Finite Group Databasebeta

← #13150404

Subgroups — #13150404

Order 1728.

718 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721815C211yesmaximal, characteristic, complemented
2C4×C9×#67C211yesmaximal, characteristic
3C3×#63486C211yesmaximal, characteristic, complemented
4#11721280C212yesmaximal
5#11721246C212yesmaximal, complemented
6D3×C4⋊Dic9C211yesmaximal, characteristic, complemented
7#11721011C212yesmaximal, complemented
8#11720960C211yesmaximal, characteristic, complemented
9#11720928C212yesmaximal, complemented
10#11720927C211yesmaximal, characteristic, complemented
11Dic3×Dic18C211yesmaximal, characteristic, complemented
12#10594942—31nomaximal
13#10589643—31nomaximal
14C22×C3×C4×C9C2211yescharacteristic
15C2×C9×#67C2211yescharacteristic
16C2×C4×D3×C9C2211yescharacteristic
17C2×C3×Dic18C2211yescharacteristic
18C2×D3×Dic9C2212yes—
19#88551C2212yes—
20C4⋊(C3⋊Dic9)C2211yescharacteristic
21C4⋊(C3⋊Dic9)C2211yescharacteristic
22C4×C3⋊Dic9C2212yes—
23C9×#217C2211yescharacteristic
24C3×#3676C2212yes—
25C9×(C2×D3)⋊C4C2211yescharacteristic
26C9×Dic3⋊C4C2211yescharacteristic
27C4×C9×Dic3C2211yescharacteristic
28C3×C4⋊Dic9C2211yescharacteristic
29C3×Dic9⋊C4C2212yes—
30(C2×D3)⋊Dic9C2212yes—
31Dic3⋊Dic9C2212yes—
32#88442C2212yes—
33#88442C2212yes—
34Dic9⋊Dic3C2212yes—
35Dic9⋊Dic3C2212yes—
36Dic3×Dic9C2212yes—
37#64135—31no—
38C3×C4×#67D311yescharacteristic
39C3×#924—31no—
40#63969—32no—
41#63956—32no—
42D3×C4⋊Dic3—31no—
43#63878—32no—
44#63870—32no—
45#63869—31no—
46#63856—31no—
47Dic3×Dic6—31no—
48C4×D4×C9—31no—
49#63497—32no—
50#63488—31no—
51#63488—61no—
52C2×C4⋊Dic9—31no—
53#63486—31no—
54#63486D311yescharacteristic, complemented
55#63451—32no—
56#63441—32no—
57#63439—32no—
58#63432—31no—
59C4×Dic18—31no—
60C23×C3×C9C2311yescharacteristic
61C22×D3×C9C2311yescharacteristic
62C2×C3×C4×C9—22no—
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3⋊Dic9C2312yes—
66C2×C3⋊Dic9C2312yes—
67C9×#67—22no—
68C2×C9×Dic3C2311yescharacteristic
69C2×C9×Dic3C2311yescharacteristic
70C2×C9×Dic3C2311yescharacteristic
71C2×C3×Dic9C2312yes—
72C2×C3×Dic9C2312yes—
73C4×D3×C9—22no—
74C3×Dic18—22no—
75D3×Dic9—24no—
76#5639—91no—
77#5541—91no—
78C22×C32×C4C2×D311yescharacteristic
79C2×C3×#67C2×D311yescharacteristic
80C2×C3×C4×D3C2×D311yescharacteristic
81C2×C3×Dic6—31no—
82C2×D3×Dic3—32no—
83#3757—32no—
84C4⋊(C3⋊Dic3)—31no—
85C4⋊(C3⋊Dic3)—31no—
86C4×C3⋊Dic3—32no—
87C3×#217—32no—
88C3×#217C2×D311yescharacteristic
89C3×(C2×D3)⋊C4C2×D311yescharacteristic
90C3×C4⋊Dic3—31no—
91C3×Dic3⋊C4—32no—
92C3×Dic3⋊C4C2×D311yescharacteristic
93C3×C4×Dic3C2×D311yescharacteristic
94#3724—32no—
95#3724—32no—
96Dic3⋊Dic3—32no—
97Dic3⋊Dic3—32no—
98Dic3⋊Dic3—32no—
99(C2×D3)⋊Dic3—32no—
100Dic32—32no—
101C2×D4×C9—31no—
102C22×C4×C9—21no—
103C22×C4×C9—31no—
104C22×C4×C9C2×D311yescharacteristic
105C22×Dic9—32no—
106C2×Dic18C2×D311yescharacteristic
107C9×C4⋊C4—31no—
108C9×C22⋊C4—31no—
109C9×C22⋊C4—31no—
110C42×C9—31no—
111#3676—32no—
112#3676—32no—
113#3676—62no—
114#3676C2×D312yes—
115C4⋊Dic9—31no—
116C4⋊Dic9—31no—
117C4⋊Dic9—32no—
118C4⋊Dic9—61no—
119C4⋊Dic9—61no—
120C4⋊Dic9C2×D311yescharacteristic
121Dic9⋊C4—32no—
122Dic9⋊C4—32no—
123Dic9⋊C4—32no—
124Dic9⋊C4—32no—
125Dic9⋊C4C2×D312yes—
126C4×Dic9—32no—
127C4×Dic9—32no—
128C4×Dic9—62no—
129C22×C3×C9—21no—
130C22×C3×C9—21no—
131C22×C3×C9—21no—
132C22×C3×C9C2411yescharacteristic
133C2×D3×C9—21no—
134C2×D3×C9—21no—
135C2×D3×C9#4112yes—
136C3×C4×C9—21no—
137C3×C4×C9#4112yes—
138C3⋊Dic9—22no—
139C3⋊Dic9—22no—
140C9×Dic3—21no—
141C9×Dic3—21no—
142C9×Dic3—21no—
143C3×Dic9—22no—
144C3×Dic9—22no—
145C3×C22⋊Q8—91no—
146C3×C4×D4—31no—
147#946—91no—
148Q8×Dic3—91no—
149#935—92no—
150C4×#67D911yescharacteristic
151#926—91no—
152#926—91no—
153#926—181no—
154C2×C4⋊Dic3—91no—
155#924—31no—
156#924—91no—
157#892—92no—
158D3×C4⋊C4—91no—
159#890—91no—
160#889—92no—
161#889—92no—
162#883—92no—
163#879—92no—
164#879—92no—
165#877—92no—
166#870—91no—
167C4×Dic6—91no—
168C23×C32C22×D311yescharacteristic
169C22×C3×D3C22×D311yescharacteristic
170C2×C32×C4—22no—
171C2×C32×C4C22×D311yescharacteristic
172C2×C32×C4C22×D311yescharacteristic
173C2×C3⋊Dic3—32no—
174C2×C3⋊Dic3—32no—
175C3×#67—22no—
176C2×C3×Dic3—32no—
177C2×C3×Dic3—32no—
178C2×C3×Dic3C22×D311yescharacteristic
179C2×C3×Dic3C22×D311yescharacteristic
180C2×C3×Dic3C22×D311yescharacteristic
181C3×C4×D3—22no—
182C3×Dic6—62no—
183D3×Dic3—64no—
184C23×C9—21no—
185C23×C9—31no—
186C23×C9C22×D311yescharacteristic
187D4×C9—62no—
188C2×C4×C9—24no—
189C2×C4×C9—22no—
190C2×C4×C9—21no—
191C2×C4×C9—21no—
192C2×C4×C9—31no—
193C2×C4×C9—31no—
194C2×C4×C9—31no—
195C2×C4×C9—62no—
196C2×C4×C9C22×D311yescharacteristic
197C2×C4×C9C22×D311yescharacteristic
198C2×Dic9—32no—
199C2×Dic9—32no—
200C2×Dic9—62no—
201C2×Dic9—62no—
202C2×Dic9—64no—
203C2×Dic9C22×D312yes—
204C2×Dic9C22×D312yes—
205Dic18—22no—
206#552—271noSylow
207C2×C3×C9—41no—
208C2×C3×C9C2×#4111yescharacteristic
209C2×C3×C9C2×#4111yescharacteristic
210C2×C3×C92−1+411yescharacteristic
211D3×C9—22no—
212C2×C3×Q8—91no—
213C2×C3×D4—31no—
214C22×C3×C4—21no—
215C22×C3×C4—31no—
216C22×C3×C4C2×D911yescharacteristic
217C22×C3×C4D3211yescharacteristic
218C2×#67C2×D911yescharacteristic
219C22×Dic3—92no—
220C2×C4×D3—92no—
221C2×C4×D3C2×D911yescharacteristic
222C2×Dic6—31no—
223C3×C4⋊C4—31no—
224C3×C4⋊C4—92no—
225C3×C4⋊C4—91no—
226C3×C22⋊C4—31no—
227C3×C22⋊C4—31no—
228C3×C22⋊C4—92no—
229C3×C42—31no—
230#217—32no—
231#217—92no—
232#217—92no—
233#217—92no—
234#217—182no—
235#217C2×D911yescharacteristic
236(C2×D3)⋊C4—92no—
237(C2×D3)⋊C4C2×D911yescharacteristic
238C4⋊Dic3—31no—
239C4⋊Dic3—92no—
240C4⋊Dic3—91no—
241C4⋊Dic3—91no—
242C4⋊Dic3—91no—
243C4⋊Dic3—91no—
244C4⋊Dic3—92no—
245C4⋊Dic3—92no—
246C4⋊Dic3—181no—
247C4⋊Dic3—181no—
248Dic3⋊C4—32no—
249Dic3⋊C4—92no—
250Dic3⋊C4—92no—
251Dic3⋊C4—92no—
252Dic3⋊C4—92no—
253Dic3⋊C4—92no—
254Dic3⋊C4—92no—
255Dic3⋊C4—92no—
256Dic3⋊C4C2×D911yescharacteristic
257C4×Dic3—92no—
258C4×Dic3—92no—
259C4×Dic3—92no—
260C4×Dic3—92no—
261C4×Dic3—182no—
262C4×Dic3C2×D911yescharacteristic
263C22×C32—21no—
264C22×C32—21no—
265C22×C32—21no—
266C22×C32C23×D311yescharacteristic
267C2×C3×D3—21no—
268C2×C3×D3—21no—
269C2×C3×D3#23512yes—
270C32×C4—21no—
271C32×C4#23712yes—
272C3⋊Dic3—62no—
273C3⋊Dic3—62no—
274C3×Dic3—21no—
275C3×Dic3—21no—
276C3×Dic3—21no—
277C3×Dic3—62no—
278C3×Dic3—62no—
279C22×C9—22no—
280C22×C9—21no—
281C22×C9—21no—
282C22×C9—21no—
283C22×C9—21no—
284C22×C9—32no—
285C22×C9—41no—
286C22×C9—41no—
287C22×C9—61no—
288C22×C9—61no—
289C22×C9C23×D311yescharacteristic
290C4×C9—22no—
291C4×C9—21no—
292C4×C9—22no—
293C4×C9—61no—
294C4×C9—61no—
295C4×C9—61no—
296C4×C9#23712yes—
297Dic9—22no—
298Dic9—22no—
299Dic9—62no—
300Dic9—62no—
301Dic9—122no—
302Dic9—122no—
303#126—272no—
304#125—271no—
305#125—272no—
306#123—272no—
307C22⋊Q8—91no—
308C22⋊Q8—271no—
309C4×Q8—271no—
310C4×D4—31no—
311#117—272no—
312#117—271no—
313C2×C4⋊C4—271no—
314C3×C9#55211yescharacteristic, complemented, Sylow
315C23×C3—21no—
316C23×C3—31no—
317C23×C3C22×D911yescharacteristic
318C23×C3C2×D3211yescharacteristic
319C22×D3C22×D911yescharacteristic
320C3×Q8—182no—
321C3×D4—62no—
322C2×C3×C4—24no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—31no—
328C2×C3×C4—31no—
329C2×C3×C4—31no—
330C2×C3×C4—62no—
331C2×C3×C4—92no—
332C2×C3×C4—92no—
333C2×C3×C4C22×D911yescharacteristic
334C2×C3×C4C22×D911yescharacteristic
335C2×C3×C4C2×D3211yescharacteristic
336C2×C3×C4C2×D3211yescharacteristic
337#67—22no—
338C2×Dic3—32no—
339C2×Dic3—32no—
340C2×Dic3—92no—
341C2×Dic3—92no—
342C2×Dic3—92no—
343C2×Dic3—92no—
344C2×Dic3—182no—
345C2×Dic3—182no—
346C2×Dic3—184no—
347C2×Dic3C22×D911yescharacteristic
348C2×Dic3C22×D911yescharacteristic
349C2×Dic3C22×D911yescharacteristic
350C4×D3—22no—
351C4×D3—184no—
352Dic6—62no—
353C2×C32—41no—
354C2×C32C2×#23511yescharacteristic
355C2×C32C2×#23711yescharacteristic
356C2×C32#101011yescharacteristic
357C3×D3—22no—
358C2×C9—21no—
359C2×C9—21no—
360C2×C9—21no—
361C2×C9—42no—
362C2×C9—41no—
363C2×C9—62no—
364C2×C9C2×#23711yescharacteristic
365C2×C9#100711yescharacteristic
366C2×C9D3×#4111yescharacteristic
367C2×Q8—91no—
368C2×D4—31no—
369C22×C4—31no—
370C22×C4—272no—
371C22×C4D3×D911yescharacteristic
372C4⋊C4—31no—
373C4⋊C4—91no—
374C4⋊C4—92no—
375C4⋊C4—272no—
376C4⋊C4—272no—
377C4⋊C4—272no—
378C4⋊C4—272no—
379C4⋊C4—272no—
380C4⋊C4—271no—
381C4⋊C4—271no—
382C22⋊C4—31no—
383C22⋊C4—31no—
384C22⋊C4—92no—
385C22⋊C4—272no—
386C22⋊C4—272no—
387C42—31no—
388C42—272no—
389C42—272no—
390C22×C3—22no—
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—21no—
397C22×C3—21no—
398C22×C3—32no—
399C22×C3—41no—
400C22×C3—41no—
401C22×C3—61no—
402C22×C3—61no—
403C22×C3C23×D911yescharacteristic
404C22×C3C22×D3211yescharacteristic
405C2×D3—21no—
406C2×D3—21no—
407C2×D3#369712yes—
408C3×C4—22no—
409C3×C4—21no—
410C3×C4—21no—
411C3×C4—22no—
412C3×C4—61no—
413C3×C4—61no—
414C3×C4—61no—
415C3×C4—182no—
416C3×C4—182no—
417C3×C4#369912yes—
418C3×C4#380512yes—
419Dic3—21no—
420Dic3—21no—
421Dic3—21no—
422Dic3—62no—
423Dic3—62no—
424Dic3—182no—
425Dic3—182no—
426Dic3—182no—
427Dic3—182no—
428Dic3—362no—
429Dic3—362no—
430C32#554111yescharacteristic
431C9—21no—
432C9#563911yescharacteristic, complemented
433C23—31no—
434C23C2×D3×D911yescharacteristic
435Q8—182no—
436D4—62no—
437C2×C4—22no—
438C2×C4—31no—
439C2×C4—31no—
440C2×C4—31no—
441C2×C4—62no—
442C2×C4—92no—
443C2×C4—92no—
444C2×C4—272no—
445C2×C4—272no—
446C2×C4—544no—
447C2×C4C2×D3×D911yescharacteristic
448C2×C4C2×D3×D911yescharacteristic
449C2×C3—21no—
450C2×C3—21no—
451C2×C3—21no—
452C2×C3—42no—
453C2×C3—41no—
454C2×C3—41no—
455C2×C3—62no—
456C2×C3C2×#369711yescharacteristic
457C2×C3C2×#369911yescharacteristic
458C2×C3#6371611yescharacteristic
459C2×C3#6429711yescharacteristic
460C2×C3D3×#23511yescharacteristic
461C2×C3C2×#380511yescharacteristic
462D3—22no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—32no—
467C22—61no—
468C22—61no—
469C22C22×D3×D911yescharacteristic
470C4—21no—
471C4—61no—
472C4—61no—
473C4—61no—
474C4—182no—
475C4—182no—
476C4—542no—
477C4—542no—
478C4#8866312yes—
479C3—21no—
480C3#1058964311yescharacteristic, complemented
481C3#1059494211yescharacteristic
482C2—41no—
483C2—62no—
484C2#1172246411yescharacteristic
485C2D3×#369711yescharacteristic
486C2C2×#8866311yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
prime radicalC2×C3×#67C2×D3
hypercenterC22×C4D3×D9
hypocenterC3×C9#552
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. #11721815C2semidirect
3. C3×#63486C2semidirect
5. #11721246C2semidirect
6. D3×C4⋊Dic9C2semidirect
7. #11721011C2semidirect
8. #11720960C2semidirect
9. #11720928C2semidirect
10. #11720927C2semidirect
11. Dic3×Dic18C2semidirect
54. #63486D3semidirect
314. C3×C9#552semidirect
432. C9#5639semidirect
480. C3#10589643semidirect

Maximal subgroups

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

subgroupindex
#117218152
C4×C9×#672
C3×#634862
#117212802
#117212462
D3×C4⋊Dic92
#117210112
#117209602
#117209282
#117209272
Dic3×Dic182
#105949423
#105896433