Finite Group Databasebeta

← #13149003

Subgroups — #13149003

Order 1728.

718 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721748C211yesmaximal, characteristic, complemented
2C3×C4⋊C4×D9C211yesmaximal, characteristic, complemented
3C9×#887C211yesmaximal, characteristic, complemented
4C4×#6192C211yesmaximal, characteristic
5#11720944C211yesmaximal, characteristic, complemented
6#11720943C212yesmaximal, complemented
7#11720938C212yesmaximal
8#11720929C211yesmaximal, characteristic, complemented
9#11720928C212yesmaximal, complemented
10#11720927C211yesmaximal, characteristic, complemented
11#11720926C212yesmaximal, complemented
12#10594333—31nomaximal
13#10589683—31nomaximal
14C2×C4×C3⋊D9C2211yescharacteristic
15C2×C3×C4×D9C2211yescharacteristic
16C2×C3×C4×D9C2212yes—
17C2×C9×Dic6C2211yescharacteristic
18C2×#6192C2211yescharacteristic
19C3×C9×C4⋊C4C2211yescharacteristic
20#88536C2212yes—
21C4⋊(C3⋊Dic9)C2211yescharacteristic
22C4×C3⋊Dic9C2212yes—
23C9×Dic3⋊C4C2212yes—
24C4×C9×Dic3C2211yescharacteristic
25C4×C9×Dic3C2212yes—
26C3×C4⋊Dic9C2211yescharacteristic
27C3×Dic9⋊C4C2212yes—
28#88445C2211yescharacteristic
29#88445C2212yes—
30(C2×D9)⋊Dic3C2211yescharacteristic
31(C2×D9)⋊Dic3C2212yes—
32Dic3⋊Dic9C2211yescharacteristic
33#88442C2212yes—
34#88442C2212yes—
35Dic9⋊Dic3C2212yes—
36Dic9⋊Dic3C2212yes—
37#64101—31no—
38C3×D3×C4⋊C4—31no—
39C3×#887D311yescharacteristic
40C4×#626—31no—
41#63871—31no—
42#63870—32no—
43#63869—31no—
44#63868—32no—
45#63856—31no—
46#63855—32no—
47#63850—32no—
48C4×Q8×C9—31no—
49#63508—31no—
50#63508—31no—
51#63456—32no—
52#63454—31no—
53#63454—61no—
54C4⋊C4×D9D311yescharacteristic, complemented
55#63451—32no—
56#63438—32no—
57C4×D36—31no—
58#63434—32no—
59#63432—31no—
60C22×C3⋊D9C2311yescharacteristic
61C22×C3×D9C2311yescharacteristic
62C2×C3×C4×C9C2312yes—
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3⋊Dic9C2311yescharacteristic
65C2×C3⋊Dic9C2312yes—
66C4×C3⋊D9—22no—
67C2×C9×Dic3C2311yescharacteristic
68C2×C9×Dic3C2312yes—
69C2×C9×Dic3C2311yescharacteristic
70C2×C3×Dic9C2312yes—
71C2×C3×Dic9C2311yescharacteristic
72C3×C4×D9—24no—
73C3×C4×D9—22no—
74C9×Dic6—22no—
75#6192—22no—
76#5581—91no—
77#5513—91no—
78C2×C4×C3⋊D3—31no—
79C2×C3×C4×D3—32no—
80C2×C3×C4×D3—31no—
81C2×C3×Dic6C2×D311yescharacteristic
82C2×#626—31no—
83C32×C4⋊C4C2×D311yescharacteristic
84#3752—32no—
85C4⋊(C3⋊Dic3)—31no—
86C4×C3⋊Dic3—32no—
87C3×C4⋊Dic3—31no—
88C3×Dic3⋊C4—32no—
89C3×Dic3⋊C4C2×D312yes—
90C3×C4×Dic3C2×D312yes—
91C3×C4×Dic3C2×D311yescharacteristic
92#3724—32no—
93#3724—32no—
94Dic3⋊Dic3—32no—
95Dic3⋊Dic3—32no—
96Dic3⋊Dic3—31no—
97#3722—32no—
98#3722—31no—
99(C2×D3)⋊Dic3—32no—
100(C2×D3)⋊Dic3—31no—
101C2×Q8×C9—31no—
102C2×D36—31no—
103C2×C4×D9—31no—
104C2×C4×D9—61no—
105C2×C4×D9C2×D311yescharacteristic
106C2×C4×D9C2×D312yes—
107C9×C4⋊C4—21no—
108C9×C4⋊C4—32no—
109C9×C4⋊C4C2×D311yescharacteristic
110C42×C9—31no—
111C42×C9—32no—
112(C2×D9)⋊C4—32no—
113(C2×D9)⋊C4—31no—
114(C2×D9)⋊C4—32no—
115(C2×D9)⋊C4—32no—
116(C2×D9)⋊C4—31no—
117(C2×D9)⋊C4—62no—
118C4⋊Dic9—31no—
119C4⋊Dic9—31no—
120C4⋊Dic9—61no—
121C4⋊Dic9C2×D311yescharacteristic
122Dic9⋊C4—32no—
123Dic9⋊C4—32no—
124Dic9⋊C4—32no—
125Dic9⋊C4—32no—
126Dic9⋊C4C2×D312yes—
127C4×Dic9—32no—
128C4×Dic9—62no—
129C22×C3×C9C2411yescharacteristic
130C2×C3⋊D9—21no—
131C2×C3⋊D9—21no—
132C2×C3⋊D9—21no—
133C2×C3×D9—21no—
134C2×C3×D9—21no—
135C2×C3×D9#4112yes—
136C3×C4×C9—22no—
137C3×C4×C9#4112yes—
138C3⋊Dic9—21no—
139C3⋊Dic9—22no—
140C9×Dic3—21no—
141C9×Dic3—22no—
142C9×Dic3—21no—
143C3×Dic9—21no—
144C3×Dic9—22no—
145C3×C4×Q8—31no—
146C2×C3×C4⋊C4—91no—
147#946—91no—
148#946—91no—
149#929—92no—
150C4×#67—91no—
151#926—92no—
152#924—91no—
153#898—92no—
154#897—91no—
155#894—92no—
156#892—91no—
157#892—91no—
158#892—181no—
159D3×C4⋊C4—31no—
160#890—91no—
161#889—92no—
162#889—92no—
163#887D911yescharacteristic, complemented
164#876—92no—
165C4×D12—91no—
166#872—92no—
167#870—91no—
168C22×C3⋊D3—31no—
169C22×C3×D3—31no—
170C2×C32×C4C22×D312yes—
171C2×C32×C4C22×D311yescharacteristic
172C2×C3⋊Dic3—31no—
173C2×C3⋊Dic3—32no—
174C4×C3⋊D3—62no—
175C2×C3×Dic3—32no—
176C2×C3×Dic3—31no—
177C2×C3×Dic3C22×D311yescharacteristic
178C2×C3×Dic3C22×D312yes—
179C2×C3×Dic3C22×D311yescharacteristic
180C3×C4×D3—64no—
181C3×C4×D3—62no—
182C3×Dic6—22no—
183#626—62no—
184C22×D9—31no—
185C22×D9—61no—
186C22×D9C22×D311yescharacteristic
187Q8×C9—62no—
188C2×C4×C9—21no—
189C2×C4×C9—22no—
190C2×C4×C9—31no—
191C2×C4×C9—31no—
192C2×C4×C9—32no—
193C2×C4×C9C22×D312yes—
194C2×C4×C9C22×D311yescharacteristic
195C2×Dic9—31no—
196C2×Dic9—32no—
197C2×Dic9—61no—
198C2×Dic9—62no—
199C2×Dic9C22×D312yes—
200C2×Dic9C22×D311yescharacteristic
201D36—62no—
202C4×D9—24no—
203C4×D9—22no—
204C4×D9—64no—
205C4×D9—62no—
206#552—271noSylow
207C2×C3×C9C2×#4111yescharacteristic
208C2×C3×C9C2×#4111yescharacteristic
209C2×C3×C92−1+411yescharacteristic
210C3⋊D9—41no—
211C3×D9—22no—
212C2×C3×Q8—31no—
213C22×C3×C4—91no—
214C22×C3×C4—92no—
215C2×#67—91no—
216C2×D12—91no—
217C2×C4×D3—32no—
218C2×C4×D3—31no—
219C2×C4×D3—91no—
220C2×C4×D3—91no—
221C2×C4×D3—181no—
222C2×Dic6C2×D911yescharacteristic
223C3×C4⋊C4—21no—
224C3×C4⋊C4—32no—
225C3×C4⋊C4—92no—
226C3×C4⋊C4—91no—
227C3×C4⋊C4C2×D911yescharacteristic
228C3×C4⋊C4D3211yescharacteristic
229C3×C42—31no—
230C3×C42—32no—
231#217—92no—
232#217—91no—
233(C2×D3)⋊C4—91no—
234(C2×D3)⋊C4—92no—
235(C2×D3)⋊C4—91no—
236(C2×D3)⋊C4—92no—
237(C2×D3)⋊C4—92no—
238(C2×D3)⋊C4—92no—
239(C2×D3)⋊C4—92no—
240(C2×D3)⋊C4—91no—
241(C2×D3)⋊C4—182no—
242C4⋊Dic3—31no—
243C4⋊Dic3—91no—
244C4⋊Dic3—91no—
245C4⋊Dic3—91no—
246C4⋊Dic3—92no—
247C4⋊Dic3—92no—
248C4⋊Dic3—181no—
249Dic3⋊C4—32no—
250Dic3⋊C4—92no—
251Dic3⋊C4—92no—
252Dic3⋊C4—92no—
253Dic3⋊C4—92no—
254Dic3⋊C4—91no—
255Dic3⋊C4—92no—
256Dic3⋊C4—92no—
257Dic3⋊C4C2×D912yes—
258C4×Dic3—92no—
259C4×Dic3—92no—
260C4×Dic3—182no—
261C4×Dic3C2×D911yescharacteristic
262C4×Dic3C2×D912yes—
263C22×C32C23×D311yescharacteristic
264C2×C3⋊D3—61no—
265C2×C3⋊D3—61no—
266C2×C3⋊D3—61no—
267C2×C3×D3—32no—
268C2×C3×D3—61no—
269C2×C3×D3—61no—
270C32×C4—22no—
271C32×C4#23512yes—
272C3⋊Dic3—61no—
273C3⋊Dic3—62no—
274C3×Dic3—22no—
275C3×Dic3—21no—
276C3×Dic3—21no—
277C3×Dic3—61no—
278C3×Dic3—62no—
279C22×C9—21no—
280C22×C9C23×D311yescharacteristic
281C2×D9—21no—
282C2×D9—21no—
283C2×D9—62no—
284C2×D9—61no—
285C2×D9—61no—
286C2×D9—61no—
287C2×D9—121no—
288C2×D9—121no—
289C2×D9#23712yes—
290C4×C9—22no—
291C4×C9—22no—
292C4×C9—42no—
293C4×C9—62no—
294C4×C9—61no—
295C4×C9—61no—
296C4×C9#23512yes—
297Dic9—21no—
298Dic9—22no—
299Dic9—62no—
300Dic9—61no—
301Dic9—62no—
302Dic9—122no—
303#126—272no—
304#125—271no—
305#125—272no—
306#123—272no—
307C22⋊Q8—271no—
308C22⋊Q8—271no—
309C4×Q8—31no—
310C4×D4—271no—
311#117—272no—
312#117—271no—
313C2×C4⋊C4—91no—
314C3×C9#55211yescharacteristic, complemented, Sylow
315C23×C3—91no—
316C22×D3—31no—
317C22×D3—91no—
318C22×D3—91no—
319C22×D3—181no—
320C3×Q8—62no—
321C2×C3×C4—21no—
322C2×C3×C4—22no—
323C2×C3×C4—31no—
324C2×C3×C4—31no—
325C2×C3×C4—32no—
326C2×C3×C4—91no—
327C2×C3×C4—92no—
328C2×C3×C4—184no—
329C2×C3×C4—182no—
330C2×C3×C4C22×D912yes—
331C2×C3×C4C22×D911yescharacteristic
332C2×C3×C4C2×D3212yes—
333C2×C3×C4C2×D3211yescharacteristic
334#67—182no—
335C2×Dic3—31no—
336C2×Dic3—32no—
337C2×Dic3—91no—
338C2×Dic3—92no—
339C2×Dic3—91no—
340C2×Dic3—92no—
341C2×Dic3—181no—
342C2×Dic3—182no—
343C2×Dic3C22×D911yescharacteristic
344C2×Dic3C22×D911yescharacteristic
345C2×Dic3C22×D912yes—
346D12—182no—
347C4×D3—64no—
348C4×D3—62no—
349C4×D3—184no—
350C4×D3—182no—
351C4×D3—182no—
352Dic6—22no—
353C2×C32C2×#23511yescharacteristic
354C2×C32#100711yescharacteristic
355C2×C32D3×#4111yescharacteristic
356C3⋊D3—121no—
357C3×D3—62no—
358C2×C9—21no—
359C2×C9—21no—
360C2×C9—21no—
361C2×C9C2×#23511yescharacteristic
362C2×C9C2×#23711yescharacteristic
363C2×C9#100711yescharacteristic
364D9—22no—
365D9—122no—
366D9—121no—
367C2×Q8—31no—
368C2×D4—271no—
369C22×C4—92no—
370C22×C4—91no—
371C22×C4—271no—
372C4⋊C4—32no—
373C4⋊C4—91no—
374C4⋊C4—92no—
375C4⋊C4—272no—
376C4⋊C4—272no—
377C4⋊C4—271no—
378C4⋊C4—272no—
379C4⋊C4—272no—
380C4⋊C4—271no—
381C4⋊C4D3×D911yescharacteristic
382C22⋊C4—271no—
383C22⋊C4—272no—
384C22⋊C4—271no—
385C22⋊C4—272no—
386C22⋊C4—272no—
387C42—32no—
388C42—31no—
389C42—272no—
390C22×C3—21no—
391C22×C3—92no—
392C22×C3—181no—
393C22×C3—181no—
394C22×C3C23×D911yescharacteristic
395C22×C3C22×D3211yescharacteristic
396C2×D3—32no—
397C2×D3—61no—
398C2×D3—61no—
399C2×D3—182no—
400C2×D3—181no—
401C2×D3—181no—
402C2×D3—181no—
403C2×D3—181no—
404C2×D3—181no—
405C2×D3—181no—
406C2×D3—361no—
407C2×D3—361no—
408C3×C4—22no—
409C3×C4—22no—
410C3×C4—22no—
411C3×C4—42no—
412C3×C4—62no—
413C3×C4—61no—
414C3×C4—61no—
415C3×C4—181no—
416C3×C4—182no—
417C3×C4#369712yes—
418C3×C4#379812yes—
419Dic3—22no—
420Dic3—21no—
421Dic3—21no—
422Dic3—61no—
423Dic3—62no—
424Dic3—182no—
425Dic3—181no—
426Dic3—182no—
427Dic3—181no—
428Dic3—182no—
429Dic3—362no—
430C32#558111yescharacteristic
431C9—21no—
432C9#551311yescharacteristic, complemented
433C23—91no—
434C23—271no—
435Q8—62no—
436D4—542no—
437C2×C4—31no—
438C2×C4—32no—
439C2×C4—31no—
440C2×C4—92no—
441C2×C4—91no—
442C2×C4—184no—
443C2×C4—182no—
444C2×C4—271no—
445C2×C4—272no—
446C2×C4—542no—
447C2×C4C2×D3×D912yes—
448C2×C4C2×D3×D911yescharacteristic
449C2×C3—21no—
450C2×C3—21no—
451C2×C3—21no—
452C2×C3—182no—
453C2×C3C2×#369711yescharacteristic
454C2×C3#6371311yescharacteristic
455C2×C3#41×D911yescharacteristic
456C2×C3C2×#379811yescharacteristic
457C2×C3D3×#23711yescharacteristic
458C2×C3#6431611yescharacteristic
459D3—62no—
460D3—362no—
461D3—361no—
462D3—361no—
463C22—92no—
464C22—181no—
465C22—181no—
466C22—541no—
467C22—541no—
468C22—541no—
469C22C22×D3×D911yescharacteristic
470C4—22no—
471C4—61no—
472C4—61no—
473C4—62no—
474C4—182no—
475C4—181no—
476C4—542no—
477C4—541no—
478C4#8864012yes—
479C3—21no—
480C3#1058968311yescharacteristic, complemented
481C3#1059433311yescharacteristic
482C2—182no—
483C2—1081no—
484C2C2×#8864011yescharacteristic
485C2D9×#23711yescharacteristic
486C2#1172250911yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
prime radicalC2×#6192C22
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#552
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #11721748C2semidirect
2. C3×C4⋊C4×D9C2semidirect
3. C9×#887C2semidirect
5. #11720944C2semidirect
6. #11720943C2semidirect
8. #11720929C2semidirect
9. #11720928C2semidirect
10. #11720927C2semidirect
11. #11720926C2semidirect
54. C4⋊C4×D9D3semidirect
163. #887D9semidirect
314. C3×C9#552semidirect
432. C9#5513semidirect
480. C3#10589683semidirect

Maximal subgroups

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

subgroupindex
#117217482
C3×C4⋊C4×D92
C9×#8872
C4×#61922
#117209442
#117209432
#117209382
#117209292
#117209282
#117209272
#117209262
#105943333
#105896833