Finite Group Databasebeta

← #13150927

Subgroups — #13150927

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#11721836C211yesmaximal, characteristic, complemented
2C9×#940C211yesmaximal, characteristic
3C3×D4×Dic9C211yesmaximal, characteristic, complemented
4#67×Dic9C212yesmaximal, complemented
5#11721280C212yesmaximal, complemented
6#11721265C212yesmaximal, complemented
7#11721263C212yesmaximal, complemented
8#11721025C211yesmaximal, characteristic, complemented
9C4×#6189C211yesmaximal, characteristic, complemented
10#11720964C211yesmaximal, characteristic, complemented
11#11720912C211yesmaximal, characteristic, complemented
12#10595198—31nomaximal
13#10589815—31nomaximal
14C2×C3×D4×C9C2211yescharacteristic
15C2×#6226C2211yescharacteristic
16C2×C9×#67C2212yes—
17C2×C9×#67C2212yes—
18C22×C3×Dic9C2212yes—
19C2×C9×D12C2211yescharacteristic
20C2×D3×Dic9C2212yes—
21C2×#6189C2211yescharacteristic
22#88551C2212yes—
23#88551C2212yes—
24C4×C3⋊Dic9C2211yescharacteristic
25C3×#3676C2212yes—
26C4×C9×Dic3C2211yescharacteristic
27C3×C4⋊Dic9C2211yescharacteristic
28C3×C4×Dic9C2211yescharacteristic
29(C2×D3)⋊Dic9C2212yes—
30(C2×D3)⋊Dic9C2212yes—
31(C2×D3)⋊Dic9C2212yes—
32Dic3⋊Dic9C2211yescharacteristic
33#88442C2211yescharacteristic
34#88442C2212yes—
35Dic9⋊Dic3C2211yescharacteristic
36Dic3×Dic9C2212yes—
37#64145—31no—
38C3×#940D311yescharacteristic
39C3×D4×Dic3—31no—
40Dic3×#67—32no—
41#63969—32no—
42#63961—32no—
43#63959—32no—
44C4×#627—31no—
45#63898—31no—
46#63860—31no—
47#63844—31no—
48C9×#127—31no—
49#63498—31no—
50#63498—32no—
51#63498—31no—
52#63498—61no—
53#63497—32no—
54#63497—32no—
55D4×Dic9—32no—
56D4×Dic9—31no—
57D4×Dic9D311yescharacteristic, complemented
58C4×Dic18—31no—
59C4×Dic18—31no—
60C23×C3×C9C2312yes—
61C22×D3×C9C2312yes—
62C3×D4×C9—22no—
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3⋊Dic9C2311yescharacteristic
65C2×C3⋊Dic9C2311yescharacteristic
66C2×C3⋊Dic9C2312yes—
67#6226—22no—
68C9×#67—24no—
69C9×#67—42no—
70C2×C9×Dic3C2311yescharacteristic
71C2×C9×Dic3C2311yescharacteristic
72C2×C3×Dic9—42no—
73C2×C3×Dic9C2311yescharacteristic
74C2×C3×Dic9C2312yes—
75C2×C3×Dic9C2311yescharacteristic
76C9×D12—22no—
77D3×Dic9—42no—
78#6189—22no—
79#5713—91no—
80#5559—91no—
81C2×C32×D4C2×D311yescharacteristic
82C2×#634—31no—
83C2×C3×#67C2×D312yes—
84C2×C3×#67C2×D312yes—
85C22×C3×Dic3—32no—
86C2×C3×D12C2×D311yescharacteristic
87C2×#627—31no—
88C2×D3×Dic3—32no—
89#3757—32no—
90#3757—32no—
91C4×C3⋊Dic3—31no—
92C3×#217—32no—
93C3×C4⋊Dic3—31no—
94C3×C4×Dic3—31no—
95C3×C4×Dic3C2×D311yescharacteristic
96#3724—32no—
97#3724—31no—
98Dic3⋊Dic3—31no—
99Dic3⋊Dic3—31no—
100(C2×D3)⋊Dic3—32no—
101(C2×D3)⋊Dic3—32no—
102(C2×D3)⋊Dic3—32no—
103Dic32—32no—
104C2×D4×C9—21no—
105C2×D4×C9—32no—
106C2×D4×C9—32no—
107C2×D4×C9—31no—
108C2×D4×C9C2×D311yescharacteristic
109C22×Dic9—32no—
110C22×Dic9C2×D312yes—
111C2×Dic18—31no—
112C2×Dic18—31no—
113C2×Dic18—61no—
114C42×C9—31no—
115#3676—32no—
116#3676—32no—
117#3676—32no—
118#3676—32no—
119#3676—32no—
120#3676—62no—
121#3676—62no—
122#3676C2×D312yes—
123C4⋊Dic9—31no—
124C4⋊Dic9C2×D311yescharacteristic
125Dic9⋊C4—31no—
126Dic9⋊C4—32no—
127Dic9⋊C4—31no—
128C4×Dic9—31no—
129C4×Dic9—32no—
130C4×Dic9—61no—
131C4×Dic9C2×D311yescharacteristic
132C22×C3×C9—22no—
133C22×C3×C9—22no—
134C22×C3×C9—22no—
135C22×C3×C9C2411yescharacteristic
136C2×D3×C9—22no—
137C2×D3×C9—22no—
138C2×D3×C9—22no—
139C3×C4×C9#4112yes—
140C3⋊Dic9—21no—
141C3⋊Dic9—22no—
142C3⋊Dic9—21no—
143C9×Dic3—21no—
144C9×Dic3#4112yes—
145C3×Dic9—21no—
146C3×Dic9—22no—
147C3×Dic9—21no—
148C3×#127—31no—
149C3×C4×D4—91no—
150#940D911yescharacteristic
151#936—91no—
152#936—92no—
153#936—91no—
154#936—91no—
155#936—181no—
156#935—92no—
157#935—92no—
158D4×Dic3—31no—
159D4×Dic3—91no—
160D4×Dic3—92no—
161#929—92no—
162C4×#67—92no—
163#893—91no—
164#887—91no—
165#885—92no—
166#883—92no—
167#875—91no—
168C4×Dic6—91no—
169C4×Dic6—91no—
170C4×Dic6—91no—
171C23×C32C22×D312yes—
172C22×C3×D3C22×D312yes—
173C32×D4—22no—
174C2×C32×C4C22×D311yescharacteristic
175C2×C3⋊Dic3—32no—
176C2×C3⋊Dic3—31no—
177C2×C3⋊Dic3—31no—
178#634—62no—
179C3×#67—24no—
180C3×#67—42no—
181C2×C3×Dic3—32no—
182C2×C3×Dic3—31no—
183C2×C3×Dic3—31no—
184C2×C3×Dic3—122no—
185C2×C3×Dic3C22×D311yescharacteristic
186C2×C3×Dic3C22×D311yescharacteristic
187C3×D12—22no—
188#627—62no—
189D3×Dic3—122no—
190C23×C9—22no—
191C23×C9—32no—
192C23×C9C22×D312yes—
193D4×C9—22no—
194D4×C9—42no—
195D4×C9—64no—
196D4×C9—62no—
197D4×C9—122no—
198C2×C4×C9—21no—
199C2×C4×C9—31no—
200C2×C4×C9—31no—
201C2×C4×C9C22×D311yescharacteristic
202C2×Dic9—31no—
203C2×Dic9—32no—
204C2×Dic9—31no—
205C2×Dic9—42no—
206C2×Dic9—61no—
207C2×Dic9—61no—
208C2×Dic9—62no—
209C2×Dic9—122no—
210C2×Dic9C22×D312yes—
211C2×Dic9C22×D311yescharacteristic
212C2×Dic9C22×D311yescharacteristic
213Dic18—62no—
214Dic18—62no—
215Dic18—122no—
216#559—271noSylow
217C2×C3×C9—42no—
218C2×C3×C9C2×#4111yescharacteristic
219C2×C3×C9C2×#4111yescharacteristic
220C2×C3×C9H(5,𝔽2)11yescharacteristic
221D3×C9—42no—
222C2×C3×D4—21no—
223C2×C3×D4—32no—
224C2×C3×D4—32no—
225C2×C3×D4—31no—
226C2×C3×D4C2×D911yescharacteristic
227C2×C3×D4D3211yescharacteristic
228C22×C3×C4—92no—
229C2×#67C2×D912yes—
230C2×#67C2×D912yes—
231C22×Dic3—32no—
232C22×Dic3—92no—
233C2×D12C2×D911yescharacteristic
234C2×C4×D3—92no—
235C2×Dic6—91no—
236C2×Dic6—91no—
237C2×Dic6—91no—
238C2×Dic6—91no—
239C2×Dic6—181no—
240C3×C4⋊C4—91no—
241C3×C22⋊C4—92no—
242C3×C42—31no—
243C3×C42—91no—
244#217—32no—
245#217—92no—
246#217—92no—
247#217—92no—
248#217—92no—
249#217—92no—
250#217—92no—
251#217—92no—
252#217—182no—
253#217—182no—
254(C2×D3)⋊C4—92no—
255(C2×D3)⋊C4—92no—
256(C2×D3)⋊C4—92no—
257C4⋊Dic3—31no—
258C4⋊Dic3—91no—
259C4⋊Dic3—91no—
260Dic3⋊C4—91no—
261Dic3⋊C4—91no—
262Dic3⋊C4—92no—
263Dic3⋊C4—91no—
264Dic3⋊C4—91no—
265Dic3⋊C4—92no—
266C4×Dic3—31no—
267C4×Dic3—92no—
268C4×Dic3—91no—
269C4×Dic3—91no—
270C4×Dic3—92no—
271C4×Dic3—181no—
272C4×Dic3C2×D911yescharacteristic
273C22×C32—22no—
274C22×C32—22no—
275C22×C32—22no—
276C22×C32C23×D311yescharacteristic
277C2×C3×D3—22no—
278C2×C3×D3—22no—
279C2×C3×D3—22no—
280C32×C4#23712yes—
281C3⋊Dic3—61no—
282C3⋊Dic3—62no—
283C3⋊Dic3—61no—
284C3×Dic3—21no—
285C3×Dic3—61no—
286C3×Dic3—62no—
287C3×Dic3—61no—
288C3×Dic3#23712yes—
289C22×C9—22no—
290C22×C9—22no—
291C22×C9—22no—
292C22×C9—21no—
293C22×C9—42no—
294C22×C9—42no—
295C22×C9—42no—
296C22×C9—62no—
297C22×C9—62no—
298C22×C9—62no—
299C22×C9C23×D311yescharacteristic
300C4×C9—22no—
301C4×C9—32no—
302C4×C9—61no—
303C4×C9#23512yes—
304Dic9—21no—
305Dic9—22no—
306Dic9—21no—
307Dic9—62no—
308Dic9—61no—
309Dic9—61no—
310Dic9—122no—
311Dic9—121no—
312Dic9—121no—
313#127—31no—
314#124—271no—
315#124—272no—
316#124—271no—
317#123—272no—
318#123—272no—
319C4×Q8—271no—
320C4×Q8—271no—
321C4×D4—91no—
322C4×D4—271no—
323C4×D4—272no—
324C3×C9#55911yescharacteristic, complemented, Sylow
325C23×C3—22no—
326C23×C3—32no—
327C23×C3C22×D912yes—
328C23×C3C2×D3212yes—
329C22×D3C22×D912yes—
330C3×D4—22no—
331C3×D4—22no—
332C3×D4—42no—
333C3×D4—64no—
334C3×D4—62no—
335C3×D4—122no—
336C2×C3×C4—21no—
337C2×C3×C4—31no—
338C2×C3×C4—31no—
339C2×C3×C4—92no—
340C2×C3×C4—91no—
341C2×C3×C4—91no—
342C2×C3×C4—362no—
343C2×C3×C4C22×D911yescharacteristic
344C2×C3×C4C2×D3211yescharacteristic
345#67—24no—
346#67—42no—
347C2×Dic3—31no—
348C2×Dic3—31no—
349C2×Dic3—32no—
350C2×Dic3—91no—
351C2×Dic3—91no—
352C2×Dic3—92no—
353C2×Dic3—91no—
354C2×Dic3—92no—
355C2×Dic3—91no—
356C2×Dic3—122no—
357C2×Dic3—181no—
358C2×Dic3—181no—
359C2×Dic3—182no—
360C2×Dic3—362no—
361C2×Dic3C22×D911yescharacteristic
362C2×Dic3C22×D911yescharacteristic
363D12—22no—
364C4×D3—362no—
365Dic6—182no—
366Dic6—182no—
367Dic6—182no—
368Dic6—182no—
369Dic6—362no—
370C2×C32—42no—
371C2×C32C2×#23711yescharacteristic
372C2×C32C2×#23711yescharacteristic
373C2×C32#100411yescharacteristic
374C3×D3—42no—
375C2×C9—21no—
376C2×C9—21no—
377C2×C9—21no—
378C2×C9—42no—
379C2×C9—82no—
380C2×C9—122no—
381C2×C9C2×#23511yescharacteristic
382C2×C9#100411yescharacteristic
383C2×C9D3×#4111yescharacteristic
384C2×Q8—271no—
385C2×Q8—271no—
386C2×D4—32no—
387C2×D4—32no—
388C2×D4—31no—
389C2×D4D3×D911yescharacteristic
390C22×C4—92no—
391C22×C4—272no—
392C4⋊C4—91no—
393C4⋊C4—271no—
394C4⋊C4—271no—
395C4⋊C4—272no—
396C4⋊C4—271no—
397C22⋊C4—92no—
398C22⋊C4—272no—
399C22⋊C4—272no—
400C22⋊C4—272no—
401C22⋊C4—272no—
402C22⋊C4—272no—
403C42—31no—
404C42—91no—
405C42—272no—
406C42—271no—
407C22×C3—22no—
408C22×C3—22no—
409C22×C3—22no—
410C22×C3—22no—
411C22×C3—22no—
412C22×C3—22no—
413C22×C3—21no—
414C22×C3—42no—
415C22×C3—42no—
416C22×C3—42no—
417C22×C3—62no—
418C22×C3—62no—
419C22×C3—62no—
420C22×C3C23×D911yescharacteristic
421C22×C3C22×D3211yescharacteristic
422C2×D3—22no—
423C2×D3—22no—
424C2×D3—22no—
425C3×C4—22no—
426C3×C4—32no—
427C3×C4—61no—
428C3×C4—181no—
429C3×C4—182no—
430C3×C4—181no—
431C3×C4#369912yes—
432C3×C4#380412yes—
433Dic3—21no—
434Dic3—61no—
435Dic3—61no—
436Dic3—62no—
437Dic3—181no—
438Dic3—181no—
439Dic3—182no—
440Dic3—181no—
441Dic3—182no—
442Dic3—181no—
443Dic3—362no—
444Dic3—361no—
445Dic3—361no—
446Dic3#369912yes—
447C32#571311yescharacteristic
448C9—21no—
449C9#555911yescharacteristic, complemented
450C23—32no—
451C23C2×D3×D912yes—
452Q8—542no—
453Q8—542no—
454D4—22no—
455D4—64no—
456D4—62no—
457D4—122no—
458C2×C4—31no—
459C2×C4—31no—
460C2×C4—91no—
461C2×C4—91no—
462C2×C4—92no—
463C2×C4—271no—
464C2×C4—272no—
465C2×C4—271no—
466C2×C4—362no—
467C2×C4—1082no—
468C2×C4C2×D3×D911yescharacteristic
469C2×C3—21no—
470C2×C3—21no—
471C2×C3—21no—
472C2×C3—42no—
473C2×C3—42no—
474C2×C3—82no—
475C2×C3—122no—
476C2×C3C2×#369911yescharacteristic
477C2×C3C2×#369911yescharacteristic
478C2×C3#6371011yescharacteristic
479C2×C3D3×#23711yescharacteristic
480C2×C3#6431311yescharacteristic
481C2×C3C2×#380411yescharacteristic
482D3—42no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—62no—
487C22—62no—
488C22—62no—
489C22C22×D3×D911yescharacteristic
490C4—32no—
491C4—61no—
492C4—181no—
493C4—181no—
494C4—182no—
495C4—541no—
496C4—541no—
497C4—542no—
498C4#8866212yes—
499C3—21no—
500C3#1058981511yescharacteristic, complemented
501C3#1059519811yescharacteristic
502C2—42no—
503C2—122no—
504C2D3×#369911yescharacteristic
505C2#1172249911yescharacteristic
506C2C2×#8866211yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC2×C3×D4×C9C22
socleC22×C32C23×D3
prime radicalC3×#940D3
hypercenterC2×D4D3×D9
hypocenterC3×C9#559
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. #11721836C2semidirect
3. C3×D4×Dic9C2semidirect
4. #67×Dic9C2semidirect
5. #11721280C2semidirect
6. #11721265C2semidirect
7. #11721263C2semidirect
8. #11721025C2semidirect
9. C4×#6189C2semidirect
10. #11720964C2semidirect
11. #11720912C2semidirect
57. D4×Dic9D3semidirect
324. C3×C9#559semidirect
449. C9#5559semidirect
500. C3#10589815semidirect

Maximal subgroups

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

subgroupindex
#117218362
C9×#9402
C3×D4×Dic92
#67×Dic92
#117212802
#117212652
#117212632
#117210252
C4×#61892
#117209642
#117209122
#105951983
#105898153