Finite Group Databasebeta

← #35315322

Subgroups — #35315322

Order 21248.

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

normal subgroups only

This group has 418 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#24907546C211yesmaximal, characteristic, complemented
2#24905777C211yesmaximal, characteristic, complemented
3#24905493C212yesmaximal, complemented
4#24905466C212yesmaximal, complemented
5C83×#2610C211yesmaximal, characteristic
6#19127754—21no—
7#19127665—21no—
8#19127470—21no—
9#19127470C2212yes—
10#19127427C2212yes—
11#19127425—21no—
12C2×#124×C83C411yescharacteristic
13C2×#117×C83C2211yescharacteristic
14#395×C83C2211yescharacteristic
15#394×C83C2211yescharacteristic
16#393×C83C411yescharacteristic
17#392×C83C411yescharacteristic
18#390×C83—21no—
19#388×C83—21no—
20#388×C83—21no—
21#386×C83—21no—
22#376×C83C411yescharacteristic
23C2×#13050574—22no—
24C4⋊(C83⋊C8)—41no—
25C22⋊(C83⋊C8)—22no—
26C22⋊(C83⋊C8)—41no—
27C22⋊(C83⋊C8)—41no—
28C4×C83⋊C8—41no—
29C22×Q8×C83C2×C411yescharacteristic
30C22×D4×C83—21no—
31C22×D4×C83C2×C411yescharacteristic
32C23×C4×C83D411yescharacteristic
33#124×C83—41no—
34#124×C83—41no—
35#117×C83—41no—
36#117×C83—41no—
37C2×C4⋊C4×C83—21no—
38C2×C4⋊C4×C83D411yescharacteristic
39C2×C4⋊C4×C83C2311yescharacteristic
40C2×C22⋊C4×C83—21no—
41C2×C22⋊C4×C83—21no—
42C2×C22⋊C4×C83—21no—
43C2×C22⋊C4×C83—21no—
44C2×C22⋊C4×C83—21no—
45C2×C22⋊C4×C83—21no—
46C2×C22⋊C4×C83C2×C411yescharacteristic
47C2×C22⋊C4×C83D411yescharacteristic
48C2×C42×C83—21no—
49C2×C42×C83C2×C411yescharacteristic
50C2×C42×C83D411yescharacteristic
51#95×C83—21no—
52#95×C83—21no—
53#95×C83—21no—
54#95×C83C2×C411yescharacteristic
55#95×C83C2×C411yescharacteristic
56#13050574—42no—
57#13050574—42no—
58C2×C83⋊C8—42no—
59C2×C83⋊C8—42no—
60C24×C83—21no—
61C24×C83C22⋊C411yescharacteristic
62C2×Q8×C83—41no—
63C2×Q8×C83—81no—
64C2×D4×C83—41no—
65C2×D4×C83—42no—
66C2×D4×C83—42no—
67C2×D4×C83—81no—
68C2×D4×C83—81no—
69C22×C4×C83—21no—
70C22×C4×C83—21no—
71C22×C4×C83—21no—
72C22×C4×C83—22no—
73C22×C4×C83—21no—
74C22×C4×C83—21no—
75C22×C4×C83—41no—
76C22×C4×C83C22⋊C411yescharacteristic
77C22×C4×C83C22⋊C411yescharacteristic
78C22×C4×C83C22⋊C411yescharacteristic
79C22×C4×C83C22×C411yescharacteristic
80C22×C4×C83C2×D411yescharacteristic
81C22×C4×C83C2×D411yescharacteristic
82C4⋊C4×C83—41no—
83C4⋊C4×C83—41no—
84C4⋊C4×C83—81no—
85C22⋊C4×C83—41no—
86C22⋊C4×C83—41no—
87C22⋊C4×C83—42no—
88C22⋊C4×C83—42no—
89C22⋊C4×C83—41no—
90C22⋊C4×C83—41no—
91C22⋊C4×C83—42no—
92C22⋊C4×C83—81no—
93C22⋊C4×C83—81no—
94C42×C83—21no—
95C42×C83—21no—
96C42×C83—21no—
97C42×C83—21no—
98C42×C83—42no—
99C23×C83—21no—
100C23×C83—21no—
101C23×C83—21no—
102C23×C83—41no—
103C23×C83—41no—
104C23×C83—41no—
105C23×C83—41no—
106C23×C83—41no—
107C23×C83—41no—
108C23×C83—41no—
109C23×C83—41no—
110C23×C83—41no—
111C23×C83C2×C22⋊C411yescharacteristic
112Q8×C83—82no—
113D4×C83—84no—
114D4×C83—82no—
115C2×C4×C83—21no—
116C2×C4×C83—21no—
117C2×C4×C83—21no—
118C2×C4×C83—21no—
119C2×C4×C83—22no—
120C2×C4×C83—21no—
121C2×C4×C83—21no—
122C2×C4×C83—21no—
123C2×C4×C83—21no—
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—41no—
131C2×C4×C83—41no—
132C2×C4×C83—41no—
133C2×C4×C83—41no—
134C2×C4×C83—41no—
135C2×C4×C83—41no—
136C2×C4×C83—41no—
137C2×C4×C83—41no—
138C2×C4×C83#9911yescharacteristic
139C2×C4×C83#9911yescharacteristic
140C2×C4×C83#13512yes—
141C83⋊C8—82no—
142C83⋊C8—82no—
143C22×C83—21no—
144C22×C83—21no—
145C22×C83—42no—
146C22×C83—41no—
147C22×C83—41no—
148C22×C83—41no—
149C22×C83—81no—
150C22×C83—81no—
151C22×C83—81no—
152C22×C83—81no—
153C22×C83—81no—
154C22×C83—81no—
155C22×C83—81no—
156C22×C83—81no—
157C22×C83C2×#9911yescharacteristic
158C22×C83#41611yescharacteristic
159C22×C83#42111yescharacteristic
160C4×C83—22no—
161C4×C83—21no—
162C4×C83—21no—
163C4×C83—42no—
164C4×C83—41no—
165C4×C83—41no—
166C4×C83—41no—
167C4×C83—41no—
168C4×C83—81no—
169C4×C83—81no—
170#11851—831nomaximal, Sylow
171C2×C83—21no—
172C2×C83—21no—
173C2×C83—81no—
174C2×C83—161no—
175C2×C83#151611yescharacteristic
176C2×C83#212211yescharacteristic
177C2×C83#213211yescharacteristic
178#2610D8311yescharacteristic
179#1822—831no—
180#1458—831no—
181#1354—832no—
182#1345—832no—
183C83#1185111yescharacteristic, complemented, Sylow
184C2×#124Dic8311yescharacteristic
185C2×#117C2×D8311yescharacteristic
186#431—1661no—
187#407—1661no—
188#395C2×D8311yescharacteristic
189#394C2×D8311yescharacteristic
190#393Dic8311yescharacteristic
191#392Dic8311yescharacteristic
192#390—21no—
193#388—21no—
194#388—21no—
195#386—21no—
196#376Dic8311yescharacteristic
197#343—832no—
198#343—1661no—
199#324—832no—
200#323—1661no—
201C22×Q8C2×Dic8311yescharacteristic
202C22×D4—21no—
203C22×D4C2×Dic8311yescharacteristic
204C23×C4C22⋊D8311yescharacteristic
205C2×M16—1662no—
206#124—41no—
207#124—41no—
208#117—41no—
209#117—41no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4C22⋊D8311yescharacteristic
212C2×C4⋊C4C22×D8311yescharacteristic
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4C2×Dic8311yescharacteristic
220C2×C22⋊C4C22⋊D8311yescharacteristic
221C2×C42—21no—
222C2×C42C2×Dic8311yescharacteristic
223C2×C42C22⋊D8311yescharacteristic
224C4⋊C8—3321no—
225C22⋊C8—3321no—
226C22⋊C8—3321no—
227C22⋊C8—1662no—
228C4×C8—3321no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95C2×Dic8311yescharacteristic
233#95C2×Dic8311yescharacteristic
234C24—21no—
235C24C22⋊Dic8311yescharacteristic
236C2×Q8—41no—
237C2×Q8—81no—
238C2×D4—41no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—81no—
242C2×D4—81no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—41no—
250C22×C4C22⋊Dic8311yescharacteristic
251C22×C4C22⋊Dic8311yescharacteristic
252C22×C4C22⋊Dic8311yescharacteristic
253C22×C4C22×Dic8311yescharacteristic
254C22×C4C2×C22⋊D8311yescharacteristic
255C22×C4C2×C22⋊D8311yescharacteristic
256M16—3322no—
257M16—3322no—
258C2×C8—3322no—
259C2×C8—3322no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—81no—
263C22⋊C4—41no—
264C22⋊C4—41no—
265C22⋊C4—42no—
266C22⋊C4—42no—
267C22⋊C4—42no—
268C22⋊C4—41no—
269C22⋊C4—41no—
270C22⋊C4—81no—
271C22⋊C4—81no—
272C42—21no—
273C42—21no—
274C42—21no—
275C42—21no—
276C42—42no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23C2×C22⋊Dic8311yescharacteristic
290Q8—82no—
291D4—84no—
292D4—82no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4#1458170711yescharacteristic
317C2×C4#1458170711yescharacteristic
318C2×C4#1458181212yes—
319C8—6642no—
320C8—6642no—
321C22—21no—
322C22—21no—
323C22—42no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22—81no—
333C22—81no—
334C22—81no—
335C22C2×#1458170711yescharacteristic
336C22#1912769711yescharacteristic
337C22#1912771811yescharacteristic
338C4—22no—
339C4—21no—
340C4—21no—
341C4—42no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—81no—
347C4—81no—
348C2—21no—
349C2—21no—
350C2—81no—
351C2—161no—
352C2#2490601911yescharacteristic
353C2#2490911311yescharacteristic
354C2#2490914811yescharacteristic
subgroupG/H
centerC22C2×#14581707
commutatorC22×C4×C83C22×C4
FrattiniC2×C4⋊C4C22×D83
FittingC83×#2610C2
socleC22×C83C2×#99
prime radical#392×C83C4
cosocleC2×C4⋊C4×C83C23
hypercenter#2610D83
hypocenterC83#11851
LCSG ▷ C22×C4×C83 ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2610

Complements

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

NH ≅ G/Ntype
1. #24907546C2semidirect
2. #24905777C2semidirect
3. #24905493C2semidirect
4. #24905466C2semidirect
183. C83#11851semidirect

Maximal subgroups

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

subgroupindex
#249075462
#249057772
#249054932
#249054662
C83×#26102
#1185183