Finite Group Databasebeta

← #10588836⋊C3 #13143323

Subgroups — #10588836⋊C3 #13143323

Order 1728.

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

normal subgroups only

This group has 632 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, complemented
2C2×D8×3−1+2C211yesmaximal, characteristic, complemented
3#63500⋊C3C211yesmaximal, characteristic, complemented
4C2×#3673⋊C3C211yesmaximal, characteristic, complemented
5#63392⋊C3C211yesmaximal, characteristic, complemented
6#63380⋊C3C211yesmaximal, characteristic, complemented
7(C2×AGL(1,ℤ/9))⋊C8C211yesmaximal, characteristic, complemented
8C3×#5156—31nomaximal
9#10588836C311yesmaximal, characteristic, complemented
10C23×AGL(1,ℤ/9)—21no—
11C2×D4×3−1+2C2211yescharacteristic
12C2×D4×3−1+2C2211yescharacteristic, complemented
13C2×#611⋊C3—21no—
14C2×#611⋊C3—21no—
15D4×AGL(1,ℤ/9)—22no—
16D4×AGL(1,ℤ/9)—22no—
17C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
18C2×C4×AGL(1,ℤ/9)C2211yescharacteristic
19D8×3−1+2—22no—
20C2×C8×3−1+2C2211yescharacteristic, complemented
21#3676⋊C3—21no—
22#3673⋊C3—22no—
23C4⋊(3−1+2⋊C4)C2211yescharacteristic, complemented
24C2×3−1+2⋊C8C2211yescharacteristic, complemented
25C2×C3×D3×D4—31no—
26C2×C32×D8D311yescharacteristic
27C3×#938—31no—
28C2×C3×#213—31no—
29C2×D4×D9C2×C311yescharacteristic, complemented
30C3×#832—31no—
31C3×#821—31no—
32C3×(C2×D3)⋊C8—31no—
33C2×C9×D8—21no—
34C2×C9×D8C2×C311yescharacteristic, complemented
35#63500C2×C311yescharacteristic, complemented
36C2×#3673C2×C311yescharacteristic, complemented
37#63392C2×C311yescharacteristic, complemented
38#63380C2×C311yescharacteristic, complemented
39(C2×D9)⋊C8C2×C311yescharacteristic, complemented
40C23×3−1+2—21no—
41C23×3−1+2—21no—
42C22×AGL(1,ℤ/9)—24no—
43C22×AGL(1,ℤ/9)—21no—
44C22×AGL(1,ℤ/9)—42no—
45C22×AGL(1,ℤ/9)—42no—
46C22×AGL(1,ℤ/9)D411yescharacteristic
47D4×3−1+2—21no—
48D4×3−1+2—21no—
49D4×3−1+2—21no—
50D4×3−1+2D412yescomplemented
51C2×C4×3−1+2C2311yescharacteristic
52#611⋊C3—24no—
53#611⋊C3—42no—
54C2×3−1+2⋊C4—21no—
55C2×3−1+2⋊C4D411yescharacteristic
56C4⋊AGL(1,ℤ/9)—21no—
57C4⋊AGL(1,ℤ/9)D412yescomplemented
58C4×AGL(1,ℤ/9)—21no—
59C4×AGL(1,ℤ/9)—21no—
60C8×3−1+2—21no—
613−1+2⋊C8—21no—
62C3×#447—91no—
63#5156—31no—
64C23×C3×D3—61no—
65C2×C32×D4C2×D311yescharacteristic
66C2×C32×D4C2×D311yescharacteristic
67C2×C3×#67—61no—
68C2×C3×#67—61no—
69C3×D3×D4—62no—
70C3×D3×D4—62no—
71C2×C3×D12—31no—
72C2×C3×C4×D3—31no—
73C23×D9—21no—
74C32×D8—22no—
75C2×C32×C8C2×D311yescharacteristic
76C3×#217—61no—
77C3×#213—62no—
78C3×C4⋊Dic3—31no—
79C2×C3×C3⋊C8—31no—
80C2×D4×C9—21no—
81C2×D4×C9—21no—
82C2×D4×C9C22×C311yescharacteristic
83C2×D4×C9C22×C311yescharacteristic, complemented
84C2×#611—21no—
85C2×#611—21no—
86D4×D9—22no—
87D4×D9—22no—
88C2×D36C22×C311yescharacteristic
89C2×C4×D9C22×C311yescharacteristic
90C9×D8—24no—
91C9×D8—22no—
92C2×C8×C9—21no—
93C2×C8×C9C22×C311yescharacteristic, complemented
94#3676—21no—
95#3673—22no—
96C4⋊Dic9C22×C311yescharacteristic, complemented
97C2×C9⋊C8C22×C311yescharacteristic, complemented
98C22×3−1+2—22no—
99C22×3−1+2—41no—
100C22×3−1+2—41no—
101C22×3−1+2—41no—
102C22×3−1+2—41no—
103C22×3−1+2—41no—
104C22×3−1+2C2×D411yescharacteristic
105C2×AGL(1,ℤ/9)—21no—
106C2×AGL(1,ℤ/9)—21no—
107C2×AGL(1,ℤ/9)—22no—
108C2×AGL(1,ℤ/9)—48no—
109C2×AGL(1,ℤ/9)—41no—
110C2×AGL(1,ℤ/9)—41no—
111C2×AGL(1,ℤ/9)D812yescomplemented
112C4×3−1+2C2×D411yescharacteristic
113C4×3−1+2C2×D411yescharacteristic
1143−1+2⋊C4—21no—
1153−1+2⋊C4—41no—
116C22×C3×D4—91no—
117C2×D3×D4—31no—
118C2×C3×D8—31no—
119C2×C3×D8—91no—
120C2×C3×D8C3×D311yescharacteristic
121C3×#121—91no—
122#938—31no—
123C2×#213—31no—
124C3×#102—91no—
125C3×#102—91no—
126C3×C22⋊C8—91no—
127#832—31no—
128#821—31no—
129(C2×D3)⋊C8—31no—
130C23×C32—21no—
131C23×C32—21no—
132C22×C3×D3—31no—
133C22×C3×D3—64no—
134C22×C3×D3—61no—
135C22×C3×D3—122no—
136C22×C3×D3—122no—
137C32×D4—21no—
138C32×D4—21no—
139C32×D4—21no—
140C32×D4#6712yes—
141C2×C32×C4C22×D311yescharacteristic
142C3×#67—64no—
143C3×#67—122no—
144C2×C3×Dic3—31no—
145C2×C3×Dic3—61no—
146C3×D12—32no—
147C3×D12—61no—
148C3×C4×D3—61no—
149C3×C4×D3—61no—
150C23×C9—21no—
151C23×C9—21no—
152C23×C9—41no—
153C23×C9—41no—
154C22×D9—24no—
155C22×D9—21no—
156C22×D9—42no—
157C22×D9—42no—
158C22×D9C3×D411yescharacteristic
159C32×C8—21no—
160C3×C3⋊C8—61no—
161D4×C9—22no—
162D4×C9—21no—
163D4×C9—21no—
164D4×C9—21no—
165D4×C9—22no—
166D4×C9—41no—
167D4×C9—41no—
168D4×C9C3×D412yescomplemented
169C2×C4×C9—21no—
170C2×C4×C9C23×C311yescharacteristic
171#611—24no—
172#611—42no—
173C2×Dic9—21no—
174C2×Dic9C3×D411yescharacteristic
175D36—21no—
176D36C3×D412yescomplemented
177C4×D9—21no—
178C4×D9—21no—
179C8×C9—22no—
180C8×C9—21no—
181C9⋊C8—21no—
182#447—91noSylow
183C2×3−1+2—42no—
184C2×3−1+2—81no—
185C2×3−1+2#12011yescharacteristic
186C2×3−1+2C2×D811yescharacteristic
187C2×3−1+2AGL(1,ℤ/8)11yescharacteristic
188AGL(1,ℤ/9)—22no—
189AGL(1,ℤ/9)—42no—
190C24×C3—181no—
191C23×D3—61no—
192C2×C3×D4—31no—
193C2×C3×D4—31no—
194C2×C3×D4—91no—
195C2×C3×D4—181no—
196C2×C3×D4—182no—
197C2×C3×D4—182no—
198C2×C3×D4—181no—
199C2×C3×D4C2×C3×D311yescharacteristic
200C2×C3×D4C2×C3×D311yescharacteristic
201C22×C3×C4—91no—
202C2×#67—61no—
203C2×#67—61no—
204D3×D4—62no—
205D3×D4—62no—
206C2×D12—31no—
207C2×C4×D3—31no—
208C3×D8—22no—
209C3×D8—62no—
210C3×D8—182no—
211C2×C3×C8—31no—
212C2×C3×C8—91no—
213C2×C3×C8C2×C3×D311yescharacteristic
214C3×C4⋊C4—91no—
215C3×C22⋊C4—181no—
216#217—61no—
217#213—62no—
218C4⋊Dic3—31no—
219C2×C3⋊C8—31no—
220C22×C32—22no—
221C22×C32—41no—
222C22×C32—41no—
223C22×C32—41no—
224C22×C32—41no—
225C22×C32—41no—
226C22×C32D3×D411yescharacteristic
227C2×C3×D3—32no—
228C2×C3×D3—61no—
229C2×C3×D3—61no—
230C2×C3×D3—62no—
231C2×C3×D3—128no—
232C2×C3×D3—121no—
233C2×C3×D3—121no—
234C32×C4D3×D411yescharacteristic
235C32×C4C2×#6711yescharacteristic
236C3×Dic3—61no—
237C3×Dic3—121no—
238C22×C9—22no—
239C22×C9—21no—
240C22×C9—42no—
241C22×C9—42no—
242C22×C9—41no—
243C22×C9—41no—
244C22×C9—41no—
245C22×C9—41no—
246C22×C9—41no—
247C22×C9—81no—
248C22×C9—81no—
249C22×C9—81no—
250C22×C9—81no—
251C22×C9C2×C3×D411yescharacteristic
252C2×D9—21no—
253C2×D9—21no—
254C2×D9—22no—
255C2×D9—48no—
256C2×D9—41no—
257C2×D9—41no—
258C2×D9C3×D812yescomplemented
259C4×C9—21no—
260C4×C9—21no—
261C4×C9C2×C3×D411yescharacteristic
262C4×C9C2×C3×D411yescharacteristic
263Dic9—21no—
264Dic9—41no—
265C22×D4—91no—
266C2×D8—91no—
267C2×D8AGL(1,ℤ/9)11yescharacteristic, complemented
268#121—91no—
269#102—91no—
270#102—91no—
271C22⋊C8—91no—
2723−1+2#44711yescharacteristic, complemented, Sylow
273C23×C3—21no—
274C23×C3—21no—
275C23×C3—61no—
276C23×C3—61no—
277C23×C3—91no—
278C23×C3—181no—
279C23×C3—184no—
280C23×C3—362no—
281C23×C3—362no—
282C22×D3—31no—
283C22×D3—64no—
284C22×D3—61no—
285C22×D3—122no—
286C22×D3—122no—
287C3×D4—21no—
288C3×D4—21no—
289C3×D4—21no—
290C3×D4—32no—
291C3×D4—61no—
292C3×D4—61no—
293C3×D4—61no—
294C3×D4—92no—
295C3×D4—181no—
296C3×D4—184no—
297C3×D4—362no—
298C3×D4C3×#6712yes—
299C2×C3×C4—31no—
300C2×C3×C4—91no—
301C2×C3×C4—181no—
302C2×C3×C4—181no—
303C2×C3×C4—181no—
304C2×C3×C4C22×C3×D311yescharacteristic
305#67—64no—
306#67—122no—
307C2×Dic3—31no—
308C2×Dic3—61no—
309D12—32no—
310D12—61no—
311C4×D3—61no—
312C4×D3—61no—
313C3×C8—21no—
314C3×C8—61no—
315C3×C8—181no—
316C3⋊C8—61no—
317C2×C32—42no—
318C2×C32—81no—
319C2×C32D3×D811yescharacteristic
320C2×C32#91111yescharacteristic
321C2×C32#93711yescharacteristic
322C3×D3—62no—
323C3×D3—122no—
324C2×C9—21no—
325C2×C9—21no—
326C2×C9—21no—
327C2×C9—42no—
328C2×C9—82no—
329C2×C9—82no—
330C2×C9—81no—
331C2×C9C3×#12011yescharacteristic
332C2×C9C2×C3×D811yescharacteristic
333C2×C9C3×AGL(1,ℤ/8)11yescharacteristic
334D9—22no—
335D9—42no—
336C24—181no—
337C2×D4—91no—
338C2×D4—181no—
339C2×D4—182no—
340C2×D4—182no—
341C2×D4—181no—
342C2×D4C2×AGL(1,ℤ/9)11yescharacteristic
343C2×D4C2×AGL(1,ℤ/9)11yescharacteristic, complemented
344C22×C4—91no—
345D8—22no—
346D8—182no—
347C2×C8—91no—
348C2×C8C2×AGL(1,ℤ/9)11yescharacteristic, complemented
349C4⋊C4—91no—
350C22⋊C4—181no—
351C22×C3—22no—
352C22×C3—31no—
353C22×C3—41no—
354C22×C3—41no—
355C22×C3—41no—
356C22×C3—41no—
357C22×C3—41no—
358C22×C3—62no—
359C22×C3—92no—
360C22×C3—121no—
361C22×C3—121no—
362C22×C3—121no—
363C22×C3—121no—
364C22×C3—121no—
365C22×C3—181no—
366C22×C3—181no—
367C22×C3—182no—
368C22×C3—361no—
369C22×C3—361no—
370C22×C3—368no—
371C22×C3C3×D3×D411yescharacteristic
372C2×D3—32no—
373C2×D3—61no—
374C2×D3—61no—
375C2×D3—62no—
376C2×D3—128no—
377C2×D3—121no—
378C2×D3—121no—
379C3×C4—31no—
380C3×C4—31no—
381C3×C4—181no—
382C3×C4—361no—
383C3×C4C3×D3×D411yescharacteristic
384C3×C4C2×C3×#6711yescharacteristic
385Dic3—61no—
386Dic3—121no—
387C32#515611yescharacteristic
388C9—21no—
389C9C3×#44711yescharacteristic, complemented
390C23—21no—
391C23—21no—
392C23—91no—
393C23—184no—
394C23—181no—
395C23—362no—
396C23—362no—
397D4—21no—
398D4—21no—
399D4—21no—
400D4—92no—
401D4—184no—
402D4—181no—
403D4—362no—
404D4#611⋊C312yescomplemented
405C2×C4—91no—
406C2×C4—181no—
407C2×C4—181no—
408C2×C4—181no—
409C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
410C8—21no—
411C8—181no—
412C2×C3—31no—
413C2×C3—31no—
414C2×C3—31no—
415C2×C3—42no—
416C2×C3—81no—
417C2×C3—122no—
418C2×C3—182no—
419C2×C3—241no—
420C2×C3—362no—
421C2×C3C3×D3×D811yescharacteristic
422C2×C3C3×#91111yescharacteristic
423C2×C3C3×#93711yescharacteristic
424D3—62no—
425D3—122no—
426C22—22no—
427C22—41no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—92no—
433C22—181no—
434C22—181no—
435C22—182no—
436C22—368no—
437C22—361no—
438C22—361no—
439C22D4×AGL(1,ℤ/9)11yescharacteristic
440C4—181no—
441C4—361no—
442C4D4×AGL(1,ℤ/9)11yescharacteristic
443C4C2×#611⋊C311yescharacteristic
444C3—31no—
445C3C3×#515611yescharacteristic
446C2—42no—
447C2—81no—
448C2—182no—
449C2—362no—
450C2D8×AGL(1,ℤ/9)11yescharacteristic
451C2#63473⋊C311yescharacteristic
452C2#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×#447
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×D4×AGL(1,ℤ/9)C2semidirect
2. C2×D8×3−1+2C2semidirect
3. #63500⋊C3C2semidirect
4. C2×#3673⋊C3C2semidirect
5. #63392⋊C3C2semidirect
6. #63380⋊C3C2semidirect
7. (C2×AGL(1,ℤ/9))⋊C8C2semidirect
9. #10588836C3semidirect
12. C2×D4×3−1+2C22semidirect
20. C2×C8×3−1+2C22semidirect
23. C4⋊(3−1+2⋊C4)C22semidirect
24. C2×3−1+2⋊C8C22semidirect
29. C2×D4×D9C2×C3semidirect
34. C2×C9×D8C2×C3semidirect
35. #63500C2×C3semidirect
36. C2×#3673C2×C3semidirect
37. #63392C2×C3semidirect
38. #63380C2×C3semidirect
39. (C2×D9)⋊C8C2×C3semidirect
50. D4×3−1+2D4semidirect
57. C4⋊AGL(1,ℤ/9)D4semidirect
83. C2×D4×C9C22×C3semidirect
93. C2×C8×C9C22×C3semidirect
96. C4⋊Dic9C22×C3semidirect
97. C2×C9⋊C8C22×C3semidirect
111. C2×AGL(1,ℤ/9)D8semidirect
168. D4×C9C3×D4semidirect
176. D36C3×D4semidirect
258. C2×D9C3×D8semidirect
267. C2×D8AGL(1,ℤ/9)semidirect
272. 3−1+2#447semidirect
343. C2×D4C2×AGL(1,ℤ/9)semidirect
348. C2×C8C2×AGL(1,ℤ/9)semidirect
389. C9C3×#447semidirect
404. D4#611⋊C3semidirect

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×D8×3−1+22
#63500⋊C32
C2×#3673⋊C32
#63392⋊C32
#63380⋊C32
(C2×AGL(1,ℤ/9))⋊C82
C3×#51563
#105888363