Finite Group Databasebeta

← #13150334

Subgroups — #13150334

Order 1728.

808 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721819C211yesmaximal, characteristic, complemented
2C3×C4×#611C211yesmaximal, characteristic, complemented
3C9×#926C211yesmaximal, characteristic, complemented
4#11721275C212yesmaximal, complemented
5#11721271C212yesmaximal, complemented
6#11721004C211yesmaximal, characteristic, complemented
7#11720992C211yesmaximal, characteristic, complemented
8D9×C4⋊Dic3C211yesmaximal, characteristic, complemented
9#11720977C212yesmaximal, complemented
10#11720935C211yesmaximal, characteristic, complemented
11#11720926C212yesmaximal, complemented
12#10594946—31nomaximal
13#10589632—31nomaximal
14C22×C3×C4×C9C2211yescharacteristic
15C2×#6233C2212yes—
16C2×C4⋊(C3⋊D9)C2211yescharacteristic
17C2×C3×#611C2211yescharacteristic
18C2×C3×C4×D9C2211yescharacteristic
19C2×#6192C2212yes—
20C2×(C3⋊D9)⋊C4C2212yes—
21C2×Dic3×D9C2212yes—
22#88536C2212yes—
23C4⋊(C3⋊Dic9)C2211yescharacteristic
24C9×#217C2212yes—
25C3×#3676C2211yescharacteristic
26C3×(C2×D9)⋊C4C2211yescharacteristic
27C9×C4⋊Dic3C2211yescharacteristic
28C9×C4⋊Dic3C2211yescharacteristic
29C4×C9×Dic3C2212yes—
30C3×Dic9⋊C4C2211yescharacteristic
31C3×C4×Dic9C2211yescharacteristic
32#88445C2212yes—
33#88445C2212yes—
34Dic3⋊Dic9C2212yes—
35#88442C2212yes—
36Dic9⋊Dic3C2212yes—
37#64139—31no—
38C3×C4×#67—31no—
39C3×#926D311yescharacteristic
40#63975—32no—
41#63971—32no—
42#63909—31no—
43#63894—31no—
44D3×C4⋊Dic3—31no—
45#63880—32no—
46#63869—31no—
47#63868—32no—
48C9×#117—31no—
49#63492—31no—
50#63492—61no—
51C4×#611D311yescharacteristic, complemented
52#63457—31no—
53#63455—31no—
54C4⋊C4×D9—31no—
55#63452—31no—
56#63446—32no—
57#63445—32no—
58#63438—32no—
59C4×D36—32no—
60C23×C3×C9C2311yescharacteristic
61C22×C3⋊D9C2312yes—
62C22×C3×D9C2311yescharacteristic
63C2×C3×C4×C9—22no—
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3×C4×C9C2311yescharacteristic
66#6233—42no—
67C2×C3⋊Dic9C2312yes—
68C4⋊(C3⋊D9)—22no—
69C3×#611—22no—
70C2×C9×Dic3C2312yes—
71C2×C9×Dic3C2312yes—
72C2×C3×Dic9C2311yescharacteristic
73C2×C3×Dic9C2311yescharacteristic
74C2×C3×Dic9C2311yescharacteristic
75C3×C4×D9—22no—
76#6192—24no—
77(C3⋊D9)⋊C4—42no—
78Dic3×D9—42no—
79#5565—91no—
80#5530—91no—
81C22×C32×C4C2×D311yescharacteristic
82C2×#638—32no—
83C2×C4⋊(C3⋊D3)—31no—
84C2×C3×#67—31no—
85C2×C3×C4×D3—31no—
86C2×#626—32no—
87C2×#624—32no—
88C2×D3×Dic3—32no—
89#3752—32no—
90C4⋊(C3⋊Dic3)—31no—
91C3×#217—31no—
92C3×#217C2×D312yes—
93C3×(C2×D3)⋊C4—31no—
94C3×C4⋊Dic3C2×D311yescharacteristic
95C3×C4⋊Dic3C2×D311yescharacteristic
96C3×Dic3⋊C4—31no—
97C3×C4×Dic3—31no—
98C3×C4×Dic3C2×D312yes—
99#3724—32no—
100Dic3⋊Dic3—32no—
101Dic3⋊Dic3—32no—
102#3722—32no—
103#3722—32no—
104C22×C4×C9—21no—
105C22×C4×C9C2×D311yescharacteristic
106C2×#611—32no—
107C2×#611—62no—
108C2×#611C2×D311yescharacteristic
109C2×D36—32no—
110C2×D36—31no—
111C2×D36—61no—
112C2×C4×D9—32no—
113C2×C4×D9—32no—
114C2×C4×D9C2×D311yescharacteristic
115C9×C4⋊C4—31no—
116C9×C4⋊C4—31no—
117C9×C22⋊C4—32no—
118C42×C9—32no—
119#3676C2×D311yescharacteristic
120(C2×D9)⋊C4—32no—
121(C2×D9)⋊C4—32no—
122(C2×D9)⋊C4—32no—
123(C2×D9)⋊C4—62no—
124(C2×D9)⋊C4C2×D311yescharacteristic
125C4⋊Dic9—31no—
126C4⋊Dic9—32no—
127C4⋊Dic9—61no—
128Dic9⋊C4—32no—
129Dic9⋊C4—32no—
130Dic9⋊C4C2×D311yescharacteristic
131C4×Dic9C2×D311yescharacteristic
132C22×C3×C9—21no—
133C22×C3×C9—21no—
134C22×C3×C9—21no—
135C22×C3×C9C2411yescharacteristic
136C2×C3⋊D9—22no—
137C2×C3⋊D9—22no—
138C2×C3⋊D9—22no—
139C2×C3×D9—21no—
140C2×C3×D9—21no—
141C2×C3×D9#4112yes—
142C3×C4×C9—21no—
143C3×C4×C9#4112yes—
144C3⋊Dic9—22no—
145C9×Dic3—22no—
146C9×Dic3—22no—
147C3×Dic9—21no—
148C3×Dic9—21no—
149C3×Dic9—21no—
150C3×C4×D4—91no—
151C3×#117—31no—
152#939—92no—
153#930—91no—
154#930—91no—
155#930—91no—
156#930—181no—
157C4×#67—31no—
158#926D911yescharacteristic, complemented
159C2×C4⋊Dic3—91no—
160#898—92no—
161#895—91no—
162#893—91no—
163D3×C4⋊C4—91no—
164#890—91no—
165#884—92no—
166#883—92no—
167#883—92no—
168#67⋊C4—92no—
169#876—92no—
170C4×D12—92no—
171C4×D12—91no—
172#870—91no—
173C23×C32C22×D311yescharacteristic
174C22×C3⋊D3—32no—
175C22×C3×D3—31no—
176C2×C32×C4—22no—
177C2×C32×C4C22×D311yescharacteristic
178C2×C32×C4C22×D311yescharacteristic
179#638—122no—
180C2×C3⋊Dic3—32no—
181C4⋊(C3⋊D3)—62no—
182C3×#67—62no—
183C2×C3×Dic3—31no—
184C2×C3×Dic3—31no—
185C2×C3×Dic3—31no—
186C2×C3×Dic3C22×D312yes—
187C2×C3×Dic3C22×D312yes—
188C3×C4×D3—62no—
189#626—64no—
190#624—122no—
191D3×Dic3—122no—
192C23×C9—21no—
193C23×C9C22×D311yescharacteristic
194C22×D9—32no—
195C22×D9—62no—
196C22×D9C22×D311yescharacteristic
197C2×C4×C9—22no—
198C2×C4×C9—21no—
199C2×C4×C9—21no—
200C2×C4×C9—32no—
201C2×C4×C9—32no—
202C2×C4×C9—42no—
203C2×C4×C9C22×D311yescharacteristic
204C2×C4×C9C22×D311yescharacteristic
205#611—22no—
206#611—122no—
207#611—124no—
208C2×Dic9—32no—
209C2×Dic9—62no—
210C2×Dic9C22×D311yescharacteristic
211C2×Dic9C22×D311yescharacteristic
212C2×Dic9C22×D311yescharacteristic
213D36—64no—
214D36—62no—
215D36—122no—
216C4×D9—22no—
217C4×D9—122no—
218C4×D9—122no—
219#553—271noSylow
220C2×C3×C9—41no—
221C2×C3×C9C2×#4111yescharacteristic
222C2×C3×C9C2×#4111yescharacteristic
223C2×C3×C9H(5,𝔽2)11yescharacteristic
224C3⋊D9—42no—
225C3×D9—22no—
226C2×C3×D4—91no—
227C22×C3×C4—21no—
228C22×C3×C4—91no—
229C22×C3×C4C2×D911yescharacteristic
230C22×C3×C4D3211yescharacteristic
231C2×#67—31no—
232C2×#67—92no—
233C2×#67—92no—
234C2×#67—92no—
235C2×#67—182no—
236C22×Dic3—92no—
237C2×D12—92no—
238C2×D12—91no—
239C2×D12—91no—
240C2×D12—181no—
241C2×C4×D3—31no—
242C2×C4×D3—92no—
243C2×C4×D3—92no—
244C2×C4×D3—92no—
245C3×C4⋊C4—31no—
246C3×C4⋊C4—31no—
247C3×C4⋊C4—91no—
248C3×C22⋊C4—32no—
249C3×C22⋊C4—91no—
250C3×C22⋊C4—91no—
251C3×C42—32no—
252C3×C42—91no—
253#217—31no—
254#217C2×D912yes—
255(C2×D3)⋊C4—31no—
256(C2×D3)⋊C4—92no—
257(C2×D3)⋊C4—92no—
258(C2×D3)⋊C4—92no—
259(C2×D3)⋊C4—92no—
260(C2×D3)⋊C4—92no—
261(C2×D3)⋊C4—92no—
262(C2×D3)⋊C4—182no—
263C4⋊Dic3—92no—
264C4⋊Dic3—91no—
265C4⋊Dic3—91no—
266C4⋊Dic3—92no—
267C4⋊Dic3—181no—
268C4⋊Dic3C2×D911yescharacteristic
269C4⋊Dic3C2×D911yescharacteristic
270Dic3⋊C4—31no—
271Dic3⋊C4—92no—
272Dic3⋊C4—92no—
273Dic3⋊C4—92no—
274Dic3⋊C4—92no—
275C4×Dic3—31no—
276C4×Dic3C2×D912yes—
277C22×C32—21no—
278C22×C32—21no—
279C22×C32—21no—
280C22×C32C23×D311yescharacteristic
281C2×C3⋊D3—62no—
282C2×C3⋊D3—62no—
283C2×C3⋊D3—62no—
284C2×C3×D3—32no—
285C2×C3×D3—61no—
286C2×C3×D3—61no—
287C32×C4—21no—
288C32×C4#23512yes—
289C3⋊Dic3—62no—
290C3×Dic3—22no—
291C3×Dic3—22no—
292C3×Dic3—61no—
293C3×Dic3—61no—
294C3×Dic3—61no—
295C22×C9—22no—
296C22×C9—21no—
297C22×C9—21no—
298C22×C9—21no—
299C22×C9—21no—
300C22×C9—41no—
301C22×C9—41no—
302C22×C9C23×D311yescharacteristic
303C2×D9—21no—
304C2×D9—21no—
305C2×D9—62no—
306C2×D9—62no—
307C2×D9—62no—
308C2×D9—122no—
309C2×D9—122no—
310C2×D9—122no—
311C2×D9#23512yes—
312C4×C9—21no—
313C4×C9—22no—
314C4×C9—41no—
315C4×C9—62no—
316C4×C9—62no—
317C4×C9#23712yes—
318Dic9—21no—
319Dic9—21no—
320Dic9—21no—
321Dic9—62no—
322Dic9—122no—
323#126—272no—
324#125—271no—
325#123—272no—
326#121—272no—
327#121—271no—
328#121—271no—
329C4×D4—91no—
330C4×D4—272no—
331C4×D4—271no—
332#117—31no—
333C2×C4⋊C4—271no—
334C3×C9#55311yescharacteristic, complemented, Sylow
335C23×C3—21no—
336C23×C3—91no—
337C23×C3C22×D911yescharacteristic
338C23×C3C2×D3211yescharacteristic
339C22×D3—31no—
340C22×D3—92no—
341C22×D3—92no—
342C22×D3—182no—
343C3×D4—182no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—32no—
349C2×C3×C4—32no—
350C2×C3×C4—42no—
351C2×C3×C4—91no—
352C2×C3×C4—91no—
353C2×C3×C4—91no—
354C2×C3×C4—182no—
355C2×C3×C4C22×D911yescharacteristic
356C2×C3×C4C22×D911yescharacteristic
357C2×C3×C4C2×D3211yescharacteristic
358C2×C3×C4C2×D3211yescharacteristic
359#67—62no—
360#67—184no—
361#67—362no—
362#67—362no—
363#67—364no—
364C2×Dic3—31no—
365C2×Dic3—31no—
366C2×Dic3—31no—
367C2×Dic3—92no—
368C2×Dic3—92no—
369C2×Dic3—182no—
370C2×Dic3—362no—
371C2×Dic3C22×D912yes—
372C2×Dic3C22×D912yes—
373D12—184no—
374D12—182no—
375D12—182no—
376D12—362no—
377C4×D3—62no—
378C4×D3—362no—
379C4×D3—362no—
380C4×D3—362no—
381C2×C32—41no—
382C2×C32C2×#23511yescharacteristic
383C2×C32D3×#4111yescharacteristic
384C2×C32#100911yescharacteristic
385C3⋊D3—122no—
386C3×D3—62no—
387C2×C9—21no—
388C2×C9—21no—
389C2×C9—21no—
390C2×C9—42no—
391C2×C9—41no—
392C2×C9C2×#23511yescharacteristic
393C2×C9C2×#23711yescharacteristic
394C2×C9#100911yescharacteristic
395D9—22no—
396D9—122no—
397D9—242no—
398C2×D4—91no—
399C2×D4—272no—
400C2×D4—272no—
401C2×D4—271no—
402C22×C4—91no—
403C22×C4—272no—
404C22×C4—272no—
405C22×C4D3×D911yescharacteristic
406C4⋊C4—31no—
407C4⋊C4—31no—
408C4⋊C4—91no—
409C4⋊C4—272no—
410C4⋊C4—272no—
411C4⋊C4—272no—
412C4⋊C4—271no—
413C22⋊C4—32no—
414C22⋊C4—91no—
415C22⋊C4—91no—
416C22⋊C4—272no—
417C22⋊C4—272no—
418C22⋊C4—272no—
419C42—32no—
420C42—91no—
421C22×C3—22no—
422C22×C3—21no—
423C22×C3—21no—
424C22×C3—21no—
425C22×C3—21no—
426C22×C3—21no—
427C22×C3—21no—
428C22×C3—21no—
429C22×C3—41no—
430C22×C3—41no—
431C22×C3—92no—
432C22×C3—181no—
433C22×C3—181no—
434C22×C3C23×D911yescharacteristic
435C22×C3C22×D3211yescharacteristic
436C2×D3—32no—
437C2×D3—61no—
438C2×D3—61no—
439C2×D3—182no—
440C2×D3—182no—
441C2×D3—182no—
442C2×D3—182no—
443C2×D3—182no—
444C2×D3—182no—
445C2×D3—362no—
446C2×D3—362no—
447C2×D3—362no—
448C3×C4—21no—
449C3×C4—21no—
450C3×C4—22no—
451C3×C4—41no—
452C3×C4—62no—
453C3×C4—62no—
454C3×C4—181no—
455C3×C4—181no—
456C3×C4—181no—
457C3×C4#369712yes—
458C3×C4#380412yes—
459Dic3—22no—
460Dic3—22no—
461Dic3—61no—
462Dic3—61no—
463Dic3—61no—
464Dic3—182no—
465Dic3—182no—
466Dic3—362no—
467C32#553011yescharacteristic
468C9—21no—
469C9#556511yescharacteristic, complemented
470C23—91no—
471C23—272no—
472C23C2×D3×D911yescharacteristic
473D4—182no—
474D4—544no—
475D4—542no—
476D4—1082no—
477C2×C4—22no—
478C2×C4—32no—
479C2×C4—32no—
480C2×C4—91no—
481C2×C4—91no—
482C2×C4—91no—
483C2×C4—182no—
484C2×C4—272no—
485C2×C4—1082no—
486C2×C4—1082no—
487C2×C4C2×D3×D911yescharacteristic
488C2×C4C2×D3×D911yescharacteristic
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—21no—
492C2×C3—42no—
493C2×C3—41no—
494C2×C3—41no—
495C2×C3—182no—
496C2×C3C2×#369711yescharacteristic
497C2×C3#41×D911yescharacteristic
498C2×C3#6371511yescharacteristic
499C2×C3D3×#23511yescharacteristic
500C2×C3#6430811yescharacteristic
501C2×C3C2×#380411yescharacteristic
502D3—62no—
503D3—362no—
504D3—362no—
505D3—722no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—92no—
510C22—181no—
511C22—181no—
512C22—542no—
513C22—542no—
514C22—542no—
515C22C22×D3×D911yescharacteristic
516C4—21no—
517C4—62no—
518C4—62no—
519C4—181no—
520C4—181no—
521C4—181no—
522C4—542no—
523C4#8865812yes—
524C3—21no—
525C3#1058963211yescharacteristic, complemented
526C3#1059494611yescharacteristic
527C2—41no—
528C2—182no—
529C2—1082no—
530C2D9×#23511yescharacteristic
531C2#1172247811yescharacteristic
532C2C2×#8865811yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
hypercenterC22×C4D3×D9
hypocenterC3×C9#553
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. #11721819C2semidirect
2. C3×C4×#611C2semidirect
3. C9×#926C2semidirect
4. #11721275C2semidirect
5. #11721271C2semidirect
6. #11721004C2semidirect
7. #11720992C2semidirect
8. D9×C4⋊Dic3C2semidirect
9. #11720977C2semidirect
10. #11720935C2semidirect
11. #11720926C2semidirect
51. C4×#611D3semidirect
158. #926D9semidirect
334. C3×C9#553semidirect
469. C9#5565semidirect
525. C3#10589632semidirect

Maximal subgroups

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

subgroupindex
#117218192
C3×C4×#6112
C9×#9262
#117212752
#117212712
#117210042
#117209922
D9×C4⋊Dic32
#117209772
#117209352
#117209262
#105949463
#105896323