Finite Group Databasebeta

← C2×#11723621 #13178041

Subgroups — C2×#11723621 #13178041

Order 1728.

2580 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#638C211yesmaximal, characteristic
2C22×Dic3×C3⋊D3C211yesmaximal, characteristic, complemented
3C2×#88864C211yesmaximal, characteristic, complemented
4C2×C32×#217C211yesmaximal, characteristic, complemented
5#11723621C218yesmaximal, direct product factor
6C2×#88806C211yesmaximal, characteristic, complemented
7C2×Dic3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
8C2×#88803C211yesmaximal, characteristic, complemented
9C2×#64092—31nomaximal
10C2×#63969—34nomaximal
11C24×C33C2211yescharacteristic
12C23×C3×C3⋊D3C2211yescharacteristic
13C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
14C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3×#638—24no—
16C2×C3×#638C2214yes—
17C22×C3×C3⋊Dic3C2211yescharacteristic
18C22×C32×Dic3C2211yescharacteristic
19C22×C32×Dic3C2211yescharacteristic
20C2×Dic3×C3⋊D3—24no—
21C2×Dic3×C3⋊D3C2214yescomplemented
22#88864C2214yescomplemented
23C32×#217C2214yescomplemented
24#88806C2214yescomplemented
25Dic3⋊(C3⋊Dic3)C2214yescomplemented
26#88803C2214yescomplemented
27C22×#638D311yescharacteristic
28C22×C4×C3⋊D3—31no—
29C22×C3×#67—34no—
30C22×D3×Dic3—34no—
31C2×C32×C22⋊C4—31no—
32C2×#3757—31no—
33C2×#3757—34no—
34C2×#3757—64no—
35C2×#3752—31no—
36C2×C4⋊(C3⋊Dic3)—31no—
37C2×#3750—31no—
38#64092—38no—
39C2×C3×#217D314yescomplemented
40#63969—332no—
41C2×#3724—34no—
42C2×Dic3⋊Dic3—34no—
43C2×(C2×D3)⋊Dic3—34no—
44C23×C33—22no—
45C23×C33—22no—
46C23×C33—22no—
47C23×C33C2312yes—
48C23×C33C2311yescharacteristic
49C22×C3×C3⋊D3—22no—
50C22×C3×C3⋊D3—22no—
51C22×C3×C3⋊D3D414yes—
52C22×C3×C3⋊D3C2312yes—
53C2×C3⋊(C3⋊Dic3)—22no—
54C2×C3⋊(C3⋊Dic3)—22no—
55C2×C3⋊(C3⋊Dic3)C2312yes—
56C2×C3⋊(C3⋊Dic3)C2312yes—
57C3×#638—28no—
58C2×C3×C3⋊Dic3—22no—
59C2×C3×C3⋊Dic3C2312yes—
60C2×C32×Dic3—22no—
61C2×C32×Dic3—22no—
62C2×C32×Dic3C2312yes—
63C2×C32×Dic3C2312yes—
64Dic3×C3⋊D3—28no—
65C2×#935—91no—
66C2×#883—94no—
67C24×C32—24no—
68C24×C32C2×D314yes—
69C24×C32C2×D311yescharacteristic
70C23×C3⋊D3C2×D311yescharacteristic
71C23×C3×D3—34no—
72C22×C32×C4—31no—
73C22×C32×C4—31no—
74C2×#638—24no—
75C2×#638C2×D314yes—
76C22×C3⋊Dic3—31no—
77C22×C3⋊Dic3—31no—
78C22×C3⋊Dic3—34no—
79C22×C3⋊Dic3—34no—
80C22×C3⋊Dic3—64no—
81C22×C3⋊Dic3—64no—
82C22×C3⋊Dic3C2×D311yescharacteristic
83C2×C4×C3⋊D3—34no—
84C2×C4×C3⋊D3—64no—
85C2×C3×#67—316no—
86C2×C3×#67—616no—
87C22×C3×Dic3—34no—
88C22×C3×Dic3C2×D314yes—
89C22×C3×Dic3C2×D314yes—
90C2×D3×Dic3—316no—
91C2×D3×Dic3—616no—
92C32×C22⋊C4—34no—
93#3757—34no—
94#3757—316no—
95#3757—616no—
96#3752—34no—
97C4⋊(C3⋊Dic3)—34no—
98#3750—34no—
99C3×#217C2×D3116yescomplemented
100#3724—316no—
101Dic3⋊Dic3—316no—
102(C2×D3)⋊Dic3—316no—
103C22×C33—22no—
104C22×C33—22no—
105C22×C33—22no—
106C22×C33—44no—
107C22×C33C2×D412yes—
108C22×C33#4112yes—
109C22×C33#4112yes—
110C22×C33C2411yescharacteristic
111C2×C3×C3⋊D3—28no—
112C2×C3×C3⋊D3—22no—
113C2×C3×C3⋊D3—22no—
114C2×C3×C3⋊D3C2×D414yes—
115C3⋊(C3⋊Dic3)—22no—
116C3⋊(C3⋊Dic3)—22no—
117C3×C3⋊Dic3—22no—
118C32×Dic3—22no—
119C32×Dic3—22no—
120C22×C3×D4—91no—
121C22×#67—34no—
122C23×Dic3—91no—
123C22×C4×D3—94no—
124C2×C3×C22⋊C4—34no—
125C2×#217—94no—
126C2×#217—91no—
127C2×#217—91no—
128C2×#217—184no—
129C2×#217C3⋊D311yescharacteristic, complemented
130#935—98no—
131C2×(C2×D3)⋊C4—94no—
132C2×C4⋊Dic3—94no—
133C2×Dic3⋊C4—94no—
134C2×Dic3⋊C4—91no—
135C2×Dic3⋊C4—91no—
136#883—932no—
137C23×C32—216no—
138C23×C32—28no—
139C23×C32—22no—
140C23×C32—22no—
141C23×C32—22no—
142C23×C32—24no—
143C23×C32—28no—
144C23×C32—28no—
145C23×C32—28no—
146C23×C32—48no—
147C23×C32—48no—
148C23×C32C22×D318yes—
149C23×C32C22×D314yes—
150C23×C32C22×D312yes—
151C23×C32C22×D311yescharacteristic
152C22×C3⋊D3—22no—
153C22×C3⋊D3—22no—
154C22×C3⋊D3#6714yes—
155C22×C3⋊D3C22×D312yes—
156C22×C3×D3—316no—
157C22×C3×D3—38no—
158C22×C3×D3—68no—
159C22×C3×D3—68no—
160C2×C32×C4—32no—
161C2×C32×C4—32no—
162C2×C32×C4—62no—
163C2×C32×C4—62no—
164#638—28no—
165C2×C3⋊Dic3—22no—
166C2×C3⋊Dic3—32no—
167C2×C3⋊Dic3—32no—
168C2×C3⋊Dic3—38no—
169C2×C3⋊Dic3—38no—
170C2×C3⋊Dic3—68no—
171C2×C3⋊Dic3—68no—
172C2×C3⋊Dic3—62no—
173C2×C3⋊Dic3—62no—
174C2×C3⋊Dic3—68no—
175C2×C3⋊Dic3—68no—
176C2×C3⋊Dic3—128no—
177C2×C3⋊Dic3—128no—
178C2×C3⋊Dic3C22×D312yes—
179C4×C3⋊D3—68no—
180C3×#67—632no—
181C2×C3×Dic3—28no—
182C2×C3×Dic3—28no—
183C2×C3×Dic3—38no—
184C2×C3×Dic3—68no—
185C2×C3×Dic3C22×D318yes—
186C2×C3×Dic3C22×D318yes—
187D3×Dic3—632no—
188C2×#123—271noSylow
189C2×C33—42no—
190C2×C33#12314yescomplemented
191C2×C33C22×D411yescharacteristic
192C2×C33C2×#4111yescharacteristic
193C2×C33C2×#4111yescharacteristic
194C3×C3⋊D3—24no—
195C24×C3—24no—
196C24×C3—91no—
197C24×C3D3214yes—
198C24×C3C2×C3⋊D311yescharacteristic
199C23×D3—34no—
200C2×C3×D4—94no—
201C2×C3×D4—184no—
202C22×C3×C4—34no—
203C22×C3×C4—34no—
204C22×C3×C4—91no—
205C2×#67—316no—
206C2×#67—616no—
207C22×Dic3—34no—
208C22×Dic3—94no—
209C22×Dic3—94no—
210C22×Dic3—91no—
211C22×Dic3—91no—
212C22×Dic3—94no—
213C22×Dic3—184no—
214C22×Dic3—184no—
215C22×Dic3—184no—
216C22×Dic3C2×C3⋊D311yescharacteristic
217C22×Dic3C2×C3⋊D311yescharacteristic
218C2×C4×D3—916no—
219C2×C4×D3—1816no—
220C3×C22⋊C4—316no—
221#217—916no—
222#217—94no—
223#217—94no—
224#217—1816no—
225#217C2×C3⋊D314yescomplemented
226(C2×D3)⋊C4—916no—
227C4⋊Dic3—916no—
228Dic3⋊C4—916no—
229Dic3⋊C4—94no—
230Dic3⋊C4—94no—
231C22×C32—216no—
232C22×C32—28no—
233C22×C32—22no—
234C22×C32—22no—
235C22×C32—22no—
236C22×C32—24no—
237C22×C32—28no—
238C22×C32—28no—
239C22×C32—28no—
240C22×C32—28no—
241C22×C32—28no—
242C22×C32—48no—
243C22×C32—48no—
244C22×C32—432no—
245C22×C32—416no—
246C22×C32—44no—
247C22×C32#23518yes—
248C22×C32D3×D418yes—
249C22×C32#23718yes—
250C22×C32#23712yes—
251C22×C32#23712yes—
252C22×C32C2×#6712yes—
253C22×C32C23×D314yes—
254C22×C32C23×D311yescharacteristic
255C2×C3⋊D3—28no—
256C2×C3⋊D3—22no—
257C2×C3⋊D3—22no—
258C2×C3⋊D3C2×#6714yes—
259C2×C3×D3—316no—
260C2×C3×D3—632no—
261C2×C3×D3—68no—
262C2×C3×D3—68no—
263C32×C4—62no—
264C32×C4—62no—
265C3⋊Dic3—22no—
266C3⋊Dic3—62no—
267C3⋊Dic3—62no—
268C3⋊Dic3—68no—
269C3⋊Dic3—68no—
270C3⋊Dic3—128no—
271C3⋊Dic3—128no—
272C3×Dic3—28no—
273C3×Dic3—28no—
274C3×Dic3—68no—
275C22×D4—91no—
276C23×C4—271no—
277#123—278no—
278C2×C4⋊C4—271no—
279C2×C4⋊C4—271no—
280C2×C22⋊C4—31no—
281C2×C22⋊C4—271no—
282C2×C22⋊C4—271no—
283C33C2×#12311yescharacteristic, complemented, Sylow
284C23×C3—216no—
285C23×C3—28no—
286C23×C3—28no—
287C23×C3—28no—
288C23×C3—22no—
289C23×C3—22no—
290C23×C3—22no—
291C23×C3—28no—
292C23×C3—24no—
293C23×C3—48no—
294C23×C3—48no—
295C23×C3—92no—
296C23×C3—94no—
297C23×C3—182no—
298C23×C3—182no—
299C23×C3C2×D3218yes—
300C23×C3C2×D3214yes—
301C23×C3C22×C3⋊D312yes—
302C23×C3C22×C3⋊D311yescharacteristic
303C22×D3—38no—
304C22×D3—316no—
305C22×D3—68no—
306C22×D3—68no—
307C3×D4—188no—
308C2×C3×C4—38no—
309C2×C3×C4—38no—
310C2×C3×C4—68no—
311C2×C3×C4—68no—
312C2×C3×C4—92no—
313C2×C3×C4—182no—
314#67—632no—
315C2×Dic3—22no—
316C2×Dic3—22no—
317C2×Dic3—38no—
318C2×Dic3—68no—
319C2×Dic3—98no—
320C2×Dic3—98no—
321C2×Dic3—92no—
322C2×Dic3—92no—
323C2×Dic3—188no—
324C2×Dic3—188no—
325C2×Dic3—188no—
326C2×Dic3—188no—
327C2×Dic3—188no—
328C2×Dic3—182no—
329C2×Dic3—182no—
330C2×Dic3—368no—
331C2×Dic3—368no—
332C2×Dic3C22×C3⋊D312yes—
333C2×Dic3C22×C3⋊D312yes—
334C4×D3—1832no—
335C2×C32—24no—
336C2×C32—24no—
337C2×C32—24no—
338C2×C32—216no—
339C2×C32—416no—
340C2×C32—48no—
341C2×C32—42no—
342C2×C32#883116yescomplemented
343C2×C32#93514yescomplemented
344C2×C32C2×#23514yes—
345C2×C32C2×D3×D414yes—
346C2×C32C2×#23714yes—
347C2×C32C2×#23711yescharacteristic
348C2×C32C2×#23711yescharacteristic
349C2×C32C22×#6711yescharacteristic
350C3⋊D3—24no—
351C3×D3—616no—
352C24—91no—
353C24D3×C3⋊D311yescharacteristic
354C2×D4—94no—
355C2×D4—184no—
356C22×C4—31no—
357C22×C4—31no—
358C22×C4—91no—
359C22×C4—274no—
360C22×C4—271no—
361C22×C4—271no—
362C22×C4—544no—
363C4⋊C4—274no—
364C4⋊C4—274no—
365C22⋊C4—34no—
366C22⋊C4—274no—
367C22⋊C4—274no—
368C22×C3—216no—
369C22×C3—28no—
370C22×C3—28no—
371C22×C3—28no—
372C22×C3—22no—
373C22×C3—22no—
374C22×C3—22no—
375C22×C3—24no—
376C22×C3—28no—
377C22×C3—28no—
378C22×C3—28no—
379C22×C3—48no—
380C22×C3—48no—
381C22×C3—432no—
382C22×C3—44no—
383C22×C3—416no—
384C22×C3—94no—
385C22×C3—188no—
386C22×C3—182no—
387C22×C3—182no—
388C22×C3#380418yes—
389C22×C3#380518yes—
390C22×C3D3×#6718yes—
391C22×C3#382812yes—
392C22×C3D4×C3⋊D312yes—
393C22×C3#383012yes—
394C22×C3C22×D3214yes—
395C22×C3C23×C3⋊D311yescharacteristic
396C2×D3—316no—
397C2×D3—68no—
398C2×D3—68no—
399C2×D3—632no—
400C3×C4—68no—
401C3×C4—68no—
402C3×C4—182no—
403Dic3—22no—
404Dic3—22no—
405Dic3—68no—
406Dic3—188no—
407Dic3—188no—
408Dic3—182no—
409Dic3—182no—
410Dic3—368no—
411Dic3—368no—
412C32—24no—
413C32C2×#88314yescomplemented
414C32C2×#93511yescharacteristic, complemented
415C23—22no—
416C23—22no—
417C23—22no—
418C23—92no—
419C23—94no—
420C23—182no—
421C23—182no—
422C23C2×D3×C3⋊D312yes—
423C23C2×D3×C3⋊D311yescharacteristic
424D4—188no—
425C2×C4—32no—
426C2×C4—32no—
427C2×C4—62no—
428C2×C4—62no—
429C2×C4—92no—
430C2×C4—182no—
431C2×C4—272no—
432C2×C4—272no—
433C2×C4—548no—
434C2×C4—542no—
435C2×C4—542no—
436C2×C3—24no—
437C2×C3—24no—
438C2×C3—24no—
439C2×C3—216no—
440C2×C3—416no—
441C2×C3—42no—
442C2×C3—48no—
443C2×C3—184no—
444C2×C3#63969116yescomplemented
445C2×C3#6409214yescomplemented
446C2×C3C2×#380414yes—
447C2×C3C2×#380514yes—
448C2×C3C2×D3×#6714yes—
449C2×C3C2×#382811yescharacteristic
450C2×C3C2×D4×C3⋊D311yescharacteristic
451C2×C3C2×#383011yescharacteristic
452D3—616no—
453C22—22no—
454C22—22no—
455C22—22no—
456C22—44no—
457C22—94no—
458C22—182no—
459C22—182no—
460C22—188no—
461C22#8902912yes—
462C22#8903112yes—
463C22C3⋊D3×#6712yes—
464C22C22×D3×C3⋊D311yescharacteristic
465C4—62no—
466C4—62no—
467C4—182no—
468C4—542no—
469C4—542no—
470C3—24no—
471C3C2×#6396914yescomplemented
472C3C2×#6409211yescharacteristic, complemented
473C2—42no—
474C2—184no—
475C2#1172362114yesdirect product factor
476C2C2×#8902911yescharacteristic
477C2C2×#8903111yescharacteristic
478C2C2×C3⋊D3×#6711yescharacteristic
subgroupG/H
centerC23C2×D3×C3⋊D3
commutator, cosocleC22×C33C24
FrattiniC22C22×D3×C3⋊D3
FittingC24×C33C22
socleC23×C33C23
prime radicalC22×C3×#638C2
hypercenterC24D3×C3⋊D3
hypocenterC33C2×#123
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
2. C22×Dic3×C3⋊D3C2semidirect
3. C2×#88864C2semidirect
4. C2×C32×#217C2semidirect
5. #11723621C2direct (N × H)
6. C2×#88806C2semidirect
7. C2×Dic3⋊(C3⋊Dic3)C2semidirect
8. C2×#88803C2semidirect
21. C2×Dic3×C3⋊D3C22semidirect
22. #88864C22semidirect
23. C32×#217C22semidirect
24. #88806C22semidirect
25. Dic3⋊(C3⋊Dic3)C22semidirect
26. #88803C22semidirect
39. C2×C3×#217D3semidirect
99. C3×#217C2×D3semidirect
129. C2×#217C3⋊D3semidirect
190. C2×C33#123semidirect
225. #217C2×C3⋊D3semidirect
283. C33C2×#123semidirect
342. C2×C32#883semidirect
343. C2×C32#935semidirect
413. C32C2×#883semidirect
414. C32C2×#935semidirect
444. C2×C3#63969semidirect
445. C2×C3#64092semidirect
471. C3C2×#63969semidirect
472. C3C2×#64092semidirect
475. C2#11723621direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#6382
C22×Dic3×C3⋊D32
C2×#888642
C2×C32×#2172
#117236212
C2×#888062
C2×Dic3⋊(C3⋊Dic3)2
C2×#888032
C2×#640923
C2×#639693