Finite Group Databasebeta

← #13148827

Subgroups — #13148827

Order 1728.

884 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88640C211yesmaximal, characteristic
2C2×D9×Dic6C211yesmaximal, characteristic, complemented
3#11721753C211yesmaximal, characteristic, complemented
4C9×#897C211yesmaximal, characteristic, complemented
5C3×#63454C211yesmaximal, characteristic, complemented
6#11720989C212yesmaximal, complemented
7#11720988C212yesmaximal, complemented
8#11720969C212yesmaximal, complemented
9#11720961C211yesmaximal, characteristic, complemented
10#11720956C212yesmaximal, complemented
11#11720929C211yesmaximal, characteristic, complemented
12#10594380—31nomaximal
13#10589730—31nomaximal
14C2×C4×C3⋊D9C2211yescharacteristic
15C2×#6226C2211yescharacteristic
16C2×C4×D3×C9C2211yescharacteristic
17C2×C3×C4×D9C2211yescharacteristic
18C2×C9×Dic6C2211yescharacteristic
19C2×#6195C2211yescharacteristic
20C2×#6194C2211yescharacteristic
21C2×#6192C2211yescharacteristic
22C2×Dic3×D9C2212yes—
23C2×#6189C2211yescharacteristic
24C2×#6189C2212yes—
25#88640—24no—
26D9×Dic6—24no—
27C3×C9×C4⋊C4C2211yescharacteristic
28#88536C2212yes—
29C4⋊(C3⋊Dic9)C2212yes—
30C9×(C2×D3)⋊C4C2212yes—
31C3×(C2×D9)⋊C4C2212yes—
32C9×C4⋊Dic3C2212yes—
33C3×C4⋊Dic9C2211yescharacteristic
34C3×C4×Dic9C2212yes—
35(C2×D3)⋊Dic9C2212yes—
36#88445C2212yes—
37Dic9⋊Dic3C2212yes—
38Dic9⋊Dic3C2212yes—
39C2×#3798—31no—
40C2×D3×Dic6—31no—
41#64106—31no—
42C3×#897D311yescharacteristic
43C3×#892—31no—
44#63918—32no—
45#63891—32no—
46#63890—32no—
47#63871—31no—
48#63857—31no—
49#63852—32no—
50C2×Q8×D9—31no—
51C2×#3697—31no—
52C9×C22⋊Q8—31no—
53#63508—31no—
54#63486—31no—
55#63459—31no—
56#63459—61no—
57#63456—32no—
58#63454D311yescharacteristic, complemented
59#63450—32no—
60#63447—32no—
61#63445—32no—
62C22×C3⋊D9C2311yescharacteristic
63C22×D3×C9C2311yescharacteristic
64C22×C3×D9C2311yescharacteristic
65C2×C3×C4×C9C2312yes—
66C2×C3×C4×C9C2311yescharacteristic
67C2×C3⋊Dic9C2312yes—
68C2×C3⋊Dic9C2311yescharacteristic
69C4×C3⋊D9—22no—
70#6226—22no—
71C2×C9×Dic3C2312yes—
72C2×C9×Dic3C2311yescharacteristic
73C2×C3×Dic9C2312yes—
74C2×C3×Dic9C2311yescharacteristic
75C4×D3×C9—22no—
76C3×C4×D9D414yes—
77C9×Dic6—22no—
78#6195—22no—
79#6194—22no—
80#6192—22no—
81Dic3×D9—24no—
82#6189—24no—
83#6189—22no—
84#5628—91no—
85#5526—91no—
86C2×C4×C3⋊D3—31no—
87C2×#634—31no—
88C2×C3×C4×D3—31no—
89C2×C3×C4×D3C2×D311yescharacteristic
90C2×C3×Dic6C2×D311yescharacteristic
91C2×#627—32no—
92C2×#627—31no—
93C2×#626—31no—
94C2×#626—31no—
95C2×#625—31no—
96C2×D3×Dic3—32no—
97#3798—64no—
98D3×Dic6—64no—
99C32×C4⋊C4C2×D311yescharacteristic
100#3752—32no—
101C4⋊(C3⋊Dic3)—32no—
102C3×(C2×D3)⋊C4—32no—
103C3×(C2×D3)⋊C4C2×D312yes—
104C3×C4⋊Dic3—31no—
105C3×C4⋊Dic3C2×D312yes—
106C3×C4×Dic3—32no—
107Dic3⋊Dic3—32no—
108Dic3⋊Dic3—32no—
109#3722—32no—
110(C2×D3)⋊Dic3—32no—
111C2×Q8×C9—31no—
112C22×C4×C9—31no—
113C2×#611—31no—
114C2×#611—31no—
115Q8×D9—64no—
116#3697—64no—
117C2×D36—31no—
118C2×C4×D9—32no—
119C2×C4×D9—31no—
120C2×C4×D9—61no—
121C2×C4×D9C2×D311yescharacteristic
122C2×Dic18—31no—
123C2×Dic18—32no—
124C2×Dic18—31no—
125C2×Dic18—61no—
126C9×C4⋊C4—21no—
127C9×C4⋊C4—32no—
128C9×C4⋊C4C2×D311yescharacteristic
129C9×C22⋊C4—32no—
130#3676—32no—
131(C2×D9)⋊C4—32no—
132(C2×D9)⋊C4—32no—
133(C2×D9)⋊C4—62no—
134(C2×D9)⋊C4C2×D312yes—
135C4⋊Dic9—32no—
136C4⋊Dic9—62no—
137C4⋊Dic9C2×D311yescharacteristic
138Dic9⋊C4—32no—
139Dic9⋊C4—32no—
140C4×Dic9C2×D312yes—
141C22×C3×C9C2411yescharacteristic
142C2×C3⋊D9—21no—
143C2×C3⋊D9—21no—
144C2×C3⋊D9—21no—
145C2×D3×C9—21no—
146C2×D3×C9—21no—
147C2×D3×C9—21no—
148C2×C3×D9—21no—
149C2×C3×D9—21no—
150C2×C3×D9C2×D412yes—
151C3×C4×C9—22no—
152C3×C4×C9C2×D412yes—
153C3⋊Dic9—22no—
154C3⋊Dic9—21no—
155C9×Dic3—22no—
156C9×Dic3—21no—
157C3×Dic9—22no—
158C3×Dic9C2×D412yes—
159C2×D3×Q8—91no—
160C2×#235—91no—
161C2×#235—91no—
162C22×Dic6—91no—
163C3×C22⋊Q8—31no—
164C3×#117—91no—
165#946—91no—
166#924—91no—
167#897—91no—
168#897—91no—
169#897—91no—
170#897—91no—
171#897—181no—
172#897D911yescharacteristic, complemented
173#894—92no—
174#892—31no—
175#888—92no—
176#886—92no—
177#886—92no—
178#885—92no—
179#883—92no—
180#875—92no—
181#869—92no—
182C22×C3⋊D3—31no—
183C22×C3×D3—31no—
184C22×C3×D3C22×D311yescharacteristic
185C2×C32×C4C22×D312yes—
186C2×C32×C4C22×D311yescharacteristic
187C2×C3⋊Dic3—32no—
188C2×C3⋊Dic3—31no—
189C4×C3⋊D3—62no—
190#634—62no—
191C2×C3×Dic3—31no—
192C2×C3×Dic3—32no—
193C2×C3×Dic3C22×D312yes—
194C2×C3×Dic3C22×D311yescharacteristic
195C3×C4×D3—22no—
196C3×C4×D3—34no—
197C3×Dic6—22no—
198#627—64no—
199#627—62no—
200#626—62no—
201#626—62no—
202#625—62no—
203D3×Dic3—64no—
204C23×C9—31no—
205C22×D9—31no—
206C22×D9—61no—
207C22×D9C22×D311yescharacteristic
208Q8×C9—62no—
209C2×C4×C9—21no—
210C2×C4×C9—22no—
211C2×C4×C9—31no—
212C2×C4×C9—32no—
213C2×C4×C9—62no—
214C2×C4×C9C22×D312yes—
215C2×C4×C9C22×D311yescharacteristic
216#611—62no—
217#611—62no—
218C2×Dic9—31no—
219C2×Dic9—32no—
220C2×Dic9—61no—
221C2×Dic9—62no—
222C2×Dic9C22×D312yes—
223C2×Dic9C22×D311yescharacteristic
224D36—62no—
225C4×D9—64no—
226C4×D9—62no—
227C4×D9—122no—
228C4×D9D1214yes—
229Dic18—62no—
230Dic18—62no—
231Dic18—64no—
232Dic18—122no—
233#536—271noSylow
234C2×C3×C9C22×D411yescharacteristic
235C2×C3×C92−1+411yescharacteristic
236C2×C3×C92−1+411yescharacteristic
237C3⋊D9—41no—
238D3×C9—41no—
239C3×D9—22no—
240C2×C3×Q8—31no—
241C22×C3×C4—31no—
242C22×C3×C4—91no—
243C2×#67—91no—
244C2×#67—91no—
245C2×#67—91no—
246C2×#67—91no—
247C22×Dic3—92no—
248D3×Q8—184no—
249#235—184no—
250#235—184no—
251C2×D12—91no—
252C2×D12—91no—
253C2×C4×D3—31no—
254C2×C4×D3—92no—
255C2×C4×D3—91no—
256C2×C4×D3—91no—
257C2×C4×D3—181no—
258C2×C4×D3C2×D911yescharacteristic
259C2×Dic6—92no—
260C2×Dic6—91no—
261C2×Dic6—91no—
262C2×Dic6—91no—
263C2×Dic6—91no—
264C2×Dic6—92no—
265C2×Dic6—181no—
266C2×Dic6—184no—
267C2×Dic6C2×D911yescharacteristic
268C3×C4⋊C4—21no—
269C3×C4⋊C4—32no—
270C3×C4⋊C4—91no—
271C3×C4⋊C4C2×D911yescharacteristic
272C3×C4⋊C4D3211yescharacteristic
273C3×C22⋊C4—32no—
274C3×C22⋊C4—92no—
275C3×C42—92no—
276#217—92no—
277(C2×D3)⋊C4—32no—
278(C2×D3)⋊C4—92no—
279(C2×D3)⋊C4—92no—
280(C2×D3)⋊C4—92no—
281(C2×D3)⋊C4—92no—
282(C2×D3)⋊C4—92no—
283(C2×D3)⋊C4—182no—
284(C2×D3)⋊C4C2×D912yes—
285C4⋊Dic3—31no—
286C4⋊Dic3—92no—
287C4⋊Dic3—92no—
288C4⋊Dic3—92no—
289C4⋊Dic3—92no—
290C4⋊Dic3—182no—
291C4⋊Dic3C2×D912yes—
292Dic3⋊C4—92no—
293Dic3⋊C4—92no—
294C4×Dic3—32no—
295C22×C32C23×D311yescharacteristic
296C2×C3⋊D3—61no—
297C2×C3⋊D3—61no—
298C2×C3⋊D3—61no—
299C2×C3×D3—21no—
300C2×C3×D3—21no—
301C2×C3×D3—21no—
302C2×C3×D3—32no—
303C2×C3×D3—61no—
304C2×C3×D3—61no—
305C32×C4—22no—
306C32×C4D3×D412yes—
307C3⋊Dic3—62no—
308C3⋊Dic3—61no—
309C3×Dic3—22no—
310C3×Dic3—21no—
311C3×Dic3—32no—
312C3×Dic3—62no—
313C22×C9—21no—
314C22×C9—61no—
315C22×C9—61no—
316C22×C9—61no—
317C22×C9C23×D311yescharacteristic
318C2×D9—21no—
319C2×D9—21no—
320C2×D9—62no—
321C2×D9—61no—
322C2×D9—61no—
323C2×D9—61no—
324C2×D9—121no—
325C2×D9—121no—
326C2×D9C2×D1212yes—
327C4×C9—22no—
328C4×C9—22no—
329C4×C9—42no—
330C4×C9—62no—
331C4×C9—61no—
332C4×C9C2×D1212yes—
333Dic9—22no—
334Dic9—62no—
335Dic9—61no—
336Dic9—122no—
337Dic9—121no—
338Dic9C2×D1212yes—
339C2×#41—271no—
340C22×Q8—271no—
341C4⋊Q8—272no—
342#124—272no—
343#123—272no—
344#123—272no—
345C22⋊Q8—31no—
346C22⋊Q8—271no—
347C22⋊Q8—271no—
348C22⋊Q8—271no—
349#117—91no—
350C3×C9#53611yescharacteristic, complemented, Sylow
351C23×C3—31no—
352C23×C3—91no—
353C22×D3—31no—
354C22×D3—91no—
355C22×D3—91no—
356C22×D3—181no—
357C22×D3C22×D911yescharacteristic
358C3×Q8—62no—
359C2×C3×C4—21no—
360C2×C3×C4—22no—
361C2×C3×C4—31no—
362C2×C3×C4—32no—
363C2×C3×C4—62no—
364C2×C3×C4—91no—
365C2×C3×C4—94no—
366C2×C3×C4—92no—
367C2×C3×C4C22×D912yes—
368C2×C3×C4C22×D911yescharacteristic
369C2×C3×C4C2×D3212yes—
370C2×C3×C4C2×D3211yescharacteristic
371#67—182no—
372#67—182no—
373#67—182no—
374#67—182no—
375C2×Dic3—31no—
376C2×Dic3—32no—
377C2×Dic3—91no—
378C2×Dic3—92no—
379C2×Dic3—91no—
380C2×Dic3—92no—
381C2×Dic3—181no—
382C2×Dic3—182no—
383C2×Dic3—184no—
384C2×Dic3C22×D911yescharacteristic
385C2×Dic3C22×D912yes—
386D12—182no—
387D12—182no—
388C4×D3—22no—
389C4×D3—34no—
390C4×D3—184no—
391C4×D3—182no—
392C4×D3—182no—
393C4×D3—362no—
394Dic6—22no—
395Dic6—184no—
396Dic6—182no—
397Dic6—182no—
398Dic6—184no—
399Dic6—182no—
400Dic6—182no—
401Dic6—362no—
402C2×C32C2×D3×D411yescharacteristic
403C2×C32#100711yescharacteristic
404C2×C32#101011yescharacteristic
405C3⋊D3—121no—
406C3×D3—41no—
407C3×D3—62no—
408C2×C9—21no—
409C2×C9—21no—
410C2×C9—21no—
411C2×C9—121no—
412C2×C9C22×D1211yescharacteristic
413C2×C9#101011yescharacteristic
414C2×C9#101011yescharacteristic
415D9—22no—
416D9—122no—
417D9—121no—
418#41—544no—
419C2×Q8—31no—
420C2×Q8—272no—
421C2×Q8—271no—
422C2×Q8—271no—
423C2×Q8—544no—
424C2×D4—271no—
425C2×D4—271no—
426C2×D4—271no—
427C22×C4—31no—
428C22×C4—91no—
429C22×C4—272no—
430C22×C4—271no—
431C4⋊C4—32no—
432C4⋊C4—91no—
433C4⋊C4—272no—
434C4⋊C4—272no—
435C4⋊C4—272no—
436C4⋊C4D3×D911yescharacteristic
437C22⋊C4—32no—
438C22⋊C4—92no—
439C22⋊C4—272no—
440C22⋊C4—272no—
441C22⋊C4—272no—
442C42—92no—
443C22×C3—21no—
444C22×C3—61no—
445C22×C3—61no—
446C22×C3—61no—
447C22×C3—92no—
448C22×C3—181no—
449C22×C3—181no—
450C22×C3C23×D911yescharacteristic
451C22×C3C22×D3211yescharacteristic
452C2×D3—21no—
453C2×D3—21no—
454C2×D3—21no—
455C2×D3—32no—
456C2×D3—61no—
457C2×D3—61no—
458C2×D3—182no—
459C2×D3—181no—
460C2×D3—181no—
461C2×D3—181no—
462C2×D3—181no—
463C2×D3—181no—
464C2×D3—181no—
465C2×D3—361no—
466C2×D3—361no—
467C3×C4—22no—
468C3×C4—22no—
469C3×C4—22no—
470C3×C4—42no—
471C3×C4—62no—
472C3×C4—61no—
473C3×C4—92no—
474C3×C4—182no—
475C3×C4D4×D912yes—
476C3×C4D3×D1212yes—
477Dic3—22no—
478Dic3—21no—
479Dic3—32no—
480Dic3—62no—
481Dic3—182no—
482Dic3—181no—
483Dic3—182no—
484Dic3—181no—
485Dic3—362no—
486Dic3—361no—
487C32#562811yescharacteristic
488C9—21no—
489C9#552611yescharacteristic, complemented
490C23—31no—
491C23—91no—
492C23—271no—
493Q8—62no—
494Q8—544no—
495Q8—542no—
496Q8—542no—
497D4—542no—
498D4—542no—
499D4—542no—
500C2×C4—32no—
501C2×C4—31no—
502C2×C4—62no—
503C2×C4—91no—
504C2×C4—94no—
505C2×C4—92no—
506C2×C4—272no—
507C2×C4—271no—
508C2×C4—544no—
509C2×C4—542no—
510C2×C4C2×D3×D912yes—
511C2×C4C2×D3×D911yescharacteristic
512C2×C3—21no—
513C2×C3—21no—
514C2×C3—21no—
515C2×C3—121no—
516C2×C3—182no—
517C2×C3C2×D4×D911yescharacteristic
518C2×C3#6371311yescharacteristic
519C2×C3#6371611yescharacteristic
520C2×C3C2×D3×D1211yescharacteristic
521C2×C3#6430911yescharacteristic
522C2×C3#6431511yescharacteristic
523D3—41no—
524D3—62no—
525D3—362no—
526D3—361no—
527D3—361no—
528C22—61no—
529C22—61no—
530C22—61no—
531C22—92no—
532C22—181no—
533C22—181no—
534C22—541no—
535C22—541no—
536C22—541no—
537C22C22×D3×D911yescharacteristic
538C4—22no—
539C4—61no—
540C4—62no—
541C4—92no—
542C4—182no—
543C4—541no—
544C4—542no—
545C4D9×D1212yes—
546C3—21no—
547C3#1058973011yescharacteristic, complemented
548C3#1059438011yescharacteristic
549C2—121no—
550C2—182no—
551C2—1081no—
552C2C2×D9×D1211yescharacteristic
553C2#1172249411yescharacteristic
554C2#1172251011yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C9×C4⋊C4C22
socleC22×C32C23×D3
prime radicalC2×#88640C2
hypercenterC4⋊C4D3×D9
hypocenterC3×C9#536
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×D9×Dic6C2semidirect
3. #11721753C2semidirect
4. C9×#897C2semidirect
5. C3×#63454C2semidirect
6. #11720989C2semidirect
7. #11720988C2semidirect
8. #11720969C2semidirect
9. #11720961C2semidirect
10. #11720956C2semidirect
11. #11720929C2semidirect
58. #63454D3semidirect
172. #897D9semidirect
350. C3×C9#536semidirect
489. C9#5526semidirect
547. C3#10589730semidirect

Maximal subgroups

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

subgroupindex
C2×#886402
C2×D9×Dic62
#117217532
C9×#8972
C3×#634542
#117209892
#117209882
#117209692
#117209612
#117209562
#117209292
#105943803
#105897303