Finite Group Databasebeta

← #35314719

Subgroups — #35314719

Order 21248.

432 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 115 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#24907202C211yesmaximal, characteristic
2#24905777C211yesmaximal, characteristic
3#24905490C214yesmaximal, complemented
4C83×#2628C211yesmaximal, characteristic
5C22×#13050574—21no—
6#19127754—21no—
7#19127470—21no—
8#19127470C2212yes—
9#19127427C2212yes—
10#19127425—21no—
11C2×C4⋊Q8×C83C411yescharacteristic
12C2×#117×C83C2211yescharacteristic
13#400×C83C2212yes—
14#399×C83C412yes—
15#395×C83—21no—
16#393×C83—21no—
17#389×C83—21no—
18#382×C83C411yescharacteristic
19#380×C83—21no—
20C2×#13050574—24no—
21C2×#13050574—22no—
22C2×#13050574—41no—
23C2×#13050574—42no—
24C22×C83⋊C8—41no—
25C4⋊(C83⋊C8)—41no—
26C22⋊(C83⋊C8)—22no—
27C4×C83⋊C8—41no—
28C22×Q8×C83—21no—
29C23×C4×C83D411yescharacteristic
30C4⋊Q8×C83—41no—
31C4⋊Q8×C83—41no—
32#117×C83—41no—
33#117×C83—41no—
34C2×C4⋊C4×C83—21no—
35C2×C4⋊C4×C83—21no—
36C2×C4⋊C4×C83—22no—
37C2×C4⋊C4×C83C2×C412yes—
38C2×C4⋊C4×C83D411yescharacteristic
39C2×C4⋊C4×C83C2311yescharacteristic
40C2×C22⋊C4×C83—22no—
41C2×C22⋊C4×C83C2×C411yescharacteristic
42C2×C22⋊C4×C83D411yescharacteristic
43C2×C42×C83C2×C411yescharacteristic
44C2×C42×C83D411yescharacteristic
45#95×C83—21no—
46#95×C83—22no—
47#95×C83—22no—
48#95×C83C2×C412yes—
49#13050574—24no—
50#13050574—44no—
51#13050574—42no—
52C2×C83⋊C8—42no—
53C2×C83⋊C8—44no—
54C2×C83⋊C8—42no—
55C24×C83C22⋊C411yescharacteristic
56C2×Q8×C83—42no—
57C2×Q8×C83—82no—
58C22×C4×C83—21no—
59C22×C4×C83—21no—
60C22×C4×C83—22no—
61C22×C4×C83—22no—
62C22×C4×C83—22no—
63C22×C4×C83C22⋊C411yescharacteristic
64C22×C4×C83C22⋊C411yescharacteristic
65C22×C4×C83C22⋊C411yescharacteristic
66C22×C4×C83SD1612yes—
67C22×C4×C83Q1612yes—
68C22×C4×C83C22×C411yescharacteristic
69C22×C4×C83C2×D411yescharacteristic
70C22×C4×C83C2×D411yescharacteristic
71C4⋊C4×C83—41no—
72C4⋊C4×C83—41no—
73C4⋊C4×C83—42no—
74C4⋊C4×C83—41no—
75C4⋊C4×C83—41no—
76C4⋊C4×C83—42no—
77C4⋊C4×C83—82no—
78C22⋊C4×C83—41no—
79C22⋊C4×C83—41no—
80C22⋊C4×C83—82no—
81C42×C83—21no—
82C42×C83—21no—
83C42×C83—21no—
84C42×C83—21no—
85C23×C83—21no—
86C23×C83—21no—
87C23×C83—41no—
88C23×C83—41no—
89C23×C83#10312yes—
90C23×C83C2×C22⋊C411yescharacteristic
91Q8×C83—84no—
92C2×C4×C83—21no—
93C2×C4×C83—21no—
94C2×C4×C83—22no—
95C2×C4×C83—22no—
96C2×C4×C83—22no—
97C2×C4×C83—22no—
98C2×C4×C83—21no—
99C2×C4×C83—21no—
100C2×C4×C83—21no—
101C2×C4×C83—21no—
102C2×C4×C83—41no—
103C2×C4×C83—41no—
104C2×C4×C83—41no—
105C2×C4×C83—41no—
106C2×C4×C83—42no—
107C2×C4×C83—42no—
108C2×C4×C83—42no—
109C2×C4×C83—42no—
110C2×C4×C83—42no—
111C2×C4×C83—42no—
112C2×C4×C83#9911yescharacteristic
113C2×C4×C83#9911yescharacteristic
114C2×C4×C83#10312yes—
115C2×C4×C83#10412yes—
116C2×C4×C83C2×SD1611yescharacteristic
117C2×C4×C83C2×Q1611yescharacteristic
118C83⋊C8—44no—
119C83⋊C8—82no—
120C22×C83—21no—
121C22×C83—21no—
122C22×C83—22no—
123C22×C83—42no—
124C22×C83—41no—
125C22×C83—41no—
126C22×C83—81no—
127C22×C83C2×#9911yescharacteristic
128C22×C83C2×#10311yescharacteristic
129C22×C83C2×#10411yescharacteristic
130C4×C83—21no—
131C4×C83—21no—
132C4×C83—41no—
133C4×C83—41no—
134C4×C83—41no—
135C4×C83—41no—
136C4×C83—82no—
137C4×C83—82no—
138C4×C83#32812yes—
139C4×C83#32812yes—
140#11723—831nomaximal, Sylow
141C2×C83—21no—
142C2×C83—21no—
143C2×C83—42no—
144C2×C83C2×#32811yescharacteristic
145C2×C83#212311yescharacteristic
146C2×C83#213411yescharacteristic
147#2628D8311yescharacteristic
148#1742—831no—
149#1458—831no—
150#1353—834no—
151C83#1172311yescharacteristic, complemented, Sylow
152C22×M16—1661no—
153C2×C4⋊Q8Dic8311yescharacteristic
154C2×#117C2×D8311yescharacteristic
155#431—1661no—
156#400C2×D8312yes—
157#399Dic8312yes—
158#395—21no—
159#393—21no—
160#389—21no—
161#382Dic8311yescharacteristic
162#380—21no—
163#343—832no—
164#343—1661no—
165#324—832no—
166#323—1661no—
167C22×Q8—21no—
168C23×C4C22⋊D8311yescharacteristic
169C2×M16—3321no—
170C2×M16—3322no—
171C2×M16—1664no—
172C2×M16—1662no—
173C22×C8—3321no—
174C4⋊Q8—41no—
175C4⋊Q8—41no—
176#117—41no—
177#117—41no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4C2×Dic8312yes—
182C2×C4⋊C4C22⋊D8311yescharacteristic
183C2×C4⋊C4C22×D8311yescharacteristic
184C2×C22⋊C4—22no—
185C2×C22⋊C4C2×Dic8311yescharacteristic
186C2×C22⋊C4C22⋊D8311yescharacteristic
187C2×C42C2×Dic8311yescharacteristic
188C2×C42C22⋊D8311yescharacteristic
189C4⋊C8—3321no—
190C22⋊C8—1662no—
191C4×C8—3321no—
192#95—21no—
193#95—22no—
194#95—22no—
195#95C2×Dic8312yes—
196C24C22⋊Dic8311yescharacteristic
197C2×Q8—42no—
198C2×Q8—82no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4C22⋊Dic8311yescharacteristic
205C22×C4C22⋊Dic8311yescharacteristic
206C22×C4C22⋊Dic8311yescharacteristic
207C22×C4#1305057712yes—
208C22×C4#1305058012yes—
209C22×C4C22×Dic8311yescharacteristic
210C22×C4C2×C22⋊D8311yescharacteristic
211C22×C4C2×C22⋊D8311yescharacteristic
212M16—3324no—
213M16—3322no—
214M16—1664no—
215C2×C8—3322no—
216C2×C8—3324no—
217C2×C8—3322no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—42no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—42no—
224C4⋊C4—82no—
225C22⋊C4—41no—
226C22⋊C4—41no—
227C22⋊C4—82no—
228C42—21no—
229C42—21no—
230C42—21no—
231C42—21no—
232C23—21no—
233C23—21no—
234C23—41no—
235C23—41no—
236C23#1458171612yes—
237C23C2×C22⋊Dic8311yescharacteristic
238Q8—84no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4#1458170711yescharacteristic
260C2×C4#1458170711yescharacteristic
261C2×C4#1458171612yes—
262C2×C4#1458171912yes—
263C2×C4C2×#1305057711yescharacteristic
264C2×C4C2×#1305058011yescharacteristic
265C8—3324no—
266C8—6642no—
267C22—21no—
268C22—21no—
269C22—22no—
270C22—42no—
271C22—41no—
272C22—41no—
273C22—81no—
274C22C2×#1458170711yescharacteristic
275C22C2×#1458171611yescharacteristic
276C22C2×#1458171911yescharacteristic
277C4—21no—
278C4—21no—
279C4—41no—
280C4—41no—
281C4—41no—
282C4—41no—
283C4—82no—
284C4—82no—
285C4#1912743812yes—
286C4#1912743812yes—
287C2—21no—
288C2—21no—
289C2—42no—
290C2C2×#1912743811yescharacteristic
291C2#2490911711yescharacteristic
292C2#2490915511yescharacteristic
subgroupG/H
centerC22C2×#14581707
commutatorC22×C4×C83C22×C4
FrattiniC2×C4⋊C4C22×D83
FittingC83×#2628C2
socleC22×C83C2×#99
prime radicalC24×C83C22⋊C4
cosocleC2×C4⋊C4×C83C23
hypercenter#2628D83
hypocenterC83#11723
LCSG ▷ C22×C4×C83 ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2628

Complements

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

NH ≅ G/Ntype
3. #24905490C2semidirect
151. C83#11723semidirect

Maximal subgroups

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

subgroupindex
#249072022
#249057772
#249054902
C83×#26282
#1172383