Finite Group Databasebeta

← #13160948

Subgroups — #13160948

Order 1728.

1388 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 1388 conjugacy classes of subgroups — too many to draw. Only its 176 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×C3×D4×C3⋊D3C211yesmaximal, characteristic, complemented
2C2×C32×#213C211yesmaximal, characteristic, complemented
3#11723522C211yesmaximal, characteristic, complemented
4C2×#88789C211yesmaximal, characteristic, complemented
5#11722202C211yesmaximal, characteristic, complemented
6#11722034C211yesmaximal, characteristic, complemented
7#11722023C211yesmaximal, characteristic, complemented
8#10592855—31nomaximal
9#10591447—34nomaximal
10C23×C3×C3⋊D3—21no—
11C2×C33×D4C2211yescharacteristic
12C2×C3×#638—21no—
13C3×D4×C3⋊D3—22no—
14C3×D4×C3⋊D3—22no—
15C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
16C2×C3×C4×C3⋊D3C2211yescharacteristic
17C2×C32×D12C2211yescharacteristic, complemented
18C2×#6290—21no—
19C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic, complemented
20C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
21C32×#213—22no—
22C2×C32×C3⋊C8C2211yescharacteristic, complemented
23(C2×D3)⋊(C3⋊Dic3)—21no—
24#88789—22no—
25C2×D4×C3⋊D3D311yescharacteristic, complemented
26C2×C3×D3×D4—34no—
27C2×C32×D8—31no—
28#64147—31no—
29C2×#3753—31no—
30C2×C3×#213D314yescomplemented
31#63908—34no—
32C2×#3713—34no—
33#63659—31no—
34#63659—34no—
35#63659—64no—
36#63648—31no—
37#63647—31no—
38#63563—34no—
39(C2×D3)⋊(C3⋊C8)—34no—
40C23×C33—21no—
41C22×C3×C3⋊D3—24no—
42C22×C3×C3⋊D3—21no—
43C22×C3×C3⋊D3—42no—
44C22×C3×C3⋊D3—42no—
45C22×C3×C3⋊D3D411yescharacteristic
46C22×C32×D3—21no—
47C33×D4—21no—
48C33×D4D412yescomplemented
49C2×C33×C4C2311yescharacteristic
50C2×C3⋊(C3⋊Dic3)—21no—
51C3×#638—24no—
52C2×C3×C3⋊Dic3D411yescharacteristic
53C3×C4⋊(C3⋊D3)—21no—
54C3×C4⋊(C3⋊D3)D412yescomplemented
55C3×C4×C3⋊D3—21no—
56C3×C4×C3⋊D3—21no—
57C32×D12—21no—
58C32×D12—21no—
59#6290—42no—
60C3⋊(C3⋊(C3⋊C8))—21no—
61C32×C3⋊C8—21no—
62#5217—91no—
63#5156—94no—
64C23×C3⋊D3—21no—
65C23×C3×D3—64no—
66C2×C32×D4—24no—
67C2×C32×D4—31no—
68C2×C32×D4C2×D314yes—
69C2×C32×D4C2×D311yescharacteristic
70C2×#638—21no—
71C2×#638—61no—
72D4×C3⋊D3—22no—
73D4×C3⋊D3—22no—
74C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
75C2×C4×C3⋊D3C2×D311yescharacteristic
76C2×C3×#67—64no—
77C3×D3×D4—68no—
78C3×D3×D4—68no—
79C2×C3×D12—34no—
80C2×C3×D12C2×D314yescomplemented
81C2×C3×C4×D3—34no—
82C2×#625—64no—
83C32×D8—62no—
84C2×C32×C8—31no—
85#3757—61no—
86#3753—62no—
87C4⋊(C3⋊Dic3)—31no—
88C4⋊(C3⋊Dic3)—34no—
89C4⋊(C3⋊Dic3)—64no—
90C2×C3⋊(C3⋊C8)—31no—
91C2×C3⋊(C3⋊C8)—34no—
92C2×C3⋊(C3⋊C8)—64no—
93C3×#213—28no—
94C2×C3×C3⋊C8C2×D314yescomplemented
95(C2×D3)⋊Dic3—64no—
96#3713—68no—
97C22×C33—22no—
98C22×C33—41no—
99C22×C33—41no—
100C22×C33C2×D411yescharacteristic
101C2×C3×C3⋊D3—21no—
102C2×C3×C3⋊D3—21no—
103C2×C3×C3⋊D3—22no—
104C2×C3×C3⋊D3—48no—
105C2×C3×C3⋊D3—41no—
106C2×C3×C3⋊D3—41no—
107C2×C3×C3⋊D3D812yescomplemented
108C2×C32×D3—41no—
109C2×C32×D3—41no—
110C2×C32×D3—41no—
111C33×C4C2×D411yescharacteristic
112C33×C4C2×D411yescharacteristic
113C3⋊(C3⋊Dic3)—41no—
114C3×C3⋊Dic3—21no—
115C22×C3×D4—91no—
116C2×D3×D4—34no—
117C2×C3×D8—34no—
118#938—94no—
119C2×#213—94no—
120C2×#213—91no—
121C2×#213C3⋊D311yescharacteristic, complemented
122#930—91no—
123#832—94no—
124#832—91no—
125#832—91no—
126#832—184no—
127#830—91no—
128#821—94no—
129(C2×D3)⋊C8—94no—
130C23×C32—21no—
131C23×C32—24no—
132C23×C32—44no—
133C23×C32—61no—
134C22×C3⋊D3—24no—
135C22×C3⋊D3—21no—
136C22×C3⋊D3—42no—
137C22×C3⋊D3—42no—
138C22×C3⋊D3#6711yescharacteristic
139C22×C3×D3—24no—
140C22×C3×D3—34no—
141C22×C3×D3—616no—
142C22×C3×D3—64no—
143C22×C3×D3—128no—
144C22×C3×D3—128no—
145C32×D4—21no—
146C32×D4—28no—
147C32×D4—24no—
148C32×D4—44no—
149C32×D4—61no—
150C32×D4—61no—
151C32×D4#6718yescomplemented
152C32×D4#6712yescomplemented
153C2×C32×C4—24no—
154C2×C32×C4C22×D314yes—
155C2×C32×C4C22×D311yescharacteristic
156#638—24no—
157#638—122no—
158C2×C3⋊Dic3—61no—
159C2×C3⋊Dic3—64no—
160C2×C3⋊Dic3—124no—
161C2×C3⋊Dic3#6711yescharacteristic
162C4⋊(C3⋊D3)—21no—
163C4⋊(C3⋊D3)#6712yescomplemented
164C4×C3⋊D3—21no—
165C4×C3⋊D3—21no—
166C3×#67—616no—
167C2×C3×Dic3—34no—
168C3×D12—24no—
169C3×D12—24no—
170C3×D12—38no—
171C3×D12—64no—
172C3×C4×D3—64no—
173C3×C4×D3—64no—
174#625—128no—
175C32×C8—61no—
176C3⋊(C3⋊C8)—61no—
177C3⋊(C3⋊C8)—64no—
178C3⋊(C3⋊C8)—124no—
179C3×C3⋊C8—24no—
180#447—271noSylow
181C2×C33—42no—
182C2×C33#12011yescharacteristic
183C2×C33C2×D811yescharacteristic
184C2×C33AGL(1,ℤ/8)11yescharacteristic
185C3×C3⋊D3—22no—
186C3×C3⋊D3—42no—
187C32×D3—81no—
188C24×C3—181no—
189C23×D3—64no—
190C2×C3×D4—24no—
191C2×C3×D4—34no—
192C2×C3×D4—91no—
193C2×C3×D4—181no—
194C2×C3×D4—182no—
195C2×C3×D4—182no—
196C2×C3×D4D3214yes—
197C2×C3×D4C2×C3⋊D311yescharacteristic
198C22×C3×C4—91no—
199C2×#67—64no—
200C2×#67—181no—
201C2×#67—184no—
202D3×D4—68no—
203D3×D4—68no—
204C2×D12—34no—
205C2×D12C2×C3⋊D311yescharacteristic, complemented
206C2×C4×D3—34no—
207C3×D8—68no—
208C2×C3×C8—34no—
209#217—184no—
210#213—22no—
211#213—182no—
212#213—188no—
213(C2×D3)⋊C4—181no—
214C4⋊Dic3—94no—
215C4⋊Dic3—91no—
216C4⋊Dic3—184no—
217C2×C3⋊C8—94no—
218C2×C3⋊C8—91no—
219C2×C3⋊C8—184no—
220C2×C3⋊C8C2×C3⋊D311yescharacteristic, complemented
221C22×C32—22no—
222C22×C32—24no—
223C22×C32—28no—
224C22×C32—48no—
225C22×C32—41no—
226C22×C32—41no—
227C22×C32—44no—
228C22×C32—44no—
229C22×C32—84no—
230C22×C32—84no—
231C22×C32—121no—
232C22×C32—121no—
233C22×C32—121no—
234C22×C32D3×D414yes—
235C22×C32C2×#6711yescharacteristic
236C2×C3⋊D3—21no—
237C2×C3⋊D3—21no—
238C2×C3⋊D3—22no—
239C2×C3⋊D3—48no—
240C2×C3⋊D3—41no—
241C2×C3⋊D3—41no—
242C2×C3⋊D3#21312yescomplemented
243C2×C3×D3—38no—
244C2×C3×D3—44no—
245C2×C3×D3—44no—
246C2×C3×D3—44no—
247C2×C3×D3—64no—
248C2×C3×D3—64no—
249C2×C3×D3—68no—
250C2×C3×D3—1232no—
251C2×C3×D3—124no—
252C2×C3×D3—124no—
253C32×C4—24no—
254C32×C4—24no—
255C32×C4D3×D414yes—
256C32×C4C2×#6714yes—
257C32×C4C2×#6711yescharacteristic
258C32×C4C2×#6711yescharacteristic
259C3⋊Dic3—21no—
260C3⋊Dic3—121no—
261C3⋊Dic3—124no—
262C3⋊Dic3—244no—
263C3×Dic3—64no—
264C22×D4—91no—
265C2×D8—31no—
266C2×D8—271no—
267#121—271no—
268#102—271no—
269#102—271no—
270C22⋊C8—271no—
271C33#44711yescharacteristic, complemented, Sylow
272C23×C3—24no—
273C23×C3—21no—
274C23×C3—44no—
275C23×C3—64no—
276C23×C3—91no—
277C23×C3—184no—
278C23×C3—181no—
279C23×C3—362no—
280C23×C3—362no—
281C22×D3—21no—
282C22×D3—34no—
283C22×D3—616no—
284C22×D3—64no—
285C22×D3—128no—
286C22×D3—128no—
287C3×D4—24no—
288C3×D4—21no—
289C3×D4—28no—
290C3×D4—44no—
291C3×D4—64no—
292C3×D4—64no—
293C3×D4—92no—
294C3×D4—184no—
295C3×D4—181no—
296C3×D4#62518yescomplemented
297C3×D4#63812yescomplemented
298C2×C3×C4—24no—
299C2×C3×C4—91no—
300C2×C3×C4—181no—
301C2×C3×C4—181no—
302C2×C3×C4C2×D3214yes—
303C2×C3×C4C22×C3⋊D311yescharacteristic
304#67—616no—
305#67—368no—
306#67—362no—
307C2×Dic3—34no—
308C2×Dic3—184no—
309C2×Dic3—181no—
310C2×Dic3—364no—
311D12—21no—
312D12—21no—
313D12—38no—
314D12—64no—
315C4×D3—64no—
316C4×D3—64no—
317C3×C8—64no—
318C3⋊C8—21no—
319C3⋊C8—184no—
320C3⋊C8—181no—
321C3⋊C8—364no—
322C2×C32—24no—
323C2×C32—24no—
324C2×C32—24no—
325C2×C32—42no—
326C2×C32—48no—
327C2×C32—88no—
328C2×C32—241no—
329C2×C32D3×D814yes—
330C2×C32#91114yes—
331C2×C32C2×#21311yescharacteristic
332C2×C32#93211yescharacteristic
333C2×C32#93714yes—
334C2×C32#95311yescharacteristic
335C3⋊D3—22no—
336C3⋊D3—42no—
337C3×D3—68no—
338C3×D3—84no—
339C3×D3—128no—
340C24—181no—
341C2×D4—31no—
342C2×D4—91no—
343C2×D4—181no—
344C2×D4—182no—
345C2×D4—182no—
346C2×D4—541no—
347C2×D4D3×C3⋊D311yescharacteristic
348C22×C4—91no—
349D8—62no—
350D8—542no—
351C2×C8—31no—
352C2×C8—271no—
353C4⋊C4—271no—
354C22⋊C4—541no—
355C22×C3—22no—
356C22×C3—28no—
357C22×C3—24no—
358C22×C3—48no—
359C22×C3—41no—
360C22×C3—41no—
361C22×C3—44no—
362C22×C3—44no—
363C22×C3—84no—
364C22×C3—84no—
365C22×C3—92no—
366C22×C3—124no—
367C22×C3—124no—
368C22×C3—124no—
369C22×C3—181no—
370C22×C3—181no—
371C22×C3—182no—
372C22×C3—368no—
373C22×C3—361no—
374C22×C3—361no—
375C22×C3D3×#6714yes—
376C22×C3D4×C3⋊D311yescharacteristic
377C2×D3—38no—
378C2×D3—41no—
379C2×D3—41no—
380C2×D3—41no—
381C2×D3—64no—
382C2×D3—64no—
383C2×D3—68no—
384C2×D3—1232no—
385C2×D3—124no—
386C2×D3—124no—
387C3×C4—24no—
388C3×C4—24no—
389C3×C4—181no—
390C3×C4C2×#62514yes—
391C3×C4D3×#6714yes—
392C3×C4D4×C3⋊D311yescharacteristic
393C3×C4C2×#63811yescharacteristic
394Dic3—64no—
395Dic3—361no—
396Dic3—364no—
397Dic3—724no—
398C32—24no—
399C32#515614yescomplemented
400C32#521711yescharacteristic, complemented
401C23—21no—
402C23—61no—
403C23—91no—
404C23—184no—
405C23—181no—
406C23—362no—
407C23—362no—
408D4—21no—
409D4—61no—
410D4—61no—
411D4—92no—
412D4—184no—
413D4—181no—
414D4—1082no—
415D4#629012yescomplemented
416C2×C4—91no—
417C2×C4—181no—
418C2×C4—181no—
419C2×C4—541no—
420C2×C4C2×D3×C3⋊D311yescharacteristic
421C8—61no—
422C8—541no—
423C2×C3—24no—
424C2×C3—24no—
425C2×C3—24no—
426C2×C3—42no—
427C2×C3—48no—
428C2×C3—88no—
429C2×C3—182no—
430C2×C3—244no—
431C2×C3—362no—
432C2×C3D3×#21314yes—
433C2×C3#6393214yes—
434C2×C3#6397614yes—
435C2×C3D8×C3⋊D311yescharacteristic
436C2×C3#6412011yescharacteristic
437C2×C3#6414611yescharacteristic
438D3—68no—
439D3—81no—
440D3—128no—
441C22—22no—
442C22—41no—
443C22—41no—
444C22—92no—
445C22—121no—
446C22—121no—
447C22—121no—
448C22—181no—
449C22—181no—
450C22—182no—
451C22—368no—
452C22—361no—
453C22—361no—
454C22C3⋊D3×#6711yescharacteristic
455C4—181no—
456C4—1081no—
457C4C2×#629011yescharacteristic
458C4C3⋊D3×#6711yescharacteristic
459C3—24no—
460C3#1059144714yescomplemented
461C3#1059285511yescharacteristic, complemented
462C2—42no—
463C2—182no—
464C2—241no—
465C2—362no—
466C2C3⋊D3×#21311yescharacteristic
467C2#1172356011yescharacteristic
468C2#1172363411yescharacteristic
subgroupG/H
centerC22C3⋊D3×#67
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC22×C33C2×D4
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#447
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×D4×C3⋊D3C2semidirect
2. C2×C32×#213C2semidirect
3. #11723522C2semidirect
4. C2×#88789C2semidirect
5. #11722202C2semidirect
6. #11722034C2semidirect
7. #11722023C2semidirect
17. C2×C32×D12C22semidirect
19. C4⋊(C3⋊(C3⋊Dic3))C22semidirect
20. C2×C3⋊(C3⋊(C3⋊C8))C22semidirect
22. C2×C32×C3⋊C8C22semidirect
25. C2×D4×C3⋊D3D3semidirect
30. C2×C3×#213D3semidirect
48. C33×D4D4semidirect
54. C3×C4⋊(C3⋊D3)D4semidirect
80. C2×C3×D12C2×D3semidirect
94. C2×C3×C3⋊C8C2×D3semidirect
107. C2×C3×C3⋊D3D8semidirect
121. C2×#213C3⋊D3semidirect
151. C32×D4#67semidirect
152. C32×D4#67semidirect
163. C4⋊(C3⋊D3)#67semidirect
205. C2×D12C2×C3⋊D3semidirect
220. C2×C3⋊C8C2×C3⋊D3semidirect
242. C2×C3⋊D3#213semidirect
271. C33#447semidirect
296. C3×D4#625semidirect
297. C3×D4#638semidirect
399. C32#5156semidirect
400. C32#5217semidirect
415. D4#6290semidirect
460. C3#10591447semidirect
461. C3#10592855semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D4×C3⋊D32
C2×C32×#2132
#117235222
C2×#887892
#117222022
#117220342
#117220232
#105928553
#105914473