Finite Group Databasebeta

← #10588416⋊C3 #13141640

Subgroups — #10588416⋊C3 #13141640

Order 1728.

511 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 511 conjugacy classes of subgroups — too many to draw. Only its 100 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×#3697⋊C3C211yesmaximal, characteristic, complemented
2#63492⋊C3C211yesmaximal, characteristic, complemented
3C2×D72⋊C3C211yesmaximal, characteristic, complemented
4C2×#3664⋊C3C211yesmaximal, characteristic, complemented
53−1+2×C22⋊C8C211yesmaximal, characteristic, complemented
6#63380⋊C3C211yesmaximal, characteristic, complemented
7#63376⋊C3C211yesmaximal, characteristic, complemented
8C3×#4734—31nomaximal
9#10588416C311yesmaximal, characteristic, complemented
10C22×C4×3−1+2C2211yescharacteristic
11C2×#611⋊C3—21no—
12C2×#611⋊C3—21no—
13#3697⋊C3—22no—
14#3697⋊C3—22no—
15C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
16C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
17C2×C4×AGL(1,ℤ/9)—21no—
18C2×3−1+2⋊Q8C2211yescharacteristic
19C2×C8×3−1+2C2211yescharacteristic
20C2×C8×3−1+2C2211yescharacteristic
21(C2×AGL(1,ℤ/9))⋊C4—21no—
22C4⋊(3−1+2⋊C4)C2211yescharacteristic
23D72⋊C3—22no—
24#3664⋊C3—22no—
25C2×C3×#235—31no—
26C3×#930—31no—
27C2×C3×D24—31no—
28C2×C3×#204—31no—
29C2×#3697C2×C311yescharacteristic, complemented
30C32×C22⋊C8D311yescharacteristic
31C3×#821—31no—
32C3×#816—31no—
33#63492C2×C311yescharacteristic, complemented
34C2×D72C2×C311yescharacteristic, complemented
35C2×#3664C2×C311yescharacteristic, complemented
36C9×C22⋊C8—21no—
37C9×C22⋊C8C2×C311yescharacteristic, complemented
38#63380C2×C311yescharacteristic, complemented
39#63376C2×C311yescharacteristic, complemented
40C23×3−1+2D411yescharacteristic
41C22×AGL(1,ℤ/9)—21no—
42C22×AGL(1,ℤ/9)—21no—
43C2×C4×3−1+2—21no—
44C2×C4×3−1+2—21no—
45C2×C4×3−1+2D411yescharacteristic
46C2×C4×3−1+2C2311yescharacteristic
47#611⋊C3—24no—
48#611⋊C3—42no—
49C2×3−1+2⋊C4—21no—
50C2×3−1+2⋊C4—21no—
51C4⋊AGL(1,ℤ/9)—21no—
52C4⋊AGL(1,ℤ/9)—21no—
53C4⋊AGL(1,ℤ/9)—21no—
54C4⋊AGL(1,ℤ/9)D412yescomplemented
55C4×AGL(1,ℤ/9)—24no—
563−1+2⋊Q8—21no—
573−1+2⋊Q8D412yescomplemented
58C8×3−1+2—21no—
59C8×3−1+2—21no—
60C3×#449—91no—
61#4734—31no—
62C22×C32×C4C2×D311yescharacteristic
63C2×C3×#67—61no—
64C2×C3×#67—61no—
65C3×#235—62no—
66C3×#235—62no—
67C2×C3×D12—31no—
68C2×C3×D12—31no—
69C2×C3×C4×D3—61no—
70C2×C3×Dic6—31no—
71C2×C32×C8C2×D311yescharacteristic
72C2×C32×C8C2×D311yescharacteristic
73C3×(C2×D3)⋊C4—61no—
74C3×C4⋊Dic3—31no—
75C3×D24—62no—
76C3×#204—62no—
77C22×C4×C9—21no—
78C22×C4×C9C22×C311yescharacteristic
79C2×#611—21no—
80C2×#611—21no—
81#3697—22no—
82#3697—22no—
83C2×D36C22×C311yescharacteristic
84C2×D36C22×C311yescharacteristic
85C2×C4×D9—21no—
86C2×Dic18C22×C311yescharacteristic
87C2×C8×C9—21no—
88C2×C8×C9—21no—
89C2×C8×C9C22×C311yescharacteristic
90C2×C8×C9C22×C311yescharacteristic
91(C2×D9)⋊C4—21no—
92C4⋊Dic9C22×C311yescharacteristic
93D72—22no—
94#3664—22no—
95C22×3−1+2—21no—
96C22×3−1+2—21no—
97C22×3−1+2—21no—
98C22×3−1+2C2×D411yescharacteristic
99C2×AGL(1,ℤ/9)—22no—
100C2×AGL(1,ℤ/9)—41no—
101C2×AGL(1,ℤ/9)—41no—
102C2×AGL(1,ℤ/9)—41no—
103C2×AGL(1,ℤ/9)—41no—
104C2×AGL(1,ℤ/9)—41no—
105C4×3−1+2—21no—
106C4×3−1+2C2×D411yescharacteristic
107C4×3−1+2C2×D411yescharacteristic
1083−1+2⋊C4—22no—
1093−1+2⋊C4—41no—
110C2×C3×#41—91no—
111C2×#235—31no—
112C2×C3×SD16—91no—
113C2×C3×D8—91no—
114C3×#121—91no—
115#930—31no—
116C2×D24—31no—
117C2×#204—31no—
118C3×#103—91no—
119C3×#102—91no—
120C3×C22⋊C8—31no—
121C3×C22⋊C8C3×D311yescharacteristic
122#821—31no—
123#816—31no—
124C23×C32D1211yescharacteristic
125C22×C3×D3—61no—
126C22×C3×D3—61no—
127C2×C32×C4—21no—
128C2×C32×C4—21no—
129C2×C32×C4D1211yescharacteristic
130C2×C32×C4C22×D311yescharacteristic
131C3×#67—64no—
132C3×#67—122no—
133C2×C3×Dic3—61no—
134C2×C3×Dic3—61no—
135C3×D12—32no—
136C3×D12—61no—
137C3×D12—61no—
138C3×D12—61no—
139C3×C4×D3—64no—
140C3×Dic6—32no—
141C3×Dic6—61no—
142C23×C9—21no—
143C23×C9C3×D411yescharacteristic
144C22×D9—21no—
145C22×D9—21no—
146C32×C8—21no—
147C32×C8—21no—
148C2×C4×C9—21no—
149C2×C4×C9—21no—
150C2×C4×C9—21no—
151C2×C4×C9—21no—
152C2×C4×C9—41no—
153C2×C4×C9—41no—
154C2×C4×C9C3×D411yescharacteristic
155C2×C4×C9C23×C311yescharacteristic
156#611—24no—
157#611—42no—
158C2×Dic9—21no—
159C2×Dic9—21no—
160D36—21no—
161D36—21no—
162D36—21no—
163D36C3×D412yescomplemented
164C4×D9—24no—
165Dic18—21no—
166Dic18C3×D412yescomplemented
167C8×C9—21no—
168C8×C9—21no—
169C8×C9—41no—
170C8×C9—41no—
171#449—91noSylow
172C2×3−1+2—41no—
173C2×3−1+2#12011yescharacteristic
174C2×3−1+2#13511yescharacteristic
175C2×3−1+2AGL(1,ℤ/8)11yescharacteristic
176AGL(1,ℤ/9)—42no—
177AGL(1,ℤ/9)—81no—
178C3×#41—182no—
179C3×#41—182no—
180C2×C3×Q8—91no—
181C2×C3×D4—91no—
182C2×C3×D4—91no—
183C2×C3×D4—181no—
184C2×C3×D4—181no—
185C22×C3×C4—31no—
186C22×C3×C4—181no—
187C22×C3×C4C2×C3×D311yescharacteristic
188C2×#67—61no—
189C2×#67—61no—
190#235—62no—
191#235—62no—
192C2×D12—31no—
193C2×D12—31no—
194C2×C4×D3—61no—
195C2×Dic6—31no—
196C3×SD16—182no—
197C3×D8—182no—
198C2×C3×C8—31no—
199C2×C3×C8—31no—
200C2×C3×C8C2×C3×D311yescharacteristic
201C2×C3×C8C2×C3×D311yescharacteristic
202C3×C4⋊C4—91no—
203C3×C22⋊C4—181no—
204(C2×D3)⋊C4—61no—
205C4⋊Dic3—31no—
206D24—62no—
207#204—62no—
208C22×C32—21no—
209C22×C32—21no—
210C22×C32—21no—
211C22×C32C2×D1211yescharacteristic
212C2×C3×D3—62no—
213C2×C3×D3—121no—
214C2×C3×D3—121no—
215C2×C3×D3—121no—
216C2×C3×D3—121no—
217C2×C3×D3—121no—
218C32×C4—21no—
219C32×C4D3×D411yescharacteristic
220C32×C4D3×D411yescharacteristic
221C3×Dic3—62no—
222C3×Dic3—121no—
223C22×C9—22no—
224C22×C9—21no—
225C22×C9—21no—
226C22×C9—21no—
227C22×C9—21no—
228C22×C9—41no—
229C22×C9—41no—
230C22×C9C2×C3×D411yescharacteristic
231C2×D9—22no—
232C2×D9—41no—
233C2×D9—41no—
234C2×D9—41no—
235C2×D9—41no—
236C2×D9—41no—
237C4×C9—21no—
238C4×C9—21no—
239C4×C9—21no—
240C4×C9—41no—
241C4×C9C2×C3×D411yescharacteristic
242C4×C9C2×C3×D411yescharacteristic
243Dic9—22no—
244Dic9—41no—
245C2×#41—91no—
246C2×SD16—91no—
247C2×D8—91no—
248#121—91no—
249#103—91no—
250#102—91no—
251C22⋊C8AGL(1,ℤ/9)11yescharacteristic, complemented
2523−1+2#44911yescharacteristic, complemented, Sylow
253C23×C3—31no—
254C23×C3—181no—
255C23×C3—181no—
256C23×C3C3×D1211yescharacteristic
257C22×D3—61no—
258C22×D3—61no—
259C3×Q8—92no—
260C3×Q8—181no—
261C3×D4—92no—
262C3×D4—181no—
263C3×D4—181no—
264C3×D4—181no—
265C3×D4—184no—
266C3×D4—362no—
267C2×C3×C4—21no—
268C2×C3×C4—21no—
269C2×C3×C4—31no—
270C2×C3×C4—31no—
271C2×C3×C4—61no—
272C2×C3×C4—61no—
273C2×C3×C4—184no—
274C2×C3×C4—181no—
275C2×C3×C4—181no—
276C2×C3×C4C3×D1211yescharacteristic
277C2×C3×C4C22×C3×D311yescharacteristic
278#67—64no—
279#67—122no—
280C2×Dic3—61no—
281C2×Dic3—61no—
282D12—32no—
283D12—61no—
284D12—61no—
285D12—61no—
286C4×D3—64no—
287Dic6—32no—
288Dic6—61no—
289C3×C8—21no—
290C3×C8—21no—
291C3×C8—61no—
292C3×C8—61no—
293C2×C32—41no—
294C2×C32#88211yescharacteristic
295C2×C32#90411yescharacteristic
296C2×C32#90811yescharacteristic
297C3×D3—122no—
298C3×D3—241no—
299C2×C9—21no—
300C2×C9—21no—
301C2×C9—21no—
302C2×C9—42no—
303C2×C9—41no—
304C2×C9C3×#12011yescharacteristic
305C2×C9C3×#13511yescharacteristic
306C2×C9C3×AGL(1,ℤ/8)11yescharacteristic
307D9—42no—
308D9—81no—
309#41—182no—
310#41—182no—
311C2×Q8—91no—
312C2×D4—91no—
313C2×D4—91no—
314C2×D4—181no—
315C2×D4—181no—
316C22×C4—181no—
317C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
318SD16—182no—
319D8—182no—
320C2×C8C2×AGL(1,ℤ/9)11yescharacteristic
321C2×C8C2×AGL(1,ℤ/9)11yescharacteristic
322C4⋊C4—91no—
323C22⋊C4—181no—
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—21no—
327C22×C3—31no—
328C22×C3—61no—
329C22×C3—61no—
330C22×C3—61no—
331C22×C3—182no—
332C22×C3—361no—
333C22×C3—361no—
334C22×C3—361no—
335C22×C3—361no—
336C22×C3—361no—
337C22×C3C2×C3×D1211yescharacteristic
338C2×D3—62no—
339C2×D3—121no—
340C2×D3—121no—
341C2×D3—121no—
342C2×D3—121no—
343C2×D3—121no—
344C3×C4—21no—
345C3×C4—31no—
346C3×C4—31no—
347C3×C4—61no—
348C3×C4—182no—
349C3×C4—361no—
350C3×C4C3×D3×D411yescharacteristic
351C3×C4C3×D3×D411yescharacteristic
352Dic3—62no—
353Dic3—121no—
354C32#473411yescharacteristic
355C9—21no—
356C9C3×#44911yescharacteristic, complemented
357C23—181no—
358C23—181no—
359C23C4⋊AGL(1,ℤ/9)11yescharacteristic
360Q8—92no—
361Q8—181no—
362D4—92no—
363D4—184no—
364D4—181no—
365D4—181no—
366D4—181no—
367D4—362no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—184no—
371C2×C4—181no—
372C2×C4—181no—
373C2×C4C4⋊AGL(1,ℤ/9)11yescharacteristic
374C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
375C8—21no—
376C8—21no—
377C2×C3—31no—
378C2×C3—31no—
379C2×C3—31no—
380C2×C3—41no—
381C2×C3—121no—
382C2×C3—362no—
383C2×C3—721no—
384C2×C3C3×#88211yescharacteristic
385C2×C3C3×#90411yescharacteristic
386C2×C3C3×#90811yescharacteristic
387D3—122no—
388D3—241no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—182no—
393C22—361no—
394C22—361no—
395C22—361no—
396C22—361no—
397C22—361no—
398C22C2×C4⋊AGL(1,ℤ/9)11yescharacteristic
399C4—21no—
400C4—182no—
401C4—361no—
402C4D4×AGL(1,ℤ/9)11yescharacteristic
403C4D4×AGL(1,ℤ/9)11yescharacteristic
404C3—31no—
405C3C3×#473411yescharacteristic
406C2—41no—
407C2—362no—
408C2—721no—
409C2#63444⋊C311yescharacteristic
410C2#63467⋊C311yescharacteristic
411C2#63470⋊C311yescharacteristic
subgroupG/H
centerC22C2×C4⋊AGL(1,ℤ/9)
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×C22⋊C8C2
socleC22×C3C2×C3×D12
hypercenterC22⋊C8AGL(1,ℤ/9)
hypocenterC9C3×#449
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊C8

Complements

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

NH ≅ G/Ntype
1. C2×#3697⋊C3C2semidirect
2. #63492⋊C3C2semidirect
3. C2×D72⋊C3C2semidirect
4. C2×#3664⋊C3C2semidirect
5. 3−1+2×C22⋊C8C2semidirect
6. #63380⋊C3C2semidirect
7. #63376⋊C3C2semidirect
9. #10588416C3semidirect
29. C2×#3697C2×C3semidirect
33. #63492C2×C3semidirect
34. C2×D72C2×C3semidirect
35. C2×#3664C2×C3semidirect
37. C9×C22⋊C8C2×C3semidirect
38. #63380C2×C3semidirect
39. #63376C2×C3semidirect
54. C4⋊AGL(1,ℤ/9)D4semidirect
57. 3−1+2⋊Q8D4semidirect
163. D36C3×D4semidirect
166. Dic18C3×D4semidirect
251. C22⋊C8AGL(1,ℤ/9)semidirect
252. 3−1+2#449semidirect
356. C9C3×#449semidirect

Maximal subgroups

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

subgroupindex
C2×#3697⋊C32
#63492⋊C32
C2×D72⋊C32
C2×#3664⋊C32
3−1+2×C22⋊C82
#63380⋊C32
#63376⋊C32
C3×#47343
#105884163