Finite Group Databasebeta

← #25113386

Subgroups — #25113386

Order 11008.

1430 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×C43⋊C8C211yesmaximal, characteristic, complemented
2Q8×C4⋊Dic43C211yesmaximal, characteristic, complemented
3#19135523C213yesmaximal, complemented
4#19135402C213yesmaximal, complemented
5#19134595C213yesmaximal, complemented
6C4×#13063246C213yesmaximal, complemented
7C2×C43×#550C211yesmaximal, characteristic
8C2×Q8×Dic43—21no—
9Q8×C43⋊C8—24no—
10C2×C4⋊(C43⋊C8)C2213yes—
11C2×#13063246C2213yes—
12C2×#13063246C2213yes—
13C2×#13063246C2216yes—
14C2×C4×C43⋊C8C2213yes—
15#14582798—23no—
16#14582769C2213yes—
17C4×C4⋊Dic43C2213yes—
18#14582661C2213yes—
19#14582661C2211yescharacteristic
20C22×#41×C43—21no—
21C43×#550—24no—
22C43×#550C418yescomplemented
23C2×#127×C43C2213yes—
24C2×#121×C43—23no—
25C2×C4×Q8×C43C2211yescharacteristic
26C2×C4×D4×C43C2213yes—
27C22×C43⋊C8C2313yes—
28C22×C43⋊C8C2311yescharacteristic
29Q8×Dic43—44no—
30C2×C4⋊Dic43—23no—
31C2×C4⋊Dic43—23no—
32C2×C4⋊Dic43C2313yes—
33C2×C4⋊Dic43C2311yescharacteristic
34C2×C4×Dic43—23no—
35C4⋊(C43⋊C8)—26no—
36#13063246—26no—
37#13063246—26no—
38#13063246—46no—
39C4×C43⋊C8—26no—
40C43⋊#95—23no—
41C2×#41×C43—28no—
42C2×#41×C43—44no—
43C2×#41×C43—44no—
44C22×Q8×C43D411yescharacteristic
45C22×D4×C43—23no—
46C22×D4×C43—23no—
47C22×D4×C43C2313yes—
48C23×C4×C43—23no—
49#127×C43—26no—
50#127×C43C2×C4112yes—
51#121×C43—224no—
52C4×Q8×C43C2×C414yes—
53C4×Q8×C43D414yes—
54C4×D4×C43—26no—
55C4×D4×C43C2×C4112yes—
56C2×C4⋊C4×C43D413yes—
57C2×C4⋊C4×C43C2311yescharacteristic
58C2×C22⋊C4×C43—23no—
59C2×C42×C43C2313yes—
60C24×C43—23no—
61#41×C43—432no—
62C2×Q8×C43C22⋊C414yes—
63C2×Q8×C43D814yes—
64C2×Q8×C43SD1614yes—
65C2×D4×C43—212no—
66C2×D4×C43—212no—
67C2×D4×C43—23no—
68C2×D4×C43—23no—
69C2×D4×C43—23no—
70C2×D4×C43—23no—
71C2×D4×C43—23no—
72C2×D4×C43—46no—
73C2×D4×C43—46no—
74C2×D4×C43—83no—
75C2×D4×C43—83no—
76C2×D4×C43C22×C416yes—
77C22×C4×C43—212no—
78C22×C4×C43—21no—
79C22×C4×C43—46no—
80C22×C4×C43—46no—
81C22×C4×C43C2×D413yes—
82C22×C4×C43C2×D413yes—
83C22×C4×C43C2411yescharacteristic
84C22×Dic43—21no—
85C22×Dic43—23no—
86C4⋊C4×C43—21no—
87C4⋊C4×C43C22⋊C4112yes—
88C4⋊C4×C43C22×C412yes—
89C22⋊C4×C43—212no—
90C42×C43C22×C416yes—
91C42×C43C2×D416yes—
92C4⋊Dic43—21no—
93C4⋊Dic43—23no—
94C4⋊Dic43—23no—
95C4⋊Dic43—46no—
96C4⋊Dic43—83no—
97C4⋊Dic43#4112yescomplemented
98C4×Dic43—46no—
99C2×C43⋊C8—21no—
100C2×C43⋊C8—21no—
101C2×C43⋊C8—23no—
102C2×C43⋊C8—23no—
103C2×C43⋊C8—23no—
104C2×C43⋊C8#4112yescomplemented
105C23×C43—26no—
106C23×C43—43no—
107C23×C43—43no—
108C23×C43—43no—
109C23×C43—43no—
110C23×C43—43no—
111C23×C43—43no—
112C23×C43C22×D411yescharacteristic
113Q8×C43#102116yescomplemented
114D4×C43—26no—
115D4×C43—26no—
116D4×C43—43no—
117D4×C43—43no—
118D4×C43—424no—
119D4×C43—812no—
120C2×C4×C43—23no—
121C2×C4×C43—23no—
122C2×C4×C43—22no—
123C2×C4×C43—41no—
124C2×C4×C43—41no—
125C2×C4×C43—424no—
126C2×C4×C43C2×C22⋊C416yes—
127C2×C4×C43C2×C22⋊C416yes—
128C2×C4×C43C2×D816yes—
129C2×C4×C43C2×SD1616yes—
130C2×C4×C43C23×C411yescharacteristic
131C2×C4×C43C22×D411yescharacteristic
132C2×C4×C43C2×#4111yescharacteristic
133C2×C4×C43C2×#4111yescharacteristic
134C2×C4×C43H(5,𝔽2)11yescharacteristic
135C2×C4×C432−1+411yescharacteristic
136C2×Dic43—22no—
137C2×Dic43—41no—
138C2×Dic43—41no—
139C2×Dic43—43no—
140C2×Dic43—43no—
141C2×Dic43—43no—
142C43⋊C8—22no—
143C43⋊C8—43no—
144#22540—431nomaximal, Sylow
145C22×C43—46no—
146C22×C43—46no—
147C22×C43—46no—
148C22×C43—83no—
149C22×C43—83no—
150C22×C43—83no—
151C22×C43—83no—
152C22×C43C22×C22⋊C411yescharacteristic
153C22×C43#54811yescharacteristic
154C22×C43#55011yescharacteristic
155C22×C43C22×D811yescharacteristic
156C22×C43C22×SD1611yescharacteristic
157C22×C43#57611yescharacteristic
158C22×C43#57711yescharacteristic
159C4×C43—26no—
160C4×C43—42no—
161C4×C43C2×#102112yes—
162C4×C43C4×#4111yescharacteristic
163C4×C43#52011yescharacteristic
164C4×C43#54811yescharacteristic
165C4×C43#55011yescharacteristic
166Dic43—42no—
167Dic43—83no—
168C2×#550D4311yescharacteristic
169C2×C8×Q8—431no—
170Q8×C4⋊C4—431no—
171#1982—433no—
172#1957—433no—
173#1802—433no—
174C4×#102—433no—
175C2×C43—86no—
176C2×C43#234111yescharacteristic
177C2×C43C22×#10211yescharacteristic
178C2×C43#289711yescharacteristic
179C2×C43#335811yescharacteristic
180C2×C43#336211yescharacteristic
181C2×C43#339111yescharacteristic
182C2×C43#339211yescharacteristic
183C22×#41—21no—
184#550—24no—
185#550Dic4318yescomplemented
186C2×#127C2×D4313yes—
187C2×#121—23no—
188C2×C4×Q8—861no—
189C2×C4×Q8C2×D4311yescharacteristic
190C2×C4×D4C2×D4313yes—
191C8×Q8—864no—
192C2×C4⋊C8—433no—
193C2×#102—433no—
194C2×#102—433no—
195C2×#102—436no—
196C2×C4×C8—433no—
197#389—863no—
198#384—433no—
199C4×C4⋊C4—433no—
200#340—433no—
201#340—431no—
202C43#2254011yescharacteristic, complemented, Sylow
203C2×#41—28no—
204C2×#41—44no—
205C2×#41—44no—
206C22×Q8C22⋊D4311yescharacteristic
207C22×D4—23no—
208C22×D4—23no—
209C22×D4C22×D4313yes—
210C23×C4—23no—
211C22×C8—431no—
212C22×C8—433no—
213#127—26no—
214#127C2×Dic43112yes—
215#121—224no—
216C4×Q8—1724no—
217C4×Q8C2×Dic4314yes—
218C4×Q8C22⋊D4314yes—
219C4×D4—26no—
220C4×D4C2×Dic43112yes—
221C2×C4⋊C4—431no—
222C2×C4⋊C4—433no—
223C2×C4⋊C4—863no—
224C2×C4⋊C4—863no—
225C2×C4⋊C4C22⋊D4313yes—
226C2×C4⋊C4C22×D4311yescharacteristic
227C2×C22⋊C4—23no—
228C2×C42—863no—
229C2×C42C22×D4313yes—
230C4⋊C8—866no—
231#102—866no—
232#102—866no—
233#102—1726no—
234C4×C8—866no—
235#95—863no—
236C24—23no—
237#41—432no—
238C2×Q8#1062188214yes—
239C2×Q8#1062188314yes—
240C2×Q8C22⋊Dic4314yes—
241C2×D4—212no—
242C2×D4—212no—
243C2×D4—23no—
244C2×D4—23no—
245C2×D4—23no—
246C2×D4—23no—
247C2×D4—23no—
248C2×D4—46no—
249C2×D4—46no—
250C2×D4—83no—
251C2×D4—83no—
252C2×D4C22×Dic4316yes—
253C22×C4—21no—
254C22×C4—212no—
255C22×C4—46no—
256C22×C4—46no—
257C22×C4—861no—
258C22×C4—863no—
259C22×C4C2×C22⋊D4313yes—
260C22×C4C2×C22⋊D4313yes—
261C22×C4C23×D4311yescharacteristic
262C2×C8—432no—
263C2×C8—861no—
264C2×C8—861no—
265C2×C8—863no—
266C2×C8—863no—
267C2×C8—863no—
268C4⋊C4—21no—
269C4⋊C4—432no—
270C4⋊C4—861no—
271C4⋊C4—863no—
272C4⋊C4—863no—
273C4⋊C4—3443no—
274C4⋊C4—1726no—
275C4⋊C4C22⋊Dic43112yes—
276C4⋊C4C22×Dic4312yes—
277C22⋊C4—212no—
278C42—1726no—
279C42C22×Dic4316yes—
280C42C2×C22⋊D4316yes—
281C23—26no—
282C23—43no—
283C23—43no—
284C23—43no—
285C23—43no—
286C23—43no—
287C23—43no—
288C23C22×C22⋊D4311yescharacteristic
289Q8#13063246116yescomplemented
290D4—26no—
291D4—26no—
292D4—43no—
293D4—43no—
294D4—424no—
295D4—812no—
296C2×C4—23no—
297C2×C4—23no—
298C2×C4—22no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—424no—
302C2×C4—862no—
303C2×C4—1723no—
304C2×C4—1723no—
305C2×C4—1723no—
306C2×C4—1721no—
307C2×C4—1721no—
308C2×C4C2×C22⋊Dic4316yes—
309C2×C4C2×C22⋊Dic4316yes—
310C2×C4C2×#1062188216yes—
311C2×C4C2×#1062188316yes—
312C2×C4C23×Dic4311yescharacteristic
313C2×C4C22×C22⋊D4311yescharacteristic
314C2×C4#41×D4311yescharacteristic
315C2×C4#41×D4311yescharacteristic
316C2×C4#1306337111yescharacteristic
317C2×C4#1306337211yescharacteristic
318C8—862no—
319C8—1723no—
320C22—46no—
321C22—46no—
322C22—46no—
323C22—83no—
324C22—83no—
325C22—83no—
326C22—83no—
327C22C22×C22⋊Dic4311yescharacteristic
328C22#1458355611yescharacteristic
329C22#1458356811yescharacteristic
330C22C22×#1062188211yescharacteristic
331C22C22×#1062188311yescharacteristic
332C22#1458370211yescharacteristic
333C22#1458371011yescharacteristic
334C4—26no—
335C4—42no—
336C4—3443no—
337C4—1722no—
338C4C2×#13063246112yes—
339C4#41×Dic4311yescharacteristic
340C4#1458338211yescharacteristic
341C4#1458355611yescharacteristic
342C4#1458356811yescharacteristic
343C2—86no—
344C2#1913743611yescharacteristic
345C2C22×#1306324611yescharacteristic
346C2#1914312111yescharacteristic
347C2#1914722611yescharacteristic
348C2#1914726211yescharacteristic
349C2#1914756711yescharacteristic
350C2#1914757611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D43
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
Fitting, prime radicalC2×C43×#550C2
socleC23×C43C22×D4
cosocleC22×C4×C43C24
hypercenterC2×#550D43
hypocenterC43#22540
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#550

Complements

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

NH ≅ G/Ntype
1. C2×Q8×C43⋊C8C2semidirect
2. Q8×C4⋊Dic43C2semidirect
3. #19135523C2semidirect
4. #19135402C2semidirect
5. #19134595C2semidirect
6. C4×#13063246C2semidirect
22. C43×#550C4semidirect
97. C4⋊Dic43#41semidirect
104. C2×C43⋊C8#41semidirect
113. Q8×C43#102semidirect
185. #550Dic43semidirect
202. C43#22540semidirect
289. Q8#13063246semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×C43⋊C82
Q8×C4⋊Dic432
#191355232
#191354022
#191345952
C4×#130632462
C2×C43×#5502
#2254043