Finite Group Databasebeta

← #35308016

Subgroups — #35308016

Order 21248.

408 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 101 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#24907459C211yesmaximal, characteristic
2#24906233C211yesmaximal, characteristic
3#24906135C211yesmaximal, characteristic
4#24906135C211yesmaximal, characteristic
5#24905858C211yesmaximal, characteristic
6#24905360C211yesmaximal, characteristic
7C2×#462×C83C211yesmaximal, characteristic
8#19127754C2211yescharacteristic
9#19127754C2211yescharacteristic
10#19127725C2211yescharacteristic
11#19127725C2211yescharacteristic
12C2×#14581711—21no—
13#19127470—21no—
14#19127451—22no—
15#19127447—22no—
16#19127447—22no—
17Q8⋊(C83⋊C8)—22no—
18C22×Q16×C83—21no—
19C2×C4⋊Q8×C83C2211yescharacteristic
20C2×C4×Q8×C83C2211yescharacteristic
21#462×C83—22no—
22#462×C83—22no—
23C2×C4⋊C8×C83C2211yescharacteristic
24C2×#103×C83—21no—
25C2×#13050574—21no—
26C2×#13050574—21no—
27C4⋊(C83⋊C8)—21no—
28C4⋊(C83⋊C8)—21no—
29C4⋊(C83⋊C8)—21no—
30C4⋊(C83⋊C8)—21no—
31C4⋊(C83⋊C8)—21no—
32C4⋊(C83⋊C8)D412yes—
33#14581711—42no—
34C22⋊(C83⋊C8)—21no—
35C22⋊(C83⋊C8)—21no—
36C4×C83⋊C8—21no—
37C4×C83⋊C8—21no—
38C22×Q8×C83—21no—
39C22×Q8×C83D411yescharacteristic
40C2×Q16×C83—24no—
41C2×Q16×C83—42no—
42C2×Q16×C83—42no—
43C22×C8×C83—21no—
44C4⋊Q8×C83—21no—
45C4⋊Q8×C83—21no—
46C4⋊Q8×C83—21no—
47C4⋊Q8×C83D412yes—
48C4×Q8×C83—21no—
49C4×Q8×C83—21no—
50C4×Q8×C83—21no—
51C4×Q8×C83—21no—
52C2×C4⋊C4×C83—21no—
53C2×C4⋊C4×C83—21no—
54C2×C4⋊C4×C83—21no—
55C2×C4⋊C4×C83D411yescharacteristic
56C2×C42×C83—21no—
57C2×C42×C83C2311yescharacteristic
58C4⋊C8×C83—21no—
59C4⋊C8×C83D412yes—
60#103×C83—24no—
61#13050574—41no—
62#13050574—41no—
63#13050574—41no—
64#13050574—41no—
65C2×C83⋊C8—22no—
66C2×C83⋊C8—41no—
67C2×C83⋊C8—41no—
68C2×C83⋊C8—41no—
69C2×Q8×C83—21no—
70C2×Q8×C83—21no—
71C2×Q8×C83—22no—
72C2×Q8×C83—21no—
73C2×Q8×C83—21no—
74C2×Q8×C83—21no—
75C2×Q8×C83—41no—
76C2×Q8×C83—41no—
77C2×Q8×C83—41no—
78C2×Q8×C83—41no—
79C2×Q8×C83—41no—
80C2×Q8×C83SD1612yes—
81C22×C4×C83—21no—
82C22×C4×C83—21no—
83C22×C4×C83—41no—
84C22×C4×C83C2×D411yescharacteristic
85C22×C4×C83#4111yescharacteristic
86C22×C4×C83#4111yescharacteristic
87Q16×C83—48no—
88C2×C8×C83—22no—
89C2×C8×C83—41no—
90C2×C8×C83—41no—
91C4⋊C4×C83—21no—
92C4⋊C4×C83—22no—
93C4⋊C4×C83—21no—
94C4⋊C4×C83—22no—
95C4⋊C4×C83—41no—
96C4⋊C4×C83—41no—
97C4⋊C4×C83—41no—
98C4⋊C4×C83—41no—
99C42×C83—41no—
100C42×C83—41no—
101C42×C83C2×D411yescharacteristic
102C42×C83C2×D411yescharacteristic
103C42×C83C2×D411yescharacteristic
104C42×C83#4111yescharacteristic
105C23×C83#12311yescharacteristic
106Q8×C83—22no—
107Q8×C83—22no—
108Q8×C83—41no—
109Q8×C83—41no—
110Q8×C83—42no—
111Q8×C83—42no—
112Q8×C83—81no—
113Q8×C83—81no—
114C2×C4×C83—22no—
115C2×C4×C83—22no—
116C2×C4×C83—41no—
117C2×C4×C83—41no—
118C2×C4×C83—41no—
119C2×C4×C83—41no—
120C2×C4×C83—42no—
121C2×C4×C83—81no—
122C2×C4×C83—81no—
123C2×C4×C83—81no—
124C2×C4×C83#12011yescharacteristic
125C2×C4×C83#12111yescharacteristic
126C2×C4×C83#12111yescharacteristic
127C2×C4×C83#12111yescharacteristic
128C2×C4×C83#12311yescharacteristic
129C2×C4×C83#12411yescharacteristic
130C2×C4×C83C2×SD1611yescharacteristic
131C2×C4×C83#13511yescharacteristic
132C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
133C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
134C8×C83—42no—
135C83⋊C8—42no—
136C83⋊C8—81no—
137C83⋊C8—81no—
138C83⋊C8—81no—
139C22×C83—21no—
140C22×C83—21no—
141C22×C83#39411yescharacteristic
142C22×C83#48111yescharacteristic
143C22×C83#48211yescharacteristic
144C4×C83—21no—
145C4×C83—21no—
146C4×C83—21no—
147C4×C83—42no—
148C4×C83—81no—
149C4×C83—82no—
150C4×C83#44911yescharacteristic
151C4×C83#45011yescharacteristic
152C4×C83#46511yescharacteristic
153C4×C83#46911yescharacteristic
154C4×C83#47312yes—
155#10447—831nomaximal, Sylow
156C2×C83—21no—
157C2×C83—21no—
158C2×C83#203411yescharacteristic
159C2×C83#204711yescharacteristic
160C2×C83#204911yescharacteristic
161C2×#462D8311yescharacteristic
162#1801—831no—
163#1570—831no—
164#1546—831no—
165#1546—831no—
166#1479—831no—
167#1301—831no—
168C83#1044711yescharacteristic, complemented, Sylow
169C22×Q16—21no—
170C2×C4⋊Q8C2×D8311yescharacteristic
171C2×C4×Q8C2×D8311yescharacteristic
172#462—22no—
173#462—22no—
174#431—831no—
175#431—831no—
176#423—831no—
177#423—831no—
178C2×C4⋊C8C2×D8311yescharacteristic
179C2×#103—21no—
180C2×#101—1661no—
181#343—1661no—
182#334—1662no—
183#332—1662no—
184#332—1662no—
185Q8⋊C8—1662no—
186C22×Q8—21no—
187C22×Q8C22⋊D8311yescharacteristic
188C2×Q16—24no—
189C2×Q16—42no—
190C2×Q16—42no—
191C2×M16—1661no—
192C2×M16—1661no—
193C22×C8—21no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8—21no—
197C4⋊Q8C22⋊D8312yes—
198C4×Q8—21no—
199C4×Q8—21no—
200C4×Q8—21no—
201C4×Q8—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4C22⋊D8311yescharacteristic
206C2×C42—21no—
207C2×C42C22×D8311yescharacteristic
208C4⋊C8—21no—
209C4⋊C8—832no—
210C4⋊C8—1661no—
211C4⋊C8—1661no—
212C4⋊C8—1661no—
213C4⋊C8—1661no—
214C4⋊C8—1661no—
215C4⋊C8C22⋊D8312yes—
216#103—24no—
217#101—3322no—
218C22⋊C8—1661no—
219C22⋊C8—1661no—
220C4×C8—1661no—
221C4×C8—1661no—
222C2×Q8—21no—
223C2×Q8—21no—
224C2×Q8—22no—
225C2×Q8—21no—
226C2×Q8—21no—
227C2×Q8—21no—
228C2×Q8—41no—
229C2×Q8—41no—
230C2×Q8—41no—
231C2×Q8—41no—
232C2×Q8—41no—
233C2×Q8#1305057812yes—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—41no—
237C22×C4C2×C22⋊D8311yescharacteristic
238C22×C4#1305058911yescharacteristic
239C22×C4#1305058911yescharacteristic
240Q16—48no—
241M16—3321no—
242M16—3321no—
243M16—3321no—
244M16—3321no—
245C2×C8—22no—
246C2×C8—41no—
247C2×C8—41no—
248C2×C8—3321no—
249C2×C8—3321no—
250C2×C8—3321no—
251C2×C8—1662no—
252C4⋊C4—21no—
253C4⋊C4—22no—
254C4⋊C4—21no—
255C4⋊C4—22no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C42—41no—
261C42—41no—
262C42D4×D8311yescharacteristic
263C42C2×C22⋊D8311yescharacteristic
264C42C2×C22⋊D8311yescharacteristic
265C42#1305058911yescharacteristic
266C23#1458177311yescharacteristic
267Q8—22no—
268Q8—22no—
269Q8—41no—
270Q8—41no—
271Q8—42no—
272Q8—42no—
273Q8—81no—
274Q8—81no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—42no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4#1458176111yescharacteristic
286C2×C4#1458176311yescharacteristic
287C2×C4#1458176411yescharacteristic
288C2×C4#1458176411yescharacteristic
289C2×C4#1458177311yescharacteristic
290C2×C4#1458177911yescharacteristic
291C2×C4C2×#1305057811yescharacteristic
292C2×C4#1458181211yescharacteristic
293C2×C4#1458181711yescharacteristic
294C2×C4#1458181811yescharacteristic
295C8—42no—
296C8—3322no—
297C8—6641no—
298C8—6641no—
299C8—6641no—
300C22—21no—
301C22—21no—
302C22#1912761511yescharacteristic
303C22#1912801711yescharacteristic
304C22#1912802211yescharacteristic
305C4—21no—
306C4—21no—
307C4—21no—
308C4—42no—
309C4—82no—
310C4—81no—
311C4#1912784611yescharacteristic
312C4#1912785211yescharacteristic
313C4#1912792811yescharacteristic
314C4#1912795211yescharacteristic
315C4#1912797312yes—
316C2—21no—
317C2—21no—
318C2#2490864911yescharacteristic
319C2#2490872811yescharacteristic
320C2#2490874011yescharacteristic
subgroupG/H
centerC22#19127615
commutator, cosocleC2×C42×C83C23
FrattiniC2×C42C22×D83
FittingC2×#462×C83C2
socleC22×C83#394
prime radicalC23×C83#123
hypercenterC2×#462D83
hypocenterC83#10447
LCSG ▷ C2×C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#462

Complements

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

NH ≅ G/Ntype
168. C83#10447semidirect

Maximal subgroups

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

subgroupindex
#249074592
#249062332
#249061352
#249058582
#249053602
C2×#462×C832
#1044783