Finite Group Databasebeta

← #13159041

Subgroups — #13159041

Order 1728.

812 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 812 conjugacy classes of subgroups — too many to draw. Only its 150 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#11723369C211yesmaximal, characteristic
2#11723356C211yesmaximal, characteristic, complemented
3#11723352C211yesmaximal, characteristic, complemented
4#11722175C211yesmaximal, characteristic, complemented
5C3×#63632C211yesmaximal, characteristic, complemented
6C32×C4×C3⋊C8C211yesmaximal, characteristic, complemented
7#11722058C211yesmaximal, characteristic, complemented
8#10592399—31nomaximal
9#10590341—34nomaximal
10#89076C2211yescharacteristic
11C3×#3828C2211yescharacteristic
12C33×C42C2211yescharacteristic
13#88855C2211yescharacteristic
14C3×#3748C2211yescharacteristic
15C2×C32×C3⋊C8C2211yescharacteristic
16C2×C32×C3⋊C8C2211yescharacteristic
17#88800C411yescharacteristic, complemented
18#88797C411yescharacteristic, complemented
19#88796C411yescharacteristic, complemented
20#88791C411yescharacteristic, complemented
21#88788—21no—
22#88787—21no—
23#88785—21no—
24C3⋊D3×C3⋊C8—21no—
25#64113—31no—
26#64110—31no—
27#64110—31no—
28#63829—34no—
29#63817—34no—
30#63814—34no—
31C32×C4×C8—31no—
32#63646—31no—
33#63632—31no—
34#63632—34no—
35#63632—64no—
36#63632D311yescharacteristic, complemented
37C3×#805—34no—
38C3×C4×C3⋊C8D314yescomplemented
39#63577—34no—
40C2×C33×C4—21no—
41C2×C33×C4C2311yescharacteristic
42#6312—21no—
43C4⋊(C3⋊(C3⋊D3))C2×C411yescharacteristic
44C4×C3⋊(C3⋊D3)—21no—
45#6308C2×C411yescharacteristic
46C3×#638—21no—
47C3×C4⋊(C3⋊D3)C2×C411yescharacteristic
48C3×C4×C3⋊D3—21no—
49C3×#634C2×C411yescharacteristic
50C3⋊(C3⋊(C3⋊C8))—21no—
51C3×C3⋊(C3⋊C8)—21no—
52C32×C3⋊C8C2×C411yescharacteristic
53C32×C3⋊C8C2×C411yescharacteristic
54C32×C3⋊C8D411yescharacteristic
55C32×C3⋊C8D411yescharacteristic
56#4824—91no—
57#4700—94no—
58#3828—31no—
59#3828—34no—
60#3828—64no—
61#3828C2×D311yescharacteristic
62C3×#235—34no—
63C2×C32×C8—31no—
64C2×C32×C8—31no—
65C32×C42—24no—
66C32×C42C2×D314yes—
67C32×C42C2×D311yescharacteristic
68#3748—31no—
69#3748—34no—
70#3748—64no—
71#3748C2×D311yescharacteristic
72#3746—31no—
73#3745—31no—
74#3744—31no—
75#3744—31no—
76#3743—61no—
77#3743—61no—
78C8×C3⋊D3—61no—
79C8×C3⋊D3—61no—
80C3×#208—34no—
81C2×C3×C3⋊C8C2×D314yes—
82C2×C3×C3⋊C8C2×D314yes—
83#3719—34no—
84#3717—34no—
85#3716—34no—
86#3714—34no—
87#3712—64no—
88#3711—64no—
89#3710—64no—
90D3×C3⋊C8—64no—
91C22×C33#4111yescharacteristic
92C2×C3⋊(C3⋊D3)—21no—
93C2×C3×C3⋊D3—21no—
94C33×C4—22no—
95C33×C4C22×C411yescharacteristic
96C33×C4C2×D411yescharacteristic
97C3⋊(C3⋊Dic3)—21no—
98C3×C3⋊Dic3—21no—
99#950—91no—
100#948—91no—
101#907—91no—
102#904—94no—
103#901—94no—
104#901—94no—
105C3×#104—91no—
106C3×C4×C8—34no—
107#822—91no—
108#819—94no—
109#805—34no—
110#805—94no—
111#805—91no—
112#805—184no—
113C4×C3⋊C8C3⋊D311yescharacteristic, complemented
114C2×C32×C4—21no—
115C2×C32×C4—24no—
116C2×C32×C4—24no—
117C2×C32×C4—44no—
118C2×C32×C4C22×D314yes—
119C2×C32×C4C22×D311yescharacteristic
120#638—21no—
121#638—61no—
122#638—64no—
123#638—124no—
124C4⋊(C3⋊D3)—31no—
125C4⋊(C3⋊D3)—34no—
126C4⋊(C3⋊D3)—64no—
127C4⋊(C3⋊D3)C4×D311yescharacteristic
128C4×C3⋊D3—21no—
129C4×C3⋊D3—61no—
130C4×C3⋊D3—64no—
131C4×C3⋊D3—124no—
132#634—31no—
133#634—34no—
134#634—64no—
135#634C4×D311yescharacteristic
136C3×#67—64no—
137C3×D12—34no—
138C3×C4×D3—64no—
139C3×Dic6—34no—
140C32×C8—31no—
141C32×C8—31no—
142C32×C8—31no—
143C32×C8—31no—
144C3⋊(C3⋊C8)—21no—
145C3⋊(C3⋊C8)—61no—
146C3⋊(C3⋊C8)—64no—
147C3⋊(C3⋊C8)—124no—
148C3×C3⋊C8—64no—
149C3×C3⋊C8C4×D314yes—
150C3×C3⋊C8C4×D314yes—
151C3×C3⋊C8D1214yes—
152C3×C3⋊C8D1214yes—
153#443—271noSylow
154C2×C33—21no—
155C2×C33C4×D411yescharacteristic
156C3⋊(C3⋊D3)—41no—
157C3×C3⋊D3—41no—
158C3×#41—91no—
159#235—34no—
160#235—94no—
161#235—91no—
162#235—184no—
163C3×M16—91no—
164C2×C3×C8—34no—
165C2×C3×C8—34no—
166C3×C42—24no—
167C3×C42D3214yes—
168C3×C42C2×C3⋊D311yescharacteristic
169#216—91no—
170#215—91no—
171#214—91no—
172#213—91no—
173#208—34no—
174#208—94no—
175#208—91no—
176#208—184no—
177#208—181no—
178C2×C3⋊C8—181no—
179C2×C3⋊C8C2×C3⋊D311yescharacteristic
180C2×C3⋊C8C2×C3⋊D311yescharacteristic
181Dic12—94no—
182D24—94no—
183#204—94no—
184#204—94no—
185ΓL(1,𝔽25)—184no—
186ΓL(1,𝔽25)—184no—
187ΓL(1,𝔽25)—181no—
188D3×C8—184no—
189D3×C8—184no—
190D3×C8—181no—
191C22×C32—24no—
192C22×C32#23514yes—
193C22×C32#23711yescharacteristic
194C2×C3⋊D3—21no—
195C2×C3⋊D3—61no—
196C2×C3⋊D3—64no—
197C2×C3⋊D3—124no—
198C2×C3×D3—64no—
199C32×C4—22no—
200C32×C4—24no—
201C32×C4—24no—
202C32×C4—28no—
203C32×C4—48no—
204C32×C4C2×C4×D314yes—
205C32×C4C2×C4×D311yescharacteristic
206C32×C4C2×D1214yes—
207C32×C4D3×D411yescharacteristic
208C3⋊Dic3—21no—
209C3⋊Dic3—61no—
210C3⋊Dic3—64no—
211C3⋊Dic3—124no—
212C3×Dic3—64no—
213#135—271no—
214#131—271no—
215#131—271no—
216#108—271no—
217#104—91no—
218#104—271no—
219C4×C8—31no—
220C33#44311yescharacteristic, complemented, Sylow
221C3×Q8—91no—
222C3×D4—91no—
223C3×D4—181no—
224C2×C3×C4—24no—
225C2×C3×C4—21no—
226C2×C3×C4—24no—
227C2×C3×C4—44no—
228C2×C3×C4—181no—
229C2×C3×C4C2×D3214yes—
230C2×C3×C4C22×C3⋊D311yescharacteristic
231#67—64no—
232#67—184no—
233#67—181no—
234#67—364no—
235D12—34no—
236D12—94no—
237D12—91no—
238D12—184no—
239C4×D3—64no—
240C4×D3—184no—
241C4×D3—181no—
242C4×D3—364no—
243Dic6—34no—
244Dic6—94no—
245Dic6—91no—
246Dic6—184no—
247C3×C8—34no—
248C3×C8—34no—
249C3×C8—34no—
250C3×C8—34no—
251C3×C8—181no—
252C3⋊C8—64no—
253C3⋊C8—184no—
254C3⋊C8—181no—
255C3⋊C8—364no—
256C3⋊C8C4×C3⋊D311yescharacteristic
257C3⋊C8C4×C3⋊D311yescharacteristic
258C3⋊C8C4⋊(C3⋊D3)11yescharacteristic
259C3⋊C8C4⋊(C3⋊D3)11yescharacteristic
260C2×C32—21no—
261C2×C32—24no—
262C2×C32—24no—
263C2×C32—44no—
264C2×C32C4×D1214yes—
265C2×C32#67⋊C411yescharacteristic
266C3⋊D3—41no—
267C3⋊D3—121no—
268C3⋊D3—124no—
269C3⋊D3—244no—
270C3×D3—124no—
271#41—91no—
272#41—271no—
273Q16—271no—
274SD16—271no—
275SD16—271no—
276D8—271no—
277M16—91no—
278M16—271no—
279M16—541no—
280M16—541no—
281C2×C8—31no—
282C2×C8—31no—
283C2×C8—541no—
284C2×C8—541no—
285C42D3×C3⋊D311yescharacteristic
286C22×C3—24no—
287C22×C3—181no—
288C22×C3#380414yes—
289C22×C3#382811yescharacteristic
290C2×D3—64no—
291C2×D3—181no—
292C2×D3—184no—
293C2×D3—364no—
294C3×C4—22no—
295C3×C4—28no—
296C3×C4—24no—
297C3×C4—24no—
298C3×C4—48no—
299C3×C4—181no—
300C3×C4C4×D3214yes—
301C3×C4D3×D1214yes—
302C3×C4C2×C4×C3⋊D311yescharacteristic
303C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
304Dic3—64no—
305Dic3—181no—
306Dic3—184no—
307Dic3—364no—
308C32—24no—
309C32#470014yescomplemented
310C32#482411yescharacteristic, complemented
311Q8—91no—
312Q8—271no—
313D4—91no—
314D4—181no—
315D4—271no—
316D4—541no—
317C2×C4—21no—
318C2×C4—181no—
319C2×C4—541no—
320C2×C4C2×D3×C3⋊D311yescharacteristic
321C8—31no—
322C8—31no—
323C8—31no—
324C8—31no—
325C8—181no—
326C8—541no—
327C2×C3—24no—
328C2×C3—21no—
329C2×C3—24no—
330C2×C3—44no—
331C2×C3—361no—
332C2×C3#6388014yes—
333C2×C3C4×C4⋊(C3⋊D3)11yescharacteristic
334D3—124no—
335D3—364no—
336D3—361no—
337D3—724no—
338C22—181no—
339C22—541no—
340C22#8903111yescharacteristic
341C4—22no—
342C4—181no—
343C4—541no—
344C4C4×D3×C3⋊D311yescharacteristic
345C4D3×C4⋊(C3⋊D3)11yescharacteristic
346C3—24no—
347C3#1059034114yescomplemented
348C3#1059239911yescharacteristic, complemented
349C2—21no—
350C2—361no—
351C2—1081no—
352C2#1172347111yescharacteristic
subgroupG/H
centerC4D3×C4⋊(C3⋊D3)
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C42C22
socleC2×C33C4×D4
prime radical#11723369C2
cosocleC2×C33×C4C23
hypercenterC42D3×C3⋊D3
hypocenterC33#443
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
2. #11723356C2semidirect
3. #11723352C2semidirect
4. #11722175C2semidirect
5. C3×#63632C2semidirect
6. C32×C4×C3⋊C8C2semidirect
7. #11722058C2semidirect
17. #88800C4semidirect
18. #88797C4semidirect
19. #88796C4semidirect
20. #88791C4semidirect
36. #63632D3semidirect
38. C3×C4×C3⋊C8D3semidirect
113. C4×C3⋊C8C3⋊D3semidirect
220. C33#443semidirect
309. C32#4700semidirect
310. C32#4824semidirect
347. C3#10590341semidirect
348. C3#10592399semidirect

Maximal subgroups

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

subgroupindex
#117233692
#117233562
#117233522
#117221752
C3×#636322
C32×C4×C3⋊C82
#117220582
#105923993
#105903413