Finite Group Databasebeta

← #35313696

Subgroups — #35313696

Order 21248.

460 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 103 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#24910175C211yesmaximal, characteristic
2#24905781C211yesmaximal, characteristic
3#24905408C212yesmaximal, complemented
4#24905396C212yesmaximal, complemented
5C83×#1459C211yesmaximal, characteristic
6C2×#14581779C411yescharacteristic
7C2×C4×Dic166C411yescharacteristic
8C2×C22⋊(C83⋊C8)—21no—
9#19127606C2211yescharacteristic
10#19127604C411yescharacteristic
11#19127589—21no—
12#19127589—21no—
13#19127574—21no—
14#19127574—21no—
15#19127570C2211yescharacteristic
16#19127547C411yescharacteristic
17#19127427C2212yes—
18#19127425—21no—
19C2×C4×D4×C83C2211yescharacteristic
20C2×C22⋊C8×C83—21no—
21#324×C83C2212yes—
22#323×C83—21no—
23C22×Dic166C2×C411yescharacteristic
24C22×C83⋊C8—41no—
25#14581779—22no—
26#14581779C814yescomplemented
27C4×Dic166—41no—
28C4×Dic166—41no—
29C2×C4⋊Dic83C2×C411yescharacteristic
30C2×Dic83⋊C4—21no—
31C2×Dic83⋊C4—21no—
32C2×Dic83⋊C4—21no—
33C2×C22⋊Dic83—21no—
34C2×C22⋊Dic83—21no—
35C2×C22⋊Dic83—21no—
36C2×C22⋊Dic83—21no—
37C2×C4×Dic83—21no—
38C2×C4×Dic83C2×C411yescharacteristic
39C2×C4×Dic83C2×C411yescharacteristic
40C22⋊(C83⋊C8)—22no—
41C22⋊(C83⋊C8)—41no—
42C83⋊#95—21no—
43C83⋊#95—21no—
44C83⋊#95—21no—
45C83⋊#95—21no—
46C22×D4×C83C2×C411yescharacteristic
47C23×C4×C83D411yescharacteristic
48C23×C4×C83D411yescharacteristic
49C22×C8×C83—41no—
50C4×D4×C83—41no—
51C4×D4×C83—41no—
52C2×C4⋊C4×C83C2311yescharacteristic
53C2×C22⋊C4×C83D411yescharacteristic
54C2×C22⋊C4×C83D411yescharacteristic
55C2×C42×C83C2×C411yescharacteristic
56C22⋊C8×C83—22no—
57C22⋊C8×C83—41no—
58C2×Dic166—21no—
59C2×Dic166—41no—
60C2×Dic166—41no—
61C2×Dic166C2×C812yescomplemented
62C22×Dic83—21no—
63C22×Dic83—21no—
64C22×Dic83—21no—
65C22×Dic83—21no—
66C2×C83⋊C8—42no—
67C2×C83⋊C8—82no—
68C4⋊Dic83—41no—
69Dic83⋊C4—41no—
70Dic83⋊C4—41no—
71Dic83⋊C4—81no—
72Dic83⋊C4—81no—
73C22⋊Dic83—24no—
74C22⋊Dic83—24no—
75C22⋊Dic83—81no—
76C22⋊Dic83—81no—
77C4×Dic83—21no—
78C4×Dic83—42no—
79C4×Dic83—41no—
80C4×Dic83C2×C812yescomplemented
81C24×C83C22⋊C411yescharacteristic
82C24×C83C22⋊C411yescharacteristic
83C2×D4×C83—41no—
84C2×D4×C83—41no—
85C2×D4×C83C2×C812yes—
86C2×D4×C83M1612yes—
87C22×C4×C83—21no—
88C22×C4×C83—21no—
89C22×C4×C83—21no—
90C22×C4×C83—21no—
91C22×C4×C83—81no—
92C22×C4×C83—81no—
93C22×C4×C83C22⋊C411yescharacteristic
94C22×C4×C83C22⋊C411yescharacteristic
95C22×C4×C83C22×C411yescharacteristic
96C22×C4×C83C2×D411yescharacteristic
97C22×C4×C83C2×D411yescharacteristic
98C2×C8×C83—42no—
99C2×C8×C83—82no—
100C4⋊C4×C83—41no—
101C22⋊C4×C83—41no—
102C22⋊C4×C83—41no—
103C42×C83—21no—
104C42×C83—21no—
105C23×C83—21no—
106C23×C83—21no—
107C23×C83—21no—
108C23×C83—21no—
109C23×C83—41no—
110C23×C83—41no—
111C23×C83—41no—
112C23×C83—41no—
113C23×C83C22⋊C812yes—
114C23×C83C22⋊C812yes—
115C23×C83C2×C22⋊C411yescharacteristic
116D4×C83—42no—
117D4×C83—42no—
118C2×C4×C83—21no—
119C2×C4×C83—21no—
120C2×C4×C83—21no—
121C2×C4×C83—21no—
122C2×C4×C83—21no—
123C2×C4×C83—41no—
124C2×C4×C83—42no—
125C2×C4×C83—42no—
126C2×C4×C83—41no—
127C2×C4×C83—41no—
128C2×C4×C83—41no—
129C2×C4×C83—81no—
130C2×C4×C83—81no—
131C2×C4×C83#9911yescharacteristic
132C2×C4×C83#9911yescharacteristic
133C2×C4×C83C22×C811yescharacteristic
134C2×C4×C83C2×M1611yescharacteristic
135C2×Dic83—22no—
136C2×Dic83—22no—
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—
146C2×Dic83—41no—
147Dic166—42no—
148Dic166—42no—
149C8×C83—82no—
150C83⋊C8—82no—
151C22×C83—21no—
152C22×C83—21no—
153C22×C83—22no—
154C22×C83—22no—
155C22×C83—42no—
156C22×C83—42no—
157C22×C83—41no—
158C22×C83—41no—
159C22×C83—41no—
160C22×C83—41no—
161C22×C83—81no—
162C22×C83—81no—
163C22×C83C2×C22⋊C811yescharacteristic
164C22×C83C2×#9911yescharacteristic
165C22×C83#41311yescharacteristic
166C4×C83—21no—
167C4×C83—21no—
168C4×C83—21no—
169C4×C83—21no—
170C4×C83—41no—
171C4×C83—41no—
172C4×C83—41no—
173Dic83—42no—
174Dic83—42no—
175Dic83—81no—
176Dic83—81no—
177#11481—831nomaximal, Sylow
178C2×C83—21no—
179C2×C83—21no—
180C2×C83—42no—
181C2×C83—42no—
182C2×C83#145911yescharacteristic
183C2×C83#212711yescharacteristic
184C2×C83#213511yescharacteristic
185#2326—831no—
186#1459—831no—
187#1459D8311yescharacteristic
188#1325—832no—
189#1319—832no—
190C83#1148111yescharacteristic, complemented, Sylow
191C2×#124—831no—
192C2×C4×Q8—831no—
193C2×C4×D4C2×D8311yescharacteristic
194C2×C22⋊C8—21no—
195C2×C22⋊C8—1661no—
196#391—831no—
197#390—831no—
198#388—1661no—
199#388—1661no—
200#385—1661no—
201#385—1661no—
202#384—831no—
203#376—831no—
204#324—832no—
205#324C2×D8312yes—
206#323—21no—
207#323—1661no—
208C22×Q8—831no—
209C22×D4C4×D8311yescharacteristic
210C23×C4D33211yescharacteristic
211C23×C4C22⋊D8311yescharacteristic
212C22×C8—41no—
213C22×C8—3321no—
214#124—834no—
215#124—1662no—
216C4×Q8—3321no—
217C4×Q8—3321no—
218C4×D4—41no—
219C4×D4—41no—
220C2×C4⋊C4—831no—
221C2×C4⋊C4—1661no—
222C2×C4⋊C4—1661no—
223C2×C4⋊C4—1661no—
224C2×C4⋊C4C22×D8311yescharacteristic
225C2×C22⋊C4—1661no—
226C2×C22⋊C4—1661no—
227C2×C22⋊C4—1661no—
228C2×C22⋊C4—1661no—
229C2×C22⋊C4D33211yescharacteristic
230C2×C22⋊C4C22⋊D8311yescharacteristic
231C2×C42—831no—
232C2×C42—831no—
233C2×C42—1661no—
234C2×C42C4×D8311yescharacteristic
235C22⋊C8—22no—
236C22⋊C8—41no—
237C22⋊C8—3321no—
238C22⋊C8—1662no—
239#95—1661no—
240#95—1661no—
241#95—1661no—
242#95—1661no—
243C24(C2×D83)⋊C411yescharacteristic
244C24(C2×D83)⋊C411yescharacteristic
245C2×Q8—3321no—
246C2×Q8—3321no—
247C2×Q8—832no—
248C2×Q8—1661no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4C8×D8312yes—
252C2×D4#1305057312yes—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—81no—
258C22×C4—81no—
259C22×C4—1661no—
260C22×C4—1661no—
261C22×C4—1661no—
262C22×C4—1661no—
263C22×C4(C2×D83)⋊C411yescharacteristic
264C22×C4(C2×D83)⋊C411yescharacteristic
265C22×C4C2×C4×D8311yescharacteristic
266C22×C4C2×C22⋊D8311yescharacteristic
267C22×C4C2×D33211yescharacteristic
268C2×C8—42no—
269C2×C8—82no—
270C2×C8—3322no—
271C2×C8—6642no—
272C4⋊C4—41no—
273C4⋊C4—3321no—
274C4⋊C4—3321no—
275C4⋊C4—3321no—
276C4⋊C4—6641no—
277C4⋊C4—6641no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—6641no—
281C22⋊C4—6641no—
282C22⋊C4—1664no—
283C22⋊C4—1664no—
284C42—21no—
285C42—21no—
286C42—3321no—
287C42—3322no—
288C42—832no—
289C42—1661no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23(C2×D83)⋊C812yes—
299C23(C2×D83)⋊C812yes—
300C23C2×(C2×D83)⋊C411yescharacteristic
301Q8—3322no—
302Q8—3322no—
303D4—42no—
304D4—42no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—3321no—
319C2×C4—3321no—
320C2×C4—3321no—
321C2×C4—3321no—
322C2×C4—3321no—
323C2×C4—3321no—
324C2×C4—3321no—
325C2×C4—3321no—
326C2×C4—3321no—
327C2×C4—3321no—
328C2×C4—1662no—
329C2×C4—1662no—
330C2×C4#1458170611yescharacteristic
331C2×C4#1458170611yescharacteristic
332C2×C4C2×C8×D8311yescharacteristic
333C2×C4C2×#1305057311yescharacteristic
334C8—82no—
335C8—6642no—
336C22—21no—
337C22—21no—
338C22—22no—
339C22—22no—
340C22—42no—
341C22—42no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—81no—
347C22—81no—
348C22C2×(C2×D83)⋊C811yescharacteristic
349C22C2×#1458170611yescharacteristic
350C22#1912768311yescharacteristic
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—3322no—
359C4—3322no—
360C4—6641no—
361C4—6641no—
362C2—21no—
363C2—21no—
364C2—42no—
365C2—42no—
366C2#2490577911yescharacteristic
367C2#2490913011yescharacteristic
368C2#2490915711yescharacteristic
subgroupG/H
centerC22C2×#14581706
commutatorC2×C4×C83C22×C8
FrattiniC2×C4⋊C4C22×D83
FittingC83×#1459C2
socleC22×C83C2×#99
prime radicalC22×D4×C83C2×C4
cosocleC2×C4⋊C4×C83C23
hypercenter#1459D83
hypocenterC83#11481
LCSG ▷ C2×C4×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1459

Complements

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

NH ≅ G/Ntype
3. #24905408C2semidirect
4. #24905396C2semidirect
26. #14581779C8semidirect
61. C2×Dic166C2×C8semidirect
80. C4×Dic83C2×C8semidirect
190. C83#11481semidirect

Maximal subgroups

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

subgroupindex
#249101752
#249057812
#249054082
#249053962
C83×#14592
#1148183