Finite Group Databasebeta

← C4×#3754⋊C3 #13142744

Subgroups — C4×#3754⋊C3 #13142744

Order 1728.

521 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×D4×H(3,𝔽3)C211yesmaximal, characteristic
2C2×#3754⋊C3C211yesmaximal, characteristic
3C4×#634⋊C3C211yesmaximal, characteristic, complemented
4#63659⋊C3C211yesmaximal, characteristic
5#63637⋊C3C211yesmaximal, characteristic, complemented
6#63635⋊C3C211yesmaximal, characteristic, complemented
7C4×(C3⋊(C3⋊C8))⋊C3C211yesmaximal, characteristic, complemented
8C4×#3754C311yesmaximal, characteristic, complemented
9C3×C4×#214—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C22×C4×H(3,𝔽3)—21no—
12C2×#634⋊C3C2211yescharacteristic
13C4⋊C4×H(3,𝔽3)C2211yescharacteristic
14C22⋊C4×H(3,𝔽3)—21no—
15C42×H(3,𝔽3)C2211yescharacteristic
16#3754⋊C3C414yesdirect product factor
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18#3750⋊C3—21no—
19C4×(C3⋊Dic3)⋊C3—21no—
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
22C32×C4×D4—21no—
23C32×C4×D4D311yescharacteristic
24C32×C4×D4C2×C311yescharacteristic
25C2×#3754C2×C311yescharacteristic
26C4×#634C2×C311yescharacteristic, complemented
27C2×C3×#214—31no—
28C3×C4×Dic6—31no—
29#63659C2×C311yescharacteristic
30#63637C2×C311yescharacteristic, complemented
31#63635C2×C311yescharacteristic, complemented
32C4×C3⋊(C3⋊C8)C2×C311yescharacteristic, complemented
33C3×#832—31no—
34C3×#810—31no—
35C3×#808—31no—
36C3×C4×C3⋊C8—31no—
37C23×H(3,𝔽3)—21no—
38D4×H(3,𝔽3)—21no—
39D4×H(3,𝔽3)C2×C412yes—
40C2×C4×H(3,𝔽3)—21no—
41C2×C4×H(3,𝔽3)—42no—
42C2×C4×H(3,𝔽3)D411yescharacteristic
43C2×C4×H(3,𝔽3)D411yescharacteristic
44C2×C4×H(3,𝔽3)C2311yescharacteristic
45C2×(C3⋊Dic3)⋊C3—21no—
46C2×(C3⋊Dic3)⋊C3—21no—
47#634⋊C3—21no—
48#634⋊C3C2×C412yescomplemented
49(C3⋊(C3⋊C8))⋊C3—21no—
50(C3⋊(C3⋊C8))⋊C3C2×C412yescomplemented
51C3×C4×SD16—91no—
52C4×#214—31no—
53C4×#214—91no—
54C2×C32×D4—21no—
55C2×C32×D4C2×D311yescharacteristic
56C2×C32×D4C22×C311yescharacteristic
57C22×C32×C4—22no—
58C22×C32×C4—21no—
59C22×C32×C4—21no—
60C2×#634C22×C311yescharacteristic
61C2×C3×Dic6—31no—
62C32×C4⋊C4—21no—
63C32×C4⋊C4C2×D311yescharacteristic
64C32×C4⋊C4C22×C311yescharacteristic
65C32×C22⋊C4—22no—
66C32×C22⋊C4—21no—
67C32×C22⋊C4—21no—
68C32×C42—21no—
69C32×C42C2×D311yescharacteristic
70C32×C42C22×C311yescharacteristic
71#3754C3×C414yescomplemented
72C4⋊(C3⋊Dic3)C22×C311yescharacteristic
73#3750—21no—
74C4×C3⋊Dic3—21no—
75C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
76C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
77C3×#214—34no—
78C3×C4⋊Dic3—31no—
79C3×Dic3⋊C4—61no—
80C3×C4×Dic3—61no—
81C2×C3×C3⋊C8—31no—
82C2×C3×C3⋊C8—31no—
83C22×H(3,𝔽3)—22no—
84C22×H(3,𝔽3)—41no—
85C22×H(3,𝔽3)—41no—
86C22×H(3,𝔽3)C2×D411yescharacteristic
87C4×H(3,𝔽3)—21no—
88C4×H(3,𝔽3)—41no—
89C4×H(3,𝔽3)SD1612yescomplemented
90C4×H(3,𝔽3)C22×C411yescharacteristic
91C4×H(3,𝔽3)#4111yescharacteristic
92(C3⋊Dic3)⋊C3—22no—
93(C3⋊Dic3)⋊C3—41no—
94C2×C3×SD16—91no—
95C3×C4×Q8—91no—
96C3×C4×D4—31no—
97C3×C4×D4—31no—
98C3×C4×D4—61no—
99C3×C4×D4C3×D311yescharacteristic
100C2×#214—31no—
101C2×#214—91no—
102C4×Dic6—31no—
103C4×Dic6—91no—
104C3×#106—91no—
105C3×#103—91no—
106C3×#102—91no—
107C3×C4×C8—91no—
108#832—31no—
109#832—91no—
110#810—31no—
111#810—91no—
112#808—31no—
113#808—91no—
114C4×C3⋊C8—31no—
115C4×C3⋊C8—91no—
116C23×C32—22no—
117C23×C32—21no—
118C23×C32—21no—
119C32×D4—22no—
120C32×D4—21no—
121C32×D4—21no—
122C32×D4—22no—
123C32×D4C4×D312yes—
124C32×D4C2×C3×C412yes—
125C2×C32×C4—22no—
126C2×C32×C4—21no—
127C2×C32×C4—21no—
128C2×C32×C4—21no—
129C2×C32×C4—21no—
130C2×C32×C4—21no—
131C2×C32×C4—44no—
132C2×C32×C4—42no—
133C2×C32×C4—42no—
134C2×C32×C4#6711yescharacteristic
135C2×C32×C4#6711yescharacteristic
136C2×C32×C4C3×D411yescharacteristic
137C2×C32×C4C3×D411yescharacteristic
138C2×C32×C4C22×D311yescharacteristic
139C2×C32×C4C23×C311yescharacteristic
140C2×C3⋊Dic3—21no—
141C2×C3⋊Dic3—21no—
142#634—21no—
143#634C2×C3×C412yescomplemented
144C2×C3×Dic3—61no—
145C2×C3×Dic3—61no—
146C3×Dic6—32no—
147C3×Dic6—61no—
148C3⋊(C3⋊C8)—21no—
149C3⋊(C3⋊C8)C2×C3×C412yescomplemented
150C3×C3⋊C8—32no—
151C3×C3⋊C8—61no—
152C4×SD16—91noSylow
153C2×H(3,𝔽3)—42no—
154C2×H(3,𝔽3)C4×D411yescharacteristic
155C2×H(3,𝔽3)C2×SD1611yescharacteristic
156C2×H(3,𝔽3)#13511yescharacteristic
157C2×C3×Q8—91no—
158C2×C3×D4—31no—
159C2×C3×D4—31no—
160C2×C3×D4—61no—
161C2×C3×D4C2×C3×D311yescharacteristic
162C22×C3×C4—21no—
163C22×C3×C4—61no—
164C22×C3×C4—61no—
165C22×C3×C4—62no—
166C2×Dic6—31no—
167C2×Dic6—91no—
168C3×SD16—94no—
169C2×C3×C8—91no—
170C2×C3×C8—91no—
171C3×C4⋊C4—31no—
172C3×C4⋊C4—31no—
173C3×C4⋊C4—61no—
174C3×C4⋊C4—91no—
175C3×C4⋊C4—181no—
176C3×C4⋊C4C2×C3×D311yescharacteristic
177C3×C22⋊C4—21no—
178C3×C22⋊C4—61no—
179C3×C22⋊C4—61no—
180C3×C22⋊C4—62no—
181C3×C42—31no—
182C3×C42—31no—
183C3×C42—61no—
184C3×C42—181no—
185C3×C42C2×C3×D311yescharacteristic
186#214—34no—
187#214—94no—
188C4⋊Dic3—31no—
189C4⋊Dic3—91no—
190Dic3⋊C4—61no—
191Dic3⋊C4—181no—
192C4×Dic3—61no—
193C4×Dic3—181no—
194C2×C3⋊C8—31no—
195C2×C3⋊C8—31no—
196C2×C3⋊C8—91no—
197C2×C3⋊C8—91no—
198C22×C32—24no—
199C22×C32—22no—
200C22×C32—22no—
201C22×C32—21no—
202C22×C32—42no—
203C22×C32—42no—
204C22×C32—41no—
205C22×C32—41no—
206C22×C32—41no—
207C22×C32—41no—
208C22×C32C2×#6711yescharacteristic
209C22×C32C2×C3×D411yescharacteristic
210C32×C4—21no—
211C32×C4—21no—
212C32×C4—22no—
213C32×C4—21no—
214C32×C4—21no—
215C32×C4—41no—
216C32×C4—42no—
217C32×C4—41no—
218C32×C4—41no—
219C32×C4#21412yescomplemented
220C32×C4C3×SD1612yescomplemented
221C32×C4C2×C4×D311yescharacteristic
222C32×C4#23511yescharacteristic
223C32×C4C22×C3×C411yescharacteristic
224C32×C4C3×#4111yescharacteristic
225C3⋊Dic3—22no—
226C3⋊Dic3—41no—
227C3×Dic3—62no—
228C3×Dic3—121no—
229C2×SD16—91no—
230C4×Q8—91no—
231C4×D4(C3⋊D3)⋊C311yescharacteristic
232#106—91no—
233#103—91no—
234#102—91no—
235C4×C8—91no—
236H(3,𝔽3)C4×SD1611yescharacteristic, complemented, Sylow
237C23×C3—21no—
238C23×C3—61no—
239C23×C3—61no—
240C23×C3—62no—
241C3×Q8—92no—
242C3×Q8—181no—
243C3×D4—21no—
244C3×D4—32no—
245C3×D4—32no—
246C3×D4—61no—
247C3×D4—61no—
248C3×D4—62no—
249C3×D4—62no—
250C3×D4C3×C4×D312yes—
251C2×C3×C4—21no—
252C2×C3×C4—31no—
253C2×C3×C4—31no—
254C2×C3×C4—31no—
255C2×C3×C4—31no—
256C2×C3×C4—31no—
257C2×C3×C4—31no—
258C2×C3×C4—42no—
259C2×C3×C4—61no—
260C2×C3×C4—61no—
261C2×C3×C4—62no—
262C2×C3×C4—61no—
263C2×C3×C4—61no—
264C2×C3×C4—61no—
265C2×C3×C4—122no—
266C2×C3×C4—122no—
267C2×C3×C4—124no—
268C2×C3×C4—181no—
269C2×C3×C4—181no—
270C2×C3×C4C3×#6711yescharacteristic
271C2×C3×C4C3×#6711yescharacteristic
272C2×C3×C4C22×C3×D311yescharacteristic
273C2×Dic3—61no—
274C2×Dic3—61no—
275C2×Dic3—181no—
276C2×Dic3—181no—
277Dic6—32no—
278Dic6—61no—
279Dic6—92no—
280Dic6—181no—
281C3×C8—92no—
282C3×C8—181no—
283C3⋊C8—32no—
284C3⋊C8—61no—
285C3⋊C8—92no—
286C3⋊C8—181no—
287C2×C32—21no—
288C2×C32—21no—
289C2×C32—21no—
290C2×C32—44no—
291C2×C32—42no—
292C2×C32—42no—
293C2×C32C4×#6711yescharacteristic
294C2×C32C2×#21411yescharacteristic
295C2×C32#95011yescharacteristic
296C2×C32C3×C4×D411yescharacteristic
297C2×C32C2×C3×SD1611yescharacteristic
298C2×C32C3×#13511yescharacteristic
299C2×Q8—91no—
300C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
301C22×C4—21no—
302SD16—94no—
303C2×C8—91no—
304C2×C8—91no—
305C4⋊C4—91no—
306C4⋊C4—181no—
307C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
308C22⋊C4—21no—
309C42—181no—
310C42C2×(C3⋊D3)⋊C311yescharacteristic
311C22×C3—22no—
312C22×C3—31no—
313C22×C3—31no—
314C22×C3—41no—
315C22×C3—41no—
316C22×C3—62no—
317C22×C3—62no—
318C22×C3—64no—
319C22×C3—61no—
320C22×C3—121no—
321C22×C3—121no—
322C22×C3—121no—
323C22×C3—121no—
324C22×C3—122no—
325C22×C3—122no—
326C22×C3C2×C3×#6711yescharacteristic
327C3×C4—21no—
328C3×C4—31no—
329C3×C4—31no—
330C3×C4—31no—
331C3×C4—31no—
332C3×C4—32no—
333C3×C4—32no—
334C3×C4—41no—
335C3×C4—61no—
336C3×C4—61no—
337C3×C4—61no—
338C3×C4—61no—
339C3×C4—62no—
340C3×C4—121no—
341C3×C4—121no—
342C3×C4—121no—
343C3×C4—122no—
344C3×C4—182no—
345C3×C4—361no—
346C3×C4C3×#21412yes—
347C3×C4C2×C3×C4×D311yescharacteristic
348C3×C4C3×#23511yescharacteristic
349Dic3—62no—
350Dic3—121no—
351Dic3—182no—
352Dic3—361no—
353C32—21no—
354C32C4×#21411yescharacteristic, complemented
355C32C3×C4×SD1611yescharacteristic, complemented
356C23—21no—
357Q8—92no—
358Q8—181no—
359D4—21no—
360D4C4×(C3⋊D3)⋊C312yes—
361C2×C4—21no—
362C2×C4—42no—
363C2×C4—181no—
364C2×C4—181no—
365C2×C4#638⋊C311yescharacteristic
366C2×C4#638⋊C311yescharacteristic
367C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
368C8—92no—
369C8—181no—
370C2×C3—31no—
371C2×C3—31no—
372C2×C3—31no—
373C2×C3—31no—
374C2×C3—31no—
375C2×C3—31no—
376C2×C3—42no—
377C2×C3—61no—
378C2×C3—61no—
379C2×C3—61no—
380C2×C3—122no—
381C2×C3—122no—
382C2×C3—124no—
383C2×C3C3×C4×#6711yescharacteristic
384C2×C3C2×C3×#21411yescharacteristic
385C2×C3C3×#95011yescharacteristic
386C22—22no—
387C22—41no—
388C22—41no—
389C22C2×#638⋊C311yescharacteristic
390C4—21no—
391C4—41no—
392C4—182no—
393C4—361no—
394C4#3754⋊C312yesdirect product factor
395C4C2×C4×(C3⋊D3)⋊C311yescharacteristic
396C4#3828⋊C311yescharacteristic
397C3—31no—
398C3—31no—
399C3—61no—
400C3C3×C4×#21411yescharacteristic
401C2—42no—
402C2C4×#638⋊C311yescharacteristic
403C2C2×#3754⋊C311yescharacteristic
404C2#64159⋊C311yescharacteristic
subgroupG/H
centerC2×C4#638⋊C3
commutatorC32×C4C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingC4×D4×H(3,𝔽3)C2
socleC22×C3C2×C3×#67
prime radicalC2×D4×H(3,𝔽3)C22
cosocleC2×C32×C4C23×C3
hypercenterC4×D4(C3⋊D3)⋊C3
hypocenterC32C3×C4×SD16
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
3. C4×#634⋊C3C2semidirect
5. #63637⋊C3C2semidirect
6. #63635⋊C3C2semidirect
7. C4×(C3⋊(C3⋊C8))⋊C3C2semidirect
8. C4×#3754C3semidirect
16. #3754⋊C3C4direct (N × H)
26. C4×#634C2×C3semidirect
30. #63637C2×C3semidirect
31. #63635C2×C3semidirect
32. C4×C3⋊(C3⋊C8)C2×C3semidirect
48. #634⋊C3C2×C4semidirect
50. (C3⋊(C3⋊C8))⋊C3C2×C4semidirect
71. #3754C3×C4semidirect
89. C4×H(3,𝔽3)SD16semidirect
143. #634C2×C3×C4semidirect
149. C3⋊(C3⋊C8)C2×C3×C4semidirect
219. C32×C4#214semidirect
220. C32×C4C3×SD16semidirect
236. H(3,𝔽3)C4×SD16semidirect
354. C32C4×#214semidirect
355. C32C3×C4×SD16semidirect
394. C4#3754⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C4×D4×H(3,𝔽3)2
C2×#3754⋊C32
C4×#634⋊C32
#63659⋊C32
#63637⋊C32
#63635⋊C32
C4×(C3⋊(C3⋊C8))⋊C32
C4×#37543
C3×C4×#2143