Finite Group Databasebeta

← #13139680

Subgroups — #13139680

Order 1728.

642 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88536C211yesmaximal, characteristic
2C2×#88445C211yesmaximal, characteristic
3C2×#88445C211yesmaximal, characteristic
4C2×Dic3⋊Dic9C211yesmaximal, characteristic, complemented
5C9×C3⋊#95C211yesmaximal, characteristic, complemented
6C3×C9⋊#95C211yesmaximal, characteristic, complemented
7#11720331C211yesmaximal, characteristic, complemented
8#10591208—31nomaximal
9#10588354—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—
18#88445—22no—
19#88445—22no—
20Dic3⋊Dic9—22no—
21C2×#3752—31no—
22C2×Dic3⋊Dic3—31no—
23C2×#3722—31no—
24C2×#3722—31no—
25C3×C3⋊#95—31no—
26C3×C3⋊#95D311yescharacteristic
27#63579—31no—
28C2×(C2×D9)⋊C4—31no—
29C2×(C2×D9)⋊C4—31no—
30C2×(C2×D9)⋊C4—31no—
31C2×(C2×D9)⋊C4—61no—
32C2×C4⋊Dic9—31no—
33C9×#95—31no—
34C9⋊#95—31no—
35C9⋊#95D311yescharacteristic, complemented
36C23×C3×C9C2311yescharacteristic
37C22×C3⋊D9—21no—
38C22×C3⋊D9—21no—
39C22×C3⋊D9—21no—
40C22×C3⋊D9—21no—
41C22×C3⋊D9—21no—
42C22×C3⋊D9—21no—
43C22×C3⋊D9D412yes—
44C2×C3×C4×C9—21no—
45C2×C3×C4×C9—21no—
46C2×C3×C4×C9—21no—
47C2×C3⋊Dic9—21no—
48C2×C3⋊Dic9—21no—
49C2×C3⋊Dic9—21no—
50C2×C9×Dic3—21no—
51C2×C9×Dic3—21no—
52C2×C9×Dic3—21no—
53C2×C9×Dic3—21no—
54C2×C9×Dic3—21no—
55C2×C9×Dic3D412yes—
56C2×C3×Dic9—21no—
57C2×C3×Dic9—21no—
58C2×C3×Dic9—21no—
59C2×C3×Dic9—21no—
60C2×C3×Dic9—21no—
61C2×C3×Dic9—21no—
62#4675—91no—
63#4674—91no—
64C23×C3⋊D3—31no—
65C22×C32×C4C2×D311yescharacteristic
66C22×C3⋊Dic3—31no—
67C22×C3×Dic3—31no—
68C22×C3×Dic3—31no—
69C22×C3×Dic3C2×D311yescharacteristic
70C22×C3×Dic3C2×D311yescharacteristic
71C23×D9—31no—
72C23×D9—61no—
73#3752—62no—
74Dic3⋊Dic3—62no—
75#3722—62no—
76#3722—62no—
77C22×C4×C9—21no—
78C22×C4×C9—31no—
79C22×C4×C9—31no—
80C22×C4×C9C2×D311yescharacteristic
81C22×Dic9—31no—
82C22×Dic9—61no—
83C22×Dic9C2×D311yescharacteristic
84C22×Dic9C2×D311yescharacteristic
85(C2×D9)⋊C4—62no—
86(C2×D9)⋊C4—62no—
87(C2×D9)⋊C4—62no—
88(C2×D9)⋊C4—64no—
89C4⋊Dic9—62no—
90C22×C3×C9C2×D411yescharacteristic
91C22×C3×C9C2×D411yescharacteristic
92C22×C3×C9#4111yescharacteristic
93C22×C3×C9#4111yescharacteristic
94C22×C3×C9#4111yescharacteristic
95C22×C3×C9#4111yescharacteristic
96C22×C3×C9#4111yescharacteristic
97C2×C3⋊D9—22no—
98C2×C3⋊D9—22no—
99C2×C3⋊D9—22no—
100C2×C3⋊D9—41no—
101C2×C3⋊D9—41no—
102C2×C3⋊D9—41no—
103C2×C3⋊D9—41no—
104C3×C4×C9—41no—
105C3⋊Dic9—41no—
106C9×Dic3—22no—
107C9×Dic3—41no—
108C3×Dic9—41no—
109C3×Dic9—41no—
110C2×(C2×D3)⋊C4—91no—
111C2×(C2×D3)⋊C4—91no—
112C2×(C2×D3)⋊C4—91no—
113C2×(C2×D3)⋊C4—91no—
114C2×(C2×D3)⋊C4—91no—
115C2×(C2×D3)⋊C4—91no—
116C2×(C2×D3)⋊C4—181no—
117C2×C4⋊Dic3—91no—
118C2×Dic3⋊C4—91no—
119C3×#95—31no—
120C3×#95—91no—
121C3⋊#95—31no—
122C3⋊#95—91no—
123C3⋊#95—91no—
124C3⋊#95D911yescharacteristic, complemented
125C23×C32C22×D311yescharacteristic
126C22×C3⋊D3—32no—
127C22×C3⋊D3—61no—
128C22×C3⋊D3—61no—
129C22×C3⋊D3—61no—
130C22×C3⋊D3—61no—
131C22×C3⋊D3—61no—
132C22×C3⋊D3—61no—
133C2×C32×C4—21no—
134C2×C32×C4—21no—
135C2×C32×C4—21no—
136C2×C3⋊Dic3—61no—
137C2×C3⋊Dic3—61no—
138C2×C3⋊Dic3—61no—
139C2×C3×Dic3—21no—
140C2×C3×Dic3—21no—
141C2×C3×Dic3—21no—
142C2×C3×Dic3—21no—
143C2×C3×Dic3—21no—
144C2×C3×Dic3—61no—
145C2×C3×Dic3—61no—
146C2×C3×Dic3—61no—
147C2×C3×Dic3—61no—
148C2×C3×Dic3—61no—
149C2×C3×Dic3—61no—
150C2×C3×Dic3D1212yes—
151C23×C9—21no—
152C23×C9C22×D311yescharacteristic
153C22×D9—32no—
154C22×D9—62no—
155C22×D9—62no—
156C22×D9—62no—
157C22×D9—61no—
158C22×D9—61no—
159C22×D9—61no—
160C22×D9—61no—
161C22×D9—61no—
162C22×D9—61no—
163C22×D9—121no—
164C22×D9—121no—
165C22×D9—121no—
166C22×D9—121no—
167C2×C4×C9—22no—
168C2×C4×C9—21no—
169C2×C4×C9—21no—
170C2×C4×C9—21no—
171C2×C4×C9—32no—
172C2×C4×C9—41no—
173C2×C4×C9—41no—
174C2×C4×C9—61no—
175C2×C4×C9—61no—
176C2×C4×C9—61no—
177C2×C4×C9—61no—
178C2×C4×C9—61no—
179C2×Dic9—21no—
180C2×Dic9—21no—
181C2×Dic9—21no—
182C2×Dic9—21no—
183C2×Dic9—21no—
184C2×Dic9—21no—
185C2×Dic9—62no—
186C2×Dic9—61no—
187C2×Dic9—61no—
188C2×Dic9—61no—
189C2×Dic9—121no—
190C2×Dic9—121no—
191#397—271noSylow
192C2×C3×C9#12111yescharacteristic
193C2×C3×C9#12311yescharacteristic
194C2×C3×C9#12311yescharacteristic
195C2×C3×C9#12311yescharacteristic
196C2×C3×C9#12411yescharacteristic
197C2×C3×C9#12611yescharacteristic
198C2×C3×C9#12611yescharacteristic
199C3⋊D9—42no—
200C23×D3—91no—
201C23×D3—91no—
202C23×D3—181no—
203C22×C3×C4—21no—
204C22×C3×C4—31no—
205C22×C3×C4—31no—
206C22×C3×C4—91no—
207C22×C3×C4—91no—
208C22×C3×C4C2×D911yescharacteristic
209C22×C3×C4D3211yescharacteristic
210C22×Dic3—31no—
211C22×Dic3—31no—
212C22×Dic3—91no—
213C22×Dic3—91no—
214C22×Dic3—181no—
215C22×Dic3C2×D911yescharacteristic
216C22×Dic3C2×D911yescharacteristic
217(C2×D3)⋊C4—182no—
218(C2×D3)⋊C4—182no—
219(C2×D3)⋊C4—182no—
220(C2×D3)⋊C4—182no—
221(C2×D3)⋊C4—182no—
222(C2×D3)⋊C4—184no—
223(C2×D3)⋊C4—182no—
224C4⋊Dic3—182no—
225Dic3⋊C4—182no—
226C22×C32C2×D1211yescharacteristic
227C22×C32#23511yescharacteristic
228C22×C32#23511yescharacteristic
229C22×C32D3×D411yescharacteristic
230C22×C32#23711yescharacteristic
231C22×C32#23711yescharacteristic
232C22×C32#23911yescharacteristic
233C2×C3⋊D3—62no—
234C2×C3⋊D3—62no—
235C2×C3⋊D3—62no—
236C2×C3⋊D3—121no—
237C2×C3⋊D3—121no—
238C2×C3⋊D3—121no—
239C2×C3⋊D3—121no—
240C32×C4—41no—
241C3⋊Dic3—121no—
242C3×Dic3—22no—
243C3×Dic3—41no—
244C3×Dic3—121no—
245C3×Dic3—121no—
246C22×C9—21no—
247C22×C9—21no—
248C22×C9—21no—
249C22×C9—21no—
250C22×C9—21no—
251C22×C9—21no—
252C22×C9—21no—
253C22×C9#23511yescharacteristic
254C22×C9#23511yescharacteristic
255C22×C9#23511yescharacteristic
256C22×C9D3×D411yescharacteristic
257C22×C9D3×D411yescharacteristic
258C22×C9#23711yescharacteristic
259C22×C9#23911yescharacteristic
260C2×D9—64no—
261C2×D9—62no—
262C2×D9—62no—
263C2×D9—62no—
264C2×D9—122no—
265C2×D9—122no—
266C2×D9—122no—
267C2×D9—122no—
268C2×D9—122no—
269C2×D9—122no—
270C2×D9—121no—
271C2×D9—121no—
272C2×D9—121no—
273C2×D9—121no—
274C4×C9—42no—
275C4×C9—41no—
276C4×C9—62no—
277C4×C9—121no—
278Dic9—41no—
279Dic9—41no—
280Dic9—122no—
281Dic9—121no—
282C2×C4⋊C4—271no—
283C2×C22⋊C4—271no—
284C2×C22⋊C4—271no—
285C2×C22⋊C4—271no—
286#95—31no—
287#95—91no—
288#95—271no—
289C3×C9#39711yescharacteristic, complemented, Sylow
290C23×C3—21no—
291C23×C3C22×D911yescharacteristic
292C23×C3C2×D3211yescharacteristic
293C22×D3—92no—
294C22×D3—92no—
295C22×D3—182no—
296C22×D3—182no—
297C22×D3—182no—
298C22×D3—181no—
299C22×D3—181no—
300C22×D3—181no—
301C22×D3—181no—
302C22×D3—181no—
303C22×D3—181no—
304C22×D3—181no—
305C22×D3—181no—
306C22×D3—181no—
307C22×D3—181no—
308C22×D3—181no—
309C22×D3—181no—
310C22×D3—361no—
311C22×D3—361no—
312C22×D3—361no—
313C22×D3—361no—
314C2×C3×C4—22no—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—32no—
322C2×C3×C4—41no—
323C2×C3×C4—41no—
324C2×C3×C4—61no—
325C2×C3×C4—61no—
326C2×C3×C4—61no—
327C2×C3×C4—61no—
328C2×C3×C4—61no—
329C2×C3×C4—181no—
330C2×C3×C4—181no—
331C2×C3×C4—181no—
332C2×C3×C4—181no—
333C2×C3×C4—181no—
334C2×C3×C4—181no—
335C2×Dic3—21no—
336C2×Dic3—21no—
337C2×Dic3—21no—
338C2×Dic3—21no—
339C2×Dic3—21no—
340C2×Dic3—61no—
341C2×Dic3—61no—
342C2×Dic3—61no—
343C2×Dic3—61no—
344C2×Dic3—61no—
345C2×Dic3—61no—
346C2×Dic3—182no—
347C2×Dic3—181no—
348C2×Dic3—181no—
349C2×Dic3—181no—
350C2×Dic3—181no—
351C2×Dic3—181no—
352C2×Dic3—181no—
353C2×Dic3—361no—
354C2×Dic3—361no—
355C2×Dic3D3612yes—
356C2×C32#87511yescharacteristic
357C2×C32#88311yescharacteristic
358C2×C32#88311yescharacteristic
359C2×C32#88611yescharacteristic
360C2×C32#89511yescharacteristic
361C2×C32#89811yescharacteristic
362C2×C32#89811yescharacteristic
363C3⋊D3—122no—
364C2×C9—21no—
365C2×C9—21no—
366C2×C9—21no—
367C2×C9—21no—
368C2×C9—21no—
369C2×C9—21no—
370C2×C9—21no—
371C2×C9#87611yescharacteristic
372C2×C9#88311yescharacteristic
373C2×C9#88411yescharacteristic
374C2×C9#88511yescharacteristic
375C2×C9#89411yescharacteristic
376C2×C9#89411yescharacteristic
377C2×C9#89811yescharacteristic
378D9—124no—
379D9—122no—
380C24—271no—
381C22×C4—31no—
382C22×C4—31no—
383C22×C4—91no—
384C22×C4—91no—
385C22×C4—271no—
386C22×C4D3×D911yescharacteristic
387C4⋊C4—542no—
388C22⋊C4—542no—
389C22⋊C4—542no—
390C22⋊C4—542no—
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—21no—
397C22×C3—21no—
398C22×C3C2×D3611yescharacteristic
399C22×C3#369711yescharacteristic
400C22×C3#369711yescharacteristic
401C22×C3D4×D911yescharacteristic
402C22×C3#369911yescharacteristic
403C22×C3#369911yescharacteristic
404C22×C3#370111yescharacteristic
405C22×C3#379611yescharacteristic
406C22×C3#379811yescharacteristic
407C22×C3#379911yescharacteristic
408C22×C3D3×D1211yescharacteristic
409C22×C3#380411yescharacteristic
410C22×C3#380411yescharacteristic
411C22×C3#381111yescharacteristic
412C2×D3—184no—
413C2×D3—182no—
414C2×D3—182no—
415C2×D3—182no—
416C2×D3—182no—
417C2×D3—182no—
418C2×D3—182no—
419C2×D3—362no—
420C2×D3—362no—
421C2×D3—362no—
422C2×D3—362no—
423C2×D3—362no—
424C2×D3—362no—
425C2×D3—361no—
426C2×D3—361no—
427C2×D3—361no—
428C2×D3—361no—
429C2×D3—361no—
430C2×D3—361no—
431C2×D3—361no—
432C2×D3—361no—
433C3×C4—42no—
434C3×C4—41no—
435C3×C4—41no—
436C3×C4—62no—
437C3×C4—121no—
438C3×C4—361no—
439C3×C4—361no—
440Dic3—22no—
441Dic3—41no—
442Dic3—121no—
443Dic3—121no—
444Dic3—362no—
445Dic3—361no—
446Dic3—361no—
447C32#467411yescharacteristic
448C9—21no—
449C9#467511yescharacteristic, complemented
450C23—272no—
451C23—541no—
452C23—541no—
453C23—541no—
454C23—541no—
455C23—541no—
456C23—541no—
457C23C2×D3×D911yescharacteristic
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—32no—
462C2×C4—61no—
463C2×C4—61no—
464C2×C4—61no—
465C2×C4—61no—
466C2×C4—61no—
467C2×C4—181no—
468C2×C4—181no—
469C2×C4—181no—
470C2×C4—181no—
471C2×C4—181no—
472C2×C4—181no—
473C2×C4—541no—
474C2×C4—541no—
475C2×C4—541no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—21no—
480C2×C3—21no—
481C2×C3—21no—
482C2×C3—21no—
483C2×C3#6343711yescharacteristic
484C2×C3#6344511yescharacteristic
485C2×C3#6344511yescharacteristic
486C2×C3#6344811yescharacteristic
487C2×C3#6345711yescharacteristic
488C2×C3#6346011yescharacteristic
489C2×C3#6346011yescharacteristic
490C2×C3#6384811yescharacteristic
491C2×C3#6385211yescharacteristic
492C2×C3#6385311yescharacteristic
493C2×C3#6386811yescharacteristic
494C2×C3#6389011yescharacteristic
495C2×C3#6391011yescharacteristic
496C2×C3#6397511yescharacteristic
497D3—364no—
498D3—362no—
499D3—362no—
500C22—542no—
501C22—542no—
502C22—542no—
503C22—1081no—
504C22—1081no—
505C22—1081no—
506C22—1081no—
507C22#8863511yescharacteristic
508C22#8863911yescharacteristic
509C22#8864011yescharacteristic
510C22D3×D3611yescharacteristic
511C22#8865811yescharacteristic
512C22#8866211yescharacteristic
513C22#8866811yescharacteristic
514C4—41no—
515C4—62no—
516C4—121no—
517C4—361no—
518C4—361no—
519C4—1081no—
520C3—21no—
521C3#1058835411yescharacteristic, complemented
522C3#1059120811yescharacteristic
523C2—1082no—
524C2#1172092011yescharacteristic
525C2#1172093411yescharacteristic
526C2#1172094011yescharacteristic
527C2#1172095711yescharacteristic
528C2#1172100511yescharacteristic
529C2#1172102011yescharacteristic
530C2#1172127511yescharacteristic
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#397
LCSG ▷ C23×C3×C9 ▷ C3×C9
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
4. C2×Dic3⋊Dic9C2semidirect
5. C9×C3⋊#95C2semidirect
6. C3×C9⋊#95C2semidirect
7. #11720331C2semidirect
35. C9⋊#95D3semidirect
124. C3⋊#95D9semidirect
289. C3×C9#397semidirect
449. C9#4675semidirect
521. C3#10588354semidirect

Maximal subgroups

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

subgroupindex
C2×#885362
C2×#884452
C2×Dic3⋊Dic92
C9×C3⋊#952
C3×C9⋊#952
#117203312
#105912083
#105883543