Finite Group Databasebeta

← C2×#11723628 #13178066

Subgroups — C2×#11723628 #13178066

Order 1728.

2996 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6312C211yesmaximal, characteristic
2C22×#6289C211yesmaximal, characteristic, complemented
3C2×C3×#3757C211yesmaximal, characteristic, complemented
4C2×C32×#217C211yesmaximal, characteristic, complemented
5#11723628C218yesmaximal, direct product factor
6C2×#88807C211yesmaximal, characteristic, complemented
7C2×Dic3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
8C2×#88804C211yesmaximal, characteristic, complemented
9C2×#64092—31nomaximal
10C2×#63975—34nomaximal
11C24×C33C2211yescharacteristic
12C23×C3⋊(C3⋊D3)C2211yescharacteristic
13C2×#6312—24no—
14C2×#6312C2214yes—
15C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C22×C3×C3⋊Dic3C2211yescharacteristic
17C22×C3×C3⋊Dic3C2211yescharacteristic
18C22×C32×Dic3C2211yescharacteristic
19C22×C32×Dic3C2211yescharacteristic
20C2×#6289—24no—
21C2×#6289C2214yescomplemented
22C3×#3757C2214yescomplemented
23C32×#217C2214yescomplemented
24#88807C2214yescomplemented
25Dic3⋊(C3⋊Dic3)C2214yescomplemented
26#88804C2214yescomplemented
27C22×#638—31no—
28C22×#638—34no—
29C22×#638—64no—
30C22×C4×C3⋊D3—31no—
31C22×#624—34no—
32C2×C32×C22⋊C4—31no—
33C2×#3757D311yescharacteristic, complemented
34C2×#3752—31no—
35C2×C4⋊(C3⋊Dic3)—31no—
36C2×#3750—31no—
37#64092—38no—
38C2×C3×#217—34no—
39C2×C3×#217D314yescomplemented
40#63975—332no—
41C2×Dic3⋊Dic3—34no—
42C2×Dic3⋊Dic3—34no—
43C2×#3722—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⋊D3)D414yes—
52C22×C3⋊(C3⋊D3)C2312yes—
53#6312—28no—
54C2×C3⋊(C3⋊Dic3)—22no—
55C2×C3⋊(C3⋊Dic3)C2312yes—
56C2×C3×C3⋊Dic3—22no—
57C2×C3×C3⋊Dic3—22no—
58C2×C3×C3⋊Dic3C2312yes—
59C2×C3×C3⋊Dic3C2312yes—
60C2×C32×Dic3—22no—
61C2×C32×Dic3—22no—
62C2×C32×Dic3C2312yes—
63C2×C32×Dic3C2312yes—
64#6289—28no—
65C2×#883—94no—
66C2×#883—91no—
67C24×C32—24no—
68C24×C32C2×D314yes—
69C24×C32C2×D311yescharacteristic
70C23×C3⋊D3—31no—
71C23×C3⋊D3—34no—
72C23×C3⋊D3—64no—
73C22×C32×C4—31no—
74C22×C32×C4—31no—
75C2×#638—34no—
76C2×#638—316no—
77C2×#638—64no—
78C2×#638—616no—
79C2×#638—616no—
80C2×#638—632no—
81C22×C3⋊Dic3—31no—
82C22×C3⋊Dic3—34no—
83C22×C3⋊Dic3—64no—
84C22×C3⋊Dic3C2×D311yescharacteristic
85C22×C3⋊Dic3C2×D311yescharacteristic
86C2×C4×C3⋊D3—34no—
87C2×C4×C3⋊D3—64no—
88C22×C3×Dic3—34no—
89C22×C3×Dic3—34no—
90C22×C3×Dic3C2×D314yes—
91C22×C3×Dic3C2×D314yes—
92C2×#624—316no—
93C2×#624—616no—
94C32×C22⋊C4—34no—
95#3757C2×D314yescomplemented
96#3752—34no—
97C4⋊(C3⋊Dic3)—34no—
98#3750—34no—
99C3×#217—316no—
100C3×#217C2×D3116yescomplemented
101Dic3⋊Dic3—316no—
102Dic3⋊Dic3—316no—
103#3722—316no—
104C22×C33—22no—
105C22×C33—22no—
106C22×C33—22no—
107C22×C33—44no—
108C22×C33C2×D412yes—
109C22×C33#4112yes—
110C22×C33#4112yes—
111C22×C33C2411yescharacteristic
112C2×C3⋊(C3⋊D3)—28no—
113C2×C3⋊(C3⋊D3)—22no—
114C2×C3⋊(C3⋊D3)—22no—
115C2×C3⋊(C3⋊D3)C2×D414yes—
116C3⋊(C3⋊Dic3)—22no—
117C3×C3⋊Dic3—22no—
118C3×C3⋊Dic3—22no—
119C32×Dic3—22no—
120C32×Dic3—22no—
121C22×#67—94no—
122C22×#67—91no—
123C22×#67—184no—
124C22×C4×D3—94no—
125C22×C4×D3—91no—
126C2×C3×C22⋊C4—34no—
127C2×C3×C22⋊C4—91no—
128C2×#217—34no—
129C2×#217C3⋊D311yescharacteristic, complemented
130C2×(C2×D3)⋊C4—94no—
131C2×(C2×D3)⋊C4—91no—
132C2×C4⋊Dic3—94no—
133C2×C4⋊Dic3—91no—
134C2×Dic3⋊C4—94no—
135C2×Dic3⋊C4—91no—
136#883—932no—
137#883—98no—
138C23×C32—216no—
139C23×C32—22no—
140C23×C32—28no—
141C23×C32—22no—
142C23×C32—22no—
143C23×C32—24no—
144C23×C32—28no—
145C23×C32—28no—
146C23×C32—28no—
147C23×C32—48no—
148C23×C32—48no—
149C23×C32C22×D318yes—
150C23×C32C22×D314yes—
151C23×C32C22×D312yes—
152C23×C32C22×D311yescharacteristic
153C22×C3⋊D3—34no—
154C22×C3⋊D3—32no—
155C22×C3⋊D3—316no—
156C22×C3⋊D3—38no—
157C22×C3⋊D3—616no—
158C22×C3⋊D3—68no—
159C22×C3⋊D3—62no—
160C22×C3⋊D3—62no—
161C22×C3⋊D3—68no—
162C22×C3⋊D3—68no—
163C22×C3⋊D3—128no—
164C22×C3⋊D3—128no—
165C2×C32×C4—32no—
166C2×C32×C4—32no—
167C2×C32×C4—62no—
168C2×C32×C4—62no—
169#638—68no—
170#638—632no—
171#638—664no—
172C2×C3⋊Dic3—22no—
173C2×C3⋊Dic3—22no—
174C2×C3⋊Dic3—32no—
175C2×C3⋊Dic3—38no—
176C2×C3⋊Dic3—68no—
177C2×C3⋊Dic3—616no—
178C2×C3⋊Dic3—68no—
179C2×C3⋊Dic3—62no—
180C2×C3⋊Dic3C22×D312yes—
181C2×C3⋊Dic3C22×D312yes—
182C4×C3⋊D3—68no—
183C2×C3×Dic3—28no—
184C2×C3×Dic3—28no—
185C2×C3×Dic3—38no—
186C2×C3×Dic3—38no—
187C2×C3×Dic3—68no—
188C2×C3×Dic3—68no—
189C2×C3×Dic3C22×D318yes—
190C2×C3×Dic3C22×D318yes—
191#624—632no—
192C2×#123—271noSylow
193C2×C33—42no—
194C2×C33#12314yescomplemented
195C2×C33C22×D411yescharacteristic
196C2×C33C2×#4111yescharacteristic
197C2×C33C2×#4111yescharacteristic
198C3⋊(C3⋊D3)—24no—
199C24×C3—24no—
200C24×C3D3214yes—
201C24×C3C2×C3⋊D311yescharacteristic
202C23×D3—94no—
203C23×D3—91no—
204C23×D3—184no—
205C22×C3×C4—34no—
206C22×C3×C4—34no—
207C22×C3×C4—91no—
208C22×C3×C4—91no—
209C2×#67—916no—
210C2×#67—94no—
211C2×#67—1816no—
212C2×#67—1816no—
213C2×#67—184no—
214C2×#67—1832no—
215C22×Dic3—34no—
216C22×Dic3—34no—
217C22×Dic3—94no—
218C22×Dic3—91no—
219C22×Dic3—184no—
220C22×Dic3C2×C3⋊D311yescharacteristic
221C22×Dic3C2×C3⋊D311yescharacteristic
222C2×C4×D3—916no—
223C2×C4×D3—94no—
224C2×C4×D3—184no—
225C2×C4×D3—1816no—
226C3×C22⋊C4—316no—
227C3×C22⋊C4—94no—
228#217—316no—
229#217C2×C3⋊D314yescomplemented
230(C2×D3)⋊C4—916no—
231(C2×D3)⋊C4—94no—
232C4⋊Dic3—916no—
233C4⋊Dic3—94no—
234Dic3⋊C4—916no—
235Dic3⋊C4—94no—
236C22×C32—216no—
237C22×C32—22no—
238C22×C32—28no—
239C22×C32—22no—
240C22×C32—22no—
241C22×C32—24no—
242C22×C32—28no—
243C22×C32—28no—
244C22×C32—28no—
245C22×C32—28no—
246C22×C32—28no—
247C22×C32—48no—
248C22×C32—48no—
249C22×C32—432no—
250C22×C32—416no—
251C22×C32—44no—
252C22×C32#23518yes—
253C22×C32#23512yes—
254C22×C32D3×D418yes—
255C22×C32D3×D412yes—
256C22×C32#23718yes—
257C22×C32#23712yes—
258C22×C32C23×D314yes—
259C22×C32C23×D311yescharacteristic
260C2×C3⋊D3—34no—
261C2×C3⋊D3—316no—
262C2×C3⋊D3—616no—
263C2×C3⋊D3—68no—
264C2×C3⋊D3—62no—
265C2×C3⋊D3—62no—
266C2×C3⋊D3—632no—
267C2×C3⋊D3—68no—
268C2×C3⋊D3—68no—
269C2×C3⋊D3—128no—
270C2×C3⋊D3—128no—
271C2×C3⋊D3—1232no—
272C32×C4—62no—
273C32×C4—62no—
274C3⋊Dic3—22no—
275C3⋊Dic3—22no—
276C3⋊Dic3—616no—
277C3⋊Dic3—68no—
278C3⋊Dic3—62no—
279C3×Dic3—28no—
280C3×Dic3—28no—
281C3×Dic3—68no—
282C3×Dic3—68no—
283C22×D4—271no—
284C23×C4—271no—
285#123—278no—
286C2×C4⋊C4—271no—
287C2×C4⋊C4—271no—
288C2×C22⋊C4—31no—
289C2×C22⋊C4—91no—
290C2×C22⋊C4—271no—
291C33C2×#12311yescharacteristic, complemented, Sylow
292C23×C3—216no—
293C23×C3—28no—
294C23×C3—28no—
295C23×C3—28no—
296C23×C3—22no—
297C23×C3—22no—
298C23×C3—22no—
299C23×C3—28no—
300C23×C3—24no—
301C23×C3—48no—
302C23×C3—48no—
303C23×C3C2×D3218yes—
304C23×C3C2×D3214yes—
305C23×C3C22×C3⋊D312yes—
306C23×C3C22×C3⋊D311yescharacteristic
307C22×D3—98no—
308C22×D3—92no—
309C22×D3—916no—
310C22×D3—94no—
311C22×D3—188no—
312C22×D3—1816no—
313C22×D3—188no—
314C22×D3—188no—
315C22×D3—182no—
316C22×D3—182no—
317C22×D3—368no—
318C22×D3—368no—
319C2×C3×C4—38no—
320C2×C3×C4—38no—
321C2×C3×C4—68no—
322C2×C3×C4—68no—
323C2×C3×C4—92no—
324C2×C3×C4—92no—
325C2×C3×C4—182no—
326C2×C3×C4—182no—
327#67—1832no—
328#67—188no—
329#67—1864no—
330C2×Dic3—22no—
331C2×Dic3—22no—
332C2×Dic3—38no—
333C2×Dic3—38no—
334C2×Dic3—68no—
335C2×Dic3—68no—
336C2×Dic3—98no—
337C2×Dic3—92no—
338C2×Dic3—188no—
339C2×Dic3—1816no—
340C2×Dic3—188no—
341C2×Dic3—182no—
342C2×Dic3C22×C3⋊D312yes—
343C2×Dic3C22×C3⋊D312yes—
344C4×D3—188no—
345C4×D3—1832no—
346C2×C32—24no—
347C2×C32—24no—
348C2×C32—24no—
349C2×C32—216no—
350C2×C32—416no—
351C2×C32—48no—
352C2×C32—42no—
353C2×C32#883116yescomplemented
354C2×C32#88314yescomplemented
355C2×C32C2×#23514yes—
356C2×C32C2×#23511yescharacteristic
357C2×C32C2×D3×D414yes—
358C2×C32C2×D3×D411yescharacteristic
359C2×C32C2×#23714yes—
360C2×C32C2×#23711yescharacteristic
361C3⋊D3—64no—
362C3⋊D3—616no—
363C3⋊D3—1216no—
364C24—271no—
365C24D3×C3⋊D311yescharacteristic
366C2×D4—274no—
367C2×D4—544no—
368C22×C4—31no—
369C22×C4—31no—
370C22×C4—91no—
371C22×C4—91no—
372C22×C4—274no—
373C22×C4—271no—
374C22×C4—544no—
375C4⋊C4—274no—
376C4⋊C4—274no—
377C22⋊C4—34no—
378C22⋊C4—94no—
379C22⋊C4—274no—
380C22×C3—216no—
381C22×C3—28no—
382C22×C3—28no—
383C22×C3—28no—
384C22×C3—22no—
385C22×C3—22no—
386C22×C3—22no—
387C22×C3—24no—
388C22×C3—28no—
389C22×C3—28no—
390C22×C3—28no—
391C22×C3—48no—
392C22×C3—48no—
393C22×C3—432no—
394C22×C3—44no—
395C22×C3—416no—
396C22×C3#380418yes—
397C22×C3#380418yes—
398C22×C3#381118yes—
399C22×C3#382812yes—
400C22×C3D4×C3⋊D312yes—
401C22×C3#383012yes—
402C22×C3C22×D3214yes—
403C22×C3C23×C3⋊D311yescharacteristic
404C2×D3—916no—
405C2×D3—94no—
406C2×D3—1816no—
407C2×D3—188no—
408C2×D3—188no—
409C2×D3—188no—
410C2×D3—182no—
411C2×D3—182no—
412C2×D3—1832no—
413C2×D3—368no—
414C2×D3—368no—
415C2×D3—3632no—
416C3×C4—68no—
417C3×C4—68no—
418C3×C4—182no—
419C3×C4—182no—
420Dic3—22no—
421Dic3—22no—
422Dic3—68no—
423Dic3—68no—
424Dic3—1816no—
425Dic3—188no—
426Dic3—182no—
427C32—24no—
428C32C2×#88314yescomplemented
429C32C2×#88311yescharacteristic, complemented
430C23—22no—
431C23—22no—
432C23—22no—
433C23—272no—
434C23—274no—
435C23—542no—
436C23—542no—
437C23C2×D3×C3⋊D312yes—
438C23C2×D3×C3⋊D311yescharacteristic
439D4—548no—
440C2×C4—32no—
441C2×C4—32no—
442C2×C4—62no—
443C2×C4—62no—
444C2×C4—92no—
445C2×C4—92no—
446C2×C4—182no—
447C2×C4—182no—
448C2×C4—272no—
449C2×C4—548no—
450C2×C4—542no—
451C2×C3—24no—
452C2×C3—24no—
453C2×C3—24no—
454C2×C3—216no—
455C2×C3—416no—
456C2×C3—42no—
457C2×C3—48no—
458C2×C3#63975116yescomplemented
459C2×C3#6409214yescomplemented
460C2×C3C2×#380414yes—
461C2×C3C2×#380414yes—
462C2×C3C2×#381114yes—
463C2×C3C2×#382811yescharacteristic
464C2×C3C2×D4×C3⋊D311yescharacteristic
465C2×C3C2×#383011yescharacteristic
466D3—184no—
467D3—1816no—
468D3—3616no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—44no—
473C22—274no—
474C22—542no—
475C22—542no—
476C22—548no—
477C22#8902812yes—
478C22#8903112yes—
479C22#8903912yes—
480C22C22×D3×C3⋊D311yescharacteristic
481C4—62no—
482C4—62no—
483C4—182no—
484C4—182no—
485C4—542no—
486C3—24no—
487C3C2×#6397514yescomplemented
488C3C2×#6409211yescharacteristic, complemented
489C2—42no—
490C2—544no—
491C2#1172362814yesdirect product factor
492C2C2×#8902811yescharacteristic
493C2C2×#8903111yescharacteristic
494C2C2×#8903911yescharacteristic
subgroupG/H
centerC23C2×D3×C3⋊D3
commutator, cosocleC22×C33C24
FrattiniC22C22×D3×C3⋊D3
FittingC24×C33C22
socleC23×C33C23
prime radicalC22×#6312C2
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×#6289C2semidirect
3. C2×C3×#3757C2semidirect
4. C2×C32×#217C2semidirect
5. #11723628C2direct (N × H)
6. C2×#88807C2semidirect
7. C2×Dic3⋊(C3⋊Dic3)C2semidirect
8. C2×#88804C2semidirect
21. C2×#6289C22semidirect
22. C3×#3757C22semidirect
23. C32×#217C22semidirect
24. #88807C22semidirect
25. Dic3⋊(C3⋊Dic3)C22semidirect
26. #88804C22semidirect
33. C2×#3757D3semidirect
39. C2×C3×#217D3semidirect
95. #3757C2×D3semidirect
100. C3×#217C2×D3semidirect
129. C2×#217C3⋊D3semidirect
194. C2×C33#123semidirect
229. #217C2×C3⋊D3semidirect
291. C33C2×#123semidirect
353. C2×C32#883semidirect
354. C2×C32#883semidirect
428. C32C2×#883semidirect
429. C32C2×#883semidirect
458. C2×C3#63975semidirect
459. C2×C3#64092semidirect
487. C3C2×#63975semidirect
488. C3C2×#64092semidirect
491. C2#11723628direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#63122
C22×#62892
C2×C3×#37572
C2×C32×#2172
#117236282
C2×#888072
C2×Dic3⋊(C3⋊Dic3)2
C2×#888042
C2×#640923
C2×#639753