Finite Group Databasebeta

← #35320658

Subgroups — #35320658

Order 21248.

914 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 914 conjugacy classes of subgroups — too many to draw. Only its 115 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#24907378C212yesmaximal
2#24905588C214yesmaximal, complemented
3C2×#390×C83C211yesmaximal, characteristic
4C2×#14581707—22no—
5C2×#14581707C2214yes—
6#19127557—22no—
7C23×D4×C83C2211yescharacteristic
8C22×C4⋊C4×C83C411yescharacteristic, complemented
9C22×C22⋊C4×C83—22no—
10C22×C42×C83C411yescharacteristic, complemented
11#390×C83—22no—
12#390×C83C412yescomplemented
13#390×C83C2212yes—
14C2×C22⋊Dic83—24no—
15C2×C22⋊Dic83—42no—
16C2×C22⋊Dic83—42no—
17#14581707—28no—
18#14581707—44no—
19C25×C83D412yes—
20C22×D4×C83—24no—
21C22×D4×C83—42no—
22C22×D4×C83—41no—
23C22×D4×C83—41no—
24C22×D4×C83C2×C411yescharacteristic
25C22×D4×C83D412yes—
26C22×D4×C83C2311yescharacteristic
27C23×C4×C83—21no—
28C23×C4×C83—21no—
29C23×C4×C83C2×C411yescharacteristic
30C2×C4⋊C4×C83—21no—
31C2×C4⋊C4×C83—41no—
32C2×C4⋊C4×C83—41no—
33C2×C4⋊C4×C83C2×C412yes—
34C2×C22⋊C4×C83—24no—
35C2×C22⋊C4×C83—24no—
36C2×C22⋊C4×C83—48no—
37C2×C42×C83—21no—
38C2×C42×C83—42no—
39C2×C42×C83C2×C412yes—
40C22×Dic83—44no—
41C22⋊Dic83—44no—
42C22⋊Dic83—44no—
43C22⋊Dic83—84no—
44C22⋊Dic83—84no—
45C24×C83—24no—
46C24×C83—22no—
47C24×C83—22no—
48C24×C83—42no—
49C24×C83—44no—
50C24×C83—44no—
51C24×C83C22⋊C412yes—
52C24×C83C22⋊C411yescharacteristic
53C24×C83C2×D412yes—
54C2×D4×C83—24no—
55C2×D4×C83—24no—
56C2×D4×C83—22no—
57C2×D4×C83—41no—
58C2×D4×C83—41no—
59C2×D4×C83—42no—
60C2×D4×C83—41no—
61C2×D4×C83—41no—
62C2×D4×C83—48no—
63C2×D4×C83—48no—
64C2×D4×C83#4114yes—
65C22×C4×C83—21no—
66C22×C4×C83—21no—
67C22×C4×C83—22no—
68C22×C4×C83—22no—
69C22×C4×C83—24no—
70C22×C4×C83—22no—
71C22×C4×C83—22no—
72C22×C4×C83—42no—
73C22×C4×C83—42no—
74C22×C4×C83—44no—
75C22×C4×C83—41no—
76C22×C4×C83—41no—
77C22×C4×C83C22⋊C411yescharacteristic
78C22×C4×C83C22×C411yescharacteristic
79C4⋊C4×C83—42no—
80C4⋊C4×C83—42no—
81C22⋊C4×C83—48no—
82C22⋊C4×C83—48no—
83C42×C83—42no—
84C42×C83—42no—
85C23×C83—22no—
86C23×C83—22no—
87C23×C83—21no—
88C23×C83—21no—
89C23×C83—22no—
90C23×C83—22no—
91C23×C83—22no—
92C23×C83—22no—
93C23×C83—22no—
94C23×C83—44no—
95C23×C83—44no—
96C23×C83—42no—
97C23×C83—42no—
98C23×C83—42no—
99C23×C83—42no—
100C23×C83—416no—
101C23×C83—44no—
102C23×C83—44no—
103C23×C83—82no—
104C23×C83—82no—
105C23×C83—88no—
106C23×C83#9911yescharacteristic
107C23×C83#9911yescharacteristic
108C23×C83C2×C22⋊C411yescharacteristic
109C23×C83#12314yes—
110D4×C83—44no—
111D4×C83—48no—
112D4×C83—44no—
113C2×C4×C83—24no—
114C2×C4×C83—24no—
115C2×C4×C83—21no—
116C2×C4×C83—21no—
117C2×C4×C83—21no—
118C2×C4×C83—21no—
119C2×C4×C83—24no—
120C2×C4×C83—48no—
121C2×C4×C83—42no—
122C2×C4×C83—42no—
123C2×C4×C83—44no—
124C2×C4×C83—42no—
125C2×C4×C83—82no—
126C2×C4×C83—82no—
127C2×C4×C83#9912yes—
128C2×C4×C83#11712yes—
129C2×Dic83—84no—
130C2×Dic83—84no—
131C2×Dic83—84no—
132C22×C83—24no—
133C22×C83—24no—
134C22×C83—21no—
135C22×C83—21no—
136C22×C83—21no—
137C22×C83—21no—
138C22×C83—21no—
139C22×C83—21no—
140C22×C83—24no—
141C22×C83—44no—
142C22×C83—44no—
143C22×C83—42no—
144C22×C83—42no—
145C22×C83—42no—
146C22×C83—42no—
147C22×C83—42no—
148C22×C83—82no—
149C22×C83—88no—
150C22×C83—88no—
151C22×C83—82no—
152C22×C83#35312yes—
153C22×C83#35412yes—
154C22×C83#38111yescharacteristic
155C22×C83C2×#9911yescharacteristic
156C22×C83C2×#9911yescharacteristic
157C4×C83—22no—
158C4×C83—22no—
159C4×C83—44no—
160C4×C83—82no—
161Dic83—164no—
162#12856—831nomaximal, Sylow
163C2×C83—21no—
164C2×C83—21no—
165C2×C83—22no—
166C2×C83—22no—
167C2×C83—44no—
168C2×C83—82no—
169C2×C83#181911yescharacteristic
170C2×C83C2×#35311yescharacteristic
171C2×C83C2×#35411yescharacteristic
172C2×#390D8311yescharacteristic
173#1781—832no—
174#1393—834no—
175C83#1285611yescharacteristic, complemented, Sylow
176C23×D4C2×D8311yescharacteristic
177C22×C4⋊C4Dic8311yescharacteristic, complemented
178C22×C22⋊C4—22no—
179C22×C42Dic8311yescharacteristic, complemented
180C2×#99—834no—
181C2×#99—1662no—
182#390—22no—
183#390Dic8312yescomplemented
184#390C2×D8312yes—
185#379—1662no—
186C25C22⋊D8312yes—
187C22×D4—24no—
188C22×D4—42no—
189C22×D4—41no—
190C22×D4—41no—
191C22×D4C2×Dic8311yescharacteristic
192C22×D4C22⋊D8312yes—
193C22×D4C22×D8311yescharacteristic
194C23×C4—21no—
195C23×C4—21no—
196C23×C4C2×Dic8311yescharacteristic
197C2×C4⋊C4—21no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4C2×Dic8312yes—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—48no—
204C2×C22⋊C4—3322no—
205C2×C22⋊C4—3322no—
206C2×C22⋊C4—1664no—
207C2×C42—21no—
208C2×C42—42no—
209C2×C42C2×Dic8312yes—
210#99—3324no—
211#99—1668no—
212C24—24no—
213C24—22no—
214C24—22no—
215C24—42no—
216C24—44no—
217C24—44no—
218C24C22⋊Dic8312yes—
219C24C22⋊Dic8311yescharacteristic
220C24C2×C22⋊D8312yes—
221C2×D4—24no—
222C2×D4—24no—
223C2×D4—22no—
224C2×D4—41no—
225C2×D4—41no—
226C2×D4—42no—
227C2×D4—41no—
228C2×D4—41no—
229C2×D4—48no—
230C2×D4—48no—
231C2×D4#1305059014yes—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—44no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—3324no—
245C22×C4C22⋊Dic8311yescharacteristic
246C22×C4C22×Dic8311yescharacteristic
247C4⋊C4—42no—
248C4⋊C4—42no—
249C22⋊C4—48no—
250C22⋊C4—48no—
251C22⋊C4—3324no—
252C22⋊C4—3324no—
253C22⋊C4—6644no—
254C22⋊C4—6644no—
255C42—42no—
256C42—42no—
257C23—21no—
258C23—21no—
259C23—22no—
260C23—22no—
261C23—22no—
262C23—22no—
263C23—22no—
264C23—22no—
265C23—22no—
266C23—44no—
267C23—44no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—42no—
272C23—416no—
273C23—44no—
274C23—44no—
275C23—82no—
276C23—82no—
277C23—88no—
278C23#1458170711yescharacteristic
279C23#1458170711yescharacteristic
280C23C2×C22⋊Dic8311yescharacteristic
281C23#1458177514yes—
282D4—44no—
283D4—48no—
284D4—44no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—24no—
292C2×C4—48no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—44no—
296C2×C4—42no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—6644no—
300C2×C4—6644no—
301C2×C4—6644no—
302C2×C4#1458170712yes—
303C2×C4#1458175112yes—
304C22—24no—
305C22—24no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—24no—
313C22—44no—
314C22—44no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22—42no—
319C22—42no—
320C22—82no—
321C22—88no—
322C22—88no—
323C22—82no—
324C22#1912749412yes—
325C22#1912749612yes—
326C22#1912756411yescharacteristic
327C22C2×#1458170711yescharacteristic
328C22C2×#1458170711yescharacteristic
329C4—22no—
330C4—22no—
331C4—44no—
332C4—82no—
333C4—13284no—
334C2—21no—
335C2—21no—
336C2—22no—
337C2—22no—
338C2—44no—
339C2—82no—
340C2#2490753711yescharacteristic
341C2C2×#1912749411yescharacteristic
342C2C2×#1912749611yescharacteristic
subgroupG/H
centerC22C2×#14581707
commutatorC22×C4×C83C22×C4
FrattiniC22×D4C22×D83
FittingC2×#390×C83C2
socleC22×C83C2×#99
prime radicalC23×D4×C83C22
cosocleC22×D4×C83C23
hypercenterC2×#390D83
hypocenterC83#12856
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
2. #24905588C2semidirect
8. C22×C4⋊C4×C83C4semidirect
10. C22×C42×C83C4semidirect
12. #390×C83C4semidirect
175. C83#12856semidirect
177. C22×C4⋊C4Dic83semidirect
179. C22×C42Dic83semidirect
183. #390Dic83semidirect

Maximal subgroups

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

subgroupindex
#249073782
#249055882
C2×#390×C832
#1285683