Finite Group Databasebeta

← C2×#64104⋊C3 #13152438

Subgroups — C2×#64104⋊C3 #13152438

Order 1728.

1583 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
2C22×C4⋊((C3⋊D3)⋊C3)C212yesmaximal, complemented
3C22×C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
4C2×C4⋊C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
5C2×#3752⋊C3C212yesmaximal, complemented
6#64104⋊C3C218yesmaximal, direct product factor
7C2×#64104C311yesmaximal, characteristic, complemented
8C2×C3×#895—31nomaximal
9C23×(C3⋊D3)⋊C3C2212yes—
10C23×(C3⋊D3)⋊C3C2211yescharacteristic
11C22×C4×H(3,𝔽3)C2212yes—
12C22×C4×H(3,𝔽3)C2211yescharacteristic
13C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
14C2×C4⋊((C3⋊D3)⋊C3)—28no—
15C2×C4⋊((C3⋊D3)⋊C3)—24no—
16C2×C4⋊((C3⋊D3)⋊C3)C2218yescomplemented
17C2×C4⋊((C3⋊D3)⋊C3)C2214yescomplemented
18C2×C4×(C3⋊D3)⋊C3—24no—
19C2×C4×(C3⋊D3)⋊C3C2214yescomplemented
20C4⋊C4×H(3,𝔽3)C2214yescomplemented
21#3752⋊C3C2218yescomplemented
22C22×C4⋊(C3⋊D3)C2×C311yescharacteristic, complemented
23C22×C4⋊(C3⋊D3)C2×C312yescomplemented
24C22×C4×C3⋊D3C2×C311yescharacteristic, complemented
25C22×C3×D12—32no—
26C22×C3×D12—31no—
27C22×C3×C4×D3—31no—
28C2×C32×C4⋊C4—21no—
29C2×C32×C4⋊C4D311yescharacteristic, complemented
30C2×C32×C4⋊C4C2×C311yescharacteristic, complemented
31C2×#3752C2×C312yescomplemented
32#64104C2×C318yescomplemented
33C2×C3×(C2×D3)⋊C4—32no—
34C3×#895—38no—
35C23×H(3,𝔽3)C2311yescharacteristic
36C22×(C3⋊D3)⋊C3—22no—
37C22×(C3⋊D3)⋊C3—22no—
38C22×(C3⋊D3)⋊C3—24no—
39C22×(C3⋊D3)⋊C3—24no—
40C22×(C3⋊D3)⋊C3—24no—
41C22×(C3⋊D3)⋊C3D414yescomplemented
42C22×(C3⋊D3)⋊C3C2314yes—
43C22×(C3⋊D3)⋊C3C2312yes—
44C2×C4×H(3,𝔽3)—24no—
45C2×C4×H(3,𝔽3)D414yescomplemented
46C2×C4×H(3,𝔽3)C2314yes—
47C2×C4×H(3,𝔽3)C2312yes—
48C2×(C3⋊Dic3)⋊C3—22no—
49C2×(C3⋊Dic3)⋊C3C2312yes—
50C4⋊((C3⋊D3)⋊C3)—216no—
51C4⋊((C3⋊D3)⋊C3)—28no—
52C4×(C3⋊D3)⋊C3—28no—
53C2×C3×#121—91no—
54C2×#895—31no—
55C2×#895—91no—
56C23×C3⋊D3C22×C312yes—
57C23×C3⋊D3C22×C311yescharacteristic
58C23×C3×D3—31no—
59C23×C3×D3—32no—
60C22×C32×C4—21no—
61C22×C32×C4—22no—
62C22×C32×C4C2×D312yes—
63C22×C32×C4C2×D311yescharacteristic
64C22×C32×C4C22×C312yes—
65C22×C32×C4C22×C311yescharacteristic
66C22×C3⋊Dic3C22×C311yescharacteristic
67C2×C4⋊(C3⋊D3)—28no—
68C2×C4⋊(C3⋊D3)—24no—
69C2×C4⋊(C3⋊D3)C22×C318yescomplemented
70C2×C4⋊(C3⋊D3)C22×C314yescomplemented
71C2×C4×C3⋊D3—24no—
72C2×C4×C3⋊D3C22×C314yescomplemented
73C22×C3×Dic3—31no—
74C2×C3×D12—38no—
75C2×C3×D12—34no—
76C2×C3×D12—68no—
77C2×C3×D12—64no—
78C2×C3×C4×D3—34no—
79C2×C3×C4×D3—64no—
80C32×C4⋊C4—24no—
81C32×C4⋊C4C2×D314yescomplemented
82C32×C4⋊C4C22×C314yescomplemented
83#3752C22×C318yescomplemented
84C3×(C2×D3)⋊C4—38no—
85C22×H(3,𝔽3)C2×D412yes—
86C22×H(3,𝔽3)C2×D412yes—
87C22×H(3,𝔽3)#4112yes—
88C22×H(3,𝔽3)C2411yescharacteristic
89C2×(C3⋊D3)⋊C3—28no—
90C2×(C3⋊D3)⋊C3—22no—
91C2×(C3⋊D3)⋊C3—22no—
92C2×(C3⋊D3)⋊C3—24no—
93C2×(C3⋊D3)⋊C3—24no—
94C2×(C3⋊D3)⋊C3—24no—
95C2×(C3⋊D3)⋊C3—48no—
96C2×(C3⋊D3)⋊C3C2×D414yescomplemented
97C4×H(3,𝔽3)—24no—
98C4×H(3,𝔽3)C2×D414yescomplemented
99(C3⋊Dic3)⋊C3—22no—
100C22×C3×D4—91no—
101C22×C3×D4—92no—
102C23×C3×C4—91no—
103C22×D12—32no—
104C22×D12—31no—
105C22×D12—91no—
106C22×D12—92no—
107C22×C4×D3—31no—
108C22×C4×D3—91no—
109C3×#121—98no—
110C2×C3×C4⋊C4—31no—
111C2×C3×C4⋊C4—31no—
112C2×C3×C4⋊C4—61no—
113C2×C3×C4⋊C4C3×D311yescharacteristic
114C2×C3×C22⋊C4—92no—
115C2×(C2×D3)⋊C4—32no—
116C2×(C2×D3)⋊C4—92no—
117#895—38no—
118#895—98no—
119C23×C32—21no—
120C23×C32C22×D311yescharacteristic
121C23×C32C23×C311yescharacteristic
122C22×C3⋊D3—22no—
123C22×C3⋊D3—22no—
124C22×C3⋊D3—24no—
125C22×C3⋊D3—24no—
126C22×C3⋊D3—24no—
127C22×C3⋊D3C3×D414yescomplemented
128C22×C3⋊D3C23×C314yes—
129C22×C3⋊D3C23×C312yes—
130C22×C3×D3—34no—
131C22×C3×D3—32no—
132C22×C3×D3—34no—
133C22×C3×D3—62no—
134C22×C3×D3—62no—
135C22×C3×D3—64no—
136C22×C3×D3—64no—
137C22×C3×D3—64no—
138C2×C32×C4—24no—
139C2×C32×C4—24no—
140C2×C32×C4—24no—
141C2×C32×C4—22no—
142C2×C32×C4—24no—
143C2×C32×C4—44no—
144C2×C32×C4D1214yescomplemented
145C2×C32×C4C3×D414yescomplemented
146C2×C32×C4C22×D314yes—
147C2×C32×C4C22×D312yes—
148C2×C32×C4C23×C314yes—
149C2×C32×C4C23×C312yes—
150C2×C3⋊Dic3—22no—
151C2×C3⋊Dic3C23×C312yes—
152C4⋊(C3⋊D3)—216no—
153C4⋊(C3⋊D3)—28no—
154C4×C3⋊D3—28no—
155C2×C3×Dic3—32no—
156C2×C3×Dic3—62no—
157C3×D12—616no—
158C3×D12—68no—
159C3×C4×D3—68no—
160C2×#121—91noSylow
161C2×H(3,𝔽3)#12114yescomplemented
162C2×H(3,𝔽3)C22×D411yescharacteristic
163C2×H(3,𝔽3)C22×D411yescharacteristic
164C2×H(3,𝔽3)C2×#4111yescharacteristic
165(C3⋊D3)⋊C3—24no—
166(C3⋊D3)⋊C3—44no—
167C24×C3—91no—
168C24×C3—92no—
169C23×D3—31no—
170C23×D3—32no—
171C23×D3—91no—
172C23×D3—92no—
173C2×C3×D4—94no—
174C2×C3×D4—98no—
175C2×C3×D4—184no—
176C2×C3×D4—188no—
177C22×C3×C4—31no—
178C22×C3×C4—31no—
179C22×C3×C4—32no—
180C22×C3×C4—32no—
181C22×C3×C4—61no—
182C22×C3×C4—62no—
183C22×C3×C4—94no—
184C22×C3×C4—91no—
185C22×C3×C4—184no—
186C22×C3×C4C2×C3×D312yes—
187C22×C3×C4C2×C3×D311yescharacteristic
188C22×Dic3—31no—
189C22×Dic3—91no—
190C2×D12—38no—
191C2×D12—34no—
192C2×D12—68no—
193C2×D12—64no—
194C2×D12—94no—
195C2×D12—98no—
196C2×D12—184no—
197C2×D12—188no—
198C2×C4×D3—34no—
199C2×C4×D3—64no—
200C2×C4×D3—94no—
201C2×C4×D3—184no—
202C3×C4⋊C4—34no—
203C3×C4⋊C4—34no—
204C3×C4⋊C4—64no—
205C3×C4⋊C4C2×C3×D314yes—
206C3×C22⋊C4—98no—
207(C2×D3)⋊C4—38no—
208(C2×D3)⋊C4—98no—
209C22×C32—21no—
210C22×C32—22no—
211C22×C32—22no—
212C22×C32—22no—
213C22×C32C2×D1212yes—
214C22×C32D3×D412yes—
215C22×C32#23912yes—
216C22×C32C2×C3×D412yes—
217C22×C32C2×C3×D412yes—
218C22×C32C3×#4112yes—
219C22×C32C23×D311yescharacteristic
220C22×C32C24×C311yescharacteristic
221C2×C3⋊D3—28no—
222C2×C3⋊D3—22no—
223C2×C3⋊D3—22no—
224C2×C3⋊D3—24no—
225C2×C3⋊D3—24no—
226C2×C3⋊D3—24no—
227C2×C3⋊D3—48no—
228C2×C3⋊D3C2×C3×D414yescomplemented
229C2×C3×D3—34no—
230C2×C3×D3—68no—
231C2×C3×D3—62no—
232C2×C3×D3—62no—
233C2×C3×D3—64no—
234C2×C3×D3—64no—
235C2×C3×D3—64no—
236C2×C3×D3—128no—
237C32×C4—24no—
238C32×C4—24no—
239C32×C4—24no—
240C32×C4—44no—
241C32×C4C2×D1214yescomplemented
242C32×C4C2×C3×D414yescomplemented
243C3⋊Dic3—22no—
244C3×Dic3—62no—
245C22×D4—92no—
246C22×D4—91no—
247C23×C4—91no—
248#121—98no—
249C2×C4⋊C4(C3⋊D3)⋊C311yescharacteristic, complemented
250C2×C22⋊C4—92no—
251H(3,𝔽3)C2×#12111yescharacteristic, complemented, Sylow
252C23×C3—31no—
253C23×C3—31no—
254C23×C3—61no—
255C23×C3—92no—
256C23×C3—94no—
257C23×C3—94no—
258C23×C3—182no—
259C23×C3—182no—
260C23×C3—184no—
261C23×C3—184no—
262C23×C3—184no—
263C23×C3C22×C3×D311yescharacteristic
264C22×D3—32no—
265C22×D3—34no—
266C22×D3—34no—
267C22×D3—62no—
268C22×D3—62no—
269C22×D3—64no—
270C22×D3—64no—
271C22×D3—64no—
272C22×D3—94no—
273C22×D3—92no—
274C22×D3—94no—
275C22×D3—182no—
276C22×D3—182no—
277C22×D3—184no—
278C22×D3—184no—
279C22×D3—184no—
280C3×D4—188no—
281C3×D4—1816no—
282C2×C3×C4—24no—
283C2×C3×C4—34no—
284C2×C3×C4—34no—
285C2×C3×C4—32no—
286C2×C3×C4—32no—
287C2×C3×C4—34no—
288C2×C3×C4—34no—
289C2×C3×C4—64no—
290C2×C3×C4—64no—
291C2×C3×C4—64no—
292C2×C3×C4—62no—
293C2×C3×C4—64no—
294C2×C3×C4—92no—
295C2×C3×C4—124no—
296C2×C3×C4—188no—
297C2×C3×C4—182no—
298C2×C3×C4C3×D1214yes—
299C2×C3×C4C22×C3×D314yes—
300C2×C3×C4C22×C3×D312yes—
301C2×Dic3—32no—
302C2×Dic3—62no—
303C2×Dic3—92no—
304C2×Dic3—182no—
305D12—616no—
306D12—68no—
307D12—188no—
308D12—1816no—
309C4×D3—68no—
310C4×D3—188no—
311C2×C32—24no—
312C2×C32—21no—
313C2×C32—21no—
314C2×C32—21no—
315C2×C32#89514yescomplemented
316C2×C32C3×#12114yescomplemented
317C2×C32C22×D1211yescharacteristic
318C2×C32C2×D3×D411yescharacteristic
319C2×C32C2×#23911yescharacteristic
320C2×C32C22×C3×D411yescharacteristic
321C2×C32C22×C3×D411yescharacteristic
322C2×C32C2×C3×#4111yescharacteristic
323C3⋊D3—24no—
324C3⋊D3—44no—
325C3×D3—64no—
326C3×D3—124no—
327C24—91no—
328C24—92no—
329C2×D4—98no—
330C2×D4—94no—
331C2×D4—188no—
332C2×D4—184no—
333C22×C4—91no—
334C22×C4—94no—
335C22×C4—184no—
336C22×C4C2×(C3⋊D3)⋊C312yes—
337C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
338C4⋊C4C2×(C3⋊D3)⋊C314yescomplemented
339C22⋊C4—98no—
340C22×C3—31no—
341C22×C3—31no—
342C22×C3—32no—
343C22×C3—32no—
344C22×C3—32no—
345C22×C3—32no—
346C22×C3—32no—
347C22×C3—32no—
348C22×C3—61no—
349C22×C3—62no—
350C22×C3—62no—
351C22×C3—62no—
352C22×C3—94no—
353C22×C3—182no—
354C22×C3—182no—
355C22×C3—188no—
356C22×C3—184no—
357C22×C3—184no—
358C22×C3—184no—
359C22×C3—368no—
360C22×C3C2×C3×D1212yes—
361C22×C3C3×D3×D412yes—
362C22×C3C3×#23912yes—
363C22×C3C23×C3×D311yescharacteristic
364C2×D3—34no—
365C2×D3—68no—
366C2×D3—62no—
367C2×D3—62no—
368C2×D3—64no—
369C2×D3—64no—
370C2×D3—64no—
371C2×D3—94no—
372C2×D3—128no—
373C2×D3—182no—
374C2×D3—182no—
375C2×D3—188no—
376C2×D3—184no—
377C2×D3—184no—
378C2×D3—184no—
379C2×D3—368no—
380C3×C4—24no—
381C3×C4—34no—
382C3×C4—34no—
383C3×C4—64no—
384C3×C4—64no—
385C3×C4—64no—
386C3×C4—124no—
387C3×C4—182no—
388C3×C4C2×C3×D1214yes—
389Dic3—62no—
390Dic3—182no—
391C32—21no—
392C32C2×#89511yescharacteristic, complemented
393C32C2×C3×#12111yescharacteristic, complemented
394C23—92no—
395C23—94no—
396C23—94no—
397C23—182no—
398C23—182no—
399C23—184no—
400C23—184no—
401C23—184no—
402C23C22×(C3⋊D3)⋊C311yescharacteristic
403D4—1816no—
404D4—188no—
405C2×C4—24no—
406C2×C4—92no—
407C2×C4—182no—
408C2×C4—188no—
409C2×C4C4⋊((C3⋊D3)⋊C3)14yescomplemented
410C2×C4C22×(C3⋊D3)⋊C314yes—
411C2×C4C22×(C3⋊D3)⋊C312yes—
412C2×C3—31no—
413C2×C3—31no—
414C2×C3—31no—
415C2×C3—31no—
416C2×C3—31no—
417C2×C3—31no—
418C2×C3—34no—
419C2×C3—34no—
420C2×C3—61no—
421C2×C3—61no—
422C2×C3—61no—
423C2×C3—64no—
424C2×C3—184no—
425C2×C3—364no—
426C2×C3C3×#89514yes—
427C2×C3C22×C3×D1211yescharacteristic
428C2×C3C2×C3×D3×D411yescharacteristic
429C2×C3C2×C3×#23911yescharacteristic
430D3—64no—
431D3—124no—
432D3—184no—
433D3—364no—
434C22—94no—
435C22—182no—
436C22—182no—
437C22—188no—
438C22—184no—
439C22—184no—
440C22—184no—
441C22—368no—
442C22C2×C4⋊((C3⋊D3)⋊C3)12yes—
443C22D4×(C3⋊D3)⋊C312yes—
444C22#8872212yes—
445C22C23×(C3⋊D3)⋊C311yescharacteristic
446C4—24no—
447C4—182no—
448C4C2×C4⋊((C3⋊D3)⋊C3)14yescomplemented
449C3—31no—
450C3—31no—
451C3—61no—
452C3C2×C3×#89511yescharacteristic
453C2—184no—
454C2—364no—
455C2#64104⋊C314yesdirect product factor
456C2C22×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
457C2C2×D4×(C3⋊D3)⋊C311yescharacteristic
458C2C2×#8872211yescharacteristic
subgroupG/H
centerC23C22×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
FrattiniC22×C3C23×C3×D3
FittingC2×C4⋊C4×H(3,𝔽3)C2
socleC23×C3C22×C3×D3
hypercenterC2×C4⋊C4(C3⋊D3)⋊C3
hypocenterC32C2×C3×#121
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×C4⋊((C3⋊D3)⋊C3)C2semidirect
2. C22×C4⋊((C3⋊D3)⋊C3)C2semidirect
3. C22×C4×(C3⋊D3)⋊C3C2semidirect
4. C2×C4⋊C4×H(3,𝔽3)C2semidirect
5. C2×#3752⋊C3C2semidirect
6. #64104⋊C3C2direct (N × H)
7. C2×#64104C3semidirect
16. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
17. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
19. C2×C4×(C3⋊D3)⋊C3C22semidirect
20. C4⋊C4×H(3,𝔽3)C22semidirect
21. #3752⋊C3C22semidirect
22. C22×C4⋊(C3⋊D3)C2×C3semidirect
23. C22×C4⋊(C3⋊D3)C2×C3semidirect
24. C22×C4×C3⋊D3C2×C3semidirect
29. C2×C32×C4⋊C4D3semidirect
30. C2×C32×C4⋊C4C2×C3semidirect
31. C2×#3752C2×C3semidirect
32. #64104C2×C3semidirect
41. C22×(C3⋊D3)⋊C3D4semidirect
45. C2×C4×H(3,𝔽3)D4semidirect
69. C2×C4⋊(C3⋊D3)C22×C3semidirect
70. C2×C4⋊(C3⋊D3)C22×C3semidirect
72. C2×C4×C3⋊D3C22×C3semidirect
81. C32×C4⋊C4C2×D3semidirect
82. C32×C4⋊C4C22×C3semidirect
83. #3752C22×C3semidirect
96. C2×(C3⋊D3)⋊C3C2×D4semidirect
98. C4×H(3,𝔽3)C2×D4semidirect
127. C22×C3⋊D3C3×D4semidirect
144. C2×C32×C4D12semidirect
145. C2×C32×C4C3×D4semidirect
161. C2×H(3,𝔽3)#121semidirect
228. C2×C3⋊D3C2×C3×D4semidirect
241. C32×C4C2×D12semidirect
242. C32×C4C2×C3×D4semidirect
249. C2×C4⋊C4(C3⋊D3)⋊C3semidirect
251. H(3,𝔽3)C2×#121semidirect
315. C2×C32#895semidirect
316. C2×C32C3×#121semidirect
338. C4⋊C4C2×(C3⋊D3)⋊C3semidirect
392. C32C2×#895semidirect
393. C32C2×C3×#121semidirect
409. C2×C4C4⋊((C3⋊D3)⋊C3)semidirect
448. C4C2×C4⋊((C3⋊D3)⋊C3)semidirect
455. C2#64104⋊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)⋊C3)2
C22×C4×(C3⋊D3)⋊C32
C2×C4⋊C4×H(3,𝔽3)2
C2×#3752⋊C32
#64104⋊C32
C2×#641043
C2×C3×#8953