Finite Group Databasebeta

← #11788630

Subgroups — #11788630

Order 1152.

745 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#949C211yesmaximal, characteristic
2C2×C3×#948C211yesmaximal, characteristic, complemented
3#10592931C211yesmaximal, characteristic, complemented
4C2×#63563C211yesmaximal, characteristic, complemented
5C2×#63558C211yesmaximal, characteristic, complemented
6#10589168C211yesmaximal, characteristic, complemented
7#10589166C211yesmaximal, characteristic, complemented
8#72091—31nomaximal
9#72018—31nomaximal
10C2×C32×#41C2211yescharacteristic
11C22×C3×D12C2211yescharacteristic
12C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
13C2×#3748C2211yescharacteristic
14C3×#949—22no—
15C3×#949C414yescomplemented
16C3×#948—22no—
17C3×#948—22no—
18C2×C3×#215—21no—
19C2×C3×#213—21no—
20C2×C3×#208—21no—
21C2×C3×#208C2211yescharacteristic
22C2×C3×#208C2211yescharacteristic
23C22×C3×C3⋊C8—21no—
24C22×C3×C3⋊C8C2211yescharacteristic
25#63662—21no—
26#63659—21no—
27#63563—21no—
28#63563—21no—
29#63558—22no—
30C2×C3×AGL(1,ℤ/8)—31no—
31C2×C3×#131—31no—
32C2×#949D311yescharacteristic
33C2×#948D311yescharacteristic, complemented
34#5232—31no—
35#5232—31no—
36#5232—61no—
37C2×#833—31no—
38C2×#832—31no—
39C2×#823—31no—
40C2×#821—31no—
41#4558—31no—
42#4554—31no—
43#4554—31no—
44#4552—31no—
45C23×C3×D3—21no—
46C32×#41—21no—
47C32×#41—21no—
48C32×#41C2×C412yes—
49C32×#41D412yes—
50C2×C32×Q8D411yescharacteristic
51C2×C32×D4—21no—
52C2×C32×D4D411yescharacteristic
53C22×C32×C4—21no—
54C22×C32×C4C2311yescharacteristic
55C22×C3⋊Dic3—21no—
56C2×C3×D12—21no—
57C2×C3×D12—21no—
58C2×C3×D12—21no—
59C2×C3×D12—21no—
60C2×C3×D12—21no—
61C2×C3×D12C2×C412yes—
62C4⋊(C3⋊Dic3)—21no—
63C4⋊(C3⋊Dic3)—21no—
64#3748—21no—
65#3748—21no—
66C2×C3⋊(C3⋊C8)—21no—
67C3×#215—24no—
68C3×#213—24no—
69C3×#208—24no—
70C3×#208—21no—
71C3×#208—21no—
72C3×#208C2×C412yes—
73C3×#208D412yes—
74C2×C3×C3⋊C8—24no—
75C2×C3×C3⋊C8—21no—
76C2×C3×C3⋊C8—21no—
77C2×C3×C3⋊C8—21no—
78C2×C3×C3⋊C8—21no—
79C2×C3×C3⋊C8D411yescharacteristic
80C2×C3×C3⋊C8D411yescharacteristic
81#1907—91noSylow
82C2×C3×#41—21no—
83C2×C3×#41C2×D311yescharacteristic
84C2×C3×#41C2×D311yescharacteristic
85C22×C3×D4—31no—
86C22×D12C2×D311yescharacteristic
87C3×AGL(1,ℤ/8)—34no—
88C3×AGL(1,ℤ/8)—62no—
89C2×C3×SD16—61no—
90C2×C3×D8—61no—
91C3×#131—62no—
92C3×#131—62no—
93C2×C3×M16—31no—
94C2×C3×M16—31no—
95C2×C3×M16—61no—
96C22×C3×C8—31no—
97C22×C3×C8—61no—
98#949—22no—
99#949Dic314yescomplemented
100#948—22no—
101#948—22no—
102C2×#215—21no—
103C2×#213—21no—
104C2×C4⋊Dic3—31no—
105C2×C4⋊Dic3—31no—
106C2×C4⋊Dic3—61no—
107C2×#208—21no—
108C2×#208—31no—
109C2×#208—31no—
110C2×#208—61no—
111C2×#208C2×D311yescharacteristic
112C2×#208C2×D311yescharacteristic
113C22×C3⋊C8—21no—
114C22×C3⋊C8C2×D311yescharacteristic
115#835—62no—
116#835—61no—
117#835—61no—
118#833—62no—
119#832—62no—
120#832—61no—
121#832—61no—
122#832—61no—
123#832—61no—
124#823—62no—
125#821—61no—
126#821—61no—
127C23×C32—21no—
128C23×C32#4111yescharacteristic
129C22×C3×D3—22no—
130C22×C3×D3—41no—
131C22×C3×D3—41no—
132C22×C3×D3—81no—
133C22×C3×D3—81no—
134C32×Q8—21no—
135C32×Q8C22⋊C412yes—
136C32×D4—22no—
137C32×D4—22no—
138C32×D4—21no—
139C32×D4C22⋊C412yes—
140C2×C32×C4—21no—
141C2×C32×C4—22no—
142C2×C32×C4—22no—
143C2×C32×C4C22×C411yescharacteristic
144C2×C32×C4C2×D411yescharacteristic
145C2×C32×C4C2×D411yescharacteristic
146C2×C32×C4C2×D411yescharacteristic
147C2×C32×C4C2×D411yescharacteristic
148C2×C32×C4#4111yescharacteristic
149C2×C3⋊Dic3—41no—
150C2×C3⋊Dic3—81no—
151C3×D12—22no—
152C3×D12—22no—
153C3×D12—41no—
154C3×D12—41no—
155C3⋊(C3⋊C8)—41no—
156C3×C3⋊C8—22no—
157C3×C3⋊C8—21no—
158C3×C3⋊C8—22no—
159C3×C3⋊C8—21no—
160C2×AGL(1,ℤ/8)—31no—
161C2×#131—31no—
162#417—91no—
163C2×#102—91no—
164C2×#100—91no—
165#343—91no—
166#341—91no—
167C24×C3—61no—
168C23×D3—21no—
169C3×#41—21no—
170C3×#41—21no—
171C3×#41—21no—
172C3×#41—21no—
173C3×#41—22no—
174C3×#41—22no—
175C3×#41—41no—
176C3×#41—41no—
177C3×#41C4×D312yes—
178C3×#41C2×Dic312yes—
179C3×#41#6712yes—
180C3×#41#6712yes—
181C2×C3×Q8—21no—
182C2×C3×Q8D1211yescharacteristic
183C2×C3×Q8#6711yescharacteristic
184C2×C3×D4—21no—
185C2×C3×D4—21no—
186C2×C3×D4—21no—
187C2×C3×D4—32no—
188C2×C3×D4—41no—
189C2×C3×D4—61no—
190C2×C3×D4—61no—
191C2×C3×D4—61no—
192C2×C3×D4—61no—
193C2×C3×D4—61no—
194C2×C3×D4D1211yescharacteristic
195C2×C3×D4#6711yescharacteristic
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—41no—
200C22×C3×C4C22×D311yescharacteristic
201C22×C3×C4C22×D311yescharacteristic
202C22×Dic3—61no—
203C22×Dic3—61no—
204C22×Dic3—121no—
205C2×D12—21no—
206C2×D12—21no—
207C2×D12—21no—
208C2×D12—21no—
209C2×D12—21no—
210C2×D12C2×Dic312yes—
211C3×SD16—64no—
212C3×D8—64no—
213C3×M16—32no—
214C3×M16—32no—
215C3×M16—64no—
216C3×M16—61no—
217C3×M16—61no—
218C2×C3×C8—31no—
219C2×C3×C8—31no—
220C2×C3×C8—64no—
221C2×C3×C8—61no—
222C2×C3×C8—61no—
223C2×C3×C8—61no—
224C2×C3×C8—61no—
225#215—24no—
226#213—24no—
227C4⋊Dic3—62no—
228C4⋊Dic3—61no—
229C4⋊Dic3—61no—
230C4⋊Dic3—61no—
231C4⋊Dic3—61no—
232C4⋊Dic3—121no—
233#208—24no—
234#208—21no—
235#208—21no—
236#208—61no—
237#208—61no—
238#208—61no—
239#208—61no—
240#208—121no—
241#208—121no—
242#208C2×Dic312yes—
243#208#6712yes—
244C2×C3⋊C8—24no—
245C2×C3⋊C8—21no—
246C2×C3⋊C8—21no—
247C2×C3⋊C8—21no—
248C2×C3⋊C8—21no—
249C2×C3⋊C8—62no—
250C2×C3⋊C8—61no—
251C2×C3⋊C8—61no—
252C2×C3⋊C8#6711yescharacteristic
253C2×C3⋊C8#6711yescharacteristic
254C22×C32—21no—
255C22×C32—21no—
256C22×C32—22no—
257C22×C32—41no—
258C22×C32—41no—
259C22×C32C4×D411yescharacteristic
260C22×C32#12111yescharacteristic
261C22×C32#12311yescharacteristic
262C2×C3×D3—42no—
263C2×C3×D3—82no—
264C2×C3×D3—81no—
265C2×C3×D3—81no—
266C2×C3×D3—82no—
267C32×C4—22no—
268C32×C4C2×C22⋊C411yescharacteristic
269C32×C4C4×D411yescharacteristic
270C32×C4#12011yescharacteristic
271C32×C4#12111yescharacteristic
272C3⋊Dic3—81no—
273C2×#41D3211yescharacteristic
274C22×D4—31no—
275AGL(1,ℤ/8)—34no—
276AGL(1,ℤ/8)—62no—
277C2×SD16—61no—
278C2×D8—61no—
279#131—62no—
280#131—62no—
281C2×M16—31no—
282C2×M16—31no—
283C2×M16—61no—
284C2×M16—91no—
285C22×C8—31no—
286C22×C8—61no—
287C2×C4⋊C4—91no—
288#103—181no—
289#102—181no—
290#102—181no—
291#102—181no—
292#100—182no—
293C23×C3—21no—
294C23×C3—21no—
295C23×C3—21no—
296C23×C3—41no—
297C23×C3—62no—
298C23×C3—121no—
299C23×C3—121no—
300C23×C3—241no—
301C23×C3—241no—
302C23×C3#23511yescharacteristic
303C23×C3#23711yescharacteristic
304C22×D3—22no—
305C22×D3—41no—
306C22×D3—41no—
307C22×D3—81no—
308C22×D3—81no—
309C3×Q8—21no—
310C3×Q8—21no—
311C3×Q8—22no—
312C3×Q8—41no—
313C3×Q8(C2×D3)⋊C412yes—
314C3×Q8#21712yes—
315C3×D4—22no—
316C3×D4—22no—
317C3×D4—22no—
318C3×D4—22no—
319C3×D4—21no—
320C3×D4—21no—
321C3×D4—22no—
322C3×D4—41no—
323C3×D4—42no—
324C3×D4—42no—
325C3×D4—62no—
326C3×D4—62no—
327C3×D4—121no—
328C3×D4—121no—
329C3×D4(C2×D3)⋊C412yes—
330C3×D4#21712yes—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—42no—
344C2×C3×C4—42no—
345C2×C3×C4—41no—
346C2×C3×C4C2×C4×D311yescharacteristic
347C2×C3×C4C2×D1211yescharacteristic
348C2×C3×C4#23511yescharacteristic
349C2×C3×C4D3×D411yescharacteristic
350C2×C3×C4D3×D411yescharacteristic
351C2×C3×C4#23711yescharacteristic
352C2×C3×C4C22×Dic311yescharacteristic
353C2×C3×C4C2×#6711yescharacteristic
354C2×C3×C4C2×#6711yescharacteristic
355C2×C3×C4C2×#6711yescharacteristic
356C2×C3×C4C2×#6711yescharacteristic
357C2×C3×C4C2×#6711yescharacteristic
358C2×Dic3—122no—
359C2×Dic3—121no—
360C2×Dic3—121no—
361C2×Dic3—242no—
362C2×Dic3—241no—
363C2×Dic3—241no—
364D12—22no—
365D12—22no—
366D12—41no—
367D12—41no—
368C3×C8—62no—
369C3×C8—61no—
370C3×C8—61no—
371C3×C8—62no—
372C3⋊C8—22no—
373C3⋊C8—21no—
374C3⋊C8—22no—
375C3⋊C8—21no—
376C3⋊C8—122no—
377C3⋊C8—121no—
378C3⋊C8—121no—
379C2×C32—21no—
380C2×C32—21no—
381C2×C32—42no—
382C2×C32#38611yescharacteristic
383C2×C32#47111yescharacteristic
384C2×C32#47211yescharacteristic
385C3×D3—82no—
386C24—61no—
387#41—21no—
388#41—21no—
389#41D3×Dic312yes—
390#41#62512yes—
391C2×Q8#62611yescharacteristic
392C2×D4—21no—
393C2×D4—32no—
394C2×D4—61no—
395C2×D4—61no—
396C2×D4—61no—
397C2×D4—61no—
398C2×D4—61no—
399C2×D4#62611yescharacteristic
400C22×C4—21no—
401C22×C4—181no—
402C22×C4C2×D3211yescharacteristic
403SD16—64no—
404D8—64no—
405M16—32no—
406M16—32no—
407M16—64no—
408M16—61no—
409M16—61no—
410M16—181no—
411M16—181no—
412C2×C8—31no—
413C2×C8—31no—
414C2×C8—64no—
415C2×C8—61no—
416C2×C8—61no—
417C2×C8—61no—
418C2×C8—61no—
419C2×C8—181no—
420C4⋊C4—181no—
421C4⋊C4—181no—
422C22×C3—21no—
423C22×C3—21no—
424C22×C3—21no—
425C22×C3—21no—
426C22×C3—22no—
427C22×C3—22no—
428C22×C3—21no—
429C22×C3—21no—
430C22×C3—21no—
431C22×C3—41no—
432C22×C3—41no—
433C22×C3—42no—
434C22×C3—41no—
435C22×C3—41no—
436C22×C3—41no—
437C22×C3—41no—
438C22×C3—81no—
439C22×C3—81no—
440C22×C3—122no—
441C22×C3—242no—
442C22×C3—242no—
443C22×C3—241no—
444C22×C3—241no—
445C22×C3C4×#6711yescharacteristic
446C22×C3#92911yescharacteristic
447C22×C3#93011yescharacteristic
448C22×C3D4×Dic311yescharacteristic
449C22×C3#93511yescharacteristic
450C22×C3#93911yescharacteristic
451C2×D3—42no—
452C2×D3—82no—
453C2×D3—81no—
454C2×D3—81no—
455C2×D3—82no—
456C3×C4—22no—
457C3×C4—22no—
458C3×C4—21no—
459C3×C4—21no—
460C3×C4—21no—
461C3×C4—21no—
462C3×C4—42no—
463C3×C4C2×(C2×D3)⋊C411yescharacteristic
464C3×C4C4×#6711yescharacteristic
465C3×C4#93011yescharacteristic
466C3×C4D4×Dic311yescharacteristic
467C3×C4#93711yescharacteristic
468C3×C4#93911yescharacteristic
469C3×C4C2×#21711yescharacteristic
470C3×C4#95311yescharacteristic
471Dic3—242no—
472Dic3—241no—
473Dic3—241no—
474C32#190711yescharacteristic, complemented, Sylow
475C23—21no—
476C23—62no—
477C23—121no—
478C23—121no—
479C23—241no—
480C23—241no—
481C23#380411yescharacteristic
482Q8—21no—
483Q8(C2×D3)⋊Dic312yes—
484D4—22no—
485D4—22no—
486D4—21no—
487D4—62no—
488D4—62no—
489D4—121no—
490D4—121no—
491D4(C2×D3)⋊Dic312yes—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—361no—
496C2×C4—721no—
497C2×C4C2×D3×Dic311yescharacteristic
498C2×C4#380411yescharacteristic
499C2×C4C2×#62511yescharacteristic
500C2×C4C2×#62611yescharacteristic
501C2×C4D3×#6711yescharacteristic
502C2×C4D3×#6711yescharacteristic
503C8—62no—
504C8—61no—
505C8—61no—
506C8—62no—
507C8—361no—
508C2×C3—21no—
509C2×C3—21no—
510C2×C3—21no—
511C2×C3—21no—
512C2×C3—21no—
513C2×C3—21no—
514C2×C3—21no—
515C2×C3—41no—
516C2×C3—41no—
517C2×C3—42no—
518C2×C3—42no—
519C2×C3—82no—
520C2×C3—242no—
521C2×C3#514111yescharacteristic
522C2×C3#514211yescharacteristic
523C2×C3#519911yescharacteristic
524C2×C3#520011yescharacteristic
525C2×C3#521311yescharacteristic
526C2×C3#522011yescharacteristic
527D3—82no—
528C22—21no—
529C22—21no—
530C22—22no—
531C22—41no—
532C22—41no—
533C22—122no—
534C22—242no—
535C22—242no—
536C22—241no—
537C22—241no—
538C22#6396111yescharacteristic
539C22Dic3×#6711yescharacteristic
540C22#6397811yescharacteristic
541C4—22no—
542C4—721no—
543C4C2×(C2×D3)⋊Dic311yescharacteristic
544C4Dic3×#6711yescharacteristic
545C4#6397611yescharacteristic
546C4#6397811yescharacteristic
547C3—21no—
548C3#7201811yescharacteristic, complemented
549C3#7209111yescharacteristic, complemented
550C2—21no—
551C2—21no—
552C2—42no—
553C2—242no—
554C2#1059167711yescharacteristic
555C2#1059167911yescharacteristic
556C2#1059171011yescharacteristic
subgroupG/H
centerC22#63978
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C2×D32
FittingC2×C32×#41C22
socleC22×C32#121
prime radicalC2×C3×#949C2
cosocleC22×C32×C4C23
hypercenterC2×#41D32
hypocenterC32#1907
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
2. C2×C3×#948C2semidirect
3. #10592931C2semidirect
4. C2×#63563C2semidirect
5. C2×#63558C2semidirect
6. #10589168C2semidirect
7. #10589166C2semidirect
15. C3×#949C4semidirect
33. C2×#948D3semidirect
99. #949Dic3semidirect
474. C32#1907semidirect
548. C3#72018semidirect
549. C3#72091semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9492
C2×C3×#9482
#105929312
C2×#635632
C2×#635582
#105891682
#105891662
#720913
#720183