Finite Group Databasebeta

← #13177659

Subgroups — #13177659

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×C3×#3830C211yesmaximal, characteristic
2C22×#6293C211yesmaximal, characteristic, complemented
3#11723976C211yesmaximal, characteristic, complemented
4C32×#935C211yesmaximal, characteristic, complemented
5#11723641C212yesmaximal, complemented
6#11723609C211yesmaximal, characteristic, complemented
7#11723605C211yesmaximal, characteristic, complemented
8#11723595C211yesmaximal, characteristic, complemented
9#11723591C212yesmaximal, complemented
10#11723418C212yesmaximal, complemented
11#11723394C212yesmaximal, complemented
12#10596119—31nomaximal
13#10595270—34nomaximal
14C2×C33×D4C2211yescharacteristic
15C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×#638C2211yescharacteristic
17C2×C3×#638C2211yescharacteristic
18C22×C3×C3⋊Dic3C2211yescharacteristic
19C22×C3×C3⋊Dic3C2211yescharacteristic
20C3×#3830—24no—
21C2×C3×C4×C3⋊D3C2211yescharacteristic
22C2×C3×#634C2211yescharacteristic
23C22×C32×Dic3C2211yescharacteristic
24C2×#6293—24no—
25C2×#6293C2212yes—
26C2×#6293C2212yes—
27#88864C2212yes—
28#88864C2211yescharacteristic
29#88857C2212yes—
30C32×#217C2212yes—
31C32×#217C2211yescharacteristic
32C32×Dic3⋊C4C2212yes—
33#88807C2212yes—
34#88806C2212yes—
35Dic3⋊(C3⋊Dic3)C2212yes—
36#88803C2212yes—
37#88803C2212yes—
38Dic3×C3⋊Dic3C2212yes—
39C2×#3830D311yescharacteristic
40C22×#634—31no—
41C2×C3×#237—34no—
42C22×#627—34no—
43C32×#123—31no—
44#64144—31no—
45#64144—34no—
46#64144—64no—
47#64138—31no—
48#64105—32no—
49#64097—32no—
50#64095—31no—
51#64094—32no—
52#64087—32no—
53#64086—31no—
54C3×#935D314yescomplemented
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×C3×C3⋊D3C2311yescharacteristic
65C33×D4—22no—
66C2×C33×C4C2311yescharacteristic
67C2×C3⋊(C3⋊Dic3)—22no—
68C2×C3⋊(C3⋊Dic3)C2312yes—
69C2×C3⋊(C3⋊Dic3)C2312yes—
70C3×#638—22no—
71C3×#638—22no—
72C2×C3×C3⋊Dic3—22no—
73C2×C3×C3⋊Dic3D414yes—
74C2×C3×C3⋊Dic3C2311yescharacteristic
75C2×C3×C3⋊Dic3C2311yescharacteristic
76C2×C3×C3⋊Dic3C2311yescharacteristic
77C3×C4×C3⋊D3—22no—
78C3×#634—22no—
79C2×C32×Dic3—22no—
80C2×C32×Dic3C2312yes—
81C2×C32×Dic3C2312yes—
82#6293—24no—
83#6293—24no—
84#5825—91no—
85#5648—94no—
86C2×C32×D4—24no—
87C2×C32×D4C2×D314yes—
88C2×C32×D4C2×D311yescharacteristic
89C22×C32×C4—31no—
90C2×#638C2×D311yescharacteristic
91C2×#638C2×D311yescharacteristic
92C22×C3⋊Dic3—31no—
93C22×C3⋊Dic3—34no—
94C22×C3⋊Dic3—64no—
95C22×C3⋊Dic3C2×D311yescharacteristic
96C22×C3⋊Dic3C2×D311yescharacteristic
97#3830—24no—
98C2×C4×C3⋊D3C2×D311yescharacteristic
99C2×#634—32no—
100C2×#634—32no—
101C2×#634—64no—
102C2×#634C2×D311yescharacteristic
103C2×C3×#67—34no—
104C2×C3×#67—34no—
105C22×C3×Dic3—34no—
106C22×C3×Dic3—34no—
107C22×C3×Dic3C2×D314yes—
108C3×#237—616no—
109C2×C3×C4×D3—34no—
110C2×C3×Dic6—34no—
111C2×#627—38no—
112C2×#627—38no—
113C2×#627—616no—
114C32×C4⋊C4—32no—
115C32×C22⋊C4—31no—
116C32×C22⋊C4—32no—
117#3757—31no—
118#3757—32no—
119#3757—34no—
120#3757—38no—
121#3757—64no—
122#3757—68no—
123#3752—32no—
124#3752—32no—
125C4⋊(C3⋊Dic3)—32no—
126#3750—32no—
127#3750—38no—
128#3750—32no—
129#3750—32no—
130#3750—68no—
131C4×C3⋊Dic3—32no—
132C3×#217C2×D314yes—
133C3×#217C2×D318yes—
134C3×Dic3⋊C4C2×D318yes—
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×C3×C3⋊D3—21no—
147C2×C3×C3⋊D3—22no—
148C33×C4—21no—
149C3⋊(C3⋊Dic3)—22no—
150C3⋊(C3⋊Dic3)—22no—
151C3×C3⋊Dic3—21no—
152C3×C3⋊Dic3C2×D412yes—
153C3×C3⋊Dic3C2×D412yes—
154C32×Dic3—22no—
155C32×Dic3—22no—
156C2×C3×#41—91no—
157C2×#237—34no—
158C22×Dic6—94no—
159C22×Dic6—91no—
160C3×#123—34no—
161#944—92no—
162#936—92no—
163#935—94no—
164#935—91no—
165#935—91no—
166#935—91no—
167#935—184no—
168#935C3⋊D311yescharacteristic, complemented
169#929—94no—
170#926—91no—
171#924—92no—
172#924—92no—
173#896—98no—
174#888—98no—
175#886—94no—
176#885—98no—
177#878—98no—
178#877—94no—
179C23×C32—24no—
180C23×C32—24no—
181C23×C32C22×D314yes—
182C23×C32C22×D314yes—
183C23×C32C22×D311yescharacteristic
184C23×C32C22×D311yescharacteristic
185C22×C3⋊D3C22×D311yescharacteristic
186C22×C3×D3—34no—
187C32×D4—22no—
188C32×D4—28no—
189C32×D4—48no—
190C2×C32×C4—24no—
191C2×C32×C4—32no—
192C2×C32×C4—32no—
193C2×C32×C4—62no—
194C2×C32×C4C22×D314yes—
195C2×C32×C4C22×D311yescharacteristic
196#638—22no—
197#638—22no—
198C2×C3⋊Dic3—22no—
199C2×C3⋊Dic3—32no—
200C2×C3⋊Dic3—32no—
201C2×C3⋊Dic3—38no—
202C2×C3⋊Dic3—38no—
203C2×C3⋊Dic3—68no—
204C2×C3⋊Dic3—68no—
205C2×C3⋊Dic3—62no—
206C2×C3⋊Dic3—68no—
207C2×C3⋊Dic3—128no—
208C2×C3⋊Dic3#6714yes—
209C2×C3⋊Dic3C22×D311yescharacteristic
210C2×C3⋊Dic3C22×D311yescharacteristic
211C2×C3⋊Dic3C22×D311yescharacteristic
212C4×C3⋊D3—22no—
213#634—22no—
214#634—64no—
215#634—64no—
216C3×#67—68no—
217C3×#67—68no—
218C2×C3×Dic3—28no—
219C2×C3×Dic3—316no—
220C2×C3×Dic3—34no—
221C2×C3×Dic3—34no—
222C2×C3×Dic3—34no—
223C2×C3×Dic3—68no—
224C2×C3×Dic3C22×D318yes—
225C2×C3×Dic3C22×D318yes—
226C3×C4×D3—68no—
227C3×Dic6—68no—
228#627—616no—
229#627—616no—
230#536—271noSylow
231C2×C33—22no—
232C2×C33—41no—
233C2×C33C22×D411yescharacteristic
234C2×C332−1+412yes—
235C3×C3⋊D3—41no—
236C3×#41—184no—
237C2×C3×Q8—91no—
238C2×C3×D4—24no—
239C2×C3×D4—91no—
240C2×C3×D4—91no—
241C2×C3×D4D3214yes—
242C2×C3×D4C2×C3⋊D311yescharacteristic
243C22×C3×C4—34no—
244C22×C3×C4—91no—
245C22×C3×C4—91no—
246C22×C3×C4—91no—
247C2×#67—34no—
248C2×#67—34no—
249C22×Dic3—34no—
250C22×Dic3—34no—
251C22×Dic3—94no—
252C22×Dic3—91no—
253C22×Dic3—184no—
254C22×Dic3C2×C3⋊D311yescharacteristic
255#237—616no—
256C2×C4×D3—34no—
257C2×Dic6—34no—
258C2×Dic6—98no—
259C2×Dic6—98no—
260C2×Dic6—92no—
261C2×Dic6—92no—
262C2×Dic6—184no—
263C2×Dic6—1816no—
264C3×C4⋊C4—38no—
265C3×C22⋊C4—34no—
266C3×C22⋊C4—38no—
267#217—94no—
268#217—98no—
269#217—92no—
270#217—91no—
271#217—92no—
272#217—92no—
273#217—184no—
274#217—188no—
275#217C2×C3⋊D312yes—
276#217C2×C3⋊D311yescharacteristic
277(C2×D3)⋊C4—98no—
278(C2×D3)⋊C4—98no—
279C4⋊Dic3—98no—
280C4⋊Dic3—92no—
281Dic3⋊C4—98no—
282Dic3⋊C4—98no—
283Dic3⋊C4—98no—
284Dic3⋊C4—92no—
285Dic3⋊C4—92no—
286Dic3⋊C4—92no—
287Dic3⋊C4—188no—
288Dic3⋊C4C2×C3⋊D312yes—
289C4×Dic3—98no—
290C4×Dic3—92no—
291C22×C32—22no—
292C22×C32—22no—
293C22×C32—21no—
294C22×C32—24no—
295C22×C32—28no—
296C22×C32—28no—
297C22×C32—24no—
298C22×C32—28no—
299C22×C32—48no—
300C22×C32—44no—
301C22×C32—48no—
302C22×C32D3×D418yes—
303C22×C32C2×#6712yes—
304C22×C32C23×D314yes—
305C22×C32C23×D311yescharacteristic
306C2×C3⋊D3—22no—
307C2×C3⋊D3—21no—
308C2×C3×D3—64no—
309C2×C3×D3—68no—
310C32×C4—21no—
311C32×C4—24no—
312C32×C4—44no—
313C32×C4—62no—
314C32×C4—62no—
315C3⋊Dic3—21no—
316C3⋊Dic3—62no—
317C3⋊Dic3—62no—
318C3⋊Dic3—68no—
319C3⋊Dic3—68no—
320C3⋊Dic3—128no—
321C3⋊Dic3—128no—
322C3⋊Dic3C2×#6712yes—
323C3⋊Dic3C2×#6712yes—
324C3×Dic3—28no—
325C3×Dic3—28no—
326C3×Dic3—38no—
327C3×Dic3—38no—
328C3×Dic3—64no—
329C2×#41—91no—
330C22×Q8—271no—
331C4⋊Q8—272no—
332#124—272no—
333#123—31no—
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—91no—
344C23×C3C2×D3214yes—
345C23×C3C2×D3214yes—
346C23×C3C22×C3⋊D311yescharacteristic
347C23×C3C22×C3⋊D311yescharacteristic
348C22×D3—34no—
349C3×Q8—182no—
350C3×D4—28no—
351C3×D4—22no—
352C3×D4—48no—
353C3×D4—182no—
354C3×D4—182no—
355C2×C3×C4—24no—
356C2×C3×C4—38no—
357C2×C3×C4—38no—
358C2×C3×C4—68no—
359C2×C3×C4—91no—
360C2×C3×C4—91no—
361C2×C3×C4—94no—
362C2×C3×C4—91no—
363C2×C3×C4—182no—
364C2×C3×C4—182no—
365C2×C3×C4C2×D3214yes—
366C2×C3×C4C22×C3⋊D311yescharacteristic
367#67—68no—
368#67—68no—
369C2×Dic3—22no—
370C2×Dic3—34no—
371C2×Dic3—34no—
372C2×Dic3—316no—
373C2×Dic3—34no—
374C2×Dic3—68no—
375C2×Dic3—98no—
376C2×Dic3—98no—
377C2×Dic3—92no—
378C2×Dic3—92no—
379C2×Dic3—188no—
380C2×Dic3—188no—
381C2×Dic3—188no—
382C2×Dic3—182no—
383C2×Dic3—368no—
384C2×Dic3C22×C3⋊D312yes—
385C2×Dic3C22×C3⋊D312yes—
386C4×D3—68no—
387Dic6—68no—
388Dic6—184no—
389Dic6—184no—
390Dic6—1816no—
391Dic6—1816no—
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×C32C2×D3×D414yes—
401C2×C32#101018yes—
402C2×C32#101012yes—
403C2×C32C22×#6711yescharacteristic
404C3⋊D3—41no—
405C3×D3—124no—
406#41—184no—
407C2×Q8—91no—
408C2×Q8—272no—
409C2×Q8—272no—
410C2×Q8—544no—
411C2×D4—91no—
412C2×D4—91no—
413C2×D4D3×C3⋊D311yescharacteristic
414C22×C4—31no—
415C22×C4—91no—
416C22×C4—91no—
417C22×C4—91no—
418C22×C4—271no—
419C4⋊C4—32no—
420C4⋊C4—272no—
421C4⋊C4—272no—
422C4⋊C4—272no—
423C4⋊C4—272no—
424C22⋊C4—31no—
425C22⋊C4—32no—
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—21no—
436C22×C3—22no—
437C22×C3—24no—
438C22×C3—28no—
439C22×C3—48no—
440C22×C3—44no—
441C22×C3—48no—
442C22×C3—181no—
443C22×C3—182no—
444C22×C3D3×#6718yes—
445C22×C3D4×C3⋊D312yes—
446C22×C3C22×D3214yes—
447C22×C3C23×C3⋊D311yescharacteristic
448C2×D3—68no—
449C2×D3—64no—
450C3×C4—24no—
451C3×C4—21no—
452C3×C4—44no—
453C3×C4—68no—
454C3×C4—68no—
455C3×C4—92no—
456C3×C4—92no—
457C3×C4—181no—
458Dic3—22no—
459Dic3—22no—
460Dic3—38no—
461Dic3—38no—
462Dic3—64no—
463Dic3—188no—
464Dic3—188no—
465Dic3—182no—
466Dic3—182no—
467Dic3—368no—
468Dic3—368no—
469C32—24no—
470C32#564814yescomplemented
471C32#582511yescharacteristic, complemented
472C23—91no—
473C23C2×D3×C3⋊D311yescharacteristic
474C23C2×D3×C3⋊D311yescharacteristic
475Q8—182no—
476Q8—544no—
477Q8—544no—
478D4—22no—
479D4—182no—
480D4—182no—
481C2×C4—32no—
482C2×C4—32no—
483C2×C4—62no—
484C2×C4—91no—
485C2×C4—91no—
486C2×C4—94no—
487C2×C4—91no—
488C2×C4—182no—
489C2×C4—182no—
490C2×C4—272no—
491C2×C4—272no—
492C2×C4—542no—
493C2×C4C2×D3×C3⋊D311yescharacteristic
494C2×C3—22no—
495C2×C3—28no—
496C2×C3—28no—
497C2×C3—24no—
498C2×C3—48no—
499C2×C3—41no—
500C2×C3—44no—
501C2×C3—84no—
502C2×C3—361no—
503C2×C3#6430918yes—
504C2×C3C2×D3×#6714yes—
505C2×C3C2×D4×C3⋊D311yescharacteristic
506C2×C3#6436712yes—
507D3—124no—
508C22—22no—
509C22—22no—
510C22—21no—
511C22—182no—
512C22—181no—
513C22C3⋊D3×#6712yes—
514C22C22×D3×C3⋊D311yescharacteristic
515C4—21no—
516C4—62no—
517C4—62no—
518C4—92no—
519C4—92no—
520C4—181no—
521C4—542no—
522C4—542no—
523C3—24no—
524C3#1059527014yescomplemented
525C3#1059611911yescharacteristic, complemented
526C2—22no—
527C2—41no—
528C2—361no—
529C2#1172458212yes—
530C2C2×C3⋊D3×#6711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
prime radicalC2×C3×#3830C2
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. C32×#935C2semidirect
5. #11723641C2semidirect
6. #11723609C2semidirect
7. #11723605C2semidirect
8. #11723595C2semidirect
9. #11723591C2semidirect
10. #11723418C2semidirect
11. #11723394C2semidirect
54. C3×#935D3semidirect
168. #935C3⋊D3semidirect
340. C33#536semidirect
470. C32#5648semidirect
471. C32#5825semidirect
524. C3#10595270semidirect
525. C3#10596119semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38302
C22×#62932
#117239762
C32×#9352
#117236412
#117236092
#117236052
#117235952
#117235912
#117234182
#117233942
#105961193
#105952703