Finite Group Databasebeta

← #13155662

Subgroups — #13155662

Order 1728.

614 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
1C2×#88812C211yesmaximal, characteristic
2Dic3×#634C211yesmaximal, characteristic
3C3×#63844C211yesmaximal, characteristic
4#11722057C211yesmaximal, characteristic
5#11722040C211yesmaximal, characteristic
6C3×#63567C211yesmaximal, characteristic
7C3×Dic3×C3⋊C8C211yesmaximal, characteristic
8#10590793—31nomaximal
9#10590476—31nomaximal
10C4×#3719—31nomaximal
11C2×C3×#634C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
14C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
15C32×C4⋊Dic3C2211yescharacteristic
16C32×C4×Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88812C414yescomplemented
19#88807—21no—
20Dic3×C3⋊Dic3—21no—
21C3×#3724—21no—
22C3×Dic32—21no—
23C4×#634—31no—
24C3×#887—31no—
25C3×C4×Dic6—31no—
26#63844D311yescharacteristic
27Dic3×Dic6—31no—
28Dic3×Dic6—31no—
29Dic3×Dic6—61no—
30C2×#3719—31no—
31C2×#3719—31no—
32C2×#3718—31no—
33#63643—31no—
34#63634—31no—
35C3×#816—31no—
36C3×C8×Dic3—31no—
37C3×#810—31no—
38C3×C4×C3⋊C8—31no—
39#63575—31no—
40#63575—31no—
41#63575—61no—
42#63567D311yescharacteristic
43#63566—31no—
44#63566—31no—
45#63566—61no—
46Dic3×C3⋊C8D311yescharacteristic
47C2×C33×C4C2311yescharacteristic
48C2×C3⋊(C3⋊Dic3)—21no—
49C2×C3×C3⋊Dic3—21no—
50C2×C3×C3⋊Dic3—21no—
51C3×#634—21no—
52C3×#634—21no—
53C3×#634C2×C412yes—
54C3×#634C2×C412yes—
55C2×C32×Dic3—21no—
56C2×C32×Dic3D411yescharacteristic
57C2×C32×Dic3D411yescharacteristic
58C3×C3⋊(C3⋊C8)C2×C412yes—
59C32×C3⋊C8—21no—
60C4×#216—91no—
61#4790—91no—
62C4×Dic12—91no—
63C2×#634C2×D311yescharacteristic
64C2×#634C2×D311yescharacteristic
65C2×C3×Dic6—31no—
66C2×C3×Dic6—31no—
67C2×C3×Dic6—31no—
68C2×C3×Dic6—31no—
69C2×C3×Dic6—61no—
70C2×C3×Dic6—61no—
71C2×C32×C8—31no—
72C32×C4⋊C4—31no—
73C32×C42—31no—
74C4⋊(C3⋊Dic3)—31no—
75C4⋊(C3⋊Dic3)—31no—
76C4⋊(C3⋊Dic3)—31no—
77C4⋊(C3⋊Dic3)—61no—
78C4⋊(C3⋊Dic3)—61no—
79C4⋊(C3⋊Dic3)—61no—
80C4⋊(C3⋊Dic3)—121no—
81#3750—61no—
82C4×C3⋊Dic3—61no—
83C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
84C3×C4⋊Dic3—21no—
85C3×C4⋊Dic3C2×D311yescharacteristic
86C3×C4⋊Dic3C2×D311yescharacteristic
87C3×Dic3⋊C4—61no—
88C3×Dic3⋊C4—61no—
89C3×C4×Dic3—21no—
90C3×C4×Dic3—61no—
91C3×C4×Dic3—61no—
92C3×C4×Dic3C2×D311yescharacteristic
93C3×C4×Dic3C2×D311yescharacteristic
94C2×C3×C3⋊C8—21no—
95C2×C3×C3⋊C8—31no—
96C2×C3×C3⋊C8—31no—
97C2×C3×C3⋊C8—61no—
98C2×C3×C3⋊C8C2×D311yescharacteristic
99C2×C3×C3⋊C8C2×D311yescharacteristic
100#3724—21no—
101Dic3⋊Dic3—62no—
102Dic3⋊Dic3—61no—
103Dic3⋊Dic3—61no—
104Dic32—21no—
105Dic32—62no—
106Dic32—61no—
107Dic32—61no—
108#3719—34no—
109#3719—34no—
110#3718—34no—
111C22×C33C2×D411yescharacteristic
112C33×C4C22×C411yescharacteristic
113C33×C4#4111yescharacteristic
114C3⋊(C3⋊Dic3)—41no—
115C3×C3⋊Dic3—22no—
116C3×C3⋊Dic3—22no—
117C32×Dic3—21no—
118C32×Dic3—41no—
119C32×Dic3Q1612yescomplemented
120C3×C4×Q8—91no—
121Q8×Dic3—91no—
122C2×#216—91no—
123C2×#216—91no—
124C2×Dic12—91no—
125#887—31no—
126C4×Dic6—31no—
127C4×Dic6—91no—
128C4×Dic6—91no—
129C4×Dic6—181no—
130C3×#103—91no—
131C3×C4×C8—91no—
132#835—91no—
133#818—91no—
134#816—31no—
135#816—91no—
136#816—91no—
137#816—181no—
138C8×Dic3—31no—
139#810—31no—
140#807—91no—
141#807—91no—
142#807—181no—
143C4×C3⋊C8—31no—
144C2×C32×C4—21no—
145C2×C32×C4—21no—
146C2×C32×C4—21no—
147C2×C32×C4—31no—
148C2×C32×C4—31no—
149C2×C32×C4—41no—
150C2×C32×C4—61no—
151C2×C32×C4C22×D311yescharacteristic
152C2×C32×C4C22×D311yescharacteristic
153C2×C32×C4C22×D311yescharacteristic
154C2×C3⋊Dic3—21no—
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—
163#634—21no—
164#634—21no—
165#634C4×D312yes—
166#634C4×D312yes—
167C2×C3×Dic3—21no—
168C2×C3×Dic3—21no—
169C2×C3×Dic3—21no—
170C2×C3×Dic3—21no—
171C2×C3×Dic3—41no—
172C2×C3×Dic3—62no—
173C2×C3×Dic3—61no—
174C2×C3×Dic3—61no—
175C2×C3×Dic3—61no—
176C2×C3×Dic3—61no—
177C2×C3×Dic3—121no—
178C2×C3×Dic3D1211yescharacteristic
179C2×C3×Dic3D1211yescharacteristic
180C2×C3×Dic3#6711yescharacteristic
181C2×C3×Dic3#6711yescharacteristic
182C3×Dic6—32no—
183C3×Dic6—32no—
184C3×Dic6—32no—
185C3×Dic6—32no—
186C3×Dic6—62no—
187C3×Dic6—62no—
188C3×Dic6—62no—
189C3×Dic6—61no—
190C3×Dic6—61no—
191C3×Dic6—61no—
192C3×Dic6—61no—
193C3×Dic6—121no—
194C32×C8—61no—
195C3⋊(C3⋊C8)C4×D312yes—
196C3×C3⋊C8—21no—
197C3×C3⋊C8—21no—
198C3×C3⋊C8—32no—
199C3×C3⋊C8—32no—
200C3×C3⋊C8—41no—
201C3×C3⋊C8—62no—
202C4×Q16—271noSylow
203C2×C33C4×D411yescharacteristic
204C2×C33C2×Q1611yescharacteristic
205C2×C33#13511yescharacteristic
206C2×C3×Q8—91no—
207C2×C3×Q8—91no—
208C2×Dic6—31no—
209C2×Dic6—31no—
210C2×Dic6—31no—
211C2×Dic6—31no—
212C2×Dic6—61no—
213C2×Dic6—61no—
214C2×C3×C8—31no—
215C2×C3×C8—31no—
216C2×C3×C8—61no—
217C2×C3×C8—91no—
218C3×C4⋊C4—31no—
219C3×C4⋊C4—31no—
220C3×C4⋊C4—61no—
221C3×C4⋊C4—181no—
222C3×C42—31no—
223C3×C42—31no—
224C3×C42—61no—
225C3×C42—181no—
226#216—94no—
227#216—94no—
228C4⋊Dic3—91no—
229C4⋊Dic3—91no—
230C4⋊Dic3—91no—
231C4⋊Dic3—181no—
232C4⋊Dic3—181no—
233C4⋊Dic3—181no—
234C4⋊Dic3—181no—
235C4⋊Dic3—361no—
236C4⋊Dic3D3211yescharacteristic
237Dic3⋊C4—61no—
238Dic3⋊C4—61no—
239Dic3⋊C4—181no—
240Dic3⋊C4—181no—
241Dic3⋊C4—182no—
242C4×Dic3—61no—
243C4×Dic3—61no—
244C4×Dic3—181no—
245C4×Dic3—181no—
246C4×Dic3—182no—
247C4×Dic3—181no—
248C4×Dic3D3211yescharacteristic
249C2×C3⋊C8—31no—
250C2×C3⋊C8—31no—
251C2×C3⋊C8—61no—
252C2×C3⋊C8D3211yescharacteristic
253Dic12—94no—
254C22×C32—21no—
255C22×C32—21no—
256C22×C32—21no—
257C22×C32—41no—
258C22×C32C2×D1211yescharacteristic
259C22×C32D3×D411yescharacteristic
260C22×C32C2×#6711yescharacteristic
261C32×C4—21no—
262C32×C4—21no—
263C32×C4—21no—
264C32×C4—21no—
265C32×C4—21no—
266C32×C4—21no—
267C32×C4—32no—
268C32×C4—41no—
269C32×C4—41no—
270C32×C4—61no—
271C32×C4—121no—
272C32×C4C2×C4×D311yescharacteristic
273C32×C4C2×C4×D311yescharacteristic
274C32×C4C2×C4×D311yescharacteristic
275C32×C4#23511yescharacteristic
276C32×C4#23511yescharacteristic
277C32×C4#23711yescharacteristic
278C3⋊Dic3—22no—
279C3⋊Dic3—22no—
280C3⋊Dic3—121no—
281C3⋊Dic3—122no—
282C3⋊Dic3—121no—
283C3⋊Dic3—121no—
284C3⋊Dic3—241no—
285C3⋊Dic3—241no—
286C3⋊Dic3—242no—
287C3×Dic3—21no—
288C3×Dic3—21no—
289C3×Dic3—22no—
290C3×Dic3—41no—
291C3×Dic3—41no—
292C3×Dic3—41no—
293C3×Dic3—64no—
294C3×Dic3—62no—
295C3×Dic3—62no—
296C3×Dic3—62no—
297C3×Dic3—62no—
298C3×Dic3—81no—
299C3×Dic3—122no—
300C3×Dic3Dic1212yescomplemented
301C3×Dic3#21612yescomplemented
302C2×Q16—271no—
303C4×Q8—91no—
304C4×Q8—271no—
305#107—271no—
306#103—91no—
307#103—271no—
308C4×C8—91no—
309C33C4×Q1611yescharacteristic, complemented, Sylow
310C3×Q8—92no—
311C3×Q8—92no—
312C3×Q8—181no—
313C3×Q8—181no—
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—31no—
318C2×C3×C4—31no—
319C2×C3×C4—31no—
320C2×C3×C4—31no—
321C2×C3×C4—41no—
322C2×C3×C4—61no—
323C2×C3×C4—61no—
324C2×C3×C4—61no—
325C2×C3×C4—61no—
326C2×C3×C4—121no—
327C2×C3×C4—181no—
328C2×C3×C4—181no—
329C2×C3×C4C2×D3211yescharacteristic
330C2×C3×C4C2×D3211yescharacteristic
331C2×C3×C4C2×D3211yescharacteristic
332C2×Dic3—21no—
333C2×Dic3—62no—
334C2×Dic3—61no—
335C2×Dic3—61no—
336C2×Dic3—61no—
337C2×Dic3—61no—
338C2×Dic3—121no—
339C2×Dic3—182no—
340C2×Dic3—181no—
341C2×Dic3—181no—
342C2×Dic3—181no—
343C2×Dic3—361no—
344C2×Dic3—361no—
345C2×Dic3—362no—
346C2×Dic3#62611yescharacteristic
347C2×Dic3#62611yescharacteristic
348Dic6—32no—
349Dic6—32no—
350Dic6—32no—
351Dic6—32no—
352Dic6—62no—
353Dic6—62no—
354Dic6—62no—
355Dic6—61no—
356Dic6—61no—
357Dic6—61no—
358Dic6—61no—
359Dic6—121no—
360C3×C8—61no—
361C3×C8—61no—
362C3×C8—92no—
363C3×C8—121no—
364C3⋊C8—21no—
365C3⋊C8—32no—
366C3⋊C8—32no—
367C3⋊C8—62no—
368C2×C32—21no—
369C2×C32—21no—
370C2×C32—21no—
371C2×C32—21no—
372C2×C32—21no—
373C2×C32—21no—
374C2×C32—21no—
375C2×C32—21no—
376C2×C32—21no—
377C2×C32—41no—
378C2×C32—41no—
379C2×C32—41no—
380C2×C32C4×D1211yescharacteristic
381C2×C32#67⋊C411yescharacteristic
382C2×C32#90411yescharacteristic
383C2×C32C2×Dic1211yescharacteristic
384C2×C32#91611yescharacteristic
385C2×C32D3×Q1611yescharacteristic
386C2×C32C4×#6711yescharacteristic
387C2×C32C2×#21611yescharacteristic
388C2×C32#95011yescharacteristic
389C2×Q8—91no—
390C2×Q8—91no—
391Q16—274no—
392C2×C8—31no—
393C2×C8—91no—
394C4⋊C4—31no—
395C4⋊C4—181no—
396C4⋊C4—271no—
397C4⋊C4—541no—
398C42—31no—
399C42—181no—
400C42—541no—
401C22×C3—21no—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—41no—
405C22×C3D3×D1211yescharacteristic
406C22×C3C2×#62611yescharacteristic
407C22×C3D3×#6711yescharacteristic
408C3×C4—21no—
409C3×C4—21no—
410C3×C4—21no—
411C3×C4—21no—
412C3×C4—21no—
413C3×C4—21no—
414C3×C4—32no—
415C3×C4—32no—
416C3×C4—41no—
417C3×C4—41no—
418C3×C4—62no—
419C3×C4—61no—
420C3×C4—61no—
421C3×C4—121no—
422C3×C4—121no—
423C3×C4—121no—
424C3×C4—182no—
425C3×C4—182no—
426C3×C4—241no—
427C3×C4#379511yescharacteristic
428C3×C4#379811yescharacteristic
429C3×C4C4×D3211yescharacteristic
430C3×C4C4×D3211yescharacteristic
431C3×C4C4×D3211yescharacteristic
432C3×C4#380411yescharacteristic
433Dic3—21no—
434Dic3—41no—
435Dic3—64no—
436Dic3—62no—
437Dic3—62no—
438Dic3—62no—
439Dic3—62no—
440Dic3—122no—
441Dic3—362no—
442Dic3—361no—
443Dic3—361no—
444Dic3—361no—
445Dic3—721no—
446Dic3—721no—
447Dic3—722no—
448Dic3#371912yescomplemented
449C32—21no—
450C32—21no—
451C32—21no—
452C32—41no—
453C32C4×Dic1211yescharacteristic, complemented
454C32#479011yescharacteristic, complemented
455C32C4×#21611yescharacteristic, complemented
456Q8—92no—
457Q8—92no—
458Q8—181no—
459Q8—181no—
460C2×C4—31no—
461C2×C4—31no—
462C2×C4—61no—
463C2×C4—181no—
464C2×C4—181no—
465C2×C4—541no—
466C2×C4D3311yescharacteristic
467C8—61no—
468C8—92no—
469C2×C3—21no—
470C2×C3—21no—
471C2×C3—21no—
472C2×C3—21no—
473C2×C3—21no—
474C2×C3—21no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—41no—
479C2×C3—41no—
480C2×C3—41no—
481C2×C3D3×Dic1211yescharacteristic
482C2×C3#6380611yescharacteristic
483C2×C3#6382911yescharacteristic
484C2×C3C2×#371911yescharacteristic
485C2×C3#6387911yescharacteristic
486C2×C3#6388011yescharacteristic
487C2×C3C4×#62611yescharacteristic
488C2×C3#6393511yescharacteristic
489C2×C3D3×#21611yescharacteristic
490C22D3×#62611yescharacteristic
491C4—32no—
492C4—61no—
493C4—121no—
494C4—182no—
495C4—182no—
496C4—1081no—
497C4#8894911yescharacteristic
498C4#8896011yescharacteristic
499C3—21no—
500C3—21no—
501C3—21no—
502C3—41no—
503C3C4×#371911yescharacteristic, complemented
504C3#1059047611yescharacteristic, complemented
505C3#1059079311yescharacteristic, complemented
506C2D3×#371911yescharacteristic
507C2#1172285311yescharacteristic
508C2#1172293411yescharacteristic
subgroupG/H
centerC22D3×#626
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socle, prime radicalC22×C33C2×D4
hypocenterC33C4×Q16
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
18. #88812C4semidirect
119. C32×Dic3Q16semidirect
300. C3×Dic3Dic12semidirect
301. C3×Dic3#216semidirect
309. C33C4×Q16semidirect
448. Dic3#3719semidirect
453. C32C4×Dic12semidirect
454. C32#4790semidirect
455. C32C4×#216semidirect
503. C3C4×#3719semidirect
504. C3#10590476semidirect
505. C3#10590793semidirect

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×#638442
#117220572
#117220402
C3×#635672
C3×Dic3×C3⋊C82
#105907933
#105904763
C4×#37193