Finite Group Databasebeta

← #11895767

Subgroups — #11895767

Order 1152.

859 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5543C211yesmaximal, characteristic, complemented
2#10593900C211yesmaximal, characteristic, complemented
3C4×#3753C211yesmaximal, characteristic, complemented
4C3×#5015C211yesmaximal, characteristic, complemented
5#10591481C212yesmaximal, complemented
6#10591430C212yesmaximal, complemented
7#10590550C212yesmaximal, complemented
8#10590467C212yesmaximal, complemented
9#10590300C211yesmaximal, characteristic, complemented
10#10590277C211yesmaximal, characteristic, complemented
11#10590259C211yesmaximal, characteristic, complemented
12#83638—31nomaximal
13#80320—31nomaximal
14C2×#3796—21no—
15C32×C4×D4C2211yescharacteristic
16C2×#3753C2211yescharacteristic
17C4×C4⋊(C3⋊D3)C2211yescharacteristic
18C3×D4×Dic3C2212yes—
19C2×C3×#213C2212yes—
20C2×C3×C4⋊Dic3—21no—
21C3×#924—21no—
22C3×#890C2212yes—
23C3×#878—22no—
24C3×#874C2211yescharacteristic
25C3×#869C2211yescharacteristic
26C3×C4×Dic6C2211yescharacteristic
27#63916—21no—
28#63911C2212yes—
29Dic3×D12C2212yes—
30#63886—22no—
31#63659C2211yescharacteristic
32#63636C2211yescharacteristic
33#63634C2211yescharacteristic
34C4×C3⋊(C3⋊C8)C2211yescharacteristic
35C3×#809C2212yes—
36C3×C4⋊(C3⋊C8)C2211yescharacteristic
37#63573C2212yes—
38Dic3⋊(C3⋊C8)C2212yes—
39#63564C2212yes—
40#63562C2212yes—
41#63562C2212yes—
42#63554C2212yes—
43C3×#554—31no—
44#5718—31no—
45#5577—31no—
46#5543D311yescharacteristic, complemented
47C3×#459—31no—
48#5152—32no—
49#5150—32no—
50#5088—31no—
51#5084—31no—
52#5070—31no—
53#5067—32no—
54#5066—32no—
55#5024—31no—
56#5015—32no—
57#5015D311yescharacteristic, complemented
58#5014—32no—
59C4×#213—31no—
60C4×#213—31no—
61C4×#213—61no—
62#4842—31no—
63#4831—31no—
64#4764—32no—
65#4763—32no—
66C2×C32×D4C2311yescharacteristic
67C22×C32×C4—21no—
68C2×C4⋊(C3⋊D3)C2311yescharacteristic
69C2×C4×C3⋊D3—21no—
70C22×C3×Dic3—22no—
71C2×C3×D12—21no—
72C2×C3×D12—21no—
73C2×C3×D12C2312yes—
74C2×C3×Dic6—21no—
75C2×C3×Dic6D411yescharacteristic
76C2×#626—22no—
77C2×#626—22no—
78C2×D3×Dic3—22no—
79#3796—44no—
80C32×C4⋊C4C2311yescharacteristic
81C32×C22⋊C4—21no—
82C32×C42C2311yescharacteristic
83#3753—22no—
84#3752—21no—
85C4⋊(C3⋊Dic3)C2311yescharacteristic
86C2×C3⋊(C3⋊C8)C2311yescharacteristic
87C2×C3⋊(C3⋊C8)C2311yescharacteristic
88C3×#217—22no—
89C3×#213—24no—
90C3×C4⋊Dic3—21no—
91C3×C4⋊Dic3—22no—
92C3×C4⋊Dic3D411yescharacteristic
93C3×C4⋊Dic3C2312yes—
94C3×Dic3⋊C4—22no—
95C3×Dic3⋊C4D412yes—
96C3×C4×Dic3C2312yes—
97C2×C3×C3⋊C8C2312yes—
98(C2×D3)⋊Dic3—22no—
99#3366—91noSylow
100C2×#239—61no—
101C2×#237—61no—
102C2×C3×D8—32no—
103C3×C4⋊Q8—31no—
104C3×#127—31no—
105C3×#125—32no—
106C3×C22⋊Q8—62no—
107C3×C22⋊Q8—61no—
108C3×C4×Q8—31no—
109C3×C4×D4—21no—
110C3×C4×D4—32no—
111C3×C4×D4C2×D311yescharacteristic
112C3×C4×D4C2×D311yescharacteristic
113C2×C3×C4⋊C4—61no—
114#940—32no—
115#939—62no—
116#938—61no—
117D4×Dic3—32no—
118D4×Dic3C2×D312yes—
119C2×#213—31no—
120C2×#213—31no—
121C2×#213—61no—
122C2×#213C2×D312yes—
123C2×C4⋊Dic3—21no—
124#924—21no—
125#895—62no—
126#895—61no—
127#890C2×D312yes—
128#878—22no—
129#874—32no—
130#874C2×D311yescharacteristic
131C4×D12—32no—
132C4×D12—31no—
133C4×D12—31no—
134C4×D12—61no—
135#869C2×D311yescharacteristic
136C4×Dic6C2×D311yescharacteristic
137C3×C4⋊C8—31no—
138C3×#102—32no—
139#832—31no—
140#832—31no—
141#832—32no—
142#832—32no—
143#832—61no—
144#821—32no—
145#817—32no—
146#815—32no—
147Dic3⋊C8—32no—
148#809—32no—
149#809—32no—
150#809—31no—
151#809—31no—
152#809—32no—
153#809—61no—
154#809C2×D312yes—
155#808—32no—
156#807—31no—
157#807—31no—
158#807—61no—
159C4⋊(C3⋊C8)—32no—
160C4⋊(C3⋊C8)C2×D311yescharacteristic
161#803—32no—
162C4×C3⋊C8—31no—
163C4×C3⋊C8—31no—
164C4×C3⋊C8—61no—
165C23×C32—21no—
166C22×C3⋊D3—21no—
167C22×C3×D3—22no—
168C32×D4—21no—
169C32×D4#4112yescomplemented
170C2×C32×C4—21no—
171C2×C32×C4—42no—
172C2×C32×C4C2×D411yescharacteristic
173C2×C32×C4C2×D411yescharacteristic
174C2×C32×C4C2411yescharacteristic
175C2×C3⋊Dic3—21no—
176C4⋊(C3⋊D3)—21no—
177C4⋊(C3⋊D3)—21no—
178C4×C3⋊D3—42no—
179C2×C3×Dic3—22no—
180C2×C3×Dic3—82no—
181C2×C3×Dic3C2×D412yes—
182C2×C3×Dic3C2×D412yes—
183C3×D12—22no—
184C3×D12—22no—
185C3×D12—42no—
186C3×D12—81no—
187C3×Dic6—22no—
188C3×Dic6—81no—
189#626—44no—
190#626—82no—
191D3×Dic3—44no—
192C3⋊(C3⋊C8)—21no—
193C3⋊(C3⋊C8)—21no—
194C3×C3⋊C8—22no—
195#554—31no—
196#550—91no—
197#490—92no—
198#486—91no—
199#478—92no—
200#476—91no—
201#459—31no—
202#459—92no—
203#446—91no—
204#442—92no—
205C4×D8—91no—
206C2×C3×Q8—31no—
207C2×C3×Q8—61no—
208C2×C3×D4—21no—
209C2×C3×D4—32no—
210C2×C3×D4—61no—
211C2×C3×D4—61no—
212C2×C3×D4C22×D311yescharacteristic
213C2×C3×D4C22×D311yescharacteristic
214C22×C3×C4—21no—
215C22×C3×C4—21no—
216C22×C3×C4—41no—
217C22×C3×C4—62no—
218C2×#67—62no—
219C2×#67—62no—
220C22×Dic3—22no—
221C22×Dic3—62no—
222#239—124no—
223#237—124no—
224C2×D12—21no—
225C2×D12—21no—
226C2×D12—31no—
227C2×D12—31no—
228C2×D12—61no—
229C2×D12—62no—
230C2×D12—62no—
231C2×D12C22×D312yes—
232C2×C4×D3—62no—
233C2×C4×D3—61no—
234C2×C4×D3—61no—
235C2×C4×D3—121no—
236C2×Dic6—21no—
237C2×Dic6#6711yescharacteristic
238C3×D8—64no—
239C2×C3×C8—32no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—32no—
242C3×C4⋊C4—31no—
243C3×C4⋊C4—32no—
244C3×C4⋊C4—62no—
245C3×C4⋊C4—62no—
246C3×C4⋊C4—61no—
247C3×C4⋊C4C22×D311yescharacteristic
248C3×C4⋊C4C22×D311yescharacteristic
249C3×C22⋊C4—21no—
250C3×C22⋊C4—21no—
251C3×C22⋊C4—41no—
252C3×C22⋊C4—62no—
253C3×C42—21no—
254C3×C42—32no—
255C3×C42C22×D311yescharacteristic
256C3×C42C22×D311yescharacteristic
257#217—22no—
258#217—62no—
259#213—24no—
260#213—64no—
261#213—62no—
262#213—62no—
263(C2×D3)⋊C4—62no—
264(C2×D3)⋊C4—61no—
265(C2×D3)⋊C4—61no—
266(C2×D3)⋊C4—121no—
267C4⋊Dic3—21no—
268C4⋊Dic3—22no—
269C4⋊Dic3—31no—
270C4⋊Dic3—31no—
271C4⋊Dic3—61no—
272C4⋊Dic3#6711yescharacteristic
273C4⋊Dic3C22×D312yes—
274Dic3⋊C4—22no—
275Dic3⋊C4#6712yes—
276C4×Dic3C22×D312yes—
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⋊C8C22×D312yes—
284C22×C32—22no—
285C22×C32—41no—
286C22×C32—41no—
287C22×C32C22×D411yescharacteristic
288C2×C3⋊D3—41no—
289C2×C3⋊D3—41no—
290C2×C3⋊D3—41no—
291C2×C3×D3—42no—
292C2×C3×D3—42no—
293C2×C3×D3—42no—
294C32×C4—21no—
295C32×C4—41no—
296C32×C4AGL(1,ℤ/8)12yes—
297C32×C4C2×#4111yescharacteristic
298C32×C42−1+411yescharacteristic
299C3⋊Dic3—41no—
300C3×Dic3—22no—
301C3×Dic3—22no—
302C3×Dic3—42no—
303C2×#41—181no—
304C2×D8—32no—
305C2×D8—91no—
306C4⋊Q8—31no—
307#127—31no—
308#127—92no—
309#125—32no—
310C22⋊Q8—62no—
311C22⋊Q8—61no—
312#121—181no—
313#121—182no—
314C4×Q8—31no—
315C4×D4—32no—
316C4×D4—92no—
317C4×D4—91no—
318C4×D4D3211yescharacteristic
319C2×C4⋊C4—61no—
320#107—91no—
321#106—92no—
322C4⋊C8—31no—
323C4⋊C8—92no—
324#102—32no—
325#102—92no—
326#102—92no—
327#102—92no—
328#102—91no—
329#102—91no—
330#97—92no—
331C4×C8—91no—
332C23×C3—21no—
333C23×C3—21no—
334C23×C3—41no—
335C23×C3—62no—
336C22×D3—22no—
337C22×D3—61no—
338C22×D3—61no—
339C22×D3—121no—
340C3×Q8—62no—
341C3×Q8—241no—
342C3×D4—21no—
343C3×D4—21no—
344C3×D4—22no—
345C3×D4—41no—
346C3×D4—62no—
347C3×D4—62no—
348C3×D4—122no—
349C3×D4—241no—
350C3×D4#23712yescomplemented
351C3×D4#23912yescomplemented
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—32no—
358C2×C3×C4—32no—
359C2×C3×C4—41no—
360C2×C3×C4—42no—
361C2×C3×C4—42no—
362C2×C3×C4—62no—
363C2×C3×C4—82no—
364C2×C3×C4—242no—
365C2×C3×C4D3×D411yescharacteristic
366C2×C3×C4D3×D411yescharacteristic
367C2×C3×C4C2×#6711yescharacteristic
368C2×C3×C4C2×#6711yescharacteristic
369C2×C3×C4C23×D311yescharacteristic
370C2×C3×C4C23×D311yescharacteristic
371#67—124no—
372#67—242no—
373C2×Dic3—22no—
374C2×Dic3—61no—
375C2×Dic3—61no—
376C2×Dic3—82no—
377C2×Dic3—121no—
378C2×Dic3—124no—
379C2×Dic3C2×#6712yes—
380C2×Dic3C2×#6712yes—
381D12—22no—
382D12—22no—
383D12—42no—
384D12—62no—
385D12—61no—
386D12—61no—
387D12—61no—
388D12—61no—
389D12—81no—
390D12—121no—
391D12—124no—
392D12—242no—
393C4×D3—124no—
394C4×D3—122no—
395C4×D3—122no—
396C4×D3—242no—
397Dic6—22no—
398Dic6—81no—
399C3×C8—62no—
400C3⋊C8—22no—
401C3⋊C8—62no—
402C3⋊C8—61no—
403C3⋊C8—61no—
404C3⋊C8—61no—
405C3⋊C8—61no—
406C3⋊C8—121no—
407C2×C32—42no—
408C2×C32#54811yescharacteristic
409C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
410C2×C32#57611yescharacteristic
411C3⋊D3—81no—
412C3×D3—82no—
413#41—364no—
414C2×Q8—31no—
415C2×Q8—61no—
416C2×D4—32no—
417C2×D4—61no—
418C2×D4—61no—
419C2×D4—91no—
420C2×D4—182no—
421C2×D4—182no—
422C2×D4C2×D3211yescharacteristic
423C22×C4—21no—
424C22×C4—62no—
425C22×C4—182no—
426C22×C4—181no—
427D8—64no—
428D8—182no—
429C2×C8—32no—
430C2×C8—91no—
431C2×C8—91no—
432C4⋊C4—31no—
433C4⋊C4—32no—
434C4⋊C4—32no—
435C4⋊C4—62no—
436C4⋊C4—62no—
437C4⋊C4—61no—
438C4⋊C4—91no—
439C4⋊C4C2×D3211yescharacteristic
440C22⋊C4—21no—
441C22⋊C4—62no—
442C22⋊C4—182no—
443C22⋊C4—181no—
444C42—32no—
445C42C2×D3211yescharacteristic
446C22×C3—22no—
447C22×C3—22no—
448C22×C3—21no—
449C22×C3—42no—
450C22×C3—41no—
451C22×C3—41no—
452C22×C3—41no—
453C22×C3—41no—
454C22×C3—81no—
455C22×C3—81no—
456C22×C3—122no—
457C22×C3—122no—
458C22×C3—122no—
459C22×C3C2×D3×D411yescharacteristic
460C22×C3C22×#6711yescharacteristic
461C2×D3—42no—
462C2×D3—42no—
463C2×D3—42no—
464C2×D3—122no—
465C2×D3—121no—
466C2×D3—121no—
467C2×D3—121no—
468C2×D3—121no—
469C2×D3—121no—
470C2×D3—121no—
471C2×D3—241no—
472C2×D3—241no—
473C3×C4—21no—
474C3×C4—21no—
475C3×C4—21no—
476C3×C4—21no—
477C3×C4—22no—
478C3×C4—41no—
479C3×C4—41no—
480C3×C4—41no—
481C3×C4—62no—
482C3×C4—62no—
483C3×C4—81no—
484C3×C4—122no—
485C3×C4#91112yes—
486C3×C4#94912yes—
487C3×C4C2×#23711yescharacteristic
488C3×C4C2×#23911yescharacteristic
489C3×C4#100711yescharacteristic
490C3×C4#101011yescharacteristic
491Dic3—22no—
492Dic3—22no—
493Dic3—42no—
494Dic3—121no—
495Dic3—121no—
496Dic3—241no—
497C32#336611yescharacteristic, complemented, Sylow
498C23—21no—
499C23—62no—
500C23—181no—
501Q8—62no—
502Q8—241no—
503D4—21no—
504D4—62no—
505D4—62no—
506D4—122no—
507D4—181no—
508D4—181no—
509D4—241no—
510D4—364no—
511D4—722no—
512D4#379612yescomplemented
513C2×C4—21no—
514C2×C4—32no—
515C2×C4—32no—
516C2×C4—42no—
517C2×C4—62no—
518C2×C4—181no—
519C2×C4—242no—
520C2×C4—364no—
521C2×C4—362no—
522C2×C4D3×#6711yescharacteristic
523C2×C4D3×#6711yescharacteristic
524C2×C4C22×D3211yescharacteristic
525C8—62no—
526C8—181no—
527C8—181no—
528C2×C3—21no—
529C2×C3—21no—
530C2×C3—21no—
531C2×C3—42no—
532C2×C3—42no—
533C2×C3—82no—
534C2×C3—242no—
535C2×C3#559811yescharacteristic
536C2×C3C2×#91111yescharacteristic
537C2×C3#577411yescharacteristic
538C2×C3#581711yescharacteristic
539C2×C3C2×#94911yescharacteristic
540C2×C3#583511yescharacteristic
541D3—82no—
542D3—242no—
543D3—241no—
544D3—241no—
545C22—22no—
546C22—41no—
547C22—41no—
548C22—122no—
549C22—122no—
550C22—122no—
551C22—361no—
552C22—361no—
553C22—361no—
554C22C2×D3×#6711yescharacteristic
555C4—21no—
556C4—41no—
557C4—62no—
558C4—62no—
559C4—122no—
560C4—361no—
561C4#6392512yes—
562C4C2×#379611yescharacteristic
563C4#6429711yescharacteristic
564C3—21no—
565C3#8032011yescharacteristic, complemented
566C3#8363811yescharacteristic, complemented
567C2—42no—
568C2—242no—
569C2—721no—
570C2#1059490011yescharacteristic
571C2C2×#6392511yescharacteristic
572C2#1059545411yescharacteristic
subgroupG/H
centerC22C2×D3×#67
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC32×C4×D4C22
socleC22×C32C22×D4
hypercenterC4×D4D32
hypocenterC32#3366
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C3×#5543C2semidirect
2. #10593900C2semidirect
3. C4×#3753C2semidirect
4. C3×#5015C2semidirect
5. #10591481C2semidirect
6. #10591430C2semidirect
7. #10590550C2semidirect
8. #10590467C2semidirect
9. #10590300C2semidirect
10. #10590277C2semidirect
11. #10590259C2semidirect
46. #5543D3semidirect
57. #5015D3semidirect
169. C32×D4#41semidirect
350. C3×D4#237semidirect
351. C3×D4#239semidirect
497. C32#3366semidirect
512. D4#3796semidirect
565. C3#80320semidirect
566. C3#83638semidirect

Maximal subgroups

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

subgroupindex
C3×#55432
#105939002
C4×#37532
C3×#50152
#105914812
#105914302
#105905502
#105904672
#105903002
#105902772
#105902592
#836383
#803203