Finite Group Databasebeta

← #13150310

Subgroups — #13150310

Order 1728.

806 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊(C3⋊Dic9)C211yesmaximal, characteristic, complemented
2C4×C9×#67C211yesmaximal, characteristic, complemented
3C3×#63492C211yesmaximal, characteristic, complemented
4#11721290C212yesmaximal, complemented
5#611⋊Dic3C212yesmaximal, complemented
6#11721037C211yesmaximal, characteristic, complemented
7Dic3×D36C211yesmaximal, characteristic, complemented
8#11720989C212yesmaximal, complemented
9#11720949C212yesmaximal, complemented
10#11720927C211yesmaximal, characteristic, complemented
11#11720906C211yesmaximal, characteristic, complemented
12#10594995—31nomaximal
13#10589667—31nomaximal
14C22×C3×C4×C9C2211yescharacteristic
15C22×C3⋊Dic9C2212yes—
16C2×C9×#67C2211yescharacteristic
17C2×C3×#611C2212yes—
18C2×C4×D3×C9C2211yescharacteristic
19C2×C3×D36C2211yescharacteristic
20C2×#6194C2212yes—
21C2×Dic3×D9C2212yes—
22C4⋊(C3⋊Dic9)C2212yes—
23C4⋊(C3⋊Dic9)C2211yescharacteristic
24C4⋊(C3⋊Dic9)C2211yescharacteristic
25C9×#217C2211yescharacteristic
26C9×(C2×D3)⋊C4C2211yescharacteristic
27C3×(C2×D9)⋊C4C2212yes—
28C9×Dic3⋊C4C2211yescharacteristic
29C4×C9×Dic3C2211yescharacteristic
30C3×C4⋊Dic9C2211yescharacteristic
31(C2×D3)⋊Dic9C2212yes—
32(C2×D9)⋊Dic3C2212yes—
33(C2×D9)⋊Dic3C2212yes—
34#88442C2212yes—
35Dic9⋊Dic3C2212yes—
36Dic3×Dic9C2212yes—
37C2×C4⋊(C3⋊Dic3)—31no—
38C3×#930—31no—
39C3×C4×#67D311yescharacteristic
40#63980—32no—
41#67⋊Dic3—32no—
42#63908—31no—
43Dic3×D12—31no—
44#63891—32no—
45#63869—31no—
46#63861—32no—
47#63841—31no—
48C4×D4×C9—31no—
49#63501—32no—
50#63492—31no—
51#63492D311yescharacteristic, complemented
52#63488—31no—
53C2×C4⋊Dic9—31no—
54C2×C4⋊Dic9—61no—
55#63460—32no—
56#63445—32no—
57#611⋊C4—32no—
58C4×D36—31no—
59#63432—31no—
60C23×C3×C9C2311yescharacteristic
61C22×D3×C9C2311yescharacteristic
62C22×C3×D9C2312yes—
63C2×C3×C4×C9—22no—
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3×C4×C9C2311yescharacteristic
66C2×C3⋊Dic9—42no—
67C2×C3⋊Dic9C2312yes—
68C2×C3⋊Dic9C2312yes—
69C9×#67—22no—
70C3×#611—24no—
71C2×C9×Dic3C2311yescharacteristic
72C2×C9×Dic3C2311yescharacteristic
73C2×C9×Dic3C2311yescharacteristic
74C2×C3×Dic9C2312yes—
75C4×D3×C9—22no—
76C3×D36—22no—
77#6194—42no—
78Dic3×D9—42no—
79#5618—91no—
80#5565—91no—
81C22×C32×C4C2×D311yescharacteristic
82C22×C3⋊Dic3—32no—
83C2×C3×#67—32no—
84C2×C3×#67C2×D311yescharacteristic
85C2×C3×D12—31no—
86C2×C3×C4×D3C2×D311yescharacteristic
87C2×#625—32no—
88C2×D3×Dic3—32no—
89C4⋊(C3⋊Dic3)—31no—
90C4⋊(C3⋊Dic3)—32no—
91C4⋊(C3⋊Dic3)—31no—
92C3×#217C2×D311yescharacteristic
93C3×(C2×D3)⋊C4—32no—
94C3×(C2×D3)⋊C4C2×D311yescharacteristic
95C3×C4⋊Dic3—31no—
96C3×Dic3⋊C4C2×D311yescharacteristic
97C3×C4×Dic3C2×D311yescharacteristic
98#3724—32no—
99Dic3⋊Dic3—32no—
100(C2×D3)⋊Dic3—32no—
101(C2×D3)⋊Dic3—32no—
102(C2×D3)⋊Dic3—32no—
103Dic32—32no—
104C2×D4×C9—31no—
105C22×C4×C9—21no—
106C22×C4×C9—31no—
107C22×C4×C9C2×D311yescharacteristic
108C2×#611—32no—
109C2×#611C2×D312yes—
110C22×Dic9—32no—
111C22×Dic9—62no—
112C2×D36C2×D311yescharacteristic
113C2×C4×D9—32no—
114C9×C4⋊C4—31no—
115C9×C22⋊C4—31no—
116C9×C22⋊C4—31no—
117C42×C9—31no—
118#3676—32no—
119(C2×D9)⋊C4—32no—
120(C2×D9)⋊C4—32no—
121(C2×D9)⋊C4C2×D312yes—
122C4⋊Dic9—31no—
123C4⋊Dic9—31no—
124C4⋊Dic9—32no—
125C4⋊Dic9—61no—
126C4⋊Dic9—61no—
127C4⋊Dic9—62no—
128C4⋊Dic9C2×D311yescharacteristic
129Dic9⋊C4—32no—
130Dic9⋊C4—32no—
131C4×Dic9—32no—
132C22×C3×C9—21no—
133C22×C3×C9—21no—
134C22×C3×C9#4112yes—
135C22×C3×C9C2411yescharacteristic
136C2×D3×C9—21no—
137C2×D3×C9—21no—
138C2×D3×C9—21no—
139C2×C3×D9—22no—
140C2×C3×D9—22no—
141C2×C3×D9—22no—
142C3×C4×C9—21no—
143C3×C4×C9#4112yes—
144C3⋊Dic9—22no—
145C3⋊Dic9—22no—
146C9×Dic3—21no—
147C9×Dic3—21no—
148C9×Dic3—21no—
149C3×Dic9—22no—
150C3×#121—91no—
151C3×C4×D4—31no—
152#939—92no—
153#939—92no—
154#938—91no—
155D4×Dic3—92no—
156D4×Dic3—91no—
157#930—31no—
158#930—91no—
159C4×#67D911yescharacteristic, complemented
160#926—91no—
161C2×C4⋊Dic3—91no—
162C2×C4⋊Dic3—91no—
163C2×C4⋊Dic3—181no—
164#898—92no—
165#892—91no—
166#890—91no—
167#886—92no—
168#883—92no—
169#67⋊C4—92no—
170#879—92no—
171C4×D12—91no—
172#870—91no—
173C23×C32C22×D311yescharacteristic
174C22×C3×D3—32no—
175C22×C3×D3C22×D311yescharacteristic
176C2×C32×C4—22no—
177C2×C32×C4C22×D311yescharacteristic
178C2×C32×C4C22×D311yescharacteristic
179C2×C3⋊Dic3—32no—
180C2×C3⋊Dic3—32no—
181C2×C3⋊Dic3—122no—
182C3×#67—22no—
183C3×#67—64no—
184C2×C3×Dic3—32no—
185C2×C3×Dic3C22×D311yescharacteristic
186C2×C3×Dic3C22×D311yescharacteristic
187C2×C3×Dic3C22×D311yescharacteristic
188C3×D12—62no—
189C3×C4×D3—22no—
190#625—122no—
191D3×Dic3—122no—
192C23×C9—21no—
193C23×C9—31no—
194C23×C9C22×D311yescharacteristic
195C22×D9C22×D312yes—
196D4×C9—62no—
197C2×C4×C9—24no—
198C2×C4×C9—22no—
199C2×C4×C9—21no—
200C2×C4×C9—21no—
201C2×C4×C9—31no—
202C2×C4×C9—31no—
203C2×C4×C9—31no—
204C2×C4×C9—62no—
205C2×C4×C9C22×D311yescharacteristic
206C2×C4×C9C22×D311yescharacteristic
207#611—24no—
208#611—122no—
209C2×Dic9—32no—
210C2×Dic9—32no—
211C2×Dic9—62no—
212C2×Dic9—62no—
213C2×Dic9—122no—
214C2×Dic9—124no—
215C2×Dic9C22×D312yes—
216D36—22no—
217C4×D9—122no—
218#553—271noSylow
219C2×C3×C9—22no—
220C2×C3×C9C2×#4111yescharacteristic
221C2×C3×C9C2×#4111yescharacteristic
222C2×C3×C9H(5,𝔽2)11yescharacteristic
223D3×C9—41no—
224C3×D9—42no—
225C2×C3×D4—31no—
226C2×C3×D4—92no—
227C2×C3×D4—91no—
228C22×C3×C4—21no—
229C22×C3×C4—31no—
230C22×C3×C4C2×D911yescharacteristic
231C22×C3×C4D3211yescharacteristic
232C2×#67—32no—
233C2×#67—92no—
234C2×#67—92no—
235C2×#67C2×D911yescharacteristic
236C22×Dic3—92no—
237C22×Dic3—92no—
238C22×Dic3—92no—
239C22×Dic3—182no—
240C2×D12—31no—
241C2×C4×D3—92no—
242C2×C4×D3C2×D911yescharacteristic
243C3×C4⋊C4—31no—
244C3×C4⋊C4—91no—
245C3×C22⋊C4—31no—
246C3×C22⋊C4—31no—
247C3×C22⋊C4—92no—
248C3×C42—31no—
249#217—92no—
250#217—92no—
251#217—92no—
252#217C2×D911yescharacteristic
253(C2×D3)⋊C4—32no—
254(C2×D3)⋊C4—92no—
255(C2×D3)⋊C4—92no—
256(C2×D3)⋊C4—92no—
257(C2×D3)⋊C4C2×D911yescharacteristic
258C4⋊Dic3—31no—
259C4⋊Dic3—92no—
260C4⋊Dic3—91no—
261C4⋊Dic3—91no—
262C4⋊Dic3—91no—
263C4⋊Dic3—91no—
264C4⋊Dic3—92no—
265C4⋊Dic3—92no—
266C4⋊Dic3—181no—
267C4⋊Dic3—181no—
268C4⋊Dic3—182no—
269Dic3⋊C4—92no—
270Dic3⋊C4—92no—
271Dic3⋊C4—92no—
272Dic3⋊C4C2×D911yescharacteristic
273C4×Dic3—92no—
274C4×Dic3—92no—
275C4×Dic3C2×D911yescharacteristic
276C22×C32—21no—
277C22×C32—21no—
278C22×C32#23512yes—
279C22×C32C23×D311yescharacteristic
280C2×C3×D3—21no—
281C2×C3×D3—21no—
282C2×C3×D3—21no—
283C2×C3×D3—62no—
284C2×C3×D3—62no—
285C2×C3×D3—62no—
286C32×C4—21no—
287C32×C4#23712yes—
288C3⋊Dic3—62no—
289C3⋊Dic3—62no—
290C3×Dic3—21no—
291C3×Dic3—21no—
292C3×Dic3—21no—
293C3×Dic3—62no—
294C22×C9—22no—
295C22×C9—22no—
296C22×C9—21no—
297C22×C9—21no—
298C22×C9—21no—
299C22×C9—22no—
300C22×C9—61no—
301C22×C9—61no—
302C22×C9—61no—
303C22×C9#23712yes—
304C22×C9C23×D311yescharacteristic
305C2×D9—22no—
306C2×D9—22no—
307C2×D9—22no—
308C4×C9—22no—
309C4×C9—21no—
310C4×C9—22no—
311C4×C9—61no—
312C4×C9—61no—
313C4×C9—61no—
314C4×C9#23712yes—
315Dic9—22no—
316Dic9—62no—
317Dic9—62no—
318Dic9—122no—
319Dic9—122no—
320#126—272no—
321#125—271no—
322#123—272no—
323#121—91no—
324#121—271no—
325#121—272no—
326C4×D4—31no—
327C4×D4—271no—
328C4×D4—272no—
329#117—271no—
330C2×C4⋊C4—271no—
331C3×C9#55311yescharacteristic, complemented, Sylow
332C23×C3—21no—
333C23×C3—31no—
334C23×C3—92no—
335C23×C3C22×D911yescharacteristic
336C23×C3C2×D3211yescharacteristic
337C22×D3—32no—
338C22×D3C22×D911yescharacteristic
339C3×D4—62no—
340C3×D4—184no—
341C3×D4—182no—
342C2×C3×C4—24no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—31no—
348C2×C3×C4—31no—
349C2×C3×C4—31no—
350C2×C3×C4—62no—
351C2×C3×C4—92no—
352C2×C3×C4C22×D911yescharacteristic
353C2×C3×C4C22×D911yescharacteristic
354C2×C3×C4C2×D3211yescharacteristic
355C2×C3×C4C2×D3211yescharacteristic
356#67—22no—
357#67—64no—
358#67—362no—
359#67—362no—
360C2×Dic3—32no—
361C2×Dic3—92no—
362C2×Dic3—92no—
363C2×Dic3—92no—
364C2×Dic3—92no—
365C2×Dic3—182no—
366C2×Dic3—182no—
367C2×Dic3—362no—
368C2×Dic3—362no—
369C2×Dic3—362no—
370C2×Dic3—364no—
371C2×Dic3C22×D911yescharacteristic
372C2×Dic3C22×D911yescharacteristic
373C2×Dic3C22×D911yescharacteristic
374D12—62no—
375C4×D3—22no—
376C4×D3—362no—
377C2×C32—22no—
378C2×C32C2×#23511yescharacteristic
379C2×C32C2×#23711yescharacteristic
380C2×C32#100911yescharacteristic
381C3×D3—41no—
382C3×D3—122no—
383C2×C9—24no—
384C2×C9—22no—
385C2×C9—21no—
386C2×C9—21no—
387C2×C9—21no—
388C2×C9—121no—
389C2×C9C2×#23711yescharacteristic
390C2×C9C2×#23711yescharacteristic
391C2×C9#100911yescharacteristic
392D9—42no—
393C2×D4—31no—
394C2×D4—92no—
395C2×D4—91no—
396C2×D4—272no—
397C22×C4—31no—
398C22×C4—272no—
399C22×C4—272no—
400C22×C4D3×D911yescharacteristic
401C4⋊C4—31no—
402C4⋊C4—91no—
403C4⋊C4—272no—
404C4⋊C4—272no—
405C4⋊C4—271no—
406C4⋊C4—271no—
407C4⋊C4—272no—
408C22⋊C4—31no—
409C22⋊C4—31no—
410C22⋊C4—92no—
411C22⋊C4—272no—
412C22⋊C4—272no—
413C22⋊C4—272no—
414C42—31no—
415C42—272no—
416C22×C3—22no—
417C22×C3—22no—
418C22×C3—21no—
419C22×C3—21no—
420C22×C3—21no—
421C22×C3—21no—
422C22×C3—22no—
423C22×C3—21no—
424C22×C3—61no—
425C22×C3—61no—
426C22×C3—61no—
427C22×C3—182no—
428C22×C3—182no—
429C22×C3—182no—
430C22×C3#369712yes—
431C22×C3C23×D911yescharacteristic
432C22×C3#379512yes—
433C22×C3C22×D3211yescharacteristic
434C2×D3—21no—
435C2×D3—21no—
436C2×D3—21no—
437C2×D3—62no—
438C2×D3—62no—
439C2×D3—62no—
440C3×C4—22no—
441C3×C4—21no—
442C3×C4—21no—
443C3×C4—22no—
444C3×C4—61no—
445C3×C4—61no—
446C3×C4—61no—
447C3×C4—182no—
448C3×C4#369912yes—
449C3×C4#380512yes—
450Dic3—21no—
451Dic3—21no—
452Dic3—21no—
453Dic3—62no—
454Dic3—182no—
455Dic3—182no—
456Dic3—182no—
457Dic3—182no—
458Dic3—362no—
459Dic3—362no—
460C32#556511yescharacteristic
461C9—21no—
462C9#561811yescharacteristic, complemented
463C23—31no—
464C23—92no—
465C23C2×D3×D911yescharacteristic
466D4—62no—
467D4—184no—
468D4—182no—
469D4—1082no—
470C2×C4—22no—
471C2×C4—31no—
472C2×C4—31no—
473C2×C4—31no—
474C2×C4—62no—
475C2×C4—92no—
476C2×C4—272no—
477C2×C4—272no—
478C2×C4—1082no—
479C2×C4—1082no—
480C2×C4C2×D3×D911yescharacteristic
481C2×C4C2×D3×D911yescharacteristic
482C2×C3—24no—
483C2×C3—22no—
484C2×C3—22no—
485C2×C3—21no—
486C2×C3—21no—
487C2×C3—21no—
488C2×C3—121no—
489C2×C3—362no—
490C2×C3C2×#369711yescharacteristic
491C2×C3C2×#369911yescharacteristic
492C2×C3#6371511yescharacteristic
493C2×C3C2×#379511yescharacteristic
494C2×C3#6430811yescharacteristic
495C2×C3C2×#380511yescharacteristic
496D3—41no—
497D3—122no—
498C22—21no—
499C22—21no—
500C22—61no—
501C22—61no—
502C22—61no—
503C22—182no—
504C22—182no—
505C22—182no—
506C22#8864112yes—
507C22C22×D3×D911yescharacteristic
508C4—21no—
509C4—61no—
510C4—61no—
511C4—61no—
512C4—182no—
513C4—542no—
514C4—542no—
515C4#8866312yes—
516C3—21no—
517C3#1058966711yescharacteristic, complemented
518C3#1059499511yescharacteristic
519C2—22no—
520C2—121no—
521C2—362no—
522C2C2×#8864111yescharacteristic
523C2#1172247811yescharacteristic
524C2C2×#8866311yescharacteristic
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. C2×C4⋊(C3⋊Dic9)C2semidirect
2. C4×C9×#67C2semidirect
3. C3×#63492C2semidirect
4. #11721290C2semidirect
5. #611⋊Dic3C2semidirect
6. #11721037C2semidirect
7. Dic3×D36C2semidirect
8. #11720989C2semidirect
9. #11720949C2semidirect
10. #11720927C2semidirect
11. #11720906C2semidirect
51. #63492D3semidirect
159. C4×#67D9semidirect
331. C3×C9#553semidirect
462. C9#5618semidirect
517. C3#10589667semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊(C3⋊Dic9)2
C4×C9×#672
C3×#634922
#117212902
#611⋊Dic32
#117210372
Dic3×D362
#117209892
#117209492
#117209272
#117209062
#105949953
#105896673