Finite Group Databasebeta

← #13178149

Subgroups — #13178149

Order 1728.

1566 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89042C212yesmaximal, complemented
2C3×C4×C4⋊(C3⋊D3)C212yesmaximal, complemented
3C3×C42×C3⋊D3C211yesmaximal, characteristic, complemented
4C4⋊#6296C211yesmaximal, characteristic, complemented
5#11723697C211yesmaximal, characteristic, complemented
6#11723696C212yesmaximal, complemented
7#11723695C212yesmaximal, complemented
8C4×#6295C212yesmaximal, complemented
9#11723672C212yesmaximal, complemented
10#10593972—31nomaximal
11#10593968—32nomaximal
12C2×C3×C4⋊(C3⋊D3)C2212yes—
13C2×C3×C4×C3⋊D3C2211yescharacteristic
14C2×C3×C4×C3⋊D3C2211yescharacteristic
15C2×C3×C4×C3⋊D3C2214yes—
16C2×C3×C4×C3⋊D3C2211yescharacteristic
17C2×#6296C2212yes—
18C2×#6295C2212yes—
19C2×#6295C2214yes—
20C2×#6295C2212yes—
21#89042—28no—
22C33×C42C2211yescharacteristic
23C3×#3752C2214yes—
24C3×C4⋊(C3⋊Dic3)C2212yes—
25C3×C4×C3⋊Dic3C2211yescharacteristic
26C3×C4×C3⋊Dic3C2211yescharacteristic
27C3×C4×C3⋊Dic3C2211yescharacteristic
28#88815C2212yes—
29#88814C2214yes—
30C2×#3798—34no—
31C2×#3798—32no—
32C4×C4⋊(C3⋊D3)D312yescomplemented
33C42×C3⋊D3D311yescharacteristic, complemented
34C3×C4×D12—32no—
35C3×C4×D12—32no—
36C3×C4×D12—62no—
37C3×C42×D3—32no—
38C3×C42×D3—61no—
39#63919—31no—
40#63918—32no—
41#63916—32no—
42#63915—32no—
43#63914—31no—
44#63912—32no—
45#63911—32no—
46#63909—32no—
47#63908—32no—
48#63906—34no—
49C4×#626—34no—
50C4×#625—32no—
51#63852—34no—
52C22×C3×C3⋊D3C2314yes—
53C22×C3×C3⋊D3C2311yescharacteristic
54C2×C33×C4C2311yescharacteristic
55C2×C33×C4C2311yescharacteristic
56C2×C33×C4C2311yescharacteristic
57C2×C3×C3⋊Dic3C2311yescharacteristic
58C2×C3×C3⋊Dic3C2314yes—
59C2×C3×C3⋊Dic3C2311yescharacteristic
60C2×C3×C3⋊Dic3C2311yescharacteristic
61C3×C4⋊(C3⋊D3)—24no—
62C3×C4×C3⋊D3—28no—
63C3×C4×C3⋊D3—22no—
64C3×C4×C3⋊D3—22no—
65C3×C4×C3⋊D3D414yescomplemented
66#6296—24no—
67#6295—28no—
68#6295—24no—
69#6295—24no—
70#5548—92no—
71#5477—91no—
72C2×C4⋊(C3⋊D3)C2×D312yes—
73C2×C4×C3⋊D3C2×D314yes—
74C2×C4×C3⋊D3C2×D311yescharacteristic
75C2×C4×C3⋊D3C2×D311yescharacteristic
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—34no—
82C2×C3×C4×D3—34no—
83C2×C3×C4×D3—32no—
84C2×C3×C4×D3—32no—
85C2×C3×C4×D3—61no—
86C2×C3×C4×D3—61no—
87C2×C3×C4×D3—61no—
88C2×C3×C4×D3—64no—
89C2×#627—34no—
90C2×#627—32no—
91C2×#626—34no—
92C2×#626—34no—
93C2×#626—34no—
94C2×#626—34no—
95C2×#625—34no—
96C2×#625—32no—
97C2×#625—32no—
98#3798—616no—
99#3798—68no—
100C32×C42—21no—
101C32×C42—22no—
102C32×C42—41no—
103C32×C42C2×D311yescharacteristic
104C32×C42C2×D312yes—
105#3752C2×D314yes—
106C4⋊(C3⋊Dic3)C2×D312yes—
107C4×C3⋊Dic3C2×D311yescharacteristic
108C4×C3⋊Dic3C2×D311yescharacteristic
109C4×C3⋊Dic3C2×D311yescharacteristic
110C3×(C2×D3)⋊C4—34no—
111C3×(C2×D3)⋊C4—34no—
112C3×(C2×D3)⋊C4—64no—
113C3×C4⋊Dic3—32no—
114C3×C4⋊Dic3—32no—
115C3×C4⋊Dic3—62no—
116C3×C4×Dic3—32no—
117C3×C4×Dic3—32no—
118C3×C4×Dic3—32no—
119C3×C4×Dic3—61no—
120C3×C4×Dic3—61no—
121C3×C4×Dic3—61no—
122#3724—32no—
123Dic3⋊Dic3—34no—
124#3722—34no—
125(C2×D3)⋊Dic3—34no—
126(C2×D3)⋊Dic3—34no—
127C22×C33C2411yescharacteristic
128C2×C3×C3⋊D3—21no—
129C2×C3×C3⋊D3—21no—
130C2×C3×C3⋊D3—24no—
131C2×C3×C3⋊D3—24no—
132C2×C3×C3⋊D3—24no—
133C2×C3×C3⋊D3C2×D412yes—
134C33×C4—21no—
135C33×C4C2×D412yes—
136C33×C4#4112yescomplemented
137C3×C3⋊Dic3—24no—
138C3×C3⋊Dic3—21no—
139C3×C3⋊Dic3C2×D412yes—
140C3×C3⋊Dic3#4112yescomplemented
141C2×#235—92no—
142C2×#235—94no—
143C3×C4×D4—92no—
144C2×C3×C42—91no—
145#940—92no—
146#938—92no—
147#930—92no—
148#930—92no—
149C4×#67—94no—
150#895—92no—
151#888—92no—
152#885—94no—
153#884—94no—
154#875—92no—
155#874—91no—
156C4×D12—32no—
157C4×D12—32no—
158C4×D12—62no—
159C4×D12—92no—
160C42×D3—32no—
161C42×D3—61no—
162#869—91no—
163C22×C3⋊D3C22×D314yes—
164C22×C3⋊D3C22×D311yescharacteristic
165C22×C3×D3—32no—
166C22×C3×D3—34no—
167C22×C3×D3—34no—
168C22×C3×D3—61no—
169C22×C3×D3—64no—
170C2×C32×C4—21no—
171C2×C32×C4—21no—
172C2×C32×C4—21no—
173C2×C32×C4—22no—
174C2×C32×C4—22no—
175C2×C32×C4—22no—
176C2×C32×C4—41no—
177C2×C32×C4—41no—
178C2×C32×C4—41no—
179C2×C32×C4C22×D312yes—
180C2×C32×C4C22×D311yescharacteristic
181C2×C32×C4C22×D311yescharacteristic
182C2×C32×C4C22×D312yes—
183C2×C32×C4C22×D311yescharacteristic
184C2×C32×C4C22×D312yes—
185C2×C3⋊Dic3C22×D311yescharacteristic
186C2×C3⋊Dic3C22×D314yes—
187C2×C3⋊Dic3C22×D311yescharacteristic
188C2×C3⋊Dic3C22×D311yescharacteristic
189C4⋊(C3⋊D3)—24no—
190C4×C3⋊D3—22no—
191C4×C3⋊D3—22no—
192C4×C3⋊D3—28no—
193C4×C3⋊D3D1214yescomplemented
194C2×C3×Dic3—32no—
195C2×C3×Dic3—32no—
196C2×C3×Dic3—32no—
197C2×C3×Dic3—34no—
198C2×C3×Dic3—34no—
199C2×C3×Dic3—61no—
200C2×C3×Dic3—61no—
201C2×C3×Dic3—61no—
202C2×C3×Dic3—64no—
203C3×D12—68no—
204C3×D12—64no—
205C3×D12—64no—
206C3×C4×D3—38no—
207C3×C4×D3—64no—
208C3×C4×D3—64no—
209C3×C4×D3—64no—
210C3×C4×D3—64no—
211C3×C4×D3—68no—
212C3×C4×D3—64no—
213C3×C4×D3—68no—
214C3×C4×D3—128no—
215#627—68no—
216#627—64no—
217#626—68no—
218#626—68no—
219#626—68no—
220#626—68no—
221#625—68no—
222#625—64no—
223#625—64no—
224#532—271noSylow
225C2×C33C22×D411yescharacteristic
226C2×C33C2×#4111yescharacteristic
227C2×C33C2×#4111yescharacteristic
228C3×C3⋊D3—22no—
229C3×C3⋊D3—44no—
230C2×C3×D4—92no—
231C22×C3×C4—91no—
232C22×C3×C4—94no—
233C22×C3×C4—91no—
234C22×C3×C4—91no—
235C2×#67—94no—
236C2×#67—94no—
237C2×#67—94no—
238C2×#67—94no—
239#235—1816no—
240#235—188no—
241C2×D12—32no—
242C2×D12—32no—
243C2×D12—62no—
244C2×D12—92no—
245C2×D12—92no—
246C2×D12—94no—
247C2×C4×D3—32no—
248C2×C4×D3—32no—
249C2×C4×D3—32no—
250C2×C4×D3—34no—
251C2×C4×D3—34no—
252C2×C4×D3—61no—
253C2×C4×D3—61no—
254C2×C4×D3—61no—
255C2×C4×D3—64no—
256C2×Dic6—92no—
257C2×Dic6—94no—
258C3×C4⋊C4—92no—
259C3×C22⋊C4—94no—
260C3×C42—21no—
261C3×C42—22no—
262C3×C42—41no—
263C3×C42—91no—
264C3×C42—91no—
265C3×C42—91no—
266C3×C42D3212yes—
267C3×C42D3211yescharacteristic
268#217—94no—
269(C2×D3)⋊C4—34no—
270(C2×D3)⋊C4—34no—
271(C2×D3)⋊C4—64no—
272(C2×D3)⋊C4—94no—
273(C2×D3)⋊C4—94no—
274C4⋊Dic3—32no—
275C4⋊Dic3—32no—
276C4⋊Dic3—62no—
277C4⋊Dic3—92no—
278Dic3⋊C4—94no—
279C4×Dic3—32no—
280C4×Dic3—32no—
281C4×Dic3—32no—
282C4×Dic3—61no—
283C4×Dic3—61no—
284C4×Dic3—61no—
285C22×C32—21no—
286C22×C32—22no—
287C22×C32—41no—
288C22×C32C23×D311yescharacteristic
289C22×C32C23×D312yes—
290C2×C3⋊D3—21no—
291C2×C3⋊D3—21no—
292C2×C3⋊D3—24no—
293C2×C3⋊D3—24no—
294C2×C3⋊D3—24no—
295C2×C3⋊D3C2×D1212yes—
296C2×C3×D3—34no—
297C2×C3×D3—62no—
298C2×C3×D3—62no—
299C2×C3×D3—62no—
300C2×C3×D3—62no—
301C2×C3×D3—62no—
302C2×C3×D3—68no—
303C2×C3×D3—64no—
304C2×C3×D3—64no—
305C2×C3×D3—64no—
306C2×C3×D3—64no—
307C2×C3×D3—64no—
308C2×C3×D3—64no—
309C2×C3×D3—124no—
310C2×C3×D3—124no—
311C32×C4—21no—
312C32×C4—22no—
313C32×C4—22no—
314C32×C4—22no—
315C32×C4—24no—
316C32×C4—24no—
317C32×C4—22no—
318C32×C4—42no—
319C32×C4—42no—
320C32×C4—42no—
321C32×C4—42no—
322C32×C4C2×D1212yes—
323C32×C4#23514yescomplemented
324C32×C4#23512yescomplemented
325C32×C4D3×D414yes—
326C3⋊Dic3—21no—
327C3⋊Dic3—24no—
328C3⋊Dic3C2×D1212yes—
329C3⋊Dic3#23512yescomplemented
330C3×Dic3—34no—
331C3×Dic3—34no—
332C3×Dic3—62no—
333C3×Dic3—62no—
334C3×Dic3—62no—
335C3×Dic3—64no—
336C3×Dic3—62no—
337C3×Dic3—64no—
338C3×Dic3—124no—
339C2×#41—272no—
340C4⋊Q8—271no—
341#127—271no—
342#124—272no—
343#121—272no—
344#121—272no—
345C4×D4—92no—
346C4×D4—272no—
347C2×C42—91no—
348C33#53211yescharacteristic, complemented, Sylow
349C23×C3—91no—
350C23×C3—94no—
351C22×D3—32no—
352C22×D3—34no—
353C22×D3—34no—
354C22×D3—61no—
355C22×D3—64no—
356C3×D4—184no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—21no—
362C2×C3×C4—22no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—91no—
367C2×C3×C4—91no—
368C2×C3×C4—94no—
369C2×C3×C4—94no—
370C2×C3×C4—91no—
371C2×C3×C4—182no—
372C2×C3×C4—182no—
373C2×C3×C4—188no—
374C2×C3×C4C2×D3212yes—
375C2×C3×C4C2×D3211yescharacteristic
376C2×C3×C4C2×D3212yes—
377C2×C3×C4C2×D3211yescharacteristic
378C2×C3×C4C2×D3212yes—
379C2×C3×C4C2×D3211yescharacteristic
380#67—188no—
381#67—188no—
382#67—188no—
383#67—188no—
384C2×Dic3—32no—
385C2×Dic3—32no—
386C2×Dic3—32no—
387C2×Dic3—34no—
388C2×Dic3—34no—
389C2×Dic3—61no—
390C2×Dic3—61no—
391C2×Dic3—61no—
392C2×Dic3—64no—
393D12—68no—
394D12—64no—
395D12—64no—
396D12—184no—
397D12—184no—
398D12—188no—
399C4×D3—38no—
400C4×D3—64no—
401C4×D3—64no—
402C4×D3—64no—
403C4×D3—64no—
404C4×D3—64no—
405C4×D3—68no—
406C4×D3—68no—
407C4×D3—128no—
408Dic6—188no—
409Dic6—184no—
410C2×C32—21no—
411C2×C32—22no—
412C2×C32—22no—
413C2×C32—21no—
414C2×C32—22no—
415C2×C32—21no—
416C2×C32—41no—
417C2×C32—41no—
418C2×C32—41no—
419C2×C32C22×D1211yescharacteristic
420C2×C32C2×#23511yescharacteristic
421C2×C32C2×#23512yes—
422C2×C32C2×#23511yescharacteristic
423C2×C32C2×D3×D412yes—
424C2×C32D3×#4112yes—
425C3⋊D3—22no—
426C3⋊D3—44no—
427C3×D3—64no—
428C3×D3—64no—
429C3×D3—128no—
430C3×D3—124no—
431C3×D3—124no—
432#41—548no—
433C2×Q8—272no—
434C2×D4—92no—
435C2×D4—272no—
436C2×D4—274no—
437C2×D4—272no—
438C22×C4—91no—
439C22×C4—91no—
440C22×C4—91no—
441C22×C4—94no—
442C4⋊C4—92no—
443C4⋊C4—272no—
444C22⋊C4—94no—
445C22⋊C4—274no—
446C42—91no—
447C42—91no—
448C42—91no—
449C42#111311yescharacteristic
450C22×C3—21no—
451C22×C3—22no—
452C22×C3—41no—
453C22×C3—92no—
454C22×C3—181no—
455C22×C3—181no—
456C22×C3—184no—
457C22×C3—184no—
458C22×C3—184no—
459C22×C3C22×D3212yes—
460C22×C3C22×D3211yescharacteristic
461C2×D3—34no—
462C2×D3—62no—
463C2×D3—62no—
464C2×D3—62no—
465C2×D3—62no—
466C2×D3—62no—
467C2×D3—68no—
468C2×D3—64no—
469C2×D3—64no—
470C2×D3—64no—
471C2×D3—64no—
472C2×D3—64no—
473C2×D3—64no—
474C2×D3—124no—
475C2×D3—124no—
476C3×C4—22no—
477C3×C4—22no—
478C3×C4—21no—
479C3×C4—22no—
480C3×C4—24no—
481C3×C4—22no—
482C3×C4—24no—
483C3×C4—42no—
484C3×C4—42no—
485C3×C4—42no—
486C3×C4—42no—
487C3×C4—92no—
488C3×C4—92no—
489C3×C4—181no—
490C3×C4—184no—
491C3×C4#379814yescomplemented
492C3×C4#379812yescomplemented
493C3×C4D3×D1214yes—
494C3×C4#380212yes—
495Dic3—34no—
496Dic3—34no—
497Dic3—62no—
498Dic3—62no—
499Dic3—62no—
500Dic3—62no—
501Dic3—64no—
502Dic3—64no—
503Dic3—124no—
504C32—22no—
505C32—21no—
506C32—41no—
507C32#547711yescharacteristic, complemented
508C32#554812yescomplemented
509C23—91no—
510C23—94no—
511Q8—544no—
512D4—184no—
513D4—544no—
514D4—544no—
515D4—548no—
516C2×C4—91no—
517C2×C4—91no—
518C2×C4—94no—
519C2×C4—91no—
520C2×C4—94no—
521C2×C4—182no—
522C2×C4—182no—
523C2×C4—188no—
524C2×C4C2×#111311yescharacteristic
525C2×C4C2×#111311yescharacteristic
526C2×C4C2×#111311yescharacteristic
527C2×C3—21no—
528C2×C3—22no—
529C2×C3—21no—
530C2×C3—22no—
531C2×C3—21no—
532C2×C3—22no—
533C2×C3—41no—
534C2×C3—41no—
535C2×C3—41no—
536C2×C3—182no—
537C2×C3—364no—
538C2×C3C2×#379812yes—
539C2×C3C2×#379811yescharacteristic
540C2×C3C2×D3×D1212yes—
541C2×C3C2×#380211yescharacteristic
542C2×C3D3×#23512yes—
543C2×C3#6430611yescharacteristic
544D3—64no—
545D3—64no—
546D3—128no—
547D3—124no—
548D3—124no—
549C22—92no—
550C22—181no—
551C22—181no—
552C22—184no—
553C22—184no—
554C22—184no—
555C22C22×#111311yescharacteristic
556C4—21no—
557C4—92no—
558C4—92no—
559C4—181no—
560C4—184no—
561C4#8904212yescomplemented
562C4C4⋊#111312yes—
563C3—21no—
564C3—22no—
565C3—41no—
566C3#1059396812yescomplemented
567C3#1059397211yescharacteristic, complemented
568C2—182no—
569C2—364no—
570C2C2×#8904211yescharacteristic
571C2C2×C4⋊#111311yescharacteristic
572C2#1172461811yescharacteristic
subgroupG/H
centerC2×C4C2×#1113
commutator, socle, cosocleC22×C33C24
FrattiniC22C22×#1113
FittingC33×C42C22
hypercenterC42#1113
hypocenterC33#532
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C2×#89042C2semidirect
2. C3×C4×C4⋊(C3⋊D3)C2semidirect
3. C3×C42×C3⋊D3C2semidirect
4. C4⋊#6296C2semidirect
5. #11723697C2semidirect
6. #11723696C2semidirect
7. #11723695C2semidirect
8. C4×#6295C2semidirect
9. #11723672C2semidirect
32. C4×C4⋊(C3⋊D3)D3semidirect
33. C42×C3⋊D3D3semidirect
65. C3×C4×C3⋊D3D4semidirect
136. C33×C4#41semidirect
140. C3×C3⋊Dic3#41semidirect
193. C4×C3⋊D3D12semidirect
323. C32×C4#235semidirect
324. C32×C4#235semidirect
329. C3⋊Dic3#235semidirect
348. C33#532semidirect
491. C3×C4#3798semidirect
492. C3×C4#3798semidirect
507. C32#5477semidirect
508. C32#5548semidirect
561. C4#89042semidirect
566. C3#10593968semidirect
567. C3#10593972semidirect

Maximal subgroups

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

subgroupindex
C2×#890422
C3×C4×C4⋊(C3⋊D3)2
C3×C42×C3⋊D32
C4⋊#62962
#117236972
#117236962
#117236952
C4×#62952
#117236722
#105939723
#105939683