Finite Group Databasebeta

← #13160513

Subgroups — #13160513

Order 1728.

1324 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3752C211yesmaximal, characteristic
2C2×Dic3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
3C2×#88803C211yesmaximal, characteristic
4C2×#88803C211yesmaximal, characteristic
5#11722201C211yesmaximal, characteristic, complemented
6C32×C3⋊#95C211yesmaximal, characteristic, complemented
7#11722061C211yesmaximal, characteristic, complemented
8#10592373—31nomaximal
9#10591296—34nomaximal
10C23×C3×C3⋊D3C2211yescharacteristic
11C22×C33×C4C2211yescharacteristic
12C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
13C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
14C22×C3×C3⋊Dic3C2211yescharacteristic
15C22×C32×Dic3C2211yescharacteristic
16C22×C32×Dic3C2211yescharacteristic
17C3×#3752—22no—
18Dic3⋊(C3⋊Dic3)—22no—
19#88803—22no—
20#88803—22no—
21C2×#3752—31no—
22C2×#3752—31no—
23C2×#3752D311yescharacteristic
24C2×C4⋊(C3⋊Dic3)—31no—
25C2×C3×(C2×D3)⋊C4—34no—
26C2×Dic3⋊Dic3—34no—
27C2×(C2×D3)⋊Dic3—34no—
28C2×(C2×D3)⋊Dic3—34no—
29C32×#95—31no—
30#63658—31no—
31#63658—31no—
32#63658—34no—
33#63658—64no—
34C3×C3⋊#95D314yescomplemented
35#63579—34no—
36C23×C33C2311yescharacteristic
37C22×C3×C3⋊D3—21no—
38C22×C3×C3⋊D3—21no—
39C22×C3×C3⋊D3—21no—
40C22×C3×C3⋊D3—21no—
41C22×C3×C3⋊D3—21no—
42C22×C3×C3⋊D3—21no—
43C22×C3×C3⋊D3D412yes—
44C2×C33×C4—21no—
45C2×C33×C4—21no—
46C2×C33×C4—21no—
47C2×C3⋊(C3⋊Dic3)—21no—
48C2×C3⋊(C3⋊Dic3)—21no—
49C2×C3⋊(C3⋊Dic3)—21no—
50C2×C3⋊(C3⋊Dic3)—21no—
51C2×C3⋊(C3⋊Dic3)—21no—
52C2×C3⋊(C3⋊Dic3)—21no—
53C2×C3×C3⋊Dic3—21no—
54C2×C3×C3⋊Dic3—21no—
55C2×C3×C3⋊Dic3—21no—
56C2×C32×Dic3—21no—
57C2×C32×Dic3—21no—
58C2×C32×Dic3—21no—
59C2×C32×Dic3—21no—
60C2×C32×Dic3—21no—
61C2×C32×Dic3D412yes—
62#4952—91no—
63#4674—94no—
64C23×C3⋊D3C2×D311yescharacteristic
65C23×C3×D3—34no—
66C22×C32×C4—24no—
67C22×C32×C4—31no—
68C22×C32×C4—31no—
69C22×C32×C4C2×D314yes—
70C22×C32×C4C2×D311yescharacteristic
71C22×C3⋊Dic3—31no—
72C22×C3⋊Dic3—31no—
73C22×C3⋊Dic3—34no—
74C22×C3⋊Dic3—34no—
75C22×C3⋊Dic3—64no—
76C22×C3⋊Dic3—64no—
77C22×C3⋊Dic3C2×D311yescharacteristic
78C22×C3×Dic3—34no—
79C22×C3×Dic3C2×D314yes—
80C22×C3×Dic3C2×D314yes—
81#3752—22no—
82#3752—62no—
83#3752—62no—
84C4⋊(C3⋊Dic3)—62no—
85C3×(C2×D3)⋊C4—68no—
86Dic3⋊Dic3—68no—
87(C2×D3)⋊Dic3—68no—
88(C2×D3)⋊Dic3—68no—
89C22×C33C2×D411yescharacteristic
90C22×C33C2×D411yescharacteristic
91C22×C33#4111yescharacteristic
92C22×C33#4111yescharacteristic
93C22×C33#4111yescharacteristic
94C22×C33#4111yescharacteristic
95C22×C33#4111yescharacteristic
96C2×C3×C3⋊D3—22no—
97C2×C3×C3⋊D3—22no—
98C2×C3×C3⋊D3—22no—
99C2×C3×C3⋊D3—41no—
100C2×C3×C3⋊D3—41no—
101C2×C3×C3⋊D3—41no—
102C2×C3×C3⋊D3—41no—
103C33×C4—41no—
104C3⋊(C3⋊Dic3)—41no—
105C3⋊(C3⋊Dic3)—41no—
106C3×C3⋊Dic3—41no—
107C32×Dic3—22no—
108C32×Dic3—41no—
109C2×C3×C22⋊C4—91no—
110C2×#217—91no—
111C2×#217—91no—
112C2×(C2×D3)⋊C4—34no—
113C2×(C2×D3)⋊C4—94no—
114C2×(C2×D3)⋊C4—94no—
115C2×C4⋊Dic3—94no—
116C2×Dic3⋊C4—91no—
117C3×#95—34no—
118C3⋊#95—94no—
119C3⋊#95—94no—
120C3⋊#95—91no—
121C3⋊#95—91no—
122C3⋊#95—184no—
123C3⋊#95C3⋊D311yescharacteristic, complemented
124C23×C32—24no—
125C23×C32C22×D314yes—
126C23×C32C22×D311yescharacteristic
127C22×C3⋊D3—21no—
128C22×C3⋊D3—21no—
129C22×C3⋊D3—21no—
130C22×C3⋊D3—21no—
131C22×C3⋊D3—21no—
132C22×C3⋊D3—21no—
133C22×C3⋊D3#6712yes—
134C22×C3×D3—38no—
135C22×C3×D3—64no—
136C22×C3×D3—64no—
137C22×C3×D3—64no—
138C22×C3×D3—64no—
139C22×C3×D3—64no—
140C22×C3×D3—64no—
141C2×C32×C4—28no—
142C2×C32×C4—24no—
143C2×C32×C4—21no—
144C2×C32×C4—21no—
145C2×C32×C4—21no—
146C2×C32×C4—24no—
147C2×C32×C4—24no—
148C2×C32×C4—32no—
149C2×C32×C4—44no—
150C2×C32×C4—44no—
151C2×C32×C4—61no—
152C2×C32×C4—61no—
153C2×C32×C4—61no—
154C2×C32×C4—61no—
155C2×C32×C4—61no—
156C2×C3⋊Dic3—21no—
157C2×C3⋊Dic3—21no—
158C2×C3⋊Dic3—21no—
159C2×C3⋊Dic3—68no—
160C2×C3⋊Dic3—68no—
161C2×C3⋊Dic3—64no—
162C2×C3⋊Dic3—61no—
163C2×C3⋊Dic3—61no—
164C2×C3⋊Dic3—61no—
165C2×C3⋊Dic3—61no—
166C2×C3⋊Dic3—61no—
167C2×C3⋊Dic3—64no—
168C2×C3⋊Dic3—61no—
169C2×C3⋊Dic3—64no—
170C2×C3⋊Dic3—64no—
171C2×C3⋊Dic3—64no—
172C2×C3⋊Dic3—64no—
173C2×C3⋊Dic3—124no—
174C2×C3⋊Dic3—124no—
175C2×C3⋊Dic3—124no—
176C2×C3⋊Dic3—124no—
177C2×C3×Dic3—24no—
178C2×C3×Dic3—24no—
179C2×C3×Dic3—24no—
180C2×C3×Dic3—24no—
181C2×C3×Dic3—24no—
182C2×C3×Dic3—64no—
183C2×C3×Dic3—64no—
184C2×C3×Dic3—64no—
185C2×C3×Dic3D1218yes—
186#397—271noSylow
187C2×C33#12111yescharacteristic
188C2×C33#12311yescharacteristic
189C2×C33#12311yescharacteristic
190C2×C33#12311yescharacteristic
191C2×C33#12411yescharacteristic
192C2×C33#12611yescharacteristic
193C2×C33#12611yescharacteristic
194C3×C3⋊D3—42no—
195C24×C3—91no—
196C23×D3—34no—
197C22×C3×C4—24no—
198C22×C3×C4—34no—
199C22×C3×C4—34no—
200C22×C3×C4—91no—
201C22×C3×C4D3214yes—
202C22×C3×C4C2×C3⋊D311yescharacteristic
203C22×Dic3—34no—
204C22×Dic3—94no—
205C22×Dic3—94no—
206C22×Dic3—91no—
207C22×Dic3—91no—
208C22×Dic3—184no—
209C22×Dic3—184no—
210C22×Dic3C2×C3⋊D311yescharacteristic
211C22×Dic3C2×C3⋊D311yescharacteristic
212C3×C22⋊C4—182no—
213#217—182no—
214#217—182no—
215(C2×D3)⋊C4—68no—
216(C2×D3)⋊C4—188no—
217(C2×D3)⋊C4—188no—
218C4⋊Dic3—188no—
219Dic3⋊C4—182no—
220C22×C32—24no—
221C22×C32—24no—
222C22×C32—24no—
223C22×C32—24no—
224C22×C32—24no—
225C22×C32—24no—
226C22×C32—24no—
227C22×C32C2×D1214yes—
228C22×C32#23514yes—
229C22×C32#23514yes—
230C22×C32#23511yescharacteristic
231C22×C32#23511yescharacteristic
232C22×C32D3×D414yes—
233C22×C32D3×D411yescharacteristic
234C22×C32#23714yes—
235C22×C32#23714yes—
236C22×C32#23711yescharacteristic
237C22×C32#23711yescharacteristic
238C22×C32#23711yescharacteristic
239C22×C32#23914yes—
240C22×C32C2×#6711yescharacteristic
241C2×C3⋊D3—22no—
242C2×C3⋊D3—22no—
243C2×C3⋊D3—22no—
244C2×C3⋊D3—41no—
245C2×C3⋊D3—41no—
246C2×C3⋊D3—41no—
247C2×C3⋊D3—41no—
248C2×C3×D3—68no—
249C2×C3×D3—68no—
250C2×C3×D3—68no—
251C2×C3×D3—124no—
252C2×C3×D3—124no—
253C2×C3×D3—124no—
254C2×C3×D3—124no—
255C32×C4—48no—
256C32×C4—44no—
257C32×C4—41no—
258C32×C4—62no—
259C32×C4—121no—
260C3⋊Dic3—41no—
261C3⋊Dic3—128no—
262C3⋊Dic3—128no—
263C3⋊Dic3—124no—
264C3⋊Dic3—121no—
265C3⋊Dic3—124no—
266C3⋊Dic3—121no—
267C3×Dic3—28no—
268C3×Dic3—44no—
269C3×Dic3—124no—
270C2×C4⋊C4—271no—
271C2×C22⋊C4—91no—
272C2×C22⋊C4—271no—
273C2×C22⋊C4—271no—
274#95—31no—
275#95—271no—
276#95—271no—
277C33#39711yescharacteristic, complemented, Sylow
278C23×C3—24no—
279C23×C3—92no—
280C23×C3—181no—
281C23×C3—181no—
282C23×C3—181no—
283C23×C3—181no—
284C23×C3—181no—
285C23×C3—181no—
286C23×C3C2×D3214yes—
287C23×C3C22×C3⋊D311yescharacteristic
288C22×D3—38no—
289C22×D3—64no—
290C22×D3—64no—
291C22×D3—64no—
292C22×D3—64no—
293C22×D3—64no—
294C22×D3—64no—
295C2×C3×C4—28no—
296C2×C3×C4—24no—
297C2×C3×C4—24no—
298C2×C3×C4—24no—
299C2×C3×C4—21no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—38no—
303C2×C3×C4—44no—
304C2×C3×C4—44no—
305C2×C3×C4—64no—
306C2×C3×C4—64no—
307C2×C3×C4—64no—
308C2×C3×C4—64no—
309C2×C3×C4—64no—
310C2×C3×C4—181no—
311C2×C3×C4—181no—
312C2×C3×C4—181no—
313C2×Dic3—21no—
314C2×Dic3—21no—
315C2×Dic3—21no—
316C2×Dic3—21no—
317C2×Dic3—21no—
318C2×Dic3—64no—
319C2×Dic3—64no—
320C2×Dic3—64no—
321C2×Dic3—188no—
322C2×Dic3—188no—
323C2×Dic3—184no—
324C2×Dic3—184no—
325C2×Dic3—184no—
326C2×Dic3—184no—
327C2×Dic3—184no—
328C2×Dic3—184no—
329C2×Dic3—181no—
330C2×Dic3—181no—
331C2×Dic3—181no—
332C2×Dic3—181no—
333C2×Dic3—181no—
334C2×Dic3—181no—
335C2×Dic3—364no—
336C2×Dic3—364no—
337C2×Dic3—364no—
338C2×Dic3—364no—
339C2×Dic3C4⋊(C3⋊D3)12yes—
340C2×C32—24no—
341C2×C32—24no—
342C2×C32—24no—
343C2×C32—24no—
344C2×C32—24no—
345C2×C32—24no—
346C2×C32—24no—
347C2×C32#87514yes—
348C2×C32#87911yescharacteristic
349C2×C32#87911yescharacteristic
350C2×C32#88314yes—
351C2×C32#88314yes—
352C2×C32#88311yescharacteristic
353C2×C32#88511yescharacteristic
354C2×C32#88614yes—
355C2×C32#89514yes—
356C2×C32#89814yes—
357C2×C32#89814yes—
358C2×C32#92911yescharacteristic
359C2×C32#93511yescharacteristic
360C2×C32#93911yescharacteristic
361C3⋊D3—42no—
362C3×D3—128no—
363C24—91no—
364C22×C4—31no—
365C22×C4—31no—
366C22×C4—91no—
367C22×C4—271no—
368C22×C4—271no—
369C22×C4D3×C3⋊D311yescharacteristic
370C4⋊C4—542no—
371C22⋊C4—182no—
372C22⋊C4—542no—
373C22⋊C4—542no—
374C22×C3—24no—
375C22×C3—24no—
376C22×C3—24no—
377C22×C3—24no—
378C22×C3—24no—
379C22×C3—24no—
380C22×C3—24no—
381C22×C3—182no—
382C22×C3—182no—
383C22×C3—182no—
384C22×C3—361no—
385C22×C3—361no—
386C22×C3—361no—
387C22×C3—361no—
388C22×C3#379514yes—
389C22×C3#379614yes—
390C22×C3#379814yes—
391C22×C3D3×D1214yes—
392C22×C3#380414yes—
393C22×C3#380514yes—
394C22×C3D3×#6714yes—
395C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
396C22×C3#382811yescharacteristic
397C22×C3#382811yescharacteristic
398C22×C3D4×C3⋊D311yescharacteristic
399C22×C3#383011yescharacteristic
400C22×C3#383011yescharacteristic
401C22×C3#383211yescharacteristic
402C2×D3—68no—
403C2×D3—68no—
404C2×D3—68no—
405C2×D3—124no—
406C2×D3—124no—
407C2×D3—124no—
408C2×D3—124no—
409C3×C4—48no—
410C3×C4—41no—
411C3×C4—44no—
412C3×C4—68no—
413C3×C4—124no—
414C3×C4—361no—
415Dic3—22no—
416Dic3—41no—
417Dic3—124no—
418Dic3—368no—
419Dic3—368no—
420Dic3—361no—
421Dic3—361no—
422Dic3—364no—
423Dic3—364no—
424C32—24no—
425C32#467414yescomplemented
426C32#495211yescharacteristic, complemented
427C23—92no—
428C23—181no—
429C23—181no—
430C23—181no—
431C23—181no—
432C23—181no—
433C23—181no—
434C23C2×D3×C3⋊D311yescharacteristic
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—32no—
439C2×C4—61no—
440C2×C4—61no—
441C2×C4—61no—
442C2×C4—61no—
443C2×C4—61no—
444C2×C4—181no—
445C2×C4—181no—
446C2×C4—181no—
447C2×C4—541no—
448C2×C4—541no—
449C2×C4—541no—
450C2×C4—541no—
451C2×C4—541no—
452C2×C4—541no—
453C2×C3—24no—
454C2×C3—24no—
455C2×C3—24no—
456C2×C3—24no—
457C2×C3—24no—
458C2×C3—24no—
459C2×C3—24no—
460C2×C3—362no—
461C2×C3#6385214yes—
462C2×C3#6385814yes—
463C2×C3#6386214yes—
464C2×C3#6388614yes—
465C2×C3#6389114yes—
466C2×C3#6390514yes—
467C2×C3#6396914yes—
468C2×C3#6408411yescharacteristic
469C2×C3#6409211yescharacteristic
470C2×C3#6409211yescharacteristic
471C2×C3#6409511yescharacteristic
472C2×C3#6410411yescharacteristic
473C2×C3#6410711yescharacteristic
474C2×C3#6410711yescharacteristic
475D3—128no—
476C22—182no—
477C22—182no—
478C22—182no—
479C22—361no—
480C22—361no—
481C22—361no—
482C22—361no—
483C22#8901411yescharacteristic
484C22#8901711yescharacteristic
485C22#8902011yescharacteristic
486C22D3×C4⋊(C3⋊D3)11yescharacteristic
487C22#8902911yescharacteristic
488C22#8903111yescharacteristic
489C22C3⋊D3×#6711yescharacteristic
490C4—41no—
491C4—62no—
492C4—121no—
493C4—361no—
494C4—1081no—
495C4—1081no—
496C3—24no—
497C3#1059129614yescomplemented
498C3#1059237311yescharacteristic, complemented
499C2—362no—
500C2#1172340711yescharacteristic
501C2#1172341611yescharacteristic
502C2#1172343311yescharacteristic
503C2#1172348011yescharacteristic
504C2#1172348211yescharacteristic
505C2#1172351811yescharacteristic
506C2#1172362111yescharacteristic
subgroupG/H
center, FrattiniC23C2×D3×C3⋊D3
commutator, socle, cosocleC23×C33C23
FittingC22×C33×C4C22
prime radicalC23×C3×C3⋊D3C22
hypercenterC22×C4D3×C3⋊D3
hypocenterC33#397
LCSG ▷ C23×C33 ▷ C33
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×Dic3⋊(C3⋊Dic3)C2semidirect
5. #11722201C2semidirect
6. C32×C3⋊#95C2semidirect
7. #11722061C2semidirect
34. C3×C3⋊#95D3semidirect
123. C3⋊#95C3⋊D3semidirect
277. C33#397semidirect
425. C32#4674semidirect
426. C32#4952semidirect
497. C3#10591296semidirect
498. C3#10592373semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37522
C2×Dic3⋊(C3⋊Dic3)2
C2×#888032
#117222012
C32×C3⋊#952
#117220612
#105923733
#105912963