Finite Group Databasebeta

← #13049520

Subgroups — #13049520

Order 1296.

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

normal subgroups only

This group has 868 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
2C3⋊D3×C3⋊Dic3C211yesmaximal, characteristic, complemented
3Dic3×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
4D3×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
5C3×#6293C211yesmaximal, characteristic, complemented
6C3×#6291C211yesmaximal, characteristic, complemented
7C32×#626C211yesmaximal, characteristic, complemented
8#89029—31nomaximal
9#89015—31nomaximal
10#88964—34nomaximal
11C2×C3×C3⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×C3⋊D3C2211yescharacteristic
13C2×C33×D3C2211yescharacteristic
14C3⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C3×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C32×C3⋊Dic3C2211yescharacteristic
17C33×Dic3C2211yescharacteristic
18C2×C3⋊(C3⋊Dic3)—31no—
19C4×C3⋊(C3⋊D3)—31no—
20C3×#638—31no—
21C3×#634—31no—
22C32×#67—31no—
23C32×D12—31no—
24#6295—34no—
25#6293D311yescharacteristic, complemented
26#6291—31no—
27#6291D311yescharacteristic, complemented
28#6290—31no—
29Dic3×C3⋊D3—34no—
30Dic3×C3⋊D3—31no—
31Dic3×C3⋊D3—34no—
32Dic3×C3⋊D3—64no—
33D3×C3⋊Dic3—31no—
34D3×C3⋊Dic3—34no—
35D3×C3⋊Dic3—31no—
36D3×C3⋊Dic3—31no—
37D3×C3⋊Dic3—61no—
38D3×C3⋊Dic3—64no—
39C3×#627—34no—
40C3×#626—34no—
41C3×#626D314yescomplemented
42C2×C34C2311yescharacteristic
43C3×C3⋊(C3⋊D3)—21no—
44C32×C3⋊D3—21no—
45C33×D3—21no—
46#3830—91no—
47#3805—94no—
48#3796—94no—
49#3796—91no—
50C22×C33—31no—
51C2×C3⋊(C3⋊D3)C2×D311yescharacteristic
52C2×C3×C3⋊D3—31no—
53C2×C3×C3⋊D3—34no—
54C2×C3×C3⋊D3—64no—
55C2×C3×C3⋊D3C2×D314yes—
56C2×C32×D3—24no—
57C2×C32×D3—31no—
58C2×C32×D3—31no—
59C2×C32×D3—61no—
60C2×C32×D3C2×D314yes—
61C2×C32×D3C2×D311yescharacteristic
62C33×C4—31no—
63C3⋊(C3⋊Dic3)—31no—
64C3⋊(C3⋊Dic3)—31no—
65C3⋊(C3⋊Dic3)—34no—
66C3⋊(C3⋊Dic3)—61no—
67C3⋊(C3⋊Dic3)—64no—
68C3⋊(C3⋊Dic3)—64no—
69C3⋊(C3⋊Dic3)—124no—
70C3⋊(C3⋊Dic3)C2×D311yescharacteristic
71C3×C3⋊Dic3—21no—
72C3×C3⋊Dic3—31no—
73C3×C3⋊Dic3—34no—
74C3×C3⋊Dic3—64no—
75C3×C3⋊Dic3C2×D311yescharacteristic
76C3×C3⋊Dic3C2×D311yescharacteristic
77C32×Dic3—24no—
78C32×Dic3—34no—
79C32×Dic3C2×D314yes—
80C32×Dic3C2×D311yescharacteristic
81C34#4111yescharacteristic, complemented, Sylow
82C32×D4—91no—
83#638—31no—
84#638—91no—
85C2×C3⋊Dic3—91no—
86C2×C3⋊Dic3—91no—
87C2×C3⋊Dic3—94no—
88C2×C3⋊Dic3—184no—
89C4×C3⋊D3—91no—
90C4×C3⋊D3—94no—
91C4×C3⋊D3—91no—
92C4×C3⋊D3—184no—
93#634—31no—
94C3×#67—34no—
95C3×#67—94no—
96C3×D12—34no—
97C3×D12—91no—
98C3×Dic6—94no—
99C3×Dic6—91no—
100#627—34no—
101#626—34no—
102#626—94no—
103#626—91no—
104#626C3⋊D311yescharacteristic, complemented
105#625—94no—
106D3×Dic3—94no—
107D3×Dic3—94no—
108D3×Dic3—94no—
109D3×Dic3—91no—
110D3×Dic3—94no—
111D3×Dic3—91no—
112D3×Dic3—184no—
113D3×Dic3—184no—
114D3×Dic3—184no—
115C2×C33—24no—
116C2×C33—21no—
117C2×C33—24no—
118C2×C33—44no—
119C2×C33—61no—
120C2×C33C22×D311yescharacteristic
121C2×C33C22×D311yescharacteristic
122C2×C33C22×D314yes—
123C3⋊(C3⋊D3)—21no—
124C3×C3⋊D3—24no—
125C3×C3⋊D3—68no—
126C3×C3⋊D3—64no—
127C3×C3⋊D3—61no—
128C32×D3—28no—
129C32×D3—24no—
130C32×D3—21no—
131C32×D3—62no—
132C32×D3—61no—
133C32×D3—61no—
134#239—271no—
135#237—271no—
136#237—274no—
137C22×C32—31no—
138C22×C32—34no—
139C22×C32—64no—
140C22×C32—91no—
141C2×C3⋊D3—31no—
142C2×C3⋊D3—34no—
143C2×C3⋊D3—64no—
144C2×C3⋊D3C2×C3⋊D311yescharacteristic
145C2×C3×D3—24no—
146C2×C3×D3—34no—
147C2×C3×D3—34no—
148C2×C3×D3—64no—
149C2×C3×D3—94no—
150C2×C3×D3—91no—
151C2×C3×D3—184no—
152C2×C3×D3D3214yes—
153C2×C3×D3C2×C3⋊D311yescharacteristic
154C32×C4—31no—
155C32×C4—34no—
156C32×C4—64no—
157C32×C4—91no—
158C3⋊Dic3—31no—
159C3⋊Dic3—34no—
160C3⋊Dic3—64no—
161C3⋊Dic3—91no—
162C3⋊Dic3—94no—
163C3⋊Dic3—94no—
164C3⋊Dic3—91no—
165C3⋊Dic3—184no—
166C3⋊Dic3—184no—
167C3⋊Dic3—184no—
168C3⋊Dic3—184no—
169C3⋊Dic3—184no—
170C3⋊Dic3—364no—
171C3⋊Dic3—364no—
172C3⋊Dic3—3612no—
173C3⋊Dic3D3211yescharacteristic
174C3×Dic3—24no—
175C3×Dic3—34no—
176C3×Dic3—34no—
177C3×Dic3—64no—
178C3×Dic3—94no—
179C3×Dic3—91no—
180C3×Dic3—184no—
181C3×Dic3D3214yes—
182C3×Dic3C2×C3⋊D311yescharacteristic
183C33—21no—
184C33—24no—
185C33—24no—
186C33—44no—
187C33#23711yescharacteristic, complemented
188C33#23714yescomplemented
189C33#23911yescharacteristic, complemented
190C3×Q8—271no—
191C3×D4—94no—
192C3×D4—271no—
193#67—31no—
194#67—94no—
195#67—271no—
196#67—274no—
197C2×Dic3—274no—
198C2×Dic3—271no—
199C2×Dic3—274no—
200C2×Dic3—271no—
201C2×Dic3—544no—
202D12—31no—
203D12—91no—
204D12—271no—
205C4×D3—271no—
206C4×D3—274no—
207C4×D3—271no—
208C4×D3—271no—
209C4×D3—271no—
210C4×D3—544no—
211C4×D3—541no—
212Dic6—94no—
213Dic6—91no—
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—68no—
223C2×C32—64no—
224C2×C32—61no—
225C2×C32—181no—
226C2×C32C2×D3214yes—
227C2×C32C2×D3214yes—
228C2×C32C2×D3211yescharacteristic
229C2×C32C22×C3⋊D311yescharacteristic
230C3⋊D3—21no—
231C3⋊D3—68no—
232C3⋊D3—61no—
233C3⋊D3—64no—
234C3×D3—28no—
235C3×D3—24no—
236C3×D3—21no—
237C3×D3—68no—
238C3×D3—64no—
239C3×D3—64no—
240C3×D3—188no—
241C3×D3—184no—
242C3×D3—181no—
243#41—811noSylow
244C22×C3—34no—
245C22×C3—31no—
246C22×C3—64no—
247C22×C3—94no—
248C22×C3—271no—
249C2×D3—31no—
250C2×D3—31no—
251C2×D3—61no—
252C2×D3—94no—
253C2×D3—91no—
254C2×D3—184no—
255C2×D3D3×C3⋊D311yescharacteristic
256C3×C4—34no—
257C3×C4—31no—
258C3×C4—64no—
259C3×C4—91no—
260C3×C4—91no—
261C3×C4—181no—
262C3×C4—271no—
263Dic3—34no—
264Dic3—94no—
265Dic3—91no—
266Dic3—184no—
267Dic3—274no—
268Dic3—271no—
269Dic3—271no—
270Dic3—544no—
271Dic3—541no—
272Dic3—544no—
273Dic3—1084no—
274Dic3D3×C3⋊D311yescharacteristic
275C32—24no—
276C32—24no—
277C32—24no—
278C32—24no—
279C32—24no—
280C32—44no—
281C32—412no—
282C32—44no—
283C32#379614yescomplemented
284C32#379611yescharacteristic, complemented
285C32#380514yescomplemented
286C32#383011yescharacteristic, complemented
287Q8—271no—
288D4—91no—
289D4—271no—
290D4—811no—
291C2×C4—811no—
292C2×C4—811no—
293C2×C4—811no—
294C2×C3—24no—
295C2×C3—24no—
296C2×C3—21no—
297C2×C3—44no—
298C2×C3—68no—
299C2×C3—64no—
300C2×C3—61no—
301C2×C3—184no—
302C2×C3—541no—
303C2×C3D3314yes—
304C2×C3C2×D3×C3⋊D311yescharacteristic
305C2×C3C2×D3×C3⋊D311yescharacteristic
306D3—21no—
307D3—62no—
308D3—61no—
309D3—61no—
310D3—188no—
311D3—184no—
312D3—181no—
313C22—31no—
314C22—91no—
315C22—271no—
316C4—31no—
317C4—91no—
318C4—271no—
319C4—811no—
320C3—21no—
321C3—24no—
322C3—24no—
323C3—44no—
324C3#8896414yescomplemented
325C3#8901511yescharacteristic, complemented
326C3#8902911yescharacteristic, complemented
327C2—61no—
328C2—181no—
329C2—541no—
330C2D32×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. C3⋊D3×C3⋊Dic3C2semidirect
3. Dic3×C3⋊(C3⋊D3)C2semidirect
4. D3×C3⋊(C3⋊Dic3)C2semidirect
5. C3×#6293C2semidirect
6. C3×#6291C2semidirect
7. C32×#626C2semidirect
25. #6293D3semidirect
27. #6291D3semidirect
41. C3×#626D3semidirect
81. C34#41semidirect
104. #626C3⋊D3semidirect
187. C33#237semidirect
188. C33#237semidirect
189. C33#239semidirect
283. C32#3796semidirect
284. C32#3796semidirect
285. C32#3805semidirect
286. C32#3830semidirect
324. C3#88964semidirect
325. C3#89015semidirect
326. C3#89029semidirect

Maximal subgroups

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

subgroupindex
#106201572
C3⋊D3×C3⋊Dic32
Dic3×C3⋊(C3⋊D3)2
D3×C3⋊(C3⋊Dic3)2
C3×#62932
C3×#62912
C32×#6262
#890293
#890153
#889643