Finite Group Databasebeta

← #13145730

Subgroups — #13145730

Order 1728.

574 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 99 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×#64133C211yesmaximal, characteristic
2C3×#63658C211yesmaximal, characteristic
3C3×#63657C211yesmaximal, characteristic
4#10589154—31nomaximal
5#10589153—31nomaximal
6#10589152—31nomaximal
7C22×C33×C4C2211yescharacteristic
8C22×C3×C3⋊Dic3—21no—
9C2×C3×#634C411yescharacteristic, complemented
10C3×#3757—21no—
11C3×C4⋊(C3⋊Dic3)C411yescharacteristic, complemented
12C3×#3750—21no—
13C2×C3×C3⋊(C3⋊C8)—21no—
14#64133D311yescharacteristic
15C3×#924—31no—
16C3×#924—31no—
17C3×#924—61no—
18#63658D311yescharacteristic
19#63657D311yescharacteristic
20C3×C3⋊#95—31no—
21C3×C3⋊#95—31no—
22C3×C3⋊#95—61no—
23C3×#830—31no—
24C3×#830—31no—
25C3×#830—61no—
26C23×C33D411yescharacteristic
27C2×C33×C4—41no—
28C2×C33×C4C2×C411yescharacteristic
29C2×C33×C4D411yescharacteristic
30C2×C3×C3⋊Dic3—22no—
31C2×C3×C3⋊Dic3—21no—
32C2×C3×C3⋊Dic3—21no—
33C2×C3×C3⋊Dic3—41no—
34C2×C3×C3⋊Dic3—41no—
35C3×#634—41no—
36C3×C3⋊(C3⋊C8)—41no—
37#4538—91no—
38#4467—91no—
39#4452—91no—
40C22×C32×C4—21no—
41C22×C32×C4—21no—
42C22×C32×C4—21no—
43C22×C32×C4—41no—
44C22×C32×C4C2×D311yescharacteristic
45C22×C32×C4C2×D311yescharacteristic
46C22×C32×C4C2×D311yescharacteristic
47C22×C3⋊Dic3—21no—
48C2×#634Dic311yescharacteristic, complemented
49C22×C3×Dic3—62no—
50C22×C3×Dic3—61no—
51C22×C3×Dic3—61no—
52C2×C3×Dic6—31no—
53C2×C3×Dic6—31no—
54C2×C3×Dic6—61no—
55#3757—21no—
56C4⋊(C3⋊Dic3)Dic311yescharacteristic, complemented
57#3750—21no—
58C2×C3⋊(C3⋊C8)—21no—
59C3×#217—62no—
60C3×#217—61no—
61C3×#217—61no—
62C3×C4⋊Dic3—31no—
63C3×C4⋊Dic3—31no—
64C3×C4⋊Dic3—61no—
65C3×Dic3⋊C4—62no—
66C3×Dic3⋊C4—61no—
67C3×Dic3⋊C4—61no—
68C2×C3×C3⋊C8—62no—
69C2×C3×C3⋊C8—61no—
70C2×C3×C3⋊C8—61no—
71C22×C33—21no—
72C22×C33—21no—
73C22×C33C22⋊C411yescharacteristic
74C22×C33SD1611yescharacteristic
75C22×C33Q1611yescharacteristic
76C33×C4—21no—
77C33×C4—21no—
78C3×C3⋊Dic3—42no—
79C3×C3⋊Dic3—41no—
80C3×C3⋊Dic3—41no—
81C3×C22⋊Q8—91no—
82#924—31no—
83#924—31no—
84#924—61no—
85C3×C22⋊C8—91no—
86C3×#95—91no—
87C3⋊#95—31no—
88C3⋊#95—31no—
89C3⋊#95—61no—
90#830—31no—
91#830—31no—
92#830—61no—
93C23×C32—21no—
94C23×C32—21no—
95C23×C32—21no—
96C23×C32—41no—
97C23×C32D1211yescharacteristic
98C23×C32#6711yescharacteristic
99C23×C32#6711yescharacteristic
100C2×C32×C4—21no—
101C2×C32×C4—21no—
102C2×C32×C4—21no—
103C2×C32×C4—21no—
104C2×C32×C4—21no—
105C2×C32×C4—21no—
106C2×C32×C4—41no—
107C2×C32×C4—41no—
108C2×C32×C4—41no—
109C2×C32×C4—41no—
110C2×C32×C4—41no—
111C2×C32×C4—42no—
112C2×C32×C4—42no—
113C2×C32×C4—42no—
114C2×C32×C4—44no—
115C2×C32×C4C4×D311yescharacteristic
116C2×C32×C4C4×D311yescharacteristic
117C2×C32×C4D1211yescharacteristic
118C2×C32×C4C2×Dic311yescharacteristic
119C2×C32×C4#6711yescharacteristic
120C2×C32×C4#6711yescharacteristic
121C2×C3⋊Dic3—22no—
122C2×C3⋊Dic3—21no—
123C2×C3⋊Dic3—21no—
124C2×C3⋊Dic3—41no—
125C2×C3⋊Dic3—41no—
126#634—41no—
127C2×C3×Dic3—62no—
128C2×C3×Dic3—62no—
129C2×C3×Dic3—64no—
130C2×C3×Dic3—62no—
131C2×C3×Dic3—62no—
132C2×C3×Dic3—61no—
133C2×C3×Dic3—61no—
134C2×C3×Dic3—61no—
135C2×C3×Dic3—61no—
136C2×C3×Dic3—122no—
137C2×C3×Dic3—121no—
138C2×C3×Dic3—121no—
139C2×C3×Dic3—122no—
140C2×C3×Dic3—121no—
141C2×C3×Dic3—121no—
142C3×Dic6—122no—
143C3×Dic6—121no—
144C3×Dic6—121no—
145C3⋊(C3⋊C8)—41no—
146C3×C3⋊C8—122no—
147C3×C3⋊C8—121no—
148C3×C3⋊C8—121no—
149#328—271noSylow
150C2×C33—21no—
151C2×C33—21no—
152C2×C33#9911yescharacteristic
153C2×C33#10311yescharacteristic
154C2×C33#10411yescharacteristic
155C2×C3×Q8—91no—
156C22×C3×C4—21no—
157C22×C3×C4—21no—
158C22×C3×C4—21no—
159C22×C3×C4—41no—
160C22×C3×C4—181no—
161C22×C3×C4D3211yescharacteristic
162C22×C3×C4D3211yescharacteristic
163C22×C3×C4D3211yescharacteristic
164C22×Dic3—62no—
165C22×Dic3—61no—
166C22×Dic3—61no—
167C2×Dic6—31no—
168C2×Dic6—31no—
169C2×Dic6—61no—
170C2×C3×C8—181no—
171C3×C4⋊C4—91no—
172C3×C4⋊C4—181no—
173C3×C22⋊C4—181no—
174#217—62no—
175#217—61no—
176#217—61no—
177C4⋊Dic3—31no—
178C4⋊Dic3—31no—
179C4⋊Dic3—61no—
180Dic3⋊C4—62no—
181Dic3⋊C4—61no—
182Dic3⋊C4—61no—
183C2×C3⋊C8—62no—
184C2×C3⋊C8—61no—
185C2×C3⋊C8—61no—
186C22×C32—21no—
187C22×C32—21no—
188C22×C32—22no—
189C22×C32—22no—
190C22×C32—21no—
191C22×C32—21no—
192C22×C32—21no—
193C22×C32—21no—
194C22×C32—21no—
195C22×C32—21no—
196C22×C32—21no—
197C22×C32—21no—
198C22×C32—21no—
199C22×C32—21no—
200C22×C32—21no—
201C22×C32—21no—
202C22×C32—21no—
203C22×C32—41no—
204C22×C32—41no—
205C22×C32—41no—
206C22×C32—41no—
207C22×C32—41no—
208C22×C32—41no—
209C22×C32—41no—
210C22×C32—42no—
211C22×C32—42no—
212C22×C32#20411yescharacteristic
213C22×C32Dic1211yescharacteristic
214C22×C32(C2×D3)⋊C411yescharacteristic
215C22×C32(C2×D3)⋊C411yescharacteristic
216C22×C32#21411yescharacteristic
217C22×C32#21511yescharacteristic
218C22×C32#21611yescharacteristic
219C22×C32#21611yescharacteristic
220C22×C32#21711yescharacteristic
221C32×C4—21no—
222C32×C4—21no—
223C32×C4—22no—
224C32×C4—22no—
225C32×C4—21no—
226C32×C4—21no—
227C32×C4—21no—
228C32×C4—21no—
229C32×C4—41no—
230C32×C4—41no—
231C32×C4—41no—
232C32×C4—41no—
233C32×C4—42no—
234C32×C4—42no—
235C3⋊Dic3—42no—
236C3⋊Dic3—41no—
237C3⋊Dic3—41no—
238C3×Dic3—122no—
239C3×Dic3—122no—
240C3×Dic3—124no—
241C3×Dic3—122no—
242C3×Dic3—122no—
243C3×Dic3—121no—
244C3×Dic3—121no—
245C3×Dic3—121no—
246C3×Dic3—121no—
247C22⋊Q8—91no—
248C22⋊C8—91no—
249#95—91no—
250C33#32811yescharacteristic, complemented, Sylow
251C23×C3—21no—
252C23×C3—21no—
253C23×C3—21no—
254C23×C3—41no—
255C23×C3#62511yescharacteristic
256C23×C3#62611yescharacteristic
257C23×C3#62611yescharacteristic
258C3×Q8—361no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—42no—
269C2×C3×C4—42no—
270C2×C3×C4—42no—
271C2×C3×C4—44no—
272C2×C3×C4—41no—
273C2×C3×C4—41no—
274C2×C3×C4—182no—
275C2×C3×C4—181no—
276C2×C3×C4—181no—
277C2×C3×C4—361no—
278C2×C3×C4—361no—
279C2×C3×C4D3×Dic311yescharacteristic
280C2×C3×C4D3×Dic311yescharacteristic
281C2×C3×C4#62411yescharacteristic
282C2×C3×C4#62511yescharacteristic
283C2×C3×C4#62611yescharacteristic
284C2×C3×C4#62611yescharacteristic
285C2×Dic3—62no—
286C2×Dic3—62no—
287C2×Dic3—64no—
288C2×Dic3—62no—
289C2×Dic3—62no—
290C2×Dic3—61no—
291C2×Dic3—61no—
292C2×Dic3—61no—
293C2×Dic3—61no—
294C2×Dic3—122no—
295C2×Dic3—121no—
296C2×Dic3—121no—
297C2×Dic3—122no—
298C2×Dic3—121no—
299C2×Dic3—121no—
300Dic6—122no—
301Dic6—121no—
302Dic6—121no—
303C3×C8—361no—
304C3⋊C8—122no—
305C3⋊C8—121no—
306C3⋊C8—121no—
307C2×C32—21no—
308C2×C32—21no—
309C2×C32—22no—
310C2×C32—22no—
311C2×C32—21no—
312C2×C32—21no—
313C2×C32—21no—
314C2×C32—21no—
315C2×C32—21no—
316C2×C32—21no—
317C2×C32—21no—
318C2×C32—21no—
319C2×C32—21no—
320C2×C32—21no—
321C2×C32—21no—
322C2×C32—21no—
323C2×C32—21no—
324C2×C32—41no—
325C2×C32—41no—
326C2×C32—41no—
327C2×C32—41no—
328C2×C32—41no—
329C2×C32—41no—
330C2×C32—41no—
331C2×C32—42no—
332C2×C32—42no—
333C2×C32#80511yescharacteristic
334C2×C32#80611yescharacteristic
335C2×C32#80611yescharacteristic
336C2×C32#81011yescharacteristic
337C2×C32#81611yescharacteristic
338C2×C32#82511yescharacteristic
339C2×C32#83411yescharacteristic
340C2×C32#83511yescharacteristic
341C2×C32#83711yescharacteristic
342C2×Q8—91no—
343C22×C4—181no—
344C22×C4#111311yescharacteristic
345C2×C8—181no—
346C4⋊C4—91no—
347C4⋊C4—181no—
348C22⋊C4—181no—
349C22×C3—22no—
350C22×C3—22no—
351C22×C3—21no—
352C22×C3—21no—
353C22×C3—21no—
354C22×C3—21no—
355C22×C3—21no—
356C22×C3—21no—
357C22×C3—21no—
358C22×C3—21no—
359C22×C3—21no—
360C22×C3—21no—
361C22×C3—21no—
362C22×C3—21no—
363C22×C3—21no—
364C22×C3—21no—
365C22×C3—21no—
366C22×C3—41no—
367C22×C3—41no—
368C22×C3—41no—
369C22×C3—41no—
370C22×C3—42no—
371C22×C3—42no—
372C22×C3—41no—
373C22×C3—41no—
374C22×C3—41no—
375C22×C3#371511yescharacteristic
376C22×C3#371611yescharacteristic
377C22×C3#371711yescharacteristic
378C22×C3#371811yescharacteristic
379C22×C3#371911yescharacteristic
380C22×C3#371911yescharacteristic
381C22×C3(C2×D3)⋊Dic311yescharacteristic
382C22×C3(C2×D3)⋊Dic311yescharacteristic
383C22×C3#372211yescharacteristic
384C3×C4—22no—
385C3×C4—22no—
386C3×C4—21no—
387C3×C4—21no—
388C3×C4—21no—
389C3×C4—21no—
390C3×C4—21no—
391C3×C4—21no—
392C3×C4—41no—
393C3×C4—41no—
394C3×C4—41no—
395C3×C4—41no—
396C3×C4—42no—
397C3×C4—42no—
398C3×C4—362no—
399C3×C4—361no—
400C3×C4—361no—
401Dic3—122no—
402Dic3—124no—
403Dic3—122no—
404Dic3—122no—
405Dic3—122no—
406Dic3—121no—
407Dic3—121no—
408Dic3—121no—
409Dic3—121no—
410C32—21no—
411C32—21no—
412C32—21no—
413C32—41no—
414C32#445211yescharacteristic, complemented
415C32#446711yescharacteristic, complemented
416C32#453811yescharacteristic, complemented
417C23#629511yescharacteristic
418Q8—361no—
419C2×C4—41no—
420C2×C4—182no—
421C2×C4—181no—
422C2×C4—181no—
423C2×C4—361no—
424C2×C4—361no—
425C2×C4#629411yescharacteristic
426C2×C4#629511yescharacteristic
427C8—361no—
428C2×C3—22no—
429C2×C3—22no—
430C2×C3—21no—
431C2×C3—21no—
432C2×C3—21no—
433C2×C3—21no—
434C2×C3—21no—
435C2×C3—21no—
436C2×C3—21no—
437C2×C3—21no—
438C2×C3—21no—
439C2×C3—21no—
440C2×C3—21no—
441C2×C3—21no—
442C2×C3—21no—
443C2×C3—21no—
444C2×C3—21no—
445C2×C3—41no—
446C2×C3—41no—
447C2×C3—41no—
448C2×C3—41no—
449C2×C3—42no—
450C2×C3—42no—
451C2×C3—41no—
452C2×C3—41no—
453C2×C3—41no—
454C2×C3#6356511yescharacteristic
455C2×C3#6356611yescharacteristic
456C2×C3#6356711yescharacteristic
457C2×C3#6356811yescharacteristic
458C2×C3#6356911yescharacteristic
459C2×C3#6357011yescharacteristic
460C2×C3#6358011yescharacteristic
461C2×C3#6358011yescharacteristic
462C2×C3#6358111yescharacteristic
463C22—21no—
464C22—21no—
465C22#8881111yescharacteristic
466C22#8881211yescharacteristic
467C22#8881411yescharacteristic
468C4—21no—
469C4—21no—
470C4—362no—
471C4—361no—
472C4—361no—
473C3—21no—
474C3—21no—
475C3—21no—
476C3—41no—
477C3#1058915211yescharacteristic, complemented
478C3#1058915311yescharacteristic, complemented
479C3#1058915411yescharacteristic, complemented
480C2—21no—
481C2—21no—
482C2#1172207611yescharacteristic
483C2#1172207711yescharacteristic
484C2#1172208311yescharacteristic
subgroupG/H
centerC22#88814
commutatorC2×C33×C4C2×C4
Frattini, hypercenterC22×C4#1113
Fitting, cosocleC22×C33×C4C22
socleC22×C33C22⋊C4
prime radicalC23×C33D4
hypocenterC33#328
LCSG ▷ C2×C33×C4 ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
9. C2×C3×#634C4semidirect
11. C3×C4⋊(C3⋊Dic3)C4semidirect
48. C2×#634Dic3semidirect
56. C4⋊(C3⋊Dic3)Dic3semidirect
250. C33#328semidirect
414. C32#4452semidirect
415. C32#4467semidirect
416. C32#4538semidirect
477. C3#10589152semidirect
478. C3#10589153semidirect
479. C3#10589154semidirect

Maximal subgroups

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

subgroupindex
C3×#641332
C3×#636582
C3×#636572
#105891543
#105891533
#105891523