Finite Group Databasebeta

← #13147653

Subgroups — #13147653

Order 1728.

1006 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88642C211yesmaximal, characteristic, complemented
2C2×#88641C211yesmaximal, characteristic, complemented
3#11721712C211yesmaximal, characteristic, complemented
4C42×D3×C9C211yesmaximal, characteristic, complemented
5C3×#63437C211yesmaximal, characteristic, complemented
6#11721034C214yesmaximal, complemented
7#11721006C211yesmaximal, characteristic, complemented
8#11720977C214yesmaximal, complemented
9#11720932C211yesmaximal, characteristic, complemented
10#10593969—31nomaximal
11#10589579—31nomaximal
12C2×C4⋊(C3⋊D9)C2211yescharacteristic
13C2×#6226C2211yescharacteristic
14C2×C4×D3×C9C2212yes—
15C2×C4×D3×C9C2211yescharacteristic
16C2×C3×D36C2211yescharacteristic
17C2×C3×Dic18C2211yescharacteristic
18C2×#6195C2212yes—
19C2×#6194C2212yes—
20C2×#6192C2214yes—
21C2×(C3⋊D9)⋊C4C2212yes—
22C2×Dic3×D9C2212yes—
23#88642—24no—
24#88641—24no—
25C3×C42×C9C2211yescharacteristic
26#88536C2214yes—
27C3×(C2×D9)⋊C4C2214yes—
28C4×C9×Dic3C2212yes—
29C4×C9×Dic3C2211yescharacteristic
30Dic3⋊Dic9C2214yes—
31C2×#3799—31no—
32C2×#3795—31no—
33#64084—31no—
34C3×#875—31no—
35C3×C42×D3D311yescharacteristic
36#63911—31no—
37#63906—34no—
38#63880—34no—
39#63866—31no—
40C2×#3697—31no—
41C2×#3697—31no—
42C2×C42×C9—31no—
43#63492—34no—
44#63437—31no—
45#63437—61no—
46#63437D311yescharacteristic, complemented
47#63436—31no—
48C4×D36—34no—
49#63431—31no—
50C22×C3⋊D9C2312yes—
51C22×D3×C9C2311yescharacteristic
52C22×C3×D9C2312yes—
53C2×C3×C4×C9C2312yes—
54C2×C3×C4×C9C2311yescharacteristic
55C2×C3⋊Dic9C2312yes—
56C4⋊(C3⋊D9)—22no—
57#6226—22no—
58C2×C9×Dic3C2311yescharacteristic
59C2×C9×Dic3C2312yes—
60C2×C3×Dic9C2312yes—
61C4×D3×C9—24no—
62C4×D3×C9D414yescomplemented
63C3×D36—22no—
64C3×Dic18—22no—
65#6195—24no—
66#6194—24no—
67#6192—28no—
68(C3⋊D9)⋊C4—24no—
69Dic3×D9—24no—
70#5668—91no—
71#5477—91no—
72C2×C4⋊(C3⋊D3)—31no—
73C2×#634—31no—
74C2×C3×D12—31no—
75C2×C3×C4×D3C2×D312yes—
76C2×C3×C4×D3C2×D311yescharacteristic
77C2×C3×Dic6—31no—
78C2×#626—32no—
79C2×#626—34no—
80C2×#625—32no—
81C2×#624—32no—
82C2×D3×Dic3—32no—
83#3799—64no—
84#3795—64no—
85C32×C42C2×D311yescharacteristic
86#3752—34no—
87C3×(C2×D3)⋊C4—34no—
88C3×C4×Dic3C2×D311yescharacteristic
89C3×C4×Dic3C2×D312yes—
90Dic3⋊Dic3—34no—
91C22×C4×C9—31no—
92C22×C4×C9—32no—
93C2×#611—32no—
94C2×#611—32no—
95#3697—64no—
96#3697—64no—
97C2×D36—34no—
98C2×D36—31no—
99C2×D36—61no—
100C2×D36C2×D311yescharacteristic
101C2×C4×D9—32no—
102C2×C4×D9—32no—
103C2×Dic18—31no—
104C2×Dic18—61no—
105C2×Dic18C2×D311yescharacteristic
106C42×C9—21no—
107C42×C9—31no—
108C42×C9—32no—
109C42×C9C2×D311yescharacteristic
110(C2×D9)⋊C4—34no—
111(C2×D9)⋊C4—64no—
112(C2×D9)⋊C4C2×D314yes—
113C4⋊Dic9—34no—
114C22×C3×C9C2411yescharacteristic
115C2×C3⋊D9—22no—
116C2×C3⋊D9—24no—
117C2×D3×C9—22no—
118C2×D3×C9C2×D412yes—
119C2×C3×D9—22no—
120C2×C3×D9—24no—
121C3×C4×C9—22no—
122C3×C4×C9C2×D412yes—
123C3⋊Dic9—22no—
124C9×Dic3C2×D412yes—
125C9×Dic3#4114yescomplemented
126C3×Dic9—22no—
127C2×#239—91no—
128C2×#237—91no—
129C2×#235—91no—
130C2×#235—91no—
131C3×#124—91no—
132C2×C3×C42—31no—
133#944—91no—
134#940—91no—
135#930—94no—
136#884—94no—
137#67⋊C4—94no—
138#875—31no—
139#875—91no—
140#875—91no—
141#875—181no—
142#874—91no—
143C4×D12—94no—
144C42×D3D911yescharacteristic, complemented
145#869—91no—
146C22×C3⋊D3—32no—
147C22×C3×D3—32no—
148C22×C3×D3C22×D311yescharacteristic
149C2×C32×C4C22×D312yes—
150C2×C32×C4C22×D311yescharacteristic
151C2×C3⋊Dic3—32no—
152C4⋊(C3⋊D3)—62no—
153#634—62no—
154C2×C3×Dic3—32no—
155C2×C3×Dic3C22×D311yescharacteristic
156C2×C3×Dic3C22×D312yes—
157C3×D12—62no—
158C3×C4×D3—24no—
159C3×C4×D3D1214yes—
160C3×Dic6—62no—
161#626—64no—
162#626—68no—
163#625—64no—
164#624—64no—
165D3×Dic3—64no—
166C23×C9—31no—
167C22×D9—32no—
168C22×D9—62no—
169C22×D9C22×D312yes—
170C2×C4×C9—21no—
171C2×C4×C9—22no—
172C2×C4×C9—32no—
173C2×C4×C9—31no—
174C2×C4×C9—34no—
175C2×C4×C9—64no—
176C2×C4×C9C22×D312yes—
177C2×C4×C9C22×D311yescharacteristic
178#611—64no—
179#611—64no—
180C2×Dic9—32no—
181C2×Dic9—62no—
182C2×Dic9C22×D312yes—
183D36—22no—
184D36—68no—
185D36—62no—
186D36—122no—
187C4×D9—64no—
188C4×D9—64no—
189Dic18—22no—
190Dic18—62no—
191Dic18—122no—
192#532—271noSylow
193C2×C3×C9C22×D411yescharacteristic
194C2×C3×C9C2×#4112yes—
195C3⋊D9—42no—
196D3×C9—22no—
197C3×D9—42no—
198C2×C3×Q8—91no—
199C2×C3×D4—91no—
200C22×C3×C4—31no—
201C22×C3×C4—32no—
202C2×#67—92no—
203C2×#67—92no—
204C2×#67—92no—
205C2×#67—94no—
206C22×Dic3—92no—
207#239—184no—
208#237—184no—
209#235—184no—
210#235—184no—
211C2×D12—31no—
212C2×D12—94no—
213C2×D12—92no—
214C2×D12—91no—
215C2×D12—91no—
216C2×D12—181no—
217C2×C4×D3—92no—
218C2×C4×D3—92no—
219C2×C4×D3—92no—
220C2×C4×D3C2×D912yes—
221C2×C4×D3C2×D911yescharacteristic
222C2×Dic6—31no—
223C2×Dic6—91no—
224C2×Dic6—91no—
225C2×Dic6—181no—
226C3×C22⋊C4—94no—
227C3×C42—21no—
228C3×C42—31no—
229C3×C42—32no—
230C3×C42C2×D911yescharacteristic
231C3×C42D3211yescharacteristic
232(C2×D3)⋊C4—34no—
233(C2×D3)⋊C4—94no—
234(C2×D3)⋊C4—94no—
235(C2×D3)⋊C4—184no—
236C4⋊Dic3—94no—
237Dic3⋊C4—94no—
238C4×Dic3C2×D912yes—
239C4×Dic3C2×D911yescharacteristic
240C22×C32C23×D311yescharacteristic
241C2×C3⋊D3—62no—
242C2×C3⋊D3—64no—
243C2×C3×D3—22no—
244C2×C3×D3—62no—
245C2×C3×D3—64no—
246C2×C3×D3C2×D1212yes—
247C32×C4—22no—
248C32×C4C2×D1212yes—
249C3⋊Dic3—62no—
250C3×Dic3—62no—
251C3×Dic3C2×D1212yes—
252C3×Dic3#23514yes—
253C22×C9—21no—
254C22×C9—32no—
255C22×C9—62no—
256C22×C9C23×D311yescharacteristic
257C2×D9—22no—
258C2×D9—24no—
259C2×D9—62no—
260C2×D9—64no—
261C2×D9—122no—
262C2×D9—124no—
263C4×C9—22no—
264C4×C9—22no—
265C4×C9—32no—
266C4×C9—34no—
267C4×C9—42no—
268C4×C9D3×D412yes—
269Dic9—22no—
270Dic9—62no—
271Dic9—122no—
272C2×#41—271no—
273C2×#41—271no—
274C4⋊Q8—271no—
275#127—271no—
276#124—91no—
277#124—271no—
278#121—274no—
279C4×D4—274no—
280C2×C42—31no—
281C3×C9#53211yescharacteristic, complemented, Sylow
282C23×C3—31no—
283C23×C3—92no—
284C22×D3—32no—
285C22×D3—92no—
286C22×D3—92no—
287C22×D3—182no—
288C22×D3C22×D911yescharacteristic
289C3×Q8—182no—
290C3×D4—182no—
291C2×C3×C4—21no—
292C2×C3×C4—22no—
293C2×C3×C4—32no—
294C2×C3×C4—31no—
295C2×C3×C4—34no—
296C2×C3×C4—64no—
297C2×C3×C4—92no—
298C2×C3×C4C22×D912yes—
299C2×C3×C4C22×D911yescharacteristic
300C2×C3×C4C2×D3212yes—
301C2×C3×C4C2×D3211yescharacteristic
302#67—184no—
303#67—184no—
304#67—184no—
305#67—188no—
306C2×Dic3—32no—
307C2×Dic3—92no—
308C2×Dic3—92no—
309C2×Dic3—182no—
310C2×Dic3—184no—
311C2×Dic3C22×D911yescharacteristic
312C2×Dic3C22×D912yes—
313D12—62no—
314D12—188no—
315D12—184no—
316D12—182no—
317D12—182no—
318D12—362no—
319C4×D3—24no—
320C4×D3—184no—
321C4×D3—184no—
322C4×D3—184no—
323C4×D3D3614yescomplemented
324Dic6—62no—
325Dic6—182no—
326Dic6—182no—
327Dic6—362no—
328C2×C32C22×D1211yescharacteristic
329C2×C32C2×#23512yes—
330C3⋊D3—122no—
331C3×D3—22no—
332C3×D3—122no—
333C2×C9—21no—
334C2×C9—22no—
335C2×C9—62no—
336C2×C9C2×D3×D411yescharacteristic
337C2×C9D3×#4112yes—
338D9—42no—
339D9—122no—
340D9—242no—
341#41—544no—
342#41—544no—
343C2×Q8—91no—
344C2×Q8—271no—
345C2×D4—91no—
346C2×D4—274no—
347C2×D4—272no—
348C2×D4—272no—
349C2×D4—271no—
350C22×C4—31no—
351C22×C4—32no—
352C22×C4—272no—
353C22×C4—272no—
354C4⋊C4—274no—
355C22⋊C4—94no—
356C22⋊C4—274no—
357C42—32no—
358C42—31no—
359C42D3×D911yescharacteristic
360C22×C3—21no—
361C22×C3—32no—
362C22×C3—62no—
363C22×C3—182no—
364C22×C3—184no—
365C22×C3C23×D911yescharacteristic
366C22×C3C22×D3211yescharacteristic
367C2×D3—22no—
368C2×D3—62no—
369C2×D3—64no—
370C2×D3—182no—
371C2×D3—184no—
372C2×D3—182no—
373C2×D3—184no—
374C2×D3—362no—
375C2×D3—364no—
376C2×D3C2×D3612yes—
377C3×C4—22no—
378C3×C4—22no—
379C3×C4—22no—
380C3×C4—32no—
381C3×C4—34no—
382C3×C4—42no—
383C3×C4—182no—
384C3×C4C2×D3612yes—
385C3×C4D3×D1212yes—
386Dic3—62no—
387Dic3—182no—
388Dic3—182no—
389Dic3—362no—
390Dic3C2×D3612yes—
391Dic3#369714yescomplemented
392C32#547711yescharacteristic
393C9—21no—
394C9#566811yescharacteristic, complemented
395C23—31no—
396C23—92no—
397C23—272no—
398Q8—182no—
399Q8—542no—
400D4—182no—
401D4—548no—
402D4—544no—
403D4—544no—
404D4—542no—
405C2×C4—32no—
406C2×C4—31no—
407C2×C4—34no—
408C2×C4—64no—
409C2×C4—92no—
410C2×C4—272no—
411C2×C4—544no—
412C2×C4—544no—
413C2×C4C2×D3×D912yes—
414C2×C4C2×D3×D911yescharacteristic
415C2×C3—21no—
416C2×C3—22no—
417C2×C3—62no—
418C2×C3—362no—
419C2×C3C22×D3611yescharacteristic
420C2×C3C2×#369712yes—
421C2×C3C2×D3×D1211yescharacteristic
422C2×C3D3×#23512yes—
423D3—22no—
424D3—122no—
425D3—362no—
426D3—362no—
427D3—722no—
428C22—32no—
429C22—62no—
430C22—182no—
431C22—184no—
432C22—542no—
433C22—544no—
434C22C22×D3×D911yescharacteristic
435C4—22no—
436C4—32no—
437C4—34no—
438C4—182no—
439C4—542no—
440C4D3×D3612yes—
441C3—21no—
442C3#1058957911yescharacteristic, complemented
443C3#1059396911yescharacteristic
444C2—62no—
445C2—362no—
446C2—1082no—
447C2C2×D3×D3611yescharacteristic
448C2D3×#369712yes—
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC3×C42×C9C22
socleC22×C32C23×D3
hypercenterC42D3×D9
hypocenterC3×C9#532
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C2×#88642C2semidirect
2. C2×#88641C2semidirect
3. #11721712C2semidirect
4. C42×D3×C9C2semidirect
5. C3×#63437C2semidirect
6. #11721034C2semidirect
7. #11721006C2semidirect
8. #11720977C2semidirect
9. #11720932C2semidirect
46. #63437D3semidirect
62. C4×D3×C9D4semidirect
125. C9×Dic3#41semidirect
144. C42×D3D9semidirect
281. C3×C9#532semidirect
323. C4×D3D36semidirect
391. Dic3#3697semidirect
394. C9#5668semidirect
442. C3#10589579semidirect

Maximal subgroups

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

subgroupindex
C2×#886422
C2×#886412
#117217122
C42×D3×C92
C3×#634372
#117210342
#117210062
#117209772
#117209322
#105939693
#105895793