Finite Group Databasebeta

← #13145739

Subgroups — #13145739

Order 1728.

658 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 67 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
1C3×#64146C211yesmaximal, characteristic
2C3×#63661C211yesmaximal, characteristic, complemented
3C3×#63660C211yesmaximal, characteristic, complemented
4#10589174—31nomaximal
5#10589173—31nomaximal
6#10589172—31nomaximal
7C23×C3×C3⋊D3C411yescharacteristic, complemented
8C2×C33×D4C2211yescharacteristic
9C2×C3×#638—21no—
10C3×#3757—21no—
11C3×#3757C411yescharacteristic, complemented
12C3×#3752—21no—
13C3×#3748—21no—
14#64146D311yescharacteristic
15C3×#937—31no—
16C3×#937—31no—
17C3×#937—61no—
18#63661D311yescharacteristic, complemented
19#63660D311yescharacteristic, complemented
20C3×#834—31no—
21C3×#834—31no—
22C3×#834—61no—
23C3×#833—31no—
24C3×#833—31no—
25C3×#833—61no—
26C23×C33C2×C411yescharacteristic
27C23×C33D411yescharacteristic
28C22×C3×C3⋊D3—21no—
29C22×C3×C3⋊D3—42no—
30C22×C3×C3⋊D3—41no—
31C33×D4—41no—
32C2×C33×C4D411yescharacteristic
33C3×#638—42no—
34C2×C3×C3⋊Dic3—21no—
35C2×C3×C3⋊Dic3—41no—
36C3×C3⋊(C3⋊C8)—41no—
37#4536—91no—
38#4475—91no—
39#4471—91no—
40C23×C3⋊D3Dic311yescharacteristic, complemented
41C23×C3×D3—31no—
42C23×C3×D3—31no—
43C23×C3×D3—61no—
44C2×C32×D4—21no—
45C2×C32×D4—21no—
46C2×C32×D4—21no—
47C2×C32×D4—41no—
48C2×C32×D4C2×D311yescharacteristic
49C2×C32×D4C2×D311yescharacteristic
50C2×C32×D4C2×D311yescharacteristic
51C2×#638—21no—
52C2×C3×#67—62no—
53C2×C3×#67—61no—
54C2×C3×#67—61no—
55#3757—21no—
56#3757Dic311yescharacteristic, complemented
57#3752—21no—
58#3748—21no—
59C3×#217—31no—
60C3×#217—31no—
61C3×#217—61no—
62C3×#217—62no—
63C3×#217—61no—
64C3×#217—61no—
65C3×(C2×D3)⋊C4—62no—
66C3×(C2×D3)⋊C4—61no—
67C3×(C2×D3)⋊C4—61no—
68C3×#208—62no—
69C3×#208—61no—
70C3×#208—61no—
71C22×C33—21no—
72C22×C33—21no—
73C22×C33—41no—
74C22×C33—41no—
75C22×C33C22⋊C411yescharacteristic
76C2×C3×C3⋊D3—22no—
77C2×C3×C3⋊D3—41no—
78C2×C3×C3⋊D3—42no—
79C2×C3×C3⋊D3—41no—
80C2×C3×C3⋊D3—42no—
81C33×C4—21no—
82C3×C3⋊Dic3—41no—
83C3×C3⋊Dic3—81no—
84C3×#120—91no—
85#937—31no—
86#937—31no—
87#937—61no—
88C3×#100—91no—
89C3×#99—91no—
90#834—31no—
91#834—31no—
92#834—61no—
93#833—31no—
94#833—31no—
95#833—61no—
96C23×C32—21no—
97C23×C32—21no—
98C23×C32—21no—
99C23×C32—21no—
100C23×C32—21no—
101C23×C32—21no—
102C23×C32—41no—
103C23×C32—41no—
104C23×C32C4×D311yescharacteristic
105C23×C32C4×D311yescharacteristic
106C23×C32D1211yescharacteristic
107C23×C32C2×Dic311yescharacteristic
108C23×C32#6711yescharacteristic
109C23×C32#6711yescharacteristic
110C22×C3⋊D3—21no—
111C22×C3⋊D3—42no—
112C22×C3⋊D3—41no—
113C22×C3×D3—62no—
114C22×C3×D3—61no—
115C22×C3×D3—61no—
116C22×C3×D3—124no—
117C22×C3×D3—122no—
118C22×C3×D3—122no—
119C22×C3×D3—122no—
120C22×C3×D3—121no—
121C22×C3×D3—121no—
122C32×D4—41no—
123C32×D4—41no—
124C32×D4—41no—
125C32×D4—42no—
126C32×D4—42no—
127C32×D4—42no—
128C32×D4—44no—
129C2×C32×C4—21no—
130C2×C32×C4—21no—
131C2×C32×C4—21no—
132C2×C32×C4—41no—
133C2×C32×C4D1211yescharacteristic
134C2×C32×C4#6711yescharacteristic
135C2×C32×C4#6711yescharacteristic
136#638—42no—
137C2×C3⋊Dic3—21no—
138C2×C3⋊Dic3—41no—
139C3×#67—122no—
140C3×#67—122no—
141C3×#67—124no—
142C2×C3×Dic3—62no—
143C2×C3×Dic3—61no—
144C2×C3×Dic3—61no—
145C2×C3×Dic3—122no—
146C2×C3×Dic3—121no—
147C2×C3×Dic3—121no—
148C3⋊(C3⋊C8)—41no—
149C3×C3⋊C8—122no—
150C3×C3⋊C8—121no—
151C3×C3⋊C8—121no—
152AΣL(1,𝔽16)—271noSylow
153C2×C33—21no—
154C2×C33—41no—
155C2×C33—41no—
156C2×C33#9911yescharacteristic
157C3×C3⋊D3—42no—
158C24×C3—91no—
159C23×D3—31no—
160C23×D3—31no—
161C23×D3—61no—
162C2×C3×D4—21no—
163C2×C3×D4—21no—
164C2×C3×D4—21no—
165C2×C3×D4—41no—
166C2×C3×D4—181no—
167C2×C3×D4D3211yescharacteristic
168C2×C3×D4D3211yescharacteristic
169C2×C3×D4D3211yescharacteristic
170C2×#67—62no—
171C2×#67—61no—
172C2×#67—61no—
173C3×M16—181no—
174C3×C22⋊C4—91no—
175C3×C22⋊C4—181no—
176C3×C22⋊C4—181no—
177#217—31no—
178#217—31no—
179#217—61no—
180#217—62no—
181#217—61no—
182#217—61no—
183(C2×D3)⋊C4—62no—
184(C2×D3)⋊C4—61no—
185(C2×D3)⋊C4—61no—
186#208—62no—
187#208—61no—
188#208—61no—
189C22×C32—21no—
190C22×C32—21no—
191C22×C32—22no—
192C22×C32—22no—
193C22×C32—21no—
194C22×C32—21no—
195C22×C32—21no—
196C22×C32—21no—
197C22×C32—21no—
198C22×C32—21no—
199C22×C32—21no—
200C22×C32—41no—
201C22×C32—41no—
202C22×C32—41no—
203C22×C32—41no—
204C22×C32—41no—
205C22×C32—42no—
206C22×C32—42no—
207C22×C32—41no—
208C22×C32—41no—
209C22×C32—41no—
210C22×C32—41no—
211C22×C32—41no—
212C22×C32—41no—
213C22×C32—42no—
214C22×C32—42no—
215C22×C32—81no—
216C22×C32—81no—
217C22×C32—81no—
218C22×C32—81no—
219C22×C32—82no—
220C22×C32—82no—
221C22×C32(C2×D3)⋊C411yescharacteristic
222C22×C32(C2×D3)⋊C411yescharacteristic
223C22×C32#21711yescharacteristic
224C2×C3⋊D3—22no—
225C2×C3⋊D3—41no—
226C2×C3⋊D3—42no—
227C2×C3⋊D3—41no—
228C2×C3⋊D3—42no—
229C2×C3×D3—64no—
230C2×C3×D3—62no—
231C2×C3×D3—62no—
232C2×C3×D3—122no—
233C2×C3×D3—121no—
234C2×C3×D3—121no—
235C2×C3×D3—122no—
236C2×C3×D3—124no—
237C2×C3×D3—124no—
238C2×C3×D3—122no—
239C2×C3×D3—122no—
240C2×C3×D3—121no—
241C2×C3×D3—121no—
242C2×C3×D3—122no—
243C2×C3×D3—122no—
244C32×C4—21no—
245C32×C4—22no—
246C32×C4—21no—
247C32×C4—21no—
248C32×C4—41no—
249C32×C4—41no—
250C32×C4—42no—
251C3⋊Dic3—41no—
252C3⋊Dic3—81no—
253C3×Dic3—122no—
254C3×Dic3—121no—
255C3×Dic3—121no—
256C3×Dic3—242no—
257C3×Dic3—241no—
258C3×Dic3—241no—
259#120—91no—
260#100—91no—
261#99—91no—
262C33AΣL(1,𝔽16)11yescharacteristic, complemented, Sylow
263C23×C3—21no—
264C23×C3—21no—
265C23×C3—21no—
266C23×C3—21no—
267C23×C3—21no—
268C23×C3—21no—
269C23×C3—41no—
270C23×C3—41no—
271C23×C3—181no—
272C23×C3—361no—
273C23×C3—362no—
274C23×C3D3×Dic311yescharacteristic
275C23×C3D3×Dic311yescharacteristic
276C23×C3#62411yescharacteristic
277C23×C3#62511yescharacteristic
278C23×C3#62611yescharacteristic
279C23×C3#62611yescharacteristic
280C22×D3—62no—
281C22×D3—61no—
282C22×D3—61no—
283C22×D3—124no—
284C22×D3—122no—
285C22×D3—122no—
286C22×D3—122no—
287C22×D3—121no—
288C22×D3—121no—
289C3×D4—41no—
290C3×D4—41no—
291C3×D4—41no—
292C3×D4—42no—
293C3×D4—42no—
294C3×D4—42no—
295C3×D4—44no—
296C3×D4—362no—
297C2×C3×C4—21no—
298C2×C3×C4—21no—
299C2×C3×C4—21no—
300C2×C3×C4—41no—
301C2×C3×C4—181no—
302C2×C3×C4—361no—
303C2×C3×C4#62511yescharacteristic
304C2×C3×C4#62611yescharacteristic
305C2×C3×C4#62611yescharacteristic
306#67—122no—
307#67—122no—
308#67—124no—
309C2×Dic3—62no—
310C2×Dic3—61no—
311C2×Dic3—61no—
312C2×Dic3—122no—
313C2×Dic3—121no—
314C2×Dic3—121no—
315C3×C8—361no—
316C3⋊C8—122no—
317C3⋊C8—121no—
318C3⋊C8—121no—
319C2×C32—21no—
320C2×C32—22no—
321C2×C32—21no—
322C2×C32—21no—
323C2×C32—21no—
324C2×C32—21no—
325C2×C32—21no—
326C2×C32—41no—
327C2×C32—41no—
328C2×C32—41no—
329C2×C32—42no—
330C2×C32—41no—
331C2×C32—41no—
332C2×C32—42no—
333C2×C32—42no—
334C2×C32—41no—
335C2×C32—41no—
336C2×C32—41no—
337C2×C32—41no—
338C2×C32—81no—
339C2×C32—81no—
340C2×C32—81no—
341C2×C32—81no—
342C2×C32—82no—
343C2×C32—82no—
344C2×C32#80611yescharacteristic
345C2×C32#80611yescharacteristic
346C2×C32#83411yescharacteristic
347C3⋊D3—42no—
348C3×D3—124no—
349C3×D3—122no—
350C3×D3—122no—
351C24—91no—
352C2×D4—181no—
353C2×D4#111311yescharacteristic
354M16—181no—
355C22⋊C4—91no—
356C22⋊C4—181no—
357C22⋊C4—181no—
358C22×C3—22no—
359C22×C3—22no—
360C22×C3—21no—
361C22×C3—21no—
362C22×C3—21no—
363C22×C3—21no—
364C22×C3—21no—
365C22×C3—21no—
366C22×C3—21no—
367C22×C3—21no—
368C22×C3—21no—
369C22×C3—41no—
370C22×C3—41no—
371C22×C3—41no—
372C22×C3—41no—
373C22×C3—42no—
374C22×C3—42no—
375C22×C3—41no—
376C22×C3—41no—
377C22×C3—41no—
378C22×C3—41no—
379C22×C3—41no—
380C22×C3—41no—
381C22×C3—42no—
382C22×C3—42no—
383C22×C3—41no—
384C22×C3—81no—
385C22×C3—81no—
386C22×C3—81no—
387C22×C3—81no—
388C22×C3—82no—
389C22×C3—82no—
390C22×C3—182no—
391C22×C3—361no—
392C22×C3—362no—
393C22×C3—362no—
394C22×C3—361no—
395C22×C3(C2×D3)⋊Dic311yescharacteristic
396C22×C3(C2×D3)⋊Dic311yescharacteristic
397C22×C3#372211yescharacteristic
398C2×D3—64no—
399C2×D3—62no—
400C2×D3—62no—
401C2×D3—122no—
402C2×D3—121no—
403C2×D3—121no—
404C2×D3—122no—
405C2×D3—124no—
406C2×D3—124no—
407C2×D3—122no—
408C2×D3—122no—
409C2×D3—121no—
410C2×D3—121no—
411C2×D3—122no—
412C2×D3—122no—
413C3×C4—22no—
414C3×C4—21no—
415C3×C4—21no—
416C3×C4—21no—
417C3×C4—41no—
418C3×C4—41no—
419C3×C4—42no—
420C3×C4—361no—
421C3×C4—721no—
422Dic3—122no—
423Dic3—121no—
424Dic3—121no—
425Dic3—242no—
426Dic3—241no—
427Dic3—241no—
428C32—21no—
429C32—21no—
430C32—21no—
431C32—41no—
432C32#447111yescharacteristic, complemented
433C32#447511yescharacteristic, complemented
434C32#453611yescharacteristic, complemented
435C23—181no—
436C23—361no—
437C23—362no—
438C23#629411yescharacteristic
439C23#629511yescharacteristic
440D4—41no—
441D4—362no—
442C2×C4—181no—
443C2×C4—361no—
444C2×C4#629511yescharacteristic
445C8—361no—
446C2×C3—22no—
447C2×C3—21no—
448C2×C3—21no—
449C2×C3—21no—
450C2×C3—21no—
451C2×C3—21no—
452C2×C3—21no—
453C2×C3—41no—
454C2×C3—41no—
455C2×C3—42no—
456C2×C3—42no—
457C2×C3—42no—
458C2×C3—41no—
459C2×C3—41no—
460C2×C3—41no—
461C2×C3—41no—
462C2×C3—41no—
463C2×C3—41no—
464C2×C3—41no—
465C2×C3—81no—
466C2×C3—81no—
467C2×C3—81no—
468C2×C3—81no—
469C2×C3—82no—
470C2×C3—82no—
471C2×C3—362no—
472C2×C3#6358011yescharacteristic
473C2×C3#6358011yescharacteristic
474C2×C3#6358111yescharacteristic
475D3—124no—
476D3—122no—
477D3—122no—
478C22—21no—
479C22—21no—
480C22—41no—
481C22—41no—
482C22—182no—
483C22—361no—
484C22—362no—
485C22—362no—
486C22—361no—
487C22#8881411yescharacteristic
488C4—21no—
489C4—361no—
490C4—721no—
491C3—21no—
492C3—21no—
493C3—21no—
494C3—41no—
495C3#1058917211yescharacteristic, complemented
496C3#1058917311yescharacteristic, complemented
497C3#1058917411yescharacteristic, complemented
498C2—21no—
499C2—41no—
500C2—41no—
501C2—362no—
502C2#1172208311yescharacteristic
subgroupG/H
centerC2#11722083
commutatorC23×C33C2×C4
Frattini, hypercenterC2×D4#1113
Fitting, cosocleC2×C33×D4C22
socleC2×C33#99
prime radicalC3×#64146C2
hypocenterC33AΣL(1,𝔽16)
LCSG ▷ C23×C33 ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C3×#63661C2semidirect
3. C3×#63660C2semidirect
7. C23×C3×C3⋊D3C4semidirect
11. C3×#3757C4semidirect
18. #63661D3semidirect
19. #63660D3semidirect
40. C23×C3⋊D3Dic3semidirect
56. #3757Dic3semidirect
262. C33AΣL(1,𝔽16)semidirect
432. C32#4471semidirect
433. C32#4475semidirect
434. C32#4536semidirect
495. C3#10589172semidirect
496. C3#10589173semidirect
497. C3#10589174semidirect

Maximal subgroups

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

subgroupindex
C3×#641462
C3×#636612
C3×#636602
#105891743
#105891733
#105891723