Finite Group Databasebeta

← #35315028

Subgroups — #35315028

Order 21248.

462 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 83 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#24912639C211yesmaximal, characteristic
2#24905804C211yesmaximal, characteristic
3#24905486C212yesmaximal, complemented
4#24905483C212yesmaximal, complemented
5C4×#99×C83C211yesmaximal, characteristic
6C2×#14581771C411yescharacteristic, complemented
7#19127758—21no—
8#19127640C411yescharacteristic, complemented
9#19127637C2211yescharacteristic
10#19127621C411yescharacteristic, complemented
11#19127613C2211yescharacteristic
12#19127585—21no—
13#19127581—21no—
14#19127574—21no—
15#19127559—21no—
16C2×C83⋊#95C411yescharacteristic, complemented
17#19127425—21no—
18#19127425C2212yes—
19C2×C4×D4×C83C2211yescharacteristic
20C2×#99×C83—21no—
21C4×C22⋊C4×C83—21no—
22#342×C83C2212yes—
23C22×Dic166C2×C411yescharacteristic
24C23×Dic83—21no—
25#14581771—41no—
26#14581771—41no—
27C2×C4⋊Dic83C2×C411yescharacteristic
28C2×Dic83⋊C4—21no—
29C2×Dic83⋊C4—21no—
30C2×Dic83⋊C4—21no—
31C2×C22⋊Dic83—21no—
32C2×C22⋊Dic83—21no—
33C2×C22⋊Dic83—21no—
34C4⋊(C83⋊C8)—41no—
35C22⋊(C83⋊C8)—22no—
36#14581703—41no—
37C83⋊#95—21no—
38C83⋊#95—21no—
39C83⋊#95—21no—
40C83⋊#95—21no—
41C83⋊#95—21no—
42C83⋊#95C2×C411yescharacteristic
43C83⋊#95C2×C411yescharacteristic
44C22×D4×C83D411yescharacteristic
45C23×C4×C83C2×C411yescharacteristic
46C23×C4×C83D411yescharacteristic
47C4×D4×C83—41no—
48C4×D4×C83—41no—
49C2×C4⋊C4×C83C2×C411yescharacteristic
50C2×C22⋊C4×C83—22no—
51C2×C22⋊C4×C83D411yescharacteristic
52C2×C22⋊C4×C83C2311yescharacteristic
53C2×C42×C83—41no—
54C2×C42×C83D411yescharacteristic
55#99×C83—24no—
56#95×C83—41no—
57C2×Dic166—41no—
58C2×Dic166—81no—
59C22×Dic83—21no—
60C22×Dic83—21no—
61C22×Dic83—21no—
62C22×Dic83—21no—
63C22×Dic83—42no—
64C22×Dic83—42no—
65C22×Dic83—81no—
66C2×C83⋊C8—42no—
67C4⋊Dic83—41no—
68Dic83⋊C4—42no—
69Dic83⋊C4—41no—
70Dic83⋊C4—41no—
71Dic83⋊C4—81no—
72C22⋊Dic83—42no—
73C22⋊Dic83—42no—
74C22⋊Dic83—81no—
75C24×C83C22⋊C411yescharacteristic
76C24×C83C2×D411yescharacteristic
77C2×D4×C83—22no—
78C2×D4×C83—22no—
79C2×D4×C83—41no—
80C22×C4×C83—21no—
81C22×C4×C83—21no—
82C22×C4×C83—21no—
83C22×C4×C83—21no—
84C22×C4×C83—42no—
85C22×C4×C83—81no—
86C22×C4×C83—81no—
87C22×C4×C83C22⋊C411yescharacteristic
88C22×C4×C83C22⋊C411yescharacteristic
89C22×C4×C83C22⋊C411yescharacteristic
90C22×C4×C83C22×C411yescharacteristic
91C22×C4×C83C2×D411yescharacteristic
92C4⋊C4×C83—41no—
93C22⋊C4×C83—24no—
94C22⋊C4×C83—41no—
95C22⋊C4×C83—41no—
96C22⋊C4×C83—42no—
97C42×C83—21no—
98C42×C83—21no—
99C42×C83—82no—
100C23×C83—21no—
101C23×C83—21no—
102C23×C83—21no—
103C23×C83—21no—
104C23×C83—41no—
105C23×C83—41no—
106C23×C83—41no—
107C23×C83—41no—
108C23×C83#10412yes—
109C23×C83C2×C22⋊C411yescharacteristic
110C23×C83#13712yes—
111D4×C83—44no—
112C2×C4×C83—21no—
113C2×C4×C83—21no—
114C2×C4×C83—21no—
115C2×C4×C83—21no—
116C2×C4×C83—21no—
117C2×C4×C83—21no—
118C2×C4×C83—41no—
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—81no—
126C2×C4×C83—81no—
127C2×C4×C83—82no—
128C2×C4×C83—82no—
129C2×C4×C83#9911yescharacteristic
130C2×C4×C83#9911yescharacteristic
131C2×Dic83—22no—
132C2×Dic83—41no—
133C2×Dic83—41no—
134C2×Dic83—41no—
135C2×Dic83—41no—
136C2×Dic83—41no—
137C2×Dic83—41no—
138C2×Dic83—41no—
139C2×Dic83—41no—
140C2×Dic83—41no—
141C2×Dic83—41no—
142C2×Dic83—41no—
143C2×Dic83—84no—
144Dic166—82no—
145C83⋊C8—82no—
146C22×C83—21no—
147C22×C83—21no—
148C22×C83—22no—
149C22×C83—22no—
150C22×C83—42no—
151C22×C83—42no—
152C22×C83—42no—
153C22×C83—41no—
154C22×C83—41no—
155C22×C83—41no—
156C22×C83—41no—
157C22×C83—81no—
158C22×C83C2×#9911yescharacteristic
159C22×C83#41911yescharacteristic
160C22×C83C2×#10411yescharacteristic
161C4×C83—21no—
162C4×C83—21no—
163C4×C83—41no—
164C4×C83—41no—
165C4×C83—41no—
166C4×C83—41no—
167C4×C83—84no—
168Dic83—42no—
169Dic83—81no—
170Dic83—81no—
171Dic83—81no—
172#11789—831nomaximal, Sylow
173C2×C83—21no—
174C2×C83—21no—
175C2×C83—42no—
176C2×C83—42no—
177C2×C83#150811yescharacteristic
178C2×C83#212111yescharacteristic
179C2×C83#212611yescharacteristic
180#2564—831no—
181C4×#99D8311yescharacteristic
182#1466—831no—
183#1352—832no—
184#1351—832no—
185C83#1178911yescharacteristic, complemented, Sylow
186C2×C22⋊Q8—831no—
187C2×C4×D4C2×D8311yescharacteristic
188#432—1661no—
189C2×#99—21no—
190#400—831no—
191#399—831no—
192#395—831no—
193#393—831no—
194#387—1661no—
195#386—1661no—
196#385—1661no—
197#380—1661no—
198C4×C22⋊C4—21no—
199C2×#95—831no—
200#342C2×D8312yes—
201#323—832no—
202#323—1661no—
203C22×Q8—831no—
204C22×D4C22⋊D8311yescharacteristic
205C23×C4—1661no—
206C23×C4C4×D8311yescharacteristic
207C23×C4C22⋊D8311yescharacteristic
208C22⋊Q8—3321no—
209C22⋊Q8—3321no—
210C4×D4—41no—
211C4×D4—41no—
212C2×C4⋊C4—831no—
213C2×C4⋊C4—1661no—
214C2×C4⋊C4—1661no—
215C2×C4⋊C4—1661no—
216C2×C4⋊C4C4×D8311yescharacteristic
217C2×C22⋊C4—22no—
218C2×C22⋊C4—1661no—
219C2×C22⋊C4—1661no—
220C2×C22⋊C4—1661no—
221C2×C22⋊C4D33211yescharacteristic
222C2×C22⋊C4C22×D8311yescharacteristic
223C2×C42—41no—
224C2×C42D33211yescharacteristic
225C4⋊C8—3321no—
226#99—24no—
227C22⋊C8—1662no—
228#97—3321no—
229#95—41no—
230#95—831no—
231#95—831no—
232#95—1661no—
233#95—1661no—
234#95—1661no—
235#95—1661no—
236#95—1661no—
237C24(C2×D83)⋊C411yescharacteristic
238C24C2×C22⋊D8311yescharacteristic
239C2×Q8—3321no—
240C2×Q8—6641no—
241C2×D4—22no—
242C2×D4—22no—
243C2×D4—41no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—42no—
249C22×C4—81no—
250C22×C4—81no—
251C22×C4—3322no—
252C22×C4—3322no—
253C22×C4—6641no—
254C22×C4—1661no—
255C22×C4—1661no—
256C22×C4—1661no—
257C22×C4—1661no—
258C22×C4(C2×D83)⋊C411yescharacteristic
259C22×C4(C2×D83)⋊C411yescharacteristic
260C22×C4(C2×D83)⋊C411yescharacteristic
261C22×C4C2×C4×D8311yescharacteristic
262C22×C4C2×D33211yescharacteristic
263C2×C8—3322no—
264C4⋊C4—41no—
265C4⋊C4—3321no—
266C4⋊C4—3322no—
267C4⋊C4—3321no—
268C4⋊C4—3321no—
269C4⋊C4—6641no—
270C22⋊C4—24no—
271C22⋊C4—41no—
272C22⋊C4—41no—
273C22⋊C4—42no—
274C22⋊C4—3322no—
275C22⋊C4—3322no—
276C22⋊C4—6641no—
277C42—21no—
278C42—21no—
279C42—82no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23#1458172012yes—
289C23C2×(C2×D83)⋊C411yescharacteristic
290C23#1458182212yes—
291Q8—6642no—
292D4—44no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—44no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4—3321no—
311C2×C4—3321no—
312C2×C4—3321no—
313C2×C4—3321no—
314C2×C4—3321no—
315C2×C4—3321no—
316C2×C4—3321no—
317C2×C4—3321no—
318C2×C4—3321no—
319C2×C4—3321no—
320C2×C4—3321no—
321C2×C4—6644no—
322C2×C4—1662no—
323C2×C4#1458170611yescharacteristic
324C2×C4#1458170611yescharacteristic
325C8—6642no—
326C22—21no—
327C22—21no—
328C22—22no—
329C22—22no—
330C22—42no—
331C22—42no—
332C22—42no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—81no—
338C22C2×#1458170611yescharacteristic
339C22#1912770911yescharacteristic
340C22C2×#1458172011yescharacteristic
341C4—21no—
342C4—21no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—84no—
348C4—3322no—
349C4—6641no—
350C4—6641no—
351C4—6641no—
352C2—21no—
353C2—21no—
354C2—42no—
355C2—42no—
356C2#2490598211yescharacteristic
357C2#2490910711yescharacteristic
358C2#2490912611yescharacteristic
subgroupG/H
centerC22C2×#14581706
commutatorC22×C4×C83C22×C4
FrattiniC2×C22⋊C4C22×D83
FittingC4×#99×C83C2
socleC22×C83C2×#99
prime radicalC22×D4×C83D4
cosocleC2×C22⋊C4×C83C23
hypercenterC4×#99D83
hypocenterC83#11789
LCSG ▷ C22×C4×C83 ▷ C23×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
3. #24905486C2semidirect
4. #24905483C2semidirect
6. C2×#14581771C4semidirect
8. #19127640C4semidirect
10. #19127621C4semidirect
16. C2×C83⋊#95C4semidirect
185. C83#11789semidirect

Maximal subgroups

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

subgroupindex
#249126392
#249058042
#249054862
#249054832
C4×#99×C832
#1178983