Finite Group Databasebeta

← #13150954

Subgroups — #13150954

Order 1728.

742 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721838C211yesmaximal, characteristic
2D4×C9×Dic3C211yesmaximal, characteristic, complemented
3C3×#63498C211yesmaximal, characteristic, complemented
4#11721251C212yesmaximal, complemented
5#11721249C212yesmaximal, complemented
6#11721247C212yesmaximal, complemented
7#11721242C212yesmaximal, complemented
8#11720969C211yesmaximal, characteristic, complemented
9#11720936C211yesmaximal, characteristic, complemented
10#11720931C211yesmaximal, characteristic, complemented
11Dic3×Dic18C211yesmaximal, characteristic, complemented
12#10595121—31nomaximal
13#10589663—31nomaximal
14C2×C3×D4×C9C2211yescharacteristic
15C2×#6233C2212yes—
16C2×C4×C3⋊D9C2211yescharacteristic
17C22×C9×Dic3C2212yes—
18C2×C3×Dic18C2211yescharacteristic
19C2×#6189C2212yes—
20#88551C2212yes—
21C4⋊(C3⋊Dic9)C2211yescharacteristic
22C9×#217C2212yes—
23C3×#3676C2212yes—
24C3×#3676C2212yes—
25C9×C4⋊Dic3C2211yescharacteristic
26C4×C9×Dic3C2211yescharacteristic
27C3×C4×Dic9C2211yescharacteristic
28#88445C2211yescharacteristic
29#88445C2212yes—
30#88445C2211yescharacteristic
31Dic3⋊Dic9C2212yes—
32#88442C2211yescharacteristic
33#88442C2211yescharacteristic
34Dic9⋊Dic3C2212yes—
35Dic9⋊Dic3C2212yes—
36Dic3×Dic9C2212yes—
37#64147—31no—
38C3×#936—31no—
39C3×D4×Dic3D311yescharacteristic
40#63966—32no—
41#63964—32no—
42#63956—32no—
43#63953—32no—
44#63918—31no—
45#63874—31no—
46#63871—31no—
47Dic3×Dic6—31no—
48C4×D4×C9—31no—
49#63500—31no—
50#63500—61no—
51#63498D311yescharacteristic, complemented
52#63491—32no—
53#63486—32no—
54#63459—31no—
55#63450—31no—
56#63447—32no—
57#63441—32no—
58#63434—31no—
59C4×Dic18—31no—
60C23×C3×C9C2312yes—
61C22×C3⋊D9C2311yescharacteristic
62C3×D4×C9—22no—
63C2×C3×C4×C9C2311yescharacteristic
64#6233—42no—
65C2×C3⋊Dic9C2312yes—
66C2×C3⋊Dic9C2311yescharacteristic
67C4×C3⋊D9—22no—
68C2×C9×Dic3—42no—
69C2×C9×Dic3C2312yes—
70C2×C9×Dic3C2311yescharacteristic
71C2×C9×Dic3C2311yescharacteristic
72C2×C3×Dic9C2311yescharacteristic
73C2×C3×Dic9C2312yes—
74C2×C3×Dic9C2311yescharacteristic
75C3×Dic18—22no—
76#6189—42no—
77#5682—91no—
78#5561—91no—
79C2×C32×D4C2×D311yescharacteristic
80C2×#638—32no—
81C2×C4×C3⋊D3—31no—
82C22×C3×Dic3C2×D312yes—
83C2×C3×Dic6—31no—
84C2×#627—32no—
85#3757—32no—
86C4⋊(C3⋊Dic3)—31no—
87C3×#217—32no—
88C3×#217—32no—
89C3×#217C2×D312yes—
90C3×C4⋊Dic3C2×D311yescharacteristic
91C3×C4×Dic3—31no—
92C3×C4×Dic3C2×D311yescharacteristic
93#3724—31no—
94#3724—31no—
95Dic3⋊Dic3—32no—
96Dic3⋊Dic3—32no—
97Dic3⋊Dic3—32no—
98#3722—32no—
99#3722—31no—
100#3722—31no—
101Dic32—32no—
102C2×D4×C9—21no—
103C2×D4×C9C2×D311yescharacteristic
104C22×C4×C9—32no—
105C2×#611—32no—
106C2×#611—62no—
107C2×C4×D9—31no—
108C2×C4×D9—61no—
109C2×Dic18—32no—
110C2×Dic18C2×D311yescharacteristic
111C9×C4⋊C4—31no—
112C9×C22⋊C4—32no—
113C42×C9—31no—
114#3676—32no—
115#3676—62no—
116#3676C2×D312yes—
117#3676C2×D312yes—
118(C2×D9)⋊C4—31no—
119(C2×D9)⋊C4—32no—
120(C2×D9)⋊C4—31no—
121C4⋊Dic9—31no—
122C4⋊Dic9—32no—
123C4⋊Dic9—61no—
124Dic9⋊C4—32no—
125Dic9⋊C4—32no—
126Dic9⋊C4—31no—
127Dic9⋊C4—31no—
128C4×Dic9—32no—
129C4×Dic9C2×D311yescharacteristic
130C22×C3×C9—22no—
131C22×C3×C9—22no—
132C22×C3×C9—22no—
133C22×C3×C9C2411yescharacteristic
134C2×C3⋊D9—21no—
135C2×C3⋊D9—21no—
136C2×C3⋊D9—21no—
137C3×C4×C9#4112yes—
138C3⋊Dic9—21no—
139C3⋊Dic9—22no—
140C9×Dic3—22no—
141C9×Dic3—21no—
142C9×Dic3—21no—
143C3×Dic9—22no—
144C3×Dic9—21no—
145C3×Dic9—21no—
146C3×#124—91no—
147C3×C4×D4—31no—
148#946—91no—
149Q8×Dic3—91no—
150#938—91no—
151#938—91no—
152#938—181no—
153#936—31no—
154D4×Dic3D911yescharacteristic, complemented
155#929—92no—
156#924—92no—
157#897—91no—
158#888—91no—
159#886—92no—
160#885—92no—
161#885—92no—
162#879—92no—
163#879—92no—
164#878—92no—
165#872—91no—
166#872—91no—
167#869—91no—
168C4×Dic6—91no—
169C23×C32C22×D312yes—
170C22×C3⋊D3—31no—
171C32×D4—22no—
172C2×C32×C4C22×D311yescharacteristic
173#638—122no—
174C2×C3⋊Dic3—32no—
175C2×C3⋊Dic3—31no—
176C4×C3⋊D3—62no—
177C2×C3×Dic3—32no—
178C2×C3×Dic3—31no—
179C2×C3×Dic3—31no—
180C2×C3×Dic3—42no—
181C2×C3×Dic3C22×D311yescharacteristic
182C2×C3×Dic3C22×D311yescharacteristic
183C2×C3×Dic3C22×D312yes—
184C3×Dic6—62no—
185#627—122no—
186C23×C9—22no—
187C23×C9C22×D312yes—
188C22×D9—31no—
189C22×D9—61no—
190D4×C9—22no—
191D4×C9—42no—
192C2×C4×C9—21no—
193C2×C4×C9—31no—
194C2×C4×C9—31no—
195C2×C4×C9—32no—
196C2×C4×C9—122no—
197C2×C4×C9C22×D311yescharacteristic
198#611—122no—
199#611—124no—
200C2×Dic9—32no—
201C2×Dic9—31no—
202C2×Dic9—62no—
203C2×Dic9—61no—
204C2×Dic9C22×D312yes—
205C2×Dic9C22×D311yescharacteristic
206C2×Dic9C22×D311yescharacteristic
207C4×D9—62no—
208C4×D9—122no—
209Dic18—22no—
210Dic18—122no—
211#542—271noSylow
212C2×C3×C9—42no—
213C2×C3×C9C2×#4111yescharacteristic
214C2×C3×C9H(5,𝔽2)11yescharacteristic
215C2×C3×C92−1+411yescharacteristic
216C3⋊D9—41no—
217C2×C3×Q8—91no—
218C2×C3×D4—21no—
219C2×C3×D4C2×D911yescharacteristic
220C2×C3×D4D3211yescharacteristic
221C22×C3×C4—32no—
222C2×#67—92no—
223C2×#67—92no—
224C2×#67—182no—
225C22×Dic3C2×D912yes—
226C2×C4×D3—91no—
227C2×C4×D3—91no—
228C2×C4×D3—181no—
229C2×Dic6—31no—
230C2×Dic6—92no—
231C2×Dic6—92no—
232C3×C4⋊C4—31no—
233C3×C22⋊C4—32no—
234C3×C22⋊C4—92no—
235C3×C22⋊C4—92no—
236C3×C42—31no—
237C3×C42—91no—
238#217—32no—
239#217—32no—
240#217—92no—
241#217—92no—
242#217—182no—
243#217C2×D912yes—
244(C2×D3)⋊C4—91no—
245(C2×D3)⋊C4—91no—
246(C2×D3)⋊C4—92no—
247(C2×D3)⋊C4—91no—
248(C2×D3)⋊C4—91no—
249(C2×D3)⋊C4—92no—
250C4⋊Dic3—92no—
251C4⋊Dic3—91no—
252C4⋊Dic3—91no—
253C4⋊Dic3—92no—
254C4⋊Dic3—92no—
255C4⋊Dic3—181no—
256C4⋊Dic3C2×D911yescharacteristic
257Dic3⋊C4—92no—
258Dic3⋊C4—91no—
259Dic3⋊C4—91no—
260Dic3⋊C4—92no—
261Dic3⋊C4—91no—
262Dic3⋊C4—91no—
263Dic3⋊C4—92no—
264C4×Dic3—31no—
265C4×Dic3—92no—
266C4×Dic3—92no—
267C4×Dic3C2×D911yescharacteristic
268C22×C32—22no—
269C22×C32—22no—
270C22×C32—22no—
271C22×C32C23×D311yescharacteristic
272C2×C3⋊D3—61no—
273C2×C3⋊D3—61no—
274C2×C3⋊D3—61no—
275C32×C4#23512yes—
276C3⋊Dic3—61no—
277C3⋊Dic3—62no—
278C3×Dic3—22no—
279C3×Dic3—21no—
280C3×Dic3—21no—
281C3×Dic3—62no—
282C3×Dic3—61no—
283C3×Dic3—61no—
284C22×C9—22no—
285C22×C9—22no—
286C22×C9—22no—
287C22×C9—21no—
288C22×C9—42no—
289C22×C9—42no—
290C22×C9—42no—
291C22×C9C23×D311yescharacteristic
292C2×D9—62no—
293C2×D9—61no—
294C2×D9—61no—
295C2×D9—61no—
296C2×D9—121no—
297C2×D9—121no—
298C4×C9—22no—
299C4×C9—62no—
300C4×C9—61no—
301C4×C9—61no—
302C4×C9#23712yes—
303Dic9—22no—
304Dic9—21no—
305Dic9—21no—
306Dic9—61no—
307Dic9—62no—
308Dic9—121no—
309Dic9—122no—
310C4⋊Q8—271no—
311#126—272no—
312#124—91no—
313#124—272no—
314#123—272no—
315C22⋊Q8—272no—
316C22⋊Q8—271no—
317#121—271no—
318C4×Q8—271no—
319C4×D4—31no—
320#117—271no—
321C3×C9#54211yescharacteristic, complemented, Sylow
322C23×C3—22no—
323C23×C3C22×D912yes—
324C23×C3C2×D3212yes—
325C22×D3—91no—
326C22×D3—91no—
327C22×D3—181no—
328C3×Q8—182no—
329C3×D4—22no—
330C3×D4—22no—
331C3×D4—42no—
332C2×C3×C4—21no—
333C2×C3×C4—31no—
334C2×C3×C4—31no—
335C2×C3×C4—32no—
336C2×C3×C4—92no—
337C2×C3×C4—91no—
338C2×C3×C4—91no—
339C2×C3×C4—122no—
340C2×C3×C4C22×D911yescharacteristic
341C2×C3×C4C2×D3211yescharacteristic
342#67—362no—
343#67—362no—
344#67—364no—
345C2×Dic3—31no—
346C2×Dic3—31no—
347C2×Dic3—32no—
348C2×Dic3—42no—
349C2×Dic3—92no—
350C2×Dic3—91no—
351C2×Dic3—92no—
352C2×Dic3—91no—
353C2×Dic3—182no—
354C2×Dic3—181no—
355C2×Dic3C22×D911yescharacteristic
356C2×Dic3C22×D911yescharacteristic
357C2×Dic3C22×D912yes—
358C4×D3—182no—
359C4×D3—182no—
360C4×D3—362no—
361Dic6—62no—
362Dic6—362no—
363Dic6—362no—
364C2×C32—42no—
365C2×C32C2×#23511yescharacteristic
366C2×C32#100411yescharacteristic
367C2×C32#101011yescharacteristic
368C3⋊D3—121no—
369C2×C9—21no—
370C2×C9—21no—
371C2×C9—21no—
372C2×C9—42no—
373C2×C9—82no—
374C2×C9C2×#23711yescharacteristic
375C2×C9#100911yescharacteristic
376C2×C9#101011yescharacteristic
377D9—122no—
378D9—121no—
379C2×Q8—91no—
380C2×Q8—272no—
381C2×D4—272no—
382C2×D4D3×D911yescharacteristic
383C22×C4—32no—
384C22×C4—271no—
385C4⋊C4—31no—
386C4⋊C4—271no—
387C4⋊C4—271no—
388C4⋊C4—272no—
389C4⋊C4—272no—
390C4⋊C4—272no—
391C4⋊C4—271no—
392C22⋊C4—32no—
393C22⋊C4—92no—
394C22⋊C4—92no—
395C22⋊C4—271no—
396C22⋊C4—271no—
397C22⋊C4—272no—
398C22⋊C4—272no—
399C42—31no—
400C42—91no—
401C42—272no—
402C22×C3—22no—
403C22×C3—22no—
404C22×C3—22no—
405C22×C3—22no—
406C22×C3—22no—
407C22×C3—22no—
408C22×C3—21no—
409C22×C3—42no—
410C22×C3—42no—
411C22×C3—42no—
412C22×C3C23×D911yescharacteristic
413C22×C3C22×D3211yescharacteristic
414C2×D3—182no—
415C2×D3—181no—
416C2×D3—181no—
417C2×D3—181no—
418C2×D3—181no—
419C2×D3—181no—
420C2×D3—181no—
421C2×D3—361no—
422C2×D3—361no—
423C3×C4—22no—
424C3×C4—61no—
425C3×C4—62no—
426C3×C4—61no—
427C3×C4—182no—
428C3×C4—181no—
429C3×C4—181no—
430C3×C4#369712yes—
431C3×C4#380412yes—
432Dic3—22no—
433Dic3—21no—
434Dic3—21no—
435Dic3—61no—
436Dic3—61no—
437Dic3—62no—
438Dic3—182no—
439Dic3—181no—
440Dic3—181no—
441Dic3—182no—
442Dic3—361no—
443Dic3—362no—
444C32#556111yescharacteristic
445C9—21no—
446C9#568211yescharacteristic, complemented
447C23—271no—
448C23C2×D3×D912yes—
449Q8—182no—
450Q8—1082no—
451D4—22no—
452D4—1082no—
453C2×C4—31no—
454C2×C4—31no—
455C2×C4—32no—
456C2×C4—91no—
457C2×C4—91no—
458C2×C4—92no—
459C2×C4—122no—
460C2×C4—272no—
461C2×C4—271no—
462C2×C4—542no—
463C2×C4C2×D3×D911yescharacteristic
464C2×C3—21no—
465C2×C3—21no—
466C2×C3—21no—
467C2×C3—42no—
468C2×C3—42no—
469C2×C3—82no—
470C2×C3C2×#369711yescharacteristic
471C2×C3#6371011yescharacteristic
472C2×C3#6371611yescharacteristic
473C2×C3#6430911yescharacteristic
474C2×C3#6431411yescharacteristic
475C2×C3C2×#380411yescharacteristic
476D3—362no—
477D3—361no—
478D3—361no—
479C22—22no—
480C22—22no—
481C22—22no—
482C22—541no—
483C22—541no—
484C22—541no—
485C22C22×D3×D911yescharacteristic
486C4—61no—
487C4—62no—
488C4—61no—
489C4—181no—
490C4—182no—
491C4—181no—
492C4—542no—
493C4—541no—
494C4#8865812yes—
495C3—21no—
496C3#1058966311yescharacteristic, complemented
497C3#1059512111yescharacteristic
498C2—42no—
499C2—1081no—
500C2#1172249411yescharacteristic
501C2#1172250011yescharacteristic
502C2C2×#8865811yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC2×C3×D4×C9C22
socleC22×C32C23×D3
prime radical#11721838C2
hypercenterC2×D4D3×D9
hypocenterC3×C9#542
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. D4×C9×Dic3C2semidirect
3. C3×#63498C2semidirect
4. #11721251C2semidirect
5. #11721249C2semidirect
6. #11721247C2semidirect
7. #11721242C2semidirect
8. #11720969C2semidirect
9. #11720936C2semidirect
10. #11720931C2semidirect
11. Dic3×Dic18C2semidirect
51. #63498D3semidirect
154. D4×Dic3D9semidirect
321. C3×C9#542semidirect
446. C9#5682semidirect
496. C3#10589663semidirect

Maximal subgroups

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

subgroupindex
#117218382
D4×C9×Dic32
C3×#634982
#117212512
#117212492
#117212472
#117212422
#117209692
#117209362
#117209312
Dic3×Dic182
#105951213
#105896633