Finite Group Databasebeta

← #13156756

Subgroups — #13156756

Order 1728.

1306 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
1C22×C3×#624C211yesmaximal, characteristic
2C2×#88814C211yesmaximal, characteristic
3C2×#88807C212yesmaximal, complemented
4C2×#88803C211yesmaximal, characteristic
5C3×#63579C212yesmaximal, complemented
6#10591375—31nomaximal
7#10591278—32nomaximal
8C23×C3×C3⋊D3C2211yescharacteristic
9C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
10C22×C3×C3⋊Dic3C2212yes—
11C22×C32×Dic3C2211yescharacteristic
12C22×C32×Dic3C2212yes—
13C2×C3×#624—22no—
14C2×C3×#624—24no—
15#88814C414yescomplemented
16#88807—24no—
17#88803—22no—
18C22×C3×C4×D3—32no—
19C22×#624D311yescharacteristic
20C2×#3752—31no—
21C2×#3750—32no—
22C2×Dic3⋊Dic3—32no—
23C2×Dic3⋊Dic3—32no—
24C2×Dic3⋊Dic3—62no—
25C2×#3722—31no—
26C2×(C2×D3)⋊Dic3—32no—
27C2×(C2×D3)⋊Dic3—32no—
28C2×(C2×D3)⋊Dic3—61no—
29C3×C3⋊#95—32no—
30C3×C3⋊#95—32no—
31#63579D312yescomplemented
32C23×C33C2311yescharacteristic
33C22×C3×C3⋊D3—22no—
34C22×C3×C3⋊D3—22no—
35C22×C3×C3⋊D3C2×C412yes—
36C22×C3×C3⋊D3D412yes—
37C22×C3×C3⋊D3Q812yes—
38C2×C3⋊(C3⋊Dic3)—21no—
39C2×C3⋊(C3⋊Dic3)—22no—
40C2×C3×C3⋊Dic3—22no—
41C2×C3×C3⋊Dic3—22no—
42C2×C3×C3⋊Dic3C2×C414yescomplemented
43C2×C32×Dic3—22no—
44C2×C32×Dic3—22no—
45C2×C32×Dic3—21no—
46C2×C32×Dic3—22no—
47C2×C32×Dic3D414yes—
48C3×#624—28no—
49#5210—91no—
50#4668—92no—
51C23×C3⋊D3C2×D311yescharacteristic
52C23×C3×D3—32no—
53C23×C3×D3—61no—
54C22×C32×C4—32no—
55C22×C32×C4—31no—
56C22×C3⋊Dic3—32no—
57C22×C3⋊Dic3—31no—
58C22×C3⋊Dic3—61no—
59C22×C3⋊Dic3—62no—
60C22×C3⋊Dic3—121no—
61C22×C3⋊Dic3C2×D312yes—
62C22×C3×Dic3—21no—
63C22×C3×Dic3—22no—
64C22×C3×Dic3—32no—
65C22×C3×Dic3—32no—
66C22×C3×Dic3—62no—
67C22×C3×Dic3C2×D312yes—
68C22×C3×Dic3C2×D312yes—
69C22×C3×Dic3C2×D312yes—
70C2×C3×C4×D3—64no—
71C2×C3×C4×D3—64no—
72C2×C3×C4×D3—64no—
73C2×#624—22no—
74C2×#624—24no—
75#3752—62no—
76#3750—64no—
77Dic3⋊Dic3—68no—
78Dic3⋊Dic3—64no—
79Dic3⋊Dic3—64no—
80#3722—34no—
81(C2×D3)⋊Dic3—38no—
82(C2×D3)⋊Dic3—64no—
83(C2×D3)⋊Dic3—64no—
84C22×C33C22×C411yescharacteristic
85C22×C33C2×D411yescharacteristic
86C22×C33C2×D412yes—
87C22×C33C2×Q811yescharacteristic
88C22×C33#4112yes—
89C2×C3×C3⋊D3—24no—
90C2×C3×C3⋊D3—22no—
91C2×C3×C3⋊D3—24no—
92C2×C3×C3⋊D3—22no—
93C2×C3×C3⋊D3C4⋊C414yescomplemented
94C3⋊(C3⋊Dic3)—41no—
95C3×C3⋊Dic3—24no—
96C32×Dic3—24no—
97C32×Dic3—41no—
98C23×C3×C4—91no—
99C22×C4×D3—32no—
100C2×#217—91no—
101C2×#217—91no—
102C2×(C2×D3)⋊C4—92no—
103C2×(C2×D3)⋊C4—92no—
104C2×(C2×D3)⋊C4—181no—
105C2×C4⋊Dic3—92no—
106C2×Dic3⋊C4—92no—
107C2×Dic3⋊C4—92no—
108C2×Dic3⋊C4—182no—
109C3×#95—92no—
110C3⋊#95—32no—
111C3⋊#95—32no—
112C23×C32—21no—
113C23×C32—22no—
114C23×C32—41no—
115C23×C32C22×D311yescharacteristic
116C23×C32C22×D312yes—
117C22×C3⋊D3—22no—
118C22×C3⋊D3—22no—
119C22×C3⋊D3Dic612yes—
120C22×C3⋊D3C4×D312yes—
121C22×C3⋊D3#6712yes—
122C22×C3×D3—34no—
123C22×C3×D3—34no—
124C22×C3×D3—34no—
125C22×C3×D3—62no—
126C22×C3×D3—62no—
127C22×C3×D3—62no—
128C22×C3×D3—62no—
129C22×C3×D3—62no—
130C22×C3×D3—62no—
131C22×C3×D3—62no—
132C22×C3×D3—122no—
133C22×C3×D3—122no—
134C2×C32×C4—34no—
135C2×C32×C4—61no—
136C2×C32×C4—62no—
137C2×C32×C4—62no—
138C2×C32×C4—62no—
139C2×C3⋊Dic3—22no—
140C2×C3⋊Dic3—22no—
141C2×C3⋊Dic3—62no—
142C2×C3⋊Dic3—61no—
143C2×C3⋊Dic3—62no—
144C2×C3⋊Dic3—64no—
145C2×C3⋊Dic3—62no—
146C2×C3⋊Dic3—62no—
147C2×C3⋊Dic3—62no—
148C2×C3⋊Dic3—122no—
149C2×C3⋊Dic3—122no—
150C2×C3⋊Dic3—122no—
151C2×C3⋊Dic3—124no—
152C2×C3⋊Dic3—122no—
153C2×C3⋊Dic3C4×D314yescomplemented
154C2×C3×Dic3—22no—
155C2×C3×Dic3—22no—
156C2×C3×Dic3—22no—
157C2×C3×Dic3—22no—
158C2×C3×Dic3—22no—
159C2×C3×Dic3—22no—
160C2×C3×Dic3—22no—
161C2×C3×Dic3—22no—
162C2×C3×Dic3—24no—
163C2×C3×Dic3—34no—
164C2×C3×Dic3—34no—
165C2×C3×Dic3—42no—
166C2×C3×Dic3—42no—
167C2×C3×Dic3—42no—
168C2×C3×Dic3—64no—
169C2×C3×Dic3—64no—
170C2×C3×Dic3—64no—
171C2×C3×Dic3—62no—
172C2×C3×Dic3—62no—
173C2×C3×Dic3—62no—
174C2×C3×Dic3—62no—
175C2×C3×Dic3D1214yes—
176C2×C3×Dic3#6714yes—
177C3×C4×D3—616no—
178#624—28no—
179#385—271noSylow
180C2×C33C2×C4⋊C411yescharacteristic
181C2×C33C4×D412yes—
182C2×C33#12011yescharacteristic
183C2×C33C22⋊Q812yes—
184C2×C33#12311yescharacteristic
185C3×C3⋊D3—24no—
186C24×C3—91no—
187C23×D3—32no—
188C23×D3—61no—
189C22×C3×C4—32no—
190C22×C3×C4—32no—
191C22×C3×C4—32no—
192C22×C3×C4—62no—
193C22×C3×C4—61no—
194C22×C3×C4—92no—
195C22×C3×C4—184no—
196C22×C3×C4—182no—
197C22×Dic3—32no—
198C22×Dic3—32no—
199C22×Dic3—62no—
200C22×Dic3—91no—
201C22×Dic3—92no—
202C22×Dic3—182no—
203C22×Dic3—181no—
204C22×Dic3—361no—
205C22×Dic3D3211yescharacteristic
206C22×Dic3D3212yes—
207C2×C4×D3—64no—
208C2×C4×D3—64no—
209C2×C4×D3—64no—
210#217—94no—
211#217—182no—
212(C2×D3)⋊C4—98no—
213(C2×D3)⋊C4—184no—
214(C2×D3)⋊C4—184no—
215C4⋊Dic3—184no—
216Dic3⋊C4—184no—
217Dic3⋊C4—184no—
218Dic3⋊C4—188no—
219C22×C32—22no—
220C22×C32—22no—
221C22×C32—21no—
222C22×C32—22no—
223C22×C32—22no—
224C22×C32—21no—
225C22×C32—22no—
226C22×C32—22no—
227C22×C32—22no—
228C22×C32—22no—
229C22×C32—22no—
230C22×C32—21no—
231C22×C32—41no—
232C22×C32—42no—
233C22×C32—41no—
234C22×C32—41no—
235C22×C32—42no—
236C22×C32C2×Dic611yescharacteristic
237C22×C32C2×C4×D312yes—
238C22×C32C2×C4×D311yescharacteristic
239C22×C32C2×D1212yes—
240C22×C32#23512yes—
241C22×C32#23512yes—
242C22×C32D3×D412yes—
243C22×C32D3×D412yes—
244C22×C32#23712yes—
245C22×C32D3×Q812yes—
246C22×C32C2×#6712yes—
247C22×C32C2×#6711yescharacteristic
248C2×C3⋊D3—24no—
249C2×C3⋊D3—24no—
250C2×C3⋊D3—22no—
251C2×C3⋊D3—22no—
252C2×C3⋊D3Dic3⋊C414yescomplemented
253C2×C3×D3—38no—
254C2×C3×D3—64no—
255C2×C3×D3—64no—
256C2×C3×D3—64no—
257C2×C3×D3—64no—
258C2×C3×D3—64no—
259C2×C3×D3—64no—
260C2×C3×D3—64no—
261C2×C3×D3—124no—
262C2×C3×D3—124no—
263C2×C3×D3—122no—
264C2×C3×D3—122no—
265C32×C4—64no—
266C32×C4—121no—
267C3⋊Dic3—24no—
268C3⋊Dic3—122no—
269C3⋊Dic3—124no—
270C3⋊Dic3—124no—
271C3⋊Dic3—122no—
272C3⋊Dic3—121no—
273C3×Dic3—24no—
274C3×Dic3—24no—
275C3×Dic3—44no—
276C3×Dic3—42no—
277C3×Dic3—42no—
278C3×Dic3—68no—
279C3×Dic3—64no—
280C3×Dic3—64no—
281C23×C4—91no—
282C2×C4⋊C4—272no—
283C2×C22⋊C4—271no—
284C2×C22⋊C4—271no—
285#95—92no—
286C33#38511yescharacteristic, complemented, Sylow
287C23×C3—22no—
288C23×C3—21no—
289C23×C3—41no—
290C23×C3—92no—
291C23×C3—92no—
292C23×C3—92no—
293C23×C3—182no—
294C23×C3—182no—
295C23×C3C2×D3212yes—
296C23×C3C2×D3211yescharacteristic
297C22×D3—34no—
298C22×D3—34no—
299C22×D3—34no—
300C22×D3—62no—
301C22×D3—62no—
302C22×D3—62no—
303C22×D3—62no—
304C22×D3—62no—
305C22×D3—62no—
306C22×D3—62no—
307C22×D3—122no—
308C22×D3—122no—
309C2×C3×C4—34no—
310C2×C3×C4—34no—
311C2×C3×C4—64no—
312C2×C3×C4—62no—
313C2×C3×C4—62no—
314C2×C3×C4—62no—
315C2×C3×C4—62no—
316C2×C3×C4—62no—
317C2×C3×C4—62no—
318C2×C3×C4—62no—
319C2×C3×C4—62no—
320C2×C3×C4—94no—
321C2×C3×C4—122no—
322C2×C3×C4—122no—
323C2×C3×C4—122no—
324C2×C3×C4—188no—
325C2×C3×C4—182no—
326C2×C3×C4—182no—
327C2×Dic3—22no—
328C2×Dic3—22no—
329C2×Dic3—22no—
330C2×Dic3—21no—
331C2×Dic3—34no—
332C2×Dic3—34no—
333C2×Dic3—64no—
334C2×Dic3—64no—
335C2×Dic3—64no—
336C2×Dic3—62no—
337C2×Dic3—62no—
338C2×Dic3—62no—
339C2×Dic3—62no—
340C2×Dic3—184no—
341C2×Dic3—182no—
342C2×Dic3—182no—
343C2×Dic3—182no—
344C2×Dic3—182no—
345C2×Dic3—182no—
346C2×Dic3—181no—
347C2×Dic3—362no—
348C2×Dic3—362no—
349C2×Dic3—362no—
350C2×Dic3—364no—
351C2×Dic3—362no—
352C2×Dic3#62614yes—
353C4×D3—616no—
354C2×C32—22no—
355C2×C32—22no—
356C2×C32—22no—
357C2×C32—22no—
358C2×C32—22no—
359C2×C32—21no—
360C2×C32—22no—
361C2×C32—22no—
362C2×C32—22no—
363C2×C32—21no—
364C2×C32—21no—
365C2×C32—22no—
366C2×C32—41no—
367C2×C32—42no—
368C2×C32—41no—
369C2×C32—42no—
370C2×C32—41no—
371C2×C32C4×D1212yes—
372C2×C32#67⋊C412yes—
373C2×C32#88212yes—
374C2×C32#88312yes—
375C2×C32D3×C4⋊C412yes—
376C2×C32#89612yes—
377C2×C32#89712yes—
378C2×C32C2×Dic3⋊C411yescharacteristic
379C2×C32#92412yes—
380C2×C32C4×#6712yes—
381C2×C32#92911yescharacteristic
382C2×C32#95311yescharacteristic
383C3⋊D3—24no—
384C3×D3—68no—
385C3×D3—124no—
386C24—91no—
387C22×C4—32no—
388C22×C4—31no—
389C22×C4—92no—
390C22×C4—184no—
391C22×C4—182no—
392C22×C4—271no—
393C4⋊C4—544no—
394C22⋊C4—274no—
395C22⋊C4—542no—
396C22×C3—22no—
397C22×C3—21no—
398C22×C3—22no—
399C22×C3—22no—
400C22×C3—22no—
401C22×C3—21no—
402C22×C3—22no—
403C22×C3—21no—
404C22×C3—22no—
405C22×C3—22no—
406C22×C3—22no—
407C22×C3—22no—
408C22×C3—41no—
409C22×C3—42no—
410C22×C3—41no—
411C22×C3—41no—
412C22×C3—42no—
413C22×C3—94no—
414C22×C3—184no—
415C22×C3—184no—
416C22×C3—182no—
417C22×C3—182no—
418C22×C3D3×Dic612yes—
419C22×C3#379512yes—
420C22×C3#379711yescharacteristic
421C22×C3#379812yes—
422C22×C3C4×D3212yes—
423C22×C3C4×D3211yescharacteristic
424C22×C3D3×D1212yes—
425C22×C3#380412yes—
426C22×C3C2×#62612yes—
427C22×C3D3×#6712yes—
428C22×C3D3×#6712yes—
429C22×C3#381111yescharacteristic
430C2×D3—38no—
431C2×D3—64no—
432C2×D3—64no—
433C2×D3—64no—
434C2×D3—64no—
435C2×D3—64no—
436C2×D3—64no—
437C2×D3—64no—
438C2×D3—124no—
439C2×D3—122no—
440C2×D3—124no—
441C2×D3—122no—
442C3×C4—64no—
443C3×C4—64no—
444C3×C4—124no—
445C3×C4—122no—
446C3×C4—122no—
447C3×C4—184no—
448Dic3—24no—
449Dic3—41no—
450Dic3—68no—
451Dic3—64no—
452Dic3—64no—
453Dic3—364no—
454Dic3—362no—
455Dic3—364no—
456Dic3—361no—
457Dic3—362no—
458C32—22no—
459C32—21no—
460C32—41no—
461C32#466812yescomplemented
462C32#521011yescharacteristic, complemented
463C23—92no—
464C23—92no—
465C23—92no—
466C23—182no—
467C23—182no—
468C23D3311yescharacteristic
469C2×C4—34no—
470C2×C4—62no—
471C2×C4—62no—
472C2×C4—61no—
473C2×C4—62no—
474C2×C4—94no—
475C2×C4—188no—
476C2×C4—182no—
477C2×C4—182no—
478C2×C4—542no—
479C2×C4—541no—
480C2×C3—21no—
481C2×C3—22no—
482C2×C3—22no—
483C2×C3—22no—
484C2×C3—22no—
485C2×C3—22no—
486C2×C3—21no—
487C2×C3—22no—
488C2×C3—22no—
489C2×C3—22no—
490C2×C3—21no—
491C2×C3—22no—
492C2×C3—41no—
493C2×C3—42no—
494C2×C3—41no—
495C2×C3—42no—
496C2×C3—41no—
497C2×C3—184no—
498C2×C3#6385112yes—
499C2×C3#6385712yes—
500C2×C3D3×Dic3⋊C412yes—
501C2×C3#6387912yes—
502C2×C3#6388012yes—
503C2×C3#6388311yescharacteristic
504C2×C3#6389512yes—
505C2×C3C4×#62612yes—
506C2×C3#6390512yes—
507C2×C3#6392311yescharacteristic
508C2×C3#6397511yescharacteristic
509C2×C3#6397712yes—
510D3—68no—
511D3—124no—
512C22—94no—
513C22—184no—
514C22—182no—
515C22—184no—
516C22—182no—
517C22#8894911yescharacteristic
518C22D3×#62612yes—
519C22#8895211yescharacteristic
520C22#8895711yescharacteristic
521C22#8896012yes—
522C4—64no—
523C4—121no—
524C4—184no—
525C4—1081no—
526C3—21no—
527C3—22no—
528C3—41no—
529C3#1059127812yescomplemented
530C3#1059137511yescharacteristic, complemented
531C2—184no—
532C2#1172291011yescharacteristic
533C2#1172293412yes—
534C2#1172294811yescharacteristic
535C2#1172297212yes—
536C2#1172299211yescharacteristic
subgroupG/H
center, Frattini, hypercenterC23D33
commutatorC22×C33C22×C4
Fitting, socle, cosocleC23×C33C23
prime radicalC23×C3×C3⋊D3C22
hypocenterC33#385
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
3. C2×#88807C2semidirect
5. C3×#63579C2semidirect
15. #88814C4semidirect
31. #63579D3semidirect
42. C2×C3×C3⋊Dic3C2×C4semidirect
93. C2×C3×C3⋊D3C4⋊C4semidirect
153. C2×C3⋊Dic3C4×D3semidirect
252. C2×C3⋊D3Dic3⋊C4semidirect
286. C33#385semidirect
461. C32#4668semidirect
462. C32#5210semidirect
529. C3#10591278semidirect
530. C3#10591375semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#6242
C2×#888142
C2×#888072
C2×#888032
C3×#635792
#105913753
#105912783