Finite Group Databasebeta

← #13138649

Subgroups — #13138649

Order 1728.

644 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 95 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×D36C211yesmaximal, characteristic, complemented
2C2×C3×D72C211yesmaximal, characteristic, complemented
3#11721037C211yesmaximal, characteristic, complemented
4C2×#88417C211yesmaximal, characteristic, complemented
5#11720543C211yesmaximal, characteristic, complemented
6C9×(C2×D3)⋊C8C211yesmaximal, characteristic, complemented
7#11720283C211yesmaximal, characteristic, complemented
8#10590944—31nomaximal
9#10588414—31nomaximal
10C22×D3×D9—21no—
11C2×C4⋊(C3⋊D9)C2211yescharacteristic
12C2×C4×D3×C9C2211yescharacteristic
13C2×C3×D36C2211yescharacteristic
14C2×C3×D36C2211yescharacteristic, complemented
15C2×#6194—21no—
16C2×#6192—21no—
17D3×D36—22no—
18D3×D36—22no—
19C2×C3×C8×C9C2211yescharacteristic, complemented
20C4⋊(C3⋊Dic9)C2211yescharacteristic, complemented
21C2×C9×C3⋊C8C2211yescharacteristic, complemented
22C3×D72—22no—
23(C2×D9)⋊Dic3—21no—
24#88417—22no—
25C2×D3×D12—31no—
26C2×C3×D24—31no—
27#63908—31no—
28C2×#3714—31no—
29C22×D36—31no—
30#63648—31no—
31C3×(C2×D3)⋊C8D311yescharacteristic
32#63563—31no—
33#63492—31no—
34C2×D72—31no—
35C2×D72D311yescharacteristic, complemented
36C9×C22⋊C8—31no—
37#63380—31no—
38#63380—31no—
39#63380—61no—
40C22×C3⋊D9—21no—
41C22×D3×C9D411yescharacteristic
42C22×C3×D9—21no—
43C22×C3×D9—21no—
44C2×D3×D9—24no—
45C2×D3×D9—42no—
46C2×D3×D9—42no—
47C2×C3×C4×C9C2311yescharacteristic
48C2×C3⋊Dic9—21no—
49C4⋊(C3⋊D9)—21no—
50C4⋊(C3⋊D9)D412yescomplemented
51C2×C9×Dic3D411yescharacteristic
52C4×D3×C9—21no—
53C4×D3×C9—21no—
54C3×D36—21no—
55C3×D36—21no—
56C3×D36—21no—
57C3×D36D412yescomplemented
58#6194—42no—
59#6192—24no—
60C3×C8×C9—21no—
61C9×C3⋊C8—21no—
62#5156—91no—
63#4732—91no—
64C22×D32—61no—
65C2×C4⋊(C3⋊D3)—31no—
66C2×C3×D12—31no—
67C2×C3×D12—31no—
68C2×C3×C4×D3C2×D311yescharacteristic
69C2×#626—61no—
70C2×#625—61no—
71D3×D12—62no—
72D3×D12—62no—
73C23×D9—61no—
74C2×C32×C8C2×D311yescharacteristic
75C4⋊(C3⋊Dic3)—31no—
76C2×C3×C3⋊C8C2×D311yescharacteristic
77C3×D24—62no—
78(C2×D3)⋊Dic3—61no—
79#3714—62no—
80C22×C4×C9—31no—
81C2×#611—61no—
82C2×D36—31no—
83C2×D36—62no—
84C2×D36—61no—
85C2×D36—62no—
86C2×D36—61no—
87C2×D36C2×D311yescharacteristic
88C2×D36C2×D311yescharacteristic, complemented
89C2×C8×C9—21no—
90C2×C8×C9—31no—
91C2×C8×C9C2×D311yescharacteristic, complemented
92(C2×D9)⋊C4—61no—
93C4⋊Dic9—31no—
94C4⋊Dic9—61no—
95D72—22no—
96D72—62no—
97C22×C3×C9C2×D411yescharacteristic
98C2×C3⋊D9—22no—
99C2×C3⋊D9—41no—
100C2×C3⋊D9—41no—
101C2×D3×C9—21no—
102C2×D3×C9—21no—
103C2×D3×C9D812yescomplemented
104C2×C3×D9—22no—
105C2×C3×D9—41no—
106C2×C3×D9—41no—
107C2×C3×D9—41no—
108C2×C3×D9—41no—
109C2×C3×D9—41no—
110D3×D9—48no—
111C3×C4×C9C2×D411yescharacteristic
112C3×C4×C9C2×D411yescharacteristic
113C3⋊Dic9—41no—
114C9×Dic3—21no—
115C2×D3×D4—91no—
116C22×D12—91no—
117C2×C3×D8—91no—
118#938—91no—
119C2×#213—91no—
120#930—91no—
121C2×D24—31no—
122C2×D24—91no—
123C3×C22⋊C8—31no—
124#832—91no—
125#821—91no—
126#821—91no—
127#821—91no—
128#821—181no—
129(C2×D3)⋊C8D911yescharacteristic, complemented
130C22×C3⋊D3—61no—
131C22×C3×D3—61no—
132C22×C3×D3—61no—
133C22×C3×D3D1211yescharacteristic
134C2×D32—64no—
135C2×D32—122no—
136C2×D32—122no—
137C2×C32×C4C22×D311yescharacteristic
138C2×C3⋊Dic3—61no—
139C4⋊(C3⋊D3)—32no—
140C4⋊(C3⋊D3)—61no—
141C2×C3×Dic3D1211yescharacteristic
142C3×D12—32no—
143C3×D12—61no—
144C3×D12—61no—
145C3×D12—61no—
146C3×C4×D3—21no—
147C3×C4×D3—21no—
148#626—64no—
149#625—122no—
150C23×C9—31no—
151C22×D9—21no—
152C22×D9—21no—
153C22×D9—64no—
154C22×D9—61no—
155C22×D9—121no—
156C22×D9—122no—
157C22×D9—122no—
158C32×C8—21no—
159C3×C3⋊C8—21no—
160C2×C4×C9—21no—
161C2×C4×C9—31no—
162C2×C4×C9—61no—
163C2×C4×C9—61no—
164C2×C4×C9C22×D311yescharacteristic
165#611—122no—
166C2×Dic9—61no—
167C2×Dic9—121no—
168D36—21no—
169D36—21no—
170D36—21no—
171D36—32no—
172D36—62no—
173D36—64no—
174D36—61no—
175D36—121no—
176D36#6712yescomplemented
177C8×C9—21no—
178C8×C9—41no—
179C8×C9—61no—
180#447—271noSylow
181C2×C3×C9#12011yescharacteristic
182C2×C3×C9C2×D811yescharacteristic
183C2×C3×C9AGL(1,ℤ/8)11yescharacteristic
184C3⋊D9—42no—
185D3×C9—22no—
186C3×D9—42no—
187C3×D9—81no—
188C23×D3—181no—
189C23×D3—181no—
190C2×C3×D4—91no—
191C2×C3×D4—91no—
192C22×C3×C4—31no—
193C2×#67—181no—
194C2×#67—181no—
195C2×#67—181no—
196D3×D4—182no—
197D3×D4—182no—
198C2×D12—31no—
199C2×D12—31no—
200C2×D12—91no—
201C2×D12—91no—
202C2×D12—181no—
203C2×D12—182no—
204C2×D12—182no—
205C2×D12—181no—
206C2×C4×D3C2×D911yescharacteristic
207C3×D8—182no—
208C2×C3×C8—21no—
209C2×C3×C8—31no—
210C2×C3×C8C2×D911yescharacteristic, complemented
211C2×C3×C8D3211yescharacteristic
212#217—181no—
213#213—182no—
214(C2×D3)⋊C4—181no—
215C4⋊Dic3—91no—
216C4⋊Dic3—91no—
217C4⋊Dic3—181no—
218C2×C3⋊C8C2×D911yescharacteristic, complemented
219D24—62no—
220D24—182no—
221C22×C32C2×D1211yescharacteristic
222C2×C3⋊D3—62no—
223C2×C3⋊D3—121no—
224C2×C3⋊D3—121no—
225C2×C3×D3—21no—
226C2×C3×D3—21no—
227C2×C3×D3—62no—
228C2×C3×D3—121no—
229C2×C3×D3—121no—
230C2×C3×D3—121no—
231C2×C3×D3—121no—
232C2×C3×D3—121no—
233C2×C3×D3D2412yes—
234D32—128no—
235C32×C4D3×D411yescharacteristic
236C32×C4D3×D411yescharacteristic
237C3⋊Dic3—121no—
238C3×Dic3—21no—
239C22×C9—21no—
240C22×C9—32no—
241C22×C9—61no—
242C22×C9—61no—
243C22×C9D3×D411yescharacteristic
244C2×D9—22no—
245C2×D9—41no—
246C2×D9—41no—
247C2×D9—41no—
248C2×D9—41no—
249C2×D9—41no—
250C2×D9—62no—
251C2×D9—122no—
252C2×D9—128no—
253C2×D9—121no—
254C2×D9—121no—
255C2×D9—241no—
256C2×D9—241no—
257C4×C9—21no—
258C4×C9—21no—
259C4×C9—61no—
260C4×C9D3×D411yescharacteristic
261C4×C9C2×#6711yescharacteristic
262Dic9—121no—
263Dic9—241no—
264C22×D4—271no—
265C2×D8—91no—
266C2×D8—271no—
267#121—271no—
268#102—271no—
269#102—271no—
270C22⋊C8—31no—
271C3×C9#44711yescharacteristic, complemented, Sylow
272C23×C3—31no—
273C23×C3—181no—
274C23×C3—181no—
275C22×D3—61no—
276C22×D3—61no—
277C22×D3—184no—
278C22×D3—184no—
279C22×D3—181no—
280C22×D3—181no—
281C22×D3—361no—
282C22×D3—362no—
283C22×D3—362no—
284C22×D3—362no—
285C22×D3—362no—
286C22×D3D3611yescharacteristic
287C3×D4—92no—
288C3×D4—181no—
289C3×D4—181no—
290C3×D4—181no—
291C2×C3×C4—21no—
292C2×C3×C4—31no—
293C2×C3×C4—61no—
294C2×C3×C4—61no—
295C2×C3×C4C22×D911yescharacteristic
296C2×C3×C4C2×D3211yescharacteristic
297#67—184no—
298#67—362no—
299#67—362no—
300C2×Dic3—181no—
301C2×Dic3—181no—
302C2×Dic3—361no—
303C2×Dic3D3611yescharacteristic
304D12—32no—
305D12—61no—
306D12—61no—
307D12—61no—
308D12—92no—
309D12—92no—
310D12—182no—
311D12—184no—
312D12—181no—
313D12—181no—
314D12—361no—
315C4×D3—21no—
316C4×D3—21no—
317C3×C8—21no—
318C3×C8—21no—
319C3×C8—41no—
320C3×C8—61no—
321C3⋊C8—21no—
322C2×C32#88211yescharacteristic
323C2×C32C2×D2411yescharacteristic
324C2×C32#90811yescharacteristic
325C3⋊D3—122no—
326C3×D3—22no—
327C3×D3—122no—
328C3×D3—241no—
329C2×C9—21no—
330C2×C9—21no—
331C2×C9—21no—
332C2×C9—62no—
333C2×C9D3×D811yescharacteristic
334C2×C9#91111yescharacteristic
335C2×C9#93711yescharacteristic
336D9—42no—
337D9—81no—
338D9—122no—
339D9—242no—
340C24—541no—
341C2×D4—91no—
342C2×D4—91no—
343C2×D4—271no—
344C2×D4—541no—
345C2×D4—542no—
346C2×D4—542no—
347C2×D4—541no—
348C22×C4—31no—
349D8—182no—
350D8—542no—
351C2×C8—31no—
352C2×C8D3×D911yescharacteristic, complemented
353C4⋊C4—271no—
354C22⋊C4—541no—
355C22×C3—21no—
356C22×C3—32no—
357C22×C3—61no—
358C22×C3—61no—
359C22×C3—182no—
360C22×C3—361no—
361C22×C3—361no—
362C22×C3—361no—
363C22×C3—361no—
364C22×C3—361no—
365C22×C3C2×D3611yescharacteristic
366C22×C3D3×D1211yescharacteristic
367C2×D3—21no—
368C2×D3—21no—
369C2×D3—62no—
370C2×D3—121no—
371C2×D3—121no—
372C2×D3—121no—
373C2×D3—121no—
374C2×D3—121no—
375C2×D3—182no—
376C2×D3—182no—
377C2×D3—362no—
378C2×D3—368no—
379C2×D3—368no—
380C2×D3—361no—
381C2×D3—361no—
382C2×D3—361no—
383C2×D3—361no—
384C2×D3—721no—
385C2×D3—721no—
386C2×D3D7212yescomplemented
387C3×C4—21no—
388C3×C4—21no—
389C3×C4—61no—
390C3×C4D4×D911yescharacteristic
391C3×C4D4×D911yescharacteristic
392C3×C4#380211yescharacteristic
393C3×C4D3×#6711yescharacteristic
394Dic3—21no—
395Dic3—361no—
396Dic3—361no—
397Dic3—721no—
398C32#473211yescharacteristic
399C9—21no—
400C9#515611yescharacteristic, complemented
401C23—31no—
402C23—181no—
403C23—181no—
404C23—544no—
405C23—541no—
406C23—1082no—
407C23—1082no—
408D4—92no—
409D4—181no—
410D4—181no—
411D4—181no—
412D4—272no—
413D4—544no—
414D4—541no—
415D4—1082no—
416C2×C4—31no—
417C2×C4—61no—
418C2×C4—61no—
419C2×C4—541no—
420C2×C4C2×D3×D911yescharacteristic
421C8—21no—
422C8—61no—
423C2×C3—21no—
424C2×C3—21no—
425C2×C3—21no—
426C2×C3—62no—
427C2×C3—362no—
428C2×C3—721no—
429C2×C3#6344411yescharacteristic
430C2×C3C2×D7211yescharacteristic
431C2×C3#6347011yescharacteristic
432C2×C3D3×D2411yescharacteristic
433C2×C3#6379511yescharacteristic
434C2×C3#6392211yescharacteristic
435D3—22no—
436D3—122no—
437D3—241no—
438D3—362no—
439D3—362no—
440D3—722no—
441C22—32no—
442C22—61no—
443C22—61no—
444C22—182no—
445C22—361no—
446C22—361no—
447C22—361no—
448C22—361no—
449C22—361no—
450C22—542no—
451C22—1088no—
452C22—1081no—
453C22—1081no—
454C22D3×D3611yescharacteristic
455C4—61no—
456C4—1081no—
457C4#8864611yescharacteristic
458C4D9×#6711yescharacteristic
459C3—21no—
460C3#1058841411yescharacteristic, complemented
461C3#1059094411yescharacteristic
462C2—62no—
463C2—362no—
464C2—721no—
465C2—1082no—
466C2D3×D7211yescharacteristic
467C2#1172080511yescharacteristic
468C2#1172104911yescharacteristic
subgroupG/H
centerC22D3×D36
commutator, cosocleC2×C3×C4×C9C23
FrattiniC2×C3×C4C2×D32
FittingC2×C3×C8×C9C22
socleC22×C32C2×D12
hypercenterC2×C8D3×D9
hypocenterC3×C9#447
LCSG ▷ C2×C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. C2×D3×D36C2semidirect
2. C2×C3×D72C2semidirect
3. #11721037C2semidirect
4. C2×#88417C2semidirect
5. #11720543C2semidirect
6. C9×(C2×D3)⋊C8C2semidirect
7. #11720283C2semidirect
14. C2×C3×D36C22semidirect
19. C2×C3×C8×C9C22semidirect
20. C4⋊(C3⋊Dic9)C22semidirect
21. C2×C9×C3⋊C8C22semidirect
35. C2×D72D3semidirect
50. C4⋊(C3⋊D9)D4semidirect
57. C3×D36D4semidirect
88. C2×D36C2×D3semidirect
91. C2×C8×C9C2×D3semidirect
103. C2×D3×C9D8semidirect
129. (C2×D3)⋊C8D9semidirect
176. D36#67semidirect
210. C2×C3×C8C2×D9semidirect
218. C2×C3⋊C8C2×D9semidirect
271. C3×C9#447semidirect
352. C2×C8D3×D9semidirect
386. C2×D3D72semidirect
400. C9#5156semidirect
460. C3#10588414semidirect

Maximal subgroups

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

subgroupindex
C2×D3×D362
C2×C3×D722
#117210372
C2×#884172
#117205432
C9×(C2×D3)⋊C82
#117202832
#105909443
#105884143