Finite Group Databasebeta

← #13156352

Subgroups — #13156352

Order 1728.

626 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723499C211yesmaximal, characteristic
2#11723450C211yesmaximal, characteristic, complemented
3#11722079C211yesmaximal, characteristic, complemented
4#11722050C211yesmaximal, characteristic, complemented
5C3×#63575C211yesmaximal, characteristic, complemented
6C3×#63563C211yesmaximal, characteristic
7C3×#63562C211yesmaximal, characteristic
8#10591479—31nomaximal
9#10590780—31nomaximal
10#10590473—31nomaximal
11C2×C32×D12C2211yescharacteristic
12C2×D3×C3⋊Dic3—21no—
13C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88807—21no—
20Dic3⋊(C3⋊Dic3)—21no—
21(C2×D3)⋊(C3⋊Dic3)—21no—
22D4×C3⋊Dic3—31no—
23#64099—31no—
24#63898—31no—
25#63898—31no—
26#63898—61no—
27#63872—31no—
28#63872—31no—
29#63872—61no—
30#63641—31no—
31C3×#832—31no—
32C3×#832—31no—
33C3×#821—31no—
34C3×#818—31no—
35C3×#809—31no—
36C3×#807—31no—
37#63575D311yescharacteristic, complemented
38#63574—31no—
39#63573—31no—
40#63573—31no—
41#63572—31no—
42#63572—31no—
43#63572—61no—
44#63563D311yescharacteristic
45#63562D311yescharacteristic
46C22×C32×D3—21no—
47C2×C33×C4C2311yescharacteristic
48C2×C3⋊(C3⋊Dic3)D411yescharacteristic
49C2×C3⋊(C3⋊Dic3)D411yescharacteristic
50C2×C3×C3⋊Dic3—21no—
51C2×C3×C3⋊Dic3—21no—
52C2×C32×Dic3—21no—
53C32×D12—21no—
54C32×D12Q812yes—
55D3×C3⋊Dic3—42no—
56C3×C3⋊(C3⋊C8)—21no—
57C32×C3⋊C8—21no—
58#4891—91no—
59#4766—91no—
60#4763—91no—
61C2×C32×D4—31no—
62C22×C3⋊Dic3—61no—
63C2×C3×D12—21no—
64C2×C3×D12C2×D311yescharacteristic
65C2×C3×D12C2×D311yescharacteristic
66C2×D3×Dic3—61no—
67C2×D3×Dic3—61no—
68C2×D3×Dic3—62no—
69C2×C32×C8—31no—
70C32×C4⋊C4—31no—
71#3757—61no—
72C4⋊(C3⋊Dic3)—61no—
73C4⋊(C3⋊Dic3)C2×D311yescharacteristic
74C4⋊(C3⋊Dic3)C2×D311yescharacteristic
75#3750—61no—
76C4×C3⋊Dic3—31no—
77C4×C3⋊Dic3—31no—
78C4×C3⋊Dic3—31no—
79C4×C3⋊Dic3—61no—
80C4×C3⋊Dic3—61no—
81C4×C3⋊Dic3—61no—
82C4×C3⋊Dic3—121no—
83C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
84C3×C4⋊Dic3—21no—
85C3×C4⋊Dic3—31no—
86C3×C4⋊Dic3—31no—
87C3×C4⋊Dic3—31no—
88C3×C4⋊Dic3—31no—
89C3×C4⋊Dic3—61no—
90C3×C4⋊Dic3—61no—
91C3×C4⋊Dic3C2×D311yescharacteristic
92C3×C4⋊Dic3C2×D311yescharacteristic
93C2×C3×C3⋊C8—21no—
94C2×C3×C3⋊C8—31no—
95C2×C3×C3⋊C8—31no—
96C2×C3×C3⋊C8—61no—
97C2×C3×C3⋊C8C2×D311yescharacteristic
98C2×C3×C3⋊C8C2×D311yescharacteristic
99Dic3⋊Dic3—61no—
100Dic3⋊Dic3—61no—
101Dic3⋊Dic3—61no—
102Dic3⋊Dic3—61no—
103Dic3⋊Dic3—62no—
104Dic3⋊Dic3—62no—
105(C2×D3)⋊Dic3—61no—
106(C2×D3)⋊Dic3—61no—
107(C2×D3)⋊Dic3—62no—
108C22×C33C2×D411yescharacteristic
109C2×C32×D3—22no—
110C2×C32×D3—41no—
111C2×C32×D3—41no—
112C33×C4C2×Q811yescharacteristic
113C33×C4#4111yescharacteristic
114C3⋊(C3⋊Dic3)—21no—
115C3⋊(C3⋊Dic3)—21no—
116C3×C3⋊Dic3—41no—
117C3×C3⋊Dic3—41no—
118C32×Dic3—41no—
119D4×Dic3—91no—
120D4×Dic3—91no—
121D4×Dic3—181no—
122#893—91no—
123#890—91no—
124#890—91no—
125#890—91no—
126#890—181no—
127C3×#107—91no—
128C3×#102—91no—
129C3×#102—91no—
130#832—31no—
131#832—31no—
132#821—31no—
133#818—31no—
134#817—91no—
135Dic3⋊C8—91no—
136Dic3⋊C8—91no—
137Dic3⋊C8—91no—
138Dic3⋊C8—181no—
139#809—31no—
140#808—91no—
141#808—91no—
142#807—31no—
143C23×C32—61no—
144C22×C3×D3—22no—
145C22×C3×D3—21no—
146C22×C3×D3—21no—
147C32×D4—32no—
148C32×D4—61no—
149C2×C32×C4—21no—
150C2×C32×C4—21no—
151C2×C32×C4—21no—
152C2×C32×C4—41no—
153C2×C32×C4—61no—
154C2×C32×C4C22×D311yescharacteristic
155C2×C32×C4C22×D311yescharacteristic
156C2×C32×C4C22×D311yescharacteristic
157C2×C3⋊Dic3—21no—
158C2×C3⋊Dic3—21no—
159C2×C3⋊Dic3—31no—
160C2×C3⋊Dic3—31no—
161C2×C3⋊Dic3—31no—
162C2×C3⋊Dic3—31no—
163C2×C3⋊Dic3—31no—
164C2×C3⋊Dic3—31no—
165C2×C3⋊Dic3—61no—
166C2×C3⋊Dic3—61no—
167C2×C3⋊Dic3—61no—
168C2×C3⋊Dic3—61no—
169C2×C3⋊Dic3—61no—
170C2×C3⋊Dic3—61no—
171C2×C3⋊Dic3—121no—
172C2×C3⋊Dic3—121no—
173C2×C3⋊Dic3—122no—
174C2×C3×Dic3—22no—
175C2×C3×Dic3—21no—
176C2×C3×Dic3—21no—
177C2×C3×Dic3—62no—
178C2×C3×Dic3—62no—
179C2×C3×Dic3—61no—
180C2×C3×Dic3—61no—
181C2×C3×Dic3—61no—
182C2×C3×Dic3—61no—
183C3×D12—22no—
184C3×D12—21no—
185C3×D12—21no—
186C3×D12—22no—
187C3×D12Dic612yes—
188C3×D12Dic612yes—
189D3×Dic3—124no—
190D3×Dic3—122no—
191D3×Dic3—122no—
192C32×C8—61no—
193C3⋊(C3⋊C8)—21no—
194C3×C3⋊C8—21no—
195C3×C3⋊C8—21no—
196C3×C3⋊C8—41no—
197C3×C3⋊C8—61no—
198C3×C3⋊C8—61no—
199C3×C3⋊C8—121no—
200#478—271noSylow
201C2×C33C22⋊Q811yescharacteristic
202C2×C33#13511yescharacteristic
203C2×C33AGL(1,ℤ/8)11yescharacteristic
204C32×D3—42no—
205C2×C3×D4—31no—
206C2×C3×D4—31no—
207C2×C3×D4—61no—
208C22×Dic3—181no—
209C22×Dic3—181no—
210C22×Dic3—182no—
211C2×D12D3211yescharacteristic
212C2×C4×D3—181no—
213C2×C3×C8—31no—
214C2×C3×C8—31no—
215C2×C3×C8—61no—
216C2×C3×C8—91no—
217C3×C4⋊C4—31no—
218C3×C4⋊C4—31no—
219C3×C4⋊C4—61no—
220C3×C4⋊C4—91no—
221C3×C4⋊C4—91no—
222#217—181no—
223#217—181no—
224#217—182no—
225(C2×D3)⋊C4—181no—
226C4⋊Dic3—31no—
227C4⋊Dic3—31no—
228C4⋊Dic3—31no—
229C4⋊Dic3—31no—
230C4⋊Dic3—61no—
231C4⋊Dic3—61no—
232C4⋊Dic3—181no—
233C4⋊Dic3—181no—
234C4⋊Dic3—181no—
235C4⋊Dic3—182no—
236C4⋊Dic3D3211yescharacteristic
237Dic3⋊C4—181no—
238Dic3⋊C4—181no—
239Dic3⋊C4—181no—
240Dic3⋊C4—182no—
241C4×Dic3—91no—
242C4×Dic3—91no—
243C4×Dic3—91no—
244C4×Dic3—181no—
245C4×Dic3—181no—
246C4×Dic3—181no—
247C4×Dic3—361no—
248C2×C3⋊C8—31no—
249C2×C3⋊C8—31no—
250C2×C3⋊C8—61no—
251C2×C3⋊C8D3211yescharacteristic
252C22×C32—21no—
253C22×C32—21no—
254C22×C32—21no—
255C22×C32—41no—
256C22×C32—62no—
257C22×C32—121no—
258C22×C32—121no—
259C22×C32D3×D411yescharacteristic
260C22×C32D3×D411yescharacteristic
261C22×C32D3×D411yescharacteristic
262C2×C3×D3—24no—
263C2×C3×D3—22no—
264C2×C3×D3—22no—
265C2×C3×D3—42no—
266C2×C3×D3—42no—
267C2×C3×D3—41no—
268C2×C3×D3—41no—
269C2×C3×D3—41no—
270C2×C3×D3—41no—
271C32×C4—21no—
272C32×C4—21no—
273C32×C4—21no—
274C32×C4—21no—
275C32×C4—21no—
276C32×C4—21no—
277C32×C4—41no—
278C32×C4—41no—
279C32×C4—121no—
280C32×C4C2×Dic611yescharacteristic
281C32×C4C2×Dic611yescharacteristic
282C32×C4#23511yescharacteristic
283C32×C4#23711yescharacteristic
284C32×C4#23711yescharacteristic
285C32×C4D3×Q811yescharacteristic
286C3⋊Dic3—41no—
287C3⋊Dic3—41no—
288C3⋊Dic3—61no—
289C3⋊Dic3—61no—
290C3⋊Dic3—62no—
291C3⋊Dic3—62no—
292C3⋊Dic3—61no—
293C3⋊Dic3—61no—
294C3⋊Dic3—61no—
295C3⋊Dic3—61no—
296C3⋊Dic3—121no—
297C3⋊Dic3—121no—
298C3⋊Dic3—121no—
299C3⋊Dic3—121no—
300C3⋊Dic3—122no—
301C3⋊Dic3—122no—
302C3×Dic3—42no—
303C3×Dic3—41no—
304C3×Dic3—41no—
305C3×Dic3—122no—
306C3×Dic3—122no—
307C3×Dic3—121no—
308C3×Dic3—121no—
309C3×Dic3—121no—
310C3×Dic3—121no—
311#125—271no—
312C4×D4—271no—
313#107—91no—
314#106—271no—
315C4⋊C8—271no—
316#102—91no—
317#102—91no—
318C33#47811yescharacteristic, complemented, Sylow
319C23×C3—62no—
320C23×C3—61no—
321C23×C3—61no—
322C22×D3—21no—
323C3×D4—32no—
324C3×D4—32no—
325C3×D4—62no—
326C3×D4—62no—
327C3×D4—61no—
328C3×D4—61no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—41no—
333C2×C3×C4—62no—
334C2×C3×C4—61no—
335C2×C3×C4—61no—
336C2×C3×C4—181no—
337C2×C3×C4—181no—
338C2×C3×C4C2×D3211yescharacteristic
339C2×C3×C4C2×D3211yescharacteristic
340C2×C3×C4C2×D3211yescharacteristic
341C2×Dic3—21no—
342C2×Dic3—62no—
343C2×Dic3—62no—
344C2×Dic3—61no—
345C2×Dic3—61no—
346C2×Dic3—61no—
347C2×Dic3—61no—
348C2×Dic3—91no—
349C2×Dic3—91no—
350C2×Dic3—91no—
351C2×Dic3—91no—
352C2×Dic3—91no—
353C2×Dic3—91no—
354C2×Dic3—181no—
355C2×Dic3—181no—
356C2×Dic3—181no—
357C2×Dic3—181no—
358C2×Dic3—181no—
359C2×Dic3—181no—
360C2×Dic3—361no—
361C2×Dic3—361no—
362C2×Dic3—364no—
363C2×Dic3—362no—
364C2×Dic3—362no—
365D12—21no—
366D12#62712yes—
367C4×D3—362no—
368C3×C8—61no—
369C3×C8—61no—
370C3×C8—121no—
371C3×C8—181no—
372C3⋊C8—21no—
373C3⋊C8—61no—
374C3⋊C8—61no—
375C3⋊C8—121no—
376C2×C32—21no—
377C2×C32—21no—
378C2×C32—21no—
379C2×C32—21no—
380C2×C32—21no—
381C2×C32—21no—
382C2×C32—21no—
383C2×C32—21no—
384C2×C32—21no—
385C2×C32—41no—
386C2×C32—41no—
387C2×C32—41no—
388C2×C32—122no—
389C2×C32#87811yescharacteristic
390C2×C32#87811yescharacteristic
391C2×C32#89611yescharacteristic
392C2×C32#91111yescharacteristic
393C2×C32#91111yescharacteristic
394C2×C32#91211yescharacteristic
395C2×C32#91411yescharacteristic
396C2×C32#91611yescharacteristic
397C2×C32#91911yescharacteristic
398C3×D3—44no—
399C3×D3—42no—
400C3×D3—42no—
401C2×D4—31no—
402C22×C4—541no—
403C2×C8—31no—
404C2×C8—91no—
405C4⋊C4—31no—
406C4⋊C4—91no—
407C4⋊C4—91no—
408C4⋊C4—541no—
409C4⋊C4—541no—
410C22⋊C4—541no—
411C42—271no—
412C22×C3—21no—
413C22×C3—21no—
414C22×C3—21no—
415C22×C3—41no—
416C22×C3—64no—
417C22×C3—62no—
418C22×C3—62no—
419C22×C3—122no—
420C22×C3—122no—
421C22×C3—121no—
422C22×C3—121no—
423C22×C3—121no—
424C22×C3—121no—
425C22×C3#380211yescharacteristic
426C22×C3#381111yescharacteristic
427C22×C3#381111yescharacteristic
428C2×D3—22no—
429C2×D3—41no—
430C2×D3—41no—
431C3×C4—21no—
432C3×C4—21no—
433C3×C4—21no—
434C3×C4—21no—
435C3×C4—21no—
436C3×C4—21no—
437C3×C4—41no—
438C3×C4—41no—
439C3×C4—122no—
440C3×C4—121no—
441C3×C4—121no—
442C3×C4—361no—
443C3×C4—361no—
444C3×C4D3×Dic611yescharacteristic
445C3×C4D3×Dic611yescharacteristic
446C3×C4#379511yescharacteristic
447C3×C4#380411yescharacteristic
448C3×C4#380511yescharacteristic
449C3×C4C2×#62711yescharacteristic
450Dic3—41no—
451Dic3—122no—
452Dic3—122no—
453Dic3—121no—
454Dic3—121no—
455Dic3—121no—
456Dic3—121no—
457Dic3—182no—
458Dic3—182no—
459Dic3—181no—
460Dic3—181no—
461Dic3—181no—
462Dic3—181no—
463Dic3—181no—
464Dic3—181no—
465Dic3—361no—
466Dic3—361no—
467Dic3—361no—
468Dic3—361no—
469Dic3—362no—
470Dic3—362no—
471C32—21no—
472C32—21no—
473C32—21no—
474C32—41no—
475C32#476311yescharacteristic, complemented
476C32#476611yescharacteristic, complemented
477C32#489111yescharacteristic, complemented
478C23—61no—
479D4—32no—
480D4—61no—
481C2×C4—61no—
482C2×C4—181no—
483C2×C4—181no—
484C2×C4—271no—
485C2×C4—271no—
486C2×C4—1082no—
487C2×C4D3311yescharacteristic
488C8—61no—
489C8—181no—
490C2×C3—21no—
491C2×C3—21no—
492C2×C3—21no—
493C2×C3—21no—
494C2×C3—21no—
495C2×C3—21no—
496C2×C3—21no—
497C2×C3—21no—
498C2×C3—21no—
499C2×C3—41no—
500C2×C3—41no—
501C2×C3—41no—
502C2×C3—124no—
503C2×C3—122no—
504C2×C3—122no—
505C2×C3#6379811yescharacteristic
506C2×C3#6381011yescharacteristic
507C2×C3#6389611yescharacteristic
508C2×C3#6389911yescharacteristic
509C2×C3#6392711yescharacteristic
510C2×C3#6393611yescharacteristic
511C2×C3#6393711yescharacteristic
512C2×C3#6395011yescharacteristic
513C2×C3#6398211yescharacteristic
514D3—42no—
515C22—62no—
516C22—121no—
517C22—121no—
518C22#8895411yescharacteristic
519C4—121no—
520C4—361no—
521C4—361no—
522C4—541no—
523C4—541no—
524C4D3×#62711yescharacteristic
525C4#8896111yescharacteristic
526C3—21no—
527C3—21no—
528C3—21no—
529C3—41no—
530C3#1059047311yescharacteristic, complemented
531C3#1059078011yescharacteristic, complemented
532C3#1059147911yescharacteristic, complemented
533C2—122no—
534C2#1172286211yescharacteristic
535C2#1172287711yescharacteristic
536C2#1172298011yescharacteristic
subgroupG/H
centerC22#88954
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
hypocenterC33#478
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
2. #11723450C2semidirect
3. #11722079C2semidirect
4. #11722050C2semidirect
5. C3×#63575C2semidirect
37. #63575D3semidirect
318. C33#478semidirect
475. C32#4763semidirect
476. C32#4766semidirect
477. C32#4891semidirect
530. C3#10590473semidirect
531. C3#10590780semidirect
532. C3#10591479semidirect

Maximal subgroups

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

subgroupindex
#117234992
#117234502
#117220792
#117220502
C3×#635752
C3×#635632
C3×#635622
#105914793
#105907803
#105904733