Finite Group Databasebeta

← #35322727

Subgroups — #35322727

Order 21248.

540 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 111 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×#19127498C214yesmaximal, complemented
2#24907459C211yesmaximal, characteristic, complemented
3C4×#14581711C211yesmaximal, characteristic, complemented
4C83×#2548C211yesmaximal, characteristic
5#19127725—21no—
6C2×#14581711C2212yes—
7C2×#14581711C2212yes—
8C4×#13050574—21no—
9#19127498—28no—
10#19127470—21no—
11#19127470—21no—
12C2×#127×C83C411yescharacteristic
13C2×#124×C83C2212yes—
14C2×C4×Q8×C83C2211yescharacteristic
15#394×C83—21no—
16#394×C83—21no—
17#390×C83—21no—
18#390×C83—21no—
19#390×C83C412yes—
20C4×C4⋊C4×C83C411yescharacteristic
21C2×#13050574—22no—
22C2×#13050574—22no—
23C4⋊(C83⋊C8)—41no—
24C4⋊(C83⋊C8)—41no—
25#14581711—24no—
26#14581711—22no—
27#14581711D414yescomplemented
28#14581703—41no—
29C4×C83⋊C8—41no—
30C22×Q8×C83C2311yescharacteristic
31C22×D4×C83—21no—
32C22×D4×C83—21no—
33C22×D4×C83C2×C412yes—
34#127×C83—41no—
35#127×C83—41no—
36#124×C83—24no—
37#124×C83—24no—
38C4×Q8×C83—41no—
39C4×Q8×C83—41no—
40C2×C4⋊C4×C83—21no—
41C2×C4⋊C4×C83C2×C411yescharacteristic
42C2×C4⋊C4×C83D411yescharacteristic
43C2×C4⋊C4×C83D411yescharacteristic
44C2×C22⋊C4×C83—22no—
45C2×C22⋊C4×C83—22no—
46C2×C22⋊C4×C83—22no—
47C2×C22⋊C4×C83—22no—
48C2×C42×C83C2×C412yes—
49C2×C42×C83C2×C411yescharacteristic
50C2×C42×C83D411yescharacteristic
51C2×C42×C83D411yescharacteristic
52#95×C83—21no—
53#13050574—24no—
54#13050574—42no—
55#13050574—42no—
56#13050574—42no—
57C2×C83⋊C8—42no—
58C2×C83⋊C8—42no—
59C24×C83—22no—
60C2×Q8×C83—41no—
61C2×Q8×C83—41no—
62C2×Q8×C83C2×D412yes—
63C2×Q8×C83#4112yes—
64C2×D4×C83—24no—
65C2×D4×C83—42no—
66C2×D4×C83—42no—
67C2×D4×C83—42no—
68C2×D4×C83—42no—
69C2×D4×C83—42no—
70C2×D4×C83—41no—
71C2×D4×C83—41no—
72C2×D4×C83—82no—
73C22×C4×C83—22no—
74C22×C4×C83C22⋊C411yescharacteristic
75C22×C4×C83C22⋊C411yescharacteristic
76C22×C4×C83C22⋊C411yescharacteristic
77C22×C4×C83C22⋊C411yescharacteristic
78C22×C4×C83C22×C411yescharacteristic
79C22×C4×C83C2×D411yescharacteristic
80C22×C4×C83C2×D411yescharacteristic
81C4⋊C4×C83—41no—
82C4⋊C4×C83—41no—
83C4⋊C4×C83—41no—
84C4⋊C4×C83—42no—
85C22⋊C4×C83—44no—
86C22⋊C4×C83—42no—
87C22⋊C4×C83—42no—
88C22⋊C4×C83—82no—
89C22⋊C4×C83—82no—
90C42×C83—21no—
91C42×C83—21no—
92C42×C83—21no—
93C42×C83—21no—
94C42×C83—21no—
95C42×C83—21no—
96C42×C83—24no—
97C23×C83—42no—
98C23×C83—42no—
99C23×C83—42no—
100C23×C83—42no—
101C23×C83—42no—
102C23×C83—42no—
103C23×C83—42no—
104C23×C83C2×C22⋊C411yescharacteristic
105Q8×C83—42no—
106Q8×C83—42no—
107D4×C83—84no—
108D4×C83—82no—
109D4×C83—84no—
110D4×C83—82no—
111C2×C4×C83—21no—
112C2×C4×C83—21no—
113C2×C4×C83—21no—
114C2×C4×C83—21no—
115C2×C4×C83—21no—
116C2×C4×C83—21no—
117C2×C4×C83—22no—
118C2×C4×C83—21no—
119C2×C4×C83—21no—
120C2×C4×C83—21no—
121C2×C4×C83—41no—
122C2×C4×C83—41no—
123C2×C4×C83—41no—
124C2×C4×C83—42no—
125C2×C4×C83—42no—
126C2×C4×C83—42no—
127C2×C4×C83#9911yescharacteristic
128C2×C4×C83#9911yescharacteristic
129C2×C4×C83C4×D412yes—
130C2×C4×C83#12112yes—
131C2×C4×C83#12411yescharacteristic
132C2×C4×C83#12711yescharacteristic
133C83⋊C8—44no—
134C83⋊C8—82no—
135C22×C83—21no—
136C22×C83—21no—
137C22×C83—82no—
138C22×C83—82no—
139C22×C83—82no—
140C22×C83—82no—
141C22×C83—82no—
142C22×C83—82no—
143C22×C83—82no—
144C22×C83#39011yescharacteristic
145C22×C83C2×#9911yescharacteristic
146C22×C83#41011yescharacteristic
147C4×C83—21no—
148C4×C83—22no—
149C4×C83—21no—
150C4×C83—22no—
151C4×C83—21no—
152C4×C83—41no—
153C4×C83—41no—
154C4×C83—41no—
155C4×C83—82no—
156C4×C83#35512yescomplemented
157#13226—831nomaximal, Sylow
158C2×C83—21no—
159C2×C83—21no—
160C2×C83—162no—
161C2×C83#196411yescharacteristic
162C2×C83C2×#35511yescharacteristic
163C2×C83#213411yescharacteristic
164#2548D8311yescharacteristic
165C2×#355—834no—
166#1801—831no—
167C4×#101—831no—
168C83#1322611yescharacteristic, complemented, Sylow
169C2×#127Dic8311yescharacteristic
170C2×#124C2×D8312yes—
171C2×C4×Q8C2×D8311yescharacteristic
172#423—1661no—
173C2×#101—832no—
174C2×#101—832no—
175C4×M16—1661no—
176#394—21no—
177#394—21no—
178#390—21no—
179#390—21no—
180#390Dic8312yes—
181C4×C4⋊C4Dic8311yescharacteristic
182#355—1668no—
183#343—1661no—
184#343—1661no—
185C22×Q8C22×D8311yescharacteristic
186C22×D4—21no—
187C22×D4—21no—
188C22×D4C2×Dic8312yes—
189C2×M16—1662no—
190C2×M16—1662no—
191#127—41no—
192#127—41no—
193#124—24no—
194#124—24no—
195C4×Q8—41no—
196C4×Q8—41no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4C2×Dic8311yescharacteristic
199C2×C4⋊C4C22⋊D8311yescharacteristic
200C2×C4⋊C4C22⋊D8311yescharacteristic
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C42C2×Dic8312yes—
206C2×C42C2×Dic8311yescharacteristic
207C2×C42C22⋊D8311yescharacteristic
208C2×C42C22⋊D8311yescharacteristic
209C4⋊C8—3321no—
210C4⋊C8—3321no—
211#101—834no—
212#101—1664no—
213#101—1662no—
214#97—3321no—
215C4×C8—3321no—
216#95—21no—
217C24—22no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8D4×D8312yes—
221C2×Q8#1305058912yes—
222C2×D4—24no—
223C2×D4—42no—
224C2×D4—42no—
225C2×D4—42no—
226C2×D4—42no—
227C2×D4—42no—
228C2×D4—41no—
229C2×D4—41no—
230C2×D4—82no—
231C22×C4—22no—
232C22×C4C22⋊Dic8311yescharacteristic
233C22×C4C22⋊Dic8311yescharacteristic
234C22×C4C22⋊Dic8311yescharacteristic
235C22×C4C22⋊Dic8311yescharacteristic
236C22×C4C22×Dic8311yescharacteristic
237C22×C4C2×C22⋊D8311yescharacteristic
238C22×C4C2×C22⋊D8311yescharacteristic
239M16—3322no—
240M16—3322no—
241M16—3322no—
242M16—1664no—
243C2×C8—3322no—
244C2×C8—3322no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—42no—
249C22⋊C4—44no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—82no—
253C22⋊C4—82no—
254C42—21no—
255C42—21no—
256C42—21no—
257C42—21no—
258C42—21no—
259C42—21no—
260C42—24no—
261C23—42no—
262C23—42no—
263C23—42no—
264C23—42no—
265C23—42no—
266C23—42no—
267C23—42no—
268C23C2×C22⋊Dic8311yescharacteristic
269Q8—42no—
270Q8—42no—
271D4—84no—
272D4—82no—
273D4—84no—
274D4—82no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4#1458170711yescharacteristic
292C2×C4#1458170711yescharacteristic
293C2×C4D4×Dic8312yes—
294C2×C4#1458176312yes—
295C2×C4#1458177911yescharacteristic
296C2×C4#1458178811yescharacteristic
297C8—3324no—
298C8—6642no—
299C22—21no—
300C22—21no—
301C22—82no—
302C22—82no—
303C22—82no—
304C22—82no—
305C22—82no—
306C22—82no—
307C22—82no—
308C22#1912760411yescharacteristic
309C22C2×#1458170711yescharacteristic
310C22#1912767511yescharacteristic
311C4—22no—
312C4—21no—
313C4—21no—
314C4—22no—
315C4—21no—
316C4—41no—
317C4—41no—
318C4—41no—
319C4—82no—
320C4#1912749812yescomplemented
321C2—21no—
322C2—21no—
323C2—162no—
324C2#2490830011yescharacteristic
325C2C2×#1912749811yescharacteristic
326C2#2490915511yescharacteristic
subgroupG/H
centerC22C2×#14581707
commutatorC22×C4×C83C22×C4
FrattiniC22×Q8C22×D83
FittingC83×#2548C2
socleC22×C83C2×#99
prime radicalC2×#127×C83C4
cosocleC22×Q8×C83C23
hypercenter#2548D83
hypocenterC83#13226
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2548

Complements

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

NH ≅ G/Ntype
1. C2×#19127498C2semidirect
2. #24907459C2semidirect
3. C4×#14581711C2semidirect
27. #14581711D4semidirect
156. C4×C83#355semidirect
168. C83#13226semidirect
320. C4#19127498semidirect

Maximal subgroups

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

subgroupindex
C2×#191274982
#249074592
C4×#145817112
C83×#25482
#1322683