Finite Group Databasebeta

← #25081833

Subgroups — #25081833

Order 11008.

728 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19145718C211yesmaximal, characteristic, complemented
2#19144059C211yesmaximal, characteristic
3C2×#14583104C211yesmaximal, characteristic, complemented
4#19134032C214yesmaximal, complemented
5#19133984C214yesmaximal, complemented
6#19133370C211yesmaximal, characteristic, complemented
7#19133266C211yesmaximal, characteristic, complemented
8#19133242C211yesmaximal, characteristic, complemented
9C43×#2972C211yesmaximal, characteristic, complemented
10C2×#13063351—21no—
11C22×#10621883—21no—
12#14583449C2211yescharacteristic
13C2×#13063325C2211yescharacteristic
14#14583304C2212yescomplemented
15#14583246C2212yescomplemented
16#14583213—22no—
17#14583180—22no—
18#14583114—22no—
19#14583104—22no—
20#14583104C2212yes—
21#14583104C2214yes—
22#14583094C2212yes—
23#14582923C2211yescharacteristic
24C2×C4⋊(C43⋊C8)C2211yescharacteristic
25#14582898—21no—
26C2×#13063248—21no—
27#14582652C2214yescomplemented
28#14582644C2212yescomplemented
29#14582644C2214yescomplemented
30#14582642C2212yescomplemented
31C2×C4×D4×C43C2211yescharacteristic
32C8×D4×C43—22no—
33C8×D4×C43C2214yes—
34C43×#431C2211yescharacteristic
35C43×#431C2211yescharacteristic
36C2×C22⋊C8×C43—22no—
37C2×#10621898—21no—
38C22×Dic86—21no—
39#13063351—44no—
40C2×#10621883—24no—
41C2×#10621883—22no—
42C2×#10621883—44no—
43C2×#10621882—22no—
44C2×#10621877—21no—
45C22×C43⋊C8—21no—
46#13063325—21no—
47#13063325—21no—
48#13063325—21no—
49#13063325C2311yescharacteristic
50#13063325C2312yes—
51#13063322C2311yescharacteristic
52#13063313—21no—
53#13063299—21no—
54#13063299—21no—
55C4×D172—21no—
56C4×D172—21no—
57C2×C4⋊Dic43—21no—
58C2×C4⋊Dic43—21no—
59#13063259—24no—
60C4⋊(C43⋊C8)—21no—
61C4⋊(C43⋊C8)—21no—
62C4⋊(C43⋊C8)C2312yes—
63C4⋊(C43⋊C8)C2312yes—
64#13063248—24no—
65#13063248—22no—
66#13063247—24no—
67#13063245—22no—
68C22×D4×C43D411yescharacteristic
69C23×C4×C43—21no—
70C22×C8×C43—24no—
71C4×D4×C43—21no—
72C4×D4×C43—21no—
73C4×D4×C43D412yes—
74C4×D4×C43C2312yes—
75C2×C4⋊C4×C43D411yescharacteristic
76C2×C22⋊C4×C43—21no—
77C2×C42×C43C2311yescharacteristic
78C4⋊C8×C43D412yes—
79C4⋊C8×C43C2312yes—
80C22⋊C8×C43—21no—
81C22⋊C8×C43—24no—
82C22⋊C8×C43D412yes—
83C4×C8×C43—21no—
84C4×C8×C43C2312yes—
85C24×C43—21no—
86C2×D4×C43—21no—
87C2×D4×C43—22no—
88C2×D4×C43D814yes—
89C2×D4×C43C2×D412yes—
90C22×C4×C43—22no—
91C22×C4×C43—22no—
92C22×C4×C43—21no—
93C22×C4×C43—82no—
94C22×C4×C43C2×D411yescharacteristic
95C22×C4×C43C2×D411yescharacteristic
96C22×C4×C43#4111yescharacteristic
97C2×C22⋊D43—41no—
98C22×Dic43—41no—
99#10621898—42no—
100#10621898—42no—
101C2×D172—21no—
102C2×D172—21no—
103C2×D172—21no—
104C2×C4×D43—41no—
105C2×Dic86—21no—
106C2×Dic86—22no—
107C2×Dic86—42no—
108C2×Dic86—42no—
109C2×Dic86—41no—
110C2×C8×C43—22no—
111C2×C8×C43—84no—
112C2×C8×C43C2×D414yes—
113C4⋊C4×C43—21no—
114C4⋊C4×C43C2×D412yes—
115C22⋊C4×C43—22no—
116C22⋊C4×C43—22no—
117C42×C43C2×D411yescharacteristic
118C42×C43C2×D411yescharacteristic
119C42×C43#4111yescharacteristic
120C42×C43C2411yescharacteristic
121#10621883—48no—
122#10621883—44no—
123#10621882—44no—
124(C2×D43)⋊C4—41no—
125(C2×D43)⋊C4—41no—
126C4⋊Dic43—22no—
127C4⋊Dic43—21no—
128C4⋊Dic43—22no—
129C4⋊Dic43—21no—
130C4⋊Dic43—41no—
131C4⋊Dic43—41no—
132#10621877—42no—
133C2×C43⋊C8—22no—
134C2×C43⋊C8—22no—
135C2×C43⋊C8—42no—
136C23×C43—22no—
137C23×C43—24no—
138C23×C43—81no—
139C23×C43—81no—
140C23×C43#12111yescharacteristic
141C22×D43—41no—
142D4×C43—22no—
143D4×C43—24no—
144D4×C43C2×D814yes—
145C2×C4×C43—22no—
146C2×C4×C43—22no—
147C2×C4×C43—41no—
148C2×C4×C43—44no—
149C2×C4×C43—82no—
150C2×C4×C43—81no—
151C2×C4×C43#12111yescharacteristic
152C2×C4×C43#12111yescharacteristic
153C2×C4×C43#12111yescharacteristic
154C2×C4×C43C2×D812yes—
155C2×C4×C43#13512yes—
156C2×C4×C43C22×D411yescharacteristic
157C2×C4×C43C22×D411yescharacteristic
158C2×C4×C43C2×#4111yescharacteristic
159C22⋊D43—82no—
160C2×Dic43—42no—
161C2×Dic43—41no—
162C2×Dic43—82no—
163D172—41no—
164D172—41no—
165D172—42no—
166D172—81no—
167C4×D43—82no—
168Dic86—42no—
169Dic86—41no—
170Dic86—42no—
171Dic86—41no—
172Dic86—82no—
173C8×C43—24no—
174C8×C43—42no—
175C43⋊C8—42no—
176C43⋊C8—42no—
177#19392—431nomaximal, Sylow
178C22×C43—21no—
179C22×C43—21no—
180C22×C43—24no—
181C22×C43—42no—
182C22×C43—42no—
183C22×C43—84no—
184C22×C43#46612yes—
185C22×C43C2×#12111yescharacteristic
186C2×D43—81no—
187C2×D43—81no—
188C2×D43—81no—
189C4×C43—21no—
190C4×C43—21no—
191C4×C43—21no—
192C4×C43—42no—
193C4×C43#46612yes—
194C4×C43C22×D811yescharacteristic
195C4×C43C2×#13511yescharacteristic
196C4×C43#57712yes—
197Dic43—82no—
198Dic43—81no—
199#3226—431no—
200#3036—431no—
201C2×#461—431no—
202#2972D4311yescharacteristic, complemented
203#1681—434no—
204#1673—434no—
205#1571—431no—
206#1546—431no—
207#1540—431no—
208C2×C43—22no—
209C2×C43—44no—
210C2×C43#304211yescharacteristic
211C2×C43C2×#46611yescharacteristic
212C2×C43#306911yescharacteristic
213D43—161no—
214C2×AGL(1,ℤ/8)—861no—
215C22×SD16—861no—
216#532—431no—
217C2×C4⋊Q8—431no—
218C2×C4×D4C2×D4311yescharacteristic
219#500—432no—
220#486—432no—
221#480—862no—
222#474—862no—
223#463—862no—
224#461—432no—
225#461—434no—
226#461—862no—
227#459—432no—
228C8×D4—22no—
229C8×D4C2×D4314yes—
230#431C2×D4311yescharacteristic
231#431C2×D4311yescharacteristic
232#423—431no—
233C2×C4⋊C8—431no—
234#417—861no—
235C2×#103—861no—
236C2×C22⋊C8—22no—
237#335—434no—
238#332—432no—
239#332—434no—
240#331—432no—
241C43#1939211yescharacteristic, complemented, Sylow
242C2×#41—861no—
243C22×Q8—861no—
244C22×D4D17211yescharacteristic
245C23×C4—21no—
246AGL(1,ℤ/8)—1724no—
247C2×SD16—864no—
248C2×SD16—862no—
249C2×SD16—1724no—
250C2×D8—862no—
251C2×M16—861no—
252C22×C8—24no—
253C22×C8—861no—
254C4⋊Q8—432no—
255C4⋊Q8—431no—
256C4⋊Q8—861no—
257C4⋊Q8—861no—
258C4⋊Q8—861no—
259#127—431no—
260#124—861no—
261#121—861no—
262#121—861no—
263C4×D4—21no—
264C4×D4—21no—
265C4×D4—861no—
266C4×D4—861no—
267C4×D4D17212yes—
268C4×D4C22×D4312yes—
269C2×C4⋊C4—861no—
270C2×C4⋊C4—861no—
271C2×C4⋊C4D17211yescharacteristic
272C2×C22⋊C4—21no—
273C2×C42C22×D4311yescharacteristic
274#107—864no—
275C4⋊C8—432no—
276C4⋊C8—432no—
277C4⋊C8—861no—
278C4⋊C8—861no—
279C4⋊C8C22⋊D4312yes—
280C4⋊C8C22×D4312yes—
281#103—864no—
282#103—862no—
283#102—864no—
284#102—862no—
285C22⋊C8—21no—
286C22⋊C8—24no—
287C22⋊C8C22⋊D4312yes—
288C4×C8—21no—
289C4×C8C22×D4312yes—
290C24—21no—
291#41—1722no—
292#41—1722no—
293C2×Q8—861no—
294C2×Q8—862no—
295C2×Q8—1722no—
296C2×Q8—1722no—
297C2×Q8—1721no—
298C2×D4—21no—
299C2×D4—22no—
300C2×D4—861no—
301C2×D4—861no—
302C2×D4—861no—
303C2×D4—1721no—
304C2×D4D34414yes—
305C2×D4C2×D17212yes—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—82no—
310C22×C4—1721no—
311C22×C4—1721no—
312C22×C4C2×D17211yescharacteristic
313C22×C4#1062189811yescharacteristic
314C22×C4C2×C22⋊D4311yescharacteristic
315SD16—1728no—
316SD16—1724no—
317D8—1724no—
318M16—1722no—
319C2×C8—22no—
320C2×C8—84no—
321C2×C8—862no—
322C2×C8—862no—
323C2×C8—1722no—
324C2×C8C2×C22⋊D4314yes—
325C4⋊C4—21no—
326C4⋊C4—862no—
327C4⋊C4—861no—
328C4⋊C4—862no—
329C4⋊C4—861no—
330C4⋊C4—1721no—
331C4⋊C4—1721no—
332C4⋊C4C2×D17212yes—
333C22⋊C4—22no—
334C22⋊C4—22no—
335C22⋊C4—1721no—
336C22⋊C4—1721no—
337C42C2×D17211yescharacteristic
338C42#1062189811yescharacteristic
339C42C2×C22⋊D4311yescharacteristic
340C42C23×D4311yescharacteristic
341C23—22no—
342C23—24no—
343C23—81no—
344C23—81no—
345C23—1721no—
346C23#1306329911yescharacteristic
347Q8—3442no—
348Q8—1722no—
349Q8—1721no—
350Q8—1722no—
351Q8—1721no—
352D4—24no—
353D4—22no—
354D4—3441no—
355D4—3442no—
356D4—1721no—
357D4—1721no—
358D4—1722no—
359D4C2×D34414yes—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—41no—
363C2×C4—44no—
364C2×C4—82no—
365C2×C4—81no—
366C2×C4—3442no—
367C2×C4—3442no—
368C2×C4—1721no—
369C2×C4—1722no—
370C2×C4#1306329911yescharacteristic
371C2×C4#1306329911yescharacteristic
372C2×C4#1306329911yescharacteristic
373C2×C4C2×D34412yes—
374C2×C4#1306334712yes—
375C2×C4C22×C22⋊D4311yescharacteristic
376C2×C4C22×D17211yescharacteristic
377C2×C4C2×#1062189811yescharacteristic
378C8—24no—
379C8—42no—
380C8—1722no—
381C8—1722no—
382C22—21no—
383C22—21no—
384C22—24no—
385C22—42no—
386C22—42no—
387C22—84no—
388C22—3441no—
389C22—3441no—
390C22—3441no—
391C22#1458313512yes—
392C22C2×#1306329911yescharacteristic
393C4—21no—
394C4—21no—
395C4—21no—
396C4—42no—
397C4—3441no—
398C4—3442no—
399C4#1458313512yes—
400C4C22×D34411yescharacteristic
401C4C2×#1306334711yescharacteristic
402C4#1458371012yes—
403C2—22no—
404C2—44no—
405C2—6881no—
406C2#1914410711yescharacteristic
407C2C2×#1458313511yescharacteristic
408C2#1914433211yescharacteristic
subgroupG/H
centerC22C2×#13063299
commutator, cosocleC42×C43C24
FrattiniC42C23×D43
FittingC43×#2972C2
socleC22×C43C2×#121
prime radical#19144059C2
hypercenter#2972D43
hypocenterC43#19392
LCSG ▷ C42×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2972

Complements

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

NH ≅ G/Ntype
1. #19145718C2semidirect
3. C2×#14583104C2semidirect
4. #19134032C2semidirect
5. #19133984C2semidirect
6. #19133370C2semidirect
7. #19133266C2semidirect
8. #19133242C2semidirect
9. C43×#2972C2semidirect
14. #14583304C22semidirect
15. #14583246C22semidirect
27. #14582652C22semidirect
28. #14582644C22semidirect
29. #14582644C22semidirect
30. #14582642C22semidirect
202. #2972D43semidirect
241. C43#19392semidirect

Maximal subgroups

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

subgroupindex
#191457182
#191440592
C2×#145831042
#191340322
#191339842
#191333702
#191332662
#191332422
C43×#29722
#1939243