Finite Group Databasebeta

← C2×#64160⋊C3 #13153711

Subgroups — C2×#64160⋊C3 #13153711

Order 1728.

1067 conjugacy classes of subgroups.

The subgroup lattice has 469 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×H(3,𝔽3)C211yesmaximal, characteristic
2C22×#634⋊C3C211yesmaximal, characteristic, complemented
3#64160⋊C3C218yesmaximal, direct product factor
4C2×#3756⋊C3C212yesmaximal, complemented
5C2×#3754⋊C3C212yesmaximal, complemented
6C2×#3748⋊C3C211yesmaximal, characteristic, complemented
7C2×#64160C311yesmaximal, characteristic, complemented
8C2×C3×#951—31nomaximal
9#41×H(3,𝔽3)—22no—
10#41×H(3,𝔽3)C2214yes—
11C2×Q8×H(3,𝔽3)C2211yescharacteristic
12C2×D4×H(3,𝔽3)—21no—
13C2×D4×H(3,𝔽3)C2211yescharacteristic
14C22×C4×H(3,𝔽3)—21no—
15C22×C4×H(3,𝔽3)C2211yescharacteristic
16C22×(C3⋊Dic3)⋊C3—21no—
17C2×#634⋊C3—22no—
18C2×#634⋊C3—21no—
19C2×#634⋊C3C2212yes—
20C2×#634⋊C3C2214yescomplemented
21#3756⋊C3C2218yescomplemented
22#3754⋊C3C2218yescomplemented
23#3748⋊C3C2214yescomplemented
24C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
25C2×C32×#41—21no—
26C2×C32×#41D311yescharacteristic
27C2×C32×#41C2×C311yescharacteristic
28C22×#634C2×C311yescharacteristic, complemented
29C22×C3×Dic6—31no—
30#64160C2×C318yescomplemented
31C2×#3756C2×C312yescomplemented
32C2×#3754C2×C312yescomplemented
33C2×#3748C2×C311yescharacteristic, complemented
34C3×#951—38no—
35C2×C3×#216—32no—
36C2×C3×#214—32no—
37C2×C3×#208—31no—
38C23×H(3,𝔽3)—21no—
39C23×H(3,𝔽3)D411yescharacteristic
40Q8×H(3,𝔽3)—21no—
41Q8×H(3,𝔽3)C2312yes—
42D4×H(3,𝔽3)—24no—
43D4×H(3,𝔽3)—21no—
44D4×H(3,𝔽3)C2312yes—
45C2×C4×H(3,𝔽3)—21no—
46C2×C4×H(3,𝔽3)—24no—
47C2×C4×H(3,𝔽3)D411yescharacteristic
48C2×C4×H(3,𝔽3)D412yes—
49C2×C4×H(3,𝔽3)C2311yescharacteristic
50C2×C4×H(3,𝔽3)C2312yes—
51C2×(C3⋊Dic3)⋊C3—22no—
52C2×(C3⋊Dic3)⋊C3—24no—
53#634⋊C3—22no—
54#634⋊C3—24no—
55#634⋊C3C2314yes—
56(C3⋊(C3⋊C8))⋊C3C2314yes—
57C2×C3×#137—91no—
58C2×#951—31no—
59C2×#951—91no—
60C32×#41—24no—
61C32×#41—24no—
62C32×#41—22no—
63C32×#41—22no—
64C32×#41C2×D314yes—
65C32×#41C22×C314yes—
66C2×C32×Q8—21no—
67C2×C32×Q8C2×D311yescharacteristic
68C2×C32×Q8C22×C311yescharacteristic
69C2×C32×D4—22no—
70C2×C32×D4—21no—
71C2×C32×D4—21no—
72C2×C32×D4—21no—
73C2×C32×D4C2×D311yescharacteristic
74C2×C32×D4C22×C311yescharacteristic
75C22×C32×C4—22no—
76C22×C32×C4—21no—
77C22×C32×C4—21no—
78C22×C32×C4—21no—
79C22×C32×C4C2×D311yescharacteristic
80C22×C32×C4C22×C311yescharacteristic
81C22×C3⋊Dic3—21no—
82C2×#634—22no—
83C2×#634—21no—
84C2×#634C22×C312yes—
85C2×#634C22×C314yescomplemented
86C22×C3×Dic3—61no—
87C2×C3×Dic6—32no—
88C2×C3×Dic6—34no—
89C2×C3×Dic6—61no—
90C2×C3×Dic6—62no—
91#3756C22×C318yescomplemented
92#3754C22×C318yescomplemented
93#3748C22×C314yescomplemented
94C2×C3⋊(C3⋊C8)C22×C312yes—
95C3×#216—38no—
96C3×#214—38no—
97C3×#208—34no—
98C2×C3×C3⋊C8—32no—
99C22×H(3,𝔽3)—22no—
100C22×H(3,𝔽3)—22no—
101C22×H(3,𝔽3)—42no—
102C22×H(3,𝔽3)C2×D412yes—
103C22×H(3,𝔽3)C2×D411yescharacteristic
104C4×H(3,𝔽3)—22no—
105C4×H(3,𝔽3)C2×D412yes—
106C4×H(3,𝔽3)C2×D411yescharacteristic
107C4×H(3,𝔽3)C2411yescharacteristic
108(C3⋊Dic3)⋊C3—24no—
109C2×C3×#41—31no—
110C2×C3×#41—31no—
111C2×C3×#41—61no—
112C2×C3×#41C3×D311yescharacteristic
113C22×C3×Q8—91no—
114C22×Dic6—31no—
115C22×Dic6—91no—
116C3×#137—98no—
117C2×C3×Q16—92no—
118C2×C3×SD16—92no—
119C2×C3×M16—91no—
120#951—38no—
121#951—98no—
122C2×#216—32no—
123C2×#216—92no—
124C2×#214—32no—
125C2×#214—92no—
126C2×#208—31no—
127C2×#208—91no—
128C23×C32—22no—
129C23×C32—21no—
130C23×C32—21no—
131C23×C32—21no—
132C23×C32#6711yescharacteristic
133C23×C32C3×D411yescharacteristic
134C32×Q8—22no—
135C32×Q8—21no—
136C32×Q8—21no—
137C32×Q8—22no—
138C32×Q8C22×D312yes—
139C32×Q8C23×C312yes—
140C32×D4—28no—
141C32×D4—22no—
142C32×D4—24no—
143C32×D4—24no—
144C32×D4—21no—
145C32×D4—21no—
146C32×D4—22no—
147C32×D4C22×D312yes—
148C32×D4C23×C312yes—
149C2×C32×C4—22no—
150C2×C32×C4—28no—
151C2×C32×C4—21no—
152C2×C32×C4—21no—
153C2×C32×C4—24no—
154C2×C32×C4—24no—
155C2×C32×C4—21no—
156C2×C32×C4—22no—
157C2×C32×C4—21no—
158C2×C32×C4—22no—
159C2×C32×C4#6711yescharacteristic
160C2×C32×C4#6712yes—
161C2×C32×C4C3×D411yescharacteristic
162C2×C32×C4C3×D412yes—
163C2×C32×C4C22×D311yescharacteristic
164C2×C32×C4C22×D312yes—
165C2×C32×C4C23×C311yescharacteristic
166C2×C32×C4C23×C312yes—
167C2×C3⋊Dic3—22no—
168C2×C3⋊Dic3—24no—
169#634—22no—
170#634—24no—
171#634C23×C314yes—
172C2×C3×Dic3—64no—
173C2×C3×Dic3—62no—
174C3×Dic6—34no—
175C3×Dic6—64no—
176C3×Dic6—62no—
177C3⋊(C3⋊C8)C23×C314yes—
178C3×C3⋊C8—34no—
179C2×#137—91noSylow
180C2×H(3,𝔽3)—22no—
181C2×H(3,𝔽3)—42no—
182C2×H(3,𝔽3)#13712yescomplemented
183C2×H(3,𝔽3)C22×D411yescharacteristic
184C3×#41—22no—
185C3×#41—34no—
186C3×#41—34no—
187C3×#41—62no—
188C3×#41—62no—
189C3×#41—64no—
190C3×#41—64no—
191C3×#41C2×C3×D314yes—
192C2×C3×Q8—31no—
193C2×C3×Q8—31no—
194C2×C3×Q8—61no—
195C2×C3×Q8—92no—
196C2×C3×Q8—94no—
197C2×C3×Q8—181no—
198C2×C3×Q8—182no—
199C2×C3×Q8C2×C3×D311yescharacteristic
200C2×C3×D4—21no—
201C2×C3×D4—31no—
202C2×C3×D4—31no—
203C2×C3×D4—61no—
204C2×C3×D4—61no—
205C2×C3×D4—62no—
206C2×C3×D4—61no—
207C2×C3×D4C2×C3×D311yescharacteristic
208C22×C3×C4—21no—
209C22×C3×C4—31no—
210C22×C3×C4—31no—
211C22×C3×C4—61no—
212C22×C3×C4—61no—
213C22×C3×C4—62no—
214C22×C3×C4—61no—
215C22×C3×C4—181no—
216C22×C3×C4C2×C3×D311yescharacteristic
217C22×Dic3—61no—
218C22×Dic3—181no—
219C2×Dic6—32no—
220C2×Dic6—34no—
221C2×Dic6—61no—
222C2×Dic6—62no—
223C2×Dic6—92no—
224C2×Dic6—94no—
225C2×Dic6—181no—
226C2×Dic6—182no—
227C3×Q16—98no—
228C3×SD16—98no—
229C3×M16—94no—
230C2×C3×C8—92no—
231#216—38no—
232#216—98no—
233#214—38no—
234#214—98no—
235#208—34no—
236#208—94no—
237C2×C3⋊C8—32no—
238C2×C3⋊C8—92no—
239C22×C32—22no—
240C22×C32—22no—
241C22×C32—24no—
242C22×C32—22no—
243C22×C32—22no—
244C22×C32—21no—
245C22×C32—22no—
246C22×C32—42no—
247C22×C32—44no—
248C22×C32—42no—
249C22×C32—42no—
250C22×C32C2×#6712yes—
251C22×C32C2×#6711yescharacteristic
252C22×C32C2×C3×D412yes—
253C22×C32C2×C3×D411yescharacteristic
254C32×C4—24no—
255C32×C4—22no—
256C32×C4—22no—
257C32×C4—21no—
258C32×C4—22no—
259C32×C4—21no—
260C32×C4C2×#6712yes—
261C32×C4C2×#6711yescharacteristic
262C32×C4C2×C3×D412yes—
263C32×C4C2×C3×D411yescharacteristic
264C32×C4C23×D311yescharacteristic
265C32×C4C24×C311yescharacteristic
266C3⋊Dic3—24no—
267C3×Dic3—64no—
268C2×#41(C3⋊D3)⋊C311yescharacteristic
269C22×Q8—91no—
270#137—98no—
271C2×Q16—92no—
272C2×SD16—92no—
273C2×M16—91no—
274H(3,𝔽3)C2×#13711yescharacteristic, complemented, Sylow
275C23×C3—21no—
276C23×C3—31no—
277C23×C3—31no—
278C23×C3—61no—
279C23×C3—61no—
280C23×C3—62no—
281C23×C3—61no—
282C23×C3C3×#6711yescharacteristic
283C3×Q8—21no—
284C3×Q8—32no—
285C3×Q8—32no—
286C3×Q8—61no—
287C3×Q8—61no—
288C3×Q8—62no—
289C3×Q8—62no—
290C3×Q8—94no—
291C3×Q8—184no—
292C3×Q8—182no—
293C3×Q8C22×C3×D312yes—
294C3×D4—24no—
295C3×D4—21no—
296C3×D4—32no—
297C3×D4—32no—
298C3×D4—61no—
299C3×D4—61no—
300C3×D4—64no—
301C3×D4—64no—
302C3×D4—68no—
303C3×D4—62no—
304C3×D4—62no—
305C3×D4C22×C3×D312yes—
306C2×C3×C4—21no—
307C2×C3×C4—24no—
308C2×C3×C4—31no—
309C2×C3×C4—32no—
310C2×C3×C4—31no—
311C2×C3×C4—32no—
312C2×C3×C4—31no—
313C2×C3×C4—32no—
314C2×C3×C4—31no—
315C2×C3×C4—32no—
316C2×C3×C4—64no—
317C2×C3×C4—64no—
318C2×C3×C4—61no—
319C2×C3×C4—61no—
320C2×C3×C4—62no—
321C2×C3×C4—68no—
322C2×C3×C4—61no—
323C2×C3×C4—62no—
324C2×C3×C4—61no—
325C2×C3×C4—62no—
326C2×C3×C4—184no—
327C2×C3×C4—182no—
328C2×C3×C4C3×#6711yescharacteristic
329C2×C3×C4C3×#6712yes—
330C2×C3×C4C22×C3×D311yescharacteristic
331C2×C3×C4C22×C3×D312yes—
332C2×Dic3—64no—
333C2×Dic3—62no—
334C2×Dic3—184no—
335C2×Dic3—182no—
336Dic6—34no—
337Dic6—64no—
338Dic6—62no—
339Dic6—94no—
340Dic6—184no—
341Dic6—182no—
342C3×C8—94no—
343C3⋊C8—34no—
344C3⋊C8—94no—
345C2×C32—22no—
346C2×C32—22no—
347C2×C32—22no—
348C2×C32—21no—
349C2×C32—42no—
350C2×C32—44no—
351C2×C32—42no—
352C2×C32—42no—
353C2×C32#95112yescomplemented
354C2×C32C3×#13712yescomplemented
355C2×C32C22×#6711yescharacteristic
356C2×C32C22×C3×D411yescharacteristic
357#41—22no—
358#41C2×(C3⋊D3)⋊C314yes—
359C2×Q8—92no—
360C2×Q8—94no—
361C2×Q8—182no—
362C2×Q8—181no—
363C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
364C2×D4—21no—
365C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
366C22×C4—21no—
367C22×C4—181no—
368C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
369Q16—98no—
370SD16—98no—
371M16—94no—
372C2×C8—92no—
373C22×C3—22no—
374C22×C3—22no—
375C22×C3—32no—
376C22×C3—32no—
377C22×C3—31no—
378C22×C3—31no—
379C22×C3—42no—
380C22×C3—62no—
381C22×C3—62no—
382C22×C3—62no—
383C22×C3—62no—
384C22×C3—64no—
385C22×C3—62no—
386C22×C3—61no—
387C22×C3—122no—
388C22×C3—122no—
389C22×C3—122no—
390C22×C3—124no—
391C22×C3C2×C3×#6712yes—
392C22×C3C2×C3×#6711yescharacteristic
393C3×C4—22no—
394C3×C4—31no—
395C3×C4—31no—
396C3×C4—32no—
397C3×C4—31no—
398C3×C4—32no—
399C3×C4—31no—
400C3×C4—62no—
401C3×C4—62no—
402C3×C4—64no—
403C3×C4—61no—
404C3×C4—62no—
405C3×C4—61no—
406C3×C4—184no—
407C3×C4C2×C3×#6712yes—
408C3×C4C2×C3×#6711yescharacteristic
409C3×C4C23×C3×D311yescharacteristic
410Dic3—64no—
411Dic3—184no—
412C32—21no—
413C32C2×#95111yescharacteristic, complemented
414C32C2×C3×#13711yescharacteristic, complemented
415C23—21no—
416C23#638⋊C311yescharacteristic
417Q8—21no—
418Q8—94no—
419Q8—184no—
420Q8—182no—
421Q8C22×(C3⋊D3)⋊C312yes—
422D4—24no—
423D4—21no—
424D4C22×(C3⋊D3)⋊C312yes—
425C2×C4—21no—
426C2×C4—24no—
427C2×C4—184no—
428C2×C4—182no—
429C2×C4#638⋊C312yes—
430C2×C4#638⋊C311yescharacteristic
431C2×C4C22×(C3⋊D3)⋊C312yes—
432C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
433C8—94no—
434C2×C3—22no—
435C2×C3—31no—
436C2×C3—31no—
437C2×C3—32no—
438C2×C3—32no—
439C2×C3—42no—
440C2×C3—62no—
441C2×C3—62no—
442C2×C3—61no—
443C2×C3—62no—
444C2×C3—122no—
445C2×C3—122no—
446C2×C3—122no—
447C2×C3—124no—
448C2×C3C3×#95112yes—
449C2×C3C22×C3×#6711yescharacteristic
450C22—22no—
451C22—22no—
452C22—42no—
453C22C2×#638⋊C312yes—
454C22C2×#638⋊C311yescharacteristic
455C4—22no—
456C4—184no—
457C4C2×#638⋊C312yes—
458C4C2×#638⋊C311yescharacteristic
459C4C23×(C3⋊D3)⋊C311yescharacteristic
460C3—31no—
461C3—31no—
462C3—61no—
463C3C2×C3×#95111yescharacteristic
464C2—22no—
465C2—42no—
466C2#64160⋊C312yesdirect product factor
467C2C22×#638⋊C311yescharacteristic
subgroupG/H
centerC22C2×#638⋊C3
commutator, cosocleC32×C4C24×C3
FrattiniC3×C4C23×C3×D3
Fitting, prime radicalC2×#41×H(3,𝔽3)C2
socleC22×C3C2×C3×#67
hypercenterC2×#41(C3⋊D3)⋊C3
hypocenterC32C2×C3×#137
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
2. C22×#634⋊C3C2semidirect
3. #64160⋊C3C2direct (N × H)
4. C2×#3756⋊C3C2semidirect
5. C2×#3754⋊C3C2semidirect
6. C2×#3748⋊C3C2semidirect
7. C2×#64160C3semidirect
20. C2×#634⋊C3C22semidirect
21. #3756⋊C3C22semidirect
22. #3754⋊C3C22semidirect
23. #3748⋊C3C22semidirect
28. C22×#634C2×C3semidirect
30. #64160C2×C3semidirect
31. C2×#3756C2×C3semidirect
32. C2×#3754C2×C3semidirect
33. C2×#3748C2×C3semidirect
85. C2×#634C22×C3semidirect
91. #3756C22×C3semidirect
92. #3754C22×C3semidirect
93. #3748C22×C3semidirect
182. C2×H(3,𝔽3)#137semidirect
274. H(3,𝔽3)C2×#137semidirect
353. C2×C32#951semidirect
354. C2×C32C3×#137semidirect
413. C32C2×#951semidirect
414. C32C2×C3×#137semidirect
466. C2#64160⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#41×H(3,𝔽3)2
C22×#634⋊C32
#64160⋊C32
C2×#3756⋊C32
C2×#3754⋊C32
C2×#3748⋊C32
C2×#641603
C2×C3×#9513