Finite Group Databasebeta

← #10588486⋊C3 #13141951

Subgroups — #10588486⋊C3 #13141951

Order 1728.

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

normal subgroups only

This group has 495 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×Q8×AGL(1,ℤ/9)C211yesmaximal, characteristic, complemented
2C2×#3675⋊C3C211yesmaximal, characteristic, complemented
3C2×#3664⋊C3C211yesmaximal, characteristic, complemented
4#63457⋊C3C211yesmaximal, characteristic
53−1+2×#103C211yesmaximal, characteristic, complemented
6(C2×AGL(1,ℤ/9))⋊C8C211yesmaximal, characteristic, complemented
7#63368⋊C3C211yesmaximal, characteristic, complemented
8C3×#4806—31nomaximal
9#10588486C311yesmaximal, characteristic, complemented
10C2×Q8×3−1+2C2211yescharacteristic
11Q8×AGL(1,ℤ/9)—22no—
12Q8×AGL(1,ℤ/9)—22no—
13C2×C4⋊AGL(1,ℤ/9)—21no—
14C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
15C2×C4×AGL(1,ℤ/9)—21no—
16C2×C4×AGL(1,ℤ/9)C2211yescharacteristic
17C2×3−1+2⋊Q8—21no—
18C2×3−1+2⋊Q8C2211yescharacteristic
19C2×C8×3−1+2C2211yescharacteristic
20C4⋊C4×3−1+2C2211yescharacteristic
21#3675⋊C3—22no—
22(C2×AGL(1,ℤ/9))⋊C4—21no—
23C2×3−1+2⋊C8C2211yescharacteristic
24#3664⋊C3—22no—
25C2×C3×D3×Q8—31no—
26C2×C3×#215—31no—
27C2×C3×#204—31no—
28C3×#895—31no—
29C2×Q8×D9C2×C311yescharacteristic, complemented
30C32×#103D311yescharacteristic
31C3×(C2×D3)⋊C8—31no—
32C3×#810—31no—
33C2×#3675C2×C311yescharacteristic, complemented
34C2×#3664C2×C311yescharacteristic, complemented
35#63457C2×C311yescharacteristic
36C9×#103—21no—
37C9×#103C2×C311yescharacteristic, complemented
38(C2×D9)⋊C8C2×C311yescharacteristic, complemented
39#63368C2×C311yescharacteristic, complemented
40C22×AGL(1,ℤ/9)—21no—
41C22×AGL(1,ℤ/9)D411yescharacteristic
42Q8×3−1+2—21no—
43Q8×3−1+2D412yescomplemented
44C2×C4×3−1+2—21no—
45C2×C4×3−1+2—21no—
46C2×C4×3−1+2C2311yescharacteristic
47C2×3−1+2⋊C4—21no—
48C2×3−1+2⋊C4D411yescharacteristic
49C4⋊AGL(1,ℤ/9)—21no—
50C4⋊AGL(1,ℤ/9)—21no—
51C4⋊AGL(1,ℤ/9)—42no—
52C4×AGL(1,ℤ/9)—24no—
53C4×AGL(1,ℤ/9)—21no—
54C4×AGL(1,ℤ/9)—21no—
553−1+2⋊Q8—24no—
563−1+2⋊Q8—21no—
573−1+2⋊Q8D412yescomplemented
58C8×3−1+2—21no—
593−1+2⋊C8—21no—
60C3×#448—91no—
61#4806—31no—
62C2×C32×Q8C2×D311yescharacteristic
63C3×D3×Q8—62no—
64C3×D3×Q8—62no—
65C2×C3×D12—31no—
66C2×C3×D12—61no—
67C2×C3×C4×D3—31no—
68C2×C3×C4×D3—61no—
69C2×C3×Dic6—31no—
70C2×C3×Dic6—61no—
71C2×C32×C8C2×D311yescharacteristic
72C32×C4⋊C4C2×D311yescharacteristic
73C3×#215—62no—
74C3×(C2×D3)⋊C4—61no—
75C2×C3×C3⋊C8—31no—
76C3×#204—62no—
77C2×Q8×C9—21no—
78C2×Q8×C9C22×C311yescharacteristic
79Q8×D9—22no—
80Q8×D9—22no—
81C2×D36—21no—
82C2×D36C22×C311yescharacteristic
83C2×C4×D9—21no—
84C2×C4×D9C22×C311yescharacteristic
85C2×Dic18—21no—
86C2×Dic18C22×C311yescharacteristic
87C2×C8×C9—21no—
88C2×C8×C9C22×C311yescharacteristic
89C9×C4⋊C4—21no—
90C9×C4⋊C4C22×C311yescharacteristic
91#3675—22no—
92(C2×D9)⋊C4—21no—
93C2×C9⋊C8C22×C311yescharacteristic
94#3664—22no—
95C22×3−1+2C2×D411yescharacteristic
96C2×AGL(1,ℤ/9)—21no—
97C2×AGL(1,ℤ/9)—21no—
98C2×AGL(1,ℤ/9)—41no—
99C2×AGL(1,ℤ/9)—41no—
100C2×AGL(1,ℤ/9)—41no—
101C2×AGL(1,ℤ/9)SD1612yescomplemented
102C4×3−1+2—22no—
103C4×3−1+2—41no—
104C4×3−1+2C2×D411yescharacteristic
105C4×3−1+2C2×D411yescharacteristic
1063−1+2⋊C4—21no—
1073−1+2⋊C4—22no—
108C22×C3×Q8—91no—
109C2×D3×Q8—31no—
110C2×C3×SD16—91no—
111C2×C3×SD16—91no—
112C3×#121—91no—
113C2×#215—31no—
114C2×#204—31no—
115#895—31no—
116C3×#103—31no—
117C3×#103—91no—
118C3×#103C3×D311yescharacteristic
119C3×C22⋊C8—91no—
120(C2×D3)⋊C8—31no—
121#810—31no—
122C22×C3×D3—31no—
123C22×C3×D3—61no—
124C32×Q8—21no—
125C32×Q8D1212yes—
126C2×C32×C4—21no—
127C2×C32×C4—21no—
128C2×C32×C4C22×D311yescharacteristic
129C2×C3×Dic3—31no—
130C2×C3×Dic3—61no—
131C3×D12—61no—
132C3×D12—61no—
133C3×D12—122no—
134C3×C4×D3—64no—
135C3×C4×D3—61no—
136C3×C4×D3—61no—
137C3×Dic6—32no—
138C3×Dic6—64no—
139C3×Dic6—61no—
140C22×D9—21no—
141C22×D9C3×D411yescharacteristic
142C32×C8—21no—
143C3×C3⋊C8—61no—
144Q8×C9—21no—
145Q8×C9—22no—
146Q8×C9—41no—
147Q8×C9C3×D412yescomplemented
148C2×C4×C9—21no—
149C2×C4×C9—21no—
150C2×C4×C9—21no—
151C2×C4×C9—41no—
152C2×C4×C9—41no—
153C2×C4×C9C23×C311yescharacteristic
154C2×Dic9—21no—
155C2×Dic9C3×D411yescharacteristic
156D36—21no—
157D36—21no—
158D36—42no—
159C4×D9—24no—
160C4×D9—21no—
161C4×D9—21no—
162Dic18—24no—
163Dic18—21no—
164Dic18C3×D412yescomplemented
165C8×C9—21no—
166C8×C9—41no—
167C9⋊C8—21no—
168#448—91noSylow
169C2×3−1+2#12011yescharacteristic
170C2×3−1+2C2×SD1611yescharacteristic
171C2×3−1+2#13711yescharacteristic
172AGL(1,ℤ/9)—22no—
173AGL(1,ℤ/9)—81no—
174C2×C3×Q8—31no—
175C2×C3×Q8—91no—
176C2×C3×Q8—181no—
177C2×C3×Q8—182no—
178C2×C3×Q8—182no—
179C2×C3×Q8C2×C3×D311yescharacteristic
180C2×C3×D4—91no—
181C2×C3×D4—181no—
182C22×C3×C4—91no—
183C22×C3×C4—181no—
184D3×Q8—62no—
185D3×Q8—62no—
186C2×D12—31no—
187C2×D12—61no—
188C2×C4×D3—31no—
189C2×C4×D3—61no—
190C2×Dic6—31no—
191C2×Dic6—61no—
192C3×SD16—182no—
193C3×SD16—182no—
194C2×C3×C8—31no—
195C2×C3×C8—91no—
196C2×C3×C8C2×C3×D311yescharacteristic
197C3×C4⋊C4—31no—
198C3×C4⋊C4C2×C3×D311yescharacteristic
199C3×C22⋊C4—181no—
200#215—62no—
201(C2×D3)⋊C4—61no—
202C2×C3⋊C8—31no—
203#204—62no—
204C22×C32D3×D411yescharacteristic
205C2×C3×D3—32no—
206C2×C3×D3—61no—
207C2×C3×D3—61no—
208C2×C3×D3—121no—
209C2×C3×D3—121no—
210C2×C3×D3—121no—
211C32×C4—22no—
212C32×C4—41no—
213C32×C4C2×D1211yescharacteristic
214C32×C4D3×D411yescharacteristic
215C3×Dic3—62no—
216C3×Dic3—61no—
217C22×C9—21no—
218C22×C9C2×C3×D411yescharacteristic
219C2×D9—21no—
220C2×D9—21no—
221C2×D9—41no—
222C2×D9—41no—
223C2×D9—41no—
224C2×D9C3×SD1612yescomplemented
225C4×C9—22no—
226C4×C9—21no—
227C4×C9—21no—
228C4×C9—42no—
229C4×C9—41no—
230C4×C9—81no—
231C4×C9C2×C3×D411yescharacteristic
232C4×C9C2×C3×D411yescharacteristic
233Dic9—21no—
234Dic9—22no—
235C22×Q8—91no—
236C2×SD16—91no—
237C2×SD16—91no—
238#121—91no—
239#103—91no—
240#103AGL(1,ℤ/9)11yescharacteristic, complemented
241C22⋊C8—91no—
2423−1+2#44811yescharacteristic, complemented, Sylow
243C23×C3—91no—
244C23×C3—181no—
245C22×D3—31no—
246C22×D3—61no—
247C3×Q8—21no—
248C3×Q8—32no—
249C3×Q8—61no—
250C3×Q8—92no—
251C3×Q8—181no—
252C3×Q8—184no—
253C3×Q8C3×D1212yes—
254C3×D4—181no—
255C3×D4—181no—
256C3×D4—362no—
257C2×C3×C4—21no—
258C2×C3×C4—21no—
259C2×C3×C4—31no—
260C2×C3×C4—61no—
261C2×C3×C4—61no—
262C2×C3×C4—91no—
263C2×C3×C4—181no—
264C2×C3×C4—181no—
265C2×C3×C4—184no—
266C2×C3×C4—181no—
267C2×C3×C4C22×C3×D311yescharacteristic
268C2×Dic3—31no—
269C2×Dic3—61no—
270D12—61no—
271D12—61no—
272D12—122no—
273C4×D3—64no—
274C4×D3—61no—
275C4×D3—61no—
276Dic6—32no—
277Dic6—64no—
278Dic6—61no—
279C3×C8—21no—
280C3×C8—61no—
281C3×C8—181no—
282C3⋊C8—61no—
283C2×C32#88211yescharacteristic
284C2×C32D3×SD1611yescharacteristic
285C2×C32#91811yescharacteristic
286C3×D3—62no—
287C3×D3—241no—
288C2×C9—21no—
289C2×C9—21no—
290C2×C9—21no—
291C2×C9C3×#12011yescharacteristic
292C2×C9C2×C3×SD1611yescharacteristic
293C2×C9C3×#13711yescharacteristic
294D9—22no—
295D9—81no—
296C2×Q8—91no—
297C2×Q8—181no—
298C2×Q8—182no—
299C2×Q8—182no—
300C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic
301C2×D4—91no—
302C2×D4—181no—
303C22×C4—91no—
304C22×C4—181no—
305SD16—182no—
306SD16—182no—
307C2×C8—91no—
308C2×C8C2×AGL(1,ℤ/9)11yescharacteristic
309C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
310C22⋊C4—181no—
311C22×C3—31no—
312C22×C3—92no—
313C22×C3—181no—
314C22×C3—181no—
315C22×C3—361no—
316C22×C3—361no—
317C22×C3—361no—
318C22×C3C3×D3×D411yescharacteristic
319C2×D3—32no—
320C2×D3—61no—
321C2×D3—61no—
322C2×D3—121no—
323C2×D3—121no—
324C2×D3—121no—
325C3×C4—22no—
326C3×C4—31no—
327C3×C4—31no—
328C3×C4—41no—
329C3×C4—62no—
330C3×C4—121no—
331C3×C4—181no—
332C3×C4—182no—
333C3×C4C2×C3×D1211yescharacteristic
334C3×C4C3×D3×D411yescharacteristic
335Dic3—62no—
336Dic3—61no—
337C32#480611yescharacteristic
338C9—21no—
339C9C3×#44811yescharacteristic, complemented
340C23—91no—
341C23—181no—
342Q8—21no—
343Q8—92no—
344Q8—184no—
345Q8—181no—
346Q8C4⋊AGL(1,ℤ/9)12yescomplemented
347D4—181no—
348D4—181no—
349D4—362no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—91no—
353C2×C4—184no—
354C2×C4—181no—
355C2×C4—181no—
356C2×C4—181no—
357C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
358C8—21no—
359C8—181no—
360C2×C3—31no—
361C2×C3—31no—
362C2×C3—31no—
363C2×C3—182no—
364C2×C3—721no—
365C2×C3C3×#88211yescharacteristic
366C2×C3C3×D3×SD1611yescharacteristic
367C2×C3C3×#91811yescharacteristic
368D3—62no—
369D3—241no—
370C22—92no—
371C22—181no—
372C22—181no—
373C22—361no—
374C22—361no—
375C22—361no—
376C22D4×AGL(1,ℤ/9)11yescharacteristic
377C4—22no—
378C4—41no—
379C4—181no—
380C4—182no—
381C4C2×C4⋊AGL(1,ℤ/9)11yescharacteristic
382C4D4×AGL(1,ℤ/9)11yescharacteristic
383C3—31no—
384C3C3×#480611yescharacteristic
385C2—182no—
386C2—721no—
387C2#63444⋊C311yescharacteristic
388C2SD16×AGL(1,ℤ/9)11yescharacteristic
389C2#63480⋊C311yescharacteristic
subgroupG/H
centerC22D4×AGL(1,ℤ/9)
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×#103C2
socleC22×C3C3×D3×D4
prime radical#63457⋊C3C2
hypercenter#103AGL(1,ℤ/9)
hypocenterC9C3×#448
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ #103

Complements

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

NH ≅ G/Ntype
1. C2×Q8×AGL(1,ℤ/9)C2semidirect
2. C2×#3675⋊C3C2semidirect
3. C2×#3664⋊C3C2semidirect
5. 3−1+2×#103C2semidirect
6. (C2×AGL(1,ℤ/9))⋊C8C2semidirect
7. #63368⋊C3C2semidirect
9. #10588486C3semidirect
29. C2×Q8×D9C2×C3semidirect
33. C2×#3675C2×C3semidirect
34. C2×#3664C2×C3semidirect
37. C9×#103C2×C3semidirect
38. (C2×D9)⋊C8C2×C3semidirect
39. #63368C2×C3semidirect
43. Q8×3−1+2D4semidirect
57. 3−1+2⋊Q8D4semidirect
101. C2×AGL(1,ℤ/9)SD16semidirect
147. Q8×C9C3×D4semidirect
164. Dic18C3×D4semidirect
224. C2×D9C3×SD16semidirect
240. #103AGL(1,ℤ/9)semidirect
242. 3−1+2#448semidirect
339. C9C3×#448semidirect
346. Q8C4⋊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
C2×Q8×AGL(1,ℤ/9)2
C2×#3675⋊C32
C2×#3664⋊C32
#63457⋊C32
3−1+2×#1032
(C2×AGL(1,ℤ/9))⋊C82
#63368⋊C32
C3×#48063
#105884863