Finite Group Databasebeta

← #10588632⋊C3 #13142520

Subgroups — #10588632⋊C3 #13142520

Order 1728.

576 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
1C2×C22⋊C4×3−1+2C211yesmaximal, characteristic
2C2×#3676⋊C3C211yesmaximal, characteristic
3C2×#3676⋊C3C211yesmaximal, characteristic
4C2×(3−1+2⋊C4)⋊C4C211yesmaximal, characteristic, complemented
53−1+2⋊#95C211yesmaximal, characteristic, complemented
63−1+2⋊#95C211yesmaximal, characteristic, complemented
73−1+2⋊#95C211yesmaximal, characteristic, complemented
8C3×#4952—31nomaximal
9#10588632C311yesmaximal, characteristic, complemented
10C24×3−1+2C2211yescharacteristic
11C22×C4×3−1+2C2211yescharacteristic
12C22×C4×3−1+2C2211yescharacteristic
13C22×3−1+2⋊C4C2211yescharacteristic
14C22×3−1+2⋊C4C2211yescharacteristic
15C22×3−1+2⋊C4C2211yescharacteristic
16C22×3−1+2⋊C4C2211yescharacteristic
17C22⋊C4×3−1+2—22no—
18#3676⋊C3—22no—
19#3676⋊C3—22no—
20(3−1+2⋊C4)⋊C4—22no—
21C2×C32×C22⋊C4D311yescharacteristic
22C2×C3×#217—31no—
23C2×C3×#217—31no—
24C2×C3×Dic3⋊C4—31no—
25C3×C3⋊#95—31no—
26C3×C3⋊#95—31no—
27C3×C3⋊#95—31no—
28C2×C9×C22⋊C4—21no—
29C2×C9×C22⋊C4C2×C311yescharacteristic
30C2×#3676C2×C311yescharacteristic
31C2×#3676C2×C311yescharacteristic
32C2×Dic9⋊C4C2×C311yescharacteristic, complemented
33C9⋊#95C2×C311yescharacteristic, complemented
34C9⋊#95C2×C311yescharacteristic, complemented
35C9⋊#95C2×C311yescharacteristic, complemented
36C23×3−1+2—21no—
37C23×3−1+2—21no—
38C23×3−1+2—21no—
39C23×3−1+2—21no—
40C23×3−1+2—21no—
41C23×3−1+2—21no—
42C23×3−1+2D412yes—
43C23×3−1+2C2311yescharacteristic
44C2×C4×3−1+2—21no—
45C2×C4×3−1+2—21no—
46C2×C4×3−1+2—21no—
47C2×C4×3−1+2—21no—
48C2×C4×3−1+2—21no—
49C2×C4×3−1+2—21no—
50C2×3−1+2⋊C4—21no—
51C2×3−1+2⋊C4—21no—
52C2×3−1+2⋊C4—21no—
53C2×3−1+2⋊C4—21no—
54C2×3−1+2⋊C4—21no—
55C2×3−1+2⋊C4—21no—
56C2×3−1+2⋊C4—21no—
57C2×3−1+2⋊C4—21no—
58C2×3−1+2⋊C4—21no—
59C2×3−1+2⋊C4—21no—
60C2×3−1+2⋊C4—21no—
61C2×3−1+2⋊C4D412yes—
62C3×#397—91no—
63#4952—31no—
64C24×C32C2×D311yescharacteristic
65C22×C32×C4C2×D311yescharacteristic
66C22×C32×C4C2×D311yescharacteristic
67C22×C3×Dic3—31no—
68C22×C3×Dic3—31no—
69C22×C3×Dic3—31no—
70C22×C3×Dic3—31no—
71C24×C9—21no—
72C24×C9C22×C311yescharacteristic
73C32×C22⋊C4—22no—
74C3×#217—62no—
75C3×#217—62no—
76C3×Dic3⋊C4—62no—
77C22×C4×C9—21no—
78C22×C4×C9—21no—
79C22×C4×C9C22×C311yescharacteristic
80C22×C4×C9C22×C311yescharacteristic
81C22×Dic9C22×C311yescharacteristic
82C22×Dic9C22×C311yescharacteristic
83C22×Dic9C22×C311yescharacteristic
84C22×Dic9C22×C311yescharacteristic
85C9×C22⋊C4—24no—
86C9×C22⋊C4—22no—
87#3676—22no—
88#3676—22no—
89Dic9⋊C4—22no—
90C22×3−1+2—22no—
91C22×3−1+2—22no—
92C22×3−1+2—22no—
93C22×3−1+2—41no—
94C22×3−1+2—41no—
95C22×3−1+2—41no—
96C22×3−1+2—41no—
97C22×3−1+2C2×D411yescharacteristic
98C22×3−1+2C2×D411yescharacteristic
99C22×3−1+2#4111yescharacteristic
100C22×3−1+2#4111yescharacteristic
101C22×3−1+2#4111yescharacteristic
102C22×3−1+2#4111yescharacteristic
103C22×3−1+2#4111yescharacteristic
104C4×3−1+2—41no—
105C4×3−1+2—41no—
1063−1+2⋊C4—22no—
1073−1+2⋊C4—41no—
1083−1+2⋊C4—41no—
1093−1+2⋊C4—41no—
110C2×C3×C4⋊C4—91no—
111C2×C3×C22⋊C4—31no—
112C2×C3×C22⋊C4—91no—
113C2×C3×C22⋊C4—91no—
114C2×C3×C22⋊C4C3×D311yescharacteristic
115C2×#217—31no—
116C2×#217—31no—
117C2×Dic3⋊C4—31no—
118C3×#95—91no—
119C3×#95—91no—
120C3×#95—91no—
121C3⋊#95—31no—
122C3⋊#95—31no—
123C3⋊#95—31no—
124C23×C32—21no—
125C23×C32—21no—
126C23×C32—21no—
127C23×C32—21no—
128C23×C32—21no—
129C23×C32—21no—
130C23×C32#6712yes—
131C23×C32C22×D311yescharacteristic
132C2×C32×C4—21no—
133C2×C32×C4—21no—
134C2×C32×C4—21no—
135C2×C32×C4—21no—
136C2×C32×C4—21no—
137C2×C32×C4—21no—
138C2×C3×Dic3—32no—
139C2×C3×Dic3—61no—
140C2×C3×Dic3—61no—
141C2×C3×Dic3—61no—
142C2×C3×Dic3—61no—
143C2×C3×Dic3—61no—
144C2×C3×Dic3—61no—
145C2×C3×Dic3—61no—
146C2×C3×Dic3—61no—
147C2×C3×Dic3—61no—
148C2×C3×Dic3—61no—
149C2×C3×Dic3—61no—
150C23×C9—22no—
151C23×C9—22no—
152C23×C9—21no—
153C23×C9—21no—
154C23×C9—21no—
155C23×C9—21no—
156C23×C9—21no—
157C23×C9—21no—
158C23×C9—22no—
159C23×C9—21no—
160C23×C9—41no—
161C23×C9—41no—
162C23×C9—41no—
163C23×C9—41no—
164C23×C9C3×D412yes—
165C23×C9C23×C311yescharacteristic
166C2×C4×C9—22no—
167C2×C4×C9—22no—
168C2×C4×C9—21no—
169C2×C4×C9—21no—
170C2×C4×C9—21no—
171C2×C4×C9—21no—
172C2×C4×C9—21no—
173C2×C4×C9—21no—
174C2×C4×C9—41no—
175C2×C4×C9—41no—
176C2×C4×C9—41no—
177C2×C4×C9—41no—
178C2×Dic9—21no—
179C2×Dic9—21no—
180C2×Dic9—21no—
181C2×Dic9—21no—
182C2×Dic9—21no—
183C2×Dic9—21no—
184C2×Dic9—21no—
185C2×Dic9—21no—
186C2×Dic9—21no—
187C2×Dic9—21no—
188C2×Dic9—21no—
189C2×Dic9C3×D412yes—
190#397—91noSylow
191C2×3−1+2—42no—
192C2×3−1+2#12111yescharacteristic
193C2×3−1+2#12311yescharacteristic
194C2×3−1+2#12311yescharacteristic
195C2×3−1+2#12311yescharacteristic
196C2×3−1+2#12411yescharacteristic
197C2×3−1+2#12611yescharacteristic
198C2×3−1+2#12611yescharacteristic
199C24×C3—31no—
200C24×C3C2×C3×D311yescharacteristic
201C22×C3×C4—31no—
202C22×C3×C4—31no—
203C22×C3×C4—91no—
204C22×C3×C4—91no—
205C22×C3×C4—91no—
206C22×C3×C4—91no—
207C22×C3×C4C2×C3×D311yescharacteristic
208C22×C3×C4C2×C3×D311yescharacteristic
209C22×Dic3—31no—
210C22×Dic3—31no—
211C22×Dic3—31no—
212C22×Dic3—31no—
213C3×C4⋊C4—182no—
214C3×C22⋊C4—22no—
215C3×C22⋊C4—62no—
216C3×C22⋊C4—182no—
217C3×C22⋊C4—182no—
218#217—62no—
219#217—62no—
220Dic3⋊C4—62no—
221C22×C32—22no—
222C22×C32—22no—
223C22×C32—22no—
224C22×C32—41no—
225C22×C32—41no—
226C22×C32—41no—
227C22×C32—41no—
228C22×C32#23511yescharacteristic
229C22×C32#23511yescharacteristic
230C22×C32D3×D411yescharacteristic
231C22×C32#23711yescharacteristic
232C22×C32#23711yescharacteristic
233C22×C32#23711yescharacteristic
234C22×C32C2×#6711yescharacteristic
235C32×C4—41no—
236C32×C4—41no—
237C3×Dic3—62no—
238C3×Dic3—121no—
239C3×Dic3—121no—
240C3×Dic3—121no—
241C22×C9—24no—
242C22×C9—22no—
243C22×C9—22no—
244C22×C9—22no—
245C22×C9—21no—
246C22×C9—21no—
247C22×C9—21no—
248C22×C9—21no—
249C22×C9—21no—
250C22×C9—21no—
251C22×C9—21no—
252C22×C9—42no—
253C22×C9—42no—
254C22×C9—42no—
255C22×C9—42no—
256C22×C9—42no—
257C22×C9—42no—
258C22×C9—41no—
259C22×C9—41no—
260C22×C9—41no—
261C22×C9—41no—
262C22×C9C2×C3×D411yescharacteristic
263C22×C9C2×C3×D411yescharacteristic
264C22×C9C3×#4111yescharacteristic
265C22×C9C3×#4111yescharacteristic
266C22×C9C3×#4111yescharacteristic
267C22×C9C3×#4111yescharacteristic
268C22×C9C3×#4111yescharacteristic
269C4×C9—42no—
270C4×C9—42no—
271C4×C9—41no—
272C4×C9—41no—
273Dic9—22no—
274Dic9—41no—
275Dic9—41no—
276Dic9—41no—
277C2×C4⋊C4—91no—
278C2×C22⋊C4—91no—
279C2×C22⋊C4—91no—
280C2×C22⋊C4AGL(1,ℤ/9)11yescharacteristic
281#95—91no—
282#95—91no—
283#95—91no—
2843−1+2#39711yescharacteristic, complemented, Sylow
285C23×C3—21no—
286C23×C3—21no—
287C23×C3—21no—
288C23×C3—21no—
289C23×C3—21no—
290C23×C3—21no—
291C23×C3—32no—
292C23×C3—31no—
293C23×C3—61no—
294C23×C3—61no—
295C23×C3—61no—
296C23×C3—61no—
297C23×C3—61no—
298C23×C3—61no—
299C23×C3C3×#6712yes—
300C23×C3C22×C3×D311yescharacteristic
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—21no—
305C2×C3×C4—21no—
306C2×C3×C4—21no—
307C2×C3×C4—61no—
308C2×C3×C4—61no—
309C2×C3×C4—61no—
310C2×C3×C4—61no—
311C2×C3×C4—61no—
312C2×C3×C4—61no—
313C2×C3×C4—92no—
314C2×C3×C4—181no—
315C2×C3×C4—181no—
316C2×C3×C4—181no—
317C2×C3×C4—181no—
318C2×C3×C4—181no—
319C2×C3×C4—181no—
320C2×C3×C4—181no—
321C2×C3×C4—181no—
322C2×C3×C4—181no—
323C2×C3×C4—181no—
324C2×C3×C4—181no—
325C2×Dic3—32no—
326C2×Dic3—61no—
327C2×Dic3—61no—
328C2×Dic3—61no—
329C2×Dic3—61no—
330C2×Dic3—61no—
331C2×Dic3—61no—
332C2×Dic3—61no—
333C2×Dic3—61no—
334C2×Dic3—61no—
335C2×Dic3—61no—
336C2×Dic3—61no—
337C2×C32—42no—
338C2×C32#87911yescharacteristic
339C2×C32#87911yescharacteristic
340C2×C32#88311yescharacteristic
341C2×C32#88511yescharacteristic
342C2×C32#92911yescharacteristic
343C2×C32#93511yescharacteristic
344C2×C32#93911yescharacteristic
345C2×C9—21no—
346C2×C9—21no—
347C2×C9—21no—
348C2×C9—21no—
349C2×C9—21no—
350C2×C9—21no—
351C2×C9—21no—
352C2×C9—44no—
353C2×C9—42no—
354C2×C9C3×#12111yescharacteristic
355C2×C9C3×#12311yescharacteristic
356C2×C9C3×#12311yescharacteristic
357C2×C9C3×#12311yescharacteristic
358C2×C9C3×#12411yescharacteristic
359C2×C9C3×#12611yescharacteristic
360C2×C9C3×#12611yescharacteristic
361C24C2×AGL(1,ℤ/9)11yescharacteristic
362C22×C4—91no—
363C22×C4—91no—
364C22×C4—91no—
365C22×C4—91no—
366C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
367C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
368C4⋊C4—182no—
369C22⋊C4—22no—
370C22⋊C4—182no—
371C22⋊C4—182no—
372C22×C3—22no—
373C22×C3—22no—
374C22×C3—22no—
375C22×C3—31no—
376C22×C3—31no—
377C22×C3—31no—
378C22×C3—31no—
379C22×C3—31no—
380C22×C3—31no—
381C22×C3—31no—
382C22×C3—41no—
383C22×C3—41no—
384C22×C3—41no—
385C22×C3—41no—
386C22×C3—62no—
387C22×C3—62no—
388C22×C3—62no—
389C22×C3—121no—
390C22×C3—121no—
391C22×C3—121no—
392C22×C3—121no—
393C22×C3C3×#23511yescharacteristic
394C22×C3C3×#23511yescharacteristic
395C22×C3C3×D3×D411yescharacteristic
396C22×C3C3×#23711yescharacteristic
397C22×C3C3×#23711yescharacteristic
398C22×C3C3×#23711yescharacteristic
399C22×C3C2×C3×#6711yescharacteristic
400C3×C4—41no—
401C3×C4—41no—
402C3×C4—121no—
403C3×C4—121no—
404C3×C4—182no—
405C3×C4—361no—
406C3×C4—361no—
407C3×C4—361no—
408Dic3—62no—
409Dic3—121no—
410Dic3—121no—
411Dic3—121no—
412C32#495211yescharacteristic
413C9—21no—
414C9C3×#39711yescharacteristic, complemented
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23#611⋊C312yes—
422C23C22×AGL(1,ℤ/9)11yescharacteristic
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—92no—
430C2×C4—181no—
431C2×C4—181no—
432C2×C4—181no—
433C2×C4—181no—
434C2×C4—181no—
435C2×C4—181no—
436C2×C4—181no—
437C2×C4—181no—
438C2×C4—181no—
439C2×C4—181no—
440C2×C4—181no—
441C2×C3—31no—
442C2×C3—31no—
443C2×C3—31no—
444C2×C3—31no—
445C2×C3—31no—
446C2×C3—31no—
447C2×C3—31no—
448C2×C3—42no—
449C2×C3—122no—
450C2×C3C3×#87911yescharacteristic
451C2×C3C3×#87911yescharacteristic
452C2×C3C3×#88311yescharacteristic
453C2×C3C3×#88511yescharacteristic
454C2×C3C3×#92911yescharacteristic
455C2×C3C3×#93511yescharacteristic
456C2×C3C3×#93911yescharacteristic
457C22—22no—
458C22—22no—
459C22—22no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22#3697⋊C311yescharacteristic
465C22#3697⋊C311yescharacteristic
466C22D4×AGL(1,ℤ/9)11yescharacteristic
467C22#3699⋊C311yescharacteristic
468C22#3699⋊C311yescharacteristic
469C22#3699⋊C311yescharacteristic
470C22C2×#611⋊C311yescharacteristic
471C4—41no—
472C4—41no—
473C4—182no—
474C4—361no—
475C4—361no—
476C4—361no—
477C3—31no—
478C3C3×#495211yescharacteristic
479C2—42no—
480C2#63441⋊C311yescharacteristic
481C2#63441⋊C311yescharacteristic
482C2#63445⋊C311yescharacteristic
483C2#63447⋊C311yescharacteristic
484C2#63491⋊C311yescharacteristic
485C2#63497⋊C311yescharacteristic
486C2#63501⋊C311yescharacteristic
subgroupG/H
centerC23C22×AGL(1,ℤ/9)
commutator, cosocleC23×C9C23×C3
Frattini, socleC23×C3C22×C3×D3
FittingC2×C22⋊C4×3−1+2C2
prime radicalC24×C32C2×D3
hypercenterC2×C22⋊C4AGL(1,ℤ/9)
hypocenterC9C3×#397
LCSG ▷ C23×C9 ▷ C9
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
4. C2×(3−1+2⋊C4)⋊C4C2semidirect
5. 3−1+2⋊#95C2semidirect
6. 3−1+2⋊#95C2semidirect
7. 3−1+2⋊#95C2semidirect
9. #10588632C3semidirect
32. C2×Dic9⋊C4C2×C3semidirect
33. C9⋊#95C2×C3semidirect
34. C9⋊#95C2×C3semidirect
35. C9⋊#95C2×C3semidirect
284. 3−1+2#397semidirect
414. C9C3×#397semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊C4×3−1+22
C2×#3676⋊C32
C2×(3−1+2⋊C4)⋊C42
3−1+2⋊#952
C3×#49523
#105886323