4×D8311yescharacteristic
289C22×C4#1305059011yescharacteristic
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—3321no—
297C2×C8—3321no—
298C2×C8—832no—
299C2×C8—1662no—
300C2×C8—1661no—
301C2×C8—1661no—
302C2×C8—1661no—
303C2×C8—1661no—
304C2×C8—1661no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—3321no—
310C4⋊C4—3321no—
311C4⋊C4—3321no—
312C4⋊C4—3321no—
313C4⋊C4—6641no—
314C4⋊C4—1661no—
315C4⋊C4—1661no—
316C22⋊C4—3321no—
317C22⋊C4—3321no—
318C42D4×D8311yescharacteristic
319C42C2×C22⋊D8311yescharacteristic
320C42#1305058911yescharacteristic
321C42#1305059011yescharacteristic
322C23—3321no—
323C23—3321no—
324C23—3321no—
325C23—3321no—
326C23—3321no—
327C23—3321no—
328C23—1662no—
329C23#1458176511yescharacteristic
330Q8—82no—
331Q8—82no—
332D4—3321no—
333D4—3321no—
334D4—1662no—
335D4—1662no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—42no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—3321no—
342C2×C4—3321no—
343C2×C4—3321no—
344C2×C4—6641no—
345C2×C4—6641no—
346C2×C4—6641no—
347C2×C4—6641no—
348C2×C4—6641no—
349C2×C4—6641no—
350C2×C4—6641no—
351C2×C4#1458176011yescharacteristic
352C2×C4#1458176411yescharacteristic
353C2×C4#1458176511yescharacteristic
354C2×C4#1458177411yescharacteristic
355C2×C4#1458177511yescharacteristic
356C2×C4#1458177811yescharacteristic
357C2×C4SD16×D8311yescharacteristic
358C2×C4Q16×D8311yescharacteristic
359C2×C4#1458181311yescharacteristic
360C2×C4#1458181511yescharacteristic
361C2×C4#1458181911yescharacteristic
362C2×C4#1458181911yescharacteristic
363C2×C4#1458182411yescharacteristic
364C2×C4#1458182411yescharacteristic
365C8—81no—
366C8—81no—
367C8—3322no—
368C8—3321no—
369C8—1662no—
370C22—3322no—
371C22—3322no—
372C22—3322no—
373C22—6641no—
374C22—6641no—
375C22—6641no—
376C22—6641no—
377C22#1912761611yescharacteristic
378C22#1912790511yescharacteristic
379C22#1912791911yescharacteristic
380C22#1912794111yescharacteristic
381C22#1912795011yescharacteristic
382C22#1912796311yescharacteristic
383C22#1912797011yescharacteristic
384C4—21no—
385C4—21no—
386C4—82no—
387C4—13281no—
388C4—6641no—
389C4#1912784511yescharacteristic
390C4#1912785511yescharacteristic
391C4#1912789911yescharacteristic
392C4#1912792511yescharacteristic
393C4#1912793311yescharacteristic
394C4#1912795011yescharacteristic
395C4#1912803511yescharacteristic
396C4#1912804011yescharacteristic
397C2—6642no—
398C2#2490686111yescharacteristic
399C2#2490691711yescharacteristic
400C2#2490696511yescharacteristic
401C2#2490699911yescharacteristic
402C2#2490866411yescharacteristic
403C2#2490875511yescharacteristic
404C2#2490877711yescharacteristic
subgroupG/H
centerC23#14581765
commutator, cosocleC2×C42×C83C23
FrattiniC2×C42C22×D83
FittingC2×#333×C83C2
socleC23×C83#121
prime radicalC22×D332D4
hypercenterC2×#333D83
hypocenterC83#10481
LCSG ▷ C2×C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#333

Complements

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

NH ≅ G/Ntype
2. #24908425C2semidirect
5. C2×#19127453C2semidirect
6. #24905347C2semidirect
7. C2×#333×C83C2semidirect
208. C2×#333D83semidirect
210. C83#10481semidirect

Maximal subgroups

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

subgroupindex
C2×#191279882
#249084252
#249082562
#249074642
C2×#191274532
#249053472
C2×#333×C832
#1048183