Finite Group Databasebeta

← #13048769

Subgroups — #13048769

Order 1296.

348 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 62 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
1C9×#638C211yesmaximal, characteristic, complemented
2C3×#6233C211yesmaximal, characteristic, complemented
3C2×C3×C3⋊Dic9C211yesmaximal, characteristic, complemented
4#10619899C211yesmaximal, characteristic, complemented
5#10619898C211yesmaximal, characteristic, complemented
6#10619896C211yesmaximal, characteristic, complemented
7#10619895C211yesmaximal, characteristic, complemented
8#89046—31nomaximal
9#88663—31nomaximal
10#88662—31nomaximal
11C22×C32×C9C2211yescharacteristic
12C2×C9×C3⋊D3C2211yescharacteristic
13C2×C3×C3⋊D9C2211yescharacteristic
14C9×C3⋊Dic3C2211yescharacteristic
15C3×C3⋊Dic9C2211yescharacteristic
16C3×C3⋊Dic9C2211yescharacteristic
17C3×C3⋊Dic9C2211yescharacteristic
18C3×#638—31no—
19C3×#638D311yescharacteristic
20C2×C3×C3⋊Dic3—31no—
21#6296—31no—
22#6295—31no—
23#6294—31no—
24#6294—31no—
25#6233D311yescharacteristic, complemented
26C2×C3⋊Dic9D311yescharacteristic, complemented
27C9×#67—31no—
28C9×#67—31no—
29C9×#67—61no—
30C3×#611—31no—
31C3×#611—61no—
32C2×C3×Dic9—31no—
33C2×C3×Dic9—61no—
34#6195—31no—
35#6194—31no—
36D3×Dic9—31no—
37D3×Dic9—31no—
38(C3⋊D9)⋊C4—31no—
39Dic3×D9—31no—
40#6189—31no—
41#6189—31no—
42C2×C32×C9—21no—
43C2×C32×C9C2311yescharacteristic
44C9×C3⋊D3—21no—
45C3×C3⋊D9—21no—
46#3805—91no—
47#3804—91no—
48#3804—91no—
49#3699—91no—
50C22×C33C2×D311yescharacteristic
51C2×C3×C3⋊D3—31no—
52C2×C3×C3⋊D3C2×D311yescharacteristic
53C3×C3⋊Dic3—31no—
54C3×C3⋊Dic3—31no—
55C3×C3⋊Dic3—31no—
56C3×C3⋊Dic3C2×D311yescharacteristic
57C22×C3×C9—21no—
58C22×C3×C9—21no—
59C22×C3×C9—21no—
60C22×C3×C9—41no—
61C22×C3×C9C2×D311yescharacteristic
62C22×C3×C9C2×D311yescharacteristic
63C2×C3⋊D9C2×D311yescharacteristic
64C2×D3×C9—31no—
65C2×D3×C9—31no—
66C2×D3×C9—61no—
67C2×C3×D9—31no—
68C2×C3×D9—61no—
69C3⋊Dic9C2×D311yescharacteristic
70C3⋊Dic9C2×D311yescharacteristic
71C3⋊Dic9C2×D311yescharacteristic
72C9×Dic3—31no—
73C9×Dic3—31no—
74C9×Dic3—61no—
75C3×Dic9—31no—
76C3×Dic9—31no—
77C3×Dic9—31no—
78C3×Dic9—61no—
79C3×Dic9—61no—
80C3×Dic9—61no—
81C32×C9#4111yescharacteristic, complemented, Sylow
82#638—31no—
83#638D911yescharacteristic, complemented
84C2×C3⋊Dic3—31no—
85C3×#67—31no—
86C3×#67—31no—
87C3×#67—61no—
88C3×#67—91no—
89C3×#67—91no—
90C3×#67—181no—
91C2×C3×Dic3—91no—
92C2×C3×Dic3—91no—
93C2×C3×Dic3—181no—
94#627—91no—
95#627—91no—
96#627—91no—
97#626—91no—
98#626—91no—
99#625—91no—
100#624—91no—
101#624—91no—
102D3×Dic3—91no—
103D3×Dic3—91no—
104D3×Dic3—91no—
105D3×Dic3—91no—
106D4×C9—91no—
107#611—31no—
108#611—61no—
109#611—91no—
110C2×Dic9—31no—
111C2×Dic9—61no—
112C2×Dic9—91no—
113C4×D9—91no—
114Dic18—91no—
115C2×C33—21no—
116C2×C33C22×D311yescharacteristic
117C3×C3⋊D3—21no—
118C3×C3⋊D3—61no—
119C2×C3×C9—21no—
120C2×C3×C9—22no—
121C2×C3×C9—21no—
122C2×C3×C9—21no—
123C2×C3×C9—21no—
124C2×C3×C9—21no—
125C2×C3×C9—41no—
126C2×C3×C9—41no—
127C2×C3×C9—42no—
128C2×C3×C9—41no—
129C2×C3×C9C22×D311yescharacteristic
130C2×C3×C9C22×D311yescharacteristic
131C3⋊D9—21no—
132D3×C9—61no—
133D3×C9—61no—
134D3×C9—121no—
135C3×D9—61no—
136C3×D9—121no—
137#237—271no—
138#237—271no—
139#235—271no—
140C22×C32—21no—
141C22×C32—21no—
142C22×C32—21no—
143C22×C32—41no—
144C22×C32C2×D911yescharacteristic
145C22×C32D3211yescharacteristic
146C22×C32D3211yescharacteristic
147C2×C3⋊D3—31no—
148C2×C3⋊D3C2×D911yescharacteristic
149C2×C3×D3—31no—
150C2×C3×D3—31no—
151C2×C3×D3—61no—
152C2×C3×D3—91no—
153C2×C3×D3—91no—
154C2×C3×D3—181no—
155C3⋊Dic3—31no—
156C3⋊Dic3—31no—
157C3⋊Dic3—31no—
158C3⋊Dic3C2×D911yescharacteristic
159C3×Dic3—31no—
160C3×Dic3—31no—
161C3×Dic3—61no—
162C3×Dic3—91no—
163C3×Dic3—91no—
164C3×Dic3—91no—
165C3×Dic3—91no—
166C3×Dic3—91no—
167C3×Dic3—91no—
168C3×Dic3—181no—
169C3×Dic3—181no—
170C3×Dic3—181no—
171C22×C9—21no—
172C22×C9—21no—
173C22×C9—41no—
174C22×C9—91no—
175C22×C9D3211yescharacteristic
176C2×D9—31no—
177C2×D9—61no—
178C4×C9—91no—
179Dic9—31no—
180Dic9—31no—
181Dic9—31no—
182Dic9—61no—
183Dic9—61no—
184Dic9—61no—
185C33#23711yescharacteristic
186C3×C9—21no—
187C3×C9—21no—
188C3×C9—21no—
189C3×C9—41no—
190C3×C9#23511yescharacteristic, complemented
191C3×C9#23711yescharacteristic, complemented
192C3×D4—91no—
193C3×D4—271no—
194C2×C3×C4—271no—
195#67—31no—
196#67—31no—
197#67—61no—
198#67—91no—
199#67—91no—
200#67—181no—
201#67—271no—
202#67—271no—
203C2×Dic3—91no—
204C2×Dic3—91no—
205C2×Dic3—181no—
206C2×Dic3—271no—
207C2×Dic3—271no—
208D12—271no—
209C4×D3—271no—
210C4×D3—271no—
211C4×D3—271no—
212C4×D3—271no—
213Dic6—271no—
214Dic6—271no—
215Dic6—271no—
216C2×C32—21no—
217C2×C32—22no—
218C2×C32—21no—
219C2×C32—21no—
220C2×C32—21no—
221C2×C32—21no—
222C2×C32—21no—
223C2×C32—41no—
224C2×C32—41no—
225C2×C32—41no—
226C2×C32—42no—
227C2×C32C22×D911yescharacteristic
228C2×C32C2×D3211yescharacteristic
229C2×C32C2×D3211yescharacteristic
230C3⋊D3—21no—
231C3⋊D3—61no—
232C3×D3—61no—
233C3×D3—61no—
234C3×D3—121no—
235C3×D3—181no—
236C3×D3—181no—
237C3×D3—361no—
238C2×C9—22no—
239C2×C9—21no—
240C2×C9—21no—
241C2×C9—21no—
242C2×C9—41no—
243C2×C9—42no—
244C2×C9—41no—
245C2×C9—181no—
246C2×C9C2×D3211yescharacteristic
247D9—61no—
248D9—121no—
249#41—811noSylow
250C22×C3—21no—
251C22×C3—21no—
252C22×C3—21no—
253C22×C3—41no—
254C22×C3—91no—
255C22×C3—271no—
256C22×C3D3×D911yescharacteristic
257C22×C3D3×D911yescharacteristic
258C22×C3#111311yescharacteristic
259C2×D3—31no—
260C2×D3—31no—
261C2×D3—61no—
262C2×D3—91no—
263C2×D3—91no—
264C2×D3—181no—
265C3×C4—91no—
266C3×C4—271no—
267C3×C4—271no—
268C3×C4—271no—
269Dic3—31no—
270Dic3—31no—
271Dic3—61no—
272Dic3—91no—
273Dic3—91no—
274Dic3—91no—
275Dic3—91no—
276Dic3—91no—
277Dic3—91no—
278Dic3—181no—
279Dic3—181no—
280Dic3—181no—
281C32—21no—
282C32—21no—
283C32—21no—
284C32—41no—
285C32#369911yescharacteristic, complemented
286C32#380411yescharacteristic
287C32#380511yescharacteristic
288C9—21no—
289C9—21no—
290C9—41no—
291C9#380411yescharacteristic, complemented
292Q8—811no—
293D4—91no—
294D4—271no—
295D4—811no—
296C2×C4—271no—
297C2×C4—811no—
298C2×C4—811no—
299C2×C3—22no—
300C2×C3—21no—
301C2×C3—21no—
302C2×C3—21no—
303C2×C3—21no—
304C2×C3—21no—
305C2×C3—21no—
306C2×C3—41no—
307C2×C3—41no—
308C2×C3—42no—
309C2×C3—41no—
310C2×C3—181no—
311C2×C3—541no—
312C2×C3C2×D3×D911yescharacteristic
313C2×C3C2×D3×D911yescharacteristic
314C2×C3C2×#111311yescharacteristic
315D3—61no—
316D3—61no—
317D3—121no—
318D3—181no—
319D3—181no—
320D3—361no—
321C22—91no—
322C22—271no—
323C22#6643111yescharacteristic
324C4—91no—
325C4—271no—
326C4—271no—
327C4—271no—
328C3—21no—
329C3—21no—
330C3—21no—
331C3—41no—
332C3#8866211yescharacteristic, complemented
333C3#8866311yescharacteristic, complemented
334C3#8904611yescharacteristic
335C2—21no—
336C2—181no—
337C2—541no—
338C2C2×#6643111yescharacteristic
subgroupG/H
centerC2C2×#66431
commutator, cosocleC2×C32×C9C23
FrattiniC2×C3C2×#1113
FittingC22×C32×C9C22
socleC2×C33C22×D3
hypercenterC22#66431
hypocenterC32×C9#41
LCSG ▷ C2×C32×C9 ▷ C32×C9
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C9×#638C2semidirect
2. C3×#6233C2semidirect
3. C2×C3×C3⋊Dic9C2semidirect
4. #10619899C2semidirect
5. #10619898C2semidirect
6. #10619896C2semidirect
7. #10619895C2semidirect
25. #6233D3semidirect
26. C2×C3⋊Dic9D3semidirect
81. C32×C9#41semidirect
83. #638D9semidirect
190. C3×C9#235semidirect
191. C3×C9#237semidirect
285. C32#3699semidirect
291. C9#3804semidirect
332. C3#88662semidirect
333. C3#88663semidirect

Maximal subgroups

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

subgroupindex
C9×#6382
C3×#62332
C2×C3×C3⋊Dic92
#106198992
#106198982
#106198962
#106198952
#890463
#886633
#886623