Finite Group Databasebeta

← #10588628⋊C3 #13142522

Subgroups — #10588628⋊C3 #13142522

Order 1728.

624 conjugacy classes of subgroups.

The subgroup lattice has 500 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×(3−1+2⋊C4)⋊C4C211yesmaximal, characteristic, complemented
6C2×(3−1+2⋊C4)⋊C4C211yesmaximal, characteristic, complemented
73−1+2⋊#95C211yesmaximal, characteristic, complemented
8C3×#4948—31nomaximal
9#10588628C311yesmaximal, 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—
20C4⋊(3−1+2⋊C4)—22no—
21(3−1+2⋊C4)⋊C4—22no—
22(3−1+2⋊C4)⋊C4—22no—
23C2×C32×C22⋊C4D311yescharacteristic
24C2×C3×#217—31no—
25C2×C3×#217—31no—
26C2×C3×C4⋊Dic3—31no—
27C2×C3×Dic3⋊C4—31no—
28C2×C3×Dic3⋊C4—31no—
29C3×C3⋊#95—31no—
30C2×C9×C22⋊C4—21no—
31C2×C9×C22⋊C4C2×C311yescharacteristic
32C2×#3676C2×C311yescharacteristic
33C2×#3676C2×C311yescharacteristic
34C2×C4⋊Dic9C2×C311yescharacteristic, complemented
35C2×Dic9⋊C4C2×C311yescharacteristic, complemented
36C2×Dic9⋊C4C2×C311yescharacteristic, complemented
37C9⋊#95C2×C311yescharacteristic, complemented
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+2—21no—
44C23×3−1+2Q812yes—
45C23×3−1+2C2311yescharacteristic
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×C4×3−1+2—21no—
51C2×C4×3−1+2D412yes—
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⋊C4—21no—
62C2×3−1+2⋊C4D412yes—
63C2×3−1+2⋊C4D412yes—
64C3×#396—91no—
65#4948—31no—
66C24×C32C2×D311yescharacteristic
67C22×C32×C4C2×D311yescharacteristic
68C22×C32×C4C2×D311yescharacteristic
69C22×C3×Dic3—31no—
70C22×C3×Dic3—31no—
71C22×C3×Dic3—31no—
72C22×C3×Dic3—31no—
73C24×C9—21no—
74C24×C9C22×C311yescharacteristic
75C32×C22⋊C4—22no—
76C3×#217—62no—
77C3×#217—62no—
78C3×C4⋊Dic3—62no—
79C3×Dic3⋊C4—62no—
80C3×Dic3⋊C4—62no—
81C22×C4×C9—21no—
82C22×C4×C9—21no—
83C22×C4×C9C22×C311yescharacteristic
84C22×C4×C9C22×C311yescharacteristic
85C22×Dic9C22×C311yescharacteristic
86C22×Dic9C22×C311yescharacteristic
87C22×Dic9C22×C311yescharacteristic
88C22×Dic9C22×C311yescharacteristic
89C9×C22⋊C4—24no—
90C9×C22⋊C4—22no—
91#3676—22no—
92#3676—22no—
93C4⋊Dic9—22no—
94Dic9⋊C4—22no—
95Dic9⋊C4—22no—
96C22×3−1+2—22no—
97C22×3−1+2—22no—
98C22×3−1+2—22no—
99C22×3−1+2—41no—
100C22×3−1+2—41no—
101C22×3−1+2—41no—
102C22×3−1+2—41no—
103C22×3−1+2C2×D411yescharacteristic
104C22×3−1+2C2×D411yescharacteristic
105C22×3−1+2C2×D411yescharacteristic
106C22×3−1+2C2×Q811yescharacteristic
107C22×3−1+2#4111yescharacteristic
108C22×3−1+2#4111yescharacteristic
109C22×3−1+2#4111yescharacteristic
110C4×3−1+2—22no—
111C4×3−1+2—41no—
1123−1+2⋊C4—22no—
1133−1+2⋊C4—22no—
1143−1+2⋊C4—41no—
1153−1+2⋊C4—41no—
116C2×C3×C4⋊C4—91no—
117C2×C3×C4⋊C4—91no—
118C2×C3×C4⋊C4—91no—
119C2×C3×C22⋊C4—31no—
120C2×C3×C22⋊C4—91no—
121C2×C3×C22⋊C4—91no—
122C2×C3×C22⋊C4C3×D311yescharacteristic
123C2×#217—31no—
124C2×#217—31no—
125C2×C4⋊Dic3—31no—
126C2×Dic3⋊C4—31no—
127C2×Dic3⋊C4—31no—
128C3×#95—91no—
129C3⋊#95—31no—
130C23×C32—21no—
131C23×C32—21no—
132C23×C32—21no—
133C23×C32—21no—
134C23×C32—21no—
135C23×C32—21no—
136C23×C32Dic612yes—
137C23×C32C22×D311yescharacteristic
138C2×C32×C4—21no—
139C2×C32×C4—21no—
140C2×C32×C4—21no—
141C2×C32×C4—21no—
142C2×C32×C4—21no—
143C2×C32×C4#6712yes—
144C2×C3×Dic3—32no—
145C2×C3×Dic3—32no—
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—
153C2×C3×Dic3—61no—
154C2×C3×Dic3—61no—
155C2×C3×Dic3—61no—
156C23×C9—22no—
157C23×C9—22no—
158C23×C9—21no—
159C23×C9—21no—
160C23×C9—21no—
161C23×C9—21no—
162C23×C9—21no—
163C23×C9—21no—
164C23×C9—22no—
165C23×C9—21no—
166C23×C9—41no—
167C23×C9—41no—
168C23×C9—41no—
169C23×C9—41no—
170C23×C9C3×Q812yes—
171C23×C9C23×C311yescharacteristic
172C2×C4×C9—21no—
173C2×C4×C9—21no—
174C2×C4×C9—22no—
175C2×C4×C9—21no—
176C2×C4×C9—21no—
177C2×C4×C9—21no—
178C2×C4×C9—22no—
179C2×C4×C9—41no—
180C2×C4×C9—41no—
181C2×C4×C9—41no—
182C2×C4×C9—41no—
183C2×C4×C9C3×D412yes—
184C2×Dic9—21no—
185C2×Dic9—21no—
186C2×Dic9—21no—
187C2×Dic9—21no—
188C2×Dic9—21no—
189C2×Dic9—21no—
190C2×Dic9—21no—
191C2×Dic9—21no—
192C2×Dic9—21no—
193C2×Dic9—21no—
194C2×Dic9C3×D412yes—
195C2×Dic9C3×D412yes—
196#396—91noSylow
197C2×3−1+2—42no—
198C2×3−1+2#12111yescharacteristic
199C2×3−1+2#12111yescharacteristic
200C2×3−1+2#12111yescharacteristic
201C2×3−1+2C22⋊Q811yescharacteristic
202C2×3−1+2C22⋊Q811yescharacteristic
203C2×3−1+2C22⋊Q811yescharacteristic
204C2×3−1+2#12611yescharacteristic
205C24×C3—31no—
206C24×C3C2×C3×D311yescharacteristic
207C22×C3×C4—31no—
208C22×C3×C4—31no—
209C22×C3×C4—91no—
210C22×C3×C4—91no—
211C22×C3×C4—91no—
212C22×C3×C4—91no—
213C22×C3×C4C2×C3×D311yescharacteristic
214C22×C3×C4C2×C3×D311yescharacteristic
215C22×Dic3—31no—
216C22×Dic3—31no—
217C22×Dic3—31no—
218C22×Dic3—31no—
219C3×C4⋊C4—182no—
220C3×C4⋊C4—182no—
221C3×C4⋊C4—182no—
222C3×C22⋊C4—22no—
223C3×C22⋊C4—62no—
224C3×C22⋊C4—182no—
225C3×C22⋊C4—182no—
226#217—62no—
227#217—62no—
228C4⋊Dic3—62no—
229Dic3⋊C4—62no—
230Dic3⋊C4—62no—
231C22×C32—22no—
232C22×C32—22no—
233C22×C32—22no—
234C22×C32—41no—
235C22×C32—41no—
236C22×C32—41no—
237C22×C32—41no—
238C22×C32C2×Dic611yescharacteristic
239C22×C32#23511yescharacteristic
240C22×C32D3×D411yescharacteristic
241C22×C32D3×D411yescharacteristic
242C22×C32#23711yescharacteristic
243C22×C32#23711yescharacteristic
244C22×C32C2×#6711yescharacteristic
245C32×C4—22no—
246C32×C4—41no—
247C3×Dic3—62no—
248C3×Dic3—62no—
249C3×Dic3—121no—
250C3×Dic3—121no—
251C22×C9—24no—
252C22×C9—22no—
253C22×C9—22no—
254C22×C9—22no—
255C22×C9—21no—
256C22×C9—21no—
257C22×C9—21no—
258C22×C9—21no—
259C22×C9—21no—
260C22×C9—21no—
261C22×C9—21no—
262C22×C9—42no—
263C22×C9—42no—
264C22×C9—42no—
265C22×C9—42no—
266C22×C9—42no—
267C22×C9—42no—
268C22×C9—41no—
269C22×C9—41no—
270C22×C9—41no—
271C22×C9—41no—
272C22×C9C2×C3×D411yescharacteristic
273C22×C9C2×C3×D411yescharacteristic
274C22×C9C2×C3×D411yescharacteristic
275C22×C9C2×C3×Q811yescharacteristic
276C22×C9C3×#4111yescharacteristic
277C22×C9C3×#4111yescharacteristic
278C22×C9C3×#4111yescharacteristic
279C4×C9—22no—
280C4×C9—42no—
281C4×C9—42no—
282C4×C9—41no—
283Dic9—22no—
284Dic9—22no—
285Dic9—41no—
286Dic9—41no—
287C2×C4⋊C4—91no—
288C2×C4⋊C4—91no—
289C2×C4⋊C4—91no—
290C2×C22⋊C4—91no—
291C2×C22⋊C4—91no—
292C2×C22⋊C4AGL(1,ℤ/9)11yescharacteristic
293#95—91no—
2943−1+2#39611yescharacteristic, complemented, Sylow
295C23×C3—21no—
296C23×C3—21no—
297C23×C3—21no—
298C23×C3—21no—
299C23×C3—21no—
300C23×C3—21no—
301C23×C3—32no—
302C23×C3—31no—
303C23×C3—61no—
304C23×C3—61no—
305C23×C3—61no—
306C23×C3—61no—
307C23×C3—61no—
308C23×C3—61no—
309C23×C3C3×Dic612yes—
310C23×C3C22×C3×D311yescharacteristic
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—32no—
317C2×C3×C4—61no—
318C2×C3×C4—61no—
319C2×C3×C4—61no—
320C2×C3×C4—61no—
321C2×C3×C4—61no—
322C2×C3×C4—92no—
323C2×C3×C4—92no—
324C2×C3×C4—181no—
325C2×C3×C4—181no—
326C2×C3×C4—181no—
327C2×C3×C4—181no—
328C2×C3×C4—181no—
329C2×C3×C4—181no—
330C2×C3×C4—181no—
331C2×C3×C4—181no—
332C2×C3×C4—181no—
333C2×C3×C4—181no—
334C2×C3×C4C3×#6712yes—
335C2×Dic3—32no—
336C2×Dic3—32no—
337C2×Dic3—61no—
338C2×Dic3—61no—
339C2×Dic3—61no—
340C2×Dic3—61no—
341C2×Dic3—61no—
342C2×Dic3—61no—
343C2×Dic3—61no—
344C2×Dic3—61no—
345C2×Dic3—61no—
346C2×Dic3—61no—
347C2×C32—42no—
348C2×C32#87811yescharacteristic
349C2×C32#87811yescharacteristic
350C2×C32#87911yescharacteristic
351C2×C32#88411yescharacteristic
352C2×C32#92411yescharacteristic
353C2×C32#93811yescharacteristic
354C2×C32#93911yescharacteristic
355C2×C9—21no—
356C2×C9—21no—
357C2×C9—21no—
358C2×C9—21no—
359C2×C9—21no—
360C2×C9—21no—
361C2×C9—21no—
362C2×C9—44no—
363C2×C9—42no—
364C2×C9C3×#12111yescharacteristic
365C2×C9C3×#12111yescharacteristic
366C2×C9C3×#12111yescharacteristic
367C2×C9C3×C22⋊Q811yescharacteristic
368C2×C9C3×C22⋊Q811yescharacteristic
369C2×C9C3×C22⋊Q811yescharacteristic
370C2×C9C3×#12611yescharacteristic
371C24C2×AGL(1,ℤ/9)11yescharacteristic
372C22×C4—91no—
373C22×C4—91no—
374C22×C4—91no—
375C22×C4—91no—
376C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
377C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
378C4⋊C4—182no—
379C4⋊C4—182no—
380C4⋊C4—182no—
381C22⋊C4—22no—
382C22⋊C4—182no—
383C22⋊C4—182no—
384C22×C3—22no—
385C22×C3—22no—
386C22×C3—22no—
387C22×C3—31no—
388C22×C3—31no—
389C22×C3—31no—
390C22×C3—31no—
391C22×C3—31no—
392C22×C3—31no—
393C22×C3—31no—
394C22×C3—41no—
395C22×C3—41no—
396C22×C3—41no—
397C22×C3—41no—
398C22×C3—62no—
399C22×C3—62no—
400C22×C3—62no—
401C22×C3—121no—
402C22×C3—121no—
403C22×C3—121no—
404C22×C3—121no—
405C22×C3C2×C3×Dic611yescharacteristic
406C22×C3C3×#23511yescharacteristic
407C22×C3C3×D3×D411yescharacteristic
408C22×C3C3×D3×D411yescharacteristic
409C22×C3C3×#23711yescharacteristic
410C22×C3C3×#23711yescharacteristic
411C22×C3C2×C3×#6711yescharacteristic
412C3×C4—22no—
413C3×C4—41no—
414C3×C4—62no—
415C3×C4—121no—
416C3×C4—182no—
417C3×C4—182no—
418C3×C4—361no—
419C3×C4—361no—
420Dic3—62no—
421Dic3—62no—
422Dic3—121no—
423Dic3—121no—
424C32#494811yescharacteristic
425C9—21no—
426C9C3×#39611yescharacteristic, complemented
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C233−1+2⋊Q812yes—
434C23C22×AGL(1,ℤ/9)11yescharacteristic
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—92no—
441C2×C4—92no—
442C2×C4—181no—
443C2×C4—181no—
444C2×C4—181no—
445C2×C4—181no—
446C2×C4—181no—
447C2×C4—181no—
448C2×C4—181no—
449C2×C4—181no—
450C2×C4—181no—
451C2×C4—181no—
452C2×C4#611⋊C312yes—
453C2×C3—31no—
454C2×C3—31no—
455C2×C3—31no—
456C2×C3—31no—
457C2×C3—31no—
458C2×C3—31no—
459C2×C3—31no—
460C2×C3—42no—
461C2×C3—122no—
462C2×C3C3×#87811yescharacteristic
463C2×C3C3×#87811yescharacteristic
464C2×C3C3×#87911yescharacteristic
465C2×C3C3×#88411yescharacteristic
466C2×C3C3×#92411yescharacteristic
467C2×C3C3×#93811yescharacteristic
468C2×C3C3×#93911yescharacteristic
469C22—22no—
470C22—22no—
471C22—22no—
472C22—41no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22C2×3−1+2⋊Q811yescharacteristic
477C22#3697⋊C311yescharacteristic
478C22D4×AGL(1,ℤ/9)11yescharacteristic
479C22D4×AGL(1,ℤ/9)11yescharacteristic
480C22#3699⋊C311yescharacteristic
481C22#3699⋊C311yescharacteristic
482C22C2×#611⋊C311yescharacteristic
483C4—22no—
484C4—41no—
485C4—182no—
486C4—182no—
487C4—361no—
488C4—361no—
489C3—31no—
490C3C3×#494811yescharacteristic
491C2—42no—
492C2#63440⋊C311yescharacteristic
493C2#63440⋊C311yescharacteristic
494C2#63441⋊C311yescharacteristic
495C2#63446⋊C311yescharacteristic
496C2#63486⋊C311yescharacteristic
497C2#63500⋊C311yescharacteristic
498C2#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×#396
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×C4⋊(3−1+2⋊C4)C2semidirect
5. C2×(3−1+2⋊C4)⋊C4C2semidirect
6. C2×(3−1+2⋊C4)⋊C4C2semidirect
7. 3−1+2⋊#95C2semidirect
9. #10588628C3semidirect
34. C2×C4⋊Dic9C2×C3semidirect
35. C2×Dic9⋊C4C2×C3semidirect
36. C2×Dic9⋊C4C2×C3semidirect
37. C9⋊#95C2×C3semidirect
294. 3−1+2#396semidirect
426. C9C3×#396semidirect

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×(3−1+2⋊C4)⋊C42
3−1+2⋊#952
C3×#49483
#105886283