Finite Group Databasebeta

← #13147459

Subgroups — #13147459

Order 1728.

736 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#6226C211yesmaximal, characteristic, complemented
2C3×C42×D9C211yesmaximal, characteristic
3C4×C9×Dic6C211yesmaximal, characteristic, complemented
4#11720969C211yesmaximal, characteristic, complemented
5C4×#6189C212yesmaximal, complemented
6#11720945C211yesmaximal, characteristic
7#11720944C211yesmaximal, characteristic
8#11720943C212yesmaximal
9#11720939C212yesmaximal
10#11720933C212yesmaximal, complemented
11#11720924C211yesmaximal, characteristic, complemented
12#10593873—31nomaximal
13#10589680—31nomaximal
14C2×#6226C2211yescharacteristic
15C2×C3×C4×D9C2211yescharacteristic
16C2×C3×C4×D9C2211yescharacteristic
17C2×C3×C4×D9C2211yescharacteristic
18C2×C9×Dic6C2211yescharacteristic
19C2×#6189C2212yes—
20C3×C42×C9C2211yescharacteristic
21C4⋊(C3⋊Dic9)C2211yescharacteristic
22#88534C2212yes—
23C4×C3⋊Dic9C2212yes—
24C9×C4⋊Dic3C2211yescharacteristic
25C9×Dic3⋊C4C2212yes—
26C4×C9×Dic3C2212yes—
27C3×C4×Dic9C2211yescharacteristic
28C3×C4×Dic9C2211yescharacteristic
29C3×C4×Dic9C2211yescharacteristic
30(C2×D9)⋊Dic3C2212yes—
31(C2×D9)⋊Dic3C2212yes—
32Dic3⋊Dic9C2212yes—
33#88442C2212yes—
34Dic9⋊Dic3C2212yes—
35Dic9⋊Dic3C2212yes—
36Dic9⋊Dic3C2212yes—
37C4×#634—31no—
38C3×C42×D3—31no—
39C3×C4×Dic6D311yescharacteristic
40#63918—31no—
41C4×#627—32no—
42#63867—32no—
43#63866—31no—
44#63857—31no—
45#63856—31no—
46#63855—32no—
47#63851—32no—
48C4×Q8×C9—31no—
49#63508—31no—
50#63506—31no—
51#63459—31no—
52#63458—32no—
53#63451—32no—
54#63450—31no—
55#63434—32no—
56C42×D9D311yescharacteristic
57C4×Dic18—31no—
58C4×Dic18—32no—
59C4×Dic18—61no—
60C22×C3×D9C2311yescharacteristic
61C2×C3×C4×C9C2311yescharacteristic
62C2×C3×C4×C9C2311yescharacteristic
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3⋊Dic9C2312yes—
65C2×C3⋊Dic9C2312yes—
66#6226—22no—
67C2×C9×Dic3C2312yes—
68C2×C9×Dic3C2312yes—
69C2×C3×Dic9C2311yescharacteristic
70C2×C3×Dic9C2311yescharacteristic
71C2×C3×Dic9C2311yescharacteristic
72C3×C4×D9—22no—
73C3×C4×D9—22no—
74C3×C4×D9Q814yescomplemented
75C9×Dic6—22no—
76#6189—24no—
77#5578—91no—
78#5470—91no—
79C2×#634—31no—
80C2×C3×C4×D3—31no—
81C2×C3×C4×D3—31no—
82C2×C3×C4×D3—31no—
83C2×C3×Dic6C2×D311yescharacteristic
84C2×#627—32no—
85C32×C42C2×D311yescharacteristic
86C4⋊(C3⋊Dic3)—31no—
87#3750—32no—
88C4×C3⋊Dic3—32no—
89C3×C4⋊Dic3C2×D311yescharacteristic
90C3×Dic3⋊C4C2×D312yes—
91C3×C4×Dic3—31no—
92C3×C4×Dic3—31no—
93C3×C4×Dic3—31no—
94C3×C4×Dic3C2×D312yes—
95#3724—32no—
96Dic3⋊Dic3—32no—
97Dic3⋊Dic3—32no—
98Dic3⋊Dic3—32no—
99Dic3⋊Dic3—32no—
100(C2×D3)⋊Dic3—32no—
101(C2×D3)⋊Dic3—32no—
102C2×Q8×C9—31no—
103C2×C4×D9C2×D311yescharacteristic
104C2×C4×D9C2×D311yescharacteristic
105C2×C4×D9C2×D311yescharacteristic
106C2×Dic18—31no—
107C2×Dic18—32no—
108C2×Dic18—61no—
109C9×C4⋊C4—32no—
110C9×C4⋊C4—31no—
111C42×C9—21no—
112C42×C9—32no—
113C42×C9C2×D311yescharacteristic
114(C2×D9)⋊C4—32no—
115(C2×D9)⋊C4—32no—
116C4⋊Dic9—31no—
117C4⋊Dic9—32no—
118C4⋊Dic9—61no—
119Dic9⋊C4—32no—
120Dic9⋊C4—32no—
121Dic9⋊C4—32no—
122Dic9⋊C4—32no—
123Dic9⋊C4—32no—
124Dic9⋊C4—62no—
125C4×Dic9—32no—
126C4×Dic9—62no—
127C4×Dic9C2×D311yescharacteristic
128C4×Dic9C2×D311yescharacteristic
129C4×Dic9C2×D311yescharacteristic
130C22×C3×C9C2411yescharacteristic
131C2×C3×D9—21no—
132C2×C3×D9—21no—
133C2×C3×D9C2×Q812yes—
134C3×C4×C9—21no—
135C3×C4×C9C2×Q812yes—
136C3×C4×C9#4112yes—
137C3⋊Dic9—22no—
138C3⋊Dic9—22no—
139C9×Dic3—22no—
140C9×Dic3—22no—
141C3×Dic9—21no—
142C3×Dic9C2×Q812yes—
143C3×Dic9#4112yes—
144C3×C4×Q8—31no—
145C2×C3×C42—91no—
146#946—91no—
147#944—91no—
148#926—92no—
149#924—91no—
150#924—91no—
151#924—92no—
152#897—91no—
153#896—92no—
154#889—92no—
155#888—91no—
156#872—92no—
157C42×D3—31no—
158#870—92no—
159#869—91no—
160#869—91no—
161C4×Dic6—92no—
162C4×Dic6—91no—
163C4×Dic6—91no—
164C4×Dic6—92no—
165C4×Dic6—181no—
166C4×Dic6D911yescharacteristic, complemented
167C22×C3×D3—31no—
168C2×C32×C4C22×D311yescharacteristic
169C2×C32×C4C22×D311yescharacteristic
170C2×C32×C4C22×D311yescharacteristic
171C2×C3⋊Dic3—32no—
172C2×C3⋊Dic3—32no—
173#634—62no—
174C2×C3×Dic3—31no—
175C2×C3×Dic3—31no—
176C2×C3×Dic3—31no—
177C2×C3×Dic3C22×D312yes—
178C2×C3×Dic3C22×D312yes—
179C3×C4×D3—34no—
180C3×C4×D3—62no—
181C3×C4×D3—62no—
182C3×Dic6—22no—
183#627—64no—
184C22×D9C22×D311yescharacteristic
185Q8×C9—62no—
186C2×C4×C9—21no—
187C2×C4×C9—21no—
188C2×C4×C9—21no—
189C2×C4×C9—32no—
190C2×C4×C9—32no—
191C2×C4×C9C22×D311yescharacteristic
192C2×C4×C9C22×D311yescharacteristic
193C2×C4×C9C22×D311yescharacteristic
194C2×Dic9—32no—
195C2×Dic9—32no—
196C2×Dic9—62no—
197C2×Dic9—62no—
198C2×Dic9C22×D311yescharacteristic
199C2×Dic9C22×D311yescharacteristic
200C2×Dic9C22×D311yescharacteristic
201C4×D9—22no—
202C4×D9—22no—
203C4×D9Dic614yescomplemented
204Dic18—64no—
205Dic18—62no—
206Dic18—64no—
207#533—271noSylow
208C2×C3×C9C22×Q811yescharacteristic
209C2×C3×C9C2×#4111yescharacteristic
210C2×C3×C9C2×#4111yescharacteristic
211C3×D9—22no—
212C2×C3×Q8—31no—
213C22×C3×C4—91no—
214C22×C3×C4—91no—
215C22×C3×C4—91no—
216C2×C4×D3—31no—
217C2×C4×D3—31no—
218C2×C4×D3—31no—
219C2×Dic6—92no—
220C2×Dic6—91no—
221C2×Dic6—91no—
222C2×Dic6—92no—
223C2×Dic6—181no—
224C2×Dic6C2×D911yescharacteristic
225C3×C4⋊C4—31no—
226C3×C4⋊C4—32no—
227C3×C42—21no—
228C3×C42—32no—
229C3×C42—91no—
230C3×C42—91no—
231C3×C42—91no—
232C3×C42C2×D911yescharacteristic
233C3×C42D3211yescharacteristic
234#217—92no—
235#217—92no—
236(C2×D3)⋊C4—92no—
237(C2×D3)⋊C4—92no—
238C4⋊Dic3—92no—
239C4⋊Dic3—91no—
240C4⋊Dic3—91no—
241C4⋊Dic3—92no—
242C4⋊Dic3—92no—
243C4⋊Dic3—92no—
244C4⋊Dic3—181no—
245C4⋊Dic3C2×D911yescharacteristic
246Dic3⋊C4—92no—
247Dic3⋊C4—92no—
248Dic3⋊C4—92no—
249Dic3⋊C4—92no—
250Dic3⋊C4—92no—
251Dic3⋊C4—92no—
252Dic3⋊C4—92no—
253Dic3⋊C4—92no—
254Dic3⋊C4—182no—
255Dic3⋊C4C2×D912yes—
256C4×Dic3—31no—
257C4×Dic3—31no—
258C4×Dic3—31no—
259C4×Dic3—92no—
260C4×Dic3—92no—
261C4×Dic3—182no—
262C4×Dic3C2×D912yes—
263C22×C32C23×D311yescharacteristic
264C2×C3×D3—32no—
265C2×C3×D3—61no—
266C2×C3×D3—61no—
267C32×C4—21no—
268C32×C4#23512yes—
269C32×C4D3×Q812yes—
270C3⋊Dic3—62no—
271C3⋊Dic3—62no—
272C3×Dic3—22no—
273C3×Dic3—22no—
274C3×Dic3—32no—
275C3×Dic3—32no—
276C3×Dic3—61no—
277C22×C9—21no—
278C22×C9C23×D311yescharacteristic
279C2×D9—21no—
280C2×D9—21no—
281C2×D9C2×Dic612yes—
282C4×C9—21no—
283C4×C9—22no—
284C4×C9—22no—
285C4×C9—41no—
286C4×C9—62no—
287C4×C9—62no—
288C4×C9C2×Dic612yes—
289C4×C9#23512yes—
290Dic9—21no—
291Dic9—64no—
292Dic9—62no—
293Dic9—62no—
294Dic9—122no—
295Dic9C2×Dic612yes—
296Dic9#23512yes—
297C4⋊Q8—271no—
298C4⋊Q8—271no—
299#125—272no—
300C22⋊Q8—272no—
301C22⋊Q8—271no—
302C22⋊Q8—271no—
303C4×Q8—31no—
304C4×Q8—272no—
305C4×Q8—271no—
306#117—272no—
307C2×C42—91no—
308C3×C9#53311yescharacteristic, complemented, Sylow
309C23×C3—91no—
310C22×D3—31no—
311C3×Q8—62no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—21no—
315C2×C3×C4—32no—
316C2×C3×C4—32no—
317C2×C3×C4—91no—
318C2×C3×C4—91no—
319C2×C3×C4—94no—
320C2×C3×C4—91no—
321C2×C3×C4—182no—
322C2×C3×C4—182no—
323C2×C3×C4C22×D911yescharacteristic
324C2×C3×C4C22×D911yescharacteristic
325C2×C3×C4C22×D911yescharacteristic
326C2×C3×C4C2×D3211yescharacteristic
327C2×C3×C4C2×D3211yescharacteristic
328C2×C3×C4C2×D3211yescharacteristic
329C2×Dic3—31no—
330C2×Dic3—31no—
331C2×Dic3—31no—
332C2×Dic3—92no—
333C2×Dic3—92no—
334C2×Dic3—92no—
335C2×Dic3—92no—
336C2×Dic3—182no—
337C2×Dic3—182no—
338C2×Dic3C22×D912yes—
339C2×Dic3C22×D912yes—
340C4×D3—34no—
341C4×D3—62no—
342C4×D3—62no—
343Dic6—22no—
344Dic6—184no—
345Dic6—184no—
346Dic6—184no—
347Dic6—182no—
348Dic6—182no—
349C2×C32C2×#23511yescharacteristic
350C2×C32C2×D3×Q811yescharacteristic
351C2×C32D3×#4111yescharacteristic
352C3×D3—62no—
353C2×C9—21no—
354C2×C9—21no—
355C2×C9—21no—
356C2×C9C22×Dic611yescharacteristic
357C2×C9C2×#23511yescharacteristic
358C2×C9C2×#23511yescharacteristic
359D9—22no—
360C2×Q8—31no—
361C2×Q8—272no—
362C2×Q8—271no—
363C22×C4—91no—
364C22×C4—91no—
365C22×C4—91no—
366C4⋊C4—32no—
367C4⋊C4—31no—
368C4⋊C4—272no—
369C4⋊C4—272no—
370C4⋊C4—272no—
371C4⋊C4—272no—
372C4⋊C4—272no—
373C4⋊C4—272no—
374C4⋊C4—271no—
375C22⋊C4—272no—
376C22⋊C4—272no—
377C42—32no—
378C42—91no—
379C42—91no—
380C42—91no—
381C42—272no—
382C42D3×D911yescharacteristic
383C22×C3—21no—
384C22×C3—92no—
385C22×C3—181no—
386C22×C3—181no—
387C22×C3C23×D911yescharacteristic
388C22×C3C22×D3211yescharacteristic
389C2×D3—32no—
390C2×D3—61no—
391C2×D3—61no—
392C3×C4—21no—
393C3×C4—21no—
394C3×C4—22no—
395C3×C4—22no—
396C3×C4—41no—
397C3×C4—62no—
398C3×C4—62no—
399C3×C4—92no—
400C3×C4—92no—
401C3×C4—181no—
402C3×C4#369712yes—
403C3×C4Q8×D912yes—
404C3×C4D3×Dic612yes—
405C3×C4#379812yes—
406Dic3—22no—
407Dic3—22no—
408Dic3—32no—
409Dic3—32no—
410Dic3—61no—
411Dic3—184no—
412Dic3—182no—
413Dic3—182no—
414Dic3—182no—
415Dic3—182no—
416Dic3—362no—
417C32#557811yescharacteristic
418C9—21no—
419C9#547011yescharacteristic, complemented
420C23—91no—
421Q8—62no—
422Q8—544no—
423Q8—542no—
424C2×C4—32no—
425C2×C4—32no—
426C2×C4—91no—
427C2×C4—91no—
428C2×C4—94no—
429C2×C4—91no—
430C2×C4—182no—
431C2×C4—182no—
432C2×C4—272no—
433C2×C4—272no—
434C2×C4C2×D3×D911yescharacteristic
435C2×C4C2×D3×D911yescharacteristic
436C2×C4C2×D3×D911yescharacteristic
437C2×C3—21no—
438C2×C3—21no—
439C2×C3—21no—
440C2×C3—182no—
441C2×C3C2×#369711yescharacteristic
442C2×C3C2×Q8×D911yescharacteristic
443C2×C3#41×D911yescharacteristic
444C2×C3C2×D3×Dic611yescharacteristic
445C2×C3C2×#379811yescharacteristic
446C2×C3D3×#23511yescharacteristic
447D3—62no—
448C22—92no—
449C22—181no—
450C22—181no—
451C22C22×D3×D911yescharacteristic
452C4—21no—
453C4—62no—
454C4—62no—
455C4—92no—
456C4—92no—
457C4—181no—
458C4—542no—
459C4—542no—
460C4D9×Dic612yes—
461C4#8864012yes—
462C3—21no—
463C3#1058968011yescharacteristic, complemented
464C3#1059387311yescharacteristic
465C2—182no—
466C2C2×D9×Dic611yescharacteristic
467C2C2×#8864011yescharacteristic
468C2D9×#23511yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
prime radicalC22×C3×D9C23
hypercenterC42D3×D9
hypocenterC3×C9#533
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C4×#6226C2semidirect
3. C4×C9×Dic6C2semidirect
4. #11720969C2semidirect
5. C4×#6189C2semidirect
10. #11720933C2semidirect
11. #11720924C2semidirect
74. C3×C4×D9Q8semidirect
166. C4×Dic6D9semidirect
203. C4×D9Dic6semidirect
308. C3×C9#533semidirect
419. C9#5470semidirect
463. C3#10589680semidirect

Maximal subgroups

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

subgroupindex
C4×#62262
C3×C42×D92
C4×C9×Dic62
#117209692
C4×#61892
#117209452
#117209442
#117209432
#117209392
#117209332
#117209242
#105938733
#105896803