Finite Group Databasebeta

← #13178287

Subgroups — #13178287

Order 1728.

1690 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89049C211yesmaximal, characteristic
2C22×#6294C211yesmaximal, characteristic, complemented
3C3×#64095C212yesmaximal, complemented
4C3×C22⋊C4×C3⋊D3C211yesmaximal, characteristic, complemented
5#11723720C212yesmaximal, complemented
6#11723700C212yesmaximal, complemented
7#11723690C214yesmaximal, complemented
8#11723689C212yesmaximal, complemented
9#10594295—31nomaximal
10D3×#886—32nomaximal
11C23×C3×C3⋊D3C2211yescharacteristic
12C22×#1113C2211yescharacteristic
13C2×C3×#638C2212yes—
14C22×C3×C3⋊Dic3C2212yescomplemented
15C2×C3×C4×C3⋊D3C2212yes—
16#89049—24no—
17C2×#6295C2212yes—
18C2×#6294—24no—
19C2×#6294C2212yes—
20C2×#6294C2211yescharacteristic
21C2×#6294C2212yescomplemented
22C33×C22⋊C4C2211yescharacteristic, complemented
23C3×#3757C2211yescharacteristic, complemented
24C3×#3752C2212yescomplemented
25C3×#3752C2214yescomplemented
26C3×C4⋊(C3⋊Dic3)C2214yescomplemented
27#88815C2214yescomplemented
28#88814C2214yescomplemented
29C2×#3811—31no—
30C2×D3×#67—32no—
31C22×#624—31no—
32C22×D3×Dic3—32no—
33#64095D312yescomplemented
34C22⋊C4×C3⋊D3D311yescharacteristic, complemented
35C3×#886—32no—
36C3×#886—32no—
37C3×#886—62no—
38C3×D3×C22⋊C4—32no—
39C3×D3×C22⋊C4—61no—
40#63966—32no—
41#63962—32no—
42#63961—32no—
43#63921—32no—
44D3×(C2×D3)⋊C4—34no—
45#63900—32no—
46#63891—34no—
47#63890—34no—
48D3×C4⋊Dic3—34no—
49#63854—34no—
50C23×C33C2311yescharacteristic
51C22×C3×C3⋊D3—24no—
52C22×C3×C3⋊D3D414yes—
53C22×C3×C3⋊D3C2312yes—
54C22×C3×C3⋊D3C2311yescharacteristic
55C22×C3×C3⋊D3C2311yescharacteristic
56C2×#1113—22no—
57C2×#1113—24no—
58C2×C33×C4C2312yes—
59C3×#638—24no—
60C2×C3×C3⋊Dic3—24no—
61C2×C3×C3⋊Dic3C2312yes—
62C2×C3×C3⋊Dic3C2314yes—
63C2×C3×C3⋊Dic3C2312yes—
64C3×C4×C3⋊D3—24no—
65#6295—24no—
66#6294—24no—
67#6294—24no—
68#6294—22no—
69D3×#123—92no—
70C2×#886—91no—
71C23×C3⋊D3C2×D311yescharacteristic
72C23×C3×D3—32no—
73C23×C3×D3—61no—
74C22×D32—32no—
75C22×D32—31no—
76C2×#638C2×D312yes—
77C22×C3⋊Dic3C2×D312yescomplemented
78C2×C4×C3⋊D3C2×D312yes—
79C2×C3×#67—32no—
80C2×C3×#67—32no—
81C2×C3×#67—62no—
82C22×C3×Dic3—32no—
83C22×C3×Dic3—32no—
84C22×C3×Dic3—62no—
85C2×C3×C4×D3—34no—
86C2×C3×C4×D3—62no—
87#3811—64no—
88D3×#67—68no—
89C2×#626—32no—
90C2×#626—32no—
91C2×#625—32no—
92C2×#624—32no—
93C2×#624—32no—
94C2×#624—31no—
95C2×#624—64no—
96C2×D3×Dic3—34no—
97C2×D3×Dic3—34no—
98C2×D3×Dic3—32no—
99C2×D3×Dic3—68no—
100C32×C22⋊C4—21no—
101C32×C22⋊C4—22no—
102C32×C22⋊C4—41no—
103C32×C22⋊C4C2×D312yescomplemented
104C32×C22⋊C4C2×D311yescharacteristic, complemented
105#3757C2×D311yescharacteristic, complemented
106#3752C2×D312yescomplemented
107#3752C2×D314yescomplemented
108C4⋊(C3⋊Dic3)C2×D314yescomplemented
109C3×#217—32no—
110C3×#217—61no—
111C3×(C2×D3)⋊C4—34no—
112C3×(C2×D3)⋊C4—34no—
113C3×(C2×D3)⋊C4—34no—
114C3×(C2×D3)⋊C4—62no—
115C3×(C2×D3)⋊C4—64no—
116C3×C4⋊Dic3—34no—
117C3×C4⋊Dic3—34no—
118C3×C4⋊Dic3—64no—
119#3724—34no—
120Dic3⋊Dic3—34no—
121Dic3⋊Dic3—34no—
122#3722—34no—
123(C2×D3)⋊Dic3—34no—
124(C2×D3)⋊Dic3—34no—
125C22×C33—22no—
126C22×C33C2×D412yes—
127C22×C33C2411yescharacteristic
128C2×C3×C3⋊D3—22no—
129C2×C3×C3⋊D3—28no—
130C2×C3×C3⋊D3—22no—
131C2×C3×C3⋊D3—24no—
132C2×C3×C3⋊D3C2×D412yes—
133C2×C3×C3⋊D3C2×D412yes—
134C2×C3×C3⋊D3#4114yes—
135#1113—44no—
136C33×C4—22no—
137C3×C3⋊Dic3—24no—
138C3×C3⋊Dic3—22no—
139C3×C3⋊Dic3—22no—
140C22×#67—91no—
141C23×Dic3—91no—
142C2×D3×D4—92no—
143C22×C4×D3—92no—
144C3×#123—92no—
145C2×C3×C22⋊C4—91no—
146#935—92no—
147#929—92no—
148C2×(C2×D3)⋊C4—92no—
149C2×C4⋊Dic3—92no—
150#894—94no—
151D3×C4⋊C4—94no—
152#886—32no—
153#886—32no—
154#886—62no—
155#886—94no—
156#886—92no—
157#883—94no—
158D3×C22⋊C4—32no—
159D3×C22⋊C4—61no—
160D3×C22⋊C4—94no—
161C23×C32—21no—
162C23×C32—22no—
163C23×C32—41no—
164C23×C32C22×D311yescharacteristic
165C23×C32C22×D312yes—
166C22×C3⋊D3—24no—
167C22×C3⋊D3D1214yes—
168C22×C3⋊D3C22×D312yes—
169C22×C3⋊D3C22×D311yescharacteristic
170C22×C3⋊D3C22×D311yescharacteristic
171C22×C3×D3—32no—
172C22×C3×D3—38no—
173C22×C3×D3—32no—
174C22×C3×D3—32no—
175C22×C3×D3—32no—
176C22×C3×D3—61no—
177C22×C3×D3—61no—
178C22×C3×D3—64no—
179C22×C3×D3—62no—
180C22×C3×D3—68no—
181C22×C3×D3—124no—
182C2×D32—68no—
183C2×D32—64no—
184C2×D32—62no—
185C2×D32—64no—
186C2×C32×C4—24no—
187C2×C32×C4—22no—
188C2×C32×C4—42no—
189C2×C32×C4C22×D312yes—
190C2×C32×C4C22×D314yes—
191#638—24no—
192C2×C3⋊Dic3—24no—
193C2×C3⋊Dic3C22×D312yes—
194C2×C3⋊Dic3C22×D314yes—
195C2×C3⋊Dic3C22×D312yes—
196C4×C3⋊D3—24no—
197C3×#67—64no—
198C3×#67—64no—
199C3×#67—124no—
200C2×C3×Dic3—34no—
201C2×C3×Dic3—34no—
202C2×C3×Dic3—34no—
203C2×C3×Dic3—32no—
204C2×C3×Dic3—32no—
205C2×C3×Dic3—62no—
206C2×C3×Dic3—64no—
207C2×C3×Dic3—62no—
208C2×C3×Dic3—64no—
209C2×C3×Dic3—64no—
210C2×C3×Dic3—124no—
211C3×C4×D3—68no—
212C3×C4×D3—124no—
213#626—64no—
214#626—64no—
215#625—64no—
216#624—64no—
217#624—64no—
218#624—62no—
219D3×Dic3—68no—
220D3×Dic3—68no—
221D3×Dic3—64no—
222C2×#123—271noSylow
223C2×C33—22no—
224C2×C33C22×D411yescharacteristic
225C2×C33C2×#4112yes—
226C3×C3⋊D3—22no—
227C3×C3⋊D3—42no—
228C3×C3⋊D3#12314yescomplemented
229C24×C3—91no—
230C23×D3—32no—
231C23×D3—61no—
232C23×D3—92no—
233C23×D3—91no—
234C2×C3×D4—92no—
235C22×C3×C4—92no—
236C22×C3×C4—92no—
237C2×#67—32no—
238C2×#67—32no—
239C2×#67—62no—
240C2×#67—92no—
241C2×#67—92no—
242C2×#67—184no—
243C22×Dic3—32no—
244C22×Dic3—32no—
245C22×Dic3—62no—
246C22×Dic3—92no—
247C22×Dic3—91no—
248C22×Dic3—92no—
249C22×Dic3—184no—
250D3×D4—188no—
251C2×D12—92no—
252C2×C4×D3—34no—
253C2×C4×D3—62no—
254C2×C4×D3—94no—
255C2×C4×D3—94no—
256C2×C4×D3—92no—
257C2×C4×D3—188no—
258C3×C4⋊C4—94no—
259C3×C22⋊C4—21no—
260C3×C22⋊C4—22no—
261C3×C22⋊C4—41no—
262C3×C22⋊C4—91no—
263C3×C22⋊C4—94no—
264C3×C22⋊C4—92no—
265C3×C22⋊C4D3212yescomplemented
266C3×C22⋊C4D3211yescharacteristic, complemented
267#217—32no—
268#217—61no—
269#217—94no—
270(C2×D3)⋊C4—34no—
271(C2×D3)⋊C4—34no—
272(C2×D3)⋊C4—34no—
273(C2×D3)⋊C4—62no—
274(C2×D3)⋊C4—64no—
275(C2×D3)⋊C4—94no—
276(C2×D3)⋊C4—94no—
277C4⋊Dic3—34no—
278C4⋊Dic3—34no—
279C4⋊Dic3—64no—
280C4⋊Dic3—94no—
281Dic3⋊C4—94no—
282Dic3⋊C4—94no—
283C22×C32—22no—
284C22×C32—24no—
285C22×C32—22no—
286C22×C32—21no—
287C22×C32—24no—
288C22×C32—22no—
289C22×C32—41no—
290C22×C32—42no—
291C22×C32—42no—
292C22×C32—44no—
293C22×C32—82no—
294C22×C32C2×D1212yes—
295C22×C32D3×D414yes—
296C22×C32C23×D311yescharacteristic
297C22×C32C23×D312yes—
298C2×C3⋊D3—22no—
299C2×C3⋊D3—28no—
300C2×C3⋊D3—22no—
301C2×C3⋊D3—24no—
302C2×C3⋊D3C2×D1212yes—
303C2×C3⋊D3C2×D1212yes—
304C2×C3⋊D3#23714yes—
305C2×C3×D3—34no—
306C2×C3×D3—38no—
307C2×C3×D3—34no—
308C2×C3×D3—62no—
309C2×C3×D3—64no—
310C2×C3×D3—62no—
311C2×C3×D3—616no—
312C2×C3×D3—64no—
313C2×C3×D3—62no—
314C2×C3×D3—62no—
315C2×C3×D3—64no—
316C2×C3×D3—64no—
317C2×C3×D3—128no—
318C2×C3×D3—122no—
319C2×C3×D3—122no—
320C2×C3×D3—124no—
321D32—128no—
322D32—124no—
323C32×C4—22no—
324C32×C4—24no—
325C32×C4—42no—
326C32×C4—44no—
327C32×C4—82no—
328C3⋊Dic3—24no—
329C3⋊Dic3—22no—
330C3⋊Dic3—22no—
331C3×Dic3—64no—
332C3×Dic3—62no—
333C3×Dic3—64no—
334C3×Dic3—62no—
335C3×Dic3—64no—
336C3×Dic3—122no—
337C3×Dic3—124no—
338C3×Dic3—122no—
339C22×D4—271no—
340C23×C4—271no—
341#123—92no—
342#123—274no—
343#123—272no—
344C2×C4⋊C4—272no—
345C2×C22⋊C4—91no—
346C2×C22⋊C4—272no—
347C33C2×#12311yescharacteristic, complemented, Sylow
348C23×C3—21no—
349C23×C3—22no—
350C23×C3—41no—
351C23×C3—91no—
352C23×C3—91no—
353C23×C3—94no—
354C23×C3—92no—
355C23×C3—184no—
356C23×C3C2×D3212yes—
357C23×C3C2×D3211yescharacteristic
358C22×D3—32no—
359C22×D3—32no—
360C22×D3—38no—
361C22×D3—32no—
362C22×D3—32no—
363C22×D3—61no—
364C22×D3—64no—
365C22×D3—61no—
366C22×D3—62no—
367C22×D3—68no—
368C22×D3—124no—
369C22×D3—184no—
370C22×D3—188no—
371C22×D3—182no—
372C22×D3—184no—
373C3×D4—184no—
374C2×C3×C4—24no—
375C2×C3×C4—22no—
376C2×C3×C4—42no—
377C2×C3×C4—92no—
378C2×C3×C4—94no—
379C2×C3×C4—92no—
380C2×C3×C4—184no—
381C2×C3×C4—184no—
382C2×C3×C4C2×D3212yes—
383C2×C3×C4C2×D3214yes—
384#67—64no—
385#67—64no—
386#67—124no—
387#67—184no—
388#67—184no—
389C2×Dic3—32no—
390C2×Dic3—34no—
391C2×Dic3—34no—
392C2×Dic3—32no—
393C2×Dic3—34no—
394C2×Dic3—62no—
395C2×Dic3—64no—
396C2×Dic3—62no—
397C2×Dic3—64no—
398C2×Dic3—64no—
399C2×Dic3—124no—
400C2×Dic3—184no—
401C2×Dic3—184no—
402C2×Dic3—182no—
403D12—184no—
404C4×D3—68no—
405C4×D3—124no—
406C4×D3—188no—
407C4×D3—188no—
408C4×D3—184no—
409C2×C32—24no—
410C2×C32—22no—
411C2×C32—22no—
412C2×C32—21no—
413C2×C32—24no—
414C2×C32—22no—
415C2×C32—41no—
416C2×C32—42no—
417C2×C32—42no—
418C2×C32—44no—
419C2×C32—82no—
420C2×C32C22×D1211yescharacteristic
421C2×C32C2×D3×D412yes—
422C2×C32C2×#23712yes—
423C2×C32D3×#4114yes—
424C3⋊D3—22no—
425C3⋊D3—42no—
426C3⋊D3#88614yescomplemented
427C3×D3—38no—
428C3×D3—64no—
429C3×D3—64no—
430C3×D3—122no—
431C3×D3—122no—
432C3×D3—122no—
433C3×D3—242no—
434C24—91no—
435C24—271no—
436C2×D4—92no—
437C2×D4—272no—
438C2×D4—544no—
439C22×C4—92no—
440C22×C4—92no—
441C22×C4—272no—
442C22×C4—271no—
443C22×C4—272no—
444C22×C4—544no—
445C4⋊C4—94no—
446C4⋊C4—274no—
447C22⋊C4—91no—
448C22⋊C4—92no—
449C22⋊C4—94no—
450C22⋊C4—274no—
451C22⋊C4#111311yescharacteristic, complemented
452C22×C3—24no—
453C22×C3—22no—
454C22×C3—22no—
455C22×C3—21no—
456C22×C3—24no—
457C22×C3—22no—
458C22×C3—42no—
459C22×C3—44no—
460C22×C3—41no—
461C22×C3—42no—
462C22×C3—82no—
463C22×C3—92no—
464C22×C3—94no—
465C22×C3—92no—
466C22×C3—188no—
467C22×C3—182no—
468C22×C3—182no—
469C22×C3—184no—
470C22×C3D3×D1214yes—
471C22×C3#380212yes—
472C22×C3C22×D3212yes—
473C22×C3C22×D3211yescharacteristic
474C2×D3—34no—
475C2×D3—38no—
476C2×D3—34no—
477C2×D3—64no—
478C2×D3—62no—
479C2×D3—62no—
480C2×D3—616no—
481C2×D3—64no—
482C2×D3—62no—
483C2×D3—62no—
484C2×D3—64no—
485C2×D3—64no—
486C2×D3—128no—
487C2×D3—122no—
488C2×D3—122no—
489C2×D3—124no—
490C2×D3—368no—
491C2×D3—364no—
492C3×C4—24no—
493C3×C4—22no—
494C3×C4—42no—
495C3×C4—44no—
496C3×C4—82no—
497C3×C4—184no—
498C3×C4—182no—
499C3×C4—182no—
500Dic3—64no—
501Dic3—62no—
502Dic3—64no—
503Dic3—62no—
504Dic3—64no—
505Dic3—122no—
506Dic3—124no—
507Dic3—122no—
508C32—22no—
509C32—21no—
510C32—41no—
511C32C2×#88611yescharacteristic, complemented
512C32D3×#12312yescomplemented
513C23—91no—
514C23—91no—
515C23—94no—
516C23—92no—
517C23—184no—
518C23—542no—
519C23—544no—
520C23C2×#111311yescharacteristic
521D4—184no—
522D4—544no—
523C2×C4—92no—
524C2×C4—94no—
525C2×C4—92no—
526C2×C4—184no—
527C2×C4—184no—
528C2×C4—544no—
529C2×C4—544no—
530C2×C4—542no—
531C2×C4C2×#111312yes—
532C2×C3—24no—
533C2×C3—22no—
534C2×C3—22no—
535C2×C3—21no—
536C2×C3—22no—
537C2×C3—24no—
538C2×C3—42no—
539C2×C3—44no—
540C2×C3—41no—
541C2×C3—42no—
542C2×C3—82no—
543C2×C3—94no—
544C2×C3—182no—
545C2×C3—362no—
546C2×C3C2×D3×D1212yes—
547C2×C3C2×#380211yescharacteristic
548C2×C3D3×#23714yes—
549C2×C3#6431212yes—
550D3—38no—
551D3—64no—
552D3—64no—
553D3—122no—
554D3—122no—
555D3—122no—
556D3—242no—
557C22—22no—
558C22—92no—
559C22—94no—
560C22—92no—
561C22—188no—
562C22—182no—
563C22—182no—
564C22—184no—
565C22—1084no—
566C22C4⋊#111312yes—
567C22C22×#111311yescharacteristic
568C4—22no—
569C4—184no—
570C4—182no—
571C4—182no—
572C3—21no—
573C3—22no—
574C3—41no—
575C3D3×#88612yescomplemented
576C3#1059429511yescharacteristic, complemented
577C2—22no—
578C2—94no—
579C2—182no—
580C2—362no—
581C2C2×C4⋊#111311yescharacteristic
582C2#1172462212yes—
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC33×C22⋊C4C22
prime radicalC2×#89049C2
hypercenterC22⋊C4#1113
hypocenterC33C2×#123
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C22×#6294C2semidirect
3. C3×#64095C2semidirect
4. C3×C22⋊C4×C3⋊D3C2semidirect
5. #11723720C2semidirect
6. #11723700C2semidirect
7. #11723690C2semidirect
8. #11723689C2semidirect
14. C22×C3×C3⋊Dic3C22semidirect
21. C2×#6294C22semidirect
22. C33×C22⋊C4C22semidirect
23. C3×#3757C22semidirect
24. C3×#3752C22semidirect
25. C3×#3752C22semidirect
26. C3×C4⋊(C3⋊Dic3)C22semidirect
27. #88815C22semidirect
28. #88814C22semidirect
33. #64095D3semidirect
34. C22⋊C4×C3⋊D3D3semidirect
77. C22×C3⋊Dic3C2×D3semidirect
103. C32×C22⋊C4C2×D3semidirect
104. C32×C22⋊C4C2×D3semidirect
105. #3757C2×D3semidirect
106. #3752C2×D3semidirect
107. #3752C2×D3semidirect
108. C4⋊(C3⋊Dic3)C2×D3semidirect
228. C3×C3⋊D3#123semidirect
265. C3×C22⋊C4D32semidirect
266. C3×C22⋊C4D32semidirect
347. C33C2×#123semidirect
426. C3⋊D3#886semidirect
451. C22⋊C4#1113semidirect
511. C32C2×#886semidirect
512. C32D3×#123semidirect
575. C3D3×#886semidirect
576. C3#10594295semidirect

Maximal subgroups

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

subgroupindex
C2×#890492
C22×#62942
C3×#640952
C3×C22⋊C4×C3⋊D32
#117237202
#117237002
#117236902
#117236892
#105942953
D3×#8863