Finite Group Databasebeta

← #13161563

Subgroups — #13161563

Order 1728.

678 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64104C211yesmaximal, characteristic
2#11723674C211yesmaximal, characteristic, complemented
3C2×#88811C211yesmaximal, characteristic, complemented
4C2×#88809C211yesmaximal, characteristic, complemented
5C3×#63637C211yesmaximal, characteristic, complemented
6C3×#63634C211yesmaximal, characteristic, complemented
7#11722075C211yesmaximal, characteristic, complemented
8#10590588—31nomaximal
9#10590522—31nomaximal
10#10590515—31nomaximal
11C2×C3×C4⋊(C3⋊D3)—21no—
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×C4×C3⋊D3C2211yescharacteristic
14C2×C3×#634C2211yescharacteristic
15C33×C4⋊C4C2211yescharacteristic
16C3×#3752—21no—
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
19C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
20#88815—21no—
21#88814—21no—
22#88811—22no—
23#88809—22no—
24#64104D311yescharacteristic
25C3×#895—31no—
26C3×#895—31no—
27C3×#895—61no—
28#63871—31no—
29#63857—31no—
30#63856—31no—
31C2×#3717—31no—
32C2×#3716—31no—
33C2×#3715—31no—
34C2×#3712—31no—
35C2×#3711—31no—
36C2×#3711—31no—
37#63637D311yescharacteristic, complemented
38#63634D311yescharacteristic, complemented
39C3×#810—31no—
40C3×#810—31no—
41C3×#810—61no—
42C3×#807—31no—
43C3×#807—31no—
44C3×#807—61no—
45#63564—31no—
46#63563—31no—
47#63562—31no—
48C22×C3×C3⋊D3—21no—
49C22×C3×C3⋊D3D411yescharacteristic
50C2×C33×C4—21no—
51C2×C33×C4C2311yescharacteristic
52C2×C3×C3⋊Dic3—21no—
53C2×C3×C3⋊Dic3—21no—
54C2×C3×C3⋊Dic3D411yescharacteristic
55C3×C4⋊(C3⋊D3)—21no—
56C3×C4⋊(C3⋊D3)—21no—
57C3×C4⋊(C3⋊D3)—42no—
58C3×C4×C3⋊D3—21no—
59C3×C4×C3⋊D3—21no—
60C3×#634—21no—
61C3×#634—21no—
62C3×C3⋊(C3⋊C8)—21no—
63C3×C3⋊(C3⋊C8)D412yescomplemented
64#5042—91no—
65#4886—91no—
66#4812—91no—
67C2×C4⋊(C3⋊D3)—21no—
68C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
69C2×C4×C3⋊D3C2×D311yescharacteristic
70C2×#634C2×D311yescharacteristic
71C2×C3×D12—31no—
72C2×C3×D12—31no—
73C2×C3×D12—61no—
74C2×C3×D12—61no—
75C2×C3×D12—61no—
76C2×C3×D12—62no—
77C2×C3×C4×D3—31no—
78C2×C3×C4×D3—31no—
79C2×C3×C4×D3—61no—
80C2×C3×Dic6—31no—
81C2×C3×Dic6—31no—
82C2×C3×Dic6—61no—
83C32×C4⋊C4—21no—
84C32×C4⋊C4—21no—
85C32×C4⋊C4—21no—
86C32×C4⋊C4—41no—
87C32×C4⋊C4C2×D311yescharacteristic
88C32×C4⋊C4C2×D311yescharacteristic
89C32×C4⋊C4C2×D311yescharacteristic
90#3752—21no—
91C4⋊(C3⋊Dic3)C2×D311yescharacteristic
92C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
93C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
94C3×(C2×D3)⋊C4—61no—
95C3×(C2×D3)⋊C4—61no—
96C3×(C2×D3)⋊C4—62no—
97C3×C4⋊Dic3—31no—
98C3×C4⋊Dic3—31no—
99C3×C4⋊Dic3—61no—
100C2×C3×C3⋊C8—31no—
101C2×C3×C3⋊C8—31no—
102C2×C3×C3⋊C8—31no—
103C2×C3×C3⋊C8—31no—
104C2×C3×C3⋊C8—61no—
105C2×C3×C3⋊C8—61no—
106#3724—61no—
107Dic3⋊Dic3—61no—
108Dic3⋊Dic3—61no—
109#3722—61no—
110(C2×D3)⋊Dic3—61no—
111(C2×D3)⋊Dic3—61no—
112#3717—62no—
113#3716—62no—
114#3715—62no—
115#3712—62no—
116#3711—62no—
117#3711—62no—
118C22×C33C2×D411yescharacteristic
119C2×C3×C3⋊D3—21no—
120C2×C3×C3⋊D3—21no—
121C2×C3×C3⋊D3—21no—
122C2×C3×C3⋊D3—41no—
123C2×C3×C3⋊D3—41no—
124C2×C3×C3⋊D3—41no—
125C33×C4—41no—
126C33×C4C2×D411yescharacteristic
127C33×C4#4111yescharacteristic
128C3×C3⋊Dic3—21no—
129C3×C3⋊Dic3—41no—
130C3×C3⋊Dic3—41no—
131C3×#121—91no—
132#946—91no—
133C2×#215—91no—
134C2×#214—91no—
135#924—91no—
136C2×#208—91no—
137C2×#204—91no—
138C2×ΓL(1,𝔽25)—91no—
139C2×ΓL(1,𝔽25)—91no—
140#897—91no—
141#895—31no—
142#895—31no—
143#895—61no—
144C3×#107—91no—
145C3×#103—91no—
146#832—91no—
147#821—91no—
148#810—31no—
149#810—31no—
150#810—61no—
151#809—91no—
152#807—31no—
153#807—31no—
154#807—61no—
155C22×C3⋊D3—21no—
156C22×C3⋊D3#6711yescharacteristic
157C22×C3×D3—31no—
158C22×C3×D3—31no—
159C22×C3×D3—61no—
160C22×C3×D3—62no—
161C22×C3×D3—61no—
162C22×C3×D3—61no—
163C2×C32×C4—21no—
164C2×C32×C4—22no—
165C2×C32×C4—21no—
166C2×C32×C4—21no—
167C2×C32×C4—21no—
168C2×C32×C4—21no—
169C2×C32×C4—21no—
170C2×C32×C4—41no—
171C2×C32×C4—41no—
172C2×C32×C4—41no—
173C2×C32×C4—42no—
174C2×C32×C4C22×D311yescharacteristic
175C2×C32×C4C22×D311yescharacteristic
176C2×C32×C4C22×D311yescharacteristic
177C2×C3⋊Dic3—21no—
178C2×C3⋊Dic3—21no—
179C2×C3⋊Dic3#6711yescharacteristic
180C4⋊(C3⋊D3)—21no—
181C4⋊(C3⋊D3)—21no—
182C4⋊(C3⋊D3)—42no—
183C4×C3⋊D3—21no—
184C4×C3⋊D3—21no—
185#634—21no—
186#634—21no—
187C2×C3×Dic3—31no—
188C2×C3×Dic3—31no—
189C2×C3×Dic3—61no—
190C2×C3×Dic3—61no—
191C2×C3×Dic3—61no—
192C2×C3×Dic3—61no—
193C2×C3×Dic3—61no—
194C2×C3×Dic3—121no—
195C2×C3×Dic3—121no—
196C3×D12—61no—
197C3×D12—61no—
198C3×D12—61no—
199C3×D12—61no—
200C3×D12—121no—
201C3×D12—121no—
202C3×D12—124no—
203C3×D12—122no—
204C3×D12—122no—
205C3×C4×D3—61no—
206C3×C4×D3—61no—
207C3×C4×D3—61no—
208C3×C4×D3—61no—
209C3×C4×D3—121no—
210C3×C4×D3—121no—
211C3×Dic6—62no—
212C3×Dic6—61no—
213C3×Dic6—61no—
214C3×Dic6—61no—
215C3×Dic6—61no—
216C3×Dic6—121no—
217C3⋊(C3⋊C8)—21no—
218C3⋊(C3⋊C8)D1212yescomplemented
219C3×C3⋊C8—32no—
220C3×C3⋊C8—32no—
221C3×C3⋊C8—62no—
222C3×C3⋊C8—61no—
223C3×C3⋊C8—61no—
224C3×C3⋊C8—121no—
225#468—271noSylow
226C2×C33#12111yescharacteristic
227C2×C33AGL(1,ℤ/8)11yescharacteristic
228C2×C33#13711yescharacteristic
229C3×C3⋊D3—41no—
230C3×C3⋊D3—81no—
231C2×C3×Q8—91no—
232C2×C3×D4—91no—
233C2×C3×D4—181no—
234C22×C3×C4—91no—
235C2×D12—31no—
236C2×D12—31no—
237C2×D12—61no—
238C2×D12—61no—
239C2×D12—61no—
240C2×D12—62no—
241C2×C4×D3—31no—
242C2×C4×D3—31no—
243C2×C4×D3—61no—
244C2×Dic6—31no—
245C2×Dic6—31no—
246C2×Dic6—61no—
247C2×C3×C8—91no—
248C2×C3×C8—91no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—41no—
253C3×C4⋊C4—91no—
254C3×C4⋊C4D3211yescharacteristic
255C3×C4⋊C4D3211yescharacteristic
256C3×C4⋊C4D3211yescharacteristic
257C3×C22⋊C4—181no—
258#217—181no—
259#215—182no—
260#214—182no—
261(C2×D3)⋊C4—61no—
262(C2×D3)⋊C4—61no—
263(C2×D3)⋊C4—62no—
264(C2×D3)⋊C4—181no—
265(C2×D3)⋊C4—181no—
266C4⋊Dic3—31no—
267C4⋊Dic3—31no—
268C4⋊Dic3—61no—
269C4⋊Dic3—181no—
270Dic3⋊C4—181no—
271Dic3⋊C4—181no—
272#208—182no—
273C2×C3⋊C8—31no—
274C2×C3⋊C8—31no—
275C2×C3⋊C8—31no—
276C2×C3⋊C8—31no—
277C2×C3⋊C8—61no—
278C2×C3⋊C8—61no—
279#204—182no—
280ΓL(1,𝔽25)—182no—
281ΓL(1,𝔽25)—182no—
282C22×C32—21no—
283C22×C32—21no—
284C22×C32—21no—
285C22×C32—41no—
286C22×C32D3×D411yescharacteristic
287C22×C32D3×D411yescharacteristic
288C22×C32C2×#6711yescharacteristic
289C2×C3⋊D3—21no—
290C2×C3⋊D3—21no—
291C2×C3⋊D3—21no—
292C2×C3⋊D3—41no—
293C2×C3⋊D3—41no—
294C2×C3⋊D3—41no—
295C2×C3×D3—62no—
296C2×C3×D3—61no—
297C2×C3×D3—61no—
298C2×C3×D3—61no—
299C2×C3×D3—61no—
300C2×C3×D3—61no—
301C2×C3×D3—61no—
302C2×C3×D3—121no—
303C2×C3×D3—121no—
304C2×C3×D3—122no—
305C2×C3×D3—122no—
306C2×C3×D3—122no—
307C2×C3×D3—121no—
308C2×C3×D3—121no—
309C2×C3×D3—121no—
310C2×C3×D3—121no—
311C2×C3×D3—121no—
312C2×C3×D3—121no—
313C32×C4—21no—
314C32×C4—21no—
315C32×C4—21no—
316C32×C4—21no—
317C32×C4—21no—
318C32×C4—21no—
319C32×C4—41no—
320C32×C4—41no—
321C32×C4—41no—
322C32×C4—42no—
323C32×C4—41no—
324C32×C4—41no—
325C32×C4—81no—
326C32×C4—81no—
327C32×C4—82no—
328C32×C4C2×D1211yescharacteristic
329C32×C4#23511yescharacteristic
330C32×C4D3×D411yescharacteristic
331C32×C4D3×D411yescharacteristic
332C32×C4#23711yescharacteristic
333C32×C4#23911yescharacteristic
334C3⋊Dic3—21no—
335C3⋊Dic3—41no—
336C3⋊Dic3—41no—
337C3×Dic3—61no—
338C3×Dic3—61no—
339C3×Dic3—121no—
340C3×Dic3—122no—
341C3×Dic3—121no—
342C3×Dic3—121no—
343C3×Dic3—121no—
344C3×Dic3—121no—
345C3×Dic3—241no—
346C2×SD16—271no—
347C2×M16—271no—
348C22⋊Q8—271no—
349#121—91no—
350#107—91no—
351#103—91no—
352#102—271no—
353C33#46811yescharacteristic, complemented, Sylow
354C23×C3—91no—
355C23×C3—181no—
356C22×D3—31no—
357C22×D3—31no—
358C22×D3—61no—
359C22×D3—62no—
360C22×D3—61no—
361C22×D3—61no—
362C3×Q8—181no—
363C3×Q8—181no—
364C3×D4—181no—
365C3×D4—181no—
366C3×D4—362no—
367C2×C3×C4—22no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—42no—
377C2×C3×C4—41no—
378C2×C3×C4—91no—
379C2×C3×C4—181no—
380C2×C3×C4—181no—
381C2×C3×C4—181no—
382C2×C3×C4—181no—
383C2×C3×C4C2×D3211yescharacteristic
384C2×C3×C4C2×D3211yescharacteristic
385C2×C3×C4C2×D3211yescharacteristic
386C2×Dic3—31no—
387C2×Dic3—31no—
388C2×Dic3—61no—
389C2×Dic3—61no—
390C2×Dic3—61no—
391C2×Dic3—61no—
392C2×Dic3—61no—
393C2×Dic3—121no—
394C2×Dic3—121no—
395D12—61no—
396D12—61no—
397D12—61no—
398D12—61no—
399D12—121no—
400D12—121no—
401D12—124no—
402D12—122no—
403D12—122no—
404C4×D3—61no—
405C4×D3—61no—
406C4×D3—61no—
407C4×D3—61no—
408C4×D3—121no—
409C4×D3—121no—
410Dic6—62no—
411Dic6—61no—
412Dic6—61no—
413Dic6—61no—
414Dic6—61no—
415Dic6—121no—
416C3×C8—92no—
417C3×C8—181no—
418C3⋊C8—32no—
419C3⋊C8—32no—
420C3⋊C8—62no—
421C3⋊C8—61no—
422C3⋊C8—61no—
423C3⋊C8—121no—
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#89511yescharacteristic
438C2×C32#91111yescharacteristic
439C2×C32#91411yescharacteristic
440C2×C32#91811yescharacteristic
441C2×C32#91811yescharacteristic
442C2×C32#93211yescharacteristic
443C2×C32#93911yescharacteristic
444C2×C32#95111yescharacteristic
445C3⋊D3—41no—
446C3⋊D3—81no—
447C3×D3—122no—
448C3×D3—121no—
449C3×D3—121no—
450C3×D3—242no—
451C3×D3—241no—
452C3×D3—241no—
453C2×Q8—91no—
454C2×D4—91no—
455C2×D4—181no—
456C22×C4—91no—
457SD16—542no—
458M16—542no—
459C2×C8—91no—
460C2×C8—91no—
461C4⋊C4—91no—
462C4⋊C4—541no—
463C4⋊C4#111311yescharacteristic
464C22⋊C4—181no—
465C22⋊C4—541no—
466C22×C3—21no—
467C22×C3—21no—
468C22×C3—21no—
469C22×C3—41no—
470C22×C3—181no—
471C22×C3—181no—
472C22×C3—181no—
473C22×C3—361no—
474C22×C3—361no—
475C22×C3—361no—
476C22×C3D3×#6711yescharacteristic
477C22×C3D3×#6711yescharacteristic
478C22×C3#381111yescharacteristic
479C2×D3—62no—
480C2×D3—61no—
481C2×D3—61no—
482C2×D3—61no—
483C2×D3—61no—
484C2×D3—61no—
485C2×D3—61no—
486C2×D3—121no—
487C2×D3—121no—
488C2×D3—122no—
489C2×D3—122no—
490C2×D3—122no—
491C2×D3—121no—
492C2×D3—121no—
493C2×D3—121no—
494C2×D3—121no—
495C2×D3—121no—
496C2×D3—121no—
497C3×C4—21no—
498C3×C4—21no—
499C3×C4—21no—
500C3×C4—21no—
501C3×C4—21no—
502C3×C4—21no—
503C3×C4—42no—
504C3×C4—41no—
505C3×C4—41no—
506C3×C4—41no—
507C3×C4—41no—
508C3×C4—41no—
509C3×C4—81no—
510C3×C4—81no—
511C3×C4—82no—
512C3×C4—181no—
513C3×C4—361no—
514C3×C4—361no—
515C3×C4#379511yescharacteristic
516C3×C4#379611yescharacteristic
517C3×C4#379911yescharacteristic
518C3×C4D3×D1211yescharacteristic
519C3×C4D3×D1211yescharacteristic
520C3×C4#380211yescharacteristic
521Dic3—61no—
522Dic3—61no—
523Dic3—121no—
524Dic3—122no—
525Dic3—121no—
526Dic3—121no—
527Dic3—121no—
528Dic3—121no—
529Dic3—241no—
530C32—21no—
531C32—21no—
532C32—21no—
533C32—41no—
534C32#481211yescharacteristic, complemented
535C32#488611yescharacteristic, complemented
536C32#504211yescharacteristic, complemented
537C23—91no—
538C23—181no—
539Q8—181no—
540Q8—181no—
541D4—181no—
542D4—181no—
543D4—362no—
544C2×C4—21no—
545C2×C4—91no—
546C2×C4—181no—
547C2×C4—181no—
548C2×C4—181no—
549C2×C4—181no—
550C2×C4C2×#111311yescharacteristic
551C8—92no—
552C8—181no—
553C2×C3—21no—
554C2×C3—21no—
555C2×C3—21no—
556C2×C3—21no—
557C2×C3—21no—
558C2×C3—21no—
559C2×C3—21no—
560C2×C3—21no—
561C2×C3—21no—
562C2×C3—41no—
563C2×C3—41no—
564C2×C3—41no—
565C2×C3—361no—
566C2×C3—721no—
567C2×C3#6388511yescharacteristic
568C2×C3#6388611yescharacteristic
569C2×C3#6391011yescharacteristic
570C2×C3#6393211yescharacteristic
571C2×C3#6393411yescharacteristic
572C2×C3#6393711yescharacteristic
573C2×C3#6394311yescharacteristic
574C2×C3#6394311yescharacteristic
575C2×C3#6394511yescharacteristic
576D3—122no—
577D3—121no—
578D3—121no—
579D3—242no—
580D3—241no—
581D3—241no—
582C22—181no—
583C22—181no—
584C22—181no—
585C22—361no—
586C22—361no—
587C22—361no—
588C22#8904911yescharacteristic
589C4—41no—
590C4—181no—
591C4—361no—
592C4—361no—
593C4#8904111yescharacteristic
594C4C4⋊#111311yescharacteristic
595C3—21no—
596C3—21no—
597C3—21no—
598C3—41no—
599C3#1059051511yescharacteristic, complemented
600C3#1059052211yescharacteristic, complemented
601C3#1059058811yescharacteristic, complemented
602C2—361no—
603C2—721no—
604C2#1172368711yescharacteristic
605C2#1172370611yescharacteristic
606C2#1172371111yescharacteristic
subgroupG/H
centerC22#89049
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC3×#64104C2
hypercenterC4⋊C4#1113
hypocenterC33#468
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #11723674C2semidirect
3. C2×#88811C2semidirect
4. C2×#88809C2semidirect
5. C3×#63637C2semidirect
6. C3×#63634C2semidirect
7. #11722075C2semidirect
37. #63637D3semidirect
38. #63634D3semidirect
63. C3×C3⋊(C3⋊C8)D4semidirect
218. C3⋊(C3⋊C8)D12semidirect
353. C33#468semidirect
534. C32#4812semidirect
535. C32#4886semidirect
536. C32#5042semidirect
599. C3#10590515semidirect
600. C3#10590522semidirect
601. C3#10590588semidirect

Maximal subgroups

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

subgroupindex
C3×#641042
#117236742
C2×#888112
C2×#888092
C3×#636372
C3×#636342
#117220752
#105905883
#105905223
#105905153