Finite Group Databasebeta

← #35301197

Subgroups — #35301197

Order 21248.

416 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 105 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#24917264C211yesmaximal, characteristic
2#24916546C211yesmaximal, characteristic
3#24907479C211yesmaximal, characteristic
4#24906027C211yesmaximal, characteristic
5#24905360C211yesmaximal, characteristic
6#24905322C211yesmaximal, characteristic
7C83×#1481C211yesmaximal, characteristic
8C2×C4×Dic166C2211yescharacteristic
9#19128005—22no—
10#19127993C412yescomplemented
11#19127983C412yescomplemented
12#19127831—22no—
13#19127828—22no—
14#19127755—22no—
15#19127754C2211yescharacteristic
16#19127744—21no—
17#19127725C2211yescharacteristic
18#19127709—21no—
19C2×#14581717—21no—
20#19127567C2211yescharacteristic
21#19127464—21no—
22#19127438—22no—
23#533×C83C2211yescharacteristic
24#431×C83C2211yescharacteristic
25C2×C4⋊C8×C83C2211yescharacteristic
26Q8⋊C8×C83—22no—
27C22×Dic166D411yescharacteristic
28C2×#13050574—21no—
29C4×Dic166—21no—
30C4×Dic166—21no—
31C4×Dic166C2×C412yes—
32C4×Dic166D412yes—
33C2×C4⋊Dic83—21no—
34C2×C4⋊Dic83D411yescharacteristic
35C2×Dic83⋊C4—21no—
36C2×C4×Dic83—21no—
37#14581725—22no—
38#14581723—22no—
39C4⋊(C83⋊C8)—21no—
40C4⋊(C83⋊C8)—21no—
41C4⋊(C83⋊C8)C2×C412yes—
42Dic83⋊C8—24no—
43#14581717—42no—
44#14581715—22no—
45#14581715—22no—
46C22⋊(C83⋊C8)D412yes—
47#14581702—24no—
48C4×C83⋊C8—21no—
49C83⋊#95—21no—
50C83⋊#95—21no—
51C22×C8×C83—21no—
52C4⋊Q8×C83C2×C411yescharacteristic
53C4⋊Q8×C83C2×C411yescharacteristic
54#125×C83—21no—
55C22⋊Q8×C83—21no—
56C22⋊Q8×C83—21no—
57C4×Q8×C83—21no—
58C4×Q8×C83—21no—
59#117×C83—21no—
60C2×C42×C83C2311yescharacteristic
61C4⋊C8×C83—21no—
62C4⋊C8×C83—21no—
63C4⋊C8×C83—21no—
64C22⋊C8×C83—21no—
65C4×C8×C83D412yes—
66C2×Dic166—21no—
67C2×Dic166—21no—
68C2×Dic166—21no—
69C2×Dic166—21no—
70C2×Dic166—21no—
71C2×Dic166C22⋊C412yes—
72C22×Dic83—21no—
73C22×Dic83—21no—
74C22×Dic83—41no—
75#13050574—41no—
76#13050574—41no—
77C2×C83⋊C8—22no—
78C2×C83⋊C8—22no—
79C4⋊Dic83—21no—
80C4⋊Dic83—41no—
81C4⋊Dic83—41no—
82C4⋊Dic83C22⋊C412yes—
83Dic83⋊C4—22no—
84Dic83⋊C4—22no—
85C4×Dic83—22no—
86C4×Dic83—22no—
87C2×Q8×C83—21no—
88C2×Q8×C83—21no—
89C22×C4×C83C2×D411yescharacteristic
90C22×C4×C83C2×D411yescharacteristic
91C22×C4×C83#4111yescharacteristic
92C2×C8×C83—22no—
93C2×C8×C83—41no—
94C2×C8×C83—41no—
95C2×C8×C83—41no—
96C4⋊C4×C83—21no—
97C4⋊C4×C83—21no—
98C4⋊C4×C83—21no—
99C4⋊C4×C83—21no—
100C4⋊C4×C83—41no—
101C4⋊C4×C83—41no—
102C22⋊C4×C83—41no—
103C42×C83—41no—
104C42×C83C22×C411yescharacteristic
105C42×C83C2×D411yescharacteristic
106C42×C83C2×D411yescharacteristic
107C42×C83#4111yescharacteristic
108C23×C83#12111yescharacteristic
109Q8×C83—42no—
110Q8×C83—81no—
111C2×C4×C83—41no—
112C2×C4×C83—41no—
113C2×C4×C83—41no—
114C2×C4×C83—41no—
115C2×C4×C83C2×C22⋊C411yescharacteristic
116C2×C4×C83C4×D411yescharacteristic
117C2×C4×C83C4×D411yescharacteristic
118C2×C4×C83#12011yescharacteristic
119C2×C4×C83#12111yescharacteristic
120C2×C4×C83#12311yescharacteristic
121C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
122C2×C4×C83#13711yescharacteristic
123C2×C4×C83#13711yescharacteristic
124C2×C4×C83#13711yescharacteristic
125C2×Dic83—22no—
126C2×Dic83—22no—
127C2×Dic83—41no—
128C2×Dic83—41no—
129C2×Dic83—81no—
130C2×Dic83—81no—
131C2×Dic83—81no—
132C2×Dic83—81no—
133Dic166—22no—
134Dic166—22no—
135Dic166—41no—
136Dic166—41no—
137C8×C83—42no—
138C8×C83—81no—
139C83⋊C8—42no—
140C83⋊C8—42no—
141C22×C83—21no—
142C22×C83—21no—
143C22×C83#38611yescharacteristic
144C22×C83#46811yescharacteristic
145C22×C83#47011yescharacteristic
146C4×C83—21no—
147C4×C83—21no—
148C4×C83—21no—
149C4×C83—21no—
150C4×C83—81no—
151C4×C83—81no—
152C4×C83#41711yescharacteristic
153C4×C83#41911yescharacteristic
154C4×C83#46811yescharacteristic
155C4×C83#47011yescharacteristic
156Dic83—42no—
157Dic83—42no—
158Dic83—161no—
159#9308—831nomaximal, Sylow
160C2×C83—21no—
161C2×C83—21no—
162C2×C83#188111yescharacteristic
163C2×C83#189411yescharacteristic
164C2×C83#190511yescharacteristic
165#3078—831no—
166#2983—831no—
167#1805—831no—
168#1518—831no—
169#1481D8311yescharacteristic
170#1301—831no—
171#1286—831no—
172C83#930811yescharacteristic, complemented, Sylow
173#533C2×D8311yescharacteristic
174C2×C4×Q8—831no—
175#479—1662no—
176#477—832no—
177#475—832no—
178#446—1662no—
179#446—1662no—
180#432—1662no—
181#431—831no—
182#431C2×D8311yescharacteristic
183#428—1661no—
184#423—831no—
185C2×C4⋊C8C2×D8311yescharacteristic
186#419—1661no—
187C2×#103—1661no—
188#382—831no—
189#340—1661no—
190#328—1662no—
191Q8⋊C8—22no—
192C22×Q8—831no—
193C2×M16—1661no—
194C22×C8—21no—
195C4⋊Q8C4×D8311yescharacteristic
196C4⋊Q8C4×D8311yescharacteristic
197#125—21no—
198C22⋊Q8—21no—
199C22⋊Q8—21no—
200C4×Q8—21no—
201C4×Q8—21no—
202C4×Q8—832no—
203C4×Q8—832no—
204C4×Q8—1661no—
205C4×Q8—1661no—
206#117—21no—
207C2×C4⋊C4—831no—
208C2×C4⋊C4—1661no—
209C2×C4⋊C4—1661no—
210C2×C42—1661no—
211C2×C42C22×D8311yescharacteristic
212#107—1662no—
213#106—1662no—
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8—21no—
217C4⋊C8—832no—
218C4⋊C8—1664no—
219C4⋊C8—1661no—
220C4⋊C8—1661no—
221#103—3322no—
222#103—1662no—
223#103—1662no—
224C22⋊C8—21no—
225C22⋊C8—832no—
226#97—1664no—
227C4×C8—1661no—
228C4×C8C22⋊D8312yes—
229#95—1661no—
230#95—1661no—
231C2×Q8—21no—
232C2×Q8—21no—
233C2×Q8—832no—
234C2×Q8—1661no—
235C2×Q8—1661no—
236C2×Q8—1661no—
237C2×Q8—1661no—
238C2×Q8—1661no—
239C22×C4—3321no—
240C22×C4—1661no—
241C22×C4—1661no—
242C22×C4D4×D8311yescharacteristic
243C22×C4D4×D8311yescharacteristic
244C22×C4#1305059011yescharacteristic
245M16—3321no—
246M16—3321no—
247C2×C8—22no—
248C2×C8—41no—
249C2×C8—41no—
250C2×C8—41no—
251C2×C8—1662no—
252C2×C8—1662no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—3321no—
260C4⋊C4—3321no—
261C4⋊C4—832no—
262C4⋊C4—1661no—
263C4⋊C4—1662no—
264C4⋊C4—1662no—
265C22⋊C4—41no—
266C42—41no—
267C42—1662no—
268C42—1662no—
269C42C2×C4×D8311yescharacteristic
270C42D4×D8311yescharacteristic
271C42C2×C22⋊D8311yescharacteristic
272C42#1305058911yescharacteristic
273C23#1458176511yescharacteristic
274Q8—42no—
275Q8—81no—
276Q8—3321no—
277Q8—3321no—
278Q8—1662no—
279Q8—1662no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—3321no—
285C2×C4—3321no—
286C2×C4—6641no—
287C2×C4—6641no—
288C2×C4—6641no—
289C2×C4—6641no—
290C2×C4—1662no—
291C2×C4—1662no—
292C2×C4C22⋊C4×D8311yescharacteristic
293C2×C4(C22⋊D83)⋊C411yescharacteristic
294C2×C4C4×C22⋊D8311yescharacteristic
295C2×C4#1458176011yescharacteristic
296C2×C4#1458176311yescharacteristic
297C2×C4#1458177411yescharacteristic
298C2×C4#1458181911yescharacteristic
299C2×C4#1458182111yescharacteristic
300C2×C4#1458182411yescharacteristic
301C2×C4#1458182411yescharacteristic
302C8—42no—
303C8—81no—
304C8—3322no—
305C8—3322no—
306C22—21no—
307C22—21no—
308C22#1912757711yescharacteristic
309C22#1912795011yescharacteristic
310C22#1912795811yescharacteristic
311C4—21no—
312C4—21no—
313C4—21no—
314C4—21no—
315C4—81no—
316C4—81no—
317C4—13281no—
318C4—3322no—
319C4—3322no—
320C4#1912770111yescharacteristic
321C4#1912770811yescharacteristic
322C4#1912794411yescharacteristic
323C4#1912795611yescharacteristic
324C2—21no—
325C2—21no—
326C2#2490782811yescharacteristic
327C2#2490789311yescharacteristic
328C2#2490795411yescharacteristic
subgroupG/H
centerC22#19127577
commutatorC42×C83C22×C4
FrattiniC2×C42C22×D83
FittingC83×#1481C2
socleC22×C83#386
prime radicalC23×C83#121
cosocleC2×C42×C83C23
hypercenter#1481D83
hypocenterC83#9308
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1481

Complements

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

NH ≅ G/Ntype
10. #19127993C4semidirect
11. #19127983C4semidirect
172. C83#9308semidirect

Maximal subgroups

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

subgroupindex
#249172642
#249165462
#249074792
#249060272
#249053602
#249053222
C83×#14812
#930883