Finite Group Databasebeta

← C2×D3×#3754 #13177660

Subgroups — C2×D3×#3754 #13177660

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×C32×D3×D4C211yesmaximal, characteristic
2C2×D3×#634C211yesmaximal, characteristic, complemented
3C2×#88861C211yesmaximal, characteristic, complemented
4C2×C3×#3754C211yesmaximal, characteristic, complemented
5D3×#3754C218yesmaximal, direct product factor
6C2×#88794C211yesmaximal, characteristic, complemented
7C2×#88792C211yesmaximal, characteristic, complemented
8C2×D3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
9C22×#3754—31nomaximal
10C2×D3×#214—34nomaximal
11C23×C32×D3—21no—
12C2×C33×D4C2211yescharacteristic
13C2×#6308C2211yescharacteristic, complemented
14C2×C3×#634C2211yescharacteristic, complemented
15C2×C32×#67—21no—
16C32×D3×D4—22no—
17C32×D3×D4C2214yes—
18C2×C32×D12C2211yescharacteristic
19C2×C32×C4×D3C2211yescharacteristic
20C2×#6293—21no—
21C2×D3×C3⋊Dic3—21no—
22D3×#634—22no—
23D3×#634C2214yescomplemented
24#88861C2214yescomplemented
25C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
26C3×#3754C2214yescomplemented
27C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
28#88794C2214yescomplemented
29#88792C2214yescomplemented
30D3×C3⋊(C3⋊C8)C2214yescomplemented
31C22×C32×D4—31no—
32C22×#634—31no—
33C2×C3×D3×D4D314yes—
34C2×D3×Dic6—34no—
35C2×#3754—31no—
36C2×#3754—31no—
37C2×#3754—38no—
38C2×#3754—34no—
39C2×#3754—31no—
40C2×#3754—64no—
41C2×#3754D311yescharacteristic, direct product factor
42C22×C3⋊(C3⋊C8)—31no—
43C2×C3×#214—34no—
44D3×#214—332no—
45C2×#3716—34no—
46C2×#3715—34no—
47C2×D3×C3⋊C8—34no—
48C23×C33—21no—
49C22×C32×D3—21no—
50C22×C32×D3—24no—
51C22×C32×D3—44no—
52C22×C32×D3D411yescharacteristic
53C33×D4—21no—
54C33×D4C2312yes—
55C2×C33×C4C2311yescharacteristic
56C2×C3⋊(C3⋊Dic3)—21no—
57#6308—21no—
58#6308C2312yescomplemented
59C2×C3×C3⋊Dic3—21no—
60C3×#634—21no—
61C3×#634C2312yescomplemented
62C32×#67—24no—
63C2×C32×Dic3D411yescharacteristic
64C32×D12—21no—
65C32×D12C2312yes—
66C32×C4×D3D412yes—
67C32×C4×D3C2312yes—
68#6293—24no—
69D3×C3⋊Dic3—24no—
70C3⋊(C3⋊(C3⋊C8))C2312yescomplemented
71C3×C3⋊(C3⋊C8)C2312yescomplemented
72C22×#214—94no—
73C2×D3×SD16—91no—
74C24×C32—61no—
75C23×C3×D3—24no—
76C2×C32×D4—24no—
77C2×C32×D4—34no—
78C2×C32×D4—31no—
79C2×C32×D4—62no—
80C2×C32×D4—61no—
81C2×C32×D4C2×D314yes—
82C2×C32×D4C2×D311yescharacteristic
83C22×C32×C4—31no—
84C22×C3⋊Dic3—61no—
85C2×#634—31no—
86C2×#634—34no—
87C2×#634—34no—
88C2×#634—62no—
89C2×#634—61no—
90C2×#634—64no—
91C2×#634C2×D311yescharacteristic, complemented
92C2×C3×#67—24no—
93C3×D3×D4—28no—
94C3×D3×D4C2×D3116yes—
95C2×C3×D12C2×D314yes—
96C2×C3×C4×D3C2×D314yes—
97C2×C3×Dic6—34no—
98C2×#627—64no—
99C2×D3×Dic3—64no—
100D3×Dic6—316no—
101D3×Dic6—68no—
102#3754—34no—
103#3754—316no—
104#3754—34no—
105#3754—34no—
106#3754—616no—
107#3754C2×D314yesdirect product factor
108C2×C3⋊(C3⋊C8)—31no—
109C2×C3⋊(C3⋊C8)—34no—
110C2×C3⋊(C3⋊C8)—34no—
111C2×C3⋊(C3⋊C8)—64no—
112C2×C3⋊(C3⋊C8)C2×D311yescharacteristic, complemented
113C3×#214—316no—
114C2×C3×C3⋊C8—34no—
115#3716—316no—
116#3715—316no—
117D3×C3⋊C8—316no—
118C22×C33—22no—
119C22×C33—42no—
120C22×C33C2×D411yescharacteristic
121C2×C32×D3—22no—
122C2×C32×D3—42no—
123C2×C32×D3—48no—
124C2×C32×D3SD1614yescomplemented
125C2×C32×D3C2×D412yes—
126C33×C4C2×D411yescharacteristic
127C33×C4C2411yescharacteristic
128C3⋊(C3⋊Dic3)—22no—
129C3×C3⋊Dic3—22no—
130C32×Dic3C2×D412yes—
131C22×C3×D4—34no—
132C2×D3×Q8—91no—
133C2×D3×D4C3⋊D311yescharacteristic
134C22×Dic6—94no—
135C2×C3×SD16—91no—
136C2×#215—91no—
137C2×#214—34no—
138C2×#214—94no—
139C2×#214—932no—
140C2×#214—94no—
141C2×#214—94no—
142C2×#214—91no—
143C2×#214—184no—
144C22×C3⋊C8—94no—
145D3×SD16—98no—
146C2×#204—91no—
147C2×D3×C8—91no—
148C23×C32—21no—
149C23×C32—24no—
150C23×C32—31no—
151C23×C32—44no—
152C23×C32—64no—
153C23×C32—61no—
154C23×C32—124no—
155C22×C3×D3—24no—
156C22×C3×D3—216no—
157C22×C3×D3—416no—
158C22×C3×D3#6714yes—
159C32×D4—21no—
160C32×D4—28no—
161C32×D4—24no—
162C32×D4—32no—
163C32×D4—44no—
164C32×D4—64no—
165C32×D4—61no—
166C32×D4C22×D318yes—
167C32×D4C22×D312yes—
168C2×C32×C4—24no—
169C2×C32×C4—31no—
170C2×C32×C4—32no—
171C2×C32×C4—32no—
172C2×C32×C4C22×D314yes—
173C2×C32×C4C22×D311yescharacteristic
174C2×C3⋊Dic3—21no—
175C2×C3⋊Dic3—64no—
176C2×C3⋊Dic3—61no—
177C2×C3⋊Dic3—64no—
178C2×C3⋊Dic3—124no—
179#634—21no—
180#634—32no—
181#634—38no—
182#634—68no—
183#634—64no—
184#634—61no—
185#634—64no—
186#634—124no—
187#634C22×D312yescomplemented
188C3×#67—216no—
189C2×C3×Dic3—64no—
190C2×C3×Dic3#6714yes—
191C3×D12—24no—
192C3×D12C22×D318yes—
193C3×C4×D3#6718yes—
194C3×C4×D3C22×D318yes—
195C3×Dic6—38no—
196C3×Dic6—64no—
197#627—616no—
198D3×Dic3—616no—
199C3⋊(C3⋊C8)—32no—
200C3⋊(C3⋊C8)—38no—
201C3⋊(C3⋊C8)—68no—
202C3⋊(C3⋊C8)C22×D312yescomplemented
203C3×C3⋊C8—38no—
204C22×SD16—271noSylow
205C2×C33—42no—
206C2×C33C2×SD1612yescomplemented
207C2×C33C22×D411yescharacteristic
208C32×D3—42no—
209C32×D3C2×SD1614yescomplemented
210C24×C3—64no—
211C23×D3—21no—
212C2×C3×Q8—91no—
213C2×C3×D4—24no—
214C2×C3×D4—316no—
215C2×C3×D4—34no—
216C2×C3×D4—68no—
217C2×C3×D4—64no—
218C2×C3×D4D3214yes—
219C2×C3×D4C2×C3⋊D311yescharacteristic
220C22×C3×C4—34no—
221C2×#67—21no—
222C22×Dic3—184no—
223D3×Q8—94no—
224D3×Q8—182no—
225D3×D4—22no—
226D3×D4C2×C3⋊D314yes—
227C2×D12C2×C3⋊D311yescharacteristic
228C2×C4×D3—181no—
229C2×C4×D3C2×C3⋊D311yescharacteristic
230C2×Dic6—34no—
231C2×Dic6—916no—
232C2×Dic6—94no—
233C2×Dic6—91no—
234C2×Dic6—184no—
235C2×Dic6—188no—
236C2×Dic6—184no—
237C2×Dic6—181no—
238C3×SD16—94no—
239C2×C3×C8—91no—
240#215—94no—
241#214—316no—
242#214—916no—
243#214—916no—
244#214—916no—
245#214—94no—
246#214—1816no—
247C2×C3⋊C8—34no—
248C2×C3⋊C8—916no—
249C2×C3⋊C8—94no—
250C2×C3⋊C8—91no—
251C2×C3⋊C8—184no—
252#204—94no—
253D3×C8—94no—
254C22×C32—22no—
255C22×C32—24no—
256C22×C32—28no—
257C22×C32—34no—
258C22×C32—32no—
259C22×C32—48no—
260C22×C32—42no—
261C22×C32—48no—
262C22×C32—62no—
263C22×C32—88no—
264C22×C32—128no—
265C22×C32—122no—
266C22×C32D3×D411yescharacteristic
267C22×C32C2×#6714yes—
268C2×C3×D3—28no—
269C2×C3×D3—48no—
270C2×C3×D3—432no—
271C2×C3×D3#214116yescomplemented
272C2×C3×D3C2×#6718yes—
273C32×C4—24no—
274C32×C4—24no—
275C32×C4—32no—
276C32×C4D3×D411yescharacteristic
277C32×C4C2×#6714yes—
278C32×C4C23×D314yes—
279C32×C4C23×D311yescharacteristic
280C3⋊Dic3—22no—
281C3⋊Dic3—62no—
282C3⋊Dic3—68no—
283C3⋊Dic3—128no—
284C3×Dic3—68no—
285C3×Dic3C2×#6718yes—
286C22×Q8—271no—
287C22×D4—31no—
288C2×SD16—91no—
289C2×SD16—271no—
290C2×SD16—278no—
291C2×SD16—271no—
292C2×SD16—271no—
293C22×C8—271no—
294C33C22×SD1611yescharacteristic, complemented, Sylow
295C23×C3—24no—
296C23×C3—21no—
297C23×C3—34no—
298C23×C3—44no—
299C23×C3—616no—
300C23×C3—64no—
301C23×C3—1216no—
302C22×D3—21no—
303C22×D3—24no—
304C22×D3—44no—
305C22×D3#63811yescharacteristic
306C3×Q8—92no—
307C3×Q8—181no—
308C3×D4—24no—
309C3×D4—21no—
310C3×D4—28no—
311C3×D4—38no—
312C3×D4—44no—
313C3×D4—616no—
314C3×D4—64no—
315C3×D4C2×D3218yes—
316C3×D4C22×C3⋊D312yes—
317C2×C3×C4—24no—
318C2×C3×C4—34no—
319C2×C3×C4—38no—
320C2×C3×C4—38no—
321C2×C3×C4—181no—
322C2×C3×C4C2×D3214yes—
323C2×C3×C4C22×C3⋊D311yescharacteristic
324#67—24no—
325C2×Dic3—64no—
326C2×Dic3—1816no—
327C2×Dic3—184no—
328C2×Dic3—181no—
329C2×Dic3—364no—
330C2×Dic3#63811yescharacteristic
331D12—21no—
332D12C22×C3⋊D312yes—
333C4×D3—184no—
334C4×D3#63812yes—
335C4×D3C22×C3⋊D312yes—
336Dic6—38no—
337Dic6—64no—
338Dic6—98no—
339Dic6—92no—
340Dic6—188no—
341Dic6—1816no—
342Dic6—184no—
343Dic6—181no—
344Dic6—184no—
345Dic6—364no—
346C3×C8—92no—
347C3⋊C8—38no—
348C3⋊C8—98no—
349C3⋊C8—92no—
350C3⋊C8—188no—
351C2×C32—24no—
352C2×C32—28no—
353C2×C32—34no—
354C2×C32—42no—
355C2×C32—48no—
356C2×C32—88no—
357C2×C32—122no—
358C2×C32D3×SD1612yescomplemented
359C2×C32C2×#21418yescomplemented
360C2×C32C2×D3×D411yescharacteristic
361C2×C32C22×#6714yes—
362C3×D3—48no—
363C3×D3C2×#214116yescomplemented
364C24—61no—
365C2×Q8—91no—
366C2×Q8—274no—
367C2×Q8—271no—
368C2×Q8—541no—
369C2×Q8—542no—
370C2×D4—34no—
371C2×D4—31no—
372C2×D4—62no—
373C2×D4—61no—
374C2×D4D3×C3⋊D311yescharacteristic
375C22×C4—31no—
376C22×C4—541no—
377SD16—94no—
378SD16—274no—
379SD16—274no—
380SD16—274no—
381C2×C8—91no—
382C2×C8—274no—
383C2×C8—271no—
384C22×C3—28no—
385C22×C3—22no—
386C22×C3—24no—
387C22×C3—38no—
388C22×C3—316no—
389C22×C3—48no—
390C22×C3—42no—
391C22×C3—48no—
392C22×C3—68no—
393C22×C3—88no—
394C22×C3—1232no—
395C22×C3—128no—
396C22×C3D3×#6714yes—
397C22×C3C2×#63811yescharacteristic
398C2×D3—22no—
399C2×D3—42no—
400C2×D3—48no—
401C2×D3#375414yesdirect product factor
402C2×D3C2×#63812yes—
403C3×C4—24no—
404C3×C4—24no—
405C3×C4—38no—
406C3×C4—182no—
407C3×C4D3×#6714yes—
408C3×C4C2×#63811yescharacteristic
409C3×C4C22×D3214yes—
410C3×C4C23×C3⋊D311yescharacteristic
411Dic3—68no—
412Dic3—188no—
413Dic3—182no—
414Dic3—368no—
415Dic3C2×#63812yes—
416C32—24no—
417C32C2×D3×SD1611yescharacteristic, complemented
418C32C22×#21414yescomplemented
419C23—21no—
420C23—31no—
421C23—64no—
422C23—61no—
423C23—124no—
424Q8—92no—
425Q8—181no—
426Q8—272no—
427Q8—544no—
428Q8—541no—
429D4—21no—
430D4—32no—
431D4—64no—
432D4—61no—
433D4C2×D3×C3⋊D312yes—
434C2×C4—31no—
435C2×C4—32no—
436C2×C4—32no—
437C2×C4—181no—
438C2×C4—544no—
439C2×C4—541no—
440C2×C4C2×D3×C3⋊D311yescharacteristic
441C8—92no—
442C8—272no—
443C2×C3—28no—
444C2×C3—24no—
445C2×C3—316no—
446C2×C3—48no—
447C2×C3—42no—
448C2×C3—88no—
449C2×C3—128no—
450C2×C3D3×#21418yescomplemented
451C2×C3C2×#375412yescomplemented
452C2×C3C2×D3×#6714yes—
453C2×C3C22×#63811yescharacteristic
454D3—42no—
455D3C2×#375414yesdirect product factor
456C22—22no—
457C22—34no—
458C22—32no—
459C22—42no—
460C22—62no—
461C22—128no—
462C22—122no—
463C22D3×#63811yescharacteristic
464C4—32no—
465C4—182no—
466C4—542no—
467C4D3×#63811yescharacteristic
468C4C22×D3×C3⋊D311yescharacteristic
469C3—24no—
470C3C2×D3×#21414yescomplemented
471C3C22×#375411yescharacteristic, complemented
472C2—34no—
473C2—42no—
474C2—122no—
475C2D3×#375412yesdirect product factor
476C2C2×D3×#63811yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC33×C4C24
FrattiniC4C22×D3×C3⋊D3
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radicalC2×C32×D3×D4C2
hypercenterC2×D4D3×C3⋊D3
hypocenterC33C22×SD16
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C2×D3×#634C2semidirect
3. C2×#88861C2semidirect
4. C2×C3×#3754C2semidirect
5. D3×#3754C2direct (N × H)
6. C2×#88794C2semidirect
7. C2×#88792C2semidirect
8. C2×D3×C3⋊(C3⋊C8)C2semidirect
13. C2×#6308C22semidirect
14. C2×C3×#634C22semidirect
23. D3×#634C22semidirect
24. #88861C22semidirect
25. C2×C3⋊(C3⋊(C3⋊C8))C22semidirect
26. C3×#3754C22semidirect
27. C2×C3×C3⋊(C3⋊C8)C22semidirect
28. #88794C22semidirect
29. #88792C22semidirect
30. D3×C3⋊(C3⋊C8)C22semidirect
41. C2×#3754D3direct (N × H)
58. #6308C23semidirect
61. C3×#634C23semidirect
70. C3⋊(C3⋊(C3⋊C8))C23semidirect
71. C3×C3⋊(C3⋊C8)C23semidirect
91. C2×#634C2×D3semidirect
107. #3754C2×D3direct (N × H)
112. C2×C3⋊(C3⋊C8)C2×D3semidirect
124. C2×C32×D3SD16semidirect
187. #634C22×D3semidirect
202. C3⋊(C3⋊C8)C22×D3semidirect
206. C2×C33C2×SD16semidirect
209. C32×D3C2×SD16semidirect
271. C2×C3×D3#214semidirect
294. C33C22×SD16semidirect
358. C2×C32D3×SD16semidirect
359. C2×C32C2×#214semidirect
363. C3×D3C2×#214semidirect
401. C2×D3#3754direct (N × H)
417. C32C2×D3×SD16semidirect
418. C32C22×#214semidirect
450. C2×C3D3×#214semidirect
451. C2×C3C2×#3754semidirect
455. D3C2×#3754direct (N × H)
470. C3C2×D3×#214semidirect
471. C3C22×#3754semidirect
475. C2D3×#3754direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×D3×D42
C2×D3×#6342
C2×#888612
C2×C3×#37542
D3×#37542
C2×#887942
C2×#887922
C2×D3×C3⋊(C3⋊C8)2
C22×#37543
C2×D3×#2143