Finite Group Databasebeta

← #13156411

Subgroups — #13156411

Order 1728.

614 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#11723689C211yesmaximal, characteristic, complemented
2C3×#63916C211yesmaximal, characteristic
3#11722053C211yesmaximal, characteristic, complemented
4#11722031C211yesmaximal, characteristic, complemented
5#11722023C211yesmaximal, characteristic, complemented
6C3×#63573C211yesmaximal, characteristic, complemented
7C3×#63563C211yesmaximal, characteristic, complemented
8#10591415—31nomaximal
9#10590649—31nomaximal
10#10590441—31nomaximal
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×#6294—21no—
14C2×C3×#626—21no—
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20#88815—21no—
21C3×(C2×D3)⋊Dic3—21no—
22C3×#938—31no—
23C3×#895—31no—
24#63916D311yescharacteristic
25#63900—31no—
26D3×C4⋊Dic3—31no—
27D3×C4⋊Dic3—31no—
28#63659—31no—
29#63647—31no—
30#63635—31no—
31C3×#832—31no—
32C3×#821—31no—
33C3×#817—31no—
34C3×#808—31no—
35#63574—31no—
36#63574—31no—
37#63574—61no—
38#63573D311yescharacteristic, complemented
39#63563D311yescharacteristic, complemented
40#63562—31no—
41#63562—31no—
42#63562—61no—
43(C2×D3)⋊(C3⋊C8)—31no—
44(C2×D3)⋊(C3⋊C8)—31no—
45(C2×D3)⋊(C3⋊C8)—61no—
46C22×C3×C3⋊D3D411yescharacteristic
47C22×C32×D3—21no—
48C2×C33×C4C2311yescharacteristic
49C2×C3×C3⋊Dic3—21no—
50C2×C3×C3⋊Dic3—21no—
51C2×C3×C3⋊Dic3D411yescharacteristic
52C3×C4×C3⋊D3—21no—
53C3×C4×C3⋊D3—21no—
54C2×C32×Dic3—21no—
55C32×D12—21no—
56C32×D12—21no—
57#6294—42no—
58C3×#626—42no—
59C3⋊(C3⋊(C3⋊C8))—21no—
60C32×C3⋊C8—21no—
61#5034—91no—
62#4863—91no—
63#4777—91no—
64C2×C32×D4—31no—
65C2×C4×C3⋊D3C2×D311yescharacteristic
66C2×C3×#67—61no—
67C2×C3×D12—21no—
68C2×C3×D12—61no—
69C2×C3×D12C2×D311yescharacteristic
70C2×C3×D12C2×D311yescharacteristic
71C2×C3×C4×D3—31no—
72C2×C3×C4×D3—31no—
73C2×C3×C4×D3—61no—
74C2×#626—21no—
75C2×#624—61no—
76C2×D3×Dic3—61no—
77C2×D3×Dic3—61no—
78C2×C32×C8—31no—
79C32×C4⋊C4—31no—
80C4⋊(C3⋊Dic3)C2×D311yescharacteristic
81C4⋊(C3⋊Dic3)C2×D311yescharacteristic
82C2×C3⋊(C3⋊C8)—31no—
83C2×C3⋊(C3⋊C8)—31no—
84C2×C3⋊(C3⋊C8)—31no—
85C2×C3⋊(C3⋊C8)—61no—
86C2×C3⋊(C3⋊C8)—61no—
87C2×C3⋊(C3⋊C8)—61no—
88C2×C3⋊(C3⋊C8)—121no—
89C3×#217—61no—
90C3×(C2×D3)⋊C4—61no—
91C3×C4⋊Dic3—21no—
92C3×C4⋊Dic3—31no—
93C3×C4⋊Dic3—31no—
94C3×C4⋊Dic3—31no—
95C3×C4⋊Dic3—31no—
96C3×C4⋊Dic3—61no—
97C3×C4⋊Dic3—61no—
98C3×C4⋊Dic3C2×D311yescharacteristic
99C3×C4⋊Dic3C2×D311yescharacteristic
100C2×C3×C3⋊C8—21no—
101C2×C3×C3⋊C8C2×D311yescharacteristic
102C2×C3×C3⋊C8C2×D311yescharacteristic
103#3724—61no—
104Dic3⋊Dic3—61no—
105Dic3⋊Dic3—61no—
106(C2×D3)⋊Dic3—21no—
107C22×C33C2×D411yescharacteristic
108C2×C3×C3⋊D3—21no—
109C2×C3×C3⋊D3—21no—
110C2×C3×C3⋊D3SD1612yes—
111C2×C32×D3—41no—
112C2×C32×D3—41no—
113C2×C32×D3—41no—
114C33×C4#4111yescharacteristic
115C33×C4#4111yescharacteristic
116C3×C3⋊Dic3—21no—
117C3×C3⋊Dic3—41no—
118C3×C3⋊Dic3—41no—
119C32×Dic3—41no—
120C3×#121—91no—
121#938—31no—
122C2×C4⋊Dic3—91no—
123#895—31no—
124D3×C4⋊C4—91no—
125D3×C4⋊C4—91no—
126C3×#106—91no—
127C3×#102—91no—
128#832—31no—
129#832—91no—
130#832—91no—
131#832—181no—
132#830—91no—
133#821—31no—
134(C2×D3)⋊C8—91no—
135(C2×D3)⋊C8—91no—
136(C2×D3)⋊C8—181no—
137#817—31no—
138#809—91no—
139#808—31no—
140#808—91no—
141#808—91no—
142#808—91no—
143#808—181no—
144C23×C32—61no—
145C22×C3⋊D3#6711yescharacteristic
146C22×C3×D3—21no—
147C22×C3×D3—21no—
148C22×C3×D3—31no—
149C22×C3×D3—31no—
150C22×C3×D3—41no—
151C22×C3×D3—61no—
152C32×D4—61no—
153C32×D4—61no—
154C2×C32×C4—21no—
155C2×C32×C4—21no—
156C2×C32×C4—21no—
157C2×C32×C4—41no—
158C2×C32×C4—61no—
159C2×C32×C4C22×D311yescharacteristic
160C2×C32×C4C22×D311yescharacteristic
161C2×C32×C4C22×D311yescharacteristic
162C2×C3⋊Dic3—21no—
163C2×C3⋊Dic3—21no—
164C2×C3⋊Dic3#6711yescharacteristic
165C4×C3⋊D3—21no—
166C4×C3⋊D3—21no—
167C3×#67—122no—
168C2×C3×Dic3—21no—
169C2×C3×Dic3—21no—
170C2×C3×Dic3—31no—
171C2×C3×Dic3—31no—
172C2×C3×Dic3—41no—
173C2×C3×Dic3—61no—
174C2×C3×Dic3—61no—
175C2×C3×Dic3—61no—
176C2×C3×Dic3—61no—
177C2×C3×Dic3—61no—
178C2×C3×Dic3—121no—
179C2×C3×Dic3—121no—
180C3×D12—22no—
181C3×D12—21no—
182C3×D12—21no—
183C3×D12—21no—
184C3×D12—21no—
185C3×D12—41no—
186C3×D12—122no—
187C3×C4×D3—61no—
188C3×C4×D3—61no—
189C3×C4×D3—61no—
190C3×C4×D3—61no—
191C3×C4×D3—121no—
192C3×C4×D3—121no—
193#626—42no—
194#624—122no—
195D3×Dic3—122no—
196D3×Dic3—122no—
197C32×C8—61no—
198C3⋊(C3⋊C8)—61no—
199C3⋊(C3⋊C8)—62no—
200C3⋊(C3⋊C8)—61no—
201C3⋊(C3⋊C8)—61no—
202C3⋊(C3⋊C8)—121no—
203C3⋊(C3⋊C8)—121no—
204C3⋊(C3⋊C8)—122no—
205C3×C3⋊C8—21no—
206C3×C3⋊C8—21no—
207C3×C3⋊C8—41no—
208#481—271noSylow
209C2×C33#12311yescharacteristic
210C2×C33C2×SD1611yescharacteristic
211C2×C33AGL(1,ℤ/8)11yescharacteristic
212C3×C3⋊D3—22no—
213C32×D3—81no—
214C2×C3×D4—31no—
215C2×C3×D4—31no—
216C2×C3×D4—61no—
217C2×C3×D4—181no—
218C22×C3×C4—91no—
219C2×#67—61no—
220C22×Dic3—181no—
221C2×D12—61no—
222C2×D12D3211yescharacteristic
223C2×C4×D3—31no—
224C2×C4×D3—31no—
225C2×C4×D3—61no—
226C2×C4×D3—181no—
227C2×C4×D3—181no—
228C2×C3×C8—31no—
229C2×C3×C8—31no—
230C2×C3×C8—61no—
231C3×C4⋊C4—31no—
232C3×C4⋊C4—31no—
233C3×C4⋊C4—61no—
234C3×C4⋊C4—91no—
235C3×C4⋊C4—91no—
236C3×C22⋊C4—181no—
237#217—61no—
238(C2×D3)⋊C4—61no—
239C4⋊Dic3—31no—
240C4⋊Dic3—31no—
241C4⋊Dic3—31no—
242C4⋊Dic3—31no—
243C4⋊Dic3—61no—
244C4⋊Dic3—61no—
245C4⋊Dic3—181no—
246C4⋊Dic3D3211yescharacteristic
247Dic3⋊C4—181no—
248Dic3⋊C4—181no—
249C2×C3⋊C8—91no—
250C2×C3⋊C8—91no—
251C2×C3⋊C8—91no—
252C2×C3⋊C8—181no—
253C2×C3⋊C8—181no—
254C2×C3⋊C8—181no—
255C2×C3⋊C8—361no—
256C2×C3⋊C8D3211yescharacteristic
257C22×C32—21no—
258C22×C32—21no—
259C22×C32—21no—
260C22×C32—41no—
261C22×C32—121no—
262C22×C32—121no—
263C22×C32—121no—
264C22×C32D3×D411yescharacteristic
265C22×C32D3×D411yescharacteristic
266C22×C32C2×#6711yescharacteristic
267C2×C3⋊D3—21no—
268C2×C3⋊D3—21no—
269C2×C3⋊D3#21412yes—
270C2×C3×D3—32no—
271C2×C3×D3—32no—
272C2×C3×D3—42no—
273C2×C3×D3—41no—
274C2×C3×D3—41no—
275C2×C3×D3—41no—
276C2×C3×D3—41no—
277C2×C3×D3—41no—
278C2×C3×D3—41no—
279C2×C3×D3—62no—
280C2×C3×D3—61no—
281C2×C3×D3—61no—
282C2×C3×D3—61no—
283C2×C3×D3—61no—
284C2×C3×D3—81no—
285C2×C3×D3—81no—
286C2×C3×D3—121no—
287C2×C3×D3—121no—
288C32×C4—21no—
289C32×C4—21no—
290C32×C4—21no—
291C32×C4—21no—
292C32×C4—21no—
293C32×C4—21no—
294C32×C4—41no—
295C32×C4—41no—
296C32×C4—121no—
297C32×C4#23511yescharacteristic
298C32×C4#23511yescharacteristic
299C32×C4#23711yescharacteristic
300C32×C4#23711yescharacteristic
301C32×C4#23711yescharacteristic
302C32×C4#23911yescharacteristic
303C3⋊Dic3—21no—
304C3⋊Dic3—41no—
305C3⋊Dic3—41no—
306C3×Dic3—41no—
307C3×Dic3—41no—
308C3×Dic3—61no—
309C3×Dic3—61no—
310C3×Dic3—81no—
311C3×Dic3—121no—
312C3×Dic3—121no—
313C3×Dic3—121no—
314C3×Dic3—121no—
315C3×Dic3—121no—
316C3×Dic3—241no—
317C3×Dic3—241no—
318#121—91no—
319C2×C4⋊C4—271no—
320#106—91no—
321#106—271no—
322#102—91no—
323#102—271no—
324C22⋊C8—271no—
325C33#48111yescharacteristic, complemented, Sylow
326C23×C3—61no—
327C23×C3—61no—
328C23×C3—91no—
329C23×C3—121no—
330C22×D3—21no—
331C22×D3—31no—
332C22×D3—31no—
333C22×D3—61no—
334C3×D4—62no—
335C3×D4—61no—
336C3×D4—61no—
337C3×D4—61no—
338C3×D4—61no—
339C3×D4—121no—
340C3×D4—362no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—41no—
345C2×C3×C4—61no—
346C2×C3×C4—61no—
347C2×C3×C4—91no—
348C2×C3×C4—121no—
349C2×C3×C4—181no—
350C2×C3×C4—181no—
351C2×C3×C4—181no—
352C2×C3×C4—181no—
353C2×C3×C4C2×D3211yescharacteristic
354C2×C3×C4C2×D3211yescharacteristic
355C2×C3×C4C2×D3211yescharacteristic
356#67—122no—
357C2×Dic3—21no—
358C2×Dic3—31no—
359C2×Dic3—31no—
360C2×Dic3—61no—
361C2×Dic3—61no—
362C2×Dic3—61no—
363C2×Dic3—61no—
364C2×Dic3—61no—
365C2×Dic3—121no—
366C2×Dic3—121no—
367C2×Dic3—362no—
368D12—21no—
369D12—21no—
370D12—122no—
371C4×D3—61no—
372C4×D3—61no—
373C4×D3—61no—
374C4×D3—61no—
375C4×D3—121no—
376C4×D3—121no—
377C4×D3—362no—
378C4×D3—362no—
379C3×C8—61no—
380C3×C8—61no—
381C3×C8—121no—
382C3⋊C8—21no—
383C3⋊C8—182no—
384C3⋊C8—181no—
385C3⋊C8—181no—
386C3⋊C8—181no—
387C3⋊C8—361no—
388C3⋊C8—361no—
389C3⋊C8—362no—
390C2×C32—21no—
391C2×C32—21no—
392C2×C32—21no—
393C2×C32—21no—
394C2×C32—21no—
395C2×C32—21no—
396C2×C32—21no—
397C2×C32—21no—
398C2×C32—21no—
399C2×C32—41no—
400C2×C32—41no—
401C2×C32—41no—
402C2×C32—241no—
403C2×C32#88311yescharacteristic
404C2×C32#89411yescharacteristic
405C2×C32#91111yescharacteristic
406C2×C32D3×SD1611yescharacteristic
407C2×C32D3×SD1611yescharacteristic
408C2×C32#91411yescharacteristic
409C2×C32C2×#21411yescharacteristic
410C2×C32#93511yescharacteristic
411C2×C32#94911yescharacteristic
412C3⋊D3—22no—
413C3×D3—62no—
414C3×D3—62no—
415C3×D3—82no—
416C3×D3—81no—
417C3×D3—81no—
418C3×D3—122no—
419C2×D4—31no—
420C2×D4—181no—
421C22×C4—91no—
422C22×C4—541no—
423C2×C8—31no—
424C2×C8—271no—
425C4⋊C4—31no—
426C4⋊C4—91no—
427C4⋊C4—91no—
428C4⋊C4—541no—
429C22⋊C4—181no—
430C22×C3—21no—
431C22×C3—21no—
432C22×C3—21no—
433C22×C3—41no—
434C22×C3—92no—
435C22×C3—122no—
436C22×C3—121no—
437C22×C3—121no—
438C22×C3—121no—
439C22×C3—121no—
440C22×C3—121no—
441C22×C3—121no—
442C22×C3—181no—
443C22×C3—181no—
444C22×C3—241no—
445C22×C3—241no—
446C22×C3#380211yescharacteristic
447C22×C3D3×#6711yescharacteristic
448C22×C3D3×#6711yescharacteristic
449C2×D3—32no—
450C2×D3—32no—
451C2×D3—41no—
452C2×D3—41no—
453C2×D3—41no—
454C2×D3—62no—
455C2×D3—61no—
456C2×D3—61no—
457C2×D3—61no—
458C2×D3—61no—
459C2×D3—121no—
460C2×D3—121no—
461C3×C4—21no—
462C3×C4—21no—
463C3×C4—21no—
464C3×C4—21no—
465C3×C4—21no—
466C3×C4—21no—
467C3×C4—41no—
468C3×C4—41no—
469C3×C4—121no—
470C3×C4—121no—
471C3×C4—181no—
472C3×C4—241no—
473C3×C4—361no—
474C3×C4—361no—
475C3×C4#379511yescharacteristic
476C3×C4#379611yescharacteristic
477C3×C4#379911yescharacteristic
478C3×C4#380411yescharacteristic
479C3×C4#380411yescharacteristic
480C3×C4#380511yescharacteristic
481Dic3—41no—
482Dic3—61no—
483Dic3—61no—
484Dic3—121no—
485Dic3—121no—
486Dic3—121no—
487Dic3—121no—
488Dic3—121no—
489Dic3—241no—
490Dic3—241no—
491C32—21no—
492C32—21no—
493C32—21no—
494C32—41no—
495C32#477711yescharacteristic, complemented
496C32#486311yescharacteristic, complemented
497C32#503411yescharacteristic, complemented
498C23—61no—
499C23—91no—
500D4—61no—
501D4—61no—
502D4—362no—
503C2×C4—61no—
504C2×C4—91no—
505C2×C4—181no—
506C2×C4—181no—
507C2×C4—181no—
508C2×C4—181no—
509C2×C4—1082no—
510C2×C4D3311yescharacteristic
511C8—61no—
512C8—541no—
513C2×C3—21no—
514C2×C3—21no—
515C2×C3—21no—
516C2×C3—21no—
517C2×C3—21no—
518C2×C3—21no—
519C2×C3—21no—
520C2×C3—21no—
521C2×C3—21no—
522C2×C3—41no—
523C2×C3—41no—
524C2×C3—41no—
525C2×C3—182no—
526C2×C3—242no—
527C2×C3—241no—
528C2×C3—241no—
529C2×C3#6379611yescharacteristic
530C2×C3#6379811yescharacteristic
531C2×C3#6385411yescharacteristic
532C2×C3#6388411yescharacteristic
533C2×C3#6392511yescharacteristic
534C2×C3D3×#21411yescharacteristic
535C2×C3D3×#21411yescharacteristic
536C2×C3#6393911yescharacteristic
537C2×C3#6396911yescharacteristic
538D3—62no—
539D3—62no—
540D3—81no—
541D3—122no—
542C22—92no—
543C22—121no—
544C22—121no—
545C22—121no—
546C22—181no—
547C22—181no—
548C22#8895311yescharacteristic
549C4—121no—
550C4—181no—
551C4—361no—
552C4—361no—
553C4#8896111yescharacteristic
554C4#8896211yescharacteristic
555C3—21no—
556C3—21no—
557C3—21no—
558C3—41no—
559C3#1059044111yescharacteristic, complemented
560C3#1059064911yescharacteristic, complemented
561C3#1059141511yescharacteristic, complemented
562C2—182no—
563C2—241no—
564C2#1172283911yescharacteristic
565C2#1172287611yescharacteristic
566C2#1172299411yescharacteristic
subgroupG/H
centerC22#88953
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC3×#63916C2
hypocenterC33#481
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
1. #11723689C2semidirect
3. #11722053C2semidirect
4. #11722031C2semidirect
5. #11722023C2semidirect
6. C3×#63573C2semidirect
7. C3×#63563C2semidirect
38. #63573D3semidirect
39. #63563D3semidirect
325. C33#481semidirect
495. C32#4777semidirect
496. C32#4863semidirect
497. C32#5034semidirect
559. C3#10590441semidirect
560. C3#10590649semidirect
561. C3#10591415semidirect

Maximal subgroups

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

subgroupindex
#117236892
C3×#639162
#117220532
#117220312
#117220232
C3×#635732
C3×#635632
#105914153
#105906493
#105904413