Finite Group Databasebeta

← #13160530

Subgroups — #13160530

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×#88859C211yesmaximal, characteristic
2C2×C3×#3750C211yesmaximal, characteristic, complemented
3C2×#88807C211yesmaximal, characteristic, complemented
4C2×#88806C211yesmaximal, characteristic, complemented
5C2×#88804C211yesmaximal, characteristic
6C2×#88804C211yesmaximal, characteristic
7C32×C3⋊#95C211yesmaximal, characteristic, complemented
8#10592375—31nomaximal
9#10591347—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#88807—22no—
20#88806—22no—
21#88804—22no—
22#88804—22no—
23C2×#3752—31no—
24C2×#3752—31no—
25C2×#3752—31no—
26C2×#3752—34no—
27C2×#3752—64no—
28C2×#3750—31no—
29C2×#3750—31no—
30C2×#3750D311yescharacteristic, complemented
31C2×C3×Dic3⋊C4—34no—
32C2×#3724—34no—
33C2×Dic3⋊Dic3—34no—
34C2×#3722—34no—
35C2×#3722—34no—
36C32×#95—31no—
37C3×C3⋊#95D314yescomplemented
38C23×C33C2311yescharacteristic
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)—21no—
45C22×C3⋊(C3⋊D3)Q812yes—
46C2×C33×C4—21no—
47C2×C33×C4—21no—
48C2×C33×C4—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⋊Dic3D412yes—
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×Dic3—21no—
64#4994—91no—
65#4676—94no—
66C23×C3⋊D3—31no—
67C23×C3⋊D3—34no—
68C23×C3⋊D3—64no—
69C22×C32×C4—24no—
70C22×C32×C4—31no—
71C22×C32×C4—31no—
72C22×C32×C4C2×D314yes—
73C22×C32×C4C2×D311yescharacteristic
74C22×C3⋊Dic3—31no—
75C22×C3⋊Dic3—34no—
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—
83#3752—62no—
84#3752—68no—
85#3752—616no—
86#3752—62no—
87#3752—62no—
88#3750—22no—
89#3750—62no—
90#3750—62no—
91C3×Dic3⋊C4—68no—
92#3724—68no—
93Dic3⋊Dic3—68no—
94#3722—68no—
95#3722—68no—
96C22×C33C2×D411yescharacteristic
97C22×C33C2×D411yescharacteristic
98C22×C33C2×D411yescharacteristic
99C22×C33C2×Q811yescharacteristic
100C22×C33#4111yescharacteristic
101C22×C33#4111yescharacteristic
102C22×C33#4111yescharacteristic
103C2×C3⋊(C3⋊D3)—22no—
104C2×C3⋊(C3⋊D3)—22no—
105C2×C3⋊(C3⋊D3)—22no—
106C2×C3⋊(C3⋊D3)—41no—
107C2×C3⋊(C3⋊D3)—41no—
108C2×C3⋊(C3⋊D3)—41no—
109C2×C3⋊(C3⋊D3)—41no—
110C33×C4—41no—
111C3⋊(C3⋊Dic3)—22no—
112C3×C3⋊Dic3—22no—
113C3×C3⋊Dic3—22no—
114C32×Dic3—41no—
115C32×Dic3—41no—
116C2×C3×C4⋊C4—91no—
117C2×(C2×D3)⋊C4—94no—
118C2×(C2×D3)⋊C4—94no—
119C2×(C2×D3)⋊C4—94no—
120C2×(C2×D3)⋊C4—91no—
121C2×(C2×D3)⋊C4—91no—
122C2×(C2×D3)⋊C4—91no—
123C2×(C2×D3)⋊C4—184no—
124C2×C4⋊Dic3—91no—
125C2×Dic3⋊C4—34no—
126C2×Dic3⋊C4—94no—
127C2×Dic3⋊C4—94no—
128C2×Dic3⋊C4—91no—
129C3×#95—34no—
130C3⋊#95C3⋊D311yescharacteristic, complemented
131C23×C32—24no—
132C23×C32C22×D314yes—
133C23×C32C22×D311yescharacteristic
134C22×C3⋊D3—32no—
135C22×C3⋊D3—38no—
136C22×C3⋊D3—68no—
137C22×C3⋊D3—68no—
138C22×C3⋊D3—68no—
139C22×C3⋊D3—61no—
140C22×C3⋊D3—64no—
141C22×C3⋊D3—61no—
142C22×C3⋊D3—64no—
143C22×C3⋊D3—61no—
144C22×C3⋊D3—61no—
145C22×C3⋊D3—61no—
146C22×C3⋊D3—61no—
147C22×C3⋊D3—64no—
148C22×C3⋊D3—64no—
149C22×C3⋊D3—64no—
150C22×C3⋊D3—64no—
151C22×C3⋊D3—124no—
152C22×C3⋊D3—124no—
153C22×C3⋊D3—124no—
154C22×C3⋊D3—124no—
155C2×C32×C4—28no—
156C2×C32×C4—24no—
157C2×C32×C4—21no—
158C2×C32×C4—21no—
159C2×C32×C4—21no—
160C2×C32×C4—24no—
161C2×C32×C4—24no—
162C2×C32×C4—44no—
163C2×C32×C4—44no—
164C2×C32×C4—61no—
165C2×C32×C4—61no—
166C2×C32×C4—61no—
167C2×C32×C4—61no—
168C2×C32×C4—61no—
169C2×C32×C4—61no—
170C2×C3⋊Dic3—21no—
171C2×C3⋊Dic3—21no—
172C2×C3⋊Dic3—21no—
173C2×C3⋊Dic3—21no—
174C2×C3⋊Dic3—32no—
175C2×C3⋊Dic3—38no—
176C2×C3⋊Dic3—68no—
177C2×C3⋊Dic3—61no—
178C2×C3⋊Dic3—61no—
179C2×C3⋊Dic3—64no—
180C2×C3⋊Dic3—64no—
181C2×C3⋊Dic3—124no—
182C2×C3⋊Dic3—124no—
183C2×C3⋊Dic3D1212yes—
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—24no—
191C2×C3×Dic3—38no—
192C2×C3×Dic3—38no—
193C2×C3×Dic3—64no—
194C2×C3×Dic3—64no—
195C2×C3×Dic3—64no—
196C2×C3×Dic3—64no—
197#396—271noSylow
198C2×C33#12111yescharacteristic
199C2×C33#12111yescharacteristic
200C2×C33#12111yescharacteristic
201C2×C33C22⋊Q811yescharacteristic
202C2×C33C22⋊Q811yescharacteristic
203C2×C33C22⋊Q811yescharacteristic
204C2×C33#12611yescharacteristic
205C3⋊(C3⋊D3)—42no—
206C23×D3—94no—
207C23×D3—91no—
208C23×D3—184no—
209C22×C3×C4—24no—
210C22×C3×C4—34no—
211C22×C3×C4—34no—
212C22×C3×C4—91no—
213C22×C3×C4—91no—
214C22×C3×C4D3214yes—
215C22×C3×C4C2×C3⋊D311yescharacteristic
216C22×Dic3—34no—
217C22×Dic3—34no—
218C22×Dic3—94no—
219C22×Dic3—91no—
220C22×Dic3—184no—
221C22×Dic3C2×C3⋊D311yescharacteristic
222C22×Dic3C2×C3⋊D311yescharacteristic
223C3×C4⋊C4—182no—
224(C2×D3)⋊C4—182no—
225(C2×D3)⋊C4—182no—
226(C2×D3)⋊C4—188no—
227(C2×D3)⋊C4—188no—
228(C2×D3)⋊C4—188no—
229(C2×D3)⋊C4—182no—
230(C2×D3)⋊C4—1816no—
231C4⋊Dic3—182no—
232Dic3⋊C4—68no—
233Dic3⋊C4—182no—
234Dic3⋊C4—188no—
235Dic3⋊C4—188no—
236C22×C32—24no—
237C22×C32—24no—
238C22×C32—24no—
239C22×C32—24no—
240C22×C32—24no—
241C22×C32—24no—
242C22×C32—24no—
243C22×C32C2×D1211yescharacteristic
244C22×C32#23514yes—
245C22×C32#23514yes—
246C22×C32#23514yes—
247C22×C32#23511yescharacteristic
248C22×C32D3×D414yes—
249C22×C32D3×D414yes—
250C22×C32D3×D414yes—
251C22×C32D3×D411yescharacteristic
252C22×C32#23711yescharacteristic
253C22×C32D3×Q814yes—
254C22×C32D3×Q811yescharacteristic
255C22×C32#23911yescharacteristic
256C22×C32C2×#6711yescharacteristic
257C2×C3⋊D3—616no—
258C2×C3⋊D3—68no—
259C2×C3⋊D3—62no—
260C2×C3⋊D3—62no—
261C2×C3⋊D3—62no—
262C2×C3⋊D3—68no—
263C2×C3⋊D3—68no—
264C2×C3⋊D3—128no—
265C2×C3⋊D3—128no—
266C2×C3⋊D3—128no—
267C2×C3⋊D3—128no—
268C2×C3⋊D3—128no—
269C2×C3⋊D3—128no—
270C2×C3⋊D3—124no—
271C2×C3⋊D3—124no—
272C2×C3⋊D3—124no—
273C2×C3⋊D3—124no—
274C2×C3⋊D3—121no—
275C2×C3⋊D3—121no—
276C2×C3⋊D3—121no—
277C2×C3⋊D3—121no—
278C32×C4—48no—
279C32×C4—44no—
280C32×C4—41no—
281C32×C4—121no—
282C32×C4—121no—
283C3⋊Dic3—22no—
284C3⋊Dic3—22no—
285C3⋊Dic3—62no—
286C3⋊Dic3—68no—
287C3⋊Dic3—128no—
288C3×Dic3—44no—
289C3×Dic3—44no—
290C3×Dic3—68no—
291C3×Dic3—68no—
292C2×C4⋊C4—91no—
293C2×C4⋊C4—271no—
294C2×C4⋊C4—271no—
295C2×C22⋊C4—271no—
296C2×C22⋊C4—271no—
297C2×C22⋊C4—271no—
298#95—31no—
299C33#39611yescharacteristic, complemented, Sylow
300C23×C3—24no—
301C23×C3C2×D3214yes—
302C23×C3C22×C3⋊D311yescharacteristic
303C22×D3—98no—
304C22×D3—92no—
305C22×D3—188no—
306C22×D3—188no—
307C22×D3—188no—
308C22×D3—184no—
309C22×D3—184no—
310C22×D3—184no—
311C22×D3—184no—
312C22×D3—184no—
313C22×D3—184no—
314C22×D3—181no—
315C22×D3—181no—
316C22×D3—181no—
317C22×D3—181no—
318C22×D3—181no—
319C22×D3—181no—
320C22×D3—364no—
321C22×D3—364no—
322C22×D3—364no—
323C22×D3—364no—
324C2×C3×C4—28no—
325C2×C3×C4—24no—
326C2×C3×C4—24no—
327C2×C3×C4—24no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—44no—
332C2×C3×C4—44no—
333C2×C3×C4—64no—
334C2×C3×C4—64no—
335C2×C3×C4—64no—
336C2×C3×C4—64no—
337C2×C3×C4—64no—
338C2×C3×C4—64no—
339C2×C3×C4—92no—
340C2×C3×C4—92no—
341C2×C3×C4—181no—
342C2×C3×C4—181no—
343C2×C3×C4—181no—
344C2×C3×C4—181no—
345C2×Dic3—21no—
346C2×Dic3—21no—
347C2×Dic3—21no—
348C2×Dic3—21no—
349C2×Dic3—21no—
350C2×Dic3—21no—
351C2×Dic3—38no—
352C2×Dic3—38no—
353C2×Dic3—64no—
354C2×Dic3—64no—
355C2×Dic3—64no—
356C2×Dic3—64no—
357C2×Dic3—98no—
358C2×Dic3—92no—
359C2×Dic3—188no—
360C2×Dic3—184no—
361C2×Dic3—184no—
362C2×Dic3—181no—
363C2×Dic3—181no—
364C2×Dic3—364no—
365C2×Dic3—364no—
366C2×C32—24no—
367C2×C32—24no—
368C2×C32—24no—
369C2×C32—24no—
370C2×C32—24no—
371C2×C32—24no—
372C2×C32—24no—
373C2×C32#87614yes—
374C2×C32#88414yes—
375C2×C32#88414yes—
376C2×C32#88414yes—
377C2×C32#89511yescharacteristic
378C2×C32#89614yes—
379C2×C32#89614yes—
380C2×C32#89614yes—
381C2×C32#89611yescharacteristic
382C2×C32#89711yescharacteristic
383C2×C32#89811yescharacteristic
384C2×C32#93011yescharacteristic
385C2×C32#93911yescharacteristic
386C2×C32#94611yescharacteristic
387C3⋊D3—1216no—
388C3⋊D3—128no—
389C3⋊D3—122no—
390C24—271no—
391C22×C4—31no—
392C22×C4—31no—
393C22×C4—91no—
394C22×C4—91no—
395C22×C4—271no—
396C22×C4D3×C3⋊D311yescharacteristic
397C4⋊C4—182no—
398C4⋊C4—542no—
399C4⋊C4—542no—
400C22⋊C4—542no—
401C22⋊C4—542no—
402C22⋊C4—542no—
403C22×C3—24no—
404C22×C3—24no—
405C22×C3—24no—
406C22×C3—24no—
407C22×C3—24no—
408C22×C3—24no—
409C22×C3—24no—
410C22×C3#379714yes—
411C22×C3#379814yes—
412C22×C3#379914yes—
413C22×C3D3×D1214yes—
414C22×C3#380414yes—
415C22×C3D3×#6714yes—
416C22×C3#381114yes—
417C22×C3#382811yescharacteristic
418C22×C3#382811yescharacteristic
419C22×C3#382811yescharacteristic
420C22×C3D4×C3⋊D311yescharacteristic
421C22×C3D4×C3⋊D311yescharacteristic
422C22×C3D4×C3⋊D311yescharacteristic
423C22×C3Q8×C3⋊D311yescharacteristic
424C2×D3—1816no—
425C2×D3—182no—
426C2×D3—182no—
427C2×D3—182no—
428C2×D3—188no—
429C2×D3—188no—
430C2×D3—188no—
431C2×D3—368no—
432C2×D3—368no—
433C2×D3—368no—
434C2×D3—368no—
435C2×D3—368no—
436C2×D3—368no—
437C2×D3—361no—
438C2×D3—361no—
439C2×D3—361no—
440C2×D3—361no—
441C2×D3—364no—
442C2×D3—364no—
443C2×D3—364no—
444C2×D3—364no—
445C3×C4—48no—
446C3×C4—41no—
447C3×C4—44no—
448C3×C4—124no—
449C3×C4—124no—
450C3×C4—182no—
451C3×C4—182no—
452Dic3—41no—
453Dic3—41no—
454Dic3—68no—
455Dic3—68no—
456Dic3—182no—
457Dic3—188no—
458Dic3—368no—
459C32—24no—
460C32#467614yescomplemented
461C32#499411yescharacteristic, complemented
462C23—272no—
463C23—541no—
464C23—541no—
465C23—541no—
466C23—541no—
467C23—541no—
468C23—541no—
469C23C2×D3×C3⋊D311yescharacteristic
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—61no—
474C2×C4—61no—
475C2×C4—61no—
476C2×C4—61no—
477C2×C4—61no—
478C2×C4—61no—
479C2×C4—92no—
480C2×C4—92no—
481C2×C4—181no—
482C2×C4—181no—
483C2×C4—181no—
484C2×C4—181no—
485C2×C4—272no—
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#6386514yes—
496C2×C3#6386814yes—
497C2×C3#6388814yes—
498C2×C3#6389514yes—
499C2×C3#6390614yes—
500C2×C3#6391014yes—
501C2×C3#6397114yes—
502C2×C3#6408511yescharacteristic
503C2×C3#6409311yescharacteristic
504C2×C3#6409311yescharacteristic
505C2×C3#6409311yescharacteristic
506C2×C3#6410511yescharacteristic
507C2×C3#6410511yescharacteristic
508C2×C3#6410511yescharacteristic
509D3—3616no—
510D3—368no—
511D3—362no—
512C22—542no—
513C22—542no—
514C22—542no—
515C22—1081no—
516C22—1081no—
517C22—1081no—
518C22—1081no—
519C22#8901911yescharacteristic
520C22#8902011yescharacteristic
521C22#8902211yescharacteristic
522C22C3⋊D3×D1211yescharacteristic
523C22#8903111yescharacteristic
524C22C3⋊D3×#6711yescharacteristic
525C22#8903911yescharacteristic
526C4—41no—
527C4—121no—
528C4—121no—
529C4—182no—
530C4—182no—
531C4—542no—
532C3—24no—
533C3#1059134714yescomplemented
534C3#1059237511yescharacteristic, complemented
535C2—1082no—
536C2#1172344211yescharacteristic
537C2#1172345311yescharacteristic
538C2#1172348911yescharacteristic
539C2#1172349511yescharacteristic
540C2#1172352011yescharacteristic
541C2#1172352511yescharacteristic
542C2#1172362411yescharacteristic
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#396
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
3. C2×#88807C2semidirect
4. C2×#88806C2semidirect
7. C32×C3⋊#95C2semidirect
30. C2×#3750D3semidirect
37. C3×C3⋊#95D3semidirect
130. C3⋊#95C3⋊D3semidirect
299. C33#396semidirect
460. C32#4676semidirect
461. C32#4994semidirect
533. C3#10591347semidirect
534. C3#10592375semidirect

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×#888072
C2×#888062
C2×#888042
C32×C3⋊#952
#105923753
#105913473