Finite Group Databasebeta

← #13147575

Subgroups — #13147575

Order 1728.

784 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721712C211yesmaximal, characteristic, complemented
2C3×C4×D36C211yesmaximal, characteristic, complemented
3C4×C9×Dic6C211yesmaximal, characteristic, complemented
4#11721006C211yesmaximal, characteristic
5#11721003C212yesmaximal, complemented
6#11721002C212yesmaximal, complemented
7#11720993C211yesmaximal, characteristic, complemented
8#11720948C211yesmaximal, characteristic, complemented
9#11720942C212yesmaximal, complemented
10#11720938C212yesmaximal, complemented
11#11720912C211yesmaximal, characteristic, complemented
12#10593978—31nomaximal
13#10589687—31nomaximal
14C2×C4⋊(C3⋊D9)C2211yescharacteristic
15C2×#6226C2211yescharacteristic
16C2×C3×D36C2211yescharacteristic
17C2×C3×C4×D9C2212yes—
18C2×C9×Dic6C2211yescharacteristic
19C2×#6192C2212yes—
20C2×#6192C2212yes—
21C2×(C3⋊D9)⋊C4C2212yes—
22C3×C42×C9C2211yescharacteristic
23#88536C2212yes—
24#88536C2212yes—
25C3×(C2×D9)⋊C4C2212yes—
26C9×C4⋊Dic3C2211yescharacteristic
27C9×Dic3⋊C4C2211yescharacteristic
28C9×Dic3⋊C4C2211yescharacteristic
29C4×C9×Dic3C2211yescharacteristic
30C4×C9×Dic3C2211yescharacteristic
31C3×C4⋊Dic9C2211yescharacteristic
32#88445C2212yes—
33(C2×D9)⋊Dic3C2212yes—
34(C2×D9)⋊Dic3C2212yes—
35#88442C2212yes—
36Dic3×Dic9C2212yes—
37#64084—31no—
38C3×C4×D12—31no—
39C3×C4×Dic6D311yescharacteristic
40#63911—31no—
41#63907—32no—
42#63904—32no—
43#63894—31no—
44#63860—31no—
45#63854—32no—
46#63850—32no—
47#63844—31no—
48C4×Q8×C9—31no—
49#63509—32no—
50#63456—32no—
51#63456—32no—
52#63455—32no—
53#63437—31no—
54#63437—31no—
55#63437—61no—
56#63436—31no—
57C4×D36—31no—
58C4×D36D311yescharacteristic, complemented
59C4×Dic18—31no—
60C22×C3⋊D9C2312yes—
61C22×C3×D9C2312yes—
62C2×C3×C4×C9C2311yescharacteristic
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3⋊Dic9C2312yes—
66C4⋊(C3⋊D9)—22no—
67#6226—22no—
68C2×C9×Dic3C2311yescharacteristic
69C2×C9×Dic3C2311yescharacteristic
70C2×C9×Dic3C2311yescharacteristic
71C2×C9×Dic3C2311yescharacteristic
72C2×C3×Dic9C2312yes—
73C3×D36—22no—
74C3×C4×D9—42no—
75C9×Dic6—22no—
76#6192—24no—
77#6192—42no—
78(C3⋊D9)⋊C4—42no—
79#5585—91no—
80#5559—91no—
81C2×C4⋊(C3⋊D3)—31no—
82C2×#634—31no—
83C2×C3×D12—31no—
84C2×C3×C4×D3—32no—
85C2×C3×Dic6C2×D311yescharacteristic
86C2×#626—32no—
87C2×#626—32no—
88C2×#624—32no—
89C32×C42C2×D311yescharacteristic
90#3752—32no—
91#3752—32no—
92C3×(C2×D3)⋊C4—32no—
93C3×C4⋊Dic3—31no—
94C3×C4⋊Dic3C2×D311yescharacteristic
95C3×Dic3⋊C4C2×D311yescharacteristic
96C3×Dic3⋊C4C2×D311yescharacteristic
97C3×C4×Dic3C2×D311yescharacteristic
98C3×C4×Dic3C2×D311yescharacteristic
99#3724—32no—
100#3722—32no—
101(C2×D3)⋊Dic3—32no—
102(C2×D3)⋊Dic3—32no—
103Dic32—32no—
104C2×Q8×C9—31no—
105C2×D36—32no—
106C2×D36—31no—
107C2×D36—32no—
108C2×D36—61no—
109C2×D36C2×D311yescharacteristic
110C2×C4×D9—32no—
111C2×C4×D9C2×D312yes—
112C2×Dic18—31no—
113C2×Dic18—61no—
114C9×C4⋊C4—31no—
115C9×C4⋊C4—31no—
116C9×C4⋊C4—31no—
117C42×C9—21no—
118C42×C9—31no—
119C42×C9—31no—
120C42×C9C2×D311yescharacteristic
121(C2×D9)⋊C4—32no—
122(C2×D9)⋊C4—32no—
123(C2×D9)⋊C4—32no—
124(C2×D9)⋊C4—32no—
125(C2×D9)⋊C4—32no—
126(C2×D9)⋊C4—62no—
127(C2×D9)⋊C4—62no—
128(C2×D9)⋊C4C2×D312yes—
129C4⋊Dic9C2×D311yescharacteristic
130Dic9⋊C4—32no—
131C4×Dic9—32no—
132C22×C3×C9C2411yescharacteristic
133C2×C3⋊D9—22no—
134C2×C3⋊D9—22no—
135C2×C3⋊D9—22no—
136C2×C3×D9—22no—
137C2×C3×D9—22no—
138C2×C3×D9—22no—
139C3×C4×C9—21no—
140C3×C4×C9—21no—
141C3×C4×C9#4112yes—
142C3⋊Dic9—22no—
143C9×Dic3—21no—
144C9×Dic3—21no—
145C9×Dic3—21no—
146C9×Dic3#4112yes—
147C3×Dic9—22no—
148C3×C4×Q8—31no—
149C3×C4×D4—91no—
150#947—92no—
151#940—91no—
152#936—91no—
153#929—92no—
154C4×#67—92no—
155#894—92no—
156#894—92no—
157#893—92no—
158#893—91no—
159#887—91no—
160#885—92no—
161#883—92no—
162#875—91no—
163#875—91no—
164#875—91no—
165#875—181no—
166#874—91no—
167C4×D12—31no—
168C4×D12—91no—
169C4×Dic6—91no—
170C4×Dic6D911yescharacteristic, complemented
171C22×C3⋊D3—32no—
172C22×C3×D3—32no—
173C2×C32×C4C22×D311yescharacteristic
174C2×C32×C4C22×D311yescharacteristic
175C2×C32×C4C22×D311yescharacteristic
176C2×C3⋊Dic3—32no—
177C4⋊(C3⋊D3)—62no—
178#634—62no—
179C2×C3×Dic3—32no—
180C2×C3×Dic3C22×D311yescharacteristic
181C2×C3×Dic3C22×D311yescharacteristic
182C2×C3×Dic3C22×D311yescharacteristic
183C2×C3×Dic3C22×D311yescharacteristic
184C3×D12—62no—
185C3×C4×D3—122no—
186C3×Dic6—22no—
187#626—64no—
188#626—122no—
189#624—122no—
190C22×D9—32no—
191C22×D9—62no—
192C22×D9C22×D312yes—
193Q8×C9—62no—
194C2×C4×C9—21no—
195C2×C4×C9—21no—
196C2×C4×C9—21no—
197C2×C4×C9—31no—
198C2×C4×C9—31no—
199C2×C4×C9—31no—
200C2×C4×C9—31no—
201C2×C4×C9C22×D311yescharacteristic
202C2×C4×C9C22×D311yescharacteristic
203C2×C4×C9C22×D311yescharacteristic
204C2×Dic9—32no—
205C2×Dic9—62no—
206C2×Dic9C22×D312yes—
207D36—22no—
208D36—64no—
209D36—62no—
210D36—122no—
211D36—122no—
212C4×D9—42no—
213C4×D9—122no—
214Dic18—62no—
215Dic18—122no—
216#559—271noSylow
217C2×C3×C9C2×#4111yescharacteristic
218C2×C3×C9C2×#4111yescharacteristic
219C2×C3×C9H(5,𝔽2)11yescharacteristic
220C3⋊D9—42no—
221C3×D9—42no—
222C2×C3×Q8—31no—
223C2×C3×D4—91no—
224C22×C3×C4—92no—
225C2×#67—92no—
226C2×#67—92no—
227C2×D12—31no—
228C2×D12—92no—
229C2×D12—91no—
230C2×D12—91no—
231C2×D12—92no—
232C2×D12—181no—
233C2×C4×D3—32no—
234C2×C4×D3—92no—
235C2×C4×D3—92no—
236C2×Dic6—91no—
237C2×Dic6—91no—
238C2×Dic6—181no—
239C2×Dic6C2×D911yescharacteristic
240C3×C4⋊C4—31no—
241C3×C4⋊C4—31no—
242C3×C4⋊C4—31no—
243C3×C4⋊C4—91no—
244C3×C22⋊C4—92no—
245C3×C42—21no—
246C3×C42—31no—
247C3×C42—31no—
248C3×C42C2×D911yescharacteristic
249C3×C42D3211yescharacteristic
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—
258(C2×D3)⋊C4—92no—
259(C2×D3)⋊C4—92no—
260(C2×D3)⋊C4—92no—
261(C2×D3)⋊C4—182no—
262(C2×D3)⋊C4—182no—
263C4⋊Dic3—31no—
264C4⋊Dic3C2×D911yescharacteristic
265Dic3⋊C4—92no—
266Dic3⋊C4—92no—
267Dic3⋊C4C2×D911yescharacteristic
268Dic3⋊C4C2×D911yescharacteristic
269C4×Dic3—92no—
270C4×Dic3—92no—
271C4×Dic3C2×D911yescharacteristic
272C4×Dic3C2×D911yescharacteristic
273C22×C32C23×D311yescharacteristic
274C2×C3⋊D3—62no—
275C2×C3⋊D3—62no—
276C2×C3⋊D3—62no—
277C2×C3×D3—62no—
278C2×C3×D3—62no—
279C2×C3×D3—62no—
280C32×C4—21no—
281C32×C4—21no—
282C32×C4#23912yes—
283C3⋊Dic3—62no—
284C3×Dic3—21no—
285C3×Dic3—21no—
286C3×Dic3—21no—
287C3×Dic3—62no—
288C3×Dic3#23512yes—
289C22×C9—21no—
290C22×C9C23×D311yescharacteristic
291C2×D9—22no—
292C2×D9—22no—
293C2×D9—22no—
294C2×D9—62no—
295C2×D9—62no—
296C2×D9—62no—
297C2×D9—122no—
298C2×D9—122no—
299C2×D9—122no—
300C4×C9—21no—
301C4×C9—21no—
302C4×C9—22no—
303C4×C9—32no—
304C4×C9—41no—
305C4×C9—41no—
306C4×C9—61no—
307C4×C9—61no—
308C4×C9—61no—
309C4×C9#23512yes—
310Dic9—22no—
311Dic9—62no—
312Dic9—122no—
313#127—271no—
314#124—271no—
315#124—272no—
316#124—271no—
317#123—272no—
318#123—272no—
319C4×Q8—31no—
320C4×Q8—271no—
321C4×D4—91no—
322C4×D4—271no—
323C4×D4—272no—
324C3×C9#55911yescharacteristic, complemented, Sylow
325C23×C3—92no—
326C22×D3—32no—
327C22×D3—92no—
328C22×D3—92no—
329C22×D3—182no—
330C3×Q8—62no—
331C3×D4—182no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—31no—
336C2×C3×C4—31no—
337C2×C3×C4—31no—
338C2×C3×C4—31no—
339C2×C3×C4—92no—
340C2×C3×C4—362no—
341C2×C3×C4C22×D911yescharacteristic
342C2×C3×C4C22×D911yescharacteristic
343C2×C3×C4C22×D911yescharacteristic
344C2×C3×C4C2×D3211yescharacteristic
345C2×C3×C4C2×D3211yescharacteristic
346C2×C3×C4C2×D3211yescharacteristic
347#67—184no—
348#67—362no—
349C2×Dic3—32no—
350C2×Dic3—92no—
351C2×Dic3—92no—
352C2×Dic3—182no—
353C2×Dic3C22×D911yescharacteristic
354C2×Dic3C22×D911yescharacteristic
355C2×Dic3C22×D911yescharacteristic
356C2×Dic3C22×D911yescharacteristic
357D12—62no—
358D12—184no—
359D12—182no—
360D12—182no—
361D12—362no—
362D12—362no—
363C4×D3—122no—
364C4×D3—362no—
365C4×D3—362no—
366Dic6—22no—
367Dic6—182no—
368Dic6—182no—
369Dic6—362no—
370C2×C32C2×#23511yescharacteristic
371C2×C32C2×#23911yescharacteristic
372C2×C32#100911yescharacteristic
373C3⋊D3—122no—
374C3×D3—122no—
375C2×C9—21no—
376C2×C9—21no—
377C2×C9—21no—
378C2×C9C2×#23511yescharacteristic
379C2×C9#100411yescharacteristic
380C2×C9D3×#4111yescharacteristic
381D9—42no—
382D9—122no—
383D9—242no—
384C2×Q8—31no—
385C2×Q8—271no—
386C2×D4—91no—
387C2×D4—272no—
388C2×D4—272no—
389C2×D4—271no—
390C22×C4—92no—
391C22×C4—272no—
392C4⋊C4—31no—
393C4⋊C4—31no—
394C4⋊C4—31no—
395C4⋊C4—91no—
396C4⋊C4—272no—
397C22⋊C4—92no—
398C22⋊C4—272no—
399C22⋊C4—272no—
400C22⋊C4—272no—
401C22⋊C4—272no—
402C22⋊C4—272no—
403C42—31no—
404C42—31no—
405C42—272no—
406C42D3×D911yescharacteristic
407C22×C3—21no—
408C22×C3—182no—
409C22×C3—182no—
410C22×C3—182no—
411C22×C3C23×D911yescharacteristic
412C22×C3C22×D3211yescharacteristic
413C2×D3—62no—
414C2×D3—62no—
415C2×D3—62no—
416C2×D3—182no—
417C2×D3—182no—
418C2×D3—182no—
419C2×D3—182no—
420C2×D3—182no—
421C2×D3—182no—
422C2×D3—362no—
423C2×D3—362no—
424C2×D3—362no—
425C3×C4—21no—
426C3×C4—21no—
427C3×C4—21no—
428C3×C4—21no—
429C3×C4—22no—
430C3×C4—32no—
431C3×C4—41no—
432C3×C4—41no—
433C3×C4—61no—
434C3×C4—61no—
435C3×C4—61no—
436C3×C4—182no—
437C3×C4#370112yes—
438C3×C4#379912yes—
439Dic3—21no—
440Dic3—21no—
441Dic3—21no—
442Dic3—62no—
443Dic3—182no—
444Dic3—182no—
445Dic3—362no—
446Dic3#369712yes—
447C32#558511yescharacteristic
448C9—21no—
449C9#555911yescharacteristic, complemented
450C23—92no—
451C23—272no—
452Q8—62no—
453Q8—542no—
454D4—182no—
455D4—544no—
456D4—542no—
457D4—1082no—
458C2×C4—31no—
459C2×C4—31no—
460C2×C4—31no—
461C2×C4—31no—
462C2×C4—92no—
463C2×C4—272no—
464C2×C4—362no—
465C2×C4—1082no—
466C2×C4C2×D3×D911yescharacteristic
467C2×C4C2×D3×D911yescharacteristic
468C2×C4C2×D3×D911yescharacteristic
469C2×C3—21no—
470C2×C3—21no—
471C2×C3—21no—
472C2×C3—362no—
473C2×C3C2×#369711yescharacteristic
474C2×C3C2×#370111yescharacteristic
475C2×C3#6371511yescharacteristic
476C2×C3C2×#379911yescharacteristic
477C2×C3D3×#23511yescharacteristic
478C2×C3#6430711yescharacteristic
479D3—122no—
480D3—362no—
481D3—362no—
482D3—722no—
483C22—182no—
484C22—182no—
485C22—182no—
486C22—542no—
487C22—542no—
488C22—542no—
489C22C22×D3×D911yescharacteristic
490C4—21no—
491C4—21no—
492C4—32no—
493C4—61no—
494C4—61no—
495C4—61no—
496C4—182no—
497C4—542no—
498C4#8863512yes—
499C3—21no—
500C3#1058968711yescharacteristic, complemented
501C3#1059397811yescharacteristic
502C2—362no—
503C2—1082no—
504C2C2×#8863511yescharacteristic
505C2D3×#369711yescharacteristic
506C2#1172247711yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
prime radical#11721006C2
hypercenterC42D3×D9
hypocenterC3×C9#559
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #11721712C2semidirect
2. C3×C4×D36C2semidirect
3. C4×C9×Dic6C2semidirect
5. #11721003C2semidirect
6. #11721002C2semidirect
7. #11720993C2semidirect
8. #11720948C2semidirect
9. #11720942C2semidirect
10. #11720938C2semidirect
11. #11720912C2semidirect
58. C4×D36D3semidirect
170. C4×Dic6D9semidirect
324. C3×C9#559semidirect
449. C9#5559semidirect
500. C3#10589687semidirect

Maximal subgroups

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

subgroupindex
#117217122
C3×C4×D362
C4×C9×Dic62
#117210062
#117210032
#117210022
#117209932
#117209482
#117209422
#117209382
#117209122
#105939783
#105896873