Finite Group Databasebeta

← #10588631⋊C3 #13142519

Subgroups — #10588631⋊C3 #13142519

Order 1728.

636 conjugacy classes of subgroups.

The subgroup lattice has 494 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×C4⋊(3−1+2⋊C4)C211yesmaximal, characteristic, complemented
5C2×C4×3−1+2⋊C4C211yesmaximal, characteristic, complemented
63−1+2⋊#95C211yesmaximal, characteristic, complemented
73−1+2⋊#95C211yesmaximal, characteristic, complemented
8C3×#4951—31nomaximal
9#10588631C311yesmaximal, 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+2C414yescomplemented
18#3676⋊C3—22no—
19#3676⋊C3—22no—
20C4⋊(3−1+2⋊C4)—22no—
21C4×3−1+2⋊C4—22no—
22C2×C32×C22⋊C4D311yescharacteristic
23C2×C3×#217—31no—
24C2×C3×#217—31no—
25C2×C3×C4⋊Dic3—31no—
26C2×C3×C4×Dic3—31no—
27C3×C3⋊#95—31no—
28C3×C3⋊#95—31no—
29C2×C9×C22⋊C4—21no—
30C2×C9×C22⋊C4C2×C311yescharacteristic
31C2×#3676C2×C311yescharacteristic
32C2×#3676C2×C311yescharacteristic
33C2×C4⋊Dic9C2×C311yescharacteristic, complemented
34C2×C4×Dic9C2×C311yescharacteristic, complemented
35C9⋊#95C2×C311yescharacteristic, complemented
36C9⋊#95C2×C311yescharacteristic, complemented
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+2—21no—
43C23×3−1+2C2×C412yes—
44C23×3−1+2C2311yescharacteristic
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+2C2×C412yes—
50C2×C4×3−1+2C2×C412yes—
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—
62C2×3−1+2⋊C4D412yes—
63C3×#388—91no—
64#4951—31no—
65C24×C32C2×D311yescharacteristic
66C22×C32×C4C2×D311yescharacteristic
67C22×C32×C4C2×D311yescharacteristic
68C22×C3×Dic3—31no—
69C22×C3×Dic3—31no—
70C22×C3×Dic3—31no—
71C22×C3×Dic3—31no—
72C24×C9—21no—
73C24×C9C22×C311yescharacteristic
74C32×C22⋊C4Dic314yes—
75C3×#217—62no—
76C3×#217—62no—
77C3×C4⋊Dic3—62no—
78C3×C4×Dic3—62no—
79C22×C4×C9—21no—
80C22×C4×C9—21no—
81C22×C4×C9C22×C311yescharacteristic
82C22×C4×C9C22×C311yescharacteristic
83C22×Dic9C22×C311yescharacteristic
84C22×Dic9C22×C311yescharacteristic
85C22×Dic9C22×C311yescharacteristic
86C22×Dic9C22×C311yescharacteristic
87C9×C22⋊C4—24no—
88C9×C22⋊C4C3×C414yescomplemented
89#3676—22no—
90#3676—22no—
91C4⋊Dic9—22no—
92C4×Dic9—22no—
93C22×3−1+2—22no—
94C22×3−1+2—22no—
95C22×3−1+2—22no—
96C22×3−1+2—41no—
97C22×3−1+2—41no—
98C22×3−1+2—41no—
99C22×3−1+2—41no—
100C22×3−1+2C22×C411yescharacteristic
101C22×3−1+2C2×D411yescharacteristic
102C22×3−1+2C2×D411yescharacteristic
103C22×3−1+2#4111yescharacteristic
104C22×3−1+2#4111yescharacteristic
105C22×3−1+2#4111yescharacteristic
106C22×3−1+2#4111yescharacteristic
107C4×3−1+2—22no—
108C4×3−1+2—22no—
1093−1+2⋊C4—22no—
1103−1+2⋊C4—22no—
1113−1+2⋊C4—41no—
1123−1+2⋊C4—41no—
113C2×C3×C4⋊C4—91no—
114C2×C3×C22⋊C4—31no—
115C2×C3×C22⋊C4—91no—
116C2×C3×C22⋊C4—91no—
117C2×C3×C22⋊C4C3×D311yescharacteristic
118C2×C3×C42—91no—
119C2×#217—31no—
120C2×#217—31no—
121C2×C4⋊Dic3—31no—
122C2×C4×Dic3—31no—
123C3×#95—91no—
124C3×#95—91no—
125C3⋊#95—31no—
126C3⋊#95—31no—
127C23×C32—21no—
128C23×C32—21no—
129C23×C32—21no—
130C23×C32—21no—
131C23×C32—21no—
132C23×C32—21no—
133C23×C32C2×Dic312yes—
134C23×C32C22×D311yescharacteristic
135C2×C32×C4—21no—
136C2×C32×C4—21no—
137C2×C32×C4—21no—
138C2×C32×C4—21no—
139C2×C32×C4C2×Dic312yes—
140C2×C32×C4C2×Dic312yes—
141C2×C3×Dic3—32no—
142C2×C3×Dic3—32no—
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—
150C2×C3×Dic3—61no—
151C2×C3×Dic3—61no—
152C2×C3×Dic3—61no—
153C23×C9—22no—
154C23×C9—22no—
155C23×C9—21no—
156C23×C9—21no—
157C23×C9—21no—
158C23×C9—21no—
159C23×C9—21no—
160C23×C9—21no—
161C23×C9—22no—
162C23×C9—21no—
163C23×C9—41no—
164C23×C9—41no—
165C23×C9—41no—
166C23×C9—41no—
167C23×C9C2×C3×C412yes—
168C23×C9C23×C311yescharacteristic
169C2×C4×C9—21no—
170C2×C4×C9—21no—
171C2×C4×C9—21no—
172C2×C4×C9—21no—
173C2×C4×C9—22no—
174C2×C4×C9—22no—
175C2×C4×C9—41no—
176C2×C4×C9—41no—
177C2×C4×C9—41no—
178C2×C4×C9—41no—
179C2×C4×C9C2×C3×C412yes—
180C2×C4×C9C2×C3×C412yes—
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×Dic9—21no—
190C2×Dic9—21no—
191C2×Dic9C3×D412yes—
192C2×Dic9C3×D412yes—
193#388—91noSylow
194C2×3−1+2—42no—
195C2×3−1+2#11711yescharacteristic
196C2×3−1+2C4×D411yescharacteristic
197C2×3−1+2C4×D411yescharacteristic
198C2×3−1+2#12111yescharacteristic
199C2×3−1+2#12311yescharacteristic
200C2×3−1+2#12411yescharacteristic
201C2×3−1+2#12611yescharacteristic
202C24×C3—31no—
203C24×C3C2×C3×D311yescharacteristic
204C22×C3×C4—31no—
205C22×C3×C4—31no—
206C22×C3×C4—91no—
207C22×C3×C4—91no—
208C22×C3×C4—91no—
209C22×C3×C4—91no—
210C22×C3×C4C2×C3×D311yescharacteristic
211C22×C3×C4C2×C3×D311yescharacteristic
212C22×Dic3—31no—
213C22×Dic3—31no—
214C22×Dic3—31no—
215C22×Dic3—31no—
216C3×C4⋊C4—182no—
217C3×C22⋊C4—34no—
218C3×C22⋊C4—182no—
219C3×C22⋊C4—182no—
220C3×C22⋊C4C3×Dic314yes—
221C3×C42—182no—
222#217—62no—
223#217—62no—
224C4⋊Dic3—62no—
225C4×Dic3—62no—
226C22×C32—22no—
227C22×C32—22no—
228C22×C32—22no—
229C22×C32—41no—
230C22×C32—41no—
231C22×C32—41no—
232C22×C32—41no—
233C22×C32#23511yescharacteristic
234C22×C32#23511yescharacteristic
235C22×C32D3×D411yescharacteristic
236C22×C32D3×D411yescharacteristic
237C22×C32#23711yescharacteristic
238C22×C32#23711yescharacteristic
239C22×C32C22×Dic311yescharacteristic
240C32×C4—22no—
241C32×C4—22no—
242C3×Dic3—62no—
243C3×Dic3—62no—
244C3×Dic3—121no—
245C3×Dic3—121no—
246C22×C9—24no—
247C22×C9—22no—
248C22×C9—22no—
249C22×C9—22no—
250C22×C9—21no—
251C22×C9—21no—
252C22×C9—21no—
253C22×C9—21no—
254C22×C9—21no—
255C22×C9—21no—
256C22×C9—21no—
257C22×C9—42no—
258C22×C9—42no—
259C22×C9—42no—
260C22×C9—42no—
261C22×C9—42no—
262C22×C9—42no—
263C22×C9—41no—
264C22×C9—41no—
265C22×C9—41no—
266C22×C9—41no—
267C22×C9C22×C3×C411yescharacteristic
268C22×C9C2×C3×D411yescharacteristic
269C22×C9C2×C3×D411yescharacteristic
270C22×C9C3×#4111yescharacteristic
271C22×C9C3×#4111yescharacteristic
272C22×C9C3×#4111yescharacteristic
273C22×C9C3×#4111yescharacteristic
274C4×C9—22no—
275C4×C9—22no—
276C4×C9—42no—
277C4×C9—42no—
278Dic9—22no—
279Dic9—22no—
280Dic9—41no—
281Dic9—41no—
282C2×C4⋊C4—91no—
283C2×C22⋊C4—91no—
284C2×C22⋊C4—91no—
285C2×C22⋊C4AGL(1,ℤ/9)11yescharacteristic
286C2×C42—91no—
287#95—91no—
288#95—91no—
2893−1+2#38811yescharacteristic, complemented, Sylow
290C23×C3—21no—
291C23×C3—21no—
292C23×C3—21no—
293C23×C3—21no—
294C23×C3—21no—
295C23×C3—21no—
296C23×C3—32no—
297C23×C3—31no—
298C23×C3—61no—
299C23×C3—61no—
300C23×C3—61no—
301C23×C3—61no—
302C23×C3—61no—
303C23×C3—61no—
304C23×C3C2×C3×Dic312yes—
305C23×C3C22×C3×D311yescharacteristic
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—32no—
311C2×C3×C4—32no—
312C2×C3×C4—61no—
313C2×C3×C4—61no—
314C2×C3×C4—61no—
315C2×C3×C4—61no—
316C2×C3×C4—92no—
317C2×C3×C4—92no—
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×C3×C4—181no—
326C2×C3×C4—181no—
327C2×C3×C4—181no—
328C2×C3×C4C2×C3×Dic312yes—
329C2×C3×C4C2×C3×Dic312yes—
330C2×Dic3—32no—
331C2×Dic3—32no—
332C2×Dic3—61no—
333C2×Dic3—61no—
334C2×Dic3—61no—
335C2×Dic3—61no—
336C2×Dic3—61no—
337C2×Dic3—61no—
338C2×Dic3—61no—
339C2×Dic3—61no—
340C2×Dic3—61no—
341C2×Dic3—61no—
342C2×C32—42no—
343C2×C32#87911yescharacteristic
344C2×C32#88311yescharacteristic
345C2×C32#88411yescharacteristic
346C2×C32#88511yescharacteristic
347C2×C32#92611yescharacteristic
348C2×C32D4×Dic311yescharacteristic
349C2×C32D4×Dic311yescharacteristic
350C2×C9—21no—
351C2×C9—21no—
352C2×C9—21no—
353C2×C9—21no—
354C2×C9—21no—
355C2×C9—21no—
356C2×C9—21no—
357C2×C9—44no—
358C2×C9—42no—
359C2×C9C3×#11711yescharacteristic
360C2×C9C3×C4×D411yescharacteristic
361C2×C9C3×C4×D411yescharacteristic
362C2×C9C3×#12111yescharacteristic
363C2×C9C3×#12311yescharacteristic
364C2×C9C3×#12411yescharacteristic
365C2×C9C3×#12611yescharacteristic
366C24C2×AGL(1,ℤ/9)11yescharacteristic
367C22×C4—91no—
368C22×C4—91no—
369C22×C4—91no—
370C22×C4—91no—
371C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
372C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
373C4⋊C4—182no—
374C22⋊C4—182no—
375C22⋊C4—182no—
376C22⋊C43−1+2⋊C414yescomplemented
377C42—182no—
378C22×C3—22no—
379C22×C3—22no—
380C22×C3—22no—
381C22×C3—31no—
382C22×C3—31no—
383C22×C3—31no—
384C22×C3—31no—
385C22×C3—31no—
386C22×C3—31no—
387C22×C3—31no—
388C22×C3—41no—
389C22×C3—41no—
390C22×C3—41no—
391C22×C3—41no—
392C22×C3—62no—
393C22×C3—62no—
394C22×C3—62no—
395C22×C3—121no—
396C22×C3—121no—
397C22×C3—121no—
398C22×C3—121no—
399C22×C3C3×#23511yescharacteristic
400C22×C3C3×#23511yescharacteristic
401C22×C3C3×D3×D411yescharacteristic
402C22×C3C3×D3×D411yescharacteristic
403C22×C3C3×#23711yescharacteristic
404C22×C3C3×#23711yescharacteristic
405C22×C3C22×C3×Dic311yescharacteristic
406C3×C4—22no—
407C3×C4—22no—
408C3×C4—62no—
409C3×C4—62no—
410C3×C4—182no—
411C3×C4—182no—
412C3×C4—361no—
413C3×C4—361no—
414Dic3—62no—
415Dic3—62no—
416Dic3—121no—
417Dic3—121no—
418C32#495111yescharacteristic
419C9—21no—
420C9C3×#38811yescharacteristic, complemented
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23C2×3−1+2⋊C412yes—
428C23C22×AGL(1,ℤ/9)11yescharacteristic
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—92no—
434C2×C4—92no—
435C2×C4—181no—
436C2×C4—181no—
437C2×C4—181no—
438C2×C4—181no—
439C2×C4—181no—
440C2×C4—181no—
441C2×C4—181no—
442C2×C4—181no—
443C2×C4—181no—
444C2×C4—181no—
445C2×C4C2×3−1+2⋊C412yes—
446C2×C4C2×3−1+2⋊C412yes—
447C2×C3—31no—
448C2×C3—31no—
449C2×C3—31no—
450C2×C3—31no—
451C2×C3—31no—
452C2×C3—31no—
453C2×C3—31no—
454C2×C3—42no—
455C2×C3—122no—
456C2×C3C3×#87911yescharacteristic
457C2×C3C3×#88311yescharacteristic
458C2×C3C3×#88411yescharacteristic
459C2×C3C3×#88511yescharacteristic
460C2×C3C3×#92611yescharacteristic
461C2×C3C3×D4×Dic311yescharacteristic
462C2×C3C3×D4×Dic311yescharacteristic
463C22—22no—
464C22—22no—
465C22—22no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22#3697⋊C311yescharacteristic
471C22#3697⋊C311yescharacteristic
472C22D4×AGL(1,ℤ/9)11yescharacteristic
473C22D4×AGL(1,ℤ/9)11yescharacteristic
474C22#3699⋊C311yescharacteristic
475C22#3699⋊C311yescharacteristic
476C22C22×3−1+2⋊C411yescharacteristic
477C4—22no—
478C4—22no—
479C4—182no—
480C4—182no—
481C4—361no—
482C4—361no—
483C3—31no—
484C3C3×#495111yescharacteristic
485C2—42no—
486C2#63441⋊C311yescharacteristic
487C2#63445⋊C311yescharacteristic
488C2#63446⋊C311yescharacteristic
489C2#63447⋊C311yescharacteristic
490C2#63488⋊C311yescharacteristic
491C2D4×3−1+2⋊C411yescharacteristic
492C2D4×3−1+2⋊C411yescharacteristic
subgroupG/H
centerC23C22×AGL(1,ℤ/9)
commutatorC22×C9C22×C3×C4
Frattini, socleC23×C3C22×C3×D3
FittingC2×C22⋊C4×3−1+2C2
prime radicalC24×C32C2×D3
cosocleC23×C9C23×C3
hypercenterC2×C22⋊C4AGL(1,ℤ/9)
hypocenterC9C3×#388
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
4. C2×C4⋊(3−1+2⋊C4)C2semidirect
5. C2×C4×3−1+2⋊C4C2semidirect
6. 3−1+2⋊#95C2semidirect
7. 3−1+2⋊#95C2semidirect
9. #10588631C3semidirect
17. C22⋊C4×3−1+2C4semidirect
33. C2×C4⋊Dic9C2×C3semidirect
34. C2×C4×Dic9C2×C3semidirect
35. C9⋊#95C2×C3semidirect
36. C9⋊#95C2×C3semidirect
88. C9×C22⋊C4C3×C4semidirect
289. 3−1+2#388semidirect
376. C22⋊C43−1+2⋊C4semidirect
420. C9C3×#388semidirect

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×C4⋊(3−1+2⋊C4)2
C2×C4×3−1+2⋊C42
3−1+2⋊#952
C3×#49513
#105886313