Finite Group Databasebeta

← #72268942

Subgroups — #72268942

Order 2592.

609 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 609 conjugacy classes of subgroups — too many to draw. Only its 69 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×A4×3−1+2C211yesmaximal, characteristic, complemented
2C2×A4⋊AGL(1,ℤ/9)C211yesmaximal, characteristic, complemented
3A4⋊(3−1+2⋊C4)C211yesmaximal, characteristic, complemented
4C3×#64269—31nomaximal
5#11722376C311yesmaximal, characteristic, complemented
6#63694⋊C3—32nomaximal
7#11721704—31nomaximal
8C2×A4×3−1+2—21no—
9C2×A4×3−1+2C2211yescharacteristic
10A4⋊AGL(1,ℤ/9)—21no—
11#10619980—41nomaximal
12C22×C32×A4D311yescharacteristic
13C2×C3×C3⋊S4—31no—
14C3×C3⋊SL(2,ℤ/4)—31no—
15C24×3−1+2D311yescharacteristic, complemented
16C22×#1094D312yescomplemented
17C22×C9⋊A4—21no—
18C22×C9⋊A4—22no—
19C22×C9×A4—21no—
20C22×C9×A4C2×C311yescharacteristic, complemented
21C2×C9⋊S4C2×C311yescharacteristic, complemented
22C2×((C22⋊C9)⋊C2)⋊C3—32no—
23C2×#611⋊C3—31no—
24C2×#611⋊C3—31no—
25C2×#611⋊C3—31no—
26C9⋊SL(2,ℤ/4)C2×C311yescharacteristic, complemented
27#1094⋊C4—32no—
28#3676⋊C3—31no—
29#3676⋊C3—31no—
30#3676⋊C3—31no—
31C22×C3×3−1+2—41no—
32C2×C3⋊AGL(1,ℤ/9)—41no—
33A4×3−1+2D411yescharacteristic, complemented
34C3⋊(3−1+2⋊C4)—41no—
35#64269—31no—
36C3×GL(2,ℤ/4)—93no—
37C3×#953—91no—
38#63694—32no—
39#63515—31no—
40C2×C32×A4—21no—
41C2×C32×A4C2×D311yescharacteristic
42C3×C3⋊S4—61no—
43C3×#638—121no—
44C23×3−1+2—21no—
45C23×3−1+2—31no—
46C23×3−1+2—31no—
47C23×3−1+2—61no—
48C23×3−1+2C2×D311yescharacteristic
49C22×AGL(1,ℤ/9)—31no—
50C2×#1094—22no—
51C2×#1094C2×D312yes—
52C2×C9⋊A4—22no—
53C2×C9⋊A4—24no—
54C2×C9⋊A4—21no—
55C2×C9⋊A4—22no—
56C2×C9×A4—22no—
57C2×C9×A4—21no—
58C2×C9×A4—21no—
59C2×C9×A4C22×C311yescharacteristic
60C9⋊S4—21no—
61((C22⋊C9)⋊C2)⋊C3—62no—
62#6233—41no—
63#611⋊C3—61no—
64#611⋊C3—62no—
65#611⋊C3—61no—
66#611⋊C3—61no—
67#611⋊C3—61no—
68#611⋊C3—122no—
69C2×3−1+2⋊C4—31no—
70C2×3−1+2⋊C4—31no—
71C2×3−1+2⋊C4—31no—
72C2×C3×3−1+2—41no—
73C2×C3×3−1+2—81no—
74C3⋊AGL(1,ℤ/9)—81no—
75C24×C32C3⋊D311yescharacteristic
76C22×C3×A4—21no—
77C22×C3×A4—33no—
78C22×C3×A4C3×D311yescharacteristic
79C2×C3⋊S4—31no—
80C2×C3×S4—93no—
81C2×C3×#67—91no—
82C2×C3×#67—91no—
83C2×C3×#67—91no—
84C3⋊SL(2,ℤ/4)—31no—
85C3×SL(2,ℤ/4)—93no—
86C24×C9—21no—
87C24×C9C3×D311yescharacteristic, complemented
88C22×C22⋊C9—22no—
89C22×C22⋊C9C3×D312yescomplemented
90C2×(C22⋊C9)⋊C2—32no—
91C3×#217—91no—
92C3×#217—91no—
93C3×#217—91no—
94C2×#611—31no—
95C2×#611—31no—
96C2×#611—31no—
97(C22⋊C9)⋊C4—32no—
98#3676—31no—
99#3676—31no—
100#3676—31no—
101C22×C33—41no—
102C2×C3×C3⋊D3—121no—
103C32×A4#6711yescharacteristic
104C3×C3⋊Dic3—121no—
105C22×3−1+2—31no—
106C22×3−1+2—31no—
107C22×3−1+2—32no—
108C22×3−1+2—31no—
109C22×3−1+2—42no—
110C22×3−1+2—61no—
111C22×3−1+2—61no—
112C22×3−1+2—61no—
113C22×3−1+2—82no—
114C22×3−1+2—81no—
115C22×3−1+2#6711yescharacteristic, complemented
116C22×3−1+2S411yescharacteristic, complemented
117C22×C3×C9—41no—
118C22×C3×C9—81no—
119C2×C3⋊D9—41no—
120C2×AGL(1,ℤ/9)—61no—
121C2×AGL(1,ℤ/9)—61no—
122C2×AGL(1,ℤ/9)—61no—
123C2×AGL(1,ℤ/9)—122no—
124#1094#6712yescomplemented
125C9⋊A4—21no—
126C9⋊A4—22no—
127C9×A4—21no—
128C9×A4C3×D411yescharacteristic, complemented
129C3⋊Dic9—41no—
1303−1+2⋊C4—61no—
1313−1+2⋊C4—61no—
1323−1+2⋊C4—61no—
1333−1+2⋊C4—122no—
134GL(2,ℤ/4)—93no—
135C3×#120—271no—
136#953—91no—
137C3×3−1+2—41noSylow
138C23×C32—21no—
139C23×C32—31no—
140C23×C32—31no—
141C23×C32—61no—
142C23×C32C2×C3⋊D311yescharacteristic
143C22×C3×D3—91no—
144C2×C3×A4—21no—
145C2×C3×A4—22no—
146C2×C3×A4—21no—
147C2×C3×A4—33no—
148C2×C3×A4—63no—
149C2×C3×A4C2×C3×D311yescharacteristic
150C3⋊S4—61no—
151C3×S4—183no—
152#638—121no—
153C3×#67—181no—
154C3×#67—182no—
155C3×#67—181no—
156C3×#67—181no—
157C3×#67—181no—
158C3×#67—363no—
159C2×C3×Dic3—91no—
160C2×C3×Dic3—91no—
161C2×C3×Dic3—91no—
162C23×C9—22no—
163C23×C9—21no—
164C23×C9—21no—
165C23×C9—31no—
166C23×C9—31no—
167C23×C9—61no—
168C23×C9—61no—
169C23×C9—61no—
170C23×C9—62no—
171C23×C9C2×C3×D311yescharacteristic
172C22×D9—31no—
173C2×C22⋊C9—24no—
174C2×C22⋊C9—22no—
175C2×C22⋊C9—22no—
176C2×C22⋊C9C2×C3×D312yes—
177(C22⋊C9)⋊C2—62no—
178#611—61no—
179#611—62no—
180#611—61no—
181#611—61no—
182#611—61no—
183#611—122no—
184C2×Dic9—31no—
185C2×Dic9—31no—
186C2×Dic9—31no—
187C2×C33—41no—
188C2×C33—81no—
189C3×C3⋊D3—241no—
190C2×3−1+2—21no—
191C2×3−1+2—31no—
192C2×3−1+2—31no—
193C2×3−1+2—42no—
194C2×3−1+2—61no—
195C2×3−1+2—82no—
196C2×3−1+2—84no—
197C2×3−1+2—82no—
198C2×3−1+2—82no—
199C2×3−1+2—81no—
200C2×3−1+2C2×S411yescharacteristic
201C2×C3×C9—41no—
202C2×C3×C9—81no—
203C2×C3×C9—81no—
204C2×C3×C9—82no—
205C3⋊D9—81no—
206AGL(1,ℤ/9)—121no—
207AGL(1,ℤ/9)—242no—
208C24×C3—31no—
209C24×C3C3×C3⋊D311yescharacteristic
210C22×A4—61no—
211C22×A4AGL(1,ℤ/9)13yescomplemented
212C2×S4—93no—
213C2×C3×D4—271no—
214C2×C3×D4—271no—
215C2×C3×D4—271no—
216C2×#67—91no—
217C2×#67—91no—
218C2×#67—91no—
219SL(2,ℤ/4)—93no—
220C3×C22⋊C4—271no—
221C3×C22⋊C4—271no—
222C3×C22⋊C4—271no—
223#217—91no—
224#217—91no—
225#217—91no—
226C22×C32—31no—
227C22×C32—31no—
228C22×C32—32no—
229C22×C32—31no—
230C22×C32—41no—
231C22×C32—61no—
232C22×C32—61no—
233C22×C32—61no—
234C22×C32—81no—
235C22×C32—123no—
236C22×C32#63811yescharacteristic
237C22×C32C3⋊S411yescharacteristic
238C2×C3⋊D3—121no—
239C2×C3×D3—181no—
240C2×C3×D3—181no—
241C2×C3×D3—181no—
242C2×C3×D3—363no—
243C3×A4—21no—
244C3×A4—33no—
245C3×A4C3×#6711yescharacteristic
246C3⋊Dic3—121no—
247C3×Dic3—181no—
248C3×Dic3—181no—
249C3×Dic3—181no—
250C3×Dic3—363no—
251C22×C9—21no—
252C22×C9—21no—
253C22×C9—31no—
254C22×C9—31no—
255C22×C9—32no—
256C22×C9—31no—
257C22×C9—42no—
258C22×C9—61no—
259C22×C9—61no—
260C22×C9—61no—
261C22×C9—62no—
262C22×C9—61no—
263C22×C9—61no—
264C22×C9—62no—
265C22×C9—62no—
266C22×C9—61no—
267C22×C9—62no—
268C22×C9—82no—
269C22×C9C3×#6711yescharacteristic, complemented
270C22×C9C3×S411yescharacteristic, complemented
271C2×D9—61no—
272C2×D9—61no—
273C2×D9—61no—
274C2×D9—122no—
275C22⋊C9—22no—
276C22⋊C9C3×#6712yescomplemented
277Dic9—61no—
278Dic9—61no—
279Dic9—61no—
280Dic9—122no—
281#120—271noSylow
282C33—41no—
2833−1+2—42no—
2843−1+2—82no—
2853−1+2—81no—
2863−1+2GL(2,ℤ/4)11yescharacteristic, complemented
287C3×C9—41no—
288C3×C9—81no—
289C23×C3—21no—
290C23×C3—31no—
291C23×C3—31no—
292C23×C3—31no—
293C23×C3—61no—
294C23×C3—61no—
295C23×C3—91no—
296C23×C3—91no—
297C23×C3—181no—
298C23×C3—271no—
299C23×C3C2×C3×C3⋊D311yescharacteristic
300C22×D3—91no—
301C2×A4—23no—
302C2×A4—62no—
303C2×A4—61no—
304C2×A4C2×AGL(1,ℤ/9)13yes—
305S4—183no—
306C3×D4—541no—
307C3×D4—541no—
308C3×D4—541no—
309C3×D4—541no—
310C3×D4—542no—
311C2×C3×C4—271no—
312C2×C3×C4—271no—
313C2×C3×C4—271no—
314#67—181no—
315#67—182no—
316#67—181no—
317#67—181no—
318#67—181no—
319#67—363no—
320C2×Dic3—91no—
321C2×Dic3—91no—
322C2×Dic3—91no—
323C2×C32—21no—
324C2×C32—31no—
325C2×C32—31no—
326C2×C32—41no—
327C2×C32—61no—
328C2×C32—81no—
329C2×C32—82no—
330C2×C32—81no—
331C2×C32—123no—
332C2×C32—243no—
333C2×C32C2×C3⋊S411yescharacteristic
334C3⋊D3—241no—
335C3×D3—361no—
336C3×D3—723no—
337C2×C9—22no—
338C2×C9—21no—
339C2×C9—21no—
340C2×C9—31no—
341C2×C9—31no—
342C2×C9—42no—
343C2×C9—61no—
344C2×C9—61no—
345C2×C9—62no—
346C2×C9—61no—
347C2×C9—82no—
348C2×C9—84no—
349C2×C9—82no—
350C2×C9C2×C3×S411yescharacteristic
351D9—121no—
352D9—242no—
353C24C3⋊AGL(1,ℤ/9)11yescharacteristic, complemented
354C2×D4—271no—
355C2×D4—271no—
356C2×D4—271no—
357C22⋊C4—271no—
358C22⋊C4—271no—
359C22⋊C4—271no—
360C22×C3—31no—
361C22×C3—31no—
362C22×C3—32no—
363C22×C3—31no—
364C22×C3—31no—
365C22×C3—31no—
366C22×C3—43no—
367C22×C3—61no—
368C22×C3—61no—
369C22×C3—61no—
370C22×C3—91no—
371C22×C3—91no—
372C22×C3—91no—
373C22×C3—92no—
374C22×C3—181no—
375C22×C3—181no—
376C22×C3—181no—
377C22×C3—241no—
378C22×C3—541no—
379C22×C3—541no—
380C22×C3—541no—
381C22×C3C3×#63811yescharacteristic
382C22×C3C3×C3⋊S411yescharacteristic
383C2×D3—181no—
384C2×D3—181no—
385C2×D3—181no—
386C2×D3—363no—
387A4—61no—
388A4#611⋊C313yescomplemented
389C3×C4—541no—
390C3×C4—541no—
391C3×C4—541no—
392Dic3—181no—
393Dic3—181no—
394Dic3—181no—
395Dic3—363no—
396C32—41no—
397C32—81no—
398C32—123no—
399C32#6426911yescharacteristic
400C9—21no—
401C9—42no—
402C9—82no—
403C9C3×GL(2,ℤ/4)11yescharacteristic, complemented
404C23—21no—
405C23—31no—
406C23—31no—
407C23—61no—
408C23—271no—
409C23C2×C3⋊AGL(1,ℤ/9)11yescharacteristic
410D4—541no—
411D4—542no—
412D4—541no—
413D4—541no—
414D4—541no—
415C2×C4—271no—
416C2×C4—271no—
417C2×C4—271no—
418C2×C3—21no—
419C2×C3—31no—
420C2×C3—31no—
421C2×C3—31no—
422C2×C3—43no—
423C2×C3—61no—
424C2×C3—61no—
425C2×C3—83no—
426C2×C3—91no—
427C2×C3—91no—
428C2×C3—181no—
429C2×C3—242no—
430C2×C3—241no—
431C2×C3—1081no—
432C2×C3C2×C3×C3⋊S411yescharacteristic
433D3—361no—
434D3—723no—
435C22—31no—
436C22—31no—
437C22—32no—
438C22—31no—
439C22—61no—
440C22—61no—
441C22—61no—
442C22—541no—
443C22—541no—
444C22—541no—
445C22#1061998011yescharacteristic, complemented
446C22A4⋊AGL(1,ℤ/9)11yescharacteristic, complemented
447C4—541no—
448C4—541no—
449C4—541no—
450C3—31no—
451C3—43no—
452C3—241no—
453C3C3×#6426911yescharacteristic
454C2—21no—
455C2—31no—
456C2—31no—
457C2—61no—
458C2—1081no—
459C2C2×A4⋊AGL(1,ℤ/9)11yescharacteristic
subgroupG/H
centerC2C2×A4⋊AGL(1,ℤ/9)
commutator, cosocleC2×C9×A4C22×C3
FrattiniC2×C3C2×C3×C3⋊S4
FittingC24×3−1+2D3
socleC23×C3C2×C3×C3⋊D3
hypercenterC22A4⋊AGL(1,ℤ/9)
hypocenterC9×A4C3×D4
LCSG ▷ C2×C9×A4 ▷ C9×A4
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C22×A4×3−1+2C2semidirect
2. C2×A4⋊AGL(1,ℤ/9)C2semidirect
3. A4⋊(3−1+2⋊C4)C2semidirect
5. #11722376C3semidirect
15. C24×3−1+2D3semidirect
16. C22×#1094D3semidirect
20. C22×C9×A4C2×C3semidirect
21. C2×C9⋊S4C2×C3semidirect
26. C9⋊SL(2,ℤ/4)C2×C3semidirect
33. A4×3−1+2D4semidirect
87. C24×C9C3×D3semidirect
89. C22×C22⋊C9C3×D3semidirect
115. C22×3−1+2#67semidirect
116. C22×3−1+2S4semidirect
124. #1094#67semidirect
128. C9×A4C3×D4semidirect
211. C22×A4AGL(1,ℤ/9)semidirect
269. C22×C9C3×#67semidirect
270. C22×C9C3×S4semidirect
276. C22⋊C9C3×#67semidirect
286. 3−1+2GL(2,ℤ/4)semidirect
353. C24C3⋊AGL(1,ℤ/9)semidirect
388. A4#611⋊C3semidirect
403. C9C3×GL(2,ℤ/4)semidirect
445. C22#10619980semidirect
446. C22A4⋊AGL(1,ℤ/9)semidirect

Maximal subgroups

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

subgroupindex
C22×A4×3−1+22
C2×A4⋊AGL(1,ℤ/9)2
A4⋊(3−1+2⋊C4)2
C3×#642693
#117223763
#63694⋊C33
#117217043
#106199804