Finite Group Databasebeta

← #13156857

Subgroups — #13156857

Order 1728.

1442 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6295C211yesmaximal, characteristic
2C2×#88807C211yesmaximal, characteristic, complemented
3C2×#88803C212yesmaximal, complemented
4C2×C3×#3722C212yesmaximal, complemented
5C2×C3×Dic32C211yesmaximal, characteristic, complemented
6#10591386—31nomaximal
7#10591383—32nomaximal
8C23×C3×C3⋊D3C2212yes—
9C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
10C22×C3×C3⋊Dic3C2211yescharacteristic
11C22×C32×Dic3C2211yescharacteristic
12C22×C32×Dic3C2212yes—
13C2×#6295—22no—
14C2×#6295—22no—
15C2×#6295C414yescomplemented
16#88807—22no—
17#88803—24no—
18C3×#3722—24no—
19C3×Dic32—22no—
20C22×#626—32no—
21C22×#625—31no—
22C2×#3752—32no—
23C2×#3750—31no—
24C2×C3×(C2×D3)⋊C4—32no—
25C2×C3×(C2×D3)⋊C4—32no—
26C2×C3×C4×Dic3—32no—
27C2×Dic3⋊Dic3—32no—
28C2×Dic3⋊Dic3—61no—
29C2×#3722D312yescomplemented
30C2×(C2×D3)⋊Dic3—32no—
31C2×(C2×D3)⋊Dic3—32no—
32C2×(C2×D3)⋊Dic3—62no—
33C2×Dic32D311yescharacteristic, complemented
34C23×C33C2311yescharacteristic
35C22×C3×C3⋊D3—22no—
36C22×C3×C3⋊D3—22no—
37C22×C3×C3⋊D3—22no—
38C22×C3×C3⋊D3—22no—
39C22×C3×C3⋊D3—22no—
40C22×C3×C3⋊D3—22no—
41C22×C3×C3⋊D3C2×C414yes—
42C2×C3⋊(C3⋊Dic3)—22no—
43C2×C3⋊(C3⋊Dic3)—21no—
44C2×C3×C3⋊Dic3C2×C412yes—
45C2×C3×C3⋊Dic3D412yes—
46C2×C3×C3⋊Dic3D412yes—
47C2×C32×Dic3—22no—
48C2×C32×Dic3—22no—
49C2×C32×Dic3—21no—
50C2×C32×Dic3—22no—
51C2×C32×Dic3D414yescomplemented
52#6295—28no—
53#4957—92no—
54#4939—91no—
55C23×C3⋊D3C2×D312yes—
56C23×C3×D3—32no—
57C23×C3×D3—32no—
58C23×C3×D3—62no—
59C22×C32×C4—32no—
60C22×C32×C4—31no—
61C22×C3⋊Dic3—31no—
62C22×C3⋊Dic3—32no—
63C22×C3⋊Dic3—62no—
64C22×C3⋊Dic3—61no—
65C22×C3⋊Dic3—121no—
66C22×C3⋊Dic3C2×D311yescharacteristic
67C22×C3×Dic3—22no—
68C22×C3×Dic3—21no—
69C22×C3×Dic3—32no—
70C22×C3×Dic3—61no—
71C22×C3×Dic3C2×D312yes—
72C22×C3×Dic3C2×D312yes—
73C22×C3×Dic3C2×D312yes—
74C2×#626—38no—
75C2×#626—64no—
76C2×#626—64no—
77C2×#625—34no—
78C2×#625—62no—
79C2×#625—62no—
80#3752—64no—
81#3750—62no—
82C3×(C2×D3)⋊C4—64no—
83C3×(C2×D3)⋊C4—64no—
84C3×C4×Dic3—64no—
85Dic3⋊Dic3—64no—
86Dic3⋊Dic3—64no—
87#3722—24no—
88(C2×D3)⋊Dic3—68no—
89(C2×D3)⋊Dic3—64no—
90(C2×D3)⋊Dic3—64no—
91Dic32—22no—
92C22×C33C22×C411yescharacteristic
93C22×C33C2×D411yescharacteristic
94C22×C33C2×D411yescharacteristic
95C22×C33C2×D412yes—
96C22×C33#4112yes—
97C2×C3×C3⋊D3—24no—
98C2×C3×C3⋊D3—24no—
99C2×C3×C3⋊D3—24no—
100C2×C3×C3⋊D3—42no—
101C2×C3×C3⋊D3—42no—
102C2×C3×C3⋊D3—42no—
103C2×C3×C3⋊D3—42no—
104C3⋊(C3⋊Dic3)—41no—
105C3×C3⋊Dic3C22⋊C414yescomplemented
106C32×Dic3—24no—
107C32×Dic3—41no—
108C22×#67—92no—
109C22×D12—91no—
110C2×C3×C22⋊C4—92no—
111C2×C3×C42—91no—
112C2×#217—92no—
113C2×(C2×D3)⋊C4—32no—
114C2×(C2×D3)⋊C4—32no—
115C2×(C2×D3)⋊C4—92no—
116C2×(C2×D3)⋊C4—92no—
117C2×(C2×D3)⋊C4—182no—
118C2×C4⋊Dic3—91no—
119C2×Dic3⋊C4—92no—
120C2×Dic3⋊C4—181no—
121C2×C4×Dic3—32no—
122C23×C32—22no—
123C23×C32—21no—
124C23×C32—41no—
125C23×C32C22×D311yescharacteristic
126C23×C32C22×D312yes—
127C22×C3⋊D3—22no—
128C22×C3⋊D3—22no—
129C22×C3⋊D3—22no—
130C22×C3⋊D3—22no—
131C22×C3⋊D3—22no—
132C22×C3⋊D3—22no—
133C22×C3⋊D3C4×D314yes—
134C22×C3×D3—34no—
135C22×C3×D3—34no—
136C22×C3×D3—64no—
137C22×C3×D3—64no—
138C22×C3×D3—64no—
139C22×C3×D3—62no—
140C22×C3×D3—62no—
141C22×C3×D3—62no—
142C22×C3×D3—62no—
143C22×C3×D3—62no—
144C22×C3×D3—62no—
145C22×C3×D3—62no—
146C22×C3×D3—62no—
147C22×C3×D3—62no—
148C22×C3×D3—62no—
149C22×C3×D3—62no—
150C22×C3×D3—62no—
151C22×C3×D3—122no—
152C22×C3×D3—122no—
153C22×C3×D3—122no—
154C22×C3×D3—122no—
155C2×C32×C4—34no—
156C2×C32×C4—61no—
157C2×C32×C4—62no—
158C2×C32×C4—62no—
159C2×C32×C4—62no—
160C2×C3⋊Dic3—61no—
161C2×C3⋊Dic3—62no—
162C2×C3⋊Dic3—64no—
163C2×C3⋊Dic3—62no—
164C2×C3⋊Dic3—62no—
165C2×C3⋊Dic3—62no—
166C2×C3⋊Dic3—62no—
167C2×C3⋊Dic3—122no—
168C2×C3⋊Dic3—122no—
169C2×C3⋊Dic3—122no—
170C2×C3⋊Dic3—122no—
171C2×C3⋊Dic3—124no—
172C2×C3⋊Dic3C4×D312yes—
173C2×C3⋊Dic3D1212yes—
174C2×C3⋊Dic3#6712yes—
175C2×C3×Dic3—22no—
176C2×C3×Dic3—22no—
177C2×C3×Dic3—22no—
178C2×C3×Dic3—22no—
179C2×C3×Dic3—22no—
180C2×C3×Dic3—22no—
181C2×C3×Dic3—22no—
182C2×C3×Dic3—22no—
183C2×C3×Dic3—24no—
184C2×C3×Dic3—34no—
185C2×C3×Dic3—34no—
186C2×C3×Dic3—34no—
187C2×C3×Dic3—42no—
188C2×C3×Dic3—42no—
189C2×C3×Dic3—42no—
190C2×C3×Dic3—62no—
191C2×C3×Dic3—62no—
192C2×C3×Dic3—62no—
193C2×C3×Dic3D1214yescomplemented
194C2×C3×Dic3#6714yescomplemented
195#626—616no—
196#625—68no—
197#390—271noSylow
198C2×C33C2×C22⋊C411yescharacteristic
199C2×C33C4×D412yes—
200C2×C33#12112yes—
201C2×C33#12411yescharacteristic
202C2×C33#12711yescharacteristic
203C3×C3⋊D3—44no—
204C24×C3—92no—
205C23×D3—32no—
206C23×D3—32no—
207C23×D3—62no—
208C22×C3×C4—32no—
209C22×C3×C4—32no—
210C22×C3×C4—32no—
211C22×C3×C4—61no—
212C22×C3×C4—62no—
213C22×C3×C4—91no—
214C2×#67—98no—
215C2×#67—184no—
216C2×#67—184no—
217C22×Dic3—32no—
218C22×Dic3—61no—
219C22×Dic3—92no—
220C22×Dic3—91no—
221C22×Dic3—181no—
222C22×Dic3—182no—
223C22×Dic3—361no—
224C22×Dic3D3211yescharacteristic
225C22×Dic3D3212yes—
226C2×D12—94no—
227C2×D12—182no—
228C2×D12—182no—
229C3×C22⋊C4—184no—
230C3×C42—182no—
231#217—184no—
232(C2×D3)⋊C4—64no—
233(C2×D3)⋊C4—64no—
234(C2×D3)⋊C4—184no—
235(C2×D3)⋊C4—184no—
236(C2×D3)⋊C4—188no—
237C4⋊Dic3—182no—
238Dic3⋊C4—184no—
239Dic3⋊C4—184no—
240C4×Dic3—64no—
241C22×C32—22no—
242C22×C32—22no—
243C22×C32—21no—
244C22×C32—22no—
245C22×C32—22no—
246C22×C32—22no—
247C22×C32—21no—
248C22×C32—22no—
249C22×C32—22no—
250C22×C32—22no—
251C22×C32—22no—
252C22×C32—21no—
253C22×C32—41no—
254C22×C32—42no—
255C22×C32—41no—
256C22×C32—41no—
257C22×C32—42no—
258C22×C32C2×C4×D311yescharacteristic
259C22×C32C2×C4×D312yes—
260C22×C32C2×D1211yescharacteristic
261C22×C32C2×D1212yes—
262C22×C32#23512yes—
263C22×C32#23512yes—
264C22×C32D3×D412yes—
265C22×C32D3×D412yes—
266C22×C32D3×D412yes—
267C22×C32#23712yes—
268C22×C32C2×#6712yes—
269C22×C32C2×#6711yescharacteristic
270C2×C3⋊D3—24no—
271C2×C3⋊D3—24no—
272C2×C3⋊D3—24no—
273C2×C3⋊D3—42no—
274C2×C3⋊D3—42no—
275C2×C3⋊D3—42no—
276C2×C3⋊D3—42no—
277C2×C3×D3—68no—
278C2×C3×D3—64no—
279C2×C3×D3—64no—
280C2×C3×D3—64no—
281C2×C3×D3—64no—
282C2×C3×D3—64no—
283C2×C3×D3—64no—
284C2×C3×D3—124no—
285C2×C3×D3—124no—
286C2×C3×D3—124no—
287C2×C3×D3—124no—
288C2×C3×D3—124no—
289C2×C3×D3—124no—
290C2×C3×D3—122no—
291C2×C3×D3—122no—
292C2×C3×D3—122no—
293C2×C3×D3—122no—
294C2×C3×D3—122no—
295C2×C3×D3—122no—
296C2×C3×D3—122no—
297C2×C3×D3—122no—
298C32×C4—64no—
299C32×C4—121no—
300C3⋊Dic3—124no—
301C3⋊Dic3—122no—
302C3⋊Dic3—124no—
303C3⋊Dic3—121no—
304C3⋊Dic3—122no—
305C3⋊Dic3(C2×D3)⋊C414yescomplemented
306C3×Dic3—24no—
307C3×Dic3—24no—
308C3×Dic3—38no—
309C3×Dic3—44no—
310C3×Dic3—42no—
311C3×Dic3—42no—
312C3×Dic3—64no—
313C22×D4—271no—
314C2×C4⋊C4—271no—
315C2×C22⋊C4—92no—
316C2×C22⋊C4—272no—
317C2×C42—91no—
318C33#39011yescharacteristic, complemented, Sylow
319C23×C3—21no—
320C23×C3—22no—
321C23×C3—41no—
322C23×C3—94no—
323C23×C3—182no—
324C23×C3—182no—
325C23×C3—182no—
326C23×C3—182no—
327C23×C3—182no—
328C23×C3—182no—
329C23×C3C2×D3211yescharacteristic
330C23×C3C2×D3212yes—
331C22×D3—34no—
332C22×D3—34no—
333C22×D3—64no—
334C22×D3—64no—
335C22×D3—64no—
336C22×D3—62no—
337C22×D3—62no—
338C22×D3—62no—
339C22×D3—62no—
340C22×D3—62no—
341C22×D3—62no—
342C22×D3—62no—
343C22×D3—62no—
344C22×D3—62no—
345C22×D3—62no—
346C22×D3—62no—
347C22×D3—62no—
348C22×D3—122no—
349C22×D3—122no—
350C22×D3—122no—
351C22×D3—122no—
352C2×C3×C4—34no—
353C2×C3×C4—34no—
354C2×C3×C4—64no—
355C2×C3×C4—62no—
356C2×C3×C4—62no—
357C2×C3×C4—62no—
358C2×C3×C4—62no—
359C2×C3×C4—62no—
360C2×C3×C4—62no—
361C2×C3×C4—62no—
362C2×C3×C4—62no—
363C2×C3×C4—92no—
364C2×C3×C4—92no—
365C2×C3×C4—92no—
366C2×C3×C4—122no—
367C2×C3×C4—122no—
368C2×C3×C4—122no—
369#67—1816no—
370C2×Dic3—22no—
371C2×Dic3—22no—
372C2×Dic3—22no—
373C2×Dic3—21no—
374C2×Dic3—34no—
375C2×Dic3—34no—
376C2×Dic3—34no—
377C2×Dic3—62no—
378C2×Dic3—62no—
379C2×Dic3—62no—
380C2×Dic3—182no—
381C2×Dic3—184no—
382C2×Dic3—182no—
383C2×Dic3—182no—
384C2×Dic3—182no—
385C2×Dic3—182no—
386C2×Dic3—181no—
387C2×Dic3—362no—
388C2×Dic3—362no—
389C2×Dic3—362no—
390C2×Dic3—364no—
391C2×Dic3—362no—
392C2×Dic3#62614yescomplemented
393D12—188no—
394C2×C32—22no—
395C2×C32—22no—
396C2×C32—22no—
397C2×C32—22no—
398C2×C32—22no—
399C2×C32—22no—
400C2×C32—22no—
401C2×C32—21no—
402C2×C32—22no—
403C2×C32—21no—
404C2×C32—22no—
405C2×C32—21no—
406C2×C32—41no—
407C2×C32—42no—
408C2×C32—42no—
409C2×C32—41no—
410C2×C32—41no—
411C2×C32C4×D1212yes—
412C2×C32#87411yescharacteristic
413C2×C32#87511yescharacteristic
414C2×C32D3×C22⋊C412yes—
415C2×C32#67⋊C412yes—
416C2×C32#88412yes—
417C2×C32#88512yes—
418C2×C32C2×(C2×D3)⋊C411yescharacteristic
419C2×C32C4×#6712yes—
420C2×C32#93012yes—
421C2×C32#93912yes—
422C2×C32#94012yes—
423C3⋊D3—44no—
424C3×D3—128no—
425C3×D3—124no—
426C3×D3—124no—
427C24—92no—
428C2×D4—274no—
429C2×D4—542no—
430C2×D4—542no—
431C22×C4—32no—
432C22×C4—31no—
433C22×C4—91no—
434C22×C4—271no—
435C4⋊C4—542no—
436C22⋊C4—184no—
437C22⋊C4—544no—
438C42—182no—
439C22×C3—21no—
440C22×C3—22no—
441C22×C3—22no—
442C22×C3—22no—
443C22×C3—22no—
444C22×C3—21no—
445C22×C3—22no—
446C22×C3—21no—
447C22×C3—22no—
448C22×C3—22no—
449C22×C3—22no—
450C22×C3—22no—
451C22×C3—41no—
452C22×C3—42no—
453C22×C3—41no—
454C22×C3—41no—
455C22×C3—42no—
456C22×C3—184no—
457C22×C3—184no—
458C22×C3—184no—
459C22×C3—362no—
460C22×C3—362no—
461C22×C3—362no—
462C22×C3—362no—
463C22×C3#379512yes—
464C22×C3#379812yes—
465C22×C3C4×D3211yescharacteristic
466C22×C3C4×D3212yes—
467C22×C3D3×D1212yes—
468C22×C3D3×D1212yes—
469C22×C3#380211yescharacteristic
470C22×C3#380412yes—
471C22×C3C2×#62612yes—
472C22×C3D3×#6712yes—
473C22×C3D3×#6712yes—
474C22×C3#381111yescharacteristic
475C2×D3—68no—
476C2×D3—64no—
477C2×D3—64no—
478C2×D3—64no—
479C2×D3—64no—
480C2×D3—64no—
481C2×D3—64no—
482C2×D3—124no—
483C2×D3—124no—
484C2×D3—124no—
485C2×D3—124no—
486C2×D3—124no—
487C2×D3—124no—
488C2×D3—122no—
489C2×D3—122no—
490C2×D3—122no—
491C2×D3—122no—
492C2×D3—122no—
493C2×D3—122no—
494C2×D3—122no—
495C2×D3—122no—
496C3×C4—64no—
497C3×C4—64no—
498C3×C4—94no—
499C3×C4—124no—
500C3×C4—122no—
501C3×C4—122no—
502Dic3—24no—
503Dic3—38no—
504Dic3—41no—
505Dic3—64no—
506Dic3—362no—
507Dic3—364no—
508Dic3—364no—
509Dic3—362no—
510Dic3—361no—
511C32—22no—
512C32—21no—
513C32—41no—
514C32#493911yescharacteristic, complemented
515C32#495712yescomplemented
516C23—94no—
517C23—182no—
518C23—182no—
519C23—182no—
520C23—182no—
521C23—182no—
522C23—182no—
523C23D3311yescharacteristic
524D4—548no—
525C2×C4—34no—
526C2×C4—62no—
527C2×C4—62no—
528C2×C4—62no—
529C2×C4—61no—
530C2×C4—92no—
531C2×C4—92no—
532C2×C4—92no—
533C2×C4—542no—
534C2×C4—541no—
535C2×C3—22no—
536C2×C3—21no—
537C2×C3—21no—
538C2×C3—22no—
539C2×C3—21no—
540C2×C3—22no—
541C2×C3—22no—
542C2×C3—22no—
543C2×C3—22no—
544C2×C3—22no—
545C2×C3—22no—
546C2×C3—22no—
547C2×C3—41no—
548C2×C3—41no—
549C2×C3—41no—
550C2×C3—42no—
551C2×C3—42no—
552C2×C3—364no—
553C2×C3#6385212yes—
554C2×C3#6387912yes—
555C2×C3#6388012yes—
556C2×C3#6388512yes—
557C2×C3C4×#62612yes—
558C2×C3#6390612yes—
559C2×C3#6391112yes—
560C2×C3#6391311yescharacteristic
561C2×C3D3×(C2×D3)⋊C412yes—
562C2×C3#6392111yescharacteristic
563C2×C3#6397812yes—
564C2×C3#6397911yescharacteristic
565D3—128no—
566D3—124no—
567D3—124no—
568C22—184no—
569C22—184no—
570C22—184no—
571C22—362no—
572C22—362no—
573C22—362no—
574C22—362no—
575C22#8894911yescharacteristic
576C22D3×#62612yes—
577C22#8895311yescharacteristic
578C22#8895511yescharacteristic
579C22#8896012yes—
580C4—64no—
581C4—94no—
582C4—121no—
583C4—1081no—
584C3—22no—
585C3—21no—
586C3—41no—
587C3#1059138312yescomplemented
588C3#1059138611yescharacteristic, complemented
589C2—364no—
590C2#1172290511yescharacteristic
591C2#1172293412yes—
592C2#1172296012yes—
593C2#1172301011yescharacteristic
594C2#1172303011yescharacteristic
subgroupG/H
center, Frattini, hypercenterC23D33
commutatorC22×C33C22×C4
Fitting, socle, cosocleC23×C33C23
prime radicalC22×#6295C2
hypocenterC33#390
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×#88807C2semidirect
3. C2×#88803C2semidirect
4. C2×C3×#3722C2semidirect
5. C2×C3×Dic32C2semidirect
15. C2×#6295C4semidirect
29. C2×#3722D3semidirect
33. C2×Dic32D3semidirect
51. C2×C32×Dic3D4semidirect
105. C3×C3⋊Dic3C22⋊C4semidirect
193. C2×C3×Dic3D12semidirect
194. C2×C3×Dic3#67semidirect
305. C3⋊Dic3(C2×D3)⋊C4semidirect
318. C33#390semidirect
392. C2×Dic3#626semidirect
514. C32#4939semidirect
515. C32#4957semidirect
587. C3#10591383semidirect
588. C3#10591386semidirect

Maximal subgroups

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

subgroupindex
C22×#62952
C2×#888072
C2×#888032
C2×C3×#37222
C2×C3×Dic322
#105913863
#105913833