Finite Group Databasebeta

← #35315014

Subgroups — #35315014

Order 21248.

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

normal subgroups only

This group has 590 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
1#24910828C211yesmaximal, characteristic
2C4×#14581707C211yesmaximal, characteristic
3#24905485C214yesmaximal, complemented
4C83×#1466C211yesmaximal, characteristic
5C23×Dic166C411yescharacteristic, complemented
6C2×#14581771—21no—
7C2×#14581707—21no—
8#19127621C412yescomplemented
9#19127613C2212yes—
10#19127607—21no—
11#19127581—21no—
12#19127573—21no—
13#19127561C411yescharacteristic, complemented
14C4×C22⋊Dic83—21no—
15#19127468C2212yes—
16C2×C4×D4×C83C2211yescharacteristic
17#432×C83—21no—
18#323×C83—21no—
19#323×C83C2212yes—
20C22×Dic166—21no—
21C22×Dic166—41no—
22C22×Dic166—41no—
23C22×Dic166—44no—
24C22×Dic166C2×C412yes—
25C23×Dic83—21no—
26#14581771—44no—
27C2×C4⋊Dic83—21no—
28C2×Dic83⋊C4—22no—
29C2×C22⋊Dic83—21no—
30C2×C22⋊Dic83—22no—
31C2×C22⋊Dic83—22no—
32C2×C4×Dic83—41no—
33#14581707—24no—
34C83⋊#95—22no—
35C83⋊#95—22no—
36C83⋊#95—41no—
37C83⋊#95C2×C412yes—
38C22×D4×C83D411yescharacteristic
39C23×C4×C83C2×C411yescharacteristic
40C23×C4×C83D411yescharacteristic
41C4×D4×C83—41no—
42C4×D4×C83—41no—
43C2×C4⋊C4×C83C2×C411yescharacteristic
44C2×C22⋊C4×C83D411yescharacteristic
45C2×C22⋊C4×C83C2311yescharacteristic
46C2×C42×C83D411yescharacteristic
47C4⋊C8×C83—41no—
48C22⋊C8×C83—22no—
49#97×C83—41no—
50C2×Dic166—24no—
51C2×Dic166—42no—
52C2×Dic166—48no—
53C2×Dic166—44no—
54C2×Dic166—41no—
55C2×Dic166—41no—
56C2×Dic166—42no—
57C2×Dic166—44no—
58C2×Dic166—44no—
59C22×Dic83—22no—
60C22×Dic83—22no—
61C22×Dic83—44no—
62C22×Dic83—42no—
63C22×Dic83—42no—
64C4⋊Dic83—42no—
65Dic83⋊C4—44no—
66C22⋊Dic83—24no—
67C22⋊Dic83—44no—
68C22⋊Dic83—42no—
69C22⋊Dic83—42no—
70C4×Dic83—82no—
71C24×C83C22⋊C411yescharacteristic
72C24×C83C2×D411yescharacteristic
73C2×D4×C83—22no—
74C2×D4×C83—22no—
75C2×D4×C83—41no—
76C22×C4×C83—21no—
77C22×C4×C83—21no—
78C22×C4×C83—21no—
79C22×C4×C83—21no—
80C22×C4×C83—42no—
81C22×C4×C83—42no—
82C22×C4×C83C22⋊C411yescharacteristic
83C22×C4×C83C22⋊C411yescharacteristic
84C22×C4×C83C22⋊C411yescharacteristic
85C22×C4×C83C22×C411yescharacteristic
86C22×C4×C83C2×D411yescharacteristic
87C2×C8×C83—42no—
88C4⋊C4×C83—41no—
89C22⋊C4×C83—41no—
90C22⋊C4×C83—41no—
91C42×C83—21no—
92C42×C83—21no—
93C23×C83—21no—
94C23×C83—21no—
95C23×C83—21no—
96C23×C83—21no—
97C23×C83—41no—
98C23×C83—41no—
99C23×C83—41no—
100C23×C83—41no—
101C23×C83#10412yes—
102C23×C83C2×C22⋊C411yescharacteristic
103C23×C83#13712yes—
104D4×C83—44no—
105C2×C4×C83—22no—
106C2×C4×C83—21no—
107C2×C4×C83—21no—
108C2×C4×C83—21no—
109C2×C4×C83—21no—
110C2×C4×C83—21no—
111C2×C4×C83—21no—
112C2×C4×C83—42no—
113C2×C4×C83—42no—
114C2×C4×C83—42no—
115C2×C4×C83—41no—
116C2×C4×C83—41no—
117C2×C4×C83—41no—
118C2×C4×C83—81no—
119C2×C4×C83#9911yescharacteristic
120C2×C4×C83#9911yescharacteristic
121C2×Dic83—24no—
122C2×Dic83—42no—
123C2×Dic83—48no—
124C2×Dic83—42no—
125C2×Dic83—44no—
126C2×Dic83—42no—
127C2×Dic83—42no—
128C2×Dic83—42no—
129C2×Dic83—82no—
130C2×Dic83—82no—
131Dic166—44no—
132Dic166—44no—
133Dic166—44no—
134Dic166—44no—
135C8×C83—82no—
136C22×C83—21no—
137C22×C83—21no—
138C22×C83—22no—
139C22×C83—22no—
140C22×C83—42no—
141C22×C83—42no—
142C22×C83—42no—
143C22×C83—41no—
144C22×C83—41no—
145C22×C83—41no—
146C22×C83—41no—
147C22×C83—81no—
148C22×C83C2×#9911yescharacteristic
149C22×C83#41911yescharacteristic
150C22×C83C2×#10411yescharacteristic
151C4×C83—21no—
152C4×C83—41no—
153C4×C83—41no—
154C4×C83—41no—
155C4×C83—41no—
156C4×C83AΣL(1,𝔽16)12yes—
157Dic83—44no—
158Dic83—82no—
159Dic83—84no—
160#11785—831nomaximal, Sylow
161C2×C83—21no—
162C2×C83—21no—
163C2×C83—42no—
164C2×C83—42no—
165C2×C83#150811yescharacteristic
166C2×C83C2×AΣL(1,𝔽16)11yescharacteristic
167C2×C83#212911yescharacteristic
168#2409—831no—
169C4×#99—831no—
170#1466D8311yescharacteristic
171#1351—834no—
172C83#1178511yescharacteristic, complemented, Sylow
173C23×Q8—831no—
174C2×C22⋊Q8—1661no—
175C2×C4×D4C2×D8311yescharacteristic
176#432—21no—
177C2×#99—1661no—
178#395—832no—
179#393—832no—
180#391—1661no—
181#386—1661no—
182#385—1661no—
183#380—831no—
184C4×C22⋊C4—1661no—
185#342—832no—
186#323—21no—
187#323C2×D8312yes—
188C22×Q8—3321no—
189C22×Q8—3321no—
190C22×Q8—3324no—
191C22×Q8—832no—
192C22×Q8—1661no—
193C22×D4D33211yescharacteristic
194C23×C4—1661no—
195C23×C4C4×D8311yescharacteristic
196C23×C4D33211yescharacteristic
197C22⋊Q8—3324no—
198C4×D4—41no—
199C4×D4—41no—
200C2×C4⋊C4—1662no—
201C2×C4⋊C4—1661no—
202C2×C4⋊C4C4×D8311yescharacteristic
203C2×C22⋊C4—1661no—
204C2×C22⋊C4—1662no—
205C2×C22⋊C4—1662no—
206C2×C22⋊C4C22⋊D8311yescharacteristic
207C2×C22⋊C4C22×D8311yescharacteristic
208C2×C42—3321no—
209C2×C42C22⋊D8311yescharacteristic
210C4⋊C8—41no—
211#99—1664no—
212C22⋊C8—22no—
213#97—41no—
214#95—3321no—
215#95—832no—
216#95—1662no—
217#95—1662no—
218C24(C2×D83)⋊C411yescharacteristic
219C24C2×D33211yescharacteristic
220C2×Q8—3322no—
221C2×Q8—3322no—
222C2×Q8—3324no—
223C2×Q8—3328no—
224C2×Q8—3324no—
225C2×Q8—3324no—
226C2×Q8—3321no—
227C2×Q8—3321no—
228C2×Q8—1664no—
229C2×D4—22no—
230C2×D4—22no—
231C2×D4—41no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—3322no—
239C22×C4—3324no—
240C22×C4—3322no—
241C22×C4—1662no—
242C22×C4—1662no—
243C22×C4(C2×D83)⋊C411yescharacteristic
244C22×C4(C2×D83)⋊C411yescharacteristic
245C22×C4(C2×D83)⋊C411yescharacteristic
246C22×C4C2×C4×D8311yescharacteristic
247C22×C4C2×C22⋊D8311yescharacteristic
248C2×C8—42no—
249C4⋊C4—41no—
250C4⋊C4—3322no—
251C4⋊C4—3324no—
252C22⋊C4—41no—
253C22⋊C4—41no—
254C22⋊C4—3322no—
255C22⋊C4—3322no—
256C22⋊C4—3324no—
257C22⋊C4—1664no—
258C42—21no—
259C42—21no—
260C42—6642no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23#1458171812yes—
270C23C2×(C2×D83)⋊C411yescharacteristic
271C23#1458182512yes—
272Q8—3324no—
273Q8—3324no—
274Q8—3324no—
275Q8—3324no—
276D4—44no—
277C2×C4—22no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—81no—
291C2×C4—3322no—
292C2×C4—3328no—
293C2×C4—3322no—
294C2×C4—3322no—
295C2×C4—3324no—
296C2×C4—3322no—
297C2×C4—3322no—
298C2×C4—6642no—
299C2×C4—6642no—
300C2×C4—1664no—
301C2×C4#1458170611yescharacteristic
302C2×C4#1458170611yescharacteristic
303C8—82no—
304C22—21no—
305C22—21no—
306C22—22no—
307C22—22no—
308C22—42no—
309C22—42no—
310C22—42no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—81no—
316C22C2×#1458170611yescharacteristic
317C22#1912771111yescharacteristic
318C22C2×#1458171811yescharacteristic
319C4—21no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—3324no—
325C4—6642no—
326C4—6644no—
327C4#1912748912yes—
328C2—21no—
329C2—21no—
330C2—42no—
331C2—42no—
332C2#2490598411yescharacteristic
333C2C2×#1912748911yescharacteristic
334C2#2490913711yescharacteristic
subgroupG/H
centerC22C2×#14581706
commutatorC22×C4×C83C22×C4
FrattiniC2×C22⋊C4C22×D83
FittingC83×#1466C2
socleC22×C83C2×#99
prime radicalC22×D4×C83D4
cosocleC2×C22⋊C4×C83C23
hypercenter#1466D83
hypocenterC83#11785
LCSG ▷ C22×C4×C83 ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1466

Complements

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

NH ≅ G/Ntype
3. #24905485C2semidirect
5. C23×Dic166C4semidirect
8. #19127621C4semidirect
13. #19127561C4semidirect
172. C83#11785semidirect

Maximal subgroups

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

subgroupindex
#249108282
C4×#145817072
#249054852
C83×#14662
#1178583