Finite Group Databasebeta

← #13156861

Subgroups — #13156861

Order 1728.

1058 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#627C211yesmaximal, characteristic
2C2×#88815C211yesmaximal, characteristic
3C2×Dic3×C3⋊Dic3C211yesmaximal, characteristic
4#11722061C212yesmaximal
5C3×#63579C212yesmaximal
6#10591396—31nomaximal
7#10591261—32nomaximal
8C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
9C22×C3×C3⋊Dic3C2212yes—
10C22×C3×C3⋊Dic3C2211yescharacteristic
11C22×C32×Dic3C2212yes—
12C22×C32×Dic3C2211yescharacteristic
13C2×C3×#627—22no—
14C2×C3×#627—22no—
15C2×C3×#627C414yescomplemented
16#88815—22no—
17Dic3×C3⋊Dic3—22no—
18C22×C3×Dic6—32no—
19C22×#627D311yescharacteristic
20C2×C4×C3⋊Dic3—31no—
21C2×#3724—31no—
22C2×Dic3⋊Dic3—32no—
23C2×Dic32—32no—
24C2×Dic32—61no—
25#63658—32no—
26C3×C3⋊#95—32no—
27C3×C3⋊#95—32no—
28#63579—32no—
29#63579—32no—
30#63579—62no—
31#63579D312yes—
32C23×C33C2311yescharacteristic
33C2×C3⋊(C3⋊Dic3)—21no—
34C2×C3⋊(C3⋊Dic3)—21no—
35C2×C3⋊(C3⋊Dic3)Q812yescomplemented
36C2×C3×C3⋊Dic3—22no—
37C2×C3×C3⋊Dic3—22no—
38C2×C3×C3⋊Dic3—22no—
39C2×C3×C3⋊Dic3C2×C412yes—
40C2×C3×C3⋊Dic3D412yes—
41C2×C3×C3⋊Dic3D412yes—
42C2×C32×Dic3—21no—
43C2×C32×Dic3—22no—
44C2×C32×Dic3—22no—
45C2×C32×Dic3—21no—
46C2×C32×Dic3C2×C414yes—
47C2×C32×Dic3Q812yescomplemented
48C3×#627—28no—
49#5229—91no—
50#4646—92no—
51C22×C32×C4—32no—
52C22×C32×C4—31no—
53C22×C3⋊Dic3—31no—
54C22×C3⋊Dic3—32no—
55C22×C3⋊Dic3—61no—
56C22×C3⋊Dic3—62no—
57C22×C3⋊Dic3—121no—
58C22×C3⋊Dic3C2×D312yes—
59C22×C3⋊Dic3C2×D311yescharacteristic
60C22×C3×Dic3—21no—
61C22×C3×Dic3—22no—
62C22×C3×Dic3—32no—
63C22×C3×Dic3—32no—
64C22×C3×Dic3—32no—
65C22×C3×Dic3—61no—
66C22×C3×Dic3—62no—
67C22×C3×Dic3C2×D312yes—
68C22×C3×Dic3C2×D312yes—
69C22×C3×Dic3C2×D312yes—
70C2×C3×Dic6—38no—
71C2×C3×Dic6—64no—
72C2×C3×Dic6—64no—
73C2×#627—22no—
74C2×#627—22no—
75C2×#627Dic314yescomplemented
76C4×C3⋊Dic3—62no—
77#3724—62no—
78Dic3⋊Dic3—64no—
79Dic32—64no—
80Dic32—64no—
81C22×C33C22×C411yescharacteristic
82C22×C33C2×D411yescharacteristic
83C22×C33C2×D411yescharacteristic
84C22×C33C2×Q811yescharacteristic
85C22×C33C2×Q811yescharacteristic
86C22×C33#4111yescharacteristic
87C22×C33#4111yescharacteristic
88C3⋊(C3⋊Dic3)—22no—
89C3×C3⋊Dic3—42no—
90C3×C3⋊Dic3C22⋊C414yes—
91C32×Dic3—24no—
92C32×Dic3—22no—
93C22×C3×Q8—91no—
94C22×Dic6—32no—
95C2×C4⋊Dic3—91no—
96C2×Dic3⋊C4—92no—
97C2×C4×Dic3—92no—
98C2×C4×Dic3—91no—
99C2×C4×Dic3—181no—
100C3×#95—92no—
101C3⋊#95—32no—
102C3⋊#95—32no—
103C3⋊#95—92no—
104C3⋊#95—92no—
105C3⋊#95—92no—
106C3⋊#95—182no—
107C23×C32—22no—
108C23×C32—21no—
109C23×C32—41no—
110C23×C32C22×D312yes—
111C23×C32C22×D311yescharacteristic
112C2×C32×C4—34no—
113C2×C32×C4—32no—
114C2×C32×C4—61no—
115C2×C32×C4—61no—
116C2×C32×C4—62no—
117C2×C32×C4—62no—
118C2×C3⋊Dic3—22no—
119C2×C3⋊Dic3—22no—
120C2×C3⋊Dic3—22no—
121C2×C3⋊Dic3—32no—
122C2×C3⋊Dic3—34no—
123C2×C3⋊Dic3—62no—
124C2×C3⋊Dic3—64no—
125C2×C3⋊Dic3—61no—
126C2×C3⋊Dic3—61no—
127C2×C3⋊Dic3—62no—
128C2×C3⋊Dic3—62no—
129C2×C3⋊Dic3—62no—
130C2×C3⋊Dic3—62no—
131C2×C3⋊Dic3—122no—
132C2×C3⋊Dic3—122no—
133C2×C3⋊Dic3—122no—
134C2×C3⋊Dic3—122no—
135C2×C3⋊Dic3—122no—
136C2×C3⋊Dic3C2×Dic312yes—
137C2×C3⋊Dic3#6712yes—
138C2×C3⋊Dic3#6712yes—
139C2×C3×Dic3—22no—
140C2×C3×Dic3—22no—
141C2×C3×Dic3—22no—
142C2×C3×Dic3—22no—
143C2×C3×Dic3—22no—
144C2×C3×Dic3—22no—
145C2×C3×Dic3—22no—
146C2×C3×Dic3—22no—
147C2×C3×Dic3—24no—
148C2×C3×Dic3—22no—
149C2×C3×Dic3—34no—
150C2×C3×Dic3—34no—
151C2×C3×Dic3—34no—
152C2×C3×Dic3—42no—
153C2×C3×Dic3—42no—
154C2×C3×Dic3—62no—
155C2×C3×Dic3—62no—
156C2×C3×Dic3—62no—
157C2×C3×Dic3—64no—
158C2×C3×Dic3—62no—
159C2×C3×Dic3—62no—
160C2×C3×Dic3—62no—
161C2×C3×Dic3—62no—
162C2×C3×Dic3—62no—
163C2×C3×Dic3—62no—
164C2×C3×Dic3—122no—
165C2×C3×Dic3—122no—
166C2×C3×Dic3Dic614yescomplemented
167C2×C3×Dic3C4×D314yes—
168C2×C3×Dic3C2×Dic314yes—
169C3×Dic6—616no—
170#627—28no—
171#391—271noSylow
172C2×C33C2×C22⋊C411yescharacteristic
173C2×C33C4×Q811yescharacteristic
174C2×C33C4×Q811yescharacteristic
175C2×C33C22⋊Q811yescharacteristic
176C2×C33C22⋊Q811yescharacteristic
177C2×C33#12411yescharacteristic
178C2×C33C4⋊Q811yescharacteristic
179C2×C3×Q8—94no—
180C2×C3×Q8—182no—
181C2×C3×Q8—182no—
182C22×C3×C4—32no—
183C22×C3×C4—32no—
184C22×C3×C4—32no—
185C22×C3×C4—61no—
186C22×C3×C4—62no—
187C22×C3×C4—91no—
188C22×C3×C4—92no—
189C22×Dic3—32no—
190C22×Dic3—32no—
191C22×Dic3—32no—
192C22×Dic3—61no—
193C22×Dic3—62no—
194C22×Dic3—91no—
195C22×Dic3—92no—
196C22×Dic3—181no—
197C22×Dic3—182no—
198C22×Dic3—361no—
199C22×Dic3D3212yes—
200C22×Dic3D3211yescharacteristic
201C2×Dic6—38no—
202C2×Dic6—64no—
203C2×Dic6—64no—
204C4⋊Dic3—182no—
205Dic3⋊C4—184no—
206C4×Dic3—182no—
207C4×Dic3—184no—
208C4×Dic3—184no—
209C22×C32—22no—
210C22×C32—22no—
211C22×C32—21no—
212C22×C32—22no—
213C22×C32—21no—
214C22×C32—22no—
215C22×C32—21no—
216C22×C32—21no—
217C22×C32—22no—
218C22×C32—21no—
219C22×C32—21no—
220C22×C32—22no—
221C22×C32—22no—
222C22×C32—21no—
223C22×C32—41no—
224C22×C32—41no—
225C22×C32—41no—
226C22×C32—41no—
227C22×C32—41no—
228C22×C32—41no—
229C22×C32—41no—
230C22×C32C2×Dic612yes—
231C22×C32C2×C4×D312yes—
232C22×C32#23512yes—
233C22×C32D3×D412yes—
234C22×C32D3×D412yes—
235C22×C32#23712yes—
236C22×C32D3×Q811yescharacteristic
237C22×C32D3×Q812yes—
238C22×C32D3×Q811yescharacteristic
239C22×C32#23911yescharacteristic
240C22×C32#23911yescharacteristic
241C22×C32C22×Dic311yescharacteristic
242C22×C32C2×#6711yescharacteristic
243C22×C32C2×#6711yescharacteristic
244C32×C4—62no—
245C32×C4—64no—
246C3⋊Dic3—42no—
247C3⋊Dic3—62no—
248C3⋊Dic3—64no—
249C3⋊Dic3—64no—
250C3⋊Dic3—124no—
251C3⋊Dic3—124no—
252C3⋊Dic3#21714yes—
253C3×Dic3—24no—
254C3×Dic3—24no—
255C3×Dic3—24no—
256C3×Dic3—24no—
257C3×Dic3—38no—
258C3×Dic3—44no—
259C3×Dic3—64no—
260C3×Dic3—124no—
261C3×Dic3—122no—
262C3×Dic3—122no—
263C22×Q8—91no—
264C2×C4⋊C4—271no—
265C2×C42—271no—
266#95—92no—
267#95—272no—
268C33#39111yescharacteristic, complemented, Sylow
269C23×C3—21no—
270C23×C3—22no—
271C23×C3—41no—
272C23×C3C2×D3212yes—
273C23×C3C2×D3211yescharacteristic
274C3×Q8—188no—
275C2×C3×C4—34no—
276C2×C3×C4—34no—
277C2×C3×C4—34no—
278C2×C3×C4—62no—
279C2×C3×C4—64no—
280C2×C3×C4—62no—
281C2×C3×C4—62no—
282C2×C3×C4—62no—
283C2×C3×C4—62no—
284C2×C3×C4—62no—
285C2×C3×C4—62no—
286C2×C3×C4—62no—
287C2×C3×C4—62no—
288C2×C3×C4—92no—
289C2×C3×C4—92no—
290C2×C3×C4—92no—
291C2×C3×C4—122no—
292C2×C3×C4—122no—
293C2×C3×C4—182no—
294C2×C3×C4—182no—
295C2×C3×C4—182no—
296C2×Dic3—21no—
297C2×Dic3—22no—
298C2×Dic3—22no—
299C2×Dic3—21no—
300C2×Dic3—34no—
301C2×Dic3—34no—
302C2×Dic3—34no—
303C2×Dic3—62no—
304C2×Dic3—62no—
305C2×Dic3—62no—
306C2×Dic3—64no—
307C2×Dic3—62no—
308C2×Dic3—62no—
309C2×Dic3—62no—
310C2×Dic3—62no—
311C2×Dic3—62no—
312C2×Dic3—62no—
313C2×Dic3—92no—
314C2×Dic3—94no—
315C2×Dic3—122no—
316C2×Dic3—122no—
317C2×Dic3—182no—
318C2×Dic3—184no—
319C2×Dic3—182no—
320C2×Dic3—182no—
321C2×Dic3—181no—
322C2×Dic3—182no—
323C2×Dic3—182no—
324C2×Dic3—181no—
325C2×Dic3—362no—
326C2×Dic3—362no—
327C2×Dic3—362no—
328C2×Dic3—362no—
329C2×Dic3—362no—
330C2×Dic3D3×Dic314yes—
331C2×Dic3#62712yescomplemented
332Dic6—616no—
333C2×C32—21no—
334C2×C32—22no—
335C2×C32—22no—
336C2×C32—22no—
337C2×C32—21no—
338C2×C32—22no—
339C2×C32—22no—
340C2×C32—22no—
341C2×C32—21no—
342C2×C32—21no—
343C2×C32—21no—
344C2×C32—21no—
345C2×C32—22no—
346C2×C32—21no—
347C2×C32—41no—
348C2×C32—41no—
349C2×C32—41no—
350C2×C32—41no—
351C2×C32—41no—
352C2×C32—41no—
353C2×C32—41no—
354C2×C32C4×Dic612yes—
355C2×C32#87812yes—
356C2×C32D3×C22⋊C412yes—
357C2×C32#88512yes—
358C2×C32#88712yes—
359C2×C32#88812yes—
360C2×C32#89612yes—
361C2×C32#94411yescharacteristic
362C2×C32Q8×Dic311yescharacteristic
363C2×C32Q8×Dic311yescharacteristic
364C2×C32#94611yescharacteristic
365C2×C32#94611yescharacteristic
366C2×C32#94711yescharacteristic
367C2×C32C2×#21711yescharacteristic
368C2×Q8—94no—
369C2×Q8—182no—
370C2×Q8—182no—
371C22×C4—31no—
372C22×C4—32no—
373C22×C4—91no—
374C22×C4—92no—
375C22×C4—271no—
376C4⋊C4—542no—
377C42—542no—
378C22×C3—21no—
379C22×C3—22no—
380C22×C3—21no—
381C22×C3—22no—
382C22×C3—21no—
383C22×C3—22no—
384C22×C3—21no—
385C22×C3—22no—
386C22×C3—22no—
387C22×C3—21no—
388C22×C3—22no—
389C22×C3—21no—
390C22×C3—22no—
391C22×C3—21no—
392C22×C3—41no—
393C22×C3—41no—
394C22×C3—41no—
395C22×C3—41no—
396C22×C3—41no—
397C22×C3—41no—
398C22×C3—41no—
399C22×C3D3×Dic612yes—
400C22×C3#379612yes—
401C22×C3#379711yescharacteristic
402C22×C3#379712yes—
403C22×C3#379811yescharacteristic
404C22×C3#379912yes—
405C22×C3C4×D3211yescharacteristic
406C22×C3#380211yescharacteristic
407C22×C3C2×D3×Dic312yes—
408C22×C3#380511yescharacteristic
409C22×C3C2×#62711yescharacteristic
410C22×C3D3×#6712yes—
411C22×C3D3×#6712yes—
412C22×C3#381111yescharacteristic
413C3×C4—64no—
414C3×C4—64no—
415C3×C4—64no—
416C3×C4—64no—
417C3×C4—94no—
418C3×C4—124no—
419C3×C4—362no—
420Dic3—24no—
421Dic3—22no—
422Dic3—38no—
423Dic3—64no—
424Dic3—124no—
425Dic3—122no—
426Dic3—122no—
427Dic3—184no—
428Dic3—182no—
429Dic3—184no—
430Dic3—364no—
431Dic3—364no—
432C32—22no—
433C32—21no—
434C32—41no—
435C32#464612yescomplemented
436C32#522911yescharacteristic, complemented
437C23D3311yescharacteristic
438Q8—188no—
439C2×C4—34no—
440C2×C4—32no—
441C2×C4—61no—
442C2×C4—62no—
443C2×C4—62no—
444C2×C4—61no—
445C2×C4—92no—
446C2×C4—92no—
447C2×C4—92no—
448C2×C4—182no—
449C2×C4—182no—
450C2×C4—182no—
451C2×C4—272no—
452C2×C4—541no—
453C2×C4—541no—
454C2×C3—22no—
455C2×C3—21no—
456C2×C3—21no—
457C2×C3—22no—
458C2×C3—22no—
459C2×C3—21no—
460C2×C3—22no—
461C2×C3—21no—
462C2×C3—21no—
463C2×C3—22no—
464C2×C3—21no—
465C2×C3—22no—
466C2×C3—22no—
467C2×C3—21no—
468C2×C3—41no—
469C2×C3—41no—
470C2×C3—41no—
471C2×C3—41no—
472C2×C3—41no—
473C2×C3—41no—
474C2×C3—41no—
475C2×C3#6383811yescharacteristic
476C2×C3#6383912yes—
477C2×C3Dic3×Dic612yes—
478C2×C3#6384312yes—
479C2×C3#6386511yescharacteristic
480C2×C3#6388712yes—
481C2×C3#6388812yes—
482C2×C3#6390712yes—
483C2×C3#6391511yescharacteristic
484C2×C3C4×#62711yescharacteristic
485C2×C3#6391911yescharacteristic
486C2×C3#6392111yescharacteristic
487C2×C3D3×#21712yes—
488C2×C3#6398211yescharacteristic
489C22D32×Dic311yescharacteristic
490C22#8895211yescharacteristic
491C22#8895311yescharacteristic
492C22D3×#62711yescharacteristic
493C22#8895811yescharacteristic
494C22#8896411yescharacteristic
495C22#8896511yescharacteristic
496C4—64no—
497C4—62no—
498C4—94no—
499C4—362no—
500C4—542no—
501C3—22no—
502C3—21no—
503C3—41no—
504C3#1059126112yescomplemented
505C3#1059139611yescharacteristic, complemented
506C2#1172290211yescharacteristic
507C2Dic3×#62711yescharacteristic
508C2#1172294111yescharacteristic
509C2#1172297611yescharacteristic
510C2#1172297711yescharacteristic
511C2#1172300711yescharacteristic
512C2#1172303511yescharacteristic
subgroupG/H
center, Frattini, hypercenterC23D33
commutatorC22×C33C22×C4
Fitting, socle, prime radical, cosocleC23×C33C23
hypocenterC33#391
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
15. C2×C3×#627C4semidirect
35. C2×C3⋊(C3⋊Dic3)Q8semidirect
47. C2×C32×Dic3Q8semidirect
75. C2×#627Dic3semidirect
166. C2×C3×Dic3Dic6semidirect
268. C33#391semidirect
331. C2×Dic3#627semidirect
435. C32#4646semidirect
436. C32#5229semidirect
504. C3#10591261semidirect
505. C3#10591396semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#6272
C2×#888152
C2×Dic3×C3⋊Dic32
#117220612
C3×#635792
#105913963
#105912613