Finite Group Databasebeta

← C3⋊D3×2−1+4⋊C3 #13181473

Subgroups — C3⋊D3×2−1+4⋊C3 #13181473

Order 1728.

1328 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 1328 conjugacy classes of subgroups — too many to draw. Only its 131 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
1C32×2−1+4⋊C3C211yesmaximal, characteristic, complemented
2#11724493C211yesmaximal, characteristic, complemented
3C3⋊D3×#41⋊C3C211yesmaximal, characteristic, complemented
4#11724490C212yesmaximal, complemented
5C2×C3⋊D3×SL(2,𝔽3)C212yesmaximal, complemented
6C3⋊D3×2−1+4C311yesmaximal, characteristic, complemented
7D3×2−1+4⋊C3—34nomaximal
8C3×D4×C3⋊D3—41nomaximal
9C32×#41⋊C3C2211yescharacteristic, complemented
10C2×C32×SL(2,𝔽3)C2212yescomplemented
11C3⋊D3×SL(2,𝔽3)C2211yescharacteristic
12C3⋊D3×SL(2,𝔽3)C2212yescomplemented
13C3⋊(C3⋊(#41⋊C3))C2211yescharacteristic, complemented
14C32×2−1+4C2×C311yescharacteristic, complemented
15#64367—32no—
16#64367—31no—
17#64367C2×C311yescharacteristic, complemented
18#41×C3⋊D3—31no—
19#41×C3⋊D3—32no—
20#41×C3⋊D3C2×C311yescharacteristic, complemented
21#64364—31no—
22#64364C2×C312yescomplemented
23C2×Q8×C3⋊D3—31no—
24C2×Q8×C3⋊D3C2×C312yescomplemented
25C3×2−1+4⋊C3—24no—
26C3×2−1+4⋊C3D314yescomplemented
27#1010⋊C3—34no—
28D3×#41⋊C3—34no—
29#1007⋊C3—38no—
30C2×D3×SL(2,𝔽3)—38no—
31C22×C3×C3⋊D3—42no—
32C33×D4—41no—
33C3×#638—42no—
34C3×C4⋊(C3⋊D3)—41no—
35C3×C4×C3⋊D3—41no—
36C32×SL(2,𝔽3)C2311yescharacteristic
37D3×2−1+4—34no—
38C2×2−1+4⋊C3—91no—
39C32×#41—31no—
40C32×#41—32no—
41C32×#41C22×C311yescharacteristic, complemented
42C2×C32×Q8—31no—
43C2×C32×Q8C22×C312yescomplemented
44#3832—31no—
45#3832—32no—
46#3832C22×C311yescharacteristic, complemented
47Q8×C3⋊D3—32no—
48Q8×C3⋊D3—31no—
49Q8×C3⋊D3—32no—
50Q8×C3⋊D3—32no—
51Q8×C3⋊D3—32no—
52Q8×C3⋊D3C22×C311yescharacteristic
53Q8×C3⋊D3C22×C312yescomplemented
54#3830—34no—
55#3830—31no—
56#3830—32no—
57#3830—32no—
58#3830—31no—
59D4×C3⋊D3—32no—
60D4×C3⋊D3—31no—
61D4×C3⋊D3A411yescharacteristic
62#3828—34no—
63#3828—32no—
64#3828—32no—
65#3828—31no—
66#3828—31no—
67C2×C4×C3⋊D3—32no—
68C2×C4×C3⋊D3—32no—
69C2×C4×C3⋊D3—31no—
70C2×#634—32no—
71C2×#634—32no—
72C2×#634—31no—
73C3×D3×D4—124no—
74C3×#41⋊C3—24no—
75C3×#41⋊C3C2×D314yescomplemented
76C2×C3×SL(2,𝔽3)—28no—
77C2×C3×SL(2,𝔽3)C2×D318yescomplemented
78D3×SL(2,𝔽3)—34no—
79D3×SL(2,𝔽3)—38no—
80C3⋊(#41⋊C3)—34no—
81C22×C33—42no—
82C2×C3×C3⋊D3—41no—
83C2×C3×C3⋊D3—42no—
84C2×C3×C3⋊D3—84no—
85C33×C4—41no—
86C3×C3⋊Dic3—41no—
87C3×2−1+4C3×D314yescomplemented
88#1010—34no—
89#1010—98no—
90#1010—94no—
91D3×#41—34no—
92D3×#41—94no—
93D3×#41—98no—
94#1007—38no—
95#1007—94no—
96C2×D3×Q8—38no—
97C2×D3×Q8—94no—
982−1+4⋊C3—24no—
992−1+4⋊C3—91no—
1002−1+4⋊C3—92no—
1012−1+4⋊C3C3⋊D311yescharacteristic, direct product factor
102C2×#41⋊C3—91no—
103C22×SL(2,𝔽3)—92no—
104C22×C3⋊D3—31no—
105C22×C3⋊D3C2×A412yes—
106C22×C3×D3—128no—
107C32×Q8—32no—
108C32×Q8—31no—
109C32×Q8C23×C311yescharacteristic
110C32×D4—31no—
111C32×D4—32no—
112C32×D4—44no—
113C32×D4—84no—
114C32×D4C2×A411yescharacteristic
115C2×C32×C4—32no—
116C2×C32×C4—32no—
117C2×C32×C4—31no—
118#638—32no—
119#638—32no—
120#638—32no—
121#638C2×A412yes—
122C2×C3⋊Dic3—32no—
123C2×C3⋊Dic3—32no—
124C2×C3⋊Dic3—31no—
125C4⋊(C3⋊D3)—31no—
126C4⋊(C3⋊D3)—32no—
127C4⋊(C3⋊D3)C2×A411yescharacteristic
128C4×C3⋊D3—34no—
129C4×C3⋊D3—32no—
130C4×C3⋊D3—31no—
131C4×C3⋊D3—32no—
132C4×C3⋊D3—32no—
133C4×C3⋊D3—31no—
134C4×C3⋊D3—31no—
135C4×C3⋊D3C2×A411yescharacteristic
136#634—34no—
137#634—31no—
138#634—32no—
139#634—31no—
140#634—32no—
141C3×#67—128no—
142C3×D12—124no—
143C3×C4×D3—124no—
144C3×SL(2,𝔽3)—24no—
145C3×SL(2,𝔽3)C22×D314yes—
146C2×2−1+4—91noSylow
147C2×C33—41no—
148C2×C33—82no—
149C3×C3⋊D3—42no—
150C3×C3⋊D3—82no—
151C3×#41—34no—
152C3×#41—38no—
153C3×#41C2×C3×D314yescomplemented
154C2×C3×Q8—34no—
155C2×C3×Q8C2×C3×D318yescomplemented
156C2×C3×D4—361no—
157#239—34no—
158#239—94no—
159#239—98no—
160D3×Q8—34no—
161D3×Q8—38no—
162D3×Q8—94no—
163D3×Q8—98no—
164D3×Q8—98no—
165D3×Q8—98no—
166D3×Q8—98no—
167#237—98no—
168#237—916no—
169#237—94no—
170#237—98no—
171#237—94no—
172D3×D4—34no—
173D3×D4—98no—
174D3×D4—94no—
175#235—94no—
176#235—98no—
177#235—916no—
178#235—98no—
179#235—94no—
180C2×C4×D3—94no—
181C2×C4×D3—98no—
182C2×C4×D3—98no—
183C2×Dic6—98no—
184C2×Dic6—98no—
185C2×Dic6—94no—
186#41⋊C3—24no—
187#41⋊C3—91no—
188#41⋊C3C2×C3⋊D311yescharacteristic, complemented
189C2×SL(2,𝔽3)—28no—
190C2×SL(2,𝔽3)—91no—
191C2×SL(2,𝔽3)—92no—
192C2×SL(2,𝔽3)C2×C3⋊D312yescomplemented
193C22×C32—31no—
194C22×C32—48no—
195C22×C32—88no—
196C22×C32C22×A412yes—
197C2×C3⋊D3—24no—
198C2×C3⋊D3—31no—
199C2×C3⋊D3—62no—
200C2×C3⋊D3C22×A412yes—
201C2×C3⋊D3C22×A411yescharacteristic
202C2×C3×D3—124no—
203C2×C3×D3—128no—
204C2×C3×D3—2416no—
205C32×C4—32no—
206C32×C4—31no—
207C32×C4—44no—
208C32×C4—84no—
209C32×C4C22×A411yescharacteristic
210C3⋊Dic3—31no—
211C3⋊Dic3—32no—
212C3⋊Dic3C22×A411yescharacteristic
213C3×Dic3—124no—
2142−1+4—91no—
2152−1+4—92no—
2162−1+4—271no—
2172−1+4—271no—
2182−1+4—272no—
2192−1+4C3×C3⋊D311yescharacteristic, complemented
220C2×#41—91no—
221C2×#41—271no—
222C2×#41—272no—
223C22×Q8—92no—
224C22×Q8—271no—
225C33—41noSylow
226C23×C3—362no—
227C22×D3—38no—
228C22×D3—94no—
229C3×Q8—38no—
230C3×Q8—34no—
231C3×Q8C22×C3×D314yes—
232C3×D4—34no—
233C3×D4—38no—
234C3×D4—41no—
235C3×D4—84no—
236C3×D4—361no—
237C3×D4—362no—
238C3×D4D3×A414yes—
239C2×C3×C4—38no—
240C2×C3×C4—38no—
241C2×C3×C4—34no—
242C2×C3×C4—361no—
243#67—38no—
244#67—98no—
245#67—98no—
246#67—98no—
247C2×Dic3—94no—
248C2×Dic3—98no—
249C2×Dic3—98no—
250D12—34no—
251D12—98no—
252D12—94no—
253C4×D3—34no—
254C4×D3—94no—
255C4×D3—94no—
256C4×D3—98no—
257C4×D3—98no—
258C4×D3—98no—
259C4×D3—916no—
260C4×D3—94no—
261Dic6—98no—
262Dic6—916no—
263Dic6—94no—
264Dic6—94no—
265Dic6—98no—
266SL(2,𝔽3)—24no—
267SL(2,𝔽3)C22×C3⋊D311yescharacteristic
268C2×C32—22no—
269C2×C32—44no—
270C2×C32—61no—
271C2×C32—88no—
272C2×C32—84no—
273C2×C32—168no—
274C2×C32C23×A411yescharacteristic
275C3⋊D3—22no—
276C3⋊D3—61no—
277C3⋊D32−1+4⋊C312yesdirect product factor
278C3×D3—128no—
279C3×D3—248no—
280#41—31no—
281#41—32no—
282#41—91no—
283#41—272no—
284#41—271no—
285#41—274no—
286#41—272no—
287#41—271no—
288#41—272no—
289#41—274no—
290#41—272no—
291#41—271no—
292#41—271no—
293#41—272no—
294#41—271no—
295#41C2×C3×C3⋊D311yescharacteristic, complemented
296C2×Q8—31no—
297C2×Q8—91no—
298C2×Q8—92no—
299C2×Q8—271no—
300C2×Q8—272no—
301C2×Q8—272no—
302C2×Q8—271no—
303C2×Q8—272no—
304C2×Q8—272no—
305C2×Q8—272no—
306C2×Q8—272no—
307C2×Q8C2×C3×C3⋊D312yescomplemented
308C2×D4—91no—
309C2×D4—272no—
310C2×D4—271no—
311C22×C4—271no—
312C22×C4—272no—
313C22×C4—272no—
314C22×C3—34no—
315C22×C3—42no—
316C22×C3—88no—
317C22×C3—361no—
318C22×C3—362no—
319C22×C3—724no—
320C22×C3C2×D3×A418yes—
321C2×D3—34no—
322C2×D3—38no—
323C2×D3—616no—
324C2×D3—94no—
325C2×D3—188no—
326C3×C4—34no—
327C3×C4—38no—
328C3×C4—41no—
329C3×C4—84no—
330C3×C4—361no—
331C3×C4C2×D3×A414yes—
332Dic3—34no—
333Dic3—98no—
334Dic3—94no—
335C32—44no—
336C32—84no—
337C32C2×2−1+4⋊C311yescharacteristic, complemented
338C23—92no—
339C23—271no—
340Q8—32no—
341Q8—31no—
342Q8—272no—
343Q8—271no—
344Q8—272no—
345Q8—274no—
346Q8—271no—
347Q8C22×C3×C3⋊D311yescharacteristic
348D4—31no—
349D4—32no—
350D4—92no—
351D4—91no—
352D4—272no—
353D4—271no—
354D4—272no—
355D4—272no—
356D4—272no—
357D4A4×C3⋊D311yescharacteristic
358C2×C4—32no—
359C2×C4—32no—
360C2×C4—31no—
361C2×C4—91no—
362C2×C4—271no—
363C2×C4—271no—
364C2×C4—272no—
365C2×C4—272no—
366C2×C4—272no—
367C2×C4—274no—
368C2×C4—271no—
369C2×C4—272no—
370C2×C4—272no—
371C2×C4—271no—
372C2×C3—28no—
373C2×C3—41no—
374C2×C3—64no—
375C2×C3—82no—
376C2×C3—84no—
377C2×C3—168no—
378C2×C3—362no—
379C2×C3—722no—
380C2×C3C22×D3×A414yes—
381D3—38no—
382D3—68no—
383D3—184no—
384C22—31no—
385C22—91no—
386C22—92no—
387C22—184no—
388C22—271no—
389C22—542no—
390C22C2×A4×C3⋊D312yes—
391C4—31no—
392C4—32no—
393C4—91no—
394C4—271no—
395C4—272no—
396C4C2×A4×C3⋊D311yescharacteristic
397C3—41no—
398C3—84no—
399C3D3×2−1+4⋊C314yescomplemented
400C2—22no—
401C2—61no—
402C2—92no—
403C2—182no—
404C2—541no—
405C2C22×A4×C3⋊D311yescharacteristic
subgroupG/H
center, FrattiniC2C22×A4×C3⋊D3
commutator, cosocle, hypocenterC32×Q8C23×C3
FittingC32×2−1+4C2×C3
socleC2×C32C23×A4
hypercenterD4A4×C3⋊D3
LCSG ▷ C32×Q8
UCSC1 ◁ C2 ◁ D4

Complements

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

NH ≅ G/Ntype
1. C32×2−1+4⋊C3C2semidirect
2. #11724493C2semidirect
3. C3⋊D3×#41⋊C3C2semidirect
4. #11724490C2semidirect
5. C2×C3⋊D3×SL(2,𝔽3)C2semidirect
6. C3⋊D3×2−1+4C3semidirect
9. C32×#41⋊C3C22semidirect
10. C2×C32×SL(2,𝔽3)C22semidirect
12. C3⋊D3×SL(2,𝔽3)C22semidirect
13. C3⋊(C3⋊(#41⋊C3))C22semidirect
14. C32×2−1+4C2×C3semidirect
17. #64367C2×C3semidirect
20. #41×C3⋊D3C2×C3semidirect
22. #64364C2×C3semidirect
24. C2×Q8×C3⋊D3C2×C3semidirect
26. C3×2−1+4⋊C3D3semidirect
41. C32×#41C22×C3semidirect
43. C2×C32×Q8C22×C3semidirect
46. #3832C22×C3semidirect
53. Q8×C3⋊D3C22×C3semidirect
75. C3×#41⋊C3C2×D3semidirect
77. C2×C3×SL(2,𝔽3)C2×D3semidirect
87. C3×2−1+4C3×D3semidirect
101. 2−1+4⋊C3C3⋊D3direct (N × H)
153. C3×#41C2×C3×D3semidirect
155. C2×C3×Q8C2×C3×D3semidirect
188. #41⋊C3C2×C3⋊D3semidirect
192. C2×SL(2,𝔽3)C2×C3⋊D3semidirect
219. 2−1+4C3×C3⋊D3semidirect
277. C3⋊D32−1+4⋊C3direct (N × H)
295. #41C2×C3×C3⋊D3semidirect
307. C2×Q8C2×C3×C3⋊D3semidirect
337. C32C2×2−1+4⋊C3semidirect
399. C3D3×2−1+4⋊C3semidirect

Maximal subgroups

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

subgroupindex
C32×2−1+4⋊C32
#117244932
C3⋊D3×#41⋊C32
#117244902
C2×C3⋊D3×SL(2,𝔽3)2
C3⋊D3×2−1+43
D3×2−1+4⋊C33
C3×D4×C3⋊D34