Finite Group Databasebeta

← #11874317

Subgroups — #11874317

Order 1152.

1209 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#919C211yesmaximal, characteristic, complemented
2#10595360C212yesmaximal, complemented
3C2×#63949C212yesmaximal, complemented
4C2×#63808C211yesmaximal, characteristic, complemented
5#10592890C211yesmaximal, characteristic, complemented
6C3×#5183C211yesmaximal, characteristic, complemented
7#10591618C212yesmaximal, complemented
8#10591610C212yesmaximal, complemented
9#10590880C211yesmaximal, characteristic, complemented
10#10590869C211yesmaximal, characteristic, complemented
11D3×C8×Dic3C211yesmaximal, characteristic, complemented
12#83903—31nomaximal
13#81799—31nomaximal
14C2×C3×#239C2212yes—
15C2×#3799C2212yes—
16C2×C32×Q16C2211yescharacteristic
17#64156C2212yes—
18C2×#3755C2212yes—
19C2×#3745C2211yescharacteristic
20C3×#944C2212yes—
21C2×C3×#216C2212yes—
22C2×C3×#215C2212yes—
23C3×#919—24no—
24C2×C3×Dic12C2211yescharacteristic
25C2×C3×D24C2211yescharacteristic
26C2×C3×D3×C8C2211yescharacteristic
27#63949—28no—
28#63911C2212yes—
29#63904—24no—
30#63886—24no—
31C4×D3×Dic3C2211yescharacteristic
32#63863C2212yes—
33C2×#3717C2212yes—
34C2×#3715C2212yes—
35C2×#3714C2212yes—
36C2×D3×C3⋊C8C2211yescharacteristic
37#63808—24no—
38C8×C3⋊Dic3C2211yescharacteristic
39C3×C8×Dic3C2211yescharacteristic
40#63553C2211yescharacteristic
41Dic3×C3⋊C8C2211yescharacteristic
42C2×C3×#135—31no—
43#5832—32no—
44C2×#950—32no—
45C2×#919—32no—
46C2×#919—31no—
47C2×#919D311yescharacteristic, complemented
48C2×#904—31no—
49#5725—32no—
50C3×#494—31no—
51#5191—31no—
52#5191—31no—
53#5191—32no—
54#5191—31no—
55#5191—61no—
56#5183D311yescharacteristic, complemented
57#5175—32no—
58#5168—32no—
59#5155—31no—
60#5151—31no—
61C2×C8×Dic3—31no—
62#5085—32no—
63#4695—31no—
64C4×D3×C8—31no—
65C2×C32×Q8C2312yes—
66C2×C4⋊(C3⋊D3)C2312yes—
67C3×#239—24no—
68C3×#239—24no—
69C2×C3×D12—22no—
70C2×C3×D12C2312yes—
71C2×C3×C4×D3—22no—
72C2×C3×C4×D3C2311yescharacteristic
73C2×C3×Dic6C2312yes—
74C2×#626—22no—
75C2×#626—24no—
76C2×#624—22no—
77C2×D3×Dic3D412yes—
78#3799—24no—
79#3799—24no—
80C32×Q16—22no—
81C2×C32×C8C2311yescharacteristic
82#3755—24no—
83#3752—24no—
84C4×C3⋊Dic3C2311yescharacteristic
85C2×C3⋊(C3⋊C8)C2311yescharacteristic
86#3745—22no—
87C3×#216—24no—
88C3×#215—24no—
89C3×Dic3⋊C4—24no—
90C3×C4×Dic3C2311yescharacteristic
91C2×C3×C3⋊C8C2311yescharacteristic
92C2×C3×C3⋊C8C2311yescharacteristic
93C3×Dic12—22no—
94C3×D24—22no—
95C3×D3×C8D414yescomplemented
96(C2×D3)⋊Dic3—24no—
97Dic32D412yes—
98#3717—24no—
99#3715—24no—
100#3714—24no—
101D3×C3⋊C8D414yescomplemented
102#3147—91noSylow
103C2×C3×#41—32no—
104C2×#239—32no—
105C2×#239C2×D312yes—
106C2×#235—32no—
107C3×#135—64no—
108C2×C3×Q16—21no—
109C2×C3×Q16—32no—
110C2×C3×Q16—31no—
111C2×C3×Q16C2×D311yescharacteristic
112C2×C3×Q16C2×D311yescharacteristic
113C2×C3×SD16—32no—
114C2×C3×D8—31no—
115C22×C3×C8—31no—
116C3×C4⋊Q8—32no—
117#950—68no—
118#947—32no—
119#947—32no—
120#947—32no—
121#947—62no—
122#944C2×D312yes—
123C2×#216C2×D312yes—
124C2×#215—32no—
125C2×#215—32no—
126C2×#215—32no—
127C2×#215—32no—
128C2×#215—62no—
129C2×#215C2×D312yes—
130#940—32no—
131#939—64no—
132#936—32no—
133C2×#214—32no—
134C2×#213—32no—
135C4×#67—64no—
136C2×C4×Dic3—31no—
137C22×C3⋊C8—31no—
138#919—24no—
139#919—68no—
140#919—64no—
141C2×Dic12C2×D311yescharacteristic
142#904—64no—
143C2×D24—32no—
144C2×D24—31no—
145C2×D24—31no—
146C2×D24—61no—
147C2×D24C2×D311yescharacteristic
148C2×#204—32no—
149C2×D3×C8—31no—
150C2×D3×C8C2×D311yescharacteristic
151#895—64no—
152#893—64no—
153#874—32no—
154C42×D3—31no—
155C3×C4×C8—31no—
156C8×Dic3—31no—
157C8×Dic3—31no—
158C8×Dic3—31no—
159C8×Dic3—31no—
160C8×Dic3—31no—
161C8×Dic3—61no—
162C8×Dic3C2×D311yescharacteristic
163C4×C3⋊C8—31no—
164C22×C3⋊D3—22no—
165C22×C3×D3—22no—
166C22×C3×D3C2×D411yescharacteristic
167C32×Q8—22no—
168C32×Q8—22no—
169C2×C32×C4—22no—
170C2×C32×C4C2411yescharacteristic
171C2×C3⋊Dic3C2×D412yes—
172C4⋊(C3⋊D3)—22no—
173C4⋊(C3⋊D3)—22no—
174C2×C3×Dic3—22no—
175C2×C3×Dic3C2×D411yescharacteristic
176C2×C3×Dic3C2×D412yes—
177C3×D12—22no—
178C3×D12—22no—
179C3×D12—44no—
180C3×C4×D3—44no—
181C3×C4×D3C2×D412yes—
182C3×C4×D3C2×D412yes—
183C3×Dic6—22no—
184C3×Dic6—22no—
185#626—44no—
186#626—48no—
187#624—44no—
188D3×Dic3—24no—
189C32×C8C2×D412yes—
190C3⋊(C3⋊C8)C2×D412yes—
191C3×C3⋊C8C2×D412yes—
192C3×C3⋊C8C2×D412yes—
193C2×#135—31no—
194C2×#135—91no—
195C2×#135—92no—
196#532—92no—
197#495—91no—
198#495—92no—
199#495—91no—
200#494—31no—
201#493—91no—
202#492—92no—
203C2×C4×C8—91no—
204C3×#41—64no—
205C3×#41—64no—
206C2×C3×Q8—22no—
207C2×C3×Q8—32no—
208C2×C3×Q8C22×D312yes—
209C2×C3×Q8C22×D312yes—
210C2×C3×D4—32no—
211C2×C3×D4—62no—
212C22×C3×C4—31no—
213C22×C3×C4—62no—
214C2×#67—62no—
215C2×#67—64no—
216C22×Dic3—32no—
217#239—24no—
218#239—24no—
219#239—64no—
220#239—64no—
221#235—64no—
222#235—64no—
223C2×D12—22no—
224C2×D12—32no—
225C2×D12—32no—
226C2×D12—62no—
227C2×D12—64no—
228C2×D12—62no—
229C2×D12C22×D312yes—
230C2×C4×D3—22no—
231C2×C4×D3—32no—
232C2×C4×D3—62no—
233C2×C4×D3—62no—
234C2×C4×D3C22×D311yescharacteristic
235C2×Dic6C22×D312yes—
236C3×Q16—22no—
237C3×Q16—22no—
238C3×Q16—42no—
239C3×Q16—64no—
240C3×Q16—62no—
241C3×SD16—64no—
242C3×D8—62no—
243C2×C3×C8—21no—
244C2×C3×C8—31no—
245C2×C3×C8—31no—
246C2×C3×C8—34no—
247C2×C3×C8C22×D311yescharacteristic
248C2×C3×C8C22×D311yescharacteristic
249C3×C4⋊C4—64no—
250C3×C42—31no—
251#217—64no—
252#216—24no—
253#215—24no—
254#215—64no—
255#215—64no—
256#215—64no—
257#215—64no—
258#215—124no—
259#214—64no—
260#213—64no—
261(C2×D3)⋊C4—64no—
262(C2×D3)⋊C4—64no—
263(C2×D3)⋊C4—64no—
264(C2×D3)⋊C4—124no—
265Dic3⋊C4—24no—
266C4×Dic3—31no—
267C4×Dic3—31no—
268C4×Dic3—32no—
269C4×Dic3—32no—
270C4×Dic3—61no—
271C4×Dic3C22×D311yescharacteristic
272C2×C3⋊C8—31no—
273C2×C3⋊C8—31no—
274C2×C3⋊C8—34no—
275C2×C3⋊C8—61no—
276C2×C3⋊C8C22×D311yescharacteristic
277C2×C3⋊C8C22×D311yescharacteristic
278Dic12—22no—
279D24—22no—
280D24—64no—
281D24—62no—
282D24—62no—
283D24—122no—
284#204—64no—
285D3×C8—34no—
286D3×C8#6714yescomplemented
287C22×C32C22×D411yescharacteristic
288C2×C3⋊D3—42no—
289C2×C3⋊D3—44no—
290C2×C3×D3—22no—
291C2×C3×D3—42no—
292C2×C3×D3—44no—
293C2×C3×D3#12712yes—
294C32×C4—42no—
295C32×C4C22×D411yescharacteristic
296C32×C4C22×D411yescharacteristic
297C3⋊Dic3—22no—
298C3×Dic3—42no—
299C3×Dic3#12712yes—
300C3×Dic3#13514yescomplemented
301C2×#41—32no—
302C2×#41—92no—
303#135—64no—
304#135—184no—
305#135—188no—
306C2×Q16—32no—
307C2×Q16—31no—
308C2×Q16D3211yescharacteristic
309C2×SD16—32no—
310C2×SD16—92no—
311C2×SD16—92no—
312C2×SD16—92no—
313C2×D8—31no—
314C2×D8—92no—
315C2×D8—91no—
316C22×C8—31no—
317C22×C8—91no—
318C4⋊Q8—32no—
319#127—92no—
320#124—92no—
321#124—92no—
322#121—184no—
323C4×D4—184no—
324C2×C42—91no—
325C4×C8—31no—
326C4×C8—91no—
327C4×C8—91no—
328C4×C8—91no—
329C23×C3—31no—
330C23×C3—62no—
331C22×D3—22no—
332C22×D3—62no—
333C22×D3—62no—
334C22×D3—122no—
335C22×D3D3×D411yescharacteristic
336C3×Q8—22no—
337C3×Q8—22no—
338C3×Q8—22no—
339C3×Q8—22no—
340C3×Q8—42no—
341C3×Q8—42no—
342C3×Q8—62no—
343C3×Q8—62no—
344C3×D4—62no—
345C3×D4—62no—
346C3×D4—124no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—21no—
350C2×C3×C4—31no—
351C2×C3×C4—32no—
352C2×C3×C4—32no—
353C2×C3×C4—32no—
354C2×C3×C4—42no—
355C2×C3×C4—62no—
356C2×C3×C4—124no—
357C2×C3×C4C23×D311yescharacteristic
358C2×C3×C4C23×D311yescharacteristic
359#67—124no—
360#67—128no—
361C2×Dic3—22no—
362C2×Dic3—32no—
363C2×Dic3—32no—
364C2×Dic3—62no—
365C2×Dic3—64no—
366C2×Dic3D3×D412yes—
367C2×Dic3D3×D411yescharacteristic
368D12—22no—
369D12—22no—
370D12—44no—
371D12—62no—
372D12—62no—
373D12—62no—
374D12—62no—
375D12—122no—
376D12—122no—
377D12—128no—
378D12—124no—
379C4×D3—44no—
380C4×D3—64no—
381C4×D3—124no—
382C4×D3—124no—
383C4×D3D3×D412yes—
384C4×D3C2×#6712yes—
385Dic6—22no—
386Dic6—22no—
387C3×C8—22no—
388C3×C8—32no—
389C3×C8—32no—
390C3×C8D3×D412yes—
391C3×C8C2×#6712yes—
392C3⋊C8—32no—
393C3⋊C8—32no—
394C3⋊C8—62no—
395C3⋊C8D3×D412yes—
396C3⋊C8C2×#6712yes—
397C2×C32C2×#12711yescharacteristic
398C2×C32C2×#13512yes—
399C3⋊D3—82no—
400C3×D3—22no—
401C3×D3—82no—
402#41—64no—
403#41—64no—
404#41—184no—
405#41—184no—
406C2×Q8—32no—
407C2×Q8C2×D3212yes—
408C2×D4—32no—
409C2×D4—62no—
410C2×D4—92no—
411C2×D4—184no—
412C2×D4—182no—
413C22×C4—31no—
414C22×C4—62no—
415C22×C4—92no—
416C22×C4—182no—
417Q16—22no—
418Q16—64no—
419Q16—62no—
420SD16—64no—
421SD16—184no—
422SD16—184no—
423SD16—184no—
424D8—62no—
425D8—184no—
426D8—182no—
427C2×C8—31no—
428C2×C8—31no—
429C2×C8—34no—
430C2×C8—94no—
431C2×C8—91no—
432C2×C8C2×D3211yescharacteristic
433C4⋊C4—64no—
434C22⋊C4—184no—
435C22⋊C4—184no—
436C42—31no—
437C42—92no—
438C42—91no—
439C22×C3—21no—
440C22×C3—32no—
441C22×C3—62no—
442C22×C3—122no—
443C22×C3—124no—
444C22×C3C2×D3×D411yescharacteristic
445C22×C3C2×D3×D411yescharacteristic
446C2×D3—22no—
447C2×D3—44no—
448C2×D3—42no—
449C2×D3—122no—
450C2×D3—124no—
451C2×D3—124no—
452C2×D3—122no—
453C2×D3—242no—
454C2×D3—244no—
455C2×D3#94012yes—
456C3×C4—21no—
457C3×C4—21no—
458C3×C4—32no—
459C3×C4—34no—
460C3×C4—42no—
461C3×C4—42no—
462C3×C4—82no—
463C3×C4—122no—
464C3×C4C2×D3×D411yescharacteristic
465C3×C4C2×D3×D411yescharacteristic
466C3×C4C2×D3×D411yescharacteristic
467C3×C4C22×#6711yescharacteristic
468Dic3—42no—
469Dic3—62no—
470Dic3—62no—
471Dic3—122no—
472Dic3#91914yescomplemented
473Dic3#94012yes—
474C32#314711yescharacteristic, complemented, Sylow
475C23—31no—
476C23—62no—
477C23—182no—
478Q8—22no—
479Q8—22no—
480Q8—62no—
481Q8—62no—
482D4—62no—
483D4—62no—
484D4—124no—
485D4—182no—
486D4—182no—
487D4—368no—
488D4—364no—
489C2×C4—22no—
490C2×C4—32no—
491C2×C4—31no—
492C2×C4—32no—
493C2×C4—32no—
494C2×C4—62no—
495C2×C4—92no—
496C2×C4—124no—
497C2×C4—184no—
498C2×C4—364no—
499C2×C4C22×D3211yescharacteristic
500C8—32no—
501C8—32no—
502C8—92no—
503C8D3×#6712yes—
504C2×C3—22no—
505C2×C3—21no—
506C2×C3—62no—
507C2×C3—242no—
508C2×C3D3×#12711yescharacteristic
509C2×C3C2×#91912yes—
510C2×C3D3×#13512yes—
511C2×C3C2×#94011yescharacteristic
512D3—22no—
513D3—82no—
514D3—242no—
515D3—242no—
516D3—482no—
517C22—32no—
518C22—62no—
519C22—124no—
520C22—122no—
521C22—364no—
522C22—362no—
523C22D32×D411yescharacteristic
524C4—34no—
525C4—32no—
526C4—42no—
527C4—122no—
528C4—182no—
529C4D32×D411yescharacteristic
530C4C2×D3×#6711yescharacteristic
531C3—21no—
532C3#8179911yescharacteristic, complemented
533C3#8390311yescharacteristic, complemented
534C2—62no—
535C2—242no—
536C2—722no—
537C2D3×#91912yes—
538C2D3×#94011yescharacteristic
subgroupG/H
centerC22D32×D4
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC2×C32×Q16C22
socleC22×C32C22×D4
hypercenterC2×Q16D32
hypocenterC32#3147
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q16

Complements

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

NH ≅ G/Ntype
1. C2×C3×#919C2semidirect
2. #10595360C2semidirect
3. C2×#63949C2semidirect
4. C2×#63808C2semidirect
5. #10592890C2semidirect
6. C3×#5183C2semidirect
7. #10591618C2semidirect
8. #10591610C2semidirect
9. #10590880C2semidirect
10. #10590869C2semidirect
11. D3×C8×Dic3C2semidirect
47. C2×#919D3semidirect
56. #5183D3semidirect
95. C3×D3×C8D4semidirect
101. D3×C3⋊C8D4semidirect
286. D3×C8#67semidirect
300. C3×Dic3#135semidirect
472. Dic3#919semidirect
474. C32#3147semidirect
532. C3#81799semidirect
533. C3#83903semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9192
#105953602
C2×#639492
C2×#638082
#105928902
C3×#51832
#105916182
#105916102
#105908802
#105908692
D3×C8×Dic32
#839033
#817993