Finite Group Databasebeta

← #13170751

Subgroups — #13170751

Order 1728.

2010 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊#1113C211yesmaximal, characteristic
2C2×#88949C212yesmaximal, complemented
3C3⋊D3×C4⋊Dic3C211yesmaximal, characteristic, complemented
4Dic3×C4⋊(C3⋊D3)C212yesmaximal, complemented
5C3×#63894C212yesmaximal, complemented
6C3×C4×#624C211yesmaximal, characteristic, complemented
7#11722934C214yesmaximal, complemented
8#11722905C212yesmaximal, complemented
9#10594605—31nomaximal
10C4×D3×D12—32nomaximal
11C22×#1113C2212yes—
12C2×C3×C4⋊(C3⋊D3)C2212yes—
13C2×C3×C4×C3⋊D3C2211yescharacteristic
14C2×#6295C2212yes—
15C4⋊#1113C418yescomplemented
16C2×Dic3×C3⋊D3C2212yes—
17C2×Dic3×C3⋊D3C2214yescomplemented
18C2×C3×#624C2211yescharacteristic
19C2×C3×#624C2211yescharacteristic
20C2×C3×#624C2214yescomplemented
21#88949—28no—
22C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic, complemented
23C32×C4⋊Dic3C2211yescharacteristic, complemented
24C32×C4×Dic3C2212yescomplemented
25#88807C2212yescomplemented
26#88803C2214yescomplemented
27C3×#3722C2214yescomplemented
28C3×Dic32C2212yescomplemented
29C2×#3802—31no—
30C2×D3×D12—32no—
31C2×C4×D32—32no—
32C2×C4×D32—34no—
33C4⋊C4×C3⋊D3—31no—
34C4×C4⋊(C3⋊D3)—32no—
35C3×#893—32no—
36C3×C4×D12—32no—
37C3×C42×D3—32no—
38#63921—32no—
39D3×(C2×D3)⋊C4—34no—
40C4×#626—34no—
41#63894D312yescomplemented
42Dic3×D12—32no—
43Dic3×D12—32no—
44Dic3×D12—62no—
45D3×C4⋊Dic3—32no—
46D3×C4⋊Dic3—61no—
47C4×#624D311yescharacteristic, complemented
48#63880—34no—
49#63879—34no—
50C22×C3×C3⋊D3C2314yes—
51C22×C3×C3⋊D3C2311yescharacteristic
52C2×#1113—24no—
53C2×#1113—24no—
54C2×#1113C2×C418yescomplemented
55C2×C33×C4C2311yescharacteristic
56C2×C3⋊(C3⋊Dic3)C2312yes—
57C2×C3×C3⋊Dic3C2311yescharacteristic
58C3×C4⋊(C3⋊D3)C2×C418yescomplemented
59C3×C4×C3⋊D3C2×C414yescomplemented
60C2×C32×Dic3C2312yes—
61C2×C32×Dic3C2312yes—
62C2×C32×Dic3C2312yes—
63#6295C2×C418yescomplemented
64Dic3×C3⋊D3—24no—
65Dic3×C3⋊D3—28no—
66C3×#624—22no—
67C3×#624—28no—
68C3×#624D414yescomplemented
69C4×D3×D4—92no—
70C2×C4×D12—91no—
71C22×D32—34no—
72C22×D32—32no—
73C2×C4⋊(C3⋊D3)C2×D312yes—
74C2×C4×C3⋊D3—34no—
75C2×C4×C3⋊D3—32no—
76C2×C4×C3⋊D3C2×D311yescharacteristic
77C2×C3×D12—32no—
78C2×C3×D12—32no—
79C2×C3×D12—62no—
80C2×C3×C4×D3—32no—
81C2×C3×C4×D3—32no—
82C2×C3×C4×D3—34no—
83C2×C3×C4×D3—34no—
84C2×C3×C4×D3—32no—
85C2×C3×C4×D3—61no—
86C2×#626—34no—
87C2×#625—32no—
88C2×#624C2×D311yescharacteristic
89C2×#624C2×D311yescharacteristic
90C2×#624C2×D314yescomplemented
91C2×D3×Dic3—34no—
92C2×D3×Dic3—34no—
93C2×D3×Dic3—34no—
94C2×D3×Dic3—62no—
95C2×D3×Dic3—64no—
96#3802—38no—
97D3×D12—316no—
98C4×D32—616no—
99C4×D32—68no—
100C32×C4⋊C4—31no—
101C32×C42—32no—
102#3752—34no—
103C4⋊(C3⋊Dic3)—32no—
104C4⋊(C3⋊Dic3)—31no—
105C4⋊(C3⋊Dic3)—61no—
106C4⋊(C3⋊Dic3)—62no—
107C4⋊(C3⋊Dic3)—121no—
108#3750—32no—
109C3×(C2×D3)⋊C4—34no—
110C3×(C2×D3)⋊C4—34no—
111C3×C4⋊Dic3—21no—
112C3×C4⋊Dic3C2×D312yescomplemented
113C3×C4×Dic3—22no—
114C3×C4×Dic3—32no—
115C3×C4×Dic3—32no—
116C3×C4×Dic3C2×D312yescomplemented
117C3×C4×Dic3C2×D312yescomplemented
118Dic3⋊Dic3—34no—
119Dic3⋊Dic3—62no—
120#3722C2×D314yescomplemented
121(C2×D3)⋊Dic3—34no—
122(C2×D3)⋊Dic3—34no—
123(C2×D3)⋊Dic3—64no—
124Dic32C2×D312yescomplemented
125C22×C33C2411yescharacteristic
126C2×C3×C3⋊D3—24no—
127C2×C3×C3⋊D3—24no—
128C2×C3×C3⋊D3C22×C412yes—
129C2×C3×C3⋊D3C22×C418yescomplemented
130C2×C3×C3⋊D3C2×D412yes—
131C2×C3×C3⋊D3#4112yes—
132#1113—216no—
133C33×C4C22×C412yescomplemented
134C3⋊(C3⋊Dic3)—22no—
135C3×C3⋊Dic3C22×C412yescomplemented
136C32×Dic3—22no—
137C32×Dic3—22no—
138C32×Dic3C2×D414yescomplemented
139C2×D3×D4—92no—
140C22×D12—91no—
141C22×C4×D3—92no—
142C22×C4×D3—94no—
143C3×C4×D4—92no—
144C2×C3×C42—91no—
145D4×Dic3—92no—
146C4×#67—94no—
147C2×(C2×D3)⋊C4—92no—
148C2×C4⋊Dic3—91no—
149#893—32no—
150D3×C4⋊C4—92no—
151D3×C4⋊C4—181no—
152#67⋊C4—94no—
153D3×C22⋊C4—94no—
154C4×D12—32no—
155C4×D12—94no—
156C4×D12—92no—
157C4×D12—92no—
158C4×D12—182no—
159C42×D3—32no—
160C22×C3⋊D3C22×D314yes—
161C22×C3⋊D3C22×D311yescharacteristic
162C22×C3×D3—32no—
163C22×C3×D3—34no—
164C22×C3×D3—34no—
165C22×C3×D3—61no—
166C22×C3×D3—64no—
167C2×D32—38no—
168C2×D32—316no—
169C2×D32—68no—
170C2×D32—68no—
171C2×D32—64no—
172C2×D32—64no—
173C2×C32×C4—22no—
174C2×C32×C4—21no—
175C2×C32×C4—32no—
176C2×C32×C4—32no—
177C2×C32×C4—32no—
178C2×C32×C4—41no—
179C2×C32×C4C22×D311yescharacteristic
180C2×C32×C4C22×D312yes—
181C2×C3⋊Dic3—32no—
182C2×C3⋊Dic3—34no—
183C2×C3⋊Dic3—64no—
184C2×C3⋊Dic3—62no—
185C2×C3⋊Dic3—122no—
186C2×C3⋊Dic3C22×D311yescharacteristic
187C4⋊(C3⋊D3)C4×D318yescomplemented
188C4×C3⋊D3—68no—
189C4×C3⋊D3—64no—
190C4×C3⋊D3C4×D314yescomplemented
191C2×C3×Dic3—22no—
192C2×C3×Dic3—22no—
193C2×C3×Dic3—22no—
194C2×C3×Dic3—32no—
195C2×C3×Dic3—61no—
196C2×C3×Dic3C22×D312yes—
197C2×C3×Dic3C22×D312yes—
198C2×C3×Dic3C22×D314yes—
199C2×C3×Dic3C22×D312yes—
200C2×C3×Dic3C22×D312yes—
201C3×D12—38no—
202C3×D12—38no—
203C3×D12—68no—
204C3×C4×D3—38no—
205C3×C4×D3—38no—
206C3×C4×D3—64no—
207C3×C4×D3—68no—
208C3×C4×D3—64no—
209C3×C4×D3—68no—
210#626—316no—
211#625—38no—
212#624—22no—
213#624—28no—
214#624D1214yescomplemented
215D3×Dic3—68no—
216D3×Dic3—68no—
217D3×Dic3—68no—
218D3×Dic3—128no—
219D3×Dic3—124no—
220C2×C4×D4—271noSylow
221C2×C33C23×C411yescharacteristic
222C2×C33C22×D411yescharacteristic
223C2×C33C2×#4111yescharacteristic
224C3×C3⋊D3—28no—
225C3×C3⋊D3C4×D414yescomplemented
226C23×D3—94no—
227C23×D3—92no—
228C2×C3×D4—92no—
229C22×C3×C4—91no—
230C22×C3×C4—91no—
231C22×C3×C4—94no—
232C22×C3×C4—91no—
233C2×#67—94no—
234C22×Dic3—94no—
235C22×Dic3—92no—
236D3×D4—916no—
237C2×D12—32no—
238C2×D12—32no—
239C2×D12—62no—
240C2×D12—98no—
241C2×D12—92no—
242C2×C4×D3—32no—
243C2×C4×D3—32no—
244C2×C4×D3—32no—
245C2×C4×D3—34no—
246C2×C4×D3—34no—
247C2×C4×D3—61no—
248C2×C4×D3—94no—
249C2×C4×D3—94no—
250C2×C4×D3—94no—
251C2×C4×D3—184no—
252C2×C4×D3—182no—
253C2×C4×D3—1816no—
254C2×C4×D3—188no—
255C3×C4⋊C4—32no—
256C3×C4⋊C4—61no—
257C3×C22⋊C4—94no—
258C3×C42—32no—
259C3×C42—32no—
260C3×C42—62no—
261C3×C42—92no—
262#217—94no—
263(C2×D3)⋊C4—34no—
264(C2×D3)⋊C4—34no—
265(C2×D3)⋊C4—94no—
266(C2×D3)⋊C4—94no—
267(C2×D3)⋊C4—184no—
268C4⋊Dic3—91no—
269C4⋊Dic3—92no—
270C4⋊Dic3—92no—
271C4⋊Dic3—182no—
272C4⋊Dic3—181no—
273C4⋊Dic3—361no—
274C4⋊Dic3D3211yescharacteristic, complemented
275Dic3⋊C4—94no—
276Dic3⋊C4—182no—
277C4×Dic3—32no—
278C4×Dic3—32no—
279C4×Dic3D3212yescomplemented
280C22×C32—21no—
281C22×C32—22no—
282C22×C32—41no—
283C22×C32C23×D311yescharacteristic
284C22×C32C23×D312yes—
285C2×C3⋊D3—24no—
286C2×C3⋊D3—24no—
287C2×C3⋊D3C2×C4×D312yes—
288C2×C3⋊D3C2×C4×D318yescomplemented
289C2×C3⋊D3C2×D1212yes—
290C2×C3⋊D3#23512yes—
291C2×C3×D3—34no—
292C2×C3×D3—34no—
293C2×C3×D3—34no—
294C2×C3×D3—38no—
295C2×C3×D3—38no—
296C2×C3×D3—62no—
297C2×C3×D3—62no—
298C2×C3×D3—62no—
299C2×C3×D3—68no—
300C2×C3×D3—64no—
301C2×C3×D3—64no—
302C2×C3×D3—64no—
303C2×C3×D3—64no—
304C2×C3×D3—124no—
305C2×C3×D3—124no—
306D32—632no—
307D32—616no—
308C32×C4—22no—
309C32×C4—24no—
310C32×C4—34no—
311C32×C4—42no—
312C32×C4—62no—
313C32×C4—62no—
314C32×C4C2×C4×D314yescomplemented
315C32×C4C2×C4×D312yescomplemented
316C3⋊Dic3—62no—
317C3⋊Dic3—64no—
318C3⋊Dic3—124no—
319C3⋊Dic3—122no—
320C3⋊Dic3—242no—
321C3⋊Dic3C2×C4×D312yescomplemented
322C3×Dic3—22no—
323C3×Dic3—24no—
324C3×Dic3—22no—
325C3×Dic3—24no—
326C3×Dic3—34no—
327C3×Dic3—42no—
328C3×Dic3—42no—
329C3×Dic3—62no—
330C3×Dic3C2×D1214yescomplemented
331C3×Dic3D3×D414yescomplemented
332C22×D4—271no—
333C23×C4—272no—
334C4×D4—92no—
335C4×D4—272no—
336C4×D4—274no—
337C2×C4⋊C4—271no—
338C2×C22⋊C4—272no—
339C2×C42—91no—
340C33C2×C4×D411yescharacteristic, complemented, Sylow
341C23×C3—91no—
342C23×C3—94no—
343C22×D3—32no—
344C22×D3—34no—
345C22×D3—34no—
346C22×D3—61no—
347C22×D3—64no—
348C22×D3—98no—
349C22×D3—916no—
350C22×D3—188no—
351C22×D3—188no—
352C22×D3—184no—
353C22×D3—184no—
354C3×D4—98no—
355C2×C3×C4—22no—
356C2×C3×C4—21no—
357C2×C3×C4—32no—
358C2×C3×C4—32no—
359C2×C3×C4—32no—
360C2×C3×C4—34no—
361C2×C3×C4—32no—
362C2×C3×C4—41no—
363C2×C3×C4—62no—
364C2×C3×C4—62no—
365C2×C3×C4—62no—
366C2×C3×C4—94no—
367C2×C3×C4—91no—
368C2×C3×C4—94no—
369C2×C3×C4—188no—
370C2×C3×C4—182no—
371C2×C3×C4C2×D3212yes—
372C2×C3×C4C2×D3211yescharacteristic
373#67—916no—
374C2×Dic3—32no—
375C2×Dic3—61no—
376C2×Dic3—94no—
377C2×Dic3—92no—
378C2×Dic3—182no—
379C2×Dic3—184no—
380C2×Dic3—184no—
381C2×Dic3—188no—
382C2×Dic3—362no—
383C2×Dic3C2×D3212yes—
384C2×Dic3C2×D3212yes—
385C2×Dic3C2×D3212yes—
386D12—38no—
387D12—38no—
388D12—68no—
389D12—98no—
390C4×D3—38no—
391C4×D3—38no—
392C4×D3—64no—
393C4×D3—68no—
394C4×D3—64no—
395C4×D3—68no—
396C4×D3—188no—
397C4×D3—188no—
398C4×D3—188no—
399C4×D3—368no—
400C4×D3—364no—
401C2×C32—21no—
402C2×C32—22no—
403C2×C32—22no—
404C2×C32—22no—
405C2×C32—21no—
406C2×C32—21no—
407C2×C32—41no—
408C2×C32—41no—
409C2×C32—41no—
410C2×C32C22×C4×D311yescharacteristic
411C2×C32C22×C4×D312yes—
412C2×C32C22×D1211yescharacteristic
413C2×C32C2×#23511yescharacteristic
414C2×C32C2×D3×D412yes—
415C2×C32D3×#4112yes—
416C3⋊D3—28no—
417C3⋊D3C4×D1214yescomplemented
418C3×D3—38no—
419C3×D3—64no—
420C3×D3—68no—
421C3×D3—68no—
422C3×D3—128no—
423C24—272no—
424C2×D4—92no—
425C2×D4—278no—
426C2×D4—272no—
427C22×C4—91no—
428C22×C4—91no—
429C22×C4—94no—
430C22×C4—91no—
431C22×C4—272no—
432C22×C4—274no—
433C22×C4—548no—
434C4⋊C4—31no—
435C4⋊C4—272no—
436C4⋊C4—271no—
437C22⋊C4—94no—
438C22⋊C4—274no—
439C42—32no—
440C42—92no—
441C22×C3—21no—
442C22×C3—22no—
443C22×C3—41no—
444C22×C3—92no—
445C22×C3—92no—
446C22×C3—92no—
447C22×C3—98no—
448C22×C3—184no—
449C22×C3—184no—
450C22×C3C22×D3212yes—
451C22×C3C22×D3211yescharacteristic
452C2×D3—34no—
453C2×D3—34no—
454C2×D3—34no—
455C2×D3—38no—
456C2×D3—38no—
457C2×D3—62no—
458C2×D3—62no—
459C2×D3—62no—
460C2×D3—68no—
461C2×D3—64no—
462C2×D3—64no—
463C2×D3—64no—
464C2×D3—64no—
465C2×D3—124no—
466C2×D3—124no—
467C2×D3—1832no—
468C2×D3—1816no—
469C3×C4—22no—
470C3×C4—24no—
471C3×C4—34no—
472C3×C4—34no—
473C3×C4—42no—
474C3×C4—64no—
475C3×C4—64no—
476C3×C4—62no—
477C3×C4—62no—
478C3×C4—92no—
479C3×C4—122no—
480C3×C4—122no—
481C3×C4C4×D3214yescomplemented
482C3×C4C4×D3212yescomplemented
483Dic3—22no—
484Dic3—22no—
485Dic3—34no—
486Dic3—62no—
487Dic3—184no—
488Dic3—182no—
489Dic3—364no—
490Dic3—362no—
491Dic3—722no—
492Dic3D3×D1214yescomplemented
493C32—21no—
494C32—22no—
495C32—41no—
496C32C2×C4×D1211yescharacteristic, complemented
497C32C4×D3×D412yescomplemented
498C23—91no—
499C23—94no—
500C23—278no—
501C23—544no—
502C23—544no—
503D4—98no—
504D4—278no—
505C2×C4—32no—
506C2×C4—32no—
507C2×C4—32no—
508C2×C4—94no—
509C2×C4—91no—
510C2×C4—94no—
511C2×C4—188no—
512C2×C4—182no—
513C2×C4—272no—
514C2×C4—548no—
515C2×C4—544no—
516C2×C4D3311yescharacteristic
517C2×C3—22no—
518C2×C3—21no—
519C2×C3—22no—
520C2×C3—21no—
521C2×C3—21no—
522C2×C3—22no—
523C2×C3—41no—
524C2×C3—41no—
525C2×C3—41no—
526C2×C3—94no—
527C2×C3—188no—
528C2×C3C2×C4×D3211yescharacteristic
529C2×C3C2×C4×D3212yes—
530C2×C3C2×D3×D1212yes—
531C2×C3D3×#23512yes—
532C2×C3D32×D411yescharacteristic
533C2×C3#6431911yescharacteristic
534D3—38no—
535D3—64no—
536D3—68no—
537D3—68no—
538D3—128no—
539C22—92no—
540C22—92no—
541C22—92no—
542C22—98no—
543C22—184no—
544C22—184no—
545C22—5416no—
546C22C2×D3311yescharacteristic
547C4—34no—
548C4—62no—
549C4—62no—
550C4—92no—
551C4—542no—
552C4#8894912yescomplemented
553C3—21no—
554C3—22no—
555C3—41no—
556C3C4×D3×D1212yescomplemented
557C3#1059460511yescharacteristic, complemented
558C2—94no—
559C2—188no—
560C2#1172438911yescharacteristic
561C2D32×D1211yescharacteristic
562C2C2×#8894911yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutatorC2×C33C23×C4
FittingC2×C33×C4C23
socle, cosocleC22×C33C24
prime radicalC2×C4⋊#1113C2
hypercenterC2×C4D33
hypocenterC33C2×C4×D4
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×#88949C2semidirect
3. C3⋊D3×C4⋊Dic3C2semidirect
4. Dic3×C4⋊(C3⋊D3)C2semidirect
5. C3×#63894C2semidirect
6. C3×C4×#624C2semidirect
7. #11722934C2semidirect
8. #11722905C2semidirect
15. C4⋊#1113C4semidirect
17. C2×Dic3×C3⋊D3C22semidirect
20. C2×C3×#624C22semidirect
22. C4⋊(C3⋊(C3⋊Dic3))C22semidirect
23. C32×C4⋊Dic3C22semidirect
24. C32×C4×Dic3C22semidirect
25. #88807C22semidirect
26. #88803C22semidirect
27. C3×#3722C22semidirect
28. C3×Dic32C22semidirect
41. #63894D3semidirect
47. C4×#624D3semidirect
54. C2×#1113C2×C4semidirect
58. C3×C4⋊(C3⋊D3)C2×C4semidirect
59. C3×C4×C3⋊D3C2×C4semidirect
63. #6295C2×C4semidirect
68. C3×#624D4semidirect
90. C2×#624C2×D3semidirect
112. C3×C4⋊Dic3C2×D3semidirect
116. C3×C4×Dic3C2×D3semidirect
117. C3×C4×Dic3C2×D3semidirect
120. #3722C2×D3semidirect
124. Dic32C2×D3semidirect
129. C2×C3×C3⋊D3C22×C4semidirect
133. C33×C4C22×C4semidirect
135. C3×C3⋊Dic3C22×C4semidirect
138. C32×Dic3C2×D4semidirect
187. C4⋊(C3⋊D3)C4×D3semidirect
190. C4×C3⋊D3C4×D3semidirect
214. #624D12semidirect
225. C3×C3⋊D3C4×D4semidirect
274. C4⋊Dic3D32semidirect
279. C4×Dic3D32semidirect
288. C2×C3⋊D3C2×C4×D3semidirect
314. C32×C4C2×C4×D3semidirect
315. C32×C4C2×C4×D3semidirect
321. C3⋊Dic3C2×C4×D3semidirect
330. C3×Dic3C2×D12semidirect
331. C3×Dic3D3×D4semidirect
340. C33C2×C4×D4semidirect
417. C3⋊D3C4×D12semidirect
481. C3×C4C4×D32semidirect
482. C3×C4C4×D32semidirect
492. Dic3D3×D12semidirect
496. C32C2×C4×D12semidirect
497. C32C4×D3×D4semidirect
552. C4#88949semidirect
556. C3C4×D3×D12semidirect
557. C3#10594605semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊#11132
C2×#889492
C3⋊D3×C4⋊Dic32
Dic3×C4⋊(C3⋊D3)2
C3×#638942
C3×C4×#6242
#117229342
#117229052
#105946053
C4×D3×D123