Finite Group Databasebeta

← C2×#11723723 #13178787

Subgroups — C2×#11723723 #13178787

Order 1728.

3842 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C24×C3×C3⋊D3C211yesmaximal, characteristic
2C22×#6294C212yesmaximal, complemented
3C2×C3×#3757C212yesmaximal, complemented
4#11723723C218yesmaximal, direct product factor
5C2×#88814C212yesmaximal, complemented
6C2×#63974—31nomaximal
7C2×D3×#217—32nomaximal
8C24×C33C2211yescharacteristic
9C23×C3×C3⋊D3—28no—
10C23×C3×C3⋊D3C418yescomplemented
11C23×C3×C3⋊D3C2211yescharacteristic
12C23×C3×C3⋊D3C2214yes—
13C23×C3×C3⋊D3C2211yescharacteristic
14C22×C3×C3⋊Dic3C2214yescomplemented
15C2×#6294—28no—
16C2×#6294C2218yescomplemented
17C3×#3757C2218yescomplemented
18#88814C2218yescomplemented
19C24×C3⋊D3D311yescharacteristic
20C24×C3×D3—32no—
21C24×C3×D3—61no—
22C22×#624—32no—
23C22×D3×Dic3—34no—
24C2×#3757D312yescomplemented
25C2×C3×#217—32no—
26C2×C3×#217—32no—
27C2×C3×#217—62no—
28#63974—38no—
29D3×#217—316no—
30C2×#3722—32no—
31C2×(C2×D3)⋊Dic3—34no—
32C23×C33—24no—
33C23×C33C2×C414yescomplemented
34C23×C33C2312yes—
35C23×C33C2311yescharacteristic
36C22×C3×C3⋊D3—24no—
37C22×C3×C3⋊D3—232no—
38C22×C3×C3⋊D3—216no—
39C22×C3×C3⋊D3C2×C414yes—
40C22×C3×C3⋊D3C2×C4116yescomplemented
41C22×C3×C3⋊D3C2×C414yescomplemented
42C22×C3×C3⋊D3D418yes—
43C22×C3×C3⋊D3C2312yes—
44C22×C3×C3⋊D3C2312yes—
45C2×C3×C3⋊Dic3—28no—
46C2×C3×C3⋊Dic3C2318yescomplemented
47#6294—216no—
48C22×#217—91no—
49C2×D3×C22⋊C4—92no—
50C24×C32—21no—
51C24×C32—22no—
52C24×C32—41no—
53C24×C32C2×D312yes—
54C24×C32C2×D311yescharacteristic
55C23×C3⋊D3—28no—
56C23×C3⋊D3Dic318yescomplemented
57C23×C3⋊D3C2×D311yescharacteristic
58C23×C3⋊D3C2×D314yes—
59C23×C3⋊D3C2×D311yescharacteristic
60C23×C3×D3—32no—
61C23×C3×D3—316no—
62C23×C3×D3—32no—
63C23×C3×D3—38no—
64C23×C3×D3—616no—
65C23×C3×D3—68no—
66C23×C3×D3—61no—
67C23×C3×D3—61no—
68C23×C3×D3—64no—
69C23×C3×D3—616no—
70C22×C3⋊Dic3C2×D314yescomplemented
71C22×C3×Dic3—34no—
72C22×C3×Dic3—34no—
73C22×C3×Dic3—64no—
74C2×#624—38no—
75C2×#624—68no—
76C2×D3×Dic3—316no—
77C2×D3×Dic3—616no—
78#3757C2×D318yescomplemented
79C3×#217—38no—
80C3×#217—38no—
81C3×#217—68no—
82#3722—38no—
83(C2×D3)⋊Dic3—316no—
84C22×C33—28no—
85C22×C33—24no—
86C22×C33C22×C412yes—
87C22×C33C22×C414yescomplemented
88C22×C33C2×D414yes—
89C22×C33C2411yescharacteristic
90C2×C3×C3⋊D3—28no—
91C2×C3×C3⋊D3—24no—
92C2×C3×C3⋊D3—232no—
93C2×C3×C3⋊D3C22⋊C4116yescomplemented
94C2×C3×C3⋊D3C22×C414yes—
95C2×C3×C3⋊D3C22×C414yescomplemented
96C2×C3×C3⋊D3C2×D418yes—
97C3×C3⋊Dic3—28no—
98C25×C3—91no—
99C24×D3—32no—
100C24×D3—61no—
101C23×Dic3—92no—
102C22×C4×D3—94no—
103C2×C3×C22⋊C4—92no—
104C2×#217—32no—
105C2×#217—32no—
106C2×#217—62no—
107C2×#217—98no—
108C2×#217—92no—
109C2×(C2×D3)⋊C4—94no—
110D3×C22⋊C4—916no—
111C23×C32—24no—
112C23×C32—28no—
113C23×C32—28no—
114C23×C32—28no—
115C23×C32—24no—
116C23×C32—22no—
117C23×C32—21no—
118C23×C32—22no—
119C23×C32—24no—
120C23×C32—44no—
121C23×C32—42no—
122C23×C32—41no—
123C23×C32—48no—
124C23×C32—48no—
125C23×C32C4×D318yescomplemented
126C23×C32C2×Dic314yescomplemented
127C23×C32C22×D314yes—
128C23×C32C22×D312yes—
129C23×C32C22×D312yes—
130C23×C32C22×D311yescharacteristic
131C22×C3⋊D3—24no—
132C22×C3⋊D3—232no—
133C22×C3⋊D3—216no—
134C22×C3⋊D3C2×Dic314yes—
135C22×C3⋊D3C2×Dic3116yescomplemented
136C22×C3⋊D3C2×Dic314yescomplemented
137C22×C3⋊D3#6718yes—
138C22×C3⋊D3C22×D312yes—
139C22×C3⋊D3C22×D312yes—
140C22×C3×D3—38no—
141C22×C3×D3—316no—
142C22×C3×D3—34no—
143C22×C3×D3—38no—
144C22×C3×D3—332no—
145C22×C3×D3—34no—
146C22×C3×D3—64no—
147C22×C3×D3—64no—
148C22×C3×D3—616no—
149C22×C3×D3—68no—
150C22×C3×D3—62no—
151C22×C3×D3—62no—
152C22×C3×D3—664no—
153C22×C3×D3—632no—
154C22×C3×D3—664no—
155C22×C3×D3—632no—
156C22×C3×D3—68no—
157C22×C3×D3—68no—
158C2×C3⋊Dic3—28no—
159C2×C3⋊Dic3C22×D318yescomplemented
160C2×C3×Dic3—38no—
161C2×C3×Dic3—38no—
162C2×C3×Dic3—68no—
163C2×C3×Dic3—68no—
164C2×C3×Dic3—68no—
165C2×C3×Dic3—128no—
166#624—616no—
167D3×Dic3—632no—
168C22×C22⋊C4—271noSylow
169C2×C33—24no—
170C2×C33C2×C22⋊C414yescomplemented
171C2×C33C23×C411yescharacteristic
172C2×C33C22×D412yes—
173C3×C3⋊D3—24no—
174C3×C3⋊D3C2×C22⋊C418yescomplemented
175C24×C3—21no—
176C24×C3—22no—
177C24×C3—41no—
178C24×C3—91no—
179C24×C3—98no—
180C24×C3—94no—
181C24×C3—91no—
182C24×C3—188no—
183C24×C3D3212yes—
184C24×C3D3211yescharacteristic
185C23×D3—32no—
186C23×D3—316no—
187C23×D3—38no—
188C23×D3—32no—
189C23×D3—68no—
190C23×D3—61no—
191C23×D3—64no—
192C23×D3—61no—
193C23×D3—616no—
194C23×D3—616no—
195C22×C3×C4—94no—
196C22×Dic3—34no—
197C22×Dic3—34no—
198C22×Dic3—64no—
199C22×Dic3—98no—
200C22×Dic3—188no—
201C2×C4×D3—916no—
202C2×C4×D3—1816no—
203C3×C22⋊C4—98no—
204#217—38no—
205#217—38no—
206#217—68no—
207#217—98no—
208(C2×D3)⋊C4—916no—
209C22×C32—28no—
210C22×C32—24no—
211C22×C32—28no—
212C22×C32—216no—
213C22×C32—216no—
214C22×C32—28no—
215C22×C32—22no—
216C22×C32—21no—
217C22×C32—24no—
218C22×C32—28no—
219C22×C32—22no—
220C22×C32—24no—
221C22×C32—28no—
222C22×C32—24no—
223C22×C32—42no—
224C22×C32—44no—
225C22×C32—44no—
226C22×C32—41no—
227C22×C32—48no—
228C22×C32—416no—
229C22×C32—416no—
230C22×C32—48no—
231C22×C32C2×C4×D314yes—
232C22×C32C2×C4×D318yescomplemented
233C22×C32D3×D418yes—
234C22×C32C22×Dic312yes—
235C22×C32C22×Dic314yescomplemented
236C22×C32C2×#6714yes—
237C22×C32C23×D312yes—
238C22×C32C23×D311yescharacteristic
239C2×C3⋊D3—28no—
240C2×C3⋊D3—24no—
241C2×C3⋊D3—232no—
242C2×C3⋊D3#217116yescomplemented
243C2×C3⋊D3C22×Dic314yes—
244C2×C3⋊D3C22×Dic314yescomplemented
245C2×C3⋊D3C2×#6718yes—
246C2×C3×D3—332no—
247C2×C3×D3—38no—
248C2×C3×D3—316no—
249C2×C3×D3—38no—
250C2×C3×D3—616no—
251C2×C3×D3—64no—
252C2×C3×D3—64no—
253C2×C3×D3—68no—
254C2×C3×D3—664no—
255C2×C3×D3—664no—
256C2×C3×D3—68no—
257C2×C3×D3—616no—
258C2×C3×D3—616no—
259C2×C3×D3—68no—
260C3⋊Dic3—28no—
261C3×Dic3—68no—
262C3×Dic3—68no—
263C3×Dic3—128no—
264C25—91no—
265C23×C4—272no—
266C2×C22⋊C4—92no—
267C2×C22⋊C4—278no—
268C2×C22⋊C4—272no—
269C33C22×C22⋊C411yescharacteristic, complemented, Sylow
270C23×C3—28no—
271C23×C3—28no—
272C23×C3—24no—
273C23×C3—24no—
274C23×C3—28no—
275C23×C3—22no—
276C23×C3—24no—
277C23×C3—22no—
278C23×C3—21no—
279C23×C3—48no—
280C23×C3—48no—
281C23×C3—44no—
282C23×C3—42no—
283C23×C3—41no—
284C23×C3—92no—
285C23×C3—94no—
286C23×C3—98no—
287C23×C3—92no—
288C23×C3—94no—
289C23×C3—916no—
290C23×C3—1816no—
291C23×C3—1832no—
292C23×C3—184no—
293C23×C3D3×Dic318yescomplemented
294C23×C3#62414yescomplemented
295C23×C3C2×D3214yes—
296C23×C3C2×D3212yes—
297C23×C3C2×D3212yes—
298C23×C3C2×D3211yescharacteristic
299C22×D3—34no—
300C22×D3—38no—
301C22×D3—316no—
302C22×D3—34no—
303C22×D3—38no—
304C22×D3—332no—
305C22×D3—64no—
306C22×D3—64no—
307C22×D3—616no—
308C22×D3—68no—
309C22×D3—62no—
310C22×D3—62no—
311C22×D3—632no—
312C22×D3—664no—
313C22×D3—664no—
314C22×D3—632no—
315C22×D3—68no—
316C22×D3—68no—
317C2×C3×C4—98no—
318C2×C3×C4—188no—
319C2×Dic3—38no—
320C2×Dic3—38no—
321C2×Dic3—68no—
322C2×Dic3—68no—
323C2×Dic3—68no—
324C2×Dic3—128no—
325C2×Dic3—1816no—
326C4×D3—1832no—
327C2×C32—24no—
328C2×C32—28no—
329C2×C32—28no—
330C2×C32—22no—
331C2×C32—21no—
332C2×C32—24no—
333C2×C32—22no—
334C2×C32—28no—
335C2×C32—24no—
336C2×C32—44no—
337C2×C32—41no—
338C2×C32—42no—
339C2×C32—48no—
340C2×C32—48no—
341C2×C32D3×C22⋊C418yescomplemented
342C2×C32C2×#21714yescomplemented
343C2×C32C22×C4×D312yes—
344C2×C32C2×D3×D414yes—
345C2×C32C23×Dic311yescharacteristic
346C2×C32C22×#6712yes—
347C3⋊D3—24no—
348C3⋊D3C2×#21718yescomplemented
349C3×D3—316no—
350C3×D3—68no—
351C3×D3—68no—
352C3×D3—68no—
353C24—91no—
354C24—98no—
355C24—94no—
356C24—91no—
357C24—188no—
358C24#111311yescharacteristic
359C22×C4—94no—
360C22×C4—278no—
361C22×C4—548no—
362C22⋊C4—98no—
363C22⋊C4—278no—
364C22×C3—28no—
365C22×C3—216no—
366C22×C3—28no—
367C22×C3—24no—
368C22×C3—216no—
369C22×C3—28no—
370C22×C3—22no—
371C22×C3—21no—
372C22×C3—22no—
373C22×C3—24no—
374C22×C3—24no—
375C22×C3—28no—
376C22×C3—28no—
377C22×C3—24no—
378C22×C3—48no—
379C22×C3—416no—
380C22×C3—48no—
381C22×C3—416no—
382C22×C3—41no—
383C22×C3—42no—
384C22×C3—44no—
385C22×C3—44no—
386C22×C3—94no—
387C22×C3—98no—
388C22×C3—916no—
389C22×C3—94no—
390C22×C3—1832no—
391C22×C3—184no—
392C22×C3—188no—
393C22×C3C2×D3×Dic314yes—
394C22×C3C2×D3×Dic318yescomplemented
395C22×C3C2×#62412yes—
396C22×C3C2×#62414yescomplemented
397C22×C3D3×#6718yes—
398C22×C3#381114yes—
399C22×C3C22×D3212yes—
400C22×C3C22×D3211yescharacteristic
401C2×D3—38no—
402C2×D3—316no—
403C2×D3—332no—
404C2×D3—38no—
405C2×D3—616no—
406C2×D3—64no—
407C2×D3—64no—
408C2×D3—68no—
409C2×D3—664no—
410C2×D3—664no—
411C2×D3—68no—
412C2×D3—616no—
413C2×D3—68no—
414C2×D3—616no—
415C3×C4—188no—
416Dic3—68no—
417Dic3—68no—
418Dic3—128no—
419C32—22no—
420C32—21no—
421C32—41no—
422C32C2×D3×C22⋊C412yescomplemented
423C32C22×#21711yescharacteristic, complemented
424C23—24no—
425C23—92no—
426C23—94no—
427C23—98no—
428C23—92no—
429C23—94no—
430C23—916no—
431C23—1816no—
432C23—1832no—
433C23—184no—
434C23#629414yescomplemented
435C23C2×#111312yes—
436C23C2×#111311yescharacteristic
437C2×C4—98no—
438C2×C4—188no—
439C2×C4—5416no—
440C2×C3—28no—
441C2×C3—28no—
442C2×C3—24no—
443C2×C3—21no—
444C2×C3—22no—
445C2×C3—24no—
446C2×C3—22no—
447C2×C3—24no—
448C2×C3—28no—
449C2×C3—48no—
450C2×C3—48no—
451C2×C3—41no—
452C2×C3—42no—
453C2×C3—44no—
454C2×C3—98no—
455C2×C3—184no—
456C2×C3D3×#21718yescomplemented
457C2×C3#6397414yescomplemented
458C2×C3C22×D3×Dic312yes—
459C2×C3C22×#62411yescharacteristic
460C2×C3C2×D3×#6714yes—
461C2×C3C2×#381112yes—
462D3—316no—
463D3—68no—
464D3—68no—
465D3—68no—
466C22—24no—
467C22—28no—
468C22—94no—
469C22—98no—
470C22—916no—
471C22—94no—
472C22—1832no—
473C22—184no—
474C22—188no—
475C22C2×#629412yes—
476C22C2×#629414yescomplemented
477C22#8904914yes—
478C22C22×#111311yescharacteristic
479C4—188no—
480C3—21no—
481C3—22no—
482C3—41no—
483C3C2×D3×#21712yescomplemented
484C3C2×#6397411yescharacteristic, complemented
485C2—24no—
486C2—98no—
487C2—184no—
488C2#1172372314yesdirect product factor
489C2C22×#629411yescharacteristic
490C2C2×#8904912yes—
subgroupG/H
centerC23C2×#1113
commutatorC2×C33C23×C4
FrattiniC22C22×#1113
FittingC24×C33C22
socleC23×C33C23
prime radicalC24×C3×C3⋊D3C2
cosocleC22×C33C24
hypercenterC24#1113
hypocenterC33C22×C22⋊C4
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
2. C22×#6294C2semidirect
3. C2×C3×#3757C2semidirect
4. #11723723C2direct (N × H)
5. C2×#88814C2semidirect
10. C23×C3×C3⋊D3C4semidirect
14. C22×C3×C3⋊Dic3C22semidirect
16. C2×#6294C22semidirect
17. C3×#3757C22semidirect
18. #88814C22semidirect
24. C2×#3757D3semidirect
33. C23×C33C2×C4semidirect
40. C22×C3×C3⋊D3C2×C4semidirect
41. C22×C3×C3⋊D3C2×C4semidirect
46. C2×C3×C3⋊Dic3C23semidirect
56. C23×C3⋊D3Dic3semidirect
70. C22×C3⋊Dic3C2×D3semidirect
78. #3757C2×D3semidirect
87. C22×C33C22×C4semidirect
93. C2×C3×C3⋊D3C22⋊C4semidirect
95. C2×C3×C3⋊D3C22×C4semidirect
125. C23×C32C4×D3semidirect
126. C23×C32C2×Dic3semidirect
135. C22×C3⋊D3C2×Dic3semidirect
136. C22×C3⋊D3C2×Dic3semidirect
159. C2×C3⋊Dic3C22×D3semidirect
170. C2×C33C2×C22⋊C4semidirect
174. C3×C3⋊D3C2×C22⋊C4semidirect
232. C22×C32C2×C4×D3semidirect
235. C22×C32C22×Dic3semidirect
242. C2×C3⋊D3#217semidirect
244. C2×C3⋊D3C22×Dic3semidirect
269. C33C22×C22⋊C4semidirect
293. C23×C3D3×Dic3semidirect
294. C23×C3#624semidirect
341. C2×C32D3×C22⋊C4semidirect
342. C2×C32C2×#217semidirect
348. C3⋊D3C2×#217semidirect
394. C22×C3C2×D3×Dic3semidirect
396. C22×C3C2×#624semidirect
422. C32C2×D3×C22⋊C4semidirect
423. C32C22×#217semidirect
434. C23#6294semidirect
456. C2×C3D3×#217semidirect
457. C2×C3#63974semidirect
476. C22C2×#6294semidirect
483. C3C2×D3×#217semidirect
484. C3C2×#63974semidirect
488. C2#11723723direct (N × H)

Maximal subgroups

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

subgroupindex
C24×C3×C3⋊D32
C22×#62942
C2×C3×#37572
#117237232
C2×#888142
C2×#639743
C2×D3×#2173