Finite Group Databasebeta

← #35313882

Subgroups — #35313882

Order 21248.

412 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 87 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
1C4×#14581790C211yesmaximal, characteristic
2#24905807C211yesmaximal, characteristic
3#24905408C212yesmaximal
4#24905396C212yesmaximal
5C83×#1467C211yesmaximal, characteristic
6C2×#14581790C411yescharacteristic
7C2×C4×Dic166C411yescharacteristic
8#19127758—21no—
9#19127608C411yescharacteristic
10#19127606C2211yescharacteristic
11#19127596—21no—
12#19127596—21no—
13#19127570C2211yescharacteristic
14C4×Dic83⋊C4—21no—
15C4×Dic83⋊C4—21no—
16C42×Dic83C411yescharacteristic
17#19127427—21no—
18#19127427C2212yes—
19C2×C4×Q8×C83C2211yescharacteristic
20#432×C83—21no—
21#324×C83—21no—
22#324×C83C2212yes—
23C22×Dic166C2×C411yescharacteristic
24#14581790—22no—
25#14581790—22no—
26C4×Dic166—41no—
27C4×Dic166—41no—
28C2×C4⋊Dic83C2×C411yescharacteristic
29C2×Dic83⋊C4—21no—
30C2×Dic83⋊C4—21no—
31C2×Dic83⋊C4—21no—
32C2×Dic83⋊C4—21no—
33C2×Dic83⋊C4—21no—
34C2×C4×Dic83—21no—
35C2×C4×Dic83—21no—
36C2×C4×Dic83—21no—
37C2×C4×Dic83C2×C411yescharacteristic
38C2×C4×Dic83C2×C411yescharacteristic
39C4⋊(C83⋊C8)—41no—
40C22⋊(C83⋊C8)—22no—
41#14581703—41no—
42C83⋊#95—21no—
43C83⋊#95—21no—
44C83⋊#95—21no—
45C83⋊#95—21no—
46C22×Q8×C83C2×C411yescharacteristic
47C4×Q8×C83—41no—
48C4×Q8×C83—41no—
49C2×C4⋊C4×C83D411yescharacteristic
50C2×C4⋊C4×C83D411yescharacteristic
51C2×C4⋊C4×C83C2311yescharacteristic
52C2×C42×C83C2×C411yescharacteristic
53C2×C42×C83D411yescharacteristic
54C2×C42×C83D411yescharacteristic
55C4⋊C8×C83—41no—
56C22⋊C8×C83—22no—
57#97×C83—41no—
58C2×Dic166—21no—
59C2×Dic166—41no—
60C2×Dic166—41no—
61C2×Dic166M1612yes—
62C22×Dic83—21no—
63C22×Dic83—21no—
64C22×Dic83—21no—
65C22×Dic83—21no—
66C2×C83⋊C8—42no—
67C4⋊Dic83—41no—
68Dic83⋊C4—24no—
69Dic83⋊C4—24no—
70Dic83⋊C4—42no—
71Dic83⋊C4—42no—
72Dic83⋊C4—42no—
73C4×Dic83—24no—
74C4×Dic83—21no—
75C4×Dic83—42no—
76C4×Dic83—41no—
77C4×Dic83—41no—
78C4×Dic83—41no—
79C4×Dic83M1612yes—
80C2×Q8×C83—21no—
81C2×Q8×C83—21no—
82C2×Q8×C83—41no—
83C2×Q8×C83—41no—
84C22×C4×C83C22⋊C411yescharacteristic
85C22×C4×C83C22⋊C411yescharacteristic
86C22×C4×C83C22⋊C411yescharacteristic
87C22×C4×C83C22⋊C411yescharacteristic
88C22×C4×C83C22×C411yescharacteristic
89C22×C4×C83C2×D411yescharacteristic
90C22×C4×C83C2×D411yescharacteristic
91C2×C8×C83—42no—
92C4⋊C4×C83—41no—
93C4⋊C4×C83—41no—
94C4⋊C4×C83—41no—
95C42×C83—21no—
96C42×C83—21no—
97C42×C83—21no—
98C42×C83—21no—
99C42×C83—21no—
100C42×C83—21no—
101C23×C83C2×C22⋊C411yescharacteristic
102Q8×C83—42no—
103Q8×C83—42no—
104C2×C4×C83—21no—
105C2×C4×C83—21no—
106C2×C4×C83—22no—
107C2×C4×C83—21no—
108C2×C4×C83—21no—
109C2×C4×C83—21no—
110C2×C4×C83—21no—
111C2×C4×C83—21no—
112C2×C4×C83—21no—
113C2×C4×C83—21no—
114C2×C4×C83—21no—
115C2×C4×C83—21no—
116C2×C4×C83—41no—
117C2×C4×C83—41no—
118C2×C4×C83—41no—
119C2×C4×C83#9911yescharacteristic
120C2×C4×C83#9911yescharacteristic
121C2×C4×C83C2×M1611yescharacteristic
122C2×C4×C83C2×M1611yescharacteristic
123C2×Dic83—22no—
124C2×Dic83—22no—
125C2×Dic83—22no—
126C2×Dic83—22no—
127C2×Dic83—22no—
128C2×Dic83—41no—
129C2×Dic83—41no—
130C2×Dic83—41no—
131C2×Dic83—41no—
132C2×Dic83—41no—
133C2×Dic83—41no—
134C2×Dic83—41no—
135Dic166—42no—
136Dic166—42no—
137C8×C83—82no—
138C83⋊C8—82no—
139C22×C83—21no—
140C22×C83—21no—
141C22×C83#40711yescharacteristic
142C22×C83C2×#9911yescharacteristic
143C22×C83#41311yescharacteristic
144C4×C83—21no—
145C4×C83—21no—
146C4×C83—21no—
147C4×C83—41no—
148C4×C83—41no—
149C4×C83—41no—
150C4×C83—41no—
151C4×C83—41no—
152C4×C83#35612yes—
153Dic83—24no—
154Dic83—42no—
155Dic83—42no—
156Dic83—81no—
157#11532—831nomaximal, Sylow
158C2×C83—21no—
159C2×C83—21no—
160C2×C83#146511yescharacteristic
161C2×C83#212611yescharacteristic
162C2×C83C2×#35611yescharacteristic
163C4×C4⋊Q8—831no—
164#1467—831no—
165#1467D8311yescharacteristic
166#1325—832no—
167#1319—832no—
168C83#1153211yescharacteristic, complemented, Sylow
169C2×C4⋊Q8—831no—
170C2×C4×Q8—831no—
171C2×C4×Q8C2×D8311yescharacteristic
172#432—21no—
173#432—1661no—
174#391—831no—
175#391—831no—
176#389—1661no—
177#389—1661no—
178#384—831no—
179C4×C4⋊C4—1661no—
180C4×C4⋊C4—1661no—
181C43—831no—
182#324—21no—
183#324—832no—
184#324—1661no—
185#324C2×D8312yes—
186C22×Q8—831no—
187C22×Q8C4×D8311yescharacteristic
188C4⋊Q8—1662no—
189C4⋊Q8—1662no—
190C4×Q8—41no—
191C4×Q8—41no—
192C4×Q8—3321no—
193C4×Q8—3321no—
194C2×C4⋊C4—831no—
195C2×C4⋊C4—1661no—
196C2×C4⋊C4—1661no—
197C2×C4⋊C4—1661no—
198C2×C4⋊C4—1661no—
199C2×C4⋊C4—1661no—
200C2×C4⋊C4D33211yescharacteristic
201C2×C4⋊C4C22⋊D8311yescharacteristic
202C2×C4⋊C4C22×D8311yescharacteristic
203C2×C42—831no—
204C2×C42—831no—
205C2×C42—1661no—
206C2×C42—1661no—
207C2×C42—1661no—
208C2×C42C4×D8311yescharacteristic
209C2×C42D33211yescharacteristic
210C2×C42C22⋊D8311yescharacteristic
211C4⋊C8—41no—
212C4⋊C8—3321no—
213C22⋊C8—22no—
214C22⋊C8—1662no—
215#97—41no—
216#97—3321no—
217#95—1661no—
218#95—1661no—
219#95—1661no—
220#95—1661no—
221C2×Q8—21no—
222C2×Q8—21no—
223C2×Q8—41no—
224C2×Q8—41no—
225C2×Q8—3321no—
226C2×Q8—3321no—
227C2×Q8—832no—
228C2×Q8—1661no—
229C22×C4—1661no—
230C22×C4—1661no—
231C22×C4—1661no—
232C22×C4—1661no—
233C22×C4(C2×D83)⋊C411yescharacteristic
234C22×C4(C2×D83)⋊C411yescharacteristic
235C22×C4(C2×D83)⋊C411yescharacteristic
236C22×C4(C2×D83)⋊C411yescharacteristic
237C22×C4C2×C4×D8311yescharacteristic
238C22×C4C2×C22⋊D8311yescharacteristic
239C22×C4C2×D33211yescharacteristic
240C2×C8—42no—
241C2×C8—3322no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—3321no—
246C4⋊C4—3322no—
247C4⋊C4—3322no—
248C4⋊C4—3322no—
249C4⋊C4—1664no—
250C4⋊C4—1664no—
251C42—21no—
252C42—21no—
253C42—21no—
254C42—21no—
255C42—21no—
256C42—21no—
257C42—3321no—
258C42—3322no—
259C42—3321no—
260C42—3321no—
261C42—832no—
262C42—1664no—
263C42—1661no—
264C23C2×(C2×D83)⋊C411yescharacteristic
265Q8—42no—
266Q8—42no—
267Q8—3322no—
268Q8—3322no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—3321no—
285C2×C4—3321no—
286C2×C4—3321no—
287C2×C4—3321no—
288C2×C4—3321no—
289C2×C4—3321no—
290C2×C4—3321no—
291C2×C4—1662no—
292C2×C4—1662no—
293C2×C4—1662no—
294C2×C4—1662no—
295C2×C4—1662no—
296C2×C4#1458170611yescharacteristic
297C2×C4#1458170611yescharacteristic
298C2×C4M16×D8311yescharacteristic
299C2×C4M16×D8311yescharacteristic
300C8—82no—
301C8—6642no—
302C22—21no—
303C22—21no—
304C22#1912766411yescharacteristic
305C22C2×#1458170611yescharacteristic
306C22#1912768311yescharacteristic
307C4—21no—
308C4—21no—
309C4—21no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—41no—
314C4—41no—
315C4—3322no—
316C4—3322no—
317C4—6641no—
318C4—1664no—
319C4#1912749912yes—
320C2—21no—
321C2—21no—
322C2#2490579911yescharacteristic
323C2#2490912611yescharacteristic
324C2C2×#1912749911yescharacteristic
subgroupG/H
centerC22C2×#14581706
commutatorC22×C4×C83C22×C4
FrattiniC2×C4⋊C4C22×D83
FittingC83×#1467C2
socleC22×C83C2×#99
prime radicalC23×C83C2×C22⋊C4
cosocleC2×C4⋊C4×C83C23
hypercenter#1467D83
hypocenterC83#11532
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1467

Complements

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

NH ≅ G/Ntype
168. C83#11532semidirect

Maximal subgroups

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

subgroupindex
C4×#145817902
#249058072
#249054082
#249053962
C83×#14672
#1153283