Finite Group Databasebeta

← C2×#11721178 #13151116

Subgroups — C2×#11721178 #13151116

Order 1728.

962 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88742C211yesmaximal, characteristic, complemented
2C2×#88639C211yesmaximal, characteristic, complemented
3C2×C9×#213C211yesmaximal, characteristic, complemented
4C2×C3×#3672C211yesmaximal, characteristic, complemented
5#11721178C218yesmaximal, direct product factor
6C2×#88424C211yesmaximal, characteristic, complemented
7C2×#88418C211yesmaximal, characteristic, complemented
8C2×(C3⋊D9)⋊C8C211yesmaximal, characteristic, complemented
9C2×#63936—31nomaximal
10C2×#63474—31nomaximal
11C2×C3×D4×C9C2211yescharacteristic
12C2×#6233—21no—
13C22×C3⋊Dic9—21no—
14#88742—22no—
15#88742C2214yescomplemented
16C2×C4×C3⋊D9C2211yescharacteristic
17C2×#6226C2211yescharacteristic
18C2×C9×D12C2211yescharacteristic
19C2×C3×Dic18C2211yescharacteristic
20C2×#6195—21no—
21C2×D3×Dic9—21no—
22#88639—22no—
23#88639C2214yescomplemented
24C9×#213C2214yescomplemented
25C3×#3672C2214yescomplemented
26C2×C9×C3⋊C8C2211yescharacteristic
27C2×C3×C9⋊C8C2211yescharacteristic
28#88424C2214yescomplemented
29#88418C2214yescomplemented
30(C3⋊D9)⋊C8C2214yescomplemented
31C2×#3830—31no—
32C2×#3796—31no—
33C2×C3×#214—31no—
34C2×C3×#213D311yescharacteristic
35#63936—38no—
36C2×#3719—31no—
37C2×#3716—31no—
38C2×#3710—31no—
39C2×#3699—31no—
40C2×#3699—31no—
41C2×#3699—61no—
42C2×C9×D8—31no—
43C2×#3672—31no—
44C2×#3672D311yescharacteristic, complemented
45#63474—38no—
46C2×C8×D9—31no—
47C2×Dic36—31no—
48C23×C3×C9—21no—
49C22×C3⋊D9D411yescharacteristic
50C22×D3×C9—21no—
51C3×D4×C9—21no—
52C3×D4×C9C2312yes—
53C2×C3×C4×C9C2311yescharacteristic
54#6233—24no—
55C2×C3⋊Dic9—21no—
56C2×C3⋊Dic9—24no—
57C2×C3⋊Dic9D411yescharacteristic
58C4×C3⋊D9D412yes—
59C4×C3⋊D9C2312yes—
60#6226—21no—
61#6226C2312yes—
62C2×C3×Dic9—21no—
63C9×D12—21no—
64C9×D12C2312yes—
65C3×Dic18—21no—
66C3×Dic18C2312yes—
67#6195—24no—
68D3×Dic9—24no—
69C9×C3⋊C8C2312yes—
70C3×C9⋊C8C2312yes—
71C2×#916—91no—
72C2×#912—91no—
73C2×C32×D4C2×D311yescharacteristic
74C2×#638—61no—
75C22×C3⋊Dic3—61no—
76#3830—34no—
77#3830—62no—
78C2×C4×C3⋊D3—31no—
79C2×#634—31no—
80C2×C3×D12C2×D311yescharacteristic
81C2×C3×Dic6—31no—
82C2×#626—61no—
83C2×D3×Dic3—61no—
84#3796—34no—
85#3796—62no—
86C3×#214—34no—
87C3×#213C2×D314yes—
88C2×C3×C3⋊C8—31no—
89C2×C3×C3⋊C8C2×D311yescharacteristic
90#3719—34no—
91#3716—34no—
92#3710—34no—
93C2×D4×C9—21no—
94C2×D4×C9—31no—
95C2×D4×C9C2×D311yescharacteristic
96C2×#611—61no—
97C2×#611—61no—
98C2×#611—62no—
99C22×Dic9—61no—
100C22×Dic9—61no—
101C22×Dic9—62no—
102#3699—34no—
103#3699—34no—
104#3699—62no—
105#3699—62no—
106#3699—64no—
107#3699—64no—
108C2×C4×D9—31no—
109C2×C4×D9—61no—
110C2×Dic18—31no—
111C2×Dic18—61no—
112C2×Dic18C2×D311yescharacteristic
113C9×D8—34no—
114C2×C8×C9—31no—
115#3672—34no—
116#3672C2×D314yescomplemented
117C2×C9⋊C8C2×D311yescharacteristic
118C8×D9—34no—
119Dic36—34no—
120C22×C3×C9—22no—
121C22×C3×C9—42no—
122C22×C3×C9C2×D411yescharacteristic
123C2×C3⋊D9—22no—
124C2×C3⋊D9C2×D412yes—
125C2×D3×C9—22no—
126C2×D3×C9—42no—
127C3×C4×C9C2×D411yescharacteristic
128C3×C4×C9C2411yescharacteristic
129C3⋊Dic9—22no—
130C3⋊Dic9C2×D412yes—
131C3×Dic9—22no—
132C2×#239—91no—
133C2×#237—91no—
134C2×#237—91no—
135C2×#237—91no—
136C2×#237—181no—
137C2×C3×SD16—91no—
138C2×C3×D8—31no—
139C2×#216—91no—
140C2×#214—31no—
141C2×#214—91no—
142C2×#213D911yescharacteristic, complemented
143#916—98no—
144#912—98no—
145C2×Dic12—91no—
146C2×#204—91no—
147C2×D3×C8—91no—
148C2×D3×C8—91no—
149C23×C32—21no—
150C22×C3⋊D3—31no—
151C22×C3×D3—21no—
152C32×D4—21no—
153C32×D4C22×D312yes—
154C2×C32×C4C22×D311yescharacteristic
155#638—64no—
156C2×C3⋊Dic3—31no—
157C2×C3⋊Dic3—64no—
158C2×C3⋊Dic3—61no—
159C4×C3⋊D3—32no—
160C4×C3⋊D3—32no—
161#634—32no—
162#634—61no—
163C2×C3×Dic3—61no—
164C3×D12—21no—
165C3×D12C22×D312yes—
166C3×Dic6—32no—
167C3×Dic6—61no—
168#626—64no—
169D3×Dic3—64no—
170C23×C9—22no—
171C23×C9—21no—
172C23×C9—61no—
173C22×D9—31no—
174C22×D9—61no—
175C3×C3⋊C8—32no—
176C3×C3⋊C8C22×D312yes—
177D4×C9—22no—
178D4×C9—21no—
179D4×C9—22no—
180D4×C9—32no—
181D4×C9—61no—
182D4×C9C22×D312yes—
183C2×C4×C9—21no—
184C2×C4×C9C22×D311yescharacteristic
185#611—64no—
186#611—64no—
187#611—68no—
188C2×Dic9—21no—
189C2×Dic9—31no—
190C2×Dic9—61no—
191C2×Dic9—64no—
192C2×Dic9—64no—
193C2×Dic9—62no—
194C2×Dic9—68no—
195C2×Dic9—61no—
196C4×D9—32no—
197C4×D9—32no—
198C4×D9—62no—
199C4×D9—62no—
200Dic18—21no—
201Dic18—32no—
202Dic18—62no—
203Dic18—62no—
204Dic18—61no—
205Dic18C22×D312yes—
206C8×C9—32no—
207C9⋊C8C22×D312yes—
208C2×#135—271noSylow
209C2×C3×C9—42no—
210C2×C3×C9#13512yescomplemented
211C2×C3×C9C22×D411yescharacteristic
212C3⋊D9—22no—
213D3×C9—42no—
214C2×C3×Q8—91no—
215C2×C3×D4—21no—
216C2×C3×D4—31no—
217C2×C3×D4C2×D911yescharacteristic
218C2×C3×D4D3211yescharacteristic
219C2×#67—181no—
220C2×#67—181no—
221C2×#67—181no—
222C2×#67—182no—
223C22×Dic3—181no—
224C22×Dic3—181no—
225C22×Dic3—181no—
226C22×Dic3—182no—
227#239—94no—
228#239—182no—
229#237—94no—
230#237—94no—
231#237—94no—
232#237—184no—
233#237—182no—
234#237—182no—
235#237—182no—
236#237—184no—
237C2×D12—181no—
238C2×D12C2×D911yescharacteristic
239C2×C4×D3—91no—
240C2×C4×D3—91no—
241C2×C4×D3—181no—
242C2×C4×D3—181no—
243C2×Dic6—31no—
244C2×Dic6—91no—
245C2×Dic6—91no—
246C2×Dic6—181no—
247C3×SD16—94no—
248C3×D8—34no—
249C2×C3×C8—31no—
250C2×C3×C8—91no—
251#216—94no—
252#214—34no—
253#214—94no—
254#213C2×D914yescomplemented
255C2×C3⋊C8—31no—
256C2×C3⋊C8C2×D911yescharacteristic
257Dic12—94no—
258#204—94no—
259D3×C8—94no—
260D3×C8—94no—
261C22×C32—22no—
262C22×C32—42no—
263C22×C32D3×D411yescharacteristic
264C2×C3⋊D3—32no—
265C2×C3⋊D3—62no—
266C2×C3×D3—22no—
267C2×C3×D3—42no—
268C32×C4D3×D411yescharacteristic
269C32×C4C23×D311yescharacteristic
270C3⋊Dic3—32no—
271C3⋊Dic3—62no—
272C3×Dic3—62no—
273C22×C9—24no—
274C22×C9—22no—
275C22×C9—21no—
276C22×C9—44no—
277C22×C9—42no—
278C22×C9—62no—
279C22×C9—122no—
280C22×C9D3×D411yescharacteristic
281C2×D9—32no—
282C2×D9—62no—
283C2×D9—62no—
284C2×D9—122no—
285C4×C9—21no—
286C4×C9—21no—
287C4×C9D3×D411yescharacteristic
288C4×C9C23×D311yescharacteristic
289Dic9—22no—
290Dic9—32no—
291Dic9—62no—
292Dic9—64no—
293Dic9—62no—
294C2×#41—271no—
295C2×#41—271no—
296#135—278no—
297C2×Q16—271no—
298C2×SD16—91no—
299C2×SD16—271no—
300C2×D8—31no—
301C22×C8—271no—
302C3×C9C2×#13511yescharacteristic, complemented, Sylow
303C23×C3—22no—
304C23×C3—21no—
305C23×C3—21no—
306C23×C3—61no—
307C22×D3—21no—
308C22×D3—91no—
309C22×D3—91no—
310C22×D3—181no—
311C3×Q8—92no—
312C3×Q8—181no—
313C3×D4—22no—
314C3×D4—21no—
315C3×D4—21no—
316C3×D4—22no—
317C3×D4—32no—
318C3×D4—61no—
319C3×D4C22×D912yes—
320C3×D4C2×D3212yes—
321C2×C3×C4—21no—
322C2×C3×C4—181no—
323C2×C3×C4C22×D911yescharacteristic
324C2×C3×C4C2×D3211yescharacteristic
325#67—184no—
326#67—184no—
327#67—184no—
328#67—188no—
329C2×Dic3—61no—
330C2×Dic3—91no—
331C2×Dic3—91no—
332C2×Dic3—181no—
333C2×Dic3—184no—
334C2×Dic3—184no—
335C2×Dic3—184no—
336C2×Dic3—182no—
337C2×Dic3—188no—
338C2×Dic3—181no—
339C2×Dic3—181no—
340D12—21no—
341D12—184no—
342D12C22×D912yes—
343C4×D3—92no—
344C4×D3—92no—
345C4×D3—92no—
346C4×D3—92no—
347C4×D3—182no—
348C4×D3—182no—
349C4×D3—184no—
350Dic6—32no—
351Dic6—61no—
352Dic6—92no—
353Dic6—92no—
354Dic6—182no—
355Dic6—182no—
356Dic6—181no—
357Dic6—181no—
358C3×C8—32no—
359C3×C8—92no—
360C3⋊C8—32no—
361C3⋊C8C22×D912yes—
362C2×C32—42no—
363C2×C32#91212yes—
364C2×C32C2×D3×D411yescharacteristic
365C3⋊D3—62no—
366C3×D3—42no—
367C2×C9—21no—
368C2×C9—22no—
369C2×C9—44no—
370C2×C9—42no—
371C2×C9—122no—
372C2×C9#91612yescomplemented
373C2×C9C2×D3×D411yescharacteristic
374D9—62no—
375D9—122no—
376#41—274no—
377#41—274no—
378#41—542no—
379#41—542no—
380C2×Q8—91no—
381C2×Q8—271no—
382C2×D4—31no—
383C2×D4—541no—
384C2×D4—541no—
385C2×D4D3×D911yescharacteristic
386C22×C4—271no—
387C22×C4—541no—
388C22×C4—541no—
389Q16—274no—
390SD16—94no—
391SD16—274no—
392D8—34no—
393C2×C8—31no—
394C2×C8—91no—
395C2×C8—274no—
396C22×C3—24no—
397C22×C3—22no—
398C22×C3—22no—
399C22×C3—21no—
400C22×C3—44no—
401C22×C3—42no—
402C22×C3—42no—
403C22×C3—62no—
404C22×C3—122no—
405C22×C3D4×D911yescharacteristic
406C22×C3#381111yescharacteristic
407C2×D3—22no—
408C2×D3—42no—
409C2×D3—92no—
410C2×D3—92no—
411C2×D3—182no—
412C2×D3—182no—
413C2×D3—182no—
414C2×D3—362no—
415C3×C4—21no—
416C3×C4—21no—
417C3×C4—182no—
418C3×C4D4×D911yescharacteristic
419C3×C4C23×D911yescharacteristic
420C3×C4#381111yescharacteristic
421C3×C4C22×D3211yescharacteristic
422Dic3—62no—
423Dic3—92no—
424Dic3—92no—
425Dic3—182no—
426Dic3—184no—
427Dic3—182no—
428Dic3—182no—
429C32C2×#91211yescharacteristic
430C9—21no—
431C9C2×#91611yescharacteristic, complemented
432C23—21no—
433C23—61no—
434C23—271no—
435Q8—92no—
436Q8—181no—
437Q8—272no—
438Q8—541no—
439D4—21no—
440D4—32no—
441D4—61no—
442D4—544no—
443D4—544no—
444D4C2×D3×D912yes—
445C2×C4—181no—
446C2×C4—271no—
447C2×C4—272no—
448C2×C4—272no—
449C2×C4—544no—
450C2×C4—544no—
451C2×C4—541no—
452C2×C4C2×D3×D911yescharacteristic
453C8—32no—
454C8—92no—
455C2×C3—21no—
456C2×C3—22no—
457C2×C3—44no—
458C2×C3—42no—
459C2×C3—42no—
460C2×C3—122no—
461C2×C3#6347412yescomplemented
462C2×C3C2×D4×D911yescharacteristic
463C2×C3#6393612yes—
464C2×C3C2×#381111yescharacteristic
465D3—42no—
466D3—182no—
467D3—182no—
468D3—362no—
469C22—22no—
470C22—42no—
471C22—62no—
472C22—122no—
473C22—272no—
474C22—542no—
475C22#8866811yescharacteristic
476C4—182no—
477C4—272no—
478C4—542no—
479C4#8866811yescharacteristic
480C4C22×D3×D911yescharacteristic
481C3—21no—
482C3C2×#6347411yescharacteristic, complemented
483C3C2×#6393611yescharacteristic
484C2—42no—
485C2—122no—
486C2—542no—
487C2#1172117812yesdirect product factor
488C2C2×#8866811yescharacteristic
subgroupG/H
centerC22#88668
commutator, cosocleC3×C4×C9C24
FrattiniC3×C4C22×D32
FittingC2×C3×D4×C9C22
socleC22×C32D3×D4
hypercenterC2×D4D3×D9
hypocenterC3×C9C2×#135
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#88742C2semidirect
2. C2×#88639C2semidirect
3. C2×C9×#213C2semidirect
4. C2×C3×#3672C2semidirect
5. #11721178C2direct (N × H)
6. C2×#88424C2semidirect
7. C2×#88418C2semidirect
8. C2×(C3⋊D9)⋊C8C2semidirect
15. #88742C22semidirect
23. #88639C22semidirect
24. C9×#213C22semidirect
25. C3×#3672C22semidirect
28. #88424C22semidirect
29. #88418C22semidirect
30. (C3⋊D9)⋊C8C22semidirect
44. C2×#3672D3semidirect
116. #3672C2×D3semidirect
142. C2×#213D9semidirect
210. C2×C3×C9#135semidirect
254. #213C2×D9semidirect
302. C3×C9C2×#135semidirect
372. C2×C9#916semidirect
431. C9C2×#916semidirect
461. C2×C3#63474semidirect
482. C3C2×#63474semidirect
487. C2#11721178direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#887422
C2×#886392
C2×C9×#2132
C2×C3×#36722
#117211782
C2×#884242
C2×#884182
C2×(C3⋊D9)⋊C82
C2×#639363
C2×#634743