Finite Group Databasebeta

← #13155992

Subgroups — #13155992

Order 1728.

690 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723697C211yesmaximal, characteristic
2C2×#88793C211yesmaximal, characteristic, complemented
3C3×#63843C211yesmaximal, characteristic, complemented
4C2×C3×#3717C211yesmaximal, characteristic, complemented
5#11722051C211yesmaximal, characteristic, complemented
6#11722033C211yesmaximal, characteristic, complemented
7C3×#63564C211yesmaximal, characteristic, complemented
8#10591617—31nomaximal
9#10590876—31nomaximal
10#10590549—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×#6295—21no—
15C2×#6295—21no—
16C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
17C3×C4×C3⋊Dic3C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20#88793—22no—
21C3×Dic3⋊Dic3—21no—
22C3×Dic32—21no—
23C3×#3717—22no—
24C2×#3755—31no—
25C3×Q8×Dic3—31no—
26C2×C3×#215—31no—
27C2×C3×#204—31no—
28C3×#887—31no—
29#63914—31no—
30#63911—31no—
31#63911—31no—
32#63843D311yescharacteristic, complemented
33C2×#3717D311yescharacteristic, complemented
34C2×#3715—31no—
35C2×#3715—31no—
36C2×#3715—61no—
37#63641—31no—
38#63636—31no—
39C3×#821—31no—
40C3×#809—31no—
41Dic3⋊(C3⋊C8)—31no—
42Dic3⋊(C3⋊C8)—31no—
43Dic3⋊(C3⋊C8)—61no—
44#63564D311yescharacteristic, complemented
45#63562—31no—
46#63562—31no—
47#63562—61no—
48C22×C3×C3⋊D3—21no—
49C22×C3×C3⋊D3—21no—
50C2×C33×C4C2311yescharacteristic
51C2×C3×C3⋊Dic3D411yescharacteristic
52C2×C3×C3⋊Dic3D411yescharacteristic
53C3×C4⋊(C3⋊D3)—21no—
54C3×C4⋊(C3⋊D3)—21no—
55C3×C4⋊(C3⋊D3)—21no—
56C3×C4⋊(C3⋊D3)—21no—
57C2×C32×Dic3—21no—
58C2×C32×Dic3—21no—
59C32×Dic6—21no—
60C32×Dic6D412yescomplemented
61#6295—42no—
62#6295—42no—
63C3⋊(C3⋊(C3⋊C8))—21no—
64C32×C3⋊C8—21no—
65#5163—91no—
66#5027—91no—
67#4762—91no—
68C2×C32×Q8—31no—
69C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
70C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
71C2×C3×D12—31no—
72C2×C3×D12—31no—
73C2×C3×D12—31no—
74C2×C3×D12—31no—
75C2×C3×D12—61no—
76C2×C3×D12—61no—
77C2×C3×Dic6—21no—
78C2×C3×Dic6C2×D311yescharacteristic
79C2×C3×Dic6C2×D311yescharacteristic
80C2×#626—61no—
81C2×#626—61no—
82C2×#626—61no—
83C2×#626—61no—
84C2×#625—61no—
85C2×#625—61no—
86C2×C32×C8—31no—
87C32×C4⋊C4—31no—
88#3755—62no—
89C4×C3⋊Dic3C2×D311yescharacteristic
90C2×C3⋊(C3⋊C8)—31no—
91C2×C3⋊(C3⋊C8)—31no—
92C2×C3⋊(C3⋊C8)—31no—
93C2×C3⋊(C3⋊C8)—61no—
94C2×C3⋊(C3⋊C8)—61no—
95C2×C3⋊(C3⋊C8)—61no—
96C2×C3⋊(C3⋊C8)—121no—
97C3×#215—62no—
98C3×C4⋊Dic3—21no—
99C3×C4⋊Dic3—61no—
100C3×C4⋊Dic3C2×D311yescharacteristic
101C3×C4⋊Dic3C2×D311yescharacteristic
102C3×Dic3⋊C4—61no—
103C3×C4×Dic3—31no—
104C3×C4×Dic3—31no—
105C3×C4×Dic3—61no—
106C3×C4×Dic3—61no—
107C3×C4×Dic3—61no—
108C2×C3×C3⋊C8—21no—
109C2×C3×C3⋊C8C2×D311yescharacteristic
110C2×C3×C3⋊C8C2×D311yescharacteristic
111C3×#204—62no—
112Dic3⋊Dic3—21no—
113Dic32—21no—
114#3717—22no—
115#3715—64no—
116#3715—62no—
117#3715—62no—
118C22×C33C2×D411yescharacteristic
119C2×C3×C3⋊D3—41no—
120C2×C3×C3⋊D3—41no—
121C2×C3×C3⋊D3—41no—
122C2×C3×C3⋊D3—41no—
123C2×C3×C3⋊D3—41no—
124C2×C3×C3⋊D3—41no—
125C33×C4C2×D411yescharacteristic
126C33×C4#4111yescharacteristic
127C3×C3⋊Dic3—21no—
128C3×C3⋊Dic3SD1612yescomplemented
129C32×Dic3—22no—
130C32×Dic3—41no—
131C2×C3×SD16—91no—
132C3×C4×Q8—91no—
133Q8×Dic3—31no—
134C2×#215—31no—
135C2×#215—91no—
136C2×#215—91no—
137C2×#215—181no—
138#940—91no—
139#940—91no—
140C2×#214—91no—
141C2×#204—31no—
142#887—31no—
143#874—91no—
144C3×#102—91no—
145#832—91no—
146#821—31no—
147Dic3⋊C8—91no—
148Dic3⋊C8—91no—
149Dic3⋊C8—181no—
150#809—31no—
151#809—91no—
152#809—91no—
153#809—181no—
154C4⋊(C3⋊C8)—91no—
155C22×C3⋊D3—21no—
156C22×C3⋊D3—21no—
157C22×C3×D3—61no—
158C22×C3×D3—61no—
159C22×C3×D3—61no—
160C22×C3×D3—61no—
161C22×C3×D3—121no—
162C22×C3×D3—121no—
163C32×Q8—32no—
164C32×Q8—61no—
165C2×C32×C4—21no—
166C2×C32×C4—21no—
167C2×C32×C4—21no—
168C2×C32×C4—41no—
169C2×C32×C4—61no—
170C2×C32×C4—61no—
171C2×C32×C4C22×D311yescharacteristic
172C2×C32×C4C22×D311yescharacteristic
173C2×C32×C4C22×D311yescharacteristic
174C2×C3⋊Dic3#6711yescharacteristic
175C2×C3⋊Dic3#6711yescharacteristic
176C4⋊(C3⋊D3)—21no—
177C4⋊(C3⋊D3)—21no—
178C4⋊(C3⋊D3)—21no—
179C4⋊(C3⋊D3)—21no—
180C2×C3×Dic3—21no—
181C2×C3×Dic3—21no—
182C2×C3×Dic3—21no—
183C2×C3×Dic3—21no—
184C2×C3×Dic3—31no—
185C2×C3×Dic3—31no—
186C2×C3×Dic3—31no—
187C2×C3×Dic3—31no—
188C2×C3×Dic3—41no—
189C2×C3×Dic3—41no—
190C2×C3×Dic3—61no—
191C2×C3×Dic3—61no—
192C3×D12—62no—
193C3×D12—62no—
194C3×D12—61no—
195C3×D12—61no—
196C3×D12—61no—
197C3×D12—61no—
198C3×D12—61no—
199C3×D12—61no—
200C3×D12—61no—
201C3×D12—61no—
202C3×D12—121no—
203C3×D12—121no—
204C3×Dic6—21no—
205C3×Dic6—21no—
206C3×Dic6—22no—
207C3×Dic6—41no—
208C3×Dic6D1212yescomplemented
209C3×Dic6#6712yescomplemented
210#626—122no—
211#626—122no—
212#626—122no—
213#626—122no—
214#625—122no—
215#625—122no—
216C32×C8—61no—
217C3⋊(C3⋊C8)—61no—
218C3⋊(C3⋊C8)—62no—
219C3⋊(C3⋊C8)—61no—
220C3⋊(C3⋊C8)—61no—
221C3⋊(C3⋊C8)—121no—
222C3⋊(C3⋊C8)—121no—
223C3⋊(C3⋊C8)—122no—
224C3×C3⋊C8—21no—
225C3×C3⋊C8—21no—
226C3×C3⋊C8—41no—
227#460—271noSylow
228C2×C33#12111yescharacteristic
229C2×C33C2×SD1611yescharacteristic
230C2×C33AGL(1,ℤ/8)11yescharacteristic
231C3×C3⋊D3—81no—
232C3×C3⋊D3—81no—
233C2×C3×Q8—31no—
234C2×C3×Q8—31no—
235C2×C3×Q8—61no—
236C2×C3×D4—91no—
237C2×C3×D4—91no—
238C2×#67—181no—
239C2×#67—181no—
240C2×#67—181no—
241C2×#67—181no—
242C2×D12—31no—
243C2×D12—31no—
244C2×D12—31no—
245C2×D12—31no—
246C2×D12—61no—
247C2×D12—61no—
248C2×D12—181no—
249C2×D12—181no—
250C2×Dic6D3211yescharacteristic
251C3×SD16—182no—
252C2×C3×C8—31no—
253C2×C3×C8—31no—
254C2×C3×C8—61no—
255C3×C4⋊C4—31no—
256C3×C4⋊C4—31no—
257C3×C4⋊C4—61no—
258C3×C4⋊C4—181no—
259C3×C42—91no—
260C3×C42—181no—
261#215—62no—
262#215—182no—
263#215—182no—
264#215—184no—
265#214—182no—
266C4⋊Dic3—61no—
267C4⋊Dic3D3211yescharacteristic
268Dic3⋊C4—61no—
269C4×Dic3—31no—
270C4×Dic3—31no—
271C4×Dic3—61no—
272C4×Dic3—61no—
273C4×Dic3—61no—
274C2×C3⋊C8—91no—
275C2×C3⋊C8—91no—
276C2×C3⋊C8—91no—
277C2×C3⋊C8—181no—
278C2×C3⋊C8—181no—
279C2×C3⋊C8—181no—
280C2×C3⋊C8—361no—
281C2×C3⋊C8D3211yescharacteristic
282#204—62no—
283C22×C32—21no—
284C22×C32—21no—
285C22×C32—21no—
286C22×C32—41no—
287C22×C32D3×D411yescharacteristic
288C22×C32D3×D411yescharacteristic
289C22×C32C2×#6711yescharacteristic
290C2×C3⋊D3—41no—
291C2×C3⋊D3—41no—
292C2×C3⋊D3—41no—
293C2×C3⋊D3—41no—
294C2×C3⋊D3—41no—
295C2×C3⋊D3—41no—
296C2×C3×D3—122no—
297C2×C3×D3—122no—
298C2×C3×D3—121no—
299C2×C3×D3—121no—
300C2×C3×D3—121no—
301C2×C3×D3—121no—
302C2×C3×D3—121no—
303C2×C3×D3—121no—
304C2×C3×D3—121no—
305C2×C3×D3—121no—
306C2×C3×D3—121no—
307C2×C3×D3—121no—
308C2×C3×D3—121no—
309C2×C3×D3—121no—
310C2×C3×D3—241no—
311C2×C3×D3—241no—
312C2×C3×D3—241no—
313C2×C3×D3—241no—
314C32×C4—21no—
315C32×C4—21no—
316C32×C4—21no—
317C32×C4—21no—
318C32×C4—21no—
319C32×C4—21no—
320C32×C4—41no—
321C32×C4—41no—
322C32×C4—62no—
323C32×C4—121no—
324C32×C4C2×D1211yescharacteristic
325C32×C4#23511yescharacteristic
326C32×C4#23511yescharacteristic
327C32×C4D3×D411yescharacteristic
328C32×C4#23711yescharacteristic
329C32×C4C2×#6711yescharacteristic
330C3⋊Dic3—21no—
331C3⋊Dic3#21512yescomplemented
332C3×Dic3—22no—
333C3×Dic3—22no—
334C3×Dic3—32no—
335C3×Dic3—32no—
336C3×Dic3—42no—
337C3×Dic3—41no—
338C3×Dic3—41no—
339C3×Dic3—62no—
340C3×Dic3—61no—
341C3×Dic3—61no—
342C3×Dic3—81no—
343C3×Dic3—121no—
344C2×SD16—91no—
345C2×SD16—271no—
346#127—271no—
347C4×Q8—91no—
348C4⋊C8—271no—
349#102—91no—
350#102—271no—
351C33#46011yescharacteristic, complemented, Sylow
352C23×C3—181no—
353C23×C3—181no—
354C22×D3—61no—
355C22×D3—61no—
356C22×D3—61no—
357C22×D3—61no—
358C22×D3—121no—
359C22×D3—121no—
360C3×Q8—32no—
361C3×Q8—32no—
362C3×Q8—62no—
363C3×Q8—61no—
364C3×Q8—61no—
365C3×Q8—121no—
366C3×D4—181no—
367C3×D4—181no—
368C3×D4—181no—
369C3×D4—181no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—41no—
374C2×C3×C4—61no—
375C2×C3×C4—61no—
376C2×C3×C4—61no—
377C2×C3×C4—61no—
378C2×C3×C4—91no—
379C2×C3×C4—91no—
380C2×C3×C4—121no—
381C2×C3×C4—121no—
382C2×C3×C4C2×D3211yescharacteristic
383C2×C3×C4C2×D3211yescharacteristic
384C2×C3×C4C2×D3211yescharacteristic
385#67—362no—
386#67—362no—
387#67—362no—
388#67—362no—
389C2×Dic3—21no—
390C2×Dic3—21no—
391C2×Dic3—31no—
392C2×Dic3—31no—
393C2×Dic3—31no—
394C2×Dic3—31no—
395C2×Dic3—61no—
396C2×Dic3—61no—
397D12—62no—
398D12—62no—
399D12—61no—
400D12—61no—
401D12—61no—
402D12—61no—
403D12—61no—
404D12—61no—
405D12—61no—
406D12—61no—
407D12—121no—
408D12—121no—
409D12—362no—
410D12—362no—
411Dic6—21no—
412Dic6#62612yescomplemented
413C3×C8—61no—
414C3×C8—61no—
415C3×C8—121no—
416C3⋊C8—21no—
417C3⋊C8—182no—
418C3⋊C8—181no—
419C3⋊C8—181no—
420C3⋊C8—181no—
421C3⋊C8—361no—
422C3⋊C8—361no—
423C3⋊C8—362no—
424C2×C32—21no—
425C2×C32—21no—
426C2×C32—21no—
427C2×C32—21no—
428C2×C32—21no—
429C2×C32—21no—
430C2×C32—21no—
431C2×C32—21no—
432C2×C32—21no—
433C2×C32—41no—
434C2×C32—41no—
435C2×C32—41no—
436C2×C32#88411yescharacteristic
437C2×C32#91111yescharacteristic
438C2×C32D3×SD1611yescharacteristic
439C2×C32D3×SD1611yescharacteristic
440C2×C32#91411yescharacteristic
441C2×C32#93011yescharacteristic
442C2×C32#93911yescharacteristic
443C2×C32C2×#21511yescharacteristic
444C2×C32#94911yescharacteristic
445C3⋊D3—81no—
446C3⋊D3—81no—
447C3×D3—242no—
448C3×D3—242no—
449C3×D3—241no—
450C3×D3—241no—
451C3×D3—241no—
452C3×D3—241no—
453C2×Q8—31no—
454C2×D4—91no—
455C2×D4—91no—
456C2×D4—541no—
457C2×D4—541no—
458SD16—182no—
459SD16—542no—
460C2×C8—31no—
461C2×C8—271no—
462C4⋊C4—31no—
463C4⋊C4—181no—
464C42—91no—
465C42—181no—
466C22×C3—21no—
467C22×C3—21no—
468C22×C3—21no—
469C22×C3—41no—
470C22×C3—361no—
471C22×C3—361no—
472C22×C3—361no—
473C22×C3—361no—
474C22×C3—361no—
475C22×C3—361no—
476C22×C3#380211yescharacteristic
477C22×C3D3×#6711yescharacteristic
478C22×C3D3×#6711yescharacteristic
479C2×D3—122no—
480C2×D3—122no—
481C2×D3—121no—
482C2×D3—121no—
483C2×D3—121no—
484C2×D3—121no—
485C2×D3—121no—
486C2×D3—121no—
487C2×D3—121no—
488C2×D3—121no—
489C2×D3—121no—
490C2×D3—121no—
491C2×D3—121no—
492C2×D3—121no—
493C2×D3—241no—
494C2×D3—241no—
495C2×D3—241no—
496C2×D3—241no—
497C3×C4—21no—
498C3×C4—21no—
499C3×C4—21no—
500C3×C4—21no—
501C3×C4—21no—
502C3×C4—21no—
503C3×C4—41no—
504C3×C4—41no—
505C3×C4—62no—
506C3×C4—62no—
507C3×C4—92no—
508C3×C4—122no—
509C3×C4—121no—
510C3×C4—121no—
511C3×C4—181no—
512C3×C4—241no—
513C3×C4#379511yescharacteristic
514C3×C4#379811yescharacteristic
515C3×C4D3×D1211yescharacteristic
516C3×C4#380411yescharacteristic
517C3×C4C2×#62611yescharacteristic
518C3×C4D3×#6711yescharacteristic
519Dic3—22no—
520Dic3—32no—
521Dic3—32no—
522Dic3—41no—
523Dic3—62no—
524Dic3—61no—
525Dic3—61no—
526Dic3—121no—
527C32—21no—
528C32—21no—
529C32—21no—
530C32—41no—
531C32#476211yescharacteristic, complemented
532C32#502711yescharacteristic, complemented
533C32#516311yescharacteristic, complemented
534C23—181no—
535C23—181no—
536Q8—32no—
537Q8—61no—
538D4—181no—
539D4—181no—
540D4—181no—
541D4—181no—
542D4—1082no—
543D4—1082no—
544C2×C4—61no—
545C2×C4—61no—
546C2×C4—91no—
547C2×C4—91no—
548C2×C4D3311yescharacteristic
549C8—61no—
550C8—541no—
551C2×C3—21no—
552C2×C3—21no—
553C2×C3—21no—
554C2×C3—21no—
555C2×C3—21no—
556C2×C3—21no—
557C2×C3—21no—
558C2×C3—21no—
559C2×C3—21no—
560C2×C3—41no—
561C2×C3—41no—
562C2×C3—41no—
563C2×C3—721no—
564C2×C3—721no—
565C2×C3#6379611yescharacteristic
566C2×C3#6379811yescharacteristic
567C2×C3#6388511yescharacteristic
568C2×C3#6390611yescharacteristic
569C2×C3#6392511yescharacteristic
570C2×C3D3×#21511yescharacteristic
571C2×C3D3×#21511yescharacteristic
572C2×C3#6393911yescharacteristic
573C2×C3#6397811yescharacteristic
574D3—242no—
575D3—242no—
576D3—241no—
577D3—241no—
578D3—241no—
579D3—241no—
580C22—361no—
581C22—361no—
582C22—361no—
583C22—361no—
584C22—361no—
585C22—361no—
586C22#8895311yescharacteristic
587C4—62no—
588C4—92no—
589C4—121no—
590C4—181no—
591C4D3×#62611yescharacteristic
592C4#8896011yescharacteristic
593C3—21no—
594C3—21no—
595C3—21no—
596C3—41no—
597C3#1059054911yescharacteristic, complemented
598C3#1059087611yescharacteristic, complemented
599C3#1059161711yescharacteristic, complemented
600C2—721no—
601C2—721no—
602C2#1172284011yescharacteristic
603C2#1172287611yescharacteristic
604C2#1172296011yescharacteristic
subgroupG/H
centerC22#88953
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radical#11723697C2
hypocenterC33#460
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
2. C2×#88793C2semidirect
3. C3×#63843C2semidirect
4. C2×C3×#3717C2semidirect
5. #11722051C2semidirect
6. #11722033C2semidirect
7. C3×#63564C2semidirect
32. #63843D3semidirect
33. C2×#3717D3semidirect
44. #63564D3semidirect
60. C32×Dic6D4semidirect
128. C3×C3⋊Dic3SD16semidirect
208. C3×Dic6D12semidirect
209. C3×Dic6#67semidirect
331. C3⋊Dic3#215semidirect
351. C33#460semidirect
412. Dic6#626semidirect
531. C32#4762semidirect
532. C32#5027semidirect
533. C32#5163semidirect
597. C3#10590549semidirect
598. C3#10590876semidirect
599. C3#10591617semidirect

Maximal subgroups

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

subgroupindex
#117236972
C2×#887932
C3×#638432
C2×C3×#37172
#117220512
#117220332
C3×#635642
#105916173
#105908763
#105905493