Finite Group Databasebeta

← #13150176

Subgroups — #13150176

Order 1728.

884 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3697C211yesmaximal, characteristic
2C2×D9×Dic6C211yesmaximal, characteristic, complemented
3#11721813C211yesmaximal, characteristic, complemented
4C9×#924C211yesmaximal, characteristic, complemented
5#11721269C212yesmaximal, complemented
6#11721257C212yesmaximal, complemented
7#11720981C212yesmaximal, complemented
8#11720945C211yesmaximal, characteristic, complemented
9#11720944C211yesmaximal, characteristic, complemented
10#11720907C211yesmaximal, characteristic, complemented
11#11720904C212yesmaximal, complemented
12#10594884—31nomaximal
13#10589730—31nomaximal
14C22×C3×C4×C9C2211yescharacteristic
15C2×#6226C2211yescharacteristic
16C2×C3×#611C2211yescharacteristic
17C2×C3×#611C2211yescharacteristic
18C3×#3697—24no—
19C2×C3×D36C2211yescharacteristic
20C2×C3×C4×D9C2211yescharacteristic
21C2×C3×C4×D9C2211yescharacteristic
22C2×C9×Dic6C2211yescharacteristic
23C2×C3×Dic18C2211yescharacteristic
24C2×Dic3×D9C2212yes—
25C2×#6189C2212yes—
26D9×Dic6—24no—
27#88551C2212yes—
28C4⋊(C3⋊Dic9)C2211yescharacteristic
29#88534C2212yes—
30C9×#217C2212yes—
31C9×C4⋊Dic3C2211yescharacteristic
32C9×Dic3⋊C4C2212yes—
33(C2×D9)⋊Dic3C2212yes—
34(C2×D9)⋊Dic3C2212yes—
35(C2×D9)⋊Dic3C2212yes—
36Dic3⋊Dic9C2212yes—
37#88442C2212yes—
38Dic3×Dic9C2212yes—
39C2×C3×#235—31no—
40C2×D3×Dic6—31no—
41#64133—31no—
42C3×#924D311yescharacteristic
43#63969—32no—
44#63959—32no—
45#63884—32no—
46#63857—31no—
47#63856—31no—
48#63841—31no—
49#63839—32no—
50C2×Q8×D9—31no—
51C2×#3697D311yescharacteristic
52C9×C22⋊Q8—31no—
53#63508—31no—
54#63486—31no—
55#63486—61no—
56#63459—31no—
57#63456—32no—
58#63454—31no—
59#63450—32no—
60#63447—32no—
61#63445—32no—
62C23×C3×C9C2311yescharacteristic
63C22×C3×D9C2311yescharacteristic
64C22×C3×D9C2311yescharacteristic
65C2×C3×C4×C9—22no—
66C2×C3×C4×C9C2311yescharacteristic
67C2×C3×C4×C9C2311yescharacteristic
68C2×C3⋊Dic9C2312yes—
69C2×C3⋊Dic9C2312yes—
70#6226—22no—
71C3×#611—22no—
72C3×#611—22no—
73C2×C9×Dic3C2312yes—
74C2×C9×Dic3C2312yes—
75C2×C3×Dic9C2311yescharacteristic
76C2×C3×Dic9C2311yescharacteristic
77C3×D36—22no—
78C3×C4×D9—22no—
79C3×C4×D9D414yes—
80C9×Dic6—22no—
81C3×Dic18—22no—
82Dic3×D9—24no—
83#6189—24no—
84#5825—91no—
85#5628—91no—
86C22×C32×C4C2×D311yescharacteristic
87C2×#634—31no—
88C2×C3×#67—31no—
89C2×C3×#67—31no—
90C3×#235—64no—
91C2×C3×D12—31no—
92C2×C3×C4×D3—31no—
93C2×C3×C4×D3—31no—
94C2×C3×Dic6—31no—
95C2×C3×Dic6C2×D311yescharacteristic
96C2×#627—32no—
97C2×D3×Dic3—32no—
98D3×Dic6—64no—
99#3757—32no—
100C4⋊(C3⋊Dic3)—31no—
101#3750—32no—
102C3×#217C2×D312yes—
103C3×C4⋊Dic3C2×D311yescharacteristic
104C3×Dic3⋊C4C2×D312yes—
105#3724—32no—
106Dic3⋊Dic3—32no—
107(C2×D3)⋊Dic3—32no—
108(C2×D3)⋊Dic3—32no—
109(C2×D3)⋊Dic3—32no—
110Dic32—32no—
111C2×Q8×C9—31no—
112C22×C4×C9—21no—
113C22×C4×C9C2×D311yescharacteristic
114C2×#611C2×D311yescharacteristic
115C2×#611C2×D311yescharacteristic
116Q8×D9—64no—
117#3697—24no—
118C2×D36C2×D311yescharacteristic
119C2×C4×D9—32no—
120C2×C4×D9C2×D311yescharacteristic
121C2×C4×D9C2×D311yescharacteristic
122C2×Dic18—31no—
123C2×Dic18—32no—
124C2×Dic18—61no—
125C2×Dic18C2×D311yescharacteristic
126C9×C4⋊C4—32no—
127C9×C4⋊C4—31no—
128C9×C22⋊C4—32no—
129#3676—32no—
130#3676—62no—
131(C2×D9)⋊C4—32no—
132(C2×D9)⋊C4—32no—
133(C2×D9)⋊C4—32no—
134C4⋊Dic9—31no—
135C4⋊Dic9—32no—
136C4⋊Dic9—61no—
137Dic9⋊C4—32no—
138Dic9⋊C4—32no—
139Dic9⋊C4—62no—
140C4×Dic9—32no—
141C22×C3×C9—21no—
142C22×C3×C9—21no—
143C22×C3×C9—21no—
144C22×C3×C9C2411yescharacteristic
145C2×C3×D9—21no—
146C2×C3×D9—21no—
147C2×C3×D9—21no—
148C2×C3×D9—21no—
149C2×C3×D9—21no—
150C2×C3×D9C2×D412yes—
151C3×C4×C9—21no—
152C3×C4×C9C2×D412yes—
153C3⋊Dic9—22no—
154C3⋊Dic9—22no—
155C9×Dic3—22no—
156C9×Dic3—22no—
157C3×Dic9—21no—
158C3×Dic9C2×D412yes—
159C2×C3×#41—91no—
160C2×D3×Q8—91no—
161C2×#235—31no—
162C22×Dic6—91no—
163C3×C22⋊Q8—31no—
164#946—91no—
165#944—92no—
166#936—92no—
167#935—92no—
168#935—92no—
169#926—91no—
170#924—91no—
171#924—91no—
172#924—91no—
173#924—91no—
174#924—181no—
175#924D911yescharacteristic, complemented
176#897—91no—
177#894—92no—
178#892—91no—
179#888—92no—
180#885—92no—
181#883—92no—
182C23×C32C22×D311yescharacteristic
183C22×C3×D3—31no—
184C22×C3×D3—31no—
185C2×C32×C4—22no—
186C2×C32×C4C22×D311yescharacteristic
187C2×C32×C4C22×D311yescharacteristic
188C2×C3⋊Dic3—32no—
189C2×C3⋊Dic3—32no—
190#634—62no—
191C3×#67—62no—
192C3×#67—62no—
193C2×C3×Dic3—31no—
194C2×C3×Dic3—31no—
195C2×C3×Dic3C22×D312yes—
196C2×C3×Dic3C22×D312yes—
197C3×D12—62no—
198C3×C4×D3—34no—
199C3×C4×D3—62no—
200C3×Dic6—22no—
201C3×Dic6—62no—
202#627—64no—
203D3×Dic3—64no—
204C23×C9—21no—
205C23×C9C22×D311yescharacteristic
206C22×D9C22×D311yescharacteristic
207C22×D9C22×D311yescharacteristic
208Q8×C9—62no—
209C2×C4×C9—22no—
210C2×C4×C9—21no—
211C2×C4×C9—21no—
212C2×C4×C9—32no—
213C2×C4×C9—32no—
214C2×C4×C9—42no—
215C2×C4×C9C22×D311yescharacteristic
216C2×C4×C9C22×D311yescharacteristic
217#611—22no—
218#611—22no—
219C2×Dic9—32no—
220C2×Dic9—32no—
221C2×Dic9—62no—
222C2×Dic9—62no—
223C2×Dic9C22×D311yescharacteristic
224C2×Dic9C22×D311yescharacteristic
225D36—22no—
226C4×D9—22no—
227C4×D9—64no—
228C4×D9#6714yes—
229Dic18—22no—
230Dic18—62no—
231Dic18—64no—
232Dic18—122no—
233#536—271noSylow
234C2×C3×C9—41no—
235C2×C3×C9C22×D411yescharacteristic
236C2×C3×C92−1+411yescharacteristic
237C2×C3×C92−1+411yescharacteristic
238C3×D9—22no—
239C3×D9—41no—
240C3×#41—184no—
241C2×C3×Q8—31no—
242C2×C3×Q8—91no—
243C2×C3×D4—91no—
244C2×C3×D4—91no—
245C2×C3×D4—91no—
246C22×C3×C4—21no—
247C22×C3×C4—91no—
248C22×C3×C4—91no—
249C22×C3×C4C2×D911yescharacteristic
250C22×C3×C4D3211yescharacteristic
251C2×#67—31no—
252C2×#67—31no—
253C22×Dic3—92no—
254D3×Q8—184no—
255#235—64no—
256C2×D12—31no—
257C2×C4×D3—31no—
258C2×C4×D3—31no—
259C2×C4×D3—92no—
260C2×Dic6—31no—
261C2×Dic6—92no—
262C2×Dic6—91no—
263C2×Dic6—91no—
264C2×Dic6—92no—
265C2×Dic6—181no—
266C2×Dic6—184no—
267C2×Dic6C2×D911yescharacteristic
268C3×C4⋊C4—32no—
269C3×C4⋊C4—31no—
270C3×C22⋊C4—32no—
271#217—92no—
272#217—92no—
273#217—92no—
274#217—92no—
275#217—92no—
276#217—182no—
277#217C2×D912yes—
278(C2×D3)⋊C4—92no—
279(C2×D3)⋊C4—92no—
280(C2×D3)⋊C4—92no—
281C4⋊Dic3—92no—
282C4⋊Dic3—91no—
283C4⋊Dic3—91no—
284C4⋊Dic3—181no—
285C4⋊Dic3C2×D911yescharacteristic
286Dic3⋊C4—92no—
287Dic3⋊C4—92no—
288Dic3⋊C4—92no—
289Dic3⋊C4—92no—
290Dic3⋊C4—92no—
291Dic3⋊C4—182no—
292Dic3⋊C4C2×D912yes—
293C4×Dic3—92no—
294C4×Dic3—92no—
295C22×C32—21no—
296C22×C32—21no—
297C22×C32—21no—
298C22×C32C23×D311yescharacteristic
299C2×C3×D3—32no—
300C2×C3×D3—61no—
301C2×C3×D3—61no—
302C2×C3×D3—61no—
303C2×C3×D3—61no—
304C2×C3×D3—61no—
305C32×C4—21no—
306C32×C4D3×D412yes—
307C3⋊Dic3—62no—
308C3⋊Dic3—62no—
309C3×Dic3—22no—
310C3×Dic3—22no—
311C3×Dic3—32no—
312C3×Dic3—61no—
313C22×C9—22no—
314C22×C9—21no—
315C22×C9—21no—
316C22×C9—21no—
317C22×C9—21no—
318C22×C9—41no—
319C22×C9—41no—
320C22×C9C23×D311yescharacteristic
321C2×D9—21no—
322C2×D9—21no—
323C2×D9—21no—
324C2×D9—21no—
325C2×D9—21no—
326C2×D9C2×#6712yes—
327C4×C9—21no—
328C4×C9—22no—
329C4×C9—41no—
330C4×C9—62no—
331C4×C9—62no—
332C4×C9C2×#6712yes—
333Dic9—21no—
334Dic9—62no—
335Dic9—62no—
336Dic9—122no—
337Dic9—122no—
338Dic9C2×#6712yes—
339C2×#41—91no—
340C22×Q8—271no—
341C4⋊Q8—272no—
342#124—272no—
343#123—272no—
344#123—272no—
345C22⋊Q8—31no—
346C22⋊Q8—271no—
347C22⋊Q8—271no—
348C22⋊Q8—271no—
349#117—271no—
350C3×C9#53611yescharacteristic, complemented, Sylow
351C23×C3—21no—
352C23×C3—91no—
353C23×C3—91no—
354C23×C3C22×D911yescharacteristic
355C23×C3C2×D3211yescharacteristic
356C22×D3—31no—
357C22×D3—31no—
358C3×Q8—62no—
359C3×Q8—182no—
360C3×D4—182no—
361C3×D4—182no—
362C3×D4—182no—
363C2×C3×C4—22no—
364C2×C3×C4—22no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—32no—
368C2×C3×C4—32no—
369C2×C3×C4—42no—
370C2×C3×C4—91no—
371C2×C3×C4—94no—
372C2×C3×C4—91no—
373C2×C3×C4—182no—
374C2×C3×C4C22×D911yescharacteristic
375C2×C3×C4C22×D911yescharacteristic
376C2×C3×C4C2×D3211yescharacteristic
377C2×C3×C4C2×D3211yescharacteristic
378#67—62no—
379#67—62no—
380C2×Dic3—31no—
381C2×Dic3—31no—
382C2×Dic3—92no—
383C2×Dic3—92no—
384C2×Dic3—92no—
385C2×Dic3—92no—
386C2×Dic3—182no—
387C2×Dic3—182no—
388C2×Dic3—184no—
389C2×Dic3C22×D912yes—
390C2×Dic3C22×D912yes—
391D12—62no—
392C4×D3—34no—
393C4×D3—62no—
394C4×D3—184no—
395Dic6—22no—
396Dic6—62no—
397Dic6—184no—
398Dic6—184no—
399Dic6—182no—
400Dic6—182no—
401Dic6—362no—
402C2×C32—41no—
403C2×C32C2×D3×D411yescharacteristic
404C2×C32#100711yescharacteristic
405C2×C32#101011yescharacteristic
406C3×D3—62no—
407C3×D3—121no—
408C2×C9—21no—
409C2×C9—21no—
410C2×C9—21no—
411C2×C9—42no—
412C2×C9—41no—
413C2×C9#101011yescharacteristic
414C2×C9#101011yescharacteristic
415C2×C9C22×#6711yescharacteristic
416D9—22no—
417D9—41no—
418#41—184no—
419C2×Q8—31no—
420C2×Q8—91no—
421C2×Q8—272no—
422C2×Q8—271no—
423C2×Q8—544no—
424C2×D4—91no—
425C2×D4—91no—
426C2×D4—91no—
427C22×C4—91no—
428C22×C4—91no—
429C22×C4—272no—
430C22×C4D3×D911yescharacteristic
431C4⋊C4—31no—
432C4⋊C4—32no—
433C4⋊C4—272no—
434C4⋊C4—272no—
435C4⋊C4—271no—
436C4⋊C4—272no—
437C22⋊C4—32no—
438C22⋊C4—272no—
439C22⋊C4—272no—
440C22⋊C4—272no—
441C22⋊C4—272no—
442C42—272no—
443C22×C3—22no—
444C22×C3—21no—
445C22×C3—21no—
446C22×C3—21no—
447C22×C3—21no—
448C22×C3—21no—
449C22×C3—21no—
450C22×C3—21no—
451C22×C3—41no—
452C22×C3—41no—
453C22×C3—92no—
454C22×C3—181no—
455C22×C3—181no—
456C22×C3—181no—
457C22×C3—181no—
458C22×C3—181no—
459C22×C3C23×D911yescharacteristic
460C22×C3C22×D3211yescharacteristic
461C2×D3—32no—
462C2×D3—61no—
463C2×D3—61no—
464C2×D3—61no—
465C2×D3—61no—
466C2×D3—61no—
467C3×C4—21no—
468C3×C4—21no—
469C3×C4—22no—
470C3×C4—41no—
471C3×C4—62no—
472C3×C4—62no—
473C3×C4—92no—
474C3×C4—181no—
475C3×C4D4×D912yes—
476C3×C4D3×#6712yes—
477Dic3—22no—
478Dic3—22no—
479Dic3—32no—
480Dic3—61no—
481Dic3—182no—
482Dic3—182no—
483Dic3—182no—
484Dic3—182no—
485Dic3—362no—
486Dic3—362no—
487C32#562811yescharacteristic
488C9—21no—
489C9#582511yescharacteristic, complemented
490C23—91no—
491C23—91no—
492C23C2×D3×D911yescharacteristic
493Q8—62no—
494Q8—182no—
495Q8—544no—
496Q8—542no—
497D4—182no—
498D4—182no—
499D4—182no—
500C2×C4—22no—
501C2×C4—32no—
502C2×C4—32no—
503C2×C4—91no—
504C2×C4—94no—
505C2×C4—91no—
506C2×C4—182no—
507C2×C4—272no—
508C2×C4—272no—
509C2×C4—544no—
510C2×C4C2×D3×D911yescharacteristic
511C2×C4C2×D3×D911yescharacteristic
512C2×C3—21no—
513C2×C3—21no—
514C2×C3—21no—
515C2×C3—42no—
516C2×C3—41no—
517C2×C3—41no—
518C2×C3—182no—
519C2×C3—361no—
520C2×C3C2×D4×D911yescharacteristic
521C2×C3#6371311yescharacteristic
522C2×C3#6371611yescharacteristic
523C2×C3#6429711yescharacteristic
524C2×C3#6429811yescharacteristic
525C2×C3C2×D3×#6711yescharacteristic
526D3—62no—
527D3—121no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—92no—
532C22—181no—
533C22—181no—
534C22—181no—
535C22—181no—
536C22—181no—
537C22C22×D3×D911yescharacteristic
538C4—21no—
539C4—62no—
540C4—62no—
541C4—92no—
542C4—181no—
543C4—542no—
544C4—542no—
545C4D9×#6712yes—
546C3—21no—
547C3#1058973011yescharacteristic, complemented
548C3#1059488411yescharacteristic
549C2—41no—
550C2—182no—
551C2—361no—
552C2#1172245811yescharacteristic
553C2#1172246511yescharacteristic
554C2C2×D9×#6711yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
prime radicalC2×C3×#3697C2
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. C2×D9×Dic6C2semidirect
3. #11721813C2semidirect
4. C9×#924C2semidirect
5. #11721269C2semidirect
6. #11721257C2semidirect
7. #11720981C2semidirect
8. #11720945C2semidirect
9. #11720944C2semidirect
10. #11720907C2semidirect
11. #11720904C2semidirect
175. #924D9semidirect
350. C3×C9#536semidirect
489. C9#5825semidirect
547. C3#10589730semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#36972
C2×D9×Dic62
#117218132
C9×#9242
#117212692
#117212572
#117209812
#117209452
#117209442
#117209072
#117209042
#105948843
#105897303