Finite Group Databasebeta

← #13156518

Subgroups — #13156518

Order 1728.

570 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723534C211yesmaximal, characteristic
2C3×#63866C211yesmaximal, characteristic
3#11722076C211yesmaximal, characteristic
4#11722040C211yesmaximal, characteristic
5#11722038C211yesmaximal, characteristic
6C3×#63566C211yesmaximal, characteristic
7C3×Dic3×C3⋊C8C211yesmaximal, characteristic
8#10591623—31nomaximal
9#10590890—31nomaximal
10#10590287—31nomaximal
11C2×C3×#634C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×#6293—21no—
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4×Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19Dic3⋊(C3⋊Dic3)—21no—
20C3×Dic3⋊Dic3—21no—
21C3×Dic3⋊Dic3—21no—
22#64078—31no—
23C3×#944—31no—
24C3×#869—31no—
25#63918—31no—
26#63918—31no—
27#63918—61no—
28#63866D311yescharacteristic
29#63662—31no—
30#63643—31no—
31C3×#835—31no—
32C3×#816—31no—
33C3×C8×Dic3—31no—
34C3×C4×C3⋊C8—31no—
35#63567—31no—
36#63566—31no—
37#63566—31no—
38#63566—31no—
39#63566—31no—
40#63566—31no—
41#63566—61no—
42#63566—61no—
43#63566D311yescharacteristic
44#63565—31no—
45Dic3×C3⋊C8D311yescharacteristic
46C2×C33×C4C2311yescharacteristic
47C2×C3⋊(C3⋊Dic3)—21no—
48C2×C3×C3⋊Dic3—21no—
49C2×C3×C3⋊Dic3—21no—
50C3×#634—21no—
51C3×#634—21no—
52C2×C32×Dic3—21no—
53C2×C32×Dic3D411yescharacteristic
54C2×C32×Dic3D411yescharacteristic
55C32×Dic6—21no—
56C32×Dic6—21no—
57#6293—42no—
58C3×C3⋊(C3⋊C8)—21no—
59C32×C3⋊C8—21no—
60#5079—91no—
61#4792—91no—
62#4681—91no—
63C2×C32×Q8—31no—
64C2×#634—61no—
65C2×#634C2×D311yescharacteristic
66C2×C3×Dic6—21no—
67C2×C3×Dic6—31no—
68C2×C3×Dic6—31no—
69C2×C3×Dic6—61no—
70C2×C3×Dic6C2×D311yescharacteristic
71C2×C3×Dic6C2×D311yescharacteristic
72C2×#627—62no—
73C2×#627—61no—
74C2×#627—61no—
75C2×C32×C8—31no—
76C32×C42—31no—
77C4⋊(C3⋊Dic3)—31no—
78C4⋊(C3⋊Dic3)—31no—
79C4⋊(C3⋊Dic3)—31no—
80C4⋊(C3⋊Dic3)—61no—
81C4⋊(C3⋊Dic3)—61no—
82C4⋊(C3⋊Dic3)—61no—
83C4⋊(C3⋊Dic3)—61no—
84C4⋊(C3⋊Dic3)—121no—
85C4⋊(C3⋊Dic3)C2×D311yescharacteristic
86C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
87C3×C4⋊Dic3—31no—
88C3×C4⋊Dic3—31no—
89C3×C4⋊Dic3—61no—
90C3×C4⋊Dic3—61no—
91C3×C4⋊Dic3—61no—
92C3×Dic3⋊C4—61no—
93C3×Dic3⋊C4—61no—
94C3×C4×Dic3—21no—
95C3×C4×Dic3C2×D311yescharacteristic
96C3×C4×Dic3C2×D311yescharacteristic
97C2×C3×C3⋊C8—21no—
98C2×C3×C3⋊C8—31no—
99C2×C3×C3⋊C8—31no—
100C2×C3×C3⋊C8—61no—
101C2×C3×C3⋊C8C2×D311yescharacteristic
102C2×C3×C3⋊C8C2×D311yescharacteristic
103Dic3⋊Dic3—21no—
104Dic3⋊Dic3—21no—
105Dic3⋊Dic3—62no—
106Dic3⋊Dic3—61no—
107Dic3⋊Dic3—61no—
108C22×C33C2×D411yescharacteristic
109C33×C4#4111yescharacteristic
110C33×C4#4111yescharacteristic
111C3⋊(C3⋊Dic3)—41no—
112C3×C3⋊Dic3—41no—
113C3×C3⋊Dic3—41no—
114C32×Dic3—41no—
115C32×Dic3SD1612yes—
116C32×Dic3Q1612yes—
117C3×C4⋊Q8—91no—
118#944—31no—
119#888—91no—
120#869—31no—
121#869—91no—
122#869—91no—
123#869—181no—
124C3×#103—91no—
125C3×C4×C8—91no—
126#835—31no—
127#835—91no—
128#835—91no—
129#835—91no—
130#835—91no—
131#835—181no—
132#816—31no—
133#816—91no—
134#816—91no—
135#816—91no—
136#816—91no—
137#816—181no—
138C8×Dic3—31no—
139#810—91no—
140C4×C3⋊C8—31no—
141C32×Q8—61no—
142C32×Q8—61no—
143C2×C32×C4—21no—
144C2×C32×C4—21no—
145C2×C32×C4—21no—
146C2×C32×C4—31no—
147C2×C32×C4—31no—
148C2×C32×C4—41no—
149C2×C32×C4—61no—
150C2×C32×C4C22×D311yescharacteristic
151C2×C32×C4C22×D311yescharacteristic
152C2×C32×C4C22×D311yescharacteristic
153C2×C3⋊Dic3—21no—
154C2×C3⋊Dic3—21no—
155C2×C3⋊Dic3—61no—
156C2×C3⋊Dic3—62no—
157C2×C3⋊Dic3—61no—
158C2×C3⋊Dic3—61no—
159C2×C3⋊Dic3—121no—
160C2×C3⋊Dic3—121no—
161C2×C3⋊Dic3—122no—
162#634—21no—
163#634—21no—
164#634—122no—
165C2×C3×Dic3—21no—
166C2×C3×Dic3—21no—
167C2×C3×Dic3—21no—
168C2×C3×Dic3—21no—
169C2×C3×Dic3—41no—
170C2×C3×Dic3—62no—
171C2×C3×Dic3—61no—
172C2×C3×Dic3—61no—
173C2×C3×Dic3—61no—
174C2×C3×Dic3—61no—
175C2×C3×Dic3—121no—
176C2×C3×Dic3D1211yescharacteristic
177C2×C3×Dic3D1211yescharacteristic
178C2×C3×Dic3#6711yescharacteristic
179C2×C3×Dic3#6711yescharacteristic
180C3×Dic6—22no—
181C3×Dic6—21no—
182C3×Dic6—21no—
183C3×Dic6—21no—
184C3×Dic6—21no—
185C3×Dic6—41no—
186C3×Dic6—62no—
187C3×Dic6—61no—
188C3×Dic6—61no—
189C3×Dic6—61no—
190C3×Dic6—61no—
191C3×Dic6—121no—
192#627—124no—
193#627—122no—
194#627—122no—
195C32×C8—61no—
196C3⋊(C3⋊C8)—21no—
197C3×C3⋊C8—21no—
198C3×C3⋊C8—21no—
199C3×C3⋊C8—41no—
200C3×C3⋊C8—61no—
201C3×C3⋊C8—61no—
202C3×C3⋊C8—121no—
203#487—271noSylow
204C2×C33#12411yescharacteristic
205C2×C33C2×SD1611yescharacteristic
206C2×C33C2×Q1611yescharacteristic
207C2×C3×Q8—31no—
208C2×C3×Q8—31no—
209C2×C3×Q8—61no—
210C2×C3×Q8—91no—
211C2×Dic6—31no—
212C2×Dic6—31no—
213C2×Dic6—61no—
214C2×Dic6—181no—
215C2×Dic6—181no—
216C2×Dic6—182no—
217C2×Dic6—181no—
218C2×Dic6D3211yescharacteristic
219C2×C3×C8—31no—
220C2×C3×C8—31no—
221C2×C3×C8—61no—
222C2×C3×C8—91no—
223C3×C4⋊C4—91no—
224C3×C4⋊C4—181no—
225C3×C4⋊C4—181no—
226C3×C42—31no—
227C3×C42—31no—
228C3×C42—61no—
229C4⋊Dic3—31no—
230C4⋊Dic3—31no—
231C4⋊Dic3—61no—
232C4⋊Dic3—61no—
233C4⋊Dic3—61no—
234C4⋊Dic3—91no—
235C4⋊Dic3—91no—
236C4⋊Dic3—91no—
237C4⋊Dic3—181no—
238C4⋊Dic3—181no—
239C4⋊Dic3—181no—
240C4⋊Dic3—181no—
241C4⋊Dic3—181no—
242C4⋊Dic3—182no—
243C4⋊Dic3—361no—
244Dic3⋊C4—61no—
245Dic3⋊C4—61no—
246Dic3⋊C4—181no—
247C4×Dic3D3211yescharacteristic
248C2×C3⋊C8—31no—
249C2×C3⋊C8—31no—
250C2×C3⋊C8—61no—
251C2×C3⋊C8D3211yescharacteristic
252C22×C32—21no—
253C22×C32—21no—
254C22×C32—21no—
255C22×C32—41no—
256C22×C32C2×D1211yescharacteristic
257C22×C32D3×D411yescharacteristic
258C22×C32C2×#6711yescharacteristic
259C32×C4—21no—
260C32×C4—21no—
261C32×C4—21no—
262C32×C4—21no—
263C32×C4—21no—
264C32×C4—21no—
265C32×C4—32no—
266C32×C4—32no—
267C32×C4—41no—
268C32×C4—41no—
269C32×C4—121no—
270C32×C4#23511yescharacteristic
271C32×C4#23511yescharacteristic
272C32×C4#23511yescharacteristic
273C32×C4#23711yescharacteristic
274C32×C4#23711yescharacteristic
275C32×C4#23711yescharacteristic
276C3⋊Dic3—41no—
277C3⋊Dic3—41no—
278C3⋊Dic3—121no—
279C3⋊Dic3—122no—
280C3⋊Dic3—121no—
281C3⋊Dic3—121no—
282C3⋊Dic3—241no—
283C3⋊Dic3—241no—
284C3⋊Dic3—242no—
285C3×Dic3—22no—
286C3×Dic3—22no—
287C3×Dic3—42no—
288C3×Dic3—41no—
289C3×Dic3—41no—
290C3×Dic3—122no—
291C3×Dic3—122no—
292C3×Dic3—121no—
293C3×Dic3—121no—
294C3×Dic3—121no—
295C3×Dic3—121no—
296C3×Dic3#20412yes—
297C3×Dic3Dic1212yes—
298C3×Dic3#21512yes—
299C3×Dic3#21612yes—
300C4⋊Q8—91no—
301C4⋊Q8—271no—
302#103—91no—
303#103—271no—
304#103—271no—
305#103—271no—
306C4×C8—91no—
307C33#48711yescharacteristic, complemented, Sylow
308C3×Q8—62no—
309C3×Q8—61no—
310C3×Q8—61no—
311C3×Q8—61no—
312C3×Q8—61no—
313C3×Q8—121no—
314C3×Q8—181no—
315C3×Q8—181no—
316C2×C3×C4—21no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—31no—
320C2×C3×C4—31no—
321C2×C3×C4—31no—
322C2×C3×C4—31no—
323C2×C3×C4—41no—
324C2×C3×C4—61no—
325C2×C3×C4—61no—
326C2×C3×C4—61no—
327C2×C3×C4—61no—
328C2×C3×C4—121no—
329C2×C3×C4—181no—
330C2×C3×C4—181no—
331C2×C3×C4C2×D3211yescharacteristic
332C2×C3×C4C2×D3211yescharacteristic
333C2×C3×C4C2×D3211yescharacteristic
334C2×Dic3—21no—
335C2×Dic3—62no—
336C2×Dic3—61no—
337C2×Dic3—61no—
338C2×Dic3—61no—
339C2×Dic3—61no—
340C2×Dic3—121no—
341C2×Dic3—182no—
342C2×Dic3—181no—
343C2×Dic3—181no—
344C2×Dic3—181no—
345C2×Dic3—361no—
346C2×Dic3—361no—
347C2×Dic3—362no—
348C2×Dic3#62611yescharacteristic
349C2×Dic3#62611yescharacteristic
350Dic6—21no—
351Dic6—21no—
352Dic6—62no—
353Dic6—61no—
354Dic6—61no—
355Dic6—61no—
356Dic6—61no—
357Dic6—121no—
358Dic6—362no—
359Dic6—362no—
360Dic6—364no—
361Dic6—362no—
362C3×C8—61no—
363C3×C8—61no—
364C3×C8—121no—
365C3×C8—181no—
366C3⋊C8—21no—
367C3⋊C8—61no—
368C3⋊C8—61no—
369C3⋊C8—121no—
370C2×C32—21no—
371C2×C32—21no—
372C2×C32—21no—
373C2×C32—21no—
374C2×C32—21no—
375C2×C32—21no—
376C2×C32—21no—
377C2×C32—21no—
378C2×C32—21no—
379C2×C32—41no—
380C2×C32—41no—
381C2×C32—41no—
382C2×C32#87511yescharacteristic
383C2×C32#88511yescharacteristic
384C2×C32C2×#20411yescharacteristic
385C2×C32C2×Dic1211yescharacteristic
386C2×C32D3×SD1611yescharacteristic
387C2×C32D3×Q1611yescharacteristic
388C2×C32#93611yescharacteristic
389C2×C32C2×#21511yescharacteristic
390C2×C32C2×#21611yescharacteristic
391C2×Q8—31no—
392C2×Q8—91no—
393C2×Q8—541no—
394C2×C8—31no—
395C2×C8—91no—
396C4⋊C4—91no—
397C4⋊C4—181no—
398C4⋊C4—181no—
399C4⋊C4—271no—
400C4⋊C4—541no—
401C42—31no—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—21no—
405C22×C3—41no—
406C22×C3D3×D1211yescharacteristic
407C22×C3C2×#62611yescharacteristic
408C22×C3D3×#6711yescharacteristic
409C3×C4—21no—
410C3×C4—21no—
411C3×C4—21no—
412C3×C4—21no—
413C3×C4—21no—
414C3×C4—21no—
415C3×C4—32no—
416C3×C4—32no—
417C3×C4—32no—
418C3×C4—32no—
419C3×C4—41no—
420C3×C4—41no—
421C3×C4—62no—
422C3×C4—62no—
423C3×C4—122no—
424C3×C4—121no—
425C3×C4—121no—
426C3×C4—361no—
427C3×C4—361no—
428C3×C4#379511yescharacteristic
429C3×C4#379511yescharacteristic
430C3×C4#379811yescharacteristic
431C3×C4#380411yescharacteristic
432C3×C4#380411yescharacteristic
433C3×C4#380511yescharacteristic
434Dic3—41no—
435Dic3—122no—
436Dic3—122no—
437Dic3—121no—
438Dic3—121no—
439Dic3—121no—
440Dic3—121no—
441Dic3—362no—
442Dic3—361no—
443Dic3—361no—
444Dic3—361no—
445Dic3—721no—
446Dic3—721no—
447Dic3—722no—
448Dic3#371712yes—
449Dic3#371912yes—
450C32—21no—
451C32—21no—
452C32—21no—
453C32—41no—
454C32#468111yescharacteristic, complemented
455C32#479211yescharacteristic, complemented
456C32#507911yescharacteristic, complemented
457Q8—61no—
458Q8—61no—
459Q8—181no—
460Q8—181no—
461Q8—1082no—
462C2×C4—31no—
463C2×C4—31no—
464C2×C4—61no—
465C2×C4—181no—
466C2×C4—181no—
467C2×C4—541no—
468C2×C4D3311yescharacteristic
469C8—61no—
470C8—181no—
471C2×C3—21no—
472C2×C3—21no—
473C2×C3—21no—
474C2×C3—21no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—21no—
480C2×C3—41no—
481C2×C3—41no—
482C2×C3—41no—
483C2×C3D3×#20411yescharacteristic
484C2×C3D3×Dic1211yescharacteristic
485C2×C3C2×#371711yescharacteristic
486C2×C3C2×#371911yescharacteristic
487C2×C3#6385211yescharacteristic
488C2×C3#6386011yescharacteristic
489C2×C3D3×#21511yescharacteristic
490C2×C3D3×#21611yescharacteristic
491C2×C3#6395911yescharacteristic
492C22D3×#62611yescharacteristic
493C4—32no—
494C4—32no—
495C4—121no—
496C4—361no—
497C4—361no—
498C4—1081no—
499C4#8896011yescharacteristic
500C4#8896111yescharacteristic
501C3—21no—
502C3—21no—
503C3—21no—
504C3—41no—
505C3#1059028711yescharacteristic, complemented
506C3#1059089011yescharacteristic, complemented
507C3#1059162311yescharacteristic, complemented
508C2D3×#371711yescharacteristic
509C2D3×#371911yescharacteristic
510C2#1172300211yescharacteristic
subgroupG/H
centerC22D3×#626
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socle, prime radicalC22×C33C2×D4
hypocenterC33#487
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
307. C33#487semidirect
454. C32#4681semidirect
455. C32#4792semidirect
456. C32#5079semidirect
505. C3#10590287semidirect
506. C3#10590890semidirect
507. C3#10591623semidirect

Maximal subgroups

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

subgroupindex
#117235342
C3×#638662
#117220762
#117220402
#117220382
C3×#635662
C3×Dic3×C3⋊C82
#105916233
#105908903
#105902873