Finite Group Databasebeta

← #13150920

Subgroups — #13150920

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
1C2×C9×#237C211yesmaximal, characteristic
2C22×#6189C211yesmaximal, characteristic, complemented
3#11721835C211yesmaximal, characteristic, complemented
4C3×#63497C211yesmaximal, characteristic, complemented
5#11721266C211yesmaximal, characteristic, complemented
6#11721265C211yesmaximal, characteristic, complemented
7#11721263C212yesmaximal, complemented
8#11721249C212yesmaximal, complemented
9#11721244C211yesmaximal, characteristic, complemented
10#11720955C212yesmaximal, complemented
11#11720903C212yesmaximal, complemented
12#10595270—31nomaximal
13#10589927—31nomaximal
14C2×C3×D4×C9C2211yescharacteristic
15C22×C3⋊Dic9C2211yescharacteristic
16C2×C9×#67C2211yescharacteristic
17C2×C9×#67C2211yescharacteristic
18C22×C9×Dic3C2211yescharacteristic
19C22×C9×Dic3C2211yescharacteristic
20C22×C3×Dic9C2211yescharacteristic
21C9×#237—24no—
22C2×C4×D3×C9C2211yescharacteristic
23C2×C9×Dic6C2211yescharacteristic
24C2×#6189—24no—
25C2×#6189C2212yes—
26C2×#6189C2212yes—
27#88551C2211yescharacteristic
28#88551C2212yes—
29#88534C2212yes—
30C3×#3676C2211yescharacteristic
31C3×#3676C2212yes—
32C3×Dic9⋊C4C2212yes—
33(C2×D3)⋊Dic9C2212yes—
34(C2×D3)⋊Dic9C2212yes—
35Dic3⋊Dic9C2212yes—
36#88442C2212yes—
37Dic9⋊Dic3C2212yes—
38Dic3×Dic9C2212yes—
39C2×C3×#237D311yescharacteristic
40C22×#627—31no—
41#64144—31no—
42C3×#935—31no—
43#63964—32no—
44#63962—31no—
45#63961—31no—
46#63959—32no—
47#63955—31no—
48#63851—32no—
49#63839—32no—
50C2×C9×#41—31no—
51C22×Dic18—31no—
52#63506—32no—
53#63498—32no—
54#63497—31no—
55#63497—31no—
56#63497—31no—
57#63497—61no—
58#63497D311yescharacteristic, complemented
59#63488—31no—
60#63486—32no—
61#63486—32no—
62C23×C3×C9C2311yescharacteristic
63C23×C3×C9C2311yescharacteristic
64C22×D3×C9C2311yescharacteristic
65C3×D4×C9—22no—
66C2×C3×C4×C9C2311yescharacteristic
67C2×C3⋊Dic9—22no—
68C2×C3⋊Dic9C2312yes—
69C2×C3⋊Dic9C2312yes—
70C9×#67—22no—
71C9×#67—22no—
72C2×C9×Dic3—22no—
73C2×C9×Dic3D414yes—
74C2×C9×Dic3C2311yescharacteristic
75C2×C9×Dic3C2311yescharacteristic
76C2×C9×Dic3C2311yescharacteristic
77C2×C3×Dic9—22no—
78C2×C3×Dic9C2312yes—
79C2×C3×Dic9C2312yes—
80C4×D3×C9—22no—
81C9×Dic6—22no—
82#6189—24no—
83#6189—24no—
84#5825—91no—
85#5648—91no—
86C2×C32×D4C2×D311yescharacteristic
87C22×C3⋊Dic3—31no—
88C2×C3×#67C2×D311yescharacteristic
89C2×C3×#67C2×D311yescharacteristic
90C22×C3×Dic3—31no—
91C22×C3×Dic3C2×D311yescharacteristic
92C22×C3×Dic3C2×D311yescharacteristic
93C3×#237—24no—
94C2×C3×C4×D3C2×D311yescharacteristic
95C2×C3×Dic6C2×D311yescharacteristic
96C2×#627—32no—
97C2×#627—32no—
98C2×#627—64no—
99#3757—31no—
100#3757—32no—
101#3750—32no—
102C3×#217—31no—
103C3×#217—32no—
104C3×Dic3⋊C4—32no—
105#3724—32no—
106Dic3⋊Dic3—32no—
107Dic3⋊Dic3—32no—
108(C2×D3)⋊Dic3—32no—
109(C2×D3)⋊Dic3—32no—
110Dic32—32no—
111C9×#41—64no—
112C2×Q8×C9—31no—
113C2×D4×C9—21no—
114C2×D4×C9—31no—
115C2×D4×C9—31no—
116C2×D4×C9C2×D311yescharacteristic
117C22×C4×C9—31no—
118C22×C4×C9—31no—
119C22×C4×C9—31no—
120C22×Dic9—31no—
121C22×Dic9—61no—
122C22×Dic9C2×D311yescharacteristic
123C2×Dic18—32no—
124C2×Dic18—32no—
125C2×Dic18—64no—
126#3676—31no—
127#3676—32no—
128#3676—32no—
129#3676—32no—
130#3676—61no—
131#3676—62no—
132#3676C2×D311yescharacteristic
133#3676C2×D312yes—
134C4⋊Dic9—32no—
135Dic9⋊C4—32no—
136Dic9⋊C4—32no—
137Dic9⋊C4—32no—
138Dic9⋊C4—62no—
139Dic9⋊C4C2×D312yes—
140C4×Dic9—32no—
141C22×C3×C9—22no—
142C22×C3×C9—21no—
143C22×C3×C9—22no—
144C22×C3×C9C2×D412yes—
145C22×C3×C9C2411yescharacteristic
146C2×D3×C9—21no—
147C2×D3×C9—22no—
148C3×C4×C9—21no—
149C3⋊Dic9—22no—
150C3⋊Dic9—22no—
151C9×Dic3—21no—
152C9×Dic3C2×D412yes—
153C9×Dic3C2×D412yes—
154C3×Dic9—22no—
155C3×Dic9—22no—
156C2×C3×#41—31no—
157C2×#237D911yescharacteristic
158C22×Dic6—91no—
159C22×Dic6—91no—
160C3×#123—91no—
161#944—92no—
162#936—92no—
163#935—31no—
164#935—91no—
165#935—91no—
166#935—91no—
167#935—91no—
168#935—181no—
169#929—91no—
170#926—91no—
171#924—92no—
172#924—92no—
173#896—92no—
174#888—92no—
175#886—91no—
176#885—92no—
177#878—92no—
178#877—91no—
179C23×C32C22×D311yescharacteristic
180C23×C32C22×D311yescharacteristic
181C22×C3×D3C22×D311yescharacteristic
182C32×D4—22no—
183C2×C32×C4C22×D311yescharacteristic
184C2×C3⋊Dic3—32no—
185C2×C3⋊Dic3—32no—
186C2×C3⋊Dic3—62no—
187C3×#67—22no—
188C3×#67—22no—
189C2×C3×Dic3—22no—
190C2×C3×Dic3—32no—
191C2×C3×Dic3—32no—
192C2×C3×Dic3—62no—
193C2×C3×Dic3#6714yes—
194C2×C3×Dic3C22×D311yescharacteristic
195C2×C3×Dic3C22×D311yescharacteristic
196C2×C3×Dic3C22×D311yescharacteristic
197C3×C4×D3—22no—
198C3×Dic6—22no—
199#627—64no—
200#627—64no—
201C23×C9—21no—
202C23×C9—21no—
203C23×C9—31no—
204C23×C9C22×D311yescharacteristic
205C23×C9C22×D311yescharacteristic
206Q8×C9—62no—
207D4×C9—22no—
208D4×C9—42no—
209D4×C9—62no—
210D4×C9—62no—
211C2×C4×C9—21no—
212C2×C4×C9—34no—
213C2×C4×C9—31no—
214C2×C4×C9—31no—
215C2×C4×C9—31no—
216C2×C4×C9—62no—
217C2×C4×C9—62no—
218C2×C4×C9C22×D311yescharacteristic
219C2×Dic9—22no—
220C2×Dic9—32no—
221C2×Dic9—32no—
222C2×Dic9—62no—
223C2×Dic9—62no—
224C2×Dic9—62no—
225C2×Dic9—122no—
226C2×Dic9C22×D312yes—
227C2×Dic9C22×D312yes—
228Dic18—64no—
229Dic18—64no—
230#536—271noSylow
231C2×C3×C9—22no—
232C2×C3×C9—41no—
233C2×C3×C9C22×D411yescharacteristic
234C2×C3×C92−1+412yes—
235D3×C9—41no—
236C3×#41—64no—
237C2×C3×Q8—31no—
238C2×C3×D4—21no—
239C2×C3×D4—31no—
240C2×C3×D4—31no—
241C2×C3×D4C2×D911yescharacteristic
242C2×C3×D4D3211yescharacteristic
243C22×C3×C4—31no—
244C22×C3×C4—31no—
245C22×C3×C4—31no—
246C22×C3×C4—91no—
247C2×#67C2×D911yescharacteristic
248C2×#67C2×D911yescharacteristic
249C22×Dic3—31no—
250C22×Dic3—91no—
251C22×Dic3—91no—
252C22×Dic3—181no—
253C22×Dic3C2×D911yescharacteristic
254C22×Dic3C2×D911yescharacteristic
255#237—24no—
256C2×C4×D3C2×D911yescharacteristic
257C2×Dic6—92no—
258C2×Dic6—92no—
259C2×Dic6—92no—
260C2×Dic6—92no—
261C2×Dic6—184no—
262C2×Dic6—184no—
263C2×Dic6C2×D911yescharacteristic
264C3×C4⋊C4—92no—
265C3×C22⋊C4—91no—
266C3×C22⋊C4—92no—
267#217—32no—
268#217—31no—
269#217—92no—
270#217—92no—
271#217—91no—
272#217—92no—
273#217—92no—
274#217—91no—
275#217—181no—
276#217—182no—
277(C2×D3)⋊C4—92no—
278(C2×D3)⋊C4—92no—
279C4⋊Dic3—92no—
280C4⋊Dic3—92no—
281Dic3⋊C4—32no—
282Dic3⋊C4—92no—
283Dic3⋊C4—92no—
284Dic3⋊C4—92no—
285Dic3⋊C4—92no—
286Dic3⋊C4—92no—
287Dic3⋊C4—92no—
288Dic3⋊C4—182no—
289C4×Dic3—92no—
290C4×Dic3—92no—
291C22×C32—22no—
292C22×C32—22no—
293C22×C32—21no—
294C22×C32C2×#6712yes—
295C22×C32C23×D311yescharacteristic
296C2×C3×D3—22no—
297C2×C3×D3—21no—
298C32×C4—21no—
299C3⋊Dic3—62no—
300C3⋊Dic3—62no—
301C3×Dic3—21no—
302C3×Dic3—62no—
303C3×Dic3—62no—
304C3×Dic3C2×#6712yes—
305C3×Dic3C2×#6712yes—
306C22×C9—22no—
307C22×C9—21no—
308C22×C9—22no—
309C22×C9—21no—
310C22×C9—22no—
311C22×C9—42no—
312C22×C9—41no—
313C22×C9—42no—
314C22×C9—61no—
315C22×C9—62no—
316C22×C9D3×D412yes—
317C22×C9C23×D311yescharacteristic
318C4×C9—21no—
319C4×C9—32no—
320C4×C9—32no—
321C4×C9—41no—
322C4×C9—61no—
323Dic9—22no—
324Dic9—22no—
325Dic9—62no—
326Dic9—62no—
327Dic9—122no—
328Dic9—122no—
329C2×#41—31no—
330C22×Q8—271no—
331C4⋊Q8—272no—
332#124—272no—
333#123—91no—
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—21no—
343C23×C3—31no—
344C23×C3C22×D911yescharacteristic
345C23×C3C22×D911yescharacteristic
346C23×C3C2×D3211yescharacteristic
347C23×C3C2×D3211yescharacteristic
348C22×D3C22×D911yescharacteristic
349C3×Q8—62no—
350C3×D4—22no—
351C3×D4—22no—
352C3×D4—42no—
353C3×D4—62no—
354C3×D4—62no—
355C2×C3×C4—21no—
356C2×C3×C4—31no—
357C2×C3×C4—31no—
358C2×C3×C4—34no—
359C2×C3×C4—31no—
360C2×C3×C4—62no—
361C2×C3×C4—62no—
362C2×C3×C4—92no—
363C2×C3×C4—92no—
364C2×C3×C4—182no—
365C2×C3×C4C22×D911yescharacteristic
366C2×C3×C4C2×D3211yescharacteristic
367#67—22no—
368#67—22no—
369C2×Dic3—22no—
370C2×Dic3—32no—
371C2×Dic3—32no—
372C2×Dic3—62no—
373C2×Dic3—92no—
374C2×Dic3—92no—
375C2×Dic3—92no—
376C2×Dic3—92no—
377C2×Dic3—182no—
378C2×Dic3—182no—
379C2×Dic3—182no—
380C2×Dic3—182no—
381C2×Dic3—362no—
382C2×Dic3#61114yes—
383C2×Dic3C22×D911yescharacteristic
384C2×Dic3C22×D911yescharacteristic
385C2×Dic3C22×D911yescharacteristic
386C4×D3—22no—
387Dic6—22no—
388Dic6—184no—
389Dic6—184no—
390Dic6—184no—
391Dic6—184no—
392C2×C32—22no—
393C2×C32—41no—
394C2×C32#101012yes—
395C2×C32C22×#6711yescharacteristic
396C3×D3—41no—
397C2×C9—22no—
398C2×C9—22no—
399C2×C9—21no—
400C2×C9—42no—
401C2×C9—41no—
402C2×C9—81no—
403C2×C9—121no—
404C2×C9C2×D3×D411yescharacteristic
405C2×C9#101012yes—
406#41—64no—
407C2×Q8—31no—
408C2×Q8—272no—
409C2×Q8—272no—
410C2×Q8—544no—
411C2×D4—31no—
412C2×D4—31no—
413C2×D4D3×D911yescharacteristic
414C22×C4—31no—
415C22×C4—31no—
416C22×C4—31no—
417C22×C4—91no—
418C22×C4—271no—
419C4⋊C4—92no—
420C4⋊C4—272no—
421C4⋊C4—272no—
422C4⋊C4—272no—
423C4⋊C4—272no—
424C22⋊C4—91no—
425C22⋊C4—92no—
426C22⋊C4—272no—
427C22⋊C4—272no—
428C22⋊C4—271no—
429C22⋊C4—272no—
430C42—272no—
431C22×C3—22no—
432C22×C3—22no—
433C22×C3—22no—
434C22×C3—21no—
435C22×C3—22no—
436C22×C3—21no—
437C22×C3—21no—
438C22×C3—22no—
439C22×C3—42no—
440C22×C3—41no—
441C22×C3—42no—
442C22×C3—62no—
443C22×C3—61no—
444C22×C3C2×#61112yes—
445C22×C3C23×D911yescharacteristic
446C22×C3D3×#6712yes—
447C22×C3C22×D3211yescharacteristic
448C2×D3—22no—
449C2×D3—21no—
450C3×C4—21no—
451C3×C4—21no—
452C3×C4—32no—
453C3×C4—32no—
454C3×C4—41no—
455C3×C4—61no—
456C3×C4—182no—
457C3×C4—182no—
458Dic3—21no—
459Dic3—62no—
460Dic3—62no—
461Dic3—182no—
462Dic3—182no—
463Dic3—182no—
464Dic3—182no—
465Dic3—362no—
466Dic3—362no—
467Dic3C2×#61112yes—
468Dic3C2×#61112yes—
469C32#582511yescharacteristic
470C9—21no—
471C9#564811yescharacteristic, complemented
472C23—31no—
473C23C2×D3×D911yescharacteristic
474C23C2×D3×D911yescharacteristic
475Q8—62no—
476Q8—544no—
477Q8—544no—
478D4—22no—
479D4—62no—
480D4—62no—
481C2×C4—31no—
482C2×C4—31no—
483C2×C4—34no—
484C2×C4—31no—
485C2×C4—62no—
486C2×C4—62no—
487C2×C4—92no—
488C2×C4—92no—
489C2×C4—182no—
490C2×C4—272no—
491C2×C4—272no—
492C2×C4—542no—
493C2×C4C2×D3×D911yescharacteristic
494C2×C3—22no—
495C2×C3—22no—
496C2×C3—21no—
497C2×C3—22no—
498C2×C3—42no—
499C2×C3—41no—
500C2×C3—41no—
501C2×C3—81no—
502C2×C3—121no—
503C2×C3#6371612yes—
504C2×C3C22×#61111yescharacteristic
505C2×C3#6430912yes—
506C2×C3C2×D3×#6711yescharacteristic
507D3—41no—
508C22—22no—
509C22—21no—
510C22—22no—
511C22—61no—
512C22—62no—
513C22D3×#61112yes—
514C22C22×D3×D911yescharacteristic
515C4—21no—
516C4—32no—
517C4—32no—
518C4—61no—
519C4—182no—
520C4—182no—
521C4—542no—
522C4—542no—
523C3—21no—
524C3#1058992711yescharacteristic, complemented
525C3#1059527011yescharacteristic
526C2—22no—
527C2—41no—
528C2—121no—
529C2#1172249412yes—
530C2C2×D3×#61111yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC2×C3×D4×C9C22
socleC22×C32C23×D3
prime radicalC2×C3×#237D3
hypercenterC2×D4D3×D9
hypocenterC3×C9#536
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C22×#6189C2semidirect
3. #11721835C2semidirect
4. C3×#63497C2semidirect
5. #11721266C2semidirect
6. #11721265C2semidirect
7. #11721263C2semidirect
8. #11721249C2semidirect
9. #11721244C2semidirect
10. #11720955C2semidirect
11. #11720903C2semidirect
58. #63497D3semidirect
340. C3×C9#536semidirect
471. C9#5648semidirect
524. C3#10589927semidirect

Maximal subgroups

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

subgroupindex
C2×C9×#2372
C22×#61892
#117218352
C3×#634972
#117212662
#117212652
#117212632
#117212492
#117212442
#117209552
#117209032
#105952703
#105899273