Finite Group Databasebeta

← #13049714

Subgroups — #13049714

Order 1296.

924 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 924 conjugacy classes of subgroups — too many to draw. Only its 107 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
1C3×#6312C211yesmaximal, characteristic, complemented
2C2×C3×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
3C32×#638C211yesmaximal, characteristic, complemented
4C3⋊#6296C211yesmaximal, characteristic, complemented
5#10620157C211yesmaximal, characteristic, complemented
6#10620155C211yesmaximal, characteristic, complemented
7#10620154C211yesmaximal, characteristic, complemented
8#89046—34nomaximal
9#89029—31nomaximal
10#89028—31nomaximal
11C22×C34C2211yescharacteristic
12C2×C3×C3⋊(C3⋊D3)C2211yescharacteristic
13C2×C32×C3⋊D3C2211yescharacteristic
14C3×C3⋊(C3⋊Dic3)C2211yescharacteristic
15C3×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C3×C3⋊(C3⋊Dic3)C2211yescharacteristic
17C32×C3⋊Dic3C2211yescharacteristic
18#6312D311yescharacteristic, complemented
19C2×C3⋊(C3⋊Dic3)D311yescharacteristic, complemented
20C3×#638—31no—
21C3×#638—34no—
22C3×#638—64no—
23C3×#638D314yescomplemented
24C2×C3×C3⋊Dic3—31no—
25C2×C3×C3⋊Dic3—34no—
26C2×C3×C3⋊Dic3—64no—
27C32×#67—31no—
28C32×#67—31no—
29C32×#67—61no—
30#6296—34no—
31#6295—34no—
32#6294—34no—
33#6294—34no—
34#6293—31no—
35#6293—31no—
36#6291—31no—
37#6290—31no—
38#6289—31no—
39Dic3×C3⋊D3—31no—
40D3×C3⋊Dic3—31no—
41D3×C3⋊Dic3—31no—
42C2×C34—21no—
43C2×C34C2311yescharacteristic
44C3×C3⋊(C3⋊D3)—21no—
45C32×C3⋊D3—21no—
46#3830—91no—
47#3805—94no—
48#3804—94no—
49#3804—91no—
50C22×C33—24no—
51C22×C33—21no—
52C22×C33—24no—
53C22×C33—44no—
54C22×C33C2×D311yescharacteristic
55C22×C33C2×D311yescharacteristic
56C22×C33C2×D314yes—
57C2×C3⋊(C3⋊D3)C2×D311yescharacteristic
58C2×C3×C3⋊D3—31no—
59C2×C3×C3⋊D3—34no—
60C2×C3×C3⋊D3—64no—
61C2×C3×C3⋊D3C2×D314yes—
62C2×C32×D3—31no—
63C2×C32×D3—31no—
64C2×C32×D3—61no—
65C3⋊(C3⋊Dic3)C2×D311yescharacteristic
66C3⋊(C3⋊Dic3)C2×D311yescharacteristic
67C3⋊(C3⋊Dic3)C2×D311yescharacteristic
68C3×C3⋊Dic3—31no—
69C3×C3⋊Dic3—31no—
70C3×C3⋊Dic3—31no—
71C3×C3⋊Dic3—34no—
72C3×C3⋊Dic3—34no—
73C3×C3⋊Dic3—34no—
74C3×C3⋊Dic3—64no—
75C3×C3⋊Dic3—64no—
76C3×C3⋊Dic3—64no—
77C3×C3⋊Dic3C2×D314yes—
78C32×Dic3—31no—
79C32×Dic3—31no—
80C32×Dic3—61no—
81C34#4111yescharacteristic, complemented, Sylow
82C32×D4—91no—
83#638—31no—
84#638—34no—
85#638—64no—
86#638—91no—
87#638C3⋊D311yescharacteristic, complemented
88C2×C3⋊Dic3—31no—
89C2×C3⋊Dic3—34no—
90C2×C3⋊Dic3—64no—
91C2×C3⋊Dic3—91no—
92C4×C3⋊D3—91no—
93#634—91no—
94C3×#67—34no—
95C3×#67—34no—
96C3×#67—64no—
97C3×#67—94no—
98C3×#67—91no—
99C3×#67—184no—
100C2×C3×Dic3—94no—
101C2×C3×Dic3—91no—
102C2×C3×Dic3—184no—
103#627—94no—
104#627—94no—
105#627—91no—
106#626—94no—
107#626—91no—
108#625—94no—
109#624—94no—
110#624—91no—
111D3×Dic3—94no—
112D3×Dic3—94no—
113D3×Dic3—94no—
114D3×Dic3—91no—
115C2×C33—21no—
116C2×C33—28no—
117C2×C33—24no—
118C2×C33—21no—
119C2×C33—21no—
120C2×C33—24no—
121C2×C33—24no—
122C2×C33—44no—
123C2×C33—41no—
124C2×C33—44no—
125C2×C33—48no—
126C2×C33C22×D311yescharacteristic
127C2×C33C22×D311yescharacteristic
128C2×C33C22×D314yes—
129C3⋊(C3⋊D3)—21no—
130C3×C3⋊D3—24no—
131C3×C3⋊D3—61no—
132C3×C3⋊D3—64no—
133C3×C3⋊D3—124no—
134C32×D3—61no—
135C32×D3—61no—
136C32×D3—121no—
137#237—271no—
138#237—274no—
139#235—271no—
140C22×C32—24no—
141C22×C32—24no—
142C22×C32—24no—
143C22×C32—24no—
144C22×C32—24no—
145C22×C32—44no—
146C22×C32—412no—
147C22×C32—44no—
148C22×C32—91no—
149C22×C32D3214yes—
150C22×C32D3214yes—
151C22×C32D3211yescharacteristic
152C22×C32C2×C3⋊D311yescharacteristic
153C2×C3⋊D3—31no—
154C2×C3⋊D3—34no—
155C2×C3⋊D3—64no—
156C2×C3⋊D3C2×C3⋊D311yescharacteristic
157C2×C3×D3—34no—
158C2×C3×D3—34no—
159C2×C3×D3—64no—
160C2×C3×D3—94no—
161C2×C3×D3—91no—
162C2×C3×D3—184no—
163C32×C4—91no—
164C3⋊Dic3—31no—
165C3⋊Dic3—31no—
166C3⋊Dic3—31no—
167C3⋊Dic3—34no—
168C3⋊Dic3—34no—
169C3⋊Dic3—34no—
170C3⋊Dic3—64no—
171C3⋊Dic3—64no—
172C3⋊Dic3—64no—
173C3⋊Dic3C2×C3⋊D311yescharacteristic
174C3×Dic3—34no—
175C3×Dic3—34no—
176C3×Dic3—64no—
177C3×Dic3—94no—
178C3×Dic3—94no—
179C3×Dic3—94no—
180C3×Dic3—91no—
181C3×Dic3—91no—
182C3×Dic3—91no—
183C3×Dic3—184no—
184C3×Dic3—184no—
185C3×Dic3—184no—
186C33—21no—
187C33—24no—
188C33—24no—
189C33—44no—
190C33#23511yescharacteristic, complemented
191C33#23711yescharacteristic, complemented
192C33#23714yescomplemented
193C3×D4—94no—
194C3×D4—271no—
195C2×C3×C4—271no—
196#67—31no—
197#67—31no—
198#67—61no—
199#67—91no—
200#67—94no—
201#67—184no—
202#67—271no—
203#67—274no—
204C2×Dic3—91no—
205C2×Dic3—94no—
206C2×Dic3—184no—
207C2×Dic3—271no—
208C2×Dic3—274no—
209D12—271no—
210C4×D3—271no—
211C4×D3—274no—
212C4×D3—271no—
213C4×D3—271no—
214Dic6—271no—
215Dic6—271no—
216Dic6—274no—
217C2×C32—28no—
218C2×C32—24no—
219C2×C32—21no—
220C2×C32—24no—
221C2×C32—24no—
222C2×C32—28no—
223C2×C32—24no—
224C2×C32—21no—
225C2×C32—24no—
226C2×C32—24no—
227C2×C32—24no—
228C2×C32—44no—
229C2×C32—412no—
230C2×C32—44no—
231C2×C32—44no—
232C2×C32—44no—
233C2×C32—48no—
234C2×C32—424no—
235C2×C32—48no—
236C2×C32—44no—
237C2×C32—181no—
238C2×C32C2×D3214yes—
239C2×C32C2×D3214yes—
240C2×C32C2×D3211yescharacteristic
241C2×C32C22×C3⋊D311yescharacteristic
242C3⋊D3—21no—
243C3⋊D3—61no—
244C3⋊D3—64no—
245C3⋊D3—124no—
246C3×D3—64no—
247C3×D3—64no—
248C3×D3—124no—
249C3×D3—184no—
250C3×D3—181no—
251C3×D3—364no—
252#41—811noSylow
253C22×C3—24no—
254C22×C3—21no—
255C22×C3—24no—
256C22×C3—44no—
257C22×C3—94no—
258C22×C3—271no—
259C22×C3D3×C3⋊D311yescharacteristic
260C22×C3D3×C3⋊D311yescharacteristic
261C22×C3#111314yes—
262C2×D3—31no—
263C2×D3—31no—
264C2×D3—61no—
265C2×D3—91no—
266C2×D3—94no—
267C2×D3—184no—
268C3×C4—94no—
269C3×C4—271no—
270C3×C4—271no—
271C3×C4—271no—
272Dic3—31no—
273Dic3—31no—
274Dic3—61no—
275Dic3—91no—
276Dic3—91no—
277Dic3—91no—
278Dic3—94no—
279Dic3—94no—
280Dic3—94no—
281Dic3—184no—
282Dic3—184no—
283Dic3—184no—
284C32—24no—
285C32—24no—
286C32—24no—
287C32—24no—
288C32—24no—
289C32—44no—
290C32—412no—
291C32—44no—
292C32#380414yescomplemented
293C32#380411yescharacteristic, complemented
294C32#380514yescomplemented
295C32#383011yescharacteristic, complemented
296Q8—811no—
297D4—91no—
298D4—271no—
299D4—811no—
300C2×C4—271no—
301C2×C4—811no—
302C2×C4—811no—
303C2×C3—28no—
304C2×C3—21no—
305C2×C3—21no—
306C2×C3—24no—
307C2×C3—21no—
308C2×C3—24no—
309C2×C3—24no—
310C2×C3—41no—
311C2×C3—44no—
312C2×C3—48no—
313C2×C3—44no—
314C2×C3—184no—
315C2×C3—541no—
316C2×C3C2×D3×C3⋊D311yescharacteristic
317C2×C3C2×D3×C3⋊D311yescharacteristic
318C2×C3C2×#111314yes—
319D3—61no—
320D3—61no—
321D3—121no—
322D3—181no—
323D3—184no—
324D3—364no—
325C22—91no—
326C22—271no—
327C22C3⋊#111311yescharacteristic
328C4—91no—
329C4—271no—
330C4—271no—
331C4—271no—
332C3—21no—
333C3—24no—
334C3—24no—
335C3—44no—
336C3#8902811yescharacteristic, complemented
337C3#8902911yescharacteristic, complemented
338C3#8904614yescomplemented
339C2—21no—
340C2—181no—
341C2—541no—
342C2C2×C3⋊#111311yescharacteristic
subgroupG/H
center, FrattiniC2C2×C3⋊#1113
commutator, socle, cosocleC2×C34C23
FittingC22×C34C22
hypercenterC22C3⋊#1113
hypocenterC34#41
LCSG ▷ C2×C34 ▷ C34
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C3×#6312C2semidirect
2. C2×C3×C3⋊(C3⋊Dic3)C2semidirect
3. C32×#638C2semidirect
4. C3⋊#6296C2semidirect
5. #10620157C2semidirect
6. #10620155C2semidirect
7. #10620154C2semidirect
18. #6312D3semidirect
19. C2×C3⋊(C3⋊Dic3)D3semidirect
23. C3×#638D3semidirect
81. C34#41semidirect
87. #638C3⋊D3semidirect
190. C33#235semidirect
191. C33#237semidirect
192. C33#237semidirect
292. C32#3804semidirect
293. C32#3804semidirect
294. C32#3805semidirect
295. C32#3830semidirect
336. C3#89028semidirect
337. C3#89029semidirect
338. C3#89046semidirect

Maximal subgroups

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

subgroupindex
C3×#63122
C2×C3×C3⋊(C3⋊Dic3)2
C32×#6382
C3⋊#62962
#106201572
#106201552
#106201542
#890463
#890293
#890283