Finite Group Databasebeta

← #35302359

Subgroups — #35302359

Order 21248.

682 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 682 conjugacy classes of subgroups — too many to draw. Only its 111 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#24916504C211yesmaximal, characteristic
2#24907387C212yesmaximal, complemented
3#24905801C211yesmaximal, characteristic
4#24905369C212yesmaximal, complemented
5C83×#2409C211yesmaximal, characteristic
6#19127779—24no—
7#19127754—21no—
8C2×#14581711—24no—
9#19127665—21no—
10#19127665C2212yes—
11#19127665C2212yes—
12#19127427—21no—
13#19127425—21no—
14C23×Q8×C83C2211yescharacteristic
15C2×C22⋊Q8×C83—21no—
16C2×C4×D4×C83C2211yescharacteristic
17#395×C83—21no—
18#393×C83—21no—
19#391×C83—21no—
20#386×C83C412yes—
21#385×C83C412yes—
22#380×C83C2211yescharacteristic
23C2×#13050574—22no—
24C2×#13050574—22no—
25C2×#13050574D414yes—
26C4⋊(C83⋊C8)—22no—
27#14581711—48no—
28C22⋊(C83⋊C8)—22no—
29C22⋊(C83⋊C8)—22no—
30C22⋊(C83⋊C8)—22no—
31C22⋊(C83⋊C8)—22no—
32C22⋊(C83⋊C8)—22no—
33C4×C83⋊C8—22no—
34C22×Q8×C83—21no—
35C22×Q8×C83—22no—
36C22×Q8×C83—21no—
37C22×Q8×C83—22no—
38C22×Q8×C83—44no—
39C22×D4×C83C2×C411yescharacteristic
40C23×C4×C83C2×C412yes—
41C23×C4×C83D411yescharacteristic
42C23×C4×C83C2311yescharacteristic
43C22⋊Q8×C83—44no—
44C4×D4×C83—22no—
45C4×D4×C83—22no—
46C2×C4⋊C4×C83—22no—
47C2×C4⋊C4×C83—21no—
48C2×C4⋊C4×C83D411yescharacteristic
49C2×C22⋊C4×C83—22no—
50C2×C22⋊C4×C83C2×C411yescharacteristic
51C2×C22⋊C4×C83C2×C412yes—
52C2×C22⋊C4×C83D411yescharacteristic
53C2×C42×C83D411yescharacteristic
54#95×C83—22no—
55#95×C83—22no—
56#95×C83—21no—
57#13050574—44no—
58#13050574—44no—
59#13050574—82no—
60C2×C83⋊C8—24no—
61C2×C83⋊C8—42no—
62C24×C83C22⋊C411yescharacteristic
63C24×C83C22×C411yescharacteristic
64C2×Q8×C83—24no—
65C2×Q8×C83—28no—
66C2×Q8×C83—42no—
67C2×Q8×C83—44no—
68C2×Q8×C83—44no—
69C2×Q8×C83—48no—
70C2×Q8×C83—41no—
71C2×Q8×C83—42no—
72C2×Q8×C83—41no—
73C2×D4×C83—22no—
74C2×D4×C83—22no—
75C2×D4×C83—41no—
76C22×C4×C83—22no—
77C22×C4×C83—24no—
78C22×C4×C83—21no—
79C22×C4×C83—21no—
80C22×C4×C83—22no—
81C22×C4×C83—44no—
82C22×C4×C83—42no—
83C22×C4×C83—42no—
84C22×C4×C83C22⋊C411yescharacteristic
85C22×C4×C83C22⋊C411yescharacteristic
86C22×C4×C83C22⋊C411yescharacteristic
87C22×C4×C83C2×D412yes—
88C22×C4×C83C2×D411yescharacteristic
89C22×C4×C83C2×D411yescharacteristic
90C22×C4×C83#4112yes—
91C4⋊C4×C83—22no—
92C4⋊C4×C83—42no—
93C4⋊C4×C83—44no—
94C22⋊C4×C83—22no—
95C22⋊C4×C83—22no—
96C22⋊C4×C83—44no—
97C22⋊C4×C83—42no—
98C42×C83—21no—
99C42×C83—21no—
100C23×C83—21no—
101C23×C83—21no—
102C23×C83—21no—
103C23×C83—21no—
104C23×C83—21no—
105C23×C83—41no—
106C23×C83—41no—
107C23×C83—41no—
108C23×C83—41no—
109C23×C83C2×C22⋊C411yescharacteristic
110C23×C83C4×D412yes—
111Q8×C83—48no—
112Q8×C83—44no—
113Q8×C83—44no—
114D4×C83—44no—
115C2×C4×C83—21no—
116C2×C4×C83—21no—
117C2×C4×C83—22no—
118C2×C4×C83—24no—
119C2×C4×C83—24no—
120C2×C4×C83—28no—
121C2×C4×C83—21no—
122C2×C4×C83—21no—
123C2×C4×C83—41no—
124C2×C4×C83—42no—
125C2×C4×C83—42no—
126C2×C4×C83—44no—
127C2×C4×C83—42no—
128C2×C4×C83—42no—
129C2×C4×C83—41no—
130C2×C4×C83—41no—
131C2×C4×C83—42no—
132C2×C4×C83—42no—
133C2×C4×C83—42no—
134C2×C4×C83—81no—
135C2×C4×C83#10112yescomplemented
136C2×C4×C83#12011yescharacteristic
137C2×C4×C83#12112yes—
138C2×C4×C83#12311yescharacteristic
139C83⋊C8—44no—
140C83⋊C8—82no—
141C22×C83—21no—
142C22×C83—21no—
143C22×C83—22no—
144C22×C83—22no—
145C22×C83—22no—
146C22×C83—42no—
147C22×C83—42no—
148C22×C83—42no—
149C22×C83—41no—
150C22×C83—41no—
151C22×C83—41no—
152C22×C83—81no—
153C22×C83#38611yescharacteristic
154C22×C83C2×#10111yescharacteristic
155C22×C83#41311yescharacteristic
156C4×C83—22no—
157C4×C83—21no—
158C4×C83—22no—
159C4×C83—44no—
160C4×C83—41no—
161C4×C83—41no—
162C4×C83—82no—
163C4×C83B(4,𝔽2)12yes—
164#9506—831nomaximal, Sylow
165C2×C83—21no—
166C2×C83—21no—
167C2×C83—42no—
168C2×C83—81no—
169C2×C83#188911yescharacteristic
170C2×C83#190111yescharacteristic
171C2×C83#191511yescharacteristic
172#2975—831no—
173#2409D8311yescharacteristic
174#1784—832no—
175#1465—831no—
176#1305—832no—
177C83#950611yescharacteristic, complemented, Sylow
178C23×Q8C2×D8311yescharacteristic
179C2×C22⋊Q8—21no—
180C2×C4×D4C2×D8311yescharacteristic
181#436—1664no—
182#431—1661no—
183C2×#101—1664no—
184#407—832no—
185#407—832no—
186#407—1661no—
187#395—21no—
188#393—21no—
189#391—21no—
190#386Dic8312yes—
191#385Dic8312yes—
192#380C2×D8311yescharacteristic
193#324—1661no—
194#323—1661no—
195C22×Q8—22no—
196C22×Q8—21no—
197C22×Q8—21no—
198C22×Q8—22no—
199C22×Q8—44no—
200C22×D4C2×Dic8311yescharacteristic
201C23×C4C2×Dic8312yes—
202C23×C4C22⋊D8311yescharacteristic
203C23×C4C22×D8311yescharacteristic
204C2×M16—834no—
205C2×M16—1662no—
206C2×M16—1662no—
207C22⋊Q8—44no—
208C4×D4—22no—
209C4×D4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4C22⋊D8311yescharacteristic
213C2×C22⋊C4—22no—
214C2×C22⋊C4C2×Dic8311yescharacteristic
215C2×C22⋊C4C2×Dic8312yes—
216C2×C22⋊C4C22⋊D8311yescharacteristic
217C2×C42C22⋊D8311yescharacteristic
218C4⋊C8—1662no—
219#101—3328no—
220C22⋊C8—1662no—
221C22⋊C8—1662no—
222C22⋊C8—1662no—
223C22⋊C8—1662no—
224C22⋊C8—1662no—
225C4×C8—1662no—
226#95—22no—
227#95—22no—
228#95—21no—
229C24C22⋊Dic8311yescharacteristic
230C24C22×Dic8311yescharacteristic
231C2×Q8—28no—
232C2×Q8—24no—
233C2×Q8—42no—
234C2×Q8—44no—
235C2×Q8—44no—
236C2×Q8—48no—
237C2×Q8—41no—
238C2×Q8—42no—
239C2×Q8—41no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—41no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—22no—
248C22×C4—44no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4C22⋊Dic8311yescharacteristic
252C22×C4C22⋊Dic8311yescharacteristic
253C22×C4C22⋊Dic8311yescharacteristic
254C22×C4D4×D8312yes—
255C22×C4C2×C22⋊D8311yescharacteristic
256C22×C4C2×C22⋊D8311yescharacteristic
257C22×C4#1305058912yes—
258M16—3324no—
259M16—3324no—
260M16—6642no—
261C2×C8—3322no—
262C2×C8—1664no—
263C4⋊C4—22no—
264C4⋊C4—42no—
265C4⋊C4—44no—
266C22⋊C4—22no—
267C22⋊C4—22no—
268C22⋊C4—44no—
269C22⋊C4—42no—
270C42—21no—
271C42—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23C2×C22⋊Dic8311yescharacteristic
282C23D4×Dic8312yes—
283Q8—48no—
284Q8—44no—
285Q8—44no—
286D4—44no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—24no—
291C2×C4—24no—
292C2×C4—28no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—44no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—81no—
307C2×C4#1458171112yescomplemented
308C2×C4#1458176011yescharacteristic
309C2×C4#1458176412yes—
310C2×C4#1458177311yescharacteristic
311C8—3324no—
312C8—6642no—
313C22—21no—
314C22—21no—
315C22—22no—
316C22—22no—
317C22—22no—
318C22—42no—
319C22—42no—
320C22—42no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—81no—
325C22#1912758111yescharacteristic
326C22C2×#1458171111yescharacteristic
327C22#1912768511yescharacteristic
328C4—22no—
329C4—21no—
330C4—22no—
331C4—44no—
332C4—41no—
333C4—41no—
334C4—82no—
335C4#1912788712yes—
336C2—21no—
337C2—21no—
338C2—42no—
339C2—81no—
340C2#2490787211yescharacteristic
341C2#2490793211yescharacteristic
342C2#2490801611yescharacteristic
subgroupG/H
centerC22#19127581
commutatorC24×C83C22×C4
FrattiniC23×C4C22×D83
FittingC83×#2409C2
socleC22×C83#386
prime radicalC22×D4×C83C2×C4
cosocleC23×C4×C83C23
hypercenter#2409D83
hypocenterC83#9506
LCSG ▷ C24×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2409

Complements

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

NH ≅ G/Ntype
2. #24907387C2semidirect
4. #24905369C2semidirect
135. C2×C4×C83#101semidirect
177. C83#9506semidirect
307. C2×C4#14581711semidirect

Maximal subgroups

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

subgroupindex
#249165042
#249073872
#249058012
#249053692
C83×#24092
#950683