Finite Group Databasebeta

← #10588838⋊C3 #13143325

Subgroups — #10588838⋊C3 #13143325

Order 1728.

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

normal subgroups only

This group has 524 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×D8×3−1+2C211yesmaximal, characteristic, complemented
3#63500⋊C3C211yesmaximal, characteristic, complemented
4C2×#3672⋊C3C211yesmaximal, characteristic, complemented
5#63392⋊C3C211yesmaximal, characteristic, complemented
6(C2×AGL(1,ℤ/9))⋊C8C211yesmaximal, characteristic, complemented
7#63376⋊C3C211yesmaximal, characteristic, complemented
8C3×#5158—31nomaximal
9#10588838C311yesmaximal, characteristic, complemented
10C2×D4×3−1+2C2211yescharacteristic
11C2×D4×3−1+2C2211yescharacteristic
12C2×#611⋊C3—21no—
13C2×#611⋊C3—21no—
14C22×3−1+2⋊C4—21no—
15#3699⋊C3—22no—
16#3699⋊C3—22no—
17C2×C4×AGL(1,ℤ/9)C2211yescharacteristic
18C2×3−1+2⋊Q8C2211yescharacteristic
19D8×3−1+2—22no—
20C2×C8×3−1+2C2211yescharacteristic
21#3676⋊C3—21no—
22#3672⋊C3—22no—
23C4⋊(3−1+2⋊C4)C2211yescharacteristic
24C2×3−1+2⋊C8C2211yescharacteristic
25C2×C3×#237—31no—
26C2×C32×D8D311yescharacteristic
27C3×#938—31no—
28C2×C3×#214—31no—
29C2×#3699C2×C311yescharacteristic, complemented
30C3×#832—31no—
31C3×(C2×D3)⋊C8—31no—
32C3×#816—31no—
33C2×C9×D8—21no—
34C2×C9×D8C2×C311yescharacteristic, complemented
35#63500C2×C311yescharacteristic, complemented
36C2×#3672C2×C311yescharacteristic, complemented
37#63392C2×C311yescharacteristic, complemented
38(C2×D9)⋊C8C2×C311yescharacteristic, complemented
39#63376C2×C311yescharacteristic, complemented
40C23×3−1+2—21no—
41C23×3−1+2—21no—
42C22×AGL(1,ℤ/9)D411yescharacteristic
43D4×3−1+2—21no—
44D4×3−1+2—21no—
45D4×3−1+2—21no—
46D4×3−1+2D412yescomplemented
47C2×C4×3−1+2C2311yescharacteristic
48#611⋊C3—24no—
49#611⋊C3—42no—
50C2×3−1+2⋊C4—24no—
51C2×3−1+2⋊C4—21no—
52C2×3−1+2⋊C4—21no—
53C2×3−1+2⋊C4D411yescharacteristic
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#5158—31no—
62C2×C32×D4C2×D311yescharacteristic
63C2×C32×D4C2×D311yescharacteristic
64C2×C3×#67—61no—
65C2×C3×#67—61no—
66C22×C3×Dic3—61no—
67C3×#237—62no—
68C3×#237—62no—
69C2×C3×C4×D3—31no—
70C2×C3×Dic6—31no—
71C32×D8—22no—
72C2×C32×C8C2×D311yescharacteristic
73C3×#217—61no—
74C3×#214—62no—
75C3×C4⋊Dic3—31no—
76C2×C3×C3⋊C8—31no—
77C2×D4×C9—21no—
78C2×D4×C9—21no—
79C2×D4×C9C22×C311yescharacteristic
80C2×D4×C9C22×C311yescharacteristic
81C2×#611—21no—
82C2×#611—21no—
83C22×Dic9—21no—
84#3699—22no—
85#3699—22no—
86C2×C4×D9C22×C311yescharacteristic
87C2×Dic18C22×C311yescharacteristic
88C9×D8—24no—
89C9×D8—22no—
90C2×C8×C9—21no—
91C2×C8×C9C22×C311yescharacteristic
92#3676—21no—
93#3672—22no—
94C4⋊Dic9C22×C311yescharacteristic
95C2×C9⋊C8C22×C311yescharacteristic
96C22×3−1+2—22no—
97C22×3−1+2—41no—
98C22×3−1+2—41no—
99C22×3−1+2—41no—
100C22×3−1+2—41no—
101C22×3−1+2—41no—
102C22×3−1+2C2×D411yescharacteristic
103C2×AGL(1,ℤ/9)—21no—
104C2×AGL(1,ℤ/9)—21no—
105C2×AGL(1,ℤ/9)—21no—
106C4×3−1+2C2×D411yescharacteristic
107C4×3−1+2C2×D411yescharacteristic
1083−1+2⋊C4—22no—
1093−1+2⋊C4—21no—
1103−1+2⋊C4—41no—
111C2×C3×#41—91no—
112C2×#237—31no—
113C2×C3×SD16—91no—
114C2×C3×D8—31no—
115C2×C3×D8C3×D311yescharacteristic
116C3×#121—91no—
117#938—31no—
118C2×#214—31no—
119C3×#103—91no—
120C3×#102—91no—
121C3×C22⋊C8—91no—
122#832—31no—
123(C2×D3)⋊C8—31no—
124#816—31no—
125C23×C32—21no—
126C23×C32—21no—
127C22×C3×D3—31no—
128C32×D4—21no—
129C32×D4—21no—
130C32×D4—21no—
131C32×D4#6712yes—
132C2×C32×C4C22×D311yescharacteristic
133C3×#67—64no—
134C3×#67—122no—
135C2×C3×Dic3—31no—
136C2×C3×Dic3—64no—
137C2×C3×Dic3—61no—
138C2×C3×Dic3—61no—
139C3×C4×D3—61no—
140C3×C4×D3—61no—
141C3×Dic6—32no—
142C3×Dic6—61no—
143C23×C9—21no—
144C23×C9—21no—
145C23×C9—41no—
146C23×C9—41no—
147C22×D9C3×D411yescharacteristic
148C32×C8—21no—
149C3×C3⋊C8—61no—
150D4×C9—22no—
151D4×C9—21no—
152D4×C9—21no—
153D4×C9—21no—
154D4×C9—22no—
155D4×C9—41no—
156D4×C9—41no—
157D4×C9C3×D412yescomplemented
158C2×C4×C9—21no—
159C2×C4×C9C23×C311yescharacteristic
160#611—24no—
161#611—42no—
162C2×Dic9—24no—
163C2×Dic9—21no—
164C2×Dic9—21no—
165C2×Dic9C3×D411yescharacteristic
166C4×D9—21no—
167C4×D9—21no—
168Dic18—21no—
169Dic18C3×D412yescomplemented
170C8×C9—22no—
171C8×C9—21no—
172C9⋊C8—21no—
173#449—91noSylow
174C2×3−1+2—42no—
175C2×3−1+2—81no—
176C2×3−1+2#12011yescharacteristic
177C2×3−1+2#13511yescharacteristic
178C2×3−1+2AGL(1,ℤ/8)11yescharacteristic
179AGL(1,ℤ/9)—41no—
180C3×#41—182no—
181C3×#41—182no—
182C2×C3×Q8—91no—
183C2×C3×D4—31no—
184C2×C3×D4—31no—
185C2×C3×D4—181no—
186C2×C3×D4—181no—
187C2×C3×D4C2×C3×D311yescharacteristic
188C2×C3×D4C2×C3×D311yescharacteristic
189C22×C3×C4—91no—
190C22×C3×C4—181no—
191C2×#67—61no—
192C2×#67—61no—
193C22×Dic3—61no—
194#237—62no—
195#237—62no—
196C2×C4×D3—31no—
197C2×Dic6—31no—
198C3×SD16—182no—
199C3×D8—22no—
200C3×D8—62no—
201C2×C3×C8—31no—
202C2×C3×C8—91no—
203C2×C3×C8C2×C3×D311yescharacteristic
204C3×C4⋊C4—91no—
205C3×C22⋊C4—181no—
206#217—61no—
207#214—62no—
208C4⋊Dic3—31no—
209C2×C3⋊C8—31no—
210C22×C32—22no—
211C22×C32—41no—
212C22×C32—41no—
213C22×C32—41no—
214C22×C32—41no—
215C22×C32—41no—
216C22×C32D3×D411yescharacteristic
217C2×C3×D3—61no—
218C2×C3×D3—61no—
219C2×C3×D3—61no—
220C32×C4D3×D411yescharacteristic
221C32×C4C2×#6711yescharacteristic
222C3×Dic3—62no—
223C3×Dic3—61no—
224C3×Dic3—121no—
225C22×C9—22no—
226C22×C9—21no—
227C22×C9—42no—
228C22×C9—42no—
229C22×C9—41no—
230C22×C9—41no—
231C22×C9—41no—
232C22×C9—41no—
233C22×C9—41no—
234C22×C9—81no—
235C22×C9—81no—
236C22×C9—81no—
237C22×C9—81no—
238C22×C9C2×C3×D411yescharacteristic
239C2×D9—21no—
240C2×D9—21no—
241C2×D9—21no—
242C4×C9—21no—
243C4×C9—21no—
244C4×C9C2×C3×D411yescharacteristic
245C4×C9C2×C3×D411yescharacteristic
246Dic9—22no—
247Dic9—21no—
248Dic9—41no—
249C2×#41—91no—
250C2×SD16—91no—
251C2×D8AGL(1,ℤ/9)11yescharacteristic, complemented
252#121—91no—
253#103—91no—
254#102—91no—
255C22⋊C8—91no—
2563−1+2#44911yescharacteristic, complemented, Sylow
257C23×C3—21no—
258C23×C3—21no—
259C23×C3—61no—
260C23×C3—61no—
261C23×C3—91no—
262C22×D3—31no—
263C3×Q8—92no—
264C3×Q8—181no—
265C3×D4—21no—
266C3×D4—21no—
267C3×D4—21no—
268C3×D4—32no—
269C3×D4—61no—
270C3×D4—61no—
271C3×D4—61no—
272C3×D4—184no—
273C3×D4—362no—
274C3×D4C3×#6712yes—
275C2×C3×C4—31no—
276C2×C3×C4—91no—
277C2×C3×C4—181no—
278C2×C3×C4—181no—
279C2×C3×C4—181no—
280C2×C3×C4—184no—
281C2×C3×C4—181no—
282C2×C3×C4C22×C3×D311yescharacteristic
283#67—64no—
284#67—122no—
285C2×Dic3—31no—
286C2×Dic3—64no—
287C2×Dic3—61no—
288C2×Dic3—61no—
289C4×D3—61no—
290C4×D3—61no—
291Dic6—32no—
292Dic6—61no—
293C3×C8—21no—
294C3×C8—61no—
295C3×C8—181no—
296C3⋊C8—61no—
297C2×C32—42no—
298C2×C32—81no—
299C2×C32#91111yescharacteristic
300C2×C32#91211yescharacteristic
301C2×C32#93711yescharacteristic
302C3×D3—121no—
303C2×C9—21no—
304C2×C9—21no—
305C2×C9—21no—
306C2×C9—42no—
307C2×C9—82no—
308C2×C9—82no—
309C2×C9—81no—
310C2×C9C3×#12011yescharacteristic
311C2×C9C3×#13511yescharacteristic
312C2×C9C3×AGL(1,ℤ/8)11yescharacteristic
313D9—41no—
314#41—182no—
315#41—182no—
316C2×Q8—91no—
317C2×D4—181no—
318C2×D4—181no—
319C2×D4C2×AGL(1,ℤ/9)11yescharacteristic
320C2×D4C2×AGL(1,ℤ/9)11yescharacteristic
321C22×C4—91no—
322C22×C4—181no—
323SD16—182no—
324D8—22no—
325C2×C8—91no—
326C2×C8C2×AGL(1,ℤ/9)11yescharacteristic
327C4⋊C4—91no—
328C22⋊C4—181no—
329C22×C3—22no—
330C22×C3—31no—
331C22×C3—41no—
332C22×C3—41no—
333C22×C3—41no—
334C22×C3—41no—
335C22×C3—41no—
336C22×C3—62no—
337C22×C3—121no—
338C22×C3—121no—
339C22×C3—121no—
340C22×C3—121no—
341C22×C3—121no—
342C22×C3—181no—
343C22×C3—181no—
344C22×C3—181no—
345C22×C3C3×D3×D411yescharacteristic
346C2×D3—61no—
347C2×D3—61no—
348C2×D3—61no—
349C3×C4—31no—
350C3×C4—31no—
351C3×C4—181no—
352C3×C4—182no—
353C3×C4—361no—
354C3×C4C3×D3×D411yescharacteristic
355C3×C4C2×C3×#6711yescharacteristic
356Dic3—62no—
357Dic3—61no—
358Dic3—121no—
359C32#515811yescharacteristic
360C9—21no—
361C9C3×#44911yescharacteristic, complemented
362C23—21no—
363C23—21no—
364C23—91no—
365Q8—92no—
366Q8—181no—
367D4—21no—
368D4—21no—
369D4—21no—
370D4—184no—
371D4—362no—
372D4#611⋊C312yescomplemented
373C2×C4—91no—
374C2×C4—184no—
375C2×C4—181no—
376C2×C4—181no—
377C2×C4—181no—
378C2×C4—181no—
379C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
380C8—21no—
381C8—181no—
382C2×C3—31no—
383C2×C3—31no—
384C2×C3—31no—
385C2×C3—42no—
386C2×C3—81no—
387C2×C3—122no—
388C2×C3—241no—
389C2×C3—361no—
390C2×C3C3×#91111yescharacteristic
391C2×C3C3×#91211yescharacteristic
392C2×C3C3×#93711yescharacteristic
393D3—121no—
394C22—22no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—181no—
401C22—181no—
402C22—181no—
403C22D4×AGL(1,ℤ/9)11yescharacteristic
404C4—181no—
405C4—182no—
406C4—361no—
407C4D4×AGL(1,ℤ/9)11yescharacteristic
408C4C2×#611⋊C311yescharacteristic
409C3—31no—
410C3C3×#515811yescharacteristic
411C2—42no—
412C2—81no—
413C2—361no—
414C2#63473⋊C311yescharacteristic
415C2#63474⋊C311yescharacteristic
416C2#63499⋊C311yescharacteristic
subgroupG/H
centerC22D4×AGL(1,ℤ/9)
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC2×D8×3−1+2C2
socleC22×C3C3×D3×D4
hypercenterC2×D8AGL(1,ℤ/9)
hypocenterC9C3×#449
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D8

Complements

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

NH ≅ G/Ntype
1. C2×#3699⋊C3C2semidirect
2. C2×D8×3−1+2C2semidirect
3. #63500⋊C3C2semidirect
4. C2×#3672⋊C3C2semidirect
5. #63392⋊C3C2semidirect
6. (C2×AGL(1,ℤ/9))⋊C8C2semidirect
7. #63376⋊C3C2semidirect
9. #10588838C3semidirect
29. C2×#3699C2×C3semidirect
34. C2×C9×D8C2×C3semidirect
35. #63500C2×C3semidirect
36. C2×#3672C2×C3semidirect
37. #63392C2×C3semidirect
38. (C2×D9)⋊C8C2×C3semidirect
39. #63376C2×C3semidirect
46. D4×3−1+2D4semidirect
57. 3−1+2⋊Q8D4semidirect
157. D4×C9C3×D4semidirect
169. Dic18C3×D4semidirect
251. C2×D8AGL(1,ℤ/9)semidirect
256. 3−1+2#449semidirect
361. C9C3×#449semidirect
372. D4#611⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×#3699⋊C32
C2×D8×3−1+22
#63500⋊C32
C2×#3672⋊C32
#63392⋊C32
(C2×AGL(1,ℤ/9))⋊C82
#63376⋊C32
C3×#51583
#105888383