Finite Group Databasebeta

← #13156727

Subgroups — #13156727

Order 1728.

1210 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#624C211yesmaximal, characteristic
2C2×#88814C211yesmaximal, characteristic
3C2×#88803C211yesmaximal, characteristic
4#11722061C212yesmaximal, complemented
5C3×#63579C212yesmaximal, complemented
6#10591371—31nomaximal
7#10591294—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—22no—
15C2×C3×#624C414yescomplemented
16#88814—22no—
17#88803—22no—
18C22×C3×C4×D3—32no—
19C22×#624D311yescharacteristic
20C2×#3752—31no—
21C2×#3722—31no—
22C2×(C2×D3)⋊Dic3—32no—
23C2×(C2×D3)⋊Dic3—32no—
24C2×(C2×D3)⋊Dic3—61no—
25#63658—32no—
26C3×C3⋊#95—32no—
27C3×C3⋊#95—32no—
28#63579—32no—
29#63579—32no—
30#63579—62no—
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⋊D3D412yes—
38C2×C3⋊(C3⋊Dic3)—21no—
39C2×C3⋊(C3⋊Dic3)—22no—
40C2×C3×C3⋊Dic3—22no—
41C2×C3×C3⋊Dic3—22no—
42C2×C3×C3⋊Dic3—22no—
43C2×C32×Dic3—22no—
44C2×C32×Dic3—22no—
45C2×C32×Dic3—21no—
46C2×C32×Dic3—22no—
47C2×C32×Dic3C2×C414yes—
48C3×#624—28no—
49#5211—91no—
50#4664—92no—
51C23×C3⋊D3C2×D311yescharacteristic
52C23×C3×D3—32no—
53C23×C3×D3—61no—
54C22×C32×C4—32no—
55C22×C32×C4—31no—
56C22×C3⋊Dic3—31no—
57C22×C3⋊Dic3—32no—
58C22×C3⋊Dic3—62no—
59C22×C3⋊Dic3—61no—
60C22×C3⋊Dic3—121no—
61C22×C3⋊Dic3C2×D312yes—
62C22×C3×Dic3—22no—
63C22×C3×Dic3—21no—
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—38no—
71C2×C3×C4×D3—64no—
72C2×C3×C4×D3—64no—
73C2×#624—22no—
74C2×#624—22no—
75C2×#624Dic314yescomplemented
76#3752—62no—
77#3722—62no—
78(C2×D3)⋊Dic3—64no—
79(C2×D3)⋊Dic3—64no—
80(C2×D3)⋊Dic3—64no—
81C22×C33C22×C411yescharacteristic
82C22×C33C2×D411yescharacteristic
83C22×C33C2×D411yescharacteristic
84C22×C33#4112yes—
85C22×C33#4112yes—
86C2×C3×C3⋊D3—24no—
87C2×C3×C3⋊D3—22no—
88C2×C3×C3⋊D3—24no—
89C2×C3×C3⋊D3—22no—
90C2×C3×C3⋊D3C22⋊C414yes—
91C3⋊(C3⋊Dic3)—41no—
92C3×C3⋊Dic3—42no—
93C32×Dic3—24no—
94C32×Dic3—41no—
95C23×C3×C4—91no—
96C22×C4×D3—32no—
97C2×#217—91no—
98C2×#217—91no—
99C2×(C2×D3)⋊C4—92no—
100C2×(C2×D3)⋊C4—92no—
101C2×(C2×D3)⋊C4—181no—
102C3×#95—92no—
103C3⋊#95—32no—
104C3⋊#95—32no—
105C3⋊#95—92no—
106C3⋊#95—92no—
107C3⋊#95—92no—
108C3⋊#95—182no—
109C23×C32—21no—
110C23×C32—22no—
111C23×C32—41no—
112C23×C32C22×D312yes—
113C23×C32C22×D311yescharacteristic
114C22×C3⋊D3—22no—
115C22×C3⋊D3—22no—
116C22×C3⋊D3C2×Dic312yes—
117C22×C3⋊D3#6712yes—
118C22×C3⋊D3#6712yes—
119C22×C3×D3—34no—
120C22×C3×D3—34no—
121C22×C3×D3—34no—
122C22×C3×D3—62no—
123C22×C3×D3—62no—
124C22×C3×D3—62no—
125C22×C3×D3—62no—
126C22×C3×D3—62no—
127C22×C3×D3—62no—
128C22×C3×D3—62no—
129C22×C3×D3—122no—
130C22×C3×D3—122no—
131C2×C32×C4—34no—
132C2×C32×C4—61no—
133C2×C32×C4—62no—
134C2×C32×C4—62no—
135C2×C32×C4—62no—
136C2×C3⋊Dic3—22no—
137C2×C3⋊Dic3—22no—
138C2×C3⋊Dic3—22no—
139C2×C3⋊Dic3—62no—
140C2×C3⋊Dic3—61no—
141C2×C3⋊Dic3—62no—
142C2×C3⋊Dic3—64no—
143C2×C3⋊Dic3—62no—
144C2×C3⋊Dic3—62no—
145C2×C3⋊Dic3—62no—
146C2×C3⋊Dic3—122no—
147C2×C3⋊Dic3—122no—
148C2×C3⋊Dic3—122no—
149C2×C3⋊Dic3—122no—
150C2×C3⋊Dic3—124no—
151C2×C3×Dic3—22no—
152C2×C3×Dic3—22no—
153C2×C3×Dic3—22no—
154C2×C3×Dic3—22no—
155C2×C3×Dic3—22no—
156C2×C3×Dic3—22no—
157C2×C3×Dic3—22no—
158C2×C3×Dic3—22no—
159C2×C3×Dic3—24no—
160C2×C3×Dic3—42no—
161C2×C3×Dic3—42no—
162C2×C3×Dic3—42no—
163C2×C3×Dic3—64no—
164C2×C3×Dic3—62no—
165C2×C3×Dic3—62no—
166C2×C3×Dic3—62no—
167C2×C3×Dic3—62no—
168C2×C3×Dic3—62no—
169C2×C3×Dic3—62no—
170C2×C3×Dic3—122no—
171C2×C3×Dic3—122no—
172C2×C3×Dic3C4×D314yes—
173C2×C3×Dic3C2×Dic314yes—
174C3×C4×D3—616no—
175#624—28no—
176#381—271noSylow
177C2×C33C2×C22⋊C411yescharacteristic
178C2×C33#11712yes—
179C2×C33#12311yescharacteristic
180C2×C33#12312yes—
181C2×C33#12311yescharacteristic
182C3×C3⋊D3—24no—
183C24×C3—91no—
184C23×D3—32no—
185C23×D3—61no—
186C22×C3×C4—32no—
187C22×C3×C4—32no—
188C22×C3×C4—32no—
189C22×C3×C4—62no—
190C22×C3×C4—61no—
191C22×C3×C4—94no—
192C22×C3×C4—92no—
193C22×C3×C4—182no—
194C22×C3×C4—182no—
195C22×Dic3—32no—
196C22×Dic3—32no—
197C22×Dic3—62no—
198C22×Dic3—92no—
199C22×Dic3—91no—
200C22×Dic3—182no—
201C22×Dic3—181no—
202C22×Dic3—361no—
203C22×Dic3D3211yescharacteristic
204C22×Dic3D3212yes—
205C2×C4×D3—38no—
206C2×C4×D3—64no—
207C2×C4×D3—64no—
208#217—182no—
209#217—182no—
210(C2×D3)⋊C4—184no—
211(C2×D3)⋊C4—184no—
212(C2×D3)⋊C4—184no—
213C22×C32—22no—
214C22×C32—22no—
215C22×C32—22no—
216C22×C32—22no—
217C22×C32—22no—
218C22×C32—21no—
219C22×C32—21no—
220C22×C32—22no—
221C22×C32—22no—
222C22×C32—22no—
223C22×C32—22no—
224C22×C32—21no—
225C22×C32—41no—
226C22×C32—41no—
227C22×C32—41no—
228C22×C32—42no—
229C22×C32—42no—
230C22×C32C2×C4×D312yes—
231C22×C32#23512yes—
232C22×C32#23512yes—
233C22×C32#23512yes—
234C22×C32#23512yes—
235C22×C32D3×D412yes—
236C22×C32D3×D412yes—
237C22×C32#23712yes—
238C22×C32#23912yes—
239C22×C32C22×Dic311yescharacteristic
240C22×C32C2×#6711yescharacteristic
241C22×C32C2×#6711yescharacteristic
242C2×C3⋊D3—24no—
243C2×C3⋊D3—22no—
244C2×C3⋊D3—24no—
245C2×C3⋊D3—22no—
246C2×C3⋊D3#21714yes—
247C2×C3×D3—38no—
248C2×C3×D3—64no—
249C2×C3×D3—64no—
250C2×C3×D3—64no—
251C2×C3×D3—64no—
252C2×C3×D3—64no—
253C2×C3×D3—64no—
254C2×C3×D3—64no—
255C2×C3×D3—124no—
256C2×C3×D3—124no—
257C2×C3×D3—122no—
258C2×C3×D3—122no—
259C32×C4—64no—
260C32×C4—121no—
261C3⋊Dic3—42no—
262C3⋊Dic3—124no—
263C3⋊Dic3—122no—
264C3⋊Dic3—124no—
265C3⋊Dic3—122no—
266C3⋊Dic3—121no—
267C3×Dic3—24no—
268C3×Dic3—24no—
269C3×Dic3—44no—
270C3×Dic3—42no—
271C3×Dic3—42no—
272C3×Dic3—124no—
273C3×Dic3—122no—
274C3×Dic3—122no—
275C23×C4—91no—
276C2×C22⋊C4—271no—
277C2×C22⋊C4—271no—
278#95—92no—
279#95—272no—
280C33#38111yescharacteristic, complemented, Sylow
281C23×C3—22no—
282C23×C3—21no—
283C23×C3—41no—
284C23×C3—92no—
285C23×C3—92no—
286C23×C3—92no—
287C23×C3—182no—
288C23×C3—182no—
289C23×C3C2×D3212yes—
290C23×C3C2×D3211yescharacteristic
291C22×D3—34no—
292C22×D3—34no—
293C22×D3—34no—
294C22×D3—62no—
295C22×D3—62no—
296C22×D3—62no—
297C22×D3—62no—
298C22×D3—62no—
299C22×D3—62no—
300C22×D3—62no—
301C22×D3—122no—
302C22×D3—122no—
303C2×C3×C4—34no—
304C2×C3×C4—34no—
305C2×C3×C4—64no—
306C2×C3×C4—62no—
307C2×C3×C4—62no—
308C2×C3×C4—62no—
309C2×C3×C4—62no—
310C2×C3×C4—62no—
311C2×C3×C4—62no—
312C2×C3×C4—62no—
313C2×C3×C4—62no—
314C2×C3×C4—122no—
315C2×C3×C4—122no—
316C2×C3×C4—122no—
317C2×C3×C4—188no—
318C2×C3×C4—182no—
319C2×C3×C4—182no—
320C2×C3×C4—182no—
321C2×Dic3—22no—
322C2×Dic3—22no—
323C2×Dic3—22no—
324C2×Dic3—21no—
325C2×Dic3—64no—
326C2×Dic3—62no—
327C2×Dic3—62no—
328C2×Dic3—62no—
329C2×Dic3—62no—
330C2×Dic3—62no—
331C2×Dic3—62no—
332C2×Dic3—122no—
333C2×Dic3—122no—
334C2×Dic3—182no—
335C2×Dic3—184no—
336C2×Dic3—182no—
337C2×Dic3—182no—
338C2×Dic3—182no—
339C2×Dic3—182no—
340C2×Dic3—181no—
341C2×Dic3—362no—
342C2×Dic3—362no—
343C2×Dic3—362no—
344C2×Dic3—362no—
345C2×Dic3—364no—
346C2×Dic3D3×Dic314yes—
347C4×D3—616no—
348C2×C32—22no—
349C2×C32—22no—
350C2×C32—22no—
351C2×C32—22no—
352C2×C32—22no—
353C2×C32—22no—
354C2×C32—21no—
355C2×C32—22no—
356C2×C32—22no—
357C2×C32—21no—
358C2×C32—22no—
359C2×C32—21no—
360C2×C32—41no—
361C2×C32—42no—
362C2×C32—41no—
363C2×C32—41no—
364C2×C32—42no—
365C2×C32#87212yes—
366C2×C32D3×C22⋊C412yes—
367C2×C32#88312yes—
368C2×C32#88312yes—
369C2×C32#89212yes—
370C2×C32#89412yes—
371C2×C32#89412yes—
372C2×C32#92612yes—
373C2×C32#92911yescharacteristic
374C2×C32#92912yes—
375C2×C32#92911yescharacteristic
376C2×C32C2×#21711yescharacteristic
377C3⋊D3—24no—
378C3×D3—68no—
379C3×D3—124no—
380C24—91no—
381C22×C4—32no—
382C22×C4—31no—
383C22×C4—94no—
384C22×C4—92no—
385C22×C4—182no—
386C22×C4—182no—
387C22×C4—271no—
388C22⋊C4—542no—
389C22⋊C4—542no—
390C22×C3—22no—
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—22no—
394C22×C3—22no—
395C22×C3—21no—
396C22×C3—22no—
397C22×C3—22no—
398C22×C3—22no—
399C22×C3—22no—
400C22×C3—22no—
401C22×C3—22no—
402C22×C3—41no—
403C22×C3—41no—
404C22×C3—41no—
405C22×C3—42no—
406C22×C3—42no—
407C22×C3—94no—
408C22×C3—184no—
409C22×C3—184no—
410C22×C3—182no—
411C22×C3—182no—
412C22×C3#379512yes—
413C22×C3#379812yes—
414C22×C3#379812yes—
415C22×C3#379912yes—
416C22×C3#379912yes—
417C22×C3C4×D3211yescharacteristic
418C22×C3#380211yescharacteristic
419C22×C3C2×D3×Dic312yes—
420C22×C3#380412yes—
421C22×C3D3×#6712yes—
422C22×C3D3×#6712yes—
423C22×C3#381111yescharacteristic
424C2×D3—38no—
425C2×D3—64no—
426C2×D3—64no—
427C2×D3—64no—
428C2×D3—64no—
429C2×D3—64no—
430C2×D3—64no—
431C2×D3—64no—
432C2×D3—124no—
433C2×D3—122no—
434C2×D3—122no—
435C2×D3—124no—
436C3×C4—64no—
437C3×C4—64no—
438C3×C4—124no—
439C3×C4—122no—
440C3×C4—122no—
441C3×C4—362no—
442Dic3—24no—
443Dic3—41no—
444Dic3—124no—
445Dic3—122no—
446Dic3—122no—
447Dic3—362no—
448Dic3—364no—
449Dic3—364no—
450Dic3—362no—
451Dic3—361no—
452C32—22no—
453C32—21no—
454C32—41no—
455C32#466412yescomplemented
456C32#521111yescharacteristic, complemented
457C23—92no—
458C23—92no—
459C23—92no—
460C23—182no—
461C23—182no—
462C23D3311yescharacteristic
463C2×C4—34no—
464C2×C4—62no—
465C2×C4—62no—
466C2×C4—61no—
467C2×C4—62no—
468C2×C4—188no—
469C2×C4—182no—
470C2×C4—182no—
471C2×C4—182no—
472C2×C4—541no—
473C2×C4—542no—
474C2×C3—22no—
475C2×C3—21no—
476C2×C3—22no—
477C2×C3—22no—
478C2×C3—22no—
479C2×C3—21no—
480C2×C3—22no—
481C2×C3—21no—
482C2×C3—22no—
483C2×C3—22no—
484C2×C3—22no—
485C2×C3—22no—
486C2×C3—41no—
487C2×C3—42no—
488C2×C3—41no—
489C2×C3—41no—
490C2×C3—42no—
491C2×C3—184no—
492C2×C3#6383612yes—
493C2×C3#6384112yes—
494C2×C3#6385012yes—
495C2×C3#6385012yes—
496C2×C3#6385412yes—
497C2×C3#6385512yes—
498C2×C3#6389711yescharacteristic
499C2×C3#6390512yes—
500C2×C3#6390512yes—
501C2×C3#6392111yescharacteristic
502C2×C3D3×#21712yes—
503C2×C3#6397511yescharacteristic
504D3—68no—
505D3—124no—
506C22—94no—
507C22—184no—
508C22—182no—
509C22—182no—
510C22—184no—
511C22D32×Dic311yescharacteristic
512C22#8895211yescharacteristic
513C22#8895311yescharacteristic
514C22#8896012yes—
515C22#8896512yes—
516C4—64no—
517C4—121no—
518C4—362no—
519C4—1081no—
520C3—21no—
521C3—22no—
522C3—41no—
523C3#1059129412yescomplemented
524C3#1059137111yescharacteristic, complemented
525C2—184no—
526C2#1172290211yescharacteristic
527C2#1172291512yes—
528C2#1172299211yescharacteristic
529C2#1172299311yescharacteristic
530C2#1172299512yes—
subgroupG/H
center, Frattini, hypercenterC23D33
commutatorC22×C33C22×C4
Fitting, socle, cosocleC23×C33C23
prime radicalC23×C3×C3⋊D3C22
hypocenterC33#381
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
4. #11722061C2semidirect
5. C3×#63579C2semidirect
15. C2×C3×#624C4semidirect
31. #63579D3semidirect
75. C2×#624Dic3semidirect
280. C33#381semidirect
455. C32#4664semidirect
456. C32#5211semidirect
523. C3#10591294semidirect
524. C3#10591371semidirect

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×#888032
#117220612
C3×#635792
#105913713
#105912943