Finite Group Databasebeta

← #13161441

Subgroups — #13161441

Order 1728.

1644 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88864C211yesmaximal, characteristic
2C2×C3×#3757C211yesmaximal, characteristic
3C2×C32×#217C211yesmaximal, characteristic
4C2×#88807C211yesmaximal, characteristic, complemented
5C2×#88806C211yesmaximal, characteristic, complemented
6C2×Dic3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
7#11722061C211yesmaximal, characteristic, complemented
8#10592648—31nomaximal
9#10591715—34nomaximal
10C24×C33C2211yescharacteristic
11C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
12C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
13C22×C3×C3⋊Dic3C2211yescharacteristic
14C22×C3×C3⋊Dic3C2211yescharacteristic
15C22×C32×Dic3C2211yescharacteristic
16C22×C32×Dic3C2211yescharacteristic
17#88864—22no—
18C3×#3757—22no—
19C32×#217—22no—
20#88807—22no—
21#88806—22no—
22Dic3⋊(C3⋊Dic3)—22no—
23C2×C32×C22⋊C4—31no—
24C2×#3757—31no—
25C2×#3757—34no—
26C2×#3757—64no—
27C2×#3757D311yescharacteristic
28C2×C4⋊(C3⋊Dic3)—31no—
29C2×#3750—31no—
30C2×#3750—31no—
31C2×C3×#217—34no—
32C2×C3×#217D314yes—
33C2×#3724—34no—
34C2×Dic3⋊Dic3—34no—
35C2×Dic3⋊Dic3—34no—
36#63658—31no—
37#63579—34no—
38C23×C33—21no—
39C23×C33—21no—
40C23×C33—21no—
41C23×C33—21no—
42C23×C33—21no—
43C23×C33—21no—
44C23×C33Q812yes—
45C23×C33C2311yescharacteristic
46C2×C3⋊(C3⋊Dic3)—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)D412yes—
52C2×C3×C3⋊Dic3—21no—
53C2×C3×C3⋊Dic3—21no—
54C2×C3×C3⋊Dic3—21no—
55C2×C3×C3⋊Dic3—21no—
56C2×C3×C3⋊Dic3—21no—
57C2×C3×C3⋊Dic3D412yes—
58C2×C32×Dic3—21no—
59C2×C32×Dic3—21no—
60C2×C32×Dic3—21no—
61C2×C32×Dic3—21no—
62C2×C32×Dic3—21no—
63C2×C32×Dic3D412yes—
64#4949—94no—
65#4949—91no—
66C24×C32—24no—
67C24×C32C2×D314yes—
68C24×C32C2×D311yescharacteristic
69C22×C32×C4—31no—
70C22×C32×C4—31no—
71C22×C3⋊Dic3—31no—
72C22×C3⋊Dic3—34no—
73C22×C3⋊Dic3—31no—
74C22×C3⋊Dic3—34no—
75C22×C3⋊Dic3—64no—
76C22×C3⋊Dic3—64no—
77C22×C3⋊Dic3C2×D311yescharacteristic
78C22×C3⋊Dic3C2×D311yescharacteristic
79C22×C3×Dic3—34no—
80C22×C3×Dic3—34no—
81C22×C3×Dic3C2×D314yes—
82C22×C3×Dic3C2×D314yes—
83C32×C22⋊C4—62no—
84#3757—22no—
85#3757—62no—
86#3757—68no—
87#3757—616no—
88C4⋊(C3⋊Dic3)—62no—
89#3750—62no—
90#3750—62no—
91C3×#217—28no—
92C3×#217—68no—
93#3724—68no—
94Dic3⋊Dic3—68no—
95Dic3⋊Dic3—68no—
96C22×C33—22no—
97C22×C33—22no—
98C22×C33—22no—
99C22×C33—41no—
100C22×C33—41no—
101C22×C33—41no—
102C22×C33—41no—
103C22×C33C2×D411yescharacteristic
104C22×C33C2×D411yescharacteristic
105C22×C33C2×D411yescharacteristic
106C22×C33C2×Q811yescharacteristic
107C22×C33#4111yescharacteristic
108C22×C33#4111yescharacteristic
109C22×C33#4111yescharacteristic
110C3⋊(C3⋊Dic3)—22no—
111C3⋊(C3⋊Dic3)—41no—
112C3×C3⋊Dic3—22no—
113C3×C3⋊Dic3—41no—
114C32×Dic3—22no—
115C32×Dic3—41no—
116C2×C3×C22⋊C4—34no—
117C2×C3×C22⋊C4—91no—
118C2×#217—34no—
119C2×#217—94no—
120C2×#217—91no—
121C2×#217—184no—
122C2×#217C3⋊D311yescharacteristic
123C2×C4⋊Dic3—94no—
124C2×C4⋊Dic3—91no—
125C2×Dic3⋊C4—94no—
126C2×Dic3⋊C4—94no—
127C2×Dic3⋊C4—91no—
128C2×Dic3⋊C4—91no—
129C3⋊#95—94no—
130C3⋊#95—91no—
131C23×C32—28no—
132C23×C32—28no—
133C23×C32—21no—
134C23×C32—24no—
135C23×C32—21no—
136C23×C32—21no—
137C23×C32—24no—
138C23×C32—21no—
139C23×C32—21no—
140C23×C32—21no—
141C23×C32—28no—
142C23×C32—24no—
143C23×C32—24no—
144C23×C32—24no—
145C23×C32—24no—
146C23×C32—24no—
147C23×C32—44no—
148C23×C32—44no—
149C23×C32—44no—
150C23×C32—44no—
151C23×C32Dic618yes—
152C23×C32Dic612yes—
153C23×C32C22×D314yes—
154C23×C32C22×D311yescharacteristic
155C2×C32×C4—32no—
156C2×C32×C4—61no—
157C2×C32×C4—61no—
158C2×C32×C4—61no—
159C2×C32×C4—61no—
160C2×C32×C4—61no—
161C2×C3⋊Dic3—21no—
162C2×C3⋊Dic3—21no—
163C2×C3⋊Dic3—21no—
164C2×C3⋊Dic3—21no—
165C2×C3⋊Dic3—21no—
166C2×C3⋊Dic3—32no—
167C2×C3⋊Dic3—38no—
168C2×C3⋊Dic3—68no—
169C2×C3⋊Dic3—61no—
170C2×C3⋊Dic3—61no—
171C2×C3⋊Dic3—68no—
172C2×C3⋊Dic3—61no—
173C2×C3⋊Dic3—64no—
174C2×C3⋊Dic3—61no—
175C2×C3⋊Dic3—61no—
176C2×C3⋊Dic3—64no—
177C2×C3⋊Dic3—64no—
178C2×C3⋊Dic3—64no—
179C2×C3⋊Dic3—64no—
180C2×C3⋊Dic3—124no—
181C2×C3⋊Dic3—124no—
182C2×C3⋊Dic3—124no—
183C2×C3⋊Dic3—124no—
184C2×C3⋊Dic3#6712yes—
185C2×C3×Dic3—24no—
186C2×C3×Dic3—24no—
187C2×C3×Dic3—24no—
188C2×C3×Dic3—24no—
189C2×C3×Dic3—24no—
190C2×C3×Dic3—38no—
191C2×C3×Dic3—64no—
192C2×C3×Dic3—64no—
193C2×C3×Dic3—64no—
194C2×C3×Dic3—64no—
195C2×C3×Dic3—64no—
196C2×C3×Dic3#6718yes—
197#399—271noSylow
198C2×C33—42no—
199C2×C33C22⋊Q811yescharacteristic
200C2×C33C22⋊Q811yescharacteristic
201C2×C33C22⋊Q811yescharacteristic
202C2×C33#12311yescharacteristic
203C2×C33#12311yescharacteristic
204C2×C33#12311yescharacteristic
205C2×C33#12711yescharacteristic
206C24×C3—24no—
207C24×C3D3214yes—
208C24×C3C2×C3⋊D311yescharacteristic
209C22×C3×C4—34no—
210C22×C3×C4—34no—
211C22×C3×C4—91no—
212C22×C3×C4—91no—
213C22×Dic3—34no—
214C22×Dic3—34no—
215C22×Dic3—94no—
216C22×Dic3—91no—
217C22×Dic3—94no—
218C22×Dic3—91no—
219C22×Dic3—184no—
220C22×Dic3—184no—
221C22×Dic3C2×C3⋊D311yescharacteristic
222C22×Dic3C2×C3⋊D311yescharacteristic
223C3×C22⋊C4—68no—
224C3×C22⋊C4—182no—
225#217—22no—
226#217—68no—
227#217—188no—
228#217—182no—
229#217—1816no—
230C4⋊Dic3—182no—
231C4⋊Dic3—188no—
232Dic3⋊C4—182no—
233Dic3⋊C4—188no—
234Dic3⋊C4—188no—
235Dic3⋊C4—182no—
236C22×C32—216no—
237C22×C32—28no—
238C22×C32—22no—
239C22×C32—22no—
240C22×C32—22no—
241C22×C32—24no—
242C22×C32—24no—
243C22×C32—24no—
244C22×C32—24no—
245C22×C32—24no—
246C22×C32—24no—
247C22×C32—24no—
248C22×C32—28no—
249C22×C32—28no—
250C22×C32—48no—
251C22×C32—48no—
252C22×C32—48no—
253C22×C32—48no—
254C22×C32—48no—
255C22×C32—48no—
256C22×C32—44no—
257C22×C32—44no—
258C22×C32—41no—
259C22×C32—41no—
260C22×C32—44no—
261C22×C32—41no—
262C22×C32—44no—
263C22×C32—41no—
264C22×C32C2×Dic614yes—
265C22×C32C2×Dic611yescharacteristic
266C22×C32#23514yes—
267C22×C32#23511yescharacteristic
268C22×C32D3×D414yes—
269C22×C32D3×D414yes—
270C22×C32D3×D411yescharacteristic
271C22×C32D3×D411yescharacteristic
272C22×C32#23714yes—
273C22×C32#23714yes—
274C22×C32#23711yescharacteristic
275C22×C32#23711yescharacteristic
276C22×C32C2×#6714yes—
277C22×C32C2×#6711yescharacteristic
278C32×C4—62no—
279C32×C4—121no—
280C3⋊Dic3—22no—
281C3⋊Dic3—41no—
282C3⋊Dic3—62no—
283C3⋊Dic3—68no—
284C3⋊Dic3—128no—
285C3⋊Dic3—128no—
286C3⋊Dic3—124no—
287C3⋊Dic3—121no—
288C3×Dic3—28no—
289C3×Dic3—44no—
290C3×Dic3—68no—
291C3×Dic3—124no—
292C2×C4⋊C4—271no—
293C2×C4⋊C4—271no—
294C2×C4⋊C4—271no—
295C2×C22⋊C4—31no—
296C2×C22⋊C4—91no—
297C2×C22⋊C4—271no—
298#95—271no—
299C33#39911yescharacteristic, complemented, Sylow
300C23×C3—28no—
301C23×C3—28no—
302C23×C3—24no—
303C23×C3—24no—
304C23×C3—24no—
305C23×C3—24no—
306C23×C3—24no—
307C23×C3—24no—
308C23×C3—21no—
309C23×C3—21no—
310C23×C3—21no—
311C23×C3—21no—
312C23×C3—21no—
313C23×C3—21no—
314C23×C3—28no—
315C23×C3—24no—
316C23×C3—44no—
317C23×C3—44no—
318C23×C3—44no—
319C23×C3—44no—
320C23×C3#62718yes—
321C23×C3#63412yes—
322C23×C3C2×D3214yes—
323C23×C3C22×C3⋊D311yescharacteristic
324C2×C3×C4—38no—
325C2×C3×C4—64no—
326C2×C3×C4—64no—
327C2×C3×C4—64no—
328C2×C3×C4—64no—
329C2×C3×C4—64no—
330C2×C3×C4—92no—
331C2×C3×C4—181no—
332C2×C3×C4—181no—
333C2×C3×C4—181no—
334C2×C3×C4—181no—
335C2×C3×C4—181no—
336C2×Dic3—21no—
337C2×Dic3—21no—
338C2×Dic3—21no—
339C2×Dic3—21no—
340C2×Dic3—21no—
341C2×Dic3—38no—
342C2×Dic3—64no—
343C2×Dic3—64no—
344C2×Dic3—64no—
345C2×Dic3—64no—
346C2×Dic3—64no—
347C2×Dic3—98no—
348C2×Dic3—92no—
349C2×Dic3—188no—
350C2×Dic3—184no—
351C2×Dic3—184no—
352C2×Dic3—181no—
353C2×Dic3—181no—
354C2×Dic3—188no—
355C2×Dic3—184no—
356C2×Dic3—184no—
357C2×Dic3—184no—
358C2×Dic3—181no—
359C2×Dic3—181no—
360C2×Dic3—181no—
361C2×Dic3—364no—
362C2×Dic3—364no—
363C2×Dic3—364no—
364C2×Dic3—364no—
365C2×Dic3#63812yes—
366C2×C32—24no—
367C2×C32—24no—
368C2×C32—24no—
369C2×C32—24no—
370C2×C32—24no—
371C2×C32—24no—
372C2×C32—24no—
373C2×C32—416no—
374C2×C32—48no—
375C2×C32—42no—
376C2×C32#87814yes—
377C2×C32#87814yes—
378C2×C32#87811yescharacteristic
379C2×C32#87811yescharacteristic
380C2×C32#88314yes—
381C2×C32#88314yes—
382C2×C32#88311yescharacteristic
383C2×C32#88311yescharacteristic
384C2×C32#92414yes—
385C2×C32#92411yescharacteristic
386C2×C32#93514yes—
387C2×C32#93511yescharacteristic
388C2×C32#94014yes—
389C2×C32#94011yescharacteristic
390C24D3×C3⋊D311yescharacteristic
391C22×C4—31no—
392C22×C4—31no—
393C22×C4—91no—
394C22×C4—91no—
395C22×C4—271no—
396C22×C4—271no—
397C4⋊C4—542no—
398C4⋊C4—542no—
399C4⋊C4—542no—
400C22⋊C4—62no—
401C22⋊C4—182no—
402C22⋊C4—542no—
403C22×C3—216no—
404C22×C3—22no—
405C22×C3—22no—
406C22×C3—22no—
407C22×C3—28no—
408C22×C3—28no—
409C22×C3—28no—
410C22×C3—24no—
411C22×C3—24no—
412C22×C3—24no—
413C22×C3—24no—
414C22×C3—24no—
415C22×C3—24no—
416C22×C3—24no—
417C22×C3—48no—
418C22×C3—48no—
419C22×C3—48no—
420C22×C3—48no—
421C22×C3—48no—
422C22×C3—48no—
423C22×C3—41no—
424C22×C3—41no—
425C22×C3—41no—
426C22×C3—41no—
427C22×C3—44no—
428C22×C3—44no—
429C22×C3—44no—
430C22×C3—44no—
431C22×C3#380414yes—
432C22×C3#380414yes—
433C22×C3#380514yes—
434C22×C3C2×#62714yes—
435C22×C3D3×#6714yes—
436C22×C3D3×#6714yes—
437C22×C3#381114yes—
438C22×C3C2×#63411yescharacteristic
439C22×C3#382811yescharacteristic
440C22×C3D4×C3⋊D311yescharacteristic
441C22×C3D4×C3⋊D311yescharacteristic
442C22×C3#383011yescharacteristic
443C22×C3#383011yescharacteristic
444C22×C3C2×#63811yescharacteristic
445C3×C4—68no—
446C3×C4—124no—
447C3×C4—182no—
448C3×C4—361no—
449Dic3—22no—
450Dic3—41no—
451Dic3—68no—
452Dic3—124no—
453Dic3—182no—
454Dic3—188no—
455Dic3—368no—
456Dic3—368no—
457Dic3—361no—
458Dic3—364no—
459C32—24no—
460C32#494914yescomplemented
461C32#494911yescharacteristic, complemented
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23#629312yes—
469C23C2×D3×C3⋊D311yescharacteristic
470C2×C4—32no—
471C2×C4—61no—
472C2×C4—61no—
473C2×C4—61no—
474C2×C4—61no—
475C2×C4—61no—
476C2×C4—92no—
477C2×C4—181no—
478C2×C4—181no—
479C2×C4—181no—
480C2×C4—181no—
481C2×C4—181no—
482C2×C4—272no—
483C2×C4—541no—
484C2×C4—541no—
485C2×C4—541no—
486C2×C4—541no—
487C2×C4—541no—
488C2×C3—24no—
489C2×C3—24no—
490C2×C3—24no—
491C2×C3—24no—
492C2×C3—24no—
493C2×C3—24no—
494C2×C3—24no—
495C2×C3—416no—
496C2×C3—42no—
497C2×C3—48no—
498C2×C3#6396414yes—
499C2×C3#6396414yes—
500C2×C3#6396914yes—
501C2×C3#6396914yes—
502C2×C3#6397514yes—
503C2×C3#6397914yes—
504C2×C3#6398214yes—
505C2×C3#6408711yescharacteristic
506C2×C3#6408711yescharacteristic
507C2×C3#6409211yescharacteristic
508C2×C3#6409211yescharacteristic
509C2×C3#6413311yescharacteristic
510C2×C3#6414411yescharacteristic
511C2×C3#6414911yescharacteristic
512C22—22no—
513C22—22no—
514C22—22no—
515C22—41no—
516C22—41no—
517C22—41no—
518C22—41no—
519C22#8902811yescharacteristic
520C22#8902911yescharacteristic
521C22#8903111yescharacteristic
522C22C2×#629311yescharacteristic
523C22D3×#63811yescharacteristic
524C22C3⋊D3×#6711yescharacteristic
525C22#8903911yescharacteristic
526C4—62no—
527C4—121no—
528C4—182no—
529C4—361no—
530C4—542no—
531C4—1081no—
532C3—24no—
533C3#1059171514yescomplemented
534C3#1059264811yescharacteristic, complemented
535C2—42no—
536C2#1172361411yescharacteristic
537C2#1172362111yescharacteristic
538C2#1172362811yescharacteristic
539C2#1172363311yescharacteristic
540C2#1172364011yescharacteristic
541C2#1172364111yescharacteristic
542C2#1172364211yescharacteristic
subgroupG/H
center, FrattiniC23C2×D3×C3⋊D3
commutator, socle, cosocleC23×C33C23
Fitting, prime radicalC24×C33C22
hypercenterC24D3×C3⋊D3
hypocenterC33#399
LCSG ▷ C23×C33 ▷ C33
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
4. C2×#88807C2semidirect
5. C2×#88806C2semidirect
6. C2×Dic3⋊(C3⋊Dic3)C2semidirect
7. #11722061C2semidirect
299. C33#399semidirect
460. C32#4949semidirect
461. C32#4949semidirect
533. C3#10591715semidirect
534. C3#10592648semidirect

Maximal subgroups

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

subgroupindex
C2×#888642
C2×C3×#37572
C2×C32×#2172
C2×#888072
C2×#888062
C2×Dic3⋊(C3⋊Dic3)2
#117220612
#105926483
#105917153