Finite Group Databasebeta

← #13147641

Subgroups — #13147641

Order 1728.

716 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#11721706C211yesmaximal, characteristic, complemented
2C9×#875C211yesmaximal, characteristic, complemented
3C3×#63434C211yesmaximal, characteristic, complemented
4#11721042C212yesmaximal, complemented
5#11721036C211yesmaximal, characteristic
6#11721024C212yesmaximal, complemented
7D12×Dic9C211yesmaximal, characteristic, complemented
8#11721021C212yesmaximal, complemented
9#11720965C212yesmaximal, complemented
10#11720944C211yesmaximal, characteristic, complemented
11Dic6×Dic9C211yesmaximal, characteristic, complemented
12#10593961—31nomaximal
13#10589784—31nomaximal
14C2×#6226C2212yes—
15C2×C9×D12C2211yescharacteristic
16C2×C3×C4×D9C2211yescharacteristic
17C2×C9×Dic6C2211yescharacteristic
18C2×#6194C2212yes—
19C2×D3×Dic9C2212yes—
20C3×C42×C9C2211yescharacteristic
21C4⋊(C3⋊Dic9)C2212yes—
22C4⋊(C3⋊Dic9)C2211yescharacteristic
23C4⋊(C3⋊Dic9)C2211yescharacteristic
24C9×(C2×D3)⋊C4C2212yes—
25C9×(C2×D3)⋊C4C2212yes—
26C3×(C2×D9)⋊C4C2211yescharacteristic
27C3×(C2×D9)⋊C4C2211yescharacteristic
28C3×Dic9⋊C4C2211yescharacteristic
29C3×Dic9⋊C4C2211yescharacteristic
30C3×C4×Dic9C2211yescharacteristic
31(C2×D3)⋊Dic9C2212yes—
32(C2×D3)⋊Dic9C2212yes—
33(C2×D9)⋊Dic3C2212yes—
34Dic3⋊Dic9C2212yes—
35#88442C2212yes—
36Dic3×Dic9C2212yes—
37#64078—31no—
38C3×#875D311yescharacteristic
39C3×#872—31no—
40#63913—32no—
41#63908—31no—
42#63896—32no—
43Dic3×D12—31no—
44#63891—32no—
45#63861—32no—
46#63856—31no—
47Dic3×Dic6—31no—
48C9×#124—31no—
49#63508—31no—
50Q8×Dic9—31no—
51#63500—31no—
52D4×Dic9—31no—
53#63448—32no—
54#63447—32no—
55#63441—32no—
56#63440—32no—
57#63434D311yescharacteristic, complemented
58#63431—31no—
59#63431—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—
67#6226—42no—
68C2×C9×Dic3C2312yes—
69C2×C3×Dic9C2311yescharacteristic
70C2×C3×Dic9C2311yescharacteristic
71C2×C3×Dic9C2311yescharacteristic
72C9×D12—22no—
73C3×C4×D9—22no—
74C9×Dic6—22no—
75#6194—42no—
76D3×Dic9—42no—
77#5682—91no—
78#5534—91no—
79C2×#634—32no—
80C2×C3×D12C2×D311yescharacteristic
81C2×C3×C4×D3—31no—
82C2×C3×Dic6C2×D311yescharacteristic
83C2×#625—32no—
84C2×D3×Dic3—32no—
85C32×C42C2×D311yescharacteristic
86C4⋊(C3⋊Dic3)—31no—
87C4⋊(C3⋊Dic3)—31no—
88C4⋊(C3⋊Dic3)—32no—
89C3×(C2×D3)⋊C4—31no—
90C3×(C2×D3)⋊C4—31no—
91C3×(C2×D3)⋊C4C2×D312yes—
92C3×(C2×D3)⋊C4C2×D312yes—
93C3×Dic3⋊C4—31no—
94C3×Dic3⋊C4—31no—
95C3×C4×Dic3—31no—
96#3724—32no—
97Dic3⋊Dic3—32no—
98(C2×D3)⋊Dic3—32no—
99(C2×D3)⋊Dic3—32no—
100(C2×D3)⋊Dic3—32no—
101Dic32—32no—
102C2×Q8×C9—31no—
103C2×D4×C9—31no—
104C2×#611—32no—
105C22×Dic9—32no—
106C2×C4×D9C2×D311yescharacteristic
107C2×Dic18—32no—
108C2×Dic18—62no—
109C9×C22⋊C4—32no—
110C9×C22⋊C4—32no—
111C42×C9—21no—
112C42×C9C2×D311yescharacteristic
113#3676—32no—
114#3676—32no—
115(C2×D9)⋊C4—32no—
116(C2×D9)⋊C4C2×D311yescharacteristic
117(C2×D9)⋊C4C2×D311yescharacteristic
118C4⋊Dic9—31no—
119C4⋊Dic9—31no—
120C4⋊Dic9—32no—
121C4⋊Dic9—32no—
122C4⋊Dic9—62no—
123C4⋊Dic9—61no—
124C4⋊Dic9—61no—
125Dic9⋊C4—32no—
126Dic9⋊C4C2×D311yescharacteristic
127Dic9⋊C4C2×D311yescharacteristic
128C4×Dic9—32no—
129C4×Dic9C2×D311yescharacteristic
130C22×C3×C9C2411yescharacteristic
131C2×D3×C9—22no—
132C2×D3×C9—22no—
133C2×D3×C9—22no—
134C2×C3×D9—21no—
135C2×C3×D9—21no—
136C2×C3×D9—21no—
137C3×C4×C9—21no—
138C3×C4×C9—21no—
139C3×C4×C9#4112yes—
140C3⋊Dic9—22no—
141C3⋊Dic9—22no—
142C9×Dic3—22no—
143C3×Dic9—21no—
144C3×Dic9—21no—
145C3×Dic9—21no—
146C3×#124—31no—
147C3×#117—91no—
148#946—91no—
149Q8×Dic3—91no—
150#938—91no—
151D4×Dic3—91no—
152#930—91no—
153#924—91no—
154#898—92no—
155#896—92no—
156#886—92no—
157#885—92no—
158#885—92no—
159#883—92no—
160#879—92no—
161#878—92no—
162#875D911yescharacteristic, complemented
163C4×D12—91no—
164#872—31no—
165#869—91no—
166#869—91no—
167#869—181no—
168C4×Dic6—91no—
169C22×C3×D3—31no—
170C22×C3×D3C22×D312yes—
171C2×C32×C4C22×D311yescharacteristic
172C2×C32×C4C22×D311yescharacteristic
173C2×C32×C4C22×D311yescharacteristic
174C2×C3⋊Dic3—32no—
175C2×C3⋊Dic3—32no—
176#634—122no—
177C2×C3×Dic3—31no—
178C2×C3×Dic3—31no—
179C2×C3×Dic3—31no—
180C2×C3×Dic3C22×D312yes—
181C3×D12—22no—
182C3×C4×D3—62no—
183C3×Dic6—22no—
184#625—122no—
185D3×Dic3—122no—
186C23×C9—32no—
187C22×D9C22×D311yescharacteristic
188Q8×C9—62no—
189D4×C9—62no—
190C2×C4×C9—21no—
191C2×C4×C9—21no—
192C2×C4×C9—21no—
193C2×C4×C9—32no—
194C2×C4×C9C22×D311yescharacteristic
195C2×C4×C9C22×D311yescharacteristic
196C2×C4×C9C22×D311yescharacteristic
197#611—122no—
198C2×Dic9—32no—
199C2×Dic9—32no—
200C2×Dic9—62no—
201C2×Dic9—62no—
202C2×Dic9—122no—
203C2×Dic9C22×D311yescharacteristic
204C2×Dic9C22×D311yescharacteristic
205C2×Dic9C22×D311yescharacteristic
206C4×D9—22no—
207Dic18—122no—
208Dic18—124no—
209#542—271noSylow
210C2×C3×C9C2×#4111yescharacteristic
211C2×C3×C9H(5,𝔽2)11yescharacteristic
212C2×C3×C92−1+411yescharacteristic
213D3×C9—42no—
214C3×D9—41no—
215C2×C3×Q8—31no—
216C2×C3×D4—31no—
217C22×C3×C4—91no—
218C2×#67—92no—
219C2×#67—92no—
220C22×Dic3—92no—
221C2×D12C2×D911yescharacteristic
222C2×C4×D3—31no—
223C2×C4×D3—92no—
224C2×Dic6—92no—
225C2×Dic6—92no—
226C2×Dic6—182no—
227C2×Dic6C2×D911yescharacteristic
228C3×C4⋊C4—91no—
229C3×C4⋊C4—91no—
230C3×C22⋊C4—32no—
231C3×C22⋊C4—32no—
232C3×C22⋊C4—91no—
233C3×C22⋊C4—91no—
234C3×C42—21no—
235C3×C42—91no—
236C3×C42C2×D911yescharacteristic
237C3×C42D3211yescharacteristic
238#217—92no—
239#217—92no—
240#217—92no—
241(C2×D3)⋊C4—31no—
242(C2×D3)⋊C4—31no—
243(C2×D3)⋊C4—92no—
244(C2×D3)⋊C4—92no—
245(C2×D3)⋊C4—92no—
246(C2×D3)⋊C4C2×D912yes—
247(C2×D3)⋊C4C2×D912yes—
248C4⋊Dic3—92no—
249C4⋊Dic3—91no—
250C4⋊Dic3—91no—
251C4⋊Dic3—92no—
252C4⋊Dic3—92no—
253C4⋊Dic3—91no—
254C4⋊Dic3—91no—
255C4⋊Dic3—181no—
256C4⋊Dic3—181no—
257C4⋊Dic3—182no—
258Dic3⋊C4—31no—
259Dic3⋊C4—31no—
260Dic3⋊C4—92no—
261Dic3⋊C4—92no—
262Dic3⋊C4—92no—
263C4×Dic3—31no—
264C4×Dic3—92no—
265C4×Dic3—92no—
266C22×C32C23×D311yescharacteristic
267C2×C3×D3—22no—
268C2×C3×D3—22no—
269C2×C3×D3—22no—
270C2×C3×D3—61no—
271C2×C3×D3—61no—
272C2×C3×D3—61no—
273C32×C4—21no—
274C32×C4—21no—
275C32×C4#23712yes—
276C3⋊Dic3—62no—
277C3⋊Dic3—62no—
278C3×Dic3—22no—
279C3×Dic3—61no—
280C3×Dic3—61no—
281C3×Dic3—61no—
282C22×C9—21no—
283C22×C9—62no—
284C22×C9—62no—
285C22×C9—62no—
286C22×C9C23×D311yescharacteristic
287C2×D9—21no—
288C2×D9—21no—
289C2×D9—21no—
290C4×C9—22no—
291C4×C9—22no—
292C4×C9—21no—
293C4×C9—21no—
294C4×C9—22no—
295C4×C9—62no—
296C4×C9#23512yes—
297Dic9—21no—
298Dic9—21no—
299Dic9—21no—
300Dic9—62no—
301Dic9—62no—
302Dic9—122no—
303Dic9—122no—
304C4⋊Q8—271no—
305#126—272no—
306#124—31no—
307#124—272no—
308#123—272no—
309C22⋊Q8—272no—
310C22⋊Q8—271no—
311#121—271no—
312C4×Q8—271no—
313C4×D4—271no—
314#117—91no—
315C3×C9#54211yescharacteristic, complemented, Sylow
316C23×C3—32no—
317C23×C3—91no—
318C22×D3—31no—
319C22×D3C22×D912yes—
320C3×Q8—62no—
321C3×D4—62no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—32no—
326C2×C3×C4—91no—
327C2×C3×C4—91no—
328C2×C3×C4—91no—
329C2×C3×C4—182no—
330C2×C3×C4C22×D911yescharacteristic
331C2×C3×C4C22×D911yescharacteristic
332C2×C3×C4C22×D911yescharacteristic
333C2×C3×C4C2×D3211yescharacteristic
334C2×C3×C4C2×D3211yescharacteristic
335C2×C3×C4C2×D3211yescharacteristic
336#67—362no—
337#67—362no—
338C2×Dic3—31no—
339C2×Dic3—31no—
340C2×Dic3—31no—
341C2×Dic3—92no—
342C2×Dic3—92no—
343C2×Dic3—92no—
344C2×Dic3—92no—
345C2×Dic3—182no—
346C2×Dic3—182no—
347C2×Dic3—362no—
348C2×Dic3C22×D912yes—
349D12—22no—
350C4×D3—62no—
351C4×D3—362no—
352Dic6—22no—
353Dic6—362no—
354Dic6—362no—
355Dic6—364no—
356C2×C32C2×#23711yescharacteristic
357C2×C32#100911yescharacteristic
358C2×C32#101011yescharacteristic
359C3×D3—42no—
360C3×D3—121no—
361C2×C9—21no—
362C2×C9—21no—
363C2×C9—21no—
364C2×C9—122no—
365C2×C9C2×#23511yescharacteristic
366C2×C9#100911yescharacteristic
367C2×C9#101011yescharacteristic
368D9—41no—
369C2×Q8—31no—
370C2×Q8—272no—
371C2×D4—31no—
372C2×D4—272no—
373C22×C4—91no—
374C22×C4—272no—
375C4⋊C4—91no—
376C4⋊C4—91no—
377C4⋊C4—272no—
378C4⋊C4—272no—
379C4⋊C4—271no—
380C4⋊C4—271no—
381C4⋊C4—272no—
382C22⋊C4—32no—
383C22⋊C4—32no—
384C22⋊C4—91no—
385C22⋊C4—91no—
386C22⋊C4—272no—
387C22⋊C4—272no—
388C22⋊C4—272no—
389C42—91no—
390C42—272no—
391C42D3×D911yescharacteristic
392C22×C3—21no—
393C22×C3—62no—
394C22×C3—62no—
395C22×C3—62no—
396C22×C3—181no—
397C22×C3—181no—
398C22×C3—181no—
399C22×C3C23×D911yescharacteristic
400C22×C3C22×D3211yescharacteristic
401C2×D3—22no—
402C2×D3—22no—
403C2×D3—22no—
404C2×D3—61no—
405C2×D3—61no—
406C2×D3—61no—
407C3×C4—22no—
408C3×C4—22no—
409C3×C4—21no—
410C3×C4—21no—
411C3×C4—21no—
412C3×C4—21no—
413C3×C4—22no—
414C3×C4—62no—
415C3×C4—181no—
416C3×C4—181no—
417C3×C4—181no—
418C3×C4#369912yes—
419C3×C4#379512yes—
420Dic3—22no—
421Dic3—61no—
422Dic3—61no—
423Dic3—61no—
424Dic3—182no—
425Dic3—182no—
426Dic3—182no—
427Dic3—182no—
428Dic3—362no—
429Dic3—362no—
430C32#568211yescharacteristic
431C9—21no—
432C9#553411yescharacteristic, complemented
433C23—32no—
434C23—91no—
435Q8—62no—
436Q8—1082no—
437D4—62no—
438D4—1082no—
439C2×C4—32no—
440C2×C4—91no—
441C2×C4—91no—
442C2×C4—91no—
443C2×C4—182no—
444C2×C4—272no—
445C2×C4—272no—
446C2×C4—1082no—
447C2×C4C2×D3×D911yescharacteristic
448C2×C4C2×D3×D911yescharacteristic
449C2×C4C2×D3×D911yescharacteristic
450C2×C3—21no—
451C2×C3—21no—
452C2×C3—21no—
453C2×C3—122no—
454C2×C3—361no—
455C2×C3C2×#369911yescharacteristic
456C2×C3#6371511yescharacteristic
457C2×C3#6371611yescharacteristic
458C2×C3C2×#379511yescharacteristic
459C2×C3#6429811yescharacteristic
460C2×C3#6430811yescharacteristic
461D3—42no—
462D3—121no—
463C22—62no—
464C22—62no—
465C22—62no—
466C22—181no—
467C22—181no—
468C22—181no—
469C22C22×D3×D911yescharacteristic
470C4—21no—
471C4—21no—
472C4—62no—
473C4—181no—
474C4—181no—
475C4—181no—
476C4—542no—
477C4—542no—
478C4#8863812yes—
479C3—21no—
480C3#1058978411yescharacteristic, complemented
481C3#1059396111yescharacteristic
482C2—122no—
483C2—361no—
484C2C2×#8863811yescharacteristic
485C2#1172246511yescharacteristic
486C2#1172247811yescharacteristic
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#542
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #11721706C2semidirect
2. C9×#875C2semidirect
3. C3×#63434C2semidirect
4. #11721042C2semidirect
6. #11721024C2semidirect
7. D12×Dic9C2semidirect
8. #11721021C2semidirect
9. #11720965C2semidirect
10. #11720944C2semidirect
11. Dic6×Dic9C2semidirect
57. #63434D3semidirect
162. #875D9semidirect
315. C3×C9#542semidirect
432. C9#5534semidirect
480. C3#10589784semidirect

Maximal subgroups

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

subgroupindex
#117217062
C9×#8752
C3×#634342
#117210422
#117210362
#117210242
D12×Dic92
#117210212
#117209652
#117209442
Dic6×Dic92
#105939613
#105897843