Finite Group Databasebeta

← #10588456⋊C3 #13141825

Subgroups — #10588456⋊C3 #13141825

Order 1728.

589 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 589 conjugacy classes of subgroups — too many to draw. Only its 104 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
1C2×D4×AGL(1,ℤ/9)C211yesmaximal, characteristic
2C2×#3672⋊C3C211yesmaximal, characteristic, complemented
3C2×#3664⋊C3C211yesmaximal, characteristic, complemented
4#63459⋊C3C211yesmaximal, characteristic, complemented
53−1+2×#102C211yesmaximal, characteristic, complemented
6(C2×AGL(1,ℤ/9))⋊C8C211yesmaximal, characteristic, complemented
7#63369⋊C3C211yesmaximal, characteristic, complemented
8C3×#4776—31nomaximal
9#10588456C311yesmaximal, characteristic, complemented
10C23×AGL(1,ℤ/9)—21no—
11C2×D4×3−1+2C2211yescharacteristic
12C2×#611⋊C3—21no—
13D4×AGL(1,ℤ/9)—22no—
14D4×AGL(1,ℤ/9)—22no—
15C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
16C2×C4×AGL(1,ℤ/9)C2211yescharacteristic
17C2×3−1+2⋊Q8C2211yescharacteristic, complemented
18C2×C8×3−1+2C2211yescharacteristic, complemented
19C4⋊C4×3−1+2C2211yescharacteristic, complemented
20#3672⋊C3—22no—
21(C2×AGL(1,ℤ/9))⋊C4—21no—
22C4⋊(3−1+2⋊C4)—21no—
23C2×3−1+2⋊C8C2211yescharacteristic, complemented
24#3664⋊C3—22no—
25C2×C3×D3×D4—31no—
26C2×C3×#214—31no—
27C2×C3×#204—31no—
28C3×#897—31no—
29C2×D4×D9C2×C311yescharacteristic
30C32×#102D311yescharacteristic
31C3×(C2×D3)⋊C8—31no—
32C3×#809—31no—
33C2×#3672C2×C311yescharacteristic, complemented
34C2×#3664C2×C311yescharacteristic, complemented
35#63459C2×C311yescharacteristic, complemented
36C9×#102—21no—
37C9×#102C2×C311yescharacteristic, complemented
38(C2×D9)⋊C8C2×C311yescharacteristic, complemented
39#63369C2×C311yescharacteristic, complemented
40C23×3−1+2—21no—
41C22×AGL(1,ℤ/9)—24no—
42C22×AGL(1,ℤ/9)—21no—
43C22×AGL(1,ℤ/9)—42no—
44C22×AGL(1,ℤ/9)—42no—
45C22×AGL(1,ℤ/9)D411yescharacteristic
46D4×3−1+2—21no—
47D4×3−1+2D412yes—
48C2×C4×3−1+2—21no—
49C2×C4×3−1+2C2311yescharacteristic
50#611⋊C3—24no—
51C2×3−1+2⋊C4—21no—
52C2×3−1+2⋊C4D411yescharacteristic
53C4⋊AGL(1,ℤ/9)—21no—
54C4⋊AGL(1,ℤ/9)D412yes—
55C4×AGL(1,ℤ/9)—21no—
56C4×AGL(1,ℤ/9)—21no—
573−1+2⋊Q8—21no—
583−1+2⋊Q8—21no—
59C8×3−1+2—21no—
603−1+2⋊C8—21no—
61C3×#450—91no—
62#4776—31no—
63C23×C3×D3—61no—
64C2×C32×D4C2×D311yescharacteristic
65C2×C3×#67—61no—
66C3×D3×D4—62no—
67C3×D3×D4—62no—
68C2×C3×D12—31no—
69C2×C3×C4×D3—31no—
70C2×C3×Dic6—31no—
71C23×D9—21no—
72C2×C32×C8C2×D311yescharacteristic
73C32×C4⋊C4C2×D311yescharacteristic
74C3×#214—62no—
75C3×(C2×D3)⋊C4—61no—
76C3×C4⋊Dic3—61no—
77C2×C3×C3⋊C8—31no—
78C3×#204—62no—
79C2×D4×C9—21no—
80C2×D4×C9C22×C311yescharacteristic
81C2×#611—21no—
82D4×D9—22no—
83D4×D9—22no—
84C2×D36C22×C311yescharacteristic
85C2×C4×D9C22×C311yescharacteristic
86C2×Dic18C22×C311yescharacteristic, complemented
87C2×C8×C9—21no—
88C2×C8×C9C22×C311yescharacteristic, complemented
89C9×C4⋊C4—21no—
90C9×C4⋊C4C22×C311yescharacteristic, complemented
91#3672—22no—
92(C2×D9)⋊C4—21no—
93C4⋊Dic9—21no—
94C2×C9⋊C8C22×C311yescharacteristic, complemented
95#3664—22no—
96C22×3−1+2—22no—
97C22×3−1+2—41no—
98C22×3−1+2—41no—
99C22×3−1+2C2×D411yescharacteristic
100C2×AGL(1,ℤ/9)—21no—
101C2×AGL(1,ℤ/9)—21no—
102C2×AGL(1,ℤ/9)—22no—
103C2×AGL(1,ℤ/9)—48no—
104C2×AGL(1,ℤ/9)—41no—
105C2×AGL(1,ℤ/9)—41no—
106C2×AGL(1,ℤ/9)SD1612yescomplemented
107C4×3−1+2—41no—
108C4×3−1+2C2×D411yescharacteristic
109C4×3−1+2C2×D411yescharacteristic
1103−1+2⋊C4—21no—
1113−1+2⋊C4—41no—
112C22×C3×D4—91no—
113C2×D3×D4—31no—
114C2×C3×SD16—91no—
115C2×C3×SD16—91no—
116C3×C22⋊Q8—91no—
117C2×#214—31no—
118C2×#204—31no—
119#897—31no—
120C3×#102—31no—
121C3×#102—91no—
122C3×#102C3×D311yescharacteristic
123C3×C22⋊C8—91no—
124(C2×D3)⋊C8—31no—
125#809—31no—
126C23×C32—21no—
127C22×C3×D3—31no—
128C22×C3×D3—64no—
129C22×C3×D3—61no—
130C22×C3×D3—122no—
131C22×C3×D3—122no—
132C32×D4—21no—
133C32×D4D1212yes—
134C2×C32×C4—21no—
135C2×C32×C4C22×D311yescharacteristic
136C3×#67—64no—
137C2×C3×Dic3—31no—
138C2×C3×Dic3—61no—
139C3×D12—32no—
140C3×D12—61no—
141C3×C4×D3—61no—
142C3×C4×D3—61no—
143C3×Dic6—61no—
144C3×Dic6—61no—
145C23×C9—21no—
146C23×C9—41no—
147C22×D9—24no—
148C22×D9—21no—
149C22×D9—42no—
150C22×D9—42no—
151C22×D9C3×D411yescharacteristic
152C32×C8—21no—
153C3×C3⋊C8—61no—
154D4×C9—21no—
155D4×C9—22no—
156D4×C9—41no—
157D4×C9C3×D412yes—
158C2×C4×C9—21no—
159C2×C4×C9—21no—
160C2×C4×C9—41no—
161C2×C4×C9C23×C311yescharacteristic
162#611—24no—
163C2×Dic9—21no—
164C2×Dic9C3×D411yescharacteristic
165D36—21no—
166D36C3×D412yes—
167C4×D9—21no—
168C4×D9—21no—
169Dic18—21no—
170Dic18—21no—
171C8×C9—21no—
172C8×C9—41no—
173C9⋊C8—21no—
174#450—91noSylow
175C2×3−1+2—42no—
176C2×3−1+2#12011yescharacteristic
177C2×3−1+2C2×SD1611yescharacteristic
178C2×3−1+2AGL(1,ℤ/8)11yescharacteristic
179AGL(1,ℤ/9)—22no—
180AGL(1,ℤ/9)—42no—
181C24×C3—181no—
182C23×D3—61no—
183C2×C3×Q8—91no—
184C2×C3×D4—31no—
185C2×C3×D4—91no—
186C2×C3×D4—181no—
187C2×C3×D4—182no—
188C2×C3×D4—182no—
189C2×C3×D4C2×C3×D311yescharacteristic
190C22×C3×C4—91no—
191C2×#67—61no—
192D3×D4—62no—
193D3×D4—62no—
194C2×D12—31no—
195C2×C4×D3—31no—
196C2×Dic6—31no—
197C3×SD16—182no—
198C3×SD16—182no—
199C2×C3×C8—31no—
200C2×C3×C8—91no—
201C2×C3×C8C2×C3×D311yescharacteristic
202C3×C4⋊C4—31no—
203C3×C4⋊C4—181no—
204C3×C4⋊C4C2×C3×D311yescharacteristic
205C3×C22⋊C4—181no—
206#214—62no—
207(C2×D3)⋊C4—61no—
208C4⋊Dic3—61no—
209C2×C3⋊C8—31no—
210#204—62no—
211C22×C32—22no—
212C22×C32—41no—
213C22×C32—41no—
214C22×C32D3×D411yescharacteristic
215C2×C3×D3—32no—
216C2×C3×D3—61no—
217C2×C3×D3—61no—
218C2×C3×D3—62no—
219C2×C3×D3—128no—
220C2×C3×D3—121no—
221C2×C3×D3—121no—
222C32×C4—41no—
223C32×C4C2×D1211yescharacteristic
224C32×C4D3×D411yescharacteristic
225C3×Dic3—61no—
226C3×Dic3—121no—
227C22×C9—22no—
228C22×C9—21no—
229C22×C9—42no—
230C22×C9—41no—
231C22×C9—41no—
232C22×C9—81no—
233C22×C9—81no—
234C22×C9C2×C3×D411yescharacteristic
235C2×D9—21no—
236C2×D9—21no—
237C2×D9—22no—
238C2×D9—48no—
239C2×D9—41no—
240C2×D9—41no—
241C2×D9C3×SD1612yescomplemented
242C4×C9—21no—
243C4×C9—21no—
244C4×C9—41no—
245C4×C9—81no—
246C4×C9C2×C3×D411yescharacteristic
247C4×C9C2×C3×D411yescharacteristic
248Dic9—21no—
249Dic9—41no—
250C22×D4—91no—
251C2×SD16—91no—
252C2×SD16—91no—
253C22⋊Q8—91no—
254#102—91no—
255#102AGL(1,ℤ/9)11yescharacteristic, complemented
256C22⋊C8—91no—
2573−1+2#45011yescharacteristic, complemented, Sylow
258C23×C3—21no—
259C23×C3—61no—
260C23×C3—91no—
261C23×C3—181no—
262C23×C3—184no—
263C23×C3—362no—
264C23×C3—362no—
265C22×D3—31no—
266C22×D3—64no—
267C22×D3—61no—
268C22×D3—122no—
269C22×D3—122no—
270C3×Q8—181no—
271C3×Q8—181no—
272C3×D4—21no—
273C3×D4—32no—
274C3×D4—61no—
275C3×D4—92no—
276C3×D4—181no—
277C3×D4—184no—
278C3×D4C3×D1212yes—
279C2×C3×C4—21no—
280C2×C3×C4—31no—
281C2×C3×C4—61no—
282C2×C3×C4—91no—
283C2×C3×C4—181no—
284C2×C3×C4—181no—
285C2×C3×C4—181no—
286C2×C3×C4C22×C3×D311yescharacteristic
287#67—64no—
288C2×Dic3—31no—
289C2×Dic3—61no—
290D12—32no—
291D12—61no—
292C4×D3—61no—
293C4×D3—61no—
294Dic6—61no—
295Dic6—61no—
296C3×C8—21no—
297C3×C8—61no—
298C3×C8—181no—
299C3⋊C8—61no—
300C2×C32—42no—
301C2×C32#88211yescharacteristic
302C2×C32#91111yescharacteristic
303C2×C32D3×SD1611yescharacteristic
304C3×D3—62no—
305C3×D3—122no—
306C2×C9—21no—
307C2×C9—21no—
308C2×C9—21no—
309C2×C9—42no—
310C2×C9—82no—
311C2×C9C3×#12011yescharacteristic
312C2×C9C2×C3×SD1611yescharacteristic
313C2×C9C3×AGL(1,ℤ/8)11yescharacteristic
314D9—22no—
315D9—42no—
316C24—181no—
317C2×Q8—91no—
318C2×D4—91no—
319C2×D4—181no—
320C2×D4—182no—
321C2×D4—182no—
322C2×D4C2×AGL(1,ℤ/9)11yescharacteristic
323C22×C4—91no—
324SD16—182no—
325SD16—182no—
326C2×C8—91no—
327C2×C8C2×AGL(1,ℤ/9)11yescharacteristic, complemented
328C4⋊C4—181no—
329C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic, complemented
330C22⋊C4—181no—
331C22×C3—22no—
332C22×C3—31no—
333C22×C3—41no—
334C22×C3—41no—
335C22×C3—62no—
336C22×C3—92no—
337C22×C3—121no—
338C22×C3—121no—
339C22×C3—181no—
340C22×C3—181no—
341C22×C3—182no—
342C22×C3—361no—
343C22×C3—361no—
344C22×C3—368no—
345C22×C3C3×D3×D411yescharacteristic
346C2×D3—32no—
347C2×D3—61no—
348C2×D3—61no—
349C2×D3—62no—
350C2×D3—128no—
351C2×D3—121no—
352C2×D3—121no—
353C3×C4—31no—
354C3×C4—31no—
355C3×C4—41no—
356C3×C4—121no—
357C3×C4—181no—
358C3×C4—361no—
359C3×C4C2×C3×D1211yescharacteristic
360C3×C4C3×D3×D411yescharacteristic
361Dic3—61no—
362Dic3—121no—
363C32#477611yescharacteristic
364C9—21no—
365C9C3×#45011yescharacteristic, complemented
366C23—21no—
367C23—91no—
368C23—184no—
369C23—181no—
370C23—362no—
371C23—362no—
372Q8—181no—
373Q8—181no—
374D4—21no—
375D4—92no—
376D4—184no—
377D4—181no—
378D4C4⋊AGL(1,ℤ/9)12yes—
379C2×C4—21no—
380C2×C4—91no—
381C2×C4—181no—
382C2×C4—181no—
383C2×C4—181no—
384C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
385C8—21no—
386C8—181no—
387C2×C3—31no—
388C2×C3—31no—
389C2×C3—31no—
390C2×C3—42no—
391C2×C3—122no—
392C2×C3—182no—
393C2×C3—362no—
394C2×C3C3×#88211yescharacteristic
395C2×C3C3×#91111yescharacteristic
396C2×C3C3×D3×SD1611yescharacteristic
397D3—62no—
398D3—122no—
399C22—22no—
400C22—41no—
401C22—41no—
402C22—92no—
403C22—181no—
404C22—181no—
405C22—182no—
406C22—368no—
407C22—361no—
408C22—361no—
409C22D4×AGL(1,ℤ/9)11yescharacteristic
410C4—41no—
411C4—181no—
412C4—361no—
413C4C2×C4⋊AGL(1,ℤ/9)11yescharacteristic
414C4D4×AGL(1,ℤ/9)11yescharacteristic
415C3—31no—
416C3C3×#477611yescharacteristic
417C2—42no—
418C2—182no—
419C2—362no—
420C2#63444⋊C311yescharacteristic
421C2#63473⋊C311yescharacteristic
422C2SD16×AGL(1,ℤ/9)11yescharacteristic
subgroupG/H
centerC22D4×AGL(1,ℤ/9)
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×#102C2
socleC22×C3C3×D3×D4
prime radicalC2×D4×AGL(1,ℤ/9)C2
hypercenter#102AGL(1,ℤ/9)
hypocenterC9C3×#450
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ #102

Complements

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

NH ≅ G/Ntype
2. C2×#3672⋊C3C2semidirect
3. C2×#3664⋊C3C2semidirect
4. #63459⋊C3C2semidirect
5. 3−1+2×#102C2semidirect
6. (C2×AGL(1,ℤ/9))⋊C8C2semidirect
7. #63369⋊C3C2semidirect
9. #10588456C3semidirect
17. C2×3−1+2⋊Q8C22semidirect
18. C2×C8×3−1+2C22semidirect
19. C4⋊C4×3−1+2C22semidirect
23. C2×3−1+2⋊C8C22semidirect
33. C2×#3672C2×C3semidirect
34. C2×#3664C2×C3semidirect
35. #63459C2×C3semidirect
37. C9×#102C2×C3semidirect
38. (C2×D9)⋊C8C2×C3semidirect
39. #63369C2×C3semidirect
86. C2×Dic18C22×C3semidirect
88. C2×C8×C9C22×C3semidirect
90. C9×C4⋊C4C22×C3semidirect
94. C2×C9⋊C8C22×C3semidirect
106. C2×AGL(1,ℤ/9)SD16semidirect
241. C2×D9C3×SD16semidirect
255. #102AGL(1,ℤ/9)semidirect
257. 3−1+2#450semidirect
327. C2×C8C2×AGL(1,ℤ/9)semidirect
329. C4⋊C4C2×AGL(1,ℤ/9)semidirect
365. C9C3×#450semidirect

Maximal subgroups

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

subgroupindex
C2×D4×AGL(1,ℤ/9)2
C2×#3672⋊C32
C2×#3664⋊C32
#63459⋊C32
3−1+2×#1022
(C2×AGL(1,ℤ/9))⋊C82
#63369⋊C32
C3×#47763
#105884563