Finite Group Databasebeta

← C2×#11720985 #13150190

Subgroups — C2×#11720985 #13150190

Order 1728.

1098 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×Dic18C211yesmaximal, characteristic, complemented
2C22×(C3⋊D9)⋊C4C211yesmaximal, characteristic
3C2×#88536C211yesmaximal, characteristic
4C2×C9×Dic3⋊C4C211yesmaximal, characteristic, complemented
5C2×#88445C211yesmaximal, characteristic
6C2×#88442C211yesmaximal, characteristic, complemented
7C2×Dic9⋊Dic3C211yesmaximal, characteristic, complemented
8#11720985C218yesmaximal, direct product factor
9C2×#63888—31nomaximal
10C2×#63458—31nomaximal
11C23×C3⋊D9C2211yescharacteristic
12C22×C3×C4×C9C2211yescharacteristic
13C22×C3⋊Dic9C2211yescharacteristic
14C22×C9×Dic3C2211yescharacteristic
15C22×C9×Dic3C2211yescharacteristic
16C22×C3×Dic9C2211yescharacteristic
17C22×C3×Dic9C2211yescharacteristic
18C2×C3×Dic18—24no—
19C2×C3×Dic18C2214yescomplemented
20C2×(C3⋊D9)⋊C4—24no—
21C2×(C3⋊D9)⋊C4C2214yes—
22#88536C2214yes—
23C9×Dic3⋊C4C2214yescomplemented
24#88445C2214yes—
25#88442C2214yescomplemented
26Dic9⋊Dic3C2214yescomplemented
27C22×C3×Dic6—31no—
28C22×#624—31no—
29C2×#3752—31no—
30C2×C3×Dic3⋊C4D311yescharacteristic
31C2×#3724—31no—
32C2×Dic3⋊Dic3—31no—
33C2×#3722—31no—
34#63888—38no—
35C22×C4×D9—31no—
36C22×Dic18D311yescharacteristic, complemented
37C2×C9×C4⋊C4—31no—
38C2×(C2×D9)⋊C4—31no—
39C2×(C2×D9)⋊C4—31no—
40C2×(C2×D9)⋊C4—61no—
41C2×Dic9⋊C4—31no—
42C2×Dic9⋊C4—31no—
43#63458—38no—
44C23×C3×C9C2311yescharacteristic
45C22×C3⋊D9—22no—
46C22×C3⋊D9—22no—
47C22×C3⋊D9Q814yescomplemented
48C22×C3⋊D9C2312yes—
49C2×C3×C4×C9—22no—
50C2×C3×C4×C9C2312yes—
51C2×C3⋊Dic9—22no—
52C2×C3⋊Dic9C2312yes—
53C2×C9×Dic3—22no—
54C2×C9×Dic3—22no—
55C2×C9×Dic3C2312yes—
56C2×C9×Dic3C2312yes—
57C2×C3×Dic9—22no—
58C2×C3×Dic9D414yes—
59C2×C3×Dic9C2312yes—
60C2×C3×Dic9C2312yes—
61C3×Dic18—28no—
62(C3⋊D9)⋊C4—28no—
63C2×#946—91no—
64C2×#896—91no—
65C23×C3⋊D3—31no—
66C22×C32×C4C2×D311yescharacteristic
67C22×C3⋊Dic3—31no—
68C22×C3×Dic3—31no—
69C22×C3×Dic3—31no—
70C22×C3×Dic3C2×D311yescharacteristic
71C22×C3×Dic3C2×D311yescharacteristic
72C2×C3×Dic6—34no—
73C2×C3×Dic6—64no—
74C2×#624—34no—
75C2×#624—64no—
76C23×D9—31no—
77C23×D9—61no—
78#3752—34no—
79C3×Dic3⋊C4C2×D314yes—
80#3724—34no—
81Dic3⋊Dic3—34no—
82#3722—34no—
83C22×C4×C9—21no—
84C22×C4×C9—31no—
85C22×C4×C9—31no—
86C22×C4×C9C2×D311yescharacteristic
87C22×Dic9—31no—
88C22×Dic9—61no—
89C22×Dic9C2×D311yescharacteristic
90C22×Dic9C2×D311yescharacteristic
91C2×C4×D9—34no—
92C2×C4×D9—64no—
93C2×Dic18—24no—
94C2×Dic18C2×D314yescomplemented
95C9×C4⋊C4—34no—
96(C2×D9)⋊C4—34no—
97(C2×D9)⋊C4—34no—
98(C2×D9)⋊C4—64no—
99Dic9⋊C4—34no—
100Dic9⋊C4—34no—
101C22×C3×C9C2×D412yes—
102C22×C3×C9C2×Q812yes—
103C22×C3×C9#4112yes—
104C22×C3×C9C2411yescharacteristic
105C2×C3⋊D9—28no—
106C2×C3⋊D9—22no—
107C2×C3⋊D9—22no—
108C2×C3⋊D9C2×Q814yescomplemented
109C3×C4×C9—22no—
110C3⋊Dic9—22no—
111C9×Dic3—22no—
112C9×Dic3—22no—
113C3×Dic9—22no—
114C3×Dic9C2×D414yes—
115C22×C3×Q8—91no—
116C22×C4×D3—91no—
117C22×C4×D3—91no—
118C22×Dic6—31no—
119C2×C3×C4⋊C4—31no—
120#946—98no—
121C2×(C2×D3)⋊C4—91no—
122C2×(C2×D3)⋊C4—91no—
123C2×(C2×D3)⋊C4—91no—
124C2×(C2×D3)⋊C4—91no—
125C2×(C2×D3)⋊C4—181no—
126C2×C4⋊Dic3—91no—
127C2×Dic3⋊C4—91no—
128C2×Dic3⋊C4—91no—
129C2×Dic3⋊C4—91no—
130C2×Dic3⋊C4D911yescharacteristic, complemented
131#896—98no—
132C23×C32C22×D311yescharacteristic
133C22×C3⋊D3—34no—
134C22×C3⋊D3—32no—
135C22×C3⋊D3—62no—
136C22×C3⋊D3—62no—
137C2×C32×C4—22no—
138C2×C32×C4C22×D312yes—
139C2×C3⋊Dic3—32no—
140C2×C3⋊Dic3—62no—
141C2×C3×Dic3—22no—
142C2×C3×Dic3—22no—
143C2×C3×Dic3—34no—
144C2×C3×Dic3—32no—
145C2×C3×Dic3—32no—
146C2×C3×Dic3—62no—
147C2×C3×Dic3C22×D312yes—
148C2×C3×Dic3C22×D312yes—
149C3×Dic6—68no—
150#624—68no—
151C23×C9—21no—
152C23×C9C22×D311yescharacteristic
153C22×D9—34no—
154C22×D9—32no—
155C22×D9—62no—
156C22×D9—64no—
157C22×D9—62no—
158C22×D9—62no—
159C22×D9—122no—
160C22×D9—122no—
161C2×C4×C9—22no—
162C2×C4×C9—22no—
163C2×C4×C9—32no—
164C2×C4×C9—32no—
165C2×C4×C9—42no—
166C2×C4×C9—62no—
167C2×C4×C9—62no—
168C2×C4×C9C22×D312yes—
169C2×Dic9—22no—
170C2×Dic9—32no—
171C2×Dic9—62no—
172C2×Dic9—62no—
173C2×Dic9—122no—
174C2×Dic9#6714yes—
175C2×Dic9C22×D312yes—
176C2×Dic9C22×D312yes—
177C4×D9—68no—
178Dic18—28no—
179C2×C22⋊Q8—271noSylow
180C2×C3×C9C22⋊Q814yescomplemented
181C2×C3×C9C22×D411yescharacteristic
182C2×C3×C9C22×Q811yescharacteristic
183C2×C3×C9C2×#4111yescharacteristic
184C3⋊D9—24no—
185C23×D3—91no—
186C23×D3—91no—
187C23×D3—181no—
188C2×C3×Q8—94no—
189C2×C3×Q8—184no—
190C22×C3×C4—21no—
191C22×C3×C4—31no—
192C22×C3×C4—31no—
193C22×C3×C4—91no—
194C22×C3×C4—91no—
195C22×C3×C4C2×D911yescharacteristic
196C22×C3×C4D3211yescharacteristic
197C22×Dic3—31no—
198C22×Dic3—31no—
199C22×Dic3—91no—
200C22×Dic3—91no—
201C22×Dic3—181no—
202C22×Dic3C2×D911yescharacteristic
203C22×Dic3C2×D911yescharacteristic
204C2×C4×D3—94no—
205C2×C4×D3—94no—
206C2×C4×D3—184no—
207C2×C4×D3—184no—
208C2×Dic6—34no—
209C2×Dic6—64no—
210C3×C4⋊C4—34no—
211(C2×D3)⋊C4—94no—
212(C2×D3)⋊C4—94no—
213(C2×D3)⋊C4—94no—
214(C2×D3)⋊C4—94no—
215(C2×D3)⋊C4—184no—
216C4⋊Dic3—94no—
217Dic3⋊C4—94no—
218Dic3⋊C4—94no—
219Dic3⋊C4—94no—
220Dic3⋊C4C2×D914yescomplemented
221C22×C32#23512yes—
222C22×C32D3×D412yes—
223C22×C32D3×Q812yes—
224C22×C32C23×D311yescharacteristic
225C2×C3⋊D3—34no—
226C2×C3⋊D3—68no—
227C2×C3⋊D3—62no—
228C2×C3⋊D3—62no—
229C32×C4—22no—
230C3⋊Dic3—62no—
231C3×Dic3—22no—
232C3×Dic3—22no—
233C3×Dic3—34no—
234C3×Dic3—62no—
235C22×C9—21no—
236C22×C9—22no—
237C22×C9—22no—
238C22×C9—22no—
239C22×C9D3×Q812yes—
240C22×C9#23912yes—
241C22×C9C2×#6712yes—
242C22×C9C23×D311yescharacteristic
243C2×D9—34no—
244C2×D9—64no—
245C2×D9—68no—
246C2×D9—62no—
247C2×D9—62no—
248C2×D9—122no—
249C2×D9—122no—
250C2×D9—128no—
251C4×C9—22no—
252C4×C9—42no—
253C4×C9—62no—
254C4×C9—62no—
255Dic9—22no—
256Dic9—62no—
257Dic9—122no—
258Dic9C2×#6714yes—
259C22×Q8—91no—
260C23×C4—271no—
261C22⋊Q8—278no—
262C2×C4⋊C4—31no—
263C2×C4⋊C4—271no—
264C2×C4⋊C4—271no—
265C2×C22⋊C4—271no—
266C2×C22⋊C4—271no—
267C3×C9C2×C22⋊Q811yescharacteristic, complemented, Sylow
268C23×C3—21no—
269C23×C3C22×D911yescharacteristic
270C23×C3C2×D3211yescharacteristic
271C22×D3—92no—
272C22×D3—94no—
273C22×D3—92no—
274C22×D3—94no—
275C22×D3—182no—
276C22×D3—184no—
277C22×D3—182no—
278C22×D3—182no—
279C22×D3—182no—
280C22×D3—182no—
281C22×D3—362no—
282C22×D3—362no—
283C3×Q8—188no—
284C2×C3×C4—22no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—32no—
288C2×C3×C4—32no—
289C2×C3×C4—42no—
290C2×C3×C4—62no—
291C2×C3×C4—62no—
292C2×C3×C4—94no—
293C2×C3×C4—92no—
294C2×C3×C4—92no—
295C2×C3×C4—182no—
296C2×C3×C4C22×D912yes—
297C2×C3×C4C2×D3212yes—
298C2×Dic3—22no—
299C2×Dic3—22no—
300C2×Dic3—32no—
301C2×Dic3—34no—
302C2×Dic3—32no—
303C2×Dic3—62no—
304C2×Dic3—92no—
305C2×Dic3—92no—
306C2×Dic3—182no—
307C2×Dic3—182no—
308C2×Dic3—182no—
309C2×Dic3—362no—
310C2×Dic3C22×D912yes—
311C2×Dic3C22×D912yes—
312C4×D3—188no—
313C4×D3—188no—
314Dic6—68no—
315C2×C32#89614yes—
316C2×C32C2×#23511yescharacteristic
317C2×C32C2×D3×D411yescharacteristic
318C2×C32C2×D3×Q811yescharacteristic
319C3⋊D3—64no—
320C2×C9—21no—
321C2×C9—21no—
322C2×C9—21no—
323C2×C9—24no—
324C2×C9#94614yescomplemented
325C2×C9C2×D3×Q811yescharacteristic
326C2×C9C2×#23911yescharacteristic
327C2×C9C22×#6711yescharacteristic
328D9—64no—
329D9—124no—
330C24—271no—
331C2×Q8—94no—
332C2×Q8—184no—
333C22×C4—31no—
334C22×C4—31no—
335C22×C4—91no—
336C22×C4—91no—
337C22×C4—274no—
338C22×C4—271no—
339C22×C4—544no—
340C22×C4D3×D911yescharacteristic
341C4⋊C4—34no—
342C4⋊C4—274no—
343C4⋊C4—274no—
344C22⋊C4—274no—
345C22⋊C4—274no—
346C22×C3—22no—
347C22×C3—21no—
348C22×C3—22no—
349C22×C3—22no—
350C22×C3#369712yes—
351C22×C3D4×D912yes—
352C22×C3Q8×D912yes—
353C22×C3C23×D911yescharacteristic
354C22×C3#379712yes—
355C22×C3#379912yes—
356C22×C3D3×#6712yes—
357C22×C3C22×D3211yescharacteristic
358C2×D3—94no—
359C2×D3—94no—
360C2×D3—184no—
361C2×D3—182no—
362C2×D3—182no—
363C2×D3—188no—
364C2×D3—182no—
365C2×D3—182no—
366C2×D3—188no—
367C2×D3—368no—
368C2×D3—362no—
369C2×D3—362no—
370C3×C4—22no—
371C3×C4—22no—
372C3×C4—42no—
373C3×C4—62no—
374C3×C4—62no—
375C3×C4—94no—
376C3×C4—182no—
377Dic3—22no—
378Dic3—22no—
379Dic3—34no—
380Dic3—62no—
381Dic3—182no—
382Dic3—182no—
383Dic3—362no—
384C32C2×#89611yescharacteristic
385C9—21no—
386C9C2×#94611yescharacteristic, complemented
387C23—274no—
388C23—272no—
389C23—542no—
390C23—542no—
391C23C2×D3×D911yescharacteristic
392Q8—188no—
393C2×C4—22no—
394C2×C4—32no—
395C2×C4—32no—
396C2×C4—62no—
397C2×C4—62no—
398C2×C4—92no—
399C2×C4—94no—
400C2×C4—92no—
401C2×C4—182no—
402C2×C4—272no—
403C2×C4—548no—
404C2×C4—542no—
405C2×C4C2×D3×D912yes—
406C2×C3—24no—
407C2×C3—21no—
408C2×C3—21no—
409C2×C3—21no—
410C2×C3#6345814yescomplemented
411C2×C3C2×#369711yescharacteristic
412C2×C3C2×D4×D911yescharacteristic
413C2×C3C2×Q8×D911yescharacteristic
414C2×C3#6388814yes—
415C2×C3C2×#379711yescharacteristic
416C2×C3C2×#379911yescharacteristic
417C2×C3C2×D3×#6711yescharacteristic
418D3—184no—
419D3—184no—
420D3—364no—
421C22—274no—
422C22—548no—
423C22—542no—
424C22—542no—
425C22(C3⋊D9)⋊Q812yes—
426C22#8864212yes—
427C22D9×#6712yes—
428C22C22×D3×D911yescharacteristic
429C4—22no—
430C4—62no—
431C4—62no—
432C4—94no—
433C4—182no—
434C4—542no—
435C3—21no—
436C3C2×#6345811yescharacteristic, complemented
437C3C2×#6388811yescharacteristic
438C2—544no—
439C2#1172098514yesdirect product factor
440C2C2×(C3⋊D9)⋊Q811yescharacteristic
441C2C2×#8864211yescharacteristic
442C2C2×D9×#6711yescharacteristic
subgroupG/H
centerC23C2×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC23×C32C22×D3
prime radicalC23×C3⋊D9C22
hypercenterC22×C4D3×D9
hypocenterC3×C9C2×C22⋊Q8
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C3×Dic18C2semidirect
4. C2×C9×Dic3⋊C4C2semidirect
6. C2×#88442C2semidirect
7. C2×Dic9⋊Dic3C2semidirect
8. #11720985C2direct (N × H)
19. C2×C3×Dic18C22semidirect
23. C9×Dic3⋊C4C22semidirect
25. #88442C22semidirect
26. Dic9⋊Dic3C22semidirect
36. C22×Dic18D3semidirect
47. C22×C3⋊D9Q8semidirect
94. C2×Dic18C2×D3semidirect
108. C2×C3⋊D9C2×Q8semidirect
130. C2×Dic3⋊C4D9semidirect
180. C2×C3×C9C22⋊Q8semidirect
220. Dic3⋊C4C2×D9semidirect
267. C3×C9C2×C22⋊Q8semidirect
324. C2×C9#946semidirect
386. C9C2×#946semidirect
410. C2×C3#63458semidirect
436. C3C2×#63458semidirect
439. C2#11720985direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×Dic182
C22×(C3⋊D9)⋊C42
C2×#885362
C2×C9×Dic3⋊C42
C2×#884452
C2×#884422
C2×Dic9⋊Dic32
#117209852
C2×#638883
C2×#634583