Finite Group Databasebeta

← C2×C3×SD16×C3⋊D3 #13179509

Subgroups — C2×C3×SD16×C3⋊D3 #13179509

Order 1728.

2060 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×Q8×C3⋊D3C211yesmaximal, characteristic, complemented
2C2×C3×D4×C3⋊D3C211yesmaximal, characteristic
3C2×C33×SD16C211yesmaximal, characteristic, complemented
4C2×C3×#3755C211yesmaximal, characteristic, complemented
5C2×C3×#3754C211yesmaximal, characteristic, complemented
6C3×SD16×C3⋊D3C218yesmaximal, direct product factor
7C2×C3×#3744C211yesmaximal, characteristic, complemented
8C2×C3×C8×C3⋊D3C211yesmaximal, characteristic, complemented
9C2×SD16×C3⋊D3C311yesmaximal, characteristic, direct product factor
10C2×C3×D3×SD16—34nomaximal
11C23×C3×C3⋊D3—21no—
12C2×C33×Q8C2211yescharacteristic, complemented
13C2×C33×D4C2211yescharacteristic
14C2×C3×#638—21no—
15C3×Q8×C3⋊D3—22no—
16C3×Q8×C3⋊D3C2214yescomplemented
17C3×D4×C3⋊D3—22no—
18C3×D4×C3⋊D3C2214yes—
19C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
20C2×C3×C4×C3⋊D3—21no—
21C2×C3×C4×C3⋊D3C2211yescharacteristic
22C2×C3×#634—21no—
23C2×C3×#634C2211yescharacteristic, complemented
24C33×SD16C2214yescomplemented
25C2×C33×C8C2211yescharacteristic, complemented
26C3×#3755C2214yescomplemented
27C3×#3754C2214yescomplemented
28C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
29C3×#3744C2214yescomplemented
30C3×C8×C3⋊D3C2214yescomplemented
31C2×Q8×C3⋊D3C2×C311yescharacteristic, complemented
32C2×D4×C3⋊D3C2×C311yescharacteristic
33C2×C3×D3×Q8—34no—
34C2×C3×D3×D4—34no—
35C2×C32×SD16—24no—
36C2×C32×SD16D314yescomplemented
37C2×C32×SD16C2×C311yescharacteristic, complemented
38C2×#3755C2×C311yescharacteristic, complemented
39C2×#3754C2×C311yescharacteristic, complemented
40SD16×C3⋊D3C2×C318yesdirect product factor
41C2×#3744C2×C311yescharacteristic, complemented
42C2×C8×C3⋊D3C2×C311yescharacteristic, complemented
43C2×C3×#215—34no—
44C2×C3×#214—34no—
45C3×D3×SD16—332no—
46C2×C3×#204—34no—
47C2×C3×D3×C8—34no—
48C23×C33—21no—
49C22×C3×C3⋊D3—21no—
50C22×C3×C3⋊D3—24no—
51C22×C3×C3⋊D3—44no—
52C22×C3×C3⋊D3D411yescharacteristic
53C33×Q8—21no—
54C33×Q8C2312yescomplemented
55C33×D4—21no—
56C33×D4C2312yes—
57C2×C33×C4—21no—
58C2×C33×C4C2311yescharacteristic
59C3×#638—24no—
60C2×C3×C3⋊Dic3—21no—
61C2×C3×C3⋊Dic3D411yescharacteristic
62C3×C4⋊(C3⋊D3)—21no—
63C3×C4⋊(C3⋊D3)C2312yes—
64C3×C4×C3⋊D3—24no—
65C3×C4×C3⋊D3D412yes—
66C3×C4×C3⋊D3C2312yes—
67C3×#634—21no—
68C3×#634—24no—
69C3×#634C2312yescomplemented
70C33×C8C2312yescomplemented
71C3×C3⋊(C3⋊C8)C2312yescomplemented
72C22×C3×SD16—91no—
73C2×D3×SD16—34no—
74C23×C3⋊D3—21no—
75C23×C3×D3—64no—
76C2×C32×Q8—24no—
77C2×C32×Q8C2×D314yescomplemented
78C2×C32×Q8C22×C311yescharacteristic, complemented
79C2×C32×D4—24no—
80C2×C32×D4C2×D314yes—
81C2×C32×D4C22×C311yescharacteristic
82C2×#638—21no—
83Q8×C3⋊D3—22no—
84Q8×C3⋊D3C22×C314yescomplemented
85D4×C3⋊D3—22no—
86D4×C3⋊D3C22×C314yes—
87C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
88C2×C4×C3⋊D3—21no—
89C2×C4×C3⋊D3C22×C311yescharacteristic
90C2×#634—21no—
91C2×#634C22×C311yescharacteristic, complemented
92C2×C3×#67—64no—
93C3×D3×Q8—316no—
94C3×D3×Q8—68no—
95C3×D3×D4—316no—
96C3×D3×D4—68no—
97C2×C3×D12—34no—
98C2×C3×C4×D3—34no—
99C2×C3×C4×D3—64no—
100C2×C3×Dic6—34no—
101C2×C3×Dic6—64no—
102C32×SD16—216no—
103C32×SD16C2×D3116yescomplemented
104C32×SD16C22×C314yescomplemented
105C2×C32×C8—24no—
106C2×C32×C8C2×D314yescomplemented
107C2×C32×C8C22×C311yescharacteristic, complemented
108#3755C22×C314yescomplemented
109#3754C22×C314yescomplemented
110C2×C3⋊(C3⋊C8)C22×C311yescharacteristic, complemented
111#3744C22×C314yescomplemented
112C8×C3⋊D3C22×C314yescomplemented
113C3×#215—316no—
114C3×#214—316no—
115C2×C3×C3⋊C8—34no—
116C3×#204—316no—
117C3×D3×C8—316no—
118C22×C33—22no—
119C22×C33—42no—
120C22×C33C2×D411yescharacteristic
121C2×C3×C3⋊D3—22no—
122C2×C3×C3⋊D3—42no—
123C2×C3×C3⋊D3—48no—
124C2×C3×C3⋊D3SD1614yesdirect product factor
125C2×C3×C3⋊D3C2×D412yes—
126C33×C4—22no—
127C33×C4C2×D411yescharacteristic
128C33×C4C2411yescharacteristic
129C3×C3⋊Dic3—22no—
130C3×C3⋊Dic3C2×D412yes—
131C22×C3×Q8—91no—
132C22×C3×D4—91no—
133C2×D3×Q8—34no—
134C2×D3×D4—34no—
135C2×C3×SD16—24no—
136C2×C3×SD16—91no—
137C2×C3×SD16—91no—
138C2×C3×SD16—98no—
139C2×C3×SD16—91no—
140C2×C3×SD16C3×D314yescomplemented
141C2×C3×SD16C3⋊D311yescharacteristic, direct product factor
142C22×C3×C8—91no—
143C2×#215—34no—
144C2×#214—34no—
145D3×SD16—332no—
146C2×#204—34no—
147C2×D3×C8—34no—
148C23×C32—21no—
149C23×C32—24no—
150C23×C32—44no—
151C22×C3⋊D3—21no—
152C22×C3⋊D3—24no—
153C22×C3⋊D3—44no—
154C22×C3⋊D3C3×D411yescharacteristic
155C22×C3×D3—34no—
156C22×C3×D3—616no—
157C22×C3×D3—64no—
158C22×C3×D3—1216no—
159C32×Q8—21no—
160C32×Q8—28no—
161C32×Q8—24no—
162C32×Q8—44no—
163C32×Q8C22×D318yescomplemented
164C32×Q8C23×C312yescomplemented
165C32×D4—21no—
166C32×D4—28no—
167C32×D4—24no—
168C32×D4—44no—
169C32×D4C22×D318yes—
170C32×D4C23×C312yes—
171C2×C32×C4—21no—
172C2×C32×C4—24no—
173C2×C32×C4—24no—
174C2×C32×C4—44no—
175C2×C32×C4C22×D314yes—
176C2×C32×C4C23×C311yescharacteristic
177#638—24no—
178C2×C3⋊Dic3—21no—
179C2×C3⋊Dic3C3×D411yescharacteristic
180C4⋊(C3⋊D3)—21no—
181C4⋊(C3⋊D3)C23×C312yes—
182C4×C3⋊D3—24no—
183C4×C3⋊D3C3×D412yes—
184C4×C3⋊D3C23×C312yes—
185#634—21no—
186#634—24no—
187#634C23×C312yescomplemented
188C3×#67—616no—
189C2×C3×Dic3—34no—
190C2×C3×Dic3—64no—
191C3×D12—38no—
192C3×D12—64no—
193C3×C4×D3—38no—
194C3×C4×D3—38no—
195C3×C4×D3—616no—
196C3×Dic6—38no—
197C3×Dic6—616no—
198C3×Dic6—64no—
199C32×C8—28no—
200C32×C8C22×D318yescomplemented
201C32×C8C23×C312yescomplemented
202C3⋊(C3⋊C8)C23×C312yescomplemented
203C3×C3⋊C8—38no—
204C22×SD16—91noSylow
205C2×C33—42no—
206C2×C33C2×SD1612yescomplemented
207C2×C33C22×D411yescharacteristic
208C3×C3⋊D3—42no—
209C3×C3⋊D3C2×SD1614yesdirect product factor
210C24×C3—181no—
211C23×D3—64no—
212C2×C3×Q8—24no—
213C2×C3×Q8—94no—
214C2×C3×Q8—91no—
215C2×C3×Q8—181no—
216C2×C3×Q8—182no—
217C2×C3×Q8C2×C3×D314yescomplemented
218C2×C3×Q8C2×C3⋊D311yescharacteristic, complemented
219C2×C3×D4—24no—
220C2×C3×D4—94no—
221C2×C3×D4—91no—
222C2×C3×D4—182no—
223C2×C3×D4—181no—
224C2×C3×D4C2×C3×D314yes—
225C2×C3×D4C2×C3⋊D311yescharacteristic
226C22×C3×C4—91no—
227C22×C3×C4—181no—
228C2×#67—64no—
229D3×Q8—316no—
230D3×Q8—68no—
231D3×D4—316no—
232D3×D4—68no—
233C2×D12—34no—
234C2×C4×D3—34no—
235C2×C4×D3—64no—
236C2×Dic6—34no—
237C2×Dic6—64no—
238C3×SD16—216no—
239C3×SD16—94no—
240C3×SD16—94no—
241C3×SD16—94no—
242C3×SD16C2×C3×D3116yescomplemented
243C3×SD16C2×C3⋊D314yesdirect product factor
244C2×C3×C8—24no—
245C2×C3×C8—91no—
246C2×C3×C8—94no—
247C2×C3×C8C2×C3×D314yescomplemented
248C2×C3×C8C2×C3⋊D311yescharacteristic, complemented
249#215—316no—
250#214—316no—
251C2×C3⋊C8—34no—
252#204—316no—
253D3×C8—316no—
254C22×C32—22no—
255C22×C32—24no—
256C22×C32—28no—
257C22×C32—48no—
258C22×C32—42no—
259C22×C32—48no—
260C22×C32—88no—
261C22×C32D3×D414yes—
262C22×C32C2×C3×D411yescharacteristic
263C2×C3⋊D3—22no—
264C2×C3⋊D3—42no—
265C2×C3⋊D3—48no—
266C2×C3⋊D3C3×SD1614yesdirect product factor
267C2×C3⋊D3C2×C3×D412yes—
268C2×C3×D3—316no—
269C2×C3×D3—38no—
270C2×C3×D3—68no—
271C2×C3×D3—1232no—
272C2×C3×D3—128no—
273C32×C4—22no—
274C32×C4—24no—
275C32×C4—24no—
276C32×C4—28no—
277C32×C4—48no—
278C32×C4D3×D414yes—
279C32×C4C2×C3×D411yescharacteristic
280C32×C4C23×D314yes—
281C32×C4C24×C311yescharacteristic
282C3⋊Dic3—22no—
283C3⋊Dic3C2×C3×D412yes—
284C3×Dic3—38no—
285C3×Dic3—68no—
286C22×Q8—91no—
287C22×D4—91no—
288C2×SD16—91no—
289C2×SD16—91no—
290C2×SD16—98no—
291C2×SD16—91no—
292C2×SD16C3×C3⋊D311yescharacteristic, direct product factor
293C22×C8—91no—
294C33C22×SD1611yescharacteristic, complemented, Sylow
295C23×C3—24no—
296C23×C3—21no—
297C23×C3—44no—
298C23×C3—91no—
299C23×C3—184no—
300C23×C3—181no—
301C23×C3—364no—
302C22×D3—34no—
303C22×D3—616no—
304C22×D3—64no—
305C22×D3—1216no—
306C3×Q8—24no—
307C3×Q8—21no—
308C3×Q8—28no—
309C3×Q8—44no—
310C3×Q8—92no—
311C3×Q8—184no—
312C3×Q8—181no—
313C3×Q8C22×C3×D318yescomplemented
314C3×Q8C22×C3⋊D312yescomplemented
315C3×D4—24no—
316C3×D4—21no—
317C3×D4—28no—
318C3×D4—44no—
319C3×D4—92no—
320C3×D4—184no—
321C3×D4—181no—
322C3×D4C22×C3×D318yes—
323C3×D4C22×C3⋊D312yes—
324C2×C3×C4—24no—
325C2×C3×C4—21no—
326C2×C3×C4—24no—
327C2×C3×C4—44no—
328C2×C3×C4—91no—
329C2×C3×C4—92no—
330C2×C3×C4—92no—
331C2×C3×C4—184no—
332C2×C3×C4—181no—
333C2×C3×C4C22×C3×D314yes—
334C2×C3×C4C22×C3⋊D311yescharacteristic
335#67—616no—
336C2×Dic3—34no—
337C2×Dic3—64no—
338D12—38no—
339D12—64no—
340C4×D3—38no—
341C4×D3—38no—
342C4×D3—616no—
343Dic6—38no—
344Dic6—616no—
345Dic6—64no—
346C3×C8—28no—
347C3×C8—92no—
348C3×C8C22×C3×D318yescomplemented
349C3×C8C22×C3⋊D312yescomplemented
350C3⋊C8—38no—
351C2×C32—28no—
352C2×C32—24no—
353C2×C32—42no—
354C2×C32—48no—
355C2×C32—88no—
356C2×C32D3×SD1618yescomplemented
357C2×C32C2×C3×SD1612yescomplemented
358C2×C32C2×D3×D414yes—
359C2×C32C22×C3×D411yescharacteristic
360C3⋊D3—42no—
361C3⋊D3C2×C3×SD1614yesdirect product factor
362C3×D3—316no—
363C3×D3—128no—
364C24—181no—
365C2×Q8—94no—
366C2×Q8—91no—
367C2×Q8—181no—
368C2×Q8—182no—
369C2×Q8C2×C3×C3⋊D311yescharacteristic, complemented
370C2×D4—94no—
371C2×D4—91no—
372C2×D4—182no—
373C2×D4—181no—
374C2×D4C2×C3×C3⋊D311yescharacteristic
375C22×C4—91no—
376C22×C4—181no—
377SD16—94no—
378SD16—94no—
379SD16—94no—
380SD16C2×C3×C3⋊D314yesdirect product factor
381C2×C8—91no—
382C2×C8—94no—
383C2×C8C2×C3×C3⋊D311yescharacteristic, complemented
384C22×C3—28no—
385C22×C3—22no—
386C22×C3—24no—
387C22×C3—48no—
388C22×C3—48no—
389C22×C3—42no—
390C22×C3—88no—
391C22×C3—92no—
392C22×C3—94no—
393C22×C3—182no—
394C22×C3—368no—
395C22×C3—362no—
396C22×C3C3×D3×D414yes—
397C22×C3D4×C3⋊D311yescharacteristic
398C2×D3—316no—
399C2×D3—38no—
400C2×D3—68no—
401C2×D3—1232no—
402C2×D3—128no—
403C3×C4—28no—
404C3×C4—22no—
405C3×C4—24no—
406C3×C4—24no—
407C3×C4—48no—
408C3×C4—92no—
409C3×C4—182no—
410C3×C4C3×D3×D414yes—
411C3×C4D4×C3⋊D311yescharacteristic
412C3×C4C23×C3×D314yes—
413C3×C4C23×C3⋊D311yescharacteristic
414Dic3—38no—
415Dic3—68no—
416C32—24no—
417C32C2×D3×SD1614yescomplemented
418C32C22×C3×SD1611yescharacteristic, complemented
419C23—21no—
420C23—91no—
421C23—184no—
422C23—181no—
423C23—364no—
424Q8—21no—
425Q8—92no—
426Q8—184no—
427Q8—181no—
428Q8C22×C3×C3⋊D312yescomplemented
429D4—21no—
430D4—92no—
431D4—184no—
432D4—181no—
433D4C22×C3×C3⋊D312yes—
434C2×C4—21no—
435C2×C4—91no—
436C2×C4—92no—
437C2×C4—92no—
438C2×C4—184no—
439C2×C4—181no—
440C2×C4C22×C3×C3⋊D311yescharacteristic
441C8—92no—
442C8C22×C3×C3⋊D312yescomplemented
443C2×C3—28no—
444C2×C3—24no—
445C2×C3—48no—
446C2×C3—42no—
447C2×C3—88no—
448C2×C3—94no—
449C2×C3—362no—
450C2×C3C3×D3×SD1618yescomplemented
451C2×C3SD16×C3⋊D312yesdirect product factor
452C2×C3C2×C3×D3×D414yes—
453C2×C3C2×D4×C3⋊D311yescharacteristic
454D3—316no—
455D3—128no—
456C22—22no—
457C22—42no—
458C22—94no—
459C22—92no—
460C22—182no—
461C22—368no—
462C22—362no—
463C22C3×D4×C3⋊D311yescharacteristic
464C4—22no—
465C4—92no—
466C4—182no—
467C4C3×D4×C3⋊D311yescharacteristic
468C4C23×C3×C3⋊D311yescharacteristic
469C3—24no—
470C3C2×C3×D3×SD1614yescomplemented
471C3C2×SD16×C3⋊D311yescharacteristic, direct product factor
472C2—42no—
473C2—94no—
474C2—362no—
475C2C3×SD16×C3⋊D312yesdirect product factor
476C2C2×C3×D4×C3⋊D311yescharacteristic
subgroupG/H
centerC22×C3D4×C3⋊D3
commutator, cosocleC32×C4C24×C3
FrattiniC4C23×C3×C3⋊D3
FittingC2×C33×SD16C2
socleC22×C33C2×D4
prime radicalC2×C3×D4×C3⋊D3C2
hypercenterC2×C3×SD16C3⋊D3
hypocenterC32C22×C3×SD16
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×SD16

Complements

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

NH ≅ G/Ntype
1. C2×C3×Q8×C3⋊D3C2semidirect
3. C2×C33×SD16C2semidirect
4. C2×C3×#3755C2semidirect
5. C2×C3×#3754C2semidirect
6. C3×SD16×C3⋊D3C2direct (N × H)
7. C2×C3×#3744C2semidirect
8. C2×C3×C8×C3⋊D3C2semidirect
9. C2×SD16×C3⋊D3C3direct (N × H)
12. C2×C33×Q8C22semidirect
16. C3×Q8×C3⋊D3C22semidirect
23. C2×C3×#634C22semidirect
24. C33×SD16C22semidirect
25. C2×C33×C8C22semidirect
26. C3×#3755C22semidirect
27. C3×#3754C22semidirect
28. C2×C3×C3⋊(C3⋊C8)C22semidirect
29. C3×#3744C22semidirect
30. C3×C8×C3⋊D3C22semidirect
31. C2×Q8×C3⋊D3C2×C3semidirect
36. C2×C32×SD16D3semidirect
37. C2×C32×SD16C2×C3semidirect
38. C2×#3755C2×C3semidirect
39. C2×#3754C2×C3semidirect
40. SD16×C3⋊D3C2×C3direct (N × H)
41. C2×#3744C2×C3semidirect
42. C2×C8×C3⋊D3C2×C3semidirect
54. C33×Q8C23semidirect
69. C3×#634C23semidirect
70. C33×C8C23semidirect
71. C3×C3⋊(C3⋊C8)C23semidirect
77. C2×C32×Q8C2×D3semidirect
78. C2×C32×Q8C22×C3semidirect
84. Q8×C3⋊D3C22×C3semidirect
91. C2×#634C22×C3semidirect
103. C32×SD16C2×D3semidirect
104. C32×SD16C22×C3semidirect
106. C2×C32×C8C2×D3semidirect
107. C2×C32×C8C22×C3semidirect
108. #3755C22×C3semidirect
109. #3754C22×C3semidirect
110. C2×C3⋊(C3⋊C8)C22×C3semidirect
111. #3744C22×C3semidirect
112. C8×C3⋊D3C22×C3semidirect
124. C2×C3×C3⋊D3SD16direct (N × H)
140. C2×C3×SD16C3×D3semidirect
141. C2×C3×SD16C3⋊D3direct (N × H)
163. C32×Q8C22×D3semidirect
164. C32×Q8C23×C3semidirect
187. #634C23×C3semidirect
200. C32×C8C22×D3semidirect
201. C32×C8C23×C3semidirect
202. C3⋊(C3⋊C8)C23×C3semidirect
206. C2×C33C2×SD16semidirect
209. C3×C3⋊D3C2×SD16direct (N × H)
217. C2×C3×Q8C2×C3×D3semidirect
218. C2×C3×Q8C2×C3⋊D3semidirect
242. C3×SD16C2×C3×D3semidirect
243. C3×SD16C2×C3⋊D3direct (N × H)
247. C2×C3×C8C2×C3×D3semidirect
248. C2×C3×C8C2×C3⋊D3semidirect
266. C2×C3⋊D3C3×SD16direct (N × H)
292. C2×SD16C3×C3⋊D3direct (N × H)
294. C33C22×SD16semidirect
313. C3×Q8C22×C3×D3semidirect
314. C3×Q8C22×C3⋊D3semidirect
348. C3×C8C22×C3×D3semidirect
349. C3×C8C22×C3⋊D3semidirect
356. C2×C32D3×SD16semidirect
357. C2×C32C2×C3×SD16semidirect
361. C3⋊D3C2×C3×SD16direct (N × H)
369. C2×Q8C2×C3×C3⋊D3semidirect
380. SD16C2×C3×C3⋊D3direct (N × H)
383. C2×C8C2×C3×C3⋊D3semidirect
417. C32C2×D3×SD16semidirect
418. C32C22×C3×SD16semidirect
428. Q8C22×C3×C3⋊D3semidirect
442. C8C22×C3×C3⋊D3semidirect
450. C2×C3C3×D3×SD16semidirect
451. C2×C3SD16×C3⋊D3direct (N × H)
470. C3C2×C3×D3×SD16semidirect
471. C3C2×SD16×C3⋊D3direct (N × H)
475. C2C3×SD16×C3⋊D3direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×Q8×C3⋊D32
C2×C3×D4×C3⋊D32
C2×C33×SD162
C2×C3×#37552
C2×C3×#37542
C3×SD16×C3⋊D32
C2×C3×#37442
C2×C3×C8×C3⋊D32
C2×SD16×C3⋊D33
C2×C3×D3×SD163