Finite Group Databasebeta

← C2×#11721269 #13151981

Subgroups — C2×#11721269 #13151981

Order 1728.

1158 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×#611C211yesmaximal, characteristic
2C22×Dic3×D9C211yesmaximal, characteristic, complemented
3C2×#88551C211yesmaximal, characteristic, complemented
4C2×C9×#217C211yesmaximal, characteristic, complemented
5#11721269C218yesmaximal, direct product factor
6C2×(C2×D9)⋊Dic3C211yesmaximal, characteristic, complemented
7C2×Dic3⋊Dic9C211yesmaximal, characteristic, complemented
8C2×#88442C211yesmaximal, characteristic, complemented
9C2×#63969—31nomaximal
10C2×#63445—31nomaximal
11C24×C3×C9C2211yescharacteristic
12C23×C3×D9C2211yescharacteristic
13C22×C3⋊Dic9C2211yescharacteristic
14C22×C3⋊Dic9C2211yescharacteristic
15C2×C3×#611—24no—
16C2×C3×#611C2214yes—
17C22×C9×Dic3C2211yescharacteristic
18C22×C9×Dic3C2211yescharacteristic
19C22×C3×Dic9C2211yescharacteristic
20C2×Dic3×D9—24no—
21C2×Dic3×D9C2214yescomplemented
22#88551C2214yescomplemented
23C9×#217C2214yescomplemented
24(C2×D9)⋊Dic3C2214yescomplemented
25Dic3⋊Dic9C2214yescomplemented
26#88442C2214yescomplemented
27C22×C3×#67—31no—
28C22×D3×Dic3—31no—
29C2×#3757—31no—
30C2×C3×#217D311yescharacteristic
31#63969—38no—
32C2×#3724—31no—
33C2×Dic3⋊Dic3—31no—
34C2×(C2×D3)⋊Dic3—31no—
35C22×#611D311yescharacteristic
36C22×C4×D9—31no—
37C2×C9×C22⋊C4—31no—
38C2×#3676—31no—
39C2×#3676—61no—
40C2×(C2×D9)⋊C4—31no—
41C2×C4⋊Dic9—31no—
42C2×Dic9⋊C4—31no—
43#63445—38no—
44C23×C3×C9—22no—
45C23×C3×C9—22no—
46C23×C3×C9—22no—
47C23×C3×C9C2312yes—
48C23×C3×C9C2311yescharacteristic
49C22×C3×D9—22no—
50C22×C3×D9—22no—
51C22×C3×D9D414yes—
52C22×C3×D9C2312yes—
53C2×C3⋊Dic9—22no—
54C2×C3⋊Dic9—22no—
55C2×C3⋊Dic9C2312yes—
56C2×C3⋊Dic9C2312yes—
57C3×#611—28no—
58C2×C9×Dic3—22no—
59C2×C9×Dic3—22no—
60C2×C9×Dic3C2312yes—
61C2×C9×Dic3C2312yes—
62C2×C3×Dic9—22no—
63C2×C3×Dic9C2312yes—
64Dic3×D9—28no—
65C2×#935—91no—
66C2×#883—91no—
67C24×C32C2×D311yescharacteristic
68C23×C3×D3—31no—
69C22×C3⋊Dic3—31no—
70C22×C3⋊Dic3—31no—
71C2×C3×#67—34no—
72C2×C3×#67—64no—
73C22×C3×Dic3—31no—
74C22×C3×Dic3C2×D311yescharacteristic
75C22×C3×Dic3C2×D311yescharacteristic
76C2×D3×Dic3—34no—
77C2×D3×Dic3—64no—
78C24×C9—21no—
79C24×C9C2×D311yescharacteristic
80C23×D9C2×D311yescharacteristic
81#3757—34no—
82C3×#217C2×D314yes—
83#3724—34no—
84Dic3⋊Dic3—34no—
85(C2×D3)⋊Dic3—34no—
86C22×C4×C9—31no—
87C22×C4×C9—31no—
88C2×#611—24no—
89C2×#611C2×D314yes—
90C22×Dic9—31no—
91C22×Dic9—31no—
92C22×Dic9—61no—
93C22×Dic9—61no—
94C22×Dic9C2×D311yescharacteristic
95C2×C4×D9—34no—
96C2×C4×D9—64no—
97C9×C22⋊C4—34no—
98#3676—34no—
99#3676—64no—
100(C2×D9)⋊C4—34no—
101C4⋊Dic9—34no—
102Dic9⋊C4—34no—
103C22×C3×C9—22no—
104C22×C3×C9—22no—
105C22×C3×C9—22no—
106C22×C3×C9—44no—
107C22×C3×C9C2×D412yes—
108C22×C3×C9#4112yes—
109C22×C3×C9#4112yes—
110C22×C3×C9C2411yescharacteristic
111C2×C3×D9—28no—
112C2×C3×D9—22no—
113C2×C3×D9—22no—
114C2×C3×D9C2×D414yes—
115C3⋊Dic9—22no—
116C3⋊Dic9—22no—
117C9×Dic3—22no—
118C9×Dic3—22no—
119C3×Dic9—22no—
120C22×C3×D4—91no—
121C22×#67—31no—
122C23×Dic3—91no—
123C22×C4×D3—91no—
124C2×C3×C22⋊C4—31no—
125C2×#217—91no—
126C2×#217—91no—
127C2×#217—91no—
128C2×#217—181no—
129C2×#217D911yescharacteristic, complemented
130#935—98no—
131C2×(C2×D3)⋊C4—91no—
132C2×C4⋊Dic3—91no—
133C2×Dic3⋊C4—91no—
134C2×Dic3⋊C4—91no—
135C2×Dic3⋊C4—91no—
136#883—98no—
137C23×C32—22no—
138C23×C32—22no—
139C23×C32—22no—
140C23×C32C22×D312yes—
141C23×C32C22×D311yescharacteristic
142C22×C3×D3—34no—
143C22×C3×D3—32no—
144C22×C3×D3—62no—
145C22×C3×D3—62no—
146C2×C3⋊Dic3—32no—
147C2×C3⋊Dic3—32no—
148C2×C3⋊Dic3—62no—
149C2×C3⋊Dic3—62no—
150C3×#67—68no—
151C2×C3×Dic3—22no—
152C2×C3×Dic3—22no—
153C2×C3×Dic3—32no—
154C2×C3×Dic3—62no—
155C2×C3×Dic3C22×D312yes—
156C2×C3×Dic3C22×D312yes—
157D3×Dic3—68no—
158C23×C9—24no—
159C23×C9—22no—
160C23×C9—22no—
161C23×C9—22no—
162C23×C9—21no—
163C23×C9—22no—
164C23×C9—42no—
165C23×C9—42no—
166C23×C9C22×D312yes—
167C23×C9C22×D311yescharacteristic
168C22×D9—22no—
169C22×D9—22no—
170C22×D9#6714yes—
171C22×D9C22×D312yes—
172C2×C4×C9—32no—
173C2×C4×C9—32no—
174C2×C4×C9—62no—
175C2×C4×C9—62no—
176#611—28no—
177C2×Dic9—22no—
178C2×Dic9—32no—
179C2×Dic9—32no—
180C2×Dic9—62no—
181C2×Dic9—62no—
182C2×Dic9—62no—
183C2×Dic9—62no—
184C2×Dic9—122no—
185C2×Dic9—122no—
186C2×Dic9C22×D312yes—
187C4×D9—68no—
188C2×#123—271noSylow
189C2×C3×C9—42no—
190C2×C3×C9#12314yescomplemented
191C2×C3×C9C22×D411yescharacteristic
192C2×C3×C9C2×#4111yescharacteristic
193C2×C3×C9C2×#4111yescharacteristic
194C3×D9—24no—
195C24×C3—21no—
196C24×C3—91no—
197C24×C3C2×D911yescharacteristic
198C24×C3D3211yescharacteristic
199C23×D3—31no—
200C2×C3×D4—94no—
201C2×C3×D4—184no—
202C22×C3×C4—31no—
203C22×C3×C4—31no—
204C22×C3×C4—91no—
205C2×#67—34no—
206C2×#67—64no—
207C22×Dic3—31no—
208C22×Dic3—91no—
209C22×Dic3—91no—
210C22×Dic3—91no—
211C22×Dic3—91no—
212C22×Dic3—94no—
213C22×Dic3—181no—
214C22×Dic3—181no—
215C22×Dic3—184no—
216C22×Dic3C2×D911yescharacteristic
217C22×Dic3C2×D911yescharacteristic
218C2×C4×D3—94no—
219C2×C4×D3—184no—
220C3×C22⋊C4—34no—
221#217—94no—
222#217—94no—
223#217—94no—
224#217—184no—
225#217C2×D914yescomplemented
226(C2×D3)⋊C4—94no—
227C4⋊Dic3—94no—
228Dic3⋊C4—94no—
229Dic3⋊C4—94no—
230Dic3⋊C4—94no—
231C22×C32—22no—
232C22×C32—22no—
233C22×C32—22no—
234C22×C32—44no—
235C22×C32#23512yes—
236C22×C32D3×D412yes—
237C22×C32#23712yes—
238C22×C32C23×D311yescharacteristic
239C2×C3×D3—34no—
240C2×C3×D3—68no—
241C2×C3×D3—62no—
242C2×C3×D3—62no—
243C3⋊Dic3—62no—
244C3⋊Dic3—62no—
245C3×Dic3—22no—
246C3×Dic3—22no—
247C3×Dic3—62no—
248C22×C9—24no—
249C22×C9—22no—
250C22×C9—22no—
251C22×C9—22no—
252C22×C9—21no—
253C22×C9—22no—
254C22×C9—22no—
255C22×C9—22no—
256C22×C9—42no—
257C22×C9—42no—
258C22×C9—48no—
259C22×C9—44no—
260C22×C9#23712yes—
261C22×C9#23712yes—
262C22×C9C2×#6712yes—
263C22×C9C23×D311yescharacteristic
264C2×D9—28no—
265C2×D9—22no—
266C2×D9—22no—
267C2×D9C2×#6714yes—
268C4×C9—62no—
269C4×C9—62no—
270Dic9—22no—
271Dic9—62no—
272Dic9—62no—
273Dic9—122no—
274Dic9—122no—
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—
283C3×C9C2×#12311yescharacteristic, complemented, Sylow
284C23×C3—24no—
285C23×C3—22no—
286C23×C3—22no—
287C23×C3—22no—
288C23×C3—22no—
289C23×C3—22no—
290C23×C3—22no—
291C23×C3—22no—
292C23×C3—21no—
293C23×C3—42no—
294C23×C3—42no—
295C23×C3—94no—
296C23×C3—92no—
297C23×C3—182no—
298C23×C3—182no—
299C23×C3C22×D912yes—
300C23×C3C22×D911yescharacteristic
301C23×C3C2×D3212yes—
302C23×C3C2×D3211yescharacteristic
303C22×D3—32no—
304C22×D3—34no—
305C22×D3—62no—
306C22×D3—62no—
307C3×D4—188no—
308C2×C3×C4—32no—
309C2×C3×C4—32no—
310C2×C3×C4—62no—
311C2×C3×C4—62no—
312C2×C3×C4—92no—
313C2×C3×C4—182no—
314#67—68no—
315C2×Dic3—22no—
316C2×Dic3—22no—
317C2×Dic3—32no—
318C2×Dic3—62no—
319C2×Dic3—92no—
320C2×Dic3—92no—
321C2×Dic3—92no—
322C2×Dic3—92no—
323C2×Dic3—182no—
324C2×Dic3—182no—
325C2×Dic3—188no—
326C2×Dic3—182no—
327C2×Dic3—182no—
328C2×Dic3—182no—
329C2×Dic3—182no—
330C2×Dic3—362no—
331C2×Dic3—362no—
332C2×Dic3C22×D912yes—
333C2×Dic3C22×D912yes—
334C4×D3—188no—
335C2×C32—42no—
336C2×C32#88314yes—
337C2×C32C2×#23511yescharacteristic
338C2×C32C2×D3×D411yescharacteristic
339C2×C32C2×#23711yescharacteristic
340C3×D3—64no—
341C2×C9—21no—
342C2×C9—21no—
343C2×C9—21no—
344C2×C9—24no—
345C2×C9—44no—
346C2×C9—42no—
347C2×C9#93514yescomplemented
348C2×C9C2×#23711yescharacteristic
349C2×C9C2×#23711yescharacteristic
350C2×C9C22×#6711yescharacteristic
351D9—24no—
352C24—91no—
353C24D3×D911yescharacteristic
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—24no—
369C22×C3—22no—
370C22×C3—22no—
371C22×C3—22no—
372C22×C3—22no—
373C22×C3—22no—
374C22×C3—22no—
375C22×C3—21no—
376C22×C3—22no—
377C22×C3—22no—
378C22×C3—22no—
379C22×C3—42no—
380C22×C3—42no—
381C22×C3—48no—
382C22×C3—44no—
383C22×C3—44no—
384C22×C3—94no—
385C22×C3—188no—
386C22×C3—182no—
387C22×C3—182no—
388C22×C3#369712yes—
389C22×C3D4×D912yes—
390C22×C3#369912yes—
391C22×C3C23×D911yescharacteristic
392C22×C3#380412yes—
393C22×C3#380512yes—
394C22×C3D3×#6712yes—
395C22×C3C22×D3211yescharacteristic
396C2×D3—34no—
397C2×D3—62no—
398C2×D3—62no—
399C2×D3—68no—
400C3×C4—62no—
401C3×C4—62no—
402C3×C4—182no—
403Dic3—22no—
404Dic3—22no—
405Dic3—62no—
406Dic3—182no—
407Dic3—182no—
408Dic3—182no—
409Dic3—182no—
410Dic3—362no—
411Dic3—362no—
412C32C2×#88311yescharacteristic
413C9—21no—
414C9C2×#93511yescharacteristic, complemented
415C23—22no—
416C23—22no—
417C23—22no—
418C23—92no—
419C23—94no—
420C23—182no—
421C23—182no—
422C23C2×D3×D912yes—
423C23C2×D3×D911yescharacteristic
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—21no—
437C2×C3—21no—
438C2×C3—21no—
439C2×C3—24no—
440C2×C3—44no—
441C2×C3—42no—
442C2×C3—42no—
443C2×C3—184no—
444C2×C3#6344514yescomplemented
445C2×C3C2×#369711yescharacteristic
446C2×C3C2×D4×D911yescharacteristic
447C2×C3C2×#369911yescharacteristic
448C2×C3#6396914yes—
449C2×C3C2×#380411yescharacteristic
450C2×C3C2×#380511yescharacteristic
451C2×C3C2×D3×#6711yescharacteristic
452D3—64no—
453C22—22no—
454C22—22no—
455C22—22no—
456C22—44no—
457C22—94no—
458C22—182no—
459C22—182no—
460C22—188no—
461C22#8865812yes—
462C22#8866312yes—
463C22D9×#6712yes—
464C22C22×D3×D911yescharacteristic
465C4—62no—
466C4—62no—
467C4—182no—
468C4—542no—
469C4—542no—
470C3—21no—
471C3C2×#6344511yescharacteristic, complemented
472C3C2×#6396911yescharacteristic
473C2—42no—
474C2—184no—
475C2#1172126914yesdirect product factor
476C2C2×#8865811yescharacteristic
477C2C2×#8866311yescharacteristic
478C2C2×D9×#6711yescharacteristic
subgroupG/H
centerC23C2×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC24×C3×C9C22
socleC23×C32C22×D3
prime radicalC22×C3×#611C2
hypercenterC24D3×D9
hypocenterC3×C9C2×#123
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
2. C22×Dic3×D9C2semidirect
3. C2×#88551C2semidirect
4. C2×C9×#217C2semidirect
5. #11721269C2direct (N × H)
6. C2×(C2×D9)⋊Dic3C2semidirect
7. C2×Dic3⋊Dic9C2semidirect
8. C2×#88442C2semidirect
21. C2×Dic3×D9C22semidirect
22. #88551C22semidirect
23. C9×#217C22semidirect
24. (C2×D9)⋊Dic3C22semidirect
25. Dic3⋊Dic9C22semidirect
26. #88442C22semidirect
129. C2×#217D9semidirect
190. C2×C3×C9#123semidirect
225. #217C2×D9semidirect
283. C3×C9C2×#123semidirect
347. C2×C9#935semidirect
414. C9C2×#935semidirect
444. C2×C3#63445semidirect
471. C3C2×#63445semidirect
475. C2#11721269direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#6112
C22×Dic3×D92
C2×#885512
C2×C9×#2172
#117212692
C2×(C2×D9)⋊Dic32
C2×Dic3⋊Dic92
C2×#884422
C2×#639693
C2×#634453