Finite Group Databasebeta

← #13145740

Subgroups — #13145740

Order 1728.

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

normal subgroups only

This group has 502 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×#64144C211yesmaximal, characteristic
2C3×#63661C211yesmaximal, characteristic
3C3×#63660C211yesmaximal, characteristic
4#10589177—31nomaximal
5#10589176—31nomaximal
6#10589175—31nomaximal
7C2×C33×D4C2211yescharacteristic
8C22×C3×C3⋊Dic3C411yescharacteristic, complemented
9C3×#3757—21no—
10C3×#3757—21no—
11C3×#3757C411yescharacteristic, complemented
12C3×#3750—21no—
13C3×#3748—21no—
14#64144D311yescharacteristic
15C3×#935—31no—
16C3×#935—31no—
17C3×#935—61no—
18#63661D311yescharacteristic
19#63660D311yescharacteristic
20C3×#834—31no—
21C3×#834—31no—
22C3×#834—61no—
23C3×#833—31no—
24C3×#833—31no—
25C3×#833—61no—
26C23×C33C2×C411yescharacteristic
27C23×C33D411yescharacteristic
28C33×D4—41no—
29C2×C33×C4D411yescharacteristic
30C2×C3×C3⋊Dic3—21no—
31C2×C3×C3⋊Dic3—21no—
32C2×C3×C3⋊Dic3—41no—
33C2×C3×C3⋊Dic3—41no—
34C3×C3⋊(C3⋊C8)—41no—
35#4539—91no—
36#4476—91no—
37#4472—91no—
38C2×C32×D4—21no—
39C2×C32×D4—21no—
40C2×C32×D4—21no—
41C2×C32×D4—41no—
42C2×C32×D4C2×D311yescharacteristic
43C2×C32×D4C2×D311yescharacteristic
44C2×C32×D4C2×D311yescharacteristic
45C22×C3⋊Dic3Dic311yescharacteristic, complemented
46C22×C3×Dic3—31no—
47C22×C3×Dic3—31no—
48C22×C3×Dic3—61no—
49#3757—21no—
50#3757—21no—
51#3757Dic311yescharacteristic, complemented
52#3750—21no—
53#3748—21no—
54C3×#217—31no—
55C3×#217—31no—
56C3×#217—61no—
57C3×#217—62no—
58C3×#217—61no—
59C3×#217—61no—
60C3×#217—62no—
61C3×#217—61no—
62C3×#217—61no—
63C3×Dic3⋊C4—62no—
64C3×Dic3⋊C4—61no—
65C3×Dic3⋊C4—61no—
66C3×#208—62no—
67C3×#208—61no—
68C3×#208—61no—
69C22×C33—21no—
70C22×C33—21no—
71C22×C33—41no—
72C22×C33—41no—
73C22×C33C22⋊C411yescharacteristic
74C33×C4—21no—
75C3×C3⋊Dic3—41no—
76C3×C3⋊Dic3—41no—
77C3×C3⋊Dic3—81no—
78C3×#123—91no—
79#935—31no—
80#935—31no—
81#935—61no—
82C3×#100—91no—
83C3×#99—91no—
84#834—31no—
85#834—31no—
86#834—61no—
87#833—31no—
88#833—31no—
89#833—61no—
90C23×C32—21no—
91C23×C32—21no—
92C23×C32—21no—
93C23×C32—21no—
94C23×C32—21no—
95C23×C32—21no—
96C23×C32—41no—
97C23×C32—41no—
98C23×C32C4×D311yescharacteristic
99C23×C32C4×D311yescharacteristic
100C23×C32D1211yescharacteristic
101C23×C32C2×Dic311yescharacteristic
102C23×C32#6711yescharacteristic
103C23×C32#6711yescharacteristic
104C32×D4—41no—
105C32×D4—41no—
106C32×D4—41no—
107C32×D4—42no—
108C32×D4—42no—
109C32×D4—42no—
110C32×D4—44no—
111C2×C32×C4—21no—
112C2×C32×C4—21no—
113C2×C32×C4—21no—
114C2×C32×C4—41no—
115C2×C32×C4D1211yescharacteristic
116C2×C32×C4#6711yescharacteristic
117C2×C32×C4#6711yescharacteristic
118C2×C3⋊Dic3—21no—
119C2×C3⋊Dic3—21no—
120C2×C3⋊Dic3—41no—
121C2×C3⋊Dic3—41no—
122C2×C3×Dic3—62no—
123C2×C3×Dic3—62no—
124C2×C3×Dic3—61no—
125C2×C3×Dic3—61no—
126C2×C3×Dic3—61no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3—122no—
129C2×C3×Dic3—121no—
130C2×C3×Dic3—121no—
131C2×C3×Dic3—122no—
132C2×C3×Dic3—121no—
133C2×C3×Dic3—121no—
134C3⋊(C3⋊C8)—41no—
135C3×C3⋊C8—122no—
136C3×C3⋊C8—121no—
137C3×C3⋊C8—121no—
138#352—271noSylow
139C2×C33—21no—
140C2×C33—41no—
141C2×C33—41no—
142C2×C33#9911yescharacteristic
143C2×C3×D4—21no—
144C2×C3×D4—21no—
145C2×C3×D4—21no—
146C2×C3×D4—41no—
147C2×C3×D4D3211yescharacteristic
148C2×C3×D4D3211yescharacteristic
149C2×C3×D4D3211yescharacteristic
150C22×C3×C4—91no—
151C22×Dic3—31no—
152C22×Dic3—31no—
153C22×Dic3—61no—
154C3×M16—181no—
155C3×C4⋊C4—181no—
156C3×C22⋊C4—91no—
157C3×C22⋊C4—181no—
158C3×C22⋊C4—181no—
159#217—31no—
160#217—31no—
161#217—61no—
162#217—62no—
163#217—61no—
164#217—61no—
165#217—62no—
166#217—61no—
167#217—61no—
168Dic3⋊C4—62no—
169Dic3⋊C4—61no—
170Dic3⋊C4—61no—
171#208—62no—
172#208—61no—
173#208—61no—
174C22×C32—21no—
175C22×C32—21no—
176C22×C32—22no—
177C22×C32—22no—
178C22×C32—21no—
179C22×C32—21no—
180C22×C32—21no—
181C22×C32—21no—
182C22×C32—21no—
183C22×C32—21no—
184C22×C32—21no—
185C22×C32—41no—
186C22×C32—41no—
187C22×C32—41no—
188C22×C32—41no—
189C22×C32—41no—
190C22×C32—42no—
191C22×C32—42no—
192C22×C32—41no—
193C22×C32—41no—
194C22×C32—41no—
195C22×C32—41no—
196C22×C32—41no—
197C22×C32—41no—
198C22×C32—42no—
199C22×C32—42no—
200C22×C32—81no—
201C22×C32—81no—
202C22×C32—81no—
203C22×C32—81no—
204C22×C32—82no—
205C22×C32—82no—
206C22×C32(C2×D3)⋊C411yescharacteristic
207C22×C32(C2×D3)⋊C411yescharacteristic
208C22×C32#21711yescharacteristic
209C32×C4—21no—
210C32×C4—22no—
211C32×C4—21no—
212C32×C4—21no—
213C32×C4—41no—
214C32×C4—41no—
215C32×C4—42no—
216C3⋊Dic3—41no—
217C3⋊Dic3—41no—
218C3⋊Dic3—81no—
219C3×Dic3—122no—
220C3×Dic3—121no—
221C3×Dic3—121no—
222C3×Dic3—122no—
223C3×Dic3—121no—
224C3×Dic3—121no—
225C3×Dic3—242no—
226C3×Dic3—241no—
227C3×Dic3—241no—
228#123—91no—
229#100—91no—
230#99—91no—
231C33#35211yescharacteristic, complemented, Sylow
232C23×C3—21no—
233C23×C3—21no—
234C23×C3—21no—
235C23×C3—21no—
236C23×C3—21no—
237C23×C3—21no—
238C23×C3—41no—
239C23×C3—41no—
240C23×C3D3×Dic311yescharacteristic
241C23×C3D3×Dic311yescharacteristic
242C23×C3#62411yescharacteristic
243C23×C3#62511yescharacteristic
244C23×C3#62611yescharacteristic
245C23×C3#62611yescharacteristic
246C3×D4—41no—
247C3×D4—41no—
248C3×D4—41no—
249C3×D4—42no—
250C3×D4—42no—
251C3×D4—42no—
252C3×D4—44no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—21no—
256C2×C3×C4—41no—
257C2×C3×C4—181no—
258C2×C3×C4—181no—
259C2×C3×C4—361no—
260C2×C3×C4—361no—
261C2×C3×C4#62511yescharacteristic
262C2×C3×C4#62611yescharacteristic
263C2×C3×C4#62611yescharacteristic
264C2×Dic3—62no—
265C2×Dic3—62no—
266C2×Dic3—61no—
267C2×Dic3—61no—
268C2×Dic3—61no—
269C2×Dic3—61no—
270C2×Dic3—122no—
271C2×Dic3—121no—
272C2×Dic3—121no—
273C2×Dic3—122no—
274C2×Dic3—121no—
275C2×Dic3—121no—
276C3×C8—361no—
277C3⋊C8—122no—
278C3⋊C8—121no—
279C3⋊C8—121no—
280C2×C32—21no—
281C2×C32—22no—
282C2×C32—21no—
283C2×C32—21no—
284C2×C32—21no—
285C2×C32—21no—
286C2×C32—21no—
287C2×C32—41no—
288C2×C32—41no—
289C2×C32—41no—
290C2×C32—42no—
291C2×C32—41no—
292C2×C32—41no—
293C2×C32—42no—
294C2×C32—42no—
295C2×C32—41no—
296C2×C32—41no—
297C2×C32—41no—
298C2×C32—41no—
299C2×C32—81no—
300C2×C32—81no—
301C2×C32—81no—
302C2×C32—81no—
303C2×C32—82no—
304C2×C32—82no—
305C2×C32#80611yescharacteristic
306C2×C32#80611yescharacteristic
307C2×C32#83411yescharacteristic
308C2×D4#111311yescharacteristic
309C22×C4—91no—
310M16—181no—
311C4⋊C4—181no—
312C22⋊C4—91no—
313C22⋊C4—181no—
314C22⋊C4—181no—
315C22×C3—22no—
316C22×C3—22no—
317C22×C3—21no—
318C22×C3—21no—
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—21no—
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—41no—
327C22×C3—41no—
328C22×C3—41no—
329C22×C3—41no—
330C22×C3—42no—
331C22×C3—42no—
332C22×C3—41no—
333C22×C3—41no—
334C22×C3—41no—
335C22×C3—41no—
336C22×C3—41no—
337C22×C3—41no—
338C22×C3—42no—
339C22×C3—42no—
340C22×C3—41no—
341C22×C3—81no—
342C22×C3—81no—
343C22×C3—81no—
344C22×C3—81no—
345C22×C3—82no—
346C22×C3—82no—
347C22×C3(C2×D3)⋊Dic311yescharacteristic
348C22×C3(C2×D3)⋊Dic311yescharacteristic
349C22×C3#372211yescharacteristic
350C3×C4—22no—
351C3×C4—21no—
352C3×C4—21no—
353C3×C4—21no—
354C3×C4—41no—
355C3×C4—41no—
356C3×C4—42no—
357C3×C4—361no—
358C3×C4—361no—
359C3×C4—721no—
360Dic3—122no—
361Dic3—122no—
362Dic3—121no—
363Dic3—121no—
364Dic3—121no—
365Dic3—121no—
366Dic3—242no—
367Dic3—241no—
368Dic3—241no—
369C32—21no—
370C32—21no—
371C32—21no—
372C32—41no—
373C32#447211yescharacteristic, complemented
374C32#447611yescharacteristic, complemented
375C32#453911yescharacteristic, complemented
376C23#629411yescharacteristic
377C23#629511yescharacteristic
378D4—41no—
379C2×C4—181no—
380C2×C4—181no—
381C2×C4—361no—
382C2×C4—361no—
383C2×C4#629511yescharacteristic
384C8—361no—
385C2×C3—22no—
386C2×C3—21no—
387C2×C3—21no—
388C2×C3—21no—
389C2×C3—21no—
390C2×C3—21no—
391C2×C3—21no—
392C2×C3—41no—
393C2×C3—41no—
394C2×C3—42no—
395C2×C3—42no—
396C2×C3—42no—
397C2×C3—41no—
398C2×C3—41no—
399C2×C3—41no—
400C2×C3—41no—
401C2×C3—41no—
402C2×C3—41no—
403C2×C3—41no—
404C2×C3—81no—
405C2×C3—81no—
406C2×C3—81no—
407C2×C3—81no—
408C2×C3—82no—
409C2×C3—82no—
410C2×C3#6358011yescharacteristic
411C2×C3#6358011yescharacteristic
412C2×C3#6358111yescharacteristic
413C22—21no—
414C22—21no—
415C22—41no—
416C22—41no—
417C22#8881411yescharacteristic
418C4—21no—
419C4—361no—
420C4—361no—
421C4—721no—
422C3—21no—
423C3—21no—
424C3—21no—
425C3—41no—
426C3#1058917511yescharacteristic, complemented
427C3#1058917611yescharacteristic, complemented
428C3#1058917711yescharacteristic, complemented
429C2—21no—
430C2—41no—
431C2—41no—
432C2#1172208311yescharacteristic
subgroupG/H
centerC2#11722083
commutatorC23×C33C2×C4
Frattini, hypercenterC2×D4#1113
Fitting, prime radical, cosocleC2×C33×D4C22
socleC2×C33#99
hypocenterC33#352
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
8. C22×C3×C3⋊Dic3C4semidirect
11. C3×#3757C4semidirect
45. C22×C3⋊Dic3Dic3semidirect
51. #3757Dic3semidirect
231. C33#352semidirect
373. C32#4472semidirect
374. C32#4476semidirect
375. C32#4539semidirect
426. C3#10589175semidirect
427. C3#10589176semidirect
428. C3#10589177semidirect

Maximal subgroups

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

subgroupindex
C3×#641442
C3×#636612
C3×#636602
#105891773
#105891763
#105891753