Finite Group Databasebeta

← #13161106

Subgroups — #13161106

Order 1728.

1492 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 1492 conjugacy classes of subgroups — too many to draw. Only its 162 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⋊D3×D12C211yesmaximal, characteristic, complemented
2C2×#88860C211yesmaximal, characteristic, complemented
3C3×#64147C211yesmaximal, characteristic, complemented
4C2×C32×#213C211yesmaximal, characteristic, complemented
5#11722036C211yesmaximal, characteristic, complemented
6#11722031C211yesmaximal, characteristic, complemented
7#11722023C211yesmaximal, characteristic, complemented
8#10592855—31nomaximal
9#10591515—34nomaximal
10C22×D3×C3⋊D3—21no—
11C2×C33×D4C2211yescharacteristic, complemented
12C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
13C2×C3×#638—21no—
14C2×C3×C4×C3⋊D3C2211yescharacteristic
15C2×C32×D12C2211yescharacteristic
16C2×#6291—21no—
17C3⋊D3×D12—22no—
18C3⋊D3×D12—22no—
19#88860—22no—
20C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
21C3×#3757—21no—
22C3×C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
23C32×#213—22no—
24C2×C32×C3⋊C8C2211yescharacteristic, complemented
25C2×D4×C3⋊D3—31no—
26C2×D3×D12—34no—
27C2×C32×D8—31no—
28#64147D311yescharacteristic, complemented
29C2×#3753—31no—
30C2×#3753—34no—
31C2×#3753—64no—
32C3×#938—34no—
33C2×C3×#213D314yescomplemented
34#63659—31no—
35#63648—31no—
36#63647—31no—
37#63564—34no—
38#63562—34no—
39(C2×D3)⋊(C3⋊C8)—34no—
40C23×C33—21no—
41C22×C3⋊(C3⋊D3)—21no—
42C22×C3×C3⋊D3D411yescharacteristic
43C22×C32×D3—21no—
44C2×D3×C3⋊D3—24no—
45C2×D3×C3⋊D3—42no—
46C2×D3×C3⋊D3—42no—
47C33×D4—21no—
48C33×D4—21no—
49C2×C33×C4C2311yescharacteristic
50C4⋊(C3⋊(C3⋊D3))—21no—
51C4⋊(C3⋊(C3⋊D3))D412yescomplemented
52C3×#638—42no—
53C2×C3×C3⋊Dic3—21no—
54C2×C3×C3⋊Dic3D411yescharacteristic
55C3×C4×C3⋊D3—21no—
56C3×C4×C3⋊D3—21no—
57C32×D12—21no—
58C32×D12D412yescomplemented
59#6291—24no—
60C3⋊(C3⋊(C3⋊C8))—21no—
61C32×C3⋊C8—21no—
62#5156—94no—
63#5036—91no—
64C23×C3⋊D3—61no—
65C22×D32—64no—
66C2×C32×D4—24no—
67C2×C32×D4—31no—
68C2×C32×D4C2×D314yescomplemented
69C2×C32×D4C2×D311yescharacteristic, complemented
70C2×#638—21no—
71C2×#638—61no—
72D4×C3⋊D3—62no—
73D4×C3⋊D3—62no—
74C2×C4⋊(C3⋊D3)—31no—
75C2×C4⋊(C3⋊D3)—34no—
76C2×C4⋊(C3⋊D3)—64no—
77C2×C4×C3⋊D3C2×D311yescharacteristic
78C2×C3×#67—64no—
79C2×C3×D12C2×D314yes—
80C2×C3×C4×D3—34no—
81C2×#626—64no—
82D3×D12—68no—
83D3×D12—68no—
84C32×D8—62no—
85C2×C32×C8—31no—
86#3757—21no—
87#3753—62no—
88#3753—68no—
89#3753—616no—
90C4⋊(C3⋊Dic3)C2×D311yescharacteristic, complemented
91C2×C3⋊(C3⋊C8)—31no—
92C2×C3⋊(C3⋊C8)—34no—
93C2×C3⋊(C3⋊C8)—64no—
94C3×#217—64no—
95C3×#213—28no—
96C3×C4⋊Dic3—34no—
97C2×C3×C3⋊C8C2×D314yescomplemented
98C22×C33—41no—
99C22×C33—41no—
100C22×C33—41no—
101C22×C33C2×D411yescharacteristic
102C2×C3⋊(C3⋊D3)—22no—
103C2×C3⋊(C3⋊D3)—41no—
104C2×C3⋊(C3⋊D3)—41no—
105C2×C3×C3⋊D3—21no—
106C2×C3×C3⋊D3—21no—
107C2×C3×C3⋊D3D812yescomplemented
108C2×C32×D3—22no—
109C2×C32×D3—41no—
110C2×C32×D3—41no—
111D3×C3⋊D3—48no—
112C33×C4C2×D411yescharacteristic
113C33×C4C2×D411yescharacteristic
114C3×C3⋊Dic3—21no—
115C3×C3⋊Dic3—41no—
116C2×D3×D4—94no—
117C22×D12—91no—
118C2×C3×D8—34no—
119C3×#121—91no—
120#938—34no—
121C2×#213—94no—
122C2×#213—91no—
123C2×#213—184no—
124C2×#213C3⋊D311yescharacteristic, complemented
125#832—94no—
126#830—91no—
127#821—94no—
128(C2×D3)⋊C8—94no—
129#809—91no—
130#809—91no—
131C23×C32—21no—
132C23×C32—24no—
133C23×C32—44no—
134C23×C32—61no—
135C22×C3⋊D3—64no—
136C22×C3⋊D3—61no—
137C22×C3⋊D3—64no—
138C22×C3⋊D3—124no—
139C22×C3⋊D3—122no—
140C22×C3⋊D3—122no—
141C22×C3⋊D3#6711yescharacteristic
142C22×C3×D3—24no—
143C22×C3×D3—34no—
144C2×D32—616no—
145C2×D32—128no—
146C2×D32—128no—
147C32×D4—28no—
148C32×D4—24no—
149C32×D4—21no—
150C32×D4—21no—
151C32×D4—24no—
152C32×D4—32no—
153C32×D4—44no—
154C32×D4—61no—
155C2×C32×C4—24no—
156C2×C32×C4C22×D314yes—
157C2×C32×C4C22×D311yescharacteristic
158#638—42no—
159#638—64no—
160C2×C3⋊Dic3—21no—
161C2×C3⋊Dic3#6711yescharacteristic
162C4⋊(C3⋊D3)—32no—
163C4⋊(C3⋊D3)—38no—
164C4⋊(C3⋊D3)—68no—
165C4⋊(C3⋊D3)—61no—
166C4⋊(C3⋊D3)—64no—
167C4⋊(C3⋊D3)—124no—
168C4×C3⋊D3—21no—
169C4×C3⋊D3—21no—
170C3×#67—128no—
171C2×C3×Dic3—34no—
172C2×C3×Dic3—64no—
173C3×D12—24no—
174C3×D12#6718yescomplemented
175C3×C4×D3—64no—
176C3×C4×D3—64no—
177#626—616no—
178C32×C8—61no—
179C3⋊(C3⋊C8)—68no—
180C3⋊(C3⋊C8)—64no—
181C3⋊(C3⋊C8)—61no—
182C3×C3⋊C8—24no—
183#447—271noSylow
184C2×C33—81no—
185C2×C33#12011yescharacteristic
186C2×C33C2×D811yescharacteristic
187C2×C33AGL(1,ℤ/8)11yescharacteristic
188C3⋊(C3⋊D3)—42no—
189C3×C3⋊D3—22no—
190C32×D3—42no—
191C23×D3—181no—
192C23×D3—184no—
193C2×C3×D4—24no—
194C2×C3×D4—34no—
195C2×C3×D4—181no—
196C2×C3×D4D3214yescomplemented
197C2×C3×D4C2×C3⋊D311yescharacteristic, complemented
198C22×C3×C4—91no—
199C2×#67—64no—
200C2×#67—184no—
201D3×D4—188no—
202D3×D4—188no—
203C2×D12—94no—
204C2×D12—91no—
205C2×D12—184no—
206C2×D12—182no—
207C2×D12—181no—
208C2×D12—182no—
209C2×D12C2×C3⋊D311yescharacteristic
210C2×C4×D3—34no—
211C3×D8—68no—
212C2×C3×C8—34no—
213C3×C4⋊C4—91no—
214C3×C22⋊C4—181no—
215#217—64no—
216#213—22no—
217#213—188no—
218#213—182no—
219#213—1816no—
220C4⋊Dic3—34no—
221C2×C3⋊C8—94no—
222C2×C3⋊C8—91no—
223C2×C3⋊C8—184no—
224C2×C3⋊C8C2×C3⋊D311yescharacteristic, complemented
225C22×C32—24no—
226C22×C32—48no—
227C22×C32—41no—
228C22×C32—44no—
229C22×C32—41no—
230C22×C32—41no—
231C22×C32—44no—
232C22×C32—44no—
233C22×C32—62no—
234C22×C32—84no—
235C22×C32—84no—
236C22×C32—121no—
237C22×C32—121no—
238C22×C32D3×D414yes—
239C22×C32C2×#6711yescharacteristic
240C2×C3⋊D3—21no—
241C2×C3⋊D3—21no—
242C2×C3⋊D3—62no—
243C2×C3⋊D3—68no—
244C2×C3⋊D3—128no—
245C2×C3⋊D3—128no—
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—28no—
254C2×C3×D3—38no—
255C2×C3×D3—44no—
256C2×C3×D3—44no—
257C2×C3×D3—64no—
258C2×C3×D3—64no—
259D32—1232no—
260C32×C4—24no—
261C32×C4—24no—
262C32×C4D3×D414yes—
263C32×C4D3×D411yescharacteristic
264C32×C4D3×D411yescharacteristic
265C32×C4C2×#6714yes—
266C3⋊Dic3—21no—
267C3⋊Dic3—41no—
268C3×Dic3—64no—
269C3×Dic3—124no—
270C22×D4—271no—
271C2×D8—31no—
272C2×D8—271no—
273#121—91no—
274#102—271no—
275#102—271no—
276C22⋊C8—271no—
277C33#44711yescharacteristic, complemented, Sylow
278C23×C3—24no—
279C23×C3—21no—
280C23×C3—44no—
281C23×C3—64no—
282C23×C3—91no—
283C22×D3—21no—
284C22×D3—34no—
285C22×D3—184no—
286C22×D3—1816no—
287C22×D3—184no—
288C22×D3—181no—
289C22×D3—364no—
290C22×D3—368no—
291C22×D3—362no—
292C22×D3—362no—
293C22×D3—368no—
294C3×D4—28no—
295C3×D4—24no—
296C3×D4—24no—
297C3×D4—21no—
298C3×D4—21no—
299C3×D4—38no—
300C3×D4—44no—
301C3×D4—64no—
302C3×D4—362no—
303C2×C3×C4—24no—
304C2×C3×C4—91no—
305C2×C3×C4—181no—
306C2×C3×C4—181no—
307C2×C3×C4—181no—
308C2×C3×C4C2×D3214yes—
309C2×C3×C4C22×C3⋊D311yescharacteristic
310#67—128no—
311#67—1816no—
312C2×Dic3—34no—
313C2×Dic3—64no—
314D12—21no—
315D12—98no—
316D12—92no—
317D12—188no—
318D12—184no—
319D12—184no—
320D12—181no—
321D12—364no—
322D12#63812yescomplemented
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—88no—
334C2×C32—84no—
335C2×C32—81no—
336C2×C32—122no—
337C2×C32D3×D814yes—
338C2×C32#91114yes—
339C2×C32C2×#21311yescharacteristic
340C2×C32#93714yes—
341C2×C32#93711yescharacteristic
342C2×C32#94911yescharacteristic
343C3⋊D3—22no—
344C3⋊D3—122no—
345C3⋊D3—128no—
346C3⋊D3—248no—
347C3×D3—48no—
348C3×D3—68no—
349C24—541no—
350C2×D4—31no—
351C2×D4—181no—
352C2×D4—271no—
353C2×D4—541no—
354C2×D4—542no—
355C2×D4—542no—
356C2×D4D3×C3⋊D311yescharacteristic, complemented
357C22×C4—91no—
358D8—62no—
359D8—542no—
360C2×C8—31no—
361C2×C8—271no—
362C4⋊C4—91no—
363C22⋊C4—181no—
364C22×C3—24no—
365C22×C3—48no—
366C22×C3—41no—
367C22×C3—41no—
368C22×C3—41no—
369C22×C3—44no—
370C22×C3—44no—
371C22×C3—44no—
372C22×C3—68no—
373C22×C3—84no—
374C22×C3—84no—
375C22×C3—92no—
376C22×C3—124no—
377C22×C3—124no—
378C22×C3—181no—
379C22×C3—181no—
380C22×C3D3×#6714yes—
381C22×C3D4×C3⋊D311yescharacteristic
382C2×D3—22no—
383C2×D3—38no—
384C2×D3—41no—
385C2×D3—41no—
386C2×D3—64no—
387C2×D3—64no—
388C2×D3—182no—
389C2×D3—188no—
390C2×D3—368no—
391C2×D3—368no—
392C2×D3—3632no—
393C2×D3—361no—
394C2×D3—361no—
395C2×D3—364no—
396C2×D3—364no—
397C2×D3—724no—
398C2×D3—724no—
399C3×C4—24no—
400C3×C4—24no—
401C3×C4—181no—
402C3×C4—361no—
403C3×C4D3×#6714yes—
404C3×C4#381114yes—
405C3×C4D4×C3⋊D311yescharacteristic
406C3×C4C2×#63811yescharacteristic
407Dic3—64no—
408Dic3—124no—
409C32—24no—
410C32#503611yescharacteristic, complemented
411C32#515614yescomplemented
412C23—21no—
413C23—61no—
414C23—91no—
415C23—544no—
416C23—541no—
417C23—1082no—
418C23—1082no—
419D4—21no—
420D4—21no—
421D4—32no—
422D4—61no—
423D4—272no—
424D4—362no—
425D4—544no—
426D4—541no—
427C2×C4—91no—
428C2×C4—181no—
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—88no—
438C2×C3—84no—
439C2×C3—81no—
440C2×C3—128no—
441C2×C3—182no—
442C2×C3D3×#21314yes—
443C2×C3#6392514yes—
444C2×C3#6398314yes—
445C2×C3D8×C3⋊D311yescharacteristic
446C2×C3#6412011yescharacteristic
447C2×C3#6414611yescharacteristic
448D3—42no—
449D3—68no—
450D3—368no—
451D3—362no—
452D3—728no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—62no—
457C22—92no—
458C22—121no—
459C22—121no—
460C22—181no—
461C22—181no—
462C22—542no—
463C22—1088no—
464C22—1081no—
465C22—1081no—
466C22C3⋊D3×#6711yescharacteristic
467C4—181no—
468C4—361no—
469C4D3×#63811yescharacteristic
470C4#8903911yescharacteristic
471C3—24no—
472C3#1059151514yescomplemented
473C3#1059285511yescharacteristic, complemented
474C2—81no—
475C2—122no—
476C2—182no—
477C2—1082no—
478C2C3⋊D3×#21311yescharacteristic
479C2#1172354611yescharacteristic
480C2#1172364311yescharacteristic
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⋊D3×D12C2semidirect
2. C2×#88860C2semidirect
3. C3×#64147C2semidirect
4. C2×C32×#213C2semidirect
5. #11722036C2semidirect
6. #11722031C2semidirect
7. #11722023C2semidirect
11. C2×C33×D4C22semidirect
20. C2×C3⋊(C3⋊(C3⋊C8))C22semidirect
22. C3×C4⋊(C3⋊Dic3)C22semidirect
24. C2×C32×C3⋊C8C22semidirect
28. #64147D3semidirect
33. C2×C3×#213D3semidirect
51. C4⋊(C3⋊(C3⋊D3))D4semidirect
58. C32×D12D4semidirect
68. C2×C32×D4C2×D3semidirect
69. C2×C32×D4C2×D3semidirect
90. C4⋊(C3⋊Dic3)C2×D3semidirect
97. C2×C3×C3⋊C8C2×D3semidirect
107. C2×C3×C3⋊D3D8semidirect
124. C2×#213C3⋊D3semidirect
174. C3×D12#67semidirect
196. C2×C3×D4D32semidirect
197. C2×C3×D4C2×C3⋊D3semidirect
224. C2×C3⋊C8C2×C3⋊D3semidirect
252. C2×C3⋊D3#213semidirect
277. C33#447semidirect
322. D12#638semidirect
356. C2×D4D3×C3⋊D3semidirect
410. C32#5036semidirect
411. C32#5156semidirect
472. C3#10591515semidirect
473. C3#10592855semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊D3×D122
C2×#888602
C3×#641472
C2×C32×#2132
#117220362
#117220312
#117220232
#105928553
#105915153