Finite Group Databasebeta

← #13160950

Subgroups — #13160950

Order 1728.

1492 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 1492 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×#88860C211yesmaximal, characteristic, complemented
3#11723526C211yesmaximal, characteristic, complemented
4C2×#88791C211yesmaximal, characteristic, complemented
5C32×#832C211yesmaximal, characteristic, complemented
6#11722033C211yesmaximal, characteristic, complemented
7#11722023C211yesmaximal, characteristic, complemented
8#10592472—31nomaximal
9#10591448—34nomaximal
10C23×C3×C3⋊D3—21no—
11C2×C33×D4C2211yescharacteristic
12C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic, complemented
13C2×C3×#638—21no—
14C3×D4×C3⋊D3—22no—
15C3×D4×C3⋊D3—22no—
16C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
17C2×C3×C4×C3⋊D3C2211yescharacteristic
18C2×#6292—21no—
19#88860—22no—
20C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
21C32×C4⋊Dic3C2211yescharacteristic, complemented
22C2×C32×C3⋊C8C2211yescharacteristic, complemented
23#88804—21no—
24#88791—22no—
25C2×D4×C3⋊D3D311yescharacteristic, complemented
26C2×C3×D3×D4—34no—
27C2×#3753—31no—
28C2×#3753—34no—
29C2×#3753—64no—
30C2×#3745—31no—
31#64104—31no—
32#63909—34no—
33C2×#3714—34no—
34C32×#102—31no—
35#63647—31no—
36#63636—31no—
37C3×#832D314yescomplemented
38#63562—34no—
39(C2×D3)⋊(C3⋊C8)—34no—
40C23×C33—21no—
41C22×C3⋊(C3⋊D3)—21no—
42C22×C3×C3⋊D3—24no—
43C22×C3×C3⋊D3—21no—
44C22×C3×C3⋊D3—42no—
45C22×C3×C3⋊D3—42no—
46C22×C3×C3⋊D3D411yescharacteristic
47C33×D4—21no—
48C33×D4D412yescomplemented
49C2×C33×C4C2311yescharacteristic
50C4⋊(C3⋊(C3⋊D3))—21no—
51C4⋊(C3⋊(C3⋊D3))—21no—
52C3×#638—24no—
53C2×C3×C3⋊Dic3D411yescharacteristic
54C3×C4⋊(C3⋊D3)—21no—
55C3×C4⋊(C3⋊D3)D412yescomplemented
56C3×C4×C3⋊D3—21no—
57C3×C4×C3⋊D3—21no—
58C2×C32×Dic3—21no—
59#6292—42no—
60C3⋊(C3⋊(C3⋊C8))—21no—
61C32×C3⋊C8—21no—
62#5217—91no—
63#4773—94no—
64C23×C3⋊D3—21no—
65C23×C3×D3—64no—
66C2×C32×D4—24no—
67C2×C32×D4C2×D314yes—
68C2×C32×D4C2×D311yescharacteristic
69C2×#638—21no—
70D4×C3⋊D3—22no—
71D4×C3⋊D3—22no—
72C2×C4⋊(C3⋊D3)—31no—
73C2×C4⋊(C3⋊D3)—34no—
74C2×C4⋊(C3⋊D3)—64no—
75C2×C4⋊(C3⋊D3)—61no—
76C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
77C2×C4×C3⋊D3C2×D311yescharacteristic
78C2×C3×#67—64no—
79C3×D3×D4—68no—
80C3×D3×D4—68no—
81C2×C3×D12—34no—
82C2×C3×C4×D3—34no—
83C2×#626—64no—
84C2×C32×C8—31no—
85C32×C4⋊C4—31no—
86#3753—616no—
87#3753—68no—
88#3753—62no—
89#3752—61no—
90C2×C3⋊(C3⋊C8)—31no—
91C2×C3⋊(C3⋊C8)—34no—
92C2×C3⋊(C3⋊C8)—64no—
93#3745—62no—
94C3×C4⋊Dic3C2×D314yescomplemented
95C2×C3×C3⋊C8C2×D314yescomplemented
96#3722—64no—
97#3714—68no—
98C22×C33—22no—
99C22×C33—41no—
100C22×C33—41no—
101C22×C33C2×D411yescharacteristic
102C2×C3⋊(C3⋊D3)—41no—
103C2×C3⋊(C3⋊D3)—41no—
104C2×C3⋊(C3⋊D3)—41no—
105C2×C3×C3⋊D3—21no—
106C2×C3×C3⋊D3—21no—
107C2×C3×C3⋊D3—22no—
108C2×C3×C3⋊D3—48no—
109C2×C3×C3⋊D3—41no—
110C2×C3×C3⋊D3—41no—
111C2×C3×C3⋊D3D812yescomplemented
112C33×C4C2×D411yescharacteristic
113C33×C4C2×D411yescharacteristic
114C3×C3⋊Dic3—21no—
115C32×Dic3—41no—
116C22×C3×D4—91no—
117C2×D3×D4—34no—
118C2×#213—94no—
119C2×#213—91no—
120C2×#213—91no—
121C2×#213—184no—
122#930—91no—
123C2×D24—94no—
124#895—94no—
125C3×#102—34no—
126#832—91no—
127#832C3⋊D311yescharacteristic, complemented
128#830—91no—
129(C2×D3)⋊C8—94no—
130#809—94no—
131C23×C32—21no—
132C23×C32—24no—
133C23×C32—44no—
134C22×C3⋊D3—24no—
135C22×C3⋊D3—21no—
136C22×C3⋊D3—42no—
137C22×C3⋊D3—42no—
138C22×C3⋊D3—61no—
139C22×C3⋊D3—64no—
140C22×C3⋊D3—124no—
141C22×C3⋊D3#6711yescharacteristic
142C22×C3×D3—34no—
143C22×C3×D3—616no—
144C22×C3×D3—64no—
145C22×C3×D3—128no—
146C22×C3×D3—128no—
147C32×D4—21no—
148C32×D4—28no—
149C32×D4—24no—
150C32×D4—44no—
151C32×D4D1218yescomplemented
152C32×D4#6712yescomplemented
153C2×C32×C4—24no—
154C2×C32×C4—61no—
155C2×C32×C4C22×D314yes—
156C2×C32×C4C22×D311yescharacteristic
157#638—24no—
158C2×C3⋊Dic3#6711yescharacteristic
159C4⋊(C3⋊D3)—21no—
160C4⋊(C3⋊D3)—68no—
161C4⋊(C3⋊D3)—64no—
162C4⋊(C3⋊D3)—61no—
163C4⋊(C3⋊D3)—61no—
164C4⋊(C3⋊D3)—64no—
165C4⋊(C3⋊D3)—124no—
166C4⋊(C3⋊D3)—122no—
167C4⋊(C3⋊D3)#6712yescomplemented
168C4×C3⋊D3—21no—
169C4×C3⋊D3—21no—
170C3×#67—616no—
171C2×C3×Dic3—24no—
172C2×C3×Dic3—34no—
173C3×D12—38no—
174C3×D12—64no—
175C3×C4×D3—64no—
176C3×C4×D3—64no—
177#626—128no—
178C32×C8—61no—
179C3⋊(C3⋊C8)—68no—
180C3⋊(C3⋊C8)—64no—
181C3⋊(C3⋊C8)—61no—
182C3×C3⋊C8—24no—
183#447—271noSylow
184C2×C33—42no—
185C2×C33#12011yescharacteristic
186C2×C33C2×D811yescharacteristic
187C2×C33AGL(1,ℤ/8)11yescharacteristic
188C3⋊(C3⋊D3)—81no—
189C3×C3⋊D3—22no—
190C3×C3⋊D3—42no—
191C24×C3—181no—
192C23×D3—64no—
193C2×C3×D4—24no—
194C2×C3×D4—91no—
195C2×C3×D4—181no—
196C2×C3×D4—182no—
197C2×C3×D4—182no—
198C2×C3×D4D3214yes—
199C2×C3×D4C2×C3⋊D311yescharacteristic
200C22×C3×C4—91no—
201C2×#67—64no—
202C2×#67—181no—
203D3×D4—68no—
204D3×D4—68no—
205C2×D12—34no—
206C2×D12—94no—
207C2×D12—91no—
208C2×D12—184no—
209C2×D12—184no—
210C2×C4×D3—34no—
211C2×C3×C8—34no—
212C3×C4⋊C4—34no—
213#213—182no—
214#213—1816no—
215#213—188no—
216#213—182no—
217(C2×D3)⋊C4—181no—
218(C2×D3)⋊C4—184no—
219C4⋊Dic3C2×C3⋊D311yescharacteristic, complemented
220C2×C3⋊C8—94no—
221C2×C3⋊C8—91no—
222C2×C3⋊C8—184no—
223C2×C3⋊C8C2×C3⋊D311yescharacteristic, complemented
224D24—188no—
225C22×C32—22no—
226C22×C32—24no—
227C22×C32—28no—
228C22×C32—48no—
229C22×C32—41no—
230C22×C32—41no—
231C22×C32—44no—
232C22×C32—44no—
233C22×C32—84no—
234C22×C32—84no—
235C22×C32D3×D414yes—
236C22×C32C2×#6711yescharacteristic
237C2×C3⋊D3—21no—
238C2×C3⋊D3—21no—
239C2×C3⋊D3—22no—
240C2×C3⋊D3—48no—
241C2×C3⋊D3—41no—
242C2×C3⋊D3—41no—
243C2×C3⋊D3—128no—
244C2×C3⋊D3—124no—
245C2×C3⋊D3—121no—
246C2×C3⋊D3—121no—
247C2×C3⋊D3—121no—
248C2×C3⋊D3—124no—
249C2×C3⋊D3—124no—
250C2×C3⋊D3—244no—
251C2×C3⋊D3—244no—
252C2×C3⋊D3#21312yescomplemented
253C2×C3×D3—38no—
254C2×C3×D3—64no—
255C2×C3×D3—64no—
256C2×C3×D3—68no—
257C2×C3×D3—1232no—
258C2×C3×D3—124no—
259C2×C3×D3—124no—
260C32×C4—24no—
261C32×C4—24no—
262C32×C4—121no—
263C32×C4C2×D1214yes—
264C32×C4D3×D414yes—
265C32×C4C2×#6711yescharacteristic
266C32×C4C2×#6711yescharacteristic
267C3⋊Dic3—21no—
268C3×Dic3—44no—
269C3×Dic3—64no—
270C22×D4—91no—
271C2×D8—271no—
272C2×D8—271no—
273#121—271no—
274#102—31no—
275#102—271no—
276C22⋊C8—271no—
277C33#44711yescharacteristic, complemented, Sylow
278C23×C3—24no—
279C23×C3—21no—
280C23×C3—44no—
281C23×C3—91no—
282C23×C3—184no—
283C23×C3—181no—
284C23×C3—362no—
285C23×C3—362no—
286C22×D3—34no—
287C22×D3—616no—
288C22×D3—64no—
289C22×D3—128no—
290C22×D3—128no—
291C22×D3—184no—
292C22×D3—181no—
293C22×D3—364no—
294C3×D4—24no—
295C3×D4—21no—
296C3×D4—28no—
297C3×D4—44no—
298C3×D4—92no—
299C3×D4—184no—
300C3×D4—181no—
301C3×D4#62618yescomplemented
302C3×D4C4⋊(C3⋊D3)12yescomplemented
303C2×C3×C4—24no—
304C2×C3×C4—64no—
305C2×C3×C4—91no—
306C2×C3×C4—181no—
307C2×C3×C4—181no—
308C2×C3×C4C2×D3214yes—
309C2×C3×C4C22×C3⋊D311yescharacteristic
310#67—616no—
311#67—362no—
312C2×Dic3—21no—
313C2×Dic3—34no—
314D12—38no—
315D12—64no—
316D12—188no—
317D12—184no—
318D12—184no—
319D12—181no—
320D12—181no—
321D12—364no—
322D12—368no—
323C4×D3—64no—
324C4×D3—64no—
325C3×C8—64no—
326C3⋊C8—21no—
327C3⋊C8—188no—
328C3⋊C8—184no—
329C3⋊C8—181no—
330C2×C32—24no—
331C2×C32—24no—
332C2×C32—24no—
333C2×C32—42no—
334C2×C32—48no—
335C2×C32—88no—
336C2×C32#88214yes—
337C2×C32D3×D814yes—
338C2×C32#91414yes—
339C2×C32C2×#21311yescharacteristic
340C2×C32#93211yescharacteristic
341C2×C32#95311yescharacteristic
342C3⋊D3—22no—
343C3⋊D3—42no—
344C3⋊D3—248no—
345C3⋊D3—241no—
346C3⋊D3—244no—
347C3×D3—68no—
348C3×D3—128no—
349C24—181no—
350C2×D4—91no—
351C2×D4—181no—
352C2×D4—182no—
353C2×D4—182no—
354C2×D4—271no—
355C2×D4—541no—
356C2×D4D3×C3⋊D311yescharacteristic
357C22×C4—91no—
358D8—542no—
359D8—542no—
360C2×C8—31no—
361C2×C8—271no—
362C4⋊C4—31no—
363C22⋊C4—541no—
364C22×C3—22no—
365C22×C3—28no—
366C22×C3—24no—
367C22×C3—48no—
368C22×C3—41no—
369C22×C3—41no—
370C22×C3—44no—
371C22×C3—44no—
372C22×C3—84no—
373C22×C3—84no—
374C22×C3—92no—
375C22×C3—181no—
376C22×C3—181no—
377C22×C3—182no—
378C22×C3—368no—
379C22×C3—361no—
380C22×C3—361no—
381C22×C3D3×#6714yes—
382C22×C3D4×C3⋊D311yescharacteristic
383C2×D3—38no—
384C2×D3—64no—
385C2×D3—64no—
386C2×D3—68no—
387C2×D3—1232no—
388C2×D3—124no—
389C2×D3—124no—
390C2×D3—368no—
391C2×D3—361no—
392C2×D3—361no—
393C2×D3—361no—
394C2×D3—364no—
395C2×D3—364no—
396C2×D3—364no—
397C2×D3—724no—
398C2×D3—724no—
399C3×C4—24no—
400C3×C4—24no—
401C3×C4—124no—
402C3×C4—181no—
403C3×C4C2×#62614yes—
404C3×C4D3×#6714yes—
405C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
406C3×C4D4×C3⋊D311yescharacteristic
407Dic3—41no—
408Dic3—64no—
409C32—24no—
410C32#477314yescomplemented
411C32#521711yescharacteristic, complemented
412C23—21no—
413C23—91no—
414C23—184no—
415C23—181no—
416C23—362no—
417C23—362no—
418C23—541no—
419D4—21no—
420D4—92no—
421D4—184no—
422D4—181no—
423D4—541no—
424D4—541no—
425D4—1082no—
426D4#629212yescomplemented
427C2×C4—61no—
428C2×C4—91no—
429C2×C4—181no—
430C2×C4—181no—
431C2×C4C2×D3×C3⋊D311yescharacteristic
432C8—61no—
433C8—541no—
434C2×C3—24no—
435C2×C3—24no—
436C2×C3—24no—
437C2×C3—48no—
438C2×C3—42no—
439C2×C3—88no—
440C2×C3—182no—
441C2×C3—362no—
442C2×C3D3×#21314yes—
443C2×C3#6393414yes—
444C2×C3#6397714yes—
445C2×C3#6409111yescharacteristic
446C2×C3D8×C3⋊D311yescharacteristic
447C2×C3#6412311yescharacteristic
448D3—68no—
449D3—128no—
450D3—728no—
451D3—724no—
452D3—721no—
453C22—22no—
454C22—41no—
455C22—41no—
456C22—92no—
457C22—181no—
458C22—181no—
459C22—182no—
460C22—368no—
461C22—361no—
462C22—361no—
463C22—1081no—
464C22—1081no—
465C22—1081no—
466C22C3⋊D3×#6711yescharacteristic
467C4—121no—
468C4—181no—
469C4C2×#629211yescharacteristic
470C4C3⋊D3×#6711yescharacteristic
471C3—24no—
472C3#1059144814yescomplemented
473C3#1059247211yescharacteristic, complemented
474C2—42no—
475C2—182no—
476C2—362no—
477C2—2161no—
478C2C3⋊D3×#21311yescharacteristic
479C2#1172356211yescharacteristic
480C2#1172363611yescharacteristic
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×#88860C2semidirect
3. #11723526C2semidirect
4. C2×#88791C2semidirect
5. C32×#832C2semidirect
6. #11722033C2semidirect
7. #11722023C2semidirect
12. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
20. C2×C3⋊(C3⋊(C3⋊C8))C22semidirect
21. C32×C4⋊Dic3C22semidirect
22. C2×C32×C3⋊C8C22semidirect
25. C2×D4×C3⋊D3D3semidirect
37. C3×#832D3semidirect
48. C33×D4D4semidirect
55. C3×C4⋊(C3⋊D3)D4semidirect
94. C3×C4⋊Dic3C2×D3semidirect
95. C2×C3×C3⋊C8C2×D3semidirect
111. C2×C3×C3⋊D3D8semidirect
127. #832C3⋊D3semidirect
151. C32×D4D12semidirect
152. C32×D4#67semidirect
167. C4⋊(C3⋊D3)#67semidirect
219. C4⋊Dic3C2×C3⋊D3semidirect
223. C2×C3⋊C8C2×C3⋊D3semidirect
252. C2×C3⋊D3#213semidirect
277. C33#447semidirect
301. C3×D4#626semidirect
302. C3×D4C4⋊(C3⋊D3)semidirect
410. C32#4773semidirect
411. C32#5217semidirect
426. D4#6292semidirect
472. C3#10591448semidirect
473. C3#10592472semidirect

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×#888602
#117235262
C2×#887912
C32×#8322
#117220332
#117220232
#105924723
#105914483