Finite Group Databasebeta

← #35303897

Subgroups — #35303897

Order 21248.

390 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 113 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×#19128104C211yesmaximal, characteristic
2#24907600C211yesmaximal, characteristic
3#24906149C211yesmaximal, characteristic
4#24906148C211yesmaximal, characteristic
5#24906143C211yesmaximal, characteristic
6#24905756C211yesmaximal, characteristic
7C83×#1548C211yesmaximal, characteristic
8C2×#14581792C2211yescharacteristic
9#19128104—21no—
10#19128104—21no—
11#19128104C414yes—
12#19127754C2211yescharacteristic
13C2×#14581725—22no—
14#19127725C2211yescharacteristic
15C2×C4×C83⋊C8C2211yescharacteristic
16#19127465—22no—
17#19127449—22no—
18#19127447—22no—
19#19127446—22no—
20#19127423—21no—
21#19127422—21no—
22C2×C4⋊Q8×C83C2211yescharacteristic
23#431×C83C2211yescharacteristic
24#423×C83C2211yescharacteristic
25#333×C83—22no—
26C22×Dic166—21no—
27C2×#13050574—21no—
28C22×C83⋊C8D412yes—
29#14581792—21no—
30#14581792—21no—
31#14581792—21no—
32#14581792C2×C412yes—
33C2×C4⋊Dic83—21no—
34C2×C4⋊Dic83—21no—
35#14581725—24no—
36#14581725—24no—
37C4⋊(C83⋊C8)—21no—
38C4⋊(C83⋊C8)—21no—
39C4⋊(C83⋊C8)D412yes—
40C22⋊(C83⋊C8)—21no—
41C4×C83⋊C8—21no—
42C4×C83⋊C8C2×C412yes—
43C4×C83⋊C8D411yescharacteristic
44C4×C83⋊C8D411yescharacteristic
45C22×Q8×C83—21no—
46C2×M16×C83—21no—
47C4⋊Q8×C83—21no—
48C4⋊Q8×C83—21no—
49C4⋊Q8×C83—21no—
50C4⋊Q8×C83C2×C412yes—
51C2×C4⋊C4×C83—21no—
52C2×C4⋊C4×C83—21no—
53C2×C42×C83C2311yescharacteristic
54C4⋊C8×C83—21no—
55C4⋊C8×C83—21no—
56C4⋊C8×C83D412yes—
57C22⋊C8×C83—21no—
58C4×C8×C83—21no—
59C2×Dic166—22no—
60C2×Dic166—41no—
61C2×Dic166—82no—
62C22×Dic83—41no—
63#13050574—42no—
64C2×C83⋊C8—22no—
65C2×C83⋊C8—22no—
66C2×C83⋊C8—22no—
67C2×C83⋊C8—41no—
68C2×C83⋊C8C22⋊C412yes—
69C2×C83⋊C8C22⋊C412yes—
70C4⋊Dic83—22no—
71C4⋊Dic83—22no—
72C4⋊Dic83—41no—
73C4⋊Dic83—41no—
74C2×Q8×C83—22no—
75C2×Q8×C83—41no—
76C2×Q8×C83—82no—
77C22×C4×C83—41no—
78C22×C4×C83C2×D411yescharacteristic
79C22×C4×C83#4111yescharacteristic
80C22×C4×C83#4111yescharacteristic
81M16×C83—42no—
82C2×C8×C83—22no—
83C2×C8×C83—41no—
84C4⋊C4×C83—22no—
85C4⋊C4×C83—22no—
86C4⋊C4×C83—41no—
87C4⋊C4×C83—41no—
88C42×C83C22×C411yescharacteristic
89C42×C83C2×D411yescharacteristic
90C42×C83C2×D411yescharacteristic
91C42×C83C2×D411yescharacteristic
92C23×C83#12411yescharacteristic
93Q8×C83—82no—
94Q8×C83—82no—
95C2×C4×C83—22no—
96C2×C4×C83—22no—
97C2×C4×C83—42no—
98C2×C4×C83—82no—
99C2×C4×C83C2×C22⋊C411yescharacteristic
100C2×C4×C83C4×D411yescharacteristic
101C2×C4×C83C4×D411yescharacteristic
102C2×C4×C83#12111yescharacteristic
103C2×C4×C83#12111yescharacteristic
104C2×C4×C83#12711yescharacteristic
105C2×C4×C83#13712yes—
106C2×C4×C83#13712yes—
107C2×Dic83—42no—
108C2×Dic83—82no—
109Dic166—82no—
110Dic166—82no—
111C8×C83—42no—
112C8×C83—81no—
113C83⋊C8—22no—
114C83⋊C8—22no—
115C83⋊C8—42no—
116C83⋊C8—81no—
117C22×C83—21no—
118C22×C83—21no—
119C22×C83#39011yescharacteristic
120C22×C83#49112yes—
121C4×C83—21no—
122C4×C83—21no—
123C4×C83—82no—
124C4×C83#41912yes—
125C4×C83#47212yes—
126C4×C83#47312yes—
127C4×C83#49711yescharacteristic
128C4×C83#49711yescharacteristic
129Dic83—82no—
130#9778—831nomaximal, Sylow
131C2×C83—22no—
132C2×C83#196011yescharacteristic
133C2×C83#197811yescharacteristic
134C2×C83#197911yescharacteristic
135C2×#500—831no—
136#1837—831no—
137#1549—831no—
138#1549—831no—
139#1548—831no—
140#1548D8311yescharacteristic
141#1452—831no—
142C83#977811yescharacteristic, complemented, Sylow
143C2×C4⋊Q8—831no—
144C2×C4⋊Q8C2×D8311yescharacteristic
145#500—834no—
146#500—1661no—
147#500—1661no—
148#431—831no—
149#431C2×D8311yescharacteristic
150C2×#107—1662no—
151#423—831no—
152#423C2×D8311yescharacteristic
153C2×C4×C8—831no—
154#341—1662no—
155#333—22no—
156#333—1662no—
157#332—1662no—
158#332—1662no—
159#322—1661no—
160#322—1661no—
161C22×Q8—21no—
162C22×Q8—1661no—
163C2×M16—21no—
164C2×M16—1661no—
165C22×C8—832no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—832no—
170C4⋊Q8—1661no—
171C4⋊Q8—1661no—
172C4⋊Q8—1661no—
173C4⋊Q8C4×D8312yes—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—1661no—
177C2×C4⋊C4—1661no—
178C2×C42C22×D8311yescharacteristic
179#107—1664no—
180#107—1664no—
181C4⋊C8—21no—
182C4⋊C8—21no—
183C4⋊C8—832no—
184C4⋊C8—1661no—
185C4⋊C8—1661no—
186C4⋊C8C22⋊D8312yes—
187C22⋊C8—21no—
188C22⋊C8—1661no—
189C4×C8—21no—
190C4×C8—832no—
191C4×C8—831no—
192C4×C8—831no—
193C4×C8—1661no—
194C2×Q8—22no—
195C2×Q8—41no—
196C2×Q8—82no—
197C2×Q8—3321no—
198C2×Q8—6642no—
199C2×Q8—1662no—
200C22×C4—41no—
201C22×C4—3321no—
202C22×C4D4×D8311yescharacteristic
203C22×C4#1305058911yescharacteristic
204C22×C4#1305059011yescharacteristic
205M16—42no—
206M16—3322no—
207C2×C8—22no—
208C2×C8—41no—
209C2×C8—3321no—
210C2×C8—832no—
211C2×C8—832no—
212C2×C8—1662no—
213C2×C8—1662no—
214C2×C8—1662no—
215C4⋊C4—22no—
216C4⋊C4—22no—
217C4⋊C4—41no—
218C4⋊C4—41no—
219C4⋊C4—3321no—
220C4⋊C4—3321no—
221C4⋊C4—1662no—
222C4⋊C4—1662no—
223C42C2×C4×D8311yescharacteristic
224C42D4×D8311yescharacteristic
225C42D4×D8311yescharacteristic
226C42C2×C22⋊D8311yescharacteristic
227C23#1458177811yescharacteristic
228Q8—82no—
229Q8—82no—
230Q8—6642no—
231Q8—6642no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—42no—
235C2×C4—82no—
236C2×C4—3322no—
237C2×C4—6642no—
238C2×C4C22⋊C4×D8311yescharacteristic
239C2×C4(C22⋊D83)⋊C411yescharacteristic
240C2×C4C4×C22⋊D8311yescharacteristic
241C2×C4#1458176411yescharacteristic
242C2×C4#1458176511yescharacteristic
243C2×C4#1458178811yescharacteristic
244C2×C4#1458182112yes—
245C2×C4#1458182412yes—
246C8—42no—
247C8—81no—
248C8—3322no—
249C8—6641no—
250C8—1662no—
251C8—1662no—
252C22—21no—
253C22—21no—
254C22#1912760111yescharacteristic
255C22#1912806812yes—
256C4—21no—
257C4—21no—
258C4—82no—
259C4—6642no—
260C4#1912770812yes—
261C4#1912797012yes—
262C4#1912797412yes—
263C4#1912809211yescharacteristic
264C4#1912809411yescharacteristic
265C2—22no—
266C2#2490828111yescharacteristic
267C2#2490837011yescharacteristic
268C2#2490837711yescharacteristic
subgroupG/H
centerC22#19127601
commutatorC42×C83C22×C4
FrattiniC2×C42C22×D83
FittingC83×#1548C2
socleC22×C83#390
prime radicalC23×C83#124
cosocleC2×C42×C83C23
hypercenter#1548D83
hypocenterC83#9778
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
142. C83#9778semidirect

Maximal subgroups

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

subgroupindex
C2×#191281042
#249076002
#249061492
#249061482
#249061432
#249057562
C83×#15482
#977883