Finite Group Databasebeta

← #11865756

Subgroups — #11865756

Order 1152.

1488 conjugacy classes of subgroups.

The subgroup lattice has 453 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64128C212yesmaximal, complemented
2#10596196C211yesmaximal, characteristic, complemented
3C4×#3832C211yesmaximal, characteristic, complemented
4C32×#462C211yesmaximal, characteristic, complemented
5#10592884C212yesmaximal, complemented
6#10592782C211yesmaximal, characteristic, complemented
7#10592726C211yesmaximal, characteristic, complemented
8#10592542C211yesmaximal, characteristic, complemented
9C3⋊D3×C4⋊C8C211yesmaximal, characteristic, complemented
10#10592518C212yesmaximal, complemented
11#10592503C212yesmaximal, complemented
12#80475—34nomaximal
13C2×#3832C2212yes—
14C2×#3832C2211yescharacteristic
15C2×C32×Q16C2212yes—
16C32×C4⋊Q8C2211yescharacteristic
17C32×C4×Q8C2211yescharacteristic
18#64156C2212yes—
19Q8×C3⋊Dic3C2211yescharacteristic
20C2×#3755C2212yes—
21C2×#3755C2212yes—
22#64128—28no—
23C2×#3745C2212yes—
24C2×C8×C3⋊D3C2212yes—
25#64104—21no—
26#64104—21no—
27#64102—21no—
28#64102—21no—
29#64102—21no—
30#64102C2211yescharacteristic
31#64101—21no—
32#64101C2211yescharacteristic
33#64083C2211yescharacteristic
34C4×C4⋊(C3⋊D3)—21no—
35C4×C4⋊(C3⋊D3)C2211yescharacteristic
36C42×C3⋊D3—21no—
37C42×C3⋊D3C2211yescharacteristic
38C32×C4⋊C8C2211yescharacteristic
39C32×#103C2212yes—
40#63662C2212yes—
41#63648C2212yes—
42#63641C2212yes—
43#63636C2212yes—
44C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
45C2×#919—38no—
46#5725—34no—
47C4×#239—34no—
48C3×#462D314yescomplemented
49#5185—38no—
50#5083—34no—
51#5027—34no—
52#4843—34no—
53D3×C4⋊C8—34no—
54#4819—38no—
55#4804—38no—
56C2×C32×Q8C2312yes—
57C2×C32×Q8C2311yescharacteristic
58#3832—22no—
59#3832—24no—
60#3832—24no—
61#3832D414yescomplemented
62C2×C4⋊(C3⋊D3)—21no—
63C2×C4⋊(C3⋊D3)—22no—
64C2×C4⋊(C3⋊D3)—21no—
65C2×C4⋊(C3⋊D3)—21no—
66C2×C4⋊(C3⋊D3)C2312yes—
67C2×C4⋊(C3⋊D3)C2311yescharacteristic
68C2×C4×C3⋊D3—21no—
69C2×C4×C3⋊D3—21no—
70C2×C4×C3⋊D3—21no—
71C2×C4×C3⋊D3—22no—
72C2×C4×C3⋊D3D411yescharacteristic
73C2×C4×C3⋊D3D411yescharacteristic
74C2×C4×C3⋊D3C2311yescharacteristic
75C32×Q16—24no—
76C2×C32×C8C2312yes—
77C32×C4⋊C4—21no—
78C32×C4⋊C4—21no—
79C32×C4⋊C4—21no—
80C32×C4⋊C4C2311yescharacteristic
81C32×C42—21no—
82C32×C42C2311yescharacteristic
83#3755—24no—
84#3755—24no—
85#3752—21no—
86#3752—21no—
87#3752—21no—
88#3752—22no—
89#3752—22no—
90C4⋊(C3⋊Dic3)—21no—
91C4⋊(C3⋊Dic3)C2311yescharacteristic
92C4×C3⋊Dic3—21no—
93C4×C3⋊Dic3—21no—
94C4×C3⋊Dic3D411yescharacteristic
95C4×C3⋊Dic3D411yescharacteristic
96C4×C3⋊Dic3C2311yescharacteristic
97C2×C3⋊(C3⋊C8)C2312yes—
98#3745—24no—
99C8×C3⋊D3—24no—
100#3035—91noSylow
101C2×#239—34no—
102C2×#239—38no—
103C2×C3×Q16C2×D318yes—
104C3×C4⋊Q8C2×D314yes—
105C3×C4×Q8C2×D314yes—
106#947—38no—
107Q8×Dic3—34no—
108C2×#215—38no—
109C2×#215—38no—
110#919—632no—
111C2×D24—38no—
112C2×D3×C8—38no—
113#895—64no—
114#895—64no—
115#893—34no—
116#893—64no—
117#893—64no—
118#893—64no—
119#892—34no—
120#892—64no—
121#874—34no—
122C4×D12—34no—
123C4×D12—64no—
124C42×D3—34no—
125C42×D3—64no—
126C3×C4⋊C8C2×D314yes—
127C3×#103C2×D318yes—
128#835—38no—
129#821—38no—
130Dic3⋊C8—38no—
131#809—38no—
132C4⋊(C3⋊C8)—34no—
133C22×C3⋊D3—21no—
134C22×C3⋊D3—22no—
135C22×C3⋊D3C2×D411yescharacteristic
136C32×Q8—21no—
137C32×Q8—22no—
138C32×Q8—22no—
139C32×Q8C2×D412yes—
140C2×C32×C4—21no—
141C2×C32×C4—21no—
142C2×C32×C4—22no—
143C2×C32×C4C2×D411yescharacteristic
144C2×C32×C4C2×D411yescharacteristic
145C2×C32×C4C2411yescharacteristic
146C2×C3⋊Dic3—21no—
147C2×C3⋊Dic3C2×D411yescharacteristic
148C2×C3⋊Dic3C2×D411yescharacteristic
149C2×C3⋊Dic3C2×D411yescharacteristic
150C4⋊(C3⋊D3)—24no—
151C4⋊(C3⋊D3)—22no—
152C4⋊(C3⋊D3)—21no—
153C4⋊(C3⋊D3)—22no—
154C4⋊(C3⋊D3)—42no—
155C4⋊(C3⋊D3)—44no—
156C4⋊(C3⋊D3)—42no—
157C4⋊(C3⋊D3)C2×D412yes—
158C4×C3⋊D3—22no—
159C4×C3⋊D3—24no—
160C4×C3⋊D3—41no—
161C4×C3⋊D3—42no—
162C4×C3⋊D3—42no—
163C4×C3⋊D3—44no—
164C4×C3⋊D3C2×D412yes—
165C4×C3⋊D3#4112yes—
166C32×C8—22no—
167C3⋊(C3⋊C8)—22no—
168C2×#135—92no—
169#532—91no—
170C4×#41—91no—
171#464—92no—
172#463—92no—
173#462C3⋊D311yescharacteristic, complemented
174#461—91no—
175#460—91no—
176#459—91no—
177C2×C4⋊C8—91no—
178#416—92no—
179C2×C3×Q8C22×D318yes—
180C2×C3×Q8C22×D314yes—
181#239—316no—
182#239—68no—
183#239—616no—
184#239—616no—
185C2×D12—34no—
186C2×D12—38no—
187C2×D12—64no—
188C2×D12—68no—
189C2×D12—64no—
190C2×D12—64no—
191C2×C4×D3—34no—
192C2×C4×D3—34no—
193C2×C4×D3—34no—
194C2×C4×D3—64no—
195C2×C4×D3—64no—
196C2×C4×D3—68no—
197C2×C4×D3—64no—
198C3×Q16—216no—
199C2×C3×C8C22×D318yes—
200C3×C4⋊C4—24no—
201C3×C4⋊C4—24no—
202C3×C4⋊C4—24no—
203C3×C4⋊C4C22×D314yes—
204C3×C42—24no—
205C3×C42C22×D314yes—
206#215—616no—
207#215—616no—
208(C2×D3)⋊C4—64no—
209(C2×D3)⋊C4—68no—
210(C2×D3)⋊C4—68no—
211(C2×D3)⋊C4—64no—
212(C2×D3)⋊C4—64no—
213C4⋊Dic3—34no—
214C4⋊Dic3—64no—
215C4×Dic3—34no—
216C4×Dic3—34no—
217C4×Dic3—34no—
218C4×Dic3—64no—
219C4×Dic3—64no—
220C2×C3⋊C8—38no—
221D24—616no—
222D3×C8—616no—
223C22×C32C22×D411yescharacteristic
224C2×C3⋊D3—21no—
225C2×C3⋊D3—21no—
226C2×C3⋊D3—22no—
227C2×C3⋊D3—41no—
228C2×C3⋊D3—41no—
229C2×C3⋊D3—42no—
230C2×C3⋊D3—42no—
231C2×C3⋊D3—42no—
232C2×C3⋊D3#12112yes—
233C32×C4—22no—
234C32×C4—21no—
235C32×C4—42no—
236C32×C4—41no—
237C32×C4#13512yescomplemented
238C32×C4C22×D411yescharacteristic
239C32×C4C2×#4111yescharacteristic
240C3⋊Dic3—21no—
241C3⋊Dic3—21no—
242C3⋊Dic3—41no—
243C3⋊Dic3#12112yes—
244C2×#41—91no—
245C2×#41—92no—
246#135—188no—
247C2×Q16C2×C3⋊D312yes—
248C2×SD16—92no—
249C2×SD16—92no—
250C2×D8—92no—
251C22×C8—92no—
252C4⋊Q8C2×C3⋊D311yescharacteristic
253#127—91no—
254#124—92no—
255#121—181no—
256#121—181no—
257C4×Q8—91no—
258C4×Q8C2×C3⋊D311yescharacteristic
259C4×D4—91no—
260C4×D4—91no—
261C4×D4—181no—
262C4×D4—181no—
263C4×D4—181no—
264C4×D4—181no—
265#117—91no—
266#117—181no—
267C2×C42—91no—
268C2×C42—181no—
269C4⋊C8—92no—
270C4⋊C8—91no—
271C4⋊C8C2×C3⋊D311yescharacteristic
272#103—92no—
273#103C2×C3⋊D312yes—
274#102—92no—
275#102—92no—
276C22×D3—34no—
277C22×D3—64no—
278C22×D3—68no—
279C3×Q8—24no—
280C3×Q8—28no—
281C3×Q8—28no—
282C3×Q8D3×D418yes—
283C2×C3×C4—24no—
284C2×C3×C4—24no—
285C2×C3×C4—28no—
286C2×C3×C4D3×D414yes—
287C2×C3×C4D3×D414yes—
288C2×C3×C4C23×D314yes—
289C2×Dic3—34no—
290C2×Dic3—34no—
291C2×Dic3—34no—
292C2×Dic3—64no—
293D12—38no—
294D12—616no—
295D12—68no—
296D12—64no—
297D12—68no—
298D12—128no—
299D12—1216no—
300D12—128no—
301C4×D3—38no—
302C4×D3—38no—
303C4×D3—616no—
304C4×D3—68no—
305C4×D3—1216no—
306C4×D3—124no—
307C4×D3—128no—
308C4×D3—128no—
309C3×C8—28no—
310C3⋊C8—68no—
311C2×C32C2×#12111yescharacteristic
312C2×C32C2×#13511yescharacteristic
313C2×C32#57511yescharacteristic
314C3⋊D3—22no—
315C3⋊D3—42no—
316C3⋊D3—82no—
317#41—94no—
318#41—182no—
319#41—184no—
320#41—184no—
321C2×Q8C22×C3⋊D312yes—
322C2×Q8C22×C3⋊D311yescharacteristic
323C2×D4—91no—
324C2×D4—92no—
325C2×D4—181no—
326C2×D4—182no—
327C2×D4—181no—
328C2×D4—181no—
329C22×C4—91no—
330C22×C4—91no—
331C22×C4—91no—
332C22×C4—181no—
333C22×C4—181no—
334C22×C4—182no—
335C22×C4—181no—
336Q16—24no—
337SD16—184no—
338SD16—184no—
339D8—184no—
340C2×C8—92no—
341C2×C8—184no—
342C2×C8C22×C3⋊D312yes—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—91no—
347C4⋊C4—181no—
348C4⋊C4C22×C3⋊D311yescharacteristic
349C22⋊C4—181no—
350C22⋊C4—182no—
351C22⋊C4—182no—
352C22⋊C4—181no—
353C22⋊C4—181no—
354C42—21no—
355C42—91no—
356C42—91no—
357C42—91no—
358C42—181no—
359C42—181no—
360C42C22×C3⋊D311yescharacteristic
361C22×C3C2×D3×D414yes—
362C2×D3—38no—
363C2×D3—64no—
364C2×D3—64no—
365C2×D3—68no—
366C2×D3—124no—
367C2×D3—124no—
368C2×D3—128no—
369C2×D3—128no—
370C2×D3—128no—
371C3×C4—24no—
372C3×C4—28no—
373C3×C4—44no—
374C3×C4—48no—
375C3×C4#91918yescomplemented
376C3×C4C2×D3×D414yes—
377C3×C4D3×#4114yes—
378Dic3—38no—
379Dic3—64no—
380Dic3—64no—
381Dic3—124no—
382C32#303511yescharacteristic, complemented, Sylow
383C23—91no—
384C23—181no—
385C23—182no—
386Q8—21no—
387Q8—22no—
388Q8—22no—
389Q8D4×C3⋊D312yes—
390D4—92no—
391D4—184no—
392D4—182no—
393D4—181no—
394D4—182no—
395D4—362no—
396D4—364no—
397D4—362no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—22no—
401C2×C4—91no—
402C2×C4—92no—
403C2×C4—92no—
404C2×C4—91no—
405C2×C4—91no—
406C2×C4—184no—
407C2×C4—182no—
408C2×C4—181no—
409C2×C4—361no—
410C2×C4—364no—
411C2×C4—362no—
412C2×C4—362no—
413C2×C4D4×C3⋊D311yescharacteristic
414C2×C4D4×C3⋊D311yescharacteristic
415C2×C4C23×C3⋊D311yescharacteristic
416C8—22no—
417C8—182no—
418C2×C3D3×#12114yes—
419C2×C3C2×#91914yes—
420C2×C3#577714yes—
421D3—68no—
422D3—128no—
423D3—248no—
424C22—92no—
425C22—181no—
426C22—181no—
427C22—182no—
428C22—361no—
429C22—361no—
430C22—362no—
431C22—362no—
432C22—362no—
433C22C2×D4×C3⋊D311yescharacteristic
434C4—21no—
435C4—22no—
436C4—41no—
437C4—42no—
438C4—92no—
439C4—181no—
440C4—181no—
441C4—361no—
442C4#6412812yescomplemented
443C4C2×D4×C3⋊D311yescharacteristic
444C4#41×C3⋊D311yescharacteristic
445C3#8047514yescomplemented
446C2—182no—
447C2—362no—
448C2—722no—
449C2C3⋊D3×#12111yescharacteristic
450C2C2×#6412811yescharacteristic
451C2#1059624811yescharacteristic
subgroupG/H
centerC22C2×D4×C3⋊D3
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C23×C3⋊D3
FittingC32×#462C2
socleC22×C32C22×D4
hypercenter#462C3⋊D3
hypocenterC32#3035
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
1. C2×#64128C2semidirect
2. #10596196C2semidirect
3. C4×#3832C2semidirect
4. C32×#462C2semidirect
5. #10592884C2semidirect
6. #10592782C2semidirect
7. #10592726C2semidirect
8. #10592542C2semidirect
9. C3⋊D3×C4⋊C8C2semidirect
10. #10592518C2semidirect
11. #10592503C2semidirect
48. C3×#462D3semidirect
61. #3832D4semidirect
173. #462C3⋊D3semidirect
237. C32×C4#135semidirect
375. C3×C4#919semidirect
382. C32#3035semidirect
442. C4#64128semidirect
445. C3#80475semidirect

Maximal subgroups

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

subgroupindex
C2×#641282
#105961962
C4×#38322
C32×#4622
#105928842
#105927822
#105927262
#105925422
C3⋊D3×C4⋊C82
#105925182
#105925032
#804753