Finite Group Databasebeta

← #13156516

Subgroups — #13156516

Order 1728.

542 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#63918C211yesmaximal, characteristic
2C3×#63866C211yesmaximal, characteristic
3#11722076C211yesmaximal, characteristic
4#11722039C211yesmaximal, characteristic
5#11722037C211yesmaximal, characteristic
6Dic3×C3⋊(C3⋊C8)C211yesmaximal, characteristic
7C3×#63565C211yesmaximal, characteristic
8#10591623—31nomaximal
9#10590560—31nomaximal
10#10590290—31nomaximal
11C2×C3×#634C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×C3×#627—21no—
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C32×C4×Dic3C2211yescharacteristic
19C3×Dic3⋊Dic3—21no—
20C3×Dic3⋊Dic3—21no—
21C3×Dic3⋊Dic3—21no—
22C3×#944—31no—
23C3×#888—31no—
24C3×#869—31no—
25C3×#869—31no—
26#63918D311yescharacteristic
27#63866D311yescharacteristic
28#63662—31no—
29#63637—31no—
30C4×C3⋊(C3⋊C8)—31no—
31C3×#835—31no—
32C3×#810—31no—
33#63567—31no—
34#63566—31no—
35#63565—31no—
36#63565—31no—
37#63565—31no—
38#63565—31no—
39#63565—31no—
40#63565—61no—
41#63565—61no—
42#63565D311yescharacteristic
43Dic3×C3⋊C8—31no—
44Dic3×C3⋊C8—31no—
45Dic3×C3⋊C8—61no—
46C2×C33×C4C2311yescharacteristic
47C2×C3×C3⋊Dic3—21no—
48C2×C3×C3⋊Dic3—21no—
49C3×#634—21no—
50C3×#634—21no—
51C2×C32×Dic3—21no—
52C2×C32×Dic3—21no—
53C2×C32×Dic3D411yescharacteristic
54C2×C32×Dic3D411yescharacteristic
55C32×Dic6—21no—
56C32×Dic6—21no—
57C3×#627—42no—
58C3⋊(C3⋊(C3⋊C8))—21no—
59C3×C3⋊(C3⋊C8)—21no—
60#5093—91no—
61#5079—91no—
62#4792—91no—
63C2×C32×Q8—31no—
64C2×#634C2×D311yescharacteristic
65C2×C3×Dic6—21no—
66C2×C3×Dic6—31no—
67C2×C3×Dic6—31no—
68C2×C3×Dic6—61no—
69C2×C3×Dic6—61no—
70C2×C3×Dic6—61no—
71C2×C3×Dic6C2×D311yescharacteristic
72C2×C3×Dic6C2×D311yescharacteristic
73C2×#627—21no—
74C32×C4⋊C4—31no—
75C32×C42—31no—
76C4⋊(C3⋊Dic3)C2×D311yescharacteristic
77C2×C3⋊(C3⋊C8)—31no—
78C2×C3⋊(C3⋊C8)—31no—
79C2×C3⋊(C3⋊C8)—31no—
80C2×C3⋊(C3⋊C8)—61no—
81C2×C3⋊(C3⋊C8)—61no—
82C2×C3⋊(C3⋊C8)—61no—
83C2×C3⋊(C3⋊C8)—121no—
84C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
85C3×C4⋊Dic3—21no—
86C3×C4⋊Dic3—31no—
87C3×C4⋊Dic3—31no—
88C3×C4⋊Dic3—61no—
89C3×C4⋊Dic3—61no—
90C3×C4⋊Dic3—61no—
91C3×C4⋊Dic3—61no—
92C3×C4⋊Dic3C2×D311yescharacteristic
93C3×C4⋊Dic3C2×D311yescharacteristic
94C3×Dic3⋊C4—61no—
95C3×Dic3⋊C4—61no—
96C3×Dic3⋊C4—61no—
97C3×C4×Dic3—21no—
98C3×C4×Dic3C2×D311yescharacteristic
99C3×C4×Dic3C2×D311yescharacteristic
100C2×C3×C3⋊C8—31no—
101C2×C3×C3⋊C8—31no—
102C2×C3×C3⋊C8—61no—
103Dic3⋊Dic3—21no—
104Dic3⋊Dic3—21no—
105Dic3⋊Dic3—21no—
106C22×C33C2×D411yescharacteristic
107C33×C4#4111yescharacteristic
108C33×C4#4111yescharacteristic
109C3×C3⋊Dic3—41no—
110C3×C3⋊Dic3—41no—
111C32×Dic3—41no—
112C32×Dic3—41no—
113C32×Dic3SD1612yes—
114C32×Dic3Q1612yes—
115C3×C4⋊Q8—91no—
116C3×C4⋊Q8—91no—
117#944—31no—
118#888—31no—
119#869—31no—
120#869—31no—
121C3×#103—91no—
122#835—31no—
123#835—91no—
124#835—91no—
125#835—91no—
126#835—91no—
127#835—181no—
128#816—91no—
129C8×Dic3—91no—
130#810—31no—
131#810—91no—
132#810—91no—
133#810—91no—
134#810—91no—
135#810—181no—
136C4×C3⋊C8—91no—
137C4×C3⋊C8—91no—
138C4×C3⋊C8—181no—
139C32×Q8—61no—
140C32×Q8—61no—
141C2×C32×C4—21no—
142C2×C32×C4—21no—
143C2×C32×C4—21no—
144C2×C32×C4—31no—
145C2×C32×C4—31no—
146C2×C32×C4—41no—
147C2×C32×C4—61no—
148C2×C32×C4—61no—
149C2×C32×C4C22×D311yescharacteristic
150C2×C32×C4C22×D311yescharacteristic
151C2×C32×C4C22×D311yescharacteristic
152C2×C3⋊Dic3—21no—
153C2×C3⋊Dic3—21no—
154#634—21no—
155#634—21no—
156C2×C3×Dic3—21no—
157C2×C3×Dic3—21no—
158C2×C3×Dic3—21no—
159C2×C3×Dic3—21no—
160C2×C3×Dic3—21no—
161C2×C3×Dic3—21no—
162C2×C3×Dic3—41no—
163C2×C3×Dic3—41no—
164C2×C3×Dic3—61no—
165C2×C3×Dic3—61no—
166C2×C3×Dic3—61no—
167C2×C3×Dic3—61no—
168C2×C3×Dic3—121no—
169C2×C3×Dic3—121no—
170C2×C3×Dic3#6711yescharacteristic
171C2×C3×Dic3#6711yescharacteristic
172C2×C3×Dic3#6711yescharacteristic
173C2×C3×Dic3#6711yescharacteristic
174C3×Dic6—22no—
175C3×Dic6—21no—
176C3×Dic6—21no—
177C3×Dic6—21no—
178C3×Dic6—21no—
179C3×Dic6—41no—
180C3×Dic6—62no—
181C3×Dic6—61no—
182C3×Dic6—61no—
183C3×Dic6—61no—
184C3×Dic6—61no—
185C3×Dic6—121no—
186C3×Dic6—122no—
187C3×Dic6—122no—
188#627—42no—
189C3⋊(C3⋊C8)—21no—
190C3⋊(C3⋊C8)—61no—
191C3⋊(C3⋊C8)—62no—
192C3⋊(C3⋊C8)—61no—
193C3⋊(C3⋊C8)—61no—
194C3⋊(C3⋊C8)—121no—
195C3⋊(C3⋊C8)—121no—
196C3⋊(C3⋊C8)—122no—
197C3×C3⋊C8—62no—
198C3×C3⋊C8—61no—
199C3×C3⋊C8—61no—
200#487—271noSylow
201C2×C33#12411yescharacteristic
202C2×C33C2×SD1611yescharacteristic
203C2×C33C2×Q1611yescharacteristic
204C2×C3×Q8—31no—
205C2×C3×Q8—31no—
206C2×C3×Q8—61no—
207C2×C3×Q8—91no—
208C2×C3×Q8—181no—
209C2×Dic6—31no—
210C2×Dic6—31no—
211C2×Dic6—61no—
212C2×Dic6—61no—
213C2×Dic6—61no—
214C2×Dic6D3211yescharacteristic
215C2×C3×C8—91no—
216C3×C4⋊C4—31no—
217C3×C4⋊C4—31no—
218C3×C4⋊C4—61no—
219C3×C4⋊C4—91no—
220C3×C4⋊C4—181no—
221C3×C4⋊C4—181no—
222C3×C4⋊C4—181no—
223C3×C42—31no—
224C3×C42—31no—
225C3×C42—61no—
226C4⋊Dic3—31no—
227C4⋊Dic3—31no—
228C4⋊Dic3—61no—
229C4⋊Dic3—61no—
230C4⋊Dic3—61no—
231C4⋊Dic3—61no—
232C4⋊Dic3D3211yescharacteristic
233Dic3⋊C4—61no—
234Dic3⋊C4—61no—
235Dic3⋊C4—61no—
236C4×Dic3D3211yescharacteristic
237C2×C3⋊C8—31no—
238C2×C3⋊C8—31no—
239C2×C3⋊C8—61no—
240C2×C3⋊C8—91no—
241C2×C3⋊C8—91no—
242C2×C3⋊C8—91no—
243C2×C3⋊C8—181no—
244C2×C3⋊C8—181no—
245C2×C3⋊C8—181no—
246C2×C3⋊C8—361no—
247C22×C32—21no—
248C22×C32—21no—
249C22×C32—21no—
250C22×C32—41no—
251C22×C32D3×D411yescharacteristic
252C22×C32C2×#6711yescharacteristic
253C22×C32C2×#6711yescharacteristic
254C32×C4—21no—
255C32×C4—21no—
256C32×C4—21no—
257C32×C4—21no—
258C32×C4—21no—
259C32×C4—21no—
260C32×C4—32no—
261C32×C4—32no—
262C32×C4—41no—
263C32×C4—41no—
264C32×C4—121no—
265C32×C4—121no—
266C32×C4#23511yescharacteristic
267C32×C4#23711yescharacteristic
268C32×C4#23711yescharacteristic
269C32×C4#23711yescharacteristic
270C32×C4#23911yescharacteristic
271C32×C4#23911yescharacteristic
272C3⋊Dic3—41no—
273C3⋊Dic3—41no—
274C3×Dic3—22no—
275C3×Dic3—22no—
276C3×Dic3—42no—
277C3×Dic3—41no—
278C3×Dic3—41no—
279C3×Dic3—41no—
280C3×Dic3—41no—
281C3×Dic3—81no—
282C3×Dic3—122no—
283C3×Dic3—121no—
284C3×Dic3—121no—
285C3×Dic3—121no—
286C3×Dic3—121no—
287C3×Dic3—241no—
288C3×Dic3#21412yes—
289C3×Dic3#21512yes—
290C3×Dic3#21612yes—
291C3×Dic3#21612yes—
292C4⋊Q8—91no—
293C4⋊Q8—91no—
294#103—91no—
295#103—271no—
296#103—271no—
297#103—271no—
298C4×C8—271no—
299C33#48711yescharacteristic, complemented, Sylow
300C3×Q8—62no—
301C3×Q8—61no—
302C3×Q8—61no—
303C3×Q8—61no—
304C3×Q8—61no—
305C3×Q8—121no—
306C3×Q8—181no—
307C3×Q8—181no—
308C3×Q8—362no—
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—31no—
313C2×C3×C4—31no—
314C2×C3×C4—31no—
315C2×C3×C4—31no—
316C2×C3×C4—41no—
317C2×C3×C4—61no—
318C2×C3×C4—61no—
319C2×C3×C4—61no—
320C2×C3×C4—61no—
321C2×C3×C4—61no—
322C2×C3×C4—61no—
323C2×C3×C4—121no—
324C2×C3×C4—121no—
325C2×C3×C4—181no—
326C2×C3×C4—181no—
327C2×C3×C4C2×D3211yescharacteristic
328C2×C3×C4C2×D3211yescharacteristic
329C2×C3×C4C2×D3211yescharacteristic
330C2×Dic3—21no—
331C2×Dic3—21no—
332C2×Dic3—61no—
333C2×Dic3—61no—
334C2×Dic3—61no—
335C2×Dic3—61no—
336C2×Dic3—121no—
337C2×Dic3—121no—
338C2×Dic3#62511yescharacteristic
339C2×Dic3#62511yescharacteristic
340Dic6—21no—
341Dic6—21no—
342Dic6—62no—
343Dic6—61no—
344Dic6—61no—
345Dic6—61no—
346Dic6—61no—
347Dic6—121no—
348Dic6—122no—
349Dic6—122no—
350C3×C8—181no—
351C3⋊C8—62no—
352C3⋊C8—61no—
353C3⋊C8—61no—
354C3⋊C8—182no—
355C3⋊C8—181no—
356C3⋊C8—181no—
357C3⋊C8—181no—
358C3⋊C8—361no—
359C3⋊C8—361no—
360C3⋊C8—362no—
361C2×C32—21no—
362C2×C32—21no—
363C2×C32—21no—
364C2×C32—21no—
365C2×C32—21no—
366C2×C32—21no—
367C2×C32—21no—
368C2×C32—21no—
369C2×C32—21no—
370C2×C32—41no—
371C2×C32—41no—
372C2×C32—41no—
373C2×C32#88511yescharacteristic
374C2×C32D3×SD1611yescharacteristic
375C2×C32D3×Q1611yescharacteristic
376C2×C32C2×#21411yescharacteristic
377C2×C32#93611yescharacteristic
378C2×C32C2×#21511yescharacteristic
379C2×C32C2×#21611yescharacteristic
380C2×C32C2×#21611yescharacteristic
381C2×C32#94711yescharacteristic
382C2×Q8—31no—
383C2×Q8—91no—
384C2×Q8—181no—
385C2×C8—91no—
386C2×C8—271no—
387C4⋊C4—31no—
388C4⋊C4—91no—
389C4⋊C4—181no—
390C4⋊C4—181no—
391C4⋊C4—181no—
392C42—31no—
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—41no—
397C22×C3C2×#62511yescharacteristic
398C22×C3D3×#6711yescharacteristic
399C22×C3D3×#6711yescharacteristic
400C3×C4—21no—
401C3×C4—21no—
402C3×C4—21no—
403C3×C4—21no—
404C3×C4—21no—
405C3×C4—21no—
406C3×C4—32no—
407C3×C4—32no—
408C3×C4—32no—
409C3×C4—32no—
410C3×C4—41no—
411C3×C4—41no—
412C3×C4—62no—
413C3×C4—62no—
414C3×C4—122no—
415C3×C4—121no—
416C3×C4—121no—
417C3×C4—121no—
418C3×C4—121no—
419C3×C4—241no—
420C3×C4—361no—
421C3×C4—361no—
422C3×C4#379611yescharacteristic
423C3×C4#379611yescharacteristic
424C3×C4#379611yescharacteristic
425C3×C4#379911yescharacteristic
426C3×C4#380411yescharacteristic
427C3×C4#380511yescharacteristic
428Dic3—41no—
429Dic3—41no—
430Dic3—122no—
431Dic3—121no—
432Dic3—121no—
433Dic3—121no—
434Dic3—121no—
435Dic3—241no—
436Dic3#371512yes—
437Dic3#371812yes—
438C32—21no—
439C32—21no—
440C32—21no—
441C32—41no—
442C32#479211yescharacteristic, complemented
443C32#507911yescharacteristic, complemented
444C32#509311yescharacteristic, complemented
445Q8—61no—
446Q8—61no—
447Q8—181no—
448Q8—181no—
449Q8—362no—
450C2×C4—31no—
451C2×C4—31no—
452C2×C4—61no—
453C2×C4—61no—
454C2×C4—181no—
455C2×C4—181no—
456C2×C4D3311yescharacteristic
457C8—181no—
458C8—541no—
459C2×C3—21no—
460C2×C3—21no—
461C2×C3—21no—
462C2×C3—21no—
463C2×C3—21no—
464C2×C3—21no—
465C2×C3—21no—
466C2×C3—21no—
467C2×C3—21no—
468C2×C3—41no—
469C2×C3—41no—
470C2×C3—41no—
471C2×C3C2×#371511yescharacteristic
472C2×C3C2×#371811yescharacteristic
473C2×C3#6386311yescharacteristic
474C2×C3#6390711yescharacteristic
475C2×C3D3×#21411yescharacteristic
476C2×C3D3×#21511yescharacteristic
477C2×C3D3×#21611yescharacteristic
478C2×C3D3×#21611yescharacteristic
479C2×C3#6395911yescharacteristic
480C22D3×#62511yescharacteristic
481C4—32no—
482C4—32no—
483C4—121no—
484C4—121no—
485C4—361no—
486C4—361no—
487C4#8896211yescharacteristic
488C4#8896411yescharacteristic
489C3—21no—
490C3—21no—
491C3—21no—
492C3—41no—
493C3#1059029011yescharacteristic, complemented
494C3#1059056011yescharacteristic, complemented
495C3#1059162311yescharacteristic, complemented
496C2D3×#371511yescharacteristic
497C2D3×#371811yescharacteristic
498C2#1172300111yescharacteristic
subgroupG/H
centerC22D3×#625
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
299. C33#487semidirect
442. C32#4792semidirect
443. C32#5079semidirect
444. C32#5093semidirect
493. C3#10590290semidirect
494. C3#10590560semidirect
495. C3#10591623semidirect

Maximal subgroups

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

subgroupindex
C3×#639182
C3×#638662
#117220762
#117220392
#117220372
Dic3×C3⋊(C3⋊C8)2
C3×#635652
#105916233
#105905603
#105902903