Finite Group Databasebeta

← #13145729

Subgroups — #13145729

Order 1728.

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

normal subgroups only

This group has 634 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×#64139C211yesmaximal, characteristic
2C3×#63658C211yesmaximal, characteristic, complemented
3C3×#63657C211yesmaximal, characteristic, complemented
4#10589151—31nomaximal
5#10589150—31nomaximal
6#10589149—31nomaximal
7C22×C33×C4C2211yescharacteristic
8C2×C3×#638—21no—
9C22×C3×C3⋊Dic3—21no—
10C2×C3×C4⋊(C3⋊D3)C411yescharacteristic, complemented
11C3×#3752—21no—
12C3×C4⋊(C3⋊Dic3)C411yescharacteristic, complemented
13C2×C3×C3⋊(C3⋊C8)—21no—
14#64139D311yescharacteristic
15C3×#930—31no—
16C3×#930—31no—
17C3×#930—61no—
18#63658D311yescharacteristic, complemented
19#63657D311yescharacteristic, complemented
20C3×C3⋊#95—31no—
21C3×C3⋊#95—31no—
22C3×C3⋊#95—61no—
23C3×#830—31no—
24C3×#830—31no—
25C3×#830—61no—
26C23×C33D411yescharacteristic
27C22×C3×C3⋊D3—21no—
28C2×C33×C4—41no—
29C2×C33×C4C2×C411yescharacteristic
30C2×C33×C4D411yescharacteristic
31C3×#638—41no—
32C3×#638—41no—
33C2×C3×C3⋊Dic3—22no—
34C2×C3×C3⋊Dic3—21no—
35C2×C3×C3⋊Dic3—41no—
36C2×C3×C3⋊Dic3—41no—
37C3×C4⋊(C3⋊D3)—41no—
38C3×C3⋊(C3⋊C8)—41no—
39#4537—91no—
40#4470—91no—
41#4451—91no—
42C22×C32×C4—21no—
43C22×C32×C4—21no—
44C22×C32×C4—21no—
45C22×C32×C4—41no—
46C22×C32×C4C2×D311yescharacteristic
47C22×C32×C4C2×D311yescharacteristic
48C22×C32×C4C2×D311yescharacteristic
49C2×#638—21no—
50C22×C3⋊Dic3—21no—
51C2×C4⋊(C3⋊D3)Dic311yescharacteristic, complemented
52C2×C3×#67—62no—
53C2×C3×#67—61no—
54C2×C3×#67—61no—
55C22×C3×Dic3—62no—
56C22×C3×Dic3—61no—
57C22×C3×Dic3—61no—
58C2×C3×D12—31no—
59C2×C3×D12—31no—
60C2×C3×D12—61no—
61#3752—21no—
62C4⋊(C3⋊Dic3)Dic311yescharacteristic, complemented
63C2×C3⋊(C3⋊C8)—21no—
64C3×(C2×D3)⋊C4—62no—
65C3×(C2×D3)⋊C4—61no—
66C3×(C2×D3)⋊C4—61no—
67C3×C4⋊Dic3—31no—
68C3×C4⋊Dic3—31no—
69C3×C4⋊Dic3—61no—
70C2×C3×C3⋊C8—62no—
71C2×C3×C3⋊C8—61no—
72C2×C3×C3⋊C8—61no—
73C22×C33—21no—
74C22×C33—21no—
75C22×C33C22⋊C411yescharacteristic
76C22×C33D811yescharacteristic
77C22×C33SD1611yescharacteristic
78C2×C3×C3⋊D3—41no—
79C2×C3×C3⋊D3—41no—
80C2×C3×C3⋊D3—41no—
81C33×C4—21no—
82C33×C4—21no—
83C3×C3⋊Dic3—42no—
84C3×C3⋊Dic3—41no—
85C3×#121—91no—
86#930—31no—
87#930—31no—
88#930—61no—
89C3×C22⋊C8—91no—
90C3×#95—91no—
91C3⋊#95—31no—
92C3⋊#95—31no—
93C3⋊#95—61no—
94#830—31no—
95#830—31no—
96#830—61no—
97C23×C32—21no—
98C23×C32—21no—
99C23×C32—21no—
100C23×C32—41no—
101C23×C32D1211yescharacteristic
102C23×C32#6711yescharacteristic
103C23×C32#6711yescharacteristic
104C22×C3⋊D3—21no—
105C22×C3×D3—62no—
106C22×C3×D3—61no—
107C22×C3×D3—61no—
108C2×C32×C4—21no—
109C2×C32×C4—21no—
110C2×C32×C4—21no—
111C2×C32×C4—21no—
112C2×C32×C4—21no—
113C2×C32×C4—21no—
114C2×C32×C4—41no—
115C2×C32×C4—41no—
116C2×C32×C4—41no—
117C2×C32×C4—41no—
118C2×C32×C4—41no—
119C2×C32×C4—42no—
120C2×C32×C4—42no—
121C2×C32×C4—42no—
122C2×C32×C4—44no—
123C2×C32×C4C4×D311yescharacteristic
124C2×C32×C4C4×D311yescharacteristic
125C2×C32×C4D1211yescharacteristic
126C2×C32×C4C2×Dic311yescharacteristic
127C2×C32×C4#6711yescharacteristic
128C2×C32×C4#6711yescharacteristic
129#638—41no—
130#638—41no—
131C2×C3⋊Dic3—22no—
132C2×C3⋊Dic3—21no—
133C2×C3⋊Dic3—41no—
134C2×C3⋊Dic3—41no—
135C4⋊(C3⋊D3)—41no—
136C3×#67—122no—
137C3×#67—122no—
138C3×#67—121no—
139C3×#67—121no—
140C3×#67—121no—
141C3×#67—121no—
142C2×C3×Dic3—62no—
143C2×C3×Dic3—64no—
144C2×C3×Dic3—62no—
145C2×C3×Dic3—62no—
146C2×C3×Dic3—61no—
147C2×C3×Dic3—61no—
148C2×C3×Dic3—122no—
149C2×C3×Dic3—121no—
150C2×C3×Dic3—121no—
151C2×C3×Dic3—122no—
152C2×C3×Dic3—121no—
153C2×C3×Dic3—121no—
154C3×D12—121no—
155C3×D12—121no—
156C3×D12—122no—
157C3⋊(C3⋊C8)—41no—
158C3×C3⋊C8—122no—
159C3×C3⋊C8—121no—
160C3×C3⋊C8—121no—
161#327—271noSylow
162C2×C33—21no—
163C2×C33—21no—
164C2×C33#9911yescharacteristic
165C2×C33#10211yescharacteristic
166C2×C33#10411yescharacteristic
167C3×C3⋊D3—81no—
168C2×C3×D4—91no—
169C2×C3×D4—181no—
170C22×C3×C4—21no—
171C22×C3×C4—21no—
172C22×C3×C4—21no—
173C22×C3×C4—41no—
174C22×C3×C4—181no—
175C22×C3×C4D3211yescharacteristic
176C22×C3×C4D3211yescharacteristic
177C22×C3×C4D3211yescharacteristic
178C2×#67—62no—
179C2×#67—61no—
180C2×#67—61no—
181C22×Dic3—62no—
182C22×Dic3—61no—
183C22×Dic3—61no—
184C2×D12—31no—
185C2×D12—31no—
186C2×D12—61no—
187C2×C3×C8—181no—
188C3×C4⋊C4—91no—
189C3×C22⋊C4—181no—
190(C2×D3)⋊C4—62no—
191(C2×D3)⋊C4—61no—
192(C2×D3)⋊C4—61no—
193C4⋊Dic3—31no—
194C4⋊Dic3—31no—
195C4⋊Dic3—61no—
196C2×C3⋊C8—62no—
197C2×C3⋊C8—61no—
198C2×C3⋊C8—61no—
199C22×C32—21no—
200C22×C32—21no—
201C22×C32—22no—
202C22×C32—22no—
203C22×C32—21no—
204C22×C32—21no—
205C22×C32—21no—
206C22×C32—21no—
207C22×C32—21no—
208C22×C32—21no—
209C22×C32—21no—
210C22×C32—21no—
211C22×C32—21no—
212C22×C32—21no—
213C22×C32—21no—
214C22×C32—21no—
215C22×C32—21no—
216C22×C32—41no—
217C22×C32—41no—
218C22×C32—41no—
219C22×C32—41no—
220C22×C32—41no—
221C22×C32—41no—
222C22×C32—41no—
223C22×C32—42no—
224C22×C32—42no—
225C22×C32#20411yescharacteristic
226C22×C32D2411yescharacteristic
227C22×C32(C2×D3)⋊C411yescharacteristic
228C22×C32(C2×D3)⋊C411yescharacteristic
229C22×C32#21311yescharacteristic
230C22×C32#21311yescharacteristic
231C22×C32#21411yescharacteristic
232C22×C32#21511yescharacteristic
233C22×C32#21711yescharacteristic
234C2×C3⋊D3—41no—
235C2×C3⋊D3—41no—
236C2×C3⋊D3—41no—
237C2×C3×D3—122no—
238C2×C3×D3—122no—
239C2×C3×D3—122no—
240C2×C3×D3—121no—
241C2×C3×D3—121no—
242C2×C3×D3—121no—
243C2×C3×D3—121no—
244C2×C3×D3—121no—
245C2×C3×D3—121no—
246C32×C4—21no—
247C32×C4—21no—
248C32×C4—22no—
249C32×C4—22no—
250C32×C4—21no—
251C32×C4—21no—
252C32×C4—21no—
253C32×C4—21no—
254C32×C4—41no—
255C32×C4—41no—
256C32×C4—41no—
257C32×C4—41no—
258C32×C4—42no—
259C32×C4—42no—
260C3⋊Dic3—42no—
261C3⋊Dic3—41no—
262C3×Dic3—124no—
263C3×Dic3—122no—
264C3×Dic3—122no—
265C3×Dic3—122no—
266C3×Dic3—121no—
267C3×Dic3—121no—
268#121—91no—
269C22⋊C8—91no—
270#95—91no—
271C33#32711yescharacteristic, complemented, Sylow
272C23×C3—21no—
273C23×C3—21no—
274C23×C3—21no—
275C23×C3—41no—
276C23×C3—181no—
277C23×C3#62511yescharacteristic
278C23×C3#62611yescharacteristic
279C23×C3#62611yescharacteristic
280C22×D3—62no—
281C22×D3—61no—
282C22×D3—61no—
283C3×D4—361no—
284C3×D4—361no—
285C3×D4—361no—
286C2×C3×C4—21no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—21no—
292C2×C3×C4—41no—
293C2×C3×C4—41no—
294C2×C3×C4—41no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4—44no—
299C2×C3×C4—41no—
300C2×C3×C4—41no—
301C2×C3×C4—182no—
302C2×C3×C4—181no—
303C2×C3×C4—361no—
304C2×C3×C4—361no—
305C2×C3×C4D3×Dic311yescharacteristic
306C2×C3×C4D3×Dic311yescharacteristic
307C2×C3×C4#62411yescharacteristic
308C2×C3×C4#62511yescharacteristic
309C2×C3×C4#62611yescharacteristic
310C2×C3×C4#62611yescharacteristic
311#67—122no—
312#67—122no—
313#67—121no—
314#67—121no—
315#67—121no—
316#67—121no—
317C2×Dic3—62no—
318C2×Dic3—64no—
319C2×Dic3—62no—
320C2×Dic3—62no—
321C2×Dic3—61no—
322C2×Dic3—61no—
323C2×Dic3—122no—
324C2×Dic3—121no—
325C2×Dic3—121no—
326C2×Dic3—122no—
327C2×Dic3—121no—
328C2×Dic3—121no—
329D12—121no—
330D12—121no—
331D12—122no—
332C3×C8—361no—
333C3⋊C8—122no—
334C3⋊C8—121no—
335C3⋊C8—121no—
336C2×C32—21no—
337C2×C32—21no—
338C2×C32—22no—
339C2×C32—22no—
340C2×C32—21no—
341C2×C32—21no—
342C2×C32—21no—
343C2×C32—21no—
344C2×C32—21no—
345C2×C32—21no—
346C2×C32—21no—
347C2×C32—21no—
348C2×C32—21no—
349C2×C32—21no—
350C2×C32—21no—
351C2×C32—21no—
352C2×C32—21no—
353C2×C32—41no—
354C2×C32—41no—
355C2×C32—41no—
356C2×C32—41no—
357C2×C32—41no—
358C2×C32—41no—
359C2×C32—41no—
360C2×C32—42no—
361C2×C32—42no—
362C2×C32#80511yescharacteristic
363C2×C32#80611yescharacteristic
364C2×C32#80611yescharacteristic
365C2×C32#80911yescharacteristic
366C2×C32#82111yescharacteristic
367C2×C32#82511yescharacteristic
368C2×C32#83211yescharacteristic
369C2×C32#83411yescharacteristic
370C2×C32#83711yescharacteristic
371C3⋊D3—81no—
372C3×D3—242no—
373C3×D3—241no—
374C3×D3—241no—
375C2×D4—91no—
376C2×D4—181no—
377C22×C4—181no—
378C22×C4#111311yescharacteristic
379C2×C8—181no—
380C4⋊C4—91no—
381C22⋊C4—181no—
382C22×C3—22no—
383C22×C3—22no—
384C22×C3—21no—
385C22×C3—21no—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3—21no—
389C22×C3—21no—
390C22×C3—21no—
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—21no—
397C22×C3—21no—
398C22×C3—21no—
399C22×C3—41no—
400C22×C3—41no—
401C22×C3—41no—
402C22×C3—41no—
403C22×C3—42no—
404C22×C3—42no—
405C22×C3—41no—
406C22×C3—41no—
407C22×C3—41no—
408C22×C3—361no—
409C22×C3—361no—
410C22×C3—361no—
411C22×C3#371311yescharacteristic
412C22×C3#371411yescharacteristic
413C22×C3#371411yescharacteristic
414C22×C3#371511yescharacteristic
415C22×C3#371611yescharacteristic
416C22×C3#371711yescharacteristic
417C22×C3(C2×D3)⋊Dic311yescharacteristic
418C22×C3(C2×D3)⋊Dic311yescharacteristic
419C22×C3#372211yescharacteristic
420C2×D3—122no—
421C2×D3—122no—
422C2×D3—122no—
423C2×D3—121no—
424C2×D3—121no—
425C2×D3—121no—
426C2×D3—121no—
427C2×D3—121no—
428C2×D3—121no—
429C3×C4—22no—
430C3×C4—22no—
431C3×C4—21no—
432C3×C4—21no—
433C3×C4—21no—
434C3×C4—21no—
435C3×C4—21no—
436C3×C4—21no—
437C3×C4—41no—
438C3×C4—41no—
439C3×C4—41no—
440C3×C4—41no—
441C3×C4—42no—
442C3×C4—42no—
443C3×C4—362no—
444C3×C4—361no—
445Dic3—124no—
446Dic3—122no—
447Dic3—122no—
448Dic3—122no—
449Dic3—121no—
450Dic3—121no—
451C32—21no—
452C32—21no—
453C32—21no—
454C32—41no—
455C32#445111yescharacteristic, complemented
456C32#447011yescharacteristic, complemented
457C32#453711yescharacteristic, complemented
458C23—181no—
459C23#629511yescharacteristic
460D4—361no—
461D4—361no—
462D4—361no—
463C2×C4—41no—
464C2×C4—182no—
465C2×C4—181no—
466C2×C4—361no—
467C2×C4—361no—
468C2×C4#629411yescharacteristic
469C2×C4#629511yescharacteristic
470C8—361no—
471C2×C3—22no—
472C2×C3—22no—
473C2×C3—21no—
474C2×C3—21no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—21no—
480C2×C3—21no—
481C2×C3—21no—
482C2×C3—21no—
483C2×C3—21no—
484C2×C3—21no—
485C2×C3—21no—
486C2×C3—21no—
487C2×C3—21no—
488C2×C3—41no—
489C2×C3—41no—
490C2×C3—41no—
491C2×C3—41no—
492C2×C3—42no—
493C2×C3—42no—
494C2×C3—41no—
495C2×C3—41no—
496C2×C3—41no—
497C2×C3—721no—
498C2×C3#6356211yescharacteristic
499C2×C3#6356311yescharacteristic
500C2×C3#6356411yescharacteristic
501C2×C3#6356811yescharacteristic
502C2×C3#6356911yescharacteristic
503C2×C3#6357011yescharacteristic
504C2×C3#6358011yescharacteristic
505C2×C3#6358011yescharacteristic
506C2×C3#6358111yescharacteristic
507D3—242no—
508D3—241no—
509D3—241no—
510C22—21no—
511C22—21no—
512C22—361no—
513C22—361no—
514C22—361no—
515C22#8881011yescharacteristic
516C22#8881111yescharacteristic
517C22#8881411yescharacteristic
518C4—21no—
519C4—21no—
520C4—362no—
521C4—361no—
522C3—21no—
523C3—21no—
524C3—21no—
525C3—41no—
526C3#1058914911yescharacteristic, complemented
527C3#1058915011yescharacteristic, complemented
528C3#1058915111yescharacteristic, complemented
529C2—21no—
530C2—21no—
531C2—721no—
532C2#1172207511yescharacteristic
533C2#1172207711yescharacteristic
534C2#1172208311yescharacteristic
subgroupG/H
centerC22#88814
commutatorC2×C33×C4C2×C4
Frattini, hypercenterC22×C4#1113
Fitting, cosocleC22×C33×C4C22
socleC22×C33C22⋊C4
prime radicalC3×#64139C2
hypocenterC33#327
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
2. C3×#63658C2semidirect
3. C3×#63657C2semidirect
10. C2×C3×C4⋊(C3⋊D3)C4semidirect
12. C3×C4⋊(C3⋊Dic3)C4semidirect
18. #63658D3semidirect
19. #63657D3semidirect
51. C2×C4⋊(C3⋊D3)Dic3semidirect
62. C4⋊(C3⋊Dic3)Dic3semidirect
271. C33#327semidirect
455. C32#4451semidirect
456. C32#4470semidirect
457. C32#4537semidirect
526. C3#10589149semidirect
527. C3#10589150semidirect
528. C3#10589151semidirect

Maximal subgroups

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

subgroupindex
C3×#641392
C3×#636582
C3×#636572
#105891513
#105891503
#105891493