Finite Group Databasebeta

← #35315328

Subgroups — #35315328

Order 21248.

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

normal subgroups only

This group has 438 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#24912943C211yesmaximal, characteristic
2#24907543C211yesmaximal, characteristic, complemented
3#24905491C212yesmaximal, complemented
4#24905467C212yesmaximal, complemented
5C2×#324×C83C211yesmaximal, characteristic, complemented
6C2×#14581766C411yescharacteristic
7#19127665—21no—
8#19127640C411yescharacteristic
9#19127640C2211yescharacteristic
10#19127618C411yescharacteristic
11#19127618C2211yescharacteristic
12#19127591—21no—
13#19127591—21no—
14#19127580—21no—
15#19127580—21no—
16C4×C22⋊Dic83C411yescharacteristic
17#19127470—21no—
18#19127470C2212yes—
19C22×C4⋊C4×C83C2211yescharacteristic
20C2×C22⋊C8×C83—21no—
21#324×C83—21no—
22#324×C83C2212yes—
23C22×D332C2×C411yescharacteristic
24C22×C22⋊D83—21no—
25C2×#13050574—22no—
26#14581766—41no—
27#14581766—41no—
28C2×C4⋊Dic83C2×C411yescharacteristic
29C2×C22⋊Dic83—21no—
30C2×(C2×D83)⋊C4—21no—
31C2×(C2×D83)⋊C4—21no—
32C2×(C2×D83)⋊C4—21no—
33C2×(C2×D83)⋊C4—21no—
34C2×(C2×D83)⋊C4—21no—
35C2×C4×Dic83—21no—
36C22⋊(C83⋊C8)—41no—
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
47C22×C8×C83—41no—
48C2×C4⋊C4×C83—41no—
49C2×C4⋊C4×C83—41no—
50C2×C4⋊C4×C83C2×C411yescharacteristic
51C2×C4⋊C4×C83D411yescharacteristic
52C2×C4⋊C4×C83D411yescharacteristic
53C2×C4⋊C4×C83C2311yescharacteristic
54C22⋊C8×C83—22no—
55C22⋊C8×C83—41no—
56C23×D83—21no—
57C2×D332—41no—
58C2×D332—81no—
59C2×C22⋊D83—42no—
60C2×C22⋊D83—42no—
61C2×C22⋊D83—41no—
62C2×C22⋊D83—41no—
63C22×Dic83—21no—
64C22×Dic83—21no—
65C22×Dic83—21no—
66#13050574—42no—
67#13050574—42no—
68C2×C83⋊C8—42no—
69C4⋊Dic83—41no—
70C22⋊Dic83—42no—
71(C2×D83)⋊C4—42no—
72(C2×D83)⋊C4—42no—
73(C2×D83)⋊C4—42no—
74(C2×D83)⋊C4—81no—
75(C2×D83)⋊C4—81no—
76C4×Dic83—81no—
77C24×C83C22⋊C411yescharacteristic
78C22×C4×C83—21no—
79C22×C4×C83—21no—
80C22×C4×C83—21no—
81C22×C4×C83—21no—
82C22×C4×C83—21no—
83C22×C4×C83—21no—
84C22×C4×C83—22no—
85C22×C4×C83—41no—
86C22×C4×C83—81no—
87C22×C4×C83—81no—
88C22×C4×C83C22⋊C411yescharacteristic
89C22×C4×C83C22⋊C411yescharacteristic
90C22×C4×C83C22⋊C411yescharacteristic
91C22×C4×C83C22×C411yescharacteristic
92C22×C4×C83C2×D411yescharacteristic
93C22×C4×C83C2×D411yescharacteristic
94C2×C8×C83—42no—
95C2×C8×C83—82no—
96C4⋊C4×C83—41no—
97C4⋊C4×C83—41no—
98C4⋊C4×C83—41no—
99C4⋊C4×C83—41no—
100C23×C83—22no—
101C23×C83—21no—
102C23×C83—21no—
103C23×C83—41no—
104C23×C83#9911yescharacteristic
105C23×C83#9911yescharacteristic
106C23×C83C2×C22⋊C411yescharacteristic
107C22×D83—41no—
108C22×D83—41no—
109C22×D83—41no—
110C22×D83—41no—
111C22×D83—41no—
112C22×D83—41no—
113C22×D83—41no—
114C2×C4×C83—21no—
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—42no—
125C2×C4×C83—42no—
126C2×C4×C83—41no—
127C2×C4×C83—41no—
128C2×C4×C83—41no—
129C2×C4×C83—41no—
130C2×C4×C83—81no—
131C2×C4×C83—81no—
132C2×C4×C83—81no—
133C2×C4×C83#13512yes—
134C22⋊D83—82no—
135C22⋊D83—82no—
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—
145D332—82no—
146C8×C83—82no—
147C83⋊C8—82no—
148C22×C83—21no—
149C22×C83—21no—
150C22×C83—21no—
151C22×C83—21no—
152C22×C83—21no—
153C22×C83—21no—
154C22×C83—21no—
155C22×C83—42no—
156C22×C83—42no—
157C22×C83#35512yes—
158C22×C83C2×#9911yescharacteristic
159C22×C83#41611yescharacteristic
160C22×C83#42111yescharacteristic
161C2×D83—81no—
162C2×D83—81no—
163C2×D83—81no—
164C2×D83—81no—
165C2×D83—81no—
166C2×D83—81no—
167C2×D83—81no—
168C4×C83—22no—
169C4×C83—41no—
170C4×C83—41no—
171C4×C83—41no—
172C4×C83—41no—
173C4×C83—41no—
174Dic83—81no—
175Dic83—81no—
176Dic83—81no—
177#11852—831nomaximal, Sylow
178C2×C83—21no—
179C2×C83—21no—
180C2×C83—22no—
181C2×C83—41no—
182C2×C83#151611yescharacteristic
183C2×C83#212111yescharacteristic
184C2×C83C2×#35511yescharacteristic
185D83—161no—
186#2591—831no—
187#1821—831no—
188C2×#324D8311yescharacteristic, complemented
189#1353—832no—
190#1345—832no—
191C83#1185211yescharacteristic, complemented, Sylow
192C2×#121—831no—
193C22×C4⋊C4C2×D8311yescharacteristic
194#407—1661no—
195C2×C22⋊C8—21no—
196#400—831no—
197#400—831no—
198#394—831no—
199#394—831no—
200#388—1661no—
201#388—1661no—
202#386—1661no—
203#386—1661no—
204C4×C22⋊C4—831no—
205#343—832no—
206#343—1661no—
207#324—21no—
208#324C2×D8312yes—
209C22×D4—831no—
210C22×D4—1661no—
211C23×C4C4×D8311yescharacteristic
212C23×C4D33211yescharacteristic
213C23×C4C22⋊D8311yescharacteristic
214C2×M16—1662no—
215C22×C8—41no—
216#121—3321no—
217#121—3321no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—831no—
221C2×C4⋊C4C4×D8311yescharacteristic
222C2×C4⋊C4D33211yescharacteristic
223C2×C4⋊C4C22⋊D8311yescharacteristic
224C2×C4⋊C4C22×D8311yescharacteristic
225C2×C22⋊C4—1661no—
226C2×C22⋊C4—1661no—
227C2×C22⋊C4—1661no—
228C2×C22⋊C4—1661no—
229C2×C22⋊C4—1661no—
230C2×C22⋊C4—1661no—
231C2×C42—1661no—
232C22⋊C8—22no—
233C22⋊C8—41no—
234C22⋊C8—3321no—
235C22⋊C8—3321no—
236#95—831no—
237#95—831no—
238#95—1661no—
239#95—1661no—
240#95—1661no—
241#95—1661no—
242C24—1661no—
243C24(C2×D83)⋊C411yescharacteristic
244C2×D4—3321no—
245C2×D4—3322no—
246C2×D4—3322no—
247C2×D4—3321no—
248C2×D4—3321no—
249C2×D4—6641no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—41no—
258C22×C4—81no—
259C22×C4—81no—
260C22×C4—1661no—
261C22×C4—1661no—
262C22×C4—1661no—
263C22×C4(C2×D83)⋊C411yescharacteristic
264C22×C4(C2×D83)⋊C411yescharacteristic
265C22×C4(C2×D83)⋊C411yescharacteristic
266C22×C4C2×C4×D8311yescharacteristic
267C22×C4C2×C22⋊D8311yescharacteristic
268C22×C4C2×D33211yescharacteristic
269M16—3322no—
270M16—3322no—
271C2×C8—42no—
272C2×C8—82no—
273C2×C8—3322no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—3321no—
279C22⋊C4—3322no—
280C22⋊C4—3322no—
281C22⋊C4—3322no—
282C22⋊C4—3322no—
283C22⋊C4—6641no—
284C22⋊C4—6641no—
285C42—6641no—
286C23—21no—
287C23—21no—
288C23—22no—
289C23—41no—
290C23—3321no—
291C23—3321no—
292C23—3321no—
293C23—3321no—
294C23—3321no—
295C23—3321no—
296C23—3321no—
297C23#1458170611yescharacteristic
298C23#1458170611yescharacteristic
299C23C2×(C2×D83)⋊C411yescharacteristic
300D4—6642no—
301D4—6642no—
302D4—6642no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—3321no—
323C2×C4—3321no—
324C2×C4—3321no—
325C2×C4—3321no—
326C2×C4—3321no—
327C2×C4—3321no—
328C2×C4—3321no—
329C2×C4—3321no—
330C2×C4—3321no—
331C2×C4#1458181412yes—
332C8—82no—
333C8—6642no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—42no—
342C22—42no—
343C22—6641no—
344C22—6641no—
345C22—6641no—
346C22—6641no—
347C22—6641no—
348C22—6641no—
349C22—6641no—
350C22#1912749712yes—
351C22C2×#1458170611yescharacteristic
352C22#1912769811yescharacteristic
353C22#1912771711yescharacteristic
354C4—22no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—6641no—
361C4—6641no—
362C4—6641no—
363C2—21no—
364C2—21no—
365C2—22no—
366C2—41no—
367C2—13281no—
368C2#2490602011yescharacteristic
369C2#2490910711yescharacteristic
370C2C2×#1912749711yescharacteristic
subgroupG/H
centerC22C2×#14581706
commutatorC22×C4×C83C22×C4
FrattiniC2×C4⋊C4C22×D83
FittingC2×#324×C83C2
socleC22×C83C2×#99
prime radicalC2×#14581766C4
cosocleC2×C4⋊C4×C83C23
hypercenterC2×#324D83
hypocenterC83#11852
LCSG ▷ C22×C4×C83 ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
2. #24907543C2semidirect
3. #24905491C2semidirect
4. #24905467C2semidirect
5. C2×#324×C83C2semidirect
188. C2×#324D83semidirect
191. C83#11852semidirect

Maximal subgroups

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

subgroupindex
#249129432
#249075432
#249054912
#249054672
C2×#324×C832
#1185283