Finite Group Databasebeta

← C4×#3744⋊C3 #13141457

Subgroups — C4×#3744⋊C3 #13141457

Order 1728.

471 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3744⋊C3C211yesmaximal, characteristic
2C4×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic
3C4×#634⋊C3C211yesmaximal, characteristic, complemented
4C4×C8×H(3,𝔽3)C211yesmaximal, characteristic, complemented
5#63648⋊C3C211yesmaximal, characteristic
6#63644⋊C3C211yesmaximal, characteristic, complemented
7#63643⋊C3C211yesmaximal, characteristic, complemented
8C4×#3744C311yesmaximal, characteristic, complemented
9C3×C4×#204—31nomaximal
10C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
11C2×C4×(C3⋊D3)⋊C3—21no—
12C2×#634⋊C3C2211yescharacteristic
13C2×C8×H(3,𝔽3)C2211yescharacteristic
14C2×C8×H(3,𝔽3)C2211yescharacteristic
15C42×H(3,𝔽3)C2211yescharacteristic
16#3752⋊C3—21no—
17C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
18C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
19#3750⋊C3—21no—
20C4×(C3⋊Dic3)⋊C3—21no—
21#3744⋊C3C414yesdirect product factor
22C2×#3744C2×C311yescharacteristic
23C4×C4⋊(C3⋊D3)C2×C311yescharacteristic
24C4×#634C2×C311yescharacteristic, complemented
25C2×C3×#204—31no—
26C3×C4×D12—31no—
27C3×C4×Dic6—31no—
28C32×C4×C8—21no—
29C32×C4×C8D311yescharacteristic, complemented
30C32×C4×C8C2×C311yescharacteristic, complemented
31#63648C2×C311yescharacteristic
32#63644C2×C311yescharacteristic, complemented
33#63643C2×C311yescharacteristic, complemented
34C3×#821—31no—
35C3×#817—31no—
36C3×#816—31no—
37C22×(C3⋊D3)⋊C3—21no—
38C2×C4×H(3,𝔽3)D411yescharacteristic
39C2×C4×H(3,𝔽3)D411yescharacteristic
40C2×C4×H(3,𝔽3)C2311yescharacteristic
41C2×(C3⋊Dic3)⋊C3—21no—
42C2×(C3⋊Dic3)⋊C3—21no—
43C2×(C3⋊Dic3)⋊C3—21no—
44C4⋊((C3⋊D3)⋊C3)—21no—
45C4⋊((C3⋊D3)⋊C3)C2×C412yes—
46C4×(C3⋊D3)⋊C3—42no—
47#634⋊C3—21no—
48#634⋊C3C2×C412yescomplemented
49C8×H(3,𝔽3)—21no—
50C8×H(3,𝔽3)C2×C412yescomplemented
51C3×C4×SD16—91no—
52C4×#204—31no—
53C4×#204—91no—
54C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
55C2×C4×C3⋊D3—21no—
56C2×#634C22×C311yescharacteristic
57C2×C3×D12—31no—
58C2×C3×C4×D3—61no—
59C2×C3×Dic6—31no—
60C2×C32×C8—21no—
61C2×C32×C8—21no—
62C2×C32×C8C2×D311yescharacteristic
63C2×C32×C8C2×D311yescharacteristic
64C2×C32×C8C22×C311yescharacteristic
65C2×C32×C8C22×C311yescharacteristic
66C32×C42—21no—
67C32×C42C2×D311yescharacteristic
68C32×C42C22×C311yescharacteristic
69#3752—21no—
70C4⋊(C3⋊Dic3)C22×C311yescharacteristic
71C4⋊(C3⋊Dic3)C22×C311yescharacteristic
72#3750—21no—
73C4×C3⋊Dic3—21no—
74#3744C3×C414yescomplemented
75C3×(C2×D3)⋊C4—61no—
76C3×C4⋊Dic3—31no—
77C3×C4⋊Dic3—31no—
78C3×Dic3⋊C4—61no—
79C3×C4×Dic3—61no—
80C3×#204—34no—
81C22×H(3,𝔽3)C2×D411yescharacteristic
82C2×(C3⋊D3)⋊C3—22no—
83C2×(C3⋊D3)⋊C3—41no—
84C2×(C3⋊D3)⋊C3—41no—
85C4×H(3,𝔽3)—21no—
86C4×H(3,𝔽3)SD1612yescomplemented
87C4×H(3,𝔽3)C22×C411yescharacteristic
88C4×H(3,𝔽3)#4111yescharacteristic
89(C3⋊Dic3)⋊C3—22no—
90(C3⋊Dic3)⋊C3—41no—
91(C3⋊Dic3)⋊C3—41no—
92C2×C3×SD16—91no—
93C3×C4×Q8—91no—
94C3×C4×D4—91no—
95C2×#204—31no—
96C2×#204—91no—
97C4×D12—31no—
98C4×D12—91no—
99C4×Dic6—31no—
100C4×Dic6—91no—
101C3×#106—91no—
102C3×#103—91no—
103C3×#102—91no—
104C3×C4×C8—31no—
105C3×C4×C8—31no—
106C3×C4×C8—61no—
107C3×C4×C8C3×D311yescharacteristic
108#821—31no—
109#821—91no—
110#817—31no—
111#817—91no—
112#816—31no—
113#816—91no—
114C22×C3⋊D3—21no—
115C22×C3×D3—61no—
116C2×C32×C4—21no—
117C2×C32×C4—21no—
118C2×C32×C4—21no—
119C2×C32×C4D1211yescharacteristic
120C2×C32×C4D1211yescharacteristic
121C2×C32×C4C3×D411yescharacteristic
122C2×C32×C4C3×D411yescharacteristic
123C2×C32×C4C22×D311yescharacteristic
124C2×C32×C4C23×C311yescharacteristic
125C2×C3⋊Dic3—21no—
126C2×C3⋊Dic3—21no—
127C2×C3⋊Dic3—21no—
128C4⋊(C3⋊D3)—21no—
129C4⋊(C3⋊D3)C2×C3×C412yes—
130C4×C3⋊D3—42no—
131#634—21no—
132#634C2×C3×C412yescomplemented
133C2×C3×Dic3—61no—
134C2×C3×Dic3—61no—
135C2×C3×Dic3—61no—
136C3×D12—32no—
137C3×D12—61no—
138C3×C4×D3—122no—
139C3×Dic6—32no—
140C3×Dic6—61no—
141C32×C8—22no—
142C32×C8—21no—
143C32×C8—21no—
144C32×C8—22no—
145C32×C8C4×D312yescomplemented
146C32×C8C2×C3×C412yescomplemented
147C4×SD16—91noSylow
148C2×H(3,𝔽3)C4×D411yescharacteristic
149C2×H(3,𝔽3)C2×SD1611yescharacteristic
150C2×H(3,𝔽3)#13511yescharacteristic
151(C3⋊D3)⋊C3—42no—
152C2×C3×Q8—91no—
153C2×C3×D4—91no—
154C22×C3×C4—181no—
155C2×D12—31no—
156C2×D12—91no—
157C2×C4×D3—61no—
158C2×C4×D3—181no—
159C2×Dic6—31no—
160C2×Dic6—91no—
161C3×SD16—94no—
162C2×C3×C8—31no—
163C2×C3×C8—31no—
164C2×C3×C8—31no—
165C2×C3×C8—31no—
166C2×C3×C8—61no—
167C2×C3×C8—61no—
168C2×C3×C8C2×C3×D311yescharacteristic
169C2×C3×C8C2×C3×D311yescharacteristic
170C3×C4⋊C4—91no—
171C3×C4⋊C4—91no—
172C3×C4⋊C4—181no—
173C3×C22⋊C4—181no—
174C3×C42—31no—
175C3×C42—31no—
176C3×C42—61no—
177C3×C42—181no—
178C3×C42C2×C3×D311yescharacteristic
179(C2×D3)⋊C4—61no—
180(C2×D3)⋊C4—181no—
181C4⋊Dic3—31no—
182C4⋊Dic3—31no—
183C4⋊Dic3—91no—
184C4⋊Dic3—91no—
185Dic3⋊C4—61no—
186Dic3⋊C4—181no—
187C4×Dic3—61no—
188C4×Dic3—181no—
189#204—34no—
190#204—94no—
191C22×C32—21no—
192C22×C32C2×D1211yescharacteristic
193C22×C32C2×C3×D411yescharacteristic
194C2×C3⋊D3—22no—
195C2×C3⋊D3—41no—
196C2×C3⋊D3—41no—
197C2×C3×D3—62no—
198C2×C3×D3—121no—
199C2×C3×D3—121no—
200C32×C4—22no—
201C32×C4—21no—
202C32×C4—21no—
203C32×C4—22no—
204C32×C4—21no—
205C32×C4—21no—
206C32×C4#20412yescomplemented
207C32×C4C3×SD1612yescomplemented
208C32×C4C2×C4×D311yescharacteristic
209C32×C4#23511yescharacteristic
210C32×C4C22×C3×C411yescharacteristic
211C32×C4C3×#4111yescharacteristic
212C3⋊Dic3—22no—
213C3⋊Dic3—41no—
214C3⋊Dic3—41no—
215C3×Dic3—62no—
216C3×Dic3—121no—
217C3×Dic3—121no—
218C2×SD16—91no—
219C4×Q8—91no—
220C4×D4—91no—
221#106—91no—
222#103—91no—
223#102—91no—
224C4×C8(C3⋊D3)⋊C311yescharacteristic, complemented
225H(3,𝔽3)C4×SD1611yescharacteristic, complemented, Sylow
226C23×C3—181no—
227C22×D3—61no—
228C22×D3—181no—
229C3×Q8—92no—
230C3×Q8—181no—
231C3×D4—92no—
232C3×D4—181no—
233C2×C3×C4—31no—
234C2×C3×C4—31no—
235C2×C3×C4—31no—
236C2×C3×C4—31no—
237C2×C3×C4—31no—
238C2×C3×C4—31no—
239C2×C3×C4—61no—
240C2×C3×C4—61no—
241C2×C3×C4—61no—
242C2×C3×C4—181no—
243C2×C3×C4—181no—
244C2×C3×C4—181no—
245C2×C3×C4—362no—
246C2×C3×C4C3×D1211yescharacteristic
247C2×C3×C4C3×D1211yescharacteristic
248C2×C3×C4C22×C3×D311yescharacteristic
249C2×Dic3—61no—
250C2×Dic3—61no—
251C2×Dic3—61no—
252C2×Dic3—181no—
253C2×Dic3—181no—
254C2×Dic3—181no—
255D12—32no—
256D12—61no—
257D12—92no—
258D12—181no—
259C4×D3—122no—
260C4×D3—362no—
261Dic6—32no—
262Dic6—61no—
263Dic6—92no—
264Dic6—181no—
265C3×C8—21no—
266C3×C8—32no—
267C3×C8—32no—
268C3×C8—61no—
269C3×C8—61no—
270C3×C8—62no—
271C3×C8—62no—
272C3×C8C3×C4×D312yes—
273C2×C32—21no—
274C2×C32—21no—
275C2×C32—21no—
276C2×C32C4×D1211yescharacteristic
277C2×C32C2×#20411yescharacteristic
278C2×C32#90411yescharacteristic
279C2×C32C3×C4×D411yescharacteristic
280C2×C32C2×C3×SD1611yescharacteristic
281C2×C32C3×#13511yescharacteristic
282C3⋊D3—42no—
283C3×D3—122no—
284C2×Q8—91no—
285C2×D4—91no—
286C22×C4—181no—
287SD16—94no—
288C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
289C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
290C4⋊C4—91no—
291C4⋊C4—91no—
292C4⋊C4—181no—
293C22⋊C4—181no—
294C42—181no—
295C42C2×(C3⋊D3)⋊C311yescharacteristic
296C22×C3—31no—
297C22×C3—31no—
298C22×C3—61no—
299C22×C3—182no—
300C22×C3—361no—
301C22×C3—361no—
302C22×C3C2×C3×D1211yescharacteristic
303C2×D3—62no—
304C2×D3—121no—
305C2×D3—121no—
306C2×D3—182no—
307C2×D3—361no—
308C2×D3—361no—
309C3×C4—21no—
310C3×C4—31no—
311C3×C4—31no—
312C3×C4—31no—
313C3×C4—31no—
314C3×C4—32no—
315C3×C4—32no—
316C3×C4—61no—
317C3×C4—61no—
318C3×C4—62no—
319C3×C4—61no—
320C3×C4—61no—
321C3×C4—62no—
322C3×C4—182no—
323C3×C4—361no—
324C3×C4—361no—
325C3×C4C3×#20412yes—
326C3×C4C2×C3×C4×D311yescharacteristic
327C3×C4C3×#23511yescharacteristic
328Dic3—62no—
329Dic3—121no—
330Dic3—121no—
331Dic3—182no—
332Dic3—361no—
333Dic3—361no—
334C32—21no—
335C32C4×#20411yescharacteristic, complemented
336C32C3×C4×SD1611yescharacteristic, complemented
337C23—181no—
338Q8—92no—
339Q8—181no—
340D4—92no—
341D4—181no—
342C2×C4—181no—
343C2×C4—181no—
344C2×C4—181no—
345C2×C4—362no—
346C2×C4C4⋊((C3⋊D3)⋊C3)11yescharacteristic
347C2×C4C4⋊((C3⋊D3)⋊C3)11yescharacteristic
348C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
349C8—21no—
350C8C4×(C3⋊D3)⋊C312yescomplemented
351C2×C3—31no—
352C2×C3—31no—
353C2×C3—31no—
354C2×C3—31no—
355C2×C3—31no—
356C2×C3—31no—
357C2×C3—61no—
358C2×C3—61no—
359C2×C3—61no—
360C2×C3—362no—
361C2×C3C3×C4×D1211yescharacteristic
362C2×C3C2×C3×#20411yescharacteristic
363C2×C3C3×#90411yescharacteristic
364D3—122no—
365D3—362no—
366C22—182no—
367C22—361no—
368C22—361no—
369C22C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
370C4—21no—
371C4—182no—
372C4—361no—
373C4—361no—
374C4#3744⋊C312yesdirect product factor
375C4C2×C4×(C3⋊D3)⋊C311yescharacteristic
376C4#3828⋊C311yescharacteristic
377C3—31no—
378C3—31no—
379C3—61no—
380C3C3×C4×#20411yescharacteristic
381C2—362no—
382C2C4×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
383C2C2×#3744⋊C311yescharacteristic
384C2#64113⋊C311yescharacteristic
subgroupG/H
centerC2×C4C4⋊((C3⋊D3)⋊C3)
commutatorC32×C4C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingC4×C8×H(3,𝔽3)C2
socleC22×C3C2×C3×D12
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C22
cosocleC2×C32×C4C23×C3
hypercenterC4×C8(C3⋊D3)⋊C3
hypocenterC32C3×C4×SD16
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C42 ◁ C4×C8

Complements

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

NH ≅ G/Ntype
3. C4×#634⋊C3C2semidirect
4. C4×C8×H(3,𝔽3)C2semidirect
6. #63644⋊C3C2semidirect
7. #63643⋊C3C2semidirect
8. C4×#3744C3semidirect
21. #3744⋊C3C4direct (N × H)
24. C4×#634C2×C3semidirect
29. C32×C4×C8D3semidirect
30. C32×C4×C8C2×C3semidirect
32. #63644C2×C3semidirect
33. #63643C2×C3semidirect
48. #634⋊C3C2×C4semidirect
50. C8×H(3,𝔽3)C2×C4semidirect
74. #3744C3×C4semidirect
86. C4×H(3,𝔽3)SD16semidirect
132. #634C2×C3×C4semidirect
145. C32×C8C4×D3semidirect
146. C32×C8C2×C3×C4semidirect
206. C32×C4#204semidirect
207. C32×C4C3×SD16semidirect
224. C4×C8(C3⋊D3)⋊C3semidirect
225. H(3,𝔽3)C4×SD16semidirect
335. C32C4×#204semidirect
336. C32C3×C4×SD16semidirect
350. C8C4×(C3⋊D3)⋊C3semidirect
374. C4#3744⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#3744⋊C32
C4×C4⋊((C3⋊D3)⋊C3)2
C4×#634⋊C32
C4×C8×H(3,𝔽3)2
#63648⋊C32
#63644⋊C32
#63643⋊C32
C4×#37443
C3×C4×#2043