Finite Group Databasebeta

← C2×#11721263 #13151976

Subgroups — C2×#11721263 #13151976

Order 1728.

1122 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×C9×#67C211yesmaximal, characteristic
2C22×#6189C211yesmaximal, characteristic, complemented
3C2×#88551C211yesmaximal, characteristic, complemented
4C2×C3×#3676C211yesmaximal, characteristic, complemented
5C2×(C2×D3)⋊Dic9C211yesmaximal, characteristic, complemented
6C2×(C2×D3)⋊Dic9C211yesmaximal, characteristic, complemented
7#11721263C218yesmaximal, direct product factor
8C2×Dic3×Dic9C211yesmaximal, characteristic, complemented
9C2×#63959—31nomaximal
10C2×#63498—31nomaximal
11C24×C3×C9C2211yescharacteristic
12C23×D3×C9C2211yescharacteristic
13C22×C3⋊Dic9C2211yescharacteristic
14C22×C3⋊Dic9C2211yescharacteristic
15C2×C9×#67—24no—
16C2×C9×#67C2214yes—
17C22×C9×Dic3C2211yescharacteristic
18C22×C3×Dic9C2211yescharacteristic
19C22×C3×Dic9C2211yescharacteristic
20C2×#6189—24no—
21C2×#6189C2214yescomplemented
22#88551C2214yescomplemented
23C3×#3676C2214yescomplemented
24(C2×D3)⋊Dic9C2214yescomplemented
25(C2×D3)⋊Dic9C2214yescomplemented
26Dic3×Dic9C2214yescomplemented
27C22×C3×#67D311yescharacteristic
28C22×#627—31no—
29C2×#3757—31no—
30C2×C3×#217—31no—
31C2×(C2×D3)⋊Dic3—31no—
32C2×(C2×D3)⋊Dic3—31no—
33#63959—38no—
34C2×Dic32—31no—
35C22×D4×C9—31no—
36C22×Dic18—31no—
37C2×#3676—31no—
38C2×#3676—31no—
39C2×#3676—31no—
40C2×#3676—61no—
41C2×#3676D311yescharacteristic, complemented
42#63498—38no—
43C2×C4×Dic9—31no—
44C23×C3×C9—22no—
45C23×C3×C9—22no—
46C23×C3×C9—22no—
47C23×C3×C9C2312yes—
48C23×C3×C9C2311yescharacteristic
49C22×D3×C9—22no—
50C22×D3×C9—22no—
51C22×D3×C9—22no—
52C22×D3×C9C2312yes—
53C2×C3⋊Dic9—22no—
54C2×C3⋊Dic9—22no—
55C2×C3⋊Dic9C2312yes—
56C2×C3⋊Dic9C2312yes—
57C9×#67—28no—
58C2×C9×Dic3D414yes—
59C2×C9×Dic3C2312yes—
60C2×C3×Dic9—22no—
61C2×C3×Dic9—22no—
62C2×C3×Dic9C2312yes—
63C2×C3×Dic9C2312yes—
64#6189—28no—
65C2×#936—91no—
66C2×#885—91no—
67C24×C32C2×D311yescharacteristic
68C23×C3×D3C2×D311yescharacteristic
69C22×C3⋊Dic3—31no—
70C22×C3⋊Dic3—31no—
71C2×C3×#67—24no—
72C2×C3×#67C2×D314yes—
73C22×C3×Dic3—31no—
74C22×C3×Dic3—31no—
75C22×C3×Dic3C2×D311yescharacteristic
76C2×#627—34no—
77C2×#627—64no—
78C24×C9—21no—
79C24×C9—31no—
80C24×C9C2×D311yescharacteristic
81#3757—34no—
82C3×#217—34no—
83(C2×D3)⋊Dic3—34no—
84(C2×D3)⋊Dic3—34no—
85Dic32—34no—
86C2×D4×C9—34no—
87C2×D4×C9—64no—
88C22×C4×C9—31no—
89C22×Dic9—31no—
90C22×Dic9—31no—
91C22×Dic9—61no—
92C22×Dic9—61no—
93C22×Dic9C2×D311yescharacteristic
94C22×Dic9C2×D311yescharacteristic
95C2×Dic18—34no—
96C2×Dic18—64no—
97#3676—34no—
98#3676—34no—
99#3676—34no—
100#3676—64no—
101#3676C2×D314yescomplemented
102C4×Dic9—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×D3×C9—22no—
112C2×D3×C9—22no—
113C2×D3×C9—22no—
114C2×D3×C9—44no—
115C3⋊Dic9—22no—
116C3⋊Dic9—22no—
117C9×Dic3C2×D414yes—
118C3×Dic9—22no—
119C3×Dic9—22no—
120C22×C3×D4—31no—
121C22×#67D911yescharacteristic
122C22×Dic6—91no—
123C22×Dic6—91no—
124C2×C3×C22⋊C4—91no—
125C2×#217—31no—
126C2×#217—91no—
127C2×#217—91no—
128C2×#217—91no—
129C2×#217—91no—
130C2×#217—181no—
131#936—98no—
132C2×(C2×D3)⋊C4—91no—
133C2×(C2×D3)⋊C4—91no—
134C2×C4×Dic3—91no—
135C2×C4×Dic3—91no—
136#885—98no—
137C23×C32—22no—
138C23×C32—22no—
139C23×C32—22no—
140C23×C32C22×D312yes—
141C23×C32C22×D311yescharacteristic
142C22×C3×D3—22no—
143C22×C3×D3—22no—
144C22×C3×D3—22no—
145C22×C3×D3C22×D312yes—
146C2×C3⋊Dic3—32no—
147C2×C3⋊Dic3—32no—
148C2×C3⋊Dic3—62no—
149C2×C3⋊Dic3—62no—
150C3×#67—28no—
151C2×C3×Dic3—32no—
152C2×C3×Dic3—32no—
153C2×C3×Dic3—62no—
154C2×C3×Dic3—62no—
155C2×C3×Dic3#6714yes—
156C2×C3×Dic3C22×D312yes—
157#627—68no—
158C23×C9—24no—
159C23×C9—22no—
160C23×C9—22no—
161C23×C9—22no—
162C23×C9—21no—
163C23×C9—22no—
164C23×C9—32no—
165C23×C9—42no—
166C23×C9—42no—
167C23×C9—62no—
168C23×C9—62no—
169C23×C9—62no—
170C23×C9C22×D312yes—
171C23×C9C22×D311yescharacteristic
172D4×C9—68no—
173C2×C4×C9—34no—
174C2×C4×C9—32no—
175C2×Dic9—22no—
176C2×Dic9—22no—
177C2×Dic9—32no—
178C2×Dic9—32no—
179C2×Dic9—62no—
180C2×Dic9—62no—
181C2×Dic9—62no—
182C2×Dic9—62no—
183C2×Dic9—122no—
184C2×Dic9—122no—
185C2×Dic9C22×D312yes—
186C2×Dic9C22×D312yes—
187Dic18—68no—
188C2×#124—271noSylow
189C2×C3×C9—42no—
190C2×C3×C9#12414yescomplemented
191C2×C3×C9C22×D411yescharacteristic
192C2×C3×C9C2×#4111yescharacteristic
193C2×C3×C9C2×#4111yescharacteristic
194D3×C9—42no—
195C24×C3—21no—
196C24×C3—31no—
197C24×C3C2×D911yescharacteristic
198C24×C3D3211yescharacteristic
199C23×D3C2×D911yescharacteristic
200C2×C3×D4—34no—
201C2×C3×D4—64no—
202C22×C3×C4—31no—
203C22×C3×C4—91no—
204C22×C3×C4—91no—
205C2×#67—24no—
206C2×#67C2×D914yes—
207C22×Dic3—31no—
208C22×Dic3—31no—
209C22×Dic3—91no—
210C22×Dic3—91no—
211C22×Dic3—91no—
212C22×Dic3—91no—
213C22×Dic3—181no—
214C22×Dic3—181no—
215C22×Dic3C2×D911yescharacteristic
216C2×Dic6—94no—
217C2×Dic6—94no—
218C2×Dic6—184no—
219C2×Dic6—184no—
220C3×C22⋊C4—94no—
221#217—34no—
222#217—94no—
223#217—94no—
224#217—94no—
225#217—94no—
226#217—184no—
227(C2×D3)⋊C4—94no—
228(C2×D3)⋊C4—94no—
229C4×Dic3—94no—
230C4×Dic3—94no—
231C22×C32—22no—
232C22×C32—22no—
233C22×C32—22no—
234C22×C32—44no—
235C22×C32#23712yes—
236C22×C32#23712yes—
237C22×C32C2×#6712yes—
238C22×C32C23×D311yescharacteristic
239C2×C3×D3—22no—
240C2×C3×D3—22no—
241C2×C3×D3—22no—
242C2×C3×D3—44no—
243C3⋊Dic3—62no—
244C3⋊Dic3—62no—
245C3×Dic3—62no—
246C3×Dic3—62no—
247C3×Dic3C2×#6714yes—
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—62no—
261C22×C9—62no—
262C22×C9—62no—
263C22×C9—124no—
264C22×C9#23512yes—
265C22×C9D3×D412yes—
266C22×C9#23712yes—
267C22×C9C23×D311yescharacteristic
268C4×C9—34no—
269Dic9—22no—
270Dic9—22no—
271Dic9—62no—
272Dic9—62no—
273Dic9—122no—
274Dic9—122no—
275C22×Q8—271no—
276C22×D4—31no—
277#124—278no—
278C2×C22⋊C4—91no—
279C2×C22⋊C4—271no—
280C2×C22⋊C4—271no—
281C2×C22⋊C4—271no—
282C2×C42—271no—
283C3×C9C2×#12411yescharacteristic, 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—32no—
294C23×C3—42no—
295C23×C3—42no—
296C23×C3—62no—
297C23×C3—62no—
298C23×C3—62no—
299C23×C3C22×D912yes—
300C23×C3C22×D911yescharacteristic
301C23×C3C2×D3212yes—
302C23×C3C2×D3211yescharacteristic
303C22×D3—22no—
304C22×D3—22no—
305C22×D3—22no—
306C22×D3C22×D912yes—
307C3×D4—68no—
308C2×C3×C4—32no—
309C2×C3×C4—34no—
310C2×C3×C4—92no—
311C2×C3×C4—92no—
312C2×C3×C4—182no—
313C2×C3×C4—182no—
314#67—28no—
315C2×Dic3—32no—
316C2×Dic3—32no—
317C2×Dic3—62no—
318C2×Dic3—62no—
319C2×Dic3—92no—
320C2×Dic3—92no—
321C2×Dic3—92no—
322C2×Dic3—92no—
323C2×Dic3—182no—
324C2×Dic3—182no—
325C2×Dic3—182no—
326C2×Dic3—182no—
327C2×Dic3—182no—
328C2×Dic3—182no—
329C2×Dic3—362no—
330C2×Dic3—362no—
331C2×Dic3#61114yes—
332C2×Dic3C22×D912yes—
333Dic6—188no—
334Dic6—188no—
335C2×C32—42no—
336C2×C32#93614yes—
337C2×C32C2×#23711yescharacteristic
338C2×C32C2×#23711yescharacteristic
339C2×C32C22×#6711yescharacteristic
340C3×D3—42no—
341C2×C9—21no—
342C2×C9—21no—
343C2×C9—21no—
344C2×C9—24no—
345C2×C9—44no—
346C2×C9—42no—
347C2×C9—122no—
348C2×C9#88514yescomplemented
349C2×C9C2×#23511yescharacteristic
350C2×C9C2×D3×D411yescharacteristic
351C2×C9C2×#23711yescharacteristic
352C24—31no—
353C24D3×D911yescharacteristic
354C2×Q8—274no—
355C2×Q8—544no—
356C2×D4—34no—
357C2×D4—64no—
358C22×C4—31no—
359C22×C4—91no—
360C22×C4—91no—
361C22×C4—271no—
362C22×C4—271no—
363C22⋊C4—94no—
364C22⋊C4—274no—
365C22⋊C4—274no—
366C22⋊C4—274no—
367C42—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—62no—
385C22×C3—62no—
386C22×C3—62no—
387C22×C3—124no—
388C22×C3#369912yes—
389C22×C3#369912yes—
390C22×C3C2×#61112yes—
391C22×C3C23×D911yescharacteristic
392C22×C3#380412yes—
393C22×C3#380512yes—
394C22×C3D3×#6712yes—
395C22×C3C22×D3211yescharacteristic
396C2×D3—22no—
397C2×D3—22no—
398C2×D3—22no—
399C2×D3—44no—
400C3×C4—34no—
401C3×C4—182no—
402C3×C4—182no—
403Dic3—62no—
404Dic3—62no—
405Dic3—182no—
406Dic3—182no—
407Dic3—182no—
408Dic3—182no—
409Dic3—362no—
410Dic3—362no—
411Dic3C2×#61114yes—
412C32C2×#93611yescharacteristic
413C9—21no—
414C9C2×#88511yescharacteristic, complemented
415C23—22no—
416C23—22no—
417C23—22no—
418C23—32no—
419C23—62no—
420C23—62no—
421C23—62no—
422C23C2×D3×D912yes—
423C23C2×D3×D911yescharacteristic
424Q8—548no—
425D4—68no—
426C2×C4—32no—
427C2×C4—34no—
428C2×C4—92no—
429C2×C4—92no—
430C2×C4—182no—
431C2×C4—182no—
432C2×C4—272no—
433C2×C4—272no—
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—122no—
444C2×C3#6349814yescomplemented
445C2×C3C2×#369911yescharacteristic
446C2×C3C2×#369911yescharacteristic
447C2×C3C22×#61111yescharacteristic
448C2×C3#6395914yes—
449C2×C3C2×#380411yescharacteristic
450C2×C3C2×#380511yescharacteristic
451C2×C3C2×D3×#6711yescharacteristic
452D3—42no—
453C22—22no—
454C22—22no—
455C22—22no—
456C22—44no—
457C22—62no—
458C22—62no—
459C22—62no—
460C22—124no—
461C22#8866212yes—
462C22#8866312yes—
463C22D3×#61112yes—
464C22C22×D3×D911yescharacteristic
465C4—34no—
466C4—182no—
467C4—182no—
468C4—542no—
469C4—542no—
470C3—21no—
471C3C2×#6349811yescharacteristic, complemented
472C3C2×#6395911yescharacteristic
473C2—42no—
474C2—122no—
475C2#1172126314yesdirect product factor
476C2C2×#8866211yescharacteristic
477C2C2×#8866311yescharacteristic
478C2C2×D3×#61111yescharacteristic
subgroupG/H
centerC23C2×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC24×C3×C9C22
socleC23×C32C22×D3
prime radicalC22×C3×#67D3
hypercenterC24D3×D9
hypocenterC3×C9C2×#124
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×#6189C2semidirect
3. C2×#88551C2semidirect
4. C2×C3×#3676C2semidirect
5. C2×(C2×D3)⋊Dic9C2semidirect
6. C2×(C2×D3)⋊Dic9C2semidirect
7. #11721263C2direct (N × H)
8. C2×Dic3×Dic9C2semidirect
21. C2×#6189C22semidirect
22. #88551C22semidirect
23. C3×#3676C22semidirect
24. (C2×D3)⋊Dic9C22semidirect
25. (C2×D3)⋊Dic9C22semidirect
26. Dic3×Dic9C22semidirect
41. C2×#3676D3semidirect
101. #3676C2×D3semidirect
190. C2×C3×C9#124semidirect
283. C3×C9C2×#124semidirect
348. C2×C9#885semidirect
414. C9C2×#885semidirect
444. C2×C3#63498semidirect
471. C3C2×#63498semidirect
475. C2#11721263direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C9×#672
C22×#61892
C2×#885512
C2×C3×#36762
C2×(C2×D3)⋊Dic92
#117212632
C2×Dic3×Dic92
C2×#639593
C2×#634983