Finite Group Databasebeta

← #10588461⋊C3 #13141830

Subgroups — #10588461⋊C3 #13141830

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×#3699⋊C3C211yesmaximal, characteristic, complemented
2C2×#3673⋊C3C211yesmaximal, characteristic, complemented
3C2×#3664⋊C3C211yesmaximal, characteristic, complemented
4#63457⋊C3C211yesmaximal, characteristic, complemented
53−1+2×#102C211yesmaximal, characteristic, complemented
6(C2×AGL(1,ℤ/9))⋊C8C211yesmaximal, characteristic, complemented
7#63368⋊C3C211yesmaximal, characteristic, complemented
8C3×#4781—31nomaximal
9#10588461C311yesmaximal, characteristic, complemented
10C2×D4×3−1+2C2211yescharacteristic
11C2×#611⋊C3—21no—
12C22×3−1+2⋊C4—21no—
13#3699⋊C3—22no—
14#3699⋊C3—22no—
15C2×C4⋊AGL(1,ℤ/9)—21no—
16C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
17C2×C4×AGL(1,ℤ/9)C2211yescharacteristic
18C2×3−1+2⋊Q8C2211yescharacteristic
19C2×C8×3−1+2C2211yescharacteristic
20C4⋊C4×3−1+2C2211yescharacteristic
21#3673⋊C3—22no—
22(C2×AGL(1,ℤ/9))⋊C4—21no—
23C2×3−1+2⋊C8C2211yescharacteristic
24#3664⋊C3—22no—
25C2×C3×#237—31no—
26C2×C3×#213—31no—
27C2×C3×#204—31no—
28C3×#895—31no—
29C2×#3699C2×C311yescharacteristic, complemented
30C32×#102D311yescharacteristic
31C3×(C2×D3)⋊C8—31no—
32C3×#810—31no—
33C2×#3673C2×C311yescharacteristic, complemented
34C2×#3664C2×C311yescharacteristic, complemented
35#63457C2×C311yescharacteristic, complemented
36C9×#102—21no—
37C9×#102C2×C311yescharacteristic, complemented
38(C2×D9)⋊C8C2×C311yescharacteristic, complemented
39#63368C2×C311yescharacteristic, complemented
40C23×3−1+2—21no—
41C22×AGL(1,ℤ/9)—21no—
42C22×AGL(1,ℤ/9)D411yescharacteristic
43D4×3−1+2—21no—
44D4×3−1+2D412yescomplemented
45C2×C4×3−1+2—21no—
46C2×C4×3−1+2C2311yescharacteristic
47#611⋊C3—24no—
48C2×3−1+2⋊C4—24no—
49C2×3−1+2⋊C4—21no—
50C2×3−1+2⋊C4D411yescharacteristic
51C4⋊AGL(1,ℤ/9)—21no—
52C4⋊AGL(1,ℤ/9)—21no—
53C4⋊AGL(1,ℤ/9)—42no—
54C4×AGL(1,ℤ/9)—21no—
55C4×AGL(1,ℤ/9)—21no—
563−1+2⋊Q8—21no—
573−1+2⋊Q8D412yescomplemented
58C8×3−1+2—21no—
593−1+2⋊C8—21no—
60C3×#449—91no—
61#4781—31no—
62C2×C32×D4C2×D311yescharacteristic
63C2×C3×#67—61no—
64C22×C3×Dic3—61no—
65C3×#237—62no—
66C3×#237—62no—
67C2×C3×D12—31no—
68C2×C3×D12—61no—
69C2×C3×C4×D3—31no—
70C2×C3×Dic6—31no—
71C2×C32×C8C2×D311yescharacteristic
72C32×C4⋊C4C2×D311yescharacteristic
73C3×#213—62no—
74C3×(C2×D3)⋊C4—61no—
75C2×C3×C3⋊C8—31no—
76C3×#204—62no—
77C2×D4×C9—21no—
78C2×D4×C9C22×C311yescharacteristic
79C2×#611—21no—
80C22×Dic9—21no—
81#3699—22no—
82#3699—22no—
83C2×D36—21no—
84C2×D36C22×C311yescharacteristic
85C2×C4×D9C22×C311yescharacteristic
86C2×Dic18C22×C311yescharacteristic
87C2×C8×C9—21no—
88C2×C8×C9C22×C311yescharacteristic
89C9×C4⋊C4—21no—
90C9×C4⋊C4C22×C311yescharacteristic
91#3673—22no—
92(C2×D9)⋊C4—21no—
93C2×C9⋊C8C22×C311yescharacteristic
94#3664—22no—
95C22×3−1+2—22no—
96C22×3−1+2—41no—
97C22×3−1+2—41no—
98C22×3−1+2C2×D411yescharacteristic
99C2×AGL(1,ℤ/9)—21no—
100C2×AGL(1,ℤ/9)—21no—
101C2×AGL(1,ℤ/9)—21no—
102C2×AGL(1,ℤ/9)—41no—
103C2×AGL(1,ℤ/9)—41no—
104C2×AGL(1,ℤ/9)—41no—
105C4×3−1+2—41no—
106C4×3−1+2C2×D411yescharacteristic
107C4×3−1+2C2×D411yescharacteristic
1083−1+2⋊C4—22no—
1093−1+2⋊C4—21no—
110C2×C3×#41—91no—
111C2×#237—31no—
112C2×C3×SD16—91no—
113C2×C3×D8—91no—
114C3×#121—91no—
115C2×#213—31no—
116C2×#204—31no—
117#895—31no—
118C3×#103—91no—
119C3×#102—31no—
120C3×#102C3×D311yescharacteristic
121C3×C22⋊C8—91no—
122(C2×D3)⋊C8—31no—
123#810—31no—
124C23×C32—21no—
125C22×C3×D3—31no—
126C22×C3×D3—61no—
127C32×D4—21no—
128C32×D4D1212yes—
129C2×C32×C4—21no—
130C2×C32×C4C22×D311yescharacteristic
131C3×#67—64no—
132C2×C3×Dic3—31no—
133C2×C3×Dic3—64no—
134C2×C3×Dic3—61no—
135C3×D12—61no—
136C3×D12—61no—
137C3×D12—122no—
138C3×C4×D3—61no—
139C3×C4×D3—61no—
140C3×Dic6—32no—
141C3×Dic6—61no—
142C23×C9—21no—
143C23×C9—41no—
144C22×D9—21no—
145C22×D9C3×D411yescharacteristic
146C32×C8—21no—
147C3×C3⋊C8—61no—
148D4×C9—21no—
149D4×C9—22no—
150D4×C9—41no—
151D4×C9C3×D412yescomplemented
152C2×C4×C9—21no—
153C2×C4×C9—21no—
154C2×C4×C9—41no—
155C2×C4×C9C23×C311yescharacteristic
156#611—24no—
157C2×Dic9—24no—
158C2×Dic9—21no—
159C2×Dic9C3×D411yescharacteristic
160D36—21no—
161D36—21no—
162D36—42no—
163C4×D9—21no—
164C4×D9—21no—
165Dic18—21no—
166Dic18C3×D412yescomplemented
167C8×C9—21no—
168C8×C9—41no—
169C9⋊C8—21no—
170#449—91noSylow
171C2×3−1+2—42no—
172C2×3−1+2#12011yescharacteristic
173C2×3−1+2#13511yescharacteristic
174C2×3−1+2AGL(1,ℤ/8)11yescharacteristic
175AGL(1,ℤ/9)—41no—
176AGL(1,ℤ/9)—81no—
177C3×#41—182no—
178C3×#41—182no—
179C2×C3×Q8—91no—
180C2×C3×D4—31no—
181C2×C3×D4—91no—
182C2×C3×D4—181no—
183C2×C3×D4—181no—
184C2×C3×D4C2×C3×D311yescharacteristic
185C22×C3×C4—91no—
186C22×C3×C4—181no—
187C2×#67—61no—
188C22×Dic3—61no—
189#237—62no—
190#237—62no—
191C2×D12—31no—
192C2×D12—61no—
193C2×C4×D3—31no—
194C2×Dic6—31no—
195C3×SD16—182no—
196C3×D8—182no—
197C2×C3×C8—31no—
198C2×C3×C8—91no—
199C2×C3×C8C2×C3×D311yescharacteristic
200C3×C4⋊C4—31no—
201C3×C4⋊C4C2×C3×D311yescharacteristic
202C3×C22⋊C4—181no—
203#213—62no—
204(C2×D3)⋊C4—61no—
205C2×C3⋊C8—31no—
206#204—62no—
207C22×C32—22no—
208C22×C32—41no—
209C22×C32—41no—
210C22×C32D3×D411yescharacteristic
211C2×C3×D3—61no—
212C2×C3×D3—61no—
213C2×C3×D3—61no—
214C2×C3×D3—121no—
215C2×C3×D3—121no—
216C2×C3×D3—121no—
217C32×C4—41no—
218C32×C4C2×D1211yescharacteristic
219C32×C4D3×D411yescharacteristic
220C3×Dic3—62no—
221C3×Dic3—61no—
222C22×C9—22no—
223C22×C9—21no—
224C22×C9—42no—
225C22×C9—41no—
226C22×C9—41no—
227C22×C9—81no—
228C22×C9—81no—
229C22×C9C2×C3×D411yescharacteristic
230C2×D9—21no—
231C2×D9—21no—
232C2×D9—21no—
233C2×D9—41no—
234C2×D9—41no—
235C2×D9—41no—
236C4×C9—21no—
237C4×C9—21no—
238C4×C9—41no—
239C4×C9—81no—
240C4×C9C2×C3×D411yescharacteristic
241C4×C9C2×C3×D411yescharacteristic
242Dic9—22no—
243Dic9—21no—
244C2×#41—91no—
245C2×SD16—91no—
246C2×D8—91no—
247#121—91no—
248#103—91no—
249#102AGL(1,ℤ/9)11yescharacteristic, complemented
250C22⋊C8—91no—
2513−1+2#44911yescharacteristic, complemented, Sylow
252C23×C3—21no—
253C23×C3—61no—
254C23×C3—91no—
255C23×C3—181no—
256C22×D3—31no—
257C22×D3—61no—
258C3×Q8—92no—
259C3×Q8—181no—
260C3×D4—21no—
261C3×D4—32no—
262C3×D4—61no—
263C3×D4—181no—
264C3×D4—181no—
265C3×D4—184no—
266C3×D4—362no—
267C3×D4C3×D1212yes—
268C2×C3×C4—21no—
269C2×C3×C4—31no—
270C2×C3×C4—61no—
271C2×C3×C4—91no—
272C2×C3×C4—181no—
273C2×C3×C4—181no—
274C2×C3×C4—181no—
275C2×C3×C4—184no—
276C2×C3×C4C22×C3×D311yescharacteristic
277#67—64no—
278C2×Dic3—31no—
279C2×Dic3—64no—
280C2×Dic3—61no—
281D12—61no—
282D12—61no—
283D12—122no—
284C4×D3—61no—
285C4×D3—61no—
286Dic6—32no—
287Dic6—61no—
288C3×C8—21no—
289C3×C8—61no—
290C3×C8—181no—
291C3⋊C8—61no—
292C2×C32—42no—
293C2×C32#88211yescharacteristic
294C2×C32#91111yescharacteristic
295C2×C32#91611yescharacteristic
296C3×D3—121no—
297C3×D3—241no—
298C2×C9—21no—
299C2×C9—21no—
300C2×C9—21no—
301C2×C9—42no—
302C2×C9—82no—
303C2×C9C3×#12011yescharacteristic
304C2×C9C3×#13511yescharacteristic
305C2×C9C3×AGL(1,ℤ/8)11yescharacteristic
306D9—41no—
307D9—81no—
308#41—182no—
309#41—182no—
310C2×Q8—91no—
311C2×D4—91no—
312C2×D4—181no—
313C2×D4—181no—
314C2×D4C2×AGL(1,ℤ/9)11yescharacteristic
315C22×C4—91no—
316C22×C4—181no—
317SD16—182no—
318D8—182no—
319C2×C8—91no—
320C2×C8C2×AGL(1,ℤ/9)11yescharacteristic
321C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
322C22⋊C4—181no—
323C22×C3—22no—
324C22×C3—31no—
325C22×C3—41no—
326C22×C3—41no—
327C22×C3—62no—
328C22×C3—121no—
329C22×C3—121no—
330C22×C3—181no—
331C22×C3—181no—
332C22×C3—181no—
333C22×C3—361no—
334C22×C3—361no—
335C22×C3—361no—
336C22×C3C3×D3×D411yescharacteristic
337C2×D3—61no—
338C2×D3—61no—
339C2×D3—61no—
340C2×D3—121no—
341C2×D3—121no—
342C2×D3—121no—
343C3×C4—31no—
344C3×C4—31no—
345C3×C4—41no—
346C3×C4—121no—
347C3×C4—181no—
348C3×C4—182no—
349C3×C4C2×C3×D1211yescharacteristic
350C3×C4C3×D3×D411yescharacteristic
351Dic3—62no—
352Dic3—61no—
353C32#478111yescharacteristic
354C9—21no—
355C9C3×#44911yescharacteristic, complemented
356C23—21no—
357C23—91no—
358C23—181no—
359Q8—92no—
360Q8—181no—
361D4—21no—
362D4—184no—
363D4—181no—
364D4—181no—
365D4—362no—
366D4C4⋊AGL(1,ℤ/9)12yescomplemented
367C2×C4—21no—
368C2×C4—91no—
369C2×C4—184no—
370C2×C4—181no—
371C2×C4—181no—
372C2×C4—181no—
373C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
374C8—21no—
375C8—181no—
376C2×C3—31no—
377C2×C3—31no—
378C2×C3—31no—
379C2×C3—42no—
380C2×C3—122no—
381C2×C3—361no—
382C2×C3—721no—
383C2×C3C3×#88211yescharacteristic
384C2×C3C3×#91111yescharacteristic
385C2×C3C3×#91611yescharacteristic
386D3—121no—
387D3—241no—
388C22—22no—
389C22—41no—
390C22—41no—
391C22—181no—
392C22—181no—
393C22—181no—
394C22—361no—
395C22—361no—
396C22—361no—
397C22D4×AGL(1,ℤ/9)11yescharacteristic
398C4—41no—
399C4—181no—
400C4—182no—
401C4C2×C4⋊AGL(1,ℤ/9)11yescharacteristic
402C4D4×AGL(1,ℤ/9)11yescharacteristic
403C3—31no—
404C3C3×#478111yescharacteristic
405C2—42no—
406C2—361no—
407C2—721no—
408C2#63444⋊C311yescharacteristic
409C2#63473⋊C311yescharacteristic
410C2#63478⋊C311yescharacteristic
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
hypercenter#102AGL(1,ℤ/9)
hypocenterC9C3×#449
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
1. C2×#3699⋊C3C2semidirect
2. C2×#3673⋊C3C2semidirect
3. C2×#3664⋊C3C2semidirect
4. #63457⋊C3C2semidirect
5. 3−1+2×#102C2semidirect
6. (C2×AGL(1,ℤ/9))⋊C8C2semidirect
7. #63368⋊C3C2semidirect
9. #10588461C3semidirect
29. C2×#3699C2×C3semidirect
33. C2×#3673C2×C3semidirect
34. C2×#3664C2×C3semidirect
35. #63457C2×C3semidirect
37. C9×#102C2×C3semidirect
38. (C2×D9)⋊C8C2×C3semidirect
39. #63368C2×C3semidirect
44. D4×3−1+2D4semidirect
57. 3−1+2⋊Q8D4semidirect
151. D4×C9C3×D4semidirect
166. Dic18C3×D4semidirect
249. #102AGL(1,ℤ/9)semidirect
251. 3−1+2#449semidirect
355. C9C3×#449semidirect
366. D4C4⋊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×#3699⋊C32
C2×#3673⋊C32
C2×#3664⋊C32
#63457⋊C32
3−1+2×#1022
(C2×AGL(1,ℤ/9))⋊C82
#63368⋊C32
C3×#47813
#105884613