Finite Group Databasebeta

← C4×#3753⋊C3 #13142747

Subgroups — C4×#3753⋊C3 #13142747

Order 1728.

563 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×D4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C2×#3753⋊C3C211yesmaximal, characteristic
3C4×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
4#63659⋊C3C211yesmaximal, characteristic, complemented
5#63636⋊C3C211yesmaximal, characteristic, complemented
6#63634⋊C3C211yesmaximal, characteristic, complemented
7C4×(C3⋊(C3⋊C8))⋊C3C211yesmaximal, characteristic, complemented
8C4×#3753C311yesmaximal, characteristic, complemented
9C3×C4×#213—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C22×C4×H(3,𝔽3)—21no—
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×C4×(C3⋊D3)⋊C3—21no—
14C4⋊C4×H(3,𝔽3)C2211yescharacteristic
15C22⋊C4×H(3,𝔽3)—21no—
16C42×H(3,𝔽3)C2211yescharacteristic
17#3753⋊C3C414yesdirect product factor
18#3752⋊C3—21no—
19C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
22C32×C4×D4—21no—
23C32×C4×D4D311yescharacteristic, complemented
24C32×C4×D4C2×C311yescharacteristic, complemented
25C2×#3753C2×C311yescharacteristic
26C4×C4⋊(C3⋊D3)C2×C311yescharacteristic, complemented
27C2×C3×#213—31no—
28C3×C4×D12—31no—
29#63659C2×C311yescharacteristic, complemented
30#63636C2×C311yescharacteristic, complemented
31#63634C2×C311yescharacteristic, complemented
32C4×C3⋊(C3⋊C8)C2×C311yescharacteristic, complemented
33C3×#832—31no—
34C3×#809—31no—
35C3×#807—31no—
36C3×C4×C3⋊C8—31no—
37C23×H(3,𝔽3)—21no—
38C22×(C3⋊D3)⋊C3—21no—
39D4×H(3,𝔽3)—21no—
40D4×H(3,𝔽3)C2×C412yescomplemented
41C2×C4×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)—42no—
43C2×C4×H(3,𝔽3)D411yescharacteristic
44C2×C4×H(3,𝔽3)D411yescharacteristic
45C2×C4×H(3,𝔽3)C2311yescharacteristic
46C2×(C3⋊Dic3)⋊C3—21no—
47C4⋊((C3⋊D3)⋊C3)—21no—
48C4⋊((C3⋊D3)⋊C3)C2×C412yescomplemented
49C4×(C3⋊D3)⋊C3—42no—
50(C3⋊(C3⋊C8))⋊C3—21no—
51(C3⋊(C3⋊C8))⋊C3C2×C412yescomplemented
52C3×C4×D8—91no—
53C4×#213—31no—
54C4×#213—91no—
55C2×C32×D4—21no—
56C2×C32×D4C2×D311yescharacteristic
57C2×C32×D4C22×C311yescharacteristic
58C22×C32×C4—22no—
59C22×C32×C4—21no—
60C22×C32×C4—21no—
61C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
62C2×C4×C3⋊D3—21no—
63C2×C3×D12—31no—
64C2×C3×C4×D3—61no—
65C32×C4⋊C4—21no—
66C32×C4⋊C4C2×D311yescharacteristic
67C32×C4⋊C4C22×C311yescharacteristic
68C32×C22⋊C4—22no—
69C32×C22⋊C4—21no—
70C32×C22⋊C4—21no—
71C32×C42—21no—
72C32×C42C2×D311yescharacteristic
73C32×C42C22×C311yescharacteristic
74#3753C3×C414yescomplemented
75#3752—21no—
76C4⋊(C3⋊Dic3)C22×C311yescharacteristic
77C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
78C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
79C3×#213—34no—
80C3×(C2×D3)⋊C4—61no—
81C3×C4⋊Dic3—31no—
82C2×C3×C3⋊C8—31no—
83C2×C3×C3⋊C8—31no—
84C22×H(3,𝔽3)—22no—
85C22×H(3,𝔽3)—41no—
86C22×H(3,𝔽3)—41no—
87C22×H(3,𝔽3)C2×D411yescharacteristic
88C2×(C3⋊D3)⋊C3—22no—
89C2×(C3⋊D3)⋊C3—41no—
90C2×(C3⋊D3)⋊C3—41no—
91C4×H(3,𝔽3)—21no—
92C4×H(3,𝔽3)—41no—
93C4×H(3,𝔽3)D812yescomplemented
94C4×H(3,𝔽3)C22×C411yescharacteristic
95C4×H(3,𝔽3)#4111yescharacteristic
96(C3⋊Dic3)⋊C3—41no—
97C2×C3×D8—91no—
98C3×C4×D4—31no—
99C3×C4×D4—31no—
100C3×C4×D4—61no—
101C3×C4×D4—91no—
102C3×C4×D4C3×D311yescharacteristic
103C2×#213—31no—
104C2×#213—91no—
105C4×D12—31no—
106C4×D12—91no—
107C3×#107—91no—
108C3×#102—91no—
109C3×#102—91no—
110C3×C4×C8—91no—
111#832—31no—
112#832—91no—
113#809—31no—
114#809—91no—
115#807—31no—
116#807—91no—
117C4×C3⋊C8—31no—
118C4×C3⋊C8—91no—
119C23×C32—22no—
120C23×C32—21no—
121C23×C32—21no—
122C22×C3⋊D3—21no—
123C22×C3×D3—61no—
124C32×D4—22no—
125C32×D4—21no—
126C32×D4—21no—
127C32×D4—22no—
128C32×D4C4×D312yescomplemented
129C32×D4C2×C3×C412yescomplemented
130C2×C32×C4—22no—
131C2×C32×C4—21no—
132C2×C32×C4—21no—
133C2×C32×C4—21no—
134C2×C32×C4—21no—
135C2×C32×C4—21no—
136C2×C32×C4—44no—
137C2×C32×C4—42no—
138C2×C32×C4—42no—
139C2×C32×C4#6711yescharacteristic
140C2×C32×C4#6711yescharacteristic
141C2×C32×C4C3×D411yescharacteristic
142C2×C32×C4C3×D411yescharacteristic
143C2×C32×C4C22×D311yescharacteristic
144C2×C32×C4C23×C311yescharacteristic
145C2×C3⋊Dic3—21no—
146C4⋊(C3⋊D3)—21no—
147C4⋊(C3⋊D3)C2×C3×C412yescomplemented
148C4×C3⋊D3—42no—
149C2×C3×Dic3—61no—
150C3×D12—32no—
151C3×D12—61no—
152C3×C4×D3—122no—
153C3⋊(C3⋊C8)—21no—
154C3⋊(C3⋊C8)C2×C3×C412yescomplemented
155C3×C3⋊C8—32no—
156C3×C3⋊C8—61no—
157C4×D8—91noSylow
158C2×H(3,𝔽3)—42no—
159C2×H(3,𝔽3)C4×D411yescharacteristic
160C2×H(3,𝔽3)C2×D811yescharacteristic
161C2×H(3,𝔽3)#13511yescharacteristic
162(C3⋊D3)⋊C3—42no—
163C2×C3×D4—31no—
164C2×C3×D4—31no—
165C2×C3×D4—61no—
166C2×C3×D4—91no—
167C2×C3×D4C2×C3×D311yescharacteristic
168C22×C3×C4—21no—
169C22×C3×C4—61no—
170C22×C3×C4—61no—
171C22×C3×C4—62no—
172C22×C3×C4—181no—
173C2×D12—31no—
174C2×D12—91no—
175C2×C4×D3—61no—
176C2×C4×D3—181no—
177C3×D8—94no—
178C2×C3×C8—91no—
179C2×C3×C8—91no—
180C3×C4⋊C4—31no—
181C3×C4⋊C4—31no—
182C3×C4⋊C4—61no—
183C3×C4⋊C4—91no—
184C3×C4⋊C4C2×C3×D311yescharacteristic
185C3×C22⋊C4—21no—
186C3×C22⋊C4—61no—
187C3×C22⋊C4—61no—
188C3×C22⋊C4—62no—
189C3×C22⋊C4—181no—
190C3×C42—31no—
191C3×C42—31no—
192C3×C42—61no—
193C3×C42C2×C3×D311yescharacteristic
194#213—34no—
195#213—94no—
196(C2×D3)⋊C4—61no—
197(C2×D3)⋊C4—181no—
198C4⋊Dic3—31no—
199C4⋊Dic3—91no—
200C2×C3⋊C8—31no—
201C2×C3⋊C8—31no—
202C2×C3⋊C8—91no—
203C2×C3⋊C8—91no—
204C22×C32—24no—
205C22×C32—22no—
206C22×C32—22no—
207C22×C32—21no—
208C22×C32—42no—
209C22×C32—42no—
210C22×C32—41no—
211C22×C32—41no—
212C22×C32—41no—
213C22×C32—41no—
214C22×C32C2×#6711yescharacteristic
215C22×C32C2×C3×D411yescharacteristic
216C2×C3⋊D3—22no—
217C2×C3⋊D3—41no—
218C2×C3⋊D3—41no—
219C2×C3×D3—62no—
220C2×C3×D3—121no—
221C2×C3×D3—121no—
222C32×C4—21no—
223C32×C4—21no—
224C32×C4—22no—
225C32×C4—21no—
226C32×C4—21no—
227C32×C4—41no—
228C32×C4—42no—
229C32×C4—41no—
230C32×C4—41no—
231C32×C4#21312yescomplemented
232C32×C4C3×D812yescomplemented
233C32×C4C2×C4×D311yescharacteristic
234C32×C4#23511yescharacteristic
235C32×C4C22×C3×C411yescharacteristic
236C32×C4C3×#4111yescharacteristic
237C3⋊Dic3—41no—
238C3×Dic3—121no—
239C2×D8—91no—
240C4×D4—91no—
241C4×D4(C3⋊D3)⋊C311yescharacteristic, complemented
242#107—91no—
243#102—91no—
244#102—91no—
245C4×C8—91no—
246H(3,𝔽3)C4×D811yescharacteristic, complemented, Sylow
247C23×C3—21no—
248C23×C3—61no—
249C23×C3—61no—
250C23×C3—62no—
251C23×C3—181no—
252C22×D3—61no—
253C22×D3—181no—
254C3×D4—21no—
255C3×D4—32no—
256C3×D4—32no—
257C3×D4—61no—
258C3×D4—61no—
259C3×D4—62no—
260C3×D4—62no—
261C3×D4—92no—
262C3×D4—181no—
263C3×D4C3×C4×D312yes—
264C2×C3×C4—21no—
265C2×C3×C4—31no—
266C2×C3×C4—31no—
267C2×C3×C4—31no—
268C2×C3×C4—31no—
269C2×C3×C4—31no—
270C2×C3×C4—31no—
271C2×C3×C4—42no—
272C2×C3×C4—61no—
273C2×C3×C4—61no—
274C2×C3×C4—62no—
275C2×C3×C4—61no—
276C2×C3×C4—61no—
277C2×C3×C4—61no—
278C2×C3×C4—122no—
279C2×C3×C4—122no—
280C2×C3×C4—124no—
281C2×C3×C4—181no—
282C2×C3×C4—362no—
283C2×C3×C4C3×#6711yescharacteristic
284C2×C3×C4C3×#6711yescharacteristic
285C2×C3×C4C22×C3×D311yescharacteristic
286C2×Dic3—61no—
287C2×Dic3—181no—
288D12—32no—
289D12—61no—
290D12—92no—
291D12—181no—
292C4×D3—122no—
293C4×D3—362no—
294C3×C8—92no—
295C3×C8—181no—
296C3⋊C8—32no—
297C3⋊C8—61no—
298C3⋊C8—92no—
299C3⋊C8—181no—
300C2×C32—21no—
301C2×C32—21no—
302C2×C32—21no—
303C2×C32—44no—
304C2×C32—42no—
305C2×C32—42no—
306C2×C32C4×#6711yescharacteristic
307C2×C32C2×#21311yescharacteristic
308C2×C32#95011yescharacteristic
309C2×C32C3×C4×D411yescharacteristic
310C2×C32C2×C3×D811yescharacteristic
311C2×C32C3×#13511yescharacteristic
312C3⋊D3—42no—
313C3×D3—122no—
314C2×D4—91no—
315C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
316C22×C4—21no—
317C22×C4—181no—
318D8—94no—
319C2×C8—91no—
320C2×C8—91no—
321C4⋊C4—91no—
322C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
323C22⋊C4—21no—
324C22⋊C4—181no—
325C42C2×(C3⋊D3)⋊C311yescharacteristic
326C22×C3—22no—
327C22×C3—31no—
328C22×C3—31no—
329C22×C3—41no—
330C22×C3—41no—
331C22×C3—62no—
332C22×C3—62no—
333C22×C3—64no—
334C22×C3—61no—
335C22×C3—121no—
336C22×C3—121no—
337C22×C3—121no—
338C22×C3—121no—
339C22×C3—122no—
340C22×C3—122no—
341C22×C3—182no—
342C22×C3—361no—
343C22×C3—361no—
344C22×C3C2×C3×#6711yescharacteristic
345C2×D3—62no—
346C2×D3—121no—
347C2×D3—121no—
348C2×D3—182no—
349C2×D3—361no—
350C2×D3—361no—
351C3×C4—21no—
352C3×C4—31no—
353C3×C4—31no—
354C3×C4—31no—
355C3×C4—31no—
356C3×C4—32no—
357C3×C4—32no—
358C3×C4—41no—
359C3×C4—61no—
360C3×C4—61no—
361C3×C4—61no—
362C3×C4—61no—
363C3×C4—62no—
364C3×C4—121no—
365C3×C4—121no—
366C3×C4—121no—
367C3×C4—122no—
368C3×C4—361no—
369C3×C4C3×#21312yes—
370C3×C4C2×C3×C4×D311yescharacteristic
371C3×C4C3×#23511yescharacteristic
372Dic3—121no—
373Dic3—361no—
374C32—21no—
375C32C4×#21311yescharacteristic, complemented
376C32C3×C4×D811yescharacteristic, complemented
377C23—21no—
378C23—181no—
379D4—21no—
380D4—92no—
381D4—181no—
382D4C4×(C3⋊D3)⋊C312yescomplemented
383C2×C4—21no—
384C2×C4—42no—
385C2×C4—181no—
386C2×C4—362no—
387C2×C4#638⋊C311yescharacteristic
388C2×C4#638⋊C311yescharacteristic
389C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
390C8—92no—
391C8—181no—
392C2×C3—31no—
393C2×C3—31no—
394C2×C3—31no—
395C2×C3—31no—
396C2×C3—31no—
397C2×C3—31no—
398C2×C3—42no—
399C2×C3—61no—
400C2×C3—61no—
401C2×C3—61no—
402C2×C3—122no—
403C2×C3—122no—
404C2×C3—124no—
405C2×C3—362no—
406C2×C3C3×C4×#6711yescharacteristic
407C2×C3C2×C3×#21311yescharacteristic
408C2×C3C3×#95011yescharacteristic
409D3—122no—
410D3—362no—
411C22—22no—
412C22—41no—
413C22—41no—
414C22—182no—
415C22—361no—
416C22—361no—
417C22C2×#638⋊C311yescharacteristic
418C4—21no—
419C4—41no—
420C4—361no—
421C4#3753⋊C312yesdirect product factor
422C4C2×C4×(C3⋊D3)⋊C311yescharacteristic
423C4#3828⋊C311yescharacteristic
424C3—31no—
425C3—31no—
426C3—61no—
427C3C3×C4×#21311yescharacteristic
428C2—42no—
429C2—362no—
430C2C4×#638⋊C311yescharacteristic
431C2C2×#3753⋊C311yescharacteristic
432C2#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×#3753⋊C3C2
cosocleC2×C32×C4C23×C3
hypercenterC4×D4(C3⋊D3)⋊C3
hypocenterC32C3×C4×D8
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
1. C4×D4×H(3,𝔽3)C2semidirect
3. C4×C4⋊((C3⋊D3)⋊C3)C2semidirect
4. #63659⋊C3C2semidirect
5. #63636⋊C3C2semidirect
6. #63634⋊C3C2semidirect
7. C4×(C3⋊(C3⋊C8))⋊C3C2semidirect
8. C4×#3753C3semidirect
17. #3753⋊C3C4direct (N × H)
23. C32×C4×D4D3semidirect
24. C32×C4×D4C2×C3semidirect
26. C4×C4⋊(C3⋊D3)C2×C3semidirect
29. #63659C2×C3semidirect
30. #63636C2×C3semidirect
31. #63634C2×C3semidirect
32. C4×C3⋊(C3⋊C8)C2×C3semidirect
40. D4×H(3,𝔽3)C2×C4semidirect
48. C4⋊((C3⋊D3)⋊C3)C2×C4semidirect
51. (C3⋊(C3⋊C8))⋊C3C2×C4semidirect
74. #3753C3×C4semidirect
93. C4×H(3,𝔽3)D8semidirect
128. C32×D4C4×D3semidirect
129. C32×D4C2×C3×C4semidirect
147. C4⋊(C3⋊D3)C2×C3×C4semidirect
154. C3⋊(C3⋊C8)C2×C3×C4semidirect
231. C32×C4#213semidirect
232. C32×C4C3×D8semidirect
241. C4×D4(C3⋊D3)⋊C3semidirect
246. H(3,𝔽3)C4×D8semidirect
375. C32C4×#213semidirect
376. C32C3×C4×D8semidirect
382. D4C4×(C3⋊D3)⋊C3semidirect
421. C4#3753⋊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×#3753⋊C32
C4×C4⋊((C3⋊D3)⋊C3)2
#63659⋊C32
#63636⋊C32
#63634⋊C32
C4×(C3⋊(C3⋊C8))⋊C32
C4×#37533
C3×C4×#2133