Finite Group Databasebeta

← #13150369

Subgroups — #13150369

Order 1728.

888 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×Dic18C211yesmaximal, characteristic, complemented
2C2×#88662C211yesmaximal, characteristic
3#11721818C211yesmaximal, characteristic, complemented
4C9×#929C211yesmaximal, characteristic, complemented
5#11721265C211yesmaximal, characteristic, complemented
6#11721255C211yesmaximal, characteristic, complemented
7#11721245C211yesmaximal, characteristic, complemented
8#11721035C212yesmaximal, complemented
9#11721018C212yesmaximal, complemented
10#11720985C212yesmaximal, complemented
11#11720904C212yesmaximal, complemented
12#10594897—31nomaximal
13#10589750—31nomaximal
14C22×C3×C4×C9C2211yescharacteristic
15C2×#6233C2211yescharacteristic
16C2×C9×#67C2211yescharacteristic
17C22×C3×Dic9C2211yescharacteristic
18C22×C3×Dic9C2211yescharacteristic
19C2×C3×Dic18—24no—
20C2×C3×Dic18C2212yes—
21C2×C3×Dic18C2212yes—
22C2×#6195C2211yescharacteristic
23#88662—24no—
24C2×D3×Dic9C2211yescharacteristic
25C2×(C3⋊D9)⋊C4C2211yescharacteristic
26C2×#6189C2211yescharacteristic
27#88551C2211yescharacteristic
28#88536C2212yes—
29#88534C2212yes—
30C9×#217C2211yescharacteristic
31C9×(C2×D3)⋊C4C2212yes—
32C9×Dic3⋊C4C2212yes—
33(C2×D3)⋊Dic9C2212yes—
34#88445C2212yes—
35Dic3⋊Dic9C2212yes—
36#88442C2212yes—
37Dic9⋊Dic3C2212yes—
38Dic3×Dic9C2212yes—
39C22×C3×Dic6—31no—
40C2×#3804—31no—
41#64138—31no—
42C3×#929D311yescharacteristic
43#63966—31no—
44#63961—31no—
45#63955—31no—
46#63907—32no—
47#63888—32no—
48#63887—32no—
49#63839—32no—
50C2×#3699—31no—
51C22×Dic18D311yescharacteristic, complemented
52C9×#123—31no—
53#63497—31no—
54#63491—31no—
55#63491—61no—
56#63458—32no—
57#63450—32no—
58#63448—31no—
59#63447—32no—
60#63440—32no—
61#63439—31no—
62C23×C3×C9C2311yescharacteristic
63C22×C3⋊D9C2311yescharacteristic
64C22×D3×C9C2311yescharacteristic
65C2×C3×C4×C9—22no—
66C2×C3×C4×C9C2312yes—
67#6233—22no—
68C2×C3⋊Dic9C2311yescharacteristic
69C2×C3⋊Dic9C2312yes—
70C9×#67—22no—
71C2×C9×Dic3C2311yescharacteristic
72C2×C9×Dic3C2312yes—
73C2×C3×Dic9—22no—
74C2×C3×Dic9D414yes—
75C2×C3×Dic9C2312yes—
76C2×C3×Dic9C2311yescharacteristic
77C2×C3×Dic9C2311yescharacteristic
78C3×Dic18—24no—
79C3×Dic18—24no—
80#6195—22no—
81D3×Dic9—22no—
82(C3⋊D9)⋊C4—22no—
83#6189—22no—
84#5816—91no—
85#5648—91no—
86C22×C32×C4C2×D311yescharacteristic
87C2×#638—31no—
88C2×C3×#67C2×D311yescharacteristic
89C22×C3×Dic3—31no—
90C22×C3×Dic3—31no—
91C2×C3×Dic6—32no—
92C2×C3×Dic6—32no—
93C2×C3×Dic6—64no—
94C2×#627—31no—
95C2×#626—31no—
96C2×#624—31no—
97#3804—64no—
98C2×D3×Dic3—31no—
99#3757—31no—
100#3752—32no—
101#3750—32no—
102C3×#217C2×D311yescharacteristic
103C3×(C2×D3)⋊C4C2×D312yes—
104C3×Dic3⋊C4C2×D312yes—
105#3724—32no—
106Dic3⋊Dic3—32no—
107Dic3⋊Dic3—32no—
108#3722—32no—
109(C2×D3)⋊Dic3—32no—
110Dic32—32no—
111C2×D4×C9—31no—
112C22×C4×C9—21no—
113C22×C4×C9C2×D311yescharacteristic
114C2×#611—31no—
115C2×#611—31no—
116C2×#611—61no—
117C22×Dic9—31no—
118C22×Dic9C2×D311yescharacteristic
119C22×Dic9C2×D311yescharacteristic
120#3699—64no—
121C2×C4×D9—31no—
122C2×Dic18—24no—
123C2×Dic18—31no—
124C2×Dic18C2×D312yes—
125C2×Dic18C2×D312yes—
126C9×C4⋊C4—32no—
127C9×C22⋊C4—31no—
128C9×C22⋊C4—32no—
129#3676—31no—
130#3676—32no—
131#3676—61no—
132(C2×D9)⋊C4—32no—
133(C2×D9)⋊C4—32no—
134(C2×D9)⋊C4—62no—
135C4⋊Dic9—32no—
136Dic9⋊C4—32no—
137Dic9⋊C4—32no—
138Dic9⋊C4—32no—
139Dic9⋊C4—62no—
140C4×Dic9—32no—
141C22×C3×C9—22no—
142C22×C3×C9C2×D412yes—
143C22×C3×C9C2411yescharacteristic
144C2×C3⋊D9—21no—
145C2×C3⋊D9—22no—
146C2×D3×C9—21no—
147C2×D3×C9—22no—
148C3×C4×C9—22no—
149C3⋊Dic9—21no—
150C3⋊Dic9—22no—
151C9×Dic3—21no—
152C9×Dic3—22no—
153C3×Dic9—22no—
154C3×Dic9C2×D412yes—
155C3×Dic9C2×D412yes—
156C22×C3×Q8—91no—
157C2×#237—91no—
158C2×#235—91no—
159C22×Dic6—31no—
160C3×#123—31no—
161#947—92no—
162#946—92no—
163#946—92no—
164#944—92no—
165#935—91no—
166#929—91no—
167#929—91no—
168#929—91no—
169#929—91no—
170#929—181no—
171#929D911yescharacteristic, complemented
172#926—91no—
173#896—92no—
174#888—92no—
175#886—91no—
176#885—92no—
177#878—92no—
178#877—91no—
179C23×C32C22×D311yescharacteristic
180C22×C3⋊D3—31no—
181C22×C3×D3C22×D311yescharacteristic
182C2×C32×C4—22no—
183C2×C32×C4C22×D312yes—
184#638—62no—
185C2×C3⋊Dic3—32no—
186C2×C3⋊Dic3—31no—
187C3×#67—22no—
188C2×C3×Dic3—34no—
189C2×C3×Dic3—31no—
190C2×C3×Dic3—31no—
191C2×C3×Dic3—32no—
192C2×C3×Dic3—62no—
193C2×C3×Dic3C22×D312yes—
194C2×C3×Dic3C22×D311yescharacteristic
195C3×Dic6—64no—
196C3×Dic6—64no—
197#627—62no—
198#626—62no—
199#624—62no—
200D3×Dic3—62no—
201C23×C9—21no—
202C23×C9—31no—
203C23×C9C22×D311yescharacteristic
204C22×D9—31no—
205C22×D9—61no—
206D4×C9—62no—
207C2×C4×C9—22no—
208C2×C4×C9—22no—
209C2×C4×C9—32no—
210C2×C4×C9—31no—
211C2×C4×C9—42no—
212C2×C4×C9C22×D312yes—
213#611—62no—
214#611—62no—
215#611—122no—
216C2×Dic9—22no—
217C2×Dic9—32no—
218C2×Dic9—31no—
219C2×Dic9—61no—
220C2×Dic9—62no—
221C2×Dic9—62no—
222C2×Dic9#6714yes—
223C2×Dic9C22×D312yes—
224C2×Dic9C22×D311yescharacteristic
225C2×Dic9C22×D311yescharacteristic
226C4×D9—62no—
227Dic18—24no—
228Dic18—24no—
229Dic18—62no—
230#536—271noSylow
231C2×C3×C9—22no—
232C2×C3×C9C22×D411yescharacteristic
233C2×C3×C92−1+412yes—
234C3⋊D9—41no—
235D3×C9—41no—
236C2×C3×Q8—92no—
237C2×C3×Q8—92no—
238C2×C3×Q8—184no—
239C2×C3×D4—31no—
240C22×C3×C4—21no—
241C22×C3×C4—91no—
242C22×C3×C4—91no—
243C22×C3×C4C2×D911yescharacteristic
244C22×C3×C4D3211yescharacteristic
245C2×#67—91no—
246C2×#67—91no—
247C2×#67—91no—
248C2×#67—181no—
249C2×#67C2×D911yescharacteristic
250C22×Dic3—31no—
251C22×Dic3—31no—
252C22×Dic3—91no—
253#237—184no—
254#235—184no—
255C2×D12—91no—
256C2×C4×D3—91no—
257C2×C4×D3—91no—
258C2×C4×D3—91no—
259C2×Dic6—32no—
260C2×Dic6—32no—
261C2×Dic6—64no—
262C2×Dic6—91no—
263C2×Dic6—91no—
264C3×C4⋊C4—32no—
265C3×C22⋊C4—31no—
266C3×C22⋊C4—32no—
267#217—92no—
268#217—91no—
269#217—91no—
270#217—181no—
271#217C2×D911yescharacteristic
272(C2×D3)⋊C4—92no—
273(C2×D3)⋊C4—92no—
274(C2×D3)⋊C4—92no—
275(C2×D3)⋊C4—92no—
276(C2×D3)⋊C4—92no—
277(C2×D3)⋊C4—182no—
278(C2×D3)⋊C4C2×D912yes—
279C4⋊Dic3—92no—
280C4⋊Dic3—92no—
281Dic3⋊C4—92no—
282Dic3⋊C4—92no—
283Dic3⋊C4—92no—
284Dic3⋊C4—92no—
285Dic3⋊C4—92no—
286Dic3⋊C4—92no—
287Dic3⋊C4—182no—
288Dic3⋊C4C2×D912yes—
289C4×Dic3—92no—
290C4×Dic3—92no—
291C22×C32—22no—
292C22×C32D3×D412yes—
293C22×C32C23×D311yescharacteristic
294C2×C3⋊D3—61no—
295C2×C3⋊D3—62no—
296C2×C3×D3—21no—
297C2×C3×D3—22no—
298C32×C4—22no—
299C3⋊Dic3—61no—
300C3⋊Dic3—62no—
301C3×Dic3—21no—
302C3×Dic3—22no—
303C3×Dic3—32no—
304C3×Dic3—32no—
305C3×Dic3—62no—
306C22×C9—22no—
307C22×C9—21no—
308C22×C9—22no—
309C22×C9—42no—
310C22×C9—62no—
311C22×C9—61no—
312C22×C9C2×#6712yes—
313C22×C9C23×D311yescharacteristic
314C2×D9—62no—
315C2×D9—61no—
316C2×D9—121no—
317C2×D9—122no—
318C4×C9—22no—
319C4×C9—42no—
320C4×C9—62no—
321C4×C9—61no—
322Dic9—22no—
323Dic9—62no—
324Dic9—61no—
325Dic9—122no—
326Dic9—121no—
327Dic9C2×#6712yes—
328Dic9C2×#6712yes—
329C2×#41—271no—
330C22×Q8—91no—
331C4⋊Q8—272no—
332#124—272no—
333#123—31no—
334#123—271no—
335#123—271no—
336#123—271no—
337C22⋊Q8—272no—
338C22⋊Q8—272no—
339#117—271no—
340C3×C9#53611yescharacteristic, complemented, Sylow
341C23×C3—21no—
342C23×C3—31no—
343C23×C3C22×D911yescharacteristic
344C23×C3C2×D3211yescharacteristic
345C22×D3—91no—
346C22×D3—91no—
347C22×D3—181no—
348C22×D3C22×D911yescharacteristic
349C3×Q8—184no—
350C3×Q8—184no—
351C3×D4—62no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—32no—
356C2×C3×C4—31no—
357C2×C3×C4—42no—
358C2×C3×C4—94no—
359C2×C3×C4—91no—
360C2×C3×C4—91no—
361C2×C3×C4—92no—
362C2×C3×C4—182no—
363C2×C3×C4C22×D912yes—
364C2×C3×C4C2×D3212yes—
365#67—22no—
366#67—182no—
367#67—182no—
368#67—182no—
369#67—362no—
370C2×Dic3—31no—
371C2×Dic3—31no—
372C2×Dic3—34no—
373C2×Dic3—32no—
374C2×Dic3—62no—
375C2×Dic3—92no—
376C2×Dic3—91no—
377C2×Dic3—92no—
378C2×Dic3—91no—
379C2×Dic3—181no—
380C2×Dic3—182no—
381C2×Dic3—182no—
382C2×Dic3C22×D912yes—
383C2×Dic3C22×D911yescharacteristic
384D12—182no—
385C4×D3—182no—
386C4×D3—182no—
387C4×D3—182no—
388Dic6—64no—
389Dic6—64no—
390Dic6—182no—
391Dic6—182no—
392C2×C32—22no—
393C2×C32C2×D3×D411yescharacteristic
394C2×C32#101012yes—
395C3⋊D3—121no—
396C3×D3—41no—
397C2×C9—22no—
398C2×C9—22no—
399C2×C9—21no—
400C2×C9—42no—
401C2×C9—121no—
402C2×C9#100712yes—
403C2×C9C22×#6711yescharacteristic
404D9—121no—
405D9—241no—
406#41—544no—
407C2×Q8—92no—
408C2×Q8—92no—
409C2×Q8—184no—
410C2×Q8—271no—
411C2×D4—31no—
412C2×D4—271no—
413C2×D4—271no—
414C22×C4—91no—
415C22×C4—91no—
416C22×C4—271no—
417C22×C4—271no—
418C22×C4D3×D911yescharacteristic
419C4⋊C4—32no—
420C4⋊C4—272no—
421C4⋊C4—272no—
422C4⋊C4—272no—
423C4⋊C4—272no—
424C22⋊C4—31no—
425C22⋊C4—32no—
426C22⋊C4—272no—
427C22⋊C4—272no—
428C22⋊C4—271no—
429C22⋊C4—272no—
430C42—272no—
431C22×C3—22no—
432C22×C3—22no—
433C22×C3—21no—
434C22×C3—22no—
435C22×C3—42no—
436C22×C3—62no—
437C22×C3—61no—
438C22×C3D4×D912yes—
439C22×C3C23×D911yescharacteristic
440C22×C3D3×#6712yes—
441C22×C3C22×D3211yescharacteristic
442C2×D3—21no—
443C2×D3—22no—
444C2×D3—182no—
445C2×D3—181no—
446C2×D3—181no—
447C2×D3—182no—
448C2×D3—361no—
449C2×D3—362no—
450C3×C4—22no—
451C3×C4—22no—
452C3×C4—42no—
453C3×C4—62no—
454C3×C4—61no—
455C3×C4—92no—
456C3×C4—92no—
457C3×C4—182no—
458Dic3—21no—
459Dic3—22no—
460Dic3—32no—
461Dic3—32no—
462Dic3—62no—
463Dic3—182no—
464Dic3—181no—
465Dic3—181no—
466Dic3—182no—
467Dic3—361no—
468Dic3—362no—
469C32#564811yescharacteristic
470C9—21no—
471C9#581611yescharacteristic, complemented
472C23—31no—
473C23—271no—
474C23C2×D3×D911yescharacteristic
475Q8—184no—
476Q8—184no—
477Q8—542no—
478D4—62no—
479D4—542no—
480D4—542no—
481C2×C4—22no—
482C2×C4—32no—
483C2×C4—31no—
484C2×C4—91no—
485C2×C4—91no—
486C2×C4—94no—
487C2×C4—92no—
488C2×C4—182no—
489C2×C4—272no—
490C2×C4—271no—
491C2×C4—542no—
492C2×C4—542no—
493C2×C4C2×D3×D912yes—
494C2×C3—22no—
495C2×C3—22no—
496C2×C3—21no—
497C2×C3—22no—
498C2×C3—42no—
499C2×C3—121no—
500C2×C3C2×D4×D911yescharacteristic
501C2×C3#6371612yes—
502C2×C3#6429712yes—
503C2×C3C2×D3×#6711yescharacteristic
504D3—41no—
505D3—361no—
506D3—361no—
507D3—721no—
508C22—22no—
509C22—61no—
510C22—62no—
511C22—541no—
512C22—542no—
513C22D9×#6712yes—
514C22C22×D3×D911yescharacteristic
515C4—22no—
516C4—61no—
517C4—62no—
518C4—92no—
519C4—92no—
520C4—182no—
521C4—541no—
522C4—542no—
523C3—21no—
524C3#1058975011yescharacteristic, complemented
525C3#1059489711yescharacteristic
526C2—22no—
527C2—121no—
528C2—1081no—
529C2#1172246412yes—
530C2C2×D9×#6711yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
prime radicalC2×#88662C2
hypercenterC22×C4D3×D9
hypocenterC3×C9#536
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C3×Dic18C2semidirect
3. #11721818C2semidirect
4. C9×#929C2semidirect
5. #11721265C2semidirect
6. #11721255C2semidirect
7. #11721245C2semidirect
8. #11721035C2semidirect
9. #11721018C2semidirect
10. #11720985C2semidirect
11. #11720904C2semidirect
51. C22×Dic18D3semidirect
171. #929D9semidirect
340. C3×C9#536semidirect
471. C9#5816semidirect
524. C3#10589750semidirect

Maximal subgroups

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

subgroupindex
C22×C3×Dic182
C2×#886622
#117218182
C9×#9292
#117212652
#117212552
#117212452
#117210352
#117210182
#117209852
#117209042
#105948973
#105897503