Finite Group Databasebeta

← #13156021

Subgroups — #13156021

Order 1728.

612 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
1C2×#88812C211yesmaximal, characteristic
2Dic3×#634C211yesmaximal, characteristic
3C3×#63866C211yesmaximal, characteristic
4C2×C3×#3719C211yesmaximal, characteristic
5#11722040C211yesmaximal, characteristic
6#11722038C211yesmaximal, characteristic
7C3×Dic3⋊(C3⋊C8)C211yesmaximal, characteristic
8#10591574—31nomaximal
9#10590800—31nomaximal
10#10590388—31nomaximal
11C2×C3×#634C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×C4×Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88812—22no—
19#88807—21no—
20Dic3×C3⋊Dic3—21no—
21C3×Dic3⋊Dic3—21no—
22C3×Dic3⋊Dic3—21no—
23C3×#3719—22no—
24C4×#634—31no—
25C3×#944—31no—
26C2×C3×#216—31no—
27C2×C3×Dic12—31no—
28C3×#869—31no—
29#63866D311yescharacteristic
30Dic3×Dic6—31no—
31Dic3×Dic6—31no—
32Dic3×Dic6—61no—
33C2×#3719—31no—
34C2×#3719—31no—
35C2×#3719D311yescharacteristic
36C2×#3718—31no—
37#63662—31no—
38#63643—31no—
39C3×Dic3⋊C8—31no—
40C3×C4⋊(C3⋊C8)—31no—
41Dic3⋊(C3⋊C8)D311yescharacteristic
42#63566—31no—
43#63566—31no—
44#63566—31no—
45#63566—31no—
46#63566—61no—
47#63566—61no—
48C2×C33×C4C2311yescharacteristic
49C2×C3⋊(C3⋊Dic3)—21no—
50C2×C3×C3⋊Dic3—21no—
51C2×C3×C3⋊Dic3—21no—
52C3×#634—21no—
53C3×#634—21no—
54C3×#634—21no—
55C3×#634D412yes—
56C2×C32×Dic3—21no—
57C2×C32×Dic3D411yescharacteristic
58C2×C32×Dic3D411yescharacteristic
59C32×Dic6—21no—
60C32×Dic6—21no—
61C3×C3⋊(C3⋊C8)—21no—
62C32×C3⋊C8—21no—
63#5182—91no—
64#5090—91no—
65#4849—91no—
66C2×C32×Q8—31no—
67C2×#634C2×D311yescharacteristic
68C2×#634C2×D311yescharacteristic
69C2×C3×Dic6—21no—
70C2×C3×Dic6—31no—
71C2×C3×Dic6—31no—
72C2×C3×Dic6—31no—
73C2×C3×Dic6—31no—
74C2×C3×Dic6—61no—
75C2×C3×Dic6—61no—
76C2×C3×Dic6C2×D311yescharacteristic
77C2×C3×Dic6C2×D311yescharacteristic
78C2×C32×C8—31no—
79C32×C42—31no—
80C4⋊(C3⋊Dic3)—31no—
81C4⋊(C3⋊Dic3)—31no—
82C4⋊(C3⋊Dic3)—31no—
83C4⋊(C3⋊Dic3)—61no—
84C4⋊(C3⋊Dic3)—61no—
85C4⋊(C3⋊Dic3)—61no—
86C4⋊(C3⋊Dic3)—121no—
87#3750—61no—
88C4×C3⋊Dic3—61no—
89C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
90C3×#216—62no—
91C3×C4⋊Dic3—61no—
92C3×C4⋊Dic3—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—21no—
99C2×C3×C3⋊C8—31no—
100C2×C3×C3⋊C8—31no—
101C2×C3×C3⋊C8—61no—
102C2×C3×C3⋊C8C2×D311yescharacteristic
103C2×C3×C3⋊C8C2×D311yescharacteristic
104C3×Dic12—62no—
105Dic3⋊Dic3—21no—
106Dic3⋊Dic3—21no—
107Dic3⋊Dic3—62no—
108Dic3⋊Dic3—61no—
109Dic3⋊Dic3—61no—
110Dic32—62no—
111Dic32—61no—
112Dic32—61no—
113#3719—22no—
114#3719—62no—
115#3719—62no—
116#3718—62no—
117C22×C33C2×D411yescharacteristic
118C33×C4C2×D411yescharacteristic
119C33×C4#4111yescharacteristic
120C3⋊(C3⋊Dic3)—41no—
121C3×C3⋊Dic3—22no—
122C3×C3⋊Dic3—41no—
123C32×Dic3—21no—
124C32×Dic3—41no—
125C32×Dic3Q1612yescomplemented
126C2×C3×Q16—91no—
127C3×C4⋊Q8—91no—
128Q8×Dic3—91no—
129#944—31no—
130C2×#216—31no—
131C2×#216—91no—
132C2×#216—91no—
133C2×Dic12—31no—
134C2×Dic12—91no—
135#869—31no—
136C4×Dic6—91no—
137C4×Dic6—91no—
138C4×Dic6—181no—
139C3×C4⋊C8—91no—
140#835—91no—
141#835—91no—
142#835—91no—
143#835—181no—
144#816—91no—
145#816—91no—
146#816—91no—
147#816—181no—
148Dic3⋊C8—31no—
149C4⋊(C3⋊C8)—31no—
150C32×Q8—61no—
151C32×Q8—61no—
152C2×C32×C4—21no—
153C2×C32×C4—21no—
154C2×C32×C4—21no—
155C2×C32×C4—31no—
156C2×C32×C4—31no—
157C2×C32×C4—41no—
158C2×C32×C4—61no—
159C2×C32×C4C22×D311yescharacteristic
160C2×C32×C4C22×D311yescharacteristic
161C2×C32×C4C22×D311yescharacteristic
162C2×C3⋊Dic3—21no—
163C2×C3⋊Dic3—21no—
164C2×C3⋊Dic3—61no—
165C2×C3⋊Dic3—62no—
166C2×C3⋊Dic3—61no—
167C2×C3⋊Dic3—61no—
168C2×C3⋊Dic3—121no—
169C2×C3⋊Dic3—121no—
170C2×C3⋊Dic3—122no—
171#634—21no—
172#634—21no—
173#634—21no—
174#634#6712yes—
175C2×C3×Dic3—21no—
176C2×C3×Dic3—21no—
177C2×C3×Dic3—21no—
178C2×C3×Dic3—21no—
179C2×C3×Dic3—41no—
180C2×C3×Dic3—62no—
181C2×C3×Dic3—61no—
182C2×C3×Dic3—61no—
183C2×C3×Dic3—61no—
184C2×C3×Dic3—61no—
185C2×C3×Dic3—121no—
186C2×C3×Dic3D1211yescharacteristic
187C2×C3×Dic3D1211yescharacteristic
188C2×C3×Dic3#6711yescharacteristic
189C2×C3×Dic3#6711yescharacteristic
190C3×Dic6—22no—
191C3×Dic6—21no—
192C3×Dic6—21no—
193C3×Dic6—21no—
194C3×Dic6—21no—
195C3×Dic6—32no—
196C3×Dic6—32no—
197C3×Dic6—41no—
198C3×Dic6—62no—
199C3×Dic6—62no—
200C3×Dic6—61no—
201C3×Dic6—61no—
202C3×Dic6—62no—
203C3×Dic6—61no—
204C3×Dic6—61no—
205C3×Dic6—61no—
206C3×Dic6—61no—
207C3×Dic6—121no—
208C32×C8—61no—
209C3⋊(C3⋊C8)—21no—
210C3×C3⋊C8—21no—
211C3×C3⋊C8—21no—
212C3×C3⋊C8—41no—
213C3×C3⋊C8—61no—
214C3×C3⋊C8—61no—
215C3×C3⋊C8—121no—
216#462—271noSylow
217C2×C33#12111yescharacteristic
218C2×C33C2×Q1611yescharacteristic
219C2×C33#13711yescharacteristic
220C2×C3×Q8—31no—
221C2×C3×Q8—31no—
222C2×C3×Q8—61no—
223C2×C3×Q8—91no—
224C2×C3×Q8—91no—
225C2×Dic6—31no—
226C2×Dic6—31no—
227C2×Dic6—31no—
228C2×Dic6—31no—
229C2×Dic6—61no—
230C2×Dic6—61no—
231C2×Dic6D3211yescharacteristic
232C3×Q16—182no—
233C2×C3×C8—31no—
234C2×C3×C8—31no—
235C2×C3×C8—61no—
236C2×C3×C8—91no—
237C3×C4⋊C4—181no—
238C3×C4⋊C4—181no—
239C3×C42—31no—
240C3×C42—31no—
241C3×C42—61no—
242#216—62no—
243#216—182no—
244#216—182no—
245C4⋊Dic3—61no—
246C4⋊Dic3—61no—
247C4⋊Dic3—91no—
248C4⋊Dic3—91no—
249C4⋊Dic3—91no—
250C4⋊Dic3—181no—
251C4⋊Dic3—181no—
252C4⋊Dic3—181no—
253C4⋊Dic3—181no—
254C4⋊Dic3—361no—
255Dic3⋊C4—61no—
256Dic3⋊C4—61no—
257Dic3⋊C4—181no—
258Dic3⋊C4—181no—
259Dic3⋊C4—182no—
260C4×Dic3—181no—
261C4×Dic3—181no—
262C4×Dic3—182no—
263C4×Dic3—181no—
264C4×Dic3D3211yescharacteristic
265C2×C3⋊C8—31no—
266C2×C3⋊C8—31no—
267C2×C3⋊C8—61no—
268C2×C3⋊C8D3211yescharacteristic
269Dic12—62no—
270Dic12—182no—
271C22×C32—21no—
272C22×C32—21no—
273C22×C32—21no—
274C22×C32—41no—
275C22×C32C2×D1211yescharacteristic
276C22×C32D3×D411yescharacteristic
277C22×C32C2×#6711yescharacteristic
278C32×C4—21no—
279C32×C4—21no—
280C32×C4—21no—
281C32×C4—21no—
282C32×C4—21no—
283C32×C4—21no—
284C32×C4—32no—
285C32×C4—41no—
286C32×C4—41no—
287C32×C4—61no—
288C32×C4—121no—
289C32×C4D3×D411yescharacteristic
290C32×C4D3×D411yescharacteristic
291C32×C4#23711yescharacteristic
292C32×C4#23711yescharacteristic
293C32×C4#23911yescharacteristic
294C32×C4C2×#6711yescharacteristic
295C3⋊Dic3—22no—
296C3⋊Dic3—41no—
297C3⋊Dic3—121no—
298C3⋊Dic3—122no—
299C3⋊Dic3—121no—
300C3⋊Dic3—121no—
301C3⋊Dic3—241no—
302C3⋊Dic3—241no—
303C3⋊Dic3—242no—
304C3×Dic3—21no—
305C3×Dic3—21no—
306C3×Dic3—22no—
307C3×Dic3—41no—
308C3×Dic3—42no—
309C3×Dic3—41no—
310C3×Dic3—41no—
311C3×Dic3—64no—
312C3×Dic3—62no—
313C3×Dic3—62no—
314C3×Dic3—122no—
315C3×Dic3—121no—
316C3×Dic3—121no—
317C3×Dic3Dic1212yescomplemented
318C3×Dic3#21612yescomplemented
319C2×Q16—91no—
320C2×Q16—271no—
321C4⋊Q8—91no—
322C4×Q8—271no—
323C4⋊C8—91no—
324#103—271no—
325#103—271no—
326C33#46211yescharacteristic, complemented, Sylow
327C3×Q8—62no—
328C3×Q8—61no—
329C3×Q8—61no—
330C3×Q8—61no—
331C3×Q8—61no—
332C3×Q8—92no—
333C3×Q8—121no—
334C3×Q8—181no—
335C3×Q8—181no—
336C3×Q8—181no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—31no—
341C2×C3×C4—31no—
342C2×C3×C4—31no—
343C2×C3×C4—31no—
344C2×C3×C4—41no—
345C2×C3×C4—61no—
346C2×C3×C4—61no—
347C2×C3×C4—61no—
348C2×C3×C4—61no—
349C2×C3×C4—121no—
350C2×C3×C4—181no—
351C2×C3×C4—181no—
352C2×C3×C4C2×D3211yescharacteristic
353C2×C3×C4C2×D3211yescharacteristic
354C2×C3×C4C2×D3211yescharacteristic
355C2×Dic3—21no—
356C2×Dic3—62no—
357C2×Dic3—61no—
358C2×Dic3—61no—
359C2×Dic3—61no—
360C2×Dic3—61no—
361C2×Dic3—121no—
362C2×Dic3—182no—
363C2×Dic3—181no—
364C2×Dic3—181no—
365C2×Dic3—181no—
366C2×Dic3—361no—
367C2×Dic3—361no—
368C2×Dic3—362no—
369C2×Dic3#62611yescharacteristic
370C2×Dic3#62611yescharacteristic
371Dic6—21no—
372Dic6—21no—
373Dic6—32no—
374Dic6—32no—
375Dic6—62no—
376Dic6—62no—
377Dic6—61no—
378Dic6—61no—
379Dic6—62no—
380Dic6—61no—
381Dic6—61no—
382Dic6—61no—
383Dic6—61no—
384Dic6—121no—
385C3×C8—61no—
386C3×C8—61no—
387C3×C8—121no—
388C3×C8—181no—
389C3⋊C8—21no—
390C3⋊C8—61no—
391C3⋊C8—61no—
392C3⋊C8—121no—
393C2×C32—21no—
394C2×C32—21no—
395C2×C32—21no—
396C2×C32—21no—
397C2×C32—21no—
398C2×C32—21no—
399C2×C32—21no—
400C2×C32—21no—
401C2×C32—21no—
402C2×C32—41no—
403C2×C32—41no—
404C2×C32—41no—
405C2×C32#89511yescharacteristic
406C2×C32C2×Dic1211yescharacteristic
407C2×C32#90911yescharacteristic
408C2×C32D3×Q1611yescharacteristic
409C2×C32#91811yescharacteristic
410C2×C32#93811yescharacteristic
411C2×C32#93911yescharacteristic
412C2×C32#94211yescharacteristic
413C2×C32C2×#21611yescharacteristic
414C2×Q8—31no—
415C2×Q8—91no—
416C2×Q8—91no—
417Q16—182no—
418Q16—542no—
419C2×C8—31no—
420C2×C8—91no—
421C4⋊C4—181no—
422C4⋊C4—181no—
423C4⋊C4—271no—
424C4⋊C4—541no—
425C42—31no—
426C42—541no—
427C22×C3—21no—
428C22×C3—21no—
429C22×C3—21no—
430C22×C3—41no—
431C22×C3D3×D1211yescharacteristic
432C22×C3C2×#62611yescharacteristic
433C22×C3D3×#6711yescharacteristic
434C3×C4—21no—
435C3×C4—21no—
436C3×C4—21no—
437C3×C4—21no—
438C3×C4—21no—
439C3×C4—21no—
440C3×C4—32no—
441C3×C4—32no—
442C3×C4—41no—
443C3×C4—41no—
444C3×C4—62no—
445C3×C4—61no—
446C3×C4—61no—
447C3×C4—121no—
448C3×C4—122no—
449C3×C4—121no—
450C3×C4—121no—
451C3×C4—182no—
452C3×C4—361no—
453C3×C4#379611yescharacteristic
454C3×C4#379611yescharacteristic
455C3×C4#380211yescharacteristic
456C3×C4#380511yescharacteristic
457C3×C4D3×#6711yescharacteristic
458C3×C4D3×#6711yescharacteristic
459Dic3—21no—
460Dic3—41no—
461Dic3—64no—
462Dic3—62no—
463Dic3—62no—
464Dic3—122no—
465Dic3—121no—
466Dic3—121no—
467Dic3—362no—
468Dic3—361no—
469Dic3—361no—
470Dic3—361no—
471Dic3—721no—
472Dic3—721no—
473Dic3—722no—
474Dic3#371912yescomplemented
475C32—21no—
476C32—21no—
477C32—21no—
478C32—41no—
479C32#484911yescharacteristic, complemented
480C32#509011yescharacteristic, complemented
481C32#518211yescharacteristic, complemented
482Q8—61no—
483Q8—61no—
484Q8—92no—
485Q8—181no—
486Q8—181no—
487Q8—181no—
488C2×C4—31no—
489C2×C4—31no—
490C2×C4—61no—
491C2×C4—181no—
492C2×C4—181no—
493C2×C4—541no—
494C2×C4D3311yescharacteristic
495C8—61no—
496C8—181no—
497C2×C3—21no—
498C2×C3—21no—
499C2×C3—21no—
500C2×C3—21no—
501C2×C3—21no—
502C2×C3—21no—
503C2×C3—21no—
504C2×C3—21no—
505C2×C3—21no—
506C2×C3—41no—
507C2×C3—41no—
508C2×C3—41no—
509C2×C3D3×Dic1211yescharacteristic
510C2×C3#6380111yescharacteristic
511C2×C3#6383311yescharacteristic
512C2×C3C2×#371911yescharacteristic
513C2×C3#6388611yescharacteristic
514C2×C3#6391611yescharacteristic
515C2×C3D3×#21611yescharacteristic
516C2×C3#6394811yescharacteristic
517C2×C3#6397011yescharacteristic
518C22D3×#62611yescharacteristic
519C4—32no—
520C4—61no—
521C4—121no—
522C4—182no—
523C4—361no—
524C4—1081no—
525C4#8895311yescharacteristic
526C4#8896411yescharacteristic
527C3—21no—
528C3—21no—
529C3—21no—
530C3—41no—
531C3#1059038811yescharacteristic, complemented
532C3#1059080011yescharacteristic, complemented
533C3#1059157411yescharacteristic, complemented
534C2D3×#371911yescharacteristic
535C2#1172288511yescharacteristic
536C2#1172296211yescharacteristic
subgroupG/H
centerC22D3×#626
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
125. C32×Dic3Q16semidirect
317. C3×Dic3Dic12semidirect
318. C3×Dic3#216semidirect
326. C33#462semidirect
474. Dic3#3719semidirect
479. C32#4849semidirect
480. C32#5090semidirect
481. C32#5182semidirect
531. C3#10590388semidirect
532. C3#10590800semidirect
533. C3#10591574semidirect

Maximal subgroups

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

subgroupindex
C2×#888122
Dic3×#6342
C3×#638662
C2×C3×#37192
#117220402
#117220382
C3×Dic3⋊(C3⋊C8)2
#105915743
#105908003
#105903883