Finite Group Databasebeta

← #35301096

Subgroups — #35301096

Order 21248.

490 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24918401C211yesmaximal, characteristic
2#24908287C211yesmaximal, characteristic
3#24906132C211yesmaximal, characteristic
4#24906027C211yesmaximal, characteristic, complemented
5#24905323C211yesmaximal, characteristic, complemented
6#24905294C211yesmaximal, characteristic, complemented
7C83×#2961C211yesmaximal, characteristic, complemented
8C2×#14581789C2211yescharacteristic
9#19128075C412yescomplemented
10#19128049C412yescomplemented
11#19127901—22no—
12#19127851—22no—
13#19127754C2211yescharacteristic
14#19127754C2211yescharacteristic
15C2×#14581714—21no—
16C2×#14581714—21no—
17#19127567C2211yescharacteristic
18#19127444—22no—
19#19127438—22no—
20#533×C83C2211yescharacteristic
21#446×C83—22no—
22#446×C83—22no—
23#424×C83—21no—
24#423×C83C2211yescharacteristic
25#405×C83—21no—
26C2×#97×C83C2211yescharacteristic
27C22×D332—21no—
28C22×D332—21no—
29C22×D332—21no—
30C2×#13050577—24no—
31#14581789—21no—
32#14581789—21no—
33#14581789—21no—
34#14581789C2×C412yes—
35C2×C4⋊Dic83—21no—
36C4⋊(C83⋊C8)—21no—
37C4⋊(C83⋊C8)—21no—
38#14581714—42no—
39#14581714—42no—
40#14581712—24no—
41C22⋊(C83⋊C8)—21no—
42C22⋊(C83⋊C8)—21no—
43C4×C83⋊C8—21no—
44C4×C83⋊C8C2×C412yes—
45C83⋊#95—21no—
46C83⋊#95—21no—
47C2×M16×C83—21no—
48C22×C8×C83D411yescharacteristic
49C22×C8×C83D411yescharacteristic
50C4⋊Q8×C83C2×C411yescharacteristic
51C4⋊Q8×C83C2×C411yescharacteristic
52#125×C83—21no—
53C22⋊Q8×C83—21no—
54C22⋊Q8×C83D412yes—
55C4×Q8×C83—21no—
56C4×Q8×C83D412yes—
57#117×C83—21no—
58C2×C42×C83C2311yescharacteristic
59C4⋊C8×C83—24no—
60C4⋊C8×C83—21no—
61C4⋊C8×C83D412yes—
62#97×C83—22no—
63#97×C83—21no—
64#97×C83—21no—
65C4×C8×C83—22no—
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—82no—
78C2×D332—82no—
79C22×Dic83—41no—
80#13050577—84no—
81C2×C83⋊C8—22no—
82C2×C83⋊C8—41no—
83C4⋊Dic83—41no—
84C4⋊Dic83—41no—
85C2×Q8×C83—21no—
86C2×Q8×C83C22⋊C412yes—
87C22×C4×C83C2×D411yescharacteristic
88C22×C4×C83C2×D411yescharacteristic
89C22×C4×C83#4111yescharacteristic
90M16×C83—41no—
91M16×C83—41no—
92C2×C8×C83—21no—
93C2×C8×C83—21no—
94C2×C8×C83—22no—
95C2×C8×C83—21no—
96C2×C8×C83—21no—
97C2×C8×C83—21no—
98C2×C8×C83SD1612yescomplemented
99C4⋊C4×C83—21no—
100C4⋊C4×C83—21no—
101C4⋊C4×C83—21no—
102C4⋊C4×C83—22no—
103C4⋊C4×C83—22no—
104C4⋊C4×C83C22⋊C412yes—
105C22⋊C4×C83—22no—
106C42×C83—22no—
107C42×C83C22×C411yescharacteristic
108C42×C83C2×D411yescharacteristic
109C42×C83C2×D411yescharacteristic
110C42×C83#4111yescharacteristic
111C23×C83#12111yescharacteristic
112C22×D83—42no—
113C22×D83—82no—
114C22×D83—82no—
115C22×D83—81no—
116C22×D83—81no—
117Q8×C83—42no—
118Q8×C83—81no—
119C2×C4×C83—22no—
120C2×C4×C83—22no—
121C2×C4×C83—41no—
122C2×C4×C83—41no—
123C2×C4×C83C2×C22⋊C411yescharacteristic
124C2×C4×C83C4×D411yescharacteristic
125C2×C4×C83C4×D411yescharacteristic
126C2×C4×C83#12011yescharacteristic
127C2×C4×C83#12111yescharacteristic
128C2×C4×C83#12311yescharacteristic
129C2×C4×C83C2×SD1611yescharacteristic
130C2×C4×C83#13511yescharacteristic
131C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
132C2×C4×C83#13711yescharacteristic
133C2×Dic83—81no—
134C2×Dic83—81no—
135C2×Dic83—81no—
136D332—44no—
137D332—42no—
138D332—42no—
139D332—82no—
140D332—82no—
141D332—82no—
142D332—81no—
143D332—81no—
144C8×C83—22no—
145C8×C83—41no—
146C8×C83—42no—
147C83⋊C8—42no—
148C83⋊C8—81no—
149C22×C83—21no—
150C22×C83—21no—
151C22×C83#38611yescharacteristic
152C22×C83#46011yescharacteristic
153C22×C83#46411yescharacteristic
154C2×D83—82no—
155C2×D83—82no—
156C2×D83—82no—
157C2×D83—162no—
158C2×D83—162no—
159C4×C83—21no—
160C4×C83—21no—
161C4×C83—21no—
162C4×C83—21no—
163C4×C83—42no—
164C4×C83—42no—
165C4×C83C4×SD1611yescharacteristic
166C4×C83#44011yescharacteristic
167C4×C83#44811yescharacteristic
168C4×C83#44911yescharacteristic
169Dic83—161no—
170#9294—831nomaximal, Sylow
171C2×C83—21no—
172C2×C83—21no—
173C2×C83#188011yescharacteristic
174C2×C83#190311yescharacteristic
175C2×C83#191011yescharacteristic
176D83—162no—
177#3208—831no—
178#2961D8311yescharacteristic, complemented
179#1961—831no—
180#1545—831no—
181#1518—831no—
182#1286—831no—
183#1275—831no—
184C83#929411yescharacteristic, complemented, Sylow
185#533C2×D8311yescharacteristic
186C2×#127—831no—
187#492—832no—
188#486—832no—
189#460—1662no—
190#450—1662no—
191#446—22no—
192#446—22no—
193#431—831no—
194#431—831no—
195#424—21no—
196#423C2×D8311yescharacteristic
197C2×#102—1661no—
198C2×#102—1661no—
199#405—21no—
200C2×#97C2×D8311yescharacteristic
201#382—831no—
202#331—1662no—
203#328—1662no—
204C22×D4—1661no—
205C22×D4—1661no—
206C22×D4—1661no—
207C2×SD16—1664no—
208C2×M16—21no—
209C22×C8C22⋊D8311yescharacteristic
210C22×C8C22⋊D8311yescharacteristic
211C4⋊Q8C4×D8311yescharacteristic
212C4⋊Q8C4×D8311yescharacteristic
213#127—832no—
214#127—1661no—
215#127—1661no—
216#127—1661no—
217#125—21no—
218C22⋊Q8—21no—
219C22⋊Q8D33212yes—
220C4×Q8—21no—
221C4×Q8C22⋊D8312yes—
222#117—21no—
223C2×C4⋊C4—1661no—
224C2×C42C22×D8311yescharacteristic
225C4⋊C8—24no—
226C4⋊C8—21no—
227C4⋊C8—1661no—
228C4⋊C8—1661no—
229C4⋊C8C22⋊D8312yes—
230#102—3322no—
231#102—3322no—
232#102—1664no—
233C22⋊C8—1661no—
234C22⋊C8—1661no—
235#97—22no—
236#97—21no—
237#97—21no—
238C4×C8—22no—
239C4×C8—832no—
240C4×C8—1661no—
241#95—1661no—
242#95—1661no—
243C24—3321no—
244C2×Q8—21no—
245C2×Q8(C2×D83)⋊C412yes—
246C2×D4—3321no—
247C2×D4—3321no—
248C2×D4—3321no—
249C2×D4—3321no—
250C2×D4—3321no—
251C2×D4—3321no—
252C2×D4—3321no—
253C2×D4—6642no—
254C2×D4—6642no—
255C2×D4—1662no—
256C2×D4—1662no—
257C2×D4—1662no—
258C22×C4—3321no—
259C22×C4C2×C22⋊D8311yescharacteristic
260C22×C4C2×D33211yescharacteristic
261C22×C4#1305059011yescharacteristic
262SD16—6644no—
263M16—41no—
264M16—41no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—22no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—3321no—
272C2×C8—1662no—
273C2×C8#1305057712yescomplemented
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—3321no—
280C4⋊C4—3321no—
281C4⋊C4(C2×D83)⋊C412yes—
282C22⋊C4—22no—
283C42—22no—
284C42C2×C4×D8311yescharacteristic
285C42C2×C22⋊D8311yescharacteristic
286C42C2×C22⋊D8311yescharacteristic
287C42#1305059011yescharacteristic
288C23—3322no—
289C23—6642no—
290C23—6642no—
291C23—6641no—
292C23—6641no—
293C23#1458176611yescharacteristic
294Q8—42no—
295Q8—81no—
296D4—3324no—
297D4—3322no—
298D4—3322no—
299D4—6642no—
300D4—6642no—
301D4—6642no—
302D4—6641no—
303D4—6641no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—6641no—
309C2×C4—6641no—
310C2×C4—6641no—
311C2×C4C2×(C2×D83)⋊C411yescharacteristic
312C2×C4C4×C22⋊D8311yescharacteristic
313C2×C4C4×C22⋊D8311yescharacteristic
314C2×C4#1458176111yescharacteristic
315C2×C4#1458176611yescharacteristic
316C2×C4#1458177511yescharacteristic
317C2×C4C2×#1305057711yescharacteristic
318C2×C4#1458181211yescharacteristic
319C2×C4#1458181711yescharacteristic
320C2×C4#1458182311yescharacteristic
321C8—22no—
322C8—41no—
323C8—42no—
324C8—3322no—
325C8—6641no—
326C22—21no—
327C22—21no—
328C22—13282no—
329C22—13282no—
330C22—6642no—
331C22—6642no—
332C22—6642no—
333C22#1912758011yescharacteristic
334C22#1912790111yescharacteristic
335C22#1912792311yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—42no—
341C4—42no—
342C4—13281no—
343C4C4×#1305057711yescharacteristic
344C4#1912779911yescharacteristic
345C4#1912784011yescharacteristic
346C4#1912784611yescharacteristic
347C2—21no—
348C2—21no—
349C2—13282no—
350C2#2490781811yescharacteristic
351C2#2490794311yescharacteristic
352C2#2490799011yescharacteristic
subgroupG/H
centerC22#19127580
commutatorC42×C83C22×C4
FrattiniC2×C42C22×D83
FittingC83×#2961C2
socleC22×C83#386
prime radicalC2×#14581789C22
cosocleC2×C42×C83C23
hypercenter#2961D83
hypocenterC83#9294
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2961

Complements

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

NH ≅ G/Ntype
4. #24906027C2semidirect
5. #24905323C2semidirect
6. #24905294C2semidirect
7. C83×#2961C2semidirect
9. #19128075C4semidirect
10. #19128049C4semidirect
98. C2×C8×C83SD16semidirect
178. #2961D83semidirect
184. C83#9294semidirect
273. C2×C8#13050577semidirect

Maximal subgroups

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

subgroupindex
#249184012
#249082872
#249061322
#249060272
#249053232
#249052942
C83×#29612
#929483