Finite Group Databasebeta

← #35302279

Subgroups — #35302279

Order 21248.

522 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 119 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#24916548C211yesmaximal, characteristic, complemented
2#24907459C211yesmaximal, characteristic, complemented
3#24905888C212yesmaximal
4#24905321C212yesmaximal, complemented
5C83×#2536C211yesmaximal, characteristic
6#19127832—24no—
7#19127758—22no—
8#19127754C2212yes—
9#19127725C2212yes—
10C2×#14581711—21no—
11#19127470—21no—
12D4⋊(C83⋊C8)—24no—
13C2×C22⋊Q8×C83—21no—
14C2×C4×Q8×C83C2211yescharacteristic
15C2×C4×D4×C83C2211yescharacteristic
16#398×C83C412yes—
17#394×C83C412yes—
18#388×C83—22no—
19#382×C83C2211yescharacteristic
20#380×C83—21no—
21C2×#13050574—22no—
22C4⋊(C83⋊C8)—24no—
23C4⋊(C83⋊C8)—22no—
24C4⋊(C83⋊C8)—22no—
25C4⋊(C83⋊C8)—22no—
26#14581711—42no—
27C22⋊(C83⋊C8)—22no—
28#14581703—24no—
29C4×C83⋊C8D414yes—
30C22×Q8×C83D411yescharacteristic
31C22×D4×C83C2×C411yescharacteristic
32C23×C4×C83—21no—
33C22⋊Q8×C83—44no—
34C4×Q8×C83—22no—
35C4×Q8×C83—22no—
36C4×D4×C83—22no—
37C4×D4×C83—22no—
38C2×C4⋊C4×C83—22no—
39C2×C4⋊C4×C83—21no—
40C2×C4⋊C4×C83C2×C411yescharacteristic
41C2×C4⋊C4×C83C2×C412yes—
42C2×C4⋊C4×C83D411yescharacteristic
43C2×C22⋊C4×C83—22no—
44C2×C22⋊C4×C83—22no—
45C2×C22⋊C4×C83—21no—
46C2×C42×C83D412yes—
47C2×C42×C83C2311yescharacteristic
48#95×C83—22no—
49#95×C83—21no—
50#95×C83C2×C412yes—
51#13050574—42no—
52#13050574—42no—
53C2×C83⋊C8—24no—
54C2×C83⋊C8—42no—
55C24×C83—21no—
56C2×Q8×C83—22no—
57C2×Q8×C83—22no—
58C2×Q8×C83—21no—
59C2×Q8×C83—21no—
60C2×D4×C83—22no—
61C2×D4×C83—42no—
62C22×C4×C83—22no—
63C22×C4×C83—21no—
64C22×C4×C83—42no—
65C22×C4×C83—42no—
66C22×C4×C83—41no—
67C22×C4×C83—41no—
68C22×C4×C83C22⋊C412yes—
69C22×C4×C83C22⋊C412yes—
70C22×C4×C83C22×C411yescharacteristic
71C22×C4×C83C2×D411yescharacteristic
72C22×C4×C83C2×D411yescharacteristic
73C4⋊C4×C83—24no—
74C4⋊C4×C83—22no—
75C4⋊C4×C83—22no—
76C4⋊C4×C83—44no—
77C4⋊C4×C83—42no—
78C22⋊C4×C83—44no—
79C22⋊C4×C83—44no—
80C22⋊C4×C83—41no—
81C22⋊C4×C83—41no—
82C42×C83—22no—
83C42×C83—22no—
84C42×C83C2×D412yes—
85C42×C83#4112yes—
86C23×C83—22no—
87C23×C83—41no—
88C23×C83—41no—
89C23×C83—41no—
90C23×C83—41no—
91C23×C83—41no—
92C23×C83—41no—
93C23×C83C2×C22⋊C411yescharacteristic
94Q8×C83—42no—
95Q8×C83—42no—
96D4×C83—24no—
97D4×C83—42no—
98C2×C4×C83—22no—
99C2×C4×C83—22no—
100C2×C4×C83—22no—
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—41no—
108C2×C4×C83—41no—
109C2×C4×C83—42no—
110C2×C4×C83—41no—
111C2×C4×C83—82no—
112C2×C4×C83—82no—
113C2×C4×C83C4×D412yes—
114C2×C4×C83#12011yescharacteristic
115C2×C4×C83#12112yes—
116C2×C4×C83#12311yescharacteristic
117C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
118C2×C4×C83AGL(1,ℤ/8)12yes—
119C2×C4×C83#13711yescharacteristic
120C83⋊C8—44no—
121C83⋊C8—82no—
122C22×C83—21no—
123C22×C83—21no—
124C22×C83—42no—
125C22×C83—42no—
126C22×C83—42no—
127C22×C83—81no—
128C22×C83—81no—
129C22×C83—81no—
130C22×C83—81no—
131C22×C83#38611yescharacteristic
132C22×C83#41711yescharacteristic
133C22×C83#41811yescharacteristic
134C4×C83—22no—
135C4×C83—21no—
136C4×C83—21no—
137C4×C83—42no—
138C4×C83—42no—
139C4×C83—82no—
140C4×C83—81no—
141C4×C83#46812yes—
142C4×C83#46912yes—
143#9490—831nomaximal, Sylow
144C2×C83—21no—
145C2×C83—21no—
146C2×C83—82no—
147C2×C83#189111yescharacteristic
148C2×C83#189511yescharacteristic
149C2×C83#191211yescharacteristic
150#2983—831no—
151#2536D8311yescharacteristic
152#1801—831no—
153#1486—832no—
154#1286—832no—
155C83#949011yescharacteristic, complemented, Sylow
156C2×C22⋊Q8—21no—
157C2×C4×Q8C2×D8311yescharacteristic
158C2×C4×D4C2×D8311yescharacteristic
159#446—1664no—
160#432—1662no—
161#431—832no—
162#423—832no—
163C2×#101—1661no—
164#398Dic8312yes—
165#394Dic8312yes—
166#388—22no—
167#382C2×D8311yescharacteristic
168#380—21no—
169#343—1661no—
170D4⋊C8—1664no—
171C22×Q8C22⋊D8311yescharacteristic
172C22×D4C2×Dic8311yescharacteristic
173C23×C4—21no—
174C2×M16—1662no—
175C22⋊Q8—44no—
176C4×Q8—22no—
177C4×Q8—22no—
178C4×D4—22no—
179C4×D4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4C2×Dic8311yescharacteristic
183C2×C4⋊C4C2×Dic8312yes—
184C2×C4⋊C4C22⋊D8311yescharacteristic
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—21no—
188C2×C42C22⋊D8312yes—
189C2×C42C22×D8311yescharacteristic
190C4⋊C8—1664no—
191C4⋊C8—1662no—
192C4⋊C8—1662no—
193C4⋊C8—1662no—
194#101—3322no—
195C22⋊C8—1662no—
196#97—1664no—
197C4×C8—834no—
198#95—22no—
199#95—21no—
200#95C2×Dic8312yes—
201C24—21no—
202C2×Q8—22no—
203C2×Q8—21no—
204C2×Q8—22no—
205C2×Q8—21no—
206C2×D4—22no—
207C2×D4—42no—
208C22×C4—22no—
209C22×C4—21no—
210C22×C4—42no—
211C22×C4—41no—
212C22×C4—42no—
213C22×C4—41no—
214C22×C4C22⋊Dic8312yes—
215C22×C4C22⋊Dic8312yes—
216C22×C4C22×Dic8311yescharacteristic
217C22×C4C2×C22⋊D8311yescharacteristic
218C22×C4C2×C22⋊D8311yescharacteristic
219M16—3322no—
220M16—3322no—
221C2×C8—3322no—
222C2×C8—1664no—
223C4⋊C4—22no—
224C4⋊C4—24no—
225C4⋊C4—22no—
226C4⋊C4—44no—
227C4⋊C4—42no—
228C22⋊C4—44no—
229C22⋊C4—44no—
230C22⋊C4—41no—
231C22⋊C4—41no—
232C42—22no—
233C42—22no—
234C42D4×D8312yes—
235C42#1305058912yes—
236C23—22no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23C2×C22⋊Dic8311yescharacteristic
244Q8—42no—
245Q8—42no—
246D4—24no—
247D4—42no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4—41no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4D4×Dic8312yes—
264C2×C4#1458176011yescharacteristic
265C2×C4#1458176412yes—
266C2×C4#1458177311yescharacteristic
267C2×C4#1458181711yescharacteristic
268C2×C4#1458181812yes—
269C2×C4#1458182211yescharacteristic
270C8—3324no—
271C8—6642no—
272C22—21no—
273C22—21no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—81no—
278C22—81no—
279C22—81no—
280C22—81no—
281C22#1912758111yescharacteristic
282C22#1912770211yescharacteristic
283C22#1912770611yescharacteristic
284C4—22no—
285C4—21no—
286C4—21no—
287C4—42no—
288C4—42no—
289C4—82no—
290C4—81no—
291C4#1912794512yes—
292C4#1912795212yes—
293C2—21no—
294C2—21no—
295C2—82no—
296C2#2490788211yescharacteristic
297C2#2490790211yescharacteristic
298C2#2490800111yescharacteristic
subgroupG/H
centerC22#19127581
commutatorC22×C4×C83C22×C4
FrattiniC2×C42C22×D83
FittingC83×#2536C2
socleC22×C83#386
prime radicalC22×D4×C83C2×C4
cosocleC2×C42×C83C23
hypercenter#2536D83
hypocenterC83#9490
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2536

Complements

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

NH ≅ G/Ntype
1. #24916548C2semidirect
2. #24907459C2semidirect
4. #24905321C2semidirect
155. C83#9490semidirect

Maximal subgroups

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

subgroupindex
#249165482
#249074592
#249058882
#249053212
C83×#25362
#949083