Finite Group Databasebeta

← #13155667

Subgroups — #13155667

Order 1728.

634 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
1C3×Dic3×D12C211yesmaximal, characteristic
2C3×Dic3×Dic6C211yesmaximal, characteristic, complemented
3C2×C3×#3715C211yesmaximal, characteristic
4#11722080C211yesmaximal, characteristic, complemented
5#11722037C211yesmaximal, characteristic, complemented
6#11722031C211yesmaximal, characteristic
7#11722019C211yesmaximal, characteristic, complemented
8#10591566—31nomaximal
9#10591440—31nomaximal
10#10590358—31nomaximal
11C2×C32×D12C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×C3×D3×Dic3—21no—
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×C4×Dic3C2211yescharacteristic
19C3×Dic3⋊Dic3—21no—
20C3×(C2×D3)⋊Dic3—21no—
21C3×Dic32—21no—
22C3×#3715C414yescomplemented
23C3×Q8×Dic3—31no—
24C2×C3×#215—31no—
25C3×D4×Dic3—31no—
26C2×C3×#214—31no—
27C3×C4×D12—31no—
28C3×C4×Dic6—31no—
29Dic3×D12D311yescharacteristic
30Dic3×Dic6D311yescharacteristic, complemented
31C2×#3715D311yescharacteristic
32#63662—31no—
33#63659—31no—
34#63630—31no—
35#63576—31no—
36#63575—31no—
37#63575—31no—
38#63565—31no—
39#63565—31no—
40#63565—61no—
41#63562—31no—
42#63562—31no—
43#63562—61no—
44#63554—31no—
45#63554—31no—
46#63554—61no—
47C22×C32×D3—21no—
48C2×C33×C4C2311yescharacteristic
49C2×C3×C3⋊Dic3—21no—
50C2×C3×C3⋊Dic3—21no—
51C2×C32×Dic3—21no—
52C2×C32×Dic3D411yescharacteristic
53C2×C32×Dic3D411yescharacteristic
54C32×D12—21no—
55C32×D12C2×C412yes—
56C32×Dic6—21no—
57C32×Dic6C2×C412yescomplemented
58C3×D3×Dic3—42no—
59C3⋊(C3⋊(C3⋊C8))—21no—
60C3×C3⋊(C3⋊C8)C2×C412yescomplemented
61#5164—91no—
62#5026—91no—
63#5018—91no—
64C2×C32×Q8—31no—
65C2×C32×D4—31no—
66C22×C3×Dic3—61no—
67C2×C3×D12—21no—
68C2×C3×D12C2×D311yescharacteristic
69C2×C3×D12C2×D311yescharacteristic
70C2×C3×C4×D3—61no—
71C2×C3×Dic6—21no—
72C2×C3×Dic6C2×D311yescharacteristic
73C2×C3×Dic6C2×D311yescharacteristic
74C2×D3×Dic3—21no—
75C32×C42—31no—
76C4⋊(C3⋊Dic3)C2×D311yescharacteristic
77C4⋊(C3⋊Dic3)C2×D311yescharacteristic
78C2×C3⋊(C3⋊C8)—31no—
79C2×C3⋊(C3⋊C8)—31no—
80C2×C3⋊(C3⋊C8)—31no—
81C2×C3⋊(C3⋊C8)—61no—
82C2×C3⋊(C3⋊C8)—61no—
83C2×C3⋊(C3⋊C8)—61no—
84C2×C3⋊(C3⋊C8)—121no—
85C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
86C3×#217—61no—
87C3×#215—34no—
88C3×#214—34no—
89C3×(C2×D3)⋊C4—61no—
90C3×C4⋊Dic3—31no—
91C3×C4⋊Dic3—31no—
92C3×C4⋊Dic3—31no—
93C3×C4⋊Dic3—31no—
94C3×C4⋊Dic3—61no—
95C3×C4⋊Dic3—61no—
96C3×C4⋊Dic3—61no—
97C3×Dic3⋊C4—61no—
98C3×C4×Dic3—21no—
99C3×C4×Dic3—61no—
100C3×C4×Dic3—61no—
101C3×C4×Dic3C2×D311yescharacteristic
102C3×C4×Dic3C2×D311yescharacteristic
103C2×C3×C3⋊C8—31no—
104C2×C3×C3⋊C8—31no—
105C2×C3×C3⋊C8—61no—
106Dic3⋊Dic3—21no—
107(C2×D3)⋊Dic3—21no—
108Dic32—21no—
109#3715Dic314yescomplemented
110C22×C33C2×D411yescharacteristic
111C2×C32×D3—22no—
112C2×C32×D3—41no—
113C2×C32×D3—41no—
114C33×C4C22×C411yescharacteristic
115C33×C4#4111yescharacteristic
116C3×C3⋊Dic3—41no—
117C3×C3⋊Dic3—41no—
118C32×Dic3—22no—
119C32×Dic3—21no—
120C32×Dic3—21no—
121C2×C3×SD16—91no—
122C3×C4×Q8—91no—
123C3×C4×D4—91no—
124Q8×Dic3—31no—
125C2×#215—31no—
126D4×Dic3—31no—
127C2×#214—31no—
128C4×D12—31no—
129C4×Dic6—31no—
130#835—91no—
131#835—91no—
132#835—181no—
133#832—91no—
134#832—91no—
135#832—181no—
136#818—91no—
137#815—91no—
138#810—91no—
139#809—91no—
140#807—91no—
141#807—91no—
142#803—91no—
143#803—91no—
144#803—181no—
145C23×C32—61no—
146C22×C3×D3—21no—
147C22×C3×D3—21no—
148C22×C3×D3—41no—
149C32×Q8—32no—
150C32×Q8—61no—
151C32×D4—32no—
152C32×D4—61no—
153C2×C32×C4—21no—
154C2×C32×C4—21no—
155C2×C32×C4—21no—
156C2×C32×C4—31no—
157C2×C32×C4—31no—
158C2×C32×C4—41no—
159C2×C32×C4—61no—
160C2×C32×C4C22×D311yescharacteristic
161C2×C32×C4C22×D311yescharacteristic
162C2×C32×C4C22×D311yescharacteristic
163C2×C3⋊Dic3—21no—
164C2×C3⋊Dic3—21no—
165C2×C3×Dic3—21no—
166C2×C3×Dic3—21no—
167C2×C3×Dic3—21no—
168C2×C3×Dic3—21no—
169C2×C3×Dic3—41no—
170C2×C3×Dic3—61no—
171C2×C3×Dic3—61no—
172C2×C3×Dic3—61no—
173C2×C3×Dic3—61no—
174C2×C3×Dic3—121no—
175C2×C3×Dic3—121no—
176C2×C3×Dic3—122no—
177C2×C3×Dic3#6711yescharacteristic
178C2×C3×Dic3#6711yescharacteristic
179C2×C3×Dic3#6711yescharacteristic
180C2×C3×Dic3#6711yescharacteristic
181C3×D12—21no—
182C3×D12—21no—
183C3×D12—22no—
184C3×D12—41no—
185C3×D12C4×D312yes—
186C3×D12C2×Dic312yes—
187C3×C4×D3—122no—
188C3×Dic6—21no—
189C3×Dic6—21no—
190C3×Dic6—22no—
191C3×Dic6—41no—
192C3×Dic6C4×D312yescomplemented
193C3×Dic6C2×Dic312yescomplemented
194D3×Dic3—42no—
195C3⋊(C3⋊C8)—61no—
196C3⋊(C3⋊C8)—62no—
197C3⋊(C3⋊C8)—61no—
198C3⋊(C3⋊C8)—61no—
199C3⋊(C3⋊C8)—121no—
200C3⋊(C3⋊C8)—121no—
201C3⋊(C3⋊C8)—122no—
202C3⋊(C3⋊C8)C2×Dic312yescomplemented
203C3×C3⋊C8—32no—
204C3×C3⋊C8—32no—
205C3×C3⋊C8—62no—
206#440—271noSylow
207C2×C33C4×D411yescharacteristic
208C2×C33AGL(1,ℤ/8)11yescharacteristic
209C2×C33#13711yescharacteristic
210C32×D3—42no—
211C2×C3×Q8—31no—
212C2×C3×Q8—31no—
213C2×C3×Q8—61no—
214C2×C3×D4—31no—
215C2×C3×D4—31no—
216C2×C3×D4—61no—
217C22×C3×C4—181no—
218C22×Dic3—61no—
219C2×D12D3211yescharacteristic
220C2×C4×D3—61no—
221C2×Dic6D3211yescharacteristic
222C3×SD16—94no—
223C2×C3×C8—91no—
224C3×C4⋊C4—91no—
225C3×C4⋊C4—91no—
226C3×C4⋊C4—181no—
227C3×C22⋊C4—181no—
228C3×C42—31no—
229C3×C42—31no—
230C3×C42—61no—
231C3×C42—181no—
232#217—61no—
233#215—34no—
234#214—34no—
235(C2×D3)⋊C4—61no—
236C4⋊Dic3—31no—
237C4⋊Dic3—31no—
238C4⋊Dic3—31no—
239C4⋊Dic3—31no—
240C4⋊Dic3—61no—
241C4⋊Dic3—61no—
242C4⋊Dic3—61no—
243Dic3⋊C4—61no—
244C4×Dic3—61no—
245C4×Dic3—61no—
246C4×Dic3D3211yescharacteristic
247C2×C3⋊C8—31no—
248C2×C3⋊C8—31no—
249C2×C3⋊C8—61no—
250C2×C3⋊C8—91no—
251C2×C3⋊C8—91no—
252C2×C3⋊C8—91no—
253C2×C3⋊C8—181no—
254C2×C3⋊C8—181no—
255C2×C3⋊C8—181no—
256C2×C3⋊C8—361no—
257C22×C32—21no—
258C22×C32—21no—
259C22×C32—21no—
260C22×C32—41no—
261C22×C32—62no—
262C22×C32—121no—
263C22×C32—121no—
264C22×C32D3×D411yescharacteristic
265C22×C32C2×#6711yescharacteristic
266C22×C32C2×#6711yescharacteristic
267C2×C3×D3—22no—
268C2×C3×D3—22no—
269C2×C3×D3—42no—
270C2×C3×D3—41no—
271C2×C3×D3—41no—
272C2×C3×D3—41no—
273C2×C3×D3—41no—
274C2×C3×D3—81no—
275C2×C3×D3—81no—
276C32×C4—21no—
277C32×C4—21no—
278C32×C4—21no—
279C32×C4—21no—
280C32×C4—21no—
281C32×C4—21no—
282C32×C4—41no—
283C32×C4—41no—
284C32×C4—61no—
285C32×C4—61no—
286C32×C4—62no—
287C32×C4C2×C4×D311yescharacteristic
288C32×C4C2×C4×D311yescharacteristic
289C32×C4#23511yescharacteristic
290C32×C4#23511yescharacteristic
291C32×C4#23711yescharacteristic
292C32×C4C22×Dic311yescharacteristic
293C3⋊Dic3—41no—
294C3⋊Dic3—41no—
295C3×Dic3—22no—
296C3×Dic3—21no—
297C3×Dic3—22no—
298C3×Dic3—21no—
299C3×Dic3—21no—
300C3×Dic3—21no—
301C3×Dic3—42no—
302C3×Dic3—41no—
303C3×Dic3—41no—
304C3×Dic3—121no—
305C3×Dic3—121no—
306C3×Dic3—121no—
307C3×Dic3—121no—
308C3×Dic3—241no—
309C3×Dic3—241no—
310C2×SD16—91no—
311C4×Q8—91no—
312C4×D4—91no—
313#107—271no—
314#103—271no—
315#102—271no—
316#97—271no—
317C33#44011yescharacteristic, complemented, Sylow
318C23×C3—61no—
319C23×C3—61no—
320C23×C3—121no—
321C22×D3—21no—
322C3×Q8—32no—
323C3×Q8—32no—
324C3×Q8—62no—
325C3×Q8—61no—
326C3×Q8—61no—
327C3×Q8—121no—
328C3×D4—32no—
329C3×D4—32no—
330C3×D4—62no—
331C3×D4—61no—
332C3×D4—61no—
333C3×D4—121no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—31no—
338C2×C3×C4—31no—
339C2×C3×C4—31no—
340C2×C3×C4—31no—
341C2×C3×C4—41no—
342C2×C3×C4—61no—
343C2×C3×C4—61no—
344C2×C3×C4—61no—
345C2×C3×C4—61no—
346C2×C3×C4—121no—
347C2×C3×C4—181no—
348C2×C3×C4—181no—
349C2×C3×C4—362no—
350C2×C3×C4C2×D3211yescharacteristic
351C2×C3×C4C2×D3211yescharacteristic
352C2×C3×C4C2×D3211yescharacteristic
353C2×Dic3—21no—
354C2×Dic3—61no—
355C2×Dic3—61no—
356C2×Dic3—61no—
357C2×Dic3—61no—
358C2×Dic3—121no—
359C2×Dic3—121no—
360C2×Dic3—122no—
361C2×Dic3#62511yescharacteristic
362C2×Dic3#62511yescharacteristic
363D12—21no—
364D12D3×Dic312yes—
365C4×D3—122no—
366Dic6—21no—
367Dic6D3×Dic312yescomplemented
368C3×C8—92no—
369C3⋊C8—32no—
370C3⋊C8—32no—
371C3⋊C8—62no—
372C3⋊C8—182no—
373C3⋊C8—181no—
374C3⋊C8—181no—
375C3⋊C8—181no—
376C3⋊C8—361no—
377C3⋊C8—361no—
378C3⋊C8—362no—
379C2×C32—21no—
380C2×C32—21no—
381C2×C32—21no—
382C2×C32—21no—
383C2×C32—21no—
384C2×C32—21no—
385C2×C32—21no—
386C2×C32—21no—
387C2×C32—21no—
388C2×C32—41no—
389C2×C32—41no—
390C2×C32—41no—
391C2×C32—122no—
392C2×C32#91411yescharacteristic
393C2×C32#91511yescharacteristic
394C2×C32C4×#6711yescharacteristic
395C2×C32C4×#6711yescharacteristic
396C2×C32#93211yescharacteristic
397C2×C32D4×Dic311yescharacteristic
398C2×C32#94211yescharacteristic
399C2×C32#94911yescharacteristic
400C2×C32#95111yescharacteristic
401C3×D3—42no—
402C3×D3—42no—
403C3×D3—82no—
404C2×Q8—31no—
405C2×D4—31no—
406C22×C4—181no—
407SD16—94no—
408C2×C8—91no—
409C2×C8—271no—
410C4⋊C4—91no—
411C4⋊C4—91no—
412C4⋊C4—181no—
413C22⋊C4—181no—
414C42—31no—
415C42—181no—
416C22×C3—21no—
417C22×C3—21no—
418C22×C3—21no—
419C22×C3—41no—
420C22×C3—62no—
421C22×C3—62no—
422C22×C3—122no—
423C22×C3—121no—
424C22×C3—121no—
425C22×C3—121no—
426C22×C3—121no—
427C22×C3—241no—
428C22×C3—241no—
429C22×C3C2×#62511yescharacteristic
430C22×C3D3×#6711yescharacteristic
431C22×C3D3×#6711yescharacteristic
432C2×D3—22no—
433C2×D3—41no—
434C2×D3—41no—
435C3×C4—21no—
436C3×C4—21no—
437C3×C4—21no—
438C3×C4—21no—
439C3×C4—21no—
440C3×C4—21no—
441C3×C4—41no—
442C3×C4—41no—
443C3×C4—62no—
444C3×C4—61no—
445C3×C4—61no—
446C3×C4—61no—
447C3×C4—61no—
448C3×C4—62no—
449C3×C4—121no—
450C3×C4—121no—
451C3×C4—122no—
452C3×C4—361no—
453C3×C4—361no—
454C3×C4#379811yescharacteristic
455C3×C4C4×D3211yescharacteristic
456C3×C4C2×D3×Dic311yescharacteristic
457C3×C4C2×D3×Dic311yescharacteristic
458C3×C4#380411yescharacteristic
459C3×C4#380411yescharacteristic
460Dic3—21no—
461Dic3—21no—
462Dic3—22no—
463Dic3—121no—
464Dic3—121no—
465Dic3—121no—
466Dic3—121no—
467Dic3—241no—
468Dic3—241no—
469C32—21no—
470C32—21no—
471C32—21no—
472C32—41no—
473C32#501811yescharacteristic, complemented
474C32#502611yescharacteristic, complemented
475C32#516411yescharacteristic, complemented
476C23—61no—
477Q8—32no—
478Q8—61no—
479D4—32no—
480D4—61no—
481C2×C4—31no—
482C2×C4—31no—
483C2×C4—61no—
484C2×C4—181no—
485C2×C4—181no—
486C2×C4—362no—
487C2×C4D3311yescharacteristic
488C8—92no—
489C8—541no—
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—122no—
503C2×C3—122no—
504C2×C3—242no—
505C2×C3#6382311yescharacteristic
506C2×C3#6382711yescharacteristic
507C2×C3C4×#62511yescharacteristic
508C2×C3#6392911yescharacteristic
509C2×C3#6393411yescharacteristic
510C2×C3#6393911yescharacteristic
511C2×C3#6394611yescharacteristic
512C2×C3Dic3×#6711yescharacteristic
513C2×C3Dic3×#6711yescharacteristic
514D3—42no—
515C22—62no—
516C22—121no—
517C22—121no—
518C22D3×#62511yescharacteristic
519C4—61no—
520C4—61no—
521C4—62no—
522C4—361no—
523C4—361no—
524C4D32×Dic311yescharacteristic
525C4#8895911yescharacteristic
526C3—21no—
527C3—21no—
528C3—21no—
529C3—41no—
530C3#1059035811yescharacteristic, complemented
531C3#1059144011yescharacteristic, complemented
532C3#1059156611yescharacteristic, complemented
533C2—122no—
534C2#1172286711yescharacteristic
535C2#1172288211yescharacteristic
536C2Dic3×#62511yescharacteristic
subgroupG/H
centerC22D3×#625
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
hypocenterC33#440
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C3×Dic3×Dic6C2semidirect
4. #11722080C2semidirect
5. #11722037C2semidirect
7. #11722019C2semidirect
22. C3×#3715C4semidirect
30. Dic3×Dic6D3semidirect
57. C32×Dic6C2×C4semidirect
60. C3×C3⋊(C3⋊C8)C2×C4semidirect
109. #3715Dic3semidirect
192. C3×Dic6C4×D3semidirect
193. C3×Dic6C2×Dic3semidirect
202. C3⋊(C3⋊C8)C2×Dic3semidirect
317. C33#440semidirect
367. Dic6D3×Dic3semidirect
473. C32#5018semidirect
474. C32#5026semidirect
475. C32#5164semidirect
530. C3#10590358semidirect
531. C3#10591440semidirect
532. C3#10591566semidirect

Maximal subgroups

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

subgroupindex
C3×Dic3×D122
C3×Dic3×Dic62
C2×C3×#37152
#117220802
#117220372
#117220312
#117220192
#105915663
#105914403
#105903583