Finite Group Databasebeta

← #13147679

Subgroups — #13147679

Order 1728.

778 conjugacy classes of subgroups.

The subgroup lattice has 512 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×#63437C211yesmaximal, characteristic, complemented
4#11721043C211yesmaximal, characteristic
5#11721042C212yesmaximal, complemented
6#11721032C212yesmaximal, complemented
7#11720998C211yesmaximal, characteristic, complemented
8#11720989C212yesmaximal, complemented
9#11720976C212yesmaximal, complemented
10#11720967C211yesmaximal, characteristic, complemented
11#11720910C211yesmaximal, characteristic, complemented
12#10593981—31nomaximal
13#10589661—31nomaximal
14C2×#6226C2211yescharacteristic
15C2×C9×D12C2211yescharacteristic
16C2×C4×D3×C9C2212yes—
17C2×C3×D36C2211yescharacteristic
18C2×C3×Dic18C2211yescharacteristic
19C2×#6194C2212yes—
20C2×#6194C2212yes—
21C2×Dic3×D9C2212yes—
22C3×C42×C9C2211yescharacteristic
23C4⋊(C3⋊Dic9)C2211yescharacteristic
24#88534C2211yescharacteristic
25#88534C2211yescharacteristic
26C4×C3⋊Dic9C2211yescharacteristic
27C4×C3⋊Dic9C2211yescharacteristic
28C9×(C2×D3)⋊C4C2212yes—
29C3×(C2×D9)⋊C4C2212yes—
30C3×(C2×D9)⋊C4C2212yes—
31C9×C4⋊Dic3C2211yescharacteristic
32(C2×D3)⋊Dic9C2212yes—
33(C2×D3)⋊Dic9C2212yes—
34(C2×D9)⋊Dic3C2212yes—
35Dic9⋊Dic3C2212yes—
36Dic3×Dic9C2212yes—
37C4×#634—31no—
38C3×#875—31no—
39C3×C4×D12D311yescharacteristic
40#63914—31no—
41#63913—32no—
42#63905—32no—
43#63898—31no—
44#63891—32no—
45#63879—32no—
46#63863—31no—
47#63843—31no—
48C4×D4×C9—31no—
49#63502—31no—
50#63498—31no—
51#63491—32no—
52C4×#611—32no—
53#63455—31no—
54#63449—31no—
55#63447—32no—
56#63445—32no—
57#63437D311yescharacteristic, complemented
58C4×Dic18—31no—
59C4×Dic18—61no—
60C22×D3×C9C2312yes—
61C22×C3×D9C2312yes—
62C2×C3×C4×C9C2311yescharacteristic
63C2×C3×C4×C9C2311yescharacteristic
64C2×C3×C4×C9C2311yescharacteristic
65C2×C3⋊Dic9C2311yescharacteristic
66C2×C3⋊Dic9C2311yescharacteristic
67C2×C3⋊Dic9C2311yescharacteristic
68C2×C3⋊Dic9C2311yescharacteristic
69#6226—22no—
70C2×C9×Dic3C2312yes—
71C2×C3×Dic9C2312yes—
72C9×D12—22no—
73C4×D3×C9—42no—
74C3×D36—22no—
75C3×Dic18—22no—
76#6194—24no—
77#6194—42no—
78Dic3×D9—42no—
79#5681—91no—
80#5559—91no—
81C2×#634—31no—
82C2×C3×D12—31no—
83C2×C3×D12C2×D311yescharacteristic
84C2×C3×C4×D3C2×D312yes—
85C2×C3×Dic6—31no—
86C2×#625—32no—
87C2×#625—32no—
88C2×D3×Dic3—32no—
89C32×C42C2×D311yescharacteristic
90C4⋊(C3⋊Dic3)—31no—
91#3750—31no—
92#3750—31no—
93C4×C3⋊Dic3—31no—
94C4×C3⋊Dic3—31no—
95C3×(C2×D3)⋊C4—32no—
96C3×(C2×D3)⋊C4—32no—
97C3×(C2×D3)⋊C4C2×D312yes—
98C3×C4⋊Dic3C2×D311yescharacteristic
99Dic3⋊Dic3—32no—
100(C2×D3)⋊Dic3—32no—
101(C2×D3)⋊Dic3—32no—
102(C2×D3)⋊Dic3—32no—
103Dic32—32no—
104C2×D4×C9—31no—
105C22×C4×C9—32no—
106C2×#611—32no—
107C2×#611—32no—
108C2×D36C2×D311yescharacteristic
109C2×C4×D9—32no—
110C2×Dic18—31no—
111C2×Dic18—61no—
112C2×Dic18C2×D311yescharacteristic
113C9×C4⋊C4—31no—
114C9×C22⋊C4—32no—
115C42×C9—21no—
116C42×C9C2×D311yescharacteristic
117#3676—32no—
118#3676—32no—
119(C2×D9)⋊C4—32no—
120(C2×D9)⋊C4C2×D312yes—
121(C2×D9)⋊C4C2×D312yes—
122C4⋊Dic9—31no—
123C4⋊Dic9—61no—
124Dic9⋊C4—31no—
125Dic9⋊C4—31no—
126Dic9⋊C4—32no—
127Dic9⋊C4—61no—
128Dic9⋊C4—61no—
129C4×Dic9—31no—
130C4×Dic9—31no—
131C4×Dic9—32no—
132C4×Dic9—61no—
133C4×Dic9—61no—
134C22×C3×C9C2411yescharacteristic
135C2×D3×C9—22no—
136C2×D3×C9—22no—
137C2×D3×C9—22no—
138C2×C3×D9—22no—
139C2×C3×D9—22no—
140C2×C3×D9—22no—
141C3×C4×C9—21no—
142C3×C4×C9—21no—
143C3×C4×C9#4112yes—
144C3⋊Dic9—21no—
145C3⋊Dic9—21no—
146C3⋊Dic9—21no—
147C3⋊Dic9#4112yes—
148C9×Dic3—22no—
149C3×Dic9—22no—
150C3×#124—91no—
151C3×C4×D4—31no—
152#947—91no—
153Q8×Dic3—91no—
154#940—91no—
155#940—91no—
156#936—91no—
157D4×Dic3—91no—
158#929—92no—
159C4×#67—92no—
160#893—91no—
161#887—91no—
162#886—92no—
163#885—92no—
164#885—92no—
165#883—92no—
166#883—92no—
167#67⋊C4—92no—
168#875—31no—
169C4×D12D911yescharacteristic, complemented
170C4×Dic6—91no—
171C4×Dic6—91no—
172C4×Dic6—181no—
173C22×C3×D3—32no—
174C22×C3×D3C22×D312yes—
175C2×C32×C4C22×D311yescharacteristic
176C2×C32×C4C22×D311yescharacteristic
177C2×C32×C4C22×D311yescharacteristic
178C2×C3⋊Dic3—31no—
179C2×C3⋊Dic3—31no—
180C2×C3⋊Dic3—31no—
181C2×C3⋊Dic3—31no—
182#634—62no—
183C2×C3×Dic3—32no—
184C2×C3×Dic3C22×D312yes—
185C3×D12—22no—
186C3×D12—62no—
187C3×C4×D3—42no—
188C3×Dic6—62no—
189#625—64no—
190#625—122no—
191D3×Dic3—122no—
192C23×C9—32no—
193C22×D9C22×D312yes—
194D4×C9—62no—
195C2×C4×C9—21no—
196C2×C4×C9—21no—
197C2×C4×C9—21no—
198C2×C4×C9—32no—
199C2×C4×C9—122no—
200C2×C4×C9C22×D311yescharacteristic
201C2×C4×C9C22×D311yescharacteristic
202C2×C4×C9C22×D311yescharacteristic
203#611—64no—
204#611—122no—
205C2×Dic9—31no—
206C2×Dic9—31no—
207C2×Dic9—31no—
208C2×Dic9—31no—
209C2×Dic9—61no—
210C2×Dic9—61no—
211C2×Dic9—61no—
212C2×Dic9—61no—
213C2×Dic9C22×D312yes—
214D36—22no—
215C4×D9—122no—
216Dic18—22no—
217Dic18—62no—
218Dic18—64no—
219#559—271noSylow
220C2×C3×C9C2×#4111yescharacteristic
221C2×C3×C9C2×#4111yescharacteristic
222C2×C3×C9H(5,𝔽2)11yescharacteristic
223D3×C9—42no—
224C3×D9—42no—
225C2×C3×Q8—91no—
226C2×C3×D4—31no—
227C2×C3×D4—91no—
228C22×C3×C4—32no—
229C2×#67—92no—
230C2×#67—92no—
231C2×#67—92no—
232C2×#67—92no—
233C22×Dic3—92no—
234C2×D12—31no—
235C2×D12C2×D911yescharacteristic
236C2×C4×D3—92no—
237C2×C4×D3C2×D912yes—
238C2×Dic6—31no—
239C2×Dic6—91no—
240C2×Dic6—91no—
241C2×Dic6—181no—
242C3×C4⋊C4—31no—
243C3×C22⋊C4—32no—
244C3×C22⋊C4—92no—
245C3×C22⋊C4—92no—
246C3×C42—21no—
247C3×C42C2×D911yescharacteristic
248C3×C42D3211yescharacteristic
249#217—92no—
250#217—92no—
251#217—92no—
252(C2×D3)⋊C4—32no—
253(C2×D3)⋊C4—32no—
254(C2×D3)⋊C4—92no—
255(C2×D3)⋊C4—92no—
256(C2×D3)⋊C4—92no—
257(C2×D3)⋊C4C2×D912yes—
258C4⋊Dic3—91no—
259C4⋊Dic3—91no—
260C4⋊Dic3—92no—
261C4⋊Dic3—181no—
262C4⋊Dic3C2×D911yescharacteristic
263Dic3⋊C4—92no—
264Dic3⋊C4—91no—
265Dic3⋊C4—91no—
266Dic3⋊C4—91no—
267Dic3⋊C4—91no—
268Dic3⋊C4—181no—
269Dic3⋊C4—181no—
270C4×Dic3—92no—
271C4×Dic3—91no—
272C4×Dic3—91no—
273C4×Dic3—91no—
274C4×Dic3—91no—
275C4×Dic3—92no—
276C4×Dic3—181no—
277C4×Dic3—181no—
278C22×C32C23×D311yescharacteristic
279C2×C3×D3—22no—
280C2×C3×D3—22no—
281C2×C3×D3—22no—
282C2×C3×D3—62no—
283C2×C3×D3—62no—
284C2×C3×D3—62no—
285C32×C4—21no—
286C32×C4—21no—
287C32×C4#23512yes—
288C3⋊Dic3—32no—
289C3⋊Dic3—61no—
290C3⋊Dic3—61no—
291C3⋊Dic3—61no—
292C3×Dic3—22no—
293C3×Dic3—62no—
294C22×C9—21no—
295C22×C9—62no—
296C22×C9—62no—
297C22×C9—62no—
298C22×C9C23×D311yescharacteristic
299C2×D9—22no—
300C2×D9—22no—
301C2×D9—22no—
302C4×C9—22no—
303C4×C9—21no—
304C4×C9—21no—
305C4×C9—22no—
306C4×C9—41no—
307C4×C9—62no—
308C4×C9#23712yes—
309Dic9—22no—
310Dic9—32no—
311Dic9—62no—
312Dic9—62no—
313Dic9—61no—
314Dic9—61no—
315Dic9—61no—
316Dic9—121no—
317Dic9—121no—
318#127—271no—
319#124—91no—
320#124—272no—
321#124—271no—
322#123—272no—
323#123—272no—
324C4×Q8—271no—
325C4×Q8—271no—
326C4×D4—31no—
327C4×D4—272no—
328C4×D4—271no—
329C3×C9#55911yescharacteristic, complemented, Sylow
330C23×C3—32no—
331C23×C3—92no—
332C22×D3—32no—
333C22×D3C22×D912yes—
334C3×Q8—182no—
335C3×D4—62no—
336C3×D4—182no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—32no—
341C2×C3×C4—92no—
342C2×C3×C4—122no—
343C2×C3×C4C22×D911yescharacteristic
344C2×C3×C4C22×D911yescharacteristic
345C2×C3×C4C22×D911yescharacteristic
346C2×C3×C4C2×D3211yescharacteristic
347C2×C3×C4C2×D3211yescharacteristic
348C2×C3×C4C2×D3211yescharacteristic
349#67—184no—
350#67—184no—
351#67—362no—
352#67—362no—
353C2×Dic3—32no—
354C2×Dic3—91no—
355C2×Dic3—91no—
356C2×Dic3—91no—
357C2×Dic3—91no—
358C2×Dic3—91no—
359C2×Dic3—91no—
360C2×Dic3—91no—
361C2×Dic3—91no—
362C2×Dic3—181no—
363C2×Dic3—181no—
364C2×Dic3—181no—
365C2×Dic3—181no—
366C2×Dic3—362no—
367C2×Dic3C22×D912yes—
368D12—22no—
369D12—62no—
370C4×D3—42no—
371C4×D3—362no—
372Dic6—62no—
373Dic6—182no—
374Dic6—182no—
375Dic6—184no—
376C2×C32C2×#23511yescharacteristic
377C2×C32#100411yescharacteristic
378C2×C32D3×#4111yescharacteristic
379C3×D3—42no—
380C3×D3—122no—
381C2×C9—21no—
382C2×C9—21no—
383C2×C9—21no—
384C2×C9—122no—
385C2×C9C2×#23711yescharacteristic
386C2×C9D3×#4111yescharacteristic
387C2×C9#100911yescharacteristic
388D9—42no—
389C2×Q8—91no—
390C2×Q8—271no—
391C2×D4—31no—
392C2×D4—91no—
393C2×D4—272no—
394C2×D4—272no—
395C22×C4—32no—
396C22×C4—272no—
397C4⋊C4—31no—
398C4⋊C4—272no—
399C4⋊C4—271no—
400C4⋊C4—271no—
401C4⋊C4—271no—
402C22⋊C4—32no—
403C22⋊C4—92no—
404C22⋊C4—92no—
405C22⋊C4—272no—
406C22⋊C4—272no—
407C22⋊C4—272no—
408C42—272no—
409C42—271no—
410C42—271no—
411C42D3×D911yescharacteristic
412C22×C3—21no—
413C22×C3—62no—
414C22×C3—62no—
415C22×C3—62no—
416C22×C3—182no—
417C22×C3—182no—
418C22×C3—182no—
419C22×C3C23×D911yescharacteristic
420C22×C3C22×D3211yescharacteristic
421C2×D3—22no—
422C2×D3—22no—
423C2×D3—22no—
424C2×D3—62no—
425C2×D3—62no—
426C2×D3—62no—
427C3×C4—22no—
428C3×C4—21no—
429C3×C4—21no—
430C3×C4—21no—
431C3×C4—21no—
432C3×C4—22no—
433C3×C4—41no—
434C3×C4—62no—
435C3×C4—182no—
436C3×C4#369712yes—
437C3×C4#379512yes—
438Dic3—22no—
439Dic3—62no—
440Dic3—92no—
441Dic3—92no—
442Dic3—182no—
443Dic3—182no—
444Dic3—181no—
445Dic3—181no—
446Dic3—181no—
447Dic3—181no—
448Dic3—181no—
449Dic3—181no—
450Dic3—361no—
451Dic3—361no—
452C32#555911yescharacteristic
453C9—21no—
454C9#568111yescharacteristic, complemented
455C23—32no—
456C23—92no—
457Q8—182no—
458Q8—542no—
459D4—62no—
460D4—182no—
461D4—544no—
462D4—1082no—
463C2×C4—32no—
464C2×C4—92no—
465C2×C4—122no—
466C2×C4—271no—
467C2×C4—271no—
468C2×C4—271no—
469C2×C4—271no—
470C2×C4—1082no—
471C2×C4C2×D3×D911yescharacteristic
472C2×C4C2×D3×D911yescharacteristic
473C2×C4C2×D3×D911yescharacteristic
474C2×C3—21no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—122no—
478C2×C3—362no—
479C2×C3C2×#369711yescharacteristic
480C2×C3#6371011yescharacteristic
481C2×C3#41×D911yescharacteristic
482C2×C3C2×#379511yescharacteristic
483C2×C3#6430611yescharacteristic
484C2×C3#6430711yescharacteristic
485D3—42no—
486D3—122no—
487C22—62no—
488C22—62no—
489C22—62no—
490C22—182no—
491C22—182no—
492C22—182no—
493C22C22×D3×D911yescharacteristic
494C4—21no—
495C4—21no—
496C4—62no—
497C4—182no—
498C4—272no—
499C4—541no—
500C4—541no—
501C4—541no—
502C4#8864112yes—
503C3—21no—
504C3#1058966111yescharacteristic, complemented
505C3#1059398111yescharacteristic
506C2—122no—
507C2—362no—
508C2C2×#8864111yescharacteristic
509C2#1172247511yescharacteristic
510C2#1172247611yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
prime radical#11721043C2
hypercenterC42D3×D9
hypocenterC3×C9#559
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C4×#6226C2semidirect
2. C4×C9×D12C2semidirect
3. C3×#63437C2semidirect
5. #11721042C2semidirect
6. #11721032C2semidirect
7. #11720998C2semidirect
8. #11720989C2semidirect
9. #11720976C2semidirect
10. #11720967C2semidirect
11. #11720910C2semidirect
57. #63437D3semidirect
169. C4×D12D9semidirect
329. C3×C9#559semidirect
454. C9#5681semidirect
504. C3#10589661semidirect

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×#634372
#117210432
#117210422
#117210322
#117209982
#117209892
#117209762
#117209672
#117209102
#105939813
#105896613