Finite Group Databasebeta

← #13049512

Subgroups — #13049512

Order 1296.

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

normal subgroups only

This group has 856 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#10620154C211yesmaximal, characteristic, complemented
2#10620153C211yesmaximal, characteristic, complemented
3#10620145C211yesmaximal, characteristic, complemented
4D3×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
5C3×#6291C211yesmaximal, characteristic, complemented
6C3×D3×C3⋊Dic3C211yesmaximal, characteristic, complemented
7C32×#625C211yesmaximal, characteristic, complemented
8#89029—31nomaximal
9#89028—31nomaximal
10#88961—34nomaximal
11C2×C3×C3⋊(C3⋊D3)C2211yescharacteristic
12C2×C33×D3C2211yescharacteristic
13C2×C33×D3C2211yescharacteristic
14C3⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C3×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C32×C3⋊Dic3C2211yescharacteristic
17C32×C3⋊Dic3C2211yescharacteristic
18#6312—31no—
19C2×C3⋊(C3⋊Dic3)—31no—
20C3×#638—31no—
21C2×C3×C3⋊Dic3—31no—
22C32×#67—31no—
23C32×#67—31no—
24#6294—34no—
25#6293—31no—
26#6293—31no—
27#6293—34no—
28#6293—61no—
29#6291D311yescharacteristic, complemented
30#6290—31no—
31#6290—34no—
32#6290—64no—
33#6289—31no—
34Dic3×C3⋊D3—31no—
35D3×C3⋊Dic3—31no—
36D3×C3⋊Dic3—34no—
37D3×C3⋊Dic3—64no—
38D3×C3⋊Dic3D311yescharacteristic, complemented
39C3×#626—34no—
40C3×#625D314yescomplemented
41C3×D3×Dic3—34no—
42C2×C34C2311yescharacteristic
43C3×C3⋊(C3⋊D3)—21no—
44C33×D3—21no—
45C33×D3—21no—
46#3830—91no—
47#3805—94no—
48#3804—94no—
49#3795—91no—
50C22×C33—31no—
51C22×C33—31no—
52C2×C3⋊(C3⋊D3)C2×D311yescharacteristic
53C2×C3×C3⋊D3—31no—
54C2×C3×C3⋊D3—34no—
55C2×C3×C3⋊D3—64no—
56C2×C32×D3—24no—
57C2×C32×D3—24no—
58C2×C32×D3C2×D314yes—
59C2×C32×D3C2×D314yes—
60C2×C32×D3C2×D311yescharacteristic
61C2×C32×D3C2×D311yescharacteristic
62C3⋊(C3⋊Dic3)—31no—
63C3⋊(C3⋊Dic3)—31no—
64C3⋊(C3⋊Dic3)—34no—
65C3⋊(C3⋊Dic3)—64no—
66C3⋊(C3⋊Dic3)—64no—
67C3⋊(C3⋊Dic3)—61no—
68C3⋊(C3⋊Dic3)—124no—
69C3⋊(C3⋊Dic3)C2×D311yescharacteristic
70C3×C3⋊Dic3—21no—
71C3×C3⋊Dic3—31no—
72C3×C3⋊Dic3—34no—
73C3×C3⋊Dic3—64no—
74C3×C3⋊Dic3C2×D311yescharacteristic
75C3×C3⋊Dic3C2×D311yescharacteristic
76C3×C3⋊Dic3C2×D314yes—
77C32×Dic3—31no—
78C32×Dic3—31no—
79C32×Dic3—34no—
80C32×Dic3—61no—
81C34#4111yescharacteristic, complemented, Sylow
82C32×D4—91no—
83#638—31no—
84#638—91no—
85#638—94no—
86#638—184no—
87C2×C3⋊Dic3—31no—
88C2×C3⋊Dic3—91no—
89C2×C3⋊Dic3—94no—
90C2×C3⋊Dic3—184no—
91C4×C3⋊D3—91no—
92#634—91no—
93#634—91no—
94C3×#67—34no—
95C3×#67—34no—
96C3×#67—94no—
97C2×C3×Dic3—94no—
98C3×D12—91no—
99C3×C4×D3—91no—
100#627—94no—
101#627—94no—
102#627—184no—
103#626—34no—
104#625—94no—
105#625—91no—
106#625—184no—
107#625C3⋊D311yescharacteristic, complemented
108#624—94no—
109D3×Dic3—34no—
110D3×Dic3—94no—
111D3×Dic3—94no—
112D3×Dic3—91no—
113D3×Dic3—91no—
114D3×Dic3—184no—
115C2×C33—21no—
116C2×C33—24no—
117C2×C33—24no—
118C2×C33—44no—
119C2×C33—61no—
120C2×C33—61no—
121C2×C33C22×D311yescharacteristic
122C2×C33C22×D311yescharacteristic
123C2×C33C22×D314yes—
124C3⋊(C3⋊D3)—21no—
125C3×C3⋊D3—61no—
126C3×C3⋊D3—64no—
127C3×C3⋊D3—124no—
128C32×D3—28no—
129C32×D3—21no—
130C32×D3—24no—
131C32×D3—21no—
132C32×D3—24no—
133C32×D3—44no—
134#237—271no—
135#237—274no—
136#235—271no—
137C22×C32—31no—
138C22×C32—31no—
139C22×C32—34no—
140C22×C32—34no—
141C22×C32—64no—
142C22×C32—64no—
143C2×C3⋊D3—31no—
144C2×C3⋊D3—34no—
145C2×C3⋊D3—64no—
146C2×C3×D3—24no—
147C2×C3×D3—24no—
148C2×C3×D3—94no—
149C2×C3×D3—91no—
150C2×C3×D3—184no—
151C2×C3×D3D3214yes—
152C2×C3×D3D3214yes—
153C2×C3×D3C2×C3⋊D311yescharacteristic
154C2×C3×D3C2×C3⋊D311yescharacteristic
155C32×C4—91no—
156C32×C4—91no—
157C3⋊Dic3—31no—
158C3⋊Dic3—34no—
159C3⋊Dic3—64no—
160C3⋊Dic3—91no—
161C3⋊Dic3—94no—
162C3⋊Dic3—94no—
163C3⋊Dic3—91no—
164C3⋊Dic3—184no—
165C3⋊Dic3—184no—
166C3⋊Dic3—184no—
167C3⋊Dic3—184no—
168C3⋊Dic3—184no—
169C3⋊Dic3—364no—
170C3⋊Dic3—364no—
171C3⋊Dic3—3612no—
172C3⋊Dic3D3211yescharacteristic
173C3⋊Dic3C2×C3⋊D311yescharacteristic
174C3×Dic3—34no—
175C3×Dic3—34no—
176C3×Dic3—34no—
177C3×Dic3—34no—
178C3×Dic3—64no—
179C3×Dic3—64no—
180C3×Dic3—94no—
181C3×Dic3—91no—
182C3×Dic3—184no—
183C33—21no—
184C33—24no—
185C33—24no—
186C33—44no—
187C33#23511yescharacteristic, complemented
188C33#23711yescharacteristic, complemented
189C33#23714yescomplemented
190C3×D4—94no—
191C3×D4—271no—
192C2×C3×C4—271no—
193#67—31no—
194#67—31no—
195#67—94no—
196#67—271no—
197#67—274no—
198#67—271no—
199#67—544no—
200C2×Dic3—94no—
201C2×Dic3—271no—
202C2×Dic3—271no—
203C2×Dic3—274no—
204C2×Dic3—544no—
205D12—91no—
206C4×D3—91no—
207C4×D3—271no—
208C4×D3—274no—
209C4×D3—271no—
210Dic6—271no—
211Dic6—271no—
212Dic6—274no—
213Dic6—541no—
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—61no—
224C2×C32—61no—
225C2×C32—64no—
226C2×C32—64no—
227C2×C32—124no—
228C2×C32C2×D3214yes—
229C2×C32C2×D3214yes—
230C2×C32C2×D3211yescharacteristic
231C2×C32C22×C3⋊D311yescharacteristic
232C3⋊D3—61no—
233C3⋊D3—64no—
234C3⋊D3—124no—
235C3×D3—28no—
236C3×D3—24no—
237C3×D3—24no—
238C3×D3—21no—
239C3×D3—21no—
240C3×D3—44no—
241C3×D3—184no—
242C3×D3—181no—
243C3×D3—364no—
244#41—811noSylow
245C22×C3—34no—
246C22×C3—34no—
247C22×C3—31no—
248C22×C3—31no—
249C22×C3—64no—
250C22×C3—64no—
251C22×C3—271no—
252C2×D3—94no—
253C2×D3—91no—
254C2×D3—184no—
255C2×D3D3×C3⋊D311yescharacteristic
256C2×D3D3×C3⋊D311yescharacteristic
257C3×C4—94no—
258C3×C4—91no—
259C3×C4—91no—
260C3×C4—181no—
261C3×C4—271no—
262Dic3—34no—
263Dic3—31no—
264Dic3—31no—
265Dic3—61no—
266Dic3—94no—
267Dic3—91no—
268Dic3—184no—
269Dic3—274no—
270Dic3—271no—
271Dic3—271no—
272Dic3—541no—
273Dic3—544no—
274Dic3—544no—
275Dic3—1084no—
276C32—24no—
277C32—24no—
278C32—24no—
279C32—24no—
280C32—24no—
281C32—44no—
282C32—412no—
283C32—44no—
284C32#379511yescharacteristic, complemented
285C32#380414yescomplemented
286C32#380514yescomplemented
287C32#383011yescharacteristic, complemented
288Q8—811no—
289D4—91no—
290D4—271no—
291D4—811no—
292C2×C4—271no—
293C2×C4—811no—
294C2×C4—811no—
295C2×C3—21no—
296C2×C3—24no—
297C2×C3—24no—
298C2×C3—44no—
299C2×C3—68no—
300C2×C3—64no—
301C2×C3—64no—
302C2×C3—61no—
303C2×C3—61no—
304C2×C3—124no—
305C2×C3—541no—
306C2×C3D3314yes—
307C2×C3C2×D3×C3⋊D311yescharacteristic
308C2×C3C2×D3×C3⋊D311yescharacteristic
309D3—21no—
310D3—21no—
311D3—184no—
312D3—181no—
313D3—364no—
314C22—31no—
315C22—31no—
316C22—271no—
317C4—91no—
318C4—91no—
319C4—271no—
320C4—811no—
321C3—24no—
322C3—24no—
323C3—21no—
324C3—44no—
325C3#8896114yescomplemented
326C3#8902811yescharacteristic, complemented
327C3#8902911yescharacteristic, complemented
328C2—61no—
329C2—61no—
330C2—541no—
331C2D32×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. #10620154C2semidirect
2. #10620153C2semidirect
3. #10620145C2semidirect
4. D3×C3⋊(C3⋊Dic3)C2semidirect
5. C3×#6291C2semidirect
6. C3×D3×C3⋊Dic3C2semidirect
7. C32×#625C2semidirect
29. #6291D3semidirect
38. D3×C3⋊Dic3D3semidirect
40. C3×#625D3semidirect
81. C34#41semidirect
107. #625C3⋊D3semidirect
187. C33#235semidirect
188. C33#237semidirect
189. C33#237semidirect
284. C32#3795semidirect
285. C32#3804semidirect
286. C32#3805semidirect
287. C32#3830semidirect
325. C3#88961semidirect
326. C3#89028semidirect
327. C3#89029semidirect

Maximal subgroups

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

subgroupindex
#106201542
#106201532
#106201452
D3×C3⋊(C3⋊Dic3)2
C3×#62912
C3×D3×C3⋊Dic32
C32×#6252
#890293
#890283
#889613