Finite Group Databasebeta

← #11787639

Subgroups — #11787639

Order 1152.

699 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5822C211yesmaximal, characteristic
2C3×#5819C211yesmaximal, characteristic, complemented
3#10592932C211yesmaximal, characteristic, complemented
4#10591110C211yesmaximal, characteristic, complemented
5#10591071C211yesmaximal, characteristic, complemented
6#10589171C211yesmaximal, characteristic, complemented
7#10589165C211yesmaximal, characteristic, complemented
8#72116—31nomaximal
9#72020—31nomaximal
10C2×C32×#41C2211yescharacteristic
11C2×C3×#235C2211yescharacteristic
12#64135C2211yescharacteristic
13C22×C3⋊(C3⋊C8)C2211yescharacteristic
14C3×#951C412yescomplemented
15C3×#950—22no—
16C3×#949C412yescomplemented
17C3×#948—22no—
18C3×#948—22no—
19C3×#942—21no—
20C3×#932—21no—
21C2×C3×#208—22no—
22C2×C3×#208C2211yescharacteristic
23C2×C3×#208C2211yescharacteristic
24C2×C3×#208C2211yescharacteristic
25#63662—21no—
26#63659—21no—
27#63568—22no—
28(C2×D3)⋊(C3⋊C8)—22no—
29C3×#575—31no—
30C3×#568—31no—
31#5822D311yescharacteristic
32#5819D311yescharacteristic, complemented
33#5236—31no—
34#5234—31no—
35#5233—31no—
36#5233—31no—
37#5233—61no—
38#5139—31no—
39#5129—31no—
40#4559—31no—
41#4558—31no—
42#4558—31no—
43#4556—31no—
44C32×#41—21no—
45C32×#41—21no—
46C32×#41C2×C412yes—
47C32×#41D412yes—
48C2×C32×Q8D411yescharacteristic
49C2×C32×D4—21no—
50C2×C32×D4D411yescharacteristic
51C22×C32×C4—21no—
52C22×C32×C4C2311yescharacteristic
53C2×C3×#67—21no—
54C3×#235—21no—
55C3×#235—21no—
56C3×#235—21no—
57C3×#235—21no—
58C2×C3×D12C2×C411yescharacteristic
59C2×C3×C4×D3—21no—
60C2×C3×Dic6C2×C411yescharacteristic
61#3757—21no—
62C4⋊(C3⋊Dic3)—21no—
63C4×C3⋊Dic3—21no—
64C2×C3⋊(C3⋊C8)—21no—
65C2×C3⋊(C3⋊C8)—21no—
66C2×C3⋊(C3⋊C8)—21no—
67C3×#216—22no—
68C3×#215—22no—
69C3×#214—22no—
70C3×#213—22no—
71C3×#208—24no—
72C3×#208—21no—
73C3×#208—21no—
74C3×#208—21no—
75C3×#208—21no—
76C3×#208C2×C412yes—
77C3×#208D412yes—
78C2×C3×C3⋊C8—24no—
79C2×C3×C3⋊C8—21no—
80C2×C3×C3⋊C8—21no—
81C2×C3×C3⋊C8D412yes—
82#1881—91noSylow
83C2×C3×#41—21no—
84C2×C3×#41—31no—
85C2×C3×#41C2×D311yescharacteristic
86C2×C3×#41C2×D311yescharacteristic
87C2×#235C2×D311yescharacteristic
88C3×#137—32no—
89C3×#137—61no—
90C3×AGL(1,ℤ/8)—32no—
91C3×AGL(1,ℤ/8)—61no—
92C3×#135—62no—
93C3×#131—62no—
94C3×#131—62no—
95C2×C3×M16—31no—
96C2×C3×M16—31no—
97C2×C3×M16—31no—
98C2×C3×M16—62no—
99#951Dic312yescomplemented
100#950—22no—
101#949Dic312yescomplemented
102#948—22no—
103#948—22no—
104#942—21no—
105#932—21no—
106#926—31no—
107#926—31no—
108#926—61no—
109C2×#208—22no—
110C2×#208C2×D311yescharacteristic
111C2×#208C2×D311yescharacteristic
112C2×#208C2×D311yescharacteristic
113C22×C3⋊C8—31no—
114C22×C3⋊C8—31no—
115C22×C3⋊C8—61no—
116#837—62no—
117#835—62no—
118#835—61no—
119#835—61no—
120#832—62no—
121#832—61no—
122#832—61no—
123#830—62no—
124#825—62no—
125(C2×D3)⋊C8—62no—
126C23×C32—21no—
127C23×C32#4111yescharacteristic
128C22×C3×D3—21no—
129C32×Q8—21no—
130C32×Q8C22⋊C412yes—
131C32×D4—22no—
132C32×D4—22no—
133C32×D4—21no—
134C32×D4C22⋊C412yes—
135C2×C32×C4—21no—
136C2×C32×C4—22no—
137C2×C32×C4—22no—
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—
145C3×#67—41no—
146C3×#67—41no—
147C2×C3×Dic3—21no—
148C3×D12—21no—
149C3×D12—21no—
150C3×C4×D3—41no—
151C3×C4×D3—41no—
152C3×Dic6—21no—
153C3×Dic6—21no—
154C3⋊(C3⋊C8)—41no—
155C3×C3⋊C8—22no—
156C3×C3⋊C8—22no—
157C3×C3⋊C8—22no—
158#575—31no—
159#568—31no—
160#421—91no—
161#416—91no—
162#408—91no—
163#344—91no—
164#341—91no—
165C3×#41—21no—
166C3×#41—21no—
167C3×#41—21no—
168C3×#41—21no—
169C3×#41—22no—
170C3×#41—22no—
171C3×#41—41no—
172C3×#41—41no—
173C3×#41—61no—
174C3×#41—61no—
175C3×#41—61no—
176C3×#41—61no—
177C3×#41C4×D312yes—
178C3×#41C2×Dic312yes—
179C3×#41#6712yes—
180C3×#41#6712yes—
181C2×C3×Q8—21no—
182C2×C3×Q8—31no—
183C2×C3×Q8D1211yescharacteristic
184C2×C3×Q8#6711yescharacteristic
185C2×C3×D4—21no—
186C2×C3×D4—21no—
187C2×C3×D4—21no—
188C2×C3×D4—31no—
189C2×C3×D4—41no—
190C2×C3×D4—61no—
191C2×C3×D4D1211yescharacteristic
192C2×C3×D4#6711yescharacteristic
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—41no—
197C22×C3×C4—61no—
198C22×C3×C4C22×D311yescharacteristic
199C22×C3×C4C22×D311yescharacteristic
200C2×#67—21no—
201#235—21no—
202#235—21no—
203#235—21no—
204#235—21no—
205C2×D12C2×Dic311yescharacteristic
206C2×C4×D3—21no—
207C2×Dic6C2×Dic311yescharacteristic
208C3×Q16—62no—
209C3×SD16—62no—
210C3×SD16—62no—
211C3×D8—62no—
212C3×M16—32no—
213C3×M16—32no—
214C3×M16—64no—
215C3×M16—61no—
216C3×M16—61no—
217C3×M16—61no—
218C3×M16—61no—
219C2×C3×C8—32no—
220C2×C3×C8—64no—
221C2×C3×C8—61no—
222C2×C3×C8—61no—
223#217—61no—
224#217—61no—
225#217—121no—
226#216—22no—
227#215—22no—
228#214—22no—
229#213—22no—
230C4⋊Dic3—62no—
231C4⋊Dic3—61no—
232C4⋊Dic3—61no—
233C4×Dic3—61no—
234C4×Dic3—61no—
235C4×Dic3—121no—
236#208—24no—
237#208—21no—
238#208—21no—
239#208—21no—
240#208—21no—
241#208C2×Dic312yes—
242#208#6712yes—
243C2×C3⋊C8—24no—
244C2×C3⋊C8—21no—
245C2×C3⋊C8—21no—
246C2×C3⋊C8—62no—
247C2×C3⋊C8—61no—
248C2×C3⋊C8—61no—
249C2×C3⋊C8—61no—
250C2×C3⋊C8—61no—
251C2×C3⋊C8—61no—
252C2×C3⋊C8—61no—
253C2×C3⋊C8—121no—
254C2×C3⋊C8—121no—
255C2×C3⋊C8#6712yes—
256C22×C32—22no—
257C22×C32—41no—
258C22×C32—81no—
259C22×C32C4×D411yescharacteristic
260C22×C32#12111yescharacteristic
261C22×C32#12311yescharacteristic
262C2×C3×D3—42no—
263C2×C3×D3—41no—
264C32×C4—22no—
265C32×C4C2×C22⋊C411yescharacteristic
266C32×C4C4×D411yescharacteristic
267C32×C4#12011yescharacteristic
268C32×C4#12111yescharacteristic
269C3⋊Dic3—81no—
270C3×Dic3—41no—
271C2×#41—31no—
272C2×#41D3211yescharacteristic
273#137—32no—
274#137—61no—
275AGL(1,ℤ/8)—32no—
276AGL(1,ℤ/8)—61no—
277#135—62no—
278#131—62no—
279#131—62no—
280C2×M16—31no—
281C2×M16—31no—
282C2×M16—31no—
283C2×M16—62no—
284C22×C8—91no—
285#117—91no—
286#104—182no—
287#103—181no—
288#102—181no—
289C22⋊C8—182no—
290C23×C3—21no—
291C23×C3—21no—
292C23×C3—21no—
293C23×C3—41no—
294C23×C3—61no—
295C23×C3#23511yescharacteristic
296C23×C3#23711yescharacteristic
297C22×D3—21no—
298C3×Q8—21no—
299C3×Q8—21no—
300C3×Q8—22no—
301C3×Q8—41no—
302C3×Q8—61no—
303C3×Q8—61no—
304C3×Q8(C2×D3)⋊C412yes—
305C3×Q8#21712yes—
306C3×D4—22no—
307C3×D4—22no—
308C3×D4—22no—
309C3×D4—22no—
310C3×D4—21no—
311C3×D4—21no—
312C3×D4—22no—
313C3×D4—41no—
314C3×D4—42no—
315C3×D4—42no—
316C3×D4—61no—
317C3×D4—61no—
318C3×D4—121no—
319C3×D4—121no—
320C3×D4(C2×D3)⋊C412yes—
321C3×D4#21712yes—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4—41no—
337C2×C3×C4—61no—
338C2×C3×C4—121no—
339C2×C3×C4—121no—
340C2×C3×C4C2×C4×D311yescharacteristic
341C2×C3×C4C2×D1211yescharacteristic
342C2×C3×C4#23511yescharacteristic
343C2×C3×C4D3×D411yescharacteristic
344C2×C3×C4D3×D411yescharacteristic
345C2×C3×C4#23711yescharacteristic
346C2×C3×C4C22×Dic311yescharacteristic
347C2×C3×C4C2×#6711yescharacteristic
348C2×C3×C4C2×#6711yescharacteristic
349C2×C3×C4C2×#6711yescharacteristic
350C2×C3×C4C2×#6711yescharacteristic
351C2×C3×C4C2×#6711yescharacteristic
352#67—41no—
353#67—41no—
354C2×Dic3—21no—
355C2×Dic3—122no—
356C2×Dic3—121no—
357C2×Dic3—121no—
358D12—21no—
359D12—21no—
360C4×D3—41no—
361C4×D3—41no—
362Dic6—21no—
363Dic6—21no—
364C3×C8—62no—
365C3×C8—62no—
366C3×C8—62no—
367C3⋊C8—22no—
368C3⋊C8—22no—
369C3⋊C8—22no—
370C3⋊C8—122no—
371C3⋊C8—121no—
372C3⋊C8—121no—
373C2×C32—21no—
374C2×C32—21no—
375C2×C32—21no—
376C2×C32—42no—
377C2×C32#38611yescharacteristic
378C3×D3—81no—
379#41—21no—
380#41—21no—
381#41—61no—
382#41—61no—
383#41—61no—
384#41—61no—
385#41D3×Dic312yes—
386#41#62512yes—
387C2×Q8—31no—
388C2×Q8#62611yescharacteristic
389C2×D4—21no—
390C2×D4—31no—
391C2×D4—61no—
392C2×D4#62611yescharacteristic
393C22×C4—21no—
394C22×C4—61no—
395C22×C4C2×D3211yescharacteristic
396Q16—62no—
397SD16—62no—
398SD16—62no—
399D8—62no—
400M16—32no—
401M16—32no—
402M16—64no—
403M16—61no—
404M16—61no—
405M16—61no—
406M16—61no—
407C2×C8—32no—
408C2×C8—64no—
409C2×C8—61no—
410C2×C8—61no—
411C2×C8—181no—
412C2×C8—181no—
413C2×C8—181no—
414C4⋊C4—181no—
415C22⋊C4—181no—
416C42—181no—
417C22×C3—22no—
418C22×C3—22no—
419C22×C3—21no—
420C22×C3—21no—
421C22×C3—21no—
422C22×C3—42no—
423C22×C3—41no—
424C22×C3—41no—
425C22×C3—42no—
426C22×C3—81no—
427C22×C3—81no—
428C22×C3—82no—
429C22×C3—122no—
430C22×C3—121no—
431C22×C3C4×#6711yescharacteristic
432C22×C3#92911yescharacteristic
433C22×C3#93011yescharacteristic
434C22×C3D4×Dic311yescharacteristic
435C22×C3#93511yescharacteristic
436C22×C3#93911yescharacteristic
437C2×D3—42no—
438C2×D3—41no—
439C3×C4—22no—
440C3×C4—22no—
441C3×C4—21no—
442C3×C4—21no—
443C3×C4—21no—
444C3×C4—21no—
445C3×C4—42no—
446C3×C4—121no—
447C3×C4C2×(C2×D3)⋊C411yescharacteristic
448C3×C4C4×#6711yescharacteristic
449C3×C4#93011yescharacteristic
450C3×C4D4×Dic311yescharacteristic
451C3×C4#93711yescharacteristic
452C3×C4#93911yescharacteristic
453C3×C4C2×#21711yescharacteristic
454C3×C4#95311yescharacteristic
455Dic3—41no—
456Dic3—242no—
457Dic3—241no—
458Dic3—241no—
459C32#188111yescharacteristic, complemented, Sylow
460C23—21no—
461C23—61no—
462C23#380411yescharacteristic
463Q8—21no—
464Q8—61no—
465Q8—61no—
466Q8(C2×D3)⋊Dic312yes—
467D4—22no—
468D4—22no—
469D4—21no—
470D4—61no—
471D4—61no—
472D4—121no—
473D4—121no—
474D4(C2×D3)⋊Dic312yes—
475C2×C4—21no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—61no—
479C2×C4—121no—
480C2×C4—121no—
481C2×C4—361no—
482C2×C4C2×D3×Dic311yescharacteristic
483C2×C4#380411yescharacteristic
484C2×C4C2×#62511yescharacteristic
485C2×C4C2×#62611yescharacteristic
486C2×C4D3×#6711yescharacteristic
487C2×C4D3×#6711yescharacteristic
488C8—62no—
489C8—62no—
490C8—62no—
491C8—361no—
492C2×C3—22no—
493C2×C3—21no—
494C2×C3—21no—
495C2×C3—21no—
496C2×C3—21no—
497C2×C3—21no—
498C2×C3—21no—
499C2×C3—21no—
500C2×C3—41no—
501C2×C3—41no—
502C2×C3—42no—
503C2×C3—42no—
504C2×C3—82no—
505C2×C3—241no—
506C2×C3#521311yescharacteristic
507C2×C3#522011yescharacteristic
508D3—81no—
509C22—22no—
510C22—41no—
511C22—81no—
512C22—122no—
513C22—121no—
514C22#6396111yescharacteristic
515C22Dic3×#6711yescharacteristic
516C22#6397811yescharacteristic
517C4—22no—
518C4—121no—
519C4—721no—
520C4C2×(C2×D3)⋊Dic311yescharacteristic
521C4Dic3×#6711yescharacteristic
522C4#6397611yescharacteristic
523C4#6397811yescharacteristic
524C3—21no—
525C3#7202011yescharacteristic, complemented
526C3#7211611yescharacteristic, complemented
527C2—21no—
528C2—21no—
529C2—21no—
530C2—42no—
531C2—241no—
532C2#1059171011yescharacteristic
subgroupG/H
centerC4#63976
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C2×D32
FittingC2×C32×#41C22
socleC2×C32#386
prime radicalC3×#5822C2
cosocleC22×C32×C4C23
hypercenterC2×#41D32
hypocenterC32#1881
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
2. C3×#5819C2semidirect
3. #10592932C2semidirect
4. #10591110C2semidirect
5. #10591071C2semidirect
6. #10589171C2semidirect
7. #10589165C2semidirect
14. C3×#951C4semidirect
16. C3×#949C4semidirect
32. #5819D3semidirect
99. #951Dic3semidirect
101. #949Dic3semidirect
459. C32#1881semidirect
525. C3#72020semidirect
526. C3#72116semidirect

Maximal subgroups

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

subgroupindex
C3×#58222
C3×#58192
#105929322
#105911102
#105910712
#105891712
#105891652
#721163
#720203