Finite Group Databasebeta

← C2×C22⋊C4×(C3⋊D3)⋊C3 #13152360

Subgroups — C2×C22⋊C4×(C3⋊D3)⋊C3 #13152360

Order 1728.

2491 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C24×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2C22×C4×(C3⋊D3)⋊C3C212yesmaximal, complemented
3C2×C22⋊C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
4C2×#3757⋊C3C211yesmaximal, characteristic, complemented
5C2×#3752⋊C3C212yesmaximal, complemented
6C22⋊C4×(C3⋊D3)⋊C3C218yesmaximal, direct product factor
7C2×C22⋊C4×C3⋊D3C311yesmaximal, characteristic, complemented
8C2×C3×D3×C22⋊C4—31nomaximal
9C24×H(3,𝔽3)C2211yescharacteristic
10C23×(C3⋊D3)⋊C3—28no—
11C23×(C3⋊D3)⋊C3C418yescomplemented
12C23×(C3⋊D3)⋊C3C2211yescharacteristic
13C23×(C3⋊D3)⋊C3C2214yes—
14C23×(C3⋊D3)⋊C3C2211yescharacteristic
15C22×C4×H(3,𝔽3)C2212yescomplemented
16C22×(C3⋊Dic3)⋊C3C2212yescomplemented
17C2×C4×(C3⋊D3)⋊C3—28no—
18C2×C4×(C3⋊D3)⋊C3C2218yescomplemented
19C22⋊C4×H(3,𝔽3)C2214yescomplemented
20#3757⋊C3C2214yescomplemented
21#3752⋊C3C2218yescomplemented
22C24×C3⋊D3C2×C311yescharacteristic
23C24×C3×D3—31no—
24C22×C4×C3⋊D3C2×C312yescomplemented
25C22×C3×C4×D3—32no—
26C2×C32×C22⋊C4—21no—
27C2×C32×C22⋊C4D311yescharacteristic, complemented
28C2×C32×C22⋊C4C2×C311yescharacteristic, complemented
29C2×#3757C2×C311yescharacteristic, complemented
30C2×#3752C2×C312yescomplemented
31C22⋊C4×C3⋊D3C2×C318yescomplemented
32C2×C3×#217—31no—
33C2×C3×(C2×D3)⋊C4—32no—
34C3×D3×C22⋊C4—38no—
35C23×H(3,𝔽3)—24no—
36C23×H(3,𝔽3)C2×C414yescomplemented
37C23×H(3,𝔽3)C2312yes—
38C23×H(3,𝔽3)C2311yescharacteristic
39C22×(C3⋊D3)⋊C3—24no—
40C22×(C3⋊D3)⋊C3—232no—
41C22×(C3⋊D3)⋊C3—216no—
42C22×(C3⋊D3)⋊C3C2×C414yes—
43C22×(C3⋊D3)⋊C3C2×C4116yescomplemented
44C22×(C3⋊D3)⋊C3C2×C414yescomplemented
45C22×(C3⋊D3)⋊C3D418yes—
46C22×(C3⋊D3)⋊C3C2312yes—
47C22×(C3⋊D3)⋊C3C2312yes—
48C2×C4×H(3,𝔽3)—24no—
49C2×C4×H(3,𝔽3)C2314yescomplemented
50C2×(C3⋊Dic3)⋊C3—24no—
51C2×(C3⋊Dic3)⋊C3C2314yescomplemented
52C4×(C3⋊D3)⋊C3—216no—
53C22×C3×C22⋊C4—91no—
54C2×D3×C22⋊C4—31no—
55C2×D3×C22⋊C4—91no—
56C24×C32—21no—
57C24×C32C2×D311yescharacteristic
58C24×C32C22×C311yescharacteristic
59C23×C3⋊D3—28no—
60C23×C3⋊D3C3×C418yescomplemented
61C23×C3⋊D3C22×C314yes—
62C23×C3⋊D3C22×C311yescharacteristic
63C23×C3⋊D3C22×C311yescharacteristic
64C23×C3×D3—31no—
65C23×C3×D3—38no—
66C23×C3×D3—34no—
67C23×C3×D3—31no—
68C23×C3×D3—68no—
69C22×C32×C4—22no—
70C22×C32×C4C2×D312yescomplemented
71C22×C32×C4C22×C312yescomplemented
72C22×C3⋊Dic3C22×C312yescomplemented
73C2×C4×C3⋊D3—28no—
74C2×C4×C3⋊D3C22×C318yescomplemented
75C22×C3×Dic3—32no—
76C2×C3×C4×D3—38no—
77C2×C3×C4×D3—68no—
78C32×C22⋊C4—24no—
79C32×C22⋊C4C2×D314yescomplemented
80C32×C22⋊C4C22×C314yescomplemented
81#3757C22×C314yescomplemented
82#3752C22×C318yescomplemented
83C3×#217—34no—
84C3×(C2×D3)⋊C4—38no—
85C22×H(3,𝔽3)—28no—
86C22×H(3,𝔽3)—24no—
87C22×H(3,𝔽3)C22×C412yes—
88C22×H(3,𝔽3)C22×C414yescomplemented
89C22×H(3,𝔽3)C2×D414yes—
90C22×H(3,𝔽3)C2411yescharacteristic
91C2×(C3⋊D3)⋊C3—28no—
92C2×(C3⋊D3)⋊C3—24no—
93C2×(C3⋊D3)⋊C3—232no—
94C2×(C3⋊D3)⋊C3C22⋊C4116yesdirect product factor
95C2×(C3⋊D3)⋊C3C22×C414yes—
96C2×(C3⋊D3)⋊C3C22×C414yescomplemented
97C2×(C3⋊D3)⋊C3C2×D418yes—
98C4×H(3,𝔽3)—24no—
99(C3⋊Dic3)⋊C3—24no—
100C25×C3—91no—
101C24×D3—31no—
102C24×D3—91no—
103C23×C3×C4—92no—
104C22×C4×D3—32no—
105C22×C4×D3—92no—
106C2×C3×C22⋊C4—31no—
107C2×C3×C22⋊C4—31no—
108C2×C3×C22⋊C4—61no—
109C2×C3×C22⋊C4—98no—
110C2×C3×C22⋊C4—92no—
111C2×C3×C22⋊C4—91no—
112C2×C3×C22⋊C4C3×D311yescharacteristic
113C2×#217—31no—
114C2×#217—91no—
115C2×(C2×D3)⋊C4—32no—
116C2×(C2×D3)⋊C4—92no—
117D3×C22⋊C4—38no—
118D3×C22⋊C4—98no—
119C23×C32—24no—
120C23×C32—24no—
121C23×C32—24no—
122C23×C32—21no—
123C23×C32—22no—
124C23×C32—44no—
125C23×C32C4×D314yescomplemented
126C23×C32C2×C3×C414yescomplemented
127C23×C32C22×D312yes—
128C23×C32C22×D311yescharacteristic
129C23×C32C23×C312yes—
130C23×C32C23×C311yescharacteristic
131C22×C3⋊D3—24no—
132C22×C3⋊D3—232no—
133C22×C3⋊D3—216no—
134C22×C3⋊D3C2×C3×C414yes—
135C22×C3⋊D3C2×C3×C4116yescomplemented
136C22×C3⋊D3C2×C3×C414yescomplemented
137C22×C3⋊D3C3×D418yes—
138C22×C3⋊D3C23×C312yes—
139C22×C3⋊D3C23×C312yes—
140C22×C3×D3—34no—
141C22×C3×D3—38no—
142C22×C3×D3—32no—
143C22×C3×D3—34no—
144C22×C3×D3—316no—
145C22×C3×D3—32no—
146C22×C3×D3—632no—
147C22×C3×D3—616no—
148C22×C3×D3—64no—
149C2×C32×C4—24no—
150C2×C32×C4—24no—
151C2×C32×C4—24no—
152C2×C32×C4—44no—
153C2×C32×C4C22×D314yescomplemented
154C2×C32×C4C23×C314yescomplemented
155C2×C3⋊Dic3—24no—
156C2×C3⋊Dic3C23×C314yescomplemented
157C4×C3⋊D3—216no—
158C2×C3×Dic3—34no—
159C2×C3×Dic3—64no—
160C3×C4×D3—616no—
161C22×C22⋊C4—91noSylow
162C2×H(3,𝔽3)—24no—
163C2×H(3,𝔽3)C2×C22⋊C414yescomplemented
164C2×H(3,𝔽3)C23×C411yescharacteristic
165C2×H(3,𝔽3)C22×D412yes—
166(C3⋊D3)⋊C3—24no—
167(C3⋊D3)⋊C3C2×C22⋊C418yesdirect product factor
168C24×C3—31no—
169C24×C3—31no—
170C24×C3—61no—
171C24×C3—91no—
172C24×C3—98no—
173C24×C3—94no—
174C24×C3—91no—
175C24×C3—188no—
176C24×C3C2×C3×D311yescharacteristic
177C23×D3—31no—
178C23×D3—38no—
179C23×D3—34no—
180C23×D3—31no—
181C23×D3—68no—
182C23×D3—91no—
183C23×D3—98no—
184C23×D3—94no—
185C23×D3—91no—
186C23×D3—188no—
187C22×C3×C4—32no—
188C22×C3×C4—32no—
189C22×C3×C4—62no—
190C22×C3×C4—98no—
191C22×C3×C4—92no—
192C22×C3×C4—188no—
193C22×C3×C4C2×C3×D312yes—
194C22×Dic3—32no—
195C22×Dic3—92no—
196C2×C4×D3—38no—
197C2×C4×D3—68no—
198C2×C4×D3—98no—
199C2×C4×D3—188no—
200C3×C22⋊C4—34no—
201C3×C22⋊C4—34no—
202C3×C22⋊C4—64no—
203C3×C22⋊C4—98no—
204C3×C22⋊C4—94no—
205C3×C22⋊C4C2×C3×D314yes—
206#217—34no—
207#217—94no—
208(C2×D3)⋊C4—38no—
209(C2×D3)⋊C4—98no—
210C22×C32—28no—
211C22×C32—28no—
212C22×C32—24no—
213C22×C32—24no—
214C22×C32—21no—
215C22×C32—22no—
216C22×C32—24no—
217C22×C32—24no—
218C22×C32—48no—
219C22×C32—44no—
220C22×C32C2×C4×D312yes—
221C22×C32C2×C4×D314yescomplemented
222C22×C32D3×D414yes—
223C22×C32C22×C3×C412yes—
224C22×C32C22×C3×C414yescomplemented
225C22×C32C2×C3×D414yes—
226C22×C32C23×D311yescharacteristic
227C22×C32C24×C311yescharacteristic
228C2×C3⋊D3—28no—
229C2×C3⋊D3—24no—
230C2×C3⋊D3—232no—
231C2×C3⋊D3C3×C22⋊C4116yescomplemented
232C2×C3⋊D3C22×C3×C414yes—
233C2×C3⋊D3C22×C3×C414yescomplemented
234C2×C3⋊D3C2×C3×D418yes—
235C2×C3×D3—316no—
236C2×C3×D3—34no—
237C2×C3×D3—38no—
238C2×C3×D3—34no—
239C2×C3×D3—632no—
240C2×C3×D3—68no—
241C2×C3×D3—64no—
242C32×C4—24no—
243C32×C4—24no—
244C32×C4—44no—
245C3⋊Dic3—24no—
246C3×Dic3—64no—
247C25—91no—
248C23×C4—92no—
249C2×C22⋊C4—91no—
250C2×C22⋊C4—92no—
251C2×C22⋊C4—98no—
252C2×C22⋊C4(C3⋊D3)⋊C311yescharacteristic, direct product factor
253H(3,𝔽3)C22×C22⋊C411yescharacteristic, complemented, Sylow
254C23×C3—24no—
255C23×C3—34no—
256C23×C3—34no—
257C23×C3—32no—
258C23×C3—32no—
259C23×C3—31no—
260C23×C3—31no—
261C23×C3—64no—
262C23×C3—64no—
263C23×C3—64no—
264C23×C3—62no—
265C23×C3—61no—
266C23×C3—92no—
267C23×C3—98no—
268C23×C3—94no—
269C23×C3—92no—
270C23×C3—94no—
271C23×C3—916no—
272C23×C3—124no—
273C23×C3—184no—
274C23×C3—1816no—
275C23×C3—1832no—
276C23×C3C3×C4×D314yes—
277C23×C3C22×C3×D312yes—
278C23×C3C22×C3×D311yescharacteristic
279C22×D3—32no—
280C22×D3—34no—
281C22×D3—38no—
282C22×D3—32no—
283C22×D3—34no—
284C22×D3—316no—
285C22×D3—632no—
286C22×D3—616no—
287C22×D3—64no—
288C22×D3—94no—
289C22×D3—98no—
290C22×D3—92no—
291C22×D3—94no—
292C22×D3—916no—
293C22×D3—92no—
294C22×D3—1816no—
295C22×D3—1832no—
296C22×D3—184no—
297C2×C3×C4—24no—
298C2×C3×C4—34no—
299C2×C3×C4—34no—
300C2×C3×C4—64no—
301C2×C3×C4—64no—
302C2×C3×C4—64no—
303C2×C3×C4—94no—
304C2×C3×C4—124no—
305C2×C3×C4—1816no—
306C2×C3×C4—184no—
307C2×C3×C4C22×C3×D314yes—
308C2×Dic3—34no—
309C2×Dic3—64no—
310C2×Dic3—94no—
311C2×Dic3—184no—
312C4×D3—616no—
313C4×D3—1816no—
314C2×C32—24no—
315C2×C32—24no—
316C2×C32—24no—
317C2×C32—21no—
318C2×C32—22no—
319C2×C32—44no—
320C2×C32D3×C22⋊C414yescomplemented
321C2×C32C2×C3×C22⋊C414yescomplemented
322C2×C32C22×C4×D311yescharacteristic
323C2×C32C2×D3×D412yes—
324C2×C32C23×C3×C411yescharacteristic
325C2×C32C22×C3×D412yes—
326C3⋊D3—24no—
327C3⋊D3C2×C3×C22⋊C418yescomplemented
328C3×D3—38no—
329C3×D3—64no—
330C24—91no—
331C24—98no—
332C24—94no—
333C24—91no—
334C24—188no—
335C24C2×(C3⋊D3)⋊C311yescharacteristic
336C22×C4—92no—
337C22×C4—98no—
338C22×C4—188no—
339C22×C4C2×(C3⋊D3)⋊C312yescomplemented
340C22⋊C4—94no—
341C22⋊C4—98no—
342C22⋊C4C2×(C3⋊D3)⋊C314yesdirect product factor
343C22×C3—28no—
344C22×C3—24no—
345C22×C3—31no—
346C22×C3—31no—
347C22×C3—32no—
348C22×C3—32no—
349C22×C3—34no—
350C22×C3—34no—
351C22×C3—34no—
352C22×C3—34no—
353C22×C3—64no—
354C22×C3—68no—
355C22×C3—64no—
356C22×C3—68no—
357C22×C3—61no—
358C22×C3—62no—
359C22×C3—64no—
360C22×C3—64no—
361C22×C3—98no—
362C22×C3—94no—
363C22×C3—916no—
364C22×C3—94no—
365C22×C3—124no—
366C22×C3—128no—
367C22×C3—184no—
368C22×C3—188no—
369C22×C3—1832no—
370C22×C3C2×C3×C4×D312yes—
371C22×C3C2×C3×C4×D314yes—
372C22×C3C3×D3×D414yes—
373C22×C3C23×C3×D311yescharacteristic
374C2×D3—316no—
375C2×D3—34no—
376C2×D3—38no—
377C2×D3—34no—
378C2×D3—632no—
379C2×D3—68no—
380C2×D3—64no—
381C2×D3—98no—
382C2×D3—94no—
383C2×D3—916no—
384C2×D3—94no—
385C2×D3—1832no—
386C2×D3—184no—
387C2×D3—188no—
388C3×C4—24no—
389C3×C4—64no—
390C3×C4—64no—
391C3×C4—124no—
392C3×C4—184no—
393Dic3—64no—
394Dic3—184no—
395C32—21no—
396C32C2×D3×C22⋊C411yescharacteristic, complemented
397C32C22×C3×C22⋊C411yescharacteristic, complemented
398C23—24no—
399C23—92no—
400C23—94no—
401C23—98no—
402C23—92no—
403C23—94no—
404C23—916no—
405C23—1816no—
406C23—1832no—
407C23—184no—
408C23C4×(C3⋊D3)⋊C314yescomplemented
409C23C22×(C3⋊D3)⋊C312yes—
410C23C22×(C3⋊D3)⋊C311yescharacteristic
411C2×C4—24no—
412C2×C4—94no—
413C2×C4—1816no—
414C2×C4—184no—
415C2×C4C22×(C3⋊D3)⋊C314yescomplemented
416C2×C3—24no—
417C2×C3—31no—
418C2×C3—31no—
419C2×C3—32no—
420C2×C3—32no—
421C2×C3—34no—
422C2×C3—34no—
423C2×C3—64no—
424C2×C3—64no—
425C2×C3—61no—
426C2×C3—62no—
427C2×C3—64no—
428C2×C3—98no—
429C2×C3—124no—
430C2×C3—184no—
431C2×C3C3×D3×C22⋊C414yes—
432C2×C3C22×C3×C4×D311yescharacteristic
433C2×C3C2×C3×D3×D412yes—
434D3—38no—
435D3—64no—
436D3—98no—
437D3—184no—
438C22—24no—
439C22—28no—
440C22—94no—
441C22—98no—
442C22—916no—
443C22—94no—
444C22—1832no—
445C22—184no—
446C22—188no—
447C22C2×C4×(C3⋊D3)⋊C312yes—
448C22C2×C4×(C3⋊D3)⋊C314yescomplemented
449C22D4×(C3⋊D3)⋊C314yes—
450C22C23×(C3⋊D3)⋊C311yescharacteristic
451C4—24no—
452C4—184no—
453C3—31no—
454C3—31no—
455C3—61no—
456C3C2×C3×D3×C22⋊C411yescharacteristic
457C2—24no—
458C2—98no—
459C2—184no—
460C2C22⋊C4×(C3⋊D3)⋊C314yesdirect product factor
461C2C22×C4×(C3⋊D3)⋊C311yescharacteristic
462C2C2×D4×(C3⋊D3)⋊C312yes—
subgroupG/H
centerC23C22×(C3⋊D3)⋊C3
commutatorC2×C32C23×C3×C4
FrattiniC22×C3C23×C3×D3
FittingC2×C22⋊C4×H(3,𝔽3)C2
socleC23×C3C22×C3×D3
prime radicalC24×(C3⋊D3)⋊C3C2
cosocleC22×C32C24×C3
hypercenterC2×C22⋊C4(C3⋊D3)⋊C3
hypocenterC32C22×C3×C22⋊C4
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C22×C4×(C3⋊D3)⋊C3C2semidirect
3. C2×C22⋊C4×H(3,𝔽3)C2semidirect
4. C2×#3757⋊C3C2semidirect
5. C2×#3752⋊C3C2semidirect
6. C22⋊C4×(C3⋊D3)⋊C3C2direct (N × H)
7. C2×C22⋊C4×C3⋊D3C3semidirect
11. C23×(C3⋊D3)⋊C3C4semidirect
15. C22×C4×H(3,𝔽3)C22semidirect
16. C22×(C3⋊Dic3)⋊C3C22semidirect
18. C2×C4×(C3⋊D3)⋊C3C22semidirect
19. C22⋊C4×H(3,𝔽3)C22semidirect
20. #3757⋊C3C22semidirect
21. #3752⋊C3C22semidirect
24. C22×C4×C3⋊D3C2×C3semidirect
27. C2×C32×C22⋊C4D3semidirect
28. C2×C32×C22⋊C4C2×C3semidirect
29. C2×#3757C2×C3semidirect
30. C2×#3752C2×C3semidirect
31. C22⋊C4×C3⋊D3C2×C3semidirect
36. C23×H(3,𝔽3)C2×C4semidirect
43. C22×(C3⋊D3)⋊C3C2×C4semidirect
44. C22×(C3⋊D3)⋊C3C2×C4semidirect
49. C2×C4×H(3,𝔽3)C23semidirect
51. C2×(C3⋊Dic3)⋊C3C23semidirect
60. C23×C3⋊D3C3×C4semidirect
70. C22×C32×C4C2×D3semidirect
71. C22×C32×C4C22×C3semidirect
72. C22×C3⋊Dic3C22×C3semidirect
74. C2×C4×C3⋊D3C22×C3semidirect
79. C32×C22⋊C4C2×D3semidirect
80. C32×C22⋊C4C22×C3semidirect
81. #3757C22×C3semidirect
82. #3752C22×C3semidirect
88. C22×H(3,𝔽3)C22×C4semidirect
94. C2×(C3⋊D3)⋊C3C22⋊C4direct (N × H)
96. C2×(C3⋊D3)⋊C3C22×C4semidirect
125. C23×C32C4×D3semidirect
126. C23×C32C2×C3×C4semidirect
135. C22×C3⋊D3C2×C3×C4semidirect
136. C22×C3⋊D3C2×C3×C4semidirect
153. C2×C32×C4C22×D3semidirect
154. C2×C32×C4C23×C3semidirect
156. C2×C3⋊Dic3C23×C3semidirect
163. C2×H(3,𝔽3)C2×C22⋊C4semidirect
167. (C3⋊D3)⋊C3C2×C22⋊C4direct (N × H)
221. C22×C32C2×C4×D3semidirect
224. C22×C32C22×C3×C4semidirect
231. C2×C3⋊D3C3×C22⋊C4semidirect
233. C2×C3⋊D3C22×C3×C4semidirect
252. C2×C22⋊C4(C3⋊D3)⋊C3direct (N × H)
253. H(3,𝔽3)C22×C22⋊C4semidirect
320. C2×C32D3×C22⋊C4semidirect
321. C2×C32C2×C3×C22⋊C4semidirect
327. C3⋊D3C2×C3×C22⋊C4semidirect
339. C22×C4C2×(C3⋊D3)⋊C3semidirect
342. C22⋊C4C2×(C3⋊D3)⋊C3direct (N × H)
396. C32C2×D3×C22⋊C4semidirect
397. C32C22×C3×C22⋊C4semidirect
408. C23C4×(C3⋊D3)⋊C3semidirect
415. C2×C4C22×(C3⋊D3)⋊C3semidirect
448. C22C2×C4×(C3⋊D3)⋊C3semidirect
460. C2C22⋊C4×(C3⋊D3)⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C24×(C3⋊D3)⋊C32
C22×C4×(C3⋊D3)⋊C32
C2×C22⋊C4×H(3,𝔽3)2
C2×#3757⋊C32
C2×#3752⋊C32
C22⋊C4×(C3⋊D3)⋊C32
C2×C22⋊C4×C3⋊D33
C2×C3×D3×C22⋊C43