Finite Group Databasebeta

← #13168278

Subgroups — #13168278

Order 1728.

1436 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3796C212yesmaximal, complemented
2C4×#6295C211yesmaximal, characteristic, complemented
3D12×C3⋊Dic3C211yesmaximal, characteristic, complemented
4#11723435C212yesmaximal, complemented
5C3×#63873C211yesmaximal, characteristic, complemented
6#11722986C214yesmaximal, complemented
7#11722955C212yesmaximal, complemented
8D3×#3724C212yesmaximal, complemented
9#10595358—32nomaximal
10#10595252—31nomaximal
11C2×C3×C4×C3⋊D3C2212yes—
12C2×C32×D12C2211yescharacteristic
13C2×C32×Dic6C2212yes—
14C2×#6295C2211yescharacteristic
15C2×D3×C3⋊Dic3C2212yes—
16C2×C3×#626C2214yes—
17C2×C3×D3×Dic3C2214yes—
18C3×#3796—28no—
19C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
20C3×C4×C3⋊Dic3C2211yescharacteristic
21#88815C2211yescharacteristic
22#88814C2212yes—
23#88806C2214yes—
24#88803C2214yes—
25(C2×D3)⋊(C3⋊Dic3)C2212yes—
26C3×#3724C2212yes—
27C3×#3724C2212yes—
28C2×C3×#239—32no—
29C2×C3×#237—32no—
30C2×#3796D312yescomplemented
31#64155—32no—
32D4×C3⋊Dic3—31no—
33C3×#944—32no—
34#63980—32no—
35C2×#3724—32no—
36#63961—34no—
37C4×#626—32no—
38C4×#625—31no—
39Dic3×D12—32no—
40Dic3×D12—61no—
41#63886—34no—
42#63884—34no—
43D3×Dic3⋊C4—34no—
44#63873D311yescharacteristic, complemented
45#63856—32no—
46#63856—32no—
47#63856—62no—
48#63850—34no—
49C22×C3×C3⋊D3C2312yes—
50C22×C32×D3C2312yes—
51C2×C33×C4C2311yescharacteristic
52C2×C3⋊(C3⋊Dic3)C2312yes—
53C2×C3×C3⋊Dic3C2311yescharacteristic
54C2×C3×C3⋊Dic3C2312yes—
55C2×C3×C3⋊Dic3C2311yescharacteristic
56C3×C4×C3⋊D3—24no—
57C2×C32×Dic3C2314yes—
58C32×D12D414yescomplemented
59C32×Dic6—24no—
60#6295—22no—
61D3×C3⋊Dic3—24no—
62C3×#626—28no—
63C3×D3×Dic3—28no—
64#5824—92no—
65#5726—91no—
66C2×C32×Q8—32no—
67C2×C32×D4—31no—
68C22×C3⋊Dic3—32no—
69C2×C4×C3⋊D3C2×D312yes—
70C2×C3×#67—34no—
71C22×C3×Dic3—34no—
72C3×#239—68no—
73C3×#237—68no—
74C2×C3×D12—21no—
75C2×C3×D12—34no—
76C2×C3×D12C2×D312yes—
77C2×C3×C4×D3—34no—
78C2×C3×C4×D3—32no—
79C2×C3×C4×D3—32no—
80C2×C3×C4×D3—62no—
81C2×C3×Dic6—22no—
82C2×C3×Dic6C2×D312yes—
83C2×C3×Dic6C2×D312yes—
84C2×#626—32no—
85C2×#626C2×D314yes—
86C2×#625—31no—
87C2×D3×Dic3—34no—
88C2×D3×Dic3—62no—
89C2×D3×Dic3C2×D314yes—
90#3796—28no—
91#3757—32no—
92#3752—34no—
93C4⋊(C3⋊Dic3)—31no—
94C4⋊(C3⋊Dic3)—32no—
95C4⋊(C3⋊Dic3)—62no—
96C4⋊(C3⋊Dic3)—61no—
97C4⋊(C3⋊Dic3)—121no—
98#3750—34no—
99C4×C3⋊Dic3C2×D311yescharacteristic
100C3×Dic3⋊C4—34no—
101C3×Dic3⋊C4—34no—
102C3×C4×Dic3—32no—
103C3×C4×Dic3—61no—
104#3724—31no—
105#3724—34no—
106#3724—34no—
107#3724—64no—
108#3724C2×D312yes—
109#3724C2×D312yes—
110Dic3⋊Dic3—32no—
111#3722—32no—
112(C2×D3)⋊Dic3—34no—
113(C2×D3)⋊Dic3—32no—
114(C2×D3)⋊Dic3—32no—
115(C2×D3)⋊Dic3—34no—
116(C2×D3)⋊Dic3—34no—
117(C2×D3)⋊Dic3—64no—
118(C2×D3)⋊Dic3—62no—
119C22×C33C2411yescharacteristic
120C2×C3×C3⋊D3—22no—
121C2×C3×C3⋊D3—22no—
122C2×C3×C3⋊D3—22no—
123C2×C32×D3—22no—
124C2×C32×D3—22no—
125C2×C32×D3C2×D414yes—
126C33×C4C2×D412yes—
127C3⋊(C3⋊Dic3)—22no—
128C3×C3⋊Dic3—22no—
129C3×C3⋊Dic3—21no—
130C3×C3⋊Dic3#4112yescomplemented
131C32×Dic3—24no—
132C2×C3×#41—92no—
133C2×#239—32no—
134C2×#237—32no—
135C3×C4⋊Q8—91no—
136#946—92no—
137#946—92no—
138#946—182no—
139#944—32no—
140#939—94no—
141#935—94no—
142D4×Dic3—92no—
143D4×Dic3—181no—
144#929—94no—
145C4×#67—92no—
146#924—92no—
147C2×Dic3⋊C4—94no—
148#895—92no—
149#894—94no—
150D3×C4⋊C4—92no—
151C4×D12—91no—
152C4×D12—91no—
153C23×C32—32no—
154C22×C3⋊D3C22×D312yes—
155C22×C3×D3—22no—
156C22×C3×D3—32no—
157C22×C3×D3—32no—
158C22×C3×D3—62no—
159C22×C3×D3C22×D314yes—
160C32×Q8—64no—
161C32×D4—34no—
162C2×C32×C4—21no—
163C2×C32×C4—22no—
164C2×C32×C4—34no—
165C2×C32×C4—41no—
166C2×C32×C4C22×D311yescharacteristic
167C2×C32×C4C22×D312yes—
168C2×C3⋊Dic3—32no—
169C2×C3⋊Dic3—34no—
170C2×C3⋊Dic3—64no—
171C2×C3⋊Dic3—62no—
172C2×C3⋊Dic3—64no—
173C2×C3⋊Dic3—122no—
174C2×C3⋊Dic3C22×D311yescharacteristic
175C2×C3⋊Dic3C22×D312yes—
176C2×C3⋊Dic3C22×D311yescharacteristic
177C4×C3⋊D3—24no—
178C3×#67—68no—
179C2×C3×Dic3—24no—
180C2×C3×Dic3—32no—
181C2×C3×Dic3—32no—
182C2×C3×Dic3—32no—
183C2×C3×Dic3—32no—
184C2×C3×Dic3—61no—
185C2×C3×Dic3—61no—
186C2×C3×Dic3—62no—
187C2×C3×Dic3—68no—
188C2×C3×Dic3C22×D314yes—
189C2×C3×Dic3C22×D314yes—
190C3×D12—24no—
191C3×D12—68no—
192C3×D12#6718yescomplemented
193C3×C4×D3—68no—
194C3×C4×D3—64no—
195C3×C4×D3—64no—
196C3×C4×D3—124no—
197C3×Dic6—24no—
198C3×Dic6—24no—
199C3×Dic6—44no—
200#626—28no—
201#626—64no—
202#625—62no—
203D3×Dic3—28no—
204D3×Dic3—68no—
205D3×Dic3—124no—
206#547—271noSylow
207C2×C33C22×D411yescharacteristic
208C2×C33C2×#4111yescharacteristic
209C2×C332−1+411yescharacteristic
210C3×C3⋊D3—42no—
211C32×D3—24no—
212C3×#41—188no—
213C2×C3×Q8—32no—
214C2×C3×Q8—32no—
215C2×C3×Q8—62no—
216C2×C3×D4—32no—
217C2×C3×D4—61no—
218C2×C3×D4—94no—
219C22×C3×C4—94no—
220C22×C3×C4—92no—
221C2×#67—34no—
222C2×#67—92no—
223C22×Dic3—34no—
224C22×Dic3—94no—
225C22×Dic3—182no—
226#239—68no—
227#237—68no—
228C2×D12—34no—
229C2×D12—91no—
230C2×D12D3211yescharacteristic
231C2×C4×D3—34no—
232C2×C4×D3—32no—
233C2×C4×D3—32no—
234C2×C4×D3—62no—
235C2×C4×D3—92no—
236C2×Dic6D3212yes—
237C3×C4⋊C4—92no—
238C3×C4⋊C4—92no—
239C3×C42—91no—
240#217—92no—
241#217—94no—
242#217—94no—
243#217—182no—
244(C2×D3)⋊C4—92no—
245(C2×D3)⋊C4—92no—
246(C2×D3)⋊C4—94no—
247(C2×D3)⋊C4—94no—
248(C2×D3)⋊C4—92no—
249(C2×D3)⋊C4—184no—
250C4⋊Dic3—91no—
251C4⋊Dic3—91no—
252C4⋊Dic3—92no—
253C4⋊Dic3—181no—
254C4⋊Dic3—182no—
255C4⋊Dic3—361no—
256Dic3⋊C4—34no—
257Dic3⋊C4—34no—
258Dic3⋊C4—92no—
259Dic3⋊C4—94no—
260Dic3⋊C4—94no—
261Dic3⋊C4—94no—
262Dic3⋊C4—184no—
263C4×Dic3—32no—
264C4×Dic3—61no—
265C22×C32—22no—
266C22×C32—21no—
267C22×C32—34no—
268C22×C32—41no—
269C22×C32—62no—
270C22×C32—62no—
271C22×C32C23×D311yescharacteristic
272C22×C32C23×D312yes—
273C2×C3⋊D3—22no—
274C2×C3⋊D3—22no—
275C2×C3⋊D3—22no—
276C2×C3×D3—24no—
277C2×C3×D3—24no—
278C2×C3×D3—24no—
279C2×C3×D3—42no—
280C2×C3×D3—42no—
281C2×C3×D3—64no—
282C2×C3×D3—62no—
283C2×C3×D3—62no—
284C2×C3×D3—62no—
285C2×C3×D3—62no—
286C2×C3×D3—62no—
287C2×C3×D3—62no—
288C2×C3×D3—122no—
289C2×C3×D3—122no—
290C2×C3×D3C2×#6718yes—
291C32×C4—22no—
292C32×C4—24no—
293C32×C4—42no—
294C32×C4—64no—
295C32×C4D3×D412yes—
296C32×C4C2×#6714yes—
297C3⋊Dic3—21no—
298C3⋊Dic3—22no—
299C3⋊Dic3—62no—
300C3⋊Dic3—64no—
301C3⋊Dic3—122no—
302C3⋊Dic3—124no—
303C3⋊Dic3—242no—
304C3⋊Dic3#23912yescomplemented
305C3×Dic3—24no—
306C3×Dic3—24no—
307C3×Dic3—34no—
308C3×Dic3—44no—
309C3×Dic3—62no—
310C3×Dic3—62no—
311C3×Dic3—62no—
312C3×Dic3—62no—
313C3×Dic3—122no—
314C3×Dic3—121no—
315C2×#41—92no—
316C4⋊Q8—91no—
317#123—274no—
318C22⋊Q8—272no—
319#121—272no—
320C4×D4—271no—
321C4×D4—271no—
322C2×C4⋊C4—272no—
323C33#54711yescharacteristic, complemented, Sylow
324C23×C3—34no—
325C23×C3—62no—
326C23×C3—92no—
327C22×D3—32no—
328C22×D3—32no—
329C22×D3—62no—
330C22×D3C2×D3212yes—
331C3×Q8—64no—
332C3×Q8—64no—
333C3×Q8—124no—
334C3×D4—38no—
335C3×D4—64no—
336C3×D4—188no—
337C2×C3×C4—21no—
338C2×C3×C4—22no—
339C2×C3×C4—34no—
340C2×C3×C4—34no—
341C2×C3×C4—41no—
342C2×C3×C4—64no—
343C2×C3×C4—91no—
344C2×C3×C4—92no—
345C2×C3×C4—91no—
346C2×C3×C4—188no—
347C2×C3×C4—184no—
348C2×C3×C4C2×D3211yescharacteristic
349C2×C3×C4C2×D3212yes—
350#67—68no—
351#67—184no—
352C2×Dic3—32no—
353C2×Dic3—32no—
354C2×Dic3—32no—
355C2×Dic3—32no—
356C2×Dic3—61no—
357C2×Dic3—61no—
358C2×Dic3—62no—
359C2×Dic3—68no—
360C2×Dic3—92no—
361C2×Dic3—94no—
362C2×Dic3—184no—
363C2×Dic3—182no—
364C2×Dic3—188no—
365C2×Dic3—362no—
366C2×Dic3—364no—
367C2×Dic3C2×D3214yes—
368D12—68no—
369D12—182no—
370D12#62514yescomplemented
371C4×D3—68no—
372C4×D3—64no—
373C4×D3—64no—
374C4×D3—124no—
375C4×D3—184no—
376Dic6—24no—
377C2×C32—21no—
378C2×C32—21no—
379C2×C32—22no—
380C2×C32—22no—
381C2×C32—21no—
382C2×C32—22no—
383C2×C32—41no—
384C2×C32—41no—
385C2×C32—41no—
386C2×C32—64no—
387C2×C32C2×D3×D411yescharacteristic
388C2×C32C2×#23911yescharacteristic
389C2×C32#100711yescharacteristic
390C2×C32D3×#4112yes—
391C2×C32#101012yes—
392C2×C32C22×#6712yes—
393C3⋊D3—42no—
394C3×D3—28no—
395C3×D3—44no—
396C3×D3—124no—
397C3×D3—122no—
398C3×D3—122no—
399#41—188no—
400C2×Q8—32no—
401C2×D4—31no—
402C2×D4—94no—
403C2×D4—271no—
404C22×C4—94no—
405C22×C4—92no—
406C22×C4—272no—
407C4⋊C4—92no—
408C4⋊C4—92no—
409C4⋊C4—274no—
410C4⋊C4—271no—
411C4⋊C4—271no—
412C22⋊C4—274no—
413C22⋊C4—272no—
414C22⋊C4—272no—
415C42—91no—
416C22×C3—21no—
417C22×C3—22no—
418C22×C3—38no—
419C22×C3—41no—
420C22×C3—64no—
421C22×C3—64no—
422C22×C3—64no—
423C22×C3—122no—
424C22×C3—122no—
425C22×C3—182no—
426C22×C3—182no—
427C22×C3—182no—
428C22×C3C22×D3211yescharacteristic
429C22×C3C22×D3212yes—
430C2×D3—22no—
431C2×D3—22no—
432C2×D3—64no—
433C2×D3—62no—
434C2×D3—62no—
435C2×D3—62no—
436C2×D3—62no—
437C2×D3—62no—
438C2×D3—62no—
439C2×D3—122no—
440C2×D3—122no—
441C2×D3C2×#62514yes—
442C3×C4—24no—
443C3×C4—22no—
444C3×C4—42no—
445C3×C4—64no—
446C3×C4—64no—
447C3×C4—92no—
448C3×C4—124no—
449C3×C4—181no—
450C3×C4—182no—
451C3×C4C2×#62512yes—
452C3×C4D3×#6714yes—
453Dic3—24no—
454Dic3—34no—
455Dic3—62no—
456Dic3—62no—
457Dic3—62no—
458Dic3—62no—
459Dic3—122no—
460Dic3—121no—
461Dic3—182no—
462Dic3—184no—
463Dic3—362no—
464Dic3—364no—
465Dic3—722no—
466C32—22no—
467C32—21no—
468C32—41no—
469C32#572611yescharacteristic, complemented
470C32#582412yescomplemented
471C23—32no—
472C23—92no—
473Q8—64no—
474D4—34no—
475D4—188no—
476D4—542no—
477C2×C4—34no—
478C2×C4—91no—
479C2×C4—91no—
480C2×C4—92no—
481C2×C4—188no—
482C2×C4—184no—
483C2×C4—272no—
484C2×C4—544no—
485C2×C4D3311yescharacteristic
486C2×C3—21no—
487C2×C3—22no—
488C2×C3—21no—
489C2×C3—22no—
490C2×C3—22no—
491C2×C3—21no—
492C2×C3—41no—
493C2×C3—41no—
494C2×C3—41no—
495C2×C3—68no—
496C2×C3—124no—
497C2×C3—362no—
498C2×C3#6430911yescharacteristic
499C2×C3#6431211yescharacteristic
500C2×C3#6431512yes—
501C2×C3D3×#23912yes—
502C2×C3C22×#62511yescharacteristic
503C2×C3C2×D3×#6712yes—
504D3—24no—
505D3—124no—
506D3—122no—
507D3—122no—
508C22—34no—
509C22—62no—
510C22—62no—
511C22—182no—
512C22—182no—
513C22—182no—
514C22C2×D3311yescharacteristic
515C4—64no—
516C4—92no—
517C4—181no—
518C4—182no—
519C4—542no—
520C4D3×#62512yes—
521C3—22no—
522C3—21no—
523C3—41no—
524C3#1059525211yescharacteristic, complemented
525C3#1059535812yescomplemented
526C2—64no—
527C2—362no—
528C2#1172437711yescharacteristic
529C2#1172439411yescharacteristic
530C2C2×D3×#62511yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutator, socle, cosocleC22×C33C24
FittingC2×C33×C4C23
hypercenterC2×C4D33
hypocenterC33#547
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3796C2semidirect
2. C4×#6295C2semidirect
3. D12×C3⋊Dic3C2semidirect
4. #11723435C2semidirect
5. C3×#63873C2semidirect
6. #11722986C2semidirect
7. #11722955C2semidirect
8. D3×#3724C2semidirect
30. C2×#3796D3semidirect
44. #63873D3semidirect
58. C32×D12D4semidirect
130. C3×C3⋊Dic3#41semidirect
192. C3×D12#67semidirect
304. C3⋊Dic3#239semidirect
323. C33#547semidirect
370. D12#625semidirect
469. C32#5726semidirect
470. C32#5824semidirect
524. C3#10595252semidirect
525. C3#10595358semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37962
C4×#62952
D12×C3⋊Dic32
#117234352
C3×#638732
#117229862
#117229552
D3×#37242
#105953583
#105952523