Finite Group Databasebeta

← #13147661

Subgroups — #13147661

Order 1728.

758 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721707C211yesmaximal, characteristic, complemented
2C9×#875C211yesmaximal, characteristic, complemented
3C3×C42×D9C211yesmaximal, characteristic, complemented
4#11721036C211yesmaximal, characteristic
5#11721031C212yesmaximal, complemented
6D12×Dic9C211yesmaximal, characteristic, complemented
7#11721012C212yesmaximal, complemented
8#11721010C212yesmaximal, complemented
9#11720963C212yesmaximal, complemented
10#11720944C211yesmaximal, characteristic, complemented
11Dic6×Dic9C211yesmaximal, characteristic, complemented
12#10593921—31nomaximal
13#10589782—31nomaximal
14C2×C9×D12C2211yescharacteristic
15C2×C3×C4×D9C2211yescharacteristic
16C2×C3×C4×D9C2211yescharacteristic
17C2×C3×C4×D9C2211yescharacteristic
18C2×C9×Dic6C2211yescharacteristic
19C2×#6194C2212yes—
20C2×D3×Dic9C2212yes—
21C3×C42×C9C2211yescharacteristic
22C4⋊(C3⋊Dic9)C2211yescharacteristic
23C4⋊(C3⋊Dic9)C2211yescharacteristic
24#88534C2212yes—
25#88534C2212yes—
26C9×(C2×D3)⋊C4C2212yes—
27C9×(C2×D3)⋊C4C2212yes—
28C3×C4×Dic9C2211yescharacteristic
29C3×C4×Dic9C2211yescharacteristic
30C3×C4×Dic9C2211yescharacteristic
31(C2×D3)⋊Dic9C2212yes—
32(C2×D3)⋊Dic9C2212yes—
33(C2×D9)⋊Dic3C2212yes—
34Dic3⋊Dic9C2212yes—
35#88442C2212yes—
36Dic3×Dic9C2212yes—
37#64079—31no—
38C3×#875D311yescharacteristic
39C3×C42×D3—31no—
40#63908—31no—
41#63905—32no—
42Dic3×D12—31no—
43#63879—32no—
44#63877—32no—
45#63859—32no—
46#63856—31no—
47Dic3×Dic6—31no—
48C9×#124—31no—
49#63508—31no—
50Q8×Dic9—31no—
51#63500—31no—
52D4×Dic9—31no—
53#63445—32no—
54#611⋊C4—32no—
55#63441—32no—
56#63439—32no—
57C42×D9D311yescharacteristic, complemented
58#63432—31no—
59#63432—61no—
60C22×D3×C9C2312yes—
61C22×C3×D9C2311yescharacteristic
62C2×C3×C4×C9C2311yescharacteristic
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3⋊Dic9C2312yes—
66C2×C3⋊Dic9C2312yes—
67C2×C9×Dic3C2312yes—
68C2×C3×Dic9C2311yescharacteristic
69C2×C3×Dic9C2311yescharacteristic
70C2×C3×Dic9C2311yescharacteristic
71C9×D12—22no—
72C3×C4×D9—22no—
73C3×C4×D9—22no—
74C3×C4×D9—22no—
75C9×Dic6—22no—
76#6194—24no—
77D3×Dic9—24no—
78#5680—91no—
79#5479—91no—
80C2×C3×D12C2×D311yescharacteristic
81C2×C3×C4×D3—31no—
82C2×C3×C4×D3—31no—
83C2×C3×C4×D3—31no—
84C2×C3×Dic6C2×D311yescharacteristic
85C2×#625—32no—
86C2×D3×Dic3—32no—
87C32×C42C2×D311yescharacteristic
88C4⋊(C3⋊Dic3)—31no—
89C4⋊(C3⋊Dic3)—31no—
90#3750—32no—
91#3750—32no—
92C3×(C2×D3)⋊C4C2×D312yes—
93C3×(C2×D3)⋊C4C2×D312yes—
94C3×C4×Dic3—31no—
95C3×C4×Dic3—31no—
96C3×C4×Dic3—31no—
97#3724—32no—
98Dic3⋊Dic3—32no—
99(C2×D3)⋊Dic3—32no—
100(C2×D3)⋊Dic3—32no—
101(C2×D3)⋊Dic3—32no—
102Dic32—32no—
103C2×Q8×C9—31no—
104C2×D4×C9—31no—
105C2×#611—32no—
106C22×Dic9—32no—
107C2×C4×D9C2×D311yescharacteristic
108C2×C4×D9C2×D311yescharacteristic
109C2×C4×D9C2×D311yescharacteristic
110C9×C22⋊C4—32no—
111C9×C22⋊C4—32no—
112C42×C9—21no—
113C42×C9C2×D311yescharacteristic
114#3676—32no—
115#3676—32no—
116(C2×D9)⋊C4—32no—
117C4⋊Dic9—31no—
118C4⋊Dic9—31no—
119C4⋊Dic9—32no—
120C4⋊Dic9—61no—
121C4⋊Dic9—61no—
122Dic9⋊C4—32no—
123Dic9⋊C4—32no—
124Dic9⋊C4—32no—
125Dic9⋊C4—62no—
126Dic9⋊C4—62no—
127C4×Dic9—32no—
128C4×Dic9C2×D311yescharacteristic
129C4×Dic9C2×D311yescharacteristic
130C4×Dic9C2×D311yescharacteristic
131C22×C3×C9C2411yescharacteristic
132C2×D3×C9—22no—
133C2×D3×C9—22no—
134C2×D3×C9—22no—
135C2×C3×D9—21no—
136C2×C3×D9—21no—
137C2×C3×D9#4112yes—
138C3×C4×C9—21no—
139C3×C4×C9—21no—
140C3×C4×C9#4112yes—
141C3⋊Dic9—22no—
142C3⋊Dic9—22no—
143C9×Dic3—22no—
144C3×Dic9—21no—
145C3×Dic9—21no—
146C3×Dic9#4112yes—
147C3×#124—31no—
148C2×C3×C42—91no—
149#946—91no—
150Q8×Dic3—91no—
151#938—91no—
152D4×Dic3—91no—
153#930—91no—
154#929—92no—
155C4×#67—92no—
156#924—91no—
157#883—92no—
158#67⋊C4—92no—
159#879—92no—
160#877—92no—
161#876—92no—
162#875D911yescharacteristic, complemented
163C4×D12—91no—
164#872—92no—
165C42×D3—31no—
166#870—91no—
167#870—91no—
168#870—181no—
169C4×Dic6—91no—
170C22×C3×D3—31no—
171C22×C3×D3C22×D312yes—
172C2×C32×C4C22×D311yescharacteristic
173C2×C32×C4C22×D311yescharacteristic
174C2×C32×C4C22×D311yescharacteristic
175C2×C3⋊Dic3—32no—
176C2×C3⋊Dic3—32no—
177C2×C3×Dic3—31no—
178C2×C3×Dic3—31no—
179C2×C3×Dic3—31no—
180C2×C3×Dic3C22×D312yes—
181C3×D12—22no—
182C3×C4×D3—62no—
183C3×C4×D3—62no—
184C3×C4×D3—62no—
185C3×Dic6—22no—
186#625—64no—
187D3×Dic3—64no—
188C23×C9—32no—
189C22×D9C22×D311yescharacteristic
190Q8×C9—62no—
191D4×C9—62no—
192C2×C4×C9—21no—
193C2×C4×C9—21no—
194C2×C4×C9—21no—
195C2×C4×C9—32no—
196C2×C4×C9C22×D311yescharacteristic
197C2×C4×C9C22×D311yescharacteristic
198C2×C4×C9C22×D311yescharacteristic
199#611—64no—
200C2×Dic9—32no—
201C2×Dic9—32no—
202C2×Dic9—62no—
203C2×Dic9—62no—
204C2×Dic9—64no—
205C2×Dic9C22×D311yescharacteristic
206C2×Dic9C22×D311yescharacteristic
207C2×Dic9C22×D311yescharacteristic
208C4×D9—22no—
209C4×D9—22no—
210C4×D9—22no—
211#529—271noSylow
212C2×C3×C9C2×#4111yescharacteristic
213C2×C3×C9C2×#4111yescharacteristic
214C2×C3×C9C2×#4111yescharacteristic
215D3×C9—42no—
216C3×D9—22no—
217C2×C3×Q8—31no—
218C2×C3×D4—31no—
219C22×C3×C4—91no—
220C22×C3×C4—91no—
221C22×C3×C4—91no—
222C2×#67—92no—
223C2×#67—92no—
224C22×Dic3—92no—
225C2×D12C2×D911yescharacteristic
226C2×C4×D3—31no—
227C2×C4×D3—31no—
228C2×C4×D3—31no—
229C2×C4×D3—92no—
230C2×Dic6C2×D911yescharacteristic
231C3×C22⋊C4—32no—
232C3×C22⋊C4—32no—
233C3×C42—21no—
234C3×C42—91no—
235C3×C42—91no—
236C3×C42—91no—
237C3×C42C2×D911yescharacteristic
238C3×C42D3211yescharacteristic
239#217—92no—
240#217—92no—
241#217—92no—
242(C2×D3)⋊C4—92no—
243(C2×D3)⋊C4—92no—
244(C2×D3)⋊C4—92no—
245(C2×D3)⋊C4C2×D912yes—
246(C2×D3)⋊C4C2×D912yes—
247C4⋊Dic3—92no—
248C4⋊Dic3—91no—
249C4⋊Dic3—91no—
250C4⋊Dic3—91no—
251C4⋊Dic3—91no—
252C4⋊Dic3—181no—
253C4⋊Dic3—181no—
254Dic3⋊C4—92no—
255Dic3⋊C4—92no—
256Dic3⋊C4—92no—
257Dic3⋊C4—92no—
258Dic3⋊C4—92no—
259Dic3⋊C4—92no—
260Dic3⋊C4—92no—
261Dic3⋊C4—182no—
262Dic3⋊C4—182no—
263C4×Dic3—31no—
264C4×Dic3—31no—
265C4×Dic3—31no—
266C4×Dic3—92no—
267C4×Dic3—92no—
268C22×C32C23×D311yescharacteristic
269C2×C3×D3—22no—
270C2×C3×D3—22no—
271C2×C3×D3—22no—
272C2×C3×D3—32no—
273C2×C3×D3—61no—
274C2×C3×D3—61no—
275C32×C4—21no—
276C32×C4—21no—
277C32×C4#23712yes—
278C3⋊Dic3—62no—
279C3⋊Dic3—62no—
280C3×Dic3—22no—
281C3×Dic3—32no—
282C3×Dic3—61no—
283C3×Dic3—61no—
284C22×C9—21no—
285C22×C9—62no—
286C22×C9—62no—
287C22×C9—62no—
288C22×C9C23×D311yescharacteristic
289C2×D9—21no—
290C2×D9—21no—
291C2×D9#23512yes—
292C4×C9—21no—
293C4×C9—21no—
294C4×C9—22no—
295C4×C9—41no—
296C4×C9—41no—
297C4×C9—62no—
298C4×C9#23512yes—
299Dic9—21no—
300Dic9—21no—
301Dic9—62no—
302Dic9—62no—
303Dic9—122no—
304Dic9—122no—
305Dic9#23512yes—
306#126—272no—
307#125—271no—
308#124—31no—
309#123—272no—
310C22⋊Q8—271no—
311#121—271no—
312C4×Q8—271no—
313C4×D4—271no—
314C4×D4—272no—
315#117—272no—
316C2×C42—91no—
317C3×C9#52911yescharacteristic, complemented, Sylow
318C23×C3—32no—
319C23×C3—91no—
320C22×D3—31no—
321C22×D3C22×D912yes—
322C3×Q8—62no—
323C3×D4—62no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—32no—
328C2×C3×C4—91no—
329C2×C3×C4—91no—
330C2×C3×C4—91no—
331C2×C3×C4—182no—
332C2×C3×C4—182no—
333C2×C3×C4—182no—
334C2×C3×C4C22×D911yescharacteristic
335C2×C3×C4C22×D911yescharacteristic
336C2×C3×C4C22×D911yescharacteristic
337C2×C3×C4C2×D3211yescharacteristic
338C2×C3×C4C2×D3211yescharacteristic
339C2×C3×C4C2×D3211yescharacteristic
340#67—184no—
341#67—184no—
342C2×Dic3—31no—
343C2×Dic3—31no—
344C2×Dic3—31no—
345C2×Dic3—92no—
346C2×Dic3—92no—
347C2×Dic3—92no—
348C2×Dic3—92no—
349C2×Dic3—182no—
350C2×Dic3—182no—
351C2×Dic3—184no—
352C2×Dic3C22×D912yes—
353D12—22no—
354C4×D3—62no—
355C4×D3—62no—
356C4×D3—62no—
357C4×D3—184no—
358Dic6—22no—
359C2×C32C2×#23711yescharacteristic
360C2×C32D3×#4111yescharacteristic
361C2×C32D3×#4111yescharacteristic
362C3×D3—42no—
363C3×D3—62no—
364C2×C9—21no—
365C2×C9—21no—
366C2×C9—21no—
367C2×C9—122no—
368C2×C9C2×#23511yescharacteristic
369C2×C9C2×#23511yescharacteristic
370C2×C9C2×#23511yescharacteristic
371D9—22no—
372C2×Q8—31no—
373C2×D4—31no—
374C2×D4—272no—
375C22×C4—91no—
376C22×C4—91no—
377C22×C4—91no—
378C22×C4—272no—
379C4⋊C4—272no—
380C4⋊C4—272no—
381C4⋊C4—272no—
382C4⋊C4—271no—
383C4⋊C4—272no—
384C4⋊C4—271no—
385C22⋊C4—32no—
386C22⋊C4—32no—
387C22⋊C4—272no—
388C22⋊C4—272no—
389C22⋊C4—272no—
390C42—91no—
391C42—91no—
392C42—91no—
393C42—272no—
394C42D3×D911yescharacteristic
395C22×C3—21no—
396C22×C3—62no—
397C22×C3—62no—
398C22×C3—62no—
399C22×C3—92no—
400C22×C3—181no—
401C22×C3—181no—
402C22×C3C23×D911yescharacteristic
403C22×C3C22×D3211yescharacteristic
404C2×D3—22no—
405C2×D3—22no—
406C2×D3—22no—
407C2×D3—32no—
408C2×D3—61no—
409C2×D3—61no—
410C3×C4—21no—
411C3×C4—21no—
412C3×C4—21no—
413C3×C4—21no—
414C3×C4—22no—
415C3×C4—41no—
416C3×C4—41no—
417C3×C4—62no—
418C3×C4—92no—
419C3×C4—181no—
420C3×C4—181no—
421C3×C4#369912yes—
422C3×C4#379512yes—
423Dic3—22no—
424Dic3—32no—
425Dic3—61no—
426Dic3—61no—
427Dic3—182no—
428Dic3—182no—
429Dic3—182no—
430Dic3—182no—
431Dic3—362no—
432Dic3—362no—
433C32#568011yescharacteristic
434C9—21no—
435C9#547911yescharacteristic, complemented
436C23—32no—
437C23—91no—
438Q8—62no—
439D4—62no—
440D4—544no—
441C2×C4—32no—
442C2×C4—91no—
443C2×C4—91no—
444C2×C4—91no—
445C2×C4—182no—
446C2×C4—182no—
447C2×C4—182no—
448C2×C4—272no—
449C2×C4—272no—
450C2×C4—544no—
451C2×C4C2×D3×D911yescharacteristic
452C2×C4C2×D3×D911yescharacteristic
453C2×C4C2×D3×D911yescharacteristic
454C2×C3—21no—
455C2×C3—21no—
456C2×C3—21no—
457C2×C3—122no—
458C2×C3—182no—
459C2×C3C2×#369911yescharacteristic
460C2×C3#41×D911yescharacteristic
461C2×C3#41×D911yescharacteristic
462C2×C3C2×#379511yescharacteristic
463C2×C3D3×#23511yescharacteristic
464C2×C3D3×#23511yescharacteristic
465D3—42no—
466D3—62no—
467C22—62no—
468C22—62no—
469C22—62no—
470C22—92no—
471C22—181no—
472C22—181no—
473C22C22×D3×D911yescharacteristic
474C4—21no—
475C4—21no—
476C4—62no—
477C4—92no—
478C4—181no—
479C4—181no—
480C4—542no—
481C4—542no—
482C4#8863812yes—
483C3—21no—
484C3#1058978211yescharacteristic, complemented
485C3#1059392111yescharacteristic
486C2—122no—
487C2—182no—
488C2C2×#8863811yescharacteristic
489C2D9×#23511yescharacteristic
490C2D9×#23511yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
prime radical#11721036C2
hypercenterC42D3×D9
hypocenterC3×C9#529
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #11721707C2semidirect
2. C9×#875C2semidirect
3. C3×C42×D9C2semidirect
5. #11721031C2semidirect
6. D12×Dic9C2semidirect
7. #11721012C2semidirect
8. #11721010C2semidirect
9. #11720963C2semidirect
10. #11720944C2semidirect
11. Dic6×Dic9C2semidirect
57. C42×D9D3semidirect
162. #875D9semidirect
317. C3×C9#529semidirect
435. C9#5479semidirect
484. C3#10589782semidirect

Maximal subgroups

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

subgroupindex
#117217072
C9×#8752
C3×C42×D92
#117210362
#117210312
D12×Dic92
#117210122
#117210102
#117209632
#117209442
Dic6×Dic92
#105939213
#105897823