Finite Group Databasebeta

← #13139853

Subgroups — #13139853

Order 1728.

682 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
1C22×#6189C211yesmaximal, characteristic, complemented
2C2×C9×(C2×D3)⋊C4C211yesmaximal, characteristic
3C2×(C2×D3)⋊Dic9C211yesmaximal, characteristic
4C2×(C2×D3)⋊Dic9C211yesmaximal, characteristic
5#11720572C211yesmaximal, characteristic, complemented
6C3×C9⋊#95C211yesmaximal, characteristic, complemented
7#11720331C211yesmaximal, characteristic, complemented
8#10591293—31nomaximal
9#10588627—31nomaximal
10C23×D3×C9C2211yescharacteristic
11C22×C3×C4×C9C2211yescharacteristic
12C22×C3⋊Dic9C2211yescharacteristic
13C22×C3⋊Dic9C2211yescharacteristic
14C22×C9×Dic3C2211yescharacteristic
15C22×C3×Dic9C2211yescharacteristic
16C22×C3×Dic9C2211yescharacteristic
17C2×#6189—22no—
18C2×#6189—22no—
19C2×#6189—22no—
20C9×(C2×D3)⋊C4—22no—
21(C2×D3)⋊Dic9—22no—
22(C2×D3)⋊Dic9—22no—
23C22×#627—31no—
24C2×C3×(C2×D3)⋊C4D311yescharacteristic
25C2×(C2×D3)⋊Dic3—31no—
26C2×(C2×D3)⋊Dic3—31no—
27C22×Dic18—31no—
28#63658—31no—
29C3×C3⋊#95—31no—
30#63579—31no—
31C2×C9×C22⋊C4—31no—
32C2×#3676—31no—
33C2×#3676—31no—
34C9⋊#95—31no—
35C9⋊#95—31no—
36C9⋊#95—61no—
37C9⋊#95D311yescharacteristic, complemented
38C23×C3×C9C2311yescharacteristic
39C22×D3×C9—21no—
40C22×D3×C9—21no—
41C22×D3×C9—21no—
42C22×D3×C9—21no—
43C22×D3×C9—21no—
44C22×D3×C9—21no—
45C22×D3×C9Q812yescomplemented
46C2×C3×C4×C9—21no—
47C2×C3×C4×C9—21no—
48C2×C3×C4×C9—21no—
49C2×C3⋊Dic9—21no—
50C2×C3⋊Dic9—21no—
51C2×C3⋊Dic9—21no—
52C2×C3⋊Dic9—21no—
53C2×C3⋊Dic9—21no—
54C2×C3⋊Dic9D412yes—
55C2×C9×Dic3—21no—
56C2×C9×Dic3—21no—
57C2×C9×Dic3D412yes—
58C2×C3×Dic9—21no—
59C2×C3×Dic9—21no—
60C2×C3×Dic9—21no—
61C2×C3×Dic9—21no—
62C2×C3×Dic9—21no—
63C2×C3×Dic9D412yes—
64#6189—28no—
65#4947—91no—
66#4673—91no—
67C23×C3×D3C2×D311yescharacteristic
68C22×C32×C4C2×D311yescharacteristic
69C22×C3⋊Dic3—31no—
70C22×C3⋊Dic3—31no—
71C22×C3×Dic3—31no—
72C22×C3×Dic3—31no—
73C22×C3×Dic3C2×D311yescharacteristic
74C2×#627—62no—
75C2×#627—62no—
76C2×#627—62no—
77C24×C9—31no—
78C3×(C2×D3)⋊C4—22no—
79(C2×D3)⋊Dic3—62no—
80(C2×D3)⋊Dic3—62no—
81C22×C4×C9—21no—
82C22×C4×C9—31no—
83C22×C4×C9C2×D311yescharacteristic
84C22×Dic9—31no—
85C22×Dic9—31no—
86C22×Dic9—61no—
87C22×Dic9—61no—
88C22×Dic9C2×D311yescharacteristic
89C22×Dic9C2×D311yescharacteristic
90C2×Dic18—62no—
91C2×Dic18—62no—
92C2×Dic18—62no—
93C9×C22⋊C4—62no—
94#3676—62no—
95#3676—62no—
96C22×C3×C9C2×D411yescharacteristic
97C22×C3×C9C2×D411yescharacteristic
98C22×C3×C9C2×D411yescharacteristic
99C22×C3×C9C2×Q811yescharacteristic
100C22×C3×C9#4111yescharacteristic
101C22×C3×C9#4111yescharacteristic
102C22×C3×C9#4111yescharacteristic
103C2×D3×C9—22no—
104C2×D3×C9—22no—
105C2×D3×C9—22no—
106C2×D3×C9—41no—
107C2×D3×C9—41no—
108C2×D3×C9—41no—
109C2×D3×C9—41no—
110C3×C4×C9—41no—
111C3⋊Dic9—22no—
112C3⋊Dic9—41no—
113C9×Dic3—22no—
114C3×Dic9—22no—
115C3×Dic9—41no—
116C22×Dic6—91no—
117C22×Dic6—91no—
118C2×C3×C22⋊C4—31no—
119C2×#217—91no—
120C2×#217—91no—
121C2×(C2×D3)⋊C4—91no—
122C2×(C2×D3)⋊C4—91no—
123C2×(C2×D3)⋊C4D911yescharacteristic
124C3×#95—91no—
125C3⋊#95—31no—
126C3⋊#95—91no—
127C3⋊#95—91no—
128C3⋊#95—91no—
129C3⋊#95—91no—
130C3⋊#95—181no—
131C23×C32C22×D311yescharacteristic
132C22×C3×D3—21no—
133C22×C3×D3—21no—
134C22×C3×D3—21no—
135C22×C3×D3—21no—
136C22×C3×D3—21no—
137C22×C3×D3—21no—
138C22×C3×D3Dic612yes—
139C2×C32×C4—21no—
140C2×C32×C4—21no—
141C2×C32×C4—21no—
142C2×C3⋊Dic3—32no—
143C2×C3⋊Dic3—61no—
144C2×C3⋊Dic3—61no—
145C2×C3⋊Dic3—61no—
146C2×C3⋊Dic3—61no—
147C2×C3⋊Dic3—61no—
148C2×C3×Dic3—21no—
149C2×C3×Dic3—21no—
150C2×C3×Dic3—32no—
151C2×C3×Dic3—61no—
152C2×C3×Dic3—61no—
153C2×C3×Dic3—61no—
154C2×C3×Dic3—61no—
155C2×C3×Dic3—61no—
156C2×C3×Dic3#6712yes—
157#627—68no—
158C23×C9—21no—
159C23×C9—32no—
160C23×C9—61no—
161C23×C9—61no—
162C23×C9—61no—
163C23×C9—61no—
164C23×C9—61no—
165C23×C9—61no—
166C23×C9C22×D311yescharacteristic
167C2×C4×C9—22no—
168C2×C4×C9—21no—
169C2×C4×C9—21no—
170C2×C4×C9—21no—
171C2×C4×C9—32no—
172C2×C4×C9—41no—
173C2×C4×C9—41no—
174C2×C4×C9—61no—
175C2×C4×C9—61no—
176C2×Dic9—21no—
177C2×Dic9—21no—
178C2×Dic9—21no—
179C2×Dic9—21no—
180C2×Dic9—21no—
181C2×Dic9—32no—
182C2×Dic9—62no—
183C2×Dic9—61no—
184C2×Dic9—61no—
185C2×Dic9—62no—
186C2×Dic9—61no—
187C2×Dic9—61no—
188C2×Dic9—61no—
189C2×Dic9—121no—
190C2×Dic9—121no—
191C2×Dic9—121no—
192C2×Dic9—121no—
193C2×Dic9D1212yes—
194Dic18—68no—
195#393—271noSylow
196C2×C3×C9#12011yescharacteristic
197C2×C3×C9C22⋊Q811yescharacteristic
198C2×C3×C9C22⋊Q811yescharacteristic
199C2×C3×C9C22⋊Q811yescharacteristic
200C2×C3×C9#12411yescharacteristic
201C2×C3×C9#12411yescharacteristic
202C2×C3×C9#12411yescharacteristic
203D3×C9—42no—
204C24×C3—31no—
205C23×D3C2×D911yescharacteristic
206C22×C3×C4—21no—
207C22×C3×C4—31no—
208C22×C3×C4—91no—
209C22×C3×C4—91no—
210C22×C3×C4C2×D911yescharacteristic
211C22×C3×C4D3211yescharacteristic
212C22×Dic3—31no—
213C22×Dic3—31no—
214C22×Dic3—91no—
215C22×Dic3—91no—
216C22×Dic3—91no—
217C22×Dic3—91no—
218C22×Dic3—181no—
219C22×Dic3—181no—
220C22×Dic3C2×D911yescharacteristic
221C2×Dic6—182no—
222C2×Dic6—182no—
223C2×Dic6—182no—
224C2×Dic6—182no—
225C2×Dic6—182no—
226C2×Dic6—182no—
227C3×C22⋊C4—62no—
228#217—182no—
229#217—182no—
230(C2×D3)⋊C4—22no—
231(C2×D3)⋊C4—182no—
232(C2×D3)⋊C4—182no—
233C22×C32C2×Dic611yescharacteristic
234C22×C32#23511yescharacteristic
235C22×C32D3×D411yescharacteristic
236C22×C32D3×D411yescharacteristic
237C22×C32#23711yescharacteristic
238C22×C32#23711yescharacteristic
239C22×C32C2×#6711yescharacteristic
240C2×C3×D3—22no—
241C2×C3×D3—22no—
242C2×C3×D3—22no—
243C2×C3×D3—41no—
244C2×C3×D3—41no—
245C2×C3×D3—41no—
246C2×C3×D3—41no—
247C32×C4—41no—
248C3⋊Dic3—62no—
249C3⋊Dic3—121no—
250C3×Dic3—22no—
251C3×Dic3—62no—
252C3×Dic3—121no—
253C22×C9—21no—
254C22×C9—21no—
255C22×C9—21no—
256C22×C9—21no—
257C22×C9—21no—
258C22×C9—21no—
259C22×C9—21no—
260C22×C9—62no—
261C22×C9—62no—
262C22×C9—62no—
263C22×C9—121no—
264C22×C9—121no—
265C22×C9—121no—
266C22×C9—121no—
267C22×C9C2×D1211yescharacteristic
268C22×C9#23511yescharacteristic
269C22×C9#23511yescharacteristic
270C22×C9D3×D411yescharacteristic
271C22×C9D3×D411yescharacteristic
272C22×C9#23711yescharacteristic
273C22×C9D3×Q811yescharacteristic
274C4×C9—42no—
275C4×C9—41no—
276C4×C9—62no—
277Dic9—22no—
278Dic9—41no—
279Dic9—62no—
280Dic9—122no—
281Dic9—122no—
282Dic9—121no—
283C22×Q8—271no—
284C2×C22⋊C4—31no—
285C2×C22⋊C4—271no—
286C2×C22⋊C4—271no—
287#95—91no—
288#95—271no—
289#95—271no—
290C3×C9#39311yescharacteristic, complemented, Sylow
291C23×C3—21no—
292C23×C3—32no—
293C23×C3—61no—
294C23×C3—61no—
295C23×C3—61no—
296C23×C3—61no—
297C23×C3—61no—
298C23×C3—61no—
299C23×C3C22×D911yescharacteristic
300C23×C3C2×D3211yescharacteristic
301C22×D3—21no—
302C22×D3—21no—
303C22×D3—21no—
304C22×D3—21no—
305C22×D3—21no—
306C22×D3—21no—
307C22×D3Dic1812yescomplemented
308C2×C3×C4—22no—
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—21no—
315C2×C3×C4—32no—
316C2×C3×C4—41no—
317C2×C3×C4—41no—
318C2×C3×C4—61no—
319C2×C3×C4—61no—
320C2×C3×C4—92no—
321C2×C3×C4—181no—
322C2×C3×C4—181no—
323C2×C3×C4—181no—
324C2×C3×C4—181no—
325C2×C3×C4—181no—
326C2×Dic3—21no—
327C2×Dic3—21no—
328C2×Dic3—32no—
329C2×Dic3—61no—
330C2×Dic3—61no—
331C2×Dic3—61no—
332C2×Dic3—61no—
333C2×Dic3—61no—
334C2×Dic3—92no—
335C2×Dic3—92no—
336C2×Dic3—182no—
337C2×Dic3—181no—
338C2×Dic3—181no—
339C2×Dic3—181no—
340C2×Dic3—181no—
341C2×Dic3—182no—
342C2×Dic3—181no—
343C2×Dic3—181no—
344C2×Dic3—181no—
345C2×Dic3—181no—
346C2×Dic3—181no—
347C2×Dic3—181no—
348C2×Dic3—361no—
349C2×Dic3—361no—
350C2×Dic3—361no—
351C2×Dic3—361no—
352C2×Dic3#61112yes—
353Dic6—188no—
354Dic6—188no—
355C2×C32#87811yescharacteristic
356C2×C32#87811yescharacteristic
357C2×C32#88511yescharacteristic
358C2×C32#88511yescharacteristic
359C2×C32#92411yescharacteristic
360C2×C32#93611yescharacteristic
361C2×C32#93711yescharacteristic
362C3×D3—42no—
363C2×C9—21no—
364C2×C9—21no—
365C2×C9—21no—
366C2×C9—21no—
367C2×C9—21no—
368C2×C9—21no—
369C2×C9—21no—
370C2×C9—122no—
371C2×C9#87511yescharacteristic
372C2×C9#88211yescharacteristic
373C2×C9#88511yescharacteristic
374C2×C9#88511yescharacteristic
375C2×C9#89611yescharacteristic
376C2×C9#89611yescharacteristic
377C2×C9#89711yescharacteristic
378C24—31no—
379C2×Q8—542no—
380C2×Q8—542no—
381C2×Q8—542no—
382C22×C4—31no—
383C22×C4—91no—
384C22×C4—91no—
385C22×C4—271no—
386C22×C4—271no—
387C22×C4D3×D911yescharacteristic
388C22⋊C4—62no—
389C22⋊C4—542no—
390C22⋊C4—542no—
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—21no—
397C22×C3—21no—
398C22×C3—62no—
399C22×C3—62no—
400C22×C3—62no—
401C22×C3—121no—
402C22×C3—121no—
403C22×C3—121no—
404C22×C3—121no—
405C22×C3C2×Dic1811yescharacteristic
406C22×C3#369711yescharacteristic
407C22×C3D4×D911yescharacteristic
408C22×C3D4×D911yescharacteristic
409C22×C3#369911yescharacteristic
410C22×C3#369911yescharacteristic
411C22×C3C2×#61111yescharacteristic
412C22×C3D3×Dic611yescharacteristic
413C22×C3#379811yescharacteristic
414C22×C3D3×D1211yescharacteristic
415C22×C3#380211yescharacteristic
416C22×C3#380411yescharacteristic
417C22×C3#380511yescharacteristic
418C22×C3D3×#6711yescharacteristic
419C2×D3—22no—
420C2×D3—22no—
421C2×D3—22no—
422C2×D3—41no—
423C2×D3—41no—
424C2×D3—41no—
425C2×D3—41no—
426C3×C4—42no—
427C3×C4—41no—
428C3×C4—41no—
429C3×C4—62no—
430C3×C4—182no—
431C3×C4—361no—
432Dic3—22no—
433Dic3—62no—
434Dic3—121no—
435Dic3—182no—
436Dic3—182no—
437Dic3—362no—
438Dic3—362no—
439Dic3—361no—
440Dic3—361no—
441C32#494711yescharacteristic
442C9—21no—
443C9#467311yescharacteristic, complemented
444C23—32no—
445C23—61no—
446C23—61no—
447C23—61no—
448C23—61no—
449C23—61no—
450C23—61no—
451C23C2×D3×D911yescharacteristic
452Q8—548no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—32no—
457C2×C4—61no—
458C2×C4—61no—
459C2×C4—92no—
460C2×C4—181no—
461C2×C4—181no—
462C2×C4—181no—
463C2×C4—181no—
464C2×C4—181no—
465C2×C4—272no—
466C2×C4—541no—
467C2×C4—541no—
468C2×C4—541no—
469C2×C4—541no—
470C2×C4—541no—
471C2×C3—21no—
472C2×C3—21no—
473C2×C3—21no—
474C2×C3—21no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—122no—
479C2×C3#6344011yescharacteristic
480C2×C3#6344011yescharacteristic
481C2×C3#6344711yescharacteristic
482C2×C3#6344711yescharacteristic
483C2×C3#6348611yescharacteristic
484C2×C3#6349811yescharacteristic
485C2×C3#6349911yescharacteristic
486C2×C3#6385111yescharacteristic
487C2×C3#6385211yescharacteristic
488C2×C3#6389311yescharacteristic
489C2×C3#6389911yescharacteristic
490C2×C3#6391511yescharacteristic
491C2×C3#6392211yescharacteristic
492C2×C3#6395911yescharacteristic
493D3—42no—
494C22—62no—
495C22—62no—
496C22—62no—
497C22—121no—
498C22—121no—
499C22—121no—
500C22—121no—
501C22D3×Dic1811yescharacteristic
502C22#8864011yescharacteristic
503C22D9×D1211yescharacteristic
504C22#8864611yescharacteristic
505C22#8866211yescharacteristic
506C22#8866311yescharacteristic
507C22D3×#61111yescharacteristic
508C4—41no—
509C4—62no—
510C4—182no—
511C4—361no—
512C4—542no—
513C4—1081no—
514C3—21no—
515C3#1058862711yescharacteristic, complemented
516C3#1059129311yescharacteristic
517C2—122no—
518C2#1172095511yescharacteristic
519C2#1172095611yescharacteristic
520C2#1172102311yescharacteristic
521C2#1172102611yescharacteristic
522C2#1172104411yescharacteristic
523C2#1172105011yescharacteristic
524C2#1172126311yescharacteristic
subgroupG/H
centerC23C2×D3×D9
commutator, cosocleC23×C3×C9C23
FrattiniC23×C3C2×D32
FittingC22×C3×C4×C9C22
socleC23×C32C22×D3
prime radicalC23×C3×D3C2×D3
hypercenterC22×C4D3×D9
hypocenterC3×C9#393
LCSG ▷ C23×C3×C9 ▷ C3×C9
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×#6189C2semidirect
5. #11720572C2semidirect
6. C3×C9⋊#95C2semidirect
7. #11720331C2semidirect
37. C9⋊#95D3semidirect
45. C22×D3×C9Q8semidirect
290. C3×C9#393semidirect
307. C22×D3Dic18semidirect
443. C9#4673semidirect
515. C3#10588627semidirect

Maximal subgroups

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

subgroupindex
C22×#61892
C2×C9×(C2×D3)⋊C42
C2×(C2×D3)⋊Dic92
#117205722
C3×C9⋊#952
#117203312
#105912933
#105886273