Finite Group Databasebeta

← #13160531

Subgroups — #13160531

Order 1728.

1668 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88859C211yesmaximal, characteristic
2C2×C3×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
3C2×#88804C211yesmaximal, characteristic
4C2×#88804C211yesmaximal, characteristic
5C2×Dic3×C3⋊Dic3C211yesmaximal, characteristic, complemented
6C32×C3⋊#95C211yesmaximal, characteristic, complemented
7#11722061C211yesmaximal, characteristic, complemented
8#10592369—31nomaximal
9#10591266—34nomaximal
10C23×C3⋊(C3⋊D3)C2211yescharacteristic
11C22×C33×C4C2211yescharacteristic
12C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
13C22×C3×C3⋊Dic3C2211yescharacteristic
14C22×C3×C3⋊Dic3C2211yescharacteristic
15C22×C32×Dic3C2211yescharacteristic
16C22×C32×Dic3C2211yescharacteristic
17#88859C414yescomplemented
18C3×C4⋊(C3⋊Dic3)—22no—
19#88804—22no—
20#88804—22no—
21Dic3×C3⋊Dic3—22no—
22C2×#3752—31no—
23C2×#3752—31no—
24C2×#3752—31no—
25C2×#3752—34no—
26C2×#3752—64no—
27C2×C4⋊(C3⋊Dic3)D311yescharacteristic, complemented
28C2×C4×C3⋊Dic3—31no—
29C2×C3×C4⋊Dic3—34no—
30C2×#3722—34no—
31C2×#3722—34no—
32C2×Dic32—34no—
33C32×#95—31no—
34#63658—31no—
35C3×C3⋊#95D314yescomplemented
36#63579—34no—
37C23×C33C2311yescharacteristic
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⋊D3)—21no—
44C22×C3⋊(C3⋊D3)C2×C412yes—
45C2×C33×C4—21no—
46C2×C33×C4—21no—
47C2×C33×C4C2×C412yes—
48C2×C3⋊(C3⋊Dic3)—21no—
49C2×C3⋊(C3⋊Dic3)—21no—
50C2×C3⋊(C3⋊Dic3)C2×C412yes—
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×C3×C3⋊Dic3D412yes—
57C2×C32×Dic3—21no—
58C2×C32×Dic3—21no—
59C2×C32×Dic3—21no—
60C2×C32×Dic3—21no—
61C2×C32×Dic3—21no—
62C2×C32×Dic3D412yes—
63#4990—91no—
64#4670—94no—
65C23×C3⋊D3—31no—
66C23×C3⋊D3—34no—
67C23×C3⋊D3—64no—
68C22×C32×C4—24no—
69C22×C32×C4—31no—
70C22×C32×C4—31no—
71C22×C32×C4C2×D314yes—
72C22×C32×C4C2×D311yescharacteristic
73C22×C3⋊Dic3—31no—
74C22×C3⋊Dic3—34no—
75C22×C3⋊Dic3—64no—
76C22×C3⋊Dic3C2×D311yescharacteristic
77C22×C3⋊Dic3C2×D311yescharacteristic
78C22×C3×Dic3—34no—
79C22×C3×Dic3—34no—
80C22×C3×Dic3C2×D314yes—
81C22×C3×Dic3C2×D314yes—
82#3752—34no—
83#3752—316no—
84#3752—62no—
85#3752—616no—
86#3752—62no—
87C4⋊(C3⋊Dic3)—22no—
88C4×C3⋊Dic3—62no—
89C3×C4⋊Dic3—68no—
90#3722—68no—
91#3722—68no—
92Dic32—68no—
93C22×C33C22×C411yescharacteristic
94C22×C33C2×D411yescharacteristic
95C22×C33C2×D411yescharacteristic
96C22×C33#4111yescharacteristic
97C22×C33#4111yescharacteristic
98C22×C33#4111yescharacteristic
99C22×C33#4111yescharacteristic
100C2×C3⋊(C3⋊D3)—22no—
101C2×C3⋊(C3⋊D3)—22no—
102C2×C3⋊(C3⋊D3)—22no—
103C2×C3⋊(C3⋊D3)—41no—
104C2×C3⋊(C3⋊D3)—41no—
105C2×C3⋊(C3⋊D3)—41no—
106C2×C3⋊(C3⋊D3)—41no—
107C33×C4—22no—
108C3⋊(C3⋊Dic3)—22no—
109C3×C3⋊Dic3—22no—
110C3×C3⋊Dic3—41no—
111C32×Dic3—22no—
112C32×Dic3—41no—
113C2×C3×C4⋊C4—91no—
114C2×(C2×D3)⋊C4—94no—
115C2×(C2×D3)⋊C4—91no—
116C2×(C2×D3)⋊C4—94no—
117C2×(C2×D3)⋊C4—94no—
118C2×(C2×D3)⋊C4—91no—
119C2×(C2×D3)⋊C4—91no—
120C2×(C2×D3)⋊C4—184no—
121C2×C4⋊Dic3—34no—
122C2×C4×Dic3—94no—
123C2×C4×Dic3—91no—
124C3×#95—34no—
125C3⋊#95—94no—
126C3⋊#95—91no—
127C3⋊#95C3⋊D311yescharacteristic, complemented
128C23×C32—24no—
129C23×C32C22×D314yes—
130C23×C32C22×D311yescharacteristic
131C22×C3⋊D3—32no—
132C22×C3⋊D3—38no—
133C22×C3⋊D3—68no—
134C22×C3⋊D3—68no—
135C22×C3⋊D3—68no—
136C22×C3⋊D3—61no—
137C22×C3⋊D3—61no—
138C22×C3⋊D3—64no—
139C22×C3⋊D3—61no—
140C22×C3⋊D3—64no—
141C22×C3⋊D3—61no—
142C22×C3⋊D3—61no—
143C22×C3⋊D3—61no—
144C22×C3⋊D3—64no—
145C22×C3⋊D3—64no—
146C22×C3⋊D3—64no—
147C22×C3⋊D3—64no—
148C22×C3⋊D3—124no—
149C22×C3⋊D3—124no—
150C22×C3⋊D3—124no—
151C22×C3⋊D3—124no—
152C2×C32×C4—21no—
153C2×C32×C4—21no—
154C2×C32×C4—28no—
155C2×C32×C4—24no—
156C2×C32×C4—24no—
157C2×C32×C4—32no—
158C2×C32×C4—44no—
159C2×C32×C4—44no—
160C2×C32×C4—61no—
161C2×C32×C4—61no—
162C2×C32×C4—61no—
163C2×C32×C4—61no—
164C2×C32×C4—61no—
165C2×C32×C4C4×D318yes—
166C2×C32×C4C4×D312yes—
167C2×C3⋊Dic3—21no—
168C2×C3⋊Dic3—21no—
169C2×C3⋊Dic3—21no—
170C2×C3⋊Dic3—21no—
171C2×C3⋊Dic3—21no—
172C2×C3⋊Dic3—32no—
173C2×C3⋊Dic3—38no—
174C2×C3⋊Dic3—68no—
175C2×C3⋊Dic3—61no—
176C2×C3⋊Dic3—61no—
177C2×C3⋊Dic3—64no—
178C2×C3⋊Dic3—64no—
179C2×C3⋊Dic3—124no—
180C2×C3⋊Dic3—124no—
181C2×C3⋊Dic3#6712yes—
182C2×C3×Dic3—24no—
183C2×C3×Dic3—24no—
184C2×C3×Dic3—24no—
185C2×C3×Dic3—24no—
186C2×C3×Dic3—24no—
187C2×C3×Dic3—38no—
188C2×C3×Dic3—64no—
189C2×C3×Dic3—64no—
190C2×C3×Dic3—64no—
191C2×C3×Dic3—64no—
192C2×C3×Dic3—64no—
193C2×C3×Dic3D1218yes—
194#388—271noSylow
195C2×C33#11711yescharacteristic
196C2×C33C4×D411yescharacteristic
197C2×C33C4×D411yescharacteristic
198C2×C33#12111yescharacteristic
199C2×C33#12311yescharacteristic
200C2×C33#12411yescharacteristic
201C2×C33#12611yescharacteristic
202C3⋊(C3⋊D3)—42no—
203C23×D3—94no—
204C23×D3—91no—
205C23×D3—184no—
206C22×C3×C4—24no—
207C22×C3×C4—34no—
208C22×C3×C4—34no—
209C22×C3×C4—91no—
210C22×C3×C4—91no—
211C22×C3×C4D3214yes—
212C22×C3×C4C2×C3⋊D311yescharacteristic
213C22×Dic3—34no—
214C22×Dic3—34no—
215C22×Dic3—94no—
216C22×Dic3—91no—
217C22×Dic3—184no—
218C22×Dic3C2×C3⋊D311yescharacteristic
219C22×Dic3C2×C3⋊D311yescharacteristic
220C3×C4⋊C4—182no—
221(C2×D3)⋊C4—916no—
222(C2×D3)⋊C4—94no—
223(C2×D3)⋊C4—1816no—
224(C2×D3)⋊C4—182no—
225(C2×D3)⋊C4—188no—
226(C2×D3)⋊C4—188no—
227(C2×D3)⋊C4—182no—
228C4⋊Dic3—68no—
229C4×Dic3—188no—
230C4×Dic3—182no—
231C22×C32—24no—
232C22×C32—24no—
233C22×C32—24no—
234C22×C32—24no—
235C22×C32—24no—
236C22×C32—24no—
237C22×C32—24no—
238C22×C32C2×C4×D314yes—
239C22×C32C2×C4×D311yescharacteristic
240C22×C32C2×D1214yes—
241C22×C32#23514yes—
242C22×C32#23511yescharacteristic
243C22×C32D3×D414yes—
244C22×C32D3×D411yescharacteristic
245C22×C32#23714yes—
246C22×C32#23714yes—
247C22×C32#23711yescharacteristic
248C22×C32#23914yes—
249C22×C32#23911yescharacteristic
250C22×C32#23911yescharacteristic
251C22×C32C2×#6711yescharacteristic
252C2×C3⋊D3—616no—
253C2×C3⋊D3—68no—
254C2×C3⋊D3—62no—
255C2×C3⋊D3—62no—
256C2×C3⋊D3—62no—
257C2×C3⋊D3—68no—
258C2×C3⋊D3—68no—
259C2×C3⋊D3—128no—
260C2×C3⋊D3—128no—
261C2×C3⋊D3—128no—
262C2×C3⋊D3—128no—
263C2×C3⋊D3—128no—
264C2×C3⋊D3—128no—
265C2×C3⋊D3—124no—
266C2×C3⋊D3—124no—
267C2×C3⋊D3—121no—
268C2×C3⋊D3—124no—
269C2×C3⋊D3—124no—
270C2×C3⋊D3—121no—
271C2×C3⋊D3—121no—
272C2×C3⋊D3—121no—
273C32×C4—22no—
274C32×C4—28no—
275C32×C4—48no—
276C32×C4—62no—
277C32×C4—121no—
278C3⋊Dic3—22no—
279C3⋊Dic3—41no—
280C3⋊Dic3—62no—
281C3⋊Dic3—68no—
282C3⋊Dic3—128no—
283C3×Dic3—28no—
284C3×Dic3—44no—
285C3×Dic3—68no—
286C3×Dic3—124no—
287C2×C4⋊C4—91no—
288C2×C22⋊C4—271no—
289C2×C22⋊C4—271no—
290C2×C22⋊C4—271no—
291C2×C42—271no—
292#95—31no—
293#95—271no—
294C33#38811yescharacteristic, complemented, Sylow
295C23×C3—24no—
296C23×C3C2×D3214yes—
297C23×C3C22×C3⋊D311yescharacteristic
298C22×D3—98no—
299C22×D3—92no—
300C22×D3—188no—
301C22×D3—188no—
302C22×D3—188no—
303C22×D3—184no—
304C22×D3—184no—
305C22×D3—184no—
306C22×D3—184no—
307C22×D3—184no—
308C22×D3—184no—
309C22×D3—181no—
310C22×D3—181no—
311C22×D3—181no—
312C22×D3—181no—
313C22×D3—181no—
314C22×D3—181no—
315C22×D3—364no—
316C22×D3—364no—
317C22×D3—364no—
318C22×D3—364no—
319C2×C3×C4—24no—
320C2×C3×C4—24no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—28no—
324C2×C3×C4—38no—
325C2×C3×C4—44no—
326C2×C3×C4—44no—
327C2×C3×C4—64no—
328C2×C3×C4—64no—
329C2×C3×C4—64no—
330C2×C3×C4—64no—
331C2×C3×C4—64no—
332C2×C3×C4—92no—
333C2×C3×C4—181no—
334C2×C3×C4—181no—
335C2×C3×C4—181no—
336C2×C3×C4—181no—
337C2×C3×C4—181no—
338C2×C3×C4#62418yes—
339C2×C3×C4C4×C3⋊D312yes—
340C2×Dic3—21no—
341C2×Dic3—21no—
342C2×Dic3—21no—
343C2×Dic3—21no—
344C2×Dic3—21no—
345C2×Dic3—38no—
346C2×Dic3—64no—
347C2×Dic3—64no—
348C2×Dic3—64no—
349C2×Dic3—64no—
350C2×Dic3—64no—
351C2×Dic3—98no—
352C2×Dic3—92no—
353C2×Dic3—188no—
354C2×Dic3—184no—
355C2×Dic3—184no—
356C2×Dic3—181no—
357C2×Dic3—181no—
358C2×Dic3—364no—
359C2×Dic3—364no—
360C2×Dic3C4⋊(C3⋊D3)12yes—
361C2×C32—24no—
362C2×C32—24no—
363C2×C32—24no—
364C2×C32—24no—
365C2×C32—24no—
366C2×C32—24no—
367C2×C32—24no—
368C2×C32C4×D1214yes—
369C2×C32#67⋊C414yes—
370C2×C32#88514yes—
371C2×C32#88614yes—
372C2×C32#89214yes—
373C2×C32#89211yescharacteristic
374C2×C32#89311yescharacteristic
375C2×C32#89411yescharacteristic
376C2×C32#89514yes—
377C2×C32#89814yes—
378C2×C32#89811yescharacteristic
379C2×C32C4×#6711yescharacteristic
380C2×C32#93911yescharacteristic
381C2×C32#94711yescharacteristic
382C3⋊D3—1216no—
383C3⋊D3—128no—
384C3⋊D3—122no—
385C24—271no—
386C22×C4—31no—
387C22×C4—31no—
388C22×C4—91no—
389C22×C4—91no—
390C22×C4—271no—
391C22×C4D3×C3⋊D311yescharacteristic
392C4⋊C4—182no—
393C22⋊C4—274no—
394C22⋊C4—542no—
395C22⋊C4—542no—
396C42—542no—
397C22×C3—24no—
398C22×C3—24no—
399C22×C3—24no—
400C22×C3—24no—
401C22×C3—24no—
402C22×C3—24no—
403C22×C3—24no—
404C22×C3#379614yes—
405C22×C3#379614yes—
406C22×C3#379914yes—
407C22×C3D3×D1214yes—
408C22×C3#380414yes—
409C22×C3C2×#62414yes—
410C22×C3D3×#6714yes—
411C22×C3C2×C4×C3⋊D311yescharacteristic
412C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
413C22×C3#382811yescharacteristic
414C22×C3D4×C3⋊D311yescharacteristic
415C22×C3#383011yescharacteristic
416C22×C3#383011yescharacteristic
417C22×C3#383211yescharacteristic
418C2×D3—1816no—
419C2×D3—182no—
420C2×D3—182no—
421C2×D3—182no—
422C2×D3—188no—
423C2×D3—188no—
424C2×D3—188no—
425C2×D3—368no—
426C2×D3—368no—
427C2×D3—368no—
428C2×D3—368no—
429C2×D3—368no—
430C2×D3—368no—
431C2×D3—361no—
432C2×D3—361no—
433C2×D3—364no—
434C2×D3—361no—
435C2×D3—361no—
436C2×D3—364no—
437C2×D3—364no—
438C2×D3—364no—
439C3×C4—22no—
440C3×C4—28no—
441C3×C4—48no—
442C3×C4—68no—
443C3×C4—124no—
444C3×C4—182no—
445C3×C4—361no—
446Dic3—22no—
447Dic3—41no—
448Dic3—68no—
449Dic3—124no—
450Dic3—182no—
451Dic3—188no—
452Dic3—368no—
453C32—24no—
454C32#467014yescomplemented
455C32#499011yescharacteristic, complemented
456C23—272no—
457C23—541no—
458C23—541no—
459C23—541no—
460C23—541no—
461C23—541no—
462C23—541no—
463C23C2×D3×C3⋊D311yescharacteristic
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—32no—
467C2×C4—61no—
468C2×C4—61no—
469C2×C4—61no—
470C2×C4—61no—
471C2×C4—61no—
472C2×C4—92no—
473C2×C4—181no—
474C2×C4—181no—
475C2×C4—181no—
476C2×C4—181no—
477C2×C4—181no—
478C2×C4—272no—
479C2×C4—541no—
480C2×C4—541no—
481C2×C4#628912yes—
482C2×C3—24no—
483C2×C3—24no—
484C2×C3—24no—
485C2×C3—24no—
486C2×C3—24no—
487C2×C3—24no—
488C2×C3—24no—
489C2×C3#6384814yes—
490C2×C3#6384914yes—
491C2×C3#6388614yes—
492C2×C3#6389014yes—
493C2×C3#6389414yes—
494C2×C3#6390714yes—
495C2×C3#6395214yes—
496C2×C3C4×C4⋊(C3⋊D3)11yescharacteristic
497C2×C3#6409011yescharacteristic
498C2×C3#6409411yescharacteristic
499C2×C3#6409511yescharacteristic
500C2×C3#6410111yescharacteristic
501C2×C3#6410411yescharacteristic
502C2×C3#6410711yescharacteristic
503D3—3616no—
504D3—368no—
505D3—362no—
506C22—542no—
507C22—542no—
508C22—542no—
509C22—1081no—
510C22—1081no—
511C22—1081no—
512C22—1081no—
513C22#8901511yescharacteristic
514C22#8901711yescharacteristic
515C22#8902211yescharacteristic
516C22D3×C4⋊(C3⋊D3)11yescharacteristic
517C22#8902811yescharacteristic
518C22C2×#628911yescharacteristic
519C22C3⋊D3×#6711yescharacteristic
520C4—22no—
521C4—62no—
522C4—121no—
523C4—182no—
524C4—361no—
525C4—542no—
526C3—24no—
527C3#1059126614yescomplemented
528C3#1059236911yescharacteristic, complemented
529C2—1082no—
530C2#1172340311yescharacteristic
531C2#1172341311yescharacteristic
532C2#1172347911yescharacteristic
533C2#1172348211yescharacteristic
534C2#1172348411yescharacteristic
535C2#1172352111yescharacteristic
536C2#1172362211yescharacteristic
subgroupG/H
center, FrattiniC23C2×D3×C3⋊D3
commutatorC22×C33C22×C4
FittingC22×C33×C4C22
socle, cosocleC23×C33C23
prime radicalC23×C3⋊(C3⋊D3)C22
hypercenterC22×C4D3×C3⋊D3
hypocenterC33#388
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×C4⋊(C3⋊Dic3)C2semidirect
5. C2×Dic3×C3⋊Dic3C2semidirect
6. C32×C3⋊#95C2semidirect
7. #11722061C2semidirect
17. #88859C4semidirect
27. C2×C4⋊(C3⋊Dic3)D3semidirect
35. C3×C3⋊#95D3semidirect
127. C3⋊#95C3⋊D3semidirect
294. C33#388semidirect
454. C32#4670semidirect
455. C32#4990semidirect
527. C3#10591266semidirect
528. C3#10592369semidirect

Maximal subgroups

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

subgroupindex
C2×#888592
C2×C3×C4⋊(C3⋊Dic3)2
C2×#888042
C2×Dic3×C3⋊Dic32
C32×C3⋊#952
#117220612
#105923693
#105912663