Finite Group Databasebeta

← #13156304

Subgroups — #13156304

Order 1728.

612 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723535C211yesmaximal, characteristic, complemented
2C3×#63898C211yesmaximal, characteristic
3#11722078C211yesmaximal, characteristic, complemented
4#11722054C211yesmaximal, characteristic, complemented
5#11722032C211yesmaximal, characteristic
6C3×#63574C211yesmaximal, characteristic, complemented
7C3×#63562C211yesmaximal, characteristic
8#10591408—31nomaximal
9#10590558—31nomaximal
10#10590472—31nomaximal
11C2×C32×D12C2211yescharacteristic
12C2×#6293—21no—
13C2×C3×D3×Dic3—21no—
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C3×C4×C3⋊Dic3C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C32×C4⋊Dic3C2211yescharacteristic
20#88807—21no—
21C3×(C2×D3)⋊Dic3—21no—
22#64097—31no—
23C3×D4×Dic3—31no—
24C3×#893—31no—
25#63918—31no—
26#63918—31no—
27#63918—61no—
28#63898D311yescharacteristic
29#63659—31no—
30#63635—31no—
31C3×#832—31no—
32C3×#809—31no—
33C3×#808—31no—
34C3×#808—31no—
35#63574D311yescharacteristic, complemented
36#63573—31no—
37#63573—31no—
38#63573—61no—
39#63572—31no—
40Dic3⋊(C3⋊C8)—31no—
41Dic3⋊(C3⋊C8)—31no—
42#63563—31no—
43#63563—31no—
44#63563—61no—
45#63562D311yescharacteristic
46C22×C32×D3—21no—
47C2×C33×C4C2311yescharacteristic
48C2×C3⋊(C3⋊Dic3)—21no—
49C2×C3×C3⋊Dic3D411yescharacteristic
50C2×C3×C3⋊Dic3D411yescharacteristic
51C2×C32×Dic3—21no—
52C2×C32×Dic3—21no—
53C32×D12—21no—
54C32×D12Q812yescomplemented
55#6293—42no—
56C3×D3×Dic3—42no—
57C3×C3⋊(C3⋊C8)—21no—
58C3×C3⋊(C3⋊C8)—21no—
59#5008—91no—
60#4870—91no—
61#4761—91no—
62C2×C32×D4—31no—
63C2×#634—61no—
64C22×C3×Dic3—61no—
65C2×C3×D12—21no—
66C2×C3×D12C2×D311yescharacteristic
67C2×C3×D12C2×D311yescharacteristic
68C2×C3×C4×D3—61no—
69C2×#627—61no—
70C2×#627—61no—
71C2×#627—62no—
72C2×D3×Dic3—21no—
73C32×C4⋊C4—31no—
74C32×C4⋊C4—31no—
75C4⋊(C3⋊Dic3)—31no—
76C4⋊(C3⋊Dic3)—31no—
77C4⋊(C3⋊Dic3)—31no—
78C4⋊(C3⋊Dic3)—61no—
79C4⋊(C3⋊Dic3)—61no—
80C4⋊(C3⋊Dic3)—61no—
81C4⋊(C3⋊Dic3)—121no—
82#3750—61no—
83C4×C3⋊Dic3C2×D311yescharacteristic
84C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
85C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
86C3×#217—61no—
87C3×(C2×D3)⋊C4—61no—
88C3×C4⋊Dic3—21no—
89C3×C4⋊Dic3—21no—
90C3×C4⋊Dic3C2×D311yescharacteristic
91C3×C4⋊Dic3C2×D311yescharacteristic
92C3×C4⋊Dic3C2×D311yescharacteristic
93C3×C4⋊Dic3C2×D311yescharacteristic
94C3×C4×Dic3—31no—
95C3×C4×Dic3—31no—
96C3×C4×Dic3—61no—
97C2×C3×C3⋊C8—31no—
98C2×C3×C3⋊C8—31no—
99C2×C3×C3⋊C8—31no—
100C2×C3×C3⋊C8—31no—
101C2×C3×C3⋊C8—61no—
102C2×C3×C3⋊C8—61no—
103Dic3⋊Dic3—61no—
104Dic3⋊Dic3—61no—
105Dic3⋊Dic3—62no—
106(C2×D3)⋊Dic3—21no—
107C22×C33C2×D411yescharacteristic
108C2×C32×D3—22no—
109C2×C32×D3—41no—
110C2×C32×D3—41no—
111C33×C4C2×Q811yescharacteristic
112C33×C4#4111yescharacteristic
113C3⋊(C3⋊Dic3)—41no—
114C3×C3⋊Dic3—21no—
115C3×C3⋊Dic3SD1612yes—
116C32×Dic3—41no—
117C32×Dic3—41no—
118C3×C4×D4—91no—
119D4×Dic3—31no—
120#893—31no—
121#888—91no—
122#888—91no—
123#888—181no—
124#869—91no—
125C3×#106—91no—
126C3×#102—91no—
127#832—31no—
128#832—91no—
129#832—91no—
130#832—181no—
131#821—91no—
132#817—91no—
133Dic3⋊C8—91no—
134Dic3⋊C8—91no—
135#809—31no—
136#808—31no—
137#808—31no—
138#808—91no—
139#808—91no—
140#808—181no—
141C4⋊(C3⋊C8)—91no—
142C23×C32—61no—
143C22×C3×D3—21no—
144C22×C3×D3—21no—
145C22×C3×D3—41no—
146C32×D4—32no—
147C32×D4—61no—
148C2×C32×C4—21no—
149C2×C32×C4—21no—
150C2×C32×C4—21no—
151C2×C32×C4—41no—
152C2×C32×C4—61no—
153C2×C32×C4—61no—
154C2×C32×C4C22×D311yescharacteristic
155C2×C32×C4C22×D311yescharacteristic
156C2×C32×C4C22×D311yescharacteristic
157C2×C3⋊Dic3—61no—
158C2×C3⋊Dic3—62no—
159C2×C3⋊Dic3—61no—
160C2×C3⋊Dic3—61no—
161C2×C3⋊Dic3—121no—
162C2×C3⋊Dic3—121no—
163C2×C3⋊Dic3—122no—
164C2×C3⋊Dic3#6711yescharacteristic
165C2×C3⋊Dic3#6711yescharacteristic
166#634—122no—
167C2×C3×Dic3—22no—
168C2×C3×Dic3—21no—
169C2×C3×Dic3—21no—
170C2×C3×Dic3—21no—
171C2×C3×Dic3—21no—
172C2×C3×Dic3—31no—
173C2×C3×Dic3—31no—
174C2×C3×Dic3—31no—
175C2×C3×Dic3—31no—
176C2×C3×Dic3—41no—
177C2×C3×Dic3—61no—
178C2×C3×Dic3—61no—
179C2×C3×Dic3—122no—
180C3×D12—21no—
181C3×D12—21no—
182C3×D12—22no—
183C3×D12—41no—
184C3×D12Dic612yescomplemented
185C3×D12Dic612yescomplemented
186C3×C4×D3—122no—
187#627—124no—
188#627—122no—
189#627—122no—
190D3×Dic3—42no—
191C3⋊(C3⋊C8)—21no—
192C3⋊(C3⋊C8)—21no—
193C3×C3⋊C8—62no—
194C3×C3⋊C8—61no—
195C3×C3⋊C8—61no—
196C3×C3⋊C8—61no—
197C3×C3⋊C8—61no—
198C3×C3⋊C8—121no—
199#476—271noSylow
200C2×C33C22⋊Q811yescharacteristic
201C2×C33C2×SD1611yescharacteristic
202C2×C33AGL(1,ℤ/8)11yescharacteristic
203C32×D3—42no—
204C2×C3×D4—31no—
205C2×C3×D4—31no—
206C2×C3×D4—61no—
207C22×C3×C4—181no—
208C22×Dic3—61no—
209C2×D12D3211yescharacteristic
210C2×C4×D3—61no—
211C2×Dic6—181no—
212C2×Dic6—181no—
213C2×Dic6—181no—
214C2×Dic6—182no—
215C2×C3×C8—91no—
216C2×C3×C8—91no—
217C3×C4⋊C4—31no—
218C3×C4⋊C4—31no—
219C3×C4⋊C4—31no—
220C3×C4⋊C4—31no—
221C3×C4⋊C4—61no—
222C3×C4⋊C4—61no—
223C3×C22⋊C4—181no—
224C3×C42—91no—
225#217—61no—
226(C2×D3)⋊C4—61no—
227C4⋊Dic3—91no—
228C4⋊Dic3—91no—
229C4⋊Dic3—91no—
230C4⋊Dic3—181no—
231C4⋊Dic3—181no—
232C4⋊Dic3—181no—
233C4⋊Dic3—181no—
234C4⋊Dic3—361no—
235C4⋊Dic3D3211yescharacteristic
236C4⋊Dic3D3211yescharacteristic
237Dic3⋊C4—181no—
238Dic3⋊C4—181no—
239Dic3⋊C4—182no—
240C4×Dic3—31no—
241C4×Dic3—31no—
242C4×Dic3—61no—
243C2×C3⋊C8—31no—
244C2×C3⋊C8—31no—
245C2×C3⋊C8—31no—
246C2×C3⋊C8—31no—
247C2×C3⋊C8—61no—
248C2×C3⋊C8—61no—
249C22×C32—21no—
250C22×C32—21no—
251C22×C32—21no—
252C22×C32—41no—
253C22×C32—62no—
254C22×C32—121no—
255C22×C32—121no—
256C22×C32D3×D411yescharacteristic
257C22×C32D3×D411yescharacteristic
258C22×C32C2×#6711yescharacteristic
259C2×C3×D3—22no—
260C2×C3×D3—22no—
261C2×C3×D3—42no—
262C2×C3×D3—41no—
263C2×C3×D3—41no—
264C2×C3×D3—41no—
265C2×C3×D3—41no—
266C2×C3×D3—81no—
267C2×C3×D3—81no—
268C32×C4—21no—
269C32×C4—21no—
270C32×C4—21no—
271C32×C4—21no—
272C32×C4—21no—
273C32×C4—21no—
274C32×C4—41no—
275C32×C4—41no—
276C32×C4—121no—
277C32×C4—121no—
278C32×C4C2×Dic611yescharacteristic
279C32×C4C2×Dic611yescharacteristic
280C32×C4#23511yescharacteristic
281C32×C4#23511yescharacteristic
282C32×C4#23711yescharacteristic
283C32×C4D3×Q811yescharacteristic
284C3⋊Dic3—21no—
285C3⋊Dic3—121no—
286C3⋊Dic3—122no—
287C3⋊Dic3—121no—
288C3⋊Dic3—121no—
289C3⋊Dic3—241no—
290C3⋊Dic3—241no—
291C3⋊Dic3—242no—
292C3⋊Dic3#21412yes—
293C3×Dic3—32no—
294C3×Dic3—32no—
295C3×Dic3—42no—
296C3×Dic3—41no—
297C3×Dic3—41no—
298C3×Dic3—41no—
299C3×Dic3—41no—
300C3×Dic3—62no—
301C3×Dic3—62no—
302C3×Dic3—61no—
303C3×Dic3—61no—
304C3×Dic3—81no—
305C4⋊Q8—271no—
306C4×D4—91no—
307#106—91no—
308#106—271no—
309C4⋊C8—271no—
310#102—91no—
311#102—271no—
312C33#47611yescharacteristic, complemented, Sylow
313C23×C3—61no—
314C23×C3—61no—
315C23×C3—121no—
316C22×D3—21no—
317C3×D4—32no—
318C3×D4—32no—
319C3×D4—62no—
320C3×D4—61no—
321C3×D4—61no—
322C3×D4—121no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—41no—
327C2×C3×C4—62no—
328C2×C3×C4—61no—
329C2×C3×C4—61no—
330C2×C3×C4—61no—
331C2×C3×C4—61no—
332C2×C3×C4—91no—
333C2×C3×C4—91no—
334C2×C3×C4—121no—
335C2×C3×C4—362no—
336C2×C3×C4C2×D3211yescharacteristic
337C2×C3×C4C2×D3211yescharacteristic
338C2×C3×C4C2×D3211yescharacteristic
339C2×Dic3—21no—
340C2×Dic3—21no—
341C2×Dic3—31no—
342C2×Dic3—31no—
343C2×Dic3—31no—
344C2×Dic3—31no—
345C2×Dic3—61no—
346C2×Dic3—61no—
347C2×Dic3—122no—
348C2×Dic3—182no—
349C2×Dic3—181no—
350C2×Dic3—181no—
351C2×Dic3—181no—
352C2×Dic3—361no—
353C2×Dic3—361no—
354C2×Dic3—362no—
355D12—21no—
356D12#62712yescomplemented
357C4×D3—122no—
358Dic6—362no—
359Dic6—364no—
360Dic6—362no—
361Dic6—362no—
362C3×C8—181no—
363C3×C8—181no—
364C3⋊C8—62no—
365C3⋊C8—61no—
366C3⋊C8—61no—
367C3⋊C8—61no—
368C3⋊C8—61no—
369C3⋊C8—121no—
370C2×C32—21no—
371C2×C32—21no—
372C2×C32—21no—
373C2×C32—21no—
374C2×C32—21no—
375C2×C32—21no—
376C2×C32—21no—
377C2×C32—21no—
378C2×C32—21no—
379C2×C32—41no—
380C2×C32—41no—
381C2×C32—41no—
382C2×C32—122no—
383C2×C32#87811yescharacteristic
384C2×C32#89611yescharacteristic
385C2×C32#91111yescharacteristic
386C2×C32D3×SD1611yescharacteristic
387C2×C32D3×SD1611yescharacteristic
388C2×C32#91411yescharacteristic
389C2×C32#92411yescharacteristic
390C2×C32C2×#21411yescharacteristic
391C2×C32#94911yescharacteristic
392C3×D3—42no—
393C3×D3—42no—
394C3×D3—82no—
395C2×Q8—541no—
396C2×D4—31no—
397C22×C4—181no—
398C2×C8—91no—
399C2×C8—91no—
400C4⋊C4—31no—
401C4⋊C4—31no—
402C4⋊C4—271no—
403C4⋊C4—541no—
404C22⋊C4—181no—
405C42—91no—
406C22×C3—21no—
407C22×C3—21no—
408C22×C3—21no—
409C22×C3—41no—
410C22×C3—62no—
411C22×C3—62no—
412C22×C3—122no—
413C22×C3—121no—
414C22×C3—121no—
415C22×C3—121no—
416C22×C3—121no—
417C22×C3—241no—
418C22×C3—241no—
419C22×C3D3×#6711yescharacteristic
420C22×C3D3×#6711yescharacteristic
421C22×C3#381111yescharacteristic
422C2×D3—22no—
423C2×D3—41no—
424C2×D3—41no—
425C3×C4—21no—
426C3×C4—21no—
427C3×C4—21no—
428C3×C4—21no—
429C3×C4—21no—
430C3×C4—21no—
431C3×C4—41no—
432C3×C4—41no—
433C3×C4—92no—
434C3×C4—122no—
435C3×C4—121no—
436C3×C4—121no—
437C3×C4—121no—
438C3×C4—121no—
439C3×C4—181no—
440C3×C4—241no—
441C3×C4D3×Dic611yescharacteristic
442C3×C4D3×Dic611yescharacteristic
443C3×C4#379511yescharacteristic
444C3×C4#379811yescharacteristic
445C3×C4#380411yescharacteristic
446C3×C4C2×#62711yescharacteristic
447Dic3—32no—
448Dic3—32no—
449Dic3—41no—
450Dic3—41no—
451Dic3—62no—
452Dic3—62no—
453Dic3—61no—
454Dic3—61no—
455Dic3—362no—
456Dic3—361no—
457Dic3—361no—
458Dic3—361no—
459Dic3—721no—
460Dic3—721no—
461Dic3—722no—
462C32—21no—
463C32—21no—
464C32—21no—
465C32—41no—
466C32#476111yescharacteristic, complemented
467C32#487011yescharacteristic, complemented
468C32#500811yescharacteristic, complemented
469C23—61no—
470Q8—1082no—
471D4—32no—
472D4—61no—
473C2×C4—61no—
474C2×C4—61no—
475C2×C4—91no—
476C2×C4—91no—
477C2×C4—362no—
478C2×C4—541no—
479C2×C4D3311yescharacteristic
480C8—181no—
481C8—181no—
482C2×C3—21no—
483C2×C3—21no—
484C2×C3—21no—
485C2×C3—21no—
486C2×C3—21no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—41no—
492C2×C3—41no—
493C2×C3—41no—
494C2×C3—122no—
495C2×C3—122no—
496C2×C3—242no—
497C2×C3#6385111yescharacteristic
498C2×C3#6389611yescharacteristic
499C2×C3#6392511yescharacteristic
500C2×C3D3×#21411yescharacteristic
501C2×C3D3×#21411yescharacteristic
502C2×C3#6393711yescharacteristic
503C2×C3#6393911yescharacteristic
504C2×C3#6394011yescharacteristic
505C2×C3#6396411yescharacteristic
506D3—42no—
507C22—62no—
508C22—121no—
509C22—121no—
510C22#8895211yescharacteristic
511C4—92no—
512C4—121no—
513C4—121no—
514C4—181no—
515C4—1081no—
516C4D3×#62711yescharacteristic
517C4#8896011yescharacteristic
518C3—21no—
519C3—21no—
520C3—21no—
521C3—41no—
522C3#1059047211yescharacteristic, complemented
523C3#1059055811yescharacteristic, complemented
524C3#1059140811yescharacteristic, complemented
525C2—122no—
526C2#1172283711yescharacteristic
527C2#1172287411yescharacteristic
528C2#1172297511yescharacteristic
subgroupG/H
centerC22#88952
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
hypocenterC33#476
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. #11723535C2semidirect
3. #11722078C2semidirect
4. #11722054C2semidirect
6. C3×#63574C2semidirect
35. #63574D3semidirect
54. C32×D12Q8semidirect
184. C3×D12Dic6semidirect
185. C3×D12Dic6semidirect
312. C33#476semidirect
356. D12#627semidirect
466. C32#4761semidirect
467. C32#4870semidirect
468. C32#5008semidirect
522. C3#10590472semidirect
523. C3#10590558semidirect
524. C3#10591408semidirect

Maximal subgroups

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

subgroupindex
#117235352
C3×#638982
#117220782
#117220542
#117220322
C3×#635742
C3×#635622
#105914083
#105905583
#105904723