Finite Group Databasebeta

← #13049521

Subgroups — #13049521

Order 1296.

836 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 836 conjugacy classes of subgroups — too many to draw. Only its 102 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#10620157C211yesmaximal, characteristic, complemented
2#10620151C211yesmaximal, characteristic, complemented
3#10620148C211yesmaximal, characteristic, complemented
4Dic3×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
5C3×#6292C211yesmaximal, characteristic, complemented
6C3×Dic3×C3⋊D3C211yesmaximal, characteristic, complemented
7C32×#624C211yesmaximal, characteristic, complemented
8#89021—31nomaximal
9#89020—31nomaximal
10#88960—34nomaximal
11C2×C3×C3⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×C3⋊D3C2211yescharacteristic
13C2×C32×C3⋊D3C2211yescharacteristic
14C3⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C3×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C33×Dic3C2211yescharacteristic
17C33×Dic3C2211yescharacteristic
18C4×C3⋊(C3⋊D3)—31no—
19#6308—31no—
20C3×C4⋊(C3⋊D3)—31no—
21C3×C4×C3⋊D3—31no—
22C32×C4×D3—31no—
23C32×C4×D3—31no—
24#6295—34no—
25#6293—31no—
26#6293—34no—
27#6293—64no—
28#6292D311yescharacteristic, complemented
29#6291—31no—
30#6290—31no—
31#6290—31no—
32#6290—34no—
33#6290—31no—
34#6290—61no—
35Dic3×C3⋊D3—31no—
36Dic3×C3⋊D3—34no—
37Dic3×C3⋊D3—64no—
38Dic3×C3⋊D3D311yescharacteristic, complemented
39C3×#626—34no—
40C3×#624D314yescomplemented
41C3×D3×Dic3—34no—
42C2×C34C2311yescharacteristic
43C3×C3⋊(C3⋊D3)—21no—
44C32×C3⋊D3—21no—
45C32×C3⋊D3—21no—
46#3828—91no—
47#3804—91no—
48#3798—94no—
49#3795—94no—
50C2×C3⋊(C3⋊D3)C2×D311yescharacteristic
51C2×C3×C3⋊D3—21no—
52C2×C3×C3⋊D3—31no—
53C2×C3×C3⋊D3—34no—
54C2×C3×C3⋊D3—64no—
55C2×C3×C3⋊D3C2×D311yescharacteristic
56C2×C3×C3⋊D3C2×D311yescharacteristic
57C2×C3×C3⋊D3C2×D314yes—
58C2×C32×D3—31no—
59C2×C32×D3—31no—
60C2×C32×D3—34no—
61C2×C32×D3—61no—
62C33×C4—31no—
63C33×C4—31no—
64C3⋊(C3⋊Dic3)—31no—
65C3⋊(C3⋊Dic3)—31no—
66C3⋊(C3⋊Dic3)—34no—
67C3⋊(C3⋊Dic3)—64no—
68C3⋊(C3⋊Dic3)—61no—
69C3⋊(C3⋊Dic3)—64no—
70C3⋊(C3⋊Dic3)—124no—
71C3⋊(C3⋊Dic3)C2×D311yescharacteristic
72C3×C3⋊Dic3—31no—
73C3×C3⋊Dic3—34no—
74C3×C3⋊Dic3—64no—
75C32×Dic3—24no—
76C32×Dic3—24no—
77C32×Dic3C2×D314yes—
78C32×Dic3C2×D314yes—
79C32×Dic3C2×D311yescharacteristic
80C32×Dic3C2×D311yescharacteristic
81C34#4111yescharacteristic, complemented, Sylow
82C2×C32×C4—91no—
83#638—91no—
84#638—91no—
85#638—91no—
86C4⋊(C3⋊D3)—31no—
87C4×C3⋊D3—31no—
88C4×C3⋊D3—91no—
89C4×C3⋊D3—94no—
90C4×C3⋊D3—184no—
91#634—91no—
92#634—94no—
93#634—184no—
94C3×#67—91no—
95C2×C3×Dic3—91no—
96C3×D12—94no—
97C3×C4×D3—34no—
98C3×C4×D3—34no—
99C3×C4×D3—94no—
100#627—94no—
101#627—91no—
102#627—184no—
103#626—34no—
104#626—94no—
105#626—91no—
106#625—94no—
107#625—94no—
108#625—94no—
109#625—184no—
110#624C3⋊D311yescharacteristic, complemented
111D3×Dic3—34no—
112D3×Dic3—94no—
113D3×Dic3—91no—
114D3×Dic3—184no—
115C2×C33—24no—
116C2×C33—24no—
117C2×C33—21no—
118C2×C33—44no—
119C2×C33C22×D311yescharacteristic
120C2×C33C22×D311yescharacteristic
121C2×C33C22×D314yes—
122C3⋊(C3⋊D3)—21no—
123C3×C3⋊D3—21no—
124C3×C3⋊D3—21no—
125C3×C3⋊D3—24no—
126C3×C3⋊D3—41no—
127C3×C3⋊D3—68no—
128C3×C3⋊D3—64no—
129C3×C3⋊D3—61no—
130C32×D3—61no—
131C32×D3—61no—
132C32×D3—64no—
133C32×D3—121no—
134#237—271no—
135#235—271no—
136#235—274no—
137C22×C32—91no—
138C22×C32—91no—
139C2×C3⋊D3—31no—
140C2×C3⋊D3—34no—
141C2×C3⋊D3—64no—
142C2×C3⋊D3D3211yescharacteristic
143C2×C3⋊D3C2×C3⋊D311yescharacteristic
144C2×C3×D3—34no—
145C2×C3×D3—34no—
146C2×C3×D3—34no—
147C2×C3×D3—34no—
148C2×C3×D3—64no—
149C2×C3×D3—64no—
150C2×C3×D3—94no—
151C2×C3×D3—91no—
152C2×C3×D3—184no—
153C32×C4—31no—
154C32×C4—31no—
155C32×C4—34no—
156C32×C4—34no—
157C32×C4—64no—
158C32×C4—64no—
159C3⋊Dic3—31no—
160C3⋊Dic3—34no—
161C3⋊Dic3—64no—
162C3⋊Dic3—91no—
163C3⋊Dic3—94no—
164C3⋊Dic3—94no—
165C3⋊Dic3—91no—
166C3⋊Dic3—184no—
167C3⋊Dic3—184no—
168C3⋊Dic3—184no—
169C3⋊Dic3—184no—
170C3⋊Dic3—184no—
171C3⋊Dic3—364no—
172C3⋊Dic3—364no—
173C3⋊Dic3—3612no—
174C3×Dic3—24no—
175C3×Dic3—24no—
176C3×Dic3—94no—
177C3×Dic3—91no—
178C3×Dic3—184no—
179C3×Dic3D3214yes—
180C3×Dic3D3214yes—
181C3×Dic3C2×C3⋊D311yescharacteristic
182C3×Dic3C2×C3⋊D311yescharacteristic
183C33—24no—
184C33—24no—
185C33—21no—
186C33—44no—
187C33#23511yescharacteristic, complemented
188C33#23514yescomplemented
189C33#23711yescharacteristic, complemented
190C3×D4—271no—
191C2×C3×C4—94no—
192C2×C3×C4—271no—
193#67—91no—
194#67—271no—
195#67—274no—
196#67—271no—
197#67—271no—
198#67—274no—
199#67—541no—
200C2×Dic3—91no—
201C2×Dic3—271no—
202D12—94no—
203D12—271no—
204C4×D3—31no—
205C4×D3—31no—
206C4×D3—94no—
207C4×D3—271no—
208C4×D3—274no—
209C4×D3—544no—
210Dic6—271no—
211Dic6—274no—
212Dic6—271no—
213Dic6—544no—
214C2×C32—24no—
215C2×C32—24no—
216C2×C32—24no—
217C2×C32—24no—
218C2×C32—24no—
219C2×C32—44no—
220C2×C32—412no—
221C2×C32—44no—
222C2×C32—181no—
223C2×C32—181no—
224C2×C32C2×D3214yes—
225C2×C32C2×D3214yes—
226C2×C32C2×D3211yescharacteristic
227C2×C32C22×C3⋊D311yescharacteristic
228C3⋊D3—21no—
229C3⋊D3—21no—
230C3⋊D3—68no—
231C3⋊D3—64no—
232C3⋊D3—61no—
233C3×D3—64no—
234C3×D3—64no—
235C3×D3—64no—
236C3×D3—64no—
237C3×D3—124no—
238C3×D3—124no—
239C3×D3—188no—
240C3×D3—184no—
241C3×D3—181no—
242#41—811noSylow
243C22×C3—94no—
244C22×C3—91no—
245C22×C3—91no—
246C22×C3—181no—
247C22×C3—271no—
248C2×D3—34no—
249C2×D3—31no—
250C2×D3—31no—
251C2×D3—61no—
252C2×D3—94no—
253C2×D3—91no—
254C2×D3—184no—
255C3×C4—34no—
256C3×C4—34no—
257C3×C4—31no—
258C3×C4—31no—
259C3×C4—64no—
260C3×C4—64no—
261C3×C4—271no—
262Dic3—94no—
263Dic3—91no—
264Dic3—184no—
265Dic3—274no—
266Dic3—271no—
267Dic3—271no—
268Dic3—544no—
269Dic3—544no—
270Dic3—541no—
271Dic3—1084no—
272Dic3D3×C3⋊D311yescharacteristic
273Dic3D3×C3⋊D311yescharacteristic
274C32—24no—
275C32—24no—
276C32—24no—
277C32—24no—
278C32—24no—
279C32—44no—
280C32—412no—
281C32—44no—
282C32#379514yescomplemented
283C32#379814yescomplemented
284C32#380411yescharacteristic, complemented
285C32#382811yescharacteristic, complemented
286Q8—811no—
287D4—271no—
288D4—811no—
289D4—811no—
290C2×C4—91no—
291C2×C4—271no—
292C2×C4—811no—
293C2×C3—24no—
294C2×C3—24no—
295C2×C3—21no—
296C2×C3—44no—
297C2×C3—184no—
298C2×C3—181no—
299C2×C3—181no—
300C2×C3—361no—
301C2×C3—541no—
302C2×C3D3314yes—
303C2×C3C2×D3×C3⋊D311yescharacteristic
304C2×C3C2×D3×C3⋊D311yescharacteristic
305D3—64no—
306D3—61no—
307D3—61no—
308D3—121no—
309D3—188no—
310D3—184no—
311D3—181no—
312C22—91no—
313C22—91no—
314C22—271no—
315C4—31no—
316C4—31no—
317C4—271no—
318C4—811no—
319C3—21no—
320C3—24no—
321C3—24no—
322C3—44no—
323C3#8896014yescomplemented
324C3#8902011yescharacteristic, complemented
325C3#8902111yescharacteristic, complemented
326C2—181no—
327C2—181no—
328C2—541no—
329C2D32×C3⋊D311yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2D32×C3⋊D3
commutator, Fitting, socle, cosocleC2×C34C23
hypocenterC34#41
LCSG ▷ C2×C34 ▷ C34
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. #10620157C2semidirect
2. #10620151C2semidirect
3. #10620148C2semidirect
4. Dic3×C3⋊(C3⋊D3)C2semidirect
5. C3×#6292C2semidirect
6. C3×Dic3×C3⋊D3C2semidirect
7. C32×#624C2semidirect
28. #6292D3semidirect
38. Dic3×C3⋊D3D3semidirect
40. C3×#624D3semidirect
81. C34#41semidirect
110. #624C3⋊D3semidirect
187. C33#235semidirect
188. C33#235semidirect
189. C33#237semidirect
282. C32#3795semidirect
283. C32#3798semidirect
284. C32#3804semidirect
285. C32#3828semidirect
323. C3#88960semidirect
324. C3#89020semidirect
325. C3#89021semidirect

Maximal subgroups

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

subgroupindex
#106201572
#106201512
#106201482
Dic3×C3⋊(C3⋊D3)2
C3×#62922
C3×Dic3×C3⋊D32
C32×#6242
#890213
#890203
#889603