Finite Group Databasebeta

← C2×#11723504 #13177194

Subgroups — C2×#11723504 #13177194

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×#6292C211yesmaximal, characteristic
2C22×#6289C211yesmaximal, characteristic, complemented
3C2×#88859C211yesmaximal, characteristic, complemented
4C2×C3×#3752C211yesmaximal, characteristic, complemented
5C2×C32×C4⋊Dic3C211yesmaximal, characteristic, complemented
6C2×#88806C211yesmaximal, characteristic, complemented
7C2×#88803C211yesmaximal, characteristic, complemented
8#11723504C218yesmaximal, direct product factor
9C2×#64103—31nomaximal
10C2×#63854—34nomaximal
11C23×C3⋊(C3⋊D3)C2211yescharacteristic
12C23×C3×C3⋊D3C2211yescharacteristic
13C22×C33×C4C2211yescharacteristic
14C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
15C22×C3×C3⋊Dic3C2211yescharacteristic
16C22×C32×Dic3C2211yescharacteristic
17C22×C32×Dic3C2211yescharacteristic
18C2×#6292—24no—
19C2×#6292C2214yes—
20C2×#6289—24no—
21C2×#6289C2214yescomplemented
22#88859C2214yescomplemented
23C3×#3752C2214yescomplemented
24C32×C4⋊Dic3C2214yescomplemented
25#88806C2214yescomplemented
26#88803C2214yescomplemented
27C22×C4⋊(C3⋊D3)—31no—
28C22×C4×C3⋊D3—31no—
29C22×#626—34no—
30C22×#624—34no—
31C2×C32×C4⋊C4—31no—
32C2×#3752—31no—
33C2×#3752—34no—
34C2×#3752—31no—
35C2×#3752—64no—
36C2×#3752D311yescharacteristic, complemented
37C2×#3750—31no—
38#64103—38no—
39C2×C3×(C2×D3)⋊C4—34no—
40C2×C3×C4⋊Dic3D314yescomplemented
41C2×#3724—34no—
42C2×(C2×D3)⋊Dic3—34no—
43#63854—332no—
44C23×C33C2311yescharacteristic
45C22×C3⋊(C3⋊D3)—22no—
46C22×C3⋊(C3⋊D3)—22no—
47C22×C3⋊(C3⋊D3)D414yes—
48C22×C3⋊(C3⋊D3)C2312yes—
49C22×C3×C3⋊D3—22no—
50C22×C3×C3⋊D3—22no—
51C22×C3×C3⋊D3—22no—
52C22×C3×C3⋊D3C2312yes—
53C2×C33×C4—22no—
54C2×C33×C4C2312yes—
55C2×C3⋊(C3⋊Dic3)—22no—
56C2×C3⋊(C3⋊Dic3)C2312yes—
57C2×C3×C3⋊Dic3—22no—
58C2×C3×C3⋊Dic3C2312yes—
59C2×C32×Dic3—22no—
60C2×C32×Dic3—22no—
61C2×C32×Dic3C2312yes—
62C2×C32×Dic3C2312yes—
63#6292—28no—
64#6289—28no—
65C2×#894—94no—
66C2×#883—91no—
67C23×C3⋊D3—31no—
68C23×C3⋊D3—34no—
69C23×C3⋊D3—64no—
70C23×C3⋊D3C2×D311yescharacteristic
71C23×C3×D3—34no—
72C22×C32×C4—24no—
73C22×C32×C4—31no—
74C22×C32×C4—31no—
75C22×C32×C4C2×D314yes—
76C22×C32×C4C2×D311yescharacteristic
77C22×C3⋊Dic3—31no—
78C22×C3⋊Dic3—34no—
79C22×C3⋊Dic3—64no—
80C22×C3⋊Dic3C2×D311yescharacteristic
81C2×C4⋊(C3⋊D3)—34no—
82C2×C4⋊(C3⋊D3)—64no—
83C2×C4×C3⋊D3—34no—
84C2×C4×C3⋊D3—64no—
85C22×C3×Dic3—34no—
86C22×C3×Dic3C2×D314yes—
87C22×C3×Dic3C2×D314yes—
88C2×#626—316no—
89C2×#626—616no—
90C2×#624—316no—
91C2×#624—616no—
92C32×C4⋊C4—34no—
93#3752—34no—
94#3752—34no—
95#3752—316no—
96#3752—616no—
97#3752C2×D314yescomplemented
98#3750—34no—
99C3×(C2×D3)⋊C4—316no—
100C3×C4⋊Dic3C2×D3116yescomplemented
101#3724—316no—
102(C2×D3)⋊Dic3—316no—
103C22×C33C2×D412yes—
104C22×C33#4112yes—
105C22×C33#4112yes—
106C22×C33C2411yescharacteristic
107C2×C3⋊(C3⋊D3)—28no—
108C2×C3⋊(C3⋊D3)—22no—
109C2×C3⋊(C3⋊D3)—22no—
110C2×C3⋊(C3⋊D3)C2×D414yes—
111C2×C3×C3⋊D3—22no—
112C2×C3×C3⋊D3—22no—
113C2×C3×C3⋊D3—22no—
114C2×C3×C3⋊D3—44no—
115C33×C4—22no—
116C3⋊(C3⋊Dic3)—22no—
117C3×C3⋊Dic3—22no—
118C32×Dic3—22no—
119C32×Dic3—22no—
120C22×#67—91no—
121C22×D12—94no—
122C22×C4×D3—94no—
123C22×C4×D3—91no—
124C2×C3×C4⋊C4—34no—
125C2×C3×C22⋊C4—91no—
126C2×#217—91no—
127C2×(C2×D3)⋊C4—34no—
128C2×(C2×D3)⋊C4—94no—
129C2×(C2×D3)⋊C4—94no—
130C2×(C2×D3)⋊C4—91no—
131C2×(C2×D3)⋊C4—184no—
132C2×C4⋊Dic3C3⋊D311yescharacteristic, complemented
133C2×Dic3⋊C4—94no—
134C2×Dic3⋊C4—91no—
135#894—932no—
136#883—98no—
137C23×C32—24no—
138C23×C32C22×D314yes—
139C23×C32C22×D311yescharacteristic
140C22×C3⋊D3—22no—
141C22×C3⋊D3—22no—
142C22×C3⋊D3—22no—
143C22×C3⋊D3—34no—
144C22×C3⋊D3—32no—
145C22×C3⋊D3—316no—
146C22×C3⋊D3—38no—
147C22×C3⋊D3—68no—
148C22×C3⋊D3—616no—
149C22×C3⋊D3—62no—
150C22×C3⋊D3—62no—
151C22×C3⋊D3—68no—
152C22×C3⋊D3—68no—
153C22×C3⋊D3—128no—
154C22×C3⋊D3—128no—
155C22×C3⋊D3C22×D312yes—
156C22×C3×D3—38no—
157C22×C3×D3—68no—
158C22×C3×D3—68no—
159C22×C3×D3—68no—
160C2×C32×C4—22no—
161C2×C32×C4—28no—
162C2×C32×C4—28no—
163C2×C32×C4—32no—
164C2×C32×C4—32no—
165C2×C32×C4—48no—
166C2×C32×C4—62no—
167C2×C32×C4—62no—
168C2×C32×C4C22×D318yes—
169C2×C32×C4C22×D312yes—
170C2×C3⋊Dic3—22no—
171C2×C3⋊Dic3—32no—
172C2×C3⋊Dic3—38no—
173C2×C3⋊Dic3—68no—
174C2×C3⋊Dic3—62no—
175C2×C3⋊Dic3—68no—
176C2×C3⋊Dic3—128no—
177C2×C3⋊Dic3C22×D312yes—
178C4⋊(C3⋊D3)—68no—
179C4×C3⋊D3—68no—
180C2×C3×Dic3—28no—
181C2×C3×Dic3—28no—
182C2×C3×Dic3—38no—
183C2×C3×Dic3—68no—
184C2×C3×Dic3C22×D318yes—
185C2×C3×Dic3C22×D318yes—
186#626—632no—
187#624—632no—
188C2×#123—271noSylow
189C2×C33#12314yescomplemented
190C2×C33C22×D411yescharacteristic
191C2×C33C2×#4111yescharacteristic
192C2×C33C2×#4111yescharacteristic
193C3⋊(C3⋊D3)—24no—
194C3×C3⋊D3—42no—
195C24×C3—91no—
196C23×D3—34no—
197C23×D3—94no—
198C23×D3—91no—
199C23×D3—184no—
200C22×C3×C4—24no—
201C22×C3×C4—34no—
202C22×C3×C4—34no—
203C22×C3×C4—91no—
204C22×C3×C4D3214yes—
205C22×C3×C4C2×C3⋊D311yescharacteristic
206C2×#67—94no—
207C2×#67—184no—
208C22×Dic3—34no—
209C22×Dic3—94no—
210C22×Dic3—91no—
211C22×Dic3—184no—
212C22×Dic3C2×C3⋊D311yescharacteristic
213C22×Dic3C2×C3⋊D311yescharacteristic
214C2×D12—916no—
215C2×D12—1816no—
216C2×C4×D3—916no—
217C2×C4×D3—94no—
218C2×C4×D3—184no—
219C2×C4×D3—1816no—
220C3×C4⋊C4—316no—
221C3×C22⋊C4—94no—
222#217—94no—
223(C2×D3)⋊C4—316no—
224(C2×D3)⋊C4—916no—
225(C2×D3)⋊C4—916no—
226(C2×D3)⋊C4—94no—
227(C2×D3)⋊C4—1816no—
228C4⋊Dic3C2×C3⋊D314yescomplemented
229Dic3⋊C4—916no—
230Dic3⋊C4—94no—
231C22×C32—24no—
232C22×C32—28no—
233C22×C32—28no—
234C22×C32—28no—
235C22×C32#23518yes—
236C22×C32#23512yes—
237C22×C32D3×D418yes—
238C22×C32D3×D412yes—
239C22×C32#23712yes—
240C22×C32#23918yes—
241C22×C32C23×D314yes—
242C22×C32C23×D311yescharacteristic
243C2×C3⋊D3—22no—
244C2×C3⋊D3—22no—
245C2×C3⋊D3—22no—
246C2×C3⋊D3—34no—
247C2×C3⋊D3—316no—
248C2×C3⋊D3—44no—
249C2×C3⋊D3—616no—
250C2×C3⋊D3—68no—
251C2×C3⋊D3—62no—
252C2×C3⋊D3—62no—
253C2×C3⋊D3—632no—
254C2×C3⋊D3—68no—
255C2×C3⋊D3—68no—
256C2×C3⋊D3—128no—
257C2×C3⋊D3—128no—
258C2×C3⋊D3—1232no—
259C2×C3×D3—68no—
260C2×C3×D3—68no—
261C2×C3×D3—68no—
262C2×C3×D3—1216no—
263C32×C4—22no—
264C32×C4—28no—
265C32×C4—48no—
266C32×C4—62no—
267C32×C4—62no—
268C3⋊Dic3—22no—
269C3⋊Dic3—62no—
270C3⋊Dic3—68no—
271C3⋊Dic3—128no—
272C3×Dic3—28no—
273C3×Dic3—28no—
274C3×Dic3—68no—
275C22×D4—271no—
276C23×C4—271no—
277#123—278no—
278C2×C4⋊C4—31no—
279C2×C4⋊C4—271no—
280C2×C22⋊C4—91no—
281C2×C22⋊C4—271no—
282C2×C22⋊C4—271no—
283C33C2×#12311yescharacteristic, complemented, Sylow
284C23×C3—24no—
285C23×C3—92no—
286C23×C3—182no—
287C23×C3—182no—
288C23×C3—182no—
289C23×C3C2×D3214yes—
290C23×C3C22×C3⋊D311yescharacteristic
291C22×D3—38no—
292C22×D3—68no—
293C22×D3—68no—
294C22×D3—68no—
295C22×D3—98no—
296C22×D3—92no—
297C22×D3—916no—
298C22×D3—94no—
299C22×D3—188no—
300C22×D3—1816no—
301C22×D3—188no—
302C22×D3—188no—
303C22×D3—182no—
304C22×D3—182no—
305C22×D3—368no—
306C22×D3—368no—
307C2×C3×C4—28no—
308C2×C3×C4—22no—
309C2×C3×C4—28no—
310C2×C3×C4—38no—
311C2×C3×C4—38no—
312C2×C3×C4—48no—
313C2×C3×C4—68no—
314C2×C3×C4—68no—
315C2×C3×C4—92no—
316C2×C3×C4—182no—
317C2×C3×C4C2×D3218yes—
318C2×C3×C4C22×C3⋊D312yes—
319#67—188no—
320C2×Dic3—22no—
321C2×Dic3—22no—
322C2×Dic3—38no—
323C2×Dic3—68no—
324C2×Dic3—98no—
325C2×Dic3—92no—
326C2×Dic3—188no—
327C2×Dic3—188no—
328C2×Dic3—182no—
329C2×Dic3—368no—
330C2×Dic3C22×C3⋊D312yes—
331C2×Dic3C22×C3⋊D312yes—
332D12—1832no—
333C4×D3—188no—
334C4×D3—1832no—
335C2×C32—24no—
336C2×C32—24no—
337C2×C32—24no—
338C2×C32—216no—
339C2×C32#88314yescomplemented
340C2×C32#894116yescomplemented
341C2×C32C2×#23514yes—
342C2×C32C2×#23511yescharacteristic
343C2×C32C2×D3×D414yes—
344C2×C32C2×D3×D411yescharacteristic
345C2×C32C2×#23711yescharacteristic
346C2×C32C2×#23914yes—
347C3⋊D3—42no—
348C3⋊D3—64no—
349C3⋊D3—616no—
350C3⋊D3—1216no—
351C3×D3—128no—
352C24—91no—
353C24—271no—
354C2×D4—274no—
355C2×D4—544no—
356C22×C4—31no—
357C22×C4—31no—
358C22×C4—91no—
359C22×C4—274no—
360C22×C4—271no—
361C22×C4—544no—
362C22×C4D3×C3⋊D311yescharacteristic
363C4⋊C4—34no—
364C4⋊C4—274no—
365C22⋊C4—94no—
366C22⋊C4—274no—
367C22⋊C4—274no—
368C22×C3—24no—
369C22×C3—28no—
370C22×C3—28no—
371C22×C3—28no—
372C22×C3—182no—
373C22×C3—182no—
374C22×C3—182no—
375C22×C3—364no—
376C22×C3#379918yes—
377C22×C3#380218yes—
378C22×C3#380418yes—
379C22×C3#382812yes—
380C22×C3D4×C3⋊D312yes—
381C22×C3#383212yes—
382C22×C3C22×D3214yes—
383C22×C3C23×C3⋊D311yescharacteristic
384C2×D3—68no—
385C2×D3—68no—
386C2×D3—68no—
387C2×D3—916no—
388C2×D3—94no—
389C2×D3—1216no—
390C2×D3—1816no—
391C2×D3—188no—
392C2×D3—188no—
393C2×D3—182no—
394C2×D3—188no—
395C2×D3—182no—
396C2×D3—1832no—
397C2×D3—368no—
398C2×D3—368no—
399C2×D3—3632no—
400C3×C4—28no—
401C3×C4—22no—
402C3×C4—48no—
403C3×C4—68no—
404C3×C4—68no—
405C3×C4—182no—
406Dic3—22no—
407Dic3—22no—
408Dic3—68no—
409Dic3—188no—
410Dic3—182no—
411Dic3—368no—
412C32—24no—
413C32C2×#88311yescharacteristic, complemented
414C32C2×#89414yescomplemented
415C23—92no—
416C23—182no—
417C23—182no—
418C23—182no—
419C23—272no—
420C23—274no—
421C23—542no—
422C23—542no—
423C23C2×D3×C3⋊D311yescharacteristic
424D4—548no—
425C2×C4—22no—
426C2×C4—32no—
427C2×C4—32no—
428C2×C4—62no—
429C2×C4—62no—
430C2×C4—92no—
431C2×C4—182no—
432C2×C4—272no—
433C2×C4—548no—
434C2×C4—542no—
435C2×C4C2×D3×C3⋊D312yes—
436C2×C3—24no—
437C2×C3—24no—
438C2×C3—24no—
439C2×C3—216no—
440C2×C3—362no—
441C2×C3#63854116yescomplemented
442C2×C3#6410314yescomplemented
443C2×C3C2×#379914yes—
444C2×C3C2×#380214yes—
445C2×C3C2×#380414yes—
446C2×C3C2×#382811yescharacteristic
447C2×C3C2×D4×C3⋊D311yescharacteristic
448C2×C3C2×#383211yescharacteristic
449D3—128no—
450D3—1816no—
451D3—184no—
452D3—3616no—
453C22—182no—
454C22—182no—
455C22—182no—
456C22—274no—
457C22—364no—
458C22—542no—
459C22—542no—
460C22—548no—
461C22#8901812yes—
462C22#8902612yes—
463C22#8903112yes—
464C22C22×D3×C3⋊D311yescharacteristic
465C4—22no—
466C4—62no—
467C4—62no—
468C4—182no—
469C4—542no—
470C3—24no—
471C3C2×#6385414yescomplemented
472C3C2×#6410311yescharacteristic, complemented
473C2—362no—
474C2—544no—
475C2#1172350414yesdirect product factor
476C2C2×#8901811yescharacteristic
477C2C2×#8902611yescharacteristic
478C2C2×#8903111yescharacteristic
subgroupG/H
centerC23C2×D3×C3⋊D3
commutator, cosocleC22×C33C24
FrattiniC22C22×D3×C3⋊D3
FittingC22×C33×C4C22
socleC23×C33C23
prime radicalC22×#6292C2
hypercenterC22×C4D3×C3⋊D3
hypocenterC33C2×#123
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C22×#6289C2semidirect
3. C2×#88859C2semidirect
4. C2×C3×#3752C2semidirect
5. C2×C32×C4⋊Dic3C2semidirect
6. C2×#88806C2semidirect
7. C2×#88803C2semidirect
8. #11723504C2direct (N × H)
21. C2×#6289C22semidirect
22. #88859C22semidirect
23. C3×#3752C22semidirect
24. C32×C4⋊Dic3C22semidirect
25. #88806C22semidirect
26. #88803C22semidirect
36. C2×#3752D3semidirect
40. C2×C3×C4⋊Dic3D3semidirect
97. #3752C2×D3semidirect
100. C3×C4⋊Dic3C2×D3semidirect
132. C2×C4⋊Dic3C3⋊D3semidirect
189. C2×C33#123semidirect
228. C4⋊Dic3C2×C3⋊D3semidirect
283. C33C2×#123semidirect
339. C2×C32#883semidirect
340. C2×C32#894semidirect
413. C32C2×#883semidirect
414. C32C2×#894semidirect
441. C2×C3#63854semidirect
442. C2×C3#64103semidirect
471. C3C2×#63854semidirect
472. C3C2×#64103semidirect
475. C2#11723504direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#62922
C22×#62892
C2×#888592
C2×C3×#37522
C2×C32×C4⋊Dic32
C2×#888062
C2×#888032
#117235042
C2×#641033
C2×#638543