Finite Group Databasebeta

← #10588730⋊C3 #13142895

Subgroups — #10588730⋊C3 #13142895

Order 1728.

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

normal subgroups only

This group has 493 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
23−1+2×C22⋊Q8C211yesmaximal, characteristic, complemented
3C2×#3675⋊C3C211yesmaximal, characteristic, complemented
4C2×#3674⋊C3C211yesmaximal, characteristic, complemented
5#63389⋊C3C211yesmaximal, characteristic, complemented
6#63369⋊C3C211yesmaximal, characteristic, complemented
7#63368⋊C3C211yesmaximal, characteristic, complemented
8C3×#5047—31nomaximal
9#10588730C311yesmaximal, characteristic, complemented
10C2×Q8×3−1+2C2211yescharacteristic
11C22×C4×3−1+2C2211yescharacteristic
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)—21no—
17C2×3−1+2⋊Q8C2211yescharacteristic
18C4⋊C4×3−1+2—21no—
19C4⋊C4×3−1+2C2211yescharacteristic
20C22⋊C4×3−1+2—21no—
21#3675⋊C3—22no—
22#3674⋊C3—22no—
23C2×3−1+2⋊C8C2211yescharacteristic
24C2×3−1+2⋊C8C2211yescharacteristic
25C2×C3×#235—31no—
26C32×C22⋊Q8D311yescharacteristic
27C2×C3×#216—31no—
28C2×C3×#215—31no—
29C2×#3697C2×C311yescharacteristic
30C3×#830—31no—
31C3×#810—31no—
32C3×#809—31no—
33C9×C22⋊Q8—21no—
34C9×C22⋊Q8C2×C311yescharacteristic, complemented
35C2×#3675C2×C311yescharacteristic, complemented
36C2×#3674C2×C311yescharacteristic, complemented
37#63389C2×C311yescharacteristic, complemented
38#63369C2×C311yescharacteristic, complemented
39#63368C2×C311yescharacteristic, complemented
40C23×3−1+2D411yescharacteristic
41C22×AGL(1,ℤ/9)—21no—
42Q8×3−1+2—21no—
43Q8×3−1+2—21no—
44C2×C4×3−1+2—21no—
45C2×C4×3−1+2—21no—
46C2×C4×3−1+2—21no—
47C2×C4×3−1+2—21no—
48C2×C4×3−1+2D411yescharacteristic
49C2×C4×3−1+2C2311yescharacteristic
50#611⋊C3—24no—
51C2×3−1+2⋊C4—21no—
52C4⋊AGL(1,ℤ/9)—21no—
53C4⋊AGL(1,ℤ/9)D412yes—
54C4×AGL(1,ℤ/9)—24no—
553−1+2⋊Q8—21no—
563−1+2⋊Q8D412yes—
573−1+2⋊C8—21no—
583−1+2⋊C8—21no—
59C3×#452—91no—
60#5047—31no—
61C2×C32×Q8C2×D311yescharacteristic
62C22×C32×C4C2×D311yescharacteristic
63C2×C3×#67—61no—
64C3×#235—62no—
65C3×#235—62no—
66C2×C3×D12—31no—
67C2×C3×C4×D3—61no—
68C2×C3×Dic6—31no—
69C32×C4⋊C4—21no—
70C32×C4⋊C4C2×D311yescharacteristic
71C32×C22⋊C4—21no—
72C3×#216—62no—
73C3×#215—62no—
74C2×C3×C3⋊C8—31no—
75C2×C3×C3⋊C8—31no—
76C2×Q8×C9—21no—
77C2×Q8×C9C22×C311yescharacteristic
78C22×C4×C9—21no—
79C22×C4×C9C22×C311yescharacteristic
80C2×#611—21no—
81#3697—22no—
82#3697—22no—
83C2×D36C22×C311yescharacteristic
84C2×C4×D9—21no—
85C2×Dic18C22×C311yescharacteristic
86C9×C4⋊C4—22no—
87C9×C4⋊C4—21no—
88C9×C4⋊C4—21no—
89C9×C4⋊C4C22×C311yescharacteristic
90C9×C22⋊C4—22no—
91C9×C22⋊C4—21no—
92#3675—22no—
93#3674—22no—
94C2×C9⋊C8C22×C311yescharacteristic
95C2×C9⋊C8C22×C311yescharacteristic
96C22×3−1+2—21no—
97C22×3−1+2—21no—
98C22×3−1+2—21no—
99C22×3−1+2C2×D411yescharacteristic
100C2×AGL(1,ℤ/9)—22no—
101C2×AGL(1,ℤ/9)—41no—
102C2×AGL(1,ℤ/9)—41no—
103C4×3−1+2—21no—
104C4×3−1+2—41no—
105C4×3−1+2—41no—
106C4×3−1+2C2×D411yescharacteristic
107C4×3−1+2C2×D411yescharacteristic
1083−1+2⋊C4—22no—
109C2×C3×#41—91no—
110C2×#235—31no—
111C2×C3×Q16—91no—
112C2×C3×SD16—91no—
113C3×C22⋊Q8—31no—
114C3×C22⋊Q8C3×D311yescharacteristic
115C2×#216—31no—
116C2×#215—31no—
117C3×#103—91no—
118C3×#102—91no—
119C3×C22⋊C8—91no—
120#830—31no—
121#810—31no—
122#809—31no—
123C23×C32#6711yescharacteristic
124C22×C3×D3—61no—
125C32×Q8—21no—
126C32×Q8—21no—
127C2×C32×C4—21no—
128C2×C32×C4—21no—
129C2×C32×C4—21no—
130C2×C32×C4—21no—
131C2×C32×C4#6711yescharacteristic
132C2×C32×C4C22×D311yescharacteristic
133C3×#67—64no—
134C2×C3×Dic3—61no—
135C3×D12—32no—
136C3×D12—61no—
137C3×C4×D3—64no—
138C3×Dic6—32no—
139C3×Dic6—61no—
140C23×C9—21no—
141C23×C9C3×D411yescharacteristic
142C22×D9—21no—
143C3×C3⋊C8—61no—
144C3×C3⋊C8—61no—
145Q8×C9—21no—
146Q8×C9—21no—
147Q8×C9—41no—
148Q8×C9—41no—
149C2×C4×C9—21no—
150C2×C4×C9—21no—
151C2×C4×C9—22no—
152C2×C4×C9—22no—
153C2×C4×C9—21no—
154C2×C4×C9—21no—
155C2×C4×C9—21no—
156C2×C4×C9—21no—
157C2×C4×C9—41no—
158C2×C4×C9—41no—
159C2×C4×C9C3×D411yescharacteristic
160C2×C4×C9C23×C311yescharacteristic
161#611—24no—
162C2×Dic9—21no—
163D36—21no—
164D36C3×D412yes—
165C4×D9—24no—
166Dic18—21no—
167Dic18C3×D412yes—
168C9⋊C8—21no—
169C9⋊C8—21no—
170#452—91noSylow
171C2×3−1+2—41no—
172C2×3−1+2#12011yescharacteristic
173C2×3−1+2#13511yescharacteristic
174C2×3−1+2#13711yescharacteristic
175AGL(1,ℤ/9)—42no—
176C3×#41—182no—
177C3×#41—182no—
178C2×C3×Q8—31no—
179C2×C3×Q8—91no—
180C2×C3×Q8C2×C3×D311yescharacteristic
181C2×C3×D4—91no—
182C2×C3×D4—181no—
183C22×C3×C4—31no—
184C22×C3×C4—181no—
185C22×C3×C4C2×C3×D311yescharacteristic
186C2×#67—61no—
187#235—62no—
188#235—62no—
189C2×D12—31no—
190C2×C4×D3—61no—
191C2×Dic6—31no—
192C3×Q16—182no—
193C3×SD16—182no—
194C2×C3×C8—91no—
195C2×C3×C8—91no—
196C3×C4⋊C4—21no—
197C3×C4⋊C4—31no—
198C3×C4⋊C4—61no—
199C3×C4⋊C4C2×C3×D311yescharacteristic
200C3×C22⋊C4—21no—
201C3×C22⋊C4—61no—
202#216—62no—
203#215—62no—
204C2×C3⋊C8—31no—
205C2×C3⋊C8—31no—
206C22×C32—21no—
207C22×C32—21no—
208C22×C32—21no—
209C22×C32C2×#6711yescharacteristic
210C2×C3×D3—62no—
211C2×C3×D3—121no—
212C2×C3×D3—121no—
213C32×C4—21no—
214C32×C4—41no—
215C32×C4—41no—
216C32×C4D3×D411yescharacteristic
217C32×C4D3×D411yescharacteristic
218C3×Dic3—62no—
219C22×C9—22no—
220C22×C9—21no—
221C22×C9—21no—
222C22×C9—21no—
223C22×C9—21no—
224C22×C9—41no—
225C22×C9—41no—
226C22×C9C2×C3×D411yescharacteristic
227C2×D9—22no—
228C2×D9—41no—
229C2×D9—41no—
230C4×C9—21no—
231C4×C9—21no—
232C4×C9—21no—
233C4×C9—41no—
234C4×C9—42no—
235C4×C9—42no—
236C4×C9—41no—
237C4×C9—41no—
238C4×C9C2×C3×D411yescharacteristic
239C4×C9C2×C3×D411yescharacteristic
240Dic9—22no—
241C2×#41—91no—
242C2×Q16—91no—
243C2×SD16—91no—
244C22⋊Q8AGL(1,ℤ/9)11yescharacteristic, complemented
245#103—91no—
246#102—91no—
247C22⋊C8—91no—
2483−1+2#45211yescharacteristic, complemented, Sylow
249C23×C3—31no—
250C23×C3—181no—
251C23×C3C3×#6711yescharacteristic
252C22×D3—61no—
253C3×Q8—21no—
254C3×Q8—21no—
255C3×Q8—61no—
256C3×Q8—61no—
257C3×Q8—92no—
258C3×Q8—181no—
259C3×D4—92no—
260C3×D4—181no—
261C3×D4—184no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—31no—
267C2×C3×C4—31no—
268C2×C3×C4—61no—
269C2×C3×C4—61no—
270C2×C3×C4—61no—
271C2×C3×C4—61no—
272C2×C3×C4—184no—
273C2×C3×C4—181no—
274C2×C3×C4C3×#6711yescharacteristic
275C2×C3×C4C22×C3×D311yescharacteristic
276#67—64no—
277C2×Dic3—61no—
278D12—32no—
279D12—61no—
280C4×D3—64no—
281Dic6—32no—
282Dic6—61no—
283C3×C8—181no—
284C3×C8—181no—
285C3⋊C8—61no—
286C3⋊C8—61no—
287C2×C32—41no—
288C2×C32#93711yescharacteristic
289C2×C32#94211yescharacteristic
290C2×C32#95011yescharacteristic
291C3×D3—122no—
292C2×C9—21no—
293C2×C9—21no—
294C2×C9—21no—
295C2×C9—42no—
296C2×C9—41no—
297C2×C9C3×#12011yescharacteristic
298C2×C9C3×#13511yescharacteristic
299C2×C9C3×#13711yescharacteristic
300D9—42no—
301#41—182no—
302#41—182no—
303C2×Q8—91no—
304C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic
305C2×D4—91no—
306C2×D4—181no—
307C22×C4—181no—
308C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
309Q16—182no—
310SD16—182no—
311C2×C8—91no—
312C2×C8—91no—
313C4⋊C4—21no—
314C4⋊C4C2×AGL(1,ℤ/9)11yescharacteristic
315C22⋊C4—21no—
316C22×C3—21no—
317C22×C3—21no—
318C22×C3—21no—
319C22×C3—31no—
320C22×C3—61no—
321C22×C3—61no—
322C22×C3—61no—
323C22×C3—182no—
324C22×C3—361no—
325C22×C3—361no—
326C22×C3C2×C3×#6711yescharacteristic
327C2×D3—62no—
328C2×D3—121no—
329C2×D3—121no—
330C3×C4—21no—
331C3×C4—31no—
332C3×C4—31no—
333C3×C4—41no—
334C3×C4—41no—
335C3×C4—61no—
336C3×C4—121no—
337C3×C4—121no—
338C3×C4—182no—
339C3×C4C3×D3×D411yescharacteristic
340C3×C4C3×D3×D411yescharacteristic
341Dic3—62no—
342C32#504711yescharacteristic
343C9—21no—
344C9C3×#45211yescharacteristic, complemented
345C23—181no—
346C23#611⋊C311yescharacteristic
347Q8—21no—
348Q8—21no—
349Q8—92no—
350Q8—181no—
351D4—92no—
352D4—184no—
353D4—181no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—184no—
359C2×C4—181no—
360C2×C4#611⋊C311yescharacteristic
361C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
362C8—181no—
363C8—181no—
364C2×C3—31no—
365C2×C3—31no—
366C2×C3—31no—
367C2×C3—41no—
368C2×C3—121no—
369C2×C3—362no—
370C2×C3C3×#93711yescharacteristic
371C2×C3C3×#94211yescharacteristic
372C2×C3C3×#95011yescharacteristic
373D3—122no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—182no—
378C22—361no—
379C22—361no—
380C22C2×#611⋊C311yescharacteristic
381C4—21no—
382C4—41no—
383C4—41no—
384C4—182no—
385C4D4×AGL(1,ℤ/9)11yescharacteristic
386C4D4×AGL(1,ℤ/9)11yescharacteristic
387C3—31no—
388C3C3×#504711yescharacteristic
389C2—41no—
390C2—362no—
391C2#63499⋊C311yescharacteristic
392C2#63505⋊C311yescharacteristic
393C2#63513⋊C311yescharacteristic
subgroupG/H
centerC22C2×#611⋊C3
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×C22⋊Q8C2
socleC22×C3C2×C3×#67
prime radicalC2×#3697⋊C3C2
hypercenterC22⋊Q8AGL(1,ℤ/9)
hypocenterC9C3×#452
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. 3−1+2×C22⋊Q8C2semidirect
3. C2×#3675⋊C3C2semidirect
4. C2×#3674⋊C3C2semidirect
5. #63389⋊C3C2semidirect
6. #63369⋊C3C2semidirect
7. #63368⋊C3C2semidirect
9. #10588730C3semidirect
34. C9×C22⋊Q8C2×C3semidirect
35. C2×#3675C2×C3semidirect
36. C2×#3674C2×C3semidirect
37. #63389C2×C3semidirect
38. #63369C2×C3semidirect
39. #63368C2×C3semidirect
244. C22⋊Q8AGL(1,ℤ/9)semidirect
248. 3−1+2#452semidirect
344. C9C3×#452semidirect

Maximal subgroups

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

subgroupindex
C2×#3697⋊C32
3−1+2×C22⋊Q82
C2×#3675⋊C32
C2×#3674⋊C32
#63389⋊C32
#63369⋊C32
#63368⋊C32
C3×#50473
#105887303