Finite Group Databasebeta

← #35313682

Subgroups — #35313682

Order 21248.

480 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 123 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#24905781C212yesmaximal
2#24905407C212yesmaximal, complemented
3#24905397C212yesmaximal, complemented
4C4×#124×C83C211yesmaximal, characteristic
5C2×C22⋊(C83⋊C8)—22no—
6#19127427C2214yes—
7#19127425—22no—
8C2×#124×C83C411yescharacteristic
9C2×C4×Q8×C83C411yescharacteristic
10C2×C4×D4×C83C2211yescharacteristic
11#391×C83C2211yescharacteristic
12#390×C83C411yescharacteristic
13#388×C83—22no—
14#382×C83C2211yescharacteristic
15C4×C22⋊C4×C83—22no—
16C43×C83C411yescharacteristic
17C22×C83⋊C8—42no—
18C22⋊(C83⋊C8)—24no—
19C22⋊(C83⋊C8)—42no—
20C22×Q8×C83C2×C411yescharacteristic
21C22×D4×C83C2×C411yescharacteristic
22C23×C4×C83D412yes—
23#124×C83—22no—
24#124×C83C814yescomplemented
25C4×Q8×C83—41no—
26C4×Q8×C83—41no—
27C4×D4×C83—41no—
28C4×D4×C83—41no—
29C2×C4⋊C4×C83—21no—
30C2×C4⋊C4×C83C2×C411yescharacteristic
31C2×C4⋊C4×C83C2311yescharacteristic
32C2×C22⋊C4×C83—22no—
33C2×C22⋊C4×C83—22no—
34C2×C22⋊C4×C83D412yes—
35C2×C42×C83—21no—
36C2×C42×C83—21no—
37C2×C42×C83—21no—
38C2×C42×C83C2×C411yescharacteristic
39C2×C42×C83C2×C411yescharacteristic
40C2×C42×C83C2×C411yescharacteristic
41#95×C83—22no—
42#95×C83—22no—
43C2×C83⋊C8—44no—
44C2×C83⋊C8—84no—
45C24×C83C22⋊C412yes—
46C2×Q8×C83—21no—
47C2×Q8×C83—41no—
48C2×Q8×C83—41no—
49C2×Q8×C83C2×C812yescomplemented
50C2×D4×C83—41no—
51C2×D4×C83—41no—
52C2×D4×C83C2×C812yes—
53C2×D4×C83M1612yes—
54C22×C4×C83—21no—
55C22×C4×C83—21no—
56C22×C4×C83—21no—
57C22×C4×C83—22no—
58C22×C4×C83—22no—
59C22×C4×C83—21no—
60C22×C4×C83—44no—
61C22×C4×C83C22⋊C411yescharacteristic
62C22×C4×C83C22⋊C411yescharacteristic
63C22×C4×C83C22×C411yescharacteristic
64C22×C4×C83C2×D412yes—
65C4⋊C4×C83—41no—
66C4⋊C4×C83—41no—
67C4⋊C4×C83—42no—
68C22⋊C4×C83—28no—
69C22⋊C4×C83—42no—
70C22⋊C4×C83—44no—
71C42×C83—21no—
72C42×C83—21no—
73C42×C83—21no—
74C42×C83—41no—
75C42×C83—42no—
76C42×C83—42no—
77C42×C83—41no—
78C42×C83—41no—
79C42×C83C2×C812yescomplemented
80C23×C83—22no—
81C23×C83—22no—
82C23×C83—42no—
83C23×C83—42no—
84C23×C83C22⋊C814yes—
85C23×C83C2×C22⋊C411yescharacteristic
86Q8×C83—42no—
87Q8×C83—42no—
88D4×C83—42no—
89D4×C83—42no—
90C2×C4×C83—21no—
91C2×C4×C83—21no—
92C2×C4×C83—22no—
93C2×C4×C83—22no—
94C2×C4×C83—22no—
95C2×C4×C83—22no—
96C2×C4×C83—21no—
97C2×C4×C83—22no—
98C2×C4×C83—41no—
99C2×C4×C83—41no—
100C2×C4×C83—41no—
101C2×C4×C83—41no—
102C2×C4×C83—41no—
103C2×C4×C83—41no—
104C2×C4×C83—44no—
105C2×C4×C83—41no—
106C2×C4×C83—42no—
107C2×C4×C83—42no—
108C2×C4×C83—41no—
109C2×C4×C83—82no—
110C2×C4×C83#9911yescharacteristic
111C2×C4×C83#9911yescharacteristic
112C2×C4×C83C22×C811yescharacteristic
113C2×C4×C83C2×M1611yescharacteristic
114C83⋊C8—84no—
115C22×C83—21no—
116C22×C83—21no—
117C22×C83—24no—
118C22×C83—44no—
119C22×C83—42no—
120C22×C83—42no—
121C22×C83—82no—
122C22×C83C2×C22⋊C811yescharacteristic
123C22×C83C2×#9911yescharacteristic
124C22×C83#41311yescharacteristic
125C4×C83—21no—
126C4×C83—21no—
127C4×C83—21no—
128C4×C83—42no—
129C4×C83—42no—
130C4×C83—42no—
131C4×C83—41no—
132C4×C83—42no—
133C4×C83—81no—
134C4×C83#35412yes—
135#11477—831nomaximal, Sylow
136C2×C83—21no—
137C2×C83—21no—
138C2×C83—44no—
139C2×C83#145911yescharacteristic
140C2×C83C2×#35411yescharacteristic
141C2×C83#213211yescharacteristic
142C4×#124D8311yescharacteristic
143#1459—832no—
144#1324—832no—
145#1319—832no—
146C83#1147711yescharacteristic, complemented, Sylow
147C2×#124Dic8311yescharacteristic
148C2×C4×Q8Dic8311yescharacteristic
149C2×C4×D4C2×D8311yescharacteristic
150C2×C22⋊C8—1662no—
151#391C2×D8311yescharacteristic
152#390Dic8311yescharacteristic
153#388—22no—
154#382C2×D8311yescharacteristic
155C4×C22⋊C4—22no—
156C43Dic8311yescharacteristic
157#324—834no—
158#323—1662no—
159C22×Q8C2×Dic8311yescharacteristic
160C22×D4C2×Dic8311yescharacteristic
161C23×C4C22⋊D8312yes—
162C22×C8—3322no—
163#124—22no—
164#124C83⋊C814yescomplemented
165C4×Q8—41no—
166C4×Q8—41no—
167C4×D4—41no—
168C4×D4—41no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4C2×Dic8311yescharacteristic
171C2×C4⋊C4C22×D8311yescharacteristic
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4C22⋊D8312yes—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42C2×Dic8311yescharacteristic
179C2×C42C2×Dic8311yescharacteristic
180C2×C42C2×Dic8311yescharacteristic
181C22⋊C8—3322no—
182C22⋊C8—1664no—
183#95—22no—
184#95—22no—
185C24C22⋊Dic8312yes—
186C2×Q8—21no—
187C2×Q8—41no—
188C2×Q8—41no—
189C2×Q8C2×C83⋊C812yescomplemented
190C2×D4—41no—
191C2×D4—41no—
192C2×D4C2×C83⋊C812yes—
193C2×D4#1305057412yes—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—21no—
200C22×C4—44no—
201C22×C4C22⋊Dic8311yescharacteristic
202C22×C4C22⋊Dic8311yescharacteristic
203C22×C4C22×Dic8311yescharacteristic
204C22×C4C2×C22⋊D8312yes—
205C2×C8—3324no—
206C2×C8—6644no—
207C4⋊C4—41no—
208C4⋊C4—41no—
209C4⋊C4—42no—
210C22⋊C4—28no—
211C22⋊C4—42no—
212C22⋊C4—44no—
213C42—21no—
214C42—21no—
215C42—21no—
216C42—41no—
217C42—42no—
218C42—42no—
219C42—41no—
220C42—41no—
221C42C2×C83⋊C812yescomplemented
222C23—22no—
223C23—22no—
224C23—42no—
225C23—42no—
226C23C22⋊(C83⋊C8)14yes—
227C23C2×C22⋊Dic8311yescharacteristic
228Q8—42no—
229Q8—42no—
230D4—42no—
231D4—42no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—44no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—82no—
252C2×C4#1458170711yescharacteristic
253C2×C4#1458170711yescharacteristic
254C2×C4C22×C83⋊C811yescharacteristic
255C2×C4C2×#1305057411yescharacteristic
256C8—6644no—
257C22—21no—
258C22—21no—
259C22—24no—
260C22—44no—
261C22—42no—
262C22—42no—
263C22—82no—
264C22C2×C22⋊(C83⋊C8)11yescharacteristic
265C22C2×#1458170711yescharacteristic
266C22#1912768511yescharacteristic
267C4—21no—
268C4—21no—
269C4—21no—
270C4—42no—
271C4—42no—
272C4—42no—
273C4—42no—
274C4—41no—
275C4—81no—
276C4#1912749612yes—
277C2—21no—
278C2—21no—
279C2—44no—
280C2#2490578111yescharacteristic
281C2C2×#1912749611yescharacteristic
282C2#2490914811yescharacteristic
subgroupG/H
centerC2×C4#14581707
commutatorC2×C4×C83C22×C8
FrattiniC2×C4⋊C4C22×D83
FittingC4×#124×C83C2
socleC22×C83C2×#99
prime radicalC22×D4×C83C2×C4
cosocleC2×C4⋊C4×C83C23
hypercenterC4×#124D83
hypocenterC83#11477
LCSG ▷ C2×C4×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#124

Complements

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

NH ≅ G/Ntype
2. #24905407C2semidirect
3. #24905397C2semidirect
24. #124×C83C8semidirect
49. C2×Q8×C83C2×C8semidirect
79. C42×C83C2×C8semidirect
146. C83#11477semidirect
164. #124C83⋊C8semidirect
189. C2×Q8C2×C83⋊C8semidirect
221. C42C2×C83⋊C8semidirect

Maximal subgroups

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

subgroupindex
#249057812
#249054072
#249053972
C4×#124×C832
#1147783