Finite Group Databasebeta

← #13178736

Subgroups — #13178736

Order 1728.

1422 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×Q8×C3⋊D3C211yesmaximal, characteristic, complemented
2C2×#89042C211yesmaximal, characteristic
3C3×#64155C212yesmaximal, complemented
4#11723694C214yesmaximal, complemented
5#11723674C212yesmaximal, complemented
6#11723670C212yesmaximal, complemented
7#11723667C211yesmaximal, characteristic, complemented
8#11723666C212yesmaximal, complemented
9#10595405—31nomaximal
10#10595397—32nomaximal
11C2×C33×Q8C2211yescharacteristic
12C3×Q8×C3⋊D3—24no—
13C2×C3×C4×C3⋊D3C2212yes—
14C2×C3×C4×C3⋊D3C2212yes—
15C2×C3×C4×C3⋊D3C2211yescharacteristic
16C2×C3×#634C2212yes—
17C2×C3×#634C2211yescharacteristic
18C2×#6296C2211yescharacteristic
19C2×#6295C2211yescharacteristic
20C2×#6295C2212yes—
21#89042—24no—
22C3×#3752C2214yes—
23C3×C4⋊(C3⋊Dic3)C2212yes—
24C3×#3750C2214yes—
25#88815C2214yes—
26#88814C2212yes—
27#88814C2214yes—
28#88813C2212yes—
29C2×Q8×C3⋊D3D311yescharacteristic, complemented
30C2×C3×D3×Q8—32no—
31C2×C3×D3×Q8—61no—
32C2×#3798—32no—
33C2×#3798—31no—
34#64155D312yescomplemented
35C3×#946—32no—
36C3×#946—32no—
37C3×#946—62no—
38#63913—32no—
39#63907—34no—
40#63905—34no—
41#63871—32no—
42#63857—32no—
43#63856—32no—
44#63853—34no—
45#63850—34no—
46#63849—31no—
47#63841—32no—
48#63840—32no—
49#63839—34no—
50C22×C3×C3⋊D3C2312yes—
51C22×C3×C3⋊D3C2311yescharacteristic
52C33×Q8—22no—
53C2×C33×C4C2312yes—
54C2×C33×C4C2311yescharacteristic
55C2×C3×C3⋊Dic3C2312yes—
56C2×C3×C3⋊Dic3C2314yes—
57C2×C3×C3⋊Dic3C2312yes—
58C2×C3×C3⋊Dic3C2311yescharacteristic
59C3×C4×C3⋊D3—24no—
60C3×C4×C3⋊D3—24no—
61C3×C4×C3⋊D3D414yes—
62C3×#634—22no—
63C3×#634—24no—
64#6296—22no—
65#6295—24no—
66#6295—22no—
67#5816—91no—
68#5628—92no—
69C2×C32×Q8—21no—
70C2×C32×Q8—22no—
71C2×C32×Q8—41no—
72C2×C32×Q8C2×D311yescharacteristic
73C2×C32×Q8C2×D312yes—
74Q8×C3⋊D3—24no—
75C2×C4×C3⋊D3C2×D312yes—
76C2×C4×C3⋊D3C2×D312yes—
77C2×C4×C3⋊D3C2×D311yescharacteristic
78C2×#634C2×D312yes—
79C2×#634C2×D311yescharacteristic
80C3×D3×Q8—68no—
81C3×D3×Q8—68no—
82C2×C3×C4×D3—32no—
83C2×C3×C4×D3—32no—
84C2×C3×C4×D3—32no—
85C2×C3×C4×D3—34no—
86C2×C3×C4×D3—61no—
87C2×C3×C4×D3—62no—
88C2×C3×C4×D3—62no—
89C2×C3×Dic6—34no—
90C2×C3×Dic6—32no—
91C2×C3×Dic6—62no—
92C2×C3×Dic6—61no—
93C2×#627—32no—
94C2×#627—31no—
95C2×#626—32no—
96C2×#626—32no—
97C2×#626—32no—
98C2×#625—32no—
99C2×#625—31no—
100#3798—68no—
101#3798—64no—
102#3752C2×D314yes—
103C4⋊(C3⋊Dic3)C2×D312yes—
104#3750C2×D314yes—
105C3×(C2×D3)⋊C4—34no—
106C3×(C2×D3)⋊C4—34no—
107C3×(C2×D3)⋊C4—64no—
108C3×C4⋊Dic3—32no—
109C3×C4⋊Dic3—32no—
110C3×C4⋊Dic3—62no—
111C3×Dic3⋊C4—34no—
112C3×Dic3⋊C4—34no—
113C3×Dic3⋊C4—64no—
114#3724—34no—
115Dic3⋊Dic3—34no—
116Dic3⋊Dic3—34no—
117#3722—32no—
118#3722—34no—
119(C2×D3)⋊Dic3—34no—
120(C2×D3)⋊Dic3—34no—
121(C2×D3)⋊Dic3—34no—
122Dic32—34no—
123Dic32—32no—
124C22×C33C2411yescharacteristic
125C2×C3×C3⋊D3—22no—
126C2×C3×C3⋊D3—22no—
127C2×C3×C3⋊D3—22no—
128C2×C3×C3⋊D3—22no—
129C2×C3×C3⋊D3C2×D412yes—
130C33×C4—22no—
131C33×C4C2×D412yes—
132C3×C3⋊Dic3—22no—
133C3×C3⋊Dic3—22no—
134C3×C3⋊Dic3—24no—
135C3×C3⋊Dic3C2×D412yes—
136C22×C3×Q8—91no—
137C2×D3×Q8—32no—
138C2×D3×Q8—61no—
139C2×#235—91no—
140C2×#235—92no—
141C3×C22⋊Q8—92no—
142#947—92no—
143#946—32no—
144#946—32no—
145#946—62no—
146#946—92no—
147#944—92no—
148#929—94no—
149#926—91no—
150#924—92no—
151#897—92no—
152#894—94no—
153#892—92no—
154#888—94no—
155#885—94no—
156#883—94no—
157C22×C3⋊D3C22×D312yes—
158C22×C3⋊D3C22×D311yescharacteristic
159C22×C3×D3—32no—
160C22×C3×D3—32no—
161C22×C3×D3—32no—
162C22×C3×D3—61no—
163C22×C3×D3—62no—
164C32×Q8—22no—
165C32×Q8—24no—
166C32×Q8—24no—
167C32×Q8—44no—
168C32×Q8—44no—
169C2×C32×C4—22no—
170C2×C32×C4—22no—
171C2×C32×C4—21no—
172C2×C32×C4—24no—
173C2×C32×C4—41no—
174C2×C32×C4—42no—
175C2×C32×C4C22×D312yes—
176C2×C32×C4C22×D314yes—
177C2×C32×C4C22×D312yes—
178C2×C32×C4C22×D311yescharacteristic
179C2×C3⋊Dic3C22×D314yes—
180C2×C3⋊Dic3C22×D312yes—
181C2×C3⋊Dic3C22×D312yes—
182C2×C3⋊Dic3C22×D311yescharacteristic
183C4×C3⋊D3—24no—
184C4×C3⋊D3—24no—
185C4×C3⋊D3#6714yes—
186#634—22no—
187#634—24no—
188C2×C3×Dic3—32no—
189C2×C3×Dic3—34no—
190C2×C3×Dic3—34no—
191C2×C3×Dic3—34no—
192C2×C3×Dic3—32no—
193C2×C3×Dic3—32no—
194C2×C3×Dic3—61no—
195C2×C3×Dic3—62no—
196C2×C3×Dic3—62no—
197C2×C3×Dic3—64no—
198C3×C4×D3—38no—
199C3×C4×D3—64no—
200C3×C4×D3—68no—
201C3×C4×D3—68no—
202C3×C4×D3—64no—
203C3×C4×D3—64no—
204C3×C4×D3—124no—
205C3×Dic6—68no—
206C3×Dic6—68no—
207C3×Dic6—64no—
208C3×Dic6—64no—
209#627—64no—
210#627—62no—
211#626—64no—
212#626—64no—
213#626—64no—
214#625—64no—
215#625—62no—
216#536—271noSylow
217C2×C33C22×D411yescharacteristic
218C2×C332−1+412yes—
219C3×C3⋊D3—22no—
220C3×C3⋊D3—42no—
221C2×C3×Q8—21no—
222C2×C3×Q8—22no—
223C2×C3×Q8—41no—
224C2×C3×Q8—92no—
225C2×C3×Q8—91no—
226C2×C3×Q8—184no—
227C2×C3×Q8D3211yescharacteristic
228C2×C3×Q8D3212yes—
229C22×C3×C4—91no—
230C22×C3×C4—92no—
231C22×C3×C4—92no—
232C2×#67—92no—
233C2×#67—92no—
234C2×#67—92no—
235D3×Q8—68no—
236D3×Q8—68no—
237#235—188no—
238#235—184no—
239C2×D12—92no—
240C2×D12—91no—
241C2×C4×D3—32no—
242C2×C4×D3—34no—
243C2×C4×D3—32no—
244C2×C4×D3—32no—
245C2×C4×D3—61no—
246C2×C4×D3—62no—
247C2×C4×D3—62no—
248C2×Dic6—34no—
249C2×Dic6—32no—
250C2×Dic6—61no—
251C2×Dic6—62no—
252C2×Dic6—92no—
253C2×Dic6—91no—
254C3×C4⋊C4—94no—
255C3×C4⋊C4—92no—
256C3×C22⋊C4—94no—
257#217—94no—
258#217—92no—
259(C2×D3)⋊C4—34no—
260(C2×D3)⋊C4—34no—
261(C2×D3)⋊C4—64no—
262(C2×D3)⋊C4—94no—
263(C2×D3)⋊C4—94no—
264(C2×D3)⋊C4—94no—
265C4⋊Dic3—32no—
266C4⋊Dic3—32no—
267C4⋊Dic3—62no—
268C4⋊Dic3—94no—
269Dic3⋊C4—34no—
270Dic3⋊C4—34no—
271Dic3⋊C4—64no—
272Dic3⋊C4—94no—
273Dic3⋊C4—94no—
274C4×Dic3—94no—
275C4×Dic3—92no—
276C22×C32—21no—
277C22×C32—22no—
278C22×C32—41no—
279C22×C32C23×D312yes—
280C22×C32C23×D311yescharacteristic
281C2×C3⋊D3—22no—
282C2×C3⋊D3—22no—
283C2×C3⋊D3—22no—
284C2×C3⋊D3—22no—
285C2×C3⋊D3C2×#6712yes—
286C2×C3×D3—34no—
287C2×C3×D3—62no—
288C2×C3×D3—64no—
289C2×C3×D3—62no—
290C2×C3×D3—62no—
291C2×C3×D3—64no—
292C2×C3×D3—62no—
293C2×C3×D3—62no—
294C2×C3×D3—62no—
295C2×C3×D3—62no—
296C2×C3×D3—62no—
297C2×C3×D3—62no—
298C2×C3×D3—122no—
299C2×C3×D3—122no—
300C32×C4—22no—
301C32×C4—24no—
302C32×C4—24no—
303C32×C4—24no—
304C32×C4—22no—
305C32×C4—42no—
306C32×C4—44no—
307C32×C4—44no—
308C32×C4D3×D414yes—
309C32×C4C2×#6712yes—
310C3⋊Dic3—22no—
311C3⋊Dic3—22no—
312C3⋊Dic3—24no—
313C3⋊Dic3C2×#6712yes—
314C3×Dic3—34no—
315C3×Dic3—62no—
316C3×Dic3—64no—
317C3×Dic3—62no—
318C3×Dic3—64no—
319C3×Dic3—64no—
320C3×Dic3—64no—
321C3×Dic3—62no—
322C3×Dic3—122no—
323C3×Dic3—124no—
324C2×#41—271no—
325C22×Q8—91no—
326C4⋊Q8—272no—
327#124—272no—
328#123—274no—
329C22⋊Q8—92no—
330C22⋊Q8—272no—
331#117—271no—
332C33#53611yescharacteristic, complemented, Sylow
333C23×C3—91no—
334C23×C3—92no—
335C22×D3—32no—
336C22×D3—32no—
337C22×D3—32no—
338C22×D3—61no—
339C22×D3—62no—
340C3×Q8—24no—
341C3×Q8—24no—
342C3×Q8—22no—
343C3×Q8—44no—
344C3×Q8—44no—
345C3×Q8—184no—
346C3×Q8—182no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—21no—
350C2×C3×C4—24no—
351C2×C3×C4—42no—
352C2×C3×C4—41no—
353C2×C3×C4—91no—
354C2×C3×C4—94no—
355C2×C3×C4—94no—
356C2×C3×C4—92no—
357C2×C3×C4—92no—
358C2×C3×C4—184no—
359C2×C3×C4—184no—
360C2×C3×C4C2×D3212yes—
361C2×C3×C4C2×D3214yes—
362C2×C3×C4C2×D3212yes—
363C2×C3×C4C2×D3211yescharacteristic
364#67—184no—
365#67—184no—
366#67—184no—
367C2×Dic3—32no—
368C2×Dic3—34no—
369C2×Dic3—34no—
370C2×Dic3—32no—
371C2×Dic3—32no—
372C2×Dic3—34no—
373C2×Dic3—61no—
374C2×Dic3—62no—
375C2×Dic3—62no—
376C2×Dic3—64no—
377D12—182no—
378D12—184no—
379C4×D3—38no—
380C4×D3—64no—
381C4×D3—68no—
382C4×D3—68no—
383C4×D3—64no—
384C4×D3—64no—
385C4×D3—124no—
386Dic6—68no—
387Dic6—68no—
388Dic6—64no—
389Dic6—64no—
390Dic6—184no—
391Dic6—182no—
392C2×C32—22no—
393C2×C32—22no—
394C2×C32—22no—
395C2×C32—22no—
396C2×C32—21no—
397C2×C32—41no—
398C2×C32—42no—
399C2×C32C2×D3×D412yes—
400C2×C32#100712yes—
401C2×C32#100712yes—
402C2×C32#101012yes—
403C2×C32C22×#6711yescharacteristic
404C3⋊D3—22no—
405C3⋊D3—42no—
406C3×D3—64no—
407C3×D3—64no—
408C3×D3—124no—
409C3×D3—122no—
410C3×D3—122no—
411#41—544no—
412C2×Q8—91no—
413C2×Q8—92no—
414C2×Q8—184no—
415C2×Q8—271no—
416C2×Q8#111311yescharacteristic
417C2×D4—272no—
418C2×D4—271no—
419C22×C4—91no—
420C22×C4—92no—
421C22×C4—92no—
422C4⋊C4—92no—
423C4⋊C4—94no—
424C4⋊C4—274no—
425C22⋊C4—94no—
426C22⋊C4—272no—
427C22⋊C4—274no—
428C42—272no—
429C22×C3—22no—
430C22×C3—21no—
431C22×C3—41no—
432C22×C3—92no—
433C22×C3—182no—
434C22×C3—182no—
435C22×C3—182no—
436C22×C3—182no—
437C22×C3C22×D3211yescharacteristic
438C22×C3C22×D3212yes—
439C2×D3—34no—
440C2×D3—62no—
441C2×D3—64no—
442C2×D3—62no—
443C2×D3—62no—
444C2×D3—64no—
445C2×D3—62no—
446C2×D3—62no—
447C2×D3—62no—
448C2×D3—62no—
449C2×D3—62no—
450C2×D3—62no—
451C2×D3—122no—
452C2×D3—122no—
453C3×C4—24no—
454C3×C4—24no—
455C3×C4—22no—
456C3×C4—22no—
457C3×C4—24no—
458C3×C4—44no—
459C3×C4—44no—
460C3×C4—42no—
461C3×C4—92no—
462C3×C4—184no—
463C3×C4—182no—
464C3×C4—182no—
465C3×C4D3×#6714yes—
466C3×C4#381112yes—
467Dic3—34no—
468Dic3—62no—
469Dic3—64no—
470Dic3—64no—
471Dic3—64no—
472Dic3—64no—
473Dic3—62no—
474Dic3—62no—
475Dic3—122no—
476Dic3—124no—
477C32—22no—
478C32—21no—
479C32—41no—
480C32#562812yescomplemented
481C32#581611yescharacteristic, complemented
482C23—91no—
483C23—92no—
484Q8—22no—
485Q8—184no—
486Q8—182no—
487Q8—542no—
488D4—542no—
489D4—544no—
490C2×C4—91no—
491C2×C4—94no—
492C2×C4—94no—
493C2×C4—92no—
494C2×C4—92no—
495C2×C4—184no—
496C2×C4—184no—
497C2×C4C2×#111312yes—
498C2×C4C2×#111311yescharacteristic
499C2×C3—22no—
500C2×C3—22no—
501C2×C3—22no—
502C2×C3—21no—
503C2×C3—22no—
504C2×C3—42no—
505C2×C3—41no—
506C2×C3—182no—
507C2×C3—362no—
508C2×C3#6431512yes—
509C2×C3#6431512yes—
510C2×C3#6431612yes—
511C2×C3C2×D3×#6712yes—
512C2×C3C2×#381111yescharacteristic
513D3—64no—
514D3—64no—
515D3—124no—
516D3—122no—
517D3—122no—
518C22—92no—
519C22—182no—
520C22—182no—
521C22—182no—
522C22—182no—
523C22C22×#111311yescharacteristic
524C4—22no—
525C4—92no—
526C4—184no—
527C4—182no—
528C4—182no—
529C4#8904912yes—
530C3—21no—
531C3—22no—
532C3—41no—
533C3#1059539712yescomplemented
534C3#1059540511yescharacteristic, complemented
535C2—182no—
536C2—362no—
537C2#1172462412yes—
538C2C2×#8904911yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC2×C33×Q8C22
prime radicalC2×#89042C2
hypercenterC2×Q8#1113
hypocenterC33#536
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×Q8×C3⋊D3C2semidirect
3. C3×#64155C2semidirect
4. #11723694C2semidirect
5. #11723674C2semidirect
6. #11723670C2semidirect
7. #11723667C2semidirect
8. #11723666C2semidirect
29. C2×Q8×C3⋊D3D3semidirect
34. #64155D3semidirect
332. C33#536semidirect
480. C32#5628semidirect
481. C32#5816semidirect
533. C3#10595397semidirect
534. C3#10595405semidirect

Maximal subgroups

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

subgroupindex
C2×C3×Q8×C3⋊D32
C2×#890422
C3×#641552
#117236942
#117236742
#117236702
#117236672
#117236662
#105954053
#105953973