Finite Group Databasebeta

← #35308014

Subgroups — #35308014

Order 21248.

408 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 97 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127912C211yesmaximal, characteristic
2#24907456C211yesmaximal, characteristic
3#24906136C211yesmaximal, characteristic
4#24906134C211yesmaximal, characteristic
5#24905857C211yesmaximal, characteristic
6#24905360C211yesmaximal, characteristic
7C83×#1571C211yesmaximal, characteristic
8C22×Dic332—21no—
9C2×#14581792C2211yescharacteristic
10C2×C4×Dic166C2211yescharacteristic
11#19127912—22no—
12#19127912—22no—
13#19127754C2211yescharacteristic
14#19127725C2211yescharacteristic
15C2×#14581717—21no—
16C2×#14581710—21no—
17#19127469—21no—
18#19127448—22no—
19#19127446—22no—
20#19127431—22no—
21#431×C83C2211yescharacteristic
22#423×C83C2211yescharacteristic
23C2×C4⋊C8×C83C2211yescharacteristic
24#335×C83—22no—
25C22×Dic166—21no—
26C22×Dic166D411yescharacteristic
27C2×Dic332—24no—
28C2×Dic332—42no—
29C2×Dic332—42no—
30C2×#13050574—21no—
31#14581792—21no—
32#14581792—21no—
33#14581792—21no—
34#14581792D412yes—
35C4×Dic166—21no—
36C4×Dic166—21no—
37C4×Dic166—21no—
38C4×Dic166—21no—
39C2×C4⋊Dic83—21no—
40C2×C4⋊Dic83—21no—
41C2×C4⋊Dic83D411yescharacteristic
42C2×Dic83⋊C4—21no—
43C2×C4×Dic83—21no—
44C4⋊(C83⋊C8)—21no—
45C4⋊(C83⋊C8)—21no—
46C4⋊(C83⋊C8)—21no—
47#14581717—24no—
48#14581710—42no—
49C22⋊(C83⋊C8)—21no—
50C4×C83⋊C8—21no—
51C2×M16×C83—21no—
52C22×C8×C83—21no—
53C2×C42×C83C2311yescharacteristic
54C4⋊C8×C83—21no—
55C4⋊C8×C83—21no—
56C4⋊C8×C83—21no—
57C4⋊C8×C83D412yes—
58C4⋊C8×C83D412yes—
59C22⋊C8×C83—21no—
60C4×C8×C83—21no—
61C2×Dic166—21no—
62C2×Dic166—21no—
63C2×Dic166—21no—
64C2×Dic166—21no—
65C2×Dic166—21no—
66C2×Dic166—22no—
67C2×Dic166—41no—
68C2×Dic166—41no—
69C2×Dic166—41no—
70C2×Dic166—41no—
71C2×Dic166—41no—
72C2×Dic166D812yes—
73C22×Dic83—21no—
74C22×Dic83—21no—
75C22×Dic83—41no—
76Dic332—48no—
77#13050574—41no—
78#13050574—41no—
79C2×C83⋊C8—41no—
80C2×C83⋊C8—41no—
81C4⋊Dic83—21no—
82C4⋊Dic83—22no—
83C4⋊Dic83—21no—
84C4⋊Dic83—22no—
85C4⋊Dic83—41no—
86C4⋊Dic83—41no—
87Dic83⋊C4—41no—
88Dic83⋊C4—41no—
89C4×Dic83—41no—
90C4×Dic83—41no—
91C22×C4×C83C2×D411yescharacteristic
92C22×C4×C83#4111yescharacteristic
93C22×C4×C83#4111yescharacteristic
94M16×C83—41no—
95M16×C83—41no—
96C2×C8×C83—22no—
97C2×C8×C83—22no—
98C2×C8×C83—41no—
99C2×C8×C83—41no—
100C2×C8×C83—41no—
101C42×C83C2×D411yescharacteristic
102C42×C83C2×D411yescharacteristic
103C42×C83C2×D411yescharacteristic
104C42×C83#4111yescharacteristic
105C23×C83#12311yescharacteristic
106C2×C4×C83—22no—
107C2×C4×C83—41no—
108C2×C4×C83#12011yescharacteristic
109C2×C4×C83#12111yescharacteristic
110C2×C4×C83#12111yescharacteristic
111C2×C4×C83#12111yescharacteristic
112C2×C4×C83#12311yescharacteristic
113C2×C4×C83#12411yescharacteristic
114C2×C4×C83C2×D811yescharacteristic
115C2×C4×C83#13511yescharacteristic
116C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
117C2×C4×C83#13711yescharacteristic
118C2×Dic83—22no—
119C2×Dic83—41no—
120C2×Dic83—41no—
121C2×Dic83—41no—
122C2×Dic83—42no—
123C2×Dic83—81no—
124C2×Dic83—81no—
125C2×Dic83—81no—
126Dic166—22no—
127Dic166—22no—
128Dic166—41no—
129Dic166—41no—
130Dic166—42no—
131Dic166—42no—
132Dic166—81no—
133Dic166—81no—
134C8×C83—42no—
135C8×C83—42no—
136C8×C83—81no—
137C83⋊C8—81no—
138C83⋊C8—81no—
139C22×C83—21no—
140C22×C83—21no—
141C22×C83#39411yescharacteristic
142C22×C83#48011yescharacteristic
143C22×C83#48211yescharacteristic
144C4×C83—21no—
145C4×C83—21no—
146C4×C83—21no—
147C4×C83#44711yescharacteristic
148C4×C83#45211yescharacteristic
149C4×C83#46611yescharacteristic
150C4×C83#46811yescharacteristic
151C4×C83#47312yes—
152Dic83—42no—
153Dic83—81no—
154Dic83—82no—
155#10446—831nomaximal, Sylow
156C2×C83—21no—
157C2×C83—21no—
158C2×C83#203311yescharacteristic
159C2×C83#204711yescharacteristic
160C2×C83#204911yescharacteristic
161C2×#462—831no—
162#1801—831no—
163#1571D8311yescharacteristic
164#1546—831no—
165#1546—831no—
166#1479—831no—
167#1301—831no—
168C83#1044611yescharacteristic, complemented, Sylow
169C22×Q16—1661no—
170C2×C4⋊Q8—831no—
171C2×C4×Q8—831no—
172#462—1662no—
173#462—1662no—
174#431—831no—
175#431C2×D8311yescharacteristic
176#423—831no—
177#423C2×D8311yescharacteristic
178C2×C4⋊C8C2×D8311yescharacteristic
179C2×#103—1661no—
180C2×#101—1661no—
181#343—1661no—
182#335—22no—
183#332—1662no—
184#332—1662no—
185Q8⋊C8—1662no—
186C22×Q8—831no—
187C22×Q8—1661no—
188C2×Q16—3322no—
189C2×Q16—3322no—
190C2×Q16—1664no—
191C2×M16—21no—
192C2×M16—1661no—
193C22×C8—21no—
194C4⋊Q8—832no—
195C4⋊Q8—1661no—
196C4⋊Q8—1661no—
197C4⋊Q8—1661no—
198C4×Q8—1661no—
199C4×Q8—1661no—
200C4×Q8—1661no—
201C4×Q8—1661no—
202C2×C4⋊C4—831no—
203C2×C4⋊C4—1661no—
204C2×C4⋊C4—1661no—
205C2×C4⋊C4—1661no—
206C2×C42—1661no—
207C2×C42C22×D8311yescharacteristic
208C4⋊C8—21no—
209C4⋊C8—21no—
210C4⋊C8—21no—
211C4⋊C8—1661no—
212C4⋊C8—1661no—
213C4⋊C8—1661no—
214C4⋊C8D33212yes—
215C4⋊C8C22⋊D8312yes—
216#103—1664no—
217#101—3322no—
218C22⋊C8—21no—
219C22⋊C8—1661no—
220C4×C8—21no—
221C4×C8—1661no—
222C2×Q8—3321no—
223C2×Q8—3321no—
224C2×Q8—3321no—
225C2×Q8—3321no—
226C2×Q8—3321no—
227C2×Q8—832no—
228C2×Q8—1662no—
229C2×Q8—1661no—
230C2×Q8—1661no—
231C2×Q8—1661no—
232C2×Q8—1661no—
233C2×Q8—1661no—
234C22×C4—3321no—
235C22×C4—1661no—
236C22×C4—1661no—
237C22×C4D4×D8311yescharacteristic
238C22×C4#1305059011yescharacteristic
239C22×C4#1305059111yescharacteristic
240Q16—3328no—
241M16—41no—
242M16—41no—
243M16—3321no—
244M16—3321no—
245C2×C8—22no—
246C2×C8—22no—
247C2×C8—41no—
248C2×C8—41no—
249C2×C8—41no—
250C2×C8—3321no—
251C2×C8—3321no—
252C4⋊C4—3321no—
253C4⋊C4—3321no—
254C4⋊C4—3321no—
255C4⋊C4—3321no—
256C4⋊C4—1661no—
257C4⋊C4—1661no—
258C4⋊C4—1662no—
259C4⋊C4—1662no—
260C42—3321no—
261C42—3321no—
262C42D4×D8311yescharacteristic
263C42C2×C22⋊D8311yescharacteristic
264C42C2×D33211yescharacteristic
265C42#1305059111yescharacteristic
266C23#1458177711yescharacteristic
267Q8—3321no—
268Q8—3321no—
269Q8—3322no—
270Q8—3322no—
271Q8—6641no—
272Q8—6641no—
273Q8—1662no—
274Q8—1662no—
275C2×C4—22no—
276C2×C4—41no—
277C2×C4—3321no—
278C2×C4—3321no—
279C2×C4—3321no—
280C2×C4—3322no—
281C2×C4—6641no—
282C2×C4—6641no—
283C2×C4—6641no—
284C2×C4—1662no—
285C2×C4#1458176011yescharacteristic
286C2×C4#1458176611yescharacteristic
287C2×C4#1458176711yescharacteristic
288C2×C4#1458176711yescharacteristic
289C2×C4#1458177711yescharacteristic
290C2×C4#1458178111yescharacteristic
291C2×C4D8×D8311yescharacteristic
292C2×C4#1458181511yescharacteristic
293C2×C4#1458181611yescharacteristic
294C2×C4#1458182411yescharacteristic
295C8—42no—
296C8—42no—
297C8—81no—
298C8—6641no—
299C8—6641no—
300C22—21no—
301C22—21no—
302C22#1912761811yescharacteristic
303C22#1912801611yescharacteristic
304C22#1912802811yescharacteristic
305C4—21no—
306C4—21no—
307C4—21no—
308C4—3322no—
309C4—6642no—
310C4—6641no—
311C4#1912783311yescharacteristic
312C4#1912786211yescharacteristic
313C4#1912793811yescharacteristic
314C4#1912794811yescharacteristic
315C4#1912797612yes—
316C2—21no—
317C2—21no—
318C2#2490864711yescharacteristic
319C2#2490873111yescharacteristic
320C2#2490874511yescharacteristic
subgroupG/H
centerC22#19127618
commutator, cosocleC2×C42×C83C23
FrattiniC2×C42C22×D83
FittingC83×#1571C2
socleC22×C83#394
prime radicalC23×C83#123
hypercenter#1571D83
hypocenterC83#10446
LCSG ▷ C2×C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1571

Complements

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

NH ≅ G/Ntype
168. C83#10446semidirect

Maximal subgroups

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

subgroupindex
C2×#191279122
#249074562
#249061362
#249061342
#249058572
#249053602
C83×#15712
#1044683