Finite Group Databasebeta

← C2×#64158⋊C3 #13153712

Subgroups — C2×#64158⋊C3 #13153712

Order 1728.

1207 conjugacy classes of subgroups.

The subgroup lattice has 497 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C22×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
3#64158⋊C3C218yesmaximal, direct product factor
4C2×#3755⋊C3C212yesmaximal, complemented
5C2×#3753⋊C3C212yesmaximal, complemented
6C2×#3748⋊C3C211yesmaximal, characteristic, complemented
7C2×#64158C311yesmaximal, characteristic, complemented
8C2×C3×#949—31nomaximal
9C23×(C3⋊D3)⋊C3—21no—
10#41×H(3,𝔽3)—22no—
11#41×H(3,𝔽3)C2214yescomplemented
12C2×Q8×H(3,𝔽3)C2211yescharacteristic, complemented
13C2×D4×H(3,𝔽3)—21no—
14C2×D4×H(3,𝔽3)C2211yescharacteristic, complemented
15C22×C4×H(3,𝔽3)—21no—
16C22×C4×H(3,𝔽3)C2211yescharacteristic
17C2×C4⋊((C3⋊D3)⋊C3)—22no—
18C2×C4⋊((C3⋊D3)⋊C3)—21no—
19C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
20C2×C4⋊((C3⋊D3)⋊C3)C2214yescomplemented
21#3755⋊C3C2218yescomplemented
22#3753⋊C3C2218yescomplemented
23#3748⋊C3C2214yescomplemented
24C2×(C3⋊(C3⋊C8))⋊C3C2212yescomplemented
25C2×C32×#41—21no—
26C2×C32×#41D311yescharacteristic, complemented
27C2×C32×#41C2×C311yescharacteristic, complemented
28C22×C4⋊(C3⋊D3)C2×C311yescharacteristic, complemented
29C22×C3×D12—31no—
30#64158C2×C318yescomplemented
31C2×#3755C2×C312yescomplemented
32C2×#3753C2×C312yescomplemented
33C2×#3748C2×C311yescharacteristic, complemented
34C3×#949—38no—
35C2×C3×#215—32no—
36C2×C3×#213—32no—
37C2×C3×#208—31no—
38C23×H(3,𝔽3)—21no—
39C23×H(3,𝔽3)D411yescharacteristic
40C22×(C3⋊D3)⋊C3—22no—
41C22×(C3⋊D3)⋊C3—24no—
42C22×(C3⋊D3)⋊C3—44no—
43Q8×H(3,𝔽3)—21no—
44Q8×H(3,𝔽3)C2312yescomplemented
45D4×H(3,𝔽3)—24no—
46D4×H(3,𝔽3)—21no—
47D4×H(3,𝔽3)C2312yescomplemented
48C2×C4×H(3,𝔽3)—21no—
49C2×C4×H(3,𝔽3)—24no—
50C2×C4×H(3,𝔽3)D411yescharacteristic
51C2×C4×H(3,𝔽3)D412yes—
52C2×C4×H(3,𝔽3)C2311yescharacteristic
53C2×C4×H(3,𝔽3)C2312yes—
54C4⋊((C3⋊D3)⋊C3)—22no—
55C4⋊((C3⋊D3)⋊C3)—24no—
56C4⋊((C3⋊D3)⋊C3)C2314yes—
57(C3⋊(C3⋊C8))⋊C3C2314yescomplemented
58C2×C3×AGL(1,ℤ/8)—91no—
59C2×#949—31no—
60C2×#949—91no—
61C23×C3⋊D3—21no—
62C23×C3×D3—61no—
63C32×#41—24no—
64C32×#41—24no—
65C32×#41—22no—
66C32×#41—22no—
67C32×#41C2×D314yescomplemented
68C32×#41C22×C314yescomplemented
69C2×C32×Q8—21no—
70C2×C32×Q8C2×D311yescharacteristic, complemented
71C2×C32×Q8C22×C311yescharacteristic, complemented
72C2×C32×D4—22no—
73C2×C32×D4—21no—
74C2×C32×D4—21no—
75C2×C32×D4—21no—
76C2×C32×D4C2×D311yescharacteristic, complemented
77C2×C32×D4C22×C311yescharacteristic, complemented
78C22×C32×C4—22no—
79C22×C32×C4—21no—
80C22×C32×C4—21no—
81C22×C32×C4—21no—
82C22×C32×C4C2×D311yescharacteristic
83C22×C32×C4C22×C311yescharacteristic
84C2×C4⋊(C3⋊D3)—22no—
85C2×C4⋊(C3⋊D3)—21no—
86C2×C4⋊(C3⋊D3)C22×C312yes—
87C2×C4⋊(C3⋊D3)C22×C314yescomplemented
88C2×C3×D12—32no—
89C2×C3×D12—34no—
90C2×C3×D12—61no—
91C2×C3×D12—62no—
92#3755C22×C318yescomplemented
93#3753C22×C318yescomplemented
94#3748C22×C314yescomplemented
95C2×C3⋊(C3⋊C8)C22×C312yescomplemented
96C3×#215—38no—
97C3×#213—38no—
98C3×#208—34no—
99C2×C3×C3⋊C8—32no—
100C22×H(3,𝔽3)—22no—
101C22×H(3,𝔽3)—22no—
102C22×H(3,𝔽3)—42no—
103C22×H(3,𝔽3)C2×D412yes—
104C22×H(3,𝔽3)C2×D411yescharacteristic
105C2×(C3⋊D3)⋊C3—24no—
106C2×(C3⋊D3)⋊C3—48no—
107C2×(C3⋊D3)⋊C3—44no—
108C4×H(3,𝔽3)—22no—
109C4×H(3,𝔽3)C2×D412yes—
110C4×H(3,𝔽3)C2×D411yescharacteristic
111C4×H(3,𝔽3)C2411yescharacteristic
112C2×C3×#41—31no—
113C2×C3×#41—31no—
114C2×C3×#41—61no—
115C2×C3×#41C3×D311yescharacteristic
116C22×C3×D4—91no—
117C22×D12—31no—
118C22×D12—91no—
119C3×AGL(1,ℤ/8)—98no—
120C2×C3×SD16—92no—
121C2×C3×D8—92no—
122C2×C3×M16—91no—
123#949—38no—
124#949—98no—
125C2×#215—32no—
126C2×#215—92no—
127C2×#213—32no—
128C2×#213—92no—
129C2×#208—31no—
130C2×#208—91no—
131C23×C32—22no—
132C23×C32—21no—
133C23×C32—21no—
134C23×C32—21no—
135C23×C32#6711yescharacteristic
136C23×C32C3×D411yescharacteristic
137C22×C3⋊D3—22no—
138C22×C3⋊D3—24no—
139C22×C3⋊D3—44no—
140C22×C3×D3—64no—
141C22×C3×D3—62no—
142C22×C3×D3—124no—
143C32×Q8—22no—
144C32×Q8—21no—
145C32×Q8—21no—
146C32×Q8—22no—
147C32×Q8C22×D312yescomplemented
148C32×Q8C23×C312yescomplemented
149C32×D4—28no—
150C32×D4—22no—
151C32×D4—24no—
152C32×D4—24no—
153C32×D4—21no—
154C32×D4—21no—
155C32×D4—22no—
156C32×D4C22×D312yescomplemented
157C32×D4C23×C312yescomplemented
158C2×C32×C4—22no—
159C2×C32×C4—28no—
160C2×C32×C4—21no—
161C2×C32×C4—21no—
162C2×C32×C4—24no—
163C2×C32×C4—24no—
164C2×C32×C4—21no—
165C2×C32×C4—22no—
166C2×C32×C4—21no—
167C2×C32×C4—22no—
168C2×C32×C4#6711yescharacteristic
169C2×C32×C4#6712yes—
170C2×C32×C4C3×D411yescharacteristic
171C2×C32×C4C3×D412yes—
172C2×C32×C4C22×D311yescharacteristic
173C2×C32×C4C22×D312yes—
174C2×C32×C4C23×C311yescharacteristic
175C2×C32×C4C23×C312yes—
176C4⋊(C3⋊D3)—22no—
177C4⋊(C3⋊D3)—24no—
178C4⋊(C3⋊D3)C23×C314yes—
179C3×D12—34no—
180C3×D12—64no—
181C3×D12—62no—
182C3⋊(C3⋊C8)C23×C314yescomplemented
183C3×C3⋊C8—34no—
184C2×AGL(1,ℤ/8)—91noSylow
185C2×H(3,𝔽3)—22no—
186C2×H(3,𝔽3)—42no—
187C2×H(3,𝔽3)AGL(1,ℤ/8)12yescomplemented
188C2×H(3,𝔽3)C22×D411yescharacteristic
189(C3⋊D3)⋊C3—44no—
190C24×C3—181no—
191C23×D3—61no—
192C23×D3—181no—
193C3×#41—22no—
194C3×#41—34no—
195C3×#41—34no—
196C3×#41—62no—
197C3×#41—62no—
198C3×#41—64no—
199C3×#41—64no—
200C3×#41C2×C3×D314yes—
201C2×C3×Q8—31no—
202C2×C3×Q8—31no—
203C2×C3×Q8—61no—
204C2×C3×Q8C2×C3×D311yescharacteristic
205C2×C3×D4—21no—
206C2×C3×D4—31no—
207C2×C3×D4—31no—
208C2×C3×D4—61no—
209C2×C3×D4—61no—
210C2×C3×D4—62no—
211C2×C3×D4—61no—
212C2×C3×D4—92no—
213C2×C3×D4—94no—
214C2×C3×D4—181no—
215C2×C3×D4—182no—
216C2×C3×D4C2×C3×D311yescharacteristic
217C22×C3×C4—21no—
218C22×C3×C4—31no—
219C22×C3×C4—31no—
220C22×C3×C4—61no—
221C22×C3×C4—61no—
222C22×C3×C4—62no—
223C22×C3×C4—61no—
224C22×C3×C4C2×C3×D311yescharacteristic
225C2×D12—32no—
226C2×D12—34no—
227C2×D12—61no—
228C2×D12—62no—
229C2×D12—92no—
230C2×D12—94no—
231C2×D12—181no—
232C2×D12—182no—
233C3×SD16—98no—
234C3×D8—98no—
235C3×M16—94no—
236C2×C3×C8—92no—
237#215—38no—
238#215—98no—
239#213—38no—
240#213—98no—
241#208—34no—
242#208—94no—
243C2×C3⋊C8—32no—
244C2×C3⋊C8—92no—
245C22×C32—22no—
246C22×C32—22no—
247C22×C32—24no—
248C22×C32—22no—
249C22×C32—22no—
250C22×C32—21no—
251C22×C32—22no—
252C22×C32—42no—
253C22×C32—44no—
254C22×C32—42no—
255C22×C32—42no—
256C22×C32C2×#6712yes—
257C22×C32C2×#6711yescharacteristic
258C22×C32C2×C3×D412yes—
259C22×C32C2×C3×D411yescharacteristic
260C2×C3⋊D3—24no—
261C2×C3⋊D3—48no—
262C2×C3⋊D3—44no—
263C2×C3×D3—64no—
264C2×C3×D3—128no—
265C2×C3×D3—124no—
266C32×C4—24no—
267C32×C4—22no—
268C32×C4—22no—
269C32×C4—21no—
270C32×C4—22no—
271C32×C4—21no—
272C32×C4C2×#6712yes—
273C32×C4C2×#6711yescharacteristic
274C32×C4C2×C3×D412yes—
275C32×C4C2×C3×D411yescharacteristic
276C32×C4C23×D311yescharacteristic
277C32×C4C24×C311yescharacteristic
278C2×#41(C3⋊D3)⋊C311yescharacteristic, complemented
279C22×D4—91no—
280AGL(1,ℤ/8)—98no—
281C2×SD16—92no—
282C2×D8—92no—
283C2×M16—91no—
284H(3,𝔽3)C2×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
285C23×C3—21no—
286C23×C3—31no—
287C23×C3—31no—
288C23×C3—61no—
289C23×C3—61no—
290C23×C3—62no—
291C23×C3—61no—
292C23×C3—184no—
293C23×C3—182no—
294C23×C3—364no—
295C23×C3C3×#6711yescharacteristic
296C22×D3—64no—
297C22×D3—62no—
298C22×D3—124no—
299C22×D3—184no—
300C22×D3—182no—
301C22×D3—364no—
302C3×Q8—21no—
303C3×Q8—32no—
304C3×Q8—32no—
305C3×Q8—61no—
306C3×Q8—61no—
307C3×Q8—62no—
308C3×Q8—62no—
309C3×Q8C22×C3×D312yes—
310C3×D4—24no—
311C3×D4—21no—
312C3×D4—32no—
313C3×D4—32no—
314C3×D4—61no—
315C3×D4—61no—
316C3×D4—64no—
317C3×D4—64no—
318C3×D4—68no—
319C3×D4—62no—
320C3×D4—62no—
321C3×D4—94no—
322C3×D4—184no—
323C3×D4—182no—
324C3×D4C22×C3×D312yes—
325C2×C3×C4—21no—
326C2×C3×C4—24no—
327C2×C3×C4—31no—
328C2×C3×C4—32no—
329C2×C3×C4—31no—
330C2×C3×C4—32no—
331C2×C3×C4—31no—
332C2×C3×C4—32no—
333C2×C3×C4—31no—
334C2×C3×C4—32no—
335C2×C3×C4—64no—
336C2×C3×C4—64no—
337C2×C3×C4—61no—
338C2×C3×C4—61no—
339C2×C3×C4—62no—
340C2×C3×C4—68no—
341C2×C3×C4—61no—
342C2×C3×C4—62no—
343C2×C3×C4—61no—
344C2×C3×C4—62no—
345C2×C3×C4C3×#6711yescharacteristic
346C2×C3×C4C3×#6712yes—
347C2×C3×C4C22×C3×D311yescharacteristic
348C2×C3×C4C22×C3×D312yes—
349D12—34no—
350D12—64no—
351D12—62no—
352D12—94no—
353D12—184no—
354D12—182no—
355C3×C8—94no—
356C3⋊C8—34no—
357C3⋊C8—94no—
358C2×C32—22no—
359C2×C32—22no—
360C2×C32—22no—
361C2×C32—21no—
362C2×C32—42no—
363C2×C32—44no—
364C2×C32—42no—
365C2×C32—42no—
366C2×C32#94912yescomplemented
367C2×C32C3×AGL(1,ℤ/8)12yescomplemented
368C2×C32C22×#6711yescharacteristic
369C2×C32C22×C3×D411yescharacteristic
370C3⋊D3—44no—
371C3×D3—124no—
372C24—181no—
373#41—22no—
374#41C2×(C3⋊D3)⋊C314yescomplemented
375C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic, complemented
376C2×D4—21no—
377C2×D4—92no—
378C2×D4—94no—
379C2×D4—182no—
380C2×D4—181no—
381C2×D4C2×(C3⋊D3)⋊C311yescharacteristic, complemented
382C22×C4—21no—
383C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
384SD16—98no—
385D8—98no—
386M16—94no—
387C2×C8—92no—
388C22×C3—22no—
389C22×C3—22no—
390C22×C3—32no—
391C22×C3—32no—
392C22×C3—31no—
393C22×C3—31no—
394C22×C3—42no—
395C22×C3—62no—
396C22×C3—62no—
397C22×C3—62no—
398C22×C3—62no—
399C22×C3—64no—
400C22×C3—62no—
401C22×C3—61no—
402C22×C3—122no—
403C22×C3—122no—
404C22×C3—122no—
405C22×C3—124no—
406C22×C3—184no—
407C22×C3—364no—
408C22×C3—368no—
409C22×C3C2×C3×#6712yes—
410C22×C3C2×C3×#6711yescharacteristic
411C2×D3—64no—
412C2×D3—128no—
413C2×D3—124no—
414C2×D3—184no—
415C2×D3—364no—
416C2×D3—368no—
417C3×C4—22no—
418C3×C4—31no—
419C3×C4—31no—
420C3×C4—32no—
421C3×C4—31no—
422C3×C4—32no—
423C3×C4—31no—
424C3×C4—62no—
425C3×C4—62no—
426C3×C4—64no—
427C3×C4—61no—
428C3×C4—62no—
429C3×C4—61no—
430C3×C4C2×C3×#6712yes—
431C3×C4C2×C3×#6711yescharacteristic
432C3×C4C23×C3×D311yescharacteristic
433C32—21no—
434C32C2×#94911yescharacteristic, complemented
435C32C2×C3×AGL(1,ℤ/8)11yescharacteristic, complemented
436C23—21no—
437C23—184no—
438C23—182no—
439C23—364no—
440C23#638⋊C311yescharacteristic
441Q8—21no—
442Q8C22×(C3⋊D3)⋊C312yescomplemented
443D4—24no—
444D4—21no—
445D4—94no—
446D4—184no—
447D4—182no—
448D4C22×(C3⋊D3)⋊C312yescomplemented
449C2×C4—21no—
450C2×C4—24no—
451C2×C4#638⋊C312yes—
452C2×C4#638⋊C311yescharacteristic
453C2×C4C22×(C3⋊D3)⋊C312yes—
454C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
455C8—94no—
456C2×C3—22no—
457C2×C3—31no—
458C2×C3—31no—
459C2×C3—32no—
460C2×C3—32no—
461C2×C3—42no—
462C2×C3—62no—
463C2×C3—62no—
464C2×C3—61no—
465C2×C3—62no—
466C2×C3—122no—
467C2×C3—122no—
468C2×C3—122no—
469C2×C3—124no—
470C2×C3—364no—
471C2×C3C3×#94912yes—
472C2×C3C22×C3×#6711yescharacteristic
473D3—124no—
474D3—364no—
475C22—22no—
476C22—22no—
477C22—42no—
478C22—184no—
479C22—368no—
480C22—364no—
481C22C2×#638⋊C312yes—
482C22C2×#638⋊C311yescharacteristic
483C4—22no—
484C4C2×#638⋊C312yes—
485C4C2×#638⋊C311yescharacteristic
486C4C23×(C3⋊D3)⋊C311yescharacteristic
487C3—31no—
488C3—31no—
489C3—61no—
490C3C2×C3×#94911yescharacteristic
491C2—22no—
492C2—42no—
493C2—364no—
494C2#64158⋊C312yesdirect product factor
495C2C22×#638⋊C311yescharacteristic
subgroupG/H
centerC22C2×#638⋊C3
commutator, cosocleC32×C4C24×C3
FrattiniC3×C4C23×C3×D3
FittingC2×#41×H(3,𝔽3)C2
socleC22×C3C2×C3×#67
hypercenterC2×#41(C3⋊D3)⋊C3
hypocenterC32C2×C3×AGL(1,ℤ/8)
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
1. C2×#41×H(3,𝔽3)C2semidirect
2. C22×C4⋊((C3⋊D3)⋊C3)C2semidirect
3. #64158⋊C3C2direct (N × H)
4. C2×#3755⋊C3C2semidirect
5. C2×#3753⋊C3C2semidirect
6. C2×#3748⋊C3C2semidirect
7. C2×#64158C3semidirect
11. #41×H(3,𝔽3)C22semidirect
12. C2×Q8×H(3,𝔽3)C22semidirect
14. C2×D4×H(3,𝔽3)C22semidirect
20. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
21. #3755⋊C3C22semidirect
22. #3753⋊C3C22semidirect
23. #3748⋊C3C22semidirect
24. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
26. C2×C32×#41D3semidirect
27. C2×C32×#41C2×C3semidirect
28. C22×C4⋊(C3⋊D3)C2×C3semidirect
30. #64158C2×C3semidirect
31. C2×#3755C2×C3semidirect
32. C2×#3753C2×C3semidirect
33. C2×#3748C2×C3semidirect
44. Q8×H(3,𝔽3)C23semidirect
47. D4×H(3,𝔽3)C23semidirect
57. (C3⋊(C3⋊C8))⋊C3C23semidirect
67. C32×#41C2×D3semidirect
68. C32×#41C22×C3semidirect
70. C2×C32×Q8C2×D3semidirect
71. C2×C32×Q8C22×C3semidirect
76. C2×C32×D4C2×D3semidirect
77. C2×C32×D4C22×C3semidirect
87. C2×C4⋊(C3⋊D3)C22×C3semidirect
92. #3755C22×C3semidirect
93. #3753C22×C3semidirect
94. #3748C22×C3semidirect
95. C2×C3⋊(C3⋊C8)C22×C3semidirect
147. C32×Q8C22×D3semidirect
148. C32×Q8C23×C3semidirect
156. C32×D4C22×D3semidirect
157. C32×D4C23×C3semidirect
182. C3⋊(C3⋊C8)C23×C3semidirect
187. C2×H(3,𝔽3)AGL(1,ℤ/8)semidirect
278. C2×#41(C3⋊D3)⋊C3semidirect
284. H(3,𝔽3)C2×AGL(1,ℤ/8)semidirect
366. C2×C32#949semidirect
367. C2×C32C3×AGL(1,ℤ/8)semidirect
374. #41C2×(C3⋊D3)⋊C3semidirect
375. C2×Q8C2×(C3⋊D3)⋊C3semidirect
381. C2×D4C2×(C3⋊D3)⋊C3semidirect
434. C32C2×#949semidirect
435. C32C2×C3×AGL(1,ℤ/8)semidirect
442. Q8C22×(C3⋊D3)⋊C3semidirect
448. D4C22×(C3⋊D3)⋊C3semidirect
494. C2#64158⋊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×C4⋊((C3⋊D3)⋊C3)2
#64158⋊C32
C2×#3755⋊C32
C2×#3753⋊C32
C2×#3748⋊C32
C2×#641583
C2×C3×#9493