Finite Group Databasebeta

← #35301063

Subgroups — #35301063

Order 21248.

490 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24916181C211yesmaximal, characteristic
2#24908437C211yesmaximal, characteristic, complemented
3#24906148C211yesmaximal, characteristic, complemented
4#24906023C211yesmaximal, characteristic
5#24905322C211yesmaximal, characteristic, complemented
6#24905293C211yesmaximal, characteristic, complemented
7C83×#3206C211yesmaximal, characteristic
8C2×#14581799—21no—
9#19127773—22no—
10#19127773—22no—
11#19127754C2211yescharacteristic
12#19127725C2211yescharacteristic
13#19127668—21no—
14C2×#14581703C2211yescharacteristic
15#19127567C2211yescharacteristic
16#19127463—21no—
17#19127463—21no—
18#19127447—22no—
19#19127435—22no—
20#532×C83C2211yescharacteristic
21C2×C4⋊Q8×C83C2211yescharacteristic
22#492×C83C412yescomplemented
23#487×C83C412yescomplemented
24#461×C83—22no—
25#448×C83—22no—
26#431×C83C2211yescharacteristic
27#14581799—42no—
28#14581799—42no—
29C2×#13050574—22no—
30C2×#13050574—21no—
31C22×C83⋊C8—22no—
32C22×C83⋊C8D411yescharacteristic
33C22×C83⋊C8D411yescharacteristic
34C2×C4⋊Dic83—21no—
35C4⋊(C83⋊C8)—21no—
36C4⋊(C83⋊C8)—21no—
37C4⋊(C83⋊C8)D412yes—
38C22⋊(C83⋊C8)—24no—
39C22⋊(C83⋊C8)—21no—
40#14581703—21no—
41#14581703—21no—
42C4×C83⋊C8—21no—
43C83⋊#95—21no—
44C83⋊#95—21no—
45C2×#41×C83—21no—
46C22×Q8×C83—21no—
47C2×SD16×C83—24no—
48C4⋊Q8×C83—21no—
49C4⋊Q8×C83—21no—
50C4⋊Q8×C83—21no—
51C4⋊Q8×C83C2×C412yes—
52C4⋊Q8×C83C2×C411yescharacteristic
53#127×C83C2×C411yescharacteristic
54#124×C83—21no—
55#121×C83—21no—
56#121×C83D412yes—
57C4×D4×C83—21no—
58C4×D4×C83D412yes—
59C2×C4⋊C4×C83—21no—
60C2×C4⋊C4×C83—21no—
61C2×C42×C83C2311yescharacteristic
62C4⋊C8×C83—21no—
63#103×C83—24no—
64C22⋊C8×C83—21no—
65C4×C8×C83C2×C412yes—
66C22×Dic83—41no—
67#13050574—44no—
68#13050574—41no—
69#13050574—41no—
70C2×C83⋊C8—21no—
71C2×C83⋊C8—21no—
72C2×C83⋊C8—22no—
73C2×C83⋊C8—21no—
74C2×C83⋊C8—21no—
75C2×C83⋊C8—21no—
76C2×C83⋊C8—44no—
77C2×C83⋊C8—41no—
78C2×C83⋊C8SD1612yescomplemented
79C4⋊Dic83—41no—
80C4⋊Dic83—41no—
81#41×C83—41no—
82#41×C83—41no—
83#41×C83—41no—
84#41×C83—41no—
85C2×Q8×C83—21no—
86C2×Q8×C83—22no—
87C2×Q8×C83—41no—
88C2×Q8×C83—82no—
89C2×D4×C83—22no—
90C2×D4×C83—21no—
91C2×D4×C83—21no—
92C2×D4×C83C22⋊C412yes—
93C22×C4×C83—22no—
94C22×C4×C83—41no—
95C22×C4×C83C2×D411yescharacteristic
96C22×C4×C83C2×D411yescharacteristic
97C22×C4×C83#4111yescharacteristic
98SD16×C83—84no—
99C2×C8×C83—22no—
100C4⋊C4×C83—21no—
101C4⋊C4×C83—22no—
102C4⋊C4×C83—22no—
103C4⋊C4×C83—41no—
104C4⋊C4×C83—41no—
105C4⋊C4×C83C22⋊C412yes—
106C22⋊C4×C83—22no—
107C22⋊C4×C83—22no—
108C42×C83C22×C411yescharacteristic
109C42×C83C2×D411yescharacteristic
110C42×C83C2×D411yescharacteristic
111C42×C83#4111yescharacteristic
112C23×C83—22no—
113C23×C83#12111yescharacteristic
114Q8×C83—44no—
115Q8×C83—41no—
116Q8×C83—41no—
117Q8×C83—82no—
118D4×C83—42no—
119D4×C83—42no—
120D4×C83—42no—
121D4×C83—41no—
122D4×C83—41no—
123D4×C83—81no—
124C2×C4×C83—22no—
125C2×C4×C83—41no—
126C2×C4×C83—41no—
127C2×C4×C83—42no—
128C2×C4×C83—42no—
129C2×C4×C83—42no—
130C2×C4×C83—82no—
131C2×C4×C83C2×C22⋊C411yescharacteristic
132C2×C4×C83C4×D411yescharacteristic
133C2×C4×C83C4×D411yescharacteristic
134C2×C4×C83#12011yescharacteristic
135C2×C4×C83#12111yescharacteristic
136C2×C4×C83#12311yescharacteristic
137C2×C4×C83C2×SD1611yescharacteristic
138C2×C4×C83#13511yescharacteristic
139C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
140C2×C4×C83#13711yescharacteristic
141C2×Dic83—81no—
142C2×Dic83—81no—
143C2×Dic83—81no—
144C8×C83—42no—
145C83⋊C8—22no—
146C83⋊C8—42no—
147C83⋊C8—41no—
148C83⋊C8—81no—
149C22×C83—21no—
150C22×C83—21no—
151C22×C83—42no—
152C22×C83—42no—
153C22×C83—42no—
154C22×C83#38611yescharacteristic
155C22×C83#46111yescharacteristic
156C22×C83#46311yescharacteristic
157C4×C83—21no—
158C4×C83—21no—
159C4×C83—21no—
160C4×C83—21no—
161C4×C83—42no—
162C4×C83—82no—
163C4×C83C4×SD1611yescharacteristic
164C4×C83#44011yescharacteristic
165C4×C83#45011yescharacteristic
166C4×C83#45211yescharacteristic
167Dic83—161no—
168#9289—831nomaximal, Sylow
169C2×C83—21no—
170C2×C83—21no—
171C2×C83—82no—
172C2×C83#187711yescharacteristic
173C2×C83#190211yescharacteristic
174C2×C83#191011yescharacteristic
175#3206D8311yescharacteristic
176#2932—831no—
177#1993—831no—
178#1549—831no—
179#1517—831no—
180#1286—831no—
181#1275—831no—
182C83#928911yescharacteristic, complemented, Sylow
183C2×#131—1661no—
184#532C2×D8311yescharacteristic
185C2×C4⋊Q8C2×D8311yescharacteristic
186#492Dic8312yescomplemented
187#487Dic8312yescomplemented
188#461—22no—
189#448—22no—
190#435—1662no—
191#435—1662no—
192#431—831no—
193#431C2×D8311yescharacteristic
194#423—831no—
195#408—1661no—
196C2×#97—831no—
197#382—831no—
198#340—1661no—
199#340—1661no—
200#332—1662no—
201#327—1662no—
202C2×#41—21no—
203C22×Q8—21no—
204C2×SD16—24no—
205#131—3322no—
206#131—3322no—
207C2×M16—1662no—
208C2×M16—1661no—
209C22×C8—831no—
210C22×C8—831no—
211C22×C8—1662no—
212C4⋊Q8—21no—
213C4⋊Q8—21no—
214C4⋊Q8—21no—
215C4⋊Q8C2×Dic8312yes—
216C4⋊Q8C2×Dic8311yescharacteristic
217#127C2×Dic8311yescharacteristic
218#124—21no—
219#121—21no—
220#121C22⋊D8312yes—
221C4×D4—21no—
222C4×D4C22⋊D8312yes—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—1661no—
226C2×C42C22×D8311yescharacteristic
227C4⋊C8—21no—
228C4⋊C8—832no—
229C4⋊C8—1661no—
230C4⋊C8—1661no—
231#103—24no—
232C22⋊C8—21no—
233C22⋊C8—1661no—
234C22⋊C8—1664no—
235#97—1661no—
236#97—1661no—
237C4×C8—1661no—
238C4×C8C2×Dic8312yes—
239#95—1661no—
240#95—1661no—
241#41—41no—
242#41—41no—
243#41—41no—
244#41—41no—
245C2×Q8—21no—
246C2×Q8—22no—
247C2×Q8—41no—
248C2×Q8—82no—
249C2×D4—22no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4C22⋊Dic8312yes—
253C22×C4—22no—
254C22×C4—41no—
255C22×C4—3321no—
256C22×C4D4×D8311yescharacteristic
257C22×C4C2×C22⋊D8311yescharacteristic
258C22×C4#1305058911yescharacteristic
259SD16—84no—
260M16—3321no—
261M16—3321no—
262M16—3324no—
263C2×C8—22no—
264C2×C8—3324no—
265C2×C8—3321no—
266C2×C8—832no—
267C2×C8—1661no—
268C2×C8—1662no—
269C2×C8—1661no—
270C2×C8—1661no—
271C2×C8—1661no—
272C2×C8—1661no—
273C4⋊C4—21no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—3321no—
279C4⋊C4—3321no—
280C4⋊C4C22⋊Dic8312yes—
281C22⋊C4—22no—
282C22⋊C4—22no—
283C42C22×Dic8311yescharacteristic
284C42D4×D8311yescharacteristic
285C42C2×C22⋊D8311yescharacteristic
286C42#1305058911yescharacteristic
287C23—22no—
288C23#1458176411yescharacteristic
289Q8—44no—
290Q8—41no—
291Q8—41no—
292Q8—82no—
293D4—42no—
294D4—42no—
295D4—42no—
296D4—41no—
297D4—41no—
298D4—81no—
299C2×C4—22no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—82no—
306C2×C4—6641no—
307C2×C4—6641no—
308C2×C4—6641no—
309C2×C4C2×C22⋊Dic8311yescharacteristic
310C2×C4D4×Dic8311yescharacteristic
311C2×C4D4×Dic8311yescharacteristic
312C2×C4#1458176011yescharacteristic
313C2×C4#1458176411yescharacteristic
314C2×C4#1458177311yescharacteristic
315C2×C4SD16×D8311yescharacteristic
316C2×C4#1458181311yescharacteristic
317C2×C4#1458181911yescharacteristic
318C2×C4#1458182111yescharacteristic
319C8—42no—
320C8—3322no—
321C8—3321no—
322C8—6641no—
323C8—1662no—
324C22—21no—
325C22—21no—
326C22—42no—
327C22—42no—
328C22—42no—
329C22#1912758111yescharacteristic
330C22#1912790411yescharacteristic
331C22#1912791811yescharacteristic
332C4—21no—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—42no—
337C4—82no—
338C4—13281no—
339C4SD16×Dic8311yescharacteristic
340C4#1912780211yescharacteristic
341C4#1912785011yescharacteristic
342C4#1912786011yescharacteristic
343C2—21no—
344C2—21no—
345C2—82no—
346C2#2490779911yescharacteristic
347C2#2490793811yescharacteristic
348C2#2490799111yescharacteristic
subgroupG/H
centerC22#19127581
commutatorC42×C83C22×C4
FrattiniC2×C42C22×D83
FittingC83×#3206C2
socleC22×C83#386
prime radical#532×C83C22
cosocleC2×C42×C83C23
hypercenter#3206D83
hypocenterC83#9289
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3206

Complements

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

NH ≅ G/Ntype
2. #24908437C2semidirect
3. #24906148C2semidirect
5. #24905322C2semidirect
6. #24905293C2semidirect
22. #492×C83C4semidirect
23. #487×C83C4semidirect
78. C2×C83⋊C8SD16semidirect
182. C83#9289semidirect
186. #492Dic83semidirect
187. #487Dic83semidirect

Maximal subgroups

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

subgroupindex
#249161812
#249084372
#249061482
#249060232
#249053222
#249052932
C83×#32062
#928983