Finite Group Databasebeta

← C4×#3752⋊C3 #13142424

Subgroups — C4×#3752⋊C3 #13142424

Order 1728.

1011 conjugacy classes of subgroups.

The subgroup lattice has 447 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2C2×C42×H(3,𝔽3)C211yesmaximal, characteristic, complemented
3C2×#3752⋊C3C212yesmaximal
4C2×C4×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
5#63658⋊C3C212yesmaximal, complemented
6C4×#3752C311yesmaximal, characteristic, complemented
7C3×C4×(C2×D3)⋊C4—31nomaximal
8C23×(C3⋊D3)⋊C3C2211yescharacteristic
9C22×C4×H(3,𝔽3)C2212yes—
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
12C22×(C3⋊Dic3)⋊C3C2212yes—
13C2×C4×(C3⋊D3)⋊C3—22no—
14C2×C4×(C3⋊D3)⋊C3—22no—
15C2×C4×(C3⋊D3)⋊C3C414yescomplemented
16C42×H(3,𝔽3)—22no—
17#3752⋊C3C418yesdirect product factor
18C4×(C3⋊Dic3)⋊C3—22no—
19C22×C4×C3⋊D3C2×C311yescharacteristic
20C22×C3×C4×D3—31no—
21C2×C32×C42—21no—
22C2×C32×C42D311yescharacteristic, complemented
23C2×C32×C42C2×C311yescharacteristic, complemented
24C2×#3752C2×C312yes—
25C2×C4×C3⋊Dic3C2×C311yescharacteristic, complemented
26C2×C3×(C2×D3)⋊C4—32no—
27C2×C3×C4×Dic3—31no—
28#63658C2×C312yescomplemented
29C3×C3⋊#95—32no—
30C23×H(3,𝔽3)C2311yescharacteristic
31C22×(C3⋊D3)⋊C3—21no—
32C22×(C3⋊D3)⋊C3—21no—
33C22×(C3⋊D3)⋊C3—22no—
34C22×(C3⋊D3)⋊C3C2×C412yes—
35C22×(C3⋊D3)⋊C3C2×C414yes—
36C2×C4×H(3,𝔽3)—22no—
37C2×C4×H(3,𝔽3)—22no—
38C2×C4×H(3,𝔽3)C2×C412yes—
39C2×C4×H(3,𝔽3)C2×C414yescomplemented
40C2×C4×H(3,𝔽3)D412yes—
41C2×C4×H(3,𝔽3)D412yes—
42C2×(C3⋊Dic3)⋊C3—21no—
43C2×(C3⋊Dic3)⋊C3—21no—
44C2×(C3⋊Dic3)⋊C3—22no—
45C2×(C3⋊Dic3)⋊C3—22no—
46C2×(C3⋊Dic3)⋊C3C2×C412yes—
47C2×(C3⋊Dic3)⋊C3C2×C414yescomplemented
48C4×(C3⋊D3)⋊C3—28no—
49C3×C4×C22⋊C4—91no—
50C4×(C2×D3)⋊C4—31no—
51C4×(C2×D3)⋊C4—91no—
52C23×C3⋊D3C22×C311yescharacteristic
53C23×C3×D3—31no—
54C22×C32×C4—22no—
55C22×C32×C4—21no—
56C22×C32×C4C2×D312yes—
57C22×C32×C4C2×D311yescharacteristic
58C22×C32×C4C22×C312yes—
59C22×C32×C4C22×C311yescharacteristic
60C22×C3⋊Dic3C22×C311yescharacteristic
61C22×C3⋊Dic3C22×C312yes—
62C2×C4×C3⋊D3—22no—
63C2×C4×C3⋊D3—22no—
64C2×C4×C3⋊D3C3×C414yescomplemented
65C22×C3×Dic3—32no—
66C22×C3×Dic3—31no—
67C2×C3×C4×D3—34no—
68C2×C3×C4×D3—62no—
69C2×C3×C4×D3—62no—
70C32×C42—24no—
71C32×C42—22no—
72C32×C42—22no—
73#3752C3×C418yescomplemented
74C4×C3⋊Dic3—22no—
75C3×(C2×D3)⋊C4—38no—
76C3×C4×Dic3—62no—
77C22×H(3,𝔽3)C22×C411yescharacteristic
78C22×H(3,𝔽3)C22×C412yes—
79C22×H(3,𝔽3)C2×D411yescharacteristic
80C22×H(3,𝔽3)C2×D411yescharacteristic
81C22×H(3,𝔽3)#4112yes—
82C2×(C3⋊D3)⋊C3—22no—
83C2×(C3⋊D3)⋊C3—22no—
84C2×(C3⋊D3)⋊C3—24no—
85C2×(C3⋊D3)⋊C3—24no—
86C2×(C3⋊D3)⋊C3C4214yescomplemented
87C4×H(3,𝔽3)—24no—
88C4×H(3,𝔽3)C22⋊C414yescomplemented
89(C3⋊Dic3)⋊C3—22no—
90(C3⋊Dic3)⋊C3—24no—
91C23×C3×C4—91no—
92C22×C4×D3—31no—
93C22×C4×D3—91no—
94C2×C3×C22⋊C4—92no—
95C2×C3×C42—31no—
96C2×C3×C42—31no—
97C2×C3×C42—61no—
98C2×C3×C42—91no—
99C2×C3×C42C3×D311yescharacteristic
100C2×(C2×D3)⋊C4—32no—
101C2×(C2×D3)⋊C4—92no—
102C2×C4×Dic3—31no—
103C2×C4×Dic3—91no—
104C3×#95—92no—
105C3⋊#95—32no—
106C3⋊#95—92no—
107C23×C32—21no—
108C23×C32C22×D311yescharacteristic
109C23×C32C23×C311yescharacteristic
110C22×C3⋊D3—21no—
111C22×C3⋊D3—21no—
112C22×C3⋊D3—22no—
113C22×C3⋊D3C2×C3×C414yes—
114C22×C3⋊D3C2×C3×C412yes—
115C22×C3×D3—34no—
116C22×C3×D3—32no—
117C22×C3×D3—61no—
118C22×C3×D3—62no—
119C22×C3×D3—61no—
120C2×C32×C4—24no—
121C2×C32×C4—24no—
122C2×C32×C4—22no—
123C2×C32×C4—22no—
124C2×C32×C4—22no—
125C2×C32×C4—22no—
126C2×C32×C4—22no—
127C2×C32×C4—24no—
128C2×C32×C4—22no—
129C2×C32×C4—22no—
130C2×C32×C4C4×D312yes—
131C2×C32×C4C4×D314yescomplemented
132C2×C32×C4D1212yes—
133C2×C32×C4#6712yes—
134C2×C32×C4C2×C3×C412yes—
135C2×C32×C4C2×C3×C414yescomplemented
136C2×C32×C4C3×D412yes—
137C2×C32×C4C3×D412yes—
138C2×C3⋊Dic3—22no—
139C2×C3⋊Dic3—21no—
140C2×C3⋊Dic3—21no—
141C2×C3⋊Dic3—22no—
142C2×C3⋊Dic3C2×C3×C412yes—
143C2×C3⋊Dic3C2×C3×C414yescomplemented
144C4×C3⋊D3—28no—
145C2×C3×Dic3—32no—
146C2×C3×Dic3—34no—
147C2×C3×Dic3—62no—
148C2×C3×Dic3—61no—
149C2×C3×Dic3—62no—
150C2×C3×Dic3—61no—
151C3×C4×D3—68no—
152C4×C22⋊C4—91noSylow
153C2×H(3,𝔽3)C2×C4211yescharacteristic
154C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
155C2×H(3,𝔽3)#11711yescharacteristic
156C2×H(3,𝔽3)C4×D412yes—
157C2×H(3,𝔽3)C4×D412yes—
158(C3⋊D3)⋊C3—24no—
159C24×C3—91no—
160C23×D3—31no—
161C23×D3—91no—
162C22×C3×C4—32no—
163C22×C3×C4—32no—
164C22×C3×C4—31no—
165C22×C3×C4—31no—
166C22×C3×C4—62no—
167C22×C3×C4—61no—
168C22×C3×C4—94no—
169C22×C3×C4—91no—
170C22×C3×C4—92no—
171C22×C3×C4—182no—
172C22×C3×C4—182no—
173C22×C3×C4C2×C3×D312yes—
174C22×C3×C4C2×C3×D311yescharacteristic
175C22×Dic3—31no—
176C22×Dic3—32no—
177C22×Dic3—91no—
178C22×Dic3—92no—
179C2×C4×D3—34no—
180C2×C4×D3—62no—
181C2×C4×D3—62no—
182C2×C4×D3—94no—
183C2×C4×D3—182no—
184C2×C4×D3—182no—
185C3×C22⋊C4—98no—
186C3×C42—22no—
187C3×C42—62no—
188C3×C42—62no—
189C3×C42—64no—
190C3×C42—182no—
191(C2×D3)⋊C4—38no—
192(C2×D3)⋊C4—98no—
193C4×Dic3—62no—
194C4×Dic3—182no—
195C22×C32—21no—
196C22×C32—22no—
197C22×C32—21no—
198C22×C32—22no—
199C22×C32—21no—
200C22×C32C2×C4×D311yescharacteristic
201C22×C32C2×C4×D312yes—
202C22×C32C2×D1211yescharacteristic
203C22×C32#23512yes—
204C22×C32C2×#6711yescharacteristic
205C22×C32C22×C3×C411yescharacteristic
206C22×C32C22×C3×C412yes—
207C22×C32C2×C3×D411yescharacteristic
208C22×C32C2×C3×D411yescharacteristic
209C22×C32C3×#4112yes—
210C2×C3⋊D3—22no—
211C2×C3⋊D3—22no—
212C2×C3⋊D3—24no—
213C2×C3⋊D3—24no—
214C2×C3⋊D3C3×C4214yescomplemented
215C2×C3×D3—34no—
216C2×C3×D3—62no—
217C2×C3×D3—64no—
218C2×C3×D3—62no—
219C2×C3×D3—64no—
220C32×C4—28no—
221C32×C4—24no—
222C32×C4—24no—
223C32×C4—24no—
224C32×C4(C2×D3)⋊C414yescomplemented
225C32×C4C3×C22⋊C414yescomplemented
226C3⋊Dic3—22no—
227C3⋊Dic3—24no—
228C3×Dic3—64no—
229C3×Dic3—62no—
230C23×C4—91no—
231C2×C22⋊C4—92no—
232C2×C42—91no—
233C2×C42(C3⋊D3)⋊C311yescharacteristic, complemented
234#95—92no—
235H(3,𝔽3)C4×C22⋊C411yescharacteristic, complemented, Sylow
236C23×C3—31no—
237C23×C3—31no—
238C23×C3—61no—
239C23×C3—94no—
240C23×C3—92no—
241C23×C3—181no—
242C23×C3—181no—
243C23×C3—182no—
244C23×C3C22×C3×D311yescharacteristic
245C22×D3—34no—
246C22×D3—32no—
247C22×D3—61no—
248C22×D3—62no—
249C22×D3—61no—
250C22×D3—94no—
251C22×D3—92no—
252C22×D3—181no—
253C22×D3—181no—
254C22×D3—182no—
255C2×C3×C4—22no—
256C2×C3×C4—22no—
257C2×C3×C4—32no—
258C2×C3×C4—32no—
259C2×C3×C4—34no—
260C2×C3×C4—34no—
261C2×C3×C4—32no—
262C2×C3×C4—32no—
263C2×C3×C4—32no—
264C2×C3×C4—32no—
265C2×C3×C4—62no—
266C2×C3×C4—62no—
267C2×C3×C4—62no—
268C2×C3×C4—62no—
269C2×C3×C4—64no—
270C2×C3×C4—64no—
271C2×C3×C4—62no—
272C2×C3×C4—64no—
273C2×C3×C4—62no—
274C2×C3×C4—62no—
275C2×C3×C4—94no—
276C2×C3×C4—92no—
277C2×C3×C4—188no—
278C2×C3×C4—182no—
279C2×C3×C4—182no—
280C2×C3×C4—181no—
281C2×C3×C4—181no—
282C2×C3×C4C3×C4×D312yes—
283C2×C3×C4C3×C4×D314yes—
284C2×C3×C4C3×D1212yes—
285C2×C3×C4C3×#6712yes—
286C2×Dic3—32no—
287C2×Dic3—34no—
288C2×Dic3—62no—
289C2×Dic3—61no—
290C2×Dic3—62no—
291C2×Dic3—61no—
292C2×Dic3—92no—
293C2×Dic3—94no—
294C2×Dic3—182no—
295C2×Dic3—181no—
296C2×Dic3—181no—
297C2×Dic3—182no—
298C4×D3—68no—
299C4×D3—188no—
300C2×C32—21no—
301C2×C32—22no—
302C2×C32—21no—
303C2×C32—21no—
304C2×C32—22no—
305C2×C32C42×D311yescharacteristic
306C2×C32#87211yescharacteristic
307C2×C32C4×D1212yes—
308C2×C32C2×(C2×D3)⋊C411yescharacteristic
309C2×C32C4×#6712yes—
310C2×C32C2×C3×C4211yescharacteristic
311C2×C32C2×C3×C22⋊C411yescharacteristic
312C2×C32C3×#11711yescharacteristic
313C2×C32C3×C4×D412yes—
314C2×C32C3×C4×D412yes—
315C3⋊D3—24no—
316C3×D3—64no—
317C24—91no—
318C22×C4—91no—
319C22×C4—92no—
320C22×C4—94no—
321C22×C4—182no—
322C22×C4—182no—
323C22×C4C2×(C3⋊D3)⋊C312yes—
324C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
325C22⋊C4—98no—
326C42—22no—
327C42—182no—
328C22×C3—31no—
329C22×C3—32no—
330C22×C3—31no—
331C22×C3—32no—
332C22×C3—31no—
333C22×C3—31no—
334C22×C3—32no—
335C22×C3—32no—
336C22×C3—31no—
337C22×C3—31no—
338C22×C3—61no—
339C22×C3—62no—
340C22×C3—61no—
341C22×C3—62no—
342C22×C3—61no—
343C22×C3—94no—
344C22×C3—182no—
345C22×C3—184no—
346C22×C3—182no—
347C22×C3—184no—
348C22×C3C2×C3×C4×D311yescharacteristic
349C22×C3C2×C3×C4×D312yes—
350C22×C3C2×C3×D1211yescharacteristic
351C22×C3C3×#23512yes—
352C22×C3C2×C3×#6711yescharacteristic
353C2×D3—34no—
354C2×D3—62no—
355C2×D3—64no—
356C2×D3—62no—
357C2×D3—64no—
358C2×D3—94no—
359C2×D3—182no—
360C2×D3—184no—
361C2×D3—182no—
362C2×D3—184no—
363C3×C4—24no—
364C3×C4—34no—
365C3×C4—34no—
366C3×C4—64no—
367C3×C4—64no—
368C3×C4—68no—
369C3×C4—64no—
370C3×C4—182no—
371C3×C4—184no—
372C3×C4C3×(C2×D3)⋊C414yes—
373Dic3—62no—
374Dic3—64no—
375Dic3—182no—
376Dic3—184no—
377C32—21no—
378C32C4×(C2×D3)⋊C411yescharacteristic, complemented
379C32C3×C4×C22⋊C411yescharacteristic, complemented
380C23—94no—
381C23—92no—
382C23—181no—
383C23—182no—
384C23—181no—
385C23C22×(C3⋊D3)⋊C311yescharacteristic
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—94no—
389C2×C4—92no—
390C2×C4—182no—
391C2×C4—181no—
392C2×C4—182no—
393C2×C4—181no—
394C2×C4—188no—
395C2×C4C4×(C3⋊D3)⋊C312yes—
396C2×C4C4×(C3⋊D3)⋊C314yescomplemented
397C2×C4C4⋊((C3⋊D3)⋊C3)12yes—
398C2×C4#638⋊C312yes—
399C2×C3—31no—
400C2×C3—31no—
401C2×C3—32no—
402C2×C3—32no—
403C2×C3—31no—
404C2×C3—31no—
405C2×C3—31no—
406C2×C3—31no—
407C2×C3—32no—
408C2×C3—32no—
409C2×C3—61no—
410C2×C3—62no—
411C2×C3—61no—
412C2×C3—61no—
413C2×C3—62no—
414C2×C3—184no—
415C2×C3C3×C42×D311yescharacteristic
416C2×C3C3×#87211yescharacteristic
417C2×C3C3×C4×D1212yes—
418C2×C3C2×C3×(C2×D3)⋊C411yescharacteristic
419C2×C3C3×C4×#6712yes—
420D3—64no—
421D3—184no—
422C22—94no—
423C22—182no—
424C22—184no—
425C22—182no—
426C22—184no—
427C22C2×C4×(C3⋊D3)⋊C311yescharacteristic
428C22C2×C4×(C3⋊D3)⋊C312yes—
429C22C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
430C22#3828⋊C312yes—
431C22C2×#638⋊C311yescharacteristic
432C4—24no—
433C4—182no—
434C4—184no—
435C4#3752⋊C314yesdirect product factor
436C3—31no—
437C3—31no—
438C3—61no—
439C3C3×C4×(C2×D3)⋊C411yescharacteristic
440C2—184no—
441C2C42×(C3⋊D3)⋊C311yescharacteristic
442C2#64081⋊C311yescharacteristic
443C2C4×C4⋊((C3⋊D3)⋊C3)12yes—
444C2C2×#3752⋊C311yescharacteristic
445C2C4×#638⋊C312yes—
subgroupG/H
centerC22×C4C2×(C3⋊D3)⋊C3
commutatorC2×C32C2×C3×C42
Frattini, socleC23×C3C22×C3×D3
FittingC2×C42×H(3,𝔽3)C2
prime radicalC23×(C3⋊D3)⋊C3C22
cosocleC23×C32C23×C3
hypercenterC2×C42(C3⋊D3)⋊C3
hypocenterC32C3×C4×C22⋊C4
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C4 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
2. C2×C42×H(3,𝔽3)C2semidirect
4. C2×C4×(C3⋊Dic3)⋊C3C2semidirect
5. #63658⋊C3C2semidirect
6. C4×#3752C3semidirect
15. C2×C4×(C3⋊D3)⋊C3C4semidirect
17. #3752⋊C3C4direct (N × H)
22. C2×C32×C42D3semidirect
23. C2×C32×C42C2×C3semidirect
25. C2×C4×C3⋊Dic3C2×C3semidirect
28. #63658C2×C3semidirect
39. C2×C4×H(3,𝔽3)C2×C4semidirect
47. C2×(C3⋊Dic3)⋊C3C2×C4semidirect
64. C2×C4×C3⋊D3C3×C4semidirect
73. #3752C3×C4semidirect
86. C2×(C3⋊D3)⋊C3C42semidirect
88. C4×H(3,𝔽3)C22⋊C4semidirect
131. C2×C32×C4C4×D3semidirect
135. C2×C32×C4C2×C3×C4semidirect
143. C2×C3⋊Dic3C2×C3×C4semidirect
214. C2×C3⋊D3C3×C42semidirect
224. C32×C4(C2×D3)⋊C4semidirect
225. C32×C4C3×C22⋊C4semidirect
233. C2×C42(C3⋊D3)⋊C3semidirect
235. H(3,𝔽3)C4×C22⋊C4semidirect
378. C32C4×(C2×D3)⋊C4semidirect
379. C32C3×C4×C22⋊C4semidirect
396. C2×C4C4×(C3⋊D3)⋊C3semidirect
435. C4#3752⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×(C3⋊D3)⋊C32
C2×C42×H(3,𝔽3)2
C2×#3752⋊C32
C2×C4×(C3⋊Dic3)⋊C32
#63658⋊C32
C4×#37523
C3×C4×(C2×D3)⋊C43