Finite Group Databasebeta

← #11788635

Subgroups — #11788635

Order 1152.

765 conjugacy classes of subgroups.

The subgroup lattice has 559 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#932C211yesmaximal, characteristic
2C2×#63815C211yesmaximal, characteristic, complemented
3C2×#63659C211yesmaximal, characteristic, complemented
4C2×C3×#833C211yesmaximal, characteristic, complemented
5#10591088C211yesmaximal, characteristic, complemented
6#10589168C211yesmaximal, characteristic, complemented
7#10589165C211yesmaximal, characteristic, complemented
8#72091—31nomaximal
9#70088—31nomaximal
10C22×C32×D4C2211yescharacteristic
11C2×C3×#235C2211yescharacteristic
12C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
13C22×C3⋊(C3⋊C8)C2211yescharacteristic
14C2×C3×#214—21no—
15C3×#932—22no—
16C3×#932C414yescomplemented
17C2×C3×#213—21no—
18C2×C3×#208C2211yescharacteristic
19C2×C3×#208C2211yescharacteristic
20C2×C3×#208C2211yescharacteristic
21C2×#3711—21no—
22#63815—22no—
23#63815—22no—
24C2×D3×C3⋊C8—21no—
25#63659—21no—
26#63659—21no—
27C3×#833—22no—
28#63566—21no—
29#63563—21no—
30C2×C3×AGL(1,ℤ/8)—31no—
31C2×#948—31no—
32C2×#932D311yescharacteristic
33C2×#907—31no—
34C2×C3×#100—31no—
35#5232—31no—
36C2×#833D311yescharacteristic, complemented
37C2×#832—31no—
38C2×#832—31no—
39C2×#832—61no—
40#5133—31no—
41#4558—31no—
42#4558—31no—
43#4554—31no—
44#4554—31no—
45C24×C32—21no—
46C2×C32×D4—21no—
47C2×C32×D4—21no—
48C2×C32×D4—21no—
49C2×C32×D4—21no—
50C2×C32×D4—21no—
51C2×C32×D4C2×C412yes—
52C22×C32×C4C2311yescharacteristic
53C22×C3⋊Dic3—21no—
54C2×C3×#67—21no—
55C3×#235—21no—
56C3×#235—21no—
57C3×#235C2×C412yes—
58C3×#235D412yes—
59C2×C3×D12D411yescharacteristic
60C2×C3×C4×D3—21no—
61C2×C3×Dic6D411yescharacteristic
62C4⋊(C3⋊Dic3)—21no—
63C4⋊(C3⋊Dic3)—21no—
64C2×C3⋊(C3⋊C8)—21no—
65C2×C3⋊(C3⋊C8)—21no—
66C2×C3⋊(C3⋊C8)D411yescharacteristic
67C2×C3⋊(C3⋊C8)D411yescharacteristic
68C3×#214—24no—
69C3×#213—24no—
70C3×#208—21no—
71C3×#208—21no—
72C3×#208—21no—
73C3×#208—21no—
74C3×#208C2×C412yes—
75C3×#208D412yes—
76C2×C3×C3⋊C8—21no—
77C2×C3×C3⋊C8—21no—
78C2×C3×C3⋊C8—21no—
79#3711—24no—
80D3×C3⋊C8—24no—
81#1907—91noSylow
82C2×C3×#41—31no—
83C22×C3×D4—21no—
84C22×C3×D4C2×D311yescharacteristic
85C22×C3×D4C2×D311yescharacteristic
86C2×#235C2×D311yescharacteristic
87C3×AGL(1,ℤ/8)—34no—
88C3×AGL(1,ℤ/8)—62no—
89C2×C3×SD16—61no—
90C2×C3×D8—61no—
91C2×C3×M16—31no—
92C2×C3×M16—31no—
93C2×C3×M16—31no—
94#948—62no—
95#948—62no—
96C2×#214—21no—
97#932—22no—
98#932Dic314yescomplemented
99C2×#213—21no—
100C2×C4⋊Dic3—31no—
101C2×C4⋊Dic3—31no—
102C2×C4⋊Dic3—61no—
103C2×#208—61no—
104C2×#208C2×D311yescharacteristic
105C2×#208C2×D311yescharacteristic
106C2×#208C2×D311yescharacteristic
107C22×C3⋊C8—31no—
108C22×C3⋊C8—31no—
109C22×C3⋊C8—61no—
110C22×C3⋊C8—61no—
111#907—62no—
112#907—62no—
113C2×ΓL(1,𝔽25)—61no—
114C2×D3×C8—61no—
115C3×#100—62no—
116#835—61no—
117#833—22no—
118#832—61no—
119#832—61no—
120#832—61no—
121#832—61no—
122#832—61no—
123#832—62no—
124#832—62no—
125#821—61no—
126#816—61no—
127C23×C32—22no—
128C23×C32—41no—
129C23×C32—41no—
130C23×C32—81no—
131C23×C32—81no—
132C23×C32#4111yescharacteristic
133C22×C3×D3—21no—
134C32×D4—22no—
135C32×D4—22no—
136C32×D4—41no—
137C32×D4—41no—
138C2×C32×C4C22×C411yescharacteristic
139C2×C32×C4C2×D411yescharacteristic
140C2×C32×C4C2×D411yescharacteristic
141C2×C32×C4C2×D411yescharacteristic
142C2×C32×C4C2×D411yescharacteristic
143C2×C32×C4#4111yescharacteristic
144C2×C3⋊Dic3—41no—
145C2×C3⋊Dic3—81no—
146C3×#67—22no—
147C3×#67—22no—
148C2×C3×Dic3—21no—
149C3×D12—21no—
150C3×D12C22⋊C412yes—
151C3×C4×D3—22no—
152C3×C4×D3—22no—
153C3×Dic6—21no—
154C3×Dic6C22⋊C412yes—
155C3⋊(C3⋊C8)—21no—
156C3⋊(C3⋊C8)—21no—
157C3×C3⋊C8—22no—
158C3×C3⋊C8—22no—
159C3×C3⋊C8—41no—
160C2×AGL(1,ℤ/8)—31no—
161C2×#131—91no—
162#417—91no—
163C2×#102—91no—
164C2×#100—31no—
165#343—91no—
166#341—91no—
167C24×C3—21no—
168C24×C3—21no—
169C24×C3—41no—
170C3×#41—32no—
171C3×#41—32no—
172C3×#41—61no—
173C3×#41—61no—
174C2×C3×Q8—31no—
175C2×C3×D4—22no—
176C2×C3×D4—22no—
177C2×C3×D4—21no—
178C2×C3×D4—21no—
179C2×C3×D4—21no—
180C2×C3×D4—21no—
181C2×C3×D4—21no—
182C2×C3×D4—21no—
183C2×C3×D4—21no—
184C2×C3×D4—21no—
185C2×C3×D4—21no—
186C2×C3×D4—21no—
187C2×C3×D4—22no—
188C2×C3×D4—31no—
189C2×C3×D4—41no—
190C2×C3×D4—41no—
191C2×C3×D4—41no—
192C2×C3×D4—61no—
193C2×C3×D4C4×D312yes—
194C2×C3×D4C2×Dic312yes—
195C22×C3×C4—21no—
196C22×C3×C4—61no—
197C22×C3×C4C22×D311yescharacteristic
198C22×C3×C4C22×D311yescharacteristic
199C2×#67—21no—
200C22×Dic3—61no—
201C22×Dic3—61no—
202C22×Dic3—121no—
203#235—21no—
204#235—21no—
205#235C2×Dic312yes—
206#235#6712yes—
207C2×D12#6711yescharacteristic
208C2×C4×D3—21no—
209C2×Dic6#6711yescharacteristic
210C3×SD16—64no—
211C3×D8—64no—
212C3×M16—32no—
213C3×M16—32no—
214C3×M16—61no—
215C3×M16—61no—
216C3×M16—61no—
217C3×M16—61no—
218C2×C3×C8—61no—
219C2×C3×C8—61no—
220C2×C3×C8—61no—
221#214—24no—
222#213—24no—
223C4⋊Dic3—62no—
224C4⋊Dic3—61no—
225C4⋊Dic3—61no—
226C4⋊Dic3—61no—
227C4⋊Dic3—61no—
228C4⋊Dic3—121no—
229#208—21no—
230#208—21no—
231#208—21no—
232#208—21no—
233#208—64no—
234#208C2×Dic312yes—
235#208#6712yes—
236C2×C3⋊C8—21no—
237C2×C3⋊C8—21no—
238C2×C3⋊C8—21no—
239C2×C3⋊C8—31no—
240C2×C3⋊C8—31no—
241C2×C3⋊C8—31no—
242C2×C3⋊C8—31no—
243C2×C3⋊C8—61no—
244C2×C3⋊C8—61no—
245C2×C3⋊C8—64no—
246C2×C3⋊C8—61no—
247C2×C3⋊C8—61no—
248C2×C3⋊C8—61no—
249C2×C3⋊C8—61no—
250C2×C3⋊C8—121no—
251C2×C3⋊C8—121no—
252ΓL(1,𝔽25)—64no—
253D3×C8—64no—
254C22×C32—21no—
255C22×C32—21no—
256C22×C32—42no—
257C22×C32—82no—
258C22×C32—82no—
259C22×C32—81no—
260C22×C32—81no—
261C22×C32C4×D411yescharacteristic
262C22×C32#12111yescharacteristic
263C22×C32#12311yescharacteristic
264C2×C3×D3—22no—
265C2×C3×D3—41no—
266C2×C3×D3—41no—
267C32×C4C2×C22⋊C411yescharacteristic
268C32×C4C4×D411yescharacteristic
269C32×C4#12011yescharacteristic
270C32×C4#12111yescharacteristic
271C3⋊Dic3—81no—
272C3×Dic3—22no—
273C2×#41—31no—
274C22×D4D3211yescharacteristic
275AGL(1,ℤ/8)—34no—
276AGL(1,ℤ/8)—62no—
277C2×SD16—61no—
278C2×D8—61no—
279#131—182no—
280#131—182no—
281C2×M16—31no—
282C2×M16—31no—
283C2×M16—31no—
284C2×M16—181no—
285C22×C8—91no—
286C22×C8—181no—
287C2×C4⋊C4—91no—
288#103—181no—
289#102—181no—
290#102—181no—
291#102—181no—
292#100—62no—
293C23×C3—22no—
294C23×C3—22no—
295C23×C3—21no—
296C23×C3—42no—
297C23×C3—42no—
298C23×C3—42no—
299C23×C3—41no—
300C23×C3—41no—
301C23×C3—41no—
302C23×C3—41no—
303C23×C3—61no—
304C23×C3—81no—
305C23×C3—81no—
306C23×C3—81no—
307C23×C3—81no—
308C23×C3—82no—
309C23×C3—82no—
310C23×C3#23511yescharacteristic
311C23×C3#23711yescharacteristic
312C22×D3—21no—
313C3×Q8—32no—
314C3×Q8—61no—
315C3×D4—24no—
316C3×D4—22no—
317C3×D4—22no—
318C3×D4—22no—
319C3×D4—22no—
320C3×D4—32no—
321C3×D4—42no—
322C3×D4—42no—
323C3×D4—42no—
324C3×D4—41no—
325C3×D4—41no—
326C3×D4—41no—
327C3×D4—41no—
328C3×D4—62no—
329C3×D4—62no—
330C3×D4—61no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—61no—
338C2×C3×C4—62no—
339C2×C3×C4—62no—
340C2×C3×C4C2×C4×D311yescharacteristic
341C2×C3×C4D3×D411yescharacteristic
342C2×C3×C4D3×D411yescharacteristic
343C2×C3×C4D3×D411yescharacteristic
344C2×C3×C4D3×D411yescharacteristic
345C2×C3×C4D3×D411yescharacteristic
346C2×C3×C4#23711yescharacteristic
347C2×C3×C4#23711yescharacteristic
348C2×C3×C4C22×Dic311yescharacteristic
349C2×C3×C4C2×#6711yescharacteristic
350C2×C3×C4C2×#6711yescharacteristic
351C2×C3×C4C2×#6711yescharacteristic
352#67—22no—
353#67—22no—
354C2×Dic3—21no—
355C2×Dic3—122no—
356C2×Dic3—121no—
357C2×Dic3—121no—
358C2×Dic3—242no—
359C2×Dic3—241no—
360C2×Dic3—241no—
361D12—21no—
362D12#21712yes—
363C4×D3—22no—
364C4×D3—22no—
365Dic6—21no—
366Dic6#21712yes—
367C3×C8—62no—
368C3×C8—62no—
369C3×C8—121no—
370C3⋊C8—22no—
371C3⋊C8—22no—
372C3⋊C8—41no—
373C3⋊C8—61no—
374C3⋊C8—61no—
375C3⋊C8—61no—
376C3⋊C8—61no—
377C3⋊C8—121no—
378C3⋊C8—121no—
379C2×C32—21no—
380C2×C32—21no—
381C2×C32—82no—
382C2×C32#38611yescharacteristic
383C2×C32#47111yescharacteristic
384C2×C32#47211yescharacteristic
385C3×D3—42no—
386C24—21no—
387#41—32no—
388#41—32no—
389#41—61no—
390#41—61no—
391C2×Q8—31no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—31no—
398C2×D4—61no—
399C2×D4D3×Dic312yes—
400C22×C4—61no—
401C22×C4—181no—
402C22×C4C2×D3211yescharacteristic
403SD16—64no—
404D8—64no—
405M16—32no—
406M16—32no—
407M16—61no—
408M16—61no—
409M16—61no—
410M16—61no—
411M16—184no—
412C2×C8—61no—
413C2×C8—61no—
414C2×C8—61no—
415C2×C8—91no—
416C2×C8—91no—
417C2×C8—184no—
418C2×C8—181no—
419C2×C8—181no—
420C4⋊C4—181no—
421C4⋊C4—181no—
422C22×C3—22no—
423C22×C3—22no—
424C22×C3—21no—
425C22×C3—21no—
426C22×C3—21no—
427C22×C3—21no—
428C22×C3—21no—
429C22×C3—21no—
430C22×C3—21no—
431C22×C3—44no—
432C22×C3—42no—
433C22×C3—42no—
434C22×C3—62no—
435C22×C3—82no—
436C22×C3—82no—
437C22×C3—84no—
438C22×C3—84no—
439C22×C3—82no—
440C22×C3—82no—
441C22×C3—82no—
442C22×C3—82no—
443C22×C3—81no—
444C22×C3—81no—
445C22×C3—81no—
446C22×C3—81no—
447C22×C3—121no—
448C22×C3—121no—
449C22×C3#88311yescharacteristic
450C22×C3#88411yescharacteristic
451C22×C3C4×#6711yescharacteristic
452C22×C3D4×Dic311yescharacteristic
453C22×C3#93511yescharacteristic
454C22×C3#93911yescharacteristic
455C2×D3—22no—
456C2×D3—41no—
457C2×D3—41no—
458C3×C4—21no—
459C3×C4—21no—
460C3×C4—21no—
461C3×C4—21no—
462C3×C4—62no—
463C3×C4D3×C22⋊C411yescharacteristic
464C3×C4#67⋊C411yescharacteristic
465C3×C4D4×Dic311yescharacteristic
466C3×C4#93711yescharacteristic
467C3×C4#93711yescharacteristic
468C3×C4#93811yescharacteristic
469C3×C4#93911yescharacteristic
470C3×C4C2×#21711yescharacteristic
471Dic3—22no—
472Dic3—242no—
473Dic3—241no—
474Dic3—241no—
475C32#190711yescharacteristic, complemented, Sylow
476C23—22no—
477C23—41no—
478C23—41no—
479C23—61no—
480C23—81no—
481C23—81no—
482C23#380411yescharacteristic
483Q8—32no—
484Q8—61no—
485D4—22no—
486D4—22no—
487D4—32no—
488D4—41no—
489D4—41no—
490D4—62no—
491D4—62no—
492D4—61no—
493C2×C4—61no—
494C2×C4—62no—
495C2×C4—62no—
496C2×C4—361no—
497C2×C4—721no—
498C2×C4C2×D3×Dic311yescharacteristic
499C2×C4#380511yescharacteristic
500C2×C4D3×#6711yescharacteristic
501C2×C4D3×#6711yescharacteristic
502C2×C4D3×#6711yescharacteristic
503C2×C4#381111yescharacteristic
504C8—62no—
505C8—62no—
506C8—121no—
507C8—181no—
508C8—181no—
509C2×C3—22no—
510C2×C3—22no—
511C2×C3—21no—
512C2×C3—21no—
513C2×C3—21no—
514C2×C3—21no—
515C2×C3—21no—
516C2×C3—21no—
517C2×C3—21no—
518C2×C3—84no—
519C2×C3—82no—
520C2×C3—82no—
521C2×C3—122no—
522C2×C3#474811yescharacteristic
523C2×C3#474911yescharacteristic
524C2×C3#495611yescharacteristic
525C2×C3#519911yescharacteristic
526C2×C3#520011yescharacteristic
527C2×C3#522011yescharacteristic
528D3—42no—
529C22—21no—
530C22—21no—
531C22—42no—
532C22—62no—
533C22—82no—
534C22—82no—
535C22—81no—
536C22—81no—
537C22—121no—
538C22—121no—
539C22#6396911yescharacteristic
540C22#6397111yescharacteristic
541C22Dic3×#6711yescharacteristic
542C4—62no—
543C4—721no—
544C4D3×#21711yescharacteristic
545C4#6397011yescharacteristic
546C4#67⋊Dic311yescharacteristic
547C4#6398311yescharacteristic
548C3—21no—
549C3#7008811yescharacteristic, complemented
550C3#7209111yescharacteristic, complemented
551C2—21no—
552C2—21no—
553C2—82no—
554C2—122no—
555C2#1059150711yescharacteristic
556C2#1059150811yescharacteristic
557C2#1059172111yescharacteristic
subgroupG/H
centerC22#63971
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C2×D32
FittingC22×C32×D4C22
socleC22×C32#121
prime radicalC2×C3×#932C2
cosocleC22×C32×C4C23
hypercenterC22×D4D32
hypocenterC32#1907
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
2. C2×#63815C2semidirect
3. C2×#63659C2semidirect
4. C2×C3×#833C2semidirect
5. #10591088C2semidirect
6. #10589168C2semidirect
7. #10589165C2semidirect
16. C3×#932C4semidirect
36. C2×#833D3semidirect
98. #932Dic3semidirect
475. C32#1907semidirect
549. C3#70088semidirect
550. C3#72091semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9322
C2×#638152
C2×#636592
C2×C3×#8332
#105910882
#105891682
#105891652
#720913
#700883