Finite Group Databasebeta

← #35297731

Subgroups — #35297731

Order 21248.

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

normal subgroups only

This group has 898 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
1C2×C4×C22⋊Dic83C211yesmaximal, characteristic
2#24907373C211yesmaximal, characteristic
3#24905305C214yesmaximal, complemented
4C83×#1779C211yesmaximal, characteristic
5C22×C22⋊Dic83C411yescharacteristic
6C22×C22⋊Dic83C411yescharacteristic
7C22×C4×Dic83C412yes—
8C2×C22⋊(C83⋊C8)—22no—
9C2×C22⋊(C83⋊C8)C2212yes—
10C4×C22⋊Dic83—22no—
11C4×C22⋊Dic83—22no—
12C2×C83⋊#95C2212yes—
13C24×C4×C83C2211yescharacteristic
14C2×C22⋊C8×C83—22no—
15C2×C22⋊C8×C83C2212yes—
16C23×Dic83C2×C412yes—
17C23×Dic83C2×C412yes—
18C22×C83⋊C8—22no—
19C2×C22⋊Dic83—24no—
20C2×C22⋊Dic83—22no—
21C2×C22⋊Dic83—22no—
22C2×C22⋊Dic83—22no—
23C2×C22⋊Dic83C814yescomplemented
24C2×C4×Dic83—22no—
25C2×C4×Dic83—22no—
26C2×C4×Dic83—44no—
27C22⋊(C83⋊C8)—24no—
28C22⋊(C83⋊C8)—44no—
29C83⋊#95—22no—
30C83⋊#95—22no—
31C25×C83C2×C411yescharacteristic
32C23×C4×C83—24no—
33C23×C4×C83—44no—
34C23×C4×C83C2×C411yescharacteristic
35C23×C4×C83D412yes—
36C23×C4×C83D412yes—
37C23×C4×C83C2311yescharacteristic
38C22×C8×C83—22no—
39C22⋊C8×C83—24no—
40C22⋊C8×C83—44no—
41C22×Dic83—22no—
42C22×Dic83—22no—
43C22×Dic83—22no—
44C22×Dic83—22no—
45C22×Dic83—42no—
46C22×Dic83—42no—
47C22×Dic83—42no—
48C22×Dic83—42no—
49C22×Dic83C2×C814yes—
50C22×Dic83M1614yes—
51C2×C83⋊C8—44no—
52C2×C83⋊C8—42no—
53C22⋊Dic83—28no—
54C22⋊Dic83—28no—
55C4×Dic83—44no—
56C4×Dic83—44no—
57C24×C83—22no—
58C24×C83—22no—
59C24×C83—42no—
60C24×C83—42no—
61C24×C83C22⋊C412yes—
62C24×C83C2×C812yes—
63C24×C83M1612yes—
64C24×C83C22×C411yescharacteristic
65C22×C4×C83—22no—
66C22×C4×C83—24no—
67C22×C4×C83—24no—
68C22×C4×C83—24no—
69C22×C4×C83—24no—
70C22×C4×C83—24no—
71C22×C4×C83—42no—
72C22×C4×C83—42no—
73C22×C4×C83—42no—
74C22×C4×C83—416no—
75C22×C4×C83C22⋊C412yes—
76C22×C4×C83C2×D411yescharacteristic
77C22×C4×C83C2×D411yescharacteristic
78C22×C4×C83#4114yes—
79C2×C8×C83—44no—
80C2×C8×C83—42no—
81C23×C83—22no—
82C23×C83—22no—
83C23×C83—22no—
84C23×C83—22no—
85C23×C83—22no—
86C23×C83—22no—
87C23×C83—22no—
88C23×C83—22no—
89C23×C83—24no—
90C23×C83—24no—
91C23×C83—44no—
92C23×C83—44no—
93C23×C83—44no—
94C23×C83—44no—
95C23×C83—44no—
96C23×C83—44no—
97C23×C83C22⋊C814yes—
98C23×C83C2×C22⋊C411yescharacteristic
99C23×C83#11712yes—
100C23×C83C22×C811yescharacteristic
101C23×C83C2×M1612yes—
102C23×C83C2×M1611yescharacteristic
103C2×C4×C83—28no—
104C2×C4×C83—22no—
105C2×C4×C83—22no—
106C2×C4×C83—22no—
107C2×C4×C83—22no—
108C2×C4×C83—44no—
109C2×C4×C83—48no—
110C2×C4×C83—48no—
111C2×C4×C83#9914yes—
112C2×C4×C83#12312yes—
113C2×C4×C83#12312yes—
114C2×Dic83—24no—
115C2×Dic83—24no—
116C2×Dic83—44no—
117C2×Dic83—44no—
118C2×Dic83—42no—
119C2×Dic83—42no—
120C2×Dic83—42no—
121C2×Dic83—42no—
122C2×Dic83—44no—
123C2×Dic83—44no—
124C8×C83—82no—
125C83⋊C8—82no—
126C22×C83—22no—
127C22×C83—22no—
128C22×C83—22no—
129C22×C83—22no—
130C22×C83—22no—
131C22×C83—22no—
132C22×C83—24no—
133C22×C83—24no—
134C22×C83—24no—
135C22×C83—48no—
136C22×C83—48no—
137C22×C83—42no—
138C22×C83—42no—
139C22×C83—42no—
140C22×C83—42no—
141C22×C83#38111yescharacteristic
142C22×C83C2×C22⋊C811yescharacteristic
143C22×C83#40711yescharacteristic
144C22×C83C2×#9912yes—
145C22×C83#41312yes—
146C22×C83#43112yes—
147C22×C83#43212yes—
148C4×C83—22no—
149C4×C83—22no—
150C4×C83—24no—
151Dic83—44no—
152Dic83—44no—
153#8700—831nomaximal, Sylow
154C2×C83—22no—
155C2×C83—22no—
156C2×C83—44no—
157C2×C83#145912yes—
158C2×C83#146512yes—
159C2×C83#181711yescharacteristic
160C2×C83#181911yescharacteristic
161C2×C83#182211yescharacteristic
162C2×C4×C22⋊C4—831no—
163#1779—831no—
164#1779D8311yescharacteristic
165#1280—834no—
166C83#870011yescharacteristic, complemented, Sylow
167C24×C4C2×D8311yescharacteristic
168C22×C22⋊C4—831no—
169C22×C22⋊C4—831no—
170C22×C42—832no—
171C2×C22⋊C8—22no—
172C2×C22⋊C8—832no—
173C2×C22⋊C8—1662no—
174C2×C22⋊C8C2×D8312yes—
175C4×C22⋊C4—1662no—
176C4×C22⋊C4—1662no—
177C2×#95—832no—
178C25C4×D8311yescharacteristic
179C23×C4—24no—
180C23×C4—44no—
181C23×C4—832no—
182C23×C4—832no—
183C23×C4C4×D8311yescharacteristic
184C23×C4D33212yes—
185C23×C4C22⋊D8312yes—
186C23×C4C22×D8311yescharacteristic
187C22×C8—22no—
188C22×C8—1662no—
189C2×C22⋊C4—834no—
190C2×C22⋊C4—1664no—
191C2×C22⋊C4—1662no—
192C2×C22⋊C4—1662no—
193C2×C22⋊C4—1662no—
194C2×C42—3324no—
195C2×C42—1662no—
196C2×C42—1662no—
197C22⋊C8—24no—
198C22⋊C8—44no—
199C22⋊C8—3324no—
200C22⋊C8—1664no—
201#95—1662no—
202#95—1662no—
203C24—22no—
204C24—22no—
205C24—42no—
206C24—42no—
207C24(C2×D83)⋊C412yes—
208C24C8×D8312yes—
209C24#1305057312yes—
210C24C2×C4×D8311yescharacteristic
211C22×C4—22no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—416no—
221C22×C4—3322no—
222C22×C4—3322no—
223C22×C4—3322no—
224C22×C4—3322no—
225C22×C4—834no—
226C22×C4—834no—
227C22×C4—1662no—
228C22×C4—1662no—
229C22×C4—1662no—
230C22×C4—1662no—
231C22×C4(C2×D83)⋊C412yes—
232C22×C4C2×C22⋊D8311yescharacteristic
233C22×C4C2×D33211yescharacteristic
234C22×C4#1305058914yes—
235C2×C8—44no—
236C2×C8—42no—
237C2×C8—3324no—
238C2×C8—3322no—
239C22⋊C4—1668no—
240C22⋊C4—1668no—
241C42—3324no—
242C42—3324no—
243C23—22no—
244C23—22no—
245C23—22no—
246C23—22no—
247C23—22no—
248C23—22no—
249C23—22no—
250C23—22no—
251C23—24no—
252C23—24no—
253C23—44no—
254C23—44no—
255C23—44no—
256C23—44no—
257C23—44no—
258C23—44no—
259C23(C2×D83)⋊C814yes—
260C23C2×(C2×D83)⋊C411yescharacteristic
261C23#1458174812yes—
262C23C2×C8×D8311yescharacteristic
263C23M16×D8312yes—
264C23C2×#1305057311yescharacteristic
265C2×C4—28no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—44no—
271C2×C4—48no—
272C2×C4—48no—
273C2×C4—3324no—
274C2×C4—3324no—
275C2×C4—3322no—
276C2×C4—3322no—
277C2×C4—3322no—
278C2×C4—3322no—
279C2×C4—3324no—
280C2×C4—3324no—
281C2×C4—1664no—
282C2×C4—1664no—
283C2×C4#1458170614yes—
284C2×C4#1458177312yes—
285C2×C4#1458177612yes—
286C8—82no—
287C8—6642no—
288C22—22no—
289C22—22no—
290C22—22no—
291C22—22no—
292C22—22no—
293C22—22no—
294C22—24no—
295C22—24no—
296C22—24no—
297C22—48no—
298C22—48no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—42no—
303C22#1912756211yescharacteristic
304C22C2×(C2×D83)⋊C811yescharacteristic
305C22#1912766411yescharacteristic
306C22C2×#1458170612yes—
307C22#1912768312yes—
308C22#1912775112yes—
309C22#1912775512yes—
310C4—22no—
311C4—22no—
312C4—24no—
313C4—3324no—
314C4—3324no—
315C2—22no—
316C2—22no—
317C2—44no—
318C2#2490577912yes—
319C2#2490579912yes—
320C2#2490752911yescharacteristic
321C2#2490753511yescharacteristic
322C2#2490754411yescharacteristic
subgroupG/H
centerC23C2×(C2×D83)⋊C4
commutatorC23×C83C22×C8
FrattiniC23×C4C22×D83
FittingC83×#1779C2
socleC23×C83C2×C22⋊C4
prime radicalC25×C83C2×C4
cosocleC23×C4×C83C23
hypercenter#1779D83
hypocenterC83#8700
LCSG ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C24×C4 ◁ #1779

Complements

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

NH ≅ G/Ntype
3. #24905305C2semidirect
23. C2×C22⋊Dic83C8semidirect
166. C83#8700semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊Dic832
#249073732
#249053052
C83×#17792
#870083