Finite Group Databasebeta

← #13147659

Subgroups — #13147659

Order 1728.

772 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#6226C211yesmaximal, characteristic, complemented
2C4×C9×D12C211yesmaximal, characteristic, complemented
3C3×C42×D9C211yesmaximal, characteristic, complemented
4#11721036C211yesmaximal, characteristic
5#11721031C212yesmaximal, complemented
6C4×#6194C212yesmaximal, complemented
7#11720964C211yesmaximal, characteristic, complemented
8#11720963C212yesmaximal, complemented
9#11720959C212yesmaximal, complemented
10#11720945C211yesmaximal, characteristic, complemented
11#11720935C211yesmaximal, characteristic, complemented
12#10593974—31nomaximal
13#10589659—31nomaximal
14C2×#6226C2211yescharacteristic
15C2×C9×D12C2211yescharacteristic
16C2×C4×D3×C9C2212yes—
17C2×C3×C4×D9C2211yescharacteristic
18C2×C3×C4×D9C2211yescharacteristic
19C2×C3×C4×D9C2211yescharacteristic
20C2×#6194C2212yes—
21C3×C42×C9C2211yescharacteristic
22C4⋊(C3⋊Dic9)C2211yescharacteristic
23#88534C2212yes—
24C4×C3⋊Dic9C2212yes—
25C9×(C2×D3)⋊C4C2212yes—
26C9×C4⋊Dic3C2211yescharacteristic
27C3×C4×Dic9C2211yescharacteristic
28C3×C4×Dic9C2211yescharacteristic
29C3×C4×Dic9C2211yescharacteristic
30(C2×D3)⋊Dic9C2212yes—
31(C2×D3)⋊Dic9C2212yes—
32(C2×D3)⋊Dic9C2212yes—
33(C2×D9)⋊Dic3C2212yes—
34Dic3⋊Dic9C2212yes—
35Dic3⋊Dic9C2212yes—
36#88442C2212yes—
37C4×#634—31no—
38C3×C4×D12D311yescharacteristic
39C3×C42×D3—31no—
40#63908—31no—
41#63905—32no—
42C4×#625—32no—
43#63869—31no—
44#63860—31no—
45#63859—32no—
46#63857—31no—
47#63855—32no—
48C4×D4×C9—31no—
49#63500—31no—
50#63498—31no—
51C4×#611—32no—
52#63488—32no—
53#63459—31no—
54#63452—31no—
55#63445—32no—
56#63441—32no—
57C42×D9D311yescharacteristic, complemented
58C4×Dic18—31no—
59C4×Dic18—61no—
60C22×D3×C9C2312yes—
61C22×C3×D9C2311yescharacteristic
62C2×C3×C4×C9C2311yescharacteristic
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3⋊Dic9C2312yes—
66C2×C3⋊Dic9C2312yes—
67#6226—22no—
68C2×C9×Dic3C2312yes—
69C2×C3×Dic9C2311yescharacteristic
70C2×C3×Dic9C2311yescharacteristic
71C2×C3×Dic9C2311yescharacteristic
72C9×D12—22no—
73C4×D3×C9—24no—
74C3×C4×D9—22no—
75C3×C4×D9—22no—
76C3×C4×D9—22no—
77#6194—24no—
78#5557—91no—
79#5479—91no—
80C2×#634—31no—
81C2×C3×D12C2×D311yescharacteristic
82C2×C3×C4×D3—31no—
83C2×C3×C4×D3—31no—
84C2×C3×C4×D3—31no—
85C2×C3×C4×D3C2×D312yes—
86C2×#625—32no—
87C32×C42C2×D311yescharacteristic
88C4⋊(C3⋊Dic3)—31no—
89#3750—32no—
90C4×C3⋊Dic3—32no—
91C3×(C2×D3)⋊C4C2×D312yes—
92C3×C4⋊Dic3C2×D311yescharacteristic
93C3×C4×Dic3—31no—
94C3×C4×Dic3—31no—
95C3×C4×Dic3—31no—
96#3724—32no—
97Dic3⋊Dic3—32no—
98Dic3⋊Dic3—32no—
99(C2×D3)⋊Dic3—32no—
100(C2×D3)⋊Dic3—32no—
101(C2×D3)⋊Dic3—32no—
102(C2×D3)⋊Dic3—32no—
103C2×D4×C9—31no—
104C22×C4×C9—32no—
105C2×#611—32no—
106C2×C4×D9C2×D311yescharacteristic
107C2×C4×D9C2×D311yescharacteristic
108C2×C4×D9C2×D311yescharacteristic
109C2×Dic18—31no—
110C2×Dic18—61no—
111C9×C4⋊C4—31no—
112C9×C22⋊C4—32no—
113C42×C9—21no—
114C42×C9C2×D311yescharacteristic
115#3676—32no—
116#3676—32no—
117#3676—32no—
118(C2×D9)⋊C4—32no—
119C4⋊Dic9—31no—
120C4⋊Dic9—32no—
121C4⋊Dic9—32no—
122C4⋊Dic9—61no—
123Dic9⋊C4—32no—
124Dic9⋊C4—32no—
125Dic9⋊C4—62no—
126C4×Dic9—32no—
127C4×Dic9—62no—
128C4×Dic9C2×D311yescharacteristic
129C4×Dic9C2×D311yescharacteristic
130C4×Dic9C2×D311yescharacteristic
131C22×C3×C9C2411yescharacteristic
132C2×D3×C9—22no—
133C2×D3×C9—22no—
134C2×D3×C9—22no—
135C2×C3×D9—21no—
136C2×C3×D9—21no—
137C2×C3×D9#4112yes—
138C3×C4×C9—21no—
139C3×C4×C9#4112yes—
140C3×C4×C9#4112yes—
141C3⋊Dic9—22no—
142C3⋊Dic9—22no—
143C9×Dic3—22no—
144C3×Dic9—21no—
145C3×Dic9—21no—
146C3×Dic9—21no—
147C3×C4×D4—31no—
148C2×C3×C42—91no—
149#938—91no—
150#936—91no—
151#930—91no—
152#929—92no—
153C4×#67—92no—
154C4×#67—92no—
155#926—92no—
156#924—91no—
157#897—91no—
158#890—91no—
159#883—92no—
160#879—92no—
161#876—92no—
162#875—91no—
163C4×D12D911yescharacteristic, complemented
164#872—92no—
165C42×D3—31no—
166#870—91no—
167C4×Dic6—91no—
168C4×Dic6—91no—
169C4×Dic6—181no—
170C22×C3×D3—31no—
171C22×C3×D3C22×D312yes—
172C2×C32×C4C22×D311yescharacteristic
173C2×C32×C4C22×D311yescharacteristic
174C2×C32×C4C22×D311yescharacteristic
175C2×C3⋊Dic3—32no—
176C2×C3⋊Dic3—32no—
177#634—62no—
178C2×C3×Dic3—31no—
179C2×C3×Dic3—31no—
180C2×C3×Dic3—31no—
181C2×C3×Dic3C22×D312yes—
182C3×D12—22no—
183C3×C4×D3—24no—
184C3×C4×D3—62no—
185C3×C4×D3—62no—
186C3×C4×D3—62no—
187#625—64no—
188C23×C9—32no—
189C22×D9C22×D311yescharacteristic
190D4×C9—62no—
191C2×C4×C9—21no—
192C2×C4×C9—21no—
193C2×C4×C9—21no—
194C2×C4×C9—32no—
195C2×C4×C9—64no—
196C2×C4×C9C22×D311yescharacteristic
197C2×C4×C9C22×D311yescharacteristic
198C2×C4×C9C22×D311yescharacteristic
199#611—64no—
200C2×Dic9—32no—
201C2×Dic9—32no—
202C2×Dic9—62no—
203C2×Dic9—62no—
204C2×Dic9C22×D311yescharacteristic
205C2×Dic9C22×D311yescharacteristic
206C2×Dic9C22×D311yescharacteristic
207C4×D9—22no—
208C4×D9—22no—
209C4×D9—22no—
210Dic18—64no—
211Dic18—62no—
212#529—271noSylow
213C2×C3×C9C2×#4111yescharacteristic
214C2×C3×C9C2×#4111yescharacteristic
215C2×C3×C9C2×#4111yescharacteristic
216D3×C9—42no—
217C3×D9—22no—
218C2×C3×D4—31no—
219C22×C3×C4—32no—
220C22×C3×C4—91no—
221C22×C3×C4—91no—
222C22×C3×C4—91no—
223C2×#67—92no—
224C2×#67—92no—
225C2×D12C2×D911yescharacteristic
226C2×C4×D3—31no—
227C2×C4×D3—31no—
228C2×C4×D3—31no—
229C2×C4×D3C2×D912yes—
230C2×Dic6—91no—
231C2×Dic6—91no—
232C2×Dic6—181no—
233C3×C4⋊C4—31no—
234C3×C22⋊C4—32no—
235C3×C42—21no—
236C3×C42—91no—
237C3×C42—91no—
238C3×C42—91no—
239C3×C42C2×D911yescharacteristic
240C3×C42D3211yescharacteristic
241#217—92no—
242#217—92no—
243#217—92no—
244#217—92no—
245(C2×D3)⋊C4—92no—
246(C2×D3)⋊C4—92no—
247(C2×D3)⋊C4—92no—
248(C2×D3)⋊C4—92no—
249(C2×D3)⋊C4C2×D912yes—
250C4⋊Dic3—92no—
251C4⋊Dic3—92no—
252C4⋊Dic3—91no—
253C4⋊Dic3—91no—
254C4⋊Dic3—181no—
255C4⋊Dic3C2×D911yescharacteristic
256Dic3⋊C4—92no—
257Dic3⋊C4—92no—
258Dic3⋊C4—92no—
259Dic3⋊C4—92no—
260Dic3⋊C4—92no—
261Dic3⋊C4—92no—
262Dic3⋊C4—182no—
263C4×Dic3—31no—
264C4×Dic3—31no—
265C4×Dic3—31no—
266C4×Dic3—92no—
267C4×Dic3—92no—
268C4×Dic3—182no—
269C22×C32C23×D311yescharacteristic
270C2×C3×D3—22no—
271C2×C3×D3—22no—
272C2×C3×D3—22no—
273C2×C3×D3—32no—
274C2×C3×D3—61no—
275C2×C3×D3—61no—
276C32×C4—21no—
277C32×C4#23512yes—
278C32×C4#23712yes—
279C3⋊Dic3—62no—
280C3⋊Dic3—62no—
281C3×Dic3—22no—
282C3×Dic3—61no—
283C3×Dic3—61no—
284C3×Dic3—61no—
285C22×C9—21no—
286C22×C9—62no—
287C22×C9—62no—
288C22×C9—62no—
289C22×C9C23×D311yescharacteristic
290C2×D9—21no—
291C2×D9—21no—
292C2×D9#23512yes—
293C4×C9—21no—
294C4×C9—22no—
295C4×C9—22no—
296C4×C9—41no—
297C4×C9—62no—
298C4×C9#23512yes—
299C4×C9#23512yes—
300Dic9—21no—
301Dic9—21no—
302Dic9—21no—
303Dic9—64no—
304Dic9—62no—
305Dic9—62no—
306Dic9—122no—
307#126—272no—
308#125—271no—
309#124—271no—
310#123—272no—
311C22⋊Q8—271no—
312#121—271no—
313C4×Q8—271no—
314C4×D4—31no—
315C4×D4—272no—
316#117—272no—
317C2×C42—91no—
318C3×C9#52911yescharacteristic, complemented, Sylow
319C23×C3—32no—
320C23×C3—91no—
321C22×D3—31no—
322C22×D3C22×D912yes—
323C3×D4—62no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—32no—
328C2×C3×C4—64no—
329C2×C3×C4—91no—
330C2×C3×C4—91no—
331C2×C3×C4—91no—
332C2×C3×C4—182no—
333C2×C3×C4—182no—
334C2×C3×C4—182no—
335C2×C3×C4C22×D911yescharacteristic
336C2×C3×C4C22×D911yescharacteristic
337C2×C3×C4C22×D911yescharacteristic
338C2×C3×C4C2×D3211yescharacteristic
339C2×C3×C4C2×D3211yescharacteristic
340C2×C3×C4C2×D3211yescharacteristic
341#67—184no—
342#67—184no—
343C2×Dic3—31no—
344C2×Dic3—31no—
345C2×Dic3—31no—
346C2×Dic3—92no—
347C2×Dic3—92no—
348C2×Dic3—92no—
349C2×Dic3—92no—
350C2×Dic3—182no—
351C2×Dic3—182no—
352C2×Dic3C22×D912yes—
353D12—22no—
354C4×D3—24no—
355C4×D3—62no—
356C4×D3—62no—
357C4×D3—62no—
358Dic6—184no—
359Dic6—182no—
360Dic6—182no—
361C2×C32C2×#23511yescharacteristic
362C2×C32C2×#23711yescharacteristic
363C2×C32D3×#4111yescharacteristic
364C3×D3—42no—
365C3×D3—62no—
366C2×C9—21no—
367C2×C9—21no—
368C2×C9—21no—
369C2×C9—122no—
370C2×C9C2×#23511yescharacteristic
371C2×C9C2×#23511yescharacteristic
372C2×C9C2×#23511yescharacteristic
373D9—22no—
374C2×Q8—271no—
375C2×D4—31no—
376C2×D4—272no—
377C22×C4—32no—
378C22×C4—91no—
379C22×C4—91no—
380C22×C4—91no—
381C4⋊C4—31no—
382C4⋊C4—272no—
383C4⋊C4—272no—
384C4⋊C4—272no—
385C4⋊C4—271no—
386C4⋊C4—272no—
387C22⋊C4—32no—
388C22⋊C4—272no—
389C22⋊C4—272no—
390C22⋊C4—272no—
391C22⋊C4—272no—
392C42—91no—
393C42—91no—
394C42—91no—
395C42—272no—
396C42D3×D911yescharacteristic
397C22×C3—21no—
398C22×C3—62no—
399C22×C3—62no—
400C22×C3—62no—
401C22×C3—92no—
402C22×C3—181no—
403C22×C3—181no—
404C22×C3C23×D911yescharacteristic
405C22×C3C22×D3211yescharacteristic
406C2×D3—22no—
407C2×D3—22no—
408C2×D3—22no—
409C2×D3—32no—
410C2×D3—61no—
411C2×D3—61no—
412C3×C4—21no—
413C3×C4—21no—
414C3×C4—22no—
415C3×C4—22no—
416C3×C4—41no—
417C3×C4—62no—
418C3×C4—181no—
419C3×C4—181no—
420C3×C4—181no—
421C3×C4#369712yes—
422C3×C4#369912yes—
423C3×C4#379512yes—
424C3×C4#379812yes—
425Dic3—22no—
426Dic3—61no—
427Dic3—61no—
428Dic3—61no—
429Dic3—184no—
430Dic3—182no—
431Dic3—182no—
432Dic3—182no—
433Dic3—182no—
434Dic3—362no—
435C32#555711yescharacteristic
436C9—21no—
437C9#547911yescharacteristic, complemented
438C23—32no—
439C23—91no—
440Q8—542no—
441D4—62no—
442D4—544no—
443C2×C4—32no—
444C2×C4—64no—
445C2×C4—91no—
446C2×C4—91no—
447C2×C4—91no—
448C2×C4—182no—
449C2×C4—182no—
450C2×C4—182no—
451C2×C4—272no—
452C2×C4—272no—
453C2×C4C2×D3×D911yescharacteristic
454C2×C4C2×D3×D911yescharacteristic
455C2×C4C2×D3×D911yescharacteristic
456C2×C3—21no—
457C2×C3—21no—
458C2×C3—21no—
459C2×C3—122no—
460C2×C3—182no—
461C2×C3C2×#369711yescharacteristic
462C2×C3C2×#369911yescharacteristic
463C2×C3#41×D911yescharacteristic
464C2×C3C2×#379511yescharacteristic
465C2×C3C2×#379811yescharacteristic
466C2×C3D3×#23511yescharacteristic
467D3—42no—
468D3—62no—
469C22—62no—
470C22—62no—
471C22—62no—
472C22—92no—
473C22—181no—
474C22—181no—
475C22C22×D3×D911yescharacteristic
476C4—21no—
477C4—62no—
478C4—181no—
479C4—181no—
480C4—181no—
481C4—542no—
482C4—542no—
483C4#8863812yes—
484C4#8864012yes—
485C3—21no—
486C3#1058965911yescharacteristic, complemented
487C3#1059397411yescharacteristic
488C2—122no—
489C2—182no—
490C2C2×#8863811yescharacteristic
491C2C2×#8864011yescharacteristic
492C2D9×#23511yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
prime radical#11721036C2
hypercenterC42D3×D9
hypocenterC3×C9#529
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C4×#6226C2semidirect
2. C4×C9×D12C2semidirect
3. C3×C42×D9C2semidirect
5. #11721031C2semidirect
6. C4×#6194C2semidirect
7. #11720964C2semidirect
8. #11720963C2semidirect
9. #11720959C2semidirect
10. #11720945C2semidirect
11. #11720935C2semidirect
57. C42×D9D3semidirect
163. C4×D12D9semidirect
318. C3×C9#529semidirect
437. C9#5479semidirect
486. C3#10589659semidirect

Maximal subgroups

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

subgroupindex
C4×#62262
C4×C9×D122
C3×C42×D92
#117210362
#117210312
C4×#61942
#117209642
#117209632
#117209592
#117209452
#117209352
#105939743
#105896593