Finite Group Databasebeta

← #13148025

Subgroups — #13148025

Order 1728.

886 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88663C211yesmaximal, characteristic
2C22×#6189C211yesmaximal, characteristic, complemented
3#11721723C211yesmaximal, characteristic, complemented
4C9×#886C211yesmaximal, characteristic, complemented
5C3×#63448C211yesmaximal, characteristic, complemented
6#11721265C211yesmaximal, characteristic, complemented
7#11721259C211yesmaximal, characteristic, complemented
8#11721044C212yesmaximal, complemented
9#11721026C212yesmaximal, complemented
10#11720999C212yesmaximal, complemented
11#11720971C212yesmaximal, complemented
12#10594084—31nomaximal
13#10589750—31nomaximal
14C22×C3⋊Dic9C2211yescharacteristic
15C2×C9×#67C2211yescharacteristic
16C2×C3×#611C2211yescharacteristic
17C22×C9×Dic3C2211yescharacteristic
18C22×C3×Dic9C2211yescharacteristic
19C2×#6194C2211yescharacteristic
20#88663—24no—
21C2×D3×Dic9C2211yescharacteristic
22C2×Dic3×D9C2211yescharacteristic
23C2×#6189—24no—
24C2×#6189C2211yescharacteristic
25C2×#6189C2212yes—
26C2×#6189C2212yes—
27C3×C9×C22⋊C4C2211yescharacteristic
28#88551C2211yescharacteristic
29#88534C2212yes—
30C4×C3⋊Dic9C2212yes—
31C9×(C2×D3)⋊C4C2212yes—
32C3×(C2×D9)⋊C4C2212yes—
33C9×C4⋊Dic3C2212yes—
34C3×C4⋊Dic9C2212yes—
35(C2×D3)⋊Dic9C2212yes—
36(C2×D9)⋊Dic3C2212yes—
37#88442C2212yes—
38#88442C2212yes—
39C22×#627—31no—
40C2×#3805—31no—
41#64086—31no—
42C3×#886—31no—
43C3×#886D311yescharacteristic
44#63961—31no—
45#63961—31no—
46#63919—32no—
47#63915—32no—
48#63899—32no—
49#63899—32no—
50C2×#3699—31no—
51C22×Dic18—31no—
52C9×#123—31no—
53#63497—31no—
54#63491—31no—
55#63458—32no—
56#63450—32no—
57#63448D311yescharacteristic, complemented
58#63447—32no—
59#63440—32no—
60#63439—31no—
61#63439—61no—
62C23×C3×C9C2311yescharacteristic
63C22×D3×C9C2311yescharacteristic
64C22×C3×D9C2311yescharacteristic
65C2×C3×C4×C9C2312yes—
66C2×C3⋊Dic9D414yes—
67C2×C3⋊Dic9C2312yes—
68C2×C3⋊Dic9C2311yescharacteristic
69C2×C3⋊Dic9C2311yescharacteristic
70C9×#67—22no—
71C3×#611—22no—
72C2×C9×Dic3—22no—
73C2×C9×Dic3C2312yes—
74C2×C9×Dic3C2311yescharacteristic
75C2×C3×Dic9—22no—
76C2×C3×Dic9C2312yes—
77C2×C3×Dic9C2311yescharacteristic
78#6194—22no—
79D3×Dic9—22no—
80Dic3×D9—22no—
81#6189—24no—
82#6189—24no—
83#6189—22no—
84#5648—91no—
85#5648—91no—
86C22×C3⋊Dic3—31no—
87C2×C3×#67—31no—
88C2×C3×#67C2×D311yescharacteristic
89C22×C3×Dic3—31no—
90C22×C3×Dic3C2×D311yescharacteristic
91C2×#627—31no—
92C2×#627—32no—
93C2×#627—32no—
94C2×#627—64no—
95C2×#625—31no—
96#3805—64no—
97C2×D3×Dic3—31no—
98C2×D3×Dic3—31no—
99C32×C22⋊C4C2×D311yescharacteristic
100#3757—31no—
101#3750—32no—
102C4×C3⋊Dic3—32no—
103C3×(C2×D3)⋊C4—32no—
104C3×(C2×D3)⋊C4C2×D312yes—
105C3×C4⋊Dic3—32no—
106C3×C4⋊Dic3C2×D312yes—
107#3724—32no—
108#3724—32no—
109(C2×D3)⋊Dic3—32no—
110(C2×D3)⋊Dic3—32no—
111C2×D4×C9—31no—
112C22×C4×C9—31no—
113C2×#611—31no—
114C2×#611C2×D311yescharacteristic
115C22×Dic9—31no—
116C22×Dic9—31no—
117C22×Dic9—61no—
118C22×Dic9C2×D311yescharacteristic
119#3699—64no—
120C2×C4×D9—31no—
121C2×Dic18—32no—
122C2×Dic18—32no—
123C2×Dic18—31no—
124C2×Dic18—64no—
125C9×C4⋊C4—32no—
126C9×C22⋊C4—21no—
127C9×C22⋊C4—32no—
128C9×C22⋊C4C2×D311yescharacteristic
129#3676—31no—
130#3676—32no—
131#3676—61no—
132(C2×D9)⋊C4—32no—
133(C2×D9)⋊C4C2×D312yes—
134C4⋊Dic9C2×D312yes—
135Dic9⋊C4—32no—
136Dic9⋊C4—32no—
137Dic9⋊C4—32no—
138Dic9⋊C4—62no—
139C4×Dic9—32no—
140C4×Dic9—62no—
141C22×C3×C9—22no—
142C22×C3×C9C2×D412yes—
143C22×C3×C9C2411yescharacteristic
144C2×D3×C9—21no—
145C2×D3×C9—22no—
146C2×C3×D9—21no—
147C2×C3×D9—22no—
148C3×C4×C9—22no—
149C3⋊Dic9—22no—
150C3⋊Dic9C2×D412yes—
151C3⋊Dic9C2×D412yes—
152C9×Dic3—22no—
153C9×Dic3—21no—
154C3×Dic9—22no—
155C3×Dic9—21no—
156C2×#237—91no—
157C2×#237—91no—
158C22×Dic6—91no—
159C22×Dic6—91no—
160C3×#123—31no—
161C3×#123—91no—
162#935—91no—
163#935—91no—
164#929—91no—
165#929—91no—
166#896—92no—
167#896—92no—
168#888—92no—
169#888—92no—
170#886—31no—
171#886D911yescharacteristic, complemented
172#885—92no—
173#885—92no—
174#878—92no—
175#878—92no—
176#877—91no—
177#877—91no—
178#877—181no—
179C23×C32C22×D311yescharacteristic
180C22×C3×D3—31no—
181C22×C3×D3C22×D311yescharacteristic
182C2×C32×C4C22×D312yes—
183C2×C3⋊Dic3—34no—
184C2×C3⋊Dic3—31no—
185C2×C3⋊Dic3—31no—
186C2×C3⋊Dic3—32no—
187C3×#67—22no—
188C3×#67—62no—
189C2×C3×Dic3—22no—
190C2×C3×Dic3—32no—
191C2×C3×Dic3—31no—
192C2×C3×Dic3—62no—
193C2×C3×Dic3C22×D312yes—
194C2×C3×Dic3C22×D311yescharacteristic
195#627—62no—
196#627—64no—
197#627—64no—
198#625—62no—
199D3×Dic3—62no—
200D3×Dic3—62no—
201C23×C9—21no—
202C23×C9—31no—
203C23×C9C22×D311yescharacteristic
204C22×D9C22×D311yescharacteristic
205D4×C9—62no—
206C2×C4×C9—22no—
207C2×C4×C9—31no—
208C2×C4×C9—32no—
209C2×C4×C9—62no—
210C2×C4×C9C22×D312yes—
211#611—22no—
212#611—62no—
213C2×Dic9—22no—
214C2×Dic9—34no—
215C2×Dic9—31no—
216C2×Dic9—31no—
217C2×Dic9—32no—
218C2×Dic9—61no—
219C2×Dic9—61no—
220C2×Dic9—64no—
221C2×Dic9—62no—
222C2×Dic9—62no—
223C2×Dic9C22×D311yescharacteristic
224C2×Dic9C22×D312yes—
225C4×D9—62no—
226Dic18—64no—
227Dic18—64no—
228Dic18—62no—
229#536—271noSylow
230C2×C3×C9—22no—
231C2×C3×C9C22×D411yescharacteristic
232C2×C3×C92−1+412yes—
233D3×C9—41no—
234C3×D9—41no—
235C2×C3×D4—31no—
236C2×C3×D4—91no—
237C22×C3×C4—31no—
238C22×C3×C4—91no—
239C2×#67—31no—
240C2×#67—91no—
241C2×#67—91no—
242C2×#67C2×D911yescharacteristic
243C22×Dic3—31no—
244C22×Dic3—91no—
245C22×Dic3—91no—
246C22×Dic3—91no—
247C22×Dic3—91no—
248C22×Dic3—181no—
249C22×Dic3C2×D911yescharacteristic
250#237—184no—
251#237—184no—
252C2×C4×D3—91no—
253C2×C4×D3—91no—
254C2×Dic6—92no—
255C2×Dic6—92no—
256C2×Dic6—91no—
257C2×Dic6—92no—
258C2×Dic6—92no—
259C2×Dic6—91no—
260C2×Dic6—184no—
261C2×Dic6—184no—
262C3×C4⋊C4—32no—
263C3×C4⋊C4—92no—
264C3×C22⋊C4—21no—
265C3×C22⋊C4—32no—
266C3×C22⋊C4—92no—
267C3×C22⋊C4C2×D911yescharacteristic
268C3×C22⋊C4D3211yescharacteristic
269#217—92no—
270#217—91no—
271#217—91no—
272#217—92no—
273#217—181no—
274(C2×D3)⋊C4—32no—
275(C2×D3)⋊C4—92no—
276(C2×D3)⋊C4—92no—
277(C2×D3)⋊C4C2×D912yes—
278C4⋊Dic3—32no—
279C4⋊Dic3C2×D912yes—
280Dic3⋊C4—92no—
281Dic3⋊C4—92no—
282Dic3⋊C4—92no—
283Dic3⋊C4—92no—
284Dic3⋊C4—92no—
285Dic3⋊C4—92no—
286Dic3⋊C4—182no—
287C4×Dic3—92no—
288C4×Dic3—92no—
289C4×Dic3—182no—
290C22×C32—22no—
291C22×C32D3×D412yes—
292C22×C32C23×D311yescharacteristic
293C2×C3×D3—21no—
294C2×C3×D3—22no—
295C2×C3×D3—61no—
296C2×C3×D3—62no—
297C32×C4—22no—
298C3⋊Dic3—32no—
299C3⋊Dic3—32no—
300C3⋊Dic3—62no—
301C3×Dic3—21no—
302C3×Dic3—22no—
303C3×Dic3—61no—
304C3×Dic3—62no—
305C22×C9—22no—
306C22×C9—21no—
307C22×C9—22no—
308C22×C9—42no—
309C22×C9—61no—
310C22×C9—62no—
311C22×C9D3×D412yes—
312C22×C9C23×D311yescharacteristic
313C2×D9—21no—
314C2×D9—22no—
315C4×C9—22no—
316C4×C9—42no—
317C4×C9—62no—
318C4×C9—61no—
319Dic9—22no—
320Dic9—21no—
321Dic9—32no—
322Dic9—32no—
323Dic9—62no—
324Dic9—62no—
325Dic9—62no—
326Dic9—122no—
327C2×#41—271no—
328C22×Q8—271no—
329C4⋊Q8—272no—
330#124—272no—
331#123—31no—
332#123—91no—
333#123—271no—
334#123—271no—
335C22⋊Q8—272no—
336C22⋊Q8—272no—
337#117—271no—
338C3×C9#53611yescharacteristic, complemented, Sylow
339C23×C3—21no—
340C23×C3—31no—
341C23×C3—91no—
342C23×C3C22×D911yescharacteristic
343C23×C3C2×D3211yescharacteristic
344C22×D3—31no—
345C22×D3C22×D911yescharacteristic
346C3×D4—62no—
347C3×D4—182no—
348C2×C3×C4—22no—
349C2×C3×C4—31no—
350C2×C3×C4—32no—
351C2×C3×C4—62no—
352C2×C3×C4—91no—
353C2×C3×C4—92no—
354C2×C3×C4—182no—
355C2×C3×C4C22×D912yes—
356C2×C3×C4C2×D3212yes—
357#67—22no—
358#67—62no—
359#67—182no—
360#67—182no—
361C2×Dic3—22no—
362C2×Dic3—31no—
363C2×Dic3—32no—
364C2×Dic3—62no—
365C2×Dic3—91no—
366C2×Dic3—91no—
367C2×Dic3—94no—
368C2×Dic3—91no—
369C2×Dic3—91no—
370C2×Dic3—94no—
371C2×Dic3—92no—
372C2×Dic3—92no—
373C2×Dic3—181no—
374C2×Dic3—181no—
375C2×Dic3—184no—
376C2×Dic3—182no—
377C2×Dic3—182no—
378C2×Dic3—182no—
379C2×Dic3C22×D911yescharacteristic
380C2×Dic3C22×D912yes—
381C4×D3—182no—
382C4×D3—182no—
383Dic6—184no—
384Dic6—184no—
385Dic6—182no—
386Dic6—184no—
387Dic6—184no—
388Dic6—182no—
389C2×C32—22no—
390C2×C32C2×D3×D411yescharacteristic
391C2×C32#101012yes—
392C3×D3—41no—
393C3×D3—121no—
394C2×C9—22no—
395C2×C9—21no—
396C2×C9—22no—
397C2×C9—42no—
398C2×C9—121no—
399C2×C9C2×D3×D411yescharacteristic
400C2×C9#101012yes—
401D9—41no—
402#41—544no—
403C2×Q8—272no—
404C2×Q8—272no—
405C2×Q8—271no—
406C2×Q8—544no—
407C2×D4—31no—
408C2×D4—91no—
409C2×D4—271no—
410C22×C4—31no—
411C22×C4—91no—
412C22×C4—271no—
413C22×C4—271no—
414C22×C4—271no—
415C4⋊C4—32no—
416C4⋊C4—92no—
417C4⋊C4—272no—
418C4⋊C4—272no—
419C4⋊C4—272no—
420C22⋊C4—32no—
421C22⋊C4—92no—
422C22⋊C4—272no—
423C22⋊C4—272no—
424C22⋊C4—271no—
425C22⋊C4D3×D911yescharacteristic
426C42—272no—
427C22×C3—22no—
428C22×C3—22no—
429C22×C3—21no—
430C22×C3—22no—
431C22×C3—42no—
432C22×C3—61no—
433C22×C3—62no—
434C22×C3—181no—
435C22×C3—182no—
436C22×C3D4×D912yes—
437C22×C3C23×D911yescharacteristic
438C22×C3#380212yes—
439C22×C3C22×D3211yescharacteristic
440C2×D3—21no—
441C2×D3—22no—
442C2×D3—61no—
443C2×D3—62no—
444C3×C4—22no—
445C3×C4—22no—
446C3×C4—42no—
447C3×C4—62no—
448C3×C4—61no—
449C3×C4—182no—
450C3×C4—181no—
451Dic3—22no—
452Dic3—21no—
453Dic3—62no—
454Dic3—61no—
455Dic3—92no—
456Dic3—92no—
457Dic3—92no—
458Dic3—92no—
459Dic3—182no—
460Dic3—182no—
461Dic3—182no—
462Dic3—182no—
463Dic3—362no—
464C32#564811yescharacteristic
465C9—21no—
466C9#564811yescharacteristic, complemented
467C23—31no—
468C23—91no—
469C23C2×D3×D911yescharacteristic
470Q8—544no—
471Q8—544no—
472Q8—542no—
473D4—62no—
474D4—182no—
475D4—542no—
476C2×C4—32no—
477C2×C4—31no—
478C2×C4—62no—
479C2×C4—92no—
480C2×C4—91no—
481C2×C4—182no—
482C2×C4—271no—
483C2×C4—271no—
484C2×C4—274no—
485C2×C4—272no—
486C2×C4—542no—
487C2×C4—542no—
488C2×C4C2×D3×D912yes—
489C2×C3—22no—
490C2×C3—22no—
491C2×C3—21no—
492C2×C3—22no—
493C2×C3—42no—
494C2×C3—121no—
495C2×C3—361no—
496C2×C3C2×D4×D911yescharacteristic
497C2×C3#6371612yes—
498C2×C3C2×#380211yescharacteristic
499C2×C3#6430912yes—
500D3—41no—
501D3—121no—
502C22—22no—
503C22—61no—
504C22—62no—
505C22—181no—
506C22—182no—
507C22#8864612yes—
508C22C22×D3×D911yescharacteristic
509C4—22no—
510C4—62no—
511C4—61no—
512C4—182no—
513C4—181no—
514C4—272no—
515C4—272no—
516C4—542no—
517C3—21no—
518C3#1058975011yescharacteristic, complemented
519C3#1059408411yescharacteristic
520C2—22no—
521C2—121no—
522C2—361no—
523C2C2×#8864611yescharacteristic
524C2#1172249412yes—
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C22⋊C4C22
socleC22×C32C23×D3
prime radicalC2×#88663C2
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
2. C22×#6189C2semidirect
3. #11721723C2semidirect
4. C9×#886C2semidirect
5. C3×#63448C2semidirect
6. #11721265C2semidirect
7. #11721259C2semidirect
8. #11721044C2semidirect
9. #11721026C2semidirect
10. #11720999C2semidirect
11. #11720971C2semidirect
57. #63448D3semidirect
171. #886D9semidirect
338. C3×C9#536semidirect
466. C9#5648semidirect
518. C3#10589750semidirect

Maximal subgroups

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

subgroupindex
C2×#886632
C22×#61892
#117217232
C9×#8862
C3×#634482
#117212652
#117212592
#117210442
#117210262
#117209992
#117209712
#105940843
#105897503