Finite Group Databasebeta

← #11887562

Subgroups — #11887562

Order 1152.

889 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5726C211yesmaximal, characteristic
2Dic6×D12C211yesmaximal, characteristic, complemented
3#10592792C211yesmaximal, characteristic, complemented
4C3×#5094C211yesmaximal, characteristic, complemented
5#10590603C212yesmaximal, complemented
6D3×#810C212yesmaximal, complemented
7#10590578C212yesmaximal, complemented
8#10590426C212yesmaximal, complemented
9#10590365C211yesmaximal, characteristic, complemented
10C4×#3715C211yesmaximal, characteristic, complemented
11#10590329C211yesmaximal, characteristic, complemented
12#83531—31nomaximal
13#81022—31nomaximal
14C2×C3×#239—21no—
15C2×D3×Dic6C2212yes—
16C32×C4⋊Q8C2211yescharacteristic
17#64078C2211yescharacteristic
18C3×#946—21no—
19C3×#895C2212yes—
20C3×#894—22no—
21C3×D3×C4⋊C4C2212yes—
22C3×C4×D12C2211yescharacteristic
23C3×C4×D12C2211yescharacteristic
24C3×C4×Dic6C2211yescharacteristic
25#63918—21no—
26#63893—22no—
27Dic3×D12—21no—
28#63857C2212yes—
29C2×#3716C2212yes—
30C2×#3715C2211yescharacteristic
31C2×D3×C3⋊C8C2212yes—
32#63637C2212yes—
33#63637C2212yes—
34C4×C3⋊(C3⋊C8)C2211yescharacteristic
35C3×#810C2212yes—
36C3×#808C2212yes—
37C3×C4⋊(C3⋊C8)C2211yescharacteristic
38#63574C2211yescharacteristic
39#63567C2212yes—
40#63565C2211yescharacteristic
41#63562C2211yescharacteristic
42(C2×D3)⋊(C3⋊C8)C2212yes—
43C3×#547—31no—
44#5726D311yescharacteristic
45Q8×D12—31no—
46D4×Dic6—31no—
47C3×#475—31no—
48#5094D311yescharacteristic, complemented
49#5093—31no—
50#5093—31no—
51#5093—61no—
52#5041—32no—
53#5040—32no—
54C4×#215—31no—
55#5019—31no—
56C4×#214—31no—
57D4×C3⋊C8—31no—
58#4971—32no—
59C2×#810—32no—
60#4864—32no—
61#4862—32no—
62#4841—31no—
63#4834—31no—
64#4806—32no—
65D3×#103—32no—
66C2×C32×Q8—21no—
67C2×#634C2312yes—
68C3×#239—44no—
69C2×C3×D12—22no—
70C2×C3×D12D411yescharacteristic
71C2×C3×D12C2311yescharacteristic
72C2×C3×C4×D3—22no—
73C2×C3×C4×D3—21no—
74C2×C3×C4×D3C2312yes—
75C2×C3×Dic6C2311yescharacteristic
76C2×#627—22no—
77C2×D3×Dic3—22no—
78D3×Dic6—24no—
79D3×Dic6—42no—
80C32×C4⋊C4—21no—
81C32×C4⋊C4C2312yes—
82C32×C42C2311yescharacteristic
83C4⋊(C3⋊Dic3)—21no—
84C4⋊(C3⋊Dic3)—21no—
85C2×C3⋊(C3⋊C8)C2311yescharacteristic
86C2×C3⋊(C3⋊C8)C2311yescharacteristic
87C3×(C2×D3)⋊C4—21no—
88C3×(C2×D3)⋊C4—22no—
89C3×(C2×D3)⋊C4D412yes—
90C3×C4⋊Dic3D411yescharacteristic
91C3×C4⋊Dic3C2311yescharacteristic
92C3×C4⋊Dic3C2311yescharacteristic
93C3×Dic3⋊C4—21no—
94C3×Dic3⋊C4—22no—
95C3×C4×Dic3—21no—
96C2×C3×C3⋊C8C2312yes—
97Dic3⋊Dic3—22no—
98(C2×D3)⋊Dic3—22no—
99#3716—42no—
100#3715—22no—
101D3×C3⋊C8—42no—
102#3298—91noSylow
103C2×C3×#41—61no—
104C2×#239—21no—
105C2×D3×Q8—32no—
106C22×Dic6—32no—
107C3×C4⋊Q8—21no—
108C3×C4⋊Q8C2×D311yescharacteristic
109C3×C4⋊Q8C2×D311yescharacteristic
110C3×#123—62no—
111C3×C22⋊Q8—61no—
112C3×#121—32no—
113C3×C4×Q8—31no—
114C3×C4×D4—31no—
115C3×C4×D4—31no—
116C2×C3×C4⋊C4—32no—
117#946—21no—
118C2×#215—31no—
119D4×Dic3—61no—
120C2×#214—31no—
121C2×#214—32no—
122#924—32no—
123C22×C3⋊C8—32no—
124C2×#204—32no—
125C2×D3×C8—32no—
126#897—32no—
127#897—62no—
128#895C2×D312yes—
129#894—22no—
130D3×C4⋊C4C2×D312yes—
131#888—61no—
132#878—62no—
133C4×D12—61no—
134C4×D12C2×D311yescharacteristic
135C4×D12C2×D311yescharacteristic
136#869—31no—
137#869—31no—
138#869—61no—
139#869—61no—
140C4×Dic6C2×D311yescharacteristic
141C3×#106—32no—
142C3×C4⋊C8—31no—
143C3×#103—32no—
144#835—31no—
145#832—31no—
146#830—32no—
147(C2×D3)⋊C8—32no—
148#816—32no—
149#810—31no—
150#810—32no—
151#810—32no—
152#810—32no—
153#810—32no—
154#810—32no—
155#810—62no—
156#810—62no—
157#810C2×D312yes—
158#809—31no—
159#808—31no—
160#808—31no—
161#808C2×D312yes—
162C4⋊(C3⋊C8)C2×D311yescharacteristic
163C4×C3⋊C8—31no—
164C4×C3⋊C8—31no—
165C4×C3⋊C8—61no—
166C22×C3×D3—21no—
167C22×C3×D3C2×D412yes—
168C32×Q8—42no—
169C2×C32×C4—22no—
170C2×C32×C4C2×D411yescharacteristic
171C2×C32×C4C2×D411yescharacteristic
172C2×C32×C4C2411yescharacteristic
173C2×C3⋊Dic3—22no—
174#634—22no—
175#634—22no—
176C2×C3×Dic3—21no—
177C2×C3×Dic3—21no—
178C2×C3×Dic3—21no—
179C2×C3×Dic3C2×D412yes—
180C3×D12—21no—
181C3×D12—21no—
182C3×D12—44no—
183C3×D12SD1614yescomplemented
184C3×C4×D3—22no—
185C3×C4×D3—22no—
186C3×C4×D3—44no—
187C3×C4×D3—42no—
188C3×Dic6—21no—
189C3×Dic6#4112yescomplemented
190#627—44no—
191D3×Dic3—44no—
192C3⋊(C3⋊C8)—21no—
193C3⋊(C3⋊C8)—21no—
194C3×C3⋊C8—22no—
195D4×Q8—91no—
196#547—31no—
197#487—91no—
198#483—92no—
199#475—31no—
200#465—92no—
201#461—91no—
202#448—92no—
203C4×SD16—91no—
204C8×D4—91no—
205C2×#103—92no—
206C3×#41—124no—
207C2×C3×Q8—21no—
208C2×C3×Q8—21no—
209C2×C3×Q8—31no—
210C2×C3×Q8—41no—
211C2×C3×D4—31no—
212C2×C3×D4—31no—
213C2×C3×D4—62no—
214C22×C3×C4—32no—
215C22×C3×C4—62no—
216C22×C3×C4—61no—
217C22×Dic3—62no—
218#239—44no—
219D3×Q8—64no—
220D3×Q8—122no—
221C2×D12—22no—
222C2×D12#6711yescharacteristic
223C2×D12C22×D311yescharacteristic
224C2×C4×D3—22no—
225C2×C4×D3—21no—
226C2×C4×D3—62no—
227C2×C4×D3C22×D312yes—
228C2×Dic6—32no—
229C2×Dic6—32no—
230C2×Dic6—62no—
231C2×Dic6—64no—
232C2×Dic6—62no—
233C2×Dic6—62no—
234C2×Dic6—122no—
235C2×Dic6C22×D311yescharacteristic
236C2×C3×C8—32no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—22no—
240C3×C4⋊C4—31no—
241C3×C4⋊C4—31no—
242C3×C4⋊C4—31no—
243C3×C4⋊C4—41no—
244C3×C4⋊C4—62no—
245C3×C4⋊C4—61no—
246C3×C4⋊C4C22×D312yes—
247C3×C4⋊C4C22×D312yes—
248C3×C22⋊C4—32no—
249C3×C22⋊C4—62no—
250C3×C22⋊C4—61no—
251C3×C42—21no—
252C3×C42—61no—
253C3×C42C22×D311yescharacteristic
254C3×C42C22×D311yescharacteristic
255#217—62no—
256#215—62no—
257#214—62no—
258#214—122no—
259(C2×D3)⋊C4—21no—
260(C2×D3)⋊C4—22no—
261(C2×D3)⋊C4—62no—
262(C2×D3)⋊C4#6712yes—
263C4⋊Dic3—61no—
264C4⋊Dic3—61no—
265C4⋊Dic3—61no—
266C4⋊Dic3—61no—
267C4⋊Dic3—62no—
268C4⋊Dic3—121no—
269C4⋊Dic3—121no—
270C4⋊Dic3#6711yescharacteristic
271C4⋊Dic3C22×D311yescharacteristic
272C4⋊Dic3C22×D311yescharacteristic
273Dic3⋊C4—21no—
274Dic3⋊C4—22no—
275Dic3⋊C4—62no—
276C4×Dic3—21no—
277C2×C3⋊C8—31no—
278C2×C3⋊C8—31no—
279C2×C3⋊C8—31no—
280C2×C3⋊C8—31no—
281C2×C3⋊C8—61no—
282C2×C3⋊C8—61no—
283C2×C3⋊C8—122no—
284C2×C3⋊C8C22×D312yes—
285#204—122no—
286D3×C8—122no—
287C22×C32C22×D411yescharacteristic
288C2×C3×D3—22no—
289C2×C3×D3—22no—
290C2×C3×D3—41no—
291C2×C3×D3—41no—
292C2×C3×D3—41no—
293C2×C3×D3C2×SD1614yes—
294C32×C4—21no—
295C32×C4—42no—
296C32×C4C2×SD1612yes—
297C32×C4C2×#4111yescharacteristic
298C32×C4H(5,𝔽2)11yescharacteristic
299C3⋊Dic3—42no—
300C3×Dic3—22no—
301C3×Dic3—22no—
302C3×Dic3—41no—
303C3×Dic3—41no—
304C2×#41—61no—
305C22×Q8—92no—
306C2×SD16—92no—
307C2×SD16—91no—
308C22×C8—92no—
309C4⋊Q8—91no—
310C4⋊Q8—181no—
311C4⋊Q8D3211yescharacteristic
312#123—62no—
313C22⋊Q8—61no—
314C22⋊Q8—92no—
315C22⋊Q8—182no—
316#121—32no—
317C4×Q8—31no—
318C4×D4—31no—
319C4×D4—31no—
320C4×D4—181no—
321C2×C4⋊C4—32no—
322#106—32no—
323#106—91no—
324C4⋊C8—31no—
325#103—32no—
326#103—92no—
327#103—91no—
328#103—92no—
329#103—92no—
330#102—91no—
331C22⋊C8—92no—
332C4×C8—91no—
333C23×C3—32no—
334C23×C3—61no—
335C22×D3—21no—
336C22×D3C2×#6712yes—
337C3×Q8—32no—
338C3×Q8—42no—
339C3×Q8—42no—
340C3×Q8—61no—
341C3×Q8—82no—
342C3×D4—34no—
343C3×D4—61no—
344C3×D4—61no—
345C3×D4—124no—
346C2×C3×C4—22no—
347C2×C3×C4—22no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—32no—
352C2×C3×C4—42no—
353C2×C3×C4—61no—
354C2×C3×C4—62no—
355C2×C3×C4—62no—
356C2×C3×C4—61no—
357C2×C3×C4—61no—
358C2×C3×C4—124no—
359C2×C3×C4—122no—
360C2×C3×C4D3×D411yescharacteristic
361C2×C3×C4D3×D411yescharacteristic
362C2×C3×C4C2×#6711yescharacteristic
363C2×C3×C4C2×#6711yescharacteristic
364C2×C3×C4C23×D311yescharacteristic
365C2×C3×C4C23×D311yescharacteristic
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—62no—
370C2×Dic3—62no—
371C2×Dic3—122no—
372C2×Dic3—124no—
373C2×Dic3C2×#6712yes—
374D12—21no—
375D12—21no—
376D12—44no—
377D12#21414yescomplemented
378C4×D3—22no—
379C4×D3—22no—
380C4×D3—44no—
381C4×D3—42no—
382C4×D3—124no—
383Dic6—21no—
384Dic6—62no—
385Dic6—62no—
386Dic6—62no—
387Dic6—62no—
388Dic6—122no—
389Dic6—122no—
390Dic6—124no—
391Dic6—124no—
392Dic6#23912yescomplemented
393C3×C8—62no—
394C3⋊C8—22no—
395C3⋊C8—61no—
396C3⋊C8—61no—
397C3⋊C8—61no—
398C3⋊C8—61no—
399C3⋊C8—121no—
400C3⋊C8—121no—
401C2×C32#54611yescharacteristic
402C2×C32C22×SD1611yescharacteristic
403C2×C32#57811yescharacteristic
404C3×D3—24no—
405C3×D3—81no—
406#41—124no—
407C2×Q8—21no—
408C2×Q8—31no—
409C2×Q8—92no—
410C2×Q8—182no—
411C2×Q8—184no—
412C2×Q8—362no—
413C2×D4—31no—
414C2×D4—31no—
415C2×D4—62no—
416C22×C4—32no—
417C22×C4—62no—
418C22×C4—61no—
419C22×C4—182no—
420SD16—182no—
421SD16—362no—
422C2×C8—32no—
423C2×C8—91no—
424C2×C8—91no—
425C2×C8—362no—
426C4⋊C4—21no—
427C4⋊C4—31no—
428C4⋊C4—31no—
429C4⋊C4—31no—
430C4⋊C4—62no—
431C4⋊C4—61no—
432C4⋊C4—182no—
433C4⋊C4—181no—
434C4⋊C4—181no—
435C4⋊C4C2×D3212yes—
436C22⋊C4—32no—
437C22⋊C4—62no—
438C22⋊C4—61no—
439C22⋊C4—182no—
440C42—61no—
441C42C2×D3211yescharacteristic
442C22×C3—21no—
443C22×C3—34no—
444C22×C3—62no—
445C22×C3—62no—
446C22×C3—121no—
447C22×C3—121no—
448C22×C3—121no—
449C22×C3C2×D3×D411yescharacteristic
450C22×C3C22×#6711yescharacteristic
451C2×D3—22no—
452C2×D3—22no—
453C2×D3—41no—
454C2×D3—41no—
455C2×D3—41no—
456C2×D3C2×#21414yes—
457C3×C4—22no—
458C3×C4—21no—
459C3×C4—21no—
460C3×C4—21no—
461C3×C4—21no—
462C3×C4—22no—
463C3×C4—42no—
464C3×C4—42no—
465C3×C4—62no—
466C3×C4—62no—
467C3×C4—82no—
468C3×C4—121no—
469C3×C4—121no—
470C3×C4D3×SD1612yes—
471C3×C4C2×#21412yes—
472C3×C4C2×#23911yescharacteristic
473C3×C4D3×#4111yescharacteristic
474C3×C4#100911yescharacteristic
475C3×C4#100911yescharacteristic
476Dic3—22no—
477Dic3—22no—
478Dic3—41no—
479Dic3—41no—
480Dic3—122no—
481Dic3—122no—
482Dic3—242no—
483C32#329811yescharacteristic, complemented, Sylow
484C23—32no—
485C23—61no—
486Q8—32no—
487Q8—42no—
488Q8—61no—
489Q8—182no—
490Q8—182no—
491Q8—364no—
492D4—34no—
493D4—61no—
494D4—61no—
495D4—124no—
496C2×C4—22no—
497C2×C4—32no—
498C2×C4—61no—
499C2×C4—62no—
500C2×C4—62no—
501C2×C4—61no—
502C2×C4—61no—
503C2×C4—124no—
504C2×C4—122no—
505C2×C4—182no—
506C2×C4—364no—
507C2×C4D3×#6711yescharacteristic
508C2×C4D3×#6711yescharacteristic
509C2×C4C22×D3211yescharacteristic
510C8—62no—
511C8—181no—
512C8—181no—
513C2×C3—21no—
514C2×C3—21no—
515C2×C3—21no—
516C2×C3—64no—
517C2×C3—241no—
518C2×C3#563011yescharacteristic
519C2×C3C2×D3×SD1611yescharacteristic
520C2×C3#577911yescharacteristic
521C2×C3C22×#21411yescharacteristic
522C2×C3#582911yescharacteristic
523C2×C3#583811yescharacteristic
524D3—24no—
525D3—81no—
526C22—34no—
527C22—62no—
528C22—62no—
529C22—121no—
530C22—121no—
531C22—121no—
532C22C2×D3×#6711yescharacteristic
533C4—21no—
534C4—42no—
535C4—62no—
536C4—62no—
537C4—121no—
538C4—121no—
539C4—362no—
540C4D3×#21412yes—
541C4D3×#23911yescharacteristic
542C4#6432011yescharacteristic
543C3—21no—
544C3#8102211yescharacteristic, complemented
545C3#8353111yescharacteristic, complemented
546C2—64no—
547C2—241no—
548C2C2×D3×#21411yescharacteristic
549C2#1059532811yescharacteristic
550C2#1059542711yescharacteristic
subgroupG/H
centerC22C2×D3×#67
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC32×C4⋊Q8C22
socleC22×C32C22×D4
prime radicalC3×#5726C2
hypercenterC4⋊Q8D32
hypocenterC32#3298
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. Dic6×D12C2semidirect
3. #10592792C2semidirect
4. C3×#5094C2semidirect
5. #10590603C2semidirect
6. D3×#810C2semidirect
7. #10590578C2semidirect
8. #10590426C2semidirect
9. #10590365C2semidirect
10. C4×#3715C2semidirect
11. #10590329C2semidirect
48. #5094D3semidirect
183. C3×D12SD16semidirect
189. C3×Dic6#41semidirect
377. D12#214semidirect
392. Dic6#239semidirect
483. C32#3298semidirect
544. C3#81022semidirect
545. C3#83531semidirect

Maximal subgroups

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

subgroupindex
C3×#57262
Dic6×D122
#105927922
C3×#50942
#105906032
D3×#8102
#105905782
#105904262
#105903652
C4×#37152
#105903292
#835313
#810223