Finite Group Databasebeta

← #13048662

Subgroups — #13048662

Order 1296.

531 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 531 conjugacy classes of subgroups — too many to draw. Only its 74 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
1C4×C3⋊#272C211yesmaximal, characteristic, complemented
2C4×C3⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
3C3×C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
4#10619887C211yesmaximal, characteristic, complemented
5#10619886C211yesmaximal, characteristic, complemented
6#10619884C211yesmaximal, characteristic, complemented
7#10619881C211yesmaximal, characteristic, complemented
8#89042—31nomaximal
9#89020—31nomaximal
10#88650—33nomaximal
11C2×C3⋊#272C2211yescharacteristic
12C2×C3⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×C3×(C3⋊D3)⋊C3C2211yescharacteristic
14C3×C4×H(3,𝔽3)C2211yescharacteristic
15C3⋊#1084C2211yescharacteristic
16C3⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C3×(C3⋊Dic3)⋊C3C2211yescharacteristic
18C4×C3⋊(C3⋊D3)D311yescharacteristic, complemented
19C3×C4×C3⋊D3—31no—
20C3×C4×C3⋊D3—31no—
21C3×C4×C3⋊D3—31no—
22C3×C4×C3⋊D3—61no—
23C3×C4×C3⋊D3D311yescharacteristic, complemented
24C32×C4×D3—31no—
25#6296—31no—
26#6295—31no—
27#6295—31no—
28#6295—31no—
29#6293—31no—
30#6292—31no—
31#6291—31no—
32#6290—31no—
33C4×#272—33no—
34C4×#272—63no—
35C4×(C3⋊D3)⋊C3—33no—
36C4×(C3⋊D3)⋊C3D313yescomplemented
37#6200—33no—
38#6200—33no—
39#6198—33no—
40#6196—33no—
41C2×C3×H(3,𝔽3)C2311yescharacteristic
42C3⋊#272—21no—
43C3⋊((C3⋊D3)⋊C3)—21no—
44C3×(C3⋊D3)⋊C3—21no—
45#3828—91no—
46#3798—93no—
47#3798—91no—
48#3798—91no—
49#3798—91no—
50C2×C3⋊(C3⋊D3)C2×D311yescharacteristic
51C2×C3×C3⋊D3—31no—
52C2×C3×C3⋊D3—31no—
53C2×C3×C3⋊D3—31no—
54C2×C3×C3⋊D3—61no—
55C2×C3×C3⋊D3C2×D311yescharacteristic
56C2×C32×D3—31no—
57C33×C4—21no—
58C33×C4C2×D311yescharacteristic
59C33×C4C2×D311yescharacteristic
60C3⋊(C3⋊Dic3)C2×D311yescharacteristic
61C3×C3⋊Dic3—31no—
62C3×C3⋊Dic3—31no—
63C3×C3⋊Dic3—31no—
64C3×C3⋊Dic3—61no—
65C3×C3⋊Dic3C2×D311yescharacteristic
66C32×Dic3—31no—
67C2×#272—33no—
68C2×#272—63no—
69C2×(C3⋊D3)⋊C3—33no—
70C2×(C3⋊D3)⋊C3C2×D313yes—
71C4×H(3,𝔽3)—23no—
72C4×H(3,𝔽3)C2×D313yes—
73#1084—33no—
74#1084—63no—
75(C3⋊Dic3)⋊C3—33no—
76(C3⋊Dic3)⋊C3C2×D313yes—
77C3×H(3,𝔽3)#4111yescharacteristic, complemented, Sylow
78C2×C32×C4—91no—
79#638—91no—
80#638—91no—
81C4⋊(C3⋊D3)—91no—
82C4×C3⋊D3—31no—
83C4×C3⋊D3—31no—
84C4×C3⋊D3—31no—
85C4×C3⋊D3—33no—
86C4×C3⋊D3—61no—
87C4×C3⋊D3—91no—
88C4×C3⋊D3—181no—
89C4×C3⋊D3C3⋊D311yescharacteristic, complemented
90#634—91no—
91C3×C4×D3—33no—
92C3×C4×D3—31no—
93C3×C4×D3—91no—
94C3×C4×D3—91no—
95C3×C4×D3—93no—
96C3×C4×D3—91no—
97C3×C4×D3—91no—
98C3×C4×D3—91no—
99C3×C4×D3—183no—
100C3×C4×D3—181no—
101C3×C4×D3—181no—
102#627—93no—
103#627—91no—
104#627—91no—
105#627—91no—
106#626—93no—
107#626—91no—
108#626—93no—
109#626—91no—
110#626—91no—
111#626—91no—
112#626—91no—
113#626—91no—
114#625—93no—
115#625—91no—
116#625—91no—
117#625—91no—
118C2×C33—21no—
119C2×C33C22×D311yescharacteristic
120C2×C33C22×D311yescharacteristic
121C3⋊(C3⋊D3)—21no—
122C3×C3⋊D3—21no—
123C3×C3⋊D3—62no—
124C3×C3⋊D3—61no—
125C3×C3⋊D3—61no—
126C3×C3⋊D3—61no—
127C32×D3—61no—
128C2×H(3,𝔽3)—23no—
129C2×H(3,𝔽3)C22×D313yes—
130#272—66no—
131#272—63no—
132(C3⋊D3)⋊C3—23no—
133(C3⋊D3)⋊C3—63no—
134#235—271no—
135#235—271no—
136#235—271no—
137#235—273no—
138C22×C32—91no—
139C2×C3⋊D3—31no—
140C2×C3⋊D3—31no—
141C2×C3⋊D3—31no—
142C2×C3⋊D3—33no—
143C2×C3⋊D3—61no—
144C2×C3⋊D3—91no—
145C2×C3⋊D3—181no—
146C2×C3⋊D3C2×C3⋊D311yescharacteristic
147C2×C3×D3—33no—
148C2×C3×D3—31no—
149C2×C3×D3—91no—
150C2×C3×D3—91no—
151C2×C3×D3—93no—
152C2×C3×D3—91no—
153C2×C3×D3—91no—
154C2×C3×D3—91no—
155C2×C3×D3—183no—
156C2×C3×D3—181no—
157C2×C3×D3—181no—
158C32×C4—21no—
159C32×C4—23no—
160C32×C4—31no—
161C32×C4—31no—
162C32×C4—61no—
163C32×C4—61no—
164C32×C4—61no—
165C32×C4—91no—
166C32×C4—121no—
167C32×C4D3211yescharacteristic
168C32×C4D3213yes—
169C32×C4C2×C3⋊D311yescharacteristic
170C3⋊Dic3—31no—
171C3⋊Dic3—31no—
172C3⋊Dic3—31no—
173C3⋊Dic3—33no—
174C3⋊Dic3—61no—
175C3⋊Dic3—91no—
176C3⋊Dic3—181no—
177C3⋊Dic3C2×C3⋊D311yescharacteristic
178C3×Dic3—33no—
179C3×Dic3—31no—
180C3×Dic3—91no—
181C3×Dic3—91no—
182C3×Dic3—93no—
183C3×Dic3—91no—
184C3×Dic3—91no—
185C3×Dic3—91no—
186C3×Dic3—183no—
187C3×Dic3—181no—
188C3×Dic3—181no—
189C33—21no—
190C33#23511yescharacteristic, complemented
191C33#23511yescharacteristic, complemented
192H(3,𝔽3)—23no—
193H(3,𝔽3)#23513yescomplemented
194C2×C3×C4—93no—
195C2×C3×C4—91no—
196C2×C3×C4—271no—
197C2×C3×C4—271no—
198#67—271no—
199#67—271no—
200#67—271no—
201#67—271no—
202#67—271no—
203#67—271no—
204#67—273no—
205#67—273no—
206D12—271no—
207D12—273no—
208D12—271no—
209D12—271no—
210C4×D3—31no—
211C4×D3—91no—
212C4×D3—91no—
213C4×D3—93no—
214C4×D3—91no—
215C4×D3—91no—
216C4×D3—91no—
217C4×D3—181no—
218C4×D3—181no—
219C4×D3—183no—
220C4×D3—273no—
221Dic6—271no—
222Dic6—271no—
223Dic6—271no—
224Dic6—273no—
225C2×C32—21no—
226C2×C32—23no—
227C2×C32—31no—
228C2×C32—31no—
229C2×C32—61no—
230C2×C32—61no—
231C2×C32—61no—
232C2×C32—121no—
233C2×C32—181no—
234C2×C32C2×D3211yescharacteristic
235C2×C32C2×D3213yes—
236C2×C32C22×C3⋊D311yescharacteristic
237C3⋊D3—21no—
238C3⋊D3—61no—
239C3⋊D3—61no—
240C3⋊D3—62no—
241C3⋊D3—61no—
242C3⋊D3—63no—
243C3⋊D3—181no—
244C3⋊D3—182no—
245C3×D3—61no—
246C3×D3—63no—
247C3×D3—181no—
248C3×D3—182no—
249C3×D3—186no—
250C3×D3—181no—
251C3×D3—182no—
252C3×D3—181no—
253C3×D3—183no—
254C3×D3—181no—
255C3×D3—181no—
256#41—811noSylow
257C22×C3—93no—
258C22×C3—91no—
259C22×C3—271no—
260C22×C3—271no—
261C2×D3—31no—
262C2×D3—91no—
263C2×D3—91no—
264C2×D3—93no—
265C2×D3—91no—
266C2×D3—91no—
267C2×D3—91no—
268C2×D3—181no—
269C2×D3—181no—
270C2×D3—183no—
271C2×D3—273no—
272C3×C4—21no—
273C3×C4—31no—
274C3×C4—33no—
275C3×C4—61no—
276C3×C4—63no—
277C3×C4—93no—
278C3×C4—91no—
279C3×C4—271no—
280C3×C4—271no—
281C3×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
282C3×C4D3×C3⋊D311yescharacteristic
283Dic3—31no—
284Dic3—91no—
285Dic3—91no—
286Dic3—93no—
287Dic3—91no—
288Dic3—91no—
289Dic3—91no—
290Dic3—181no—
291Dic3—181no—
292Dic3—183no—
293Dic3—273no—
294C32—21no—
295C32—23no—
296C32—31no—
297C32—31no—
298C32—61no—
299C32—61no—
300C32—61no—
301C32—121no—
302C32#379811yescharacteristic
303C32#379813yescomplemented
304C32#382811yescharacteristic, complemented
305Q8—811no—
306D4—811no—
307D4—811no—
308D4—811no—
309C2×C4—91no—
310C2×C4—271no—
311C2×C4—271no—
312C2×C3—21no—
313C2×C3—31no—
314C2×C3—33no—
315C2×C3—61no—
316C2×C3—63no—
317C2×C3—183no—
318C2×C3—181no—
319C2×C3—541no—
320C2×C3—541no—
321C2×C3C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
322C2×C3C2×D3×C3⋊D311yescharacteristic
323D3—61no—
324D3—181no—
325D3—182no—
326D3—181no—
327D3—183no—
328D3—182no—
329D3—186no—
330D3—181no—
331D3—181no—
332D3—181no—
333D3—543no—
334C22—91no—
335C22—271no—
336C22—271no—
337C4—91no—
338C4—271no—
339C4—271no—
340C4((C3⋊D3)⋊C3)⋊D311yescharacteristic
341C3—21no—
342C3—31no—
343C3—33no—
344C3—61no—
345C3—63no—
346C3#8865011yescharacteristic, complemented
347C3#8902011yescharacteristic
348C2—181no—
349C2—541no—
350C2—541no—
351C2C2×((C3⋊D3)⋊C3)⋊D311yescharacteristic
subgroupG/H
center, hypercenterC4((C3⋊D3)⋊C3)⋊D3
commutator, cosocleC2×C3×H(3,𝔽3)C23
FrattiniC2×C3C2×D3×C3⋊D3
FittingC3×C4×H(3,𝔽3)C22
socleC2×C32C2×D32
hypocenterC3×H(3,𝔽3)#41
LCSG ▷ C2×C3×H(3,𝔽3) ▷ C3×H(3,𝔽3)
UCSC1 ◁ C4

Complements

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

NH ≅ G/Ntype
1. C4×C3⋊#272C2semidirect
2. C4×C3⋊((C3⋊D3)⋊C3)C2semidirect
3. C3×C4×(C3⋊D3)⋊C3C2semidirect
4. #10619887C2semidirect
5. #10619886C2semidirect
6. #10619884C2semidirect
7. #10619881C2semidirect
18. C4×C3⋊(C3⋊D3)D3semidirect
23. C3×C4×C3⋊D3D3semidirect
36. C4×(C3⋊D3)⋊C3D3semidirect
77. C3×H(3,𝔽3)#41semidirect
89. C4×C3⋊D3C3⋊D3semidirect
190. C33#235semidirect
191. C33#235semidirect
193. H(3,𝔽3)#235semidirect
303. C32#3798semidirect
304. C32#3828semidirect
346. C3#88650semidirect

Maximal subgroups

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

subgroupindex
C4×C3⋊#2722
C4×C3⋊((C3⋊D3)⋊C3)2
C3×C4×(C3⋊D3)⋊C32
#106198872
#106198862
#106198842
#106198812
#890423
#890203
#886503