Finite Group Databasebeta

← #13049518

Subgroups — #13049518

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#10620156C211yesmaximal, characteristic, complemented
2#10620153C211yesmaximal, characteristic, complemented
3#10620145C211yesmaximal, characteristic, complemented
4Dic3×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
5C3×#6291C211yesmaximal, characteristic, complemented
6C3×#6289C211yesmaximal, characteristic, complemented
7C32×D3×Dic3C211yesmaximal, characteristic, complemented
8#89031—31nomaximal
9#89020—31nomaximal
10#88960—34nomaximal
11C2×C3×C3⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×C3⋊(C3⋊D3)C2211yescharacteristic
13C2×C33×D3C2211yescharacteristic
14C3⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C32×C3⋊Dic3C2211yescharacteristic
16C32×C3⋊Dic3C2211yescharacteristic
17C33×Dic3C2211yescharacteristic
18#6312—31no—
19C4×C3⋊(C3⋊D3)—31no—
20C3×#638—31no—
21C3×C4×C3⋊D3—31no—
22C2×C32×Dic3—31no—
23C32×C4×D3—31no—
24#6295—34no—
25#6293—31no—
26#6293—31no—
27#6293—34no—
28#6293—61no—
29#6292—31no—
30#6292—31no—
31#6291D311yescharacteristic, complemented
32#6290—31no—
33#6290—34no—
34#6290—64no—
35#6289D311yescharacteristic, complemented
36Dic3×C3⋊D3—31no—
37Dic3×C3⋊D3—34no—
38Dic3×C3⋊D3—64no—
39C3×#626—34no—
40C3×#624—34no—
41C3×D3×Dic3D314yescomplemented
42C2×C34C2311yescharacteristic
43C3×C3⋊(C3⋊D3)—21no—
44C3×C3⋊(C3⋊D3)—21no—
45C33×D3—21no—
46#3828—91no—
47#3804—94no—
48#3798—94no—
49#3795—91no—
50C22×C33—31no—
51C2×C3⋊(C3⋊D3)C2×D311yescharacteristic
52C2×C3⋊(C3⋊D3)C2×D311yescharacteristic
53C2×C3×C3⋊D3—31no—
54C2×C3×C3⋊D3—31no—
55C2×C3×C3⋊D3—34no—
56C2×C3×C3⋊D3—34no—
57C2×C3×C3⋊D3—64no—
58C2×C3×C3⋊D3—64no—
59C2×C32×D3—24no—
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)—64no—
67C3⋊(C3⋊Dic3)—64no—
68C3⋊(C3⋊Dic3)—61no—
69C3⋊(C3⋊Dic3)—124no—
70C3×C3⋊Dic3—21no—
71C3×C3⋊Dic3C2×D311yescharacteristic
72C3×C3⋊Dic3C2×D311yescharacteristic
73C3×C3⋊Dic3C2×D314yes—
74C32×Dic3—24no—
75C32×Dic3—31no—
76C32×Dic3—31no—
77C32×Dic3—34no—
78C32×Dic3—61no—
79C32×Dic3C2×D314yes—
80C32×Dic3C2×D311yescharacteristic
81C34#4111yescharacteristic, complemented, Sylow
82C2×C32×C4—91no—
83#638—31no—
84#638—91no—
85#638—94no—
86#638—184no—
87C4⋊(C3⋊D3)—91no—
88C4×C3⋊D3—31no—
89C4×C3⋊D3—91no—
90C4×C3⋊D3—94no—
91C4×C3⋊D3—184no—
92#634—91no—
93#634—91no—
94C3×#67—94no—
95C2×C3×Dic3—34no—
96C3×D12—91no—
97C3×C4×D3—34no—
98C3×C4×D3—94no—
99C3×C4×D3—91no—
100#627—94no—
101#627—94no—
102#627—184no—
103#626—34no—
104#626—94no—
105#626—94no—
106#625—94no—
107#625—91no—
108#625—91no—
109#625—184no—
110#624—34no—
111D3×Dic3—94no—
112D3×Dic3—91no—
113D3×Dic3—184no—
114D3×Dic3C3⋊D311yescharacteristic, complemented
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)—21no—
125C3×C3⋊D3—68no—
126C3×C3⋊D3—64no—
127C3×C3⋊D3—61no—
128C3×C3⋊D3—61no—
129C3×C3⋊D3—64no—
130C3×C3⋊D3—124no—
131C32×D3—21no—
132C32×D3—24no—
133C32×D3—44no—
134#237—271no—
135#235—271no—
136#235—274no—
137C22×C32—31no—
138C22×C32—34no—
139C22×C32—64no—
140C2×C3⋊D3—31no—
141C2×C3⋊D3—31no—
142C2×C3⋊D3—34no—
143C2×C3⋊D3—34no—
144C2×C3⋊D3—64no—
145C2×C3⋊D3—64no—
146C2×C3×D3—24no—
147C2×C3×D3—94no—
148C2×C3×D3—94no—
149C2×C3×D3—91no—
150C2×C3×D3—91no—
151C2×C3×D3—184no—
152C2×C3×D3—184no—
153C2×C3×D3D3214yes—
154C2×C3×D3C2×C3⋊D311yescharacteristic
155C32×C4—31no—
156C32×C4—34no—
157C32×C4—64no—
158C32×C4—91no—
159C32×C4—91no—
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—24no—
175C3×Dic3—34no—
176C3×Dic3—34no—
177C3×Dic3—34no—
178C3×Dic3—34no—
179C3×Dic3—64no—
180C3×Dic3—64no—
181C3×Dic3D3214yes—
182C3×Dic3C2×C3⋊D311yescharacteristic
183C33—21no—
184C33—24no—
185C33—24no—
186C33—44no—
187C33#23511yescharacteristic, complemented
188C33#23514yescomplemented
189C33#23711yescharacteristic, complemented
190C3×D4—271no—
191C2×C3×C4—94no—
192C2×C3×C4—271no—
193#67—94no—
194#67—271no—
195#67—271no—
196#67—271no—
197#67—274no—
198#67—271no—
199#67—544no—
200C2×Dic3—31no—
201C2×Dic3—271no—
202D12—91no—
203D12—274no—
204C4×D3—31no—
205C4×D3—94no—
206C4×D3—91no—
207C4×D3—271no—
208C4×D3—274no—
209C4×D3—544no—
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—61no—
223C2×C32—64no—
224C2×C32—124no—
225C2×C32C2×D3214yes—
226C2×C32C2×D3214yes—
227C2×C32C2×D3211yescharacteristic
228C2×C32C22×C3⋊D311yescharacteristic
229C3⋊D3—68no—
230C3⋊D3—61no—
231C3⋊D3—61no—
232C3⋊D3—64no—
233C3⋊D3—64no—
234C3⋊D3—124no—
235C3×D3—24no—
236C3×D3—21no—
237C3×D3—44no—
238C3×D3—188no—
239C3×D3—184no—
240C3×D3—184no—
241C3×D3—181no—
242C3×D3—181no—
243C3×D3—364no—
244#41—811noSylow
245C22×C3—34no—
246C22×C3—31no—
247C22×C3—64no—
248C22×C3—271no—
249C22×C3—271no—
250C2×D3—94no—
251C2×D3—94no—
252C2×D3—91no—
253C2×D3—91no—
254C2×D3—184no—
255C2×D3—184no—
256C2×D3D3×C3⋊D311yescharacteristic
257C3×C4—34no—
258C3×C4—31no—
259C3×C4—64no—
260C3×C4—94no—
261C3×C4—91no—
262C3×C4—91no—
263C3×C4—181no—
264Dic3—34no—
265Dic3—31no—
266Dic3—31no—
267Dic3—61no—
268Dic3—274no—
269Dic3—271no—
270Dic3—271no—
271Dic3—541no—
272Dic3—544no—
273Dic3—544no—
274Dic3—1084no—
275Dic3D3×C3⋊D311yescharacteristic
276C32—24no—
277C32—24no—
278C32—24no—
279C32—24no—
280C32—24no—
281C32—44no—
282C32—412no—
283C32—44no—
284C32#379511yescharacteristic, complemented
285C32#379814yescomplemented
286C32#380414yescomplemented
287C32#382811yescharacteristic, complemented
288Q8—811no—
289D4—271no—
290D4—811no—
291D4—811no—
292C2×C4—91no—
293C2×C4—271no—
294C2×C4—811no—
295C2×C3—21no—
296C2×C3—24no—
297C2×C3—24no—
298C2×C3—44no—
299C2×C3—64no—
300C2×C3—61no—
301C2×C3—124no—
302C2×C3—541no—
303C2×C3—541no—
304C2×C3D3314yes—
305C2×C3C2×D3×C3⋊D311yescharacteristic
306C2×C3C2×D3×C3⋊D311yescharacteristic
307D3—21no—
308D3—188no—
309D3—184no—
310D3—184no—
311D3—181no—
312D3—181no—
313D3—364no—
314C22—31no—
315C22—271no—
316C22—271no—
317C4—31no—
318C4—91no—
319C4—91no—
320C4—811no—
321C3—24no—
322C3—24no—
323C3—21no—
324C3—44no—
325C3#8896014yescomplemented
326C3#8902011yescharacteristic, complemented
327C3#8903111yescharacteristic, complemented
328C2—61no—
329C2—541no—
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. #10620156C2semidirect
2. #10620153C2semidirect
3. #10620145C2semidirect
4. Dic3×C3⋊(C3⋊D3)C2semidirect
5. C3×#6291C2semidirect
6. C3×#6289C2semidirect
7. C32×D3×Dic3C2semidirect
31. #6291D3semidirect
35. #6289D3semidirect
41. C3×D3×Dic3D3semidirect
81. C34#41semidirect
114. D3×Dic3C3⋊D3semidirect
187. C33#235semidirect
188. C33#235semidirect
189. C33#237semidirect
284. C32#3795semidirect
285. C32#3798semidirect
286. C32#3804semidirect
287. C32#3828semidirect
325. C3#88960semidirect
326. C3#89020semidirect
327. C3#89031semidirect

Maximal subgroups

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

subgroupindex
#106201562
#106201532
#106201452
Dic3×C3⋊(C3⋊D3)2
C3×#62912
C3×#62892
C32×D3×Dic32
#890313
#890203
#889603