Finite Group Databasebeta

← #13139883

Subgroups — #13139883

Order 1728.

846 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6192C211yesmaximal, characteristic
2C2×#88536C211yesmaximal, characteristic, complemented
3C2×C3×(C2×D9)⋊C4C211yesmaximal, characteristic, complemented
4C2×C9×C4⋊Dic3C211yesmaximal, characteristic, complemented
5C2×#88445C211yesmaximal, characteristic, complemented
6C2×(C2×D9)⋊Dic3C211yesmaximal, characteristic, complemented
7#11720331C211yesmaximal, characteristic, complemented
8#10591392—31nomaximal
9#10588671—31nomaximal
10C23×C3⋊D9C2211yescharacteristic
11C23×C3×D9C2211yescharacteristic
12C22×C3×C4×C9C2211yescharacteristic
13C22×C3⋊Dic9C2211yescharacteristic
14C22×C9×Dic3C2211yescharacteristic
15C22×C9×Dic3C2211yescharacteristic
16C22×C3×Dic9C2211yescharacteristic
17C2×#6192—22no—
18C2×#6192—22no—
19C2×#6192—22no—
20#88536—22no—
21C3×(C2×D9)⋊C4—22no—
22C9×C4⋊Dic3—22no—
23#88445—22no—
24(C2×D9)⋊Dic3—22no—
25C22×#626—31no—
26C2×#3752—31no—
27C2×C3×(C2×D3)⋊C4—31no—
28C2×C3×C4⋊Dic3D311yescharacteristic
29C2×#3722—31no—
30C2×(C2×D3)⋊Dic3—31no—
31C22×D36—31no—
32#63579—31no—
33C2×C9×C4⋊C4—31no—
34C2×(C2×D9)⋊C4—31no—
35C2×(C2×D9)⋊C4—31no—
36C2×(C2×D9)⋊C4—31no—
37C2×(C2×D9)⋊C4—61no—
38C2×(C2×D9)⋊C4D311yescharacteristic, complemented
39C9⋊#95—31no—
40C23×C3×C9C2311yescharacteristic
41C22×C3⋊D9—21no—
42C22×C3⋊D9—21no—
43C22×C3⋊D9—21no—
44C22×C3⋊D9—21no—
45C22×C3⋊D9—21no—
46C22×C3⋊D9—21no—
47C22×C3⋊D9D412yes—
48C22×C3×D9—21no—
49C22×C3×D9—21no—
50C22×C3×D9—21no—
51C22×C3×D9—21no—
52C22×C3×D9—21no—
53C22×C3×D9—21no—
54C22×C3×D9D412yes—
55C2×C3×C4×C9—21no—
56C2×C3×C4×C9—21no—
57C2×C3×C4×C9D412yescomplemented
58C2×C3⋊Dic9—21no—
59C2×C3⋊Dic9—21no—
60C2×C3⋊Dic9—21no—
61C2×C9×Dic3—21no—
62C2×C9×Dic3—21no—
63C2×C9×Dic3—21no—
64C2×C9×Dic3—21no—
65C2×C9×Dic3—21no—
66C2×C9×Dic3D412yes—
67C2×C3×Dic9—21no—
68C2×C3×Dic9—21no—
69C2×C3×Dic9—21no—
70#6192—28no—
71#4991—91no—
72#4961—91no—
73C23×C3⋊D3—31no—
74C23×C3×D3—31no—
75C22×C32×C4C2×D311yescharacteristic
76C22×C3⋊Dic3—31no—
77C22×C3×Dic3—31no—
78C22×C3×Dic3C2×D311yescharacteristic
79C22×C3×Dic3C2×D311yescharacteristic
80C2×#626—62no—
81C2×#626—62no—
82C2×#626—62no—
83C23×D9—31no—
84C23×D9—61no—
85C23×D9C2×D311yescharacteristic
86#3752—62no—
87C3×(C2×D3)⋊C4—62no—
88C3×C4⋊Dic3—22no—
89#3722—62no—
90(C2×D3)⋊Dic3—62no—
91C22×C4×C9—21no—
92C22×C4×C9—31no—
93C22×C4×C9—31no—
94C22×C4×C9C2×D311yescharacteristic
95C22×Dic9—31no—
96C22×Dic9—61no—
97C22×Dic9C2×D311yescharacteristic
98C2×D36—62no—
99C2×D36—62no—
100C2×D36—62no—
101C9×C4⋊C4—62no—
102(C2×D9)⋊C4—22no—
103(C2×D9)⋊C4—62no—
104(C2×D9)⋊C4—64no—
105(C2×D9)⋊C4—62no—
106(C2×D9)⋊C4—62no—
107C22×C3×C9C2×D411yescharacteristic
108C22×C3×C9C2×D411yescharacteristic
109C22×C3×C9C2×D411yescharacteristic
110C22×C3×C9C2×D411yescharacteristic
111C22×C3×C9#4111yescharacteristic
112C22×C3×C9#4111yescharacteristic
113C22×C3×C9#4111yescharacteristic
114C2×C3⋊D9—22no—
115C2×C3⋊D9—22no—
116C2×C3⋊D9—22no—
117C2×C3⋊D9—41no—
118C2×C3⋊D9—41no—
119C2×C3⋊D9—41no—
120C2×C3⋊D9—41no—
121C2×C3×D9—22no—
122C2×C3×D9—22no—
123C2×C3×D9—22no—
124C2×C3×D9—41no—
125C2×C3×D9—41no—
126C2×C3×D9—41no—
127C2×C3×D9—41no—
128C3×C4×C9—22no—
129C3⋊Dic9—41no—
130C9×Dic3—22no—
131C9×Dic3—41no—
132C3×Dic9—41no—
133C22×#67—91no—
134C22×D12—91no—
135C2×C3×C4⋊C4—31no—
136C2×C3×C22⋊C4—91no—
137C2×#217—91no—
138C2×(C2×D3)⋊C4—31no—
139C2×(C2×D3)⋊C4—91no—
140C2×(C2×D3)⋊C4—91no—
141C2×(C2×D3)⋊C4—91no—
142C2×(C2×D3)⋊C4—91no—
143C2×(C2×D3)⋊C4—91no—
144C2×(C2×D3)⋊C4—181no—
145C2×C4⋊Dic3D911yescharacteristic, complemented
146C3⋊#95—91no—
147C3⋊#95—91no—
148C23×C32C22×D311yescharacteristic
149C22×C3⋊D3—32no—
150C22×C3⋊D3—61no—
151C22×C3⋊D3—61no—
152C22×C3⋊D3—61no—
153C22×C3⋊D3—61no—
154C22×C3⋊D3—61no—
155C22×C3⋊D3—61no—
156C22×C3×D3—32no—
157C22×C3×D3—61no—
158C22×C3×D3—61no—
159C22×C3×D3—61no—
160C22×C3×D3—61no—
161C22×C3×D3—61no—
162C22×C3×D3—61no—
163C2×C32×C4—21no—
164C2×C32×C4—21no—
165C2×C32×C4D1212yes—
166C2×C3⋊Dic3—61no—
167C2×C3⋊Dic3—61no—
168C2×C3⋊Dic3—61no—
169C2×C3×Dic3—21no—
170C2×C3×Dic3—21no—
171C2×C3×Dic3—21no—
172C2×C3×Dic3—21no—
173C2×C3×Dic3—21no—
174C2×C3×Dic3—61no—
175C2×C3×Dic3—61no—
176C2×C3×Dic3—61no—
177C2×C3×Dic3#6712yes—
178#626—68no—
179C23×C9—21no—
180C23×C9C22×D311yescharacteristic
181C22×D9—21no—
182C22×D9—21no—
183C22×D9—21no—
184C22×D9—21no—
185C22×D9—21no—
186C22×D9—21no—
187C22×D9—32no—
188C22×D9—62no—
189C22×D9—62no—
190C22×D9—62no—
191C22×D9—61no—
192C22×D9—61no—
193C22×D9—61no—
194C22×D9—61no—
195C22×D9—61no—
196C22×D9—61no—
197C22×D9—121no—
198C22×D9—121no—
199C22×D9—121no—
200C22×D9—121no—
201C22×D9D1212yes—
202C2×C4×C9—21no—
203C2×C4×C9—21no—
204C2×C4×C9—22no—
205C2×C4×C9—32no—
206C2×C4×C9—41no—
207C2×C4×C9—41no—
208C2×C4×C9—61no—
209C2×C4×C9—61no—
210C2×C4×C9—61no—
211C2×C4×C9—61no—
212C2×C4×C9—61no—
213C2×C4×C9#6712yescomplemented
214C2×Dic9—21no—
215C2×Dic9—21no—
216C2×Dic9—21no—
217C2×Dic9—62no—
218C2×Dic9—61no—
219C2×Dic9—61no—
220C2×Dic9—61no—
221C2×Dic9—121no—
222C2×Dic9—121no—
223D36—68no—
224#394—271noSylow
225C2×C3×C9#12011yescharacteristic
226C2×C3×C9#12111yescharacteristic
227C2×C3×C9#12111yescharacteristic
228C2×C3×C9#12111yescharacteristic
229C2×C3×C9#12311yescharacteristic
230C2×C3×C9#12311yescharacteristic
231C2×C3×C9#12411yescharacteristic
232C3⋊D9—42no—
233C3×D9—42no—
234C24×C3—91no—
235C23×D3—31no—
236C23×D3—91no—
237C23×D3—91no—
238C23×D3—181no—
239C22×C3×C4—21no—
240C22×C3×C4—31no—
241C22×C3×C4—31no—
242C22×C3×C4—91no—
243C22×C3×C4C2×D911yescharacteristic
244C22×C3×C4D3211yescharacteristic
245C2×#67—182no—
246C2×#67—182no—
247C2×#67—182no—
248C22×Dic3—31no—
249C22×Dic3—91no—
250C22×Dic3—91no—
251C22×Dic3—181no—
252C22×Dic3C2×D911yescharacteristic
253C22×Dic3C2×D911yescharacteristic
254C2×D12—182no—
255C2×D12—182no—
256C2×D12—182no—
257C3×C4⋊C4—62no—
258C3×C22⋊C4—182no—
259#217—182no—
260(C2×D3)⋊C4—62no—
261(C2×D3)⋊C4—182no—
262(C2×D3)⋊C4—182no—
263(C2×D3)⋊C4—184no—
264(C2×D3)⋊C4—182no—
265(C2×D3)⋊C4—182no—
266(C2×D3)⋊C4—182no—
267C4⋊Dic3—22no—
268C22×C32C2×D1211yescharacteristic
269C22×C32#23511yescharacteristic
270C22×C32D3×D411yescharacteristic
271C22×C32D3×D411yescharacteristic
272C22×C32#23911yescharacteristic
273C22×C32#23911yescharacteristic
274C22×C32C2×#6711yescharacteristic
275C2×C3⋊D3—62no—
276C2×C3⋊D3—62no—
277C2×C3⋊D3—62no—
278C2×C3⋊D3—121no—
279C2×C3⋊D3—121no—
280C2×C3⋊D3—121no—
281C2×C3⋊D3—121no—
282C2×C3×D3—62no—
283C2×C3×D3—62no—
284C2×C3×D3—62no—
285C2×C3×D3—121no—
286C2×C3×D3—121no—
287C2×C3×D3—121no—
288C2×C3×D3—121no—
289C32×C4—22no—
290C3⋊Dic3—121no—
291C3×Dic3—22no—
292C3×Dic3—41no—
293C3×Dic3—121no—
294C22×C9—21no—
295C22×C9—21no—
296C22×C9—21no—
297C22×C9—21no—
298C22×C9—21no—
299C22×C9—21no—
300C22×C9—21no—
301C22×C9C2×D1211yescharacteristic
302C22×C9#23511yescharacteristic
303C22×C9D3×D411yescharacteristic
304C22×C9D3×D411yescharacteristic
305C22×C9#23711yescharacteristic
306C22×C9#23711yescharacteristic
307C22×C9C2×#6711yescharacteristic
308C2×D9—22no—
309C2×D9—22no—
310C2×D9—22no—
311C2×D9—41no—
312C2×D9—41no—
313C2×D9—41no—
314C2×D9—41no—
315C2×D9—64no—
316C2×D9—62no—
317C2×D9—62no—
318C2×D9—62no—
319C2×D9—122no—
320C2×D9—122no—
321C2×D9—122no—
322C2×D9—122no—
323C2×D9—122no—
324C2×D9—122no—
325C2×D9—121no—
326C2×D9—121no—
327C2×D9—121no—
328C2×D9—121no—
329C4×C9—22no—
330C4×C9—42no—
331C4×C9—62no—
332C4×C9—121no—
333Dic9—41no—
334Dic9—122no—
335Dic9—121no—
336C22×D4—271no—
337C2×C4⋊C4—31no—
338C2×C22⋊C4—91no—
339C2×C22⋊C4—271no—
340C2×C22⋊C4—271no—
341C2×C22⋊C4—271no—
342#95—271no—
343C3×C9#39411yescharacteristic, complemented, Sylow
344C23×C3—21no—
345C23×C3—92no—
346C23×C3—181no—
347C23×C3—181no—
348C23×C3—181no—
349C23×C3—181no—
350C23×C3—181no—
351C23×C3—181no—
352C23×C3C22×D911yescharacteristic
353C23×C3C2×D3211yescharacteristic
354C22×D3—32no—
355C22×D3—61no—
356C22×D3—61no—
357C22×D3—61no—
358C22×D3—61no—
359C22×D3—61no—
360C22×D3—61no—
361C22×D3—92no—
362C22×D3—92no—
363C22×D3—182no—
364C22×D3—182no—
365C22×D3—182no—
366C22×D3—181no—
367C22×D3—181no—
368C22×D3—181no—
369C22×D3—181no—
370C22×D3—181no—
371C22×D3—181no—
372C22×D3—181no—
373C22×D3—181no—
374C22×D3—181no—
375C22×D3—181no—
376C22×D3—181no—
377C22×D3—181no—
378C22×D3—361no—
379C22×D3—361no—
380C22×D3—361no—
381C22×D3—361no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—21no—
386C2×C3×C4—22no—
387C2×C3×C4—32no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—61no—
391C2×C3×C4—61no—
392C2×C3×C4—61no—
393C2×C3×C4—61no—
394C2×C3×C4—61no—
395C2×C3×C4—181no—
396C2×C3×C4—181no—
397C2×C3×C4—181no—
398C2×C3×C4D3612yescomplemented
399C2×C3×C4#62612yes—
400#67—188no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3—61no—
407C2×Dic3—61no—
408C2×Dic3—61no—
409C2×Dic3—182no—
410C2×Dic3—181no—
411C2×Dic3—181no—
412C2×Dic3—181no—
413C2×Dic3—181no—
414C2×Dic3—181no—
415C2×Dic3—181no—
416C2×Dic3—361no—
417C2×Dic3—361no—
418C2×Dic3#61112yes—
419D12—188no—
420C2×C32#89411yescharacteristic
421C2×C32#89411yescharacteristic
422C2×C32#89511yescharacteristic
423C2×C32#89511yescharacteristic
424C2×C32#93011yescharacteristic
425C2×C32#93711yescharacteristic
426C2×C32#94711yescharacteristic
427C3⋊D3—122no—
428C3×D3—122no—
429C2×C9—21no—
430C2×C9—21no—
431C2×C9—21no—
432C2×C9—21no—
433C2×C9—21no—
434C2×C9—21no—
435C2×C9—21no—
436C2×C9#88211yescharacteristic
437C2×C9#88311yescharacteristic
438C2×C9#88511yescharacteristic
439C2×C9#88611yescharacteristic
440C2×C9#93011yescharacteristic
441C2×C9#93811yescharacteristic
442C2×C9#93911yescharacteristic
443D9—42no—
444D9—124no—
445D9—122no—
446C24—91no—
447C24—271no—
448C2×D4—542no—
449C2×D4—542no—
450C2×D4—542no—
451C22×C4—31no—
452C22×C4—31no—
453C22×C4—91no—
454C22×C4—271no—
455C22×C4D3×D911yescharacteristic
456C4⋊C4—62no—
457C22⋊C4—182no—
458C22⋊C4—542no—
459C22⋊C4—542no—
460C22⋊C4—542no—
461C22×C3—21no—
462C22×C3—21no—
463C22×C3—21no—
464C22×C3—21no—
465C22×C3—21no—
466C22×C3—21no—
467C22×C3—21no—
468C22×C3—182no—
469C22×C3—182no—
470C22×C3—182no—
471C22×C3—361no—
472C22×C3—361no—
473C22×C3—361no—
474C22×C3—361no—
475C22×C3C2×D3611yescharacteristic
476C22×C3#369711yescharacteristic
477C22×C3D4×D911yescharacteristic
478C22×C3D4×D911yescharacteristic
479C22×C3#370111yescharacteristic
480C22×C3#370111yescharacteristic
481C22×C3C2×#61111yescharacteristic
482C22×C3#379611yescharacteristic
483C22×C3#379911yescharacteristic
484C22×C3D3×D1211yescharacteristic
485C22×C3#380211yescharacteristic
486C22×C3#380411yescharacteristic
487C22×C3C2×#62611yescharacteristic
488C22×C3D3×#6711yescharacteristic
489C2×D3—62no—
490C2×D3—62no—
491C2×D3—62no—
492C2×D3—121no—
493C2×D3—121no—
494C2×D3—121no—
495C2×D3—121no—
496C2×D3—184no—
497C2×D3—182no—
498C2×D3—182no—
499C2×D3—182no—
500C2×D3—182no—
501C2×D3—182no—
502C2×D3—182no—
503C2×D3—362no—
504C2×D3—362no—
505C2×D3—362no—
506C2×D3—362no—
507C2×D3—362no—
508C2×D3—362no—
509C2×D3—361no—
510C2×D3—361no—
511C2×D3—361no—
512C2×D3—361no—
513C2×D3—361no—
514C2×D3—361no—
515C2×D3—361no—
516C2×D3—361no—
517C3×C4—22no—
518C3×C4—22no—
519C3×C4—42no—
520C3×C4—62no—
521C3×C4—121no—
522C3×C4—361no—
523Dic3—22no—
524Dic3—41no—
525Dic3—121no—
526Dic3—362no—
527Dic3—361no—
528Dic3—361no—
529C32#499111yescharacteristic
530C9—21no—
531C9#496111yescharacteristic, complemented
532C23—92no—
533C23—181no—
534C23—181no—
535C23—181no—
536C23—181no—
537C23—181no—
538C23—181no—
539C23—272no—
540C23—541no—
541C23—541no—
542C23—541no—
543C23—541no—
544C23—541no—
545C23—541no—
546C23C2×D3×D911yescharacteristic
547D4—548no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—32no—
551C2×C4—61no—
552C2×C4—61no—
553C2×C4—61no—
554C2×C4—61no—
555C2×C4—61no—
556C2×C4—181no—
557C2×C4—181no—
558C2×C4—181no—
559C2×C4—541no—
560C2×C4—541no—
561C2×C4—541no—
562C2×C4#619212yescomplemented
563C2×C3—21no—
564C2×C3—21no—
565C2×C3—21no—
566C2×C3—21no—
567C2×C3—21no—
568C2×C3—21no—
569C2×C3—21no—
570C2×C3—362no—
571C2×C3#6345611yescharacteristic
572C2×C3#6345611yescharacteristic
573C2×C3#6345711yescharacteristic
574C2×C3#6345711yescharacteristic
575C2×C3#6349211yescharacteristic
576C2×C3#6349911yescharacteristic
577C2×C3#6350911yescharacteristic
578C2×C3#6385411yescharacteristic
579C2×C3#6389011yescharacteristic
580C2×C3#6390711yescharacteristic
581C2×C3#6390911yescharacteristic
582C2×C3#6391611yescharacteristic
583C2×C3#6392211yescharacteristic
584C2×C3#6397811yescharacteristic
585D3—122no—
586D3—364no—
587D3—362no—
588D3—362no—
589C22—182no—
590C22—182no—
591C22—182no—
592C22—361no—
593C22—361no—
594C22—361no—
595C22—361no—
596C22—542no—
597C22—542no—
598C22—542no—
599C22—1081no—
600C22—1081no—
601C22—1081no—
602C22—1081no—
603C22#8863411yescharacteristic
604C22#8863511yescharacteristic
605C22D9×D1211yescharacteristic
606C22#8864611yescharacteristic
607C22#8865811yescharacteristic
608C22C2×#619211yescharacteristic
609C22D3×#61111yescharacteristic
610C4—22no—
611C4—62no—
612C4—121no—
613C4—361no—
614C4—1081no—
615C3—21no—
616C3#1058867111yescharacteristic, complemented
617C3#1059139211yescharacteristic
618C2—362no—
619C2—1082no—
620C2#1172094211yescharacteristic
621C2#1172098811yescharacteristic
622C2#1172100311yescharacteristic
623C2#1172100411yescharacteristic
624C2#1172100711yescharacteristic
625C2#1172105011yescharacteristic
626C2#1172127711yescharacteristic
subgroupG/H
centerC23C2×D3×D9
commutator, cosocleC23×C3×C9C23
FrattiniC23×C3C2×D32
FittingC22×C3×C4×C9C22
socleC23×C32C22×D3
prime radicalC22×#6192C2
hypercenterC22×C4D3×D9
hypocenterC3×C9#394
LCSG ▷ C23×C3×C9 ▷ C3×C9
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×#88536C2semidirect
3. C2×C3×(C2×D9)⋊C4C2semidirect
4. C2×C9×C4⋊Dic3C2semidirect
5. C2×#88445C2semidirect
6. C2×(C2×D9)⋊Dic3C2semidirect
7. #11720331C2semidirect
38. C2×(C2×D9)⋊C4D3semidirect
57. C2×C3×C4×C9D4semidirect
145. C2×C4⋊Dic3D9semidirect
213. C2×C4×C9#67semidirect
343. C3×C9#394semidirect
398. C2×C3×C4D36semidirect
531. C9#4961semidirect
562. C2×C4#6192semidirect
616. C3#10588671semidirect

Maximal subgroups

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

subgroupindex
C22×#61922
C2×#885362
C2×C3×(C2×D9)⋊C42
C2×C9×C4⋊Dic32
C2×#884452
C2×(C2×D9)⋊Dic32
#117203312
#105913923
#105886713