Finite Group Databasebeta

← #13161489

Subgroups — #13161489

Order 1728.

590 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64078C211yesmaximal, characteristic
2C3×C4×#634C211yesmaximal, characteristic
3C3×C4⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic
4#11722079C212yesmaximal
5#11722076C212yesmaximal
6#10590307—31nomaximal
7#10590301—31nomaximal
8#10590285—31nomaximal
9C2×C3×#634—21no—
10C2×C3×#634C2211yescharacteristic
11C33×C42C2211yescharacteristic
12C3×C4⋊(C3⋊Dic3)—21no—
13C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2212yes—
15C3×#3750—21no—
16C3×C4×C3⋊Dic3—21no—
17C2×C3×C3⋊(C3⋊C8)C2212yes—
18#64078D311yescharacteristic
19C4×#634D311yescharacteristic
20C3×#869—31no—
21C3×#869—31no—
22C3×#869—61no—
23C3×C4×Dic6—31no—
24C3×C4×Dic6—31no—
25C3×C4×Dic6—61no—
26C4⋊(C3⋊(C3⋊C8))D311yescharacteristic
27C3×C4⋊(C3⋊C8)—31no—
28C3×C4⋊(C3⋊C8)—31no—
29C3×C4⋊(C3⋊C8)—61no—
30#63574—32no—
31#63573—32no—
32#63573—32no—
33#63567—32no—
34#63566—32no—
35#63565—32no—
36C2×C33×C4D411yescharacteristic
37C2×C33×C4D411yescharacteristic
38C2×C33×C4C2311yescharacteristic
39C2×C3×C3⋊Dic3—21no—
40C2×C3×C3⋊Dic3—21no—
41C2×C3×C3⋊Dic3—22no—
42C3×#634—21no—
43C3×#634—42no—
44C3×#634Q812yescomplemented
45C3×C3⋊(C3⋊C8)—22no—
46#5094—91no—
47#5020—91no—
48#4851—91no—
49C2×#634—21no—
50C2×#634C2×D311yescharacteristic
51C2×C3×Dic6—31no—
52C2×C3×Dic6—31no—
53C2×C3×Dic6—61no—
54C2×C3×Dic6—62no—
55C2×C3×Dic6—61no—
56C2×C3×Dic6—61no—
57C32×C42—21no—
58C32×C42—21no—
59C32×C42—21no—
60C32×C42—41no—
61C32×C42C2×D311yescharacteristic
62C32×C42C2×D311yescharacteristic
63C32×C42C2×D311yescharacteristic
64C4⋊(C3⋊Dic3)—21no—
65C4⋊(C3⋊Dic3)C2×D311yescharacteristic
66C4⋊(C3⋊Dic3)C2×D312yes—
67#3750—21no—
68C4×C3⋊Dic3—21no—
69C2×C3⋊(C3⋊C8)C2×D312yes—
70C3×C4⋊Dic3—32no—
71C3×C4⋊Dic3—32no—
72C3×C4⋊Dic3—31no—
73C3×C4⋊Dic3—31no—
74C3×C4⋊Dic3—61no—
75C3×C4⋊Dic3—62no—
76C3×C4⋊Dic3—62no—
77C3×C4⋊Dic3—61no—
78C3×C4⋊Dic3—61no—
79C3×Dic3⋊C4—62no—
80C3×Dic3⋊C4—61no—
81C3×Dic3⋊C4—61no—
82C3×C4×Dic3—62no—
83C3×C4×Dic3—61no—
84C3×C4×Dic3—61no—
85C2×C3×C3⋊C8—32no—
86C2×C3×C3⋊C8—32no—
87C2×C3×C3⋊C8—62no—
88C22×C33C2×D411yescharacteristic
89C33×C4—21no—
90C33×C4SD1612yes—
91C33×C4C2×Q811yescharacteristic
92C33×C4#4111yescharacteristic
93C3×C3⋊Dic3—22no—
94C3×C3⋊Dic3—42no—
95C3×C3⋊Dic3—41no—
96C3×C4⋊Q8—91no—
97C3×C4×Q8—91no—
98#869—31no—
99#869—31no—
100#869—61no—
101C4×Dic6—31no—
102C4×Dic6—31no—
103C4×Dic6—61no—
104C3×C4⋊C8—91no—
105#835—92no—
106#817—92no—
107#816—92no—
108#810—92no—
109#808—92no—
110#808—92no—
111C4⋊(C3⋊C8)—31no—
112C4⋊(C3⋊C8)—31no—
113C4⋊(C3⋊C8)—61no—
114C2×C32×C4—21no—
115C2×C32×C4—21no—
116C2×C32×C4—21no—
117C2×C32×C4—21no—
118C2×C32×C4—21no—
119C2×C32×C4—21no—
120C2×C32×C4—21no—
121C2×C32×C4—21no—
122C2×C32×C4—21no—
123C2×C32×C4—41no—
124C2×C32×C4—41no—
125C2×C32×C4—41no—
126C2×C32×C4D1211yescharacteristic
127C2×C32×C4D1211yescharacteristic
128C2×C32×C4#6711yescharacteristic
129C2×C32×C4#6711yescharacteristic
130C2×C32×C4#6711yescharacteristic
131C2×C32×C4#6711yescharacteristic
132C2×C32×C4C22×D311yescharacteristic
133C2×C32×C4C22×D311yescharacteristic
134C2×C32×C4C22×D311yescharacteristic
135C2×C3⋊Dic3—21no—
136C2×C3⋊Dic3—21no—
137C2×C3⋊Dic3—22no—
138#634—21no—
139#634—42no—
140#634Dic612yescomplemented
141C2×C3×Dic3—62no—
142C2×C3×Dic3—61no—
143C2×C3×Dic3—61no—
144C2×C3×Dic3—62no—
145C2×C3×Dic3—64no—
146C2×C3×Dic3—61no—
147C2×C3×Dic3—62no—
148C2×C3×Dic3—61no—
149C2×C3×Dic3—62no—
150C3×Dic6—32no—
151C3×Dic6—32no—
152C3×Dic6—62no—
153C3×Dic6—62no—
154C3×Dic6—61no—
155C3×Dic6—61no—
156C3×Dic6—124no—
157C3×Dic6—122no—
158C3×Dic6—122no—
159C3⋊(C3⋊C8)—22no—
160C3×C3⋊C8—62no—
161C3×C3⋊C8—62no—
162C3×C3⋊C8—122no—
163#475—271noSylow
164C2×C33C22⋊Q811yescharacteristic
165C2×C33C2×SD1611yescharacteristic
166C2×C33#13711yescharacteristic
167C2×C3×Q8—91no—
168C2×C3×Q8—181no—
169C2×Dic6—31no—
170C2×Dic6—31no—
171C2×Dic6—61no—
172C2×Dic6—62no—
173C2×Dic6—61no—
174C2×Dic6—61no—
175C2×C3×C8—92no—
176C3×C4⋊C4—91no—
177C3×C4⋊C4—92no—
178C3×C4⋊C4—181no—
179C3×C4⋊C4—181no—
180C3×C42—21no—
181C3×C42—21no—
182C3×C42—21no—
183C3×C42—41no—
184C3×C42—181no—
185C3×C42D3211yescharacteristic
186C3×C42D3211yescharacteristic
187C3×C42D3211yescharacteristic
188C4⋊Dic3—31no—
189C4⋊Dic3—32no—
190C4⋊Dic3—32no—
191C4⋊Dic3—31no—
192C4⋊Dic3—61no—
193C4⋊Dic3—62no—
194C4⋊Dic3—62no—
195C4⋊Dic3—61no—
196C4⋊Dic3—61no—
197Dic3⋊C4—62no—
198Dic3⋊C4—61no—
199Dic3⋊C4—61no—
200C4×Dic3—62no—
201C4×Dic3—61no—
202C4×Dic3—61no—
203C2×C3⋊C8—32no—
204C2×C3⋊C8—32no—
205C2×C3⋊C8—62no—
206C22×C32—21no—
207C22×C32—21no—
208C22×C32—21no—
209C22×C32—41no—
210C22×C32C2×D1211yescharacteristic
211C22×C32C2×#6711yescharacteristic
212C22×C32C2×#6711yescharacteristic
213C32×C4—21no—
214C32×C4—22no—
215C32×C4—21no—
216C32×C4—21no—
217C32×C4—22no—
218C32×C4—22no—
219C32×C4—22no—
220C32×C4—21no—
221C32×C4—21no—
222C32×C4—21no—
223C32×C4—21no—
224C32×C4—21no—
225C32×C4—21no—
226C32×C4—42no—
227C32×C4—41no—
228C32×C4—41no—
229C32×C4—41no—
230C32×C4—41no—
231C32×C4—42no—
232C32×C4#20412yes—
233C32×C4#21412yes—
234C32×C4#21512yes—
235C32×C4C2×Dic611yescharacteristic
236C32×C4#23511yescharacteristic
237C32×C4#23711yescharacteristic
238C32×C4D3×Q811yescharacteristic
239C32×C4D3×Q811yescharacteristic
240C32×C4#23911yescharacteristic
241C3⋊Dic3—22no—
242C3⋊Dic3—42no—
243C3⋊Dic3—41no—
244C3×Dic3—64no—
245C3×Dic3—62no—
246C3×Dic3—62no—
247C3×Dic3—124no—
248C3×Dic3—122no—
249C3×Dic3—122no—
250C3×Dic3—121no—
251C3×Dic3—122no—
252C3×Dic3—121no—
253C4⋊Q8—91no—
254C4×Q8—91no—
255#106—272no—
256C4⋊C8—91no—
257#103—272no—
258C33#47511yescharacteristic, complemented, Sylow
259C3×Q8—92no—
260C3×Q8—181no—
261C3×Q8—362no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—21no—
267C2×C3×C4—21no—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—21no—
271C2×C3×C4—41no—
272C2×C3×C4—41no—
273C2×C3×C4—41no—
274C2×C3×C4—181no—
275C2×C3×C4—181no—
276C2×C3×C4—182no—
277C2×C3×C4#62511yescharacteristic
278C2×C3×C4#62511yescharacteristic
279C2×C3×C4#62611yescharacteristic
280C2×C3×C4#62611yescharacteristic
281C2×C3×C4#62611yescharacteristic
282C2×C3×C4#62611yescharacteristic
283C2×C3×C4C2×D3211yescharacteristic
284C2×C3×C4C2×D3211yescharacteristic
285C2×C3×C4C2×D3211yescharacteristic
286C2×Dic3—62no—
287C2×Dic3—61no—
288C2×Dic3—61no—
289C2×Dic3—62no—
290C2×Dic3—64no—
291C2×Dic3—61no—
292C2×Dic3—62no—
293C2×Dic3—61no—
294C2×Dic3—62no—
295Dic6—32no—
296Dic6—32no—
297Dic6—62no—
298Dic6—62no—
299Dic6—61no—
300Dic6—61no—
301Dic6—124no—
302Dic6—122no—
303Dic6—122no—
304C3×C8—182no—
305C3⋊C8—62no—
306C3⋊C8—62no—
307C3⋊C8—122no—
308C2×C32—21no—
309C2×C32—21no—
310C2×C32—21no—
311C2×C32—21no—
312C2×C32—21no—
313C2×C32—21no—
314C2×C32—21no—
315C2×C32—21no—
316C2×C32—21no—
317C2×C32—41no—
318C2×C32—41no—
319C2×C32—41no—
320C2×C32#89711yescharacteristic
321C2×C32C2×#20411yescharacteristic
322C2×C32#90911yescharacteristic
323C2×C32#92411yescharacteristic
324C2×C32C2×#21411yescharacteristic
325C2×C32C2×#21511yescharacteristic
326C2×C32#94211yescharacteristic
327C2×C32#94611yescharacteristic
328C2×C32#95111yescharacteristic
329C2×Q8—91no—
330C2×Q8—181no—
331C2×C8—92no—
332C4⋊C4—92no—
333C4⋊C4—91no—
334C4⋊C4—181no—
335C4⋊C4—181no—
336C42—181no—
337C42#111311yescharacteristic
338C22×C3—21no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3—41no—
342C22×C3C2×#62511yescharacteristic
343C22×C3C2×#62611yescharacteristic
344C22×C3C2×#62611yescharacteristic
345C3×C4—22no—
346C3×C4—21no—
347C3×C4—21no—
348C3×C4—21no—
349C3×C4—21no—
350C3×C4—21no—
351C3×C4—21no—
352C3×C4—21no—
353C3×C4—21no—
354C3×C4—21no—
355C3×C4—22no—
356C3×C4—22no—
357C3×C4—22no—
358C3×C4—41no—
359C3×C4—41no—
360C3×C4—42no—
361C3×C4—41no—
362C3×C4—41no—
363C3×C4—42no—
364C3×C4—182no—
365C3×C4—362no—
366C3×C4—361no—
367C3×C4#371512yes—
368C3×C4#371612yes—
369C3×C4#371712yes—
370C3×C4D3×Dic611yescharacteristic
371C3×C4D3×Dic611yescharacteristic
372C3×C4#379511yescharacteristic
373C3×C4#379611yescharacteristic
374C3×C4#379711yescharacteristic
375C3×C4#379911yescharacteristic
376Dic3—64no—
377Dic3—62no—
378Dic3—62no—
379Dic3—124no—
380Dic3—122no—
381Dic3—122no—
382Dic3—121no—
383Dic3—122no—
384Dic3—121no—
385C32—21no—
386C32—21no—
387C32—21no—
388C32—41no—
389C32#485111yescharacteristic, complemented
390C32#502011yescharacteristic, complemented
391C32#509411yescharacteristic, complemented
392Q8—92no—
393Q8—181no—
394Q8—362no—
395C2×C4—181no—
396C2×C4—181no—
397C2×C4—182no—
398C2×C4#629511yescharacteristic
399C2×C4#629511yescharacteristic
400C2×C4C2×#111311yescharacteristic
401C8—182no—
402C2×C3—21no—
403C2×C3—21no—
404C2×C3—21no—
405C2×C3—21no—
406C2×C3—21no—
407C2×C3—21no—
408C2×C3—21no—
409C2×C3—21no—
410C2×C3—21no—
411C2×C3—41no—
412C2×C3—41no—
413C2×C3—41no—
414C2×C3C2×#371511yescharacteristic
415C2×C3#6382711yescharacteristic
416C2×C3C2×#371611yescharacteristic
417C2×C3#6383111yescharacteristic
418C2×C3C2×#371711yescharacteristic
419C2×C3#6383311yescharacteristic
420C2×C3#6385611yescharacteristic
421C2×C3#6385711yescharacteristic
422C2×C3#6387111yescharacteristic
423C22C2×#629511yescharacteristic
424C4—21no—
425C4—182no—
426C4—362no—
427C4—361no—
428C4#8881112yes—
429C4#8904011yescharacteristic
430C4#8904111yescharacteristic
431C3—21no—
432C3—21no—
433C3—21no—
434C3—41no—
435C3#1059028511yescharacteristic, complemented
436C3#1059030111yescharacteristic, complemented
437C3#1059030711yescharacteristic, complemented
438C2C2×#8881111yescharacteristic
439C2#1172366311yescharacteristic
440C2#1172367411yescharacteristic
subgroupG/H
centerC22C2×#6295
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC33×C42C22
socle, prime radicalC22×C33C2×D4
hypercenterC42#1113
hypocenterC33#475
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
44. C3×#634Q8semidirect
140. #634Dic6semidirect
258. C33#475semidirect
389. C32#4851semidirect
390. C32#5020semidirect
391. C32#5094semidirect
435. C3#10590285semidirect
436. C3#10590301semidirect
437. C3#10590307semidirect

Maximal subgroups

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

subgroupindex
C3×#640782
C3×C4×#6342
C3×C4⋊(C3⋊(C3⋊C8))2
#117220792
#117220762
#105903073
#105903013
#105902853