Finite Group Databasebeta

← #35314933

Subgroups — #35314933

Order 21248.

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

normal subgroups only

This group has 462 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#24912843C211yesmaximal, characteristic
2#24905793C211yesmaximal, characteristic
3#24905492C212yesmaximal, complemented
4#24905468C212yesmaximal, complemented
5C2×#343×C83C211yesmaximal, characteristic
6C2×#14581771C411yescharacteristic
7#19127665—21no—
8#19127631C411yescharacteristic
9#19127631C2211yescharacteristic
10#19127621C411yescharacteristic
11#19127621C2211yescharacteristic
12#19127596—21no—
13#19127596—21no—
14#19127574—21no—
15#19127574—21no—
16C4×C22⋊Dic83C411yescharacteristic
17#19127427—21no—
18#19127427C2212yes—
19C22×M16×C83—21no—
20C22×C4⋊C4×C83C2211yescharacteristic
21#343×C83—21no—
22#343×C83C2212yes—
23C22×Dic166C2×C411yescharacteristic
24C2×#13050574—41no—
25#14581771—41no—
26#14581771—41no—
27C2×C4⋊Dic83C2×C411yescharacteristic
28C2×Dic83⋊C4—21no—
29C2×Dic83⋊C4—21no—
30C2×Dic83⋊C4—21no—
31C2×Dic83⋊C4—21no—
32C2×Dic83⋊C4—21no—
33C2×C22⋊Dic83—21no—
34C2×C22⋊Dic83—21no—
35C2×C4×Dic83—21no—
36C22⋊(C83⋊C8)—22no—
37C22⋊(C83⋊C8)—41no—
38C83⋊#95—21no—
39C83⋊#95—21no—
40C83⋊#95—21no—
41C83⋊#95—21no—
42C83⋊#95C2×C411yescharacteristic
43C83⋊#95C2×C411yescharacteristic
44C23×C4×C83C2×C411yescharacteristic
45C23×C4×C83D411yescharacteristic
46C23×C4×C83D411yescharacteristic
47C2×M16×C83—24no—
48C2×M16×C83—22no—
49C2×M16×C83—41no—
50C2×M16×C83—42no—
51C22×C8×C83—41no—
52C2×C4⋊C4×C83—41no—
53C2×C4⋊C4×C83—41no—
54C2×C4⋊C4×C83C2×C411yescharacteristic
55C2×C4⋊C4×C83D411yescharacteristic
56C2×C4⋊C4×C83D411yescharacteristic
57C2×C4⋊C4×C83C2311yescharacteristic
58C2×Dic166—41no—
59C2×Dic166—81no—
60C22×Dic83—21no—
61C22×Dic83—21no—
62C22×Dic83—21no—
63C22×Dic83—21no—
64#13050574—82no—
65C2×C83⋊C8—42no—
66C4⋊Dic83—41no—
67Dic83⋊C4—42no—
68Dic83⋊C4—42no—
69Dic83⋊C4—41no—
70Dic83⋊C4—41no—
71Dic83⋊C4—81no—
72Dic83⋊C4—81no—
73C22⋊Dic83—24no—
74C22⋊Dic83—41no—
75C22⋊Dic83—41no—
76C4×Dic83—81no—
77C24×C83C22⋊C411yescharacteristic
78C22×C4×C83—21no—
79C22×C4×C83—21no—
80C22×C4×C83—22no—
81C22×C4×C83—21no—
82C22×C4×C83—21no—
83C22×C4×C83—21no—
84C22×C4×C83—21no—
85C22×C4×C83—21no—
86C22×C4×C83—21no—
87C22×C4×C83—81no—
88C22×C4×C83—81no—
89C22×C4×C83C22⋊C411yescharacteristic
90C22×C4×C83C22⋊C411yescharacteristic
91C22×C4×C83C22⋊C411yescharacteristic
92C22×C4×C83C22×C411yescharacteristic
93C22×C4×C83C2×D411yescharacteristic
94C22×C4×C83C2×D411yescharacteristic
95M16×C83—24no—
96M16×C83—44no—
97M16×C83—42no—
98C2×C8×C83—44no—
99C2×C8×C83—42no—
100C4⋊C4×C83—41no—
101C4⋊C4×C83—41no—
102C4⋊C4×C83—41no—
103C4⋊C4×C83—41no—
104C23×C83—22no—
105C23×C83—21no—
106C23×C83—21no—
107C23×C83—41no—
108C23×C83#9911yescharacteristic
109C23×C83#9911yescharacteristic
110C23×C83C2×C22⋊C411yescharacteristic
111C2×C4×C83—22no—
112C2×C4×C83—22no—
113C2×C4×C83—22no—
114C2×C4×C83—22no—
115C2×C4×C83—21no—
116C2×C4×C83—21no—
117C2×C4×C83—21no—
118C2×C4×C83—21no—
119C2×C4×C83—21no—
120C2×C4×C83—21no—
121C2×C4×C83—21no—
122C2×C4×C83—21no—
123C2×C4×C83—42no—
124C2×C4×C83—41no—
125C2×C4×C83—41no—
126C2×C4×C83—41no—
127C2×C4×C83—41no—
128C2×C4×C83—81no—
129C2×C4×C83—81no—
130C2×C4×C83—81no—
131C2×C4×C83#10412yes—
132C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
133C2×C4×C83#13711yescharacteristic
134C2×Dic83—22no—
135C2×Dic83—22no—
136C2×Dic83—41no—
137C2×Dic83—41no—
138C2×Dic83—41no—
139C2×Dic83—41no—
140C2×Dic83—41no—
141C2×Dic83—41no—
142C2×Dic83—41no—
143C2×Dic83—41no—
144C2×Dic83—41no—
145C2×Dic83—41no—
146Dic166—82no—
147C8×C83—44no—
148C83⋊C8—82no—
149C22×C83—21no—
150C22×C83—21no—
151C22×C83—21no—
152C22×C83—21no—
153C22×C83—21no—
154C22×C83—21no—
155C22×C83—21no—
156C22×C83—42no—
157C22×C83—42no—
158C22×C83#35612yes—
159C22×C83C2×#9911yescharacteristic
160C22×C83#41711yescharacteristic
161C22×C83C2×#10411yescharacteristic
162C4×C83—21no—
163C4×C83—22no—
164C4×C83—21no—
165C4×C83—41no—
166C4×C83—41no—
167C4×C83—41no—
168C4×C83—41no—
169Dic83—42no—
170Dic83—42no—
171Dic83—81no—
172Dic83—81no—
173#11768—831nomaximal, Sylow
174C2×C83—21no—
175C2×C83—21no—
176C2×C83—22no—
177C2×C83—41no—
178C2×C83#150411yescharacteristic
179C2×C83#212111yescharacteristic
180C2×C83C2×#35611yescharacteristic
181#2583—831no—
182C2×#343D8311yescharacteristic
183#1463—831no—
184#1354—832no—
185#1346—832no—
186C83#1176811yescharacteristic, complemented, Sylow
187C22×M16—21no—
188C2×C22⋊Q8—831no—
189C22×C4⋊C4C2×D8311yescharacteristic
190#407—1661no—
191#398—831no—
192#398—831no—
193#395—831no—
194#395—831no—
195#389—1661no—
196#389—1661no—
197#385—1661no—
198#385—1661no—
199C4×C22⋊C4—831no—
200#343—21no—
201#343C2×D8312yes—
202#324—832no—
203#324—1661no—
204C22×Q8—831no—
205C23×C4C4×D8311yescharacteristic
206C23×C4D33211yescharacteristic
207C23×C4C22⋊D8311yescharacteristic
208C2×M16—24no—
209C2×M16—22no—
210C2×M16—41no—
211C2×M16—42no—
212C2×M16—3321no—
213C22×C8—41no—
214C22⋊Q8—3321no—
215C22⋊Q8—3321no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—831no—
219C2×C4⋊C4—1661no—
220C2×C4⋊C4—1661no—
221C2×C4⋊C4—1661no—
222C2×C4⋊C4—1661no—
223C2×C4⋊C4—1661no—
224C2×C4⋊C4C4×D8311yescharacteristic
225C2×C4⋊C4D33211yescharacteristic
226C2×C4⋊C4C22⋊D8311yescharacteristic
227C2×C4⋊C4C22×D8311yescharacteristic
228C2×C22⋊C4—1661no—
229C2×C22⋊C4—1661no—
230C2×C42—1661no—
231C22⋊C8—3321no—
232C22⋊C8—1662no—
233#95—831no—
234#95—831no—
235#95—1661no—
236#95—1661no—
237#95—1661no—
238#95—1661no—
239C24(C2×D83)⋊C411yescharacteristic
240C2×Q8—3321no—
241C2×Q8—6641no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—81no—
252C22×C4—81no—
253C22×C4—1661no—
254C22×C4—1661no—
255C22×C4—1661no—
256C22×C4—1661no—
257C22×C4(C2×D83)⋊C411yescharacteristic
258C22×C4(C2×D83)⋊C411yescharacteristic
259C22×C4(C2×D83)⋊C411yescharacteristic
260C22×C4C2×C4×D8311yescharacteristic
261C22×C4C2×C22⋊D8311yescharacteristic
262C22×C4C2×D33211yescharacteristic
263M16—24no—
264M16—44no—
265M16—42no—
266M16—6642no—
267C2×C8—44no—
268C2×C8—42no—
269C2×C8—3322no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—3321no—
275C4⋊C4—3322no—
276C4⋊C4—3321no—
277C4⋊C4—3321no—
278C4⋊C4—3322no—
279C4⋊C4—6641no—
280C4⋊C4—6641no—
281C22⋊C4—3321no—
282C22⋊C4—3321no—
283C22⋊C4—1664no—
284C42—6641no—
285C23—21no—
286C23—21no—
287C23—22no—
288C23—41no—
289C23#1458170611yescharacteristic
290C23#1458170611yescharacteristic
291C23C2×(C2×D83)⋊C411yescharacteristic
292Q8—6642no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
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—3321no—
322C2×C4—3321no—
323C2×C4—1662no—
324C2×C4—1662no—
325C2×C4#1458172012yes—
326C2×C4#1458181811yescharacteristic
327C2×C4#1458182311yescharacteristic
328C8—44no—
329C8—6642no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—42no—
338C22—42no—
339C22#1912749912yes—
340C22C2×#1458170611yescharacteristic
341C22#1912770011yescharacteristic
342C22C2×#1458172011yescharacteristic
343C4—21no—
344C4—21no—
345C4—22no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—3322no—
351C4—3322no—
352C4—6641no—
353C4—6641no—
354C2—21no—
355C2—21no—
356C2—22no—
357C2—41no—
358C2#2490596511yescharacteristic
359C2#2490911011yescharacteristic
360C2C2×#1912749911yescharacteristic
subgroupG/H
centerC22C2×#14581706
commutatorC22×C4×C83C22×C4
FrattiniC2×C4⋊C4C22×D83
FittingC2×#343×C83C2
socleC22×C83C2×#99
prime radicalC24×C83C22⋊C4
cosocleC2×C4⋊C4×C83C23
hypercenterC2×#343D83
hypocenterC83#11768
LCSG ▷ C22×C4×C83 ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ C2×#343

Complements

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

NH ≅ G/Ntype
3. #24905492C2semidirect
4. #24905468C2semidirect
186. C83#11768semidirect

Maximal subgroups

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

subgroupindex
#249128432
#249057932
#249054922
#249054682
C2×#343×C832
#1176883