Finite Group Databasebeta

← #35313465

Subgroups — #35313465

Order 21248.

666 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 666 conjugacy classes of subgroups — too many to draw. Only its 127 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
1C22×(C2×D83)⋊C8C211yesmaximal, characteristic
2#24909074C211yesmaximal, characteristic, complemented
3#24905386C214yesmaximal, complemented
4C83×#2112C211yesmaximal, characteristic, complemented
5C23×C4×D83C411yescharacteristic
6C23×C83⋊C8C411yescharacteristic
7C2×#14581732—21no—
8C2×(C2×D83)⋊C8—21no—
9C2×(C2×D83)⋊C8—21no—
10C2×(C2×D83)⋊C8—21no—
11C2×(C2×D83)⋊C8—21no—
12C2×(C2×D83)⋊C8C412yes—
13C2×(C2×D83)⋊C8C2212yes—
14C22⋊(C83⋊C16)—21no—
15C22⋊(C83⋊C16)C2212yescomplemented
16C23×C8×C83C2211yescharacteristic
17C2×M32×C83—21no—
18C22⋊C16×C83—21no—
19C22⋊C16×C83C2212yescomplemented
20C24×D83C811yescharacteristic
21C23×Dic83C811yescharacteristic
22C22×C4×D83—21no—
23C22×C4×D83—21no—
24C22×C4×D83—21no—
25C22×C4×D83—21no—
26C22×C4×D83—42no—
27C22×C4×D83—42no—
28C22×C4×D83C812yes—
29C22×C4×D83C2×C412yes—
30C22×C83⋊C8—21no—
31C22×C83⋊C8—21no—
32C22×C83⋊C8—42no—
33C22×C83⋊C8C2×C412yes—
34#14581732—42no—
35C2×C83⋊C16—22no—
36(C2×D83)⋊C8—22no—
37(C2×D83)⋊C8—22no—
38(C2×D83)⋊C8—22no—
39(C2×D83)⋊C8—22no—
40C23×C4×C83C2×C411yescharacteristic
41C22×C8×C83—42no—
42C22×C8×C83C2×C411yescharacteristic
43C22×C8×C83D411yescharacteristic
44C22×C8×C83D411yescharacteristic
45C22×C8×C83D411yescharacteristic
46C22×C8×C83D411yescharacteristic
47C22×C8×C83C2311yescharacteristic
48M32×C83—42no—
49C2×C16×C83—22no—
50C23×D83—21no—
51C23×D83—21no—
52C23×D83—24no—
53C23×D83—42no—
54C23×D83—42no—
55C23×D83C2×C812yes—
56C22×Dic83—21no—
57C22×Dic83—21no—
58C22×Dic83—42no—
59C22×Dic83C2×C812yes—
60C2×C4×D83—22no—
61C2×C4×D83—22no—
62C2×C4×D83—22no—
63C2×C4×D83—22no—
64C2×C4×D83—44no—
65C2×C4×D83—44no—
66C2×C4×D83—42no—
67C2×C4×D83—42no—
68C2×C4×D83—42no—
69C2×C4×D83—42no—
70C2×C4×D83—44no—
71C2×C4×D83—42no—
72C2×C4×D83—42no—
73C2×C83⋊C8—22no—
74C2×C83⋊C8—42no—
75C2×C83⋊C8—42no—
76C2×C83⋊C8—42no—
77C83⋊C16—42no—
78C24×C83C2×C811yescharacteristic
79C22×C4×C83—22no—
80C22×C4×C83—22no—
81C22×C4×C83C22⋊C411yescharacteristic
82C22×C4×C83C22⋊C411yescharacteristic
83C22×C4×C83C2×C811yescharacteristic
84C22×C4×C83M1611yescharacteristic
85C22×C4×C83M1611yescharacteristic
86C22×C4×C83C22×C411yescharacteristic
87C2×C8×C83—22no—
88C2×C8×C83—22no—
89C2×C8×C83—41no—
90C2×C8×C83—41no—
91C2×C8×C83—41no—
92C2×C8×C83—41no—
93C2×C8×C83C22⋊C411yescharacteristic
94C2×C8×C83C22⋊C411yescharacteristic
95C2×C8×C83C2×D411yescharacteristic
96C2×C8×C83C2×D411yescharacteristic
97C16×C83—42no—
98C23×C83—22no—
99C23×C83—22no—
100C23×C83—22no—
101C23×C83C22⋊C811yescharacteristic
102C23×C83C22⋊C811yescharacteristic
103C23×C83C22×C811yescharacteristic
104C22×D83—24no—
105C22×D83—24no—
106C22×D83—22no—
107C22×D83—22no—
108C22×D83—42no—
109C22×D83—416no—
110C22×D83—42no—
111C22×D83—42no—
112C22×D83—42no—
113C22×D83—42no—
114C22×D83—42no—
115C22×D83M3214yescomplemented
116C2×C4×C83—22no—
117C2×C4×C83—22no—
118C2×C4×C83—22no—
119C2×C4×C83—21no—
120C2×C4×C83—21no—
121C2×C4×C83—21no—
122C2×C4×C83—21no—
123C2×C4×C83C22⋊C811yescharacteristic
124C2×C4×C83C22⋊C811yescharacteristic
125C2×C4×C83#9912yes—
126C2×C4×C83#10012yes—
127C2×C4×C83C2×C22⋊C411yescharacteristic
128C2×C4×C83C2×M1611yescharacteristic
129C2×Dic83—22no—
130C2×Dic83—42no—
131C2×Dic83—42no—
132C2×Dic83—42no—
133C4×D83—44no—
134C4×D83—44no—
135C4×D83—44no—
136C4×D83—42no—
137C4×D83—42no—
138C8×C83—21no—
139C8×C83—21no—
140C8×C83—21no—
141C8×C83—21no—
142C83⋊C8—42no—
143C22×C83—28no—
144C22×C83—21no—
145C22×C83—21no—
146C22×C83—21no—
147C22×C83—21no—
148C22×C83—21no—
149C22×C83—21no—
150C22×C83#32312yes—
151C22×C83#34912yes—
152C22×C83C2×C22⋊C811yescharacteristic
153C22×C83C2×M3212yes—
154C2×D83—24no—
155C2×D83—48no—
156C2×D83—48no—
157C2×D83—44no—
158C2×D83—42no—
159C2×D83—42no—
160C2×D83—44no—
161C4×C83—21no—
162C4×C83—21no—
163C4×C83#32312yes—
164C4×C83C2×#9911yescharacteristic
165C4×C83C2×#10011yescharacteristic
166Dic83—42no—
167#11418—831nomaximal, Sylow
168C2×C83—22no—
169C2×C83—22no—
170C2×C83—22no—
171C2×C83C2×#32311yescharacteristic
172C2×C83#211211yescharacteristic
173C2×C83C2×#34911yescharacteristic
174D83—44no—
175C22×C22⋊C8—831no—
176#2112—831no—
177#2112D8311yescharacteristic, complemented
178#1314—834no—
179C83#1141811yescharacteristic, complemented, Sylow
180C24×C4—831no—
181C23×C8—831no—
182C23×C8C2×D8311yescharacteristic
183C2×M32—21no—
184C2×M32—1661no—
185C2×C22⋊C8—832no—
186C2×C22⋊C8—832no—
187C2×C22⋊C8—1661no—
188C2×C22⋊C8—1661no—
189C2×C22⋊C8—1661no—
190C2×C22⋊C8—1661no—
191C22⋊C16—21no—
192C22⋊C16—832no—
193C22⋊C16—1661no—
194C22⋊C16C2×D8312yescomplemented
195C25—831no—
196C23×C4—3322no—
197C23×C4—3322no—
198C23×C4—831no—
199C23×C4—832no—
200C23×C4—832no—
201C23×C4—1661no—
202C23×C4—1661no—
203C23×C4—1661no—
204C23×C4—1661no—
205C23×C4C4×D8311yescharacteristic
206C22×C8—42no—
207C22×C8—3322no—
208C22×C8—832no—
209C22×C8—1661no—
210C22×C8—1661no—
211C22×C8C4×D8311yescharacteristic
212C22×C8D33211yescharacteristic
213C22×C8D33211yescharacteristic
214C22×C8C22⋊D8311yescharacteristic
215C22×C8C22⋊D8311yescharacteristic
216C22×C8C22×D8311yescharacteristic
217M32—42no—
218M32—3322no—
219C2×C16—22no—
220C2×C16—1662no—
221C22⋊C8—1662no—
222C22⋊C8—1662no—
223C22⋊C8—1662no—
224C22⋊C8—1662no—
225C24—3322no—
226C24—3322no—
227C24—832no—
228C24—1664no—
229C24—1661no—
230C24—1661no—
231C24C8×D8311yescharacteristic
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—3324no—
235C22×C4—3322no—
236C22×C4—3322no—
237C22×C4—3324no—
238C22×C4—3322no—
239C22×C4—3322no—
240C22×C4—3324no—
241C22×C4—3322no—
242C22×C4—3322no—
243C22×C4—3322no—
244C22×C4—832no—
245C22×C4—1661no—
246C22×C4—1661no—
247C22×C4—1662no—
248C22×C4—1662no—
249C22×C4—1662no—
250C22×C4—1662no—
251C22×C4(C2×D83)⋊C411yescharacteristic
252C22×C4(C2×D83)⋊C411yescharacteristic
253C22×C4C8×D8311yescharacteristic
254C22×C4#1305057311yescharacteristic
255C22×C4#1305057311yescharacteristic
256C22×C4C2×C4×D8311yescharacteristic
257C2×C8—22no—
258C2×C8—22no—
259C2×C8—41no—
260C2×C8—41no—
261C2×C8—41no—
262C2×C8—41no—
263C2×C8—3322no—
264C2×C8—3322no—
265C2×C8—3322no—
266C2×C8—1662no—
267C2×C8(C2×D83)⋊C411yescharacteristic
268C2×C8(C2×D83)⋊C411yescharacteristic
269C2×C8C2×C22⋊D8311yescharacteristic
270C2×C8C2×D33211yescharacteristic
271C16—42no—
272C16—3322no—
273C23—22no—
274C23—22no—
275C23—22no—
276C23—3322no—
277C23—3322no—
278C23—3322no—
279C23—33216no—
280C23—3322no—
281C23—3322no—
282C23—3322no—
283C23—834no—
284C23—1664no—
285C23—1664no—
286C23—1662no—
287C23—1662no—
288C23(C2×D83)⋊C811yescharacteristic
289C23(C2×D83)⋊C811yescharacteristic
290C23C2×C8×D8311yescharacteristic
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—3324no—
299C2×C4—3324no—
300C2×C4—3322no—
301C2×C4—3322no—
302C2×C4—3322no—
303C2×C4—3322no—
304C2×C4—3324no—
305C2×C4—3322no—
306C2×C4—1662no—
307C2×C4(C2×D83)⋊C811yescharacteristic
308C2×C4(C2×D83)⋊C811yescharacteristic
309C2×C4#1458170612yes—
310C2×C4#1458170812yes—
311C2×C4C2×(C2×D83)⋊C411yescharacteristic
312C2×C4C2×#1305057311yescharacteristic
313C8—21no—
314C8—21no—
315C8—21no—
316C8—21no—
317C8—3322no—
318C22—28no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—3324no—
326C22—3324no—
327C22—3328no—
328C22—3328no—
329C22—3322no—
330C22—3322no—
331C22—1664no—
332C22#1912742412yes—
333C22#1912748212yes—
334C22C2×(C2×D83)⋊C811yescharacteristic
335C22M32×D8312yes—
336C4—21no—
337C4—21no—
338C4—3322no—
339C4#1912742412yes—
340C4C2×#1458170611yescharacteristic
341C4C2×#1458170811yescharacteristic
342C2—22no—
343C2—22no—
344C2—22no—
345C2—3324no—
346C2C2×#1912742411yescharacteristic
347C2#2490907311yescharacteristic
348C2C2×#1912748211yescharacteristic
subgroupG/H
centerC2×C4C2×(C2×D83)⋊C4
commutatorC23×C83C22×C8
FrattiniC22×C8C22×D83
FittingC83×#2112C2
socleC22×C83C2×C22⋊C8
prime radicalC24×D83C8
cosocleC22×C8×C83C23
hypercenter#2112D83
hypocenterC83#11418
LCSG ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C23×C8 ◁ #2112

Complements

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

NH ≅ G/Ntype
2. #24909074C2semidirect
3. #24905386C2semidirect
4. C83×#2112C2semidirect
15. C22⋊(C83⋊C16)C22semidirect
19. C22⋊C16×C83C22semidirect
115. C22×D83M32semidirect
177. #2112D83semidirect
179. C83#11418semidirect
194. C22⋊C16C2×D83semidirect

Maximal subgroups

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

subgroupindex
C22×(C2×D83)⋊C82
#249090742
#249053862
C83×#21122
#1141883