Finite Group Databasebeta

← #35293709

Subgroups — #35293709

Order 21248.

494 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 87 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
1#24909001C211yesmaximal, characteristic, complemented
2#24908996C212yesmaximal, complemented
3#24905289C211yesmaximal, characteristic, complemented
4#24905284C212yesmaximal, complemented
5C83×#2733C211yesmaximal, characteristic
6C2×#14581725—21no—
7C2×#14581723—22no—
8C2×#14581716—21no—
9C2×#14581713—22no—
10#19127571C2211yescharacteristic
11#19127456C2211yescharacteristic
12#19127456C2212yes—
13C2×#124×C83—21no—
14#394×C83—21no—
15#392×C83—22no—
16#391×C83C2211yescharacteristic
17#390×C83C2212yes—
18#388×C83—22no—
19C22×C83⋊C8—22no—
20C22×C83⋊C8—21no—
21C2×C4⋊Dic83—22no—
22C2×C4⋊Dic83—21no—
23#14581725—42no—
24#14581723—44no—
25#14581716—42no—
26#14581713—44no—
27C22×Q8×C83D411yescharacteristic
28C22×D4×C83—21no—
29C22×D4×C83—21no—
30C22×D4×C83D412yes—
31#124×C83—42no—
32#124×C83—42no—
33C2×C4⋊C4×C83D412yes—
34C2×C4⋊C4×C83D411yescharacteristic
35C2×C22⋊C4×C83—22no—
36C2×C22⋊C4×C83—22no—
37C2×C22⋊C4×C83—22no—
38C2×C22⋊C4×C83—22no—
39C2×C42×C83—21no—
40C2×C42×C83C2311yescharacteristic
41#95×C83—21no—
42#95×C83—22no—
43C22×Dic83—41no—
44C2×C83⋊C8—24no—
45C2×C83⋊C8—22no—
46C2×C83⋊C8—42no—
47C2×C83⋊C8—41no—
48C2×C83⋊C8—42no—
49C2×C83⋊C8—41no—
50C4⋊Dic83—42no—
51C4⋊Dic83—41no—
52C4⋊Dic83—42no—
53C4⋊Dic83—41no—
54C24×C83—22no—
55C2×Q8×C83—21no—
56C2×Q8×C83—41no—
57C2×Q8×C83—41no—
58C2×Q8×C83SD1612yes—
59C2×D4×C83—22no—
60C2×D4×C83—22no—
61C2×D4×C83—42no—
62C2×D4×C83—42no—
63C2×D4×C83—42no—
64C2×D4×C83—81no—
65C2×D4×C83—81no—
66C2×D4×C83—82no—
67C2×D4×C83—81no—
68C22×C4×C83—21no—
69C22×C4×C83—21no—
70C22×C4×C83—22no—
71C22×C4×C83C2×D411yescharacteristic
72C22×C4×C83C2×D412yes—
73C4⋊C4×C83—42no—
74C4⋊C4×C83—41no—
75C22⋊C4×C83—44no—
76C22⋊C4×C83—42no—
77C22⋊C4×C83—42no—
78C22⋊C4×C83—42no—
79C22⋊C4×C83—42no—
80C22⋊C4×C83—82no—
81C42×C83—41no—
82C42×C83—42no—
83C23×C83—42no—
84C23×C83—42no—
85C23×C83—42no—
86C23×C83—42no—
87C23×C83—42no—
88C23×C83—42no—
89C23×C83—42no—
90C23×C83#12011yescharacteristic
91Q8×C83—42no—
92Q8×C83—81no—
93D4×C83—44no—
94D4×C83—82no—
95D4×C83—84no—
96D4×C83—84no—
97C2×C4×C83—22no—
98C2×C4×C83—21no—
99C2×C4×C83—22no—
100C2×C4×C83—21no—
101C2×C4×C83—22no—
102C2×C4×C83—41no—
103C2×C4×C83—41no—
104C2×C4×C83—42no—
105C2×C4×C83—42no—
106C2×C4×C83—42no—
107C2×C4×C83—42no—
108C2×C4×C83—41no—
109C2×C4×C83C2×SD1611yescharacteristic
110C2×C4×C83#13512yes—
111C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
112C2×C4×C83#13712yes—
113C2×Dic83—81no—
114C2×Dic83—82no—
115C83⋊C8—42no—
116C83⋊C8—44no—
117C22×C83—82no—
118C22×C83—82no—
119C22×C83—82no—
120C22×C83—82no—
121C22×C83—82no—
122C22×C83—82no—
123C22×C83—82no—
124C22×C83#45011yescharacteristic
125C22×C83#45212yes—
126C22×C83#45311yescharacteristic
127C22×C83#45312yes—
128C22×C83#45811yescharacteristic
129C4×C83—21no—
130C4×C83—21no—
131C4×C83—22no—
132C4×C83—22no—
133C4×C83—42no—
134C4×C83—82no—
135C4×C83—81no—
136Dic83—161no—
137#7830—831nomaximal, Sylow
138C2×C83—162no—
139C2×C83#161311yescharacteristic
140C2×C83#161812yes—
141C2×C83#163112yes—
142C2×C83#163411yescharacteristic
143C2×C83#165511yescharacteristic
144#2733D8311yescharacteristic
145#2095—831no—
146#2094—832no—
147#1272—831no—
148#1270—832no—
149C83#783011yescharacteristic, complemented, Sylow
150C2×#124—21no—
151C2×#107—1661no—
152C2×#106—1662no—
153C2×#103—1661no—
154C2×#102—1662no—
155#394—21no—
156#392—22no—
157#391C2×D8311yescharacteristic
158#390C2×D8312yes—
159#388—22no—
160#384—831no—
161#336—832no—
162#336—831no—
163C22×Q8C22⋊D8311yescharacteristic
164C22×D4—21no—
165C22×D4—21no—
166C22×D4C22⋊D8312yes—
167C22×C8—1662no—
168C22×C8—1661no—
169#124—42no—
170#124—42no—
171C2×C4⋊C4—1662no—
172C2×C4⋊C4—1661no—
173C2×C4⋊C4C22⋊D8312yes—
174C2×C4⋊C4C22⋊D8311yescharacteristic
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C42—21no—
180C2×C42C22×D8311yescharacteristic
181#107—3322no—
182#106—3324no—
183#103—3322no—
184#102—3324no—
185#95—21no—
186#95—22no—
187C24—22no—
188C2×Q8—21no—
189C2×Q8—41no—
190C2×Q8—41no—
191C2×Q8#1305057812yes—
192C2×D4—22no—
193C2×D4—22no—
194C2×D4—42no—
195C2×D4—42no—
196C2×D4—42no—
197C2×D4—81no—
198C2×D4—81no—
199C2×D4—82no—
200C2×D4—81no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4—3321no—
205C22×C4C2×C22⋊D8311yescharacteristic
206C22×C4C2×C22⋊D8312yes—
207C2×C8—3322no—
208C2×C8—3321no—
209C2×C8—3322no—
210C2×C8—3321no—
211C2×C8—1664no—
212C2×C8—1662no—
213C4⋊C4—42no—
214C4⋊C4—41no—
215C4⋊C4—3322no—
216C4⋊C4—3321no—
217C4⋊C4—3322no—
218C4⋊C4—3321no—
219C22⋊C4—44no—
220C22⋊C4—42no—
221C22⋊C4—42no—
222C22⋊C4—42no—
223C22⋊C4—42no—
224C22⋊C4—82no—
225C42—41no—
226C42—42no—
227C23—42no—
228C23—42no—
229C23—42no—
230C23—42no—
231C23—42no—
232C23—42no—
233C23—42no—
234C23#1458176111yescharacteristic
235Q8—42no—
236Q8—81no—
237D4—44no—
238D4—82no—
239D4—84no—
240D4—84no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—6642no—
254C2×C4—6641no—
255C2×C4C2×#1305057811yescharacteristic
256C2×C4#1458181212yes—
257C2×C4#1458181811yescharacteristic
258C2×C4#1458182212yes—
259C8—3324no—
260C8—3322no—
261C22—82no—
262C22—82no—
263C22—82no—
264C22—82no—
265C22—82no—
266C22—82no—
267C22—82no—
268C22#1912785211yescharacteristic
269C22#1912786312yes—
270C22#1912786912yes—
271C22#1912786911yescharacteristic
272C22#1912789111yescharacteristic
273C4—22no—
274C4—22no—
275C4—21no—
276C4—21no—
277C4—42no—
278C4—81no—
279C4—82no—
280C4—13281no—
281C2—162no—
282C2#2490648311yescharacteristic
283C2#2490651112yes—
284C2#2490659012yes—
285C2#2490660811yescharacteristic
286C2#2490675211yescharacteristic
subgroupG/H
centerC23#14581761
commutator, cosocleC2×C42×C83C23
FrattiniC2×C42C22×D83
Fitting, prime radicalC83×#2733C2
socleC23×C83#120
hypercenter#2733D83
hypocenterC83#7830
LCSG ▷ C2×C42×C83 ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2733

Complements

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

NH ≅ G/Ntype
1. #24909001C2semidirect
2. #24908996C2semidirect
3. #24905289C2semidirect
4. #24905284C2semidirect
149. C83#7830semidirect

Maximal subgroups

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

subgroupindex
#249090012
#249089962
#249052892
#249052842
C83×#27332
#783083