Finite Group Databasebeta

← #10588462⋊C3 #13141831

Subgroups — #10588462⋊C3 #13141831

Order 1728.

415 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 415 conjugacy classes of subgroups — too many to draw. Only its 90 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
1#63500⋊C3C211yesmaximal, characteristic, complemented
2C2×#3673⋊C3C211yesmaximal, characteristic, complemented
3C2×#3663⋊C3C211yesmaximal, characteristic, complemented
4#63457⋊C3C211yesmaximal, characteristic, complemented
53−1+2×#102C211yesmaximal, characteristic, complemented
6#63380⋊C3C211yesmaximal, characteristic, complemented
7#63367⋊C3C211yesmaximal, characteristic, complemented
8C3×#4782—31nomaximal
9#10588462C311yesmaximal, characteristic, complemented
10C2×D4×3−1+2C2211yescharacteristic
11C2×#611⋊C3—21no—
12C2×C4⋊AGL(1,ℤ/9)—21no—
13C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
14C2×C4×AGL(1,ℤ/9)C2211yescharacteristic
15C2×C8×3−1+2C2211yescharacteristic
16C4⋊C4×3−1+2C2211yescharacteristic
17#3676⋊C3—21no—
18#3673⋊C3—22no—
19(C2×AGL(1,ℤ/9))⋊C4—21no—
20C4⋊(3−1+2⋊C4)C2211yescharacteristic
21C2×3−1+2⋊C8C2211yescharacteristic
22#3663⋊C3—22no—
23C3×#938—31no—
24C2×C3×#213—31no—
25C2×C3×ΓL(1,𝔽25)—31no—
26C3×#895—31no—
27C32×#102D311yescharacteristic
28C3×#821—31no—
29C3×#808—31no—
30#63500C2×C311yescharacteristic, complemented
31C2×#3673C2×C311yescharacteristic, complemented
32C2×#3663C2×C311yescharacteristic, complemented
33#63457C2×C311yescharacteristic, complemented
34C9×#102—21no—
35C9×#102C2×C311yescharacteristic, complemented
36#63380C2×C311yescharacteristic, complemented
37#63367C2×C311yescharacteristic, complemented
38C23×3−1+2—21no—
39C22×AGL(1,ℤ/9)—21no—
40C22×AGL(1,ℤ/9)D411yescharacteristic
41D4×3−1+2—21no—
42D4×3−1+2—21no—
43C2×C4×3−1+2—21no—
44C2×C4×3−1+2C2311yescharacteristic
45#611⋊C3—42no—
46C2×3−1+2⋊C4—21no—
47C2×3−1+2⋊C4D411yescharacteristic
48C4⋊AGL(1,ℤ/9)—21no—
49C4⋊AGL(1,ℤ/9)—21no—
50C4⋊AGL(1,ℤ/9)—42no—
51C4×AGL(1,ℤ/9)—21no—
52C4×AGL(1,ℤ/9)—21no—
53C8×3−1+2—21no—
543−1+2⋊C8D412yescomplemented
55C3×#469—91no—
56#4782—31no—
57C2×C32×D4C2×D311yescharacteristic
58C2×C3×#67—61no—
59C2×C3×D12—31no—
60C2×C3×D12—61no—
61C2×C3×C4×D3—31no—
62C2×C32×C8C2×D311yescharacteristic
63C32×C4⋊C4C2×D311yescharacteristic
64C3×#217—61no—
65C3×#213—62no—
66C3×(C2×D3)⋊C4—61no—
67C3×C4⋊Dic3—31no—
68C2×C3×C3⋊C8—31no—
69C3×ΓL(1,𝔽25)—62no—
70C2×D4×C9—21no—
71C2×D4×C9C22×C311yescharacteristic
72C2×#611—21no—
73C2×D36—21no—
74C2×D36C22×C311yescharacteristic
75C2×C4×D9C22×C311yescharacteristic
76C2×C8×C9—21no—
77C2×C8×C9C22×C311yescharacteristic
78C9×C4⋊C4—21no—
79C9×C4⋊C4C22×C311yescharacteristic
80#3676—21no—
81#3673—22no—
82(C2×D9)⋊C4—21no—
83C4⋊Dic9C22×C311yescharacteristic
84C2×C9⋊C8C22×C311yescharacteristic
85#3663—22no—
86C22×3−1+2—41no—
87C22×3−1+2—41no—
88C22×3−1+2—41no—
89C22×3−1+2C2×D411yescharacteristic
90C2×AGL(1,ℤ/9)—21no—
91C2×AGL(1,ℤ/9)—21no—
92C2×AGL(1,ℤ/9)—21no—
93C2×AGL(1,ℤ/9)—41no—
94C2×AGL(1,ℤ/9)—41no—
95C2×AGL(1,ℤ/9)—41no—
96C4×3−1+2—41no—
97C4×3−1+2C2×D411yescharacteristic
98C4×3−1+2#4111yescharacteristic
993−1+2⋊C4—21no—
1003−1+2⋊C4—41no—
101C2×C3×D8—91no—
102C2×C3×M16—91no—
103C3×#121—91no—
104C3×#121—91no—
105#938—31no—
106C2×#213—31no—
107C2×ΓL(1,𝔽25)—31no—
108#895—31no—
109C3×#106—91no—
110C3×#102—31no—
111C3×#102—91no—
112C3×#102C3×D311yescharacteristic
113#821—31no—
114#808—31no—
115C23×C32—21no—
116C22×C3×D3—31no—
117C22×C3×D3—61no—
118C32×D4—21no—
119C32×D4—21no—
120C2×C32×C4—21no—
121C2×C32×C4C22×D311yescharacteristic
122C3×#67—122no—
123C2×C3×Dic3—31no—
124C2×C3×Dic3—61no—
125C3×D12—61no—
126C3×D12—61no—
127C3×D12—122no—
128C3×C4×D3—61no—
129C3×C4×D3—61no—
130C23×C9—21no—
131C23×C9—41no—
132C22×D9—21no—
133C22×D9C3×D411yescharacteristic
134C32×C8—21no—
135C3×C3⋊C8—32no—
136D4×C9—22no—
137D4×C9—21no—
138D4×C9—21no—
139D4×C9—41no—
140C2×C4×C9—21no—
141C2×C4×C9—21no—
142C2×C4×C9—41no—
143C2×C4×C9C23×C311yescharacteristic
144#611—42no—
145C2×Dic9—21no—
146C2×Dic9C3×D411yescharacteristic
147D36—21no—
148D36—21no—
149D36—42no—
150C4×D9—21no—
151C4×D9—21no—
152C8×C9—21no—
153C8×C9—41no—
154C9⋊C8C3×D412yescomplemented
155#469—91noSylow
156C2×3−1+2—81no—
157C2×3−1+2#12111yescharacteristic
158C2×3−1+2AGL(1,ℤ/8)11yescharacteristic
159C2×3−1+2AGL(1,ℤ/8)11yescharacteristic
160AGL(1,ℤ/9)—41no—
161AGL(1,ℤ/9)—81no—
162C2×C3×D4—31no—
163C2×C3×D4—91no—
164C2×C3×D4—181no—
165C2×C3×D4—181no—
166C2×C3×D4C2×C3×D311yescharacteristic
167C22×C3×C4—91no—
168C2×#67—61no—
169C2×D12—31no—
170C2×D12—61no—
171C2×C4×D3—31no—
172C3×D8—182no—
173C3×M16—182no—
174C2×C3×C8—31no—
175C2×C3×C8—91no—
176C2×C3×C8C2×C3×D311yescharacteristic
177C3×C4⋊C4—31no—
178C3×C4⋊C4—91no—
179C3×C4⋊C4C2×C3×D311yescharacteristic
180C3×C22⋊C4—181no—
181C3×C22⋊C4—181no—
182#217—61no—
183#213—62no—
184(C2×D3)⋊C4—61no—
185C4⋊Dic3—31no—
186C2×C3⋊C8—31no—
187ΓL(1,𝔽25)—62no—
188C22×C32—41no—
189C22×C32—41no—
190C22×C32—41no—
191C22×C32D3×D411yescharacteristic
192C2×C3×D3—61no—
193C2×C3×D3—61no—
194C2×C3×D3—61no—
195C2×C3×D3—121no—
196C2×C3×D3—121no—
197C2×C3×D3—121no—
198C32×C4—41no—
199C32×C4#23511yescharacteristic
200C32×C4D3×D411yescharacteristic
201C3×Dic3—61no—
202C3×Dic3—121no—
203C22×C9—21no—
204C22×C9—42no—
205C22×C9—41no—
206C22×C9—41no—
207C22×C9—41no—
208C22×C9—81no—
209C22×C9—81no—
210C22×C9C2×C3×D411yescharacteristic
211C2×D9—21no—
212C2×D9—21no—
213C2×D9—21no—
214C2×D9—41no—
215C2×D9—41no—
216C2×D9—41no—
217C4×C9—21no—
218C4×C9—21no—
219C4×C9—41no—
220C4×C9—81no—
221C4×C9C2×C3×D411yescharacteristic
222C4×C9C3×#4111yescharacteristic
223Dic9—21no—
224Dic9—41no—
225C2×D8—91no—
226C2×M16—91no—
227#121—91no—
228#121—91no—
229#106—91no—
230#102—91no—
231#102AGL(1,ℤ/9)11yescharacteristic, complemented
2323−1+2#46911yescharacteristic, complemented, Sylow
233C23×C3—21no—
234C23×C3—61no—
235C23×C3—91no—
236C23×C3—181no—
237C22×D3—31no—
238C22×D3—61no—
239C3×D4—21no—
240C3×D4—21no—
241C3×D4—61no—
242C3×D4—61no—
243C3×D4—181no—
244C3×D4—181no—
245C3×D4—362no—
246C3×D4—362no—
247C2×C3×C4—21no—
248C2×C3×C4—31no—
249C2×C3×C4—61no—
250C2×C3×C4—91no—
251C2×C3×C4—181no—
252C2×C3×C4—181no—
253C2×C3×C4—181no—
254C2×C3×C4C22×C3×D311yescharacteristic
255#67—122no—
256C2×Dic3—31no—
257C2×Dic3—61no—
258D12—61no—
259D12—61no—
260D12—122no—
261C4×D3—61no—
262C4×D3—61no—
263C3×C8—21no—
264C3×C8—61no—
265C3×C8—92no—
266C3⋊C8—32no—
267C2×C32—81no—
268C2×C32#88411yescharacteristic
269C2×C32#91111yescharacteristic
270C2×C32#91411yescharacteristic
271C3×D3—121no—
272C3×D3—241no—
273C2×C9—21no—
274C2×C9—21no—
275C2×C9—21no—
276C2×C9—82no—
277C2×C9—81no—
278C2×C9C3×#12111yescharacteristic
279C2×C9C3×AGL(1,ℤ/8)11yescharacteristic
280C2×C9C3×AGL(1,ℤ/8)11yescharacteristic
281D9—41no—
282D9—81no—
283C2×D4—91no—
284C2×D4—181no—
285C2×D4—181no—
286C2×D4C2×AGL(1,ℤ/9)11yescharacteristic
287C22×C4—91no—
288D8—182no—
289M16—182no—
290C2×C8—91no—
291C2×C8C2×AGL(1,ℤ/9)11yescharacteristic
292C4⋊C4—91no—
293C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
294C22⋊C4—181no—
295C22⋊C4—181no—
296C22×C3—31no—
297C22×C3—41no—
298C22×C3—41no—
299C22×C3—41no—
300C22×C3—121no—
301C22×C3—121no—
302C22×C3—121no—
303C22×C3—181no—
304C22×C3—181no—
305C22×C3—181no—
306C22×C3—361no—
307C22×C3—361no—
308C22×C3—361no—
309C22×C3C3×D3×D411yescharacteristic
310C2×D3—61no—
311C2×D3—61no—
312C2×D3—61no—
313C2×D3—121no—
314C2×D3—121no—
315C2×D3—121no—
316C3×C4—31no—
317C3×C4—31no—
318C3×C4—41no—
319C3×C4—121no—
320C3×C4—181no—
321C3×C4—361no—
322C3×C4C3×#23511yescharacteristic
323C3×C4C3×D3×D411yescharacteristic
324Dic3—61no—
325Dic3—121no—
326C32#478211yescharacteristic
327C9—21no—
328C9C3×#46911yescharacteristic, complemented
329C23—21no—
330C23—91no—
331C23—181no—
332D4—21no—
333D4—21no—
334D4—181no—
335D4—181no—
336D4—362no—
337D4—362no—
338C2×C4—21no—
339C2×C4—91no—
340C2×C4—181no—
341C2×C4—181no—
342C2×C4—181no—
343C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
344C8—21no—
345C8—92no—
346C2×C3—31no—
347C2×C3—31no—
348C2×C3—31no—
349C2×C3—81no—
350C2×C3—241no—
351C2×C3—361no—
352C2×C3—721no—
353C2×C3C3×#88411yescharacteristic
354C2×C3C3×#91111yescharacteristic
355C2×C3C3×#91411yescharacteristic
356D3—121no—
357D3—241no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—181no—
362C22—181no—
363C22—181no—
364C22—361no—
365C22—361no—
366C22—361no—
367C22D4×AGL(1,ℤ/9)11yescharacteristic
368C4—41no—
369C4—181no—
370C4—361no—
371C4#3697⋊C311yescharacteristic
372C4D4×AGL(1,ℤ/9)11yescharacteristic
373C3—31no—
374C3C3×#478211yescharacteristic
375C2—81no—
376C2—361no—
377C2—721no—
378C2#63446⋊C311yescharacteristic
379C2#63473⋊C311yescharacteristic
380C2#63476⋊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×#469
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. #63500⋊C3C2semidirect
2. C2×#3673⋊C3C2semidirect
3. C2×#3663⋊C3C2semidirect
4. #63457⋊C3C2semidirect
5. 3−1+2×#102C2semidirect
6. #63380⋊C3C2semidirect
7. #63367⋊C3C2semidirect
9. #10588462C3semidirect
30. #63500C2×C3semidirect
31. C2×#3673C2×C3semidirect
32. C2×#3663C2×C3semidirect
33. #63457C2×C3semidirect
35. C9×#102C2×C3semidirect
36. #63380C2×C3semidirect
37. #63367C2×C3semidirect
54. 3−1+2⋊C8D4semidirect
154. C9⋊C8C3×D4semidirect
231. #102AGL(1,ℤ/9)semidirect
232. 3−1+2#469semidirect
328. C9C3×#469semidirect

Maximal subgroups

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

subgroupindex
#63500⋊C32
C2×#3673⋊C32
C2×#3663⋊C32
#63457⋊C32
3−1+2×#1022
#63380⋊C32
#63367⋊C32
C3×#47823
#105884623