Finite Group Databasebeta

← #13141117

Subgroups — #13141117

Order 1728.

642 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×#88551C211yesmaximal, characteristic
2C2×C9×#217C211yesmaximal, characteristic
3C2×C3×#3676C211yesmaximal, characteristic
4C2×Dic3⋊Dic9C211yesmaximal, characteristic, complemented
5#11720331C211yesmaximal, characteristic, complemented
6#11720331C211yesmaximal, characteristic, complemented
7#11720331C211yesmaximal, characteristic, complemented
8#10591717—31nomaximal
9#10588633—31nomaximal
10C24×C3×C9C2211yescharacteristic
11C22×C3⋊Dic9C2211yescharacteristic
12C22×C3⋊Dic9C2211yescharacteristic
13C22×C9×Dic3C2211yescharacteristic
14C22×C9×Dic3C2211yescharacteristic
15C22×C3×Dic9C2211yescharacteristic
16C22×C3×Dic9C2211yescharacteristic
17#88551—22no—
18C9×#217—22no—
19C3×#3676—22no—
20Dic3⋊Dic9—22no—
21C2×#3757—31no—
22C2×C3×#217—31no—
23C2×C3×#217D311yescharacteristic
24C2×Dic3⋊Dic3—31no—
25#63579—31no—
26#63579—31no—
27#63579—31no—
28C2×C9×C22⋊C4—31no—
29C2×#3676—31no—
30C2×#3676—61no—
31C2×#3676D311yescharacteristic
32C2×C4⋊Dic9—31no—
33C9⋊#95—31no—
34C9⋊#95—31no—
35C9⋊#95—31no—
36C23×C3×C9—21no—
37C23×C3×C9—21no—
38C23×C3×C9—21no—
39C23×C3×C9—21no—
40C23×C3×C9—21no—
41C23×C3×C9—21no—
42C23×C3×C9D412yes—
43C23×C3×C9C2311yescharacteristic
44C2×C3⋊Dic9—21no—
45C2×C3⋊Dic9—21no—
46C2×C3⋊Dic9—21no—
47C2×C3⋊Dic9—21no—
48C2×C3⋊Dic9—21no—
49C2×C3⋊Dic9—21no—
50C2×C9×Dic3—21no—
51C2×C9×Dic3—21no—
52C2×C9×Dic3—21no—
53C2×C9×Dic3—21no—
54C2×C9×Dic3—21no—
55C2×C9×Dic3D412yes—
56C2×C3×Dic9—21no—
57C2×C3×Dic9—21no—
58C2×C3×Dic9—21no—
59C2×C3×Dic9—21no—
60C2×C3×Dic9—21no—
61C2×C3×Dic9—21no—
62#4953—91no—
63#4952—91no—
64C24×C32C2×D311yescharacteristic
65C22×C3⋊Dic3—31no—
66C22×C3⋊Dic3—31no—
67C22×C3×Dic3—31no—
68C22×C3×Dic3—31no—
69C22×C3×Dic3C2×D311yescharacteristic
70C22×C3×Dic3C2×D311yescharacteristic
71C24×C9—21no—
72C24×C9C2×D311yescharacteristic
73#3757—62no—
74C3×#217—22no—
75C3×#217—62no—
76Dic3⋊Dic3—62no—
77C22×C4×C9—31no—
78C22×C4×C9—31no—
79C22×Dic9—31no—
80C22×Dic9—31no—
81C22×Dic9—61no—
82C22×Dic9—61no—
83C22×Dic9C2×D311yescharacteristic
84C22×Dic9C2×D311yescharacteristic
85C9×C22⋊C4—62no—
86#3676—22no—
87#3676—64no—
88#3676—62no—
89C4⋊Dic9—62no—
90C22×C3×C9—22no—
91C22×C3×C9—22no—
92C22×C3×C9—22no—
93C22×C3×C9—41no—
94C22×C3×C9—41no—
95C22×C3×C9—41no—
96C22×C3×C9—41no—
97C22×C3×C9C2×D411yescharacteristic
98C22×C3×C9C2×D411yescharacteristic
99C22×C3×C9#4111yescharacteristic
100C22×C3×C9#4111yescharacteristic
101C22×C3×C9#4111yescharacteristic
102C22×C3×C9#4111yescharacteristic
103C22×C3×C9#4111yescharacteristic
104C3⋊Dic9—41no—
105C3⋊Dic9—41no—
106C9×Dic3—22no—
107C9×Dic3—41no—
108C3×Dic9—41no—
109C3×Dic9—41no—
110C2×C3×C22⋊C4—31no—
111C2×C3×C22⋊C4—91no—
112C2×#217—31no—
113C2×#217—91no—
114C2×#217—91no—
115C2×#217—181no—
116C2×#217D911yescharacteristic
117C2×C4⋊Dic3—91no—
118C2×Dic3⋊C4—91no—
119C3⋊#95—91no—
120C3⋊#95—91no—
121C3⋊#95—91no—
122C3⋊#95—91no—
123C3⋊#95—91no—
124C3⋊#95—91no—
125C23×C32—21no—
126C23×C32—21no—
127C23×C32—21no—
128C23×C32—21no—
129C23×C32—21no—
130C23×C32—21no—
131C23×C32D1212yes—
132C23×C32C22×D311yescharacteristic
133C2×C3⋊Dic3—61no—
134C2×C3⋊Dic3—61no—
135C2×C3⋊Dic3—61no—
136C2×C3⋊Dic3—61no—
137C2×C3⋊Dic3—61no—
138C2×C3⋊Dic3—61no—
139C2×C3×Dic3—21no—
140C2×C3×Dic3—21no—
141C2×C3×Dic3—21no—
142C2×C3×Dic3—21no—
143C2×C3×Dic3—21no—
144C2×C3×Dic3—61no—
145C2×C3×Dic3—61no—
146C2×C3×Dic3—61no—
147C2×C3×Dic3—61no—
148C2×C3×Dic3—61no—
149C2×C3×Dic3—61no—
150C2×C3×Dic3#6712yes—
151C23×C9—22no—
152C23×C9—22no—
153C23×C9—21no—
154C23×C9—21no—
155C23×C9—21no—
156C23×C9—21no—
157C23×C9—21no—
158C23×C9—21no—
159C23×C9—22no—
160C23×C9—21no—
161C23×C9—41no—
162C23×C9—41no—
163C23×C9—41no—
164C23×C9—41no—
165C23×C9#6712yes—
166C23×C9C22×D311yescharacteristic
167C2×C4×C9—32no—
168C2×C4×C9—61no—
169C2×C4×C9—61no—
170C2×C4×C9—61no—
171C2×C4×C9—61no—
172C2×C4×C9—61no—
173C2×Dic9—21no—
174C2×Dic9—21no—
175C2×Dic9—21no—
176C2×Dic9—21no—
177C2×Dic9—21no—
178C2×Dic9—21no—
179C2×Dic9—62no—
180C2×Dic9—62no—
181C2×Dic9—61no—
182C2×Dic9—61no—
183C2×Dic9—61no—
184C2×Dic9—61no—
185C2×Dic9—61no—
186C2×Dic9—61no—
187C2×Dic9—121no—
188C2×Dic9—121no—
189C2×Dic9—121no—
190C2×Dic9—121no—
191#397—271noSylow
192C2×C3×C9—42no—
193C2×C3×C9#12111yescharacteristic
194C2×C3×C9#12311yescharacteristic
195C2×C3×C9#12311yescharacteristic
196C2×C3×C9#12311yescharacteristic
197C2×C3×C9#12411yescharacteristic
198C2×C3×C9#12611yescharacteristic
199C2×C3×C9#12611yescharacteristic
200C24×C3—21no—
201C24×C3C2×D911yescharacteristic
202C24×C3D3211yescharacteristic
203C22×C3×C4—31no—
204C22×C3×C4—31no—
205C22×C3×C4—91no—
206C22×C3×C4—91no—
207C22×Dic3—31no—
208C22×Dic3—31no—
209C22×Dic3—91no—
210C22×Dic3—91no—
211C22×Dic3—91no—
212C22×Dic3—91no—
213C22×Dic3—181no—
214C22×Dic3—181no—
215C22×Dic3C2×D911yescharacteristic
216C22×Dic3C2×D911yescharacteristic
217C3×C22⋊C4—62no—
218C3×C22⋊C4—182no—
219#217—22no—
220#217—62no—
221#217—184no—
222#217—182no—
223#217—182no—
224C4⋊Dic3—182no—
225Dic3⋊C4—182no—
226C22×C32—22no—
227C22×C32—22no—
228C22×C32—22no—
229C22×C32—41no—
230C22×C32—41no—
231C22×C32—41no—
232C22×C32—41no—
233C22×C32C2×D1211yescharacteristic
234C22×C32#23511yescharacteristic
235C22×C32#23711yescharacteristic
236C22×C32#23711yescharacteristic
237C22×C32#23711yescharacteristic
238C22×C32#23711yescharacteristic
239C22×C32C2×#6711yescharacteristic
240C3⋊Dic3—121no—
241C3⋊Dic3—121no—
242C3×Dic3—22no—
243C3×Dic3—41no—
244C3×Dic3—121no—
245C3×Dic3—121no—
246C22×C9—24no—
247C22×C9—22no—
248C22×C9—22no—
249C22×C9—22no—
250C22×C9—21no—
251C22×C9—21no—
252C22×C9—21no—
253C22×C9—21no—
254C22×C9—21no—
255C22×C9—21no—
256C22×C9—21no—
257C22×C9—42no—
258C22×C9—42no—
259C22×C9—42no—
260C22×C9—42no—
261C22×C9—42no—
262C22×C9—42no—
263C22×C9—41no—
264C22×C9—41no—
265C22×C9—41no—
266C22×C9—41no—
267C22×C9#23511yescharacteristic
268C22×C9#23511yescharacteristic
269C22×C9D3×D411yescharacteristic
270C22×C9#23711yescharacteristic
271C22×C9#23711yescharacteristic
272C22×C9#23711yescharacteristic
273C22×C9C2×#6711yescharacteristic
274C4×C9—62no—
275C4×C9—121no—
276Dic9—41no—
277Dic9—41no—
278Dic9—122no—
279Dic9—122no—
280Dic9—121no—
281Dic9—121no—
282C2×C4⋊C4—271no—
283C2×C22⋊C4—31no—
284C2×C22⋊C4—91no—
285C2×C22⋊C4—271no—
286#95—271no—
287#95—271no—
288#95—271no—
289C3×C9#39711yescharacteristic, complemented, Sylow
290C23×C3—22no—
291C23×C3—22no—
292C23×C3—21no—
293C23×C3—21no—
294C23×C3—21no—
295C23×C3—21no—
296C23×C3—21no—
297C23×C3—21no—
298C23×C3—21no—
299C23×C3—21no—
300C23×C3—21no—
301C23×C3—21no—
302C23×C3—21no—
303C23×C3—21no—
304C23×C3—22no—
305C23×C3—21no—
306C23×C3—41no—
307C23×C3—41no—
308C23×C3—41no—
309C23×C3—41no—
310C23×C3D3612yes—
311C23×C3C22×D911yescharacteristic
312C23×C3#62612yes—
313C23×C3C2×D3211yescharacteristic
314C2×C3×C4—32no—
315C2×C3×C4—61no—
316C2×C3×C4—61no—
317C2×C3×C4—61no—
318C2×C3×C4—61no—
319C2×C3×C4—61no—
320C2×C3×C4—181no—
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—21no—
329C2×Dic3—21no—
330C2×Dic3—21no—
331C2×Dic3—61no—
332C2×Dic3—61no—
333C2×Dic3—61no—
334C2×Dic3—61no—
335C2×Dic3—61no—
336C2×Dic3—61no—
337C2×Dic3—182no—
338C2×Dic3—182no—
339C2×Dic3—181no—
340C2×Dic3—181no—
341C2×Dic3—181no—
342C2×Dic3—181no—
343C2×Dic3—181no—
344C2×Dic3—181no—
345C2×Dic3—181no—
346C2×Dic3—181no—
347C2×Dic3—181no—
348C2×Dic3—181no—
349C2×Dic3—181no—
350C2×Dic3—181no—
351C2×Dic3—361no—
352C2×Dic3—361no—
353C2×Dic3—361no—
354C2×Dic3—361no—
355C2×Dic3#61112yes—
356C2×C32—42no—
357C2×C32#87911yescharacteristic
358C2×C32#87911yescharacteristic
359C2×C32#88611yescharacteristic
360C2×C32#88611yescharacteristic
361C2×C32#93011yescharacteristic
362C2×C32#93511yescharacteristic
363C2×C32#93611yescharacteristic
364C2×C9—21no—
365C2×C9—21no—
366C2×C9—21no—
367C2×C9—21no—
368C2×C9—21no—
369C2×C9—21no—
370C2×C9—21no—
371C2×C9—44no—
372C2×C9—42no—
373C2×C9#87911yescharacteristic
374C2×C9#87911yescharacteristic
375C2×C9#88311yescharacteristic
376C2×C9#88511yescharacteristic
377C2×C9#92911yescharacteristic
378C2×C9#93511yescharacteristic
379C2×C9#93911yescharacteristic
380C24D3×D911yescharacteristic
381C22×C4—31no—
382C22×C4—31no—
383C22×C4—91no—
384C22×C4—91no—
385C22×C4—271no—
386C22×C4—271no—
387C4⋊C4—542no—
388C22⋊C4—62no—
389C22⋊C4—182no—
390C22⋊C4—542no—
391C22×C3—24no—
392C22×C3—22no—
393C22×C3—22no—
394C22×C3—22no—
395C22×C3—22no—
396C22×C3—22no—
397C22×C3—22no—
398C22×C3—21no—
399C22×C3—21no—
400C22×C3—21no—
401C22×C3—21no—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—21no—
405C22×C3—42no—
406C22×C3—42no—
407C22×C3—42no—
408C22×C3—42no—
409C22×C3—42no—
410C22×C3—42no—
411C22×C3—41no—
412C22×C3—41no—
413C22×C3—41no—
414C22×C3—41no—
415C22×C3—41no—
416C22×C3—41no—
417C22×C3—41no—
418C22×C3—41no—
419C22×C3C2×D3611yescharacteristic
420C22×C3#369711yescharacteristic
421C22×C3#369911yescharacteristic
422C22×C3#369911yescharacteristic
423C22×C3#369911yescharacteristic
424C22×C3#369911yescharacteristic
425C22×C3C2×#61111yescharacteristic
426C22×C3#380411yescharacteristic
427C22×C3#380411yescharacteristic
428C22×C3#380411yescharacteristic
429C22×C3#380511yescharacteristic
430C22×C3#380511yescharacteristic
431C22×C3C2×#62611yescharacteristic
432C22×C3D3×#6711yescharacteristic
433C3×C4—62no—
434C3×C4—121no—
435C3×C4—361no—
436C3×C4—361no—
437Dic3—22no—
438Dic3—41no—
439Dic3—121no—
440Dic3—121no—
441Dic3—362no—
442Dic3—362no—
443Dic3—361no—
444Dic3—361no—
445Dic3—361no—
446Dic3—361no—
447C32#495311yescharacteristic
448C9—21no—
449C9#495211yescharacteristic, complemented
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23#619212yes—
457C23C2×D3×D911yescharacteristic
458C2×C4—32no—
459C2×C4—61no—
460C2×C4—61no—
461C2×C4—61no—
462C2×C4—61no—
463C2×C4—61no—
464C2×C4—181no—
465C2×C4—181no—
466C2×C4—181no—
467C2×C4—181no—
468C2×C4—181no—
469C2×C4—181no—
470C2×C4—541no—
471C2×C4—541no—
472C2×C4—541no—
473C2×C4—541no—
474C2×C4—541no—
475C2×C4—541no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—21no—
480C2×C3—21no—
481C2×C3—21no—
482C2×C3—21no—
483C2×C3—44no—
484C2×C3—42no—
485C2×C3—42no—
486C2×C3#6344111yescharacteristic
487C2×C3#6344111yescharacteristic
488C2×C3#6344811yescharacteristic
489C2×C3#6344811yescharacteristic
490C2×C3#6349211yescharacteristic
491C2×C3#6349711yescharacteristic
492C2×C3#6349811yescharacteristic
493C2×C3#6395611yescharacteristic
494C2×C3#6395611yescharacteristic
495C2×C3#6395911yescharacteristic
496C2×C3#6396211yescharacteristic
497C2×C3#6396611yescharacteristic
498C2×C3#6396911yescharacteristic
499C2×C3#6397811yescharacteristic
500C22—22no—
501C22—22no—
502C22—22no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22#8865811yescharacteristic
508C22C2×#619211yescharacteristic
509C22#8866211yescharacteristic
510C22#8866211yescharacteristic
511C22#8866311yescharacteristic
512C22#8866311yescharacteristic
513C22D3×#61111yescharacteristic
514C4—62no—
515C4—121no—
516C4—361no—
517C4—361no—
518C4—1081no—
519C4—1081no—
520C3—21no—
521C3#1058863311yescharacteristic, complemented
522C3#1059171711yescharacteristic
523C2—42no—
524C2#1172124611yescharacteristic
525C2#1172124711yescharacteristic
526C2#1172125511yescharacteristic
527C2#1172126011yescharacteristic
528C2#1172126311yescharacteristic
529C2#1172127711yescharacteristic
530C2#1172128011yescharacteristic
subgroupG/H
centerC23C2×D3×D9
commutator, cosocleC23×C3×C9C23
FrattiniC23×C3C2×D32
FittingC24×C3×C9C22
socleC23×C32C22×D3
prime radicalC24×C32C2×D3
hypercenterC24D3×D9
hypocenterC3×C9#397
LCSG ▷ C23×C3×C9 ▷ C3×C9
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
4. C2×Dic3⋊Dic9C2semidirect
5. #11720331C2semidirect
6. #11720331C2semidirect
7. #11720331C2semidirect
289. C3×C9#397semidirect
449. C9#4952semidirect
521. C3#10588633semidirect

Maximal subgroups

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

subgroupindex
C2×#885512
C2×C9×#2172
C2×C3×#36762
C2×Dic3⋊Dic92
#117203312
#105917173
#105886333