Finite Group Databasebeta

← #35296170

Subgroups — #35296170

Order 21248.

738 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 147 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
1C2×Q8×Dic166C211yesmaximal, characteristic
2#24907459C211yesmaximal, characteristic
3#24906147C212yesmaximal
4#24905893C212yesmaximal
5C83×#1461C211yesmaximal, characteristic
6Q8×Dic166—24no—
7Q8×Dic166—24no—
8C2×#14581792C412yes—
9C2×#14581792C2211yescharacteristic
10C2×#14581791—21no—
11C2×#14581791—22no—
12C2×Q8×Dic83—21no—
13C2×C4×Dic166C412yes—
14C2×C4×Dic166C2211yescharacteristic
15#19127725C2212yes—
16C2×#14581711—21no—
17#19127470—21no—
18#19127448—24no—
19#19127433—24no—
20C2×C4×Q8×C83C2211yescharacteristic
21#431×C83C2212yes—
22#324×C83—22no—
23C22×Dic166—22no—
24C22×Dic166—21no—
25C22×Dic166C2×C411yescharacteristic
26C2×#13050574—22no—
27#14581792—24no—
28#14581792—24no—
29#14581792—22no—
30#14581792D414yes—
31#14581791—216no—
32#14581791—28no—
33Q8×Dic83—28no—
34C4×Dic166—24no—
35C4×Dic166—24no—
36C4×Dic166—22no—
37C4×Dic166D414yes—
38C2×C4⋊Dic83—22no—
39C2×C4⋊Dic83—21no—
40C2×C4⋊Dic83C2×C412yes—
41C2×C4⋊Dic83C2×C411yescharacteristic
42C2×Dic83⋊C4—22no—
43C2×Dic83⋊C4—21no—
44C2×C4×Dic83—22no—
45C2×C4×Dic83—21no—
46C4⋊(C83⋊C8)—22no—
47C4⋊(C83⋊C8)—22no—
48#14581711—42no—
49C22×Q8×C83D411yescharacteristic
50C4×Q8×C83—22no—
51C4×Q8×C83—22no—
52C2×C4⋊C4×C83D411yescharacteristic
53C2×C4⋊C4×C83D412yes—
54C2×C42×C83C2×C412yes—
55C2×C42×C83C2311yescharacteristic
56C4⋊C8×C83—22no—
57C22⋊C8×C83—22no—
58C4×C8×C83—22no—
59C2×Dic166—21no—
60C2×Dic166—24no—
61C2×Dic166—28no—
62C2×Dic166—24no—
63C2×Dic166—82no—
64C2×Dic166—84no—
65C2×Dic166C22⋊C412yes—
66C22×Dic83—21no—
67C22×Dic83—21no—
68C22×Dic83—22no—
69#13050574—42no—
70#13050574—42no—
71C2×C83⋊C8—42no—
72C4⋊Dic83—28no—
73C4⋊Dic83—21no—
74C4⋊Dic83—22no—
75C4⋊Dic83—24no—
76C4⋊Dic83C22⋊C412yes—
77C4⋊Dic83C22⋊C414yes—
78Dic83⋊C4—28no—
79Dic83⋊C4—24no—
80C4×Dic83—28no—
81C4×Dic83—24no—
82C2×Q8×C83—22no—
83C2×Q8×C83—22no—
84C2×Q8×C83SD1612yes—
85C2×Q8×C83Q1612yes—
86C22×C4×C83C22⋊C412yes—
87C22×C4×C83C22⋊C412yes—
88C22×C4×C83C22×C411yescharacteristic
89C22×C4×C83C2×D411yescharacteristic
90C22×C4×C83C2×D411yescharacteristic
91C2×C8×C83—42no—
92C4⋊C4×C83—24no—
93C4⋊C4×C83—22no—
94C42×C83—22no—
95C42×C83—22no—
96C42×C83C2×D412yes—
97C42×C83C2×D412yes—
98C23×C83C2×C22⋊C411yescharacteristic
99Q8×C83—24no—
100Q8×C83—24no—
101C2×C4×C83—22no—
102C2×C4×C83—22no—
103C2×C4×C83—22no—
104C2×C4×C83—22no—
105C2×C4×C83—41no—
106C2×C4×C83—42no—
107C2×C4×C83#10314yes—
108C2×C4×C83C2×C22⋊C411yescharacteristic
109C2×C4×C83C2×C22⋊C411yescharacteristic
110C2×C4×C83#12012yes—
111C2×C4×C83#12012yes—
112C2×C4×C83C2×SD1611yescharacteristic
113C2×C4×C83C2×Q1611yescharacteristic
114C2×C4×C83#13712yes—
115C2×Dic83—24no—
116C2×Dic83—22no—
117C2×Dic83—24no—
118C2×Dic83—22no—
119C2×Dic83—44no—
120C2×Dic83—42no—
121Dic166—24no—
122Dic166—22no—
123Dic166—84no—
124Dic166—88no—
125Dic166#10114yescomplemented
126C8×C83—82no—
127C83⋊C8—82no—
128C22×C83—21no—
129C22×C83—21no—
130C22×C83#37911yescharacteristic
131C22×C83C2×#10311yescharacteristic
132C22×C83#41911yescharacteristic
133C4×C83—22no—
134C4×C83—21no—
135C4×C83—24no—
136C4×C83—42no—
137C4×C83C2×#10112yes—
138C4×C83#44812yes—
139C4×C83#45112yes—
140Dic83—24no—
141Dic83—44no—
142Dic83—42no—
143#8396—831nomaximal, Sylow
144C2×C83—21no—
145C2×C83—21no—
146C2×C83#178411yescharacteristic
147C2×C83#178711yescharacteristic
148C2×C83#179511yescharacteristic
149C2×Q82—831no—
150#1801—831no—
151#1549—832no—
152#1487—832no—
153#1461D8311yescharacteristic
154C83#839611yescharacteristic, complemented, Sylow
155Q82—1664no—
156Q82—1664no—
157C2×C4⋊Q8—832no—
158C2×C4⋊Q8—831no—
159C2×C4⋊Q8—1661no—
160C2×C4⋊Q8—1662no—
161C2×C4×Q8—832no—
162C2×C4×Q8—831no—
163C2×C4×Q8—1661no—
164C2×C4×Q8C2×D8311yescharacteristic
165#431C2×D8312yes—
166#423—832no—
167C2×#101—1661no—
168#343—1661no—
169#332—1664no—
170Q8⋊C8—1664no—
171#324—22no—
172C22×Q8—831no—
173C22×Q8—1662no—
174C22×Q8—1661no—
175C22×Q8D33211yescharacteristic
176C2×M16—1662no—
177C4⋊Q8—834no—
178C4⋊Q8—1664no—
179C4⋊Q8—1664no—
180C4⋊Q8—16616no—
181C4⋊Q8—1668no—
182C4⋊Q8—1662no—
183C4×Q8—22no—
184C4×Q8—22no—
185C4×Q8—834no—
186C4×Q8—1664no—
187C4×Q8—1664no—
188C4×Q8—1668no—
189C4×Q8—1662no—
190C2×C4⋊C4—831no—
191C2×C4⋊C4—832no—
192C2×C4⋊C4—1662no—
193C2×C4⋊C4—1662no—
194C2×C4⋊C4—1661no—
195C2×C4⋊C4—1661no—
196C2×C4⋊C4D33211yescharacteristic
197C2×C4⋊C4C22⋊D8312yes—
198C2×C42—1662no—
199C2×C42—1661no—
200C2×C42C4×D8312yes—
201C2×C42C22×D8311yescharacteristic
202C4⋊C8—22no—
203C4⋊C8—1662no—
204C4⋊C8—1662no—
205#101—3322no—
206C22⋊C8—22no—
207C4×C8—22no—
208C2×Q8—22no—
209C2×Q8—22no—
210C2×Q8—832no—
211C2×Q8—6642no—
212C2×Q8—6644no—
213C2×Q8—1664no—
214C2×Q8—1668no—
215C2×Q8—1664no—
216C2×Q8—1661no—
217C2×Q8#1305057912yes—
218C2×Q8Dic33212yes—
219C22×C4—1662no—
220C22×C4—1661no—
221C22×C4—1661no—
222C22×C4(C2×D83)⋊C412yes—
223C22×C4(C2×D83)⋊C412yes—
224C22×C4C2×C4×D8311yescharacteristic
225C22×C4C2×C22⋊D8311yescharacteristic
226C22×C4C2×D33211yescharacteristic
227M16—3322no—
228M16—3322no—
229C2×C8—42no—
230C2×C8—3322no—
231C4⋊C4—24no—
232C4⋊C4—22no—
233C4⋊C4—832no—
234C4⋊C4—834no—
235C4⋊C4—1668no—
236C4⋊C4—1668no—
237C4⋊C4—1664no—
238C4⋊C4—1664no—
239C4⋊C4—1661no—
240C4⋊C4—1662no—
241C42—22no—
242C42—22no—
243C42—1668no—
244C42—1664no—
245C42D4×D8312yes—
246C42D4×D8312yes—
247C23C2×(C2×D83)⋊C411yescharacteristic
248Q8—24no—
249Q8—24no—
250Q8—834no—
251Q8—6644no—
252Q8—6648no—
253Q8—1664no—
254Q8—1662no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—3324no—
262C2×C4—3322no—
263C2×C4—1664no—
264C2×C4—1662no—
265C2×C4—1664no—
266C2×C4—1662no—
267C2×C4#1458171714yes—
268C2×C4C22⋊C4×D8311yescharacteristic
269C2×C4C22⋊C4×D8311yescharacteristic
270C2×C4#1458176012yes—
271C2×C4#1458176212yes—
272C2×C4C2×#1305057911yescharacteristic
273C2×C4C2×Dic33211yescharacteristic
274C2×C4#1458182512yes—
275C8—82no—
276C8—6642no—
277C22—21no—
278C22—21no—
279C22#1912755611yescharacteristic
280C22C2×#1458171711yescharacteristic
281C22#1912771111yescharacteristic
282C4—21no—
283C4—24no—
284C4—22no—
285C4—42no—
286C4—3324no—
287C4—3322no—
288C4—1664no—
289C4#101×D8312yes—
290C4#1912784212yes—
291C4#1912785912yes—
292C2—21no—
293C2—21no—
294C2#2490738511yescharacteristic
295C2#2490739911yescharacteristic
296C2#2490743111yescharacteristic
subgroupG/H
centerC22#19127556
commutatorC22×C4×C83C22×C4
FrattiniC2×C42C22×D83
FittingC83×#1461C2
socleC22×C83#379
prime radicalC23×C83C2×C22⋊C4
cosocleC2×C42×C83C23
hypercenter#1461D83
hypocenterC83#8396
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1461

Complements

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

NH ≅ G/Ntype
125. Dic166#101semidirect
154. C83#8396semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×Dic1662
#249074592
#249061472
#249058932
C83×#14612
#839683