Finite Group Databasebeta

← #13156010

Subgroups — #13156010

Order 1728.

650 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723691C211yesmaximal, characteristic
2C2×#88793C211yesmaximal, characteristic
3C3×#63873C211yesmaximal, characteristic, complemented
4C2×C3×#3717C211yesmaximal, characteristic
5#11722051C211yesmaximal, characteristic, complemented
6#11722037C211yesmaximal, characteristic, complemented
7C3×#63566C211yesmaximal, characteristic, complemented
8#10591565—31nomaximal
9#10591564—31nomaximal
10#10590748—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×#6294—21no—
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C3×C4×C3⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88814—21no—
20#88793—22no—
21C3×#3724—21no—
22C3×#3724—21no—
23C3×#3717—22no—
24C2×#3755—31no—
25C3×#944—31no—
26C3×#944—31no—
27C2×C3×#215—31no—
28C2×C3×#204—31no—
29#63898—31no—
30#63894—31no—
31Dic3×D12—31no—
32#63873D311yescharacteristic, complemented
33C2×#3717D311yescharacteristic
34C2×#3715—31no—
35C2×#3715—31no—
36C2×#3715—61no—
37#63662—31no—
38#63641—31no—
39C3×#835—31no—
40C3×#816—31no—
41Dic3⋊(C3⋊C8)—31no—
42Dic3⋊(C3⋊C8)—31no—
43Dic3⋊(C3⋊C8)—61no—
44#63566D311yescharacteristic, complemented
45#63565—31no—
46#63565—31no—
47#63565—61no—
48C22×C3×C3⋊D3—21no—
49C2×C33×C4C2311yescharacteristic
50C2×C3×C3⋊Dic3—21no—
51C2×C3×C3⋊Dic3D411yescharacteristic
52C2×C3×C3⋊Dic3D411yescharacteristic
53C3×C4⋊(C3⋊D3)—21no—
54C3×C4⋊(C3⋊D3)D412yes—
55C2×C32×Dic3—21no—
56C2×C32×Dic3—21no—
57C32×Dic6—21no—
58C32×Dic6—21no—
59C32×Dic6—21no—
60C32×Dic6—21no—
61#6294—42no—
62C3⋊(C3⋊(C3⋊C8))—21no—
63C32×C3⋊C8—21no—
64#5162—91no—
65#5083—91no—
66#4818—91no—
67C2×C32×Q8—31no—
68C2×C32×Q8—31no—
69C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
70C2×C3×D12—31no—
71C2×C3×D12—31no—
72C2×C3×D12—61no—
73C2×C3×Dic6—21no—
74C2×C3×Dic6—21no—
75C2×C3×Dic6C2×D311yescharacteristic
76C2×C3×Dic6C2×D311yescharacteristic
77C2×C3×Dic6C2×D311yescharacteristic
78C2×C3×Dic6C2×D311yescharacteristic
79C2×#624—61no—
80C2×D3×Dic3—61no—
81C2×D3×Dic3—61no—
82C2×C32×C8—31no—
83#3755—62no—
84C4⋊(C3⋊Dic3)C2×D311yescharacteristic
85C4×C3⋊Dic3C2×D311yescharacteristic
86C2×C3⋊(C3⋊C8)—31no—
87C2×C3⋊(C3⋊C8)—31no—
88C2×C3⋊(C3⋊C8)—31no—
89C2×C3⋊(C3⋊C8)—61no—
90C2×C3⋊(C3⋊C8)—61no—
91C2×C3⋊(C3⋊C8)—61no—
92C2×C3⋊(C3⋊C8)—121no—
93C3×#215—62no—
94C3×C4⋊Dic3—31no—
95C3×C4⋊Dic3—31no—
96C3×C4⋊Dic3—61no—
97C3×Dic3⋊C4—61no—
98C3×Dic3⋊C4—61no—
99C3×Dic3⋊C4—61no—
100C3×Dic3⋊C4—61no—
101C3×C4×Dic3—31no—
102C3×C4×Dic3—31no—
103C3×C4×Dic3—61no—
104C2×C3×C3⋊C8—21no—
105C2×C3×C3⋊C8C2×D311yescharacteristic
106C2×C3×C3⋊C8C2×D311yescharacteristic
107C3×#204—62no—
108#3724—21no—
109#3724—21no—
110#3722—61no—
111(C2×D3)⋊Dic3—61no—
112(C2×D3)⋊Dic3—61no—
113#3717—22no—
114#3715—62no—
115#3715—62no—
116#3715—64no—
117C22×C33C2×D411yescharacteristic
118C2×C3×C3⋊D3—22no—
119C2×C3×C3⋊D3—41no—
120C2×C3×C3⋊D3—41no—
121C33×C4C2×D411yescharacteristic
122C33×C4#4111yescharacteristic
123C3×C3⋊Dic3—21no—
124C3×C3⋊Dic3—41no—
125C3×C3⋊Dic3SD1612yescomplemented
126C32×Dic3—41no—
127C32×Dic3—41no—
128C2×C3×SD16—91no—
129C3×C4⋊Q8—91no—
130#944—31no—
131#944—31no—
132C2×#215—31no—
133C2×#215—91no—
134C2×#215—91no—
135C2×#215—181no—
136D4×Dic3—91no—
137C2×#214—91no—
138C2×#204—31no—
139#893—91no—
140C4×D12—91no—
141C3×#103—91no—
142#835—31no—
143#835—91no—
144#835—91no—
145#835—181no—
146#816—31no—
147Dic3⋊C8—91no—
148Dic3⋊C8—91no—
149Dic3⋊C8—181no—
150#810—91no—
151C4⋊(C3⋊C8)—91no—
152C22×C3⋊D3—21no—
153C22×C3×D3—61no—
154C22×C3×D3—61no—
155C22×C3×D3—121no—
156C32×Q8—61no—
157C32×Q8—61no—
158C32×Q8—61no—
159C32×Q8—61no—
160C2×C32×C4—21no—
161C2×C32×C4—21no—
162C2×C32×C4—21no—
163C2×C32×C4—41no—
164C2×C32×C4—61no—
165C2×C32×C4—61no—
166C2×C32×C4C22×D311yescharacteristic
167C2×C32×C4C22×D311yescharacteristic
168C2×C32×C4C22×D311yescharacteristic
169C2×C3⋊Dic3—21no—
170C2×C3⋊Dic3#6711yescharacteristic
171C2×C3⋊Dic3#6711yescharacteristic
172C4⋊(C3⋊D3)—21no—
173C4⋊(C3⋊D3)#6712yes—
174C2×C3×Dic3—21no—
175C2×C3×Dic3—21no—
176C2×C3×Dic3—21no—
177C2×C3×Dic3—21no—
178C2×C3×Dic3—31no—
179C2×C3×Dic3—31no—
180C2×C3×Dic3—31no—
181C2×C3×Dic3—31no—
182C2×C3×Dic3—41no—
183C2×C3×Dic3—41no—
184C2×C3×Dic3—61no—
185C2×C3×Dic3—61no—
186C2×C3×Dic3—61no—
187C2×C3×Dic3—61no—
188C2×C3×Dic3—121no—
189C3×D12—32no—
190C3×D12—32no—
191C3×D12—62no—
192C3×D12—61no—
193C3×D12—61no—
194C3×D12—121no—
195C3×Dic6—22no—
196C3×Dic6—22no—
197C3×Dic6—21no—
198C3×Dic6—21no—
199C3×Dic6—21no—
200C3×Dic6—21no—
201C3×Dic6—21no—
202C3×Dic6—21no—
203C3×Dic6—21no—
204C3×Dic6—21no—
205C3×Dic6—41no—
206C3×Dic6—41no—
207#624—122no—
208D3×Dic3—122no—
209D3×Dic3—122no—
210C32×C8—61no—
211C3⋊(C3⋊C8)—61no—
212C3⋊(C3⋊C8)—62no—
213C3⋊(C3⋊C8)—61no—
214C3⋊(C3⋊C8)—61no—
215C3⋊(C3⋊C8)—121no—
216C3⋊(C3⋊C8)—121no—
217C3⋊(C3⋊C8)—122no—
218C3×C3⋊C8—21no—
219C3×C3⋊C8—21no—
220C3×C3⋊C8—41no—
221#461—271noSylow
222C2×C33#12111yescharacteristic
223C2×C33C2×SD1611yescharacteristic
224C2×C33#13711yescharacteristic
225C3×C3⋊D3—42no—
226C2×C3×Q8—31no—
227C2×C3×Q8—31no—
228C2×C3×Q8—31no—
229C2×C3×Q8—31no—
230C2×C3×Q8—61no—
231C2×C3×Q8—61no—
232C2×C3×D4—91no—
233C22×Dic3—181no—
234C2×D12—31no—
235C2×D12—31no—
236C2×D12—61no—
237C2×C4×D3—181no—
238C2×C4×D3—181no—
239C2×Dic6D3211yescharacteristic
240C2×Dic6D3211yescharacteristic
241C3×SD16—182no—
242C2×C3×C8—31no—
243C2×C3×C8—31no—
244C2×C3×C8—61no—
245C3×C4⋊C4—91no—
246C3×C4⋊C4—181no—
247C3×C4⋊C4—181no—
248C3×C42—91no—
249#217—181no—
250#215—62no—
251#215—182no—
252#215—182no—
253#215—184no—
254#214—182no—
255(C2×D3)⋊C4—181no—
256(C2×D3)⋊C4—181no—
257C4⋊Dic3—31no—
258C4⋊Dic3—31no—
259C4⋊Dic3—61no—
260Dic3⋊C4—61no—
261Dic3⋊C4—61no—
262Dic3⋊C4—61no—
263Dic3⋊C4—61no—
264C4×Dic3—31no—
265C4×Dic3—31no—
266C4×Dic3—61no—
267C2×C3⋊C8—91no—
268C2×C3⋊C8—91no—
269C2×C3⋊C8—91no—
270C2×C3⋊C8—181no—
271C2×C3⋊C8—181no—
272C2×C3⋊C8—181no—
273C2×C3⋊C8—361no—
274C2×C3⋊C8D3211yescharacteristic
275#204—62no—
276C22×C32—21no—
277C22×C32—21no—
278C22×C32—21no—
279C22×C32—41no—
280C22×C32D3×D411yescharacteristic
281C22×C32D3×D411yescharacteristic
282C22×C32C2×#6711yescharacteristic
283C2×C3⋊D3—22no—
284C2×C3⋊D3—41no—
285C2×C3⋊D3—41no—
286C2×C3×D3—62no—
287C2×C3×D3—62no—
288C2×C3×D3—122no—
289C2×C3×D3—121no—
290C2×C3×D3—121no—
291C2×C3×D3—121no—
292C2×C3×D3—121no—
293C2×C3×D3—241no—
294C2×C3×D3—241no—
295C32×C4—21no—
296C32×C4—21no—
297C32×C4—21no—
298C32×C4—21no—
299C32×C4—21no—
300C32×C4—21no—
301C32×C4—41no—
302C32×C4—41no—
303C32×C4—121no—
304C32×C4—121no—
305C32×C4#23511yescharacteristic
306C32×C4D3×D411yescharacteristic
307C32×C4D3×D411yescharacteristic
308C32×C4#23711yescharacteristic
309C32×C4#23711yescharacteristic
310C32×C4C2×#6711yescharacteristic
311C3⋊Dic3—21no—
312C3⋊Dic3—41no—
313C3⋊Dic3#21512yescomplemented
314C3×Dic3—32no—
315C3×Dic3—32no—
316C3×Dic3—42no—
317C3×Dic3—42no—
318C3×Dic3—41no—
319C3×Dic3—41no—
320C3×Dic3—41no—
321C3×Dic3—41no—
322C3×Dic3—62no—
323C3×Dic3—61no—
324C3×Dic3—61no—
325C3×Dic3—121no—
326C3×Dic3—121no—
327C3×Dic3—121no—
328C3×Dic3—241no—
329C2×SD16—91no—
330C2×SD16—271no—
331C4⋊Q8—91no—
332C4×D4—271no—
333C4⋊C8—271no—
334#103—91no—
335#103—271no—
336C33#46111yescharacteristic, complemented, Sylow
337C23×C3—181no—
338C22×D3—61no—
339C22×D3—61no—
340C22×D3—121no—
341C3×Q8—62no—
342C3×Q8—62no—
343C3×Q8—61no—
344C3×Q8—61no—
345C3×Q8—61no—
346C3×Q8—61no—
347C3×Q8—61no—
348C3×Q8—61no—
349C3×Q8—61no—
350C3×Q8—61no—
351C3×Q8—121no—
352C3×Q8—121no—
353C3×D4—92no—
354C3×D4—181no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—41no—
359C2×C3×C4—61no—
360C2×C3×C4—61no—
361C2×C3×C4—61no—
362C2×C3×C4—61no—
363C2×C3×C4—91no—
364C2×C3×C4—91no—
365C2×C3×C4—121no—
366C2×C3×C4—121no—
367C2×C3×C4—181no—
368C2×C3×C4C2×D3211yescharacteristic
369C2×C3×C4C2×D3211yescharacteristic
370C2×C3×C4C2×D3211yescharacteristic
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—31no—
374C2×Dic3—31no—
375C2×Dic3—31no—
376C2×Dic3—31no—
377C2×Dic3—61no—
378C2×Dic3—61no—
379C2×Dic3—61no—
380C2×Dic3—61no—
381C2×Dic3—121no—
382C2×Dic3—362no—
383D12—32no—
384D12—32no—
385D12—62no—
386D12—61no—
387D12—61no—
388D12—121no—
389C4×D3—362no—
390C4×D3—362no—
391Dic6—21no—
392Dic6—21no—
393Dic6—21no—
394Dic6—21no—
395C3×C8—61no—
396C3×C8—61no—
397C3×C8—121no—
398C3⋊C8—21no—
399C3⋊C8—182no—
400C3⋊C8—181no—
401C3⋊C8—181no—
402C3⋊C8—181no—
403C3⋊C8—361no—
404C3⋊C8—361no—
405C3⋊C8—362no—
406C2×C32—21no—
407C2×C32—21no—
408C2×C32—21no—
409C2×C32—21no—
410C2×C32—21no—
411C2×C32—21no—
412C2×C32—21no—
413C2×C32—21no—
414C2×C32—21no—
415C2×C32—41no—
416C2×C32—41no—
417C2×C32—41no—
418C2×C32#88411yescharacteristic
419C2×C32D3×SD1611yescharacteristic
420C2×C32D3×SD1611yescharacteristic
421C2×C32#91511yescharacteristic
422C2×C32#91811yescharacteristic
423C2×C32#93811yescharacteristic
424C2×C32#93911yescharacteristic
425C2×C32C2×#21511yescharacteristic
426C2×C32#94211yescharacteristic
427C3⋊D3—42no—
428C3×D3—122no—
429C3×D3—122no—
430C3×D3—242no—
431C2×Q8—31no—
432C2×Q8—31no—
433C2×D4—91no—
434C22×C4—541no—
435SD16—182no—
436SD16—542no—
437C2×C8—31no—
438C2×C8—271no—
439C4⋊C4—91no—
440C4⋊C4—181no—
441C4⋊C4—181no—
442C22⋊C4—541no—
443C42—91no—
444C22×C3—21no—
445C22×C3—21no—
446C22×C3—21no—
447C22×C3—41no—
448C22×C3—182no—
449C22×C3—361no—
450C22×C3—361no—
451C22×C3#380211yescharacteristic
452C22×C3D3×#6711yescharacteristic
453C22×C3D3×#6711yescharacteristic
454C2×D3—62no—
455C2×D3—62no—
456C2×D3—122no—
457C2×D3—121no—
458C2×D3—121no—
459C2×D3—121no—
460C2×D3—121no—
461C2×D3—241no—
462C2×D3—241no—
463C3×C4—21no—
464C3×C4—21no—
465C3×C4—21no—
466C3×C4—21no—
467C3×C4—21no—
468C3×C4—21no—
469C3×C4—41no—
470C3×C4—41no—
471C3×C4—92no—
472C3×C4—122no—
473C3×C4—122no—
474C3×C4—121no—
475C3×C4—121no—
476C3×C4—121no—
477C3×C4—121no—
478C3×C4—181no—
479C3×C4—361no—
480C3×C4#379511yescharacteristic
481C3×C4#380411yescharacteristic
482C3×C4#380511yescharacteristic
483C3×C4D3×#6711yescharacteristic
484C3×C4D3×#6711yescharacteristic
485C3×C4#381111yescharacteristic
486Dic3—32no—
487Dic3—32no—
488Dic3—41no—
489Dic3—41no—
490Dic3—62no—
491Dic3—61no—
492Dic3—61no—
493Dic3—121no—
494Dic3—121no—
495Dic3—121no—
496Dic3—241no—
497C32—21no—
498C32—21no—
499C32—21no—
500C32—41no—
501C32#481811yescharacteristic, complemented
502C32#508311yescharacteristic, complemented
503C32#516211yescharacteristic, complemented
504C23—181no—
505Q8—61no—
506Q8—61no—
507Q8—61no—
508Q8—61no—
509D4—92no—
510D4—181no—
511C2×C4—61no—
512C2×C4—61no—
513C2×C4—91no—
514C2×C4—91no—
515C2×C4—181no—
516C2×C4—1082no—
517C2×C4D3311yescharacteristic
518C8—61no—
519C8—541no—
520C2×C3—21no—
521C2×C3—21no—
522C2×C3—21no—
523C2×C3—21no—
524C2×C3—21no—
525C2×C3—21no—
526C2×C3—21no—
527C2×C3—21no—
528C2×C3—21no—
529C2×C3—41no—
530C2×C3—41no—
531C2×C3—41no—
532C2×C3—362no—
533C2×C3#6379611yescharacteristic
534C2×C3#6381111yescharacteristic
535C2×C3#6388511yescharacteristic
536C2×C3D3×#21511yescharacteristic
537C2×C3D3×#21511yescharacteristic
538C2×C3#6394611yescharacteristic
539C2×C3#6394811yescharacteristic
540C2×C3#6395811yescharacteristic
541C2×C3#6397011yescharacteristic
542D3—122no—
543D3—122no—
544D3—242no—
545C22—182no—
546C22—361no—
547C22—361no—
548C22#8895311yescharacteristic
549C4—92no—
550C4—121no—
551C4—121no—
552C4—181no—
553C4—361no—
554C4#8895211yescharacteristic
555C4#8896111yescharacteristic
556C3—21no—
557C3—21no—
558C3—21no—
559C3—41no—
560C3#1059074811yescharacteristic, complemented
561C3#1059156411yescharacteristic, complemented
562C3#1059156511yescharacteristic, complemented
563C2—362no—
564C2#1172284011yescharacteristic
565C2#1172289211yescharacteristic
566C2#1172296511yescharacteristic
subgroupG/H
centerC22#88953
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypocenterC33#461
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. C3×#63873C2semidirect
5. #11722051C2semidirect
6. #11722037C2semidirect
7. C3×#63566C2semidirect
32. #63873D3semidirect
44. #63566D3semidirect
125. C3×C3⋊Dic3SD16semidirect
313. C3⋊Dic3#215semidirect
336. C33#461semidirect
501. C32#4818semidirect
502. C32#5083semidirect
503. C32#5162semidirect
560. C3#10590748semidirect
561. C3#10591564semidirect
562. C3#10591565semidirect

Maximal subgroups

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

subgroupindex
#117236912
C2×#887932
C3×#638732
C2×C3×#37172
#117220512
#117220372
C3×#635662
#105915653
#105915643
#105907483