Finite Group Databasebeta

← #11871311

Subgroups — #11871311

Order 1152.

1332 conjugacy classes of subgroups.

The subgroup lattice has 411 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×C3⋊Dic3C211yesmaximal, characteristic
2#10596093C211yesmaximal, characteristic, complemented
3C32×#450C211yesmaximal, characteristic
4#10592917C211yesmaximal, characteristic
5#10592863C212yesmaximal, complemented
6#10592861C212yesmaximal, complemented
7#10592743C211yesmaximal, characteristic, complemented
8#10592458C212yesmaximal, complemented
9#10592456C212yesmaximal, complemented
10#10592421C211yesmaximal, characteristic, complemented
11#10592418C211yesmaximal, characteristic, complemented
12#80134—34nomaximal
13C22×C32×D4C2211yescharacteristic
14C23×C3⋊Dic3—21no—
15C2×C32×SD16C2212yes—
16C32×C22⋊Q8C2211yescharacteristic
17C2×#3757—21no—
18Q8×C3⋊Dic3C2211yescharacteristic, complemented
19#64153C2211yescharacteristic, complemented
20D4×C3⋊Dic3—21no—
21D4×C3⋊Dic3—21no—
22D4×C3⋊Dic3C2212yes—
23D4×C3⋊Dic3C2212yes—
24C2×#3754C2212yes—
25C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
26#64133C2211yescharacteristic
27C2×C4×C3⋊Dic3C2211yescharacteristic
28#64098—21no—
29#64097C2211yescharacteristic, complemented
30#64096—21no—
31#64096C2211yescharacteristic, complemented
32#64087—21no—
33#64086—21no—
34C32×#102C2212yes—
35C32×C22⋊C8C2211yescharacteristic
36#63662C2212yescomplemented
37#63659C2212yes—
38#63657C2211yescharacteristic
39#63645C2212yescomplemented
40#63643C2212yescomplemented
41#63642C2212yescomplemented
42#63641C2212yescomplemented
43#63634C2212yescomplemented
44C2×D4×Dic3—34no—
45#5622—34no—
46C3×#450D314yes—
47#5218—34no—
48#5164—38no—
49#5162—38no—
50#5044—34no—
51#4759—38no—
52#4757—38no—
53#4722—34no—
54#4719—34no—
55C24×C32—21no—
56C2×C32×Q8C2311yescharacteristic
57C2×C32×D4—24no—
58C2×C32×D4—21no—
59C2×C32×D4C2312yes—
60C22×C32×C4C2311yescharacteristic
61C22×C3⋊Dic3—21no—
62C22×C3⋊Dic3—22no—
63C22×C3⋊Dic3—22no—
64C22×C3⋊Dic3—44no—
65C22×C3⋊Dic3D411yescharacteristic
66C22×C3⋊Dic3D411yescharacteristic
67C2×#634—21no—
68C2×#634C2311yescharacteristic
69C32×SD16—24no—
70C2×C32×C8C2312yes—
71C32×C4⋊C4—21no—
72C32×C4⋊C4C2311yescharacteristic
73C32×C22⋊C4—21no—
74#3757—21no—
75#3757—22no—
76#3757—22no—
77#3754—24no—
78C4⋊(C3⋊Dic3)—21no—
79C4⋊(C3⋊Dic3)—21no—
80C4⋊(C3⋊Dic3)C2311yescharacteristic
81C4⋊(C3⋊Dic3)C2312yes—
82#3750—21no—
83#3750—21no—
84#3750—21no—
85#3750—21no—
86#3750—21no—
87C4×C3⋊Dic3—21no—
88C4×C3⋊Dic3—21no—
89C4×C3⋊Dic3D411yescharacteristic
90C4×C3⋊Dic3D411yescharacteristic
91C4×C3⋊Dic3C2311yescharacteristic
92C4×C3⋊Dic3C2311yescharacteristic
93C2×C3⋊(C3⋊C8)C2312yes—
94#3110—91noSylow
95C22×C3×D4C2×D314yes—
96C23×Dic3—64no—
97C2×C3×SD16C2×D318yes—
98C3×C22⋊Q8C2×D314yes—
99C2×#217—64no—
100Q8×Dic3—34no—
101#944—34no—
102D4×Dic3—38no—
103D4×Dic3—38no—
104D4×Dic3—64no—
105D4×Dic3—64no—
106C2×#214—38no—
107C2×C4⋊Dic3—34no—
108#924—34no—
109C2×C4×Dic3—34no—
110#889—64no—
111#888—34no—
112#887—34no—
113#887—64no—
114#878—64no—
115#877—64no—
116C3×#102C2×D318yes—
117C3×C22⋊C8C2×D314yes—
118#835—38no—
119#832—38no—
120#830—34no—
121#818—38no—
122#816—38no—
123#815—38no—
124Dic3⋊C8—38no—
125#807—38no—
126C23×C32—24no—
127C23×C32—22no—
128C23×C32—42no—
129C23×C32—42no—
130C23×C32C2×D411yescharacteristic
131C32×Q8—22no—
132C32×D4—24no—
133C32×D4—22no—
134C32×D4#4114yes—
135C2×C32×C4—22no—
136C2×C32×C4—21no—
137C2×C32×C4—21no—
138C2×C32×C4C2×D411yescharacteristic
139C2×C32×C4C2411yescharacteristic
140C2×C3⋊Dic3—22no—
141C2×C3⋊Dic3—22no—
142C2×C3⋊Dic3—21no—
143C2×C3⋊Dic3—21no—
144C2×C3⋊Dic3—22no—
145C2×C3⋊Dic3—44no—
146C2×C3⋊Dic3—42no—
147C2×C3⋊Dic3—44no—
148C2×C3⋊Dic3C2×D411yescharacteristic
149C2×C3⋊Dic3C2×D411yescharacteristic
150C2×C3⋊Dic3C2×D411yescharacteristic
151C2×C3⋊Dic3C2×D411yescharacteristic
152#634—22no—
153#634—42no—
154C32×C8—22no—
155C3⋊(C3⋊C8)—22no—
156#533—91no—
157C2×C4×D4—91no—
158#484—91no—
159#484—91no—
160#474—92no—
161#461—92no—
162#450—91no—
163#450C3⋊D311yescharacteristic
164#440—92no—
165#440—92no—
166#432—91no—
167C24×C3—24no—
168C2×C3×Q8C22×D314yes—
169C2×C3×D4—216no—
170C2×C3×D4—24no—
171C2×C3×D4C22×D318yes—
172C22×C3×C4C22×D314yes—
173C22×Dic3—34no—
174C22×Dic3—34no—
175C22×Dic3—68no—
176C22×Dic3—68no—
177C22×Dic3—64no—
178C22×Dic3—1216no—
179C2×Dic6—34no—
180C2×Dic6—64no—
181C3×SD16—216no—
182C2×C3×C8C22×D318yes—
183C3×C4⋊C4—24no—
184C3×C4⋊C4C22×D314yes—
185C3×C22⋊C4—24no—
186#217—68no—
187#217—68no—
188#217—64no—
189#214—616no—
190C4⋊Dic3—38no—
191C4⋊Dic3—34no—
192C4⋊Dic3—64no—
193C4⋊Dic3—64no—
194Dic3⋊C4—64no—
195Dic3⋊C4—64no—
196Dic3⋊C4—64no—
197Dic3⋊C4—64no—
198Dic3⋊C4—64no—
199C4×Dic3—34no—
200C4×Dic3—34no—
201C4×Dic3—34no—
202C4×Dic3—34no—
203C4×Dic3—64no—
204C4×Dic3—64no—
205C2×C3⋊C8—38no—
206C22×C32—21no—
207C22×C32—21no—
208C22×C32—24no—
209C22×C32—48no—
210C22×C32—42no—
211C22×C32—42no—
212C22×C32#13712yes—
213C22×C32C22×D411yescharacteristic
214C32×C4—21no—
215C32×C4—41no—
216C32×C4—41no—
217C32×C4C2×#4112yes—
218C3⋊Dic3—21no—
219C3⋊Dic3—21no—
220C3⋊Dic3—21no—
221C3⋊Dic3—41no—
222C3⋊Dic3—41no—
223C3⋊Dic3—42no—
224C3⋊Dic3AGL(1,ℤ/8)12yes—
225C22×D4C2×C3⋊D311yescharacteristic
226C23×C4—181no—
227C2×SD16—92no—
228C2×SD16C2×C3⋊D312yes—
229C4⋊Q8—91no—
230C4⋊Q8—91no—
231#125—181no—
232C22⋊Q8—91no—
233C22⋊Q8—181no—
234C22⋊Q8C2×C3⋊D311yescharacteristic
235C4×Q8—91no—
236C4×Q8—91no—
237C4×Q8—181no—
238C4×D4—92no—
239C4×D4—92no—
240C4×D4—181no—
241C4×D4—181no—
242#117—181no—
243C2×C4⋊C4—91no—
244C2×C22⋊C4—181no—
245C2×C42—91no—
246#107—92no—
247#107—92no—
248C4⋊C8—92no—
249#103—92no—
250#103—92no—
251#102—92no—
252#102C2×C3⋊D312yes—
253C22⋊C8—91no—
254C22⋊C8C2×C3⋊D311yescharacteristic
255#97—92no—
256C23×C3—216no—
257C23×C3—28no—
258C23×C3—48no—
259C23×C3—48no—
260C23×C3D3×D414yes—
261C3×Q8—28no—
262C3×D4—216no—
263C3×D4—28no—
264C3×D4#237116yes—
265C2×C3×C4—28no—
266C2×C3×C4—24no—
267C2×C3×C4—24no—
268C2×C3×C4D3×D414yes—
269C2×C3×C4C23×D314yes—
270C2×Dic3—34no—
271C2×Dic3—34no—
272C2×Dic3—34no—
273C2×Dic3—34no—
274C2×Dic3—68no—
275C2×Dic3—68no—
276C2×Dic3—64no—
277C2×Dic3—64no—
278C2×Dic3—68no—
279C2×Dic3—1216no—
280C2×Dic3—1216no—
281C2×Dic3—128no—
282Dic6—68no—
283Dic6—128no—
284C3×C8—28no—
285C3⋊C8—68no—
286C2×C32—22no—
287C2×C32—44no—
288C2×C32#52511yescharacteristic
289C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
290C2×C32C2×#13711yescharacteristic
291C24—21no—
292C2×Q8—91no—
293C2×Q8—181no—
294C2×Q8C22×C3⋊D311yescharacteristic
295C2×D4—24no—
296C2×D4—21no—
297C2×D4C22×C3⋊D312yes—
298C22×C4—91no—
299C22×C4—91no—
300C22×C4—182no—
301C22×C4—182no—
302C22×C4—181no—
303C22×C4—364no—
304C22×C4C22×C3⋊D311yescharacteristic
305SD16—24no—
306SD16—184no—
307C2×C8—92no—
308C2×C8C22×C3⋊D312yes—
309C4⋊C4—21no—
310C4⋊C4—91no—
311C4⋊C4—92no—
312C4⋊C4—181no—
313C4⋊C4—181no—
314C4⋊C4—181no—
315C4⋊C4—181no—
316C4⋊C4—181no—
317C4⋊C4—181no—
318C4⋊C4—181no—
319C4⋊C4C22×C3⋊D311yescharacteristic
320C22⋊C4—21no—
321C22⋊C4—182no—
322C22⋊C4—182no—
323C22⋊C4—181no—
324C42—91no—
325C42—91no—
326C42—91no—
327C42—91no—
328C42—181no—
329C42—181no—
330C22×C3—24no—
331C22×C3—24no—
332C22×C3—216no—
333C22×C3—432no—
334C22×C3—48no—
335C22×C3—48no—
336C22×C3#91518yes—
337C22×C3C2×D3×D414yes—
338C3×C4—24no—
339C3×C4—44no—
340C3×C4—44no—
341C3×C4C2×#23718yes—
342Dic3—38no—
343Dic3—64no—
344Dic3—64no—
345Dic3—64no—
346Dic3—124no—
347Dic3—128no—
348Dic3—124no—
349C32#311011yescharacteristic, complemented, Sylow
350C23—24no—
351C23—22no—
352C23—42no—
353C23—42no—
354C23D4×C3⋊D311yescharacteristic
355Q8—22no—
356Q8—182no—
357Q8—362no—
358D4—24no—
359D4—22no—
360D4#383014yes—
361C2×C4—22no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—91no—
365C2×C4—91no—
366C2×C4—91no—
367C2×C4—91no—
368C2×C4—182no—
369C2×C4—182no—
370C2×C4—181no—
371C2×C4—181no—
372C2×C4—182no—
373C2×C4—364no—
374C2×C4—364no—
375C2×C4—362no—
376C2×C4D4×C3⋊D311yescharacteristic
377C2×C4C23×C3⋊D311yescharacteristic
378C8—22no—
379C8—182no—
380C2×C3—28no—
381C2×C3—416no—
382C2×C3#558614yes—
383C2×C3C2×#91514yes—
384C2×C3D3×AGL(1,ℤ/8)14yes—
385C22—21no—
386C22—21no—
387C22—24no—
388C22—48no—
389C22—42no—
390C22—42no—
391C22#6412412yes—
392C22C2×D4×C3⋊D311yescharacteristic
393C4—21no—
394C4—41no—
395C4—41no—
396C4—92no—
397C4—181no—
398C4—181no—
399C4—181no—
400C4—362no—
401C4—361no—
402C4—361no—
403C4C2×#383012yes—
404C3#8013414yescomplemented
405C2—22no—
406C2—44no—
407C2#1059605711yescharacteristic
408C2C2×#6412411yescharacteristic
409C2C3⋊D3×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D4×C3⋊D3
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C23×C3⋊D3
FittingC32×#450C2
socleC22×C32C22×D4
prime radicalC22×C32×D4C22
hypercenter#450C3⋊D3
hypocenterC32#3110
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
2. #10596093C2semidirect
5. #10592863C2semidirect
6. #10592861C2semidirect
7. #10592743C2semidirect
8. #10592458C2semidirect
9. #10592456C2semidirect
10. #10592421C2semidirect
11. #10592418C2semidirect
18. Q8×C3⋊Dic3C22semidirect
19. #64153C22semidirect
29. #64097C22semidirect
31. #64096C22semidirect
36. #63662C22semidirect
39. #63645C22semidirect
40. #63643C22semidirect
41. #63642C22semidirect
42. #63641C22semidirect
43. #63634C22semidirect
349. C32#3110semidirect
404. C3#80134semidirect

Maximal subgroups

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

subgroupindex
C2×D4×C3⋊Dic32
#105960932
C32×#4502
#105929172
#105928632
#105928612
#105927432
#105924582
#105924562
#105924212
#105924182
#801343