Finite Group Databasebeta

← #35314741

Subgroups — #35314741

Order 21248.

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

normal subgroups only

This group has 470 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#24910747C211yesmaximal, characteristic
2#24905781C211yesmaximal, characteristic
3#24905483C212yesmaximal, complemented
4#24905462C212yesmaximal, complemented
5C4×#99×C83C211yesmaximal, characteristic
6C2×C4×Dic166C411yescharacteristic, complemented
7C2×C22⋊(C83⋊C8)—21no—
8#19127639—21no—
9#19127626C411yescharacteristic, complemented
10#19127626C2211yescharacteristic
11#19127615—21no—
12#19127613C411yescharacteristic, complemented
13#19127613C2211yescharacteristic
14C4×C22⋊Dic83—21no—
15C4×C22⋊Dic83—21no—
16#19127547C411yescharacteristic, complemented
17#19127427—21no—
18#19127425C2212yes—
19C2×C4×D4×C83C2211yescharacteristic
20C2×#99×C83—21no—
21C4×C22⋊C4×C83—21no—
22#342×C83C2212yes—
23C22×Dic166C2×C411yescharacteristic
24C22×C83⋊C8—41no—
25C4×Dic166—41no—
26C4×Dic166—41no—
27C2×C4⋊Dic83C2×C411yescharacteristic
28C2×Dic83⋊C4—21no—
29C2×C22⋊Dic83—21no—
30C2×C22⋊Dic83—21no—
31C2×C22⋊Dic83—21no—
32C2×C22⋊Dic83—21no—
33C2×C4×Dic83—21no—
34C2×C4×Dic83—21no—
35C22⋊(C83⋊C8)—22no—
36C22⋊(C83⋊C8)—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×C83D411yescharacteristic
46C23×C4×C83D411yescharacteristic
47C4×D4×C83—41no—
48C4×D4×C83—41no—
49C2×C4⋊C4×C83D411yescharacteristic
50C2×C22⋊C4×C83—22no—
51C2×C22⋊C4×C83C2×C411yescharacteristic
52C2×C22⋊C4×C83C2311yescharacteristic
53C2×C42×C83—41no—
54C2×C42×C83C2×C411yescharacteristic
55#99×C83—24no—
56#95×C83—41no—
57C2×Dic166—41no—
58C2×Dic166—42no—
59C22×Dic83—21no—
60C22×Dic83—21no—
61C22×Dic83—21no—
62C22×Dic83—21no—
63C2×C83⋊C8—42no—
64C2×C83⋊C8—82no—
65C4⋊Dic83—41no—
66Dic83⋊C4—42no—
67C22⋊Dic83—42no—
68C22⋊Dic83—42no—
69C22⋊Dic83—42no—
70C22⋊Dic83—42no—
71C4×Dic83—42no—
72C4×Dic83—41no—
73C4×Dic83—41no—
74C24×C83C22⋊C411yescharacteristic
75C24×C83C2×D411yescharacteristic
76C2×D4×C83—22no—
77C2×D4×C83—22no—
78C2×D4×C83—41no—
79C22×C4×C83—21no—
80C22×C4×C83—21no—
81C22×C4×C83—21no—
82C22×C4×C83—21no—
83C22×C4×C83—42no—
84C22×C4×C83—42no—
85C22×C4×C83—42no—
86C22×C4×C83C22⋊C411yescharacteristic
87C22×C4×C83C22⋊C411yescharacteristic
88C22×C4×C83C22⋊C411yescharacteristic
89C22×C4×C83C22×C411yescharacteristic
90C22×C4×C83C2×D411yescharacteristic
91C4⋊C4×C83—41no—
92C22⋊C4×C83—24no—
93C22⋊C4×C83—41no—
94C22⋊C4×C83—41no—
95C22⋊C4×C83—42no—
96C42×C83—21no—
97C42×C83—21no—
98C42×C83—82no—
99C23×C83—21no—
100C23×C83—21no—
101C23×C83—21no—
102C23×C83—21no—
103C23×C83—41no—
104C23×C83—41no—
105C23×C83—41no—
106C23×C83—41no—
107C23×C83#10412yes—
108C23×C83C2×C22⋊C411yescharacteristic
109C23×C83#13512yes—
110D4×C83—44no—
111C2×C4×C83—22no—
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—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—81no—
126C2×C4×C83—82no—
127C2×C4×C83—82no—
128C2×C4×C83#9911yescharacteristic
129C2×C4×C83#9911yescharacteristic
130C2×Dic83—22no—
131C2×Dic83—41no—
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—
142Dic166—42no—
143Dic166—42no—
144C83⋊C8—82no—
145C22×C83—21no—
146C22×C83—21no—
147C22×C83—22no—
148C22×C83—22no—
149C22×C83—42no—
150C22×C83—42no—
151C22×C83—41no—
152C22×C83—41no—
153C22×C83—41no—
154C22×C83—41no—
155C22×C83—81no—
156C22×C83—81no—
157C22×C83C2×#9911yescharacteristic
158C22×C83#41611yescharacteristic
159C22×C83C2×#10411yescharacteristic
160C4×C83—21no—
161C4×C83—41no—
162C4×C83—41no—
163C4×C83—41no—
164C4×C83—41no—
165C4×C83—84no—
166C4×C83#35412yes—
167Dic83—42no—
168Dic83—81no—
169Dic83—81no—
170Dic83—81no—
171#11728—831nomaximal, Sylow
172C2×C83—21no—
173C2×C83—21no—
174C2×C83—42no—
175C2×C83—42no—
176C2×C83#150111yescharacteristic
177C2×C83#212011yescharacteristic
178C2×C83C2×#35411yescharacteristic
179#2398—831no—
180C4×#99D8311yescharacteristic
181#1459—831no—
182#1351—832no—
183#1344—832no—
184C83#1172811yescharacteristic, complemented, Sylow
185C2×C4×Q8—831no—
186C2×C4×D4C2×D8311yescharacteristic
187C2×#99—21no—
188C2×C22⋊C8—1661no—
189#400—1661no—
190#397—831no—
191#397—831no—
192#394—1661no—
193#393—831no—
194#393—831no—
195C4×C22⋊C4—21no—
196C4×C22⋊C4—1661no—
197C4×C22⋊C4—1661no—
198#376—831no—
199#342C2×D8312yes—
200#324—1661no—
201#323—832no—
202C22×Q8—831no—
203C22×D4C22⋊D8311yescharacteristic
204C23×C4D33211yescharacteristic
205C23×C4C22⋊D8311yescharacteristic
206C22×C8—3321no—
207C4×Q8—3321no—
208C4×Q8—3321no—
209C4×D4—41no—
210C4×D4—41no—
211C2×C4⋊C4—831no—
212C2×C4⋊C4—1661no—
213C2×C4⋊C4D33211yescharacteristic
214C2×C22⋊C4—22no—
215C2×C22⋊C4—1661no—
216C2×C22⋊C4—1661no—
217C2×C22⋊C4—1661no—
218C2×C22⋊C4—1661no—
219C2×C22⋊C4C4×D8311yescharacteristic
220C2×C22⋊C4C22×D8311yescharacteristic
221C2×C42—41no—
222C2×C42—1661no—
223C2×C42—1661no—
224C2×C42C4×D8311yescharacteristic
225#99—24no—
226C22⋊C8—3321no—
227C22⋊C8—1662no—
228#95—41no—
229#95—831no—
230#95—831no—
231#95—1661no—
232#95—1661no—
233#95—1661no—
234#95—1661no—
235#95—1661no—
236C24(C2×D83)⋊C411yescharacteristic
237C24C2×C22⋊D8311yescharacteristic
238C2×Q8—3321no—
239C2×Q8—3322no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—41no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—1661no—
251C22×C4—1661no—
252C22×C4—1661no—
253C22×C4—1661no—
254C22×C4(C2×D83)⋊C411yescharacteristic
255C22×C4(C2×D83)⋊C411yescharacteristic
256C22×C4(C2×D83)⋊C411yescharacteristic
257C22×C4C2×C4×D8311yescharacteristic
258C22×C4C2×D33211yescharacteristic
259C2×C8—3322no—
260C2×C8—6642no—
261C4⋊C4—41no—
262C4⋊C4—3321no—
263C4⋊C4—3322no—
264C22⋊C4—24no—
265C22⋊C4—41no—
266C22⋊C4—41no—
267C22⋊C4—42no—
268C22⋊C4—3322no—
269C22⋊C4—3322no—
270C22⋊C4—3322no—
271C22⋊C4—3322no—
272C42—21no—
273C42—21no—
274C42—82no—
275C42—3322no—
276C42—3321no—
277C42—3321no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23#1458172012yes—
287C23C2×(C2×D83)⋊C411yescharacteristic
288C23#1458181212yes—
289Q8—3322no—
290Q8—3322no—
291D4—44no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—44no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—3321no—
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—1662no—
321C2×C4#1458170611yescharacteristic
322C2×C4#1458170611yescharacteristic
323C8—6642no—
324C22—21no—
325C22—21no—
326C22—22no—
327C22—22no—
328C22—42no—
329C22—42no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—81no—
335C22—81no—
336C22C2×#1458170611yescharacteristic
337C22#1912769511yescharacteristic
338C22C2×#1458172011yescharacteristic
339C4—21no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—84no—
345C4—3322no—
346C4—6641no—
347C4—6641no—
348C4—6641no—
349C4#1912749512yes—
350C2—21no—
351C2—21no—
352C2—42no—
353C2—42no—
354C2#2490595211yescharacteristic
355C2#2490910211yescharacteristic
356C2C2×#1912749511yescharacteristic
subgroupG/H
centerC2×C4#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#11728
LCSG ▷ C22×C4×C83 ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
3. #24905483C2semidirect
4. #24905462C2semidirect
6. C2×C4×Dic166C4semidirect
9. #19127626C4semidirect
12. #19127613C4semidirect
16. #19127547C4semidirect
184. C83#11728semidirect

Maximal subgroups

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

subgroupindex
#249107472
#249057812
#249054832
#249054622
C4×#99×C832
#1172883