Finite Group Databasebeta

← #13150232

Subgroups — #13150232

Order 1728.

974 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88663C212yesmaximal, complemented
2C2×#88534C211yesmaximal, characteristic, complemented
3C9×#930C211yesmaximal, characteristic, complemented
4C3×#63492C211yesmaximal, characteristic, complemented
5#11721290C212yesmaximal, complemented
6#11721026C212yesmaximal, complemented
7#11721016C212yesmaximal, complemented
8#11720999C212yesmaximal, complemented
9#11720982C212yesmaximal, complemented
10#10594930—31nomaximal
11#10589699—31nomaximal
12C22×C3×C4×C9C2211yescharacteristic
13C22×C3⋊Dic9C2211yescharacteristic
14C22×C3⋊Dic9C2211yescharacteristic
15C2×C9×#67C2212yes—
16C2×C3×#611C2212yes—
17C2×C9×D12C2211yescharacteristic
18C2×C3×D36C2211yescharacteristic
19C2×#6194C2212yes—
20C2×#6194C2212yes—
21#88663—28no—
22C2×D3×Dic9C2212yes—
23C2×Dic3×D9C2212yes—
24C2×#6189C2212yes—
25#88534C2212yes—
26#88534C2212yes—
27C9×(C2×D3)⋊C4C2212yes—
28C3×(C2×D9)⋊C4C2212yes—
29C9×C4⋊Dic3C2211yescharacteristic
30C3×C4⋊Dic9C2211yescharacteristic
31(C2×D3)⋊Dic9C2212yes—
32(C2×D9)⋊Dic3C2212yes—
33#88442C2212yes—
34C2×#3805—32no—
35C2×#3750—31no—
36C3×#930—31no—
37C3×#930D311yescharacteristic
38#63980—32no—
39#63899—32no—
40#63899—32no—
41#63885—32no—
42#63885—32no—
43C2×#3699—32no—
44C9×#121—31no—
45#63501—32no—
46#63501—32no—
47#63492D311yescharacteristic, complemented
48C2×Dic9⋊C4—31no—
49C2×Dic9⋊C4—61no—
50#63458—32no—
51#63446—32no—
52#63440—32no—
53C23×C3×C9C2311yescharacteristic
54C22×D3×C9C2312yes—
55C22×C3×D9C2312yes—
56C2×C3×C4×C9—41no—
57C2×C3×C4×C9C2311yescharacteristic
58C2×C3×C4×C9C2311yescharacteristic
59C2×C3⋊Dic9—41no—
60C2×C3⋊Dic9D414yescomplemented
61C2×C3⋊Dic9C2312yes—
62C2×C3⋊Dic9C2312yes—
63C9×#67—24no—
64C3×#611—24no—
65C2×C9×Dic3C2312yes—
66C2×C3×Dic9C2312yes—
67C9×D12—41no—
68C3×D36—41no—
69#6194—24no—
70#6194—42no—
71D3×Dic9—24no—
72Dic3×D9—24no—
73#6189—24no—
74#5597—91no—
75#5597—91no—
76C22×C32×C4C2×D311yescharacteristic
77C22×C3⋊Dic3—31no—
78C22×C3⋊Dic3—31no—
79C2×C3×#67—32no—
80C2×C3×#67C2×D312yes—
81C2×C3×D12—31no—
82C2×C3×D12C2×D311yescharacteristic
83C2×#627—32no—
84C2×#625—32no—
85C2×#625—32no—
86#3805—68no—
87C2×D3×Dic3—32no—
88C2×D3×Dic3—32no—
89#3750—32no—
90#3750—32no—
91C3×(C2×D3)⋊C4—32no—
92C3×(C2×D3)⋊C4C2×D312yes—
93C3×C4⋊Dic3—31no—
94C3×C4⋊Dic3C2×D311yescharacteristic
95#3724—32no—
96(C2×D3)⋊Dic3—32no—
97(C2×D3)⋊Dic3—32no—
98C2×D4×C9—32no—
99C2×D4×C9—31no—
100C22×C4×C9—21no—
101C22×C4×C9C2×D311yescharacteristic
102C2×#611—32no—
103C2×#611—32no—
104C2×#611C2×D312yes—
105C22×Dic9—31no—
106C22×Dic9—31no—
107C22×Dic9—32no—
108C22×Dic9—61no—
109C22×Dic9—61no—
110#3699—68no—
111C2×D36C2×D311yescharacteristic
112C2×C4×D9—32no—
113C2×Dic18—32no—
114C9×C4⋊C4—31no—
115C9×C22⋊C4—32no—
116#3676—32no—
117(C2×D9)⋊C4—32no—
118(C2×D9)⋊C4C2×D312yes—
119C4⋊Dic9C2×D311yescharacteristic
120Dic9⋊C4—32no—
121Dic9⋊C4—32no—
122Dic9⋊C4—32no—
123Dic9⋊C4—62no—
124Dic9⋊C4—62no—
125C22×C3×C9—21no—
126C22×C3×C9—21no—
127C22×C3×C9C2×D412yes—
128C22×C3×C9C2411yescharacteristic
129C2×D3×C9—22no—
130C2×D3×C9—22no—
131C2×D3×C9—22no—
132C2×C3×D9—22no—
133C2×C3×D9—22no—
134C2×C3×D9—22no—
135C3×C4×C9—21no—
136C3×C4×C9—21no—
137C3⋊Dic9—22no—
138C3⋊Dic9C2×D414yes—
139C9×Dic3—22no—
140C3×Dic9—22no—
141C2×#237—92no—
142C2×#237—92no—
143C3×#121—31no—
144C3×#121—91no—
145#939—92no—
146#939—92no—
147#939—92no—
148#939—92no—
149#930—31no—
150#930D911yescharacteristic, complemented
151C2×Dic3⋊C4—91no—
152C2×Dic3⋊C4—91no—
153C2×Dic3⋊C4—181no—
154#896—92no—
155#896—92no—
156#884—92no—
157#884—92no—
158#878—92no—
159#878—92no—
160C23×C32C22×D311yescharacteristic
161C22×C3×D3—32no—
162C22×C3×D3C22×D312yes—
163C2×C32×C4—41no—
164C2×C32×C4C22×D311yescharacteristic
165C2×C32×C4C22×D311yescharacteristic
166C2×C3⋊Dic3—34no—
167C2×C3⋊Dic3—32no—
168C2×C3⋊Dic3—32no—
169C2×C3⋊Dic3—121no—
170C3×#67—24no—
171C3×#67—64no—
172C2×C3×Dic3—32no—
173C2×C3×Dic3C22×D312yes—
174C3×D12—41no—
175C3×D12—121no—
176#627—64no—
177#625—64no—
178#625—122no—
179D3×Dic3—64no—
180D3×Dic3—64no—
181C23×C9—21no—
182C23×C9—32no—
183C23×C9C22×D311yescharacteristic
184C22×D9C22×D312yes—
185D4×C9—64no—
186D4×C9—121no—
187C2×C4×C9—21no—
188C2×C4×C9—21no—
189C2×C4×C9—32no—
190C2×C4×C9—41no—
191C2×C4×C9—42no—
192C2×C4×C9C22×D311yescharacteristic
193C2×C4×C9C22×D311yescharacteristic
194#611—24no—
195#611—64no—
196#611—122no—
197C2×Dic9—34no—
198C2×Dic9—32no—
199C2×Dic9—32no—
200C2×Dic9—62no—
201C2×Dic9—64no—
202C2×Dic9—62no—
203C2×Dic9—64no—
204C2×Dic9—121no—
205C2×Dic9—122no—
206C2×Dic9C22×D312yes—
207D36—41no—
208C4×D9—64no—
209Dic18—64no—
210#537—271noSylow
211C2×C3×C9—22no—
212C2×C3×C9C22×D411yescharacteristic
213C2×C3×C9H(5,𝔽2)11yescharacteristic
214C2×C3×C92−1+411yescharacteristic
215D3×C9—42no—
216C3×D9—42no—
217C2×C3×D4—32no—
218C2×C3×D4—31no—
219C2×C3×D4—92no—
220C2×C3×D4—91no—
221C22×C3×C4—21no—
222C22×C3×C4C2×D911yescharacteristic
223C22×C3×C4D3211yescharacteristic
224C2×#67—32no—
225C2×#67—92no—
226C2×#67—92no—
227C2×#67—92no—
228C2×#67—92no—
229C2×#67C2×D912yes—
230C22×Dic3—92no—
231C22×Dic3—91no—
232C22×Dic3—91no—
233C22×Dic3—91no—
234C22×Dic3—91no—
235C22×Dic3—92no—
236C22×Dic3—181no—
237C22×Dic3—181no—
238#237—188no—
239#237—188no—
240C2×D12—31no—
241C2×D12C2×D911yescharacteristic
242C2×C4×D3—92no—
243C2×C4×D3—92no—
244C2×Dic6—92no—
245C2×Dic6—92no—
246C3×C4⋊C4—31no—
247C3×C4⋊C4—91no—
248C3×C22⋊C4—32no—
249C3×C22⋊C4—92no—
250#217—92no—
251#217—92no—
252(C2×D3)⋊C4—32no—
253(C2×D3)⋊C4—92no—
254(C2×D3)⋊C4—92no—
255(C2×D3)⋊C4C2×D912yes—
256C4⋊Dic3—31no—
257C4⋊Dic3C2×D911yescharacteristic
258Dic3⋊C4—92no—
259Dic3⋊C4—92no—
260Dic3⋊C4—92no—
261Dic3⋊C4—92no—
262Dic3⋊C4—92no—
263Dic3⋊C4—92no—
264Dic3⋊C4—182no—
265Dic3⋊C4—182no—
266C22×C32—21no—
267C22×C32—21no—
268C22×C32D3×D412yes—
269C22×C32C23×D311yescharacteristic
270C2×C3×D3—22no—
271C2×C3×D3—22no—
272C2×C3×D3—22no—
273C2×C3×D3—62no—
274C2×C3×D3—62no—
275C2×C3×D3—62no—
276C32×C4—21no—
277C32×C4—21no—
278C3⋊Dic3—34no—
279C3⋊Dic3—62no—
280C3×Dic3—22no—
281C3×Dic3—62no—
282C22×C9—22no—
283C22×C9—22no—
284C22×C9—21no—
285C22×C9—21no—
286C22×C9—21no—
287C22×C9—22no—
288C22×C9—62no—
289C22×C9—62no—
290C22×C9—62no—
291C22×C9D3×D412yes—
292C22×C9C23×D311yescharacteristic
293C2×D9—22no—
294C2×D9—22no—
295C2×D9—22no—
296C4×C9—21no—
297C4×C9—21no—
298C4×C9—41no—
299C4×C9—41no—
300C4×C9—62no—
301Dic9—22no—
302Dic9—34no—
303Dic9—64no—
304Dic9—62no—
305Dic9—122no—
306C2×#41—272no—
307C22⋊Q8—272no—
308C22⋊Q8—272no—
309#121—31no—
310#121—91no—
311#121—272no—
312#121—272no—
313#121—272no—
314C2×C4⋊C4—271no—
315C3×C9#53711yescharacteristic, complemented, Sylow
316C23×C3—21no—
317C23×C3—32no—
318C23×C3—92no—
319C23×C3C22×D911yescharacteristic
320C23×C3C2×D3211yescharacteristic
321C22×D3—32no—
322C22×D3C22×D912yes—
323C3×D4—64no—
324C3×D4—121no—
325C3×D4—184no—
326C3×D4—361no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—32no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4—42no—
333C2×C3×C4—92no—
334C2×C3×C4C22×D911yescharacteristic
335C2×C3×C4C22×D911yescharacteristic
336C2×C3×C4C2×D3211yescharacteristic
337C2×C3×C4C2×D3211yescharacteristic
338#67—24no—
339#67—64no—
340#67—184no—
341#67—184no—
342#67—362no—
343#67—362no—
344C2×Dic3—32no—
345C2×Dic3—92no—
346C2×Dic3—94no—
347C2×Dic3—92no—
348C2×Dic3—94no—
349C2×Dic3—92no—
350C2×Dic3—92no—
351C2×Dic3—182no—
352C2×Dic3—184no—
353C2×Dic3—182no—
354C2×Dic3—184no—
355C2×Dic3—184no—
356C2×Dic3—361no—
357C2×Dic3—361no—
358C2×Dic3—362no—
359C2×Dic3C22×D912yes—
360D12—41no—
361D12—121no—
362C4×D3—184no—
363C4×D3—184no—
364Dic6—184no—
365Dic6—184no—
366C2×C32—22no—
367C2×C32C2×D3×D411yescharacteristic
368C2×C32#100411yescharacteristic
369C2×C32#101011yescharacteristic
370C3×D3—42no—
371C3×D3—122no—
372C2×C9—24no—
373C2×C9—22no—
374C2×C9—21no—
375C2×C9—21no—
376C2×C9—21no—
377C2×C9—122no—
378C2×C9C2×D3×D411yescharacteristic
379C2×C9#100411yescharacteristic
380C2×C9#101011yescharacteristic
381D9—42no—
382#41—548no—
383C2×Q8—272no—
384C2×D4—32no—
385C2×D4—31no—
386C2×D4—92no—
387C2×D4—91no—
388C2×D4—272no—
389C2×D4—272no—
390C22×C4—272no—
391C22×C4—272no—
392C22×C4—271no—
393C22×C4—271no—
394C22×C4D3×D911yescharacteristic
395C4⋊C4—31no—
396C4⋊C4—91no—
397C4⋊C4—272no—
398C4⋊C4—272no—
399C4⋊C4—272no—
400C22⋊C4—32no—
401C22⋊C4—92no—
402C22⋊C4—272no—
403C22⋊C4—272no—
404C22×C3—22no—
405C22×C3—22no—
406C22×C3—21no—
407C22×C3—21no—
408C22×C3—21no—
409C22×C3—21no—
410C22×C3—21no—
411C22×C3—22no—
412C22×C3—62no—
413C22×C3—62no—
414C22×C3—62no—
415C22×C3—182no—
416C22×C3—182no—
417C22×C3—182no—
418C22×C3D4×D912yes—
419C22×C3C23×D911yescharacteristic
420C22×C3#380212yes—
421C22×C3C22×D3211yescharacteristic
422C2×D3—22no—
423C2×D3—22no—
424C2×D3—22no—
425C2×D3—62no—
426C2×D3—62no—
427C2×D3—62no—
428C3×C4—21no—
429C3×C4—21no—
430C3×C4—21no—
431C3×C4—21no—
432C3×C4—41no—
433C3×C4—41no—
434C3×C4—62no—
435C3×C4—182no—
436Dic3—22no—
437Dic3—62no—
438Dic3—94no—
439Dic3—94no—
440Dic3—184no—
441Dic3—182no—
442Dic3—182no—
443Dic3—362no—
444C32#559711yescharacteristic
445C9—21no—
446C9#559711yescharacteristic, complemented
447C23—32no—
448C23—92no—
449C23C2×D3×D911yescharacteristic
450Q8—544no—
451D4—64no—
452D4—121no—
453D4—184no—
454D4—361no—
455D4—544no—
456D4—1082no—
457C2×C4—32no—
458C2×C4—41no—
459C2×C4—92no—
460C2×C4—272no—
461C2×C4—274no—
462C2×C4—272no—
463C2×C4—544no—
464C2×C4—544no—
465C2×C4—1081no—
466C2×C4C2×D3×D911yescharacteristic
467C2×C4C2×D3×D911yescharacteristic
468C2×C3—24no—
469C2×C3—22no—
470C2×C3—22no—
471C2×C3—21no—
472C2×C3—21no—
473C2×C3—21no—
474C2×C3—122no—
475C2×C3—362no—
476C2×C3C2×D4×D911yescharacteristic
477C2×C3#6371011yescharacteristic
478C2×C3#6371611yescharacteristic
479C2×C3#6429811yescharacteristic
480C2×C3C2×#380211yescharacteristic
481C2×C3#6433011yescharacteristic
482D3—42no—
483D3—122no—
484C22—21no—
485C22—21no—
486C22—62no—
487C22—62no—
488C22—62no—
489C22—182no—
490C22—182no—
491C22—182no—
492C22#8864612yes—
493C22C22×D3×D911yescharacteristic
494C4—21no—
495C4—21no—
496C4—62no—
497C4—182no—
498C4—274no—
499C4—542no—
500C3—21no—
501C3#1058969911yescharacteristic, complemented
502C3#1059493011yescharacteristic
503C2—22no—
504C2—122no—
505C2—362no—
506C2#1172246511yescharacteristic
507C2C2×#8864611yescharacteristic
508C2#1172253511yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC22×C32C23×D3
hypercenterC22×C4D3×D9
hypocenterC3×C9#537
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#88663C2semidirect
2. C2×#88534C2semidirect
3. C9×#930C2semidirect
4. C3×#63492C2semidirect
5. #11721290C2semidirect
6. #11721026C2semidirect
7. #11721016C2semidirect
8. #11720999C2semidirect
9. #11720982C2semidirect
47. #63492D3semidirect
60. C2×C3⋊Dic9D4semidirect
150. #930D9semidirect
315. C3×C9#537semidirect
446. C9#5597semidirect
501. C3#10589699semidirect

Maximal subgroups

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

subgroupindex
C2×#886632
C2×#885342
C9×#9302
C3×#634922
#117212902
#117210262
#117210162
#117209992
#117209822
#105949303
#105896993