Finite Group Databasebeta

← #35296142

Subgroups — #35296142

Order 21248.

706 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 706 conjugacy classes of subgroups — too many to draw. Only its 159 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#24907459C211yesmaximal, characteristic
2#24906135C212yesmaximal
3#24905892C212yesmaximal
4#24905787C211yesmaximal, characteristic
5C2×#557×C83C211yesmaximal, characteristic
6#19127754C2212yes—
7#19127725C2212yes—
8C2×#14581711—21no—
9#19127470—21no—
10#19127447—24no—
11Q8⋊(C83⋊C8)—24no—
12#19127427—22no—
13#557×C83—24no—
14#557×C83—24no—
15C2×C4⋊Q8×C83—21no—
16C2×C4⋊Q8×C83C2211yescharacteristic
17C2×#125×C83—22no—
18C2×#125×C83C412yes—
19C2×C4×Q8×C83—21no—
20C2×C4×Q8×C83C412yes—
21C2×C4×Q8×C83C2211yescharacteristic
22C2×C4×Q8×C83C2211yescharacteristic
23C2×#13050574—22no—
24C4⋊(C83⋊C8)—22no—
25C4⋊(C83⋊C8)—22no—
26C4⋊(C83⋊C8)—22no—
27#14581711—42no—
28C22⋊(C83⋊C8)—22no—
29C4×C83⋊C8—22no—
30C22×Q8×C83—21no—
31C22×Q8×C83C2×C411yescharacteristic
32C22×Q8×C83D411yescharacteristic
33C4⋊Q8×C83—28no—
34C4⋊Q8×C83—22no—
35C4⋊Q8×C83D414yes—
36#125×C83—216no—
37#125×C83—24no—
38#125×C83—24no—
39C4×Q8×C83—28no—
40C4×Q8×C83—24no—
41C4×Q8×C83—24no—
42C4×Q8×C83—22no—
43C4×Q8×C83—22no—
44C4×Q8×C83—22no—
45C4×Q8×C83D414yes—
46C2×C4⋊C4×C83—22no—
47C2×C4⋊C4×C83—22no—
48C2×C4⋊C4×C83—22no—
49C2×C4⋊C4×C83—21no—
50C2×C4⋊C4×C83—21no—
51C2×C4⋊C4×C83C2×C412yes—
52C2×C4⋊C4×C83C2×C411yescharacteristic
53C2×C4⋊C4×C83D411yescharacteristic
54C2×C4⋊C4×C83D412yes—
55C2×C42×C83—22no—
56C2×C42×C83—21no—
57C2×C42×C83C2×C412yes—
58C2×C42×C83C2311yescharacteristic
59#13050574—42no—
60#13050574—42no—
61C2×C83⋊C8—42no—
62C2×C83⋊C8—42no—
63C2×Q8×C83—24no—
64C2×Q8×C83—21no—
65C2×Q8×C83—22no—
66C2×Q8×C83—24no—
67C2×Q8×C83—22no—
68C2×Q8×C83—21no—
69C2×Q8×C83—21no—
70C2×Q8×C83—42no—
71C2×Q8×C83—41no—
72C2×Q8×C83—41no—
73C2×Q8×C83C22⋊C412yes—
74C22×C4×C83—21no—
75C22×C4×C83—21no—
76C22×C4×C83—22no—
77C22×C4×C83C22⋊C412yes—
78C22×C4×C83C22⋊C412yes—
79C22×C4×C83C22×C411yescharacteristic
80C22×C4×C83C2×D411yescharacteristic
81C22×C4×C83C2×D411yescharacteristic
82C4⋊C4×C83—28no—
83C4⋊C4×C83—28no—
84C4⋊C4×C83—28no—
85C4⋊C4×C83—21no—
86C4⋊C4×C83—22no—
87C4⋊C4×C83—24no—
88C4⋊C4×C83—24no—
89C4⋊C4×C83—22no—
90C4⋊C4×C83—24no—
91C4⋊C4×C83C22⋊C414yes—
92C4⋊C4×C83C22⋊C412yes—
93C42×C83—28no—
94C42×C83—22no—
95C42×C83—22no—
96C42×C83—24no—
97C42×C83C2×D412yes—
98C42×C83C2×D412yes—
99C23×C83C2×C22⋊C411yescharacteristic
100Q8×C83—24no—
101Q8×C83—22no—
102Q8×C83—42no—
103Q8×C83—44no—
104Q8×C83—42no—
105Q8×C83—44no—
106Q8×C83#10114yescomplemented
107C2×C4×C83—24no—
108C2×C4×C83—22no—
109C2×C4×C83—22no—
110C2×C4×C83—22no—
111C2×C4×C83—22no—
112C2×C4×C83—22no—
113C2×C4×C83—24no—
114C2×C4×C83—22no—
115C2×C4×C83—22no—
116C2×C4×C83—41no—
117C2×C4×C83—42no—
118C2×C4×C83—42no—
119C2×C4×C83—42no—
120C2×C4×C83—42no—
121C2×C4×C83C2×C22⋊C411yescharacteristic
122C2×C4×C83C2×C22⋊C411yescharacteristic
123C2×C4×C83#12012yes—
124C2×C4×C83#12012yes—
125C2×C4×C83#13512yes—
126C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
127C2×C4×C83#13711yescharacteristic
128C83⋊C8—82no—
129C83⋊C8—82no—
130C22×C83—21no—
131C22×C83—21no—
132C22×C83#37911yescharacteristic
133C22×C83#41611yescharacteristic
134C22×C83#41711yescharacteristic
135C4×C83—22no—
136C4×C83—24no—
137C4×C83—21no—
138C4×C83—42no—
139C4×C83—42no—
140C4×C83—42no—
141C4×C83—44no—
142C4×C83C2×#10112yes—
143C4×C83#44912yes—
144C4×C83#45212yes—
145#8390—831nomaximal, Sylow
146C2×C83—21no—
147C2×C83—21no—
148C2×C83#178411yescharacteristic
149C2×C83#178811yescharacteristic
150C2×C83#179611yescharacteristic
151C2×#557D8311yescharacteristic
152#1801—831no—
153#1546—832no—
154#1487—832no—
155#1461—831no—
156C83#839011yescharacteristic, complemented, Sylow
157#557—24no—
158#557—24no—
159C2×C4⋊Q8—21no—
160C2×C4⋊Q8C2×D8311yescharacteristic
161C2×#125—22no—
162C2×#125Dic8312yes—
163C2×C4×Q8—21no—
164C2×C4×Q8Dic8312yes—
165C2×C4×Q8C2×D8311yescharacteristic
166C2×C4×Q8C2×D8311yescharacteristic
167#431—832no—
168#423—832no—
169C2×#101—1661no—
170#343—1661no—
171#332—1664no—
172Q8⋊C8—1664no—
173#324—1662no—
174C22×Q8—21no—
175C22×Q8C2×Dic8311yescharacteristic
176C22×Q8C22⋊D8311yescharacteristic
177C2×M16—1662no—
178C4⋊Q8—28no—
179C4⋊Q8—22no—
180C4⋊Q8C22⋊D8314yes—
181#125—216no—
182#125—24no—
183#125—24no—
184C4×Q8—28no—
185C4×Q8—24no—
186C4×Q8—24no—
187C4×Q8—22no—
188C4×Q8—22no—
189C4×Q8—22no—
190C4×Q8C22⋊D8314yes—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4C2×Dic8312yes—
197C2×C4⋊C4C2×Dic8311yescharacteristic
198C2×C4⋊C4C22⋊D8311yescharacteristic
199C2×C4⋊C4C22⋊D8312yes—
200C2×C42—22no—
201C2×C42—21no—
202C2×C42C2×Dic8312yes—
203C2×C42C22×D8311yescharacteristic
204C4⋊C8—1662no—
205C4⋊C8—1662no—
206C4⋊C8—1662no—
207#101—3322no—
208C22⋊C8—1662no—
209C4×C8—1662no—
210C2×Q8—24no—
211C2×Q8—22no—
212C2×Q8—24no—
213C2×Q8—22no—
214C2×Q8—21no—
215C2×Q8—21no—
216C2×Q8—21no—
217C2×Q8—42no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8C22⋊Dic8312yes—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4C22⋊Dic8312yes—
225C22×C4C22⋊Dic8312yes—
226C22×C4C22×Dic8311yescharacteristic
227C22×C4C2×C22⋊D8311yescharacteristic
228C22×C4C2×C22⋊D8311yescharacteristic
229M16—3322no—
230M16—3322no—
231C2×C8—3322no—
232C2×C8—3322no—
233C4⋊C4—28no—
234C4⋊C4—28no—
235C4⋊C4—28no—
236C4⋊C4—24no—
237C4⋊C4—24no—
238C4⋊C4—22no—
239C4⋊C4—24no—
240C4⋊C4—21no—
241C4⋊C4—22no—
242C4⋊C4C22⋊Dic8314yes—
243C4⋊C4C22⋊Dic8312yes—
244C42—28no—
245C42—24no—
246C42—22no—
247C42—22no—
248C42C2×C22⋊D8312yes—
249C42C2×C22⋊D8312yes—
250C23C2×C22⋊Dic8311yescharacteristic
251Q8—24no—
252Q8—22no—
253Q8—42no—
254Q8—44no—
255Q8—42no—
256Q8—44no—
257Q8#1458171114yescomplemented
258C2×C4—24no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—24no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—41no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4C2×C22⋊Dic8311yescharacteristic
273C2×C4C2×C22⋊Dic8311yescharacteristic
274C2×C4#1458176112yes—
275C2×C4#1458176112yes—
276C2×C4#1458181212yes—
277C2×C4#1458181711yescharacteristic
278C2×C4#1458182211yescharacteristic
279C8—6642no—
280C8—6642no—
281C22—21no—
282C22—21no—
283C22#1912755711yescharacteristic
284C22#1912769711yescharacteristic
285C22#1912770211yescharacteristic
286C4—24no—
287C4—21no—
288C4—22no—
289C4—42no—
290C4—44no—
291C4—42no—
292C4—42no—
293C4C2×#1458171112yes—
294C4#1912784612yes—
295C4#1912786312yes—
296C2—21no—
297C2—21no—
298C2#2490738711yescharacteristic
299C2#2490740311yescharacteristic
300C2#2490743811yescharacteristic
subgroupG/H
centerC22#19127557
commutatorC22×C4×C83C22×C4
FrattiniC2×C42C22×D83
FittingC2×#557×C83C2
socleC22×C83#379
prime radicalC23×C83C2×C22⋊C4
cosocleC2×C42×C83C23
hypercenterC2×#557D83
hypocenterC83#8390
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
106. Q8×C83#101semidirect
156. C83#8390semidirect
257. Q8#14581711semidirect

Maximal subgroups

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

subgroupindex
#249074592
#249061352
#249058922
#249057872
C2×#557×C832
#839083