Finite Group Databasebeta

← #25104652

Subgroups — #25104652

Order 11008.

834 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583256C211yesmaximal, characteristic
2C4×#13063313C211yesmaximal, characteristic
3#19135561C211yesmaximal, characteristic, complemented
4#19135404C211yesmaximal, characteristic
5#19135291C212yesmaximal
6#19135284C212yesmaximal, complemented
7#19134725C211yesmaximal, characteristic, complemented
8#19134428C212yesmaximal, complemented
9#19134423C212yesmaximal
10C4×#13063236C211yesmaximal, characteristic, complemented
11C4×#125×C43C211yesmaximal, characteristic, complemented
12C2×#13063313C2211yescharacteristic
13C2×C4×Dic86C2211yescharacteristic
14C2×C4×D172C2211yescharacteristic
15#14583256C418yescomplemented
16C2×#13063248C2212yes—
17C2×#13063248C2212yes—
18C2×#13063245C2212yes—
19C2×#13063245C2212yes—
20C2×#13063236C2212yes—
21C2×#13063236C2212yes—
22C2×C4×C43⋊C8C2211yescharacteristic
23C2×C4×C43⋊C8C2211yescharacteristic
24#14582805C2211yescharacteristic
25#14582801C2211yescharacteristic
26#14582790—22no—
27#14582765C2211yescharacteristic
28C4×(C2×D43)⋊C4—22no—
29#14582661C2212yes—
30#14582661C2212yes—
31#14582661C2212yes—
32#14582661C2212yes—
33C2×#125×C43C2211yescharacteristic
34C43×#389C2212yes—
35C43×#387—22no—
36C43×#382C2211yescharacteristic
37C4×C4⋊C4×C43—22no—
38C4×C4⋊C4×C43C2212yes—
39C43×C43C2211yescharacteristic
40C22×Dic86C2311yescharacteristic
41C22×D172C2311yescharacteristic
42C22×C4×D43—21no—
43C22×C43⋊C8C2312yes—
44C22×C43⋊C8C2312yes—
45#13063313—22no—
46#13063313C2×C414yes—
47C4×Dic86—24no—
48C4×D172—24no—
49C2×C4⋊Dic43C2311yescharacteristic
50C2×C4⋊Dic43C2311yescharacteristic
51C2×Dic43⋊C4—21no—
52C2×(C2×D43)⋊C4—22no—
53C2×(C2×D43)⋊C4—21no—
54C2×(C2×D43)⋊C4—22no—
55C2×C4×Dic43—21no—
56#13063248—24no—
57#13063248C2×C418yescomplemented
58#13063245—24no—
59#13063245C2×C418yes—
60#13063236—24no—
61#13063236—24no—
62C4×C43⋊C8—22no—
63C4×C43⋊C8C2×C414yescomplemented
64C43⋊#95—22no—
65C43⋊#95—22no—
66#125×C43—22no—
67#125×C43C2×C414yescomplemented
68C2×C4⋊C4×C43—22no—
69C2×C4⋊C4×C43—22no—
70C2×C4⋊C4×C43C2312yes—
71C2×C4⋊C4×C43C2312yes—
72C2×C42×C43—22no—
73C2×C42×C43D412yes—
74C2×C42×C43D412yes—
75C2×C42×C43C2311yescharacteristic
76C2×C42×C43C2311yescharacteristic
77C2×C42×C43C2311yescharacteristic
78#95×C43—22no—
79#95×C43—22no—
80#95×C43—22no—
81C23×D43—21no—
82C22×C4×C43—22no—
83C22×C4×C43—22no—
84C22×C4×C43C2×D412yes—
85C22×C4×C43C2×D412yes—
86C22×C4×C43C2×D411yescharacteristic
87C22×C4×C43C2×D411yescharacteristic
88C22×C4×C43C2411yescharacteristic
89C22×Dic43—21no—
90C22×Dic43—21no—
91C22×Dic43—21no—
92C2×D172—21no—
93C2×D172—22no—
94C2×D172—22no—
95C2×D172C22×C412yes—
96C2×C4×D43—44no—
97C2×C4×D43—42no—
98C2×Dic86—22no—
99C2×Dic86—22no—
100C2×Dic86—21no—
101C2×Dic86C22×C412yes—
102C4⋊C4×C43—28no—
103C4⋊C4×C43—22no—
104C4⋊C4×C43—22no—
105C4⋊C4×C43—22no—
106C4⋊C4×C43—44no—
107C4⋊C4×C43C22×C414yes—
108C42×C43—22no—
109C42×C43—42no—
110C42×C43—42no—
111C42×C43—44no—
112C42×C43C22×C412yes—
113C42×C43#4112yes—
114C42×C43#4114yes—
115(C2×D43)⋊C4—28no—
116(C2×D43)⋊C4—44no—
117(C2×D43)⋊C4—42no—
118C4⋊Dic43—22no—
119C4⋊Dic43—22no—
120Dic43⋊C4—42no—
121C4×Dic43—42no—
122C2×C43⋊C8—22no—
123C2×C43⋊C8—22no—
124C2×C43⋊C8—22no—
125C2×C43⋊C8—22no—
126C2×C43⋊C8C22×C414yes—
127C2×C43⋊C8#4114yes—
128C23×C43C22×D411yescharacteristic
129C22×D43—22no—
130C22×D43—41no—
131C22×D43—41no—
132C22×D43—42no—
133C22×D43—42no—
134C2×C4×C43—22no—
135C2×C4×C43—22no—
136C2×C4×C43—22no—
137C2×C4×C43—22no—
138C2×C4×C43—24no—
139C2×C4×C43—22no—
140C2×C4×C43—22no—
141C2×C4×C43—22no—
142C2×C4×C43—42no—
143C2×C4×C43—42no—
144C2×C4×C43—42no—
145C2×C4×C43—42no—
146C2×C4×C43—42no—
147C2×C4×C43C4×D414yes—
148C2×C4×C43#12412yes—
149C2×C4×C43#12412yes—
150C2×C4×C43C23×C411yescharacteristic
151C2×C4×C43C2×#4112yes—
152C2×C4×C43C2×#4111yescharacteristic
153C2×C4×C43C2×#4112yes—
154C2×Dic43—22no—
155C2×Dic43—41no—
156C2×Dic43—41no—
157C2×Dic43—41no—
158C2×Dic43—41no—
159C2×Dic43—42no—
160C2×Dic43—42no—
161D172—24no—
162D172—42no—
163C4×D43—84no—
164Dic86—24no—
165Dic86—42no—
166C43⋊C8—24no—
167C43⋊C8—24no—
168#21648—431nomaximal, Sylow
169C22×C43C2×C4×D411yescharacteristic
170C22×C43C2×#12411yescharacteristic
171C22×C43#53211yescharacteristic
172C22×C43#57512yes—
173C22×C43#57512yes—
174C2×D43—44no—
175C2×D43—42no—
176C2×D43—81no—
177C2×D43—81no—
178C2×D43—82no—
179C4×C43—22no—
180C4×C43—22no—
181C4×C43—24no—
182C4×C43—42no—
183C4×C43—44no—
184C4×C43—82no—
185C4×C43C4×#4112yes—
186C4×C43#52912yes—
187Dic43—42no—
188Dic43—81no—
189Dic43—81no—
190C2×#488—431no—
191C4×#125D4311yescharacteristic, complemented
192C4×#124—431no—
193#1991—431no—
194#1958—431no—
195#1936—432no—
196#1935—432no—
197#1835—431no—
198#1763—432no—
199#1762—432no—
200C4×#97—431no—
201C2×C43C4×#12411yescharacteristic
202C2×C43#294312yes—
203C2×C43#314011yescharacteristic
204C2×C43#314211yescharacteristic
205C2×C43#318312yes—
206D43—82no—
207C2×#125C2×D4311yescharacteristic
208C2×#124—431no—
209C2×C4×Q8—431no—
210C2×C4×D4—431no—
211#488—438no—
212C2×#103—432no—
213C2×#103—432no—
214C2×#102—432no—
215C2×#102—432no—
216C2×#97—432no—
217C2×#97—432no—
218C2×C4×C8—431no—
219C2×C4×C8—431no—
220#391—431no—
221#390—431no—
222#389C2×D4312yes—
223#388—862no—
224#387—22no—
225#382—431no—
226#382C2×D4311yescharacteristic
227C4×C4⋊C4—22no—
228C4×C4⋊C4C2×D4312yes—
229C4×C22⋊C4—862no—
230C43C2×D4311yescharacteristic
231#340—432no—
232#340—432no—
233#340—432no—
234#340—432no—
235C43#2164811yescharacteristic, complemented, Sylow
236C22×Q8—431no—
237C22×D4—431no—
238C23×C4—861no—
239C22×C8—432no—
240C22×C8—432no—
241#125—22no—
242#125C4×D4314yescomplemented
243#124—434no—
244#124—862no—
245C4×Q8—864no—
246C4×D4—864no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—431no—
250C2×C4⋊C4—431no—
251C2×C4⋊C4—861no—
252C2×C4⋊C4C22×D4312yes—
253C2×C4⋊C4C22×D4312yes—
254C2×C22⋊C4—862no—
255C2×C22⋊C4—861no—
256C2×C22⋊C4—862no—
257C2×C42—22no—
258C2×C42—861no—
259C2×C42C22⋊D4312yes—
260C2×C42C22⋊D4312yes—
261C2×C42C22×D4311yescharacteristic
262C2×C42C22×D4311yescharacteristic
263C2×C42C22×D4311yescharacteristic
264#103—438no—
265#103—864no—
266#102—438no—
267#102—864no—
268#97—864no—
269#97—864no—
270C4×C8—434no—
271C4×C8—862no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—862no—
276#95—862no—
277C24—861no—
278C2×Q8—432no—
279C2×Q8—861no—
280C2×Q8—862no—
281C2×Q8—862no—
282C2×D4—432no—
283C2×D4—862no—
284C2×D4—861no—
285C2×D4—862no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—861no—
289C22×C4—861no—
290C22×C4—861no—
291C22×C4—1724no—
292C22×C4—1722no—
293C22×C4C2×C22⋊D4312yes—
294C22×C4C2×C22⋊D4312yes—
295C22×C4C2×C22⋊D4311yescharacteristic
296C22×C4C2×C22⋊D4311yescharacteristic
297C22×C4C23×D4311yescharacteristic
298C2×C8—434no—
299C2×C8—434no—
300C2×C8—862no—
301C2×C8—862no—
302C2×C8—862no—
303C2×C8—862no—
304C4⋊C4—28no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—44no—
309C4⋊C4—862no—
310C4⋊C4—862no—
311C4⋊C4—1722no—
312C4⋊C4C2×C4×D4314yes—
313C22⋊C4—868no—
314C22⋊C4—1724no—
315C22⋊C4—1722no—
316C42—22no—
317C42—42no—
318C42—42no—
319C42—44no—
320C42—1722no—
321C42C2×C4×D4312yes—
322C42#1062189812yes—
323C42#1062190214yes—
324C23—862no—
325C23—1721no—
326C23—1722no—
327C23—1722no—
328C23—1721no—
329C23C22×C22⋊D4311yescharacteristic
330Q8—864no—
331Q8—1722no—
332D4—864no—
333D4—1722no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—24no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—862no—
348C2×C4—3444no—
349C2×C4—1721no—
350C2×C4—1721no—
351C2×C4—1722no—
352C2×C4—1722no—
353C2×C4—1721no—
354C2×C4—1721no—
355C2×C4C4×C22⋊D4314yes—
356C2×C4#1306331412yes—
357C2×C4#1306331412yes—
358C2×C4C22×C4×D4311yescharacteristic
359C2×C4#41×D4312yes—
360C2×C4C2×#1062189811yescharacteristic
361C2×C4C2×#1062190212yes—
362C8—864no—
363C8—864no—
364C22—3441no—
365C22—3441no—
366C22—3442no—
367C22—1722no—
368C22—1724no—
369C22C2×C4×C22⋊D4311yescharacteristic
370C22C2×#1306331411yescharacteristic
371C22#1458344511yescharacteristic
372C22#1458369712yes—
373C22#1458369712yes—
374C4—22no—
375C4—22no—
376C4—24no—
377C4—42no—
378C4—44no—
379C4—82no—
380C4—3441no—
381C4—3441no—
382C4—1722no—
383C4C4×#1062190212yes—
384C4#1458343612yes—
385C2—3442no—
386C2C4×#1306331411yescharacteristic
387C2#1914339312yes—
388C2#1914498411yescharacteristic
389C2#1914499811yescharacteristic
390C2#1914531812yes—
subgroupG/H
centerC23C22×C22⋊D43
commutatorC2×C4×C43C23×C4
FrattiniC22×C4C23×D43
FittingC4×#125×C43C2
socleC23×C43C22×D4
prime radicalC22×D172C23
cosocleC22×C4×C43C24
hypercenterC4×#125D43
hypocenterC43#21648
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C43 ◁ C4×#125

Complements

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

NH ≅ G/Ntype
3. #19135561C2semidirect
6. #19135284C2semidirect
7. #19134725C2semidirect
8. #19134428C2semidirect
10. C4×#13063236C2semidirect
11. C4×#125×C43C2semidirect
15. #14583256C4semidirect
57. #13063248C2×C4semidirect
63. C4×C43⋊C8C2×C4semidirect
67. #125×C43C2×C4semidirect
191. C4×#125D43semidirect
235. C43#21648semidirect
242. #125C4×D43semidirect

Maximal subgroups

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

subgroupindex
C2×#145832562
C4×#130633132
#191355612
#191354042
#191352912
#191352842
#191347252
#191344282
#191344232
C4×#130632362
C4×#125×C432
#2164843