Finite Group Databasebeta

← #10588415⋊C3 #13141639

Subgroups — #10588415⋊C3 #13141639

Order 1728.

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

normal subgroups only

This group has 481 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
2#63486⋊C3C211yesmaximal, characteristic, complemented
3C2×#3664⋊C3C211yesmaximal, characteristic, complemented
4C2×Dic36⋊C3C211yesmaximal, characteristic, complemented
53−1+2×C22⋊C8C211yesmaximal, characteristic, complemented
6#63380⋊C3C211yesmaximal, characteristic, complemented
7#63376⋊C3C211yesmaximal, characteristic, complemented
8C3×#4733—31nomaximal
9#10588415C311yesmaximal, characteristic, complemented
10C22×C4×3−1+2C2211yescharacteristic
11C2×#611⋊C3—21no—
12#3697⋊C3—22no—
13#3697⋊C3—22no—
14C2×C4⋊AGL(1,ℤ/9)C2211yescharacteristic
15C2×C4×AGL(1,ℤ/9)—21no—
16C2×3−1+2⋊Q8C2211yescharacteristic
17C2×3−1+2⋊Q8C2211yescharacteristic
18C2×C8×3−1+2C2211yescharacteristic
19C2×C8×3−1+2C2211yescharacteristic
20#3676⋊C3—21no—
21C4⋊(3−1+2⋊C4)C2211yescharacteristic
22(3−1+2⋊C4)⋊C4—21no—
23#3664⋊C3—22no—
24Dic36⋊C3—22no—
25C2×C3×#235—31no—
26C3×#924—31no—
27C2×C3×Dic12—31no—
28C2×C3×#204—31no—
29C2×#3697C2×C311yescharacteristic
30C32×C22⋊C8D311yescharacteristic
31C3×#821—31no—
32C3×#816—31no—
33#63486C2×C311yescharacteristic, complemented
34C2×#3664C2×C311yescharacteristic, complemented
35C2×Dic36C2×C311yescharacteristic, complemented
36C9×C22⋊C8—21no—
37C9×C22⋊C8C2×C311yescharacteristic, complemented
38#63380C2×C311yescharacteristic, complemented
39#63376C2×C311yescharacteristic, complemented
40C23×3−1+2D411yescharacteristic
41C22×AGL(1,ℤ/9)—21no—
42C2×C4×3−1+2—21no—
43C2×C4×3−1+2—21no—
44C2×C4×3−1+2D411yescharacteristic
45C2×C4×3−1+2C2311yescharacteristic
46#611⋊C3—24no—
47C2×3−1+2⋊C4—21no—
48C2×3−1+2⋊C4—21no—
49C2×3−1+2⋊C4—21no—
50C4⋊AGL(1,ℤ/9)—21no—
51C4⋊AGL(1,ℤ/9)D412yes—
52C4×AGL(1,ℤ/9)—24no—
533−1+2⋊Q8—21no—
543−1+2⋊Q8—21no—
553−1+2⋊Q8—21no—
563−1+2⋊Q8D412yes—
57C8×3−1+2—21no—
58C8×3−1+2—21no—
59C3×#452—91no—
60#4733—31no—
61C22×C32×C4C2×D311yescharacteristic
62C2×C3×#67—61no—
63C3×#235—62no—
64C3×#235—62no—
65C2×C3×D12—31no—
66C2×C3×C4×D3—61no—
67C2×C3×Dic6—31no—
68C2×C3×Dic6—31no—
69C2×C32×C8C2×D311yescharacteristic
70C2×C32×C8C2×D311yescharacteristic
71C3×#217—61no—
72C3×C4⋊Dic3—31no—
73C3×Dic3⋊C4—61no—
74C3×Dic12—62no—
75C3×#204—62no—
76C22×C4×C9—21no—
77C22×C4×C9C22×C311yescharacteristic
78C2×#611—21no—
79#3697—22no—
80#3697—22no—
81C2×D36C22×C311yescharacteristic
82C2×C4×D9—21no—
83C2×Dic18C22×C311yescharacteristic
84C2×Dic18C22×C311yescharacteristic
85C2×C8×C9—21no—
86C2×C8×C9—21no—
87C2×C8×C9C22×C311yescharacteristic
88C2×C8×C9C22×C311yescharacteristic
89#3676—21no—
90C4⋊Dic9C22×C311yescharacteristic
91Dic9⋊C4—21no—
92#3664—22no—
93Dic36—22no—
94C22×3−1+2—21no—
95C22×3−1+2—21no—
96C22×3−1+2—21no—
97C22×3−1+2C2×D411yescharacteristic
98C2×AGL(1,ℤ/9)—22no—
99C2×AGL(1,ℤ/9)—41no—
100C2×AGL(1,ℤ/9)—41no—
101C4×3−1+2—21no—
102C4×3−1+2C2×D411yescharacteristic
103C4×3−1+2C2×D411yescharacteristic
1043−1+2⋊C4—22no—
1053−1+2⋊C4—41no—
1063−1+2⋊C4—41no—
107C2×C3×#41—91no—
108C2×#235—31no—
109C2×C3×Q16—91no—
110C2×C3×SD16—91no—
111C3×C22⋊Q8—91no—
112#924—31no—
113C2×Dic12—31no—
114C2×#204—31no—
115C3×#103—91no—
116C3×#102—91no—
117C3×C22⋊C8—31no—
118C3×C22⋊C8C3×D311yescharacteristic
119#821—31no—
120#816—31no—
121C23×C32D1211yescharacteristic
122C22×C3×D3—61no—
123C2×C32×C4—21no—
124C2×C32×C4—21no—
125C2×C32×C4D1211yescharacteristic
126C2×C32×C4C22×D311yescharacteristic
127C3×#67—64no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3—61no—
130C2×C3×Dic3—61no—
131C3×D12—32no—
132C3×D12—61no—
133C3×C4×D3—64no—
134C3×Dic6—32no—
135C3×Dic6—61no—
136C3×Dic6—61no—
137C3×Dic6—61no—
138C23×C9—21no—
139C23×C9C3×D411yescharacteristic
140C22×D9—21no—
141C32×C8—21no—
142C32×C8—21no—
143C2×C4×C9—21no—
144C2×C4×C9—21no—
145C2×C4×C9—21no—
146C2×C4×C9—21no—
147C2×C4×C9—41no—
148C2×C4×C9—41no—
149C2×C4×C9C3×D411yescharacteristic
150C2×C4×C9C23×C311yescharacteristic
151#611—24no—
152C2×Dic9—21no—
153C2×Dic9—21no—
154C2×Dic9—21no—
155D36—21no—
156D36C3×D412yes—
157C4×D9—24no—
158Dic18—21no—
159Dic18—21no—
160Dic18—21no—
161Dic18C3×D412yes—
162C8×C9—21no—
163C8×C9—21no—
164C8×C9—41no—
165C8×C9—41no—
166#452—91noSylow
167C2×3−1+2—41no—
168C2×3−1+2#12011yescharacteristic
169C2×3−1+2#13511yescharacteristic
170C2×3−1+2#13711yescharacteristic
171AGL(1,ℤ/9)—42no—
172C3×#41—182no—
173C3×#41—182no—
174C2×C3×Q8—91no—
175C2×C3×Q8—91no—
176C2×C3×D4—91no—
177C2×C3×D4—181no—
178C22×C3×C4—31no—
179C22×C3×C4—181no—
180C22×C3×C4C2×C3×D311yescharacteristic
181C2×#67—61no—
182#235—62no—
183#235—62no—
184C2×D12—31no—
185C2×C4×D3—61no—
186C2×Dic6—31no—
187C2×Dic6—31no—
188C3×Q16—182no—
189C3×SD16—182no—
190C2×C3×C8—31no—
191C2×C3×C8—31no—
192C2×C3×C8C2×C3×D311yescharacteristic
193C2×C3×C8C2×C3×D311yescharacteristic
194C3×C4⋊C4—91no—
195C3×C4⋊C4—181no—
196C3×C22⋊C4—181no—
197#217—61no—
198C4⋊Dic3—31no—
199Dic3⋊C4—61no—
200Dic12—62no—
201#204—62no—
202C22×C32—21no—
203C22×C32—21no—
204C22×C32—21no—
205C22×C32C2×D1211yescharacteristic
206C2×C3×D3—62no—
207C2×C3×D3—121no—
208C2×C3×D3—121no—
209C32×C4—21no—
210C32×C4D3×D411yescharacteristic
211C32×C4D3×D411yescharacteristic
212C3×Dic3—62no—
213C3×Dic3—121no—
214C3×Dic3—121no—
215C22×C9—22no—
216C22×C9—21no—
217C22×C9—21no—
218C22×C9—21no—
219C22×C9—21no—
220C22×C9—41no—
221C22×C9—41no—
222C22×C9C2×C3×D411yescharacteristic
223C2×D9—22no—
224C2×D9—41no—
225C2×D9—41no—
226C4×C9—21no—
227C4×C9—21no—
228C4×C9—21no—
229C4×C9—41no—
230C4×C9C2×C3×D411yescharacteristic
231C4×C9C2×C3×D411yescharacteristic
232Dic9—22no—
233Dic9—41no—
234Dic9—41no—
235C2×#41—91no—
236C2×Q16—91no—
237C2×SD16—91no—
238C22⋊Q8—91no—
239#103—91no—
240#102—91no—
241C22⋊C8AGL(1,ℤ/9)11yescharacteristic, complemented
2423−1+2#45211yescharacteristic, complemented, Sylow
243C23×C3—31no—
244C23×C3—181no—
245C23×C3C3×D1211yescharacteristic
246C22×D3—61no—
247C3×Q8—92no—
248C3×Q8—181no—
249C3×Q8—181no—
250C3×Q8—181no—
251C3×D4—92no—
252C3×D4—181no—
253C3×D4—184no—
254C2×C3×C4—21no—
255C2×C3×C4—21no—
256C2×C3×C4—31no—
257C2×C3×C4—31no—
258C2×C3×C4—61no—
259C2×C3×C4—61no—
260C2×C3×C4—184no—
261C2×C3×C4—181no—
262C2×C3×C4—181no—
263C2×C3×C4—181no—
264C2×C3×C4C3×D1211yescharacteristic
265C2×C3×C4C22×C3×D311yescharacteristic
266#67—64no—
267C2×Dic3—61no—
268C2×Dic3—61no—
269C2×Dic3—61no—
270D12—32no—
271D12—61no—
272C4×D3—64no—
273Dic6—32no—
274Dic6—61no—
275Dic6—61no—
276Dic6—61no—
277C3×C8—21no—
278C3×C8—21no—
279C3×C8—61no—
280C3×C8—61no—
281C2×C32—41no—
282C2×C32#88211yescharacteristic
283C2×C32#90411yescharacteristic
284C2×C32#90911yescharacteristic
285C3×D3—122no—
286C2×C9—21no—
287C2×C9—21no—
288C2×C9—21no—
289C2×C9—42no—
290C2×C9—41no—
291C2×C9C3×#12011yescharacteristic
292C2×C9C3×#13511yescharacteristic
293C2×C9C3×#13711yescharacteristic
294D9—42no—
295#41—182no—
296#41—182no—
297C2×Q8—91no—
298C2×Q8—91no—
299C2×D4—91no—
300C2×D4—181no—
301C22×C4—181no—
302C22×C4C2×AGL(1,ℤ/9)11yescharacteristic
303Q16—182no—
304SD16—182no—
305C2×C8C2×AGL(1,ℤ/9)11yescharacteristic
306C2×C8C2×AGL(1,ℤ/9)11yescharacteristic
307C4⋊C4—91no—
308C4⋊C4—181no—
309C22⋊C4—181no—
310C22×C3—21no—
311C22×C3—21no—
312C22×C3—21no—
313C22×C3—31no—
314C22×C3—61no—
315C22×C3—61no—
316C22×C3—61no—
317C22×C3—182no—
318C22×C3—361no—
319C22×C3—361no—
320C22×C3C2×C3×D1211yescharacteristic
321C2×D3—62no—
322C2×D3—121no—
323C2×D3—121no—
324C3×C4—21no—
325C3×C4—31no—
326C3×C4—31no—
327C3×C4—61no—
328C3×C4—182no—
329C3×C4—361no—
330C3×C4—361no—
331C3×C4C3×D3×D411yescharacteristic
332C3×C4C3×D3×D411yescharacteristic
333Dic3—62no—
334Dic3—121no—
335Dic3—121no—
336C32#473311yescharacteristic
337C9—21no—
338C9C3×#45211yescharacteristic, complemented
339C23—181no—
340C23C4⋊AGL(1,ℤ/9)11yescharacteristic
341Q8—92no—
342Q8—181no—
343Q8—181no—
344Q8—181no—
345D4—92no—
346D4—184no—
347D4—181no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—184no—
351C2×C4—181no—
352C2×C4—181no—
353C2×C4—181no—
354C2×C4C4⋊AGL(1,ℤ/9)11yescharacteristic
355C2×C4C22×AGL(1,ℤ/9)11yescharacteristic
356C8—21no—
357C8—21no—
358C2×C3—31no—
359C2×C3—31no—
360C2×C3—31no—
361C2×C3—41no—
362C2×C3—121no—
363C2×C3—362no—
364C2×C3C3×#88211yescharacteristic
365C2×C3C3×#90411yescharacteristic
366C2×C3C3×#90911yescharacteristic
367D3—122no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—182no—
372C22—361no—
373C22—361no—
374C22C2×C4⋊AGL(1,ℤ/9)11yescharacteristic
375C4—21no—
376C4—182no—
377C4—361no—
378C4—361no—
379C4D4×AGL(1,ℤ/9)11yescharacteristic
380C4D4×AGL(1,ℤ/9)11yescharacteristic
381C3—31no—
382C3C3×#473311yescharacteristic
383C2—41no—
384C2—362no—
385C2#63444⋊C311yescharacteristic
386C2#63467⋊C311yescharacteristic
387C2#63471⋊C311yescharacteristic
subgroupG/H
centerC22C2×C4⋊AGL(1,ℤ/9)
commutator, cosocleC2×C4×C9C23×C3
FrattiniC2×C3×C4C22×C3×D3
Fitting3−1+2×C22⋊C8C2
socleC22×C3C2×C3×D12
prime radicalC2×#3697⋊C3C2
hypercenterC22⋊C8AGL(1,ℤ/9)
hypocenterC9C3×#452
LCSG ▷ C2×C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊C8

Complements

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

NH ≅ G/Ntype
2. #63486⋊C3C2semidirect
3. C2×#3664⋊C3C2semidirect
4. C2×Dic36⋊C3C2semidirect
5. 3−1+2×C22⋊C8C2semidirect
6. #63380⋊C3C2semidirect
7. #63376⋊C3C2semidirect
9. #10588415C3semidirect
33. #63486C2×C3semidirect
34. C2×#3664C2×C3semidirect
35. C2×Dic36C2×C3semidirect
37. C9×C22⋊C8C2×C3semidirect
38. #63380C2×C3semidirect
39. #63376C2×C3semidirect
241. C22⋊C8AGL(1,ℤ/9)semidirect
242. 3−1+2#452semidirect
338. 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
#63486⋊C32
C2×#3664⋊C32
C2×Dic36⋊C32
3−1+2×C22⋊C82
#63380⋊C32
#63376⋊C32
C3×#47333
#105884153