Finite Group Databasebeta

← #35320414

Subgroups — #35320414

Order 21248.

432 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 93 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#24910181C211yesmaximal, characteristic
2C2×#19127499C212yesmaximal
3#24907457C212yesmaximal
4#24905590C211yesmaximal, characteristic
5C83×#1394C211yesmaximal, characteristic
6C2×#14581790C2211yescharacteristic
7C2×Q8×Dic83C2211yescharacteristic
8C2×#14581757C411yescharacteristic
9#19127724—22no—
10C2×#14581711C2212yes—
11#19127607C411yescharacteristic
12#19127588—22no—
13#19127583—22no—
14#19127566C411yescharacteristic
15#19127547C411yescharacteristic
16#19127499—24no—
17#19127469—22no—
18C2×#101×C83C2212yes—
19#391×C83C2211yescharacteristic
20C22×Dic166C2×C411yescharacteristic
21C2×#13050574—22no—
22#14581790—22no—
23#14581790D412yes—
24#14581790Q812yes—
25Q8×Dic83—42no—
26#14581757—42no—
27C2×C4⋊Dic83C2×C411yescharacteristic
28C2×C4⋊Dic83D412yes—
29C2×Dic83⋊C4—21no—
30C2×Dic83⋊C4—22no—
31C2×C4×Dic83—21no—
32C2×C4×Dic83C2×C411yescharacteristic
33C2×C4×Dic83C2×C411yescharacteristic
34C2×C4×Dic83D412yes—
35Dic83⋊C8—44no—
36#14581711—22no—
37#14581711—22no—
38C83⋊#95—22no—
39C83⋊#95—21no—
40C83⋊#95—22no—
41C83⋊#95—21no—
42C22×Q8×C83C2311yescharacteristic
43C2×M16×C83—22no—
44C2×C4⋊C4×C83C2×C411yescharacteristic
45C2×C42×C83C2×C411yescharacteristic
46#101×C83—22no—
47#101×C83—22no—
48#95×C83—22no—
49C2×Dic166—21no—
50C2×Dic166—41no—
51C2×Dic166—41no—
52C2×Dic166C4⋊C412yes—
53C22×Dic83—21no—
54C22×Dic83—21no—
55C22×Dic83C22⋊C412yes—
56C22×Dic83C22⋊C411yescharacteristic
57C22×Dic83C22⋊C411yescharacteristic
58#13050574—42no—
59#13050574—42no—
60C2×C83⋊C8—42no—
61C4⋊Dic83—41no—
62C4⋊Dic83—42no—
63Dic83⋊C4—24no—
64Dic83⋊C4—24no—
65Dic83⋊C4—42no—
66C4×Dic83—21no—
67C4×Dic83—24no—
68C4×Dic83—22no—
69C4×Dic83—42no—
70C4×Dic83C4⋊C412yes—
71C2×Q8×C83—41no—
72C2×Q8×C83—41no—
73C2×Q8×C83C2×D411yescharacteristic
74C2×Q8×C83C2×Q811yescharacteristic
75C2×Q8×C83#4112yes—
76C22×C4×C83—21no—
77C22×C4×C83—21no—
78C22×C4×C83C22×C411yescharacteristic
79C22×C4×C83C2×D412yes—
80M16×C83—42no—
81M16×C83—42no—
82C2×C8×C83—42no—
83C4⋊C4×C83—41no—
84C42×C83—41no—
85C23×C83C2×C22⋊C411yescharacteristic
86Q8×C83—41no—
87Q8×C83—41no—
88Q8×C83—42no—
89C2×C4×C83—21no—
90C2×C4×C83—22no—
91C2×C4×C83—21no—
92C2×C4×C83—22no—
93C2×C4×C83—41no—
94C2×C4×C83—41no—
95C2×C4×C83—42no—
96C2×C4×C83—41no—
97C2×C4×C83—41no—
98C2×C4×C83C2×C4⋊C411yescharacteristic
99C2×C4×C83#11711yescharacteristic
100C2×C4×C83#12112yes—
101C2×C4×C83C22⋊Q812yes—
102C2×Dic83—22no—
103C2×Dic83—22no—
104C2×Dic83—21no—
105C2×Dic83—22no—
106C2×Dic83—41no—
107C2×Dic83—41no—
108C2×Dic83—41no—
109C2×Dic83—41no—
110C2×Dic83—41no—
111C2×Dic83—42no—
112C2×Dic83—41no—
113C2×Dic83#9912yes—
114Dic166—42no—
115Dic166—42no—
116C8×C83—82no—
117C83⋊C8—82no—
118C22×C83—21no—
119C22×C83—21no—
120C22×C83#38011yescharacteristic
121C22×C83C2×#9911yescharacteristic
122C22×C83#41011yescharacteristic
123C4×C83—22no—
124C4×C83—21no—
125C4×C83—21no—
126C4×C83—22no—
127C4×C83—81no—
128C4×C83—81no—
129Dic83—22no—
130Dic83—42no—
131Dic83—42no—
132Dic83—41no—
133Dic83—42no—
134#12804—831nomaximal, Sylow
135C2×C83—21no—
136C2×C83—21no—
137C2×C83#179911yescharacteristic
138C2×C83#213311yescharacteristic
139C2×C83#213511yescharacteristic
140#2327—831no—
141C2×#356—832no—
142#1801—832no—
143#1394—831no—
144#1394D8311yescharacteristic
145C83#1280411yescharacteristic, complemented, Sylow
146C2×C4⋊Q8—831no—
147C2×C4×Q8—831no—
148C2×C4×Q8—831no—
149#423—1662no—
150C2×#101—832no—
151C2×#101C2×D8312yes—
152#391—831no—
153#391C2×D8311yescharacteristic
154#387—1662no—
155#387—1662no—
156#382—831no—
157#376—831no—
158#356—1664no—
159#343—1662no—
160C22×Q8—831no—
161C22×Q8C22×D8311yescharacteristic
162C2×M16—22no—
163C2×M16—1662no—
164C4⋊Q8—832no—
165C4⋊Q8—832no—
166C4⋊Q8—1662no—
167C4×Q8—3322no—
168C4×Q8—3322no—
169C2×C4⋊C4—831no—
170C2×C4⋊C4—832no—
171C2×C4⋊C4—1661no—
172C2×C4⋊C4—1662no—
173C2×C4⋊C4C4×D8311yescharacteristic
174C2×C42—831no—
175C2×C42—831no—
176C2×C42—832no—
177C2×C42—1661no—
178C2×C42C4×D8311yescharacteristic
179C4⋊C8—3324no—
180#101—22no—
181#101—22no—
182#101—1662no—
183#101—1662no—
184#95—22no—
185#95—1661no—
186#95—1661no—
187#95—1662no—
188#95—1662no—
189C2×Q8—41no—
190C2×Q8—41no—
191C2×Q8—3321no—
192C2×Q8—3321no—
193C2×Q8—832no—
194C2×Q8—1661no—
195C2×Q8D4×D8311yescharacteristic
196C2×Q8Q8×D8311yescharacteristic
197C2×Q8#1305059012yes—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—831no—
201C22×C4—832no—
202C22×C4—831no—
203C22×C4—1661no—
204C22×C4—1661no—
205C22×C4C2×C4×D8311yescharacteristic
206C22×C4D4×D8312yes—
207M16—42no—
208M16—42no—
209M16—3322no—
210M16—3322no—
211C2×C8—42no—
212C2×C8—3322no—
213C4⋊C4—41no—
214C4⋊C4—3321no—
215C4⋊C4—3322no—
216C4⋊C4—3322no—
217C4⋊C4—1664no—
218C4⋊C4—1664no—
219C42—41no—
220C42—3322no—
221C42—832no—
222C42—1664no—
223C42—1662no—
224C42—1661no—
225C23C22⋊C4×D8311yescharacteristic
226Q8—41no—
227Q8—41no—
228Q8—42no—
229Q8—3322no—
230Q8—3322no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—3321no—
241C2×C4—3321no—
242C2×C4—3321no—
243C2×C4—3321no—
244C2×C4—3322no—
245C2×C4—3321no—
246C2×C4—3321no—
247C2×C4—832no—
248C2×C4—1662no—
249C2×C4—1662no—
250C2×C4—1661no—
251C2×C4—1662no—
252C2×C4C4⋊C4×D8311yescharacteristic
253C2×C4#1458174911yescharacteristic
254C2×C4#1458176512yes—
255C2×C4#1458177012yes—
256C8—82no—
257C8—6642no—
258C22—21no—
259C22—21no—
260C22#1912756011yescharacteristic
261C22#99×D8311yescharacteristic
262C22#1912767311yescharacteristic
263C4—22no—
264C4—22no—
265C4—21no—
266C4—21no—
267C4—81no—
268C4—81no—
269C4—3322no—
270C4—3321no—
271C4—3322no—
272C4—3322no—
273C4—1662no—
274C2—21no—
275C2—21no—
276C2#2490745011yescharacteristic
277C2#2490914911yescharacteristic
278C2#2490915611yescharacteristic
subgroupG/H
centerC22#99×D83
commutatorC22×C4×C83C22×C4
FrattiniC22×Q8C22×D83
FittingC83×#1394C2
socleC22×C83C2×#99
prime radicalC23×C83C2×C22⋊C4
cosocleC22×Q8×C83C23
hypercenter#1394D83
hypocenterC83#12804
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #1394

Complements

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

NH ≅ G/Ntype
145. C83#12804semidirect

Maximal subgroups

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

subgroupindex
#249101812
C2×#191274992
#249074572
#249055902
C83×#13942
#1280483