Finite Group Databasebeta

← C2×#11721044 #13150478

Subgroups — C2×#11721044 #13150478

Order 1728.

1098 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6194C211yesmaximal, characteristic
2C22×#6189C211yesmaximal, characteristic, complemented
3C2×C4×C3⋊Dic9C211yesmaximal, characteristic, complemented
4C2×C9×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
5C2×C3×(C2×D9)⋊C4C211yesmaximal, characteristic, complemented
6C2×(C2×D3)⋊Dic9C211yesmaximal, characteristic, complemented
7C2×(C2×D9)⋊Dic3C211yesmaximal, characteristic, complemented
8#11721044C218yesmaximal, direct product factor
9C2×#63915—31nomaximal
10C2×#63447—31nomaximal
11C23×D3×C9C2211yescharacteristic
12C23×C3×D9C2211yescharacteristic
13C22×C3×C4×C9C2211yescharacteristic
14C22×C3⋊Dic9C2211yescharacteristic
15C22×C3⋊Dic9C2211yescharacteristic
16C22×C9×Dic3C2211yescharacteristic
17C22×C3×Dic9C2211yescharacteristic
18C2×#6194—24no—
19C2×#6194C2214yes—
20C2×#6189—24no—
21C2×#6189C2214yescomplemented
22C4×C3⋊Dic9C2214yescomplemented
23C9×(C2×D3)⋊C4C2214yescomplemented
24C3×(C2×D9)⋊C4C2214yescomplemented
25(C2×D3)⋊Dic9C2214yescomplemented
26(C2×D9)⋊Dic3C2214yescomplemented
27C22×#627—31no—
28C22×#625—31no—
29C2×C4×C3⋊Dic3—31no—
30C2×C3×(C2×D3)⋊C4—31no—
31C2×C3×(C2×D3)⋊C4D311yescharacteristic
32C2×(C2×D3)⋊Dic3—31no—
33C2×(C2×D3)⋊Dic3—31no—
34#63915—38no—
35C22×#611—31no—
36C22×Dic18—31no—
37C2×C9×C22⋊C4—31no—
38C2×#3676—31no—
39C2×(C2×D9)⋊C4—31no—
40C2×(C2×D9)⋊C4D311yescharacteristic, complemented
41C2×C4×Dic9—31no—
42C2×C4×Dic9—61no—
43#63447—38no—
44C23×C3×C9C2311yescharacteristic
45C22×D3×C9—22no—
46C22×D3×C9—22no—
47C22×D3×C9—22no—
48C22×D3×C9C2312yes—
49C22×C3×D9—22no—
50C22×C3×D9—22no—
51C22×C3×D9—22no—
52C22×C3×D9C2312yes—
53C2×C3×C4×C9—22no—
54C2×C3×C4×C9C2312yes—
55C2×C3⋊Dic9—22no—
56C2×C3⋊Dic9D414yes—
57C2×C3⋊Dic9C2312yes—
58C2×C3⋊Dic9C2312yes—
59C2×C9×Dic3—22no—
60C2×C9×Dic3C2312yes—
61C2×C3×Dic9—22no—
62C2×C3×Dic9C2312yes—
63#6194—28no—
64#6189—28no—
65C2×#885—91no—
66C2×#885—91no—
67C23×C3×D3—31no—
68C23×C3×D3C2×D311yescharacteristic
69C22×C32×C4C2×D311yescharacteristic
70C22×C3⋊Dic3—31no—
71C22×C3⋊Dic3—31no—
72C22×C3×Dic3—31no—
73C22×C3×Dic3C2×D311yescharacteristic
74C2×#627—34no—
75C2×#627—64no—
76C2×#625—34no—
77C2×#625—64no—
78C24×C9—31no—
79C23×D9C2×D311yescharacteristic
80C4×C3⋊Dic3—34no—
81C3×(C2×D3)⋊C4—34no—
82C3×(C2×D3)⋊C4C2×D314yes—
83(C2×D3)⋊Dic3—34no—
84(C2×D3)⋊Dic3—34no—
85C22×C4×C9—21no—
86C22×C4×C9—31no—
87C22×C4×C9C2×D311yescharacteristic
88C2×#611—34no—
89C2×#611—64no—
90C22×Dic9—31no—
91C22×Dic9—31no—
92C22×Dic9—61no—
93C22×Dic9—61no—
94C22×Dic9C2×D311yescharacteristic
95C2×Dic18—34no—
96C2×Dic18—64no—
97C9×C22⋊C4—34no—
98#3676—34no—
99(C2×D9)⋊C4—34no—
100(C2×D9)⋊C4C2×D314yescomplemented
101C4×Dic9—34no—
102C4×Dic9—64no—
103C22×C3×C9C2×D412yes—
104C22×C3×C9#4112yes—
105C22×C3×C9#4112yes—
106C22×C3×C9C2411yescharacteristic
107C2×D3×C9—22no—
108C2×D3×C9—22no—
109C2×D3×C9—22no—
110C2×D3×C9—44no—
111C2×C3×D9—22no—
112C2×C3×D9—22no—
113C2×C3×D9—22no—
114C2×C3×D9—44no—
115C3×C4×C9—22no—
116C3⋊Dic9—22no—
117C3⋊Dic9C2×D414yes—
118C9×Dic3—22no—
119C3×Dic9—22no—
120C22×#67—91no—
121C22×#67—91no—
122C22×Dic6—91no—
123C22×Dic6—91no—
124C2×C3×C22⋊C4—31no—
125C2×C3×C22⋊C4—91no—
126C2×#217—91no—
127C2×#217—91no—
128C2×(C2×D3)⋊C4—31no—
129C2×(C2×D3)⋊C4—91no—
130C2×(C2×D3)⋊C4—91no—
131C2×(C2×D3)⋊C4D911yescharacteristic, complemented
132C2×C4×Dic3—91no—
133C2×C4×Dic3—91no—
134C2×C4×Dic3—181no—
135#885—98no—
136#885—98no—
137C23×C32C22×D311yescharacteristic
138C22×C3×D3—22no—
139C22×C3×D3—22no—
140C22×C3×D3—22no—
141C22×C3×D3—32no—
142C22×C3×D3—62no—
143C22×C3×D3—62no—
144C22×C3×D3—62no—
145C22×C3×D3C22×D312yes—
146C2×C32×C4—22no—
147C2×C32×C4C22×D312yes—
148C2×C3⋊Dic3—34no—
149C2×C3⋊Dic3—32no—
150C2×C3⋊Dic3—32no—
151C2×C3⋊Dic3—62no—
152C2×C3×Dic3—22no—
153C2×C3×Dic3—32no—
154C2×C3×Dic3—62no—
155C2×C3×Dic3C22×D312yes—
156#627—68no—
157#625—68no—
158C23×C9—21no—
159C23×C9—32no—
160C23×C9—62no—
161C23×C9—62no—
162C23×C9—62no—
163C23×C9C22×D311yescharacteristic
164C22×D9—22no—
165C22×D9—22no—
166C22×D9—22no—
167C22×D9C22×D312yes—
168C2×C4×C9—24no—
169C2×C4×C9—22no—
170C2×C4×C9—22no—
171C2×C4×C9—32no—
172C2×C4×C9—62no—
173C2×C4×C9C22×D312yes—
174#611—68no—
175C2×Dic9—22no—
176C2×Dic9—34no—
177C2×Dic9—32no—
178C2×Dic9—32no—
179C2×Dic9—62no—
180C2×Dic9—62no—
181C2×Dic9—64no—
182C2×Dic9—64no—
183C2×Dic9—62no—
184C2×Dic9C22×D312yes—
185Dic18—68no—
186C2×#124—271noSylow
187C2×C3×C9#12414yescomplemented
188C2×C3×C9C22×D411yescharacteristic
189C2×C3×C9C2×#4111yescharacteristic
190C2×C3×C9C2×#4111yescharacteristic
191D3×C9—42no—
192C3×D9—42no—
193C24×C3—31no—
194C24×C3—91no—
195C23×D3—31no—
196C23×D3C2×D911yescharacteristic
197C22×C3×C4—21no—
198C22×C3×C4—31no—
199C22×C3×C4—91no—
200C22×C3×C4C2×D911yescharacteristic
201C22×C3×C4D3211yescharacteristic
202C2×#67—94no—
203C2×#67—94no—
204C2×#67—184no—
205C2×#67—184no—
206C22×Dic3—31no—
207C22×Dic3—91no—
208C22×Dic3—91no—
209C22×Dic3—91no—
210C22×Dic3—91no—
211C22×Dic3—181no—
212C22×Dic3—181no—
213C22×Dic3C2×D911yescharacteristic
214C2×Dic6—94no—
215C2×Dic6—94no—
216C2×Dic6—184no—
217C2×Dic6—184no—
218C3×C22⋊C4—34no—
219C3×C22⋊C4—94no—
220#217—94no—
221#217—94no—
222(C2×D3)⋊C4—34no—
223(C2×D3)⋊C4—94no—
224(C2×D3)⋊C4—94no—
225(C2×D3)⋊C4C2×D914yescomplemented
226C4×Dic3—94no—
227C4×Dic3—94no—
228C4×Dic3—184no—
229C22×C32#23512yes—
230C22×C32D3×D412yes—
231C22×C32#23712yes—
232C22×C32C23×D311yescharacteristic
233C2×C3×D3—22no—
234C2×C3×D3—22no—
235C2×C3×D3—22no—
236C2×C3×D3—44no—
237C2×C3×D3—62no—
238C2×C3×D3—62no—
239C2×C3×D3—62no—
240C2×C3×D3—124no—
241C32×C4—22no—
242C3⋊Dic3—34no—
243C3⋊Dic3—62no—
244C3×Dic3—22no—
245C3×Dic3—62no—
246C22×C9—21no—
247C22×C9—22no—
248C22×C9—22no—
249C22×C9—22no—
250C22×C9—62no—
251C22×C9—62no—
252C22×C9—62no—
253C22×C9—124no—
254C22×C9#23512yes—
255C22×C9D3×D412yes—
256C22×C9#23712yes—
257C22×C9C23×D311yescharacteristic
258C2×D9—22no—
259C2×D9—22no—
260C2×D9—22no—
261C2×D9—44no—
262C4×C9—24no—
263C4×C9—22no—
264C4×C9—62no—
265Dic9—22no—
266Dic9—34no—
267Dic9—64no—
268Dic9—64no—
269Dic9—62no—
270C22×Q8—271no—
271C22×D4—271no—
272#124—278no—
273C2×C22⋊C4—31no—
274C2×C22⋊C4—91no—
275C2×C22⋊C4—271no—
276C2×C22⋊C4—271no—
277C2×C42—271no—
278C3×C9C2×#12411yescharacteristic, complemented, Sylow
279C23×C3—21no—
280C23×C3—32no—
281C23×C3—62no—
282C23×C3—62no—
283C23×C3—62no—
284C23×C3—92no—
285C23×C3—182no—
286C23×C3—182no—
287C23×C3—182no—
288C23×C3C22×D911yescharacteristic
289C23×C3C2×D3211yescharacteristic
290C22×D3—22no—
291C22×D3—22no—
292C22×D3—22no—
293C22×D3—32no—
294C22×D3—62no—
295C22×D3—62no—
296C22×D3—62no—
297C22×D3C22×D912yes—
298C2×C3×C4—24no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—32no—
303C2×C3×C4—62no—
304C2×C3×C4—92no—
305C2×C3×C4—182no—
306C2×C3×C4C22×D912yes—
307C2×C3×C4C2×D3212yes—
308#67—188no—
309#67—188no—
310C2×Dic3—22no—
311C2×Dic3—32no—
312C2×Dic3—62no—
313C2×Dic3—92no—
314C2×Dic3—94no—
315C2×Dic3—92no—
316C2×Dic3—94no—
317C2×Dic3—92no—
318C2×Dic3—92no—
319C2×Dic3—182no—
320C2×Dic3—182no—
321C2×Dic3—184no—
322C2×Dic3—184no—
323C2×Dic3—182no—
324C2×Dic3—182no—
325C2×Dic3C22×D912yes—
326Dic6—188no—
327Dic6—188no—
328C2×C32#88514yes—
329C2×C32C2×#23511yescharacteristic
330C2×C32C2×D3×D411yescharacteristic
331C2×C32C2×#23711yescharacteristic
332C3×D3—42no—
333C3×D3—122no—
334C2×C9—21no—
335C2×C9—21no—
336C2×C9—21no—
337C2×C9—24no—
338C2×C9—122no—
339C2×C9#88514yescomplemented
340C2×C9C2×#23511yescharacteristic
341C2×C9C2×D3×D411yescharacteristic
342C2×C9C2×#23711yescharacteristic
343D9—42no—
344C24—31no—
345C24—91no—
346C2×Q8—274no—
347C2×Q8—544no—
348C2×D4—274no—
349C2×D4—544no—
350C22×C4—31no—
351C22×C4—91no—
352C22×C4—271no—
353C22×C4—271no—
354C22×C4D3×D911yescharacteristic
355C22⋊C4—34no—
356C22⋊C4—94no—
357C22⋊C4—274no—
358C22⋊C4—274no—
359C42—274no—
360C22×C3—22no—
361C22×C3—21no—
362C22×C3—22no—
363C22×C3—22no—
364C22×C3—62no—
365C22×C3—62no—
366C22×C3—62no—
367C22×C3—124no—
368C22×C3—182no—
369C22×C3—182no—
370C22×C3—182no—
371C22×C3—364no—
372C22×C3#369712yes—
373C22×C3D4×D912yes—
374C22×C3#369912yes—
375C22×C3C23×D911yescharacteristic
376C22×C3#379812yes—
377C22×C3#380212yes—
378C22×C3#380512yes—
379C22×C3C22×D3211yescharacteristic
380C2×D3—22no—
381C2×D3—22no—
382C2×D3—22no—
383C2×D3—44no—
384C2×D3—62no—
385C2×D3—62no—
386C2×D3—62no—
387C2×D3—124no—
388C3×C4—24no—
389C3×C4—22no—
390C3×C4—22no—
391C3×C4—62no—
392C3×C4—182no—
393Dic3—22no—
394Dic3—62no—
395Dic3—94no—
396Dic3—94no—
397Dic3—184no—
398Dic3—184no—
399Dic3—182no—
400Dic3—182no—
401C32C2×#88511yescharacteristic
402C9—21no—
403C9C2×#88511yescharacteristic, complemented
404C23—32no—
405C23—62no—
406C23—62no—
407C23—62no—
408C23—92no—
409C23—182no—
410C23—182no—
411C23—182no—
412C23C2×D3×D911yescharacteristic
413Q8—548no—
414D4—548no—
415C2×C4—22no—
416C2×C4—32no—
417C2×C4—62no—
418C2×C4—92no—
419C2×C4—182no—
420C2×C4—272no—
421C2×C4—274no—
422C2×C4—272no—
423C2×C4—542no—
424C2×C4C2×D3×D912yes—
425C2×C3—21no—
426C2×C3—24no—
427C2×C3—21no—
428C2×C3—21no—
429C2×C3—122no—
430C2×C3—362no—
431C2×C3#6344714yescomplemented
432C2×C3C2×#369711yescharacteristic
433C2×C3C2×D4×D911yescharacteristic
434C2×C3C2×#369911yescharacteristic
435C2×C3#6391514yes—
436C2×C3C2×#379811yescharacteristic
437C2×C3C2×#380211yescharacteristic
438C2×C3C2×#380511yescharacteristic
439D3—42no—
440D3—122no—
441C22—62no—
442C22—62no—
443C22—62no—
444C22—124no—
445C22—182no—
446C22—182no—
447C22—182no—
448C22—364no—
449C22#8864012yes—
450C22#8864612yes—
451C22#8866312yes—
452C22C22×D3×D911yescharacteristic
453C4—22no—
454C4—62no—
455C4—182no—
456C4—274no—
457C4—542no—
458C3—21no—
459C3C2×#6344711yescharacteristic, complemented
460C3C2×#6391511yescharacteristic
461C2—122no—
462C2—362no—
463C2#1172104414yesdirect product factor
464C2C2×#8864011yescharacteristic
465C2C2×#8864611yescharacteristic
466C2C2×#8866311yescharacteristic
subgroupG/H
centerC23C2×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC23×C32C22×D3
prime radicalC22×#6194C2
hypercenterC22×C4D3×D9
hypocenterC3×C9C2×#124
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C22×#6189C2semidirect
3. C2×C4×C3⋊Dic9C2semidirect
4. C2×C9×(C2×D3)⋊C4C2semidirect
5. C2×C3×(C2×D9)⋊C4C2semidirect
6. C2×(C2×D3)⋊Dic9C2semidirect
7. C2×(C2×D9)⋊Dic3C2semidirect
8. #11721044C2direct (N × H)
21. C2×#6189C22semidirect
22. C4×C3⋊Dic9C22semidirect
23. C9×(C2×D3)⋊C4C22semidirect
24. C3×(C2×D9)⋊C4C22semidirect
25. (C2×D3)⋊Dic9C22semidirect
26. (C2×D9)⋊Dic3C22semidirect
40. C2×(C2×D9)⋊C4D3semidirect
100. (C2×D9)⋊C4C2×D3semidirect
131. C2×(C2×D3)⋊C4D9semidirect
187. C2×C3×C9#124semidirect
225. (C2×D3)⋊C4C2×D9semidirect
278. C3×C9C2×#124semidirect
339. C2×C9#885semidirect
403. C9C2×#885semidirect
431. C2×C3#63447semidirect
459. C3C2×#63447semidirect
463. C2#11721044direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#61942
C22×#61892
C2×C4×C3⋊Dic92
C2×C9×(C2×D3)⋊C42
C2×C3×(C2×D9)⋊C42
C2×(C2×D3)⋊Dic92
C2×(C2×D9)⋊Dic32
#117210442
C2×#639153
C2×#634473