Finite Group Databasebeta

← #11788639

Subgroups — #11788639

Order 1152.

709 conjugacy classes of subgroups.

The subgroup lattice has 537 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#901C211yesmaximal, characteristic, complemented
2C2×#63823C211yesmaximal, characteristic
3C2×C3×#821C211yesmaximal, characteristic, complemented
4#10591087C211yesmaximal, characteristic, complemented
5C2×#63558C211yesmaximal, characteristic, complemented
6#10589418C211yesmaximal, characteristic, complemented
7#10589167C211yesmaximal, characteristic, complemented
8#72018—31nomaximal
9#70218—31nomaximal
10C2×C3×#235C2211yescharacteristic
11C22×C3×D12C2211yescharacteristic
12C22×C32×C8C2211yescharacteristic
13C2×#3748C2211yescharacteristic
14C2×#3748C2211yescharacteristic
15C2×C3×C4⋊Dic3C2211yescharacteristic
16C2×C3×#208C2211yescharacteristic
17C3×#901—22no—
18C3×#901—22no—
19C2×C3×ΓL(1,𝔽25)—21no—
20C2×C3×D3×C8—21no—
21C2×#3715—21no—
22#63823—22no—
23#63823C414yescomplemented
24C2×#3713—21no—
25C3×#821—21no—
26C3×#821—21no—
27#63565—21no—
28#63562—21no—
29#63558—22no—
30C2×C3×#131—31no—
31C2×#949—31no—
32C2×#932—31no—
33C2×#901D311yescharacteristic, complemented
34C2×C3×#102—31no—
35#5232—31no—
36C2×#833—31no—
37C2×#823—31no—
38C2×#821D311yescharacteristic, complemented
39#4966—31no—
40#4554—31no—
41#4552—31no—
42#4552—31no—
43#4552—61no—
44#4530—31no—
45C23×C3×D3—21no—
46C22×C32×C4C2311yescharacteristic
47C2×C3×#67—21no—
48C22×C3×Dic3—21no—
49C3×#235—21no—
50C3×#235—21no—
51C3×#235C2×C412yes—
52C3×#235D412yes—
53C2×C3×D12—21no—
54C2×C3×D12—21no—
55C2×C3×D12—21no—
56C2×C3×D12—21no—
57C2×C3×D12—21no—
58C2×C3×D12C2×C412yes—
59C2×C3×D12D411yescharacteristic
60C2×C3×C4×D3—21no—
61C2×C3×Dic6D411yescharacteristic
62C2×C32×C8—21no—
63C2×C32×C8—21no—
64C2×C32×C8D411yescharacteristic
65C2×C32×C8D411yescharacteristic
66#3748—21no—
67#3748—21no—
68#3748—21no—
69#3748C2×C412yes—
70C2×C3⋊(C3⋊C8)—21no—
71C2×C3⋊(C3⋊C8)—21no—
72C3×C4⋊Dic3—21no—
73C3×C4⋊Dic3—21no—
74C3×#208—21no—
75C3×#208D412yes—
76C2×C3×C3⋊C8—21no—
77C3×ΓL(1,𝔽25)—24no—
78C3×D3×C8—24no—
79#3715—24no—
80#3713—24no—
81#1907—91noSylow
82C2×C3×#41—31no—
83C22×C3×D4—31no—
84C2×#235C2×D311yescharacteristic
85C22×D12C2×D311yescharacteristic
86C3×#131—62no—
87C3×#131—62no—
88C2×C3×M16—31no—
89C2×C3×M16—61no—
90C22×C3×C8—21no—
91C22×C3×C8—61no—
92C22×C3×C8C2×D311yescharacteristic
93C22×C3×C8C2×D311yescharacteristic
94C2×C3×C4⋊C4—31no—
95#949—34no—
96#949—62no—
97C2×#215—61no—
98C2×#214—61no—
99#932—34no—
100#932—62no—
101C2×#213—61no—
102C2×#213—61no—
103C2×C4⋊Dic3C2×D311yescharacteristic
104C2×#208—31no—
105C2×#208—31no—
106C2×#208—31no—
107C2×#208—31no—
108C2×#208—61no—
109C2×#208—61no—
110C2×#208C2×D311yescharacteristic
111#901—22no—
112#901—22no—
113C2×ΓL(1,𝔽25)—21no—
114C2×D3×C8—21no—
115C3×#102—61no—
116C3×#102—61no—
117#835—61no—
118#833—62no—
119#832—61no—
120#823—62no—
121#821—21no—
122#821—21no—
123#810—61no—
124#809—61no—
125C23×C32#4111yescharacteristic
126C22×C3×D3—21no—
127C22×C3×D3—22no—
128C22×C3×D3—41no—
129C22×C3×D3—41no—
130C22×C3×D3—81no—
131C22×C3×D3—81no—
132C2×C32×C4C22×C411yescharacteristic
133C2×C32×C4C2×D411yescharacteristic
134C2×C32×C4C2×D411yescharacteristic
135C2×C32×C4C2×D411yescharacteristic
136C2×C32×C4C2×D411yescharacteristic
137C2×C32×C4#4111yescharacteristic
138C3×#67—22no—
139C3×#67—22no—
140C2×C3×Dic3—21no—
141C2×C3×Dic3—41no—
142C2×C3×Dic3—81no—
143C3×D12—22no—
144C3×D12—22no—
145C3×D12—21no—
146C3×D12—41no—
147C3×D12—41no—
148C3×D12C22⋊C412yes—
149C3×C4×D3—22no—
150C3×C4×D3—22no—
151C3×Dic6—21no—
152C3×Dic6C22⋊C412yes—
153C32×C8—21no—
154C32×C8—21no—
155C3⋊(C3⋊C8)—22no—
156C3⋊(C3⋊C8)—41no—
157C3×C3⋊C8—22no—
158C2×AGL(1,ℤ/8)—91no—
159C2×#131—31no—
160#417—91no—
161C2×#102—31no—
162C2×#100—91no—
163#343—91no—
164#341—91no—
165C24×C3—61no—
166C23×D3—21no—
167C3×#41—32no—
168C3×#41—32no—
169C3×#41—61no—
170C3×#41—61no—
171C2×C3×Q8—31no—
172C2×C3×D4—32no—
173C2×C3×D4—31no—
174C2×C3×D4—61no—
175C2×C3×D4—61no—
176C2×C3×D4—61no—
177C2×C3×D4—61no—
178C2×C3×D4—61no—
179C2×C3×D4—61no—
180C22×C3×C4—21no—
181C22×C3×C4—61no—
182C22×C3×C4—61no—
183C22×C3×C4C22×D311yescharacteristic
184C22×C3×C4C22×D311yescharacteristic
185C2×#67—21no—
186C22×Dic3—21no—
187#235—21no—
188#235—21no—
189#235C4×D312yes—
190#235#6712yes—
191C2×D12—21no—
192C2×D12—21no—
193C2×D12—21no—
194C2×D12—21no—
195C2×D12—21no—
196C2×D12C4×D312yes—
197C2×D12D1211yescharacteristic
198C2×C4×D3—21no—
199C2×Dic6D1211yescharacteristic
200C3×M16—32no—
201C3×M16—64no—
202C3×M16—61no—
203C2×C3×C8—21no—
204C2×C3×C8—21no—
205C2×C3×C8—21no—
206C2×C3×C8—21no—
207C2×C3×C8—21no—
208C2×C3×C8—21no—
209C2×C3×C8—41no—
210C2×C3×C8—41no—
211C2×C3×C8—64no—
212C2×C3×C8—61no—
213C2×C3×C8#6711yescharacteristic
214C2×C3×C8#6711yescharacteristic
215C2×C3×C8#6711yescharacteristic
216C2×C3×C8#6711yescharacteristic
217C3×C4⋊C4—61no—
218C3×C4⋊C4—61no—
219#215—64no—
220#214—64no—
221#213—64no—
222#213—64no—
223C4⋊Dic3—21no—
224C4⋊Dic3—21no—
225#208—21no—
226#208—32no—
227#208—32no—
228#208—62no—
229#208—61no—
230#208—61no—
231#208—62no—
232#208—61no—
233#208—61no—
234#208—61no—
235#208—61no—
236#208—121no—
237#208—121no—
238#208#6712yes—
239C2×C3⋊C8—21no—
240C2×C3⋊C8—61no—
241C2×C3⋊C8—61no—
242C2×C3⋊C8—61no—
243C2×C3⋊C8—61no—
244C2×C3⋊C8—121no—
245C2×C3⋊C8—121no—
246ΓL(1,𝔽25)—24no—
247D3×C8—24no—
248C22×C32—21no—
249C22×C32—21no—
250C22×C32C4×D411yescharacteristic
251C22×C32#12111yescharacteristic
252C22×C32#12311yescharacteristic
253C2×C3×D3—22no—
254C2×C3×D3—41no—
255C2×C3×D3—41no—
256C2×C3×D3—42no—
257C2×C3×D3—82no—
258C2×C3×D3—81no—
259C2×C3×D3—81no—
260C2×C3×D3—82no—
261C32×C4C2×C22⋊C411yescharacteristic
262C32×C4C4×D411yescharacteristic
263C32×C4#12011yescharacteristic
264C32×C4#12111yescharacteristic
265C3×Dic3—22no—
266C3×Dic3—81no—
267C2×#41—31no—
268C22×D4—31no—
269AGL(1,ℤ/8)—94no—
270AGL(1,ℤ/8)—182no—
271C2×SD16—181no—
272C2×D8—181no—
273#131—62no—
274#131—62no—
275C2×M16—31no—
276C2×M16—61no—
277C2×M16—91no—
278C2×M16—91no—
279C22×C8—61no—
280C22×C8D3211yescharacteristic
281C2×C4⋊C4—31no—
282#103—181no—
283#102—61no—
284#102—61no—
285#102—181no—
286#100—182no—
287C23×C3—21no—
288C23×C3—61no—
289C23×C3—62no—
290C23×C3—121no—
291C23×C3—121no—
292C23×C3—241no—
293C23×C3—241no—
294C23×C3#23511yescharacteristic
295C23×C3#23711yescharacteristic
296C22×D3—21no—
297C22×D3—22no—
298C22×D3—41no—
299C22×D3—41no—
300C22×D3—81no—
301C22×D3—81no—
302C3×Q8—32no—
303C3×Q8—61no—
304C3×D4—32no—
305C3×D4—62no—
306C3×D4—62no—
307C3×D4—61no—
308C3×D4—62no—
309C3×D4—62no—
310C3×D4—121no—
311C3×D4—121no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—21no—
318C2×C3×C4—61no—
319C2×C3×C4—62no—
320C2×C3×C4—62no—
321C2×C3×C4—121no—
322C2×C3×C4—241no—
323C2×C3×C4C2×C4×D311yescharacteristic
324C2×C3×C4C2×C4×D311yescharacteristic
325C2×C3×C4C2×D1211yescharacteristic
326C2×C3×C4#23511yescharacteristic
327C2×C3×C4#23511yescharacteristic
328C2×C3×C4D3×D411yescharacteristic
329C2×C3×C4D3×D411yescharacteristic
330C2×C3×C4D3×D411yescharacteristic
331C2×C3×C4C2×#6711yescharacteristic
332C2×C3×C4C2×#6711yescharacteristic
333C2×C3×C4C2×#6711yescharacteristic
334C2×C3×C4C2×#6711yescharacteristic
335#67—22no—
336#67—22no—
337C2×Dic3—21no—
338C2×Dic3—41no—
339C2×Dic3—81no—
340D12—22no—
341D12—22no—
342D12—21no—
343D12—41no—
344D12—41no—
345D12(C2×D3)⋊C412yes—
346C4×D3—22no—
347C4×D3—22no—
348Dic6—21no—
349Dic6(C2×D3)⋊C412yes—
350C3×C8—21no—
351C3×C8—21no—
352C3×C8—21no—
353C3×C8—21no—
354C3×C8—41no—
355C3×C8—41no—
356C3×C8—62no—
357C3⋊C8—22no—
358C3⋊C8—62no—
359C3⋊C8—62no—
360C3⋊C8—122no—
361C3⋊C8—122no—
362C3⋊C8—121no—
363C3⋊C8—121no—
364C2×C32—21no—
365C2×C32—21no—
366C2×C32#38611yescharacteristic
367C2×C32#47111yescharacteristic
368C2×C32#47211yescharacteristic
369C3×D3—42no—
370C3×D3—82no—
371C24—61no—
372#41—32no—
373#41—32no—
374#41—61no—
375#41—61no—
376C2×Q8—31no—
377C2×D4—32no—
378C2×D4—31no—
379C2×D4—61no—
380C2×D4—61no—
381C2×D4—61no—
382C2×D4—61no—
383C2×D4—61no—
384C2×D4—61no—
385C22×C4—61no—
386C22×C4—61no—
387C22×C4C2×D3211yescharacteristic
388SD16—184no—
389D8—184no—
390M16—32no—
391M16—64no—
392M16—61no—
393M16—92no—
394M16—181no—
395M16—181no—
396M16—181no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—64no—
400C2×C8—61no—
401C2×C8—181no—
402C2×C8—181no—
403C2×C8#62511yescharacteristic
404C2×C8#62511yescharacteristic
405C4⋊C4—61no—
406C4⋊C4—61no—
407C22×C3—21no—
408C22×C3—21no—
409C22×C3—21no—
410C22×C3—21no—
411C22×C3—21no—
412C22×C3—21no—
413C22×C3—21no—
414C22×C3—41no—
415C22×C3—41no—
416C22×C3—62no—
417C22×C3—121no—
418C22×C3—121no—
419C22×C3—122no—
420C22×C3—242no—
421C22×C3—242no—
422C22×C3—241no—
423C22×C3—241no—
424C22×C3#67⋊C411yescharacteristic
425C22×C3#88311yescharacteristic
426C22×C3C4×#6711yescharacteristic
427C22×C3#92911yescharacteristic
428C22×C3#93011yescharacteristic
429C22×C3#93811yescharacteristic
430C2×D3—22no—
431C2×D3—41no—
432C2×D3—41no—
433C2×D3—42no—
434C2×D3—82no—
435C2×D3—81no—
436C2×D3—81no—
437C2×D3—82no—
438C3×C4—21no—
439C3×C4—21no—
440C3×C4—21no—
441C3×C4—21no—
442C3×C4—62no—
443C3×C4—241no—
444C3×C4D3×C22⋊C411yescharacteristic
445C3×C4#88411yescharacteristic
446C3×C4C2×(C2×D3)⋊C411yescharacteristic
447C3×C4C4×#6711yescharacteristic
448C3×C4C4×#6711yescharacteristic
449C3×C4#93011yescharacteristic
450C3×C4#93711yescharacteristic
451C3×C4#95311yescharacteristic
452Dic3—22no—
453Dic3—81no—
454C32#190711yescharacteristic, complemented, Sylow
455C23—61no—
456C23—62no—
457C23—121no—
458C23—121no—
459C23—241no—
460C23—241no—
461C23#379511yescharacteristic
462Q8—32no—
463Q8—61no—
464D4—32no—
465D4—62no—
466D4—62no—
467D4—62no—
468D4—61no—
469D4—62no—
470D4—121no—
471D4—121no—
472C2×C4—61no—
473C2×C4—62no—
474C2×C4—62no—
475C2×C4—121no—
476C2×C4—241no—
477C2×C4#379811yescharacteristic
478C2×C4C4×D3211yescharacteristic
479C2×C4D3×D1211yescharacteristic
480C2×C4C2×#62511yescharacteristic
481C2×C4D3×#6711yescharacteristic
482C2×C4D3×#6711yescharacteristic
483C8—21no—
484C8—21no—
485C8—62no—
486C8—182no—
487C8—361no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—21no—
492C2×C3—21no—
493C2×C3—21no—
494C2×C3—21no—
495C2×C3—41no—
496C2×C3—41no—
497C2×C3—122no—
498C2×C3—242no—
499C2×C3#495611yescharacteristic
500C2×C3#514111yescharacteristic
501C2×C3#514211yescharacteristic
502C2×C3#516011yescharacteristic
503C2×C3#517711yescharacteristic
504C2×C3#521311yescharacteristic
505D3—42no—
506D3—82no—
507C22—21no—
508C22—21no—
509C22—62no—
510C22—121no—
511C22—121no—
512C22—122no—
513C22—242no—
514C22—242no—
515C22—241no—
516C22—241no—
517C22#6387911yescharacteristic
518C22#6390511yescharacteristic
519C22#6390811yescharacteristic
520C4—62no—
521C4—241no—
522C4C4×#62511yescharacteristic
523C4#6390611yescharacteristic
524C4D3×(C2×D3)⋊C411yescharacteristic
525C4#6397611yescharacteristic
526C3—21no—
527C3#7021811yescharacteristic, complemented
528C3#7201811yescharacteristic, complemented
529C2—21no—
530C2—21no—
531C2—122no—
532C2—242no—
533C2#1059092511yescharacteristic
534C2#1059092611yescharacteristic
535C2#1059138111yescharacteristic
subgroupG/H
centerC22#63908
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C2×D32
FittingC22×C32×C8C22
socleC22×C32#121
prime radicalC2×#63823C2
cosocleC22×C32×C4C23
hypercenterC22×C8D32
hypocenterC32#1907
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8

Complements

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

NH ≅ G/Ntype
1. C2×C3×#901C2semidirect
3. C2×C3×#821C2semidirect
4. #10591087C2semidirect
5. C2×#63558C2semidirect
6. #10589418C2semidirect
7. #10589167C2semidirect
23. #63823C4semidirect
33. C2×#901D3semidirect
38. C2×#821D3semidirect
454. C32#1907semidirect
527. C3#70218semidirect
528. C3#72018semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9012
C2×#638232
C2×C3×#8212
#105910872
C2×#635582
#105894182
#105891672
#720183
#702183