Finite Group Databasebeta

← #35323065

Subgroups — #35323065

Order 21248.

616 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 107 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
2#24907455C211yesmaximal, characteristic
3#24905580C214yesmaximal
4C4×#99×C83C211yesmaximal, characteristic
5C23×Dic166C411yescharacteristic, complemented
6C2×#14581771—21no—
7#19127725—21no—
8C2×#14581709C2212yes—
9#19127621—21no—
10#19127613—21no—
11#19127607—21no—
12#19127581C2212yes—
13#19127573C412yescomplemented
14#19127561C411yescharacteristic, complemented
15#19127470—21no—
16C2×C4×D4×C83C2211yescharacteristic
17C2×#99×C83C2212yes—
18C4×C22⋊C4×C83—21no—
19#342×C83—21no—
20C22×Dic166—21no—
21C22×Dic166—21no—
22C22×Dic166—41no—
23C22×Dic166—41no—
24C22×Dic166—44no—
25C23×Dic83C2×C412yes—
26C2×#13050574—22no—
27#14581771—44no—
28C2×C4⋊Dic83—21no—
29C2×Dic83⋊C4—22no—
30C2×C22⋊Dic83—22no—
31C2×C22⋊Dic83C2×C412yes—
32C4⋊(C83⋊C8)—41no—
33C4⋊(C83⋊C8)—41no—
34#14581709—24no—
35C83⋊#95—21no—
36C83⋊#95—22no—
37C83⋊#95—22no—
38C22×D4×C83C2311yescharacteristic
39C23×C4×C83C2×C411yescharacteristic
40C23×C4×C83D411yescharacteristic
41C4×D4×C83—41no—
42C4×D4×C83—41no—
43C2×C4⋊C4×C83D411yescharacteristic
44C2×C22⋊C4×C83—22no—
45C2×C22⋊C4×C83C2×C411yescharacteristic
46C2×C22⋊C4×C83D411yescharacteristic
47C2×C42×C83—41no—
48C2×C42×C83D411yescharacteristic
49#99×C83—22no—
50#99×C83Q814yescomplemented
51#95×C83—41no—
52C2×Dic166—24no—
53C2×Dic166—44no—
54C2×Dic166—42no—
55C2×Dic166—42no—
56C2×Dic166—48no—
57C2×Dic166—44no—
58C2×Dic166—41no—
59C2×Dic166—41no—
60C2×Dic166—44no—
61C22×Dic83—22no—
62C22×Dic83—22no—
63C22×Dic83—44no—
64C22×Dic83—42no—
65#13050574—42no—
66#13050574—42no—
67C2×C83⋊C8—42no—
68C4⋊Dic83—42no—
69Dic83⋊C4—44no—
70C22⋊Dic83—44no—
71C22⋊Dic83—42no—
72C24×C83C22×C411yescharacteristic
73C24×C83C2×D411yescharacteristic
74C2×D4×C83—41no—
75C2×D4×C83—41no—
76C2×D4×C83C2×Q812yes—
77C2×D4×C83#4112yes—
78C22×C4×C83—21no—
79C22×C4×C83—21no—
80C22×C4×C83—21no—
81C22×C4×C83—21no—
82C22×C4×C83—42no—
83C22×C4×C83—42no—
84C22×C4×C83—42no—
85C22×C4×C83C22⋊C411yescharacteristic
86C22×C4×C83C22⋊C411yescharacteristic
87C22×C4×C83C22⋊C411yescharacteristic
88C22×C4×C83C22⋊C411yescharacteristic
89C22×C4×C83C2×D411yescharacteristic
90C4⋊C4×C83—41no—
91C22⋊C4×C83—24no—
92C22⋊C4×C83—41no—
93C22⋊C4×C83—41no—
94C22⋊C4×C83—42no—
95C42×C83—21no—
96C42×C83—21no—
97C42×C83—82no—
98C23×C83—21no—
99C23×C83—21no—
100C23×C83—21no—
101C23×C83—21no—
102C23×C83—41no—
103C23×C83—41no—
104C23×C83—41no—
105C23×C83—41no—
106C23×C83C2×C22⋊C411yescharacteristic
107C23×C83C4×Q812yes—
108C23×C83C22⋊Q812yes—
109D4×C83—42no—
110D4×C83—42no—
111C2×C4×C83—21no—
112C2×C4×C83—21no—
113C2×C4×C83—22no—
114C2×C4×C83—21no—
115C2×C4×C83—21no—
116C2×C4×C83—21no—
117C2×C4×C83—42no—
118C2×C4×C83—42no—
119C2×C4×C83—42no—
120C2×C4×C83—41no—
121C2×C4×C83—41no—
122C2×C4×C83—41no—
123C2×C4×C83—44no—
124C2×C4×C83—81no—
125C2×C4×C83—82no—
126C2×C4×C83—82no—
127C2×C4×C83#9911yescharacteristic
128C2×C4×C83#9911yescharacteristic
129C2×C4×C83#12411yescharacteristic
130C2×C4×C83C4⋊Q811yescharacteristic
131C2×Dic83—24no—
132C2×Dic83—42no—
133C2×Dic83—44no—
134C2×Dic83—42no—
135C2×Dic83—44no—
136C2×Dic83—42no—
137C2×Dic83—42no—
138C2×Dic83—42no—
139Dic166—44no—
140Dic166—48no—
141Dic166—44no—
142C83⋊C8—82no—
143C22×C83—21no—
144C22×C83—21no—
145C22×C83—22no—
146C22×C83—22no—
147C22×C83—42no—
148C22×C83—42no—
149C22×C83—42no—
150C22×C83—41no—
151C22×C83—41no—
152C22×C83—41no—
153C22×C83—41no—
154C22×C83—81no—
155C22×C83#39111yescharacteristic
156C22×C83C2×#9911yescharacteristic
157C22×C83#41011yescharacteristic
158C4×C83—21no—
159C4×C83—21no—
160C4×C83—21no—
161C4×C83—41no—
162C4×C83—41no—
163C4×C83—41no—
164C4×C83—84no—
165C4×C83AΣL(1,𝔽16)12yes—
166Dic83—44no—
167Dic83—82no—
168#13296—831nomaximal, Sylow
169C2×C83—21no—
170C2×C83—21no—
171C2×C83—42no—
172C2×C83—42no—
173C2×C83#199611yescharacteristic
174C2×C83C2×AΣL(1,𝔽16)11yescharacteristic
175C2×C83#212311yescharacteristic
176#2409—831no—
177#1800—831no—
178C4×#99D8311yescharacteristic
179#1390—834no—
180C83#1329611yescharacteristic, complemented, Sylow
181C23×Q8—831no—
182C2×C22⋊Q8—1661no—
183C2×C4×D4C2×D8311yescharacteristic
184#423—1661no—
185C2×#100—832no—
186C2×#99C2×D8312yes—
187#395—1661no—
188#393—1661no—
189#391—1661no—
190#386—832no—
191#385—832no—
192#380—831no—
193C4×C22⋊C4—21no—
194#343—1661no—
195#342—21no—
196C22×Q8—3321no—
197C22×Q8—3321no—
198C22×Q8—3324no—
199C22×Q8—1661no—
200C22×Q8—1661no—
201C22×D4C22×D8311yescharacteristic
202C23×C4—832no—
203C23×C4C4×D8311yescharacteristic
204C23×C4C22⋊D8311yescharacteristic
205C2×M16—1662no—
206C22⋊Q8—3324no—
207C4×D4—41no—
208C4×D4—41no—
209C2×C4⋊C4—1662no—
210C2×C4⋊C4—1661no—
211C2×C4⋊C4D33211yescharacteristic
212C2×C22⋊C4—22no—
213C2×C22⋊C4—832no—
214C2×C22⋊C4—1662no—
215C2×C22⋊C4C4×D8311yescharacteristic
216C2×C22⋊C4C22⋊D8311yescharacteristic
217C2×C42—41no—
218C2×C42D33211yescharacteristic
219C4⋊C8—3321no—
220C4⋊C8—3321no—
221#100—1664no—
222#99—22no—
223#99Dic16614yescomplemented
224#95—41no—
225#95—1661no—
226#95—1662no—
227#95—1662no—
228C24C2×C4×D8311yescharacteristic
229C24C2×C22⋊D8311yescharacteristic
230C2×Q8—3322no—
231C2×Q8—3322no—
232C2×Q8—3324no—
233C2×Q8—3328no—
234C2×Q8—3324no—
235C2×Q8—3324no—
236C2×Q8—3321no—
237C2×Q8—3321no—
238C2×Q8—1664no—
239C2×D4—41no—
240C2×D4—41no—
241C2×D4C2×Dic16612yes—
242C2×D4#1305059012yes—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—3322no—
251C22×C4—3324no—
252C22×C4—1662no—
253C22×C4—1662no—
254C22×C4(C2×D83)⋊C411yescharacteristic
255C22×C4(C2×D83)⋊C411yescharacteristic
256C22×C4(C2×D83)⋊C411yescharacteristic
257C22×C4(C2×D83)⋊C411yescharacteristic
258C22×C4C2×D33211yescharacteristic
259M16—3322no—
260M16—3322no—
261C2×C8—3322no—
262C4⋊C4—41no—
263C4⋊C4—3322no—
264C4⋊C4—3324no—
265C22⋊C4—24no—
266C22⋊C4—41no—
267C22⋊C4—41no—
268C22⋊C4—42no—
269C22⋊C4—3324no—
270C22⋊C4—3322no—
271C42—21no—
272C42—21no—
273C42—82no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23C2×(C2×D83)⋊C411yescharacteristic
283C23C4×Dic16612yes—
284C23#1458177112yes—
285Q8—3324no—
286Q8—3328no—
287Q8—3324no—
288D4—42no—
289D4—42no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—44no—
303C2×C4—81no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—3322no—
307C2×C4—3324no—
308C2×C4—3322no—
309C2×C4—3324no—
310C2×C4—3322no—
311C2×C4—3322no—
312C2×C4—3322no—
313C2×C4—1664no—
314C2×C4#1458170611yescharacteristic
315C2×C4#1458170611yescharacteristic
316C2×C4#1458178011yescharacteristic
317C2×C4#1458179211yescharacteristic
318C8—6642no—
319C22—21no—
320C22—21no—
321C22—22no—
322C22—22no—
323C22—42no—
324C22—42no—
325C22—42no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—81no—
331C22#1912760711yescharacteristic
332C22C2×#1458170611yescharacteristic
333C22#1912767411yescharacteristic
334C4—21no—
335C4—21no—
336C4—21no—
337C4—41no—
338C4—41no—
339C4—41no—
340C4—84no—
341C4—3324no—
342C4—6642no—
343C4#1912748712yes—
344C2—21no—
345C2—21no—
346C2—42no—
347C2—42no—
348C2#2490844811yescharacteristic
349C2C2×#1912748711yescharacteristic
350C2#2490911611yescharacteristic
subgroupG/H
centerC22C2×#14581706
commutatorC24×C83C22×C4
FrattiniC22×D4C22×D83
FittingC4×#99×C83C2
socleC22×C83C2×#99
prime radical, cosocleC22×D4×C83C23
hypercenterC4×#99D83
hypocenterC83#13296
LCSG ▷ C24×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
5. C23×Dic166C4semidirect
13. #19127573C4semidirect
14. #19127561C4semidirect
50. #99×C83Q8semidirect
180. C83#13296semidirect
223. #99Dic166semidirect

Maximal subgroups

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

subgroupindex
#249108282
#249074552
#249055802
C4×#99×C832
#1329683