Finite Group Databasebeta

← #13156446

Subgroups — #13156446

Order 1728.

566 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×D3×C4⋊Dic3C211yesmaximal, characteristic
2C3×#63856C211yesmaximal, characteristic
3#11722079C211yesmaximal, characteristic, complemented
4#11722053C211yesmaximal, characteristic, complemented
5#11722037C211yesmaximal, characteristic, complemented
6(C2×D3)⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic
7C3×#63565C211yesmaximal, characteristic, complemented
8#10591549—31nomaximal
9#10591101—31nomaximal
10#10590516—31nomaximal
11C2×C32×C4×D3C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×C3×D3×Dic3—21no—
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C3×#3724—21no—
20C3×Dic3⋊Dic3—21no—
21C3×(C2×D3)⋊Dic3—21no—
22C3×#946—31no—
23C2×C3×C4⋊Dic3—31no—
24C3×#924—31no—
25C3×D3×C4⋊C4—31no—
26D3×C4⋊Dic3D311yescharacteristic
27#63856D311yescharacteristic
28#63662—31no—
29#63657—31no—
30#63635—31no—
31C3×#835—31no—
32C3×#810—31no—
33#63574—31no—
34#63574—31no—
35#63574—31no—
36#63574—61no—
37#63573—31no—
38#63573—31no—
39#63565—31no—
40#63565—31no—
41#63565—61no—
42#63565D311yescharacteristic, complemented
43(C2×D3)⋊(C3⋊C8)—31no—
44(C2×D3)⋊(C3⋊C8)—31no—
45(C2×D3)⋊(C3⋊C8)—61no—
46C22×C32×D3D411yescharacteristic
47C2×C33×C4C2311yescharacteristic
48C2×C3×C3⋊Dic3—21no—
49C2×C3×C3⋊Dic3—21no—
50C2×C32×Dic3—21no—
51C2×C32×Dic3—21no—
52C2×C32×Dic3D411yescharacteristic
53C32×C4×D3—21no—
54C32×C4×D3—21no—
55C32×Dic6—21no—
56C32×Dic6—21no—
57C3×D3×Dic3—42no—
58C3⋊(C3⋊(C3⋊C8))—21no—
59C3×C3⋊(C3⋊C8)—21no—
60#5043—91no—
61#4971—91no—
62#4805—91no—
63C2×C32×Q8—31no—
64C22×C32×C4—31no—
65C22×C3×Dic3—61no—
66C2×C3×C4×D3—21no—
67C2×C3×C4×D3—61no—
68C2×C3×C4×D3C2×D311yescharacteristic
69C2×C3×C4×D3C2×D311yescharacteristic
70C2×C3×Dic6—21no—
71C2×C3×Dic6C2×D311yescharacteristic
72C2×C3×Dic6C2×D311yescharacteristic
73C2×D3×Dic3—21no—
74C32×C4⋊C4—31no—
75C4⋊(C3⋊Dic3)C2×D311yescharacteristic
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×#217—61no—
86C3×(C2×D3)⋊C4—61no—
87C3×C4⋊Dic3—21no—
88C3×C4⋊Dic3—31no—
89C3×C4⋊Dic3—31no—
90C3×C4⋊Dic3—31no—
91C3×C4⋊Dic3—31no—
92C3×C4⋊Dic3—61no—
93C3×C4⋊Dic3—61no—
94C3×C4⋊Dic3—61no—
95C3×C4⋊Dic3C2×D311yescharacteristic
96C3×C4⋊Dic3C2×D311yescharacteristic
97C3×Dic3⋊C4—61no—
98C3×Dic3⋊C4—61no—
99C3×Dic3⋊C4—61no—
100C2×C3×C3⋊C8—31no—
101C2×C3×C3⋊C8—31no—
102C2×C3×C3⋊C8—61no—
103#3724—21no—
104Dic3⋊Dic3—21no—
105(C2×D3)⋊Dic3—21no—
106C22×C33C2×D411yescharacteristic
107C2×C32×D3—21no—
108C2×C32×D3—21no—
109C2×C32×D3SD1612yes—
110C33×C4#4111yescharacteristic
111C33×C4#4111yescharacteristic
112C3×C3⋊Dic3—41no—
113C3×C3⋊Dic3—41no—
114C32×Dic3—21no—
115C32×Dic3—41no—
116C32×Dic3—41no—
117C3×C22⋊Q8—91no—
118C2×C3×C4⋊C4—91no—
119#946—31no—
120C2×C4⋊Dic3—31no—
121#924—31no—
122D3×C4⋊C4—31no—
123C3×#103—91no—
124#835—31no—
125#835—91no—
126#835—91no—
127#835—181no—
128#830—91no—
129#830—91no—
130#830—181no—
131(C2×D3)⋊C8—91no—
132#817—91no—
133#810—31no—
134#810—91no—
135#808—91no—
136#808—91no—
137#808—91no—
138#808—91no—
139#808—91no—
140#808—181no—
141C23×C32—31no—
142C22×C3×D3—21no—
143C22×C3×D3#6711yescharacteristic
144C22×C3×D3#6711yescharacteristic
145C32×Q8—61no—
146C32×Q8—61no—
147C2×C32×C4—21no—
148C2×C32×C4—21no—
149C2×C32×C4—21no—
150C2×C32×C4—31no—
151C2×C32×C4—41no—
152C2×C32×C4—61no—
153C2×C32×C4—61no—
154C2×C32×C4—61no—
155C2×C32×C4—61no—
156C2×C32×C4C22×D311yescharacteristic
157C2×C32×C4C22×D311yescharacteristic
158C2×C32×C4C22×D311yescharacteristic
159C2×C3⋊Dic3—21no—
160C2×C3⋊Dic3—21no—
161C2×C3×Dic3—21no—
162C2×C3×Dic3—21no—
163C2×C3×Dic3—21no—
164C2×C3×Dic3—21no—
165C2×C3×Dic3—21no—
166C2×C3×Dic3—41no—
167C2×C3×Dic3—41no—
168C2×C3×Dic3—61no—
169C2×C3×Dic3—61no—
170C2×C3×Dic3—61no—
171C2×C3×Dic3—61no—
172C2×C3×Dic3—121no—
173C2×C3×Dic3—121no—
174C2×C3×Dic3—122no—
175C2×C3×Dic3#6711yescharacteristic
176C2×C3×Dic3#6711yescharacteristic
177C3×C4×D3—21no—
178C3×C4×D3—21no—
179C3×C4×D3—21no—
180C3×C4×D3—21no—
181C3×C4×D3—41no—
182C3×C4×D3—41no—
183C3×C4×D3—122no—
184C3×Dic6—22no—
185C3×Dic6—21no—
186C3×Dic6—21no—
187C3×Dic6—21no—
188C3×Dic6—21no—
189C3×Dic6—41no—
190D3×Dic3—42no—
191C3⋊(C3⋊C8)—21no—
192C3⋊(C3⋊C8)—61no—
193C3⋊(C3⋊C8)—62no—
194C3⋊(C3⋊C8)—61no—
195C3⋊(C3⋊C8)—61no—
196C3⋊(C3⋊C8)—121no—
197C3⋊(C3⋊C8)—121no—
198C3⋊(C3⋊C8)—122no—
199C3×C3⋊C8—61no—
200C3×C3⋊C8—61no—
201C3×C3⋊C8—121no—
202#483—271noSylow
203C2×C33#12311yescharacteristic
204C2×C33C2×SD1611yescharacteristic
205C2×C33#13711yescharacteristic
206C32×D3—22no—
207C2×C3×Q8—31no—
208C2×C3×Q8—31no—
209C2×C3×Q8—61no—
210C22×C3×C4—31no—
211C22×C3×C4—31no—
212C22×C3×C4—61no—
213C22×C3×C4—181no—
214C22×Dic3—61no—
215C2×C4×D3—61no—
216C2×C4×D3D3211yescharacteristic
217C2×Dic6D3211yescharacteristic
218C2×C3×C8—91no—
219C3×C4⋊C4—31no—
220C3×C4⋊C4—31no—
221C3×C4⋊C4—61no—
222C3×C4⋊C4—91no—
223C3×C4⋊C4—91no—
224C3×C4⋊C4—181no—
225C3×C4⋊C4—181no—
226C3×C22⋊C4—181no—
227#217—61no—
228(C2×D3)⋊C4—61no—
229C4⋊Dic3—31no—
230C4⋊Dic3—31no—
231C4⋊Dic3—31no—
232C4⋊Dic3—31no—
233C4⋊Dic3—61no—
234C4⋊Dic3—61no—
235C4⋊Dic3—61no—
236C4⋊Dic3D3211yescharacteristic
237Dic3⋊C4—61no—
238Dic3⋊C4—61no—
239Dic3⋊C4—61no—
240C2×C3⋊C8—31no—
241C2×C3⋊C8—31no—
242C2×C3⋊C8—61no—
243C2×C3⋊C8—91no—
244C2×C3⋊C8—91no—
245C2×C3⋊C8—91no—
246C2×C3⋊C8—181no—
247C2×C3⋊C8—181no—
248C2×C3⋊C8—181no—
249C2×C3⋊C8—361no—
250C22×C32—21no—
251C22×C32—21no—
252C22×C32—21no—
253C22×C32—32no—
254C22×C32—41no—
255C22×C32—61no—
256C22×C32—61no—
257C22×C32D3×D411yescharacteristic
258C22×C32C2×#6711yescharacteristic
259C22×C32C2×#6711yescharacteristic
260C2×C3×D3—21no—
261C2×C3×D3—21no—
262C2×C3×D3—21no—
263C2×C3×D3—21no—
264C2×C3×D3—22no—
265C2×C3×D3—41no—
266C2×C3×D3—41no—
267C2×C3×D3#21412yes—
268C2×C3×D3#21512yes—
269C32×C4—21no—
270C32×C4—21no—
271C32×C4—21no—
272C32×C4—21no—
273C32×C4—21no—
274C32×C4—21no—
275C32×C4—41no—
276C32×C4—41no—
277C32×C4—61no—
278C32×C4—121no—
279C32×C4—121no—
280C32×C4#23511yescharacteristic
281C32×C4#23511yescharacteristic
282C32×C4#23511yescharacteristic
283C32×C4#23711yescharacteristic
284C32×C4#23711yescharacteristic
285C32×C4#23711yescharacteristic
286C3⋊Dic3—41no—
287C3⋊Dic3—41no—
288C3×Dic3—21no—
289C3×Dic3—21no—
290C3×Dic3—41no—
291C3×Dic3—42no—
292C3×Dic3—41no—
293C3×Dic3—41no—
294C3×Dic3—41no—
295C3×Dic3—41no—
296C3×Dic3—81no—
297C3×Dic3—121no—
298C3×Dic3—121no—
299C3×Dic3—121no—
300C3×Dic3—121no—
301C3×Dic3—241no—
302C3×Dic3—241no—
303C22⋊Q8—91no—
304C2×C4⋊C4—91no—
305#106—271no—
306#106—271no—
307#103—91no—
308#103—271no—
309C22⋊C8—271no—
310C33#48311yescharacteristic, complemented, Sylow
311C23×C3—31no—
312C23×C3—31no—
313C23×C3—61no—
314C22×D3#62511yescharacteristic
315C3×Q8—62no—
316C3×Q8—61no—
317C3×Q8—61no—
318C3×Q8—61no—
319C3×Q8—61no—
320C3×Q8—121no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—31no—
325C2×C3×C4—31no—
326C2×C3×C4—41no—
327C2×C3×C4—61no—
328C2×C3×C4—61no—
329C2×C3×C4—61no—
330C2×C3×C4—61no—
331C2×C3×C4—61no—
332C2×C3×C4—61no—
333C2×C3×C4—61no—
334C2×C3×C4—61no—
335C2×C3×C4—61no—
336C2×C3×C4—121no—
337C2×C3×C4—121no—
338C2×C3×C4—121no—
339C2×C3×C4—121no—
340C2×C3×C4—181no—
341C2×C3×C4—181no—
342C2×C3×C4—362no—
343C2×C3×C4C2×D3211yescharacteristic
344C2×C3×C4C2×D3211yescharacteristic
345C2×C3×C4C2×D3211yescharacteristic
346C2×Dic3—21no—
347C2×Dic3—21no—
348C2×Dic3—61no—
349C2×Dic3—61no—
350C2×Dic3—61no—
351C2×Dic3—61no—
352C2×Dic3—121no—
353C2×Dic3—121no—
354C2×Dic3—122no—
355C2×Dic3#62511yescharacteristic
356C4×D3—21no—
357C4×D3—21no—
358C4×D3—122no—
359Dic6—21no—
360Dic6—21no—
361C3×C8—181no—
362C3⋊C8—61no—
363C3⋊C8—61no—
364C3⋊C8—121no—
365C3⋊C8—182no—
366C3⋊C8—181no—
367C3⋊C8—181no—
368C3⋊C8—181no—
369C3⋊C8—361no—
370C3⋊C8—361no—
371C3⋊C8—362no—
372C2×C32—21no—
373C2×C32—21no—
374C2×C32—21no—
375C2×C32—21no—
376C2×C32—21no—
377C2×C32—21no—
378C2×C32—21no—
379C2×C32—21no—
380C2×C32—21no—
381C2×C32—41no—
382C2×C32—41no—
383C2×C32—41no—
384C2×C32—62no—
385C2×C32#88311yescharacteristic
386C2×C32D3×SD1611yescharacteristic
387C2×C32#91811yescharacteristic
388C2×C32#92911yescharacteristic
389C2×C32C2×#21411yescharacteristic
390C2×C32#93511yescharacteristic
391C2×C32C2×#21511yescharacteristic
392C2×C32#94211yescharacteristic
393C2×C32#95111yescharacteristic
394C3×D3—22no—
395C3×D3—22no—
396C3×D3—42no—
397C2×Q8—31no—
398C22×C4—31no—
399C22×C4—181no—
400C2×C8—91no—
401C2×C8—271no—
402C4⋊C4—31no—
403C4⋊C4—91no—
404C4⋊C4—91no—
405C4⋊C4—181no—
406C4⋊C4—181no—
407C22⋊C4—181no—
408C22×C3—21no—
409C22×C3—21no—
410C22×C3—21no—
411C22×C3—32no—
412C22×C3—32no—
413C22×C3—41no—
414C22×C3—62no—
415C22×C3—61no—
416C22×C3—61no—
417C22×C3—61no—
418C22×C3—61no—
419C22×C3—121no—
420C22×C3—121no—
421C22×C3C2×#62511yescharacteristic
422C22×C3D3×#6711yescharacteristic
423C22×C3D3×#6711yescharacteristic
424C2×D3—21no—
425C2×D3—21no—
426C2×D3#371512yes—
427C3×C4—21no—
428C3×C4—21no—
429C3×C4—21no—
430C3×C4—21no—
431C3×C4—21no—
432C3×C4—21no—
433C3×C4—41no—
434C3×C4—41no—
435C3×C4—61no—
436C3×C4—61no—
437C3×C4—121no—
438C3×C4—122no—
439C3×C4—121no—
440C3×C4—121no—
441C3×C4—121no—
442C3×C4—121no—
443C3×C4—241no—
444C3×C4—361no—
445C3×C4—361no—
446C3×C4#379511yescharacteristic
447C3×C4#379811yescharacteristic
448C3×C4#380411yescharacteristic
449C3×C4#380411yescharacteristic
450C3×C4#380411yescharacteristic
451C3×C4#380511yescharacteristic
452Dic3—21no—
453Dic3—41no—
454Dic3—41no—
455Dic3—121no—
456Dic3—121no—
457Dic3—121no—
458Dic3—121no—
459Dic3—241no—
460Dic3—241no—
461C32—21no—
462C32—21no—
463C32—21no—
464C32—41no—
465C32#480511yescharacteristic, complemented
466C32#497111yescharacteristic, complemented
467C32#504311yescharacteristic, complemented
468C23—31no—
469Q8—61no—
470Q8—61no—
471C2×C4—31no—
472C2×C4—61no—
473C2×C4—61no—
474C2×C4—61no—
475C2×C4—61no—
476C2×C4—181no—
477C2×C4—181no—
478C2×C4—362no—
479C2×C4D3311yescharacteristic
480C8—181no—
481C8—541no—
482C2×C3—21no—
483C2×C3—21no—
484C2×C3—21no—
485C2×C3—21no—
486C2×C3—21no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—41no—
492C2×C3—41no—
493C2×C3—41no—
494C2×C3—62no—
495C2×C3—62no—
496C2×C3—122no—
497C2×C3C2×#371511yescharacteristic
498C2×C3#6382711yescharacteristic
499C2×C3#6390511yescharacteristic
500C2×C3D3×#21411yescharacteristic
501C2×C3D3×#21511yescharacteristic
502C2×C3#6394311yescharacteristic
503C2×C3#6394811yescharacteristic
504C2×C3#6396111yescharacteristic
505C2×C3#6396911yescharacteristic
506D3—22no—
507C22—32no—
508C22—61no—
509C22—61no—
510C22D3×#62511yescharacteristic
511C4—61no—
512C4—121no—
513C4—121no—
514C4—361no—
515C4—361no—
516C4#8895911yescharacteristic
517C4#8896111yescharacteristic
518C3—21no—
519C3—21no—
520C3—21no—
521C3—41no—
522C3#1059051611yescharacteristic, complemented
523C3#1059110111yescharacteristic, complemented
524C3#1059154911yescharacteristic, complemented
525C2—62no—
526C2D3×#371511yescharacteristic
527C2#1172288311yescharacteristic
528C2#1172298511yescharacteristic
subgroupG/H
centerC22D3×#625
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC22×C32×D3D4
hypocenterC33#483
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
3. #11722079C2semidirect
4. #11722053C2semidirect
5. #11722037C2semidirect
7. C3×#63565C2semidirect
42. #63565D3semidirect
310. C33#483semidirect
465. C32#4805semidirect
466. C32#4971semidirect
467. C32#5043semidirect
522. C3#10590516semidirect
523. C3#10591101semidirect
524. C3#10591549semidirect

Maximal subgroups

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

subgroupindex
C3×D3×C4⋊Dic32
C3×#638562
#117220792
#117220532
#117220372
(C2×D3)⋊(C3⋊(C3⋊C8))2
C3×#635652
#105915493
#105911013
#105905163