Finite Group Databasebeta

← #13177663

Subgroups — #13177663

Order 1728.

1932 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×C32×#237C211yesmaximal, characteristic
2C22×#6293C211yesmaximal, characteristic, complemented
3#11723976C211yesmaximal, characteristic, complemented
4C3×#64144C211yesmaximal, characteristic, complemented
5#11723640C212yesmaximal, complemented
6#11723603C211yesmaximal, characteristic, complemented
7#11723602C211yesmaximal, characteristic, complemented
8#11723600C212yesmaximal, complemented
9#11723594C211yesmaximal, characteristic, complemented
10#11723415C212yesmaximal, complemented
11#11723393C212yesmaximal, complemented
12#10596296—31nomaximal
13#10595270—34nomaximal
14C2×C33×D4C2211yescharacteristic
15C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C22×C3×C3⋊Dic3C2211yescharacteristic
17C2×C32×#67C2211yescharacteristic
18C2×C32×#67C2211yescharacteristic
19C22×C32×Dic3C2211yescharacteristic
20C22×C32×Dic3C2211yescharacteristic
21C32×#237—24no—
22C2×C32×C4×D3C2211yescharacteristic
23C2×C32×Dic6C2211yescharacteristic
24C2×#6293—24no—
25C2×#6293C2212yes—
26C2×#6293C2212yes—
27#88864C2211yescharacteristic
28#88864C2212yes—
29#88857C2212yes—
30C3×#3757C2211yescharacteristic
31C3×#3757C2212yes—
32C3×#3750C2212yes—
33#88807C2212yes—
34#88806C2212yes—
35Dic3⋊(C3⋊Dic3)C2212yes—
36(C2×D3)⋊(C3⋊Dic3)C2212yes—
37(C2×D3)⋊(C3⋊Dic3)C2212yes—
38Dic3×C3⋊Dic3C2212yes—
39C2×C32×#41—31no—
40C22×#634—31no—
41C2×C3×#237D314yes—
42C22×#627—34no—
43#64153—32no—
44#64145—32no—
45#64144—31no—
46#64144—31no—
47#64144—31no—
48#64144—34no—
49#64144—64no—
50#64144D311yescharacteristic, complemented
51#64135—31no—
52#64133—32no—
53#64133—32no—
54C3×#935—34no—
55#63964—38no—
56#63962—34no—
57#63961—34no—
58#63959—38no—
59#63955—34no—
60#63851—38no—
61#63839—38no—
62C23×C33C2311yescharacteristic
63C23×C33C2311yescharacteristic
64C22×C32×D3C2311yescharacteristic
65C33×D4—22no—
66C2×C33×C4C2311yescharacteristic
67C2×C3⋊(C3⋊Dic3)—22no—
68C2×C3⋊(C3⋊Dic3)C2312yes—
69C2×C3⋊(C3⋊Dic3)C2312yes—
70C2×C3×C3⋊Dic3—22no—
71C2×C3×C3⋊Dic3C2312yes—
72C2×C3×C3⋊Dic3C2312yes—
73C32×#67—22no—
74C32×#67—22no—
75C2×C32×Dic3—22no—
76C2×C32×Dic3D414yes—
77C2×C32×Dic3C2311yescharacteristic
78C2×C32×Dic3C2311yescharacteristic
79C2×C32×Dic3C2311yescharacteristic
80C32×C4×D3—22no—
81C32×Dic6—22no—
82#6293—24no—
83#6293—24no—
84#5825—94no—
85#5648—91no—
86C32×#41—64no—
87C2×C32×Q8—31no—
88C2×C32×D4—24no—
89C2×C32×D4—31no—
90C2×C32×D4—31no—
91C2×C32×D4C2×D314yes—
92C2×C32×D4C2×D311yescharacteristic
93C22×C32×C4—31no—
94C22×C32×C4—31no—
95C22×C32×C4—31no—
96C22×C3⋊Dic3—31no—
97C22×C3⋊Dic3—34no—
98C22×C3⋊Dic3—64no—
99C22×C3⋊Dic3C2×D311yescharacteristic
100C2×#634—32no—
101C2×#634—32no—
102C2×#634—64no—
103C2×C3×#67C2×D314yes—
104C2×C3×#67C2×D314yes—
105C22×C3×Dic3—34no—
106C22×C3×Dic3C2×D314yes—
107C22×C3×Dic3C2×D314yes—
108C3×#237—216no—
109C2×C3×C4×D3C2×D314yes—
110C2×C3×Dic6C2×D314yes—
111C2×#627—38no—
112C2×#627—38no—
113C2×#627—616no—
114#3757—32no—
115#3757—31no—
116#3757—34no—
117#3757—38no—
118#3757—32no—
119#3757—32no—
120#3757—64no—
121#3757—68no—
122#3757C2×D312yes—
123#3757C2×D311yescharacteristic
124C4⋊(C3⋊Dic3)—32no—
125#3750—32no—
126#3750—38no—
127#3750—32no—
128#3750—32no—
129#3750—68no—
130#3750C2×D312yes—
131C4×C3⋊Dic3—32no—
132C3×#217—34no—
133C3×#217—38no—
134C3×Dic3⋊C4—38no—
135#3724—38no—
136Dic3⋊Dic3—38no—
137Dic3⋊Dic3—38no—
138(C2×D3)⋊Dic3—38no—
139(C2×D3)⋊Dic3—38no—
140Dic32—38no—
141C22×C33—22no—
142C22×C33—21no—
143C22×C33—22no—
144C22×C33C2×D412yes—
145C22×C33C2411yescharacteristic
146C2×C32×D3—21no—
147C2×C32×D3—22no—
148C33×C4—21no—
149C3⋊(C3⋊Dic3)—22no—
150C3⋊(C3⋊Dic3)—22no—
151C3×C3⋊Dic3—22no—
152C3×C3⋊Dic3—22no—
153C32×Dic3—21no—
154C32×Dic3C2×D412yes—
155C32×Dic3C2×D412yes—
156C2×C3×#41—34no—
157C2×#237C3⋊D311yescharacteristic
158C22×Dic6—94no—
159C22×Dic6—91no—
160C3×#123—91no—
161#944—98no—
162#936—98no—
163#935—34no—
164#935—94no—
165#935—94no—
166#935—94no—
167#935—91no—
168#935—184no—
169#929—91no—
170#926—94no—
171#924—98no—
172#924—98no—
173#896—92no—
174#888—92no—
175#886—91no—
176#885—92no—
177#878—92no—
178#877—91no—
179C23×C32—24no—
180C23×C32—24no—
181C23×C32—31no—
182C23×C32C22×D314yes—
183C23×C32C22×D314yes—
184C23×C32C22×D311yescharacteristic
185C23×C32C22×D311yescharacteristic
186C22×C3×D3C22×D314yes—
187C32×Q8—62no—
188C32×D4—22no—
189C32×D4—28no—
190C32×D4—48no—
191C32×D4—62no—
192C32×D4—62no—
193C2×C32×C4—24no—
194C2×C32×C4—34no—
195C2×C32×C4—31no—
196C2×C32×C4—31no—
197C2×C32×C4—31no—
198C2×C32×C4—62no—
199C2×C32×C4—62no—
200C2×C32×C4C22×D314yes—
201C2×C32×C4C22×D311yescharacteristic
202C2×C3⋊Dic3—22no—
203C2×C3⋊Dic3—32no—
204C2×C3⋊Dic3—32no—
205C2×C3⋊Dic3—38no—
206C2×C3⋊Dic3—38no—
207C2×C3⋊Dic3—68no—
208C2×C3⋊Dic3—68no—
209C2×C3⋊Dic3—62no—
210C2×C3⋊Dic3—68no—
211C2×C3⋊Dic3—128no—
212C2×C3⋊Dic3C22×D312yes—
213C2×C3⋊Dic3C22×D312yes—
214#634—64no—
215#634—64no—
216C3×#67—28no—
217C3×#67—28no—
218C2×C3×Dic3—28no—
219C2×C3×Dic3—38no—
220C2×C3×Dic3—38no—
221C2×C3×Dic3—68no—
222C2×C3×Dic3#67116yes—
223C2×C3×Dic3C22×D314yes—
224C2×C3×Dic3C22×D314yes—
225C2×C3×Dic3C22×D314yes—
226C3×C4×D3—28no—
227C3×Dic6—28no—
228#627—616no—
229#627—616no—
230#536—271noSylow
231C2×C33—22no—
232C2×C33—41no—
233C2×C33C22×D411yescharacteristic
234C2×C332−1+412yes—
235C32×D3—41no—
236C3×#41—616no—
237C2×C3×Q8—34no—
238C2×C3×D4—24no—
239C2×C3×D4—34no—
240C2×C3×D4—34no—
241C2×C3×D4D3214yes—
242C2×C3×D4C2×C3⋊D311yescharacteristic
243C22×C3×C4—34no—
244C22×C3×C4—34no—
245C22×C3×C4—34no—
246C22×C3×C4—91no—
247C2×#67C2×C3⋊D311yescharacteristic
248C2×#67C2×C3⋊D311yescharacteristic
249C22×Dic3—34no—
250C22×Dic3—94no—
251C22×Dic3—91no—
252C22×Dic3—184no—
253C22×Dic3C2×C3⋊D311yescharacteristic
254C22×Dic3C2×C3⋊D311yescharacteristic
255#237—24no—
256C2×C4×D3C2×C3⋊D311yescharacteristic
257C2×Dic6—98no—
258C2×Dic6—98no—
259C2×Dic6—92no—
260C2×Dic6—92no—
261C2×Dic6—1816no—
262C2×Dic6—184no—
263C2×Dic6C2×C3⋊D311yescharacteristic
264C3×C4⋊C4—92no—
265C3×C22⋊C4—91no—
266C3×C22⋊C4—92no—
267#217—38no—
268#217—34no—
269#217—98no—
270#217—98no—
271#217—98no—
272#217—94no—
273#217—91no—
274#217—92no—
275#217—184no—
276#217—188no—
277(C2×D3)⋊C4—92no—
278(C2×D3)⋊C4—92no—
279C4⋊Dic3—98no—
280C4⋊Dic3—92no—
281Dic3⋊C4—38no—
282Dic3⋊C4—98no—
283Dic3⋊C4—98no—
284Dic3⋊C4—98no—
285Dic3⋊C4—92no—
286Dic3⋊C4—92no—
287Dic3⋊C4—92no—
288Dic3⋊C4—188no—
289C4×Dic3—98no—
290C4×Dic3—92no—
291C22×C32—22no—
292C22×C32—21no—
293C22×C32—22no—
294C22×C32—24no—
295C22×C32—28no—
296C22×C32—28no—
297C22×C32—28no—
298C22×C32—24no—
299C22×C32—48no—
300C22×C32—44no—
301C22×C32—48no—
302C22×C32—61no—
303C22×C32—62no—
304C22×C32D3×D412yes—
305C22×C32C2×#6718yes—
306C22×C32C23×D314yes—
307C22×C32C23×D311yescharacteristic
308C2×C3×D3—28no—
309C2×C3×D3—24no—
310C32×C4—21no—
311C32×C4—24no—
312C32×C4—32no—
313C32×C4—32no—
314C32×C4—44no—
315C32×C4—61no—
316C3⋊Dic3—22no—
317C3⋊Dic3—22no—
318C3⋊Dic3—62no—
319C3⋊Dic3—62no—
320C3⋊Dic3—68no—
321C3⋊Dic3—68no—
322C3⋊Dic3—128no—
323C3⋊Dic3—128no—
324C3×Dic3—24no—
325C3×Dic3—68no—
326C3×Dic3—68no—
327C3×Dic3C2×#6718yes—
328C3×Dic3C2×#6718yes—
329C2×#41—31no—
330C22×Q8—271no—
331C4⋊Q8—272no—
332#124—272no—
333#123—91no—
334#123—271no—
335#123—271no—
336#123—271no—
337C22⋊Q8—272no—
338C22⋊Q8—272no—
339#117—271no—
340C33#53611yescharacteristic, complemented, Sylow
341C23×C3—24no—
342C23×C3—24no—
343C23×C3—34no—
344C23×C3C2×D3214yes—
345C23×C3C2×D3214yes—
346C23×C3C22×C3⋊D311yescharacteristic
347C23×C3C22×C3⋊D311yescharacteristic
348C22×D3C22×C3⋊D311yescharacteristic
349C3×Q8—68no—
350C3×D4—28no—
351C3×D4—22no—
352C3×D4—48no—
353C3×D4—68no—
354C3×D4—68no—
355C2×C3×C4—24no—
356C2×C3×C4—34no—
357C2×C3×C4—34no—
358C2×C3×C4—316no—
359C2×C3×C4—34no—
360C2×C3×C4—68no—
361C2×C3×C4—68no—
362C2×C3×C4—92no—
363C2×C3×C4—92no—
364C2×C3×C4—182no—
365C2×C3×C4C2×D3214yes—
366C2×C3×C4C22×C3⋊D311yescharacteristic
367#67—22no—
368#67—22no—
369C2×Dic3—22no—
370C2×Dic3—38no—
371C2×Dic3—38no—
372C2×Dic3—68no—
373C2×Dic3—98no—
374C2×Dic3—98no—
375C2×Dic3—92no—
376C2×Dic3—92no—
377C2×Dic3—188no—
378C2×Dic3—188no—
379C2×Dic3—188no—
380C2×Dic3—182no—
381C2×Dic3—368no—
382C2×Dic3#63814yes—
383C2×Dic3C22×C3⋊D311yescharacteristic
384C2×Dic3C22×C3⋊D311yescharacteristic
385C2×Dic3C22×C3⋊D311yescharacteristic
386C4×D3—22no—
387Dic6—22no—
388Dic6—1816no—
389Dic6—1816no—
390Dic6—184no—
391Dic6—184no—
392C2×C32—22no—
393C2×C32—28no—
394C2×C32—24no—
395C2×C32—28no—
396C2×C32—48no—
397C2×C32—41no—
398C2×C32—44no—
399C2×C32—84no—
400C2×C32—121no—
401C2×C32C2×D3×D411yescharacteristic
402C2×C32#101018yes—
403C2×C32#101012yes—
404C2×C32C22×#6714yes—
405C3×D3—44no—
406#41—64no—
407C2×Q8—31no—
408C2×Q8—272no—
409C2×Q8—272no—
410C2×Q8—544no—
411C2×D4—31no—
412C2×D4—31no—
413C2×D4D3×C3⋊D311yescharacteristic
414C22×C4—31no—
415C22×C4—31no—
416C22×C4—31no—
417C22×C4—91no—
418C22×C4—271no—
419C4⋊C4—92no—
420C4⋊C4—272no—
421C4⋊C4—272no—
422C4⋊C4—272no—
423C4⋊C4—272no—
424C22⋊C4—91no—
425C22⋊C4—92no—
426C22⋊C4—272no—
427C22⋊C4—272no—
428C22⋊C4—271no—
429C22⋊C4—272no—
430C42—272no—
431C22×C3—28no—
432C22×C3—22no—
433C22×C3—28no—
434C22×C3—24no—
435C22×C3—22no—
436C22×C3—21no—
437C22×C3—24no—
438C22×C3—28no—
439C22×C3—48no—
440C22×C3—44no—
441C22×C3—48no—
442C22×C3—68no—
443C22×C3—64no—
444C22×C3D3×#6718yes—
445C22×C3C2×#63812yes—
446C22×C3C22×D3214yes—
447C22×C3C23×C3⋊D311yescharacteristic
448C2×D3—22no—
449C2×D3—21no—
450C3×C4—24no—
451C3×C4—21no—
452C3×C4—38no—
453C3×C4—38no—
454C3×C4—44no—
455C3×C4—64no—
456C3×C4—182no—
457C3×C4—182no—
458Dic3—21no—
459Dic3—68no—
460Dic3—68no—
461Dic3—188no—
462Dic3—188no—
463Dic3—182no—
464Dic3—182no—
465Dic3—368no—
466Dic3—368no—
467Dic3C2×#63812yes—
468Dic3C2×#63812yes—
469C32—24no—
470C32#564811yescharacteristic, complemented
471C32#582514yescomplemented
472C23—31no—
473C23C2×D3×C3⋊D311yescharacteristic
474C23C2×D3×C3⋊D311yescharacteristic
475Q8—62no—
476Q8—544no—
477Q8—544no—
478D4—22no—
479D4—62no—
480D4—62no—
481C2×C4—31no—
482C2×C4—31no—
483C2×C4—34no—
484C2×C4—31no—
485C2×C4—62no—
486C2×C4—62no—
487C2×C4—92no—
488C2×C4—92no—
489C2×C4—182no—
490C2×C4—272no—
491C2×C4—272no—
492C2×C4—542no—
493C2×C4C2×D3×C3⋊D311yescharacteristic
494C2×C3—28no—
495C2×C3—22no—
496C2×C3—28no—
497C2×C3—24no—
498C2×C3—48no—
499C2×C3—44no—
500C2×C3—41no—
501C2×C3—84no—
502C2×C3—124no—
503C2×C3#6430918yes—
504C2×C3C2×D3×#6714yes—
505C2×C3#6436712yes—
506C2×C3C22×#63811yescharacteristic
507D3—41no—
508C22—22no—
509C22—22no—
510C22—21no—
511C22—62no—
512C22—61no—
513C22D3×#63812yes—
514C22C22×D3×C3⋊D311yescharacteristic
515C4—21no—
516C4—32no—
517C4—32no—
518C4—61no—
519C4—182no—
520C4—182no—
521C4—542no—
522C4—542no—
523C3—24no—
524C3#1059527014yescomplemented
525C3#1059629611yescharacteristic, complemented
526C2—22no—
527C2—41no—
528C2—121no—
529C2#1172458212yes—
530C2C2×D3×#63811yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
prime radicalC2×C32×#237C2
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#536
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C22×#6293C2semidirect
3. #11723976C2semidirect
4. C3×#64144C2semidirect
5. #11723640C2semidirect
6. #11723603C2semidirect
7. #11723602C2semidirect
8. #11723600C2semidirect
9. #11723594C2semidirect
10. #11723415C2semidirect
11. #11723393C2semidirect
50. #64144D3semidirect
340. C33#536semidirect
470. C32#5648semidirect
471. C32#5825semidirect
524. C3#10595270semidirect
525. C3#10596296semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#2372
C22×#62932
#117239762
C3×#641442
#117236402
#117236032
#117236022
#117236002
#117235942
#117234152
#117233932
#105962963
#105952703