Finite Group Databasebeta

← #13156339

Subgroups — #13156339

Order 1728.

600 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Dic3×C4⋊(C3⋊D3)C211yesmaximal, characteristic
2C3×#63869C211yesmaximal, characteristic, complemented
3#11722075C211yesmaximal, characteristic
4#11722055C211yesmaximal, characteristic, complemented
5#11722054C211yesmaximal, characteristic, complemented
6C3×Dic3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
7C3×#63564C211yesmaximal, characteristic
8#10590767—31nomaximal
9#10590469—31nomaximal
10#10590308—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×Dic3×C3⋊D3—21no—
13C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C32×C4×Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88803—21no—
20C3×#3724—21no—
21C3×Dic3⋊Dic3—21no—
22C4×C4⋊(C3⋊D3)—31no—
23C3×#890—31no—
24C3×#870—31no—
25Dic3×D12—31no—
26Dic3×D12—31no—
27Dic3×D12—61no—
28#63869D311yescharacteristic, complemented
29#63645—31no—
30#63635—31no—
31C3×#821—31no—
32C3×Dic3⋊C8—31no—
33C3×#809—31no—
34C3×C4⋊(C3⋊C8)—31no—
35#63575—31no—
36#63575—31no—
37#63575—61no—
38#63573—31no—
39#63573—31no—
40#63573—61no—
41Dic3⋊(C3⋊C8)D311yescharacteristic, complemented
42#63564—31no—
43#63564D311yescharacteristic
44#63563—31no—
45#63562—31no—
46C22×C3×C3⋊D3—21no—
47C2×C33×C4C2311yescharacteristic
48C2×C3⋊(C3⋊Dic3)—21no—
49C2×C3×C3⋊Dic3—21no—
50C3×C4⋊(C3⋊D3)—21no—
51C3×C4⋊(C3⋊D3)Q812yes—
52C2×C32×Dic3—21no—
53C2×C32×Dic3D411yescharacteristic
54C2×C32×Dic3D411yescharacteristic
55Dic3×C3⋊D3—42no—
56C3×C3⋊(C3⋊C8)—21no—
57C32×C3⋊C8—21no—
58#5066—91no—
59#4847—91no—
60#4766—91no—
61C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
62C2×C4×C3⋊D3—61no—
63C2×C3×D12—31no—
64C2×C3×D12—31no—
65C2×C3×D12—61no—
66C2×D3×Dic3—62no—
67C2×D3×Dic3—61no—
68C2×D3×Dic3—61no—
69C2×C32×C8—31no—
70C32×C4⋊C4—31no—
71C32×C42—31no—
72#3752—61no—
73C4⋊(C3⋊Dic3)—31no—
74C4⋊(C3⋊Dic3)—31no—
75C4⋊(C3⋊Dic3)—31no—
76C4⋊(C3⋊Dic3)—61no—
77C4⋊(C3⋊Dic3)—61no—
78C4⋊(C3⋊Dic3)—61no—
79C4⋊(C3⋊Dic3)—121no—
80C4⋊(C3⋊Dic3)C2×D311yescharacteristic
81C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
82C3×C4⋊Dic3—21no—
83C3×C4⋊Dic3—31no—
84C3×C4⋊Dic3—31no—
85C3×C4⋊Dic3—61no—
86C3×C4⋊Dic3—61no—
87C3×C4⋊Dic3C2×D311yescharacteristic
88C3×C4⋊Dic3C2×D311yescharacteristic
89C3×Dic3⋊C4—61no—
90C3×Dic3⋊C4—61no—
91C3×Dic3⋊C4—61no—
92C3×C4×Dic3—21no—
93C3×C4×Dic3C2×D311yescharacteristic
94C3×C4×Dic3C2×D311yescharacteristic
95C2×C3×C3⋊C8—21no—
96C2×C3×C3⋊C8—31no—
97C2×C3×C3⋊C8—31no—
98C2×C3×C3⋊C8—61no—
99C2×C3×C3⋊C8C2×D311yescharacteristic
100C2×C3×C3⋊C8C2×D311yescharacteristic
101#3724—21no—
102Dic3⋊Dic3—21no—
103(C2×D3)⋊Dic3—62no—
104(C2×D3)⋊Dic3—61no—
105(C2×D3)⋊Dic3—61no—
106C22×C33C2×D411yescharacteristic
107C2×C3×C3⋊D3—22no—
108C2×C3×C3⋊D3—41no—
109C2×C3×C3⋊D3—41no—
110C33×C4C2×Q811yescharacteristic
111C33×C4#4111yescharacteristic
112C3⋊(C3⋊Dic3)—41no—
113C3×C3⋊Dic3—41no—
114C32×Dic3—21no—
115C32×Dic3—21no—
116C32×Dic3—41no—
117C3×#125—91no—
118D4×Dic3—91no—
119#890—31no—
120C4×D12—91no—
121C4×D12—91no—
122C4×D12—181no—
123#870—31no—
124C3×C4⋊C8—91no—
125C3×#102—91no—
126#832—91no—
127#821—31no—
128#821—91no—
129#818—91no—
130#818—91no—
131#818—181no—
132#817—91no—
133Dic3⋊C8—31no—
134#809—31no—
135#809—91no—
136#808—91no—
137#808—91no—
138#808—181no—
139#807—91no—
140C4⋊(C3⋊C8)—31no—
141C22×C3⋊D3—21no—
142C22×C3×D3—62no—
143C22×C3×D3—61no—
144C22×C3×D3—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—61no—
157C2×C3⋊Dic3—62no—
158C2×C3⋊Dic3—61no—
159C2×C3⋊Dic3—61no—
160C2×C3⋊Dic3—121no—
161C2×C3⋊Dic3—121no—
162C2×C3⋊Dic3—122no—
163C4⋊(C3⋊D3)—21no—
164C4⋊(C3⋊D3)Dic612yes—
165C4×C3⋊D3—122no—
166C2×C3×Dic3—21no—
167C2×C3×Dic3—21no—
168C2×C3×Dic3—21no—
169C2×C3×Dic3—21no—
170C2×C3×Dic3—41no—
171C2×C3×Dic3—61no—
172C2×C3×Dic3—61no—
173C2×C3×Dic3—121no—
174C2×C3×Dic3D1211yescharacteristic
175C2×C3×Dic3D1211yescharacteristic
176C2×C3×Dic3#6711yescharacteristic
177C2×C3×Dic3#6711yescharacteristic
178C3×D12—32no—
179C3×D12—32no—
180C3×D12—62no—
181C3×D12—62no—
182C3×D12—61no—
183C3×D12—61no—
184D3×Dic3—124no—
185D3×Dic3—122no—
186D3×Dic3—122no—
187C32×C8—61no—
188C3⋊(C3⋊C8)—21no—
189C3×C3⋊C8—22no—
190C3×C3⋊C8—21no—
191C3×C3⋊C8—21no—
192C3×C3⋊C8—62no—
193C3×C3⋊C8—61no—
194C3×C3⋊C8—61no—
195#478—271noSylow
196C2×C33C22⋊Q811yescharacteristic
197C2×C33#13511yescharacteristic
198C2×C33AGL(1,ℤ/8)11yescharacteristic
199C3×C3⋊D3—42no—
200C2×C3×D4—91no—
201C22×Dic3—181no—
202C2×D12—31no—
203C2×D12—31no—
204C2×D12—61no—
205C2×C4×D3—181no—
206C2×C4×D3—181no—
207C2×C4×D3—182no—
208C2×C3×C8—31no—
209C2×C3×C8—31no—
210C2×C3×C8—61no—
211C2×C3×C8—91no—
212C3×C4⋊C4—31no—
213C3×C4⋊C4—31no—
214C3×C4⋊C4—61no—
215C3×C4⋊C4—91no—
216C3×C4⋊C4—181no—
217C3×C4⋊C4—181no—
218C3×C42—31no—
219C3×C42—31no—
220C3×C42—61no—
221#217—181no—
222(C2×D3)⋊C4—181no—
223(C2×D3)⋊C4—181no—
224(C2×D3)⋊C4—182no—
225C4⋊Dic3—31no—
226C4⋊Dic3—31no—
227C4⋊Dic3—61no—
228C4⋊Dic3—61no—
229C4⋊Dic3—91no—
230C4⋊Dic3—91no—
231C4⋊Dic3—91no—
232C4⋊Dic3—181no—
233C4⋊Dic3—181no—
234C4⋊Dic3—181no—
235C4⋊Dic3—361no—
236C4⋊Dic3D3211yescharacteristic
237Dic3⋊C4—61no—
238Dic3⋊C4—61no—
239Dic3⋊C4—61no—
240C4×Dic3D3211yescharacteristic
241C2×C3⋊C8—31no—
242C2×C3⋊C8—31no—
243C2×C3⋊C8—61no—
244C2×C3⋊C8D3211yescharacteristic
245C22×C32—21no—
246C22×C32—21no—
247C22×C32—21no—
248C22×C32—41no—
249C22×C32C2×D1211yescharacteristic
250C22×C32D3×D411yescharacteristic
251C22×C32C2×#6711yescharacteristic
252C2×C3⋊D3—22no—
253C2×C3⋊D3—41no—
254C2×C3⋊D3—41no—
255C2×C3×D3—64no—
256C2×C3×D3—62no—
257C2×C3×D3—62no—
258C2×C3×D3—122no—
259C2×C3×D3—122no—
260C2×C3×D3—121no—
261C2×C3×D3—121no—
262C2×C3×D3—121no—
263C2×C3×D3—121no—
264C32×C4—21no—
265C32×C4—21no—
266C32×C4—21no—
267C32×C4—21no—
268C32×C4—21no—
269C32×C4—21no—
270C32×C4—41no—
271C32×C4—41no—
272C32×C4—61no—
273C32×C4—61no—
274C32×C4—121no—
275C32×C4C2×Dic611yescharacteristic
276C32×C4#23711yescharacteristic
277C32×C4#23711yescharacteristic
278C32×C4D3×Q811yescharacteristic
279C32×C4D3×Q811yescharacteristic
280C32×C4#23911yescharacteristic
281C3⋊Dic3—41no—
282C3⋊Dic3—121no—
283C3⋊Dic3—122no—
284C3⋊Dic3—121no—
285C3⋊Dic3—121no—
286C3⋊Dic3—241no—
287C3⋊Dic3—241no—
288C3⋊Dic3—242no—
289C3×Dic3—22no—
290C3×Dic3—21no—
291C3×Dic3—21no—
292C3×Dic3—21no—
293C3×Dic3—21no—
294C3×Dic3—41no—
295C3×Dic3—41no—
296C3×Dic3—41no—
297C3×Dic3—81no—
298C3×Dic3—121no—
299C3×Dic3—121no—
300C3×Dic3—241no—
301#125—91no—
302C4×D4—271no—
303#107—271no—
304#106—271no—
305C4⋊C8—91no—
306#102—91no—
307#102—271no—
308C33#47811yescharacteristic, complemented, Sylow
309C23×C3—181no—
310C22×D3—62no—
311C22×D3—61no—
312C22×D3—61no—
313C3×D4—92no—
314C3×D4—181no—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—21no—
318C2×C3×C4—31no—
319C2×C3×C4—31no—
320C2×C3×C4—31no—
321C2×C3×C4—31no—
322C2×C3×C4—41no—
323C2×C3×C4—61no—
324C2×C3×C4—61no—
325C2×C3×C4—61no—
326C2×C3×C4—61no—
327C2×C3×C4—121no—
328C2×C3×C4—181no—
329C2×C3×C4C2×D3211yescharacteristic
330C2×C3×C4C2×D3211yescharacteristic
331C2×C3×C4C2×D3211yescharacteristic
332C2×Dic3—21no—
333C2×Dic3—61no—
334C2×Dic3—61no—
335C2×Dic3—121no—
336C2×Dic3—182no—
337C2×Dic3—181no—
338C2×Dic3—181no—
339C2×Dic3—181no—
340C2×Dic3—361no—
341C2×Dic3—361no—
342C2×Dic3—362no—
343C2×Dic3—362no—
344C2×Dic3#62611yescharacteristic
345C2×Dic3#62611yescharacteristic
346D12—32no—
347D12—32no—
348D12—62no—
349D12—62no—
350D12—61no—
351D12—61no—
352C4×D3—364no—
353C4×D3—362no—
354C4×D3—362no—
355C3×C8—62no—
356C3×C8—61no—
357C3×C8—61no—
358C3×C8—181no—
359C3⋊C8—21no—
360C3⋊C8—62no—
361C3⋊C8—61no—
362C3⋊C8—61no—
363C2×C32—21no—
364C2×C32—21no—
365C2×C32—21no—
366C2×C32—21no—
367C2×C32—21no—
368C2×C32—21no—
369C2×C32—21no—
370C2×C32—21no—
371C2×C32—21no—
372C2×C32—41no—
373C2×C32—41no—
374C2×C32—41no—
375C2×C32#87811yescharacteristic
376C2×C32#89711yescharacteristic
377C2×C32#90411yescharacteristic
378C2×C32#90811yescharacteristic
379C2×C32#91211yescharacteristic
380C2×C32#91411yescharacteristic
381C2×C32#94611yescharacteristic
382C2×C32#94911yescharacteristic
383C2×C32#95011yescharacteristic
384C3⋊D3—42no—
385C3×D3—124no—
386C3×D3—122no—
387C3×D3—122no—
388C2×D4—91no—
389C22×C4—541no—
390C2×C8—31no—
391C2×C8—91no—
392C4⋊C4—31no—
393C4⋊C4—91no—
394C4⋊C4—181no—
395C4⋊C4—181no—
396C4⋊C4—271no—
397C22⋊C4—541no—
398C42—31no—
399C22×C3—21no—
400C22×C3—21no—
401C22×C3—21no—
402C22×C3—41no—
403C22×C3—182no—
404C22×C3—361no—
405C22×C3—361no—
406C22×C3D3×D1211yescharacteristic
407C22×C3C2×#62611yescharacteristic
408C22×C3D3×#6711yescharacteristic
409C2×D3—64no—
410C2×D3—62no—
411C2×D3—62no—
412C2×D3—122no—
413C2×D3—122no—
414C2×D3—121no—
415C2×D3—121no—
416C2×D3—121no—
417C2×D3—121no—
418C3×C4—21no—
419C3×C4—21no—
420C3×C4—21no—
421C3×C4—21no—
422C3×C4—21no—
423C3×C4—21no—
424C3×C4—41no—
425C3×C4—41no—
426C3×C4—62no—
427C3×C4—61no—
428C3×C4—61no—
429C3×C4—61no—
430C3×C4—61no—
431C3×C4—121no—
432C3×C4—121no—
433C3×C4—121no—
434C3×C4—241no—
435C3×C4—361no—
436C3×C4D3×Dic611yescharacteristic
437C3×C4D3×Dic611yescharacteristic
438C3×C4#379611yescharacteristic
439C3×C4#379611yescharacteristic
440C3×C4#379711yescharacteristic
441C3×C4#380511yescharacteristic
442Dic3—21no—
443Dic3—21no—
444Dic3—41no—
445Dic3—121no—
446Dic3—121no—
447Dic3—241no—
448Dic3—362no—
449Dic3—361no—
450Dic3—361no—
451Dic3—361no—
452Dic3—721no—
453Dic3—721no—
454Dic3—722no—
455C32—21no—
456C32—21no—
457C32—21no—
458C32—41no—
459C32#476611yescharacteristic, complemented
460C32#484711yescharacteristic, complemented
461C32#506611yescharacteristic, complemented
462C23—181no—
463D4—92no—
464D4—181no—
465C2×C4—31no—
466C2×C4—31no—
467C2×C4—61no—
468C2×C4—181no—
469C2×C4—541no—
470C2×C4—1082no—
471C2×C4D3311yescharacteristic
472C8—61no—
473C8—181no—
474C2×C3—21no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—21no—
480C2×C3—21no—
481C2×C3—21no—
482C2×C3—21no—
483C2×C3—41no—
484C2×C3—41no—
485C2×C3—41no—
486C2×C3—362no—
487C2×C3#6379411yescharacteristic
488C2×C3#6380711yescharacteristic
489C2×C3#6382511yescharacteristic
490C2×C3#6382911yescharacteristic
491C2×C3#6387111yescharacteristic
492C2×C3#6388711yescharacteristic
493C2×C3#6389311yescharacteristic
494C2×C3#6393311yescharacteristic
495C2×C3#6393911yescharacteristic
496D3—124no—
497D3—122no—
498D3—122no—
499C22—182no—
500C22—361no—
501C22—361no—
502C22D3×#62611yescharacteristic
503C4—61no—
504C4—61no—
505C4—121no—
506C4—361no—
507C4—1081no—
508C4#8895711yescharacteristic
509C4#8896411yescharacteristic
510C3—21no—
511C3—21no—
512C3—21no—
513C3—41no—
514C3#1059030811yescharacteristic, complemented
515C3#1059046911yescharacteristic, complemented
516C3#1059076711yescharacteristic, complemented
517C2—362no—
518C2#1172285411yescharacteristic
519C2#1172287111yescharacteristic
520C2#1172297411yescharacteristic
subgroupG/H
centerC22D3×#626
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
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
4. #11722055C2semidirect
5. #11722054C2semidirect
6. C3×Dic3⋊(C3⋊C8)C2semidirect
28. #63869D3semidirect
41. Dic3⋊(C3⋊C8)D3semidirect
308. C33#478semidirect
459. C32#4766semidirect
460. C32#4847semidirect
461. C32#5066semidirect
514. C3#10590308semidirect
515. C3#10590469semidirect
516. C3#10590767semidirect

Maximal subgroups

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

subgroupindex
Dic3×C4⋊(C3⋊D3)2
C3×#638692
#117220752
#117220552
#117220542
C3×Dic3⋊(C3⋊C8)2
C3×#635642
#105907673
#105904693
#105903083