Finite Group Databasebeta

← #13048332

Subgroups — #13048332

Order 1296.

390 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 51 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#10619901C211yesmaximal, characteristic, complemented
2D3×#1084C211yesmaximal, characteristic, complemented
3#10619884C211yesmaximal, characteristic, complemented
4#10619881C211yesmaximal, characteristic, complemented
5#6291⋊C3C211yesmaximal, characteristic, complemented
6#6290⋊C3C211yesmaximal, characteristic, complemented
7C3×((C3⋊D3)⋊C3)⋊C4C211yesmaximal, characteristic, complemented
8#88960—31nomaximal
9#88959—31nomaximal
10#88672—31nomaximal
11C2×C3⋊((C3⋊D3)⋊C3)C2211yescharacteristic
12C2×D3×H(3,𝔽3)C2211yescharacteristic
13C2×C3×(C3⋊D3)⋊C3C2211yescharacteristic
14C3⋊#1084C2211yescharacteristic
15C3×#1084C2211yescharacteristic
16C3⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17C3×(C3⋊Dic3)⋊C3C2211yescharacteristic
18#6296—31no—
19#6295—31no—
20#6293—31no—
21#6291—31no—
22#6291D311yescharacteristic, complemented
23#6290D311yescharacteristic, complemented
24#6289—31no—
25Dic3×C3⋊D3—31no—
26C3×#626—31no—
27C3×#625—31no—
28C3×#624—31no—
29C3×D3×Dic3—31no—
30C3×D3×Dic3—31no—
31C3×D3×Dic3—31no—
32C3×D3×Dic3—61no—
33C2×#1084—31no—
34#638⋊C3—31no—
35#638⋊C3—31no—
36#6198—31no—
37#6198—61no—
38((C3⋊D3)⋊C3)⋊C4—31no—
39((C3⋊D3)⋊C3)⋊C4D311yescharacteristic, complemented
40#6196—31no—
41#6196—61no—
42C2×C3×H(3,𝔽3)C2311yescharacteristic
43C3⋊((C3⋊D3)⋊C3)—21no—
44D3×H(3,𝔽3)—21no—
45C3×(C3⋊D3)⋊C3—21no—
46#3804—91no—
47#3804—91no—
48#3804—91no—
49#3798—91no—
50#3795—91no—
51C2×C3⋊(C3⋊D3)C2×D311yescharacteristic
52C2×C3×C3⋊D3—31no—
53C2×C3×C3⋊D3C2×D311yescharacteristic
54C2×C32×D3—21no—
55C2×C32×D3—31no—
56C2×C32×D3C2×D311yescharacteristic
57C2×C32×D3C2×D311yescharacteristic
58C3⋊(C3⋊Dic3)C2×D311yescharacteristic
59C3×C3⋊Dic3—31no—
60C3×C3⋊Dic3—31no—
61C3×C3⋊Dic3—31no—
62C3×C3⋊Dic3—61no—
63C3×C3⋊Dic3C2×D311yescharacteristic
64C32×Dic3—31no—
65C32×Dic3—31no—
66C32×Dic3—31no—
67C32×Dic3—61no—
68C22×H(3,𝔽3)—31no—
69C2×(C3⋊D3)⋊C3—21no—
70C2×(C3⋊D3)⋊C3—31no—
71C2×(C3⋊D3)⋊C3—61no—
72C2×(C3⋊D3)⋊C3C2×D311yescharacteristic
73#1084—31no—
74#1084—61no—
75#1084—61no—
76#1084—121no—
77#1084C2×D311yescharacteristic
78(C3⋊Dic3)⋊C3—21no—
79(C3⋊Dic3)⋊C3—31no—
80(C3⋊Dic3)⋊C3—61no—
81(C3⋊Dic3)⋊C3C2×D311yescharacteristic
82C3×H(3,𝔽3)#4111yescharacteristic, complemented, Sylow
83#638—31no—
84#638—31no—
85#638—91no—
86C4×C3⋊D3—91no—
87#634—91no—
88C3×#67—91no—
89C3×#67—91no—
90C3×#67—91no—
91C2×C3×Dic3—91no—
92C2×C3×Dic3—91no—
93C2×C3×Dic3—91no—
94C2×C3×Dic3—181no—
95C3×D12—91no—
96C3×C4×D3—91no—
97C3×C4×D3—91no—
98C3×C4×D3—91no—
99#627—91no—
100#627—91no—
101#627—91no—
102#627—91no—
103#627—181no—
104#626—31no—
105#626—91no—
106#626—91no—
107#626—91no—
108#626—91no—
109#626—181no—
110#625—31no—
111#625—91no—
112#625—91no—
113#624—31no—
114#624—91no—
115#624—91no—
116#624—181no—
117D3×Dic3—31no—
118D3×Dic3—31no—
119D3×Dic3—31no—
120D3×Dic3—61no—
121D3×Dic3—91no—
122D3×Dic3—91no—
123D3×Dic3—181no—
124C2×C33—21no—
125C2×C33C22×D311yescharacteristic
126C2×C33C22×D311yescharacteristic
127C3⋊(C3⋊D3)—21no—
128C3×C3⋊D3—21no—
129C3×C3⋊D3—61no—
130C32×D3—22no—
131C32×D3—21no—
132C32×D3—21no—
133C32×D3—61no—
134C2×H(3,𝔽3)—21no—
135C2×H(3,𝔽3)—21no—
136C2×H(3,𝔽3)—41no—
137C2×H(3,𝔽3)—61no—
138C2×H(3,𝔽3)C22×D311yescharacteristic
139(C3⋊D3)⋊C3—21no—
140(C3⋊D3)⋊C3—41no—
141(C3⋊D3)⋊C3—61no—
142(C3⋊D3)⋊C3—121no—
143#237—271no—
144#237—271no—
145#235—271no—
146#235—271no—
147C22×C32—31no—
148C22×C32—31no—
149C22×C32—61no—
150C22×C32—91no—
151C2×C3⋊D3—31no—
152C2×C3⋊D3—31no—
153C2×C3⋊D3—61no—
154C2×C3⋊D3—91no—
155C2×C3⋊D3—181no—
156C2×C3⋊D3D3211yescharacteristic
157C2×C3×D3—31no—
158C2×C3×D3—31no—
159C2×C3×D3—31no—
160C2×C3×D3—31no—
161C2×C3×D3—61no—
162C2×C3×D3—61no—
163C2×C3×D3—91no—
164C2×C3×D3—91no—
165C2×C3×D3—91no—
166C2×C3×D3—181no—
167C2×C3×D3D3211yescharacteristic
168C32×C4—91no—
169C32×C4—91no—
170C3⋊Dic3—31no—
171C3⋊Dic3—31no—
172C3⋊Dic3—31no—
173C3⋊Dic3—31no—
174C3⋊Dic3—61no—
175C3⋊Dic3—61no—
176C3⋊Dic3—91no—
177C3⋊Dic3—181no—
178C3⋊Dic3D3211yescharacteristic
179C3×Dic3—31no—
180C3×Dic3—31no—
181C3×Dic3—31no—
182C3×Dic3—31no—
183C3×Dic3—31no—
184C3×Dic3—31no—
185C3×Dic3—61no—
186C3×Dic3—61no—
187C3×Dic3—61no—
188C3×Dic3—61no—
189C3×Dic3—61no—
190C3×Dic3—91no—
191C3×Dic3—91no—
192C3×Dic3—91no—
193C3×Dic3—91no—
194C3×Dic3—91no—
195C3×Dic3—91no—
196C3×Dic3—121no—
197C3×Dic3—181no—
198C3×Dic3—181no—
199C3×Dic3—181no—
200C3×Dic3—181no—
201C3×Dic3—361no—
202C33—21no—
203C33#23711yescharacteristic, complemented
204C33#23711yescharacteristic, complemented
205H(3,𝔽3)—21no—
206H(3,𝔽3)—21no—
207H(3,𝔽3)—41no—
208H(3,𝔽3)#23511yescharacteristic, complemented
209C3×D4—271no—
210C3×D4—271no—
211C2×C3×C4—271no—
212C2×C3×C4—271no—
213#67—91no—
214#67—91no—
215#67—91no—
216#67—271no—
217#67—271no—
218#67—271no—
219#67—271no—
220#67—271no—
221#67—541no—
222C2×Dic3—91no—
223C2×Dic3—91no—
224C2×Dic3—91no—
225C2×Dic3—181no—
226C2×Dic3—271no—
227D12—91no—
228D12—271no—
229C4×D3—91no—
230C4×D3—91no—
231C4×D3—91no—
232C4×D3—271no—
233C4×D3—271no—
234C4×D3—271no—
235C4×D3—541no—
236Dic6—271no—
237Dic6—271no—
238Dic6—271no—
239Dic6—271no—
240Dic6—541no—
241C2×C32—21no—
242C2×C32—21no—
243C2×C32—21no—
244C2×C32—31no—
245C2×C32—31no—
246C2×C32—41no—
247C2×C32—61no—
248C2×C32—61no—
249C2×C32—61no—
250C2×C32—62no—
251C2×C32—61no—
252C2×C32—61no—
253C2×C32—121no—
254C2×C32—181no—
255C2×C32C2×D3211yescharacteristic
256C2×C32C2×D3211yescharacteristic
257C2×C32C2×D3211yescharacteristic
258C3⋊D3—21no—
259C3⋊D3—61no—
260C3⋊D3—61no—
261C3⋊D3—121no—
262C3⋊D3—181no—
263C3⋊D3—182no—
264C3×D3—21no—
265C3×D3—61no—
266C3×D3—61no—
267C3×D3—62no—
268C3×D3—61no—
269C3×D3—61no—
270C3×D3—121no—
271C3×D3—181no—
272C3×D3—181no—
273C3×D3—182no—
274C3×D3—181no—
275#41—811noSylow
276C22×C3—31no—
277C22×C3—91no—
278C22×C3—91no—
279C22×C3—91no—
280C22×C3—91no—
281C22×C3—181no—
282C22×C3—181no—
283C22×C3—271no—
284C2×D3—31no—
285C2×D3—91no—
286C2×D3—91no—
287C2×D3—91no—
288C2×D3—181no—
289C2×D3—271no—
290C2×D3—541no—
291C2×D3((C3⋊D3)⋊C3)⋊C211yescharacteristic
292C3×C4—91no—
293C3×C4—91no—
294C3×C4—91no—
295C3×C4—91no—
296C3×C4—181no—
297C3×C4—181no—
298C3×C4—271no—
299C3×C4—271no—
300Dic3—31no—
301Dic3—31no—
302Dic3—31no—
303Dic3—61no—
304Dic3—91no—
305Dic3—91no—
306Dic3—91no—
307Dic3—91no—
308Dic3—91no—
309Dic3—91no—
310Dic3—181no—
311Dic3—181no—
312Dic3—181no—
313Dic3—181no—
314Dic3—271no—
315Dic3—361no—
316Dic3—541no—
317C32—21no—
318C32—21no—
319C32—21no—
320C32—31no—
321C32—31no—
322C32—41no—
323C32—61no—
324C32—61no—
325C32—61no—
326C32—121no—
327C32#379511yescharacteristic, complemented
328C32#380411yescharacteristic, complemented
329C32#380511yescharacteristic
330Q8—811no—
331D4—271no—
332D4—271no—
333D4—811no—
334C2×C4—271no—
335C2×C4—271no—
336C2×C4—811no—
337C2×C3—21no—
338C2×C3—31no—
339C2×C3—31no—
340C2×C3—61no—
341C2×C3—61no—
342C2×C3—61no—
343C2×C3—61no—
344C2×C3—121no—
345C2×C3—181no—
346C2×C3—181no—
347C2×C3—182no—
348C2×C3—181no—
349C2×C3—181no—
350C2×C3—361no—
351C2×C3—541no—
352C2×C3C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
353C2×C3D3311yescharacteristic
354D3—21no—
355D3—61no—
356D3—181no—
357D3—182no—
358D3—181no—
359D3—181no—
360D3—541no—
361D3—1081no—
362C22—31no—
363C22—91no—
364C22—271no—
365C4—91no—
366C4—91no—
367C4—271no—
368C4—271no—
369C3—21no—
370C3—31no—
371C3—31no—
372C3—61no—
373C3—61no—
374C3—61no—
375C3—121no—
376C3#8867211yescharacteristic, complemented
377C3#8896111yescharacteristic
378C2—61no—
379C2—181no—
380C2—541no—
381C2D3×((C3⋊D3)⋊C3)⋊C211yescharacteristic
subgroupG/H
center, hypercenterC2D3×((C3⋊D3)⋊C3)⋊C2
commutator, Fitting, cosocleC2×C3×H(3,𝔽3)C23
FrattiniC2×C3D33
socleC2×C32C2×D32
hypocenterC3×H(3,𝔽3)#41
LCSG ▷ C2×C3×H(3,𝔽3) ▷ C3×H(3,𝔽3)
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. #10619901C2semidirect
2. D3×#1084C2semidirect
3. #10619884C2semidirect
4. #10619881C2semidirect
5. #6291⋊C3C2semidirect
6. #6290⋊C3C2semidirect
7. C3×((C3⋊D3)⋊C3)⋊C4C2semidirect
22. #6291D3semidirect
23. #6290D3semidirect
39. ((C3⋊D3)⋊C3)⋊C4D3semidirect
82. C3×H(3,𝔽3)#41semidirect
203. C33#237semidirect
204. C33#237semidirect
208. H(3,𝔽3)#235semidirect
327. C32#3795semidirect
328. C32#3804semidirect
376. C3#88672semidirect

Maximal subgroups

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

subgroupindex
#106199012
D3×#10842
#106198842
#106198812
#6291⋊C32
#6290⋊C32
C3×((C3⋊D3)⋊C3)⋊C42
#889603
#889593
#886723