Finite Group Databasebeta

← #13160528

Subgroups — #13160528

Order 1728.

1548 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×#88859C211yesmaximal, characteristic
2C2×C3×#3750C211yesmaximal, characteristic, complemented
3C2×#88804C211yesmaximal, characteristic
4C2×#88804C211yesmaximal, characteristic
5C32×C3⋊#95C211yesmaximal, characteristic, complemented
6#11722061C211yesmaximal, characteristic, complemented
7#11722061C211yesmaximal, characteristic, complemented
8#10592374—31nomaximal
9#10591204—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#88859—22no—
18C3×#3750—22no—
19#88804—22no—
20#88804—22no—
21C2×#3752—31no—
22C2×#3752—31no—
23C2×#3752—31no—
24C2×#3752—34no—
25C2×#3752—64no—
26C2×#3750D311yescharacteristic, complemented
27C2×C3×Dic3⋊C4—34no—
28C2×#3722—34no—
29C2×#3722—34no—
30C32×#95—31no—
31#63658—31no—
32#63658—31no—
33C3×C3⋊#95D314yescomplemented
34#63579—34no—
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⋊D3)D412yes—
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⋊Dic3D412yes—
56C2×C32×Dic3—21no—
57C2×C32×Dic3—21no—
58C2×C32×Dic3—21no—
59C2×C32×Dic3—21no—
60C2×C32×Dic3—21no—
61C2×C32×Dic3—21no—
62#4992—91no—
63#4675—94no—
64C23×C3⋊D3—31no—
65C23×C3⋊D3—34no—
66C23×C3⋊D3—64no—
67C22×C32×C4—24no—
68C22×C32×C4—31no—
69C22×C32×C4—31no—
70C22×C32×C4C2×D314yes—
71C22×C32×C4C2×D311yescharacteristic
72C22×C3⋊Dic3—31no—
73C22×C3⋊Dic3—34no—
74C22×C3⋊Dic3—64no—
75C22×C3⋊Dic3C2×D311yescharacteristic
76C22×C3⋊Dic3C2×D311yescharacteristic
77C22×C3×Dic3—34no—
78C22×C3×Dic3—34no—
79C22×C3×Dic3C2×D314yes—
80C22×C3×Dic3C2×D314yes—
81#3752—62no—
82#3752—62no—
83#3752—68no—
84#3752—616no—
85#3752—62no—
86#3750—22no—
87C3×Dic3⋊C4—68no—
88#3722—68no—
89#3722—68no—
90C22×C33C2×D411yescharacteristic
91C22×C33C2×D411yescharacteristic
92C22×C33#4111yescharacteristic
93C22×C33#4111yescharacteristic
94C22×C33#4111yescharacteristic
95C22×C33#4111yescharacteristic
96C22×C33#4111yescharacteristic
97C2×C3⋊(C3⋊D3)—22no—
98C2×C3⋊(C3⋊D3)—22no—
99C2×C3⋊(C3⋊D3)—22no—
100C2×C3⋊(C3⋊D3)—41no—
101C2×C3⋊(C3⋊D3)—41no—
102C2×C3⋊(C3⋊D3)—41no—
103C2×C3⋊(C3⋊D3)—41no—
104C33×C4—41no—
105C3⋊(C3⋊Dic3)—41no—
106C3×C3⋊Dic3—22no—
107C3×C3⋊Dic3—41no—
108C32×Dic3—41no—
109C32×Dic3—41no—
110C2×C3×C4⋊C4—91no—
111C2×(C2×D3)⋊C4—94no—
112C2×(C2×D3)⋊C4—94no—
113C2×(C2×D3)⋊C4—94no—
114C2×(C2×D3)⋊C4—91no—
115C2×(C2×D3)⋊C4—91no—
116C2×(C2×D3)⋊C4—91no—
117C2×(C2×D3)⋊C4—184no—
118C2×Dic3⋊C4—34no—
119C3×#95—34no—
120C3⋊#95—94no—
121C3⋊#95—94no—
122C3⋊#95—91no—
123C3⋊#95—91no—
124C3⋊#95C3⋊D311yescharacteristic, complemented
125C23×C32—24no—
126C23×C32C22×D314yes—
127C23×C32C22×D311yescharacteristic
128C22×C3⋊D3—32no—
129C22×C3⋊D3—38no—
130C22×C3⋊D3—68no—
131C22×C3⋊D3—68no—
132C22×C3⋊D3—68no—
133C22×C3⋊D3—64no—
134C22×C3⋊D3—61no—
135C22×C3⋊D3—61no—
136C22×C3⋊D3—64no—
137C22×C3⋊D3—61no—
138C22×C3⋊D3—61no—
139C22×C3⋊D3—61no—
140C22×C3⋊D3—61no—
141C22×C3⋊D3—64no—
142C22×C3⋊D3—64no—
143C22×C3⋊D3—64no—
144C22×C3⋊D3—64no—
145C22×C3⋊D3—124no—
146C22×C3⋊D3—124no—
147C22×C3⋊D3—124no—
148C22×C3⋊D3—124no—
149C2×C32×C4—28no—
150C2×C32×C4—24no—
151C2×C32×C4—21no—
152C2×C32×C4—21no—
153C2×C32×C4—21no—
154C2×C32×C4—24no—
155C2×C32×C4—24no—
156C2×C32×C4—44no—
157C2×C32×C4—44no—
158C2×C32×C4—61no—
159C2×C32×C4—61no—
160C2×C32×C4—61no—
161C2×C32×C4—61no—
162C2×C32×C4—61no—
163C2×C32×C4—61no—
164C2×C3⋊Dic3—21no—
165C2×C3⋊Dic3—21no—
166C2×C3⋊Dic3—21no—
167C2×C3⋊Dic3—21no—
168C2×C3⋊Dic3—21no—
169C2×C3⋊Dic3—68no—
170C2×C3⋊Dic3—61no—
171C2×C3⋊Dic3—64no—
172C2×C3⋊Dic3—61no—
173C2×C3⋊Dic3—61no—
174C2×C3⋊Dic3—64no—
175C2×C3⋊Dic3—64no—
176C2×C3⋊Dic3—124no—
177C2×C3⋊Dic3—124no—
178C2×C3⋊Dic3#6712yes—
179C2×C3×Dic3—24no—
180C2×C3×Dic3—24no—
181C2×C3×Dic3—24no—
182C2×C3×Dic3—24no—
183C2×C3×Dic3—24no—
184C2×C3×Dic3—24no—
185C2×C3×Dic3—38no—
186C2×C3×Dic3—64no—
187C2×C3×Dic3—64no—
188C2×C3×Dic3—64no—
189C2×C3×Dic3—64no—
190C2×C3×Dic3—64no—
191#397—271noSylow
192C2×C33#12111yescharacteristic
193C2×C33#12311yescharacteristic
194C2×C33#12311yescharacteristic
195C2×C33#12311yescharacteristic
196C2×C33#12411yescharacteristic
197C2×C33#12611yescharacteristic
198C2×C33#12611yescharacteristic
199C3⋊(C3⋊D3)—42no—
200C23×D3—94no—
201C23×D3—91no—
202C23×D3—184no—
203C22×C3×C4—24no—
204C22×C3×C4—34no—
205C22×C3×C4—34no—
206C22×C3×C4—91no—
207C22×C3×C4—91no—
208C22×C3×C4D3214yes—
209C22×C3×C4C2×C3⋊D311yescharacteristic
210C22×Dic3—34no—
211C22×Dic3—34no—
212C22×Dic3—94no—
213C22×Dic3—91no—
214C22×Dic3—184no—
215C22×Dic3C2×C3⋊D311yescharacteristic
216C22×Dic3C2×C3⋊D311yescharacteristic
217C3×C4⋊C4—182no—
218(C2×D3)⋊C4—182no—
219(C2×D3)⋊C4—182no—
220(C2×D3)⋊C4—188no—
221(C2×D3)⋊C4—188no—
222(C2×D3)⋊C4—188no—
223(C2×D3)⋊C4—182no—
224(C2×D3)⋊C4—1816no—
225Dic3⋊C4—68no—
226C22×C32—24no—
227C22×C32—24no—
228C22×C32—24no—
229C22×C32—24no—
230C22×C32—24no—
231C22×C32—24no—
232C22×C32—24no—
233C22×C32#23514yes—
234C22×C32#23514yes—
235C22×C32#23514yes—
236C22×C32#23511yescharacteristic
237C22×C32#23511yescharacteristic
238C22×C32D3×D414yes—
239C22×C32D3×D414yes—
240C22×C32D3×D411yescharacteristic
241C22×C32#23714yes—
242C22×C32#23711yescharacteristic
243C22×C32#23914yes—
244C22×C32#23911yescharacteristic
245C22×C32#23911yescharacteristic
246C22×C32C2×#6711yescharacteristic
247C2×C3⋊D3—616no—
248C2×C3⋊D3—68no—
249C2×C3⋊D3—62no—
250C2×C3⋊D3—62no—
251C2×C3⋊D3—62no—
252C2×C3⋊D3—68no—
253C2×C3⋊D3—68no—
254C2×C3⋊D3—128no—
255C2×C3⋊D3—128no—
256C2×C3⋊D3—128no—
257C2×C3⋊D3—128no—
258C2×C3⋊D3—128no—
259C2×C3⋊D3—128no—
260C2×C3⋊D3—124no—
261C2×C3⋊D3—124no—
262C2×C3⋊D3—124no—
263C2×C3⋊D3—124no—
264C2×C3⋊D3—121no—
265C2×C3⋊D3—121no—
266C2×C3⋊D3—121no—
267C2×C3⋊D3—121no—
268C32×C4—48no—
269C32×C4—44no—
270C32×C4—41no—
271C32×C4—121no—
272C32×C4—121no—
273C3⋊Dic3—22no—
274C3⋊Dic3—41no—
275C3⋊Dic3—128no—
276C3⋊Dic3—124no—
277C3⋊Dic3—121no—
278C3×Dic3—44no—
279C3×Dic3—44no—
280C3×Dic3—68no—
281C3×Dic3—124no—
282C2×C4⋊C4—91no—
283C2×C22⋊C4—271no—
284C2×C22⋊C4—271no—
285C2×C22⋊C4—271no—
286#95—31no—
287#95—271no—
288#95—271no—
289C33#39711yescharacteristic, complemented, Sylow
290C23×C3—24no—
291C23×C3C2×D3214yes—
292C23×C3C22×C3⋊D311yescharacteristic
293C22×D3—98no—
294C22×D3—92no—
295C22×D3—188no—
296C22×D3—188no—
297C22×D3—188no—
298C22×D3—184no—
299C22×D3—184no—
300C22×D3—184no—
301C22×D3—184no—
302C22×D3—184no—
303C22×D3—184no—
304C22×D3—181no—
305C22×D3—181no—
306C22×D3—181no—
307C22×D3—181no—
308C22×D3—181no—
309C22×D3—181no—
310C22×D3—364no—
311C22×D3—364no—
312C22×D3—364no—
313C22×D3—364no—
314C2×C3×C4—28no—
315C2×C3×C4—24no—
316C2×C3×C4—24no—
317C2×C3×C4—24no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—44no—
322C2×C3×C4—44no—
323C2×C3×C4—64no—
324C2×C3×C4—64no—
325C2×C3×C4—64no—
326C2×C3×C4—64no—
327C2×C3×C4—64no—
328C2×C3×C4—64no—
329C2×C3×C4—92no—
330C2×C3×C4—181no—
331C2×C3×C4—181no—
332C2×C3×C4—181no—
333C2×C3×C4—181no—
334C2×C3×C4—181no—
335C2×Dic3—21no—
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—188no—
348C2×Dic3—184no—
349C2×Dic3—184no—
350C2×Dic3—184no—
351C2×Dic3—181no—
352C2×Dic3—181no—
353C2×Dic3—181no—
354C2×Dic3—364no—
355C2×Dic3—364no—
356C2×C32—24no—
357C2×C32—24no—
358C2×C32—24no—
359C2×C32—24no—
360C2×C32—24no—
361C2×C32—24no—
362C2×C32—24no—
363C2×C32#87614yes—
364C2×C32#88314yes—
365C2×C32#88414yes—
366C2×C32#88514yes—
367C2×C32#89414yes—
368C2×C32#89414yes—
369C2×C32#89411yescharacteristic
370C2×C32#89411yescharacteristic
371C2×C32#89814yes—
372C2×C32#89811yescharacteristic
373C2×C32#89811yescharacteristic
374C2×C32#92911yescharacteristic
375C2×C32#93811yescharacteristic
376C2×C32#94711yescharacteristic
377C3⋊D3—1216no—
378C3⋊D3—128no—
379C3⋊D3—122no—
380C24—271no—
381C22×C4—31no—
382C22×C4—31no—
383C22×C4—91no—
384C22×C4—91no—
385C22×C4—271no—
386C22×C4D3×C3⋊D311yescharacteristic
387C4⋊C4—182no—
388C22⋊C4—542no—
389C22⋊C4—542no—
390C22⋊C4—542no—
391C22×C3—24no—
392C22×C3—24no—
393C22×C3—24no—
394C22×C3—24no—
395C22×C3—24no—
396C22×C3—24no—
397C22×C3—24no—
398C22×C3#379614yes—
399C22×C3#379814yes—
400C22×C3#379914yes—
401C22×C3#379914yes—
402C22×C3#380414yes—
403C22×C3D3×#6714yes—
404C22×C3#381114yes—
405C22×C3#382811yescharacteristic
406C22×C3#382811yescharacteristic
407C22×C3#382811yescharacteristic
408C22×C3D4×C3⋊D311yescharacteristic
409C22×C3D4×C3⋊D311yescharacteristic
410C22×C3#383011yescharacteristic
411C22×C3#383211yescharacteristic
412C2×D3—1816no—
413C2×D3—182no—
414C2×D3—182no—
415C2×D3—182no—
416C2×D3—188no—
417C2×D3—188no—
418C2×D3—188no—
419C2×D3—368no—
420C2×D3—368no—
421C2×D3—368no—
422C2×D3—368no—
423C2×D3—368no—
424C2×D3—368no—
425C2×D3—361no—
426C2×D3—361no—
427C2×D3—361no—
428C2×D3—364no—
429C2×D3—364no—
430C2×D3—361no—
431C2×D3—364no—
432C2×D3—364no—
433C3×C4—48no—
434C3×C4—41no—
435C3×C4—44no—
436C3×C4—124no—
437C3×C4—124no—
438C3×C4—182no—
439C3×C4—361no—
440Dic3—41no—
441Dic3—41no—
442Dic3—68no—
443Dic3—124no—
444Dic3—368no—
445Dic3—361no—
446Dic3—364no—
447C32—24no—
448C32#467514yescomplemented
449C32#499211yescharacteristic, complemented
450C23—272no—
451C23—541no—
452C23—541no—
453C23—541no—
454C23—541no—
455C23—541no—
456C23—541no—
457C23C2×D3×C3⋊D311yescharacteristic
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—61no—
462C2×C4—61no—
463C2×C4—61no—
464C2×C4—61no—
465C2×C4—61no—
466C2×C4—61no—
467C2×C4—92no—
468C2×C4—181no—
469C2×C4—181no—
470C2×C4—181no—
471C2×C4—181no—
472C2×C4—181no—
473C2×C4—541no—
474C2×C4—541no—
475C2×C4—541no—
476C2×C3—24no—
477C2×C3—24no—
478C2×C3—24no—
479C2×C3—24no—
480C2×C3—24no—
481C2×C3—24no—
482C2×C3—24no—
483C2×C3#6384814yes—
484C2×C3#6385014yes—
485C2×C3#6385314yes—
486C2×C3#6386814yes—
487C2×C3#6389714yes—
488C2×C3#6390714yes—
489C2×C3#6395814yes—
490C2×C3#6408511yescharacteristic
491C2×C3#6409211yescharacteristic
492C2×C3#6409311yescharacteristic
493C2×C3#6409411yescharacteristic
494C2×C3#6410311yescharacteristic
495C2×C3#6410311yescharacteristic
496C2×C3#6410711yescharacteristic
497D3—3616no—
498D3—362no—
499D3—368no—
500C22—542no—
501C22—542no—
502C22—542no—
503C22—1081no—
504C22—1081no—
505C22—1081no—
506C22—1081no—
507C22#8901511yescharacteristic
508C22#8901811yescharacteristic
509C22#8902011yescharacteristic
510C22#8902211yescharacteristic
511C22#8903111yescharacteristic
512C22C3⋊D3×#6711yescharacteristic
513C22#8903911yescharacteristic
514C4—41no—
515C4—121no—
516C4—121no—
517C4—182no—
518C4—361no—
519C4—1081no—
520C3—24no—
521C3#1059120414yescomplemented
522C3#1059237411yescharacteristic, complemented
523C2—1082no—
524C2#1172341111yescharacteristic
525C2#1172341711yescharacteristic
526C2#1172342011yescharacteristic
527C2#1172345311yescharacteristic
528C2#1172349811yescharacteristic
529C2#1172352111yescharacteristic
530C2#1172362311yescharacteristic
subgroupG/H
center, FrattiniC23C2×D3×C3⋊D3
commutator, socle, cosocleC23×C33C23
FittingC22×C33×C4C22
prime radicalC23×C3⋊(C3⋊D3)C22
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×C3×#3750C2semidirect
5. C32×C3⋊#95C2semidirect
6. #11722061C2semidirect
7. #11722061C2semidirect
26. C2×#3750D3semidirect
33. C3×C3⋊#95D3semidirect
124. C3⋊#95C3⋊D3semidirect
289. C33#397semidirect
448. C32#4675semidirect
449. C32#4992semidirect
521. C3#10591204semidirect
522. C3#10592374semidirect

Maximal subgroups

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

subgroupindex
C2×#888592
C2×C3×#37502
C2×#888042
C32×C3⋊#952
#117220612
#105923743
#105912043