Finite Group Databasebeta

← #13139907

Subgroups — #13139907

Order 1728.

806 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
1C22×#6194C211yesmaximal, characteristic
2C2×#88534C211yesmaximal, characteristic, complemented
3C2×C9×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
4C2×C3×(C2×D9)⋊C4C211yesmaximal, characteristic, complemented
5C2×(C2×D3)⋊Dic9C211yesmaximal, characteristic, complemented
6C2×(C2×D9)⋊Dic3C211yesmaximal, characteristic, complemented
7#11720331C211yesmaximal, characteristic, complemented
8#10591388—31nomaximal
9#10588639—31nomaximal
10C23×D3×C9C2211yescharacteristic
11C23×C3×D9C2211yescharacteristic
12C22×C3×C4×C9C2211yescharacteristic
13C22×C3⋊Dic9C2211yescharacteristic
14C22×C3⋊Dic9C2211yescharacteristic
15C22×C9×Dic3C2211yescharacteristic
16C22×C3×Dic9C2211yescharacteristic
17C2×#6194—22no—
18C2×#6194—22no—
19C2×#6194—22no—
20#88534—22no—
21C9×(C2×D3)⋊C4—22no—
22C3×(C2×D9)⋊C4—22no—
23(C2×D3)⋊Dic9—22no—
24(C2×D9)⋊Dic3—22no—
25C22×#625—31no—
26C2×#3750—31no—
27C2×C3×(C2×D3)⋊C4—31no—
28C2×C3×(C2×D3)⋊C4D311yescharacteristic
29C2×(C2×D3)⋊Dic3—31no—
30C2×(C2×D3)⋊Dic3—31no—
31C22×#611—31no—
32#63579—31no—
33C2×C9×C22⋊C4—31no—
34C2×#3676—31no—
35C2×(C2×D9)⋊C4—31no—
36C2×(C2×D9)⋊C4D311yescharacteristic, complemented
37C2×Dic9⋊C4—31no—
38C2×Dic9⋊C4—61no—
39C9⋊#95—31no—
40C23×C3×C9C2311yescharacteristic
41C22×D3×C9—21no—
42C22×D3×C9—21no—
43C22×D3×C9—21no—
44C22×D3×C9—21no—
45C22×D3×C9—21no—
46C22×D3×C9—21no—
47C22×D3×C9D412yes—
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×C9—21no—
58C2×C3⋊Dic9—21no—
59C2×C3⋊Dic9—21no—
60C2×C3⋊Dic9—21no—
61C2×C3⋊Dic9—21no—
62C2×C3⋊Dic9D412yes—
63C2×C3⋊Dic9D412yescomplemented
64C2×C9×Dic3—21no—
65C2×C9×Dic3—21no—
66C2×C9×Dic3—21no—
67C2×C3×Dic9—21no—
68C2×C3×Dic9—21no—
69C2×C3×Dic9—21no—
70#6194—28no—
71#4959—91no—
72#4959—91no—
73C23×C3×D3—31no—
74C23×C3×D3C2×D311yescharacteristic
75C22×C32×C4C2×D311yescharacteristic
76C22×C3⋊Dic3—31no—
77C22×C3⋊Dic3—31no—
78C22×C3×Dic3—31no—
79C22×C3×Dic3C2×D311yescharacteristic
80C2×#625—62no—
81C2×#625—62no—
82C2×#625—62no—
83C24×C9—31no—
84C23×D9C2×D311yescharacteristic
85#3750—62no—
86C3×(C2×D3)⋊C4—22no—
87C3×(C2×D3)⋊C4—62no—
88(C2×D3)⋊Dic3—62no—
89(C2×D3)⋊Dic3—62no—
90C22×C4×C9—21no—
91C22×C4×C9—31no—
92C22×C4×C9C2×D311yescharacteristic
93C2×#611—62no—
94C2×#611—62no—
95C2×#611—62no—
96C22×Dic9—31no—
97C22×Dic9—31no—
98C22×Dic9—61no—
99C22×Dic9—61no—
100C22×Dic9C2×D311yescharacteristic
101C9×C22⋊C4—62no—
102#3676—62no—
103(C2×D9)⋊C4—22no—
104(C2×D9)⋊C4—62no—
105Dic9⋊C4—62no—
106Dic9⋊C4—64no—
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×D3×C9—22no—
115C2×D3×C9—22no—
116C2×D3×C9—22no—
117C2×D3×C9—41no—
118C2×D3×C9—41no—
119C2×D3×C9—41no—
120C2×D3×C9—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—41no—
129C3⋊Dic9—22no—
130C3⋊Dic9—22no—
131C9×Dic3—41no—
132C3×Dic9—41no—
133C22×#67—91no—
134C22×#67—91no—
135C2×C3×C22⋊C4—31no—
136C2×C3×C22⋊C4—91no—
137C2×#217—91no—
138C2×#217—91no—
139C2×(C2×D3)⋊C4—31no—
140C2×(C2×D3)⋊C4—91no—
141C2×(C2×D3)⋊C4—91no—
142C2×(C2×D3)⋊C4D911yescharacteristic, complemented
143C2×Dic3⋊C4—91no—
144C2×Dic3⋊C4—91no—
145C2×Dic3⋊C4—181no—
146C3⋊#95—91no—
147C3⋊#95—91no—
148C23×C32C22×D311yescharacteristic
149C22×C3×D3—21no—
150C22×C3×D3—21no—
151C22×C3×D3—21no—
152C22×C3×D3—21no—
153C22×C3×D3—21no—
154C22×C3×D3—21no—
155C22×C3×D3—32no—
156C22×C3×D3—61no—
157C22×C3×D3—61no—
158C22×C3×D3—61no—
159C22×C3×D3—61no—
160C22×C3×D3—61no—
161C22×C3×D3—61no—
162C22×C3×D3#6712yes—
163C2×C32×C4—21no—
164C2×C32×C4—21no—
165C2×C32×C4—21no—
166C2×C3⋊Dic3—32no—
167C2×C3⋊Dic3—32no—
168C2×C3⋊Dic3—61no—
169C2×C3⋊Dic3—61no—
170C2×C3⋊Dic3—61no—
171C2×C3⋊Dic3—61no—
172C2×C3×Dic3—21no—
173C2×C3×Dic3—21no—
174C2×C3×Dic3—21no—
175C2×C3×Dic3—61no—
176C2×C3×Dic3—61no—
177C2×C3×Dic3—61no—
178#625—68no—
179C23×C9—21no—
180C23×C9—32no—
181C23×C9—61no—
182C23×C9—61no—
183C23×C9—61no—
184C23×C9—61no—
185C23×C9—61no—
186C23×C9—61no—
187C23×C9C22×D311yescharacteristic
188C22×D9—21no—
189C22×D9—21no—
190C22×D9—21no—
191C22×D9—21no—
192C22×D9—21no—
193C22×D9—21no—
194C22×D9#6712yes—
195C2×C4×C9—22no—
196C2×C4×C9—21no—
197C2×C4×C9—21no—
198C2×C4×C9—21no—
199C2×C4×C9—41no—
200C2×C4×C9—41no—
201C2×C4×C9—61no—
202C2×C4×C9—61no—
203C2×C4×C9—61no—
204#611—68no—
205C2×Dic9—21no—
206C2×Dic9—21no—
207C2×Dic9—21no—
208C2×Dic9—32no—
209C2×Dic9—32no—
210C2×Dic9—62no—
211C2×Dic9—62no—
212C2×Dic9—62no—
213C2×Dic9—62no—
214C2×Dic9—61no—
215C2×Dic9—61no—
216C2×Dic9—61no—
217C2×Dic9—61no—
218C2×Dic9—121no—
219C2×Dic9—121no—
220#394—271noSylow
221C2×C3×C9#12011yescharacteristic
222C2×C3×C9#12111yescharacteristic
223C2×C3×C9#12111yescharacteristic
224C2×C3×C9#12111yescharacteristic
225C2×C3×C9#12311yescharacteristic
226C2×C3×C9#12311yescharacteristic
227C2×C3×C9#12411yescharacteristic
228D3×C9—42no—
229C3×D9—42no—
230C24×C3—31no—
231C24×C3—91no—
232C23×D3—31no—
233C23×D3C2×D911yescharacteristic
234C22×C3×C4—21no—
235C22×C3×C4—31no—
236C22×C3×C4—91no—
237C22×C3×C4C2×D911yescharacteristic
238C22×C3×C4D3211yescharacteristic
239C2×#67—182no—
240C2×#67—182no—
241C2×#67—182no—
242C2×#67—182no—
243C2×#67—182no—
244C2×#67—182no—
245C22×Dic3—31no—
246C22×Dic3—91no—
247C22×Dic3—91no—
248C22×Dic3—91no—
249C22×Dic3—91no—
250C22×Dic3—181no—
251C22×Dic3—181no—
252C22×Dic3C2×D911yescharacteristic
253C3×C22⋊C4—62no—
254C3×C22⋊C4—182no—
255#217—182no—
256#217—182no—
257(C2×D3)⋊C4—22no—
258(C2×D3)⋊C4—62no—
259(C2×D3)⋊C4—182no—
260(C2×D3)⋊C4—182no—
261Dic3⋊C4—182no—
262Dic3⋊C4—182no—
263Dic3⋊C4—184no—
264C22×C32#23511yescharacteristic
265C22×C32#23511yescharacteristic
266C22×C32D3×D411yescharacteristic
267C22×C32D3×D411yescharacteristic
268C22×C32D3×D411yescharacteristic
269C22×C32#23711yescharacteristic
270C22×C32C2×#6711yescharacteristic
271C2×C3×D3—22no—
272C2×C3×D3—22no—
273C2×C3×D3—22no—
274C2×C3×D3—41no—
275C2×C3×D3—41no—
276C2×C3×D3—41no—
277C2×C3×D3—41no—
278C2×C3×D3—62no—
279C2×C3×D3—62no—
280C2×C3×D3—62no—
281C2×C3×D3—121no—
282C2×C3×D3—121no—
283C2×C3×D3—121no—
284C2×C3×D3—121no—
285C32×C4—41no—
286C3⋊Dic3—62no—
287C3⋊Dic3—62no—
288C3×Dic3—41no—
289C3×Dic3—121no—
290C22×C9—21no—
291C22×C9—21no—
292C22×C9—21no—
293C22×C9—21no—
294C22×C9—21no—
295C22×C9—21no—
296C22×C9—21no—
297C22×C9—62no—
298C22×C9—62no—
299C22×C9—62no—
300C22×C9—121no—
301C22×C9—121no—
302C22×C9—121no—
303C22×C9—121no—
304C22×C9#23511yescharacteristic
305C22×C9#23511yescharacteristic
306C22×C9D3×D411yescharacteristic
307C22×C9D3×D411yescharacteristic
308C22×C9D3×D411yescharacteristic
309C22×C9#23711yescharacteristic
310C22×C9C2×#6711yescharacteristic
311C2×D9—22no—
312C2×D9—22no—
313C2×D9—22no—
314C2×D9—41no—
315C2×D9—41no—
316C2×D9—41no—
317C2×D9—41no—
318C4×C9—42no—
319C4×C9—41no—
320C4×C9—121no—
321Dic9—41no—
322Dic9—64no—
323Dic9—62no—
324Dic9—62no—
325Dic9—122no—
326C22×D4—271no—
327C2×C4⋊C4—271no—
328C2×C22⋊C4—31no—
329C2×C22⋊C4—91no—
330C2×C22⋊C4—271no—
331C2×C22⋊C4—271no—
332#95—271no—
333C3×C9#39411yescharacteristic, complemented, Sylow
334C23×C3—21no—
335C23×C3—32no—
336C23×C3—61no—
337C23×C3—61no—
338C23×C3—61no—
339C23×C3—61no—
340C23×C3—61no—
341C23×C3—61no—
342C23×C3—92no—
343C23×C3—181no—
344C23×C3—181no—
345C23×C3—181no—
346C23×C3—181no—
347C23×C3—181no—
348C23×C3—181no—
349C23×C3C22×D911yescharacteristic
350C23×C3C2×D3211yescharacteristic
351C22×D3—21no—
352C22×D3—21no—
353C22×D3—21no—
354C22×D3—21no—
355C22×D3—21no—
356C22×D3—21no—
357C22×D3—32no—
358C22×D3—61no—
359C22×D3—61no—
360C22×D3—61no—
361C22×D3—61no—
362C22×D3—61no—
363C22×D3—61no—
364C22×D3#61112yes—
365C2×C3×C4—22no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—61no—
375C2×C3×C4—61no—
376C2×C3×C4—61no—
377C2×C3×C4—181no—
378C2×C3×C4—181no—
379C2×C3×C4—181no—
380#67—188no—
381#67—188no—
382C2×Dic3—21no—
383C2×Dic3—21no—
384C2×Dic3—21no—
385C2×Dic3—61no—
386C2×Dic3—61no—
387C2×Dic3—61no—
388C2×Dic3—92no—
389C2×Dic3—92no—
390C2×Dic3—92no—
391C2×Dic3—92no—
392C2×Dic3—182no—
393C2×Dic3—182no—
394C2×Dic3—182no—
395C2×Dic3—182no—
396C2×Dic3—181no—
397C2×Dic3—181no—
398C2×Dic3—181no—
399C2×Dic3—181no—
400C2×Dic3—181no—
401C2×Dic3—181no—
402C2×Dic3—181no—
403C2×Dic3—181no—
404C2×Dic3—361no—
405C2×Dic3—361no—
406C2×C32#88311yescharacteristic
407C2×C32#88411yescharacteristic
408C2×C32#88411yescharacteristic
409C2×C32#88511yescharacteristic
410C2×C32#92911yescharacteristic
411C2×C32#93711yescharacteristic
412C2×C32#93911yescharacteristic
413C3×D3—42no—
414C3×D3—122no—
415C2×C9—21no—
416C2×C9—21no—
417C2×C9—21no—
418C2×C9—21no—
419C2×C9—21no—
420C2×C9—21no—
421C2×C9—21no—
422C2×C9—122no—
423C2×C9#88311yescharacteristic
424C2×C9#88411yescharacteristic
425C2×C9#88411yescharacteristic
426C2×C9#88511yescharacteristic
427C2×C9#92911yescharacteristic
428C2×C9#93711yescharacteristic
429C2×C9#93911yescharacteristic
430D9—42no—
431C24—31no—
432C24—91no—
433C2×D4—542no—
434C2×D4—542no—
435C2×D4—542no—
436C22×C4—31no—
437C22×C4—91no—
438C22×C4—271no—
439C22×C4—271no—
440C22×C4D3×D911yescharacteristic
441C4⋊C4—542no—
442C22⋊C4—62no—
443C22⋊C4—182no—
444C22⋊C4—542no—
445C22⋊C4—542no—
446C22×C3—21no—
447C22×C3—21no—
448C22×C3—21no—
449C22×C3—21no—
450C22×C3—21no—
451C22×C3—21no—
452C22×C3—21no—
453C22×C3—62no—
454C22×C3—62no—
455C22×C3—62no—
456C22×C3—121no—
457C22×C3—121no—
458C22×C3—121no—
459C22×C3—121no—
460C22×C3—182no—
461C22×C3—182no—
462C22×C3—182no—
463C22×C3—361no—
464C22×C3—361no—
465C22×C3—361no—
466C22×C3—361no—
467C22×C3#369711yescharacteristic
468C22×C3#369711yescharacteristic
469C22×C3D4×D911yescharacteristic
470C22×C3D4×D911yescharacteristic
471C22×C3D4×D911yescharacteristic
472C22×C3#369911yescharacteristic
473C22×C3C2×#61111yescharacteristic
474C22×C3#379511yescharacteristic
475C22×C3#379511yescharacteristic
476C22×C3#379811yescharacteristic
477C22×C3#380211yescharacteristic
478C22×C3D3×#6711yescharacteristic
479C22×C3D3×#6711yescharacteristic
480C22×C3#381111yescharacteristic
481C2×D3—22no—
482C2×D3—22no—
483C2×D3—22no—
484C2×D3—41no—
485C2×D3—41no—
486C2×D3—41no—
487C2×D3—41no—
488C2×D3—62no—
489C2×D3—62no—
490C2×D3—62no—
491C2×D3—121no—
492C2×D3—121no—
493C2×D3—121no—
494C2×D3—121no—
495C3×C4—42no—
496C3×C4—41no—
497C3×C4—41no—
498C3×C4—121no—
499C3×C4—361no—
500Dic3—41no—
501Dic3—121no—
502Dic3—184no—
503Dic3—182no—
504Dic3—182no—
505Dic3—182no—
506Dic3—182no—
507Dic3—362no—
508C32#495911yescharacteristic
509C9—21no—
510C9#495911yescharacteristic, complemented
511C23—32no—
512C23—61no—
513C23—61no—
514C23—61no—
515C23—61no—
516C23—61no—
517C23—61no—
518C23—92no—
519C23—181no—
520C23—181no—
521C23—181no—
522C23—181no—
523C23—181no—
524C23—181no—
525C23C2×D3×D911yescharacteristic
526D4—548no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—61no—
531C2×C4—61no—
532C2×C4—61no—
533C2×C4—181no—
534C2×C4—181no—
535C2×C4—181no—
536C2×C4—272no—
537C2×C4—272no—
538C2×C4—541no—
539C2×C4—541no—
540C2×C4—541no—
541C2×C4—541no—
542C2×C3—21no—
543C2×C3—21no—
544C2×C3—21no—
545C2×C3—21no—
546C2×C3—21no—
547C2×C3—21no—
548C2×C3—21no—
549C2×C3—122no—
550C2×C3—362no—
551C2×C3#6344511yescharacteristic
552C2×C3#6344611yescharacteristic
553C2×C3#6344611yescharacteristic
554C2×C3#6344711yescharacteristic
555C2×C3#6349111yescharacteristic
556C2×C3#6349911yescharacteristic
557C2×C3#6350111yescharacteristic
558C2×C3#6388511yescharacteristic
559C2×C3#6388511yescharacteristic
560C2×C3#6390511yescharacteristic
561C2×C3#6390511yescharacteristic
562C2×C3#6391211yescharacteristic
563C2×C3#6391311yescharacteristic
564C2×C3#6398311yescharacteristic
565D3—42no—
566D3—122no—
567C22—62no—
568C22—62no—
569C22—62no—
570C22—121no—
571C22—121no—
572C22—121no—
573C22—121no—
574C22—182no—
575C22—182no—
576C22—182no—
577C22—361no—
578C22—361no—
579C22—361no—
580C22—361no—
581C22#8863811yescharacteristic
582C22#8864011yescharacteristic
583C22#8864111yescharacteristic
584C22#8864611yescharacteristic
585C22D3×#61111yescharacteristic
586C22D9×#6711yescharacteristic
587C22#8866811yescharacteristic
588C4—41no—
589C4—121no—
590C4—361no—
591C4—542no—
592C4—542no—
593C3—21no—
594C3#1058863911yescharacteristic, complemented
595C3#1059138811yescharacteristic
596C2—122no—
597C2—362no—
598C2#1172098211yescharacteristic
599C2#1172101611yescharacteristic
600C2#1172103111yescharacteristic
601C2#1172103211yescharacteristic
602C2#1172104111yescharacteristic
603C2#1172104211yescharacteristic
604C2#1172129211yescharacteristic
subgroupG/H
centerC23C2×D3×D9
commutator, cosocleC23×C3×C9C23
FrattiniC23×C3C2×D32
FittingC22×C3×C4×C9C22
socleC23×C32C22×D3
prime radicalC22×#6194C2
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×#88534C2semidirect
3. C2×C9×(C2×D3)⋊C4C2semidirect
4. C2×C3×(C2×D9)⋊C4C2semidirect
5. C2×(C2×D3)⋊Dic9C2semidirect
6. C2×(C2×D9)⋊Dic3C2semidirect
7. #11720331C2semidirect
36. C2×(C2×D9)⋊C4D3semidirect
63. C2×C3⋊Dic9D4semidirect
142. C2×(C2×D3)⋊C4D9semidirect
333. C3×C9#394semidirect
510. C9#4959semidirect
594. C3#10588639semidirect

Maximal subgroups

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

subgroupindex
C22×#61942
C2×#885342
C2×C9×(C2×D3)⋊C42
C2×C3×(C2×D9)⋊C42
C2×(C2×D3)⋊Dic92
C2×(C2×D9)⋊Dic32
#117203312
#105913883
#105886393