Finite Group Databasebeta

← #35296162

Subgroups — #35296162

Order 21248.

738 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 151 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, complemented
2#24906144C212yesmaximal, complemented
3#24905888C212yesmaximal
4#24905787C211yesmaximal, characteristic, complemented
5C2×#556×C83C211yesmaximal, characteristic
6#19127754C2212yes—
7#19127725C2212yes—
8C2×#14581711—21no—
9#19127470—21no—
10#19127450—24no—
11D4⋊(C83⋊C8)—24no—
12#19127427—22no—
13#556×C83—24no—
14#556×C83—24no—
15C2×C4⋊Q8×C83C2211yescharacteristic
16C2×#126×C83—22no—
17C2×#124×C83C412yes—
18C2×C22⋊Q8×C83—21no—
19C2×C4×Q8×C83C412yes—
20C2×C4×Q8×C83C2211yescharacteristic
21C2×C4×D4×C83C2211yescharacteristic
22C2×#117×C83—21no—
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×C83C2×C412yes—
31C22×Q8×C83D411yescharacteristic
32C22×D4×C83C2×C411yescharacteristic
33C23×C4×C83—21no—
34C4⋊Q8×C83—22no—
35C4⋊Q8×C83D414yes—
36#126×C83—216no—
37#124×C83—24no—
38#124×C83—24no—
39C22⋊Q8×C83—28no—
40C4×Q8×C83—24no—
41C4×Q8×C83—24no—
42C4×Q8×C83—22no—
43C4×Q8×C83—22no—
44C4×D4×C83—22no—
45C4×D4×C83D414yes—
46#117×C83—28no—
47C2×C4⋊C4×C83—22no—
48C2×C4⋊C4×C83—21no—
49C2×C4⋊C4×C83—21no—
50C2×C4⋊C4×C83C2×C411yescharacteristic
51C2×C4⋊C4×C83D411yescharacteristic
52C2×C4⋊C4×C83D412yes—
53C2×C22⋊C4×C83—22no—
54C2×C22⋊C4×C83—22no—
55C2×C22⋊C4×C83—21no—
56C2×C42×C83—22no—
57C2×C42×C83C2×C412yes—
58C2×C42×C83C2311yescharacteristic
59#13050574—42no—
60#13050574—42no—
61C2×C83⋊C8—42no—
62C2×C83⋊C8—42no—
63C24×C83—21no—
64C2×Q8×C83—22no—
65C2×Q8×C83—22no—
66C2×Q8×C83—22no—
67C2×Q8×C83—21no—
68C2×Q8×C83—21no—
69C2×Q8×C83—44no—
70C2×Q8×C83C22⋊C414yes—
71C2×D4×C83—21no—
72C2×D4×C83—42no—
73C2×D4×C83C22⋊C412yes—
74C22×C4×C83—21no—
75C22×C4×C83—22no—
76C22×C4×C83—24no—
77C22×C4×C83—42no—
78C22×C4×C83—41no—
79C22×C4×C83—41no—
80C22×C4×C83C22⋊C412yes—
81C22×C4×C83C22⋊C412yes—
82C22×C4×C83C22×C411yescharacteristic
83C22×C4×C83C2×D411yescharacteristic
84C22×C4×C83C2×D411yescharacteristic
85C4⋊C4×C83—28no—
86C4⋊C4×C83—21no—
87C4⋊C4×C83—24no—
88C4⋊C4×C83—24no—
89C4⋊C4×C83—22no—
90C4⋊C4×C83—24no—
91C4⋊C4×C83C22⋊C412yes—
92C22⋊C4×C83—28no—
93C22⋊C4×C83—28no—
94C22⋊C4×C83—24no—
95C42×C83—28no—
96C42×C83—22no—
97C42×C83—22no—
98C42×C83C2×D412yes—
99C42×C83C2×D412yes—
100C23×C83—22no—
101C23×C83—42no—
102C23×C83—42no—
103C23×C83—41no—
104C23×C83—41no—
105C23×C83C2×C22⋊C411yescharacteristic
106Q8×C83—44no—
107Q8×C83—44no—
108Q8×C83—42no—
109Q8×C83—42no—
110D4×C83—24no—
111D4×C83—42no—
112C2×C4×C83—22no—
113C2×C4×C83—22no—
114C2×C4×C83—22no—
115C2×C4×C83—24no—
116C2×C4×C83—22no—
117C2×C4×C83—22no—
118C2×C4×C83—22no—
119C2×C4×C83—41no—
120C2×C4×C83—44no—
121C2×C4×C83—41no—
122C2×C4×C83—41no—
123C2×C4×C83—44no—
124C2×C4×C83—44no—
125C2×C4×C83—42no—
126C2×C4×C83C2×C22⋊C411yescharacteristic
127C2×C4×C83C2×C22⋊C411yescharacteristic
128C2×C4×C83#12012yes—
129C2×C4×C83#12012yes—
130C2×C4×C83#13512yes—
131C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
132C2×C4×C83#13711yescharacteristic
133C83⋊C8—82no—
134C83⋊C8—82no—
135C22×C83—21no—
136C22×C83—21no—
137C22×C83—42no—
138C22×C83—42no—
139C22×C83—42no—
140C22×C83—82no—
141C22×C83—82no—
142C22×C83#37911yescharacteristic
143C22×C83#41611yescharacteristic
144C22×C83#41711yescharacteristic
145C4×C83—22no—
146C4×C83—21no—
147C4×C83—42no—
148C4×C83—44no—
149C4×C83—42no—
150C4×C83—42no—
151C4×C83#41312yes—
152C4×C83#44912yes—
153C4×C83#45212yes—
154#8394—831nomaximal, Sylow
155C2×C83—21no—
156C2×C83—21no—
157C2×C83—82no—
158C2×C83#178511yescharacteristic
159C2×C83#178811yescharacteristic
160C2×C83#179611yescharacteristic
161C2×#556D8311yescharacteristic
162#1801—831no—
163#1548—832no—
164#1486—832no—
165#1461—831no—
166C83#839411yescharacteristic, complemented, Sylow
167#556—24no—
168#556—24no—
169C2×C4⋊Q8C2×D8311yescharacteristic
170C2×#126—22no—
171C2×#124Dic8312yes—
172C2×C22⋊Q8—21no—
173C2×C4×Q8Dic8312yes—
174C2×C4×Q8C2×D8311yescharacteristic
175C2×C4×D4C2×D8311yescharacteristic
176C2×#117—21no—
177#431—832no—
178#423—832no—
179C2×#101—1661no—
180#343—1661no—
181#333—1664no—
182D4⋊C8—1664no—
183#324—1662no—
184C22×Q8C2×Dic8312yes—
185C22×Q8C22⋊D8311yescharacteristic
186C22×D4C2×Dic8311yescharacteristic
187C23×C4—21no—
188C2×M16—1662no—
189C4⋊Q8—22no—
190C4⋊Q8C22⋊D8314yes—
191#126—216no—
192#124—24no—
193#124—24no—
194C22⋊Q8—28no—
195C4×Q8—24no—
196C4×Q8—24no—
197C4×Q8—22no—
198C4×Q8—22no—
199C4×D4—22no—
200C4×D4C22⋊D8314yes—
201#117—28no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4C2×Dic8311yescharacteristic
206C2×C4⋊C4C22⋊D8311yescharacteristic
207C2×C4⋊C4C22⋊D8312yes—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—21no—
211C2×C42—22no—
212C2×C42C2×Dic8312yes—
213C2×C42C22×D8311yescharacteristic
214C4⋊C8—1662no—
215C4⋊C8—1662no—
216C4⋊C8—1662no—
217#101—3322no—
218C22⋊C8—1662no—
219C4×C8—1662no—
220C24—21no—
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8—21no—
224C2×Q8—21no—
225C2×Q8—22no—
226C2×Q8—44no—
227C2×Q8C22⋊Dic8314yes—
228C2×D4—21no—
229C2×D4—42no—
230C2×D4C22⋊Dic8312yes—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—42no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4C22⋊Dic8312yes—
238C22×C4C22⋊Dic8312yes—
239C22×C4C22×Dic8311yescharacteristic
240C22×C4C2×C22⋊D8311yescharacteristic
241C22×C4C2×C22⋊D8311yescharacteristic
242M16—3322no—
243M16—3322no—
244C2×C8—3322no—
245C2×C8—3322no—
246C4⋊C4—28no—
247C4⋊C4—24no—
248C4⋊C4—24no—
249C4⋊C4—22no—
250C4⋊C4—24no—
251C4⋊C4—21no—
252C4⋊C4C22⋊Dic8312yes—
253C22⋊C4—28no—
254C22⋊C4—28no—
255C22⋊C4—24no—
256C42—28no—
257C42—22no—
258C42—22no—
259C42C2×C22⋊D8312yes—
260C42C2×C22⋊D8312yes—
261C23—22no—
262C23—42no—
263C23—42no—
264C23—41no—
265C23—41no—
266C23C2×C22⋊Dic8311yescharacteristic
267Q8—44no—
268Q8—42no—
269Q8—42no—
270Q8—44no—
271D4—24no—
272D4—42no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—24no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—41no—
281C2×C4—44no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—42no—
287C2×C4C2×C22⋊Dic8311yescharacteristic
288C2×C4C2×C22⋊Dic8311yescharacteristic
289C2×C4#1458176112yes—
290C2×C4#1458176112yes—
291C2×C4#1458181212yes—
292C2×C4#1458181711yescharacteristic
293C2×C4#1458182211yescharacteristic
294C8—6642no—
295C8—6642no—
296C22—21no—
297C22—21no—
298C22—42no—
299C22—42no—
300C22—42no—
301C22—82no—
302C22—82no—
303C22#1912755711yescharacteristic
304C22#1912769711yescharacteristic
305C22#1912770211yescharacteristic
306C4—21no—
307C4—22no—
308C4—44no—
309C4—42no—
310C4—42no—
311C4—42no—
312C4#1912768512yes—
313C4#1912784612yes—
314C4#1912786312yes—
315C2—21no—
316C2—21no—
317C2—82no—
318C2#2490739111yescharacteristic
319C2#2490740311yescharacteristic
320C2#2490743811yescharacteristic
subgroupG/H
centerC22#19127557
commutatorC22×C4×C83C22×C4
FrattiniC2×C42C22×D83
FittingC2×#556×C83C2
socleC22×C83#379
prime radicalC22×D4×C83C2×C4
cosocleC2×C42×C83C23
hypercenterC2×#556D83
hypocenterC83#8394
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#556

Complements

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

NH ≅ G/Ntype
1. #24907459C2semidirect
2. #24906144C2semidirect
4. #24905787C2semidirect
166. C83#8394semidirect

Maximal subgroups

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

subgroupindex
#249074592
#249061442
#249058882
#249057872
C2×#556×C832
#839483