Finite Group Databasebeta

← #10588842⋊C3 #13143380

Subgroups — #10588842⋊C3 #13143380

Order 1728.

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

normal subgroups only

This group has 394 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×SD16×3−1+2C211yesmaximal, characteristic
2#63506⋊C3C211yesmaximal, characteristic, complemented
3D4×3−1+2⋊C4C211yesmaximal, characteristic
4C2×#3672⋊C3C211yesmaximal, characteristic
5#63395⋊C3C211yesmaximal, characteristic, complemented
6#63376⋊C3C211yesmaximal, characteristic, complemented
7(3−1+2⋊C4)⋊C8C211yesmaximal, characteristic, complemented
8C3×#5162—31nomaximal
9#10588842C311yesmaximal, characteristic, complemented
10C2×Q8×3−1+2C2211yescharacteristic
11C2×D4×3−1+2C2211yescharacteristic
12C22×3−1+2⋊C4—21no—
13C2×3−1+2⋊Q8C2211yescharacteristic
14SD16×3−1+2—22no—
15C2×C8×3−1+2C2211yescharacteristic
16#3676⋊C3—21no—
17#3672⋊C3—22no—
18C4⋊(3−1+2⋊C4)C2211yescharacteristic
19(3−1+2⋊C4)⋊C4—21no—
20(3−1+2⋊C4)⋊C4—21no—
21C4×3−1+2⋊C4C2211yescharacteristic
22C2×3−1+2⋊C8C2211yescharacteristic
23C2×C32×SD16D311yescharacteristic
24C3×#944—31no—
25C3×D4×Dic3—31no—
26C2×C3×#214—31no—
27C3×#835—31no—
28C3×#816—31no—
29C3×Dic3⋊C8—31no—
30C2×C9×SD16—21no—
31C2×C9×SD16C2×C311yescharacteristic
32#63506C2×C311yescharacteristic, complemented
33D4×Dic9C2×C311yescharacteristic
34C2×#3672C2×C311yescharacteristic
35#63395C2×C311yescharacteristic, complemented
36#63376C2×C311yescharacteristic, complemented
37Dic9⋊C8C2×C311yescharacteristic, complemented
38C23×3−1+2—21no—
39Q8×3−1+2—21no—
40Q8×3−1+2—21no—
41D4×3−1+2—21no—
42D4×3−1+2D412yes—
43C2×C4×3−1+2—21no—
44C2×C4×3−1+2C2311yescharacteristic
45C2×3−1+2⋊C4—21no—
46C2×3−1+2⋊C4—21no—
47C2×3−1+2⋊C4—42no—
48C2×3−1+2⋊C4D411yescharacteristic
49C2×3−1+2⋊C4D411yescharacteristic
503−1+2⋊Q8—21no—
513−1+2⋊Q8—21no—
52C8×3−1+2—21no—
533−1+2⋊C8—21no—
54C3×#461—91no—
55#5162—31no—
56C2×C32×Q8C2×D311yescharacteristic
57C2×C32×D4C2×D311yescharacteristic
58C22×C3×Dic3—61no—
59C2×C3×Dic6—31no—
60C32×SD16—22no—
61C2×C32×C8C2×D311yescharacteristic
62C3×#217—61no—
63C3×#214—62no—
64C3×C4⋊Dic3—31no—
65C3×Dic3⋊C4—61no—
66C3×Dic3⋊C4—61no—
67C3×C4×Dic3—31no—
68C2×C3×C3⋊C8—31no—
69C2×Q8×C9—21no—
70C2×Q8×C9C22×C311yescharacteristic
71C2×D4×C9—21no—
72C2×D4×C9C22×C311yescharacteristic
73C22×Dic9—21no—
74C2×Dic18C22×C311yescharacteristic
75C9×SD16—24no—
76C9×SD16—22no—
77C2×C8×C9—21no—
78C2×C8×C9C22×C311yescharacteristic
79#3676—21no—
80#3672—22no—
81C4⋊Dic9C22×C311yescharacteristic
82Dic9⋊C4—21no—
83Dic9⋊C4—21no—
84C4×Dic9C22×C311yescharacteristic
85C2×C9⋊C8C22×C311yescharacteristic
86C22×3−1+2—22no—
87C22×3−1+2—41no—
88C22×3−1+2—41no—
89C22×3−1+2C2×D411yescharacteristic
90C4×3−1+2—41no—
91C4×3−1+2C2×D411yescharacteristic
92C4×3−1+2#4111yescharacteristic
933−1+2⋊C4—21no—
943−1+2⋊C4—41no—
953−1+2⋊C4—41no—
963−1+2⋊C4SD1612yescomplemented
97C2×C3×SD16—31no—
98C2×C3×SD16—91no—
99C2×C3×SD16C3×D311yescharacteristic
100C3×C4⋊Q8—91no—
101C3×C4×D4—91no—
102#944—31no—
103D4×Dic3—31no—
104C2×#214—31no—
105C3×C4⋊C8—91no—
106C3×#103—91no—
107C3×#103—91no—
108#835—31no—
109#816—31no—
110Dic3⋊C8—31no—
111C23×C32—21no—
112C32×Q8—21no—
113C32×Q8—21no—
114C32×D4—21no—
115C32×D4#6712yes—
116C2×C32×C4—21no—
117C2×C32×C4C22×D311yescharacteristic
118C2×C3×Dic3—31no—
119C2×C3×Dic3—31no—
120C2×C3×Dic3—61no—
121C2×C3×Dic3—61no—
122C2×C3×Dic3—122no—
123C3×Dic6—61no—
124C3×Dic6—61no—
125C23×C9—21no—
126C23×C9—41no—
127C32×C8—21no—
128C3×C3⋊C8—61no—
129Q8×C9—22no—
130Q8×C9—21no—
131Q8×C9—21no—
132Q8×C9—41no—
133D4×C9—21no—
134D4×C9—22no—
135D4×C9—41no—
136D4×C9C3×D412yes—
137C2×C4×C9—21no—
138C2×C4×C9—21no—
139C2×C4×C9—41no—
140C2×C4×C9C23×C311yescharacteristic
141C2×Dic9—21no—
142C2×Dic9—21no—
143C2×Dic9—42no—
144C2×Dic9C3×D411yescharacteristic
145C2×Dic9C3×D411yescharacteristic
146Dic18—21no—
147Dic18—21no—
148C8×C9—22no—
149C8×C9—21no—
150C9⋊C8—21no—
151#461—91noSylow
152C2×3−1+2—42no—
153C2×3−1+2#12111yescharacteristic
154C2×3−1+2C2×SD1611yescharacteristic
155C2×3−1+2#13711yescharacteristic
156C2×C3×Q8—31no—
157C2×C3×Q8—91no—
158C2×C3×Q8C2×C3×D311yescharacteristic
159C2×C3×D4—31no—
160C2×C3×D4C2×C3×D311yescharacteristic
161C22×C3×C4—181no—
162C22×Dic3—61no—
163C2×Dic6—31no—
164C3×SD16—22no—
165C3×SD16—62no—
166C3×SD16—182no—
167C2×C3×C8—31no—
168C2×C3×C8—91no—
169C2×C3×C8C2×C3×D311yescharacteristic
170C3×C4⋊C4—91no—
171C3×C4⋊C4—181no—
172C3×C4⋊C4—181no—
173C3×C22⋊C4—181no—
174C3×C42—91no—
175#217—61no—
176#214—62no—
177C4⋊Dic3—31no—
178Dic3⋊C4—61no—
179Dic3⋊C4—61no—
180C4×Dic3—31no—
181C2×C3⋊C8—31no—
182C22×C32—22no—
183C22×C32—41no—
184C22×C32—41no—
185C22×C32D3×D411yescharacteristic
186C32×C4—41no—
187C32×C4#23711yescharacteristic
188C32×C4C2×#6711yescharacteristic
189C3×Dic3—32no—
190C3×Dic3—61no—
191C3×Dic3—121no—
192C3×Dic3—121no—
193C22×C9—22no—
194C22×C9—21no—
195C22×C9—42no—
196C22×C9—41no—
197C22×C9—41no—
198C22×C9—81no—
199C22×C9—81no—
200C22×C9C2×C3×D411yescharacteristic
201C4×C9—21no—
202C4×C9—21no—
203C4×C9—42no—
204C4×C9—41no—
205C4×C9C2×C3×D411yescharacteristic
206C4×C9C3×#4111yescharacteristic
207Dic9—21no—
208Dic9—41no—
209Dic9—41no—
210Dic9C3×SD1612yescomplemented
211C2×SD16—91no—
212C2×SD16AGL(1,ℤ/9)11yescharacteristic
213C4⋊Q8—91no—
214C4×D4—91no—
215C4⋊C8—91no—
216#103—91no—
217#103—91no—
2183−1+2#46111yescharacteristic, complemented, Sylow
219C23×C3—21no—
220C23×C3—61no—
221C3×Q8—21no—
222C3×Q8—21no—
223C3×Q8—61no—
224C3×Q8—61no—
225C3×Q8—181no—
226C3×Q8—181no—
227C3×D4—21no—
228C3×D4—32no—
229C3×D4—61no—
230C3×D4C3×#6712yes—
231C2×C3×C4—21no—
232C2×C3×C4—31no—
233C2×C3×C4—61no—
234C2×C3×C4—91no—
235C2×C3×C4—91no—
236C2×C3×C4—181no—
237C2×C3×C4—181no—
238C2×C3×C4—362no—
239C2×C3×C4C22×C3×D311yescharacteristic
240C2×Dic3—31no—
241C2×Dic3—31no—
242C2×Dic3—61no—
243C2×Dic3—61no—
244C2×Dic3—122no—
245Dic6—61no—
246Dic6—61no—
247C3×C8—21no—
248C3×C8—61no—
249C3×C8—181no—
250C3⋊C8—61no—
251C2×C32—42no—
252C2×C32D3×SD1611yescharacteristic
253C2×C32#91511yescharacteristic
254C2×C32#93911yescharacteristic
255C2×C9—21no—
256C2×C9—21no—
257C2×C9—21no—
258C2×C9—42no—
259C2×C9—82no—
260C2×C9C3×#12111yescharacteristic
261C2×C9C2×C3×SD1611yescharacteristic
262C2×C9C3×#13711yescharacteristic
263C2×Q8—91no—
264C2×Q8C2×AGL(1,ℤ/9)11yescharacteristic
265C2×D4C2×AGL(1,ℤ/9)11yescharacteristic
266C22×C4—181no—
267SD16—22no—
268SD16—182no—
269C2×C8—91no—
270C2×C8C2×AGL(1,ℤ/9)11yescharacteristic
271C4⋊C4—91no—
272C4⋊C4—181no—
273C4⋊C4—181no—
274C22⋊C4—181no—
275C42—91no—
276C22×C3—22no—
277C22×C3—31no—
278C22×C3—41no—
279C22×C3—41no—
280C22×C3—62no—
281C22×C3—121no—
282C22×C3—121no—
283C22×C3C3×D3×D411yescharacteristic
284C3×C4—31no—
285C3×C4—31no—
286C3×C4—41no—
287C3×C4—92no—
288C3×C4—121no—
289C3×C4—181no—
290C3×C4—361no—
291C3×C4—361no—
292C3×C4C3×#23711yescharacteristic
293C3×C4C2×C3×#6711yescharacteristic
294Dic3—32no—
295Dic3—61no—
296Dic3—121no—
297Dic3—121no—
298C32#516211yescharacteristic
299C9—21no—
300C9C3×#46111yescharacteristic, complemented
301C23—21no—
302Q8—21no—
303Q8—21no—
304Q8—181no—
305Q8—181no—
306D4—21no—
307D4#611⋊C312yes—
308C2×C4—21no—
309C2×C4—91no—
310C2×C4—91no—
311C2×C4—181no—
312C2×C4—181no—
313C2×C4—362no—
314C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
315C8—21no—
316C8—181no—
317C2×C3—31no—
318C2×C3—31no—
319C2×C3—31no—
320C2×C3—42no—
321C2×C3—122no—
322C2×C3C3×D3×SD1611yescharacteristic
323C2×C3C3×#91511yescharacteristic
324C2×C3C3×#93911yescharacteristic
325C22—22no—
326C22—41no—
327C22—41no—
328C22D4×AGL(1,ℤ/9)11yescharacteristic
329C4—41no—
330C4—92no—
331C4—181no—
332C4—361no—
333C4—361no—
334C4#3699⋊C311yescharacteristic
335C4C2×#611⋊C311yescharacteristic
336C3—31no—
337C3C3×#516211yescharacteristic
338C2—42no—
339C2SD16×AGL(1,ℤ/9)11yescharacteristic
340C2#63477⋊C311yescharacteristic
341C2#63501⋊C311yescharacteristic
subgroupG/H
centerC22D4×AGL(1,ℤ/9)
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC2×SD16×3−1+2C2
socleC22×C3C3×D3×D4
prime radicalC2×C32×D4C2×D3
hypercenterC2×SD16AGL(1,ℤ/9)
hypocenterC9C3×#461
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×SD16

Complements

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

NH ≅ G/Ntype
2. #63506⋊C3C2semidirect
5. #63395⋊C3C2semidirect
6. #63376⋊C3C2semidirect
7. (3−1+2⋊C4)⋊C8C2semidirect
9. #10588842C3semidirect
32. #63506C2×C3semidirect
35. #63395C2×C3semidirect
36. #63376C2×C3semidirect
37. Dic9⋊C8C2×C3semidirect
96. 3−1+2⋊C4SD16semidirect
210. Dic9C3×SD16semidirect
218. 3−1+2#461semidirect
300. C9C3×#461semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×3−1+22
#63506⋊C32
D4×3−1+2⋊C42
C2×#3672⋊C32
#63395⋊C32
#63376⋊C32
(3−1+2⋊C4)⋊C82
C3×#51623
#105888423