Finite Group Databasebeta

← #13156332

Subgroups — #13156332

Order 1728.

554 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×Dic3×D12C211yesmaximal, characteristic
2C3×#63869C211yesmaximal, characteristic, complemented
3#11722080C211yesmaximal, characteristic, complemented
4#11722053C211yesmaximal, characteristic, complemented
5Dic3⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
6#11722031C211yesmaximal, characteristic
7C3×#63562C211yesmaximal, characteristic
8#10591411—31nomaximal
9#10590469—31nomaximal
10#10590270—31nomaximal
11C2×C32×D12C2211yescharacteristic
12C2×C3×D3×Dic3—21no—
13C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C32×C4×Dic3C2211yescharacteristic
19C3×#3724—21no—
20C3×Dic3⋊Dic3—21no—
21C3×(C2×D3)⋊Dic3—21no—
22C3×D4×Dic3—31no—
23C3×#890—31no—
24C3×C4×D12—31no—
25C3×#870—31no—
26Dic3×D12D311yescharacteristic
27#63869D311yescharacteristic, complemented
28#63659—31no—
29#63635—31no—
30C4⋊(C3⋊(C3⋊C8))—31no—
31C3×#832—31no—
32C3×#809—31no—
33#63576—31no—
34#63575—31no—
35#63575—31no—
36#63574—31no—
37#63574—31no—
38#63574—61no—
39Dic3⋊(C3⋊C8)—31no—
40Dic3⋊(C3⋊C8)—31no—
41Dic3⋊(C3⋊C8)—61no—
42#63562—31no—
43#63562—31no—
44#63562—61no—
45#63562D311yescharacteristic
46C22×C32×D3—21no—
47C2×C33×C4C2311yescharacteristic
48C2×C3×C3⋊Dic3—21no—
49C2×C3×C3⋊Dic3—21no—
50C2×C32×Dic3—21no—
51C2×C32×Dic3D411yescharacteristic
52C2×C32×Dic3D411yescharacteristic
53C32×D12—21no—
54C32×D12Q812yes—
55C3×D3×Dic3—42no—
56C3⋊(C3⋊(C3⋊C8))—21no—
57C3×C3⋊(C3⋊C8)—21no—
58#5066—91no—
59#5009—91no—
60#4766—91no—
61C2×C32×D4—31no—
62C22×C3×Dic3—61no—
63C2×C3×D12—21no—
64C2×C3×D12C2×D311yescharacteristic
65C2×C3×D12C2×D311yescharacteristic
66C2×C3×C4×D3—61no—
67C2×D3×Dic3—21no—
68C32×C4⋊C4—31no—
69C32×C42—31no—
70C4⋊(C3⋊Dic3)C2×D311yescharacteristic
71C4⋊(C3⋊Dic3)C2×D311yescharacteristic
72C2×C3⋊(C3⋊C8)—31no—
73C2×C3⋊(C3⋊C8)—31no—
74C2×C3⋊(C3⋊C8)—31no—
75C2×C3⋊(C3⋊C8)—61no—
76C2×C3⋊(C3⋊C8)—61no—
77C2×C3⋊(C3⋊C8)—61no—
78C2×C3⋊(C3⋊C8)—121no—
79C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
80C3×#217—61no—
81C3×(C2×D3)⋊C4—61no—
82C3×C4⋊Dic3—21no—
83C3×C4⋊Dic3—31no—
84C3×C4⋊Dic3—31no—
85C3×C4⋊Dic3—31no—
86C3×C4⋊Dic3—31no—
87C3×C4⋊Dic3—61no—
88C3×C4⋊Dic3—61no—
89C3×C4⋊Dic3—61no—
90C3×C4⋊Dic3C2×D311yescharacteristic
91C3×C4⋊Dic3C2×D311yescharacteristic
92C3×Dic3⋊C4—61no—
93C3×Dic3⋊C4—61no—
94C3×Dic3⋊C4—61no—
95C3×C4×Dic3—21no—
96C3×C4×Dic3C2×D311yescharacteristic
97C3×C4×Dic3C2×D311yescharacteristic
98C2×C3×C3⋊C8—31no—
99C2×C3×C3⋊C8—31no—
100C2×C3×C3⋊C8—61no—
101#3724—21no—
102Dic3⋊Dic3—21no—
103(C2×D3)⋊Dic3—21no—
104C22×C33C2×D411yescharacteristic
105C2×C32×D3—22no—
106C2×C32×D3—41no—
107C2×C32×D3—41no—
108C33×C4C2×Q811yescharacteristic
109C33×C4#4111yescharacteristic
110C3×C3⋊Dic3—41no—
111C3×C3⋊Dic3—41no—
112C32×Dic3—21no—
113C32×Dic3—21no—
114C32×Dic3—41no—
115C3×#125—91no—
116C3×C4×D4—91no—
117D4×Dic3—31no—
118#890—31no—
119C4×D12—31no—
120#870—31no—
121C3×#102—91no—
122#832—31no—
123#832—91no—
124#832—91no—
125#832—181no—
126#818—91no—
127Dic3⋊C8—91no—
128#809—31no—
129#809—91no—
130#808—91no—
131#808—91no—
132#808—91no—
133#808—181no—
134#807—91no—
135#807—91no—
136C4⋊(C3⋊C8)—91no—
137C4⋊(C3⋊C8)—91no—
138C4⋊(C3⋊C8)—181no—
139C23×C32—61no—
140C22×C3×D3—21no—
141C22×C3×D3—21no—
142C22×C3×D3—41no—
143C32×D4—32no—
144C32×D4—61no—
145C2×C32×C4—21no—
146C2×C32×C4—21no—
147C2×C32×C4—21no—
148C2×C32×C4—31no—
149C2×C32×C4—31no—
150C2×C32×C4—41no—
151C2×C32×C4—61no—
152C2×C32×C4C22×D311yescharacteristic
153C2×C32×C4C22×D311yescharacteristic
154C2×C32×C4C22×D311yescharacteristic
155C2×C3⋊Dic3—21no—
156C2×C3⋊Dic3—21no—
157C2×C3×Dic3—21no—
158C2×C3×Dic3—21no—
159C2×C3×Dic3—21no—
160C2×C3×Dic3—21no—
161C2×C3×Dic3—41no—
162C2×C3×Dic3—61no—
163C2×C3×Dic3—61no—
164C2×C3×Dic3—61no—
165C2×C3×Dic3—61no—
166C2×C3×Dic3—121no—
167C2×C3×Dic3—121no—
168C2×C3×Dic3—122no—
169C2×C3×Dic3#6711yescharacteristic
170C2×C3×Dic3#6711yescharacteristic
171C2×C3×Dic3#6711yescharacteristic
172C2×C3×Dic3#6711yescharacteristic
173C3×D12—21no—
174C3×D12—21no—
175C3×D12—22no—
176C3×D12—41no—
177C3×D12Dic612yes—
178C3×D12Dic612yes—
179C3×C4×D3—122no—
180D3×Dic3—42no—
181C3⋊(C3⋊C8)—21no—
182C3⋊(C3⋊C8)—61no—
183C3⋊(C3⋊C8)—62no—
184C3⋊(C3⋊C8)—61no—
185C3⋊(C3⋊C8)—61no—
186C3⋊(C3⋊C8)—121no—
187C3⋊(C3⋊C8)—121no—
188C3⋊(C3⋊C8)—122no—
189C3×C3⋊C8—61no—
190C3×C3⋊C8—61no—
191C3×C3⋊C8—121no—
192#478—271noSylow
193C2×C33C22⋊Q811yescharacteristic
194C2×C33#13511yescharacteristic
195C2×C33AGL(1,ℤ/8)11yescharacteristic
196C32×D3—42no—
197C2×C3×D4—31no—
198C2×C3×D4—31no—
199C2×C3×D4—61no—
200C22×C3×C4—181no—
201C22×Dic3—61no—
202C2×D12D3211yescharacteristic
203C2×C4×D3—61no—
204C2×C3×C8—91no—
205C3×C4⋊C4—31no—
206C3×C4⋊C4—31no—
207C3×C4⋊C4—61no—
208C3×C4⋊C4—91no—
209C3×C4⋊C4—91no—
210C3×C4⋊C4—181no—
211C3×C4⋊C4—181no—
212C3×C22⋊C4—181no—
213C3×C42—31no—
214C3×C42—31no—
215C3×C42—61no—
216#217—61no—
217(C2×D3)⋊C4—61no—
218C4⋊Dic3—31no—
219C4⋊Dic3—31no—
220C4⋊Dic3—31no—
221C4⋊Dic3—31no—
222C4⋊Dic3—61no—
223C4⋊Dic3—61no—
224C4⋊Dic3—61no—
225C4⋊Dic3D3211yescharacteristic
226Dic3⋊C4—61no—
227Dic3⋊C4—61no—
228Dic3⋊C4—61no—
229C4×Dic3D3211yescharacteristic
230C2×C3⋊C8—31no—
231C2×C3⋊C8—31no—
232C2×C3⋊C8—61no—
233C2×C3⋊C8—91no—
234C2×C3⋊C8—91no—
235C2×C3⋊C8—91no—
236C2×C3⋊C8—181no—
237C2×C3⋊C8—181no—
238C2×C3⋊C8—181no—
239C2×C3⋊C8—361no—
240C22×C32—21no—
241C22×C32—21no—
242C22×C32—21no—
243C22×C32—41no—
244C22×C32—62no—
245C22×C32—121no—
246C22×C32—121no—
247C22×C32D3×D411yescharacteristic
248C22×C32C2×#6711yescharacteristic
249C22×C32C2×#6711yescharacteristic
250C2×C3×D3—22no—
251C2×C3×D3—22no—
252C2×C3×D3—42no—
253C2×C3×D3—41no—
254C2×C3×D3—41no—
255C2×C3×D3—41no—
256C2×C3×D3—41no—
257C2×C3×D3—81no—
258C2×C3×D3—81no—
259C32×C4—21no—
260C32×C4—21no—
261C32×C4—21no—
262C32×C4—21no—
263C32×C4—21no—
264C32×C4—21no—
265C32×C4—41no—
266C32×C4—41no—
267C32×C4—61no—
268C32×C4—61no—
269C32×C4—121no—
270C32×C4C2×Dic611yescharacteristic
271C32×C4C2×Dic611yescharacteristic
272C32×C4#23511yescharacteristic
273C32×C4#23711yescharacteristic
274C32×C4D3×Q811yescharacteristic
275C32×C4#23911yescharacteristic
276C3⋊Dic3—41no—
277C3⋊Dic3—41no—
278C3×Dic3—22no—
279C3×Dic3—21no—
280C3×Dic3—21no—
281C3×Dic3—21no—
282C3×Dic3—21no—
283C3×Dic3—41no—
284C3×Dic3—41no—
285C3×Dic3—41no—
286C3×Dic3—81no—
287C3×Dic3—121no—
288C3×Dic3—121no—
289C3×Dic3—121no—
290C3×Dic3—121no—
291C3×Dic3—241no—
292C3×Dic3—241no—
293#125—91no—
294C4×D4—91no—
295#107—271no—
296#106—271no—
297C4⋊C8—271no—
298#102—91no—
299#102—271no—
300C33#47811yescharacteristic, complemented, Sylow
301C23×C3—61no—
302C23×C3—61no—
303C23×C3—121no—
304C22×D3—21no—
305C3×D4—32no—
306C3×D4—32no—
307C3×D4—62no—
308C3×D4—61no—
309C3×D4—61no—
310C3×D4—121no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—31no—
315C2×C3×C4—31no—
316C2×C3×C4—31no—
317C2×C3×C4—31no—
318C2×C3×C4—41no—
319C2×C3×C4—61no—
320C2×C3×C4—61no—
321C2×C3×C4—61no—
322C2×C3×C4—61no—
323C2×C3×C4—121no—
324C2×C3×C4—181no—
325C2×C3×C4—181no—
326C2×C3×C4—362no—
327C2×C3×C4C2×D3211yescharacteristic
328C2×C3×C4C2×D3211yescharacteristic
329C2×C3×C4C2×D3211yescharacteristic
330C2×Dic3—21no—
331C2×Dic3—61no—
332C2×Dic3—61no—
333C2×Dic3—61no—
334C2×Dic3—61no—
335C2×Dic3—121no—
336C2×Dic3—121no—
337C2×Dic3—122no—
338C2×Dic3#62511yescharacteristic
339C2×Dic3#62511yescharacteristic
340D12—21no—
341D12#62712yes—
342C4×D3—122no—
343C3×C8—181no—
344C3⋊C8—61no—
345C3⋊C8—61no—
346C3⋊C8—121no—
347C3⋊C8—182no—
348C3⋊C8—181no—
349C3⋊C8—181no—
350C3⋊C8—181no—
351C3⋊C8—361no—
352C3⋊C8—361no—
353C3⋊C8—362no—
354C2×C32—21no—
355C2×C32—21no—
356C2×C32—21no—
357C2×C32—21no—
358C2×C32—21no—
359C2×C32—21no—
360C2×C32—21no—
361C2×C32—21no—
362C2×C32—21no—
363C2×C32—41no—
364C2×C32—41no—
365C2×C32—41no—
366C2×C32—122no—
367C2×C32#87811yescharacteristic
368C2×C32#91211yescharacteristic
369C2×C32#91411yescharacteristic
370C2×C32#92411yescharacteristic
371C2×C32#93211yescharacteristic
372C2×C32#94611yescharacteristic
373C2×C32#94911yescharacteristic
374C2×C32#95011yescharacteristic
375C2×C32#95011yescharacteristic
376C3×D3—42no—
377C3×D3—42no—
378C3×D3—82no—
379C2×D4—31no—
380C22×C4—181no—
381C2×C8—91no—
382C2×C8—271no—
383C4⋊C4—31no—
384C4⋊C4—91no—
385C4⋊C4—91no—
386C4⋊C4—181no—
387C4⋊C4—181no—
388C22⋊C4—181no—
389C42—31no—
390C22×C3—21no—
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—41no—
394C22×C3—62no—
395C22×C3—62no—
396C22×C3—122no—
397C22×C3—121no—
398C22×C3—121no—
399C22×C3—121no—
400C22×C3—121no—
401C22×C3—241no—
402C22×C3—241no—
403C22×C3C2×#62511yescharacteristic
404C22×C3D3×#6711yescharacteristic
405C22×C3D3×#6711yescharacteristic
406C2×D3—22no—
407C2×D3—41no—
408C2×D3—41no—
409C3×C4—21no—
410C3×C4—21no—
411C3×C4—21no—
412C3×C4—21no—
413C3×C4—21no—
414C3×C4—21no—
415C3×C4—41no—
416C3×C4—41no—
417C3×C4—62no—
418C3×C4—61no—
419C3×C4—61no—
420C3×C4—61no—
421C3×C4—61no—
422C3×C4—121no—
423C3×C4—121no—
424C3×C4—121no—
425C3×C4—241no—
426C3×C4—361no—
427C3×C4—361no—
428C3×C4D3×Dic611yescharacteristic
429C3×C4D3×Dic611yescharacteristic
430C3×C4#379611yescharacteristic
431C3×C4#379911yescharacteristic
432C3×C4#380411yescharacteristic
433C3×C4C2×#62711yescharacteristic
434Dic3—21no—
435Dic3—21no—
436Dic3—41no—
437Dic3—121no—
438Dic3—121no—
439Dic3—121no—
440Dic3—121no—
441Dic3—241no—
442Dic3—241no—
443C32—21no—
444C32—21no—
445C32—21no—
446C32—41no—
447C32#476611yescharacteristic, complemented
448C32#500911yescharacteristic, complemented
449C32#506611yescharacteristic, complemented
450C23—61no—
451D4—32no—
452D4—61no—
453C2×C4—31no—
454C2×C4—31no—
455C2×C4—61no—
456C2×C4—181no—
457C2×C4—181no—
458C2×C4—362no—
459C2×C4D3311yescharacteristic
460C8—181no—
461C8—541no—
462C2×C3—21no—
463C2×C3—21no—
464C2×C3—21no—
465C2×C3—21no—
466C2×C3—21no—
467C2×C3—21no—
468C2×C3—21no—
469C2×C3—21no—
470C2×C3—21no—
471C2×C3—41no—
472C2×C3—41no—
473C2×C3—41no—
474C2×C3—122no—
475C2×C3—122no—
476C2×C3—242no—
477C2×C3#6382211yescharacteristic
478C2×C3#6382311yescharacteristic
479C2×C3#6385611yescharacteristic
480C2×C3#6388711yescharacteristic
481C2×C3#6393311yescharacteristic
482C2×C3#6393311yescharacteristic
483C2×C3#6393411yescharacteristic
484C2×C3#6393911yescharacteristic
485C2×C3#6396411yescharacteristic
486D3—42no—
487C22—62no—
488C22—121no—
489C22—121no—
490C22D3×#62511yescharacteristic
491C4—61no—
492C4—61no—
493C4—121no—
494C4—361no—
495C4—361no—
496C4D3×#62711yescharacteristic
497C4#8896211yescharacteristic
498C3—21no—
499C3—21no—
500C3—21no—
501C3—41no—
502C3#1059027011yescharacteristic, complemented
503C3#1059046911yescharacteristic, complemented
504C3#1059141111yescharacteristic, complemented
505C2—122no—
506C2#1172285011yescharacteristic
507C2#1172286711yescharacteristic
508C2#1172296911yescharacteristic
subgroupG/H
centerC22D3×#625
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. C3×#63869C2semidirect
3. #11722080C2semidirect
4. #11722053C2semidirect
5. Dic3⋊(C3⋊(C3⋊C8))C2semidirect
27. #63869D3semidirect
300. C33#478semidirect
447. C32#4766semidirect
448. C32#5009semidirect
449. C32#5066semidirect
502. C3#10590270semidirect
503. C3#10590469semidirect
504. C3#10591411semidirect

Maximal subgroups

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

subgroupindex
C3×Dic3×D122
C3×#638692
#117220802
#117220532
Dic3⋊(C3⋊(C3⋊C8))2
#117220312
C3×#635622
#105914113
#105904693
#105902703