Finite Group Databasebeta

← #35320397

Subgroups — #35320397

Order 21248.

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

normal subgroups only

This group has 524 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#24910176C211yesmaximal, characteristic
2C2×#19127496C211yesmaximal, characteristic
3#24907449C212yesmaximal
4#24905587C212yesmaximal
5C2×#354×C83C211yesmaximal, characteristic
6C2×Q8×Dic83C411yescharacteristic, complemented
7C2×D4×Dic83C2211yescharacteristic
8C2×#14581707C2212yes—
9#19127608C411yescharacteristic, complemented
10#19127604C2211yescharacteristic
11#19127594—22no—
12#19127576—22no—
13#19127571C411yescharacteristic, complemented
14#19127559—22no—
15#19127547C411yescharacteristic, complemented
16#19127496—22no—
17#19127467—22no—
18C2×#124×C83C2211yescharacteristic
19C2×#99×C83C2212yes—
20#354×C83—22no—
21C23×Dic83D412yes—
22Q8×Dic83—42no—
23D4×Dic83—42no—
24C2×C4⋊Dic83—21no—
25C2×C4⋊Dic83—22no—
26C2×C4⋊Dic83C2×C411yescharacteristic
27C2×C4⋊Dic83C2×C411yescharacteristic
28C2×Dic83⋊C4—42no—
29C2×C22⋊Dic83—22no—
30C2×C22⋊Dic83—22no—
31C2×C22⋊Dic83D412yes—
32C2×C4×Dic83—21no—
33C2×C4×Dic83C2×C411yescharacteristic
34C2×C4×Dic83C2×C411yescharacteristic
35#14581707—22no—
36#14581707—22no—
37C83⋊#95—21no—
38C83⋊#95—22no—
39C83⋊#95—21no—
40C83⋊#95—42no—
41C22×Q8×C83C2×C411yescharacteristic
42C22×D4×C83C2311yescharacteristic
43#124×C83—22no—
44#124×C83D412yes—
45#124×C83Q812yes—
46C2×C22⋊C4×C83—22no—
47C2×C22⋊C4×C83—22no—
48C2×C42×C83C2×C411yescharacteristic
49#99×C83—22no—
50#99×C83—22no—
51C22×Dic83—22no—
52C22×Dic83—22no—
53C22×Dic83—21no—
54C22×Dic83—21no—
55C22×Dic83—42no—
56C22×Dic83—82no—
57C22×Dic83C22⋊C412yes—
58C22×Dic83C22⋊C411yescharacteristic
59C22×Dic83C22⋊C411yescharacteristic
60C4⋊Dic83—41no—
61C4⋊Dic83—42no—
62C4⋊Dic83—41no—
63C4⋊Dic83—82no—
64Dic83⋊C4—84no—
65C22⋊Dic83—42no—
66C22⋊Dic83—42no—
67C22⋊Dic83—42no—
68C22⋊Dic83—82no—
69C4×Dic83—22no—
70C4×Dic83—42no—
71C4×Dic83—41no—
72C24×C83C2×D412yes—
73C2×Q8×C83—21no—
74C2×Q8×C83—41no—
75C2×Q8×C83—41no—
76C2×Q8×C83C4⋊C412yescomplemented
77C2×D4×C83—41no—
78C2×D4×C83—41no—
79C2×D4×C83C2×D411yescharacteristic
80C2×D4×C83C2×Q811yescharacteristic
81C2×D4×C83#4112yes—
82C22×C4×C83—21no—
83C22×C4×C83—21no—
84C22×C4×C83—42no—
85C22×C4×C83C22×C411yescharacteristic
86C22⋊C4×C83—24no—
87C22⋊C4×C83—24no—
88C22⋊C4×C83—42no—
89C22⋊C4×C83—42no—
90C42×C83—21no—
91C42×C83C4⋊C412yescomplemented
92C23×C83—22no—
93C23×C83—22no—
94C23×C83—42no—
95C23×C83—42no—
96C23×C83C2×C22⋊C411yescharacteristic
97C23×C83#12112yes—
98C23×C83C22⋊Q812yes—
99Q8×C83—42no—
100Q8×C83—42no—
101D4×C83—41no—
102D4×C83—41no—
103D4×C83—42no—
104C2×C4×C83—21no—
105C2×C4×C83—21no—
106C2×C4×C83—22no—
107C2×C4×C83—22no—
108C2×C4×C83—41no—
109C2×C4×C83—41no—
110C2×C4×C83—41no—
111C2×C4×C83—41no—
112C2×C4×C83—82no—
113C2×C4×C83—82no—
114C2×C4×C83—82no—
115C2×C4×C83C2×C4⋊C411yescharacteristic
116C2×C4×C83#11711yescharacteristic
117C2×Dic83—21no—
118C2×Dic83—22no—
119C2×Dic83—41no—
120C2×Dic83—44no—
121C2×Dic83—42no—
122C2×Dic83—41no—
123C2×Dic83—41no—
124C2×Dic83—42no—
125C2×Dic83—41no—
126C2×Dic83—41no—
127C2×Dic83—41no—
128C2×Dic83—82no—
129C2×Dic83—82no—
130C2×Dic83—82no—
131C2×Dic83—82no—
132C2×Dic83#9912yes—
133C22×C83—21no—
134C22×C83—21no—
135C22×C83—22no—
136C22×C83—22no—
137C22×C83—42no—
138C22×C83—42no—
139C22×C83—42no—
140C22×C83—42no—
141C22×C83—82no—
142C22×C83#38011yescharacteristic
143C22×C83C2×#9911yescharacteristic
144C22×C83#41011yescharacteristic
145C4×C83—21no—
146C4×C83—21no—
147C4×C83—42no—
148C4×C83—42no—
149C4×C83—162no—
150Dic83—22no—
151Dic83—42no—
152Dic83—41no—
153Dic83—81no—
154Dic83—81no—
155Dic83—162no—
156#12800—831nomaximal, Sylow
157C2×C83—21no—
158C2×C83—21no—
159C2×C83—42no—
160C2×C83—42no—
161C2×C83#179911yescharacteristic
162C2×C83#212711yescharacteristic
163C2×C83#212911yescharacteristic
164#2326—831no—
165C2×#354—831no—
166C2×#354D8311yescharacteristic
167#1799—832no—
168#1393—832no—
169C83#1280011yescharacteristic, complemented, Sylow
170C2×#124C2×D8311yescharacteristic
171C2×C4×Q8—831no—
172C2×C4×D4—831no—
173C2×#99—832no—
174C2×#99C2×D8312yes—
175#391—831no—
176#390—831no—
177#388—1662no—
178#385—1662no—
179#384—831no—
180#380—1662no—
181#376—831no—
182#354—22no—
183#354—1662no—
184#342—1662no—
185C22×Q8C4×D8311yescharacteristic
186C22×D4C22×D8311yescharacteristic
187C23×C4—832no—
188#124—22no—
189#124Dic16612yes—
190#124C22⋊D8312yes—
191C4×Q8—3322no—
192C4×D4—3322no—
193C2×C4⋊C4—3322no—
194C2×C4⋊C4—831no—
195C2×C4⋊C4—831no—
196C2×C4⋊C4—1661no—
197C2×C4⋊C4—1662no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—832no—
201C2×C22⋊C4—1662no—
202C2×C22⋊C4—1662no—
203C2×C42—831no—
204C2×C42—831no—
205C2×C42—1661no—
206C2×C42C4×D8311yescharacteristic
207#99—22no—
208#99—22no—
209#99—1662no—
210#99—1662no—
211#95—3322no—
212#95—1661no—
213#95—1661no—
214#95—1662no—
215C24D4×D8312yes—
216C2×Q8—21no—
217C2×Q8—41no—
218C2×Q8—41no—
219C2×Q8Dic83⋊C412yescomplemented
220C2×D4—41no—
221C2×D4—41no—
222C2×D4C2×C22⋊D8311yescharacteristic
223C2×D4C2×Dic16611yescharacteristic
224C2×D4#1305058912yes—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—42no—
228C22×C4—3322no—
229C22×C4—831no—
230C22×C4—831no—
231C22×C4—832no—
232C22×C4—6642no—
233C22×C4—1661no—
234C22×C4—1661no—
235C22×C4—1662no—
236C22×C4—1662no—
237C22×C4C2×C4×D8311yescharacteristic
238C4⋊C4—3321no—
239C4⋊C4—3321no—
240C4⋊C4—3322no—
241C4⋊C4—6642no—
242C4⋊C4—6644no—
243C22⋊C4—24no—
244C22⋊C4—24no—
245C22⋊C4—42no—
246C22⋊C4—42no—
247C22⋊C4—3322no—
248C22⋊C4—3322no—
249C22⋊C4—3322no—
250C22⋊C4—6642no—
251C42—21no—
252C42—3321no—
253C42—3322no—
254C42—1662no—
255C42Dic83⋊C412yescomplemented
256C23—22no—
257C23—22no—
258C23—42no—
259C23—42no—
260C23C22⋊C4×D8311yescharacteristic
261C23#1458176412yes—
262C23#1458176912yes—
263Q8—42no—
264Q8—42no—
265D4—41no—
266D4—41no—
267D4—42no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—3324no—
280C2×C4—3322no—
281C2×C4—3321no—
282C2×C4—3321no—
283C2×C4—3322no—
284C2×C4—3321no—
285C2×C4—3321no—
286C2×C4—3321no—
287C2×C4—3321no—
288C2×C4—832no—
289C2×C4—6642no—
290C2×C4—6642no—
291C2×C4—6642no—
292C2×C4—6642no—
293C2×C4—1661no—
294C2×C4—1662no—
295C2×C4C2×Dic83⋊C411yescharacteristic
296C2×C4#1458174811yescharacteristic
297C22—21no—
298C22—21no—
299C22—22no—
300C22—22no—
301C22—42no—
302C22—42no—
303C22—42no—
304C22—42no—
305C22—82no—
306C22#1912755911yescharacteristic
307C22#99×D8311yescharacteristic
308C22#1912767311yescharacteristic
309C4—21no—
310C4—21no—
311C4—42no—
312C4—42no—
313C4—162no—
314C4—13282no—
315C4—3322no—
316C4—3321no—
317C4—6641no—
318C4—6641no—
319C4—1662no—
320C2—21no—
321C2—21no—
322C2—42no—
323C2—42no—
324C2#2490744911yescharacteristic
325C2#2490913111yescharacteristic
326C2#2490913611yescharacteristic
subgroupG/H
centerC22#99×D83
commutatorC22×C4×C83C22×C4
FrattiniC22×D4C22×D83
FittingC2×#354×C83C2
socleC22×C83C2×#99
prime radical, cosocleC22×D4×C83C23
hypercenterC2×#354D83
hypocenterC83#12800
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×#354

Complements

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

NH ≅ G/Ntype
6. C2×Q8×Dic83C4semidirect
9. #19127608C4semidirect
13. #19127571C4semidirect
15. #19127547C4semidirect
76. C2×Q8×C83C4⋊C4semidirect
91. C42×C83C4⋊C4semidirect
169. C83#12800semidirect
219. C2×Q8Dic83⋊C4semidirect
255. C42Dic83⋊C4semidirect

Maximal subgroups

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

subgroupindex
#249101762
C2×#191274962
#249074492
#249055872
C2×#354×C832
#1280083