Finite Group Databasebeta

← #13160532

Subgroups — #13160532

Order 1728.

1956 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6292C211yesmaximal, characteristic
2C2×#88859C211yesmaximal, characteristic, complemented
3C2×C3×#3752C211yesmaximal, characteristic, complemented
4C2×#88807C211yesmaximal, characteristic, complemented
5C2×#88804C211yesmaximal, characteristic, complemented
6C2×#88803C211yesmaximal, characteristic, complemented
7C32×C3⋊#95C211yesmaximal, characteristic, complemented
8#10592371—31nomaximal
9#10591312—34nomaximal
10C23×C3⋊(C3⋊D3)C2211yescharacteristic
11C23×C3×C3⋊D3C2211yescharacteristic
12C22×C33×C4C2211yescharacteristic
13C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
14C22×C3×C3⋊Dic3C2211yescharacteristic
15C22×C32×Dic3C2211yescharacteristic
16C22×C32×Dic3C2211yescharacteristic
17C2×#6292—22no—
18C2×#6292—22no—
19C2×#6292—22no—
20#88859—22no—
21C3×#3752—22no—
22#88807—22no—
23#88804—22no—
24#88803—22no—
25C22×C4⋊(C3⋊D3)—31no—
26C22×#626—34no—
27C2×#3752—31no—
28C2×#3752—31no—
29C2×#3752—34no—
30C2×#3752—31no—
31C2×#3752—64no—
32C2×#3752D311yescharacteristic, complemented
33C2×#3750—31no—
34C2×C3×(C2×D3)⋊C4—34no—
35C2×Dic3⋊Dic3—34no—
36C2×#3722—34no—
37C2×(C2×D3)⋊Dic3—34no—
38C32×#95—31no—
39C3×C3⋊#95D314yescomplemented
40C23×C33C2311yescharacteristic
41C22×C3⋊(C3⋊D3)—21no—
42C22×C3⋊(C3⋊D3)—21no—
43C22×C3⋊(C3⋊D3)—21no—
44C22×C3⋊(C3⋊D3)—21no—
45C22×C3⋊(C3⋊D3)—21no—
46C22×C3⋊(C3⋊D3)—21no—
47C22×C3⋊(C3⋊D3)D412yes—
48C22×C3×C3⋊D3—21no—
49C22×C3×C3⋊D3—21no—
50C22×C3×C3⋊D3—21no—
51C22×C3×C3⋊D3—21no—
52C22×C3×C3⋊D3—21no—
53C22×C3×C3⋊D3—21no—
54C22×C3×C3⋊D3D412yes—
55C2×C33×C4—21no—
56C2×C33×C4—21no—
57C2×C33×C4—21no—
58C2×C3⋊(C3⋊Dic3)—21no—
59C2×C3⋊(C3⋊Dic3)—21no—
60C2×C3⋊(C3⋊Dic3)—21no—
61C2×C3×C3⋊Dic3—21no—
62C2×C3×C3⋊Dic3—21no—
63C2×C3×C3⋊Dic3D412yescomplemented
64C2×C32×Dic3—21no—
65C2×C32×Dic3—21no—
66C2×C32×Dic3—21no—
67C2×C32×Dic3—21no—
68C2×C32×Dic3—21no—
69C2×C32×Dic3D412yes—
70#6292—28no—
71#4961—91no—
72#4672—94no—
73C23×C3⋊D3—31no—
74C23×C3⋊D3—34no—
75C23×C3⋊D3—64no—
76C23×C3⋊D3C2×D311yescharacteristic
77C23×C3×D3—34no—
78C22×C32×C4—24no—
79C22×C32×C4—31no—
80C22×C32×C4—31no—
81C22×C32×C4C2×D314yes—
82C22×C32×C4C2×D311yescharacteristic
83C22×C3⋊Dic3—31no—
84C22×C3⋊Dic3—34no—
85C22×C3⋊Dic3—64no—
86C22×C3⋊Dic3C2×D311yescharacteristic
87C2×C4⋊(C3⋊D3)—62no—
88C2×C4⋊(C3⋊D3)—62no—
89C2×C4⋊(C3⋊D3)—62no—
90C22×C3×Dic3—34no—
91C22×C3×Dic3C2×D314yes—
92C22×C3×Dic3C2×D314yes—
93C2×#626—68no—
94C2×#626—68no—
95C2×#626—68no—
96#3752—22no—
97#3752—62no—
98#3752—68no—
99#3752—616no—
100#3752—62no—
101#3752—62no—
102#3750—62no—
103C3×(C2×D3)⋊C4—68no—
104Dic3⋊Dic3—68no—
105#3722—68no—
106(C2×D3)⋊Dic3—68no—
107C22×C33C2×D411yescharacteristic
108C22×C33C2×D411yescharacteristic
109C22×C33C2×D411yescharacteristic
110C22×C33C2×D411yescharacteristic
111C22×C33#4111yescharacteristic
112C22×C33#4111yescharacteristic
113C22×C33#4111yescharacteristic
114C2×C3⋊(C3⋊D3)—22no—
115C2×C3⋊(C3⋊D3)—22no—
116C2×C3⋊(C3⋊D3)—22no—
117C2×C3⋊(C3⋊D3)—41no—
118C2×C3⋊(C3⋊D3)—41no—
119C2×C3⋊(C3⋊D3)—41no—
120C2×C3⋊(C3⋊D3)—41no—
121C2×C3×C3⋊D3—22no—
122C2×C3×C3⋊D3—22no—
123C2×C3×C3⋊D3—22no—
124C2×C3×C3⋊D3—41no—
125C2×C3×C3⋊D3—41no—
126C2×C3×C3⋊D3—41no—
127C2×C3×C3⋊D3—41no—
128C33×C4—41no—
129C3⋊(C3⋊Dic3)—41no—
130C3×C3⋊Dic3—22no—
131C32×Dic3—22no—
132C32×Dic3—41no—
133C22×#67—91no—
134C22×D12—94no—
135C2×C3×C22⋊C4—91no—
136C2×#217—91no—
137C2×(C2×D3)⋊C4—34no—
138C2×(C2×D3)⋊C4—94no—
139C2×(C2×D3)⋊C4—94no—
140C2×(C2×D3)⋊C4—94no—
141C2×(C2×D3)⋊C4—91no—
142C2×(C2×D3)⋊C4—91no—
143C2×(C2×D3)⋊C4—184no—
144C2×C4⋊Dic3—91no—
145C2×Dic3⋊C4—94no—
146C3×#95—34no—
147C3⋊#95C3⋊D311yescharacteristic, complemented
148C23×C32—24no—
149C23×C32C22×D314yes—
150C23×C32C22×D311yescharacteristic
151C22×C3⋊D3—21no—
152C22×C3⋊D3—21no—
153C22×C3⋊D3—21no—
154C22×C3⋊D3—21no—
155C22×C3⋊D3—21no—
156C22×C3⋊D3—21no—
157C22×C3⋊D3—32no—
158C22×C3⋊D3—38no—
159C22×C3⋊D3—68no—
160C22×C3⋊D3—68no—
161C22×C3⋊D3—68no—
162C22×C3⋊D3—61no—
163C22×C3⋊D3—64no—
164C22×C3⋊D3—61no—
165C22×C3⋊D3—61no—
166C22×C3⋊D3—61no—
167C22×C3⋊D3—61no—
168C22×C3⋊D3—64no—
169C22×C3⋊D3—61no—
170C22×C3⋊D3—64no—
171C22×C3⋊D3—64no—
172C22×C3⋊D3—64no—
173C22×C3⋊D3—64no—
174C22×C3⋊D3—124no—
175C22×C3⋊D3—124no—
176C22×C3⋊D3—124no—
177C22×C3⋊D3—124no—
178C22×C3⋊D3D1212yes—
179C22×C3×D3—38no—
180C22×C3×D3—64no—
181C22×C3×D3—64no—
182C22×C3×D3—64no—
183C22×C3×D3—64no—
184C22×C3×D3—64no—
185C22×C3×D3—64no—
186C2×C32×C4—28no—
187C2×C32×C4—24no—
188C2×C32×C4—21no—
189C2×C32×C4—21no—
190C2×C32×C4—21no—
191C2×C32×C4—24no—
192C2×C32×C4—24no—
193C2×C32×C4—32no—
194C2×C32×C4—44no—
195C2×C32×C4—44no—
196C2×C32×C4—61no—
197C2×C32×C4—61no—
198C2×C32×C4—61no—
199C2×C32×C4—61no—
200C2×C32×C4—61no—
201C2×C3⋊Dic3—21no—
202C2×C3⋊Dic3—21no—
203C2×C3⋊Dic3—68no—
204C2×C3⋊Dic3—61no—
205C2×C3⋊Dic3—61no—
206C2×C3⋊Dic3—64no—
207C2×C3⋊Dic3—61no—
208C2×C3⋊Dic3—64no—
209C2×C3⋊Dic3—64no—
210C2×C3⋊Dic3—124no—
211C2×C3⋊Dic3—124no—
212C2×C3⋊Dic3#6712yescomplemented
213C4⋊(C3⋊D3)—68no—
214C2×C3×Dic3—24no—
215C2×C3×Dic3—24no—
216C2×C3×Dic3—24no—
217C2×C3×Dic3—24no—
218C2×C3×Dic3—24no—
219C2×C3×Dic3—38no—
220C2×C3×Dic3—64no—
221C2×C3×Dic3—64no—
222C2×C3×Dic3D1218yes—
223#626—632no—
224#394—271noSylow
225C2×C33#12011yescharacteristic
226C2×C33#12111yescharacteristic
227C2×C33#12111yescharacteristic
228C2×C33#12111yescharacteristic
229C2×C33#12311yescharacteristic
230C2×C33#12311yescharacteristic
231C2×C33#12411yescharacteristic
232C3⋊(C3⋊D3)—42no—
233C3×C3⋊D3—42no—
234C24×C3—91no—
235C23×D3—34no—
236C23×D3—94no—
237C23×D3—91no—
238C23×D3—184no—
239C22×C3×C4—24no—
240C22×C3×C4—34no—
241C22×C3×C4—34no—
242C22×C3×C4—91no—
243C22×C3×C4D3214yes—
244C22×C3×C4C2×C3⋊D311yescharacteristic
245C2×#67—182no—
246C2×#67—182no—
247C2×#67—182no—
248C22×Dic3—34no—
249C22×Dic3—94no—
250C22×Dic3—91no—
251C22×Dic3—184no—
252C22×Dic3C2×C3⋊D311yescharacteristic
253C22×Dic3C2×C3⋊D311yescharacteristic
254C2×D12—188no—
255C2×D12—188no—
256C2×D12—188no—
257C3×C22⋊C4—182no—
258#217—182no—
259(C2×D3)⋊C4—68no—
260(C2×D3)⋊C4—182no—
261(C2×D3)⋊C4—188no—
262(C2×D3)⋊C4—188no—
263(C2×D3)⋊C4—182no—
264(C2×D3)⋊C4—1816no—
265(C2×D3)⋊C4—188no—
266C4⋊Dic3—182no—
267Dic3⋊C4—188no—
268C22×C32—24no—
269C22×C32—24no—
270C22×C32—24no—
271C22×C32—24no—
272C22×C32—24no—
273C22×C32—24no—
274C22×C32—24no—
275C22×C32C2×D1214yes—
276C22×C32C2×D1211yescharacteristic
277C22×C32#23514yes—
278C22×C32#23514yes—
279C22×C32#23511yescharacteristic
280C22×C32D3×D414yes—
281C22×C32D3×D414yes—
282C22×C32D3×D414yes—
283C22×C32D3×D411yescharacteristic
284C22×C32D3×D411yescharacteristic
285C22×C32#23711yescharacteristic
286C22×C32#23711yescharacteristic
287C22×C32#23914yes—
288C22×C32C2×#6711yescharacteristic
289C2×C3⋊D3—22no—
290C2×C3⋊D3—22no—
291C2×C3⋊D3—22no—
292C2×C3⋊D3—41no—
293C2×C3⋊D3—41no—
294C2×C3⋊D3—41no—
295C2×C3⋊D3—41no—
296C2×C3⋊D3—616no—
297C2×C3⋊D3—62no—
298C2×C3⋊D3—62no—
299C2×C3⋊D3—68no—
300C2×C3⋊D3—62no—
301C2×C3⋊D3—68no—
302C2×C3⋊D3—68no—
303C2×C3⋊D3—128no—
304C2×C3⋊D3—128no—
305C2×C3⋊D3—128no—
306C2×C3⋊D3—128no—
307C2×C3⋊D3—128no—
308C2×C3⋊D3—128no—
309C2×C3⋊D3—124no—
310C2×C3⋊D3—124no—
311C2×C3⋊D3—121no—
312C2×C3⋊D3—121no—
313C2×C3⋊D3—124no—
314C2×C3⋊D3—124no—
315C2×C3⋊D3—121no—
316C2×C3⋊D3—121no—
317C2×C3×D3—68no—
318C2×C3×D3—68no—
319C2×C3×D3—68no—
320C2×C3×D3—124no—
321C2×C3×D3—124no—
322C2×C3×D3—124no—
323C2×C3×D3—124no—
324C32×C4—48no—
325C32×C4—44no—
326C32×C4—41no—
327C32×C4—62no—
328C32×C4—121no—
329C3⋊Dic3—22no—
330C3⋊Dic3—128no—
331C3⋊Dic3—124no—
332C3⋊Dic3—121no—
333C3×Dic3—28no—
334C3×Dic3—44no—
335C3×Dic3—68no—
336C22×D4—271no—
337C2×C4⋊C4—271no—
338C2×C22⋊C4—91no—
339C2×C22⋊C4—271no—
340C2×C22⋊C4—271no—
341C2×C22⋊C4—271no—
342#95—31no—
343C33#39411yescharacteristic, complemented, Sylow
344C23×C3—24no—
345C23×C3—92no—
346C23×C3—181no—
347C23×C3—181no—
348C23×C3—181no—
349C23×C3—181no—
350C23×C3—181no—
351C23×C3—181no—
352C23×C3C2×D3214yes—
353C23×C3C22×C3⋊D311yescharacteristic
354C22×D3—38no—
355C22×D3—64no—
356C22×D3—64no—
357C22×D3—64no—
358C22×D3—64no—
359C22×D3—64no—
360C22×D3—64no—
361C22×D3—98no—
362C22×D3—92no—
363C22×D3—188no—
364C22×D3—188no—
365C22×D3—188no—
366C22×D3—184no—
367C22×D3—184no—
368C22×D3—184no—
369C22×D3—184no—
370C22×D3—184no—
371C22×D3—184no—
372C22×D3—181no—
373C22×D3—181no—
374C22×D3—181no—
375C22×D3—181no—
376C22×D3—181no—
377C22×D3—181no—
378C22×D3—364no—
379C22×D3—364no—
380C22×D3—364no—
381C22×D3—364no—
382C2×C3×C4—28no—
383C2×C3×C4—24no—
384C2×C3×C4—24no—
385C2×C3×C4—24no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—38no—
390C2×C3×C4—44no—
391C2×C3×C4—44no—
392C2×C3×C4—64no—
393C2×C3×C4—64no—
394C2×C3×C4—64no—
395C2×C3×C4—64no—
396C2×C3×C4—64no—
397C2×C3×C4—92no—
398C2×C3×C4—181no—
399C2×C3×C4—181no—
400#67—188no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3—38no—
407C2×Dic3—64no—
408C2×Dic3—64no—
409C2×Dic3—188no—
410C2×Dic3—184no—
411C2×Dic3—184no—
412C2×Dic3—184no—
413C2×Dic3—181no—
414C2×Dic3—181no—
415C2×Dic3—181no—
416C2×Dic3—364no—
417C2×Dic3—364no—
418C2×Dic3C4⋊(C3⋊D3)12yes—
419D12—1832no—
420C2×C32—24no—
421C2×C32—24no—
422C2×C32—24no—
423C2×C32—24no—
424C2×C32—24no—
425C2×C32—24no—
426C2×C32—24no—
427C2×C32#87514yes—
428C2×C32#88214yes—
429C2×C32#88211yescharacteristic
430C2×C32#88311yescharacteristic
431C2×C32#88414yes—
432C2×C32#88414yes—
433C2×C32#88511yescharacteristic
434C2×C32#88611yescharacteristic
435C2×C32#89414yes—
436C2×C32#89414yes—
437C2×C32#89514yes—
438C2×C32#93011yescharacteristic
439C2×C32#93811yescharacteristic
440C2×C32#93911yescharacteristic
441C3⋊D3—42no—
442C3⋊D3—1216no—
443C3⋊D3—128no—
444C3⋊D3—122no—
445C3×D3—128no—
446C24—91no—
447C24—271no—
448C2×D4—542no—
449C2×D4—542no—
450C2×D4—542no—
451C22×C4—31no—
452C22×C4—31no—
453C22×C4—91no—
454C22×C4—271no—
455C22×C4D3×C3⋊D311yescharacteristic
456C4⋊C4—542no—
457C22⋊C4—182no—
458C22⋊C4—542no—
459C22⋊C4—542no—
460C22⋊C4—542no—
461C22×C3—24no—
462C22×C3—24no—
463C22×C3—24no—
464C22×C3—24no—
465C22×C3—24no—
466C22×C3—24no—
467C22×C3—24no—
468C22×C3—182no—
469C22×C3—182no—
470C22×C3—182no—
471C22×C3—361no—
472C22×C3—361no—
473C22×C3—361no—
474C22×C3—361no—
475C22×C3#379614yes—
476C22×C3#379814yes—
477C22×C3D3×D1214yes—
478C22×C3D3×D1214yes—
479C22×C3#380414yes—
480C22×C3D3×#6714yes—
481C22×C3#381114yes—
482C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
483C22×C3#382811yescharacteristic
484C22×C3#382811yescharacteristic
485C22×C3D4×C3⋊D311yescharacteristic
486C22×C3D4×C3⋊D311yescharacteristic
487C22×C3D4×C3⋊D311yescharacteristic
488C22×C3#383211yescharacteristic
489C2×D3—68no—
490C2×D3—68no—
491C2×D3—68no—
492C2×D3—124no—
493C2×D3—124no—
494C2×D3—124no—
495C2×D3—124no—
496C2×D3—1816no—
497C2×D3—182no—
498C2×D3—182no—
499C2×D3—182no—
500C2×D3—188no—
501C2×D3—188no—
502C2×D3—188no—
503C2×D3—368no—
504C2×D3—368no—
505C2×D3—368no—
506C2×D3—368no—
507C2×D3—368no—
508C2×D3—368no—
509C2×D3—361no—
510C2×D3—361no—
511C2×D3—361no—
512C2×D3—361no—
513C2×D3—364no—
514C2×D3—364no—
515C2×D3—364no—
516C2×D3—364no—
517C3×C4—48no—
518C3×C4—41no—
519C3×C4—44no—
520C3×C4—68no—
521C3×C4—124no—
522C3×C4—182no—
523Dic3—22no—
524Dic3—41no—
525Dic3—68no—
526Dic3—368no—
527Dic3—361no—
528Dic3—364no—
529C32—24no—
530C32#467214yescomplemented
531C32#496111yescharacteristic, complemented
532C23—92no—
533C23—181no—
534C23—181no—
535C23—181no—
536C23—181no—
537C23—181no—
538C23—181no—
539C23—272no—
540C23—541no—
541C23—541no—
542C23—541no—
543C23—541no—
544C23—541no—
545C23—541no—
546C23C2×D3×C3⋊D311yescharacteristic
547D4—548no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—32no—
552C2×C4—61no—
553C2×C4—61no—
554C2×C4—61no—
555C2×C4—61no—
556C2×C4—61no—
557C2×C4—92no—
558C2×C4—181no—
559C2×C4—181no—
560C2×C4—541no—
561C2×C4—541no—
562C2×C4—541no—
563C2×C3—24no—
564C2×C3—24no—
565C2×C3—24no—
566C2×C3—24no—
567C2×C3—24no—
568C2×C3—24no—
569C2×C3—24no—
570C2×C3—362no—
571C2×C3#6385214yes—
572C2×C3#6385314yes—
573C2×C3#6388614yes—
574C2×C3#6389014yes—
575C2×C3#6390614yes—
576C2×C3#6392314yes—
577C2×C3#6395814yes—
578C2×C3#6408411yescharacteristic
579C2×C3#6409111yescharacteristic
580C2×C3#6409311yescharacteristic
581C2×C3#6409311yescharacteristic
582C2×C3#6410311yescharacteristic
583C2×C3#6410311yescharacteristic
584C2×C3#6410411yescharacteristic
585D3—128no—
586D3—3616no—
587D3—368no—
588D3—362no—
589C22—182no—
590C22—182no—
591C22—182no—
592C22—361no—
593C22—361no—
594C22—361no—
595C22—361no—
596C22—542no—
597C22—542no—
598C22—542no—
599C22—1081no—
600C22—1081no—
601C22—1081no—
602C22—1081no—
603C22#8901711yescharacteristic
604C22#8902011yescharacteristic
605C22D3×C4⋊(C3⋊D3)11yescharacteristic
606C22C3⋊D3×D1211yescharacteristic
607C22#8903111yescharacteristic
608C22C3⋊D3×#6711yescharacteristic
609C22#8903911yescharacteristic
610C4—41no—
611C4—62no—
612C4—121no—
613C4—182no—
614C4—1081no—
615C3—24no—
616C3#1059131214yescomplemented
617C3#1059237111yescharacteristic, complemented
618C2—362no—
619C2—1082no—
620C2#1172341611yescharacteristic
621C2#1172342111yescharacteristic
622C2#1172348211yescharacteristic
623C2#1172349311yescharacteristic
624C2#1172352011yescharacteristic
625C2#1172354211yescharacteristic
626C2#1172362311yescharacteristic
subgroupG/H
center, FrattiniC23C2×D3×C3⋊D3
commutator, socle, cosocleC23×C33C23
FittingC22×C33×C4C22
prime radicalC22×#6292C2
hypercenterC22×C4D3×C3⋊D3
hypocenterC33#394
LCSG ▷ C23×C33 ▷ C33
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×#88859C2semidirect
3. C2×C3×#3752C2semidirect
4. C2×#88807C2semidirect
5. C2×#88804C2semidirect
6. C2×#88803C2semidirect
7. C32×C3⋊#95C2semidirect
32. C2×#3752D3semidirect
39. C3×C3⋊#95D3semidirect
63. C2×C3×C3⋊Dic3D4semidirect
147. C3⋊#95C3⋊D3semidirect
212. C2×C3⋊Dic3#67semidirect
343. C33#394semidirect
530. C32#4672semidirect
531. C32#4961semidirect
616. C3#10591312semidirect
617. C3#10592371semidirect

Maximal subgroups

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

subgroupindex
C22×#62922
C2×#888592
C2×C3×#37522
C2×#888072
C2×#888042
C2×#888032
C32×C3⋊#952
#105923713
#105913123