Finite Group Databasebeta

← #35320413

Subgroups — #35320413

Order 21248.

432 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 101 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#24910183C211yesmaximal, characteristic
2C2×#19127500C211yesmaximal, characteristic
3#24907457C212yesmaximal
4#24905589C212yesmaximal
5C2×#356×C83C211yesmaximal, characteristic
6C2×Q8×Dic83C411yescharacteristic
7C2×Q8×Dic83C2211yescharacteristic
8#19127724—22no—
9C2×#14581711C2212yes—
10#19127608C411yescharacteristic
11#19127608C2211yescharacteristic
12#19127587—22no—
13#19127587—22no—
14#19127567C411yescharacteristic
15#19127547C411yescharacteristic
16#19127500—22no—
17#19127469—22no—
18C2×C4⋊Q8×C83C2211yescharacteristic
19C2×#101×C83C2212yes—
20#356×C83—22no—
21C2×#13050574—22no—
22Q8×Dic83—42no—
23Q8×Dic83—42no—
24C2×C4⋊Dic83—21no—
25C2×C4⋊Dic83C2×C411yescharacteristic
26C2×C4⋊Dic83C2×C411yescharacteristic
27C2×C4⋊Dic83D412yes—
28C2×C4×Dic83—21no—
29C2×C4×Dic83C2×C411yescharacteristic
30C2×C4×Dic83C2×C411yescharacteristic
31C2×C4×Dic83D412yes—
32Dic83⋊C8—44no—
33#14581711—22no—
34#14581711—22no—
35C83⋊#95—22no—
36C83⋊#95—22no—
37C83⋊#95—21no—
38C83⋊#95—21no—
39C83⋊#95—22no—
40C22×Q8×C83C2×C411yescharacteristic
41C22×Q8×C83C2311yescharacteristic
42C2×M16×C83—22no—
43C4⋊Q8×C83—22no—
44C4⋊Q8×C83D412yes—
45C4⋊Q8×C83Q812yes—
46C2×C4⋊C4×C83—22no—
47C2×C42×C83C2×C411yescharacteristic
48#101×C83—22no—
49#101×C83—22no—
50C22×Dic83—21no—
51C22×Dic83—21no—
52C22×Dic83C22⋊C412yes—
53C22×Dic83C22⋊C411yescharacteristic
54C22×Dic83C22⋊C411yescharacteristic
55#13050574—42no—
56#13050574—42no—
57C2×C83⋊C8—42no—
58C4⋊Dic83—41no—
59C4⋊Dic83—42no—
60C4⋊Dic83—42no—
61C4⋊Dic83—41no—
62C4×Dic83—24no—
63C4×Dic83—22no—
64C4×Dic83—42no—
65C4×Dic83—41no—
66C2×Q8×C83—21no—
67C2×Q8×C83—41no—
68C2×Q8×C83—41no—
69C2×Q8×C83—41no—
70C2×Q8×C83—41no—
71C2×Q8×C83C4⋊C412yes—
72C2×Q8×C83C2×D411yescharacteristic
73C2×Q8×C83C2×Q811yescharacteristic
74C2×Q8×C83#4112yes—
75C22×C4×C83—21no—
76C22×C4×C83—21no—
77C22×C4×C83C22×C411yescharacteristic
78C22×C4×C83C2×D412yes—
79M16×C83—42no—
80M16×C83—42no—
81C2×C8×C83—42no—
82C4⋊C4×C83—24no—
83C4⋊C4×C83—24no—
84C42×C83—21no—
85C42×C83C4⋊C412yes—
86C23×C83C2×C22⋊C411yescharacteristic
87Q8×C83—41no—
88Q8×C83—41no—
89Q8×C83—42no—
90Q8×C83—42no—
91Q8×C83—42no—
92C2×C4×C83—22no—
93C2×C4×C83—21no—
94C2×C4×C83—22no—
95C2×C4×C83—21no—
96C2×C4×C83—22no—
97C2×C4×C83—22no—
98C2×C4×C83—41no—
99C2×C4×C83—41no—
100C2×C4×C83—41no—
101C2×C4×C83—41no—
102C2×C4×C83C2×C4⋊C411yescharacteristic
103C2×C4×C83#11711yescharacteristic
104C2×C4×C83#12112yes—
105C2×C4×C83C22⋊Q812yes—
106C2×Dic83—21no—
107C2×Dic83—22no—
108C2×Dic83—41no—
109C2×Dic83—42no—
110C2×Dic83—41no—
111C2×Dic83—41no—
112C2×Dic83—41no—
113C2×Dic83—42no—
114C2×Dic83—41no—
115C2×Dic83—41no—
116C2×Dic83#9912yes—
117C8×C83—82no—
118C83⋊C8—82no—
119C22×C83—21no—
120C22×C83—21no—
121C22×C83#38011yescharacteristic
122C22×C83C2×#9911yescharacteristic
123C22×C83#41011yescharacteristic
124C4×C83—22no—
125C4×C83—21no—
126C4×C83—21no—
127C4×C83—22no—
128C4×C83—42no—
129C4×C83—42no—
130Dic83—22no—
131Dic83—41no—
132Dic83—42no—
133Dic83—81no—
134Dic83—81no—
135#12804—831nomaximal, Sylow
136C2×C83—21no—
137C2×C83—21no—
138C2×C83#179911yescharacteristic
139C2×C83#213311yescharacteristic
140C2×C83#213511yescharacteristic
141#2327—831no—
142C2×#356—831no—
143C2×#356D8311yescharacteristic
144#1801—832no—
145#1394—832no—
146C83#1280411yescharacteristic, complemented, Sylow
147C2×C4⋊Q8C2×D8311yescharacteristic
148C2×C4×Q8—831no—
149C2×C4×Q8—831no—
150#423—1662no—
151C2×#101—832no—
152C2×#101C2×D8312yes—
153#391—831no—
154#391—831no—
155#387—1662no—
156#387—1662no—
157#382—831no—
158#376—831no—
159#356—22no—
160#356—1662no—
161#343—1662no—
162C22×Q8C4×D8311yescharacteristic
163C22×Q8C22×D8311yescharacteristic
164C2×M16—22no—
165C2×M16—1662no—
166C4⋊Q8—22no—
167C4⋊Q8Dic16612yes—
168C4⋊Q8C22⋊D8312yes—
169C4×Q8—3322no—
170C4×Q8—3322no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—831no—
173C2×C4⋊C4—832no—
174C2×C4⋊C4—831no—
175C2×C4⋊C4—1661no—
176C2×C42—831no—
177C2×C42—831no—
178C2×C42—832no—
179C2×C42—1661no—
180C2×C42C4×D8311yescharacteristic
181C4⋊C8—3324no—
182#101—22no—
183#101—22no—
184#101—1662no—
185#101—1662no—
186#95—1661no—
187#95—1661no—
188#95—1662no—
189#95—1662no—
190#95—1662no—
191C2×Q8—21no—
192C2×Q8—41no—
193C2×Q8—41no—
194C2×Q8—41no—
195C2×Q8—41no—
196C2×Q8Dic83⋊C412yes—
197C2×Q8C2×C22⋊D8311yescharacteristic
198C2×Q8C2×Dic16611yescharacteristic
199C2×Q8#1305058912yes—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—831no—
203C22×C4—832no—
204C22×C4—831no—
205C22×C4—1661no—
206C22×C4—1661no—
207C22×C4C2×C4×D8311yescharacteristic
208C22×C4D4×D8312yes—
209M16—42no—
210M16—42no—
211M16—3322no—
212M16—3322no—
213C2×C8—42no—
214C2×C8—3322no—
215C4⋊C4—24no—
216C4⋊C4—24no—
217C4⋊C4—3321no—
218C4⋊C4—3321no—
219C4⋊C4—3322no—
220C4⋊C4—3322no—
221C42—21no—
222C42—3321no—
223C42—3322no—
224C42—1664no—
225C42—1662no—
226C42Dic83⋊C412yes—
227C23C22⋊C4×D8311yescharacteristic
228Q8—41no—
229Q8—41no—
230Q8—42no—
231Q8—42no—
232Q8—42no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—3322no—
244C2×C4—3321no—
245C2×C4—3321no—
246C2×C4—3321no—
247C2×C4—3322no—
248C2×C4—3321no—
249C2×C4—3321no—
250C2×C4—3321no—
251C2×C4—832no—
252C2×C4—1661no—
253C2×C4—1662no—
254C2×C4C2×Dic83⋊C411yescharacteristic
255C2×C4#1458174811yescharacteristic
256C2×C4#1458176412yes—
257C2×C4#1458176912yes—
258C8—82no—
259C8—6642no—
260C22—21no—
261C22—21no—
262C22#1912755911yescharacteristic
263C22#99×D8311yescharacteristic
264C22#1912767311yescharacteristic
265C4—22no—
266C4—22no—
267C4—21no—
268C4—21no—
269C4—42no—
270C4—42no—
271C4—3321no—
272C4—3322no—
273C4—6641no—
274C4—6641no—
275C4—1662no—
276C2—21no—
277C2—21no—
278C2#2490744911yescharacteristic
279C2#2490915111yescharacteristic
280C2#2490915611yescharacteristic
subgroupG/H
centerC22#99×D83
commutatorC22×C4×C83C22×C4
FrattiniC22×Q8C22×D83
FittingC2×#356×C83C2
socleC22×C83C2×#99
prime radicalC23×C83C2×C22⋊C4
cosocleC22×Q8×C83C23
hypercenterC2×#356D83
hypocenterC83#12804
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ C2×#356

Complements

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

NH ≅ G/Ntype
146. C83#12804semidirect

Maximal subgroups

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

subgroupindex
#249101832
C2×#191275002
#249074572
#249055892
C2×#356×C832
#1280483