Finite Group Databasebeta

← #13156014

Subgroups — #13156014

Order 1728.

594 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
1C2×#88798C211yesmaximal, characteristic
2C3×#63866C211yesmaximal, characteristic
3C3×Dic3×Dic6C211yesmaximal, characteristic
4C2×C3×#3718C211yesmaximal, characteristic
5#11722076C211yesmaximal, characteristic
6Dic3⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic
7#11722037C211yesmaximal, characteristic
8#10591625—31nomaximal
9#10591574—31nomaximal
10#10590384—31nomaximal
11C2×C3×#634C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4×Dic3C2211yescharacteristic
18#88798—22no—
19C3×Dic3⋊Dic3—21no—
20C3×Dic3⋊Dic3—21no—
21C3×Dic3⋊Dic3—21no—
22C3×Dic32—21no—
23C3×#3718—22no—
24C2×#3756—31no—
25C3×Q8×Dic3—31no—
26C3×#944—31no—
27C2×C3×#216—31no—
28C2×C3×#216—31no—
29C3×#869—31no—
30C3×C4×Dic6—31no—
31#63866D311yescharacteristic
32Dic3×Dic6D311yescharacteristic
33C2×#3718—31no—
34C2×#3718—31no—
35C2×#3718—61no—
36C2×#3718D311yescharacteristic
37#63662—31no—
38C4⋊(C3⋊(C3⋊C8))—31no—
39Dic3⋊(C3⋊C8)—31no—
40Dic3⋊(C3⋊C8)—31no—
41Dic3⋊(C3⋊C8)—61no—
42#63567—31no—
43#63566—31no—
44#63565—31no—
45#63565—31no—
46#63565—31no—
47#63565—61no—
48C2×C33×C4C2311yescharacteristic
49C2×C3×C3⋊Dic3—21no—
50C2×C3×C3⋊Dic3—21no—
51C3×#634—21no—
52C3×#634—21no—
53C2×C32×Dic3—21no—
54C2×C32×Dic3—21no—
55C2×C32×Dic3D411yescharacteristic
56C2×C32×Dic3D411yescharacteristic
57C32×Dic6—21no—
58C32×Dic6—21no—
59C32×Dic6—21no—
60C32×Dic6D412yes—
61C3⋊(C3⋊(C3⋊C8))—21no—
62C3×C3⋊(C3⋊C8)—21no—
63#5182—91no—
64#5090—91no—
65#5031—91no—
66C2×C32×Q8—31no—
67C2×C32×Q8—31no—
68C2×#634C2×D311yescharacteristic
69C2×C3×Dic6—21no—
70C2×C3×Dic6—21no—
71C2×C3×Dic6—31no—
72C2×C3×Dic6—31no—
73C2×C3×Dic6—61no—
74C2×C3×Dic6C2×D311yescharacteristic
75C2×C3×Dic6C2×D311yescharacteristic
76C2×C3×Dic6C2×D311yescharacteristic
77C2×C3×Dic6C2×D311yescharacteristic
78C32×C42—31no—
79#3756—62no—
80C4⋊(C3⋊Dic3)C2×D311yescharacteristic
81C2×C3⋊(C3⋊C8)—31no—
82C2×C3⋊(C3⋊C8)—31no—
83C2×C3⋊(C3⋊C8)—31no—
84C2×C3⋊(C3⋊C8)—61no—
85C2×C3⋊(C3⋊C8)—61no—
86C2×C3⋊(C3⋊C8)—61no—
87C2×C3⋊(C3⋊C8)—121no—
88C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
89C3×#216—62no—
90C3×#216—62no—
91C3×C4⋊Dic3—31no—
92C3×C4⋊Dic3—31no—
93C3×C4⋊Dic3—61no—
94C3×C4⋊Dic3—61no—
95C3×C4⋊Dic3—61no—
96C3×C4⋊Dic3—61no—
97C3×Dic3⋊C4—61no—
98C3×Dic3⋊C4—61no—
99C3×Dic3⋊C4—61no—
100C3×C4×Dic3—21no—
101C3×C4×Dic3—61no—
102C3×C4×Dic3—61no—
103C3×C4×Dic3C2×D311yescharacteristic
104C3×C4×Dic3C2×D311yescharacteristic
105C2×C3×C3⋊C8—31no—
106C2×C3×C3⋊C8—31no—
107C2×C3×C3⋊C8—61no—
108Dic3⋊Dic3—21no—
109Dic3⋊Dic3—21no—
110Dic3⋊Dic3—21no—
111Dic32—21no—
112#3718—22no—
113#3718—64no—
114#3718—62no—
115#3718—62no—
116C22×C33C2×D411yescharacteristic
117C33×C4C2×D411yescharacteristic
118C33×C4#4111yescharacteristic
119C3×C3⋊Dic3—41no—
120C3×C3⋊Dic3—41no—
121C32×Dic3—22no—
122C32×Dic3—21no—
123C32×Dic3—41no—
124C32×Dic3Q1612yescomplemented
125C2×C3×Q16—91no—
126C3×C4⋊Q8—91no—
127C3×C4×Q8—91no—
128Q8×Dic3—31no—
129#944—31no—
130C2×#216—31no—
131C2×#216—31no—
132C2×#216—91no—
133C2×#216—91no—
134C2×#216—91no—
135C2×#216—181no—
136#869—31no—
137C4×Dic6—31no—
138#835—91no—
139#835—91no—
140#835—91no—
141#835—181no—
142#816—91no—
143Dic3⋊C8—91no—
144#810—91no—
145#810—91no—
146C4⋊(C3⋊C8)—91no—
147C4⋊(C3⋊C8)—91no—
148C4⋊(C3⋊C8)—181no—
149C32×Q8—32no—
150C32×Q8—61no—
151C32×Q8—61no—
152C32×Q8—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×C4—61no—
161C2×C32×C4C22×D311yescharacteristic
162C2×C32×C4C22×D311yescharacteristic
163C2×C32×C4C22×D311yescharacteristic
164C2×C3⋊Dic3—21no—
165C2×C3⋊Dic3—21no—
166#634—21no—
167#634—21no—
168C2×C3×Dic3—21no—
169C2×C3×Dic3—21no—
170C2×C3×Dic3—21no—
171C2×C3×Dic3—21no—
172C2×C3×Dic3—21no—
173C2×C3×Dic3—21no—
174C2×C3×Dic3—41no—
175C2×C3×Dic3—41no—
176C2×C3×Dic3—61no—
177C2×C3×Dic3—61no—
178C2×C3×Dic3—61no—
179C2×C3×Dic3—61no—
180C2×C3×Dic3—121no—
181C2×C3×Dic3—121no—
182C2×C3×Dic3#6711yescharacteristic
183C2×C3×Dic3#6711yescharacteristic
184C2×C3×Dic3#6711yescharacteristic
185C2×C3×Dic3#6711yescharacteristic
186C3×Dic6—21no—
187C3×Dic6—21no—
188C3×Dic6—22no—
189C3×Dic6—21no—
190C3×Dic6—21no—
191C3×Dic6—21no—
192C3×Dic6—21no—
193C3×Dic6—22no—
194C3×Dic6—41no—
195C3×Dic6—41no—
196C3×Dic6—62no—
197C3×Dic6—61no—
198C3×Dic6—61no—
199C3×Dic6—61no—
200C3×Dic6—61no—
201C3×Dic6—121no—
202C3×Dic6D1212yes—
203C3×Dic6#6712yes—
204C3⋊(C3⋊C8)—21no—
205C3⋊(C3⋊C8)—61no—
206C3⋊(C3⋊C8)—62no—
207C3⋊(C3⋊C8)—61no—
208C3⋊(C3⋊C8)—61no—
209C3⋊(C3⋊C8)—121no—
210C3⋊(C3⋊C8)—121no—
211C3⋊(C3⋊C8)—122no—
212C3×C3⋊C8—61no—
213C3×C3⋊C8—61no—
214C3×C3⋊C8—121no—
215#462—271noSylow
216C2×C33#12111yescharacteristic
217C2×C33C2×Q1611yescharacteristic
218C2×C33#13711yescharacteristic
219C2×C3×Q8—31no—
220C2×C3×Q8—31no—
221C2×C3×Q8—31no—
222C2×C3×Q8—31no—
223C2×C3×Q8—61no—
224C2×C3×Q8—61no—
225C2×C3×Q8—91no—
226C2×Dic6—31no—
227C2×Dic6—31no—
228C2×Dic6—61no—
229C2×Dic6D3211yescharacteristic
230C2×Dic6D3211yescharacteristic
231C3×Q16—182no—
232C2×C3×C8—91no—
233C3×C4⋊C4—91no—
234C3×C4⋊C4—181no—
235C3×C4⋊C4—181no—
236C3×C4⋊C4—181no—
237C3×C42—31no—
238C3×C42—31no—
239C3×C42—61no—
240C3×C42—181no—
241#216—62no—
242#216—62no—
243#216—182no—
244#216—182no—
245#216—184no—
246#216—182no—
247C4⋊Dic3—31no—
248C4⋊Dic3—31no—
249C4⋊Dic3—61no—
250C4⋊Dic3—61no—
251C4⋊Dic3—61no—
252C4⋊Dic3—61no—
253Dic3⋊C4—61no—
254Dic3⋊C4—61no—
255Dic3⋊C4—61no—
256C4×Dic3—61no—
257C4×Dic3—61no—
258C4×Dic3D3211yescharacteristic
259C2×C3⋊C8—31no—
260C2×C3⋊C8—31no—
261C2×C3⋊C8—61no—
262C2×C3⋊C8—91no—
263C2×C3⋊C8—91no—
264C2×C3⋊C8—91no—
265C2×C3⋊C8—181no—
266C2×C3⋊C8—181no—
267C2×C3⋊C8—181no—
268C2×C3⋊C8—361no—
269C22×C32—21no—
270C22×C32—21no—
271C22×C32—21no—
272C22×C32—41no—
273C22×C32D3×D411yescharacteristic
274C22×C32C2×#6711yescharacteristic
275C22×C32C2×#6711yescharacteristic
276C32×C4—21no—
277C32×C4—21no—
278C32×C4—21no—
279C32×C4—21no—
280C32×C4—21no—
281C32×C4—21no—
282C32×C4—32no—
283C32×C4—41no—
284C32×C4—41no—
285C32×C4—61no—
286C32×C4—62no—
287C32×C4—121no—
288C32×C4C2×D1211yescharacteristic
289C32×C4#23511yescharacteristic
290C32×C4D3×D411yescharacteristic
291C32×C4#23711yescharacteristic
292C32×C4#23711yescharacteristic
293C32×C4C2×#6711yescharacteristic
294C3⋊Dic3—41no—
295C3⋊Dic3—41no—
296C3×Dic3—22no—
297C3×Dic3—22no—
298C3×Dic3—21no—
299C3×Dic3—21no—
300C3×Dic3—22no—
301C3×Dic3—42no—
302C3×Dic3—41no—
303C3×Dic3—42no—
304C3×Dic3—41no—
305C3×Dic3—41no—
306C3×Dic3—122no—
307C3×Dic3—121no—
308C3×Dic3—121no—
309C3×Dic3—121no—
310C3×Dic3—121no—
311C3×Dic3—241no—
312C3×Dic3#21612yescomplemented
313C3×Dic3#21612yescomplemented
314C2×Q16—91no—
315C2×Q16—271no—
316C4⋊Q8—91no—
317C4×Q8—91no—
318C4⋊C8—271no—
319#103—271no—
320#103—271no—
321C33#46211yescharacteristic, complemented, Sylow
322C3×Q8—32no—
323C3×Q8—32no—
324C3×Q8—62no—
325C3×Q8—62no—
326C3×Q8—61no—
327C3×Q8—61no—
328C3×Q8—61no—
329C3×Q8—61no—
330C3×Q8—61no—
331C3×Q8—61no—
332C3×Q8—121no—
333C3×Q8—121no—
334C3×Q8—181no—
335C3×Q8—181no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—31no—
340C2×C3×C4—31no—
341C2×C3×C4—31no—
342C2×C3×C4—31no—
343C2×C3×C4—41no—
344C2×C3×C4—61no—
345C2×C3×C4—61no—
346C2×C3×C4—61no—
347C2×C3×C4—61no—
348C2×C3×C4—61no—
349C2×C3×C4—61no—
350C2×C3×C4—121no—
351C2×C3×C4—121no—
352C2×C3×C4—181no—
353C2×C3×C4—181no—
354C2×C3×C4C2×D3211yescharacteristic
355C2×C3×C4C2×D3211yescharacteristic
356C2×C3×C4C2×D3211yescharacteristic
357C2×Dic3—21no—
358C2×Dic3—21no—
359C2×Dic3—61no—
360C2×Dic3—61no—
361C2×Dic3—61no—
362C2×Dic3—61no—
363C2×Dic3—121no—
364C2×Dic3—121no—
365C2×Dic3#62511yescharacteristic
366C2×Dic3#62511yescharacteristic
367Dic6—21no—
368Dic6—21no—
369Dic6—21no—
370Dic6—62no—
371Dic6—61no—
372Dic6—61no—
373Dic6—61no—
374Dic6—61no—
375Dic6—121no—
376Dic6#62612yes—
377C3×C8—181no—
378C3⋊C8—61no—
379C3⋊C8—61no—
380C3⋊C8—121no—
381C3⋊C8—182no—
382C3⋊C8—181no—
383C3⋊C8—181no—
384C3⋊C8—181no—
385C3⋊C8—361no—
386C3⋊C8—361no—
387C3⋊C8—362no—
388C2×C32—21no—
389C2×C32—21no—
390C2×C32—21no—
391C2×C32—21no—
392C2×C32—21no—
393C2×C32—21no—
394C2×C32—21no—
395C2×C32—21no—
396C2×C32—21no—
397C2×C32—41no—
398C2×C32—41no—
399C2×C32—41no—
400C2×C32D3×Q1611yescharacteristic
401C2×C32#91811yescharacteristic
402C2×C32#93011yescharacteristic
403C2×C32#93811yescharacteristic
404C2×C32#93911yescharacteristic
405C2×C32#94211yescharacteristic
406C2×C32C2×#21611yescharacteristic
407C2×C32C2×#21611yescharacteristic
408C2×C32#95111yescharacteristic
409C2×Q8—31no—
410C2×Q8—31no—
411C2×Q8—91no—
412Q16—182no—
413Q16—542no—
414C2×C8—91no—
415C2×C8—271no—
416C4⋊C4—91no—
417C4⋊C4—181no—
418C4⋊C4—181no—
419C4⋊C4—181no—
420C42—31no—
421C42—181no—
422C22×C3—21no—
423C22×C3—21no—
424C22×C3—21no—
425C22×C3—41no—
426C22×C3C2×#62511yescharacteristic
427C22×C3D3×#6711yescharacteristic
428C22×C3D3×#6711yescharacteristic
429C3×C4—21no—
430C3×C4—21no—
431C3×C4—21no—
432C3×C4—21no—
433C3×C4—21no—
434C3×C4—21no—
435C3×C4—32no—
436C3×C4—32no—
437C3×C4—41no—
438C3×C4—41no—
439C3×C4—62no—
440C3×C4—62no—
441C3×C4—61no—
442C3×C4—61no—
443C3×C4—62no—
444C3×C4—121no—
445C3×C4—122no—
446C3×C4—122no—
447C3×C4—121no—
448C3×C4—121no—
449C3×C4—361no—
450C3×C4—361no—
451C3×C4#379511yescharacteristic
452C3×C4D3×D1211yescharacteristic
453C3×C4#380411yescharacteristic
454C3×C4#380511yescharacteristic
455C3×C4C2×#62611yescharacteristic
456C3×C4D3×#6711yescharacteristic
457Dic3—21no—
458Dic3—22no—
459Dic3—41no—
460Dic3—122no—
461Dic3—121no—
462Dic3—121no—
463Dic3—121no—
464Dic3—121no—
465Dic3—241no—
466Dic3#371812yescomplemented
467C32—21no—
468C32—21no—
469C32—21no—
470C32—41no—
471C32#503111yescharacteristic, complemented
472C32#509011yescharacteristic, complemented
473C32#518211yescharacteristic, complemented
474Q8—32no—
475Q8—61no—
476Q8—61no—
477Q8—61no—
478Q8—181no—
479Q8—181no—
480C2×C4—31no—
481C2×C4—31no—
482C2×C4—61no—
483C2×C4—61no—
484C2×C4—181no—
485C2×C4—181no—
486C2×C4D3311yescharacteristic
487C8—181no—
488C8—541no—
489C2×C3—21no—
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—41no—
499C2×C3—41no—
500C2×C3—41no—
501C2×C3#6382711yescharacteristic
502C2×C3C2×#371811yescharacteristic
503C2×C3#6390811yescharacteristic
504C2×C3D3×#21611yescharacteristic
505C2×C3D3×#21611yescharacteristic
506C2×C3#6394311yescharacteristic
507C2×C3#6394811yescharacteristic
508C2×C3#6397011yescharacteristic
509C2×C3#6397811yescharacteristic
510C22D3×#62511yescharacteristic
511C4—32no—
512C4—61no—
513C4—62no—
514C4—121no—
515C4—361no—
516C4—361no—
517C4D3×#62611yescharacteristic
518C4#8896111yescharacteristic
519C3—21no—
520C3—21no—
521C3—21no—
522C3—41no—
523C3#1059038411yescharacteristic, complemented
524C3#1059157411yescharacteristic, complemented
525C3#1059162511yescharacteristic, complemented
526C2D3×#371811yescharacteristic
527C2#1172288311yescharacteristic
528C2#1172295611yescharacteristic
subgroupG/H
centerC22D3×#625
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socle, prime radicalC22×C33C2×D4
hypocenterC33#462
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
124. C32×Dic3Q16semidirect
312. C3×Dic3#216semidirect
313. C3×Dic3#216semidirect
321. C33#462semidirect
466. Dic3#3718semidirect
471. C32#5031semidirect
472. C32#5090semidirect
473. C32#5182semidirect
523. C3#10590384semidirect
524. C3#10591574semidirect
525. C3#10591625semidirect

Maximal subgroups

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

subgroupindex
C2×#887982
C3×#638662
C3×Dic3×Dic62
C2×C3×#37182
#117220762
Dic3⋊(C3⋊(C3⋊C8))2
#117220372
#105916253
#105915743
#105903843