Finite Group Databasebeta

← #13155802

Subgroups — #13155802

Order 1728.

814 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3797C211yesmaximal, characteristic, complemented
2#11723696C211yesmaximal, characteristic
3C2×#88793C211yesmaximal, characteristic, complemented
4C2×C3×#3717C211yesmaximal, characteristic, complemented
5#11722037C211yesmaximal, characteristic, complemented
6#11722023C211yesmaximal, characteristic, complemented
7C3×#63566C211yesmaximal, characteristic, complemented
8#10591584—31nomaximal
9#10591583—31nomaximal
10#10590824—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C2×#6295—21no—
16C2×C3×#627—21no—
17C2×C3×#624—21no—
18C3×#3797—22no—
19C3×#3797—22no—
20C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
21C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
22C2×C32×C3⋊C8C2211yescharacteristic
23#88814—21no—
24#88793—22no—
25C3×#3717—22no—
26C2×C3×D3×Q8—31no—
27C2×C3×D3×Q8—31no—
28C2×#3797D311yescharacteristic, complemented
29C2×#3755—31no—
30C2×C3×#215—31no—
31C2×C3×#204—31no—
32#63916—31no—
33#63909—31no—
34#63908—31no—
35C2×#3717D311yescharacteristic, complemented
36C2×#3715—31no—
37C2×#3715—31no—
38C2×#3715—61no—
39#63662—31no—
40#63647—31no—
41C3×#835—31no—
42C3×#816—31no—
43#63566D311yescharacteristic, complemented
44#63565—31no—
45#63565—31no—
46#63565—61no—
47(C2×D3)⋊(C3⋊C8)—31no—
48(C2×D3)⋊(C3⋊C8)—31no—
49(C2×D3)⋊(C3⋊C8)—61no—
50C22×C3×C3⋊D3—21no—
51C22×C3×C3⋊D3D411yescharacteristic
52C2×C33×C4C2311yescharacteristic
53C2×C3×C3⋊Dic3—21no—
54C2×C3×C3⋊Dic3D411yescharacteristic
55C3×C4⋊(C3⋊D3)—21no—
56C3×C4⋊(C3⋊D3)—21no—
57C3×C4×C3⋊D3—21no—
58C3×C4×C3⋊D3—21no—
59C2×C32×Dic3—21no—
60C2×C32×Dic3—21no—
61C32×Dic6—21no—
62C32×Dic6—21no—
63C32×Dic6D412yescomplemented
64C32×Dic6D412yescomplemented
65#6295—42no—
66C3×#627—24no—
67C3×#624—24no—
68C3⋊(C3⋊(C3⋊C8))—21no—
69C32×C3⋊C8—21no—
70#5227—91no—
71#5170—91no—
72#4806—91no—
73C2×C32×Q8—31no—
74C2×C32×Q8—31no—
75C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
76C2×C4×C3⋊D3C2×D311yescharacteristic
77C3×D3×Q8—62no—
78C3×D3×Q8—62no—
79C3×D3×Q8—62no—
80C3×D3×Q8—62no—
81C2×C3×D12—31no—
82C2×C3×D12—31no—
83C2×C3×D12—61no—
84C2×C3×C4×D3—31no—
85C2×C3×C4×D3—31no—
86C2×C3×C4×D3—61no—
87C2×C3×C4×D3—61no—
88C2×C3×C4×D3—61no—
89C2×C3×Dic6—21no—
90C2×C3×Dic6—21no—
91C2×C3×Dic6—61no—
92C2×C3×Dic6—61no—
93C2×C3×Dic6C2×D311yescharacteristic
94C2×C3×Dic6C2×D311yescharacteristic
95C2×C3×Dic6C2×D311yescharacteristic
96C2×C3×Dic6C2×D311yescharacteristic
97C2×#627—21no—
98C2×#626—61no—
99C2×#626—61no—
100C2×#625—61no—
101C2×#624—21no—
102#3797—22no—
103#3797—22no—
104C2×C32×C8—31no—
105#3755—62no—
106C4⋊(C3⋊Dic3)C2×D311yescharacteristic
107C2×C3⋊(C3⋊C8)—31no—
108C2×C3⋊(C3⋊C8)—31no—
109C2×C3⋊(C3⋊C8)—31no—
110C2×C3⋊(C3⋊C8)—61no—
111C2×C3⋊(C3⋊C8)—61no—
112C2×C3⋊(C3⋊C8)—61no—
113C2×C3⋊(C3⋊C8)—121no—
114C3×#215—62no—
115C3×C4⋊Dic3—31no—
116C3×C4⋊Dic3—31no—
117C3×C4⋊Dic3—61no—
118C2×C3×C3⋊C8—21no—
119C2×C3×C3⋊C8C2×D311yescharacteristic
120C2×C3×C3⋊C8C2×D311yescharacteristic
121C3×#204—62no—
122#3722—61no—
123(C2×D3)⋊Dic3—61no—
124(C2×D3)⋊Dic3—61no—
125#3717—22no—
126#3715—64no—
127#3715—62no—
128#3715—62no—
129C22×C33C2×D411yescharacteristic
130C2×C3×C3⋊D3—21no—
131C2×C3×C3⋊D3—21no—
132C2×C3×C3⋊D3—41no—
133C2×C3×C3⋊D3—41no—
134C2×C3×C3⋊D3—41no—
135C2×C3×C3⋊D3SD1612yescomplemented
136C33×C4C2×D411yescharacteristic
137C33×C4C2×D411yescharacteristic
138C3×C3⋊Dic3—21no—
139C3×C3⋊Dic3—41no—
140C32×Dic3—22no—
141C32×Dic3—22no—
142C22×C3×Q8—91no—
143C2×D3×Q8—31no—
144C2×D3×Q8—31no—
145C2×C3×SD16—91no—
146C2×#215—31no—
147C2×#215—91no—
148C2×#215—91no—
149C2×#215—181no—
150#938—91no—
151C2×#214—91no—
152#930—91no—
153C2×#204—31no—
154#895—91no—
155C3×#103—91no—
156#835—31no—
157#835—91no—
158#835—91no—
159#835—181no—
160#830—91no—
161(C2×D3)⋊C8—91no—
162(C2×D3)⋊C8—91no—
163(C2×D3)⋊C8—181no—
164#816—31no—
165#810—91no—
166C22×C3⋊D3—21no—
167C22×C3⋊D3#6711yescharacteristic
168C22×C3×D3—31no—
169C22×C3×D3—31no—
170C22×C3×D3—61no—
171C22×C3×D3—61no—
172C22×C3×D3—61no—
173C22×C3×D3—121no—
174C32×Q8—32no—
175C32×Q8—32no—
176C32×Q8—61no—
177C32×Q8—61no—
178C2×C32×C4—21no—
179C2×C32×C4—21no—
180C2×C32×C4—21no—
181C2×C32×C4—41no—
182C2×C32×C4—61no—
183C2×C32×C4—61no—
184C2×C32×C4C22×D311yescharacteristic
185C2×C32×C4C22×D311yescharacteristic
186C2×C32×C4C22×D311yescharacteristic
187C2×C3⋊Dic3—21no—
188C2×C3⋊Dic3#6711yescharacteristic
189C4⋊(C3⋊D3)—21no—
190C4⋊(C3⋊D3)—21no—
191C4×C3⋊D3—21no—
192C4×C3⋊D3—21no—
193C2×C3×Dic3—21no—
194C2×C3×Dic3—21no—
195C2×C3×Dic3—21no—
196C2×C3×Dic3—21no—
197C2×C3×Dic3—31no—
198C2×C3×Dic3—31no—
199C2×C3×Dic3—41no—
200C2×C3×Dic3—41no—
201C2×C3×Dic3—61no—
202C2×C3×Dic3—61no—
203C2×C3×Dic3—61no—
204C2×C3×Dic3—121no—
205C3×D12—62no—
206C3×D12—61no—
207C3×D12—61no—
208C3×D12—61no—
209C3×D12—61no—
210C3×D12—121no—
211C3×C4×D3—64no—
212C3×C4×D3—64no—
213C3×C4×D3—61no—
214C3×C4×D3—61no—
215C3×C4×D3—61no—
216C3×C4×D3—61no—
217C3×C4×D3—121no—
218C3×C4×D3—121no—
219C3×Dic6—21no—
220C3×Dic6—21no—
221C3×Dic6—21no—
222C3×Dic6—21no—
223C3×Dic6—22no—
224C3×Dic6—22no—
225C3×Dic6—41no—
226C3×Dic6—41no—
227C3×Dic6—64no—
228C3×Dic6—64no—
229C3×Dic6D1212yescomplemented
230C3×Dic6#6712yescomplemented
231C3×Dic6#6712yescomplemented
232C3×Dic6#6712yescomplemented
233#627—24no—
234#626—122no—
235#626—122no—
236#625—122no—
237#624—24no—
238C32×C8—61no—
239C3⋊(C3⋊C8)—61no—
240C3⋊(C3⋊C8)—62no—
241C3⋊(C3⋊C8)—61no—
242C3⋊(C3⋊C8)—61no—
243C3⋊(C3⋊C8)—121no—
244C3⋊(C3⋊C8)—121no—
245C3⋊(C3⋊C8)—122no—
246C3×C3⋊C8—21no—
247C3×C3⋊C8—21no—
248C3×C3⋊C8—41no—
249#448—271noSylow
250C2×C33#12011yescharacteristic
251C2×C33C2×SD1611yescharacteristic
252C2×C33#13711yescharacteristic
253C3×C3⋊D3—22no—
254C3×C3⋊D3—81no—
255C2×C3×Q8—31no—
256C2×C3×Q8—31no—
257C2×C3×Q8—31no—
258C2×C3×Q8—31no—
259C2×C3×Q8—61no—
260C2×C3×Q8—61no—
261C2×C3×Q8—181no—
262C2×C3×Q8—182no—
263C2×C3×Q8—182no—
264C2×C3×D4—91no—
265C22×C3×C4—91no—
266C22×C3×C4—181no—
267C2×#67—181no—
268C2×#67—181no—
269D3×Q8—62no—
270D3×Q8—62no—
271D3×Q8—62no—
272D3×Q8—62no—
273C2×D12—31no—
274C2×D12—31no—
275C2×D12—61no—
276C2×D12—181no—
277C2×C4×D3—31no—
278C2×C4×D3—31no—
279C2×C4×D3—61no—
280C2×C4×D3—61no—
281C2×C4×D3—61no—
282C2×Dic6—61no—
283C2×Dic6—61no—
284C2×Dic6D3211yescharacteristic
285C2×Dic6D3211yescharacteristic
286C3×SD16—182no—
287C2×C3×C8—31no—
288C2×C3×C8—31no—
289C2×C3×C8—61no—
290C3×C4⋊C4—91no—
291#217—181no—
292#215—62no—
293#215—182no—
294#215—182no—
295#215—184no—
296#214—182no—
297(C2×D3)⋊C4—181no—
298(C2×D3)⋊C4—181no—
299C4⋊Dic3—31no—
300C4⋊Dic3—31no—
301C4⋊Dic3—61no—
302C2×C3⋊C8—91no—
303C2×C3⋊C8—91no—
304C2×C3⋊C8—91no—
305C2×C3⋊C8—181no—
306C2×C3⋊C8—181no—
307C2×C3⋊C8—181no—
308C2×C3⋊C8—361no—
309C2×C3⋊C8D3211yescharacteristic
310#204—62no—
311C22×C32—21no—
312C22×C32—21no—
313C22×C32—21no—
314C22×C32—41no—
315C22×C32D3×D411yescharacteristic
316C22×C32D3×D411yescharacteristic
317C22×C32C2×#6711yescharacteristic
318C2×C3⋊D3—21no—
319C2×C3⋊D3—21no—
320C2×C3⋊D3—41no—
321C2×C3⋊D3—41no—
322C2×C3⋊D3—41no—
323C2×C3⋊D3#21512yescomplemented
324C2×C3×D3—32no—
325C2×C3×D3—32no—
326C2×C3×D3—62no—
327C2×C3×D3—61no—
328C2×C3×D3—61no—
329C2×C3×D3—61no—
330C2×C3×D3—61no—
331C2×C3×D3—121no—
332C2×C3×D3—121no—
333C2×C3×D3—122no—
334C2×C3×D3—121no—
335C2×C3×D3—121no—
336C2×C3×D3—121no—
337C2×C3×D3—121no—
338C2×C3×D3—121no—
339C2×C3×D3—121no—
340C2×C3×D3—241no—
341C2×C3×D3—241no—
342C32×C4—21no—
343C32×C4—21no—
344C32×C4—21no—
345C32×C4—21no—
346C32×C4—21no—
347C32×C4—21no—
348C32×C4—41no—
349C32×C4—41no—
350C32×C4—62no—
351C32×C4—62no—
352C32×C4C2×D1211yescharacteristic
353C32×C4D3×D411yescharacteristic
354C32×C4D3×D411yescharacteristic
355C32×C4C2×#6711yescharacteristic
356C32×C4C2×#6711yescharacteristic
357C32×C4C2×#6711yescharacteristic
358C3⋊Dic3—21no—
359C3⋊Dic3—41no—
360C3×Dic3—22no—
361C3×Dic3—22no—
362C3×Dic3—22no—
363C3×Dic3—22no—
364C3×Dic3—42no—
365C3×Dic3—42no—
366C3×Dic3—61no—
367C3×Dic3—61no—
368C3×Dic3—121no—
369C3×Dic3—121no—
370C3×Dic3—121no—
371C3×Dic3—241no—
372C22×Q8—91no—
373C2×SD16—91no—
374C2×SD16—271no—
375#121—271no—
376#103—91no—
377#103—271no—
378C22⋊C8—271no—
379C33#44811yescharacteristic, complemented, Sylow
380C23×C3—91no—
381C23×C3—181no—
382C22×D3—31no—
383C22×D3—31no—
384C22×D3—61no—
385C22×D3—61no—
386C22×D3—61no—
387C22×D3—121no—
388C3×Q8—32no—
389C3×Q8—32no—
390C3×Q8—32no—
391C3×Q8—32no—
392C3×Q8—62no—
393C3×Q8—62no—
394C3×Q8—61no—
395C3×Q8—61no—
396C3×Q8—61no—
397C3×Q8—61no—
398C3×Q8—121no—
399C3×Q8—121no—
400C3×Q8—184no—
401C3×D4—181no—
402C3×D4—181no—
403C2×C3×C4—21no—
404C2×C3×C4—21no—
405C2×C3×C4—21no—
406C2×C3×C4—41no—
407C2×C3×C4—61no—
408C2×C3×C4—61no—
409C2×C3×C4—61no—
410C2×C3×C4—61no—
411C2×C3×C4—91no—
412C2×C3×C4—121no—
413C2×C3×C4—121no—
414C2×C3×C4—184no—
415C2×C3×C4—181no—
416C2×C3×C4—181no—
417C2×C3×C4—181no—
418C2×C3×C4C2×D3211yescharacteristic
419C2×C3×C4C2×D3211yescharacteristic
420C2×C3×C4C2×D3211yescharacteristic
421#67—362no—
422#67—362no—
423C2×Dic3—21no—
424C2×Dic3—21no—
425C2×Dic3—31no—
426C2×Dic3—31no—
427C2×Dic3—61no—
428C2×Dic3—61no—
429C2×Dic3—61no—
430C2×Dic3—121no—
431D12—62no—
432D12—61no—
433D12—61no—
434D12—61no—
435D12—61no—
436D12—121no—
437D12—362no—
438C4×D3—64no—
439C4×D3—64no—
440C4×D3—61no—
441C4×D3—61no—
442C4×D3—61no—
443C4×D3—61no—
444C4×D3—121no—
445C4×D3—121no—
446Dic6—21no—
447Dic6—21no—
448Dic6—64no—
449Dic6—64no—
450Dic6#62512yescomplemented
451Dic6#62612yescomplemented
452C3×C8—61no—
453C3×C8—61no—
454C3×C8—121no—
455C3⋊C8—21no—
456C3⋊C8—182no—
457C3⋊C8—181no—
458C3⋊C8—181no—
459C3⋊C8—181no—
460C3⋊C8—361no—
461C3⋊C8—361no—
462C3⋊C8—362no—
463C2×C32—21no—
464C2×C32—21no—
465C2×C32—21no—
466C2×C32—21no—
467C2×C32—21no—
468C2×C32—21no—
469C2×C32—21no—
470C2×C32—21no—
471C2×C32—21no—
472C2×C32—41no—
473C2×C32—41no—
474C2×C32—41no—
475C2×C32#88211yescharacteristic
476C2×C32D3×SD1611yescharacteristic
477C2×C32D3×SD1611yescharacteristic
478C2×C32#91511yescharacteristic
479C2×C32#91811yescharacteristic
480C2×C32#93711yescharacteristic
481C2×C32C2×#21511yescharacteristic
482C2×C32#94211yescharacteristic
483C2×C32#95311yescharacteristic
484C3⋊D3—22no—
485C3⋊D3—81no—
486C3×D3—62no—
487C3×D3—62no—
488C3×D3—122no—
489C3×D3—242no—
490C3×D3—241no—
491C3×D3—241no—
492C2×Q8—31no—
493C2×Q8—31no—
494C2×Q8—181no—
495C2×Q8—182no—
496C2×Q8—182no—
497C2×D4—91no—
498C2×D4—541no—
499C22×C4—91no—
500C22×C4—181no—
501SD16—182no—
502SD16—542no—
503C2×C8—31no—
504C2×C8—271no—
505C4⋊C4—91no—
506C22⋊C4—541no—
507C22×C3—21no—
508C22×C3—21no—
509C22×C3—21no—
510C22×C3—41no—
511C22×C3—92no—
512C22×C3—181no—
513C22×C3—181no—
514C22×C3—361no—
515C22×C3—361no—
516C22×C3—361no—
517C22×C3#380211yescharacteristic
518C22×C3D3×#6711yescharacteristic
519C22×C3D3×#6711yescharacteristic
520C2×D3—32no—
521C2×D3—32no—
522C2×D3—62no—
523C2×D3—61no—
524C2×D3—61no—
525C2×D3—61no—
526C2×D3—61no—
527C2×D3—121no—
528C2×D3—121no—
529C2×D3—122no—
530C2×D3—121no—
531C2×D3—121no—
532C2×D3—121no—
533C2×D3—121no—
534C2×D3—121no—
535C2×D3—121no—
536C2×D3—241no—
537C2×D3—241no—
538C3×C4—21no—
539C3×C4—21no—
540C3×C4—21no—
541C3×C4—21no—
542C3×C4—21no—
543C3×C4—21no—
544C3×C4—41no—
545C3×C4—41no—
546C3×C4—62no—
547C3×C4—62no—
548C3×C4—62no—
549C3×C4—62no—
550C3×C4—122no—
551C3×C4—122no—
552C3×C4—181no—
553C3×C4—361no—
554C3×C4D3×D1211yescharacteristic
555C3×C4C2×#62511yescharacteristic
556C3×C4C2×#62611yescharacteristic
557C3×C4D3×#6711yescharacteristic
558C3×C4D3×#6711yescharacteristic
559C3×C4D3×#6711yescharacteristic
560Dic3—22no—
561Dic3—22no—
562Dic3—61no—
563Dic3—61no—
564Dic3—121no—
565Dic3—121no—
566Dic3—121no—
567Dic3—241no—
568C32—21no—
569C32—21no—
570C32—21no—
571C32—41no—
572C32#480611yescharacteristic, complemented
573C32#517011yescharacteristic, complemented
574C32#522711yescharacteristic, complemented
575C23—91no—
576C23—181no—
577Q8—32no—
578Q8—32no—
579Q8—61no—
580Q8—61no—
581Q8—184no—
582D4—181no—
583D4—181no—
584D4—1082no—
585C2×C4—61no—
586C2×C4—61no—
587C2×C4—91no—
588C2×C4—184no—
589C2×C4—181no—
590C2×C4—181no—
591C2×C4—181no—
592C2×C4D3311yescharacteristic
593C8—61no—
594C8—541no—
595C2×C3—21no—
596C2×C3—21no—
597C2×C3—21no—
598C2×C3—21no—
599C2×C3—21no—
600C2×C3—21no—
601C2×C3—21no—
602C2×C3—21no—
603C2×C3—21no—
604C2×C3—41no—
605C2×C3—41no—
606C2×C3—41no—
607C2×C3—182no—
608C2×C3—721no—
609C2×C3#6379611yescharacteristic
610C2×C3#6381111yescharacteristic
611C2×C3#6392211yescharacteristic
612C2×C3D3×#21511yescharacteristic
613C2×C3D3×#21511yescharacteristic
614C2×C3#6394611yescharacteristic
615C2×C3#6394811yescharacteristic
616C2×C3#6397611yescharacteristic
617C2×C3#6397711yescharacteristic
618D3—62no—
619D3—62no—
620D3—122no—
621D3—242no—
622D3—241no—
623D3—241no—
624C22—92no—
625C22—181no—
626C22—181no—
627C22—361no—
628C22—361no—
629C22—361no—
630C22#8895311yescharacteristic
631C4—62no—
632C4—62no—
633C4—181no—
634C4—361no—
635C4D3×#62511yescharacteristic
636C4D3×#62611yescharacteristic
637C3—21no—
638C3—21no—
639C3—21no—
640C3—41no—
641C3#1059082411yescharacteristic, complemented
642C3#1059158311yescharacteristic, complemented
643C3#1059158411yescharacteristic, complemented
644C2—182no—
645C2—721no—
646C2#1172284011yescharacteristic
647C2#1172289211yescharacteristic
648C2#1172294711yescharacteristic
subgroupG/H
centerC22#88953
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radical#11723696C2
hypocenterC33#448
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3797C2semidirect
3. C2×#88793C2semidirect
4. C2×C3×#3717C2semidirect
5. #11722037C2semidirect
6. #11722023C2semidirect
7. C3×#63566C2semidirect
28. C2×#3797D3semidirect
35. C2×#3717D3semidirect
43. #63566D3semidirect
63. C32×Dic6D4semidirect
64. C32×Dic6D4semidirect
135. C2×C3×C3⋊D3SD16semidirect
229. C3×Dic6D12semidirect
230. C3×Dic6#67semidirect
231. C3×Dic6#67semidirect
232. C3×Dic6#67semidirect
323. C2×C3⋊D3#215semidirect
379. C33#448semidirect
450. Dic6#625semidirect
451. Dic6#626semidirect
572. C32#4806semidirect
573. C32#5170semidirect
574. C32#5227semidirect
641. C3#10590824semidirect
642. C3#10591583semidirect
643. C3#10591584semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37972
#117236962
C2×#887932
C2×C3×#37172
#117220372
#117220232
C3×#635662
#105915843
#105915833
#105908243