Finite Group Databasebeta

← #13139740

Subgroups — #13139740

Order 1728.

690 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88536C211yesmaximal, characteristic
2C2×C9×Dic3⋊C4C211yesmaximal, characteristic, complemented
3C2×#88445C211yesmaximal, characteristic
4C2×#88445C211yesmaximal, characteristic
5C2×Dic3⋊Dic9C211yesmaximal, characteristic, complemented
6C2×#88442C211yesmaximal, characteristic, complemented
7C3×C9⋊#95C211yesmaximal, characteristic, complemented
8#10591347—31nomaximal
9#10588674—31nomaximal
10C23×C3⋊D9C2211yescharacteristic
11C22×C3×C4×C9C2211yescharacteristic
12C22×C3⋊Dic9C2211yescharacteristic
13C22×C9×Dic3C2211yescharacteristic
14C22×C9×Dic3C2211yescharacteristic
15C22×C3×Dic9C2211yescharacteristic
16C22×C3×Dic9C2211yescharacteristic
17#88536—22no—
18C9×Dic3⋊C4—22no—
19#88445—22no—
20#88445—22no—
21Dic3⋊Dic9—22no—
22#88442—22no—
23C2×#3752—31no—
24C2×C3×Dic3⋊C4D311yescharacteristic
25C2×#3724—31no—
26C2×Dic3⋊Dic3—31no—
27C2×#3722—31no—
28C2×#3722—31no—
29C3×C3⋊#95—31no—
30C2×C9×C4⋊C4—31no—
31C2×(C2×D9)⋊C4—31no—
32C2×(C2×D9)⋊C4—31no—
33C2×(C2×D9)⋊C4—31no—
34C2×(C2×D9)⋊C4—61no—
35C2×C4⋊Dic9—31no—
36C2×Dic9⋊C4—31no—
37C9⋊#95D311yescharacteristic, complemented
38C23×C3×C9C2311yescharacteristic
39C22×C3⋊D9—21no—
40C22×C3⋊D9—21no—
41C22×C3⋊D9—21no—
42C22×C3⋊D9—21no—
43C22×C3⋊D9—21no—
44C22×C3⋊D9—21no—
45C22×C3⋊D9Q812yes—
46C2×C3×C4×C9—21no—
47C2×C3×C4×C9—21no—
48C2×C3×C4×C9—21no—
49C2×C3⋊Dic9—21no—
50C2×C3⋊Dic9—21no—
51C2×C3⋊Dic9D412yes—
52C2×C9×Dic3—21no—
53C2×C9×Dic3—21no—
54C2×C9×Dic3—21no—
55C2×C9×Dic3—21no—
56C2×C9×Dic3D412yes—
57C2×C9×Dic3D412yes—
58C2×C3×Dic9—21no—
59C2×C3×Dic9—21no—
60C2×C3×Dic9—21no—
61C2×C3×Dic9—21no—
62C2×C3×Dic9—21no—
63C2×C3×Dic9—21no—
64#4994—91no—
65#4676—91no—
66C23×C3⋊D3—31no—
67C22×C32×C4C2×D311yescharacteristic
68C22×C3⋊Dic3—31no—
69C22×C3×Dic3—31no—
70C22×C3×Dic3—31no—
71C22×C3×Dic3C2×D311yescharacteristic
72C22×C3×Dic3C2×D311yescharacteristic
73C23×D9—31no—
74C23×D9—61no—
75#3752—62no—
76C3×Dic3⋊C4—22no—
77#3724—62no—
78Dic3⋊Dic3—62no—
79#3722—62no—
80#3722—62no—
81C22×C4×C9—21no—
82C22×C4×C9—31no—
83C22×C4×C9—31no—
84C22×C4×C9C2×D311yescharacteristic
85C22×Dic9—31no—
86C22×Dic9—61no—
87C22×Dic9C2×D311yescharacteristic
88C22×Dic9C2×D311yescharacteristic
89C9×C4⋊C4—62no—
90(C2×D9)⋊C4—62no—
91(C2×D9)⋊C4—62no—
92(C2×D9)⋊C4—62no—
93(C2×D9)⋊C4—64no—
94C4⋊Dic9—62no—
95Dic9⋊C4—62no—
96C22×C3×C9C2×D411yescharacteristic
97C22×C3×C9C2×D411yescharacteristic
98C22×C3×C9C2×D411yescharacteristic
99C22×C3×C9C2×Q811yescharacteristic
100C22×C3×C9#4111yescharacteristic
101C22×C3×C9#4111yescharacteristic
102C22×C3×C9#4111yescharacteristic
103C2×C3⋊D9—22no—
104C2×C3⋊D9—22no—
105C2×C3⋊D9—22no—
106C2×C3⋊D9—41no—
107C2×C3⋊D9—41no—
108C2×C3⋊D9—41no—
109C2×C3⋊D9—41no—
110C3×C4×C9—41no—
111C3⋊Dic9—22no—
112C9×Dic3—22no—
113C9×Dic3—22no—
114C3×Dic9—41no—
115C3×Dic9—41no—
116C2×C3×C4⋊C4—31no—
117C2×(C2×D3)⋊C4—91no—
118C2×(C2×D3)⋊C4—91no—
119C2×(C2×D3)⋊C4—91no—
120C2×(C2×D3)⋊C4—91no—
121C2×(C2×D3)⋊C4—91no—
122C2×(C2×D3)⋊C4—91no—
123C2×(C2×D3)⋊C4—181no—
124C2×C4⋊Dic3—91no—
125C2×Dic3⋊C4—91no—
126C2×Dic3⋊C4—91no—
127C2×Dic3⋊C4—91no—
128C2×Dic3⋊C4D911yescharacteristic, complemented
129C3×#95—91no—
130C3⋊#95—31no—
131C23×C32C22×D311yescharacteristic
132C22×C3⋊D3—32no—
133C22×C3⋊D3—61no—
134C22×C3⋊D3—61no—
135C22×C3⋊D3—61no—
136C22×C3⋊D3—61no—
137C22×C3⋊D3—61no—
138C22×C3⋊D3—61no—
139C2×C32×C4—21no—
140C2×C32×C4—21no—
141C2×C32×C4—21no—
142C2×C3⋊Dic3—32no—
143C2×C3⋊Dic3—61no—
144C2×C3⋊Dic3—61no—
145C2×C3×Dic3—21no—
146C2×C3×Dic3—21no—
147C2×C3×Dic3—21no—
148C2×C3×Dic3—21no—
149C2×C3×Dic3—61no—
150C2×C3×Dic3—61no—
151C2×C3×Dic3—61no—
152C2×C3×Dic3—61no—
153C2×C3×Dic3—61no—
154C2×C3×Dic3—61no—
155C2×C3×Dic3D1212yes—
156C2×C3×Dic3#6712yes—
157C23×C9—21no—
158C23×C9C22×D311yescharacteristic
159C22×D9—32no—
160C22×D9—62no—
161C22×D9—62no—
162C22×D9—62no—
163C22×D9—61no—
164C22×D9—61no—
165C22×D9—61no—
166C22×D9—61no—
167C22×D9—61no—
168C22×D9—61no—
169C22×D9—121no—
170C22×D9—121no—
171C22×D9—121no—
172C22×D9—121no—
173C2×C4×C9—22no—
174C2×C4×C9—21no—
175C2×C4×C9—21no—
176C2×C4×C9—21no—
177C2×C4×C9—32no—
178C2×C4×C9—32no—
179C2×C4×C9—41no—
180C2×C4×C9—41no—
181C2×C4×C9—61no—
182C2×C4×C9—61no—
183C2×C4×C9—61no—
184C2×C4×C9—61no—
185C2×Dic9—21no—
186C2×Dic9—21no—
187C2×Dic9—21no—
188C2×Dic9—21no—
189C2×Dic9—21no—
190C2×Dic9—21no—
191C2×Dic9—32no—
192C2×Dic9—62no—
193C2×Dic9—61no—
194C2×Dic9—61no—
195C2×Dic9—121no—
196C2×Dic9—121no—
197#396—271noSylow
198C2×C3×C9#12111yescharacteristic
199C2×C3×C9#12111yescharacteristic
200C2×C3×C9#12111yescharacteristic
201C2×C3×C9C22⋊Q811yescharacteristic
202C2×C3×C9C22⋊Q811yescharacteristic
203C2×C3×C9C22⋊Q811yescharacteristic
204C2×C3×C9#12611yescharacteristic
205C3⋊D9—42no—
206C23×D3—91no—
207C23×D3—91no—
208C23×D3—181no—
209C22×C3×C4—21no—
210C22×C3×C4—31no—
211C22×C3×C4—31no—
212C22×C3×C4—91no—
213C22×C3×C4—91no—
214C22×C3×C4C2×D911yescharacteristic
215C22×C3×C4D3211yescharacteristic
216C22×Dic3—31no—
217C22×Dic3—31no—
218C22×Dic3—91no—
219C22×Dic3—91no—
220C22×Dic3—181no—
221C22×Dic3C2×D911yescharacteristic
222C22×Dic3C2×D911yescharacteristic
223C3×C4⋊C4—62no—
224(C2×D3)⋊C4—182no—
225(C2×D3)⋊C4—182no—
226(C2×D3)⋊C4—182no—
227(C2×D3)⋊C4—182no—
228(C2×D3)⋊C4—184no—
229(C2×D3)⋊C4—182no—
230(C2×D3)⋊C4—182no—
231C4⋊Dic3—182no—
232Dic3⋊C4—22no—
233Dic3⋊C4—182no—
234Dic3⋊C4—182no—
235Dic3⋊C4—182no—
236C22×C32C2×D1211yescharacteristic
237C22×C32#23511yescharacteristic
238C22×C32D3×D411yescharacteristic
239C22×C32#23711yescharacteristic
240C22×C32D3×Q811yescharacteristic
241C22×C32#23911yescharacteristic
242C22×C32C2×#6711yescharacteristic
243C2×C3⋊D3—62no—
244C2×C3⋊D3—62no—
245C2×C3⋊D3—62no—
246C2×C3⋊D3—121no—
247C2×C3⋊D3—121no—
248C2×C3⋊D3—121no—
249C2×C3⋊D3—121no—
250C32×C4—41no—
251C3⋊Dic3—62no—
252C3×Dic3—22no—
253C3×Dic3—22no—
254C3×Dic3—121no—
255C3×Dic3—121no—
256C22×C9—21no—
257C22×C9—21no—
258C22×C9—21no—
259C22×C9—21no—
260C22×C9—21no—
261C22×C9—21no—
262C22×C9—21no—
263C22×C9#23511yescharacteristic
264C22×C9#23511yescharacteristic
265C22×C9#23511yescharacteristic
266C22×C9D3×D411yescharacteristic
267C22×C9D3×D411yescharacteristic
268C22×C9D3×D411yescharacteristic
269C22×C9D3×Q811yescharacteristic
270C2×D9—64no—
271C2×D9—62no—
272C2×D9—62no—
273C2×D9—62no—
274C2×D9—122no—
275C2×D9—122no—
276C2×D9—122no—
277C2×D9—122no—
278C2×D9—122no—
279C2×D9—122no—
280C2×D9—121no—
281C2×D9—121no—
282C2×D9—121no—
283C2×D9—121no—
284C4×C9—42no—
285C4×C9—41no—
286C4×C9—62no—
287C4×C9—62no—
288Dic9—41no—
289Dic9—41no—
290Dic9—62no—
291Dic9—122no—
292C2×C4⋊C4—31no—
293C2×C4⋊C4—271no—
294C2×C4⋊C4—271no—
295C2×C22⋊C4—271no—
296C2×C22⋊C4—271no—
297C2×C22⋊C4—271no—
298#95—91no—
299C3×C9#39611yescharacteristic, complemented, Sylow
300C23×C3—21no—
301C23×C3C22×D911yescharacteristic
302C23×C3C2×D3211yescharacteristic
303C22×D3—92no—
304C22×D3—92no—
305C22×D3—182no—
306C22×D3—182no—
307C22×D3—182no—
308C22×D3—181no—
309C22×D3—181no—
310C22×D3—181no—
311C22×D3—181no—
312C22×D3—181no—
313C22×D3—181no—
314C22×D3—181no—
315C22×D3—181no—
316C22×D3—181no—
317C22×D3—181no—
318C22×D3—181no—
319C22×D3—181no—
320C22×D3—361no—
321C22×D3—361no—
322C22×D3—361no—
323C22×D3—361no—
324C2×C3×C4—22no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—32no—
332C2×C3×C4—32no—
333C2×C3×C4—41no—
334C2×C3×C4—41no—
335C2×C3×C4—61no—
336C2×C3×C4—61no—
337C2×C3×C4—61no—
338C2×C3×C4—61no—
339C2×C3×C4—181no—
340C2×C3×C4—181no—
341C2×C3×C4—181no—
342C2×C3×C4—181no—
343C2×C3×C4—181no—
344C2×C3×C4—181no—
345C2×Dic3—21no—
346C2×Dic3—21no—
347C2×Dic3—21no—
348C2×Dic3—21no—
349C2×Dic3—61no—
350C2×Dic3—61no—
351C2×Dic3—61no—
352C2×Dic3—61no—
353C2×Dic3—61no—
354C2×Dic3—61no—
355C2×Dic3—92no—
356C2×Dic3—92no—
357C2×Dic3—182no—
358C2×Dic3—181no—
359C2×Dic3—181no—
360C2×Dic3—181no—
361C2×Dic3—181no—
362C2×Dic3—361no—
363C2×Dic3—361no—
364C2×Dic3D3612yes—
365C2×Dic3#61112yes—
366C2×C32#89511yescharacteristic
367C2×C32#89611yescharacteristic
368C2×C32#89711yescharacteristic
369C2×C32#89811yescharacteristic
370C2×C32#93011yescharacteristic
371C2×C32#93911yescharacteristic
372C2×C32#94611yescharacteristic
373C3⋊D3—122no—
374C2×C9—21no—
375C2×C9—21no—
376C2×C9—21no—
377C2×C9—21no—
378C2×C9—21no—
379C2×C9—21no—
380C2×C9—21no—
381C2×C9#87611yescharacteristic
382C2×C9#88411yescharacteristic
383C2×C9#88411yescharacteristic
384C2×C9#88411yescharacteristic
385C2×C9#89611yescharacteristic
386C2×C9#89611yescharacteristic
387C2×C9#89611yescharacteristic
388D9—124no—
389D9—122no—
390C24—271no—
391C22×C4—31no—
392C22×C4—31no—
393C22×C4—91no—
394C22×C4—91no—
395C22×C4—271no—
396C22×C4D3×D911yescharacteristic
397C4⋊C4—62no—
398C4⋊C4—542no—
399C4⋊C4—542no—
400C22⋊C4—542no—
401C22⋊C4—542no—
402C22⋊C4—542no—
403C22×C3—21no—
404C22×C3—21no—
405C22×C3—21no—
406C22×C3—21no—
407C22×C3—21no—
408C22×C3—21no—
409C22×C3—21no—
410C22×C3C2×D3611yescharacteristic
411C22×C3#369711yescharacteristic
412C22×C3D4×D911yescharacteristic
413C22×C3#369911yescharacteristic
414C22×C3Q8×D911yescharacteristic
415C22×C3#370111yescharacteristic
416C22×C3C2×#61111yescharacteristic
417C22×C3#379711yescharacteristic
418C22×C3#379811yescharacteristic
419C22×C3#379911yescharacteristic
420C22×C3D3×D1211yescharacteristic
421C22×C3#380411yescharacteristic
422C22×C3D3×#6711yescharacteristic
423C22×C3#381111yescharacteristic
424C2×D3—184no—
425C2×D3—182no—
426C2×D3—182no—
427C2×D3—182no—
428C2×D3—182no—
429C2×D3—182no—
430C2×D3—182no—
431C2×D3—362no—
432C2×D3—362no—
433C2×D3—362no—
434C2×D3—362no—
435C2×D3—362no—
436C2×D3—362no—
437C2×D3—361no—
438C2×D3—361no—
439C2×D3—361no—
440C2×D3—361no—
441C2×D3—361no—
442C2×D3—361no—
443C2×D3—361no—
444C2×D3—361no—
445C3×C4—42no—
446C3×C4—41no—
447C3×C4—41no—
448C3×C4—62no—
449C3×C4—62no—
450C3×C4—361no—
451C3×C4—361no—
452Dic3—22no—
453Dic3—22no—
454Dic3—121no—
455Dic3—121no—
456Dic3—182no—
457Dic3—182no—
458Dic3—362no—
459C32#499411yescharacteristic
460C9—21no—
461C9#467611yescharacteristic, complemented
462C23—272no—
463C23—541no—
464C23—541no—
465C23—541no—
466C23—541no—
467C23—541no—
468C23—541no—
469C23C2×D3×D911yescharacteristic
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—32no—
474C2×C4—32no—
475C2×C4—61no—
476C2×C4—61no—
477C2×C4—61no—
478C2×C4—61no—
479C2×C4—181no—
480C2×C4—181no—
481C2×C4—181no—
482C2×C4—181no—
483C2×C4—181no—
484C2×C4—181no—
485C2×C4—272no—
486C2×C4—541no—
487C2×C4—541no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—21no—
492C2×C3—21no—
493C2×C3—21no—
494C2×C3—21no—
495C2×C3#6345711yescharacteristic
496C2×C3#6345811yescharacteristic
497C2×C3#6345911yescharacteristic
498C2×C3#6346011yescharacteristic
499C2×C3#6349211yescharacteristic
500C2×C3#6350111yescharacteristic
501C2×C3#6350811yescharacteristic
502C2×C3#6386511yescharacteristic
503C2×C3#6386811yescharacteristic
504C2×C3#6388811yescharacteristic
505C2×C3#6389511yescharacteristic
506C2×C3#6390611yescharacteristic
507C2×C3#6391011yescharacteristic
508C2×C3#6397111yescharacteristic
509D3—364no—
510D3—362no—
511D3—362no—
512C22—542no—
513C22—542no—
514C22—542no—
515C22—1081no—
516C22—1081no—
517C22—1081no—
518C22—1081no—
519C22#8863511yescharacteristic
520C22(C3⋊D9)⋊Q811yescharacteristic
521C22#8864011yescharacteristic
522C22D3×D3611yescharacteristic
523C22#8866211yescharacteristic
524C22D3×#61111yescharacteristic
525C22#8866811yescharacteristic
526C4—41no—
527C4—62no—
528C4—62no—
529C4—361no—
530C4—361no—
531C4—542no—
532C3—21no—
533C3#1058867411yescharacteristic, complemented
534C3#1059134711yescharacteristic
535C2—1082no—
536C2#1172092311yescharacteristic
537C2#1172093411yescharacteristic
538C2#1172098611yescharacteristic
539C2#1172099511yescharacteristic
540C2#1172100511yescharacteristic
541C2#1172103411yescharacteristic
542C2#1172128211yescharacteristic
subgroupG/H
centerC23C2×D3×D9
commutator, cosocleC23×C3×C9C23
FrattiniC23×C3C2×D32
FittingC22×C3×C4×C9C22
socleC23×C32C22×D3
prime radicalC23×C3⋊D9C22
hypercenterC22×C4D3×D9
hypocenterC3×C9#396
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×C9×Dic3⋊C4C2semidirect
5. C2×Dic3⋊Dic9C2semidirect
6. C2×#88442C2semidirect
7. C3×C9⋊#95C2semidirect
37. C9⋊#95D3semidirect
128. C2×Dic3⋊C4D9semidirect
299. C3×C9#396semidirect
461. C9#4676semidirect
533. C3#10588674semidirect

Maximal subgroups

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

subgroupindex
C2×#885362
C2×C9×Dic3⋊C42
C2×#884452
C2×Dic3⋊Dic92
C2×#884422
C3×C9⋊#952
#105913473
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