Finite Group Databasebeta

← #35308051

Subgroups — #35308051

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#24907455C211yesmaximal, characteristic
2#24906233C211yesmaximal, characteristic, complemented
3#24906148C211yesmaximal, characteristic, complemented
4#24906144C211yesmaximal, characteristic, complemented
5#24905854C211yesmaximal, characteristic
6#24905360C211yesmaximal, characteristic, complemented
7C2×#476×C83C211yesmaximal, characteristic
8#19127754C2211yescharacteristic
9#19127754C2211yescharacteristic
10#19127725C2211yescharacteristic
11#19127725C2211yescharacteristic
12C2×#14581709—21no—
13#19127470—21no—
14#19127451—22no—
15#19127450—22no—
16#19127447—22no—
17D4⋊(C83⋊C8)—22no—
18C2×C4⋊Q8×C83C2211yescharacteristic
19C2×C4×D4×C83C2211yescharacteristic
20#476×C83—22no—
21#476×C83—22no—
22C2×#106×C83—21no—
23C2×C4⋊C8×C83C2211yescharacteristic
24C2×#102×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#14581709—42no—
34C22⋊(C83⋊C8)—21no—
35C22⋊(C83⋊C8)—21no—
36C4×C83⋊C8—21no—
37C4×C83⋊C8—21no—
38C22×Q8×C83—21no—
39C22×D4×C83D411yescharacteristic
40C23×C4×C83—21no—
41C22×C8×C83—21no—
42C4⋊Q8×C83—21no—
43C4⋊Q8×C83—21no—
44C4⋊Q8×C83—21no—
45C4⋊Q8×C83D412yes—
46C4×D4×C83—21no—
47C4×D4×C83—21no—
48C4×D4×C83—21no—
49C4×D4×C83—21no—
50C2×C4⋊C4×C83—21no—
51C2×C4⋊C4×C83—21no—
52C2×C4⋊C4×C83D411yescharacteristic
53C2×C22⋊C4×C83—21no—
54C2×C42×C83C2311yescharacteristic
55#106×C83—24no—
56C4⋊C8×C83—21no—
57C4⋊C8×C83D412yes—
58#102×C83—24no—
59#13050574—41no—
60#13050574—41no—
61#13050574—41no—
62#13050574—41no—
63C2×C83⋊C8—22no—
64C2×C83⋊C8—41no—
65C2×C83⋊C8—41no—
66C2×C83⋊C8—41no—
67C24×C83—21no—
68C2×Q8×C83—22no—
69C2×Q8×C83—41no—
70C2×Q8×C83—82no—
71C2×D4×C83—21no—
72C2×D4×C83—21no—
73C2×D4×C83—21no—
74C2×D4×C83—21no—
75C2×D4×C83—21no—
76C2×D4×C83SD1612yes—
77C22×C4×C83—21no—
78C22×C4×C83—41no—
79C22×C4×C83—41no—
80C22×C4×C83—41no—
81C22×C4×C83—82no—
82C22×C4×C83C2×D411yescharacteristic
83C22×C4×C83#4111yescharacteristic
84C22×C4×C83#4111yescharacteristic
85C2×C8×C83—22no—
86C2×C8×C83—41no—
87C2×C8×C83—41no—
88C4⋊C4×C83—22no—
89C4⋊C4×C83—21no—
90C4⋊C4×C83—21no—
91C4⋊C4×C83—22no—
92C4⋊C4×C83—41no—
93C4⋊C4×C83—41no—
94C22⋊C4×C83—41no—
95C22⋊C4×C83—41no—
96C42×C83C2×D411yescharacteristic
97C42×C83C2×D411yescharacteristic
98C42×C83C2×D411yescharacteristic
99C42×C83#4111yescharacteristic
100C23×C83—22no—
101C23×C83—41no—
102C23×C83—41no—
103C23×C83—81no—
104C23×C83—81no—
105C23×C83#12311yescharacteristic
106Q8×C83—82no—
107Q8×C83—82no—
108D4×C83—22no—
109D4×C83—22no—
110D4×C83—41no—
111D4×C83—41no—
112C2×C4×C83—22no—
113C2×C4×C83—41no—
114C2×C4×C83—41no—
115C2×C4×C83—42no—
116C2×C4×C83—82no—
117C2×C4×C83—82no—
118C2×C4×C83—82no—
119C2×C4×C83—81no—
120C2×C4×C83#12011yescharacteristic
121C2×C4×C83#12111yescharacteristic
122C2×C4×C83#12111yescharacteristic
123C2×C4×C83#12111yescharacteristic
124C2×C4×C83#12311yescharacteristic
125C2×C4×C83#12411yescharacteristic
126C2×C4×C83C2×SD1611yescharacteristic
127C2×C4×C83#13511yescharacteristic
128C2×C4×C83#13711yescharacteristic
129C2×C4×C83#13711yescharacteristic
130C8×C83—42no—
131C83⋊C8—42no—
132C83⋊C8—81no—
133C83⋊C8—81no—
134C83⋊C8—81no—
135C22×C83—21no—
136C22×C83—21no—
137C22×C83—42no—
138C22×C83—82no—
139C22×C83—82no—
140C22×C83—81no—
141C22×C83—81no—
142C22×C83#39411yescharacteristic
143C22×C83#48311yescharacteristic
144C22×C83#48511yescharacteristic
145C4×C83—21no—
146C4×C83—21no—
147C4×C83—21no—
148C4×C83—82no—
149C4×C83—81no—
150C4×C83#44811yescharacteristic
151C4×C83#45211yescharacteristic
152C4×C83#46511yescharacteristic
153C4×C83#47011yescharacteristic
154C4×C83#47112yes—
155#10452—831nomaximal, Sylow
156C2×C83—21no—
157C2×C83—21no—
158C2×C83—82no—
159C2×C83#203511yescharacteristic
160C2×C83#204611yescharacteristic
161C2×C83#204811yescharacteristic
162C2×#476D8311yescharacteristic
163#1800—831no—
164#1570—831no—
165#1549—831no—
166#1548—831no—
167#1478—831no—
168#1301—831no—
169C83#1045211yescharacteristic, complemented, Sylow
170C2×C4⋊Q8C2×D8311yescharacteristic
171C2×C4×D4C2×D8311yescharacteristic
172#476—22no—
173#476—22no—
174#431—831no—
175#431—831no—
176C2×#106—21no—
177#423—831no—
178#423—831no—
179C2×C4⋊C8C2×D8311yescharacteristic
180C2×#102—21no—
181C2×#100—1661no—
182#343—1661no—
183#334—1662no—
184#333—1662no—
185#332—1662no—
186D4⋊C8—1662no—
187C22×Q8—21no—
188C22×D4C22⋊D8311yescharacteristic
189C23×C4—21no—
190C2×M16—1661no—
191C2×M16—1661no—
192C22×C8—21no—
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8C22⋊D8312yes—
197C4×D4—21no—
198C4×D4—21no—
199C4×D4—21no—
200C4×D4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4C22⋊D8311yescharacteristic
204C2×C22⋊C4—21no—
205C2×C42C22×D8311yescharacteristic
206#106—24no—
207C4⋊C8—21no—
208C4⋊C8—832no—
209C4⋊C8—1661no—
210C4⋊C8—1661no—
211C4⋊C8—1661no—
212C4⋊C8—1661no—
213C4⋊C8—1661no—
214C4⋊C8C22⋊D8312yes—
215#102—24no—
216#100—3322no—
217C22⋊C8—1661no—
218C22⋊C8—1661no—
219C4×C8—1661no—
220C4×C8—1661no—
221C24—21no—
222C2×Q8—22no—
223C2×Q8—41no—
224C2×Q8—82no—
225C2×D4—21no—
226C2×D4—21no—
227C2×D4—21no—
228C2×D4—21no—
229C2×D4—21no—
230C2×D4#1305057712yes—
231C22×C4—21no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—82no—
236C22×C4C2×C22⋊D8311yescharacteristic
237C22×C4#1305058911yescharacteristic
238C22×C4#1305058911yescharacteristic
239M16—3321no—
240M16—3321no—
241M16—3321no—
242M16—3321no—
243C2×C8—22no—
244C2×C8—41no—
245C2×C8—41no—
246C2×C8—3321no—
247C2×C8—3321no—
248C2×C8—3321no—
249C2×C8—1662no—
250C4⋊C4—22no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—22no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C22⋊C4—41no—
257C22⋊C4—41no—
258C42D4×D8311yescharacteristic
259C42C2×C22⋊D8311yescharacteristic
260C42C2×C22⋊D8311yescharacteristic
261C42#1305058911yescharacteristic
262C23—22no—
263C23—41no—
264C23—41no—
265C23—81no—
266C23—81no—
267C23#1458177311yescharacteristic
268Q8—82no—
269Q8—82no—
270D4—22no—
271D4—22no—
272D4—41no—
273D4—41no—
274C2×C4—22no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—42no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4—81no—
282C2×C4#1458176111yescharacteristic
283C2×C4#1458176311yescharacteristic
284C2×C4#1458176411yescharacteristic
285C2×C4#1458176411yescharacteristic
286C2×C4#1458177311yescharacteristic
287C2×C4#1458177911yescharacteristic
288C2×C4C2×#1305057711yescharacteristic
289C2×C4#1458181211yescharacteristic
290C2×C4#1458182211yescharacteristic
291C2×C4#1458182311yescharacteristic
292C8—42no—
293C8—3322no—
294C8—6641no—
295C8—6641no—
296C8—6641no—
297C22—21no—
298C22—21no—
299C22—42no—
300C22—82no—
301C22—82no—
302C22—81no—
303C22—81no—
304C22#1912761511yescharacteristic
305C22#1912802911yescharacteristic
306C22#1912803911yescharacteristic
307C4—21no—
308C4—21no—
309C4—21no—
310C4—81no—
311C4—82no—
312C4#1912784011yescharacteristic
313C4#1912786311yescharacteristic
314C4#1912793211yescharacteristic
315C4#1912795711yescharacteristic
316C4#1912796112yes—
317C2—21no—
318C2—21no—
319C2—82no—
320C2#2490865611yescharacteristic
321C2#2490872111yescharacteristic
322C2#2490873311yescharacteristic
subgroupG/H
centerC22#19127615
commutator, cosocleC2×C42×C83C23
FrattiniC2×C42C22×D83
FittingC2×#476×C83C2
socleC22×C83#394
prime radicalC22×D4×C83D4
hypercenterC2×#476D83
hypocenterC83#10452
LCSG ▷ C2×C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#476

Complements

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

NH ≅ G/Ntype
2. #24906233C2semidirect
3. #24906148C2semidirect
4. #24906144C2semidirect
6. #24905360C2semidirect
169. C83#10452semidirect

Maximal subgroups

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

subgroupindex
#249074552
#249062332
#249061482
#249061442
#249058542
#249053602
C2×#476×C832
#1045283