Finite Group Databasebeta

← #13147578

Subgroups — #13147578

Order 1728.

810 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C42×C3⋊D9C211yesmaximal, characteristic, complemented
2C3×C4×D36C211yesmaximal, characteristic, complemented
3C4×C9×Dic6C211yesmaximal, characteristic, complemented
4#11721007C211yesmaximal, characteristic
5C4×#6192C212yesmaximal, complemented
6#11720951C211yesmaximal, characteristic, complemented
7#11720950C212yesmaximal, complemented
8#11720943C212yesmaximal, complemented
9#11720941C212yesmaximal, complemented
10#11720937C211yesmaximal, characteristic, complemented
11#11720929C211yesmaximal, characteristic, complemented
12#10593875—31nomaximal
13#10589682—31nomaximal
14C2×C4×C3⋊D9C2211yescharacteristic
15C2×C4×C3⋊D9C2211yescharacteristic
16C2×C4×C3⋊D9C2211yescharacteristic
17C2×C3×D36C2211yescharacteristic
18C2×C3×C4×D9C2212yes—
19C2×C9×Dic6C2211yescharacteristic
20C2×#6192C2212yes—
21C3×C42×C9C2211yescharacteristic
22C4×C3⋊Dic9C2211yescharacteristic
23C4×C3⋊Dic9C2211yescharacteristic
24C4×C3⋊Dic9C2211yescharacteristic
25C3×(C2×D9)⋊C4C2212yes—
26C9×C4⋊Dic3C2211yescharacteristic
27C9×Dic3⋊C4C2212yes—
28C4×C9×Dic3C2212yes—
29C3×C4⋊Dic9C2211yescharacteristic
30#88445C2212yes—
31(C2×D9)⋊Dic3C2212yes—
32(C2×D9)⋊Dic3C2212yes—
33(C2×D9)⋊Dic3C2212yes—
34Dic3⋊Dic9C2212yes—
35Dic9⋊Dic3C2212yes—
36Dic9⋊Dic3C2212yes—
37C42×C3⋊D3—31no—
38C3×C4×D12—31no—
39C3×C4×Dic6D311yescharacteristic
40#63916—31no—
41C4×#626—32no—
42#63872—31no—
43#63871—31no—
44#63863—31no—
45#63862—32no—
46#63855—32no—
47#63853—32no—
48C4×Q8×C9—31no—
49#63509—31no—
50#63508—31no—
51#63460—32no—
52#63457—31no—
53#63456—32no—
54#63452—31no—
55C4×D36—32no—
56C4×D36D311yescharacteristic, complemented
57#63434—32no—
58C42×D9—31no—
59C42×D9—61no—
60C22×C3⋊D9C2311yescharacteristic
61C22×C3×D9C2312yes—
62C2×C3×C4×C9C2311yescharacteristic
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3⋊Dic9C2311yescharacteristic
66C2×C3⋊Dic9C2311yescharacteristic
67C2×C3⋊Dic9C2311yescharacteristic
68C4×C3⋊D9—22no—
69C4×C3⋊D9—22no—
70C4×C3⋊D9—22no—
71C2×C9×Dic3C2312yes—
72C2×C9×Dic3C2312yes—
73C2×C3×Dic9C2312yes—
74C3×D36—22no—
75C3×C4×D9—24no—
76C9×Dic6—22no—
77#6192—24no—
78#5580—91no—
79#5557—91no—
80C2×C4×C3⋊D3—31no—
81C2×C4×C3⋊D3—31no—
82C2×C4×C3⋊D3—31no—
83C2×C3×D12—31no—
84C2×C3×C4×D3—32no—
85C2×C3×Dic6C2×D311yescharacteristic
86C2×#626—32no—
87C32×C42C2×D311yescharacteristic
88C4×C3⋊Dic3—31no—
89C4×C3⋊Dic3—31no—
90C4×C3⋊Dic3—31no—
91C3×(C2×D3)⋊C4—32no—
92C3×C4⋊Dic3—31no—
93C3×C4⋊Dic3C2×D311yescharacteristic
94C3×Dic3⋊C4C2×D312yes—
95C3×C4×Dic3C2×D312yes—
96Dic3⋊Dic3—32no—
97Dic3⋊Dic3—32no—
98Dic3⋊Dic3—32no—
99#3722—32no—
100(C2×D3)⋊Dic3—32no—
101(C2×D3)⋊Dic3—32no—
102(C2×D3)⋊Dic3—32no—
103C2×Q8×C9—31no—
104C2×D36—32no—
105C2×D36C2×D311yescharacteristic
106C2×C4×D9—31no—
107C2×C4×D9—31no—
108C2×C4×D9—31no—
109C2×C4×D9—61no—
110C2×C4×D9—61no—
111C2×C4×D9—61no—
112C2×C4×D9C2×D312yes—
113C9×C4⋊C4—32no—
114C9×C4⋊C4—31no—
115C42×C9—21no—
116C42×C9—32no—
117C42×C9C2×D311yescharacteristic
118(C2×D9)⋊C4—32no—
119(C2×D9)⋊C4—32no—
120(C2×D9)⋊C4—32no—
121(C2×D9)⋊C4—32no—
122(C2×D9)⋊C4C2×D312yes—
123C4⋊Dic9—32no—
124C4⋊Dic9C2×D311yescharacteristic
125Dic9⋊C4—32no—
126Dic9⋊C4—32no—
127C4×Dic9—31no—
128C4×Dic9—31no—
129C4×Dic9—31no—
130C4×Dic9—61no—
131C4×Dic9—61no—
132C4×Dic9—61no—
133C22×C3×C9C2411yescharacteristic
134C2×C3⋊D9—21no—
135C2×C3⋊D9—21no—
136C2×C3⋊D9#4112yes—
137C2×C3×D9—22no—
138C2×C3×D9—22no—
139C2×C3×D9—22no—
140C3×C4×C9—21no—
141C3×C4×C9#4112yes—
142C3×C4×C9#4112yes—
143C3⋊Dic9—21no—
144C3⋊Dic9—21no—
145C3⋊Dic9—21no—
146C9×Dic3—22no—
147C9×Dic3—22no—
148C3×Dic9—22no—
149C3×C4×Q8—31no—
150C3×C4×D4—91no—
151#947—91no—
152#946—91no—
153#938—91no—
154#936—91no—
155C4×#67—92no—
156#926—92no—
157#898—92no—
158#897—91no—
159#895—91no—
160#894—92no—
161#890—91no—
162#890—91no—
163#883—92no—
164#879—92no—
165C4×D12—31no—
166C4×D12—92no—
167#872—92no—
168C42×D3—91no—
169C42×D3—91no—
170C42×D3—181no—
171C4×Dic6D911yescharacteristic, complemented
172C22×C3⋊D3—31no—
173C22×C3×D3—32no—
174C2×C32×C4C22×D311yescharacteristic
175C2×C32×C4C22×D311yescharacteristic
176C2×C32×C4C22×D311yescharacteristic
177C2×C3⋊Dic3—31no—
178C2×C3⋊Dic3—31no—
179C2×C3⋊Dic3—31no—
180C4×C3⋊D3—62no—
181C4×C3⋊D3—62no—
182C4×C3⋊D3—62no—
183C2×C3×Dic3—32no—
184C2×C3×Dic3C22×D312yes—
185C2×C3×Dic3C22×D312yes—
186C3×D12—62no—
187C3×C4×D3—64no—
188C3×Dic6—22no—
189#626—64no—
190C22×D9—31no—
191C22×D9—61no—
192C22×D9C22×D312yes—
193Q8×C9—62no—
194C2×C4×C9—21no—
195C2×C4×C9—21no—
196C2×C4×C9—21no—
197C2×C4×C9—32no—
198C2×C4×C9—32no—
199C2×C4×C9C22×D311yescharacteristic
200C2×C4×C9C22×D311yescharacteristic
201C2×C4×C9C22×D311yescharacteristic
202C2×Dic9—31no—
203C2×Dic9—31no—
204C2×Dic9—31no—
205C2×Dic9—61no—
206C2×Dic9—61no—
207C2×Dic9—61no—
208C2×Dic9C22×D312yes—
209D36—22no—
210D36—64no—
211C4×D9—24no—
212C4×D9—62no—
213C4×D9—64no—
214C4×D9—64no—
215C4×D9—64no—
216C4×D9—62no—
217C4×D9—62no—
218#529—271noSylow
219C2×C3×C9C2×#4111yescharacteristic
220C2×C3×C9C2×#4111yescharacteristic
221C2×C3×C9C2×#4111yescharacteristic
222C3⋊D9—22no—
223C3×D9—42no—
224C2×C3×Q8—31no—
225C2×C3×D4—91no—
226C22×C3×C4—92no—
227C2×#67—92no—
228C2×D12—31no—
229C2×D12—92no—
230C2×C4×D3—32no—
231C2×C4×D3—91no—
232C2×C4×D3—91no—
233C2×C4×D3—91no—
234C2×C4×D3—91no—
235C2×C4×D3—91no—
236C2×C4×D3—91no—
237C2×C4×D3—181no—
238C2×C4×D3—181no—
239C2×C4×D3—181no—
240C2×Dic6C2×D911yescharacteristic
241C3×C4⋊C4—31no—
242C3×C4⋊C4—32no—
243C3×C4⋊C4—91no—
244C3×C22⋊C4—92no—
245C3×C42—21no—
246C3×C42—32no—
247C3×C42C2×D911yescharacteristic
248C3×C42D3211yescharacteristic
249#217—92no—
250#217—92no—
251#217—92no—
252(C2×D3)⋊C4—32no—
253(C2×D3)⋊C4—92no—
254(C2×D3)⋊C4—92no—
255(C2×D3)⋊C4—92no—
256(C2×D3)⋊C4—92no—
257(C2×D3)⋊C4—92no—
258C4⋊Dic3—31no—
259C4⋊Dic3—92no—
260C4⋊Dic3—92no—
261C4⋊Dic3—92no—
262C4⋊Dic3C2×D911yescharacteristic
263Dic3⋊C4—92no—
264Dic3⋊C4—92no—
265Dic3⋊C4—92no—
266Dic3⋊C4C2×D912yes—
267C4×Dic3—91no—
268C4×Dic3—91no—
269C4×Dic3—91no—
270C4×Dic3—91no—
271C4×Dic3—91no—
272C4×Dic3—91no—
273C4×Dic3—181no—
274C4×Dic3—181no—
275C4×Dic3—181no—
276C4×Dic3C2×D912yes—
277C22×C32C23×D311yescharacteristic
278C2×C3⋊D3—32no—
279C2×C3⋊D3—61no—
280C2×C3⋊D3—61no—
281C2×C3×D3—62no—
282C2×C3×D3—62no—
283C2×C3×D3—62no—
284C32×C4—21no—
285C32×C4#23512yes—
286C32×C4#23912yes—
287C3⋊Dic3—61no—
288C3⋊Dic3—61no—
289C3⋊Dic3—61no—
290C3×Dic3—22no—
291C3×Dic3—22no—
292C3×Dic3—62no—
293C22×C9—21no—
294C22×C9C23×D311yescharacteristic
295C2×D9—22no—
296C2×D9—22no—
297C2×D9—22no—
298C2×D9—32no—
299C2×D9—62no—
300C2×D9—62no—
301C2×D9—62no—
302C2×D9—61no—
303C2×D9—61no—
304C4×C9—22no—
305C4×C9—21no—
306C4×C9—22no—
307C4×C9—22no—
308C4×C9—62no—
309C4×C9—62no—
310C4×C9#23512yes—
311C4×C9#23712yes—
312Dic9—22no—
313Dic9—62no—
314Dic9—62no—
315Dic9—62no—
316Dic9—61no—
317Dic9—61no—
318Dic9—61no—
319#126—272no—
320#125—271no—
321#124—271no—
322#123—272no—
323C22⋊Q8—271no—
324#121—271no—
325C4×Q8—31no—
326C4×D4—91no—
327C4×D4—272no—
328#117—272no—
329C2×C42—271no—
330C3×C9#52911yescharacteristic, complemented, Sylow
331C23×C3—92no—
332C22×D3—32no—
333C22×D3—91no—
334C22×D3—91no—
335C22×D3—181no—
336C3×Q8—62no—
337C3×D4—182no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—32no—
342C2×C3×C4—32no—
343C2×C3×C4—92no—
344C2×C3×C4—184no—
345C2×C3×C4C22×D911yescharacteristic
346C2×C3×C4C22×D911yescharacteristic
347C2×C3×C4C22×D911yescharacteristic
348C2×C3×C4C2×D3211yescharacteristic
349C2×C3×C4C2×D3211yescharacteristic
350C2×C3×C4C2×D3211yescharacteristic
351#67—184no—
352C2×Dic3—32no—
353C2×Dic3—91no—
354C2×Dic3—91no—
355C2×Dic3—91no—
356C2×Dic3—91no—
357C2×Dic3—91no—
358C2×Dic3—91no—
359C2×Dic3—181no—
360C2×Dic3—181no—
361C2×Dic3—181no—
362C2×Dic3C22×D912yes—
363C2×Dic3C22×D912yes—
364D12—62no—
365D12—184no—
366C4×D3—64no—
367C4×D3—182no—
368C4×D3—182no—
369C4×D3—184no—
370C4×D3—184no—
371C4×D3—184no—
372C4×D3—182no—
373C4×D3—182no—
374C4×D3—182no—
375C4×D3—182no—
376Dic6—22no—
377C2×C32C2×#23511yescharacteristic
378C2×C32C2×#23911yescharacteristic
379C2×C32D3×#4111yescharacteristic
380C3⋊D3—62no—
381C3×D3—122no—
382C2×C9—21no—
383C2×C9—21no—
384C2×C9—21no—
385C2×C9C2×#23511yescharacteristic
386C2×C9C2×#23711yescharacteristic
387C2×C9D3×#4111yescharacteristic
388D9—42no—
389D9—64no—
390D9—62no—
391C2×Q8—31no—
392C2×D4—91no—
393C2×D4—272no—
394C22×C4—92no—
395C22×C4—271no—
396C22×C4—271no—
397C22×C4—271no—
398C4⋊C4—31no—
399C4⋊C4—32no—
400C4⋊C4—91no—
401C4⋊C4—272no—
402C4⋊C4—272no—
403C4⋊C4—272no—
404C22⋊C4—92no—
405C22⋊C4—272no—
406C22⋊C4—272no—
407C22⋊C4—272no—
408C22⋊C4—272no—
409C42—32no—
410C42—271no—
411C42—271no—
412C42—271no—
413C42D3×D911yescharacteristic
414C22×C3—21no—
415C22×C3—182no—
416C22×C3—182no—
417C22×C3—182no—
418C22×C3C23×D911yescharacteristic
419C22×C3C22×D3211yescharacteristic
420C2×D3—62no—
421C2×D3—62no—
422C2×D3—62no—
423C2×D3—92no—
424C2×D3—92no—
425C2×D3—182no—
426C2×D3—182no—
427C2×D3—182no—
428C2×D3—181no—
429C2×D3—181no—
430C2×D3—181no—
431C2×D3—181no—
432C3×C4—22no—
433C3×C4—21no—
434C3×C4—21no—
435C3×C4—22no—
436C3×C4—22no—
437C3×C4—62no—
438C3×C4—62no—
439C3×C4—182no—
440C3×C4#369712yes—
441C3×C4#370112yes—
442C3×C4#379612yes—
443C3×C4#379812yes—
444Dic3—22no—
445Dic3—22no—
446Dic3—62no—
447Dic3—182no—
448Dic3—182no—
449Dic3—182no—
450Dic3—181no—
451Dic3—181no—
452Dic3—181no—
453Dic3—181no—
454Dic3—181no—
455Dic3—181no—
456C32#558011yescharacteristic
457C9—21no—
458C9#555711yescharacteristic, complemented
459C23—92no—
460C23—271no—
461Q8—62no—
462D4—182no—
463D4—544no—
464C2×C4—32no—
465C2×C4—32no—
466C2×C4—92no—
467C2×C4—184no—
468C2×C4—271no—
469C2×C4—271no—
470C2×C4—271no—
471C2×C4—542no—
472C2×C4—542no—
473C2×C4—542no—
474C2×C4C2×D3×D911yescharacteristic
475C2×C4C2×D3×D911yescharacteristic
476C2×C4C2×D3×D911yescharacteristic
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—21no—
480C2×C3—362no—
481C2×C3C2×#369711yescharacteristic
482C2×C3C2×#370111yescharacteristic
483C2×C3#41×D911yescharacteristic
484C2×C3C2×#379611yescharacteristic
485C2×C3C2×#379811yescharacteristic
486C2×C3#6430611yescharacteristic
487D3—122no—
488D3—184no—
489D3—182no—
490D3—182no—
491C22—182no—
492C22—182no—
493C22—182no—
494C22—272no—
495C22—541no—
496C22—541no—
497C22C22×D3×D911yescharacteristic
498C4—21no—
499C4—62no—
500C4—62no—
501C4—182no—
502C4—541no—
503C4—541no—
504C4—541no—
505C4#8863412yes—
506C4#8864012yes—
507C3—21no—
508C3#1058968211yescharacteristic, complemented
509C3#1059387511yescharacteristic
510C2—362no—
511C2—542no—
512C2C2×#8863411yescharacteristic
513C2C2×#8864011yescharacteristic
514C2#1172247511yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
prime radical#11721007C2
hypercenterC42D3×D9
hypocenterC3×C9#529
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C42×C3⋊D9C2semidirect
2. C3×C4×D36C2semidirect
3. C4×C9×Dic6C2semidirect
5. C4×#6192C2semidirect
6. #11720951C2semidirect
7. #11720950C2semidirect
8. #11720943C2semidirect
9. #11720941C2semidirect
10. #11720937C2semidirect
11. #11720929C2semidirect
56. C4×D36D3semidirect
171. C4×Dic6D9semidirect
330. C3×C9#529semidirect
458. C9#5557semidirect
508. C3#10589682semidirect

Maximal subgroups

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

subgroupindex
C42×C3⋊D92
C3×C4×D362
C4×C9×Dic62
#117210072
C4×#61922
#117209512
#117209502
#117209432
#117209412
#117209372
#117209292
#105938753
#105896823