Finite Group Databasebeta

← #13155687

Subgroups — #13155687

Order 1728.

746 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723484C211yesmaximal, characteristic
2C2×#88797C211yesmaximal, characteristic
3C3×#63844C211yesmaximal, characteristic, complemented
4#11722076C211yesmaximal, characteristic, complemented
5#11722035C211yesmaximal, characteristic
6C3×#63575C211yesmaximal, characteristic, complemented
7C3×#63554C211yesmaximal, characteristic, complemented
8#10590773—31nomaximal
9#10590457—31nomaximal
10#10590362—31nomaximal
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×#6289—21no—
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C32×C4×Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88804—21no—
20#88797C414yescomplemented
21C3×#3724—21no—
22C3×Dic32—21no—
23C2×#3744—31no—
24C4×C4⋊(C3⋊D3)—31no—
25C3×#887—31no—
26C3×C4×Dic6—31no—
27#63894—31no—
28#63894—31no—
29#63894—61no—
30#63844D311yescharacteristic, complemented
31C2×#3717—31no—
32C2×#3717—31no—
33C2×#3717—61no—
34#63636—31no—
35C3×#818—31no—
36C3×#815—31no—
37C3×#807—31no—
38C3×#803—31no—
39#63575D311yescharacteristic, complemented
40#63567—31no—
41#63566—31no—
42#63565—31no—
43#63564—31no—
44#63564—31no—
45#63564—61no—
46#63554D311yescharacteristic, complemented
47C22×C3⋊(C3⋊D3)—21no—
48C2×C33×C4C2311yescharacteristic
49C4⋊(C3⋊(C3⋊D3))—21no—
50C4⋊(C3⋊(C3⋊D3))C2×C412yes—
51C2×C3×C3⋊Dic3—21no—
52C2×C3×C3⋊Dic3—21no—
53C3×#634—21no—
54C3×#634C2×C412yescomplemented
55C2×C32×Dic3—21no—
56C2×C32×Dic3D411yescharacteristic
57C2×C32×Dic3D411yescharacteristic
58#6289—42no—
59C3×C3⋊(C3⋊C8)—21no—
60C32×C3⋊C8C2×C412yescomplemented
61#5026—91no—
62#4889—91no—
63#4710—91no—
64C2×C4⋊(C3⋊D3)—31no—
65C2×C4⋊(C3⋊D3)—31no—
66C2×C4⋊(C3⋊D3)—31no—
67C2×C4⋊(C3⋊D3)—61no—
68C2×C4⋊(C3⋊D3)—61no—
69C2×C4⋊(C3⋊D3)—61no—
70C2×C4⋊(C3⋊D3)—121no—
71C2×C4×C3⋊D3—61no—
72C2×#634C2×D311yescharacteristic
73C2×C3×Dic6—31no—
74C2×C3×Dic6—31no—
75C2×C3×Dic6—61no—
76C2×#624—62no—
77C2×#624—61no—
78C2×#624—61no—
79C2×C32×C8—31no—
80C32×C4⋊C4—31no—
81C32×C42—31no—
82#3752—61no—
83C4⋊(C3⋊Dic3)C2×D311yescharacteristic
84C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
85#3744—34no—
86C3×C4⋊Dic3—21no—
87C3×C4⋊Dic3—31no—
88C3×C4⋊Dic3—31no—
89C3×C4⋊Dic3—61no—
90C3×C4⋊Dic3C2×D311yescharacteristic
91C3×C4⋊Dic3C2×D311yescharacteristic
92C3×Dic3⋊C4—61no—
93C3×Dic3⋊C4—61no—
94C3×C4×Dic3—21no—
95C3×C4×Dic3—61no—
96C3×C4×Dic3—61no—
97C3×C4×Dic3C2×D311yescharacteristic
98C3×C4×Dic3C2×D311yescharacteristic
99C2×C3×C3⋊C8—21no—
100C2×C3×C3⋊C8—31no—
101C2×C3×C3⋊C8—31no—
102C2×C3×C3⋊C8—61no—
103C2×C3×C3⋊C8C2×D311yescharacteristic
104C2×C3×C3⋊C8C2×D311yescharacteristic
105#3724—21no—
106#3722—62no—
107#3722—61no—
108#3722—61no—
109Dic32—21no—
110#3717—34no—
111#3717—34no—
112#3717—64no—
113C22×C33C2×D411yescharacteristic
114C2×C3⋊(C3⋊D3)—22no—
115C2×C3⋊(C3⋊D3)—41no—
116C2×C3⋊(C3⋊D3)—41no—
117C33×C4C22×C411yescharacteristic
118C33×C4#4111yescharacteristic
119C3×C3⋊Dic3—22no—
120C3×C3⋊Dic3—41no—
121C32×Dic3—21no—
122C32×Dic3—21no—
123C32×Dic3—41no—
124C3×C4×Q8—91no—
125C2×#215—91no—
126C2×#204—91no—
127C2×#204—91no—
128C2×#204—181no—
129#893—91no—
130#887—31no—
131C4×D12—91no—
132C4×D12—91no—
133C4×D12—181no—
134C4×Dic6—31no—
135C3×#107—91no—
136C3×#97—91no—
137#835—91no—
138#821—91no—
139#818—31no—
140#816—91no—
141#815—31no—
142#810—91no—
143#809—91no—
144#809—91no—
145#809—181no—
146#807—31no—
147#803—31no—
148C22×C3⋊D3—61no—
149C22×C3⋊D3—62no—
150C22×C3⋊D3—61no—
151C22×C3⋊D3—61no—
152C22×C3⋊D3—121no—
153C22×C3⋊D3—121no—
154C22×C3⋊D3—122no—
155C2×C32×C4—21no—
156C2×C32×C4—21no—
157C2×C32×C4—21no—
158C2×C32×C4—31no—
159C2×C32×C4—31no—
160C2×C32×C4—41no—
161C2×C32×C4—61no—
162C2×C32×C4C22×D311yescharacteristic
163C2×C32×C4C22×D311yescharacteristic
164C2×C32×C4C22×D311yescharacteristic
165C2×C3⋊Dic3—21no—
166C2×C3⋊Dic3—21no—
167C4⋊(C3⋊D3)—32no—
168C4⋊(C3⋊D3)—32no—
169C4⋊(C3⋊D3)—32no—
170C4⋊(C3⋊D3)—62no—
171C4⋊(C3⋊D3)—62no—
172C4⋊(C3⋊D3)—62no—
173C4⋊(C3⋊D3)—61no—
174C4⋊(C3⋊D3)—62no—
175C4⋊(C3⋊D3)—61no—
176C4⋊(C3⋊D3)—61no—
177C4⋊(C3⋊D3)—122no—
178C4⋊(C3⋊D3)—121no—
179C4⋊(C3⋊D3)—121no—
180C4⋊(C3⋊D3)—122no—
181C4×C3⋊D3—122no—
182#634—21no—
183#634C4×D312yescomplemented
184C2×C3×Dic3—21no—
185C2×C3×Dic3—21no—
186C2×C3×Dic3—21no—
187C2×C3×Dic3—21no—
188C2×C3×Dic3—41no—
189C2×C3×Dic3—61no—
190C2×C3×Dic3—61no—
191C2×C3×Dic3—62no—
192C2×C3×Dic3—61no—
193C2×C3×Dic3—61no—
194C2×C3×Dic3—121no—
195C2×C3×Dic3D1211yescharacteristic
196C2×C3×Dic3D1211yescharacteristic
197C2×C3×Dic3#6711yescharacteristic
198C2×C3×Dic3#6711yescharacteristic
199C3×Dic6—32no—
200C3×Dic6—32no—
201C3×Dic6—62no—
202C3×Dic6—61no—
203C3×Dic6—61no—
204C3×Dic6—121no—
205#624—124no—
206#624—122no—
207#624—122no—
208C32×C8—32no—
209C3⋊(C3⋊C8)—21no—
210C3×C3⋊C8—22no—
211C3×C3⋊C8—61no—
212C3×C3⋊C8—61no—
213C3×C3⋊C8—121no—
214C3×C3⋊C8C4×D312yescomplemented
215C3×C3⋊C8C4×D312yescomplemented
216#440—271noSylow
217C2×C33C4×D411yescharacteristic
218C2×C33AGL(1,ℤ/8)11yescharacteristic
219C2×C33#13711yescharacteristic
220C3⋊(C3⋊D3)—42no—
221C2×C3×Q8—91no—
222C2×D12—91no—
223C2×D12—91no—
224C2×D12—91no—
225C2×D12—181no—
226C2×D12—181no—
227C2×D12—181no—
228C2×D12—361no—
229C2×C4×D3—181no—
230C2×C4×D3—181no—
231C2×C4×D3—182no—
232C2×C4×D3—181no—
233C2×Dic6—31no—
234C2×Dic6—31no—
235C2×Dic6—61no—
236C2×C3×C8—31no—
237C2×C3×C8—31no—
238C2×C3×C8—61no—
239C2×C3×C8—91no—
240C3×C4⋊C4—31no—
241C3×C4⋊C4—31no—
242C3×C4⋊C4—61no—
243C3×C4⋊C4—91no—
244C3×C4⋊C4—181no—
245C3×C42—31no—
246C3×C42—31no—
247C3×C42—61no—
248C3×C42—181no—
249#215—94no—
250(C2×D3)⋊C4—181no—
251(C2×D3)⋊C4—181no—
252(C2×D3)⋊C4—182no—
253(C2×D3)⋊C4—181no—
254C4⋊Dic3—31no—
255C4⋊Dic3—31no—
256C4⋊Dic3—61no—
257C4⋊Dic3D3211yescharacteristic
258Dic3⋊C4—61no—
259Dic3⋊C4—61no—
260C4×Dic3—61no—
261C4×Dic3—61no—
262C4×Dic3D3211yescharacteristic
263C2×C3⋊C8—31no—
264C2×C3⋊C8—31no—
265C2×C3⋊C8—61no—
266C2×C3⋊C8D3211yescharacteristic
267#204—94no—
268#204—94no—
269#204—184no—
270C22×C32—21no—
271C22×C32—21no—
272C22×C32—21no—
273C22×C32—41no—
274C22×C32C2×D1211yescharacteristic
275C22×C32D3×D411yescharacteristic
276C22×C32C2×#6711yescharacteristic
277C2×C3⋊D3—62no—
278C2×C3⋊D3—64no—
279C2×C3⋊D3—62no—
280C2×C3⋊D3—62no—
281C2×C3⋊D3—122no—
282C2×C3⋊D3—122no—
283C2×C3⋊D3—124no—
284C2×C3⋊D3—121no—
285C2×C3⋊D3—121no—
286C2×C3⋊D3—122no—
287C2×C3⋊D3—122no—
288C2×C3⋊D3—121no—
289C2×C3⋊D3—121no—
290C2×C3⋊D3—121no—
291C2×C3⋊D3—121no—
292C2×C3⋊D3—241no—
293C2×C3⋊D3—241no—
294C2×C3⋊D3—241no—
295C2×C3⋊D3—241no—
296C2×C3⋊D3—242no—
297C2×C3⋊D3—242no—
298C32×C4—21no—
299C32×C4—21no—
300C32×C4—21no—
301C32×C4—21no—
302C32×C4—21no—
303C32×C4—21no—
304C32×C4—41no—
305C32×C4—41no—
306C32×C4—61no—
307C32×C4—61no—
308C32×C4—121no—
309C32×C4C2×C4×D311yescharacteristic
310C32×C4C2×C4×D311yescharacteristic
311C32×C4C2×C4×D311yescharacteristic
312C32×C4#23511yescharacteristic
313C32×C4#23511yescharacteristic
314C32×C4#23911yescharacteristic
315C3⋊Dic3—22no—
316C3⋊Dic3—41no—
317C3×Dic3—22no—
318C3×Dic3—21no—
319C3×Dic3—21no—
320C3×Dic3—21no—
321C3×Dic3—21no—
322C3×Dic3—41no—
323C3×Dic3—41no—
324C3×Dic3—41no—
325C3×Dic3—62no—
326C3×Dic3—62no—
327C3×Dic3—81no—
328C3×Dic3—122no—
329C3×Dic3—122no—
330C3×Dic3—121no—
331C3×Dic3—121no—
332C2×SD16—271no—
333C4×Q8—91no—
334C4×D4—271no—
335#107—91no—
336#103—271no—
337#102—271no—
338#97—91no—
339C33#44011yescharacteristic, complemented, Sylow
340C22×D3—182no—
341C22×D3—181no—
342C22×D3—181no—
343C22×D3—181no—
344C22×D3—361no—
345C22×D3—361no—
346C22×D3—362no—
347C3×Q8—92no—
348C3×Q8—181no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—31no—
353C2×C3×C4—31no—
354C2×C3×C4—31no—
355C2×C3×C4—31no—
356C2×C3×C4—41no—
357C2×C3×C4—61no—
358C2×C3×C4—61no—
359C2×C3×C4—61no—
360C2×C3×C4—61no—
361C2×C3×C4—121no—
362C2×C3×C4—181no—
363C2×C3×C4—181no—
364C2×C3×C4C2×D3211yescharacteristic
365C2×C3×C4C2×D3211yescharacteristic
366C2×C3×C4C2×D3211yescharacteristic
367C2×Dic3—21no—
368C2×Dic3—61no—
369C2×Dic3—61no—
370C2×Dic3—62no—
371C2×Dic3—61no—
372C2×Dic3—61no—
373C2×Dic3—121no—
374C2×Dic3#62611yescharacteristic
375C2×Dic3#62611yescharacteristic
376D12—92no—
377D12—92no—
378D12—92no—
379D12—182no—
380D12—182no—
381D12—182no—
382D12—182no—
383D12—181no—
384D12—181no—
385D12—181no—
386D12—362no—
387D12—361no—
388D12—361no—
389D12—362no—
390C4×D3—362no—
391C4×D3—364no—
392C4×D3—362no—
393C4×D3—362no—
394Dic6—32no—
395Dic6—32no—
396Dic6—62no—
397Dic6—61no—
398Dic6—61no—
399Dic6—121no—
400C3×C8—32no—
401C3×C8—32no—
402C3×C8—62no—
403C3×C8—181no—
404C3⋊C8—61no—
405C3⋊C8—61no—
406C3⋊C8—121no—
407C3⋊C8#62412yescomplemented
408C2×C32—21no—
409C2×C32—21no—
410C2×C32—21no—
411C2×C32—21no—
412C2×C32—21no—
413C2×C32—21no—
414C2×C32—21no—
415C2×C32—21no—
416C2×C32—21no—
417C2×C32—41no—
418C2×C32—41no—
419C2×C32—41no—
420C2×C32C4×D1211yescharacteristic
421C2×C32#89311yescharacteristic
422C2×C32#90811yescharacteristic
423C2×C32#90911yescharacteristic
424C2×C32#91111yescharacteristic
425C2×C32#91811yescharacteristic
426C2×C32C4×#6711yescharacteristic
427C2×C32#94211yescharacteristic
428C2×C32#94911yescharacteristic
429C3⋊D3—122no—
430C3⋊D3—124no—
431C3⋊D3—122no—
432C3⋊D3—122no—
433C3⋊D3—242no—
434C3⋊D3—242no—
435C3⋊D3—244no—
436C2×Q8—91no—
437C2×D4—271no—
438C22×C4—541no—
439SD16—274no—
440C2×C8—31no—
441C2×C8—91no—
442C4⋊C4—31no—
443C4⋊C4—91no—
444C4⋊C4—181no—
445C22⋊C4—541no—
446C42—31no—
447C42—181no—
448C22×C3—21no—
449C22×C3—21no—
450C22×C3—21no—
451C22×C3—41no—
452C22×C3D3×D1211yescharacteristic
453C22×C3C2×#62611yescharacteristic
454C22×C3D3×#6711yescharacteristic
455C2×D3—184no—
456C2×D3—182no—
457C2×D3—182no—
458C2×D3—182no—
459C2×D3—362no—
460C2×D3—362no—
461C2×D3—364no—
462C2×D3—362no—
463C2×D3—362no—
464C2×D3—361no—
465C2×D3—361no—
466C2×D3—361no—
467C2×D3—361no—
468C2×D3—361no—
469C2×D3—361no—
470C2×D3—721no—
471C2×D3—721no—
472C2×D3—721no—
473C2×D3—721no—
474C2×D3—722no—
475C2×D3—722no—
476C3×C4—21no—
477C3×C4—21no—
478C3×C4—21no—
479C3×C4—21no—
480C3×C4—21no—
481C3×C4—21no—
482C3×C4—41no—
483C3×C4—41no—
484C3×C4—62no—
485C3×C4—61no—
486C3×C4—61no—
487C3×C4—61no—
488C3×C4—61no—
489C3×C4—121no—
490C3×C4—121no—
491C3×C4—121no—
492C3×C4—182no—
493C3×C4—241no—
494C3×C4—361no—
495C3×C4#379811yescharacteristic
496C3×C4#379911yescharacteristic
497C3×C4#379911yescharacteristic
498C3×C4C4×D3211yescharacteristic
499C3×C4C4×D3211yescharacteristic
500C3×C4C2×#62411yescharacteristic
501Dic3—21no—
502Dic3—21no—
503Dic3—41no—
504Dic3—62no—
505Dic3—62no—
506Dic3—122no—
507Dic3—122no—
508Dic3—121no—
509Dic3—121no—
510C32—21no—
511C32—21no—
512C32—21no—
513C32—41no—
514C32#471011yescharacteristic, complemented
515C32#488911yescharacteristic, complemented
516C32#502611yescharacteristic, complemented
517C23—541no—
518Q8—92no—
519Q8—181no—
520D4—272no—
521D4—541no—
522C2×C4—31no—
523C2×C4—31no—
524C2×C4—61no—
525C2×C4—181no—
526C2×C4—181no—
527C2×C4—1082no—
528C2×C4D3311yescharacteristic
529C8—32no—
530C8—181no—
531C2×C3—21no—
532C2×C3—21no—
533C2×C3—21no—
534C2×C3—21no—
535C2×C3—21no—
536C2×C3—21no—
537C2×C3—21no—
538C2×C3—21no—
539C2×C3—21no—
540C2×C3—41no—
541C2×C3—41no—
542C2×C3—41no—
543C2×C3#6379511yescharacteristic
544C2×C3#6380111yescharacteristic
545C2×C3#6382511yescharacteristic
546C2×C3#6383311yescharacteristic
547C2×C3#6389411yescharacteristic
548C2×C3C4×#62611yescharacteristic
549C2×C3#6390411yescharacteristic
550C2×C3#6392511yescharacteristic
551C2×C3#6394811yescharacteristic
552D3—364no—
553D3—362no—
554D3—362no—
555D3—362no—
556D3—722no—
557D3—722no—
558D3—724no—
559C22—542no—
560C22—1081no—
561C22—1081no—
562C22D3×#62611yescharacteristic
563C4—61no—
564C4—61no—
565C4—121no—
566C4—182no—
567C4—361no—
568C4D3×#62411yescharacteristic
569C4#8896511yescharacteristic
570C3—21no—
571C3—21no—
572C3—21no—
573C3—41no—
574C3#1059036211yescharacteristic, complemented
575C3#1059045711yescharacteristic, complemented
576C3#1059077311yescharacteristic, complemented
577C2—1082no—
578C2#1172287011yescharacteristic
579C2#1172288511yescharacteristic
580C2#1172293311yescharacteristic
subgroupG/H
centerC22D3×#626
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socleC22×C33C2×D4
prime radicalC2×C4⋊(C3⋊(C3⋊D3))C22
hypocenterC33#440
LCSG ▷ 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×#63844C2semidirect
4. #11722076C2semidirect
6. C3×#63575C2semidirect
7. C3×#63554C2semidirect
20. #88797C4semidirect
30. #63844D3semidirect
39. #63575D3semidirect
46. #63554D3semidirect
54. C3×#634C2×C4semidirect
60. C32×C3⋊C8C2×C4semidirect
183. #634C4×D3semidirect
214. C3×C3⋊C8C4×D3semidirect
215. C3×C3⋊C8C4×D3semidirect
339. C33#440semidirect
407. C3⋊C8#624semidirect
514. C32#4710semidirect
515. C32#4889semidirect
516. C32#5026semidirect
574. C3#10590362semidirect
575. C3#10590457semidirect
576. C3#10590773semidirect

Maximal subgroups

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

subgroupindex
#117234842
C2×#887972
C3×#638442
#117220762
#117220352
C3×#635752
C3×#635542
#105907733
#105904573
#105903623