Finite Group Databasebeta

← #13048331

Subgroups — #13048331

Order 1296.

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

normal subgroups only

This group has 392 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#10619904C211yesmaximal, characteristic, complemented
2Dic3×#272C211yesmaximal, characteristic, complemented
3#10619884C211yesmaximal, characteristic, complemented
4#10619881C211yesmaximal, characteristic, complemented
5C3×#6200C211yesmaximal, characteristic, complemented
6#6289⋊C3C211yesmaximal, characteristic, complemented
7Dic3×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
8#88960—31nomaximal
9#88959—31nomaximal
10#88650—31nomaximal
11C2×C3×#272C2211yescharacteristic
12C2×C3⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×C3×(C3⋊D3)⋊C3C2211yescharacteristic
14C3⋊#1084C2211yescharacteristic
15C3⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16Dic3×H(3,𝔽3)C2211yescharacteristic
17C3×(C3⋊Dic3)⋊C3C2211yescharacteristic
18#6296—31no—
19#6295—31no—
20#6293—31no—
21#6291—31no—
22#6291—31no—
23#6290—31no—
24#6289D311yescharacteristic, complemented
25Dic3×C3⋊D3D311yescharacteristic, complemented
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—
33C4×#272—31no—
34C4×(C3⋊D3)⋊C3—31no—
35C4×(C3⋊D3)⋊C3—31no—
36#6200—31no—
37#6200D311yescharacteristic, complemented
38#6198—31no—
39#6198—61no—
40#6196—31no—
41#6196—61no—
42C2×C3×H(3,𝔽3)C2311yescharacteristic
43C3×#272—21no—
44C3⋊((C3⋊D3)⋊C3)—21no—
45C3×(C3⋊D3)⋊C3—21no—
46#3804—91no—
47#3804—91no—
48#3798—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—31no—
55C2×C32×D3—31no—
56C2×C32×D3—31no—
57C2×C32×D3—61no—
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—21no—
65C32×Dic3—31no—
66C32×Dic3C2×D311yescharacteristic
67C32×Dic3C2×D311yescharacteristic
68C2×#272C2×D311yescharacteristic
69C2×(C3⋊D3)⋊C3—21no—
70C2×(C3⋊D3)⋊C3—31no—
71C2×(C3⋊D3)⋊C3—61no—
72C2×(C3⋊D3)⋊C3C2×D311yescharacteristic
73C4×H(3,𝔽3)—31no—
74#1084—31no—
75#1084—61no—
76#1084—61no—
77#1084—121no—
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—91no—
84#638—91no—
85C4×C3⋊D3—31no—
86C4×C3⋊D3—31no—
87#634—91no—
88C3×#67—91no—
89C3×#67—91no—
90C2×C3×Dic3—91no—
91C2×C3×Dic3—91no—
92C3×D12—91no—
93C3×C4×D3—91no—
94C3×C4×D3—91no—
95C3×C4×D3—91no—
96C3×C4×D3—91no—
97C3×C4×D3—91no—
98C3×C4×D3—181no—
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—91no—
110#626—181no—
111#626—181no—
112#625—31no—
113#625—91no—
114#625—91no—
115#625—91no—
116#625—181no—
117#624—31no—
118#624—91no—
119D3×Dic3—31no—
120D3×Dic3—31no—
121D3×Dic3—31no—
122D3×Dic3—61no—
123D3×Dic3—91no—
124C2×C33—21no—
125C2×C33C22×D311yescharacteristic
126C2×C33C22×D311yescharacteristic
127C3⋊(C3⋊D3)—21no—
128C3×C3⋊D3—21no—
129C3×C3⋊D3—61no—
130C32×D3—62no—
131C32×D3—61no—
132C32×D3—61no—
133C32×D3—61no—
134C2×H(3,𝔽3)—21no—
135C2×H(3,𝔽3)—21no—
136C2×H(3,𝔽3)—41no—
137C2×H(3,𝔽3)C22×D311yescharacteristic
138#272—21no—
139(C3⋊D3)⋊C3—21no—
140(C3⋊D3)⋊C3—41no—
141(C3⋊D3)⋊C3—61no—
142(C3⋊D3)⋊C3—121no—
143#237—271no—
144#235—271no—
145#235—271no—
146#235—271no—
147C22×C32—91no—
148C22×C32—91no—
149C2×C3⋊D3—31no—
150C2×C3⋊D3—31no—
151C2×C3⋊D3—61no—
152C2×C3⋊D3—91no—
153C2×C3⋊D3—181no—
154C2×C3⋊D3D3211yescharacteristic
155C2×C3×D3—31no—
156C2×C3×D3—31no—
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—61no—
164C2×C3×D3—61no—
165C2×C3×D3—61no—
166C2×C3×D3—91no—
167C2×C3×D3—91no—
168C2×C3×D3—91no—
169C2×C3×D3—121no—
170C2×C3×D3—181no—
171C32×C4—31no—
172C32×C4—31no—
173C32×C4—61no—
174C32×C4—91no—
175C3⋊Dic3—31no—
176C3⋊Dic3—31no—
177C3⋊Dic3—31no—
178C3⋊Dic3—31no—
179C3⋊Dic3—61no—
180C3⋊Dic3—61no—
181C3⋊Dic3—91no—
182C3⋊Dic3—181no—
183C3⋊Dic3D3211yescharacteristic
184C3×Dic3—31no—
185C3×Dic3—31no—
186C3×Dic3—31no—
187C3×Dic3—31no—
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—181no—
197C3×Dic3—181no—
198C3×Dic3—181no—
199C3×Dic3—181no—
200C3×Dic3—361no—
201C3×Dic3D3211yescharacteristic
202C33—21no—
203C33#23511yescharacteristic, complemented
204C33#23511yescharacteristic, complemented
205H(3,𝔽3)—21no—
206H(3,𝔽3)—21no—
207H(3,𝔽3)—41no—
208H(3,𝔽3)#23711yescharacteristic, complemented
209C3×D4—271no—
210C2×C3×C4—271no—
211C2×C3×C4—271no—
212C2×C3×C4—271no—
213#67—91no—
214#67—91no—
215#67—271no—
216#67—271no—
217#67—271no—
218#67—271no—
219#67—271no—
220#67—271no—
221#67—541no—
222#67—541no—
223C2×Dic3—91no—
224C2×Dic3—91no—
225D12—91no—
226D12—271no—
227D12—271no—
228C4×D3—91no—
229C4×D3—91no—
230C4×D3—91no—
231C4×D3—91no—
232C4×D3—91no—
233C4×D3—181no—
234C4×D3—271no—
235C4×D3—271no—
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—121no—
251C2×C32—181no—
252C2×C32—181no—
253C2×C32C2×D3211yescharacteristic
254C2×C32C2×D3211yescharacteristic
255C2×C32C2×D3211yescharacteristic
256C3⋊D3—21no—
257C3⋊D3—61no—
258C3⋊D3—61no—
259C3⋊D3—121no—
260C3⋊D3—181no—
261C3⋊D3—361no—
262C3×D3—62no—
263C3×D3—61no—
264C3×D3—61no—
265C3×D3—62no—
266C3×D3—61no—
267C3×D3—61no—
268C3×D3—61no—
269C3×D3—61no—
270C3×D3—121no—
271C3×D3—121no—
272C3×D3—121no—
273C3×D3—122no—
274C3×D3—181no—
275C3×D3—181no—
276C3×D3—181no—
277C3×D3—361no—
278#41—811noSylow
279C22×C3—91no—
280C22×C3—91no—
281C22×C3—91no—
282C22×C3—91no—
283C22×C3—181no—
284C22×C3—181no—
285C22×C3—271no—
286C2×D3—31no—
287C2×D3—31no—
288C2×D3—31no—
289C2×D3—61no—
290C2×D3—91no—
291C2×D3—91no—
292C2×D3—91no—
293C2×D3—181no—
294C2×D3—271no—
295C2×D3—541no—
296C3×C4—31no—
297C3×C4—91no—
298C3×C4—91no—
299C3×C4—91no—
300C3×C4—91no—
301C3×C4—181no—
302C3×C4—181no—
303C3×C4—271no—
304C3×C4—271no—
305Dic3—31no—
306Dic3—91no—
307Dic3—91no—
308Dic3—91no—
309Dic3—91no—
310Dic3—91no—
311Dic3—91no—
312Dic3—181no—
313Dic3—181no—
314Dic3—181no—
315Dic3—181no—
316Dic3—271no—
317Dic3—361no—
318Dic3—541no—
319Dic3((C3⋊D3)⋊C3)⋊C211yescharacteristic
320C32—21no—
321C32—21no—
322C32—21no—
323C32—31no—
324C32—31no—
325C32—41no—
326C32—61no—
327C32—61no—
328C32—61no—
329C32—121no—
330C32#379511yescharacteristic, complemented
331C32#379811yescharacteristic
332C32#380411yescharacteristic, complemented
333Q8—811no—
334D4—271no—
335D4—811no—
336D4—811no—
337C2×C4—271no—
338C2×C4—271no—
339C2×C4—271no—
340C2×C3—21no—
341C2×C3—31no—
342C2×C3—31no—
343C2×C3—61no—
344C2×C3—61no—
345C2×C3—61no—
346C2×C3—121no—
347C2×C3—181no—
348C2×C3—181no—
349C2×C3—181no—
350C2×C3—181no—
351C2×C3—361no—
352C2×C3—361no—
353C2×C3—541no—
354C2×C3C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
355C2×C3D3311yescharacteristic
356D3—61no—
357D3—61no—
358D3—62no—
359D3—61no—
360D3—181no—
361D3—181no—
362D3—181no—
363D3—361no—
364D3—541no—
365D3—1081no—
366C22—91no—
367C22—91no—
368C22—271no—
369C4—31no—
370C4—91no—
371C4—271no—
372C4—271no—
373C3—21no—
374C3—31no—
375C3—31no—
376C3—61no—
377C3—61no—
378C3—61no—
379C3—121no—
380C3#8865011yescharacteristic, complemented
381C3#8896011yescharacteristic
382C2—181no—
383C2—181no—
384C2—541no—
385C2D3×((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. #10619904C2semidirect
2. Dic3×#272C2semidirect
3. #10619884C2semidirect
4. #10619881C2semidirect
5. C3×#6200C2semidirect
6. #6289⋊C3C2semidirect
7. Dic3×(C3⋊D3)⋊C3C2semidirect
24. #6289D3semidirect
25. Dic3×C3⋊D3D3semidirect
37. #6200D3semidirect
82. C3×H(3,𝔽3)#41semidirect
203. C33#235semidirect
204. C33#235semidirect
208. H(3,𝔽3)#237semidirect
330. C32#3795semidirect
332. C32#3804semidirect
380. C3#88650semidirect

Maximal subgroups

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

subgroupindex
#106199042
Dic3×#2722
#106198842
#106198812
C3×#62002
#6289⋊C32
Dic3×(C3⋊D3)⋊C32
#889603
#889593
#886503