Finite Group Databasebeta

← C2×#11720980 #13150173

Subgroups — C2×#11720980 #13150173

Order 1728.

1490 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C3⋊D9C211yesmaximal, characteristic
2C22×(C3⋊D9)⋊C4C211yesmaximal, characteristic
3C22×(C3⋊D9)⋊C4C211yesmaximal, characteristic
4C2×C9×Dic3⋊C4C211yesmaximal, characteristic, complemented
5C2×C3×Dic9⋊C4C211yesmaximal, characteristic, complemented
6C2×Dic3⋊Dic9C211yesmaximal, characteristic, complemented
7C2×Dic9⋊Dic3C211yesmaximal, characteristic, complemented
8#11720980C218yesmaximal, direct product factor
9C2×#63883—31nomaximal
10C2×C4⋊C4×D9—31nomaximal
11C23×C3⋊D9C2211yescharacteristic
12C22×C3×C4×C9C2211yescharacteristic
13C22×C3⋊Dic9C2211yescharacteristic
14C2×C4×C3⋊D9—24no—
15C2×C4×C3⋊D9C2214yes—
16C22×C9×Dic3C2211yescharacteristic
17C22×C9×Dic3C2211yescharacteristic
18C22×C3×Dic9C2211yescharacteristic
19C22×C3×Dic9C2211yescharacteristic
20C2×(C3⋊D9)⋊C4—24no—
21C2×(C3⋊D9)⋊C4C418yescomplemented
22C2×(C3⋊D9)⋊C4C2214yes—
23C2×(C3⋊D9)⋊C4C2214yes—
24C9×Dic3⋊C4C2214yescomplemented
25C3×Dic9⋊C4C2214yescomplemented
26Dic3⋊Dic9C2214yescomplemented
27Dic9⋊Dic3C2214yescomplemented
28C22×C4×C3⋊D3—31no—
29C22×#624—31no—
30C22×#624—31no—
31C2×C3×Dic3⋊C4—31no—
32C2×C3×Dic3⋊C4D311yescharacteristic
33C2×Dic3⋊Dic3—31no—
34C2×Dic3⋊Dic3—31no—
35#63883—38no—
36C22×C4×D9—31no—
37C22×C4×D9—31no—
38C22×C4×D9—31no—
39C22×C4×D9—61no—
40C2×C9×C4⋊C4—31no—
41C2×C4⋊Dic9—31no—
42C2×Dic9⋊C4—31no—
43C2×Dic9⋊C4D311yescharacteristic, complemented
44C4⋊C4×D9—38no—
45C23×C3×C9C2311yescharacteristic
46C22×C3⋊D9C2×C414yes—
47C22×C3⋊D9D414yes—
48C22×C3⋊D9Q814yes—
49C22×C3⋊D9C2312yes—
50C2×C3×C4×C9—22no—
51C2×C3×C4×C9C2312yes—
52C2×C3⋊Dic9—22no—
53C2×C3⋊Dic9C2312yes—
54C4×C3⋊D9—28no—
55C2×C9×Dic3—22no—
56C2×C9×Dic3C2×C414yescomplemented
57C2×C9×Dic3C2312yes—
58C2×C9×Dic3C2312yes—
59C2×C3×Dic9—22no—
60C2×C3×Dic9C2×C414yescomplemented
61C2×C3×Dic9C2312yes—
62C2×C3×Dic9C2312yes—
63(C3⋊D9)⋊C4—28no—
64(C3⋊D9)⋊C4C2×C4116yescomplemented
65C2×D3×C4⋊C4—91no—
66C2×D3×C4⋊C4—91no—
67C23×C3⋊D3—31no—
68C22×C32×C4C2×D311yescharacteristic
69C22×C3⋊Dic3—31no—
70C2×C4×C3⋊D3—34no—
71C2×C4×C3⋊D3—64no—
72C22×C3×Dic3—31no—
73C22×C3×Dic3—31no—
74C22×C3×Dic3C2×D311yescharacteristic
75C22×C3×Dic3C2×D311yescharacteristic
76C2×#624—38no—
77C2×#624—34no—
78C2×#624—34no—
79C2×#624—64no—
80C23×D9—31no—
81C23×D9—61no—
82C3×Dic3⋊C4—34no—
83C3×Dic3⋊C4C2×D314yes—
84Dic3⋊Dic3—34no—
85Dic3⋊Dic3—34no—
86C22×C4×C9—21no—
87C22×C4×C9—31no—
88C22×C4×C9—31no—
89C22×C4×C9C2×D311yescharacteristic
90C22×Dic9—31no—
91C22×Dic9—61no—
92C22×Dic9C2×D311yescharacteristic
93C22×Dic9C2×D311yescharacteristic
94C2×C4×D9—38no—
95C2×C4×D9—34no—
96C2×C4×D9—34no—
97C2×C4×D9—34no—
98C2×C4×D9—64no—
99C2×C4×D9—64no—
100C2×C4×D9—64no—
101C2×C4×D9—68no—
102C9×C4⋊C4—34no—
103C4⋊Dic9—34no—
104Dic9⋊C4—34no—
105Dic9⋊C4C2×D314yescomplemented
106C22×C3×C9C22×C412yes—
107C22×C3×C9C2×D412yes—
108C22×C3×C9C2×Q812yes—
109C22×C3×C9C2411yescharacteristic
110C2×C3⋊D9C4⋊C4116yescomplemented
111C2×C3⋊D9C22×C414yes—
112C2×C3⋊D9C2×D414yes—
113C2×C3⋊D9C2×Q814yes—
114C3×C4×C9—22no—
115C3⋊Dic9—22no—
116C9×Dic3—22no—
117C9×Dic3C22×C414yescomplemented
118C3×Dic9—22no—
119C3×Dic9C22×C414yescomplemented
120C22×C4×D3—91no—
121C22×C4×D3—91no—
122C22×C4×D3—91no—
123C22×C4×D3—91no—
124C22×C4×D3—91no—
125C22×C4×D3—91no—
126C22×C4×D3—181no—
127C2×C3×C4⋊C4—31no—
128C2×C3×C4⋊C4—91no—
129C2×C4⋊Dic3—91no—
130C2×C4⋊Dic3—91no—
131C2×Dic3⋊C4—31no—
132C2×Dic3⋊C4—91no—
133C2×Dic3⋊C4—91no—
134C2×Dic3⋊C4D911yescharacteristic, complemented
135D3×C4⋊C4—98no—
136D3×C4⋊C4—98no—
137C23×C32C22×D311yescharacteristic
138C22×C3⋊D3—34no—
139C22×C3⋊D3—34no—
140C22×C3⋊D3—34no—
141C22×C3⋊D3—32no—
142C2×C32×C4—22no—
143C2×C32×C4C22×D312yes—
144C2×C3⋊Dic3—32no—
145C2×C3⋊Dic3—62no—
146C4×C3⋊D3—68no—
147C2×C3×Dic3—22no—
148C2×C3×Dic3—34no—
149C2×C3×Dic3—32no—
150C2×C3×Dic3—32no—
151C2×C3×Dic3—62no—
152C2×C3×Dic3C4×D314yes—
153C2×C3×Dic3C22×D312yes—
154C2×C3×Dic3C22×D312yes—
155#624—316no—
156#624—68no—
157C23×C9—21no—
158C23×C9C22×D311yescharacteristic
159C22×D9—34no—
160C22×D9—34no—
161C22×D9—34no—
162C22×D9—32no—
163C22×D9—64no—
164C22×D9—62no—
165C22×D9—64no—
166C22×D9—64no—
167C2×C4×C9—24no—
168C2×C4×C9—22no—
169C2×C4×C9—22no—
170C2×C4×C9—34no—
171C2×C4×C9—32no—
172C2×C4×C9—32no—
173C2×C4×C9—62no—
174C2×C4×C9C22×D312yes—
175C2×Dic9—22no—
176C2×Dic9—32no—
177C2×Dic9—62no—
178C2×Dic9—64no—
179C2×Dic9—62no—
180C2×Dic9C4×D314yescomplemented
181C2×Dic9C22×D312yes—
182C2×Dic9C22×D312yes—
183C4×D9—316no—
184C4×D9—68no—
185C4×D9—68no—
186C4×D9—616no—
187C22×C4⋊C4—271noSylow
188C2×C3×C9C2×C4⋊C414yescomplemented
189C2×C3×C9C23×C411yescharacteristic
190C2×C3×C9C22×D411yescharacteristic
191C2×C3×C9C22×Q811yescharacteristic
192C3⋊D9C2×C4⋊C418yescomplemented
193C23×D3—91no—
194C23×D3—91no—
195C23×D3—181no—
196C22×C3×C4—21no—
197C22×C3×C4—31no—
198C22×C3×C4—31no—
199C22×C3×C4—91no—
200C22×C3×C4—91no—
201C22×C3×C4C2×D911yescharacteristic
202C22×C3×C4D3211yescharacteristic
203C22×Dic3—31no—
204C22×Dic3—31no—
205C22×Dic3—91no—
206C22×Dic3—91no—
207C22×Dic3—181no—
208C22×Dic3C2×D911yescharacteristic
209C22×Dic3C2×D911yescharacteristic
210C2×C4×D3—94no—
211C2×C4×D3—98no—
212C2×C4×D3—98no—
213C2×C4×D3—94no—
214C2×C4×D3—94no—
215C2×C4×D3—94no—
216C2×C4×D3—94no—
217C2×C4×D3—94no—
218C2×C4×D3—184no—
219C2×C4×D3—184no—
220C2×C4×D3—184no—
221C2×C4×D3—184no—
222C2×C4×D3—184no—
223C2×C4×D3—188no—
224C3×C4⋊C4—34no—
225C3×C4⋊C4—94no—
226C4⋊Dic3—94no—
227C4⋊Dic3—94no—
228Dic3⋊C4—34no—
229Dic3⋊C4—94no—
230Dic3⋊C4—94no—
231Dic3⋊C4C2×D914yescomplemented
232C22×C32C2×C4×D312yes—
233C22×C32D3×D412yes—
234C22×C32D3×Q812yes—
235C22×C32C23×D311yescharacteristic
236C2×C3⋊D3—316no—
237C2×C3⋊D3—34no—
238C2×C3⋊D3—34no—
239C2×C3⋊D3—34no—
240C32×C4—22no—
241C3⋊Dic3—62no—
242C3×Dic3—22no—
243C3×Dic3—34no—
244C3×Dic3—62no—
245C3×Dic3C2×C4×D314yes—
246C22×C9—21no—
247C22×C9—22no—
248C22×C9—22no—
249C22×C9—22no—
250C22×C9C2×C4×D312yes—
251C22×C9D3×D412yes—
252C22×C9D3×Q812yes—
253C22×C9C23×D311yescharacteristic
254C2×D9—316no—
255C2×D9—34no—
256C2×D9—34no—
257C2×D9—34no—
258C2×D9—616no—
259C2×D9—64no—
260C2×D9—64no—
261C2×D9—64no—
262C4×C9—24no—
263C4×C9—22no—
264C4×C9—34no—
265C4×C9—62no—
266Dic9—22no—
267Dic9—64no—
268Dic9—62no—
269Dic9C2×C4×D314yescomplemented
270C23×C4—271no—
271C23×C4—271no—
272C23×C4—271no—
273C2×C4⋊C4—31no—
274C2×C4⋊C4—91no—
275C2×C4⋊C4—278no—
276C2×C4⋊C4—271no—
277C2×C4⋊C4—271no—
278C3×C9C22×C4⋊C411yescharacteristic, complemented, Sylow
279C23×C3—21no—
280C23×C3C22×D911yescharacteristic
281C23×C3C2×D3211yescharacteristic
282C22×D3—92no—
283C22×D3—94no—
284C22×D3—94no—
285C22×D3—94no—
286C22×D3—92no—
287C22×D3—94no—
288C22×D3—94no—
289C22×D3—94no—
290C22×D3—182no—
291C22×D3—184no—
292C22×D3—184no—
293C22×D3—184no—
294C2×C3×C4—24no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—32no—
299C2×C3×C4—34no—
300C2×C3×C4—32no—
301C2×C3×C4—62no—
302C2×C3×C4—94no—
303C2×C3×C4—92no—
304C2×C3×C4—92no—
305C2×C3×C4—182no—
306C2×C3×C4C22×D912yes—
307C2×C3×C4C2×D3212yes—
308C2×Dic3—22no—
309C2×Dic3—32no—
310C2×Dic3—34no—
311C2×Dic3—32no—
312C2×Dic3—62no—
313C2×Dic3—92no—
314C2×Dic3—92no—
315C2×Dic3—182no—
316C2×Dic3—184no—
317C2×Dic3—182no—
318C2×Dic3—182no—
319C2×Dic3C4×D914yescomplemented
320C2×Dic3C22×D912yes—
321C2×Dic3C22×D912yes—
322C4×D3—916no—
323C4×D3—916no—
324C4×D3—188no—
325C4×D3—188no—
326C4×D3—188no—
327C4×D3—188no—
328C4×D3—1816no—
329C2×C32D3×C4⋊C414yes—
330C2×C32C22×C4×D311yescharacteristic
331C2×C32C2×D3×D411yescharacteristic
332C2×C32C2×D3×Q811yescharacteristic
333C3⋊D3—38no—
334C2×C9—21no—
335C2×C9—21no—
336C2×C9—21no—
337C2×C9—24no—
338C2×C9D3×C4⋊C414yescomplemented
339C2×C9C22×C4×D311yescharacteristic
340C2×C9C2×D3×D411yescharacteristic
341C2×C9C2×D3×Q811yescharacteristic
342D9—38no—
343D9—68no—
344C24—271no—
345C22×C4—31no—
346C22×C4—31no—
347C22×C4—91no—
348C22×C4—91no—
349C22×C4—274no—
350C22×C4—278no—
351C22×C4—274no—
352C22×C4—271no—
353C22×C4—274no—
354C22×C4—544no—
355C22×C4—544no—
356C22×C4D3×D911yescharacteristic
357C4⋊C4—34no—
358C4⋊C4—94no—
359C4⋊C4—274no—
360C4⋊C4—274no—
361C22×C3—22no—
362C22×C3—21no—
363C22×C3—22no—
364C22×C3—22no—
365C22×C3C2×C4×D912yes—
366C22×C3D4×D912yes—
367C22×C3Q8×D912yes—
368C22×C3C23×D911yescharacteristic
369C22×C3#379712yes—
370C22×C3C4×D3212yes—
371C22×C3#381112yes—
372C22×C3C22×D3211yescharacteristic
373C2×D3—94no—
374C2×D3—94no—
375C2×D3—94no—
376C2×D3—916no—
377C2×D3—94no—
378C2×D3—94no—
379C2×D3—916no—
380C2×D3—94no—
381C2×D3—184no—
382C2×D3—1816no—
383C2×D3—184no—
384C2×D3—184no—
385C3×C4—24no—
386C3×C4—22no—
387C3×C4—22no—
388C3×C4—34no—
389C3×C4—62no—
390C3×C4—94no—
391C3×C4—182no—
392Dic3—22no—
393Dic3—34no—
394Dic3—62no—
395Dic3—184no—
396Dic3—182no—
397Dic3—182no—
398Dic3C2×C4×D914yescomplemented
399C32C2×D3×C4⋊C411yescharacteristic
400C9—21no—
401C9C2×D3×C4⋊C411yescharacteristic, complemented
402C23—274no—
403C23—272no—
404C23—274no—
405C23—274no—
406C23C2×D3×D911yescharacteristic
407C2×C4—22no—
408C2×C4—32no—
409C2×C4—34no—
410C2×C4—32no—
411C2×C4—62no—
412C2×C4—92no—
413C2×C4—94no—
414C2×C4—92no—
415C2×C4—182no—
416C2×C4—2716no—
417C2×C4—272no—
418C2×C4—548no—
419C2×C4—548no—
420C2×C4—542no—
421C2×C4C2×D3×D912yes—
422C2×C3—24no—
423C2×C3—21no—
424C2×C3—21no—
425C2×C3—21no—
426C2×C3C4⋊C4×D914yescomplemented
427C2×C3C22×C4×D911yescharacteristic
428C2×C3C2×D4×D911yescharacteristic
429C2×C3C2×Q8×D911yescharacteristic
430C2×C3#6388314yes—
431C2×C3C2×#379711yescharacteristic
432C2×C3C2×C4×D3211yescharacteristic
433C2×C3C2×#381111yescharacteristic
434D3—98no—
435D3—98no—
436D3—188no—
437C22—2716no—
438C22—274no—
439C22—274no—
440C22—274no—
441C22(C3⋊D9)⋊Q812yes—
442C22C4×D3×D912yes—
443C22#8866812yes—
444C22C22×D3×D911yescharacteristic
445C4—22no—
446C4—34no—
447C4—62no—
448C4—94no—
449C4—182no—
450C4—542no—
451C3—21no—
452C3C2×C4⋊C4×D911yescharacteristic, complemented
453C3C2×#6388311yescharacteristic
454C2—278no—
455C2#1172098014yesdirect product factor
456C2C2×(C3⋊D9)⋊Q811yescharacteristic
457C2C2×C4×D3×D911yescharacteristic
458C2C2×#8866811yescharacteristic
subgroupG/H
centerC23C2×D3×D9
commutatorC2×C3×C9C23×C4
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC23×C32C22×D3
prime radicalC23×C3⋊D9C22
cosocleC22×C3×C9C24
hypercenterC22×C4D3×D9
hypocenterC3×C9C22×C4⋊C4
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
4. C2×C9×Dic3⋊C4C2semidirect
5. C2×C3×Dic9⋊C4C2semidirect
6. C2×Dic3⋊Dic9C2semidirect
7. C2×Dic9⋊Dic3C2semidirect
8. #11720980C2direct (N × H)
21. C2×(C3⋊D9)⋊C4C4semidirect
24. C9×Dic3⋊C4C22semidirect
25. C3×Dic9⋊C4C22semidirect
26. Dic3⋊Dic9C22semidirect
27. Dic9⋊Dic3C22semidirect
43. C2×Dic9⋊C4D3semidirect
56. C2×C9×Dic3C2×C4semidirect
60. C2×C3×Dic9C2×C4semidirect
64. (C3⋊D9)⋊C4C2×C4semidirect
105. Dic9⋊C4C2×D3semidirect
110. C2×C3⋊D9C4⋊C4semidirect
117. C9×Dic3C22×C4semidirect
119. C3×Dic9C22×C4semidirect
134. C2×Dic3⋊C4D9semidirect
180. C2×Dic9C4×D3semidirect
188. C2×C3×C9C2×C4⋊C4semidirect
192. C3⋊D9C2×C4⋊C4semidirect
231. Dic3⋊C4C2×D9semidirect
269. Dic9C2×C4×D3semidirect
278. C3×C9C22×C4⋊C4semidirect
319. C2×Dic3C4×D9semidirect
338. C2×C9D3×C4⋊C4semidirect
398. Dic3C2×C4×D9semidirect
401. C9C2×D3×C4⋊C4semidirect
426. C2×C3C4⋊C4×D9semidirect
452. C3C2×C4⋊C4×D9semidirect
455. C2#11720980direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×C3⋊D92
C22×(C3⋊D9)⋊C42
C2×C9×Dic3⋊C42
C2×C3×Dic9⋊C42
C2×Dic3⋊Dic92
C2×Dic9⋊Dic32
#117209802
C2×#638833
C2×C4⋊C4×D93