Finite Group Databasebeta

← #10801780

Subgroups — #10801780

Order 768.

1066 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#466C211yesmaximal, characteristic
2C2×#5152C211yesmaximal, characteristic, complemented
3C2×#5101C211yesmaximal, characteristic, complemented
4#82633C212yesmaximal, complemented
5#72039C211yesmaximal, characteristic, complemented
6#71970C211yesmaximal, characteristic, complemented
7#71682C212yesmaximal, complemented
8#70613C211yesmaximal, characteristic, complemented
9#70605C211yesmaximal, characteristic, complemented
10#70185C212yesmaximal, complemented
11Dic3×#102C212yesmaximal, complemented
12#21529—31nomaximal, Sylow
13C22×C3×D8C2211yescharacteristic
14C23×C3×C8C2211yescharacteristic
15C2×C3×#121C2212yes—
16C2×D4×Dic3C2212yes—
17C2×#926C2211yescharacteristic
18C22×#208C2211yescharacteristic
19C3×#466C418yescomplemented
20C2×C3×#107C2211yescharacteristic
21C2×C3×#102C2212yes—
22#5221—22no—
23#5221C2212yes—
24#5220—22no—
25C2×#832C2212yes—
26C2×#832C2212yes—
27#5212C2212yes—
28#5152—24no—
29C2×#817C2211yescharacteristic
30C2×#817C2211yescharacteristic
31#5101—24no—
32C2×#815C2211yescharacteristic
33C2×#815C2211yescharacteristic
34C2×C8×Dic3C2211yescharacteristic
35C2×C8×Dic3C2211yescharacteristic
36#4984C2212yes—
37Dic3×C4⋊C4C2212yes—
38#4951—22no—
39#4950—22no—
40#4551C2212yes—
41#4529C2212yes—
42#4529C2212yes—
43C2×#466D311yescharacteristic
44C2×#442—31no—
45C2×#405—31no—
46#2324—32no—
47#1972—31no—
48#1943—31no—
49#1934—32no—
50#1895—32no—
51#1807—31no—
52#1776—31no—
53C4×#102—32no—
54C22×C3×D4—22no—
55C22×C3×D4C2312yes—
56C23×C3×C4C2311yescharacteristic
57C23×Dic3—22no—
58C2×C3×D8—22no—
59C2×C3×D8—22no—
60C2×C3×D8C2×C414yes—
61C22×C3×C8—22no—
62C22×C3×C8—22no—
63C22×C3×C8C2×C414yescomplemented
64C22×C3×C8C2311yescharacteristic
65C22×C3×C8C2311yescharacteristic
66C3×#121—24no—
67C3×#121C2×C418yescomplemented
68C2×C3×C4⋊C4C2312yes—
69C2×C3×C22⋊C4—22no—
70C2×#217—22no—
71C2×#217—22no—
72C2×#217—22no—
73C2×#217—22no—
74C2×#217D412yes—
75D4×Dic3—24no—
76D4×Dic3—24no—
77#926—22no—
78#926—22no—
79C2×C4⋊Dic3—22no—
80C2×C4⋊Dic3D412yes—
81C2×C4⋊Dic3C2312yes—
82C2×C4⋊Dic3C2312yes—
83C2×C4×Dic3—22no—
84C2×C4×Dic3C2311yescharacteristic
85C2×C4×Dic3C2311yescharacteristic
86C2×#208—22no—
87C2×#208—22no—
88C2×#208—22no—
89C22×C3⋊C8C2311yescharacteristic
90C22×C3⋊C8C2311yescharacteristic
91C3×#107C2×C414yescomplemented
92C3×#102C2×C418yescomplemented
93#832—24no—
94#832—24no—
95C3⋊#95—22no—
96C3⋊#95—22no—
97C3⋊#95—22no—
98#817—22no—
99#817—22no—
100#815—22no—
101#815D414yescomplemented
102C8×Dic3—22no—
103C8×Dic3—22no—
104C22×D8C2×D311yescharacteristic
105C22×M16—31no—
106C23×C8C2×D311yescharacteristic
107C2×#121C2×D312yes—
108C2×C4×D4—32no—
109C2×#117—31no—
110#466Dic318yescomplemented
111#442—64no—
112C2×#107C2×D311yescharacteristic
113C2×#106—31no—
114C2×#106—31no—
115C2×#102—32no—
116C2×#102—32no—
117C2×#102C2×D312yes—
118#405—64no—
119C2×#97—31no—
120C2×#97—31no—
121C2×C4×C8—31no—
122C2×C4×C8—31no—
123#390—32no—
124#390—62no—
125#389—32no—
126#388—62no—
127#386—62no—
128#385—62no—
129#380—32no—
130C4×C4⋊C4—32no—
131#340—32no—
132#340—32no—
133#340—32no—
134C24×C3—22no—
135C24×C3C2×D411yescharacteristic
136C2×C3×D4—28no—
137C2×C3×D4—22no—
138C2×C3×D4—22no—
139C2×C3×D4—22no—
140C2×C3×D4—22no—
141C2×C3×D4—22no—
142C2×C3×D4—82no—
143C2×C3×D4—82no—
144C2×C3×D4C22×C414yes—
145C22×C3×C4—21no—
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—21no—
149C22×C3×C4—22no—
150C22×C3×C4C22×C412yes—
151C22×C3×C4C2×D411yescharacteristic
152C22×C3×C4#4112yes—
153C22×C3×C4C2411yescharacteristic
154C22×Dic3—22no—
155C22×Dic3—22no—
156C22×Dic3—44no—
157C22×Dic3—82no—
158C22×Dic3—82no—
159C22×Dic3C2×D412yes—
160C22×Dic3C2×D412yes—
161C3×D8—28no—
162C2×C3×C8—21no—
163C2×C3×C8—21no—
164C2×C3×C8—28no—
165C2×C3×C8C22×C412yes—
166C2×C3×C8C22×C412yes—
167C2×C3×C8C2×D412yes—
168C2×C3×C8#4112yes—
169C3×C4⋊C4—22no—
170C3×C4⋊C4C22×C414yes—
171C3×C22⋊C4—28no—
172#217—24no—
173#217—44no—
174#217—82no—
175#217—82no—
176#217—82no—
177C4⋊Dic3—24no—
178C4⋊Dic3—22no—
179C4⋊Dic3—22no—
180C4⋊Dic3—22no—
181C4⋊Dic3—22no—
182C4⋊Dic3—82no—
183C4×Dic3—21no—
184C4×Dic3—21no—
185C4×Dic3—82no—
186C4×Dic3C2×D412yes—
187C4×Dic3#4112yes—
188#208—28no—
189C2×C3⋊C8—21no—
190C2×C3⋊C8—21no—
191C2×C3⋊C8—21no—
192C2×C3⋊C8—21no—
193C2×C3⋊C8C2×D412yes—
194C2×C3⋊C8#4112yes—
195C22×D4—22no—
196C22×D4C22×D312yes—
197C23×C4—62no—
198C23×C4C22×D311yescharacteristic
199C2×D8—22no—
200C2×D8—22no—
201C2×D8C2×Dic314yes—
202C2×M16—62no—
203C2×M16—62no—
204C2×M16—62no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—31no—
208C22×C8—31no—
209C22×C8C2×Dic314yescomplemented
210C22×C8C22×D311yescharacteristic
211C22×C8C22×D311yescharacteristic
212#121—24no—
213#121C2×Dic318yescomplemented
214C4×D4—64no—
215C4×D4—64no—
216#117—62no—
217#117—62no—
218C2×C4⋊C4—32no—
219C2×C4⋊C4—32no—
220C2×C4⋊C4—32no—
221C2×C4⋊C4—62no—
222C2×C4⋊C4C22×D312yes—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—32no—
225C2×C22⋊C4—62no—
226C2×C22⋊C4—62no—
227C2×C22⋊C4—62no—
228C2×C22⋊C4—62no—
229C2×C42—31no—
230C2×C42—31no—
231C2×C42—62no—
232#107C2×Dic314yescomplemented
233#106—62no—
234#106—62no—
235#102—64no—
236#102—64no—
237#102C2×Dic318yescomplemented
238#97—34no—
239#97—62no—
240C4×C8—62no—
241C4×C8—62no—
242#95—62no—
243#95—62no—
244#95—62no—
245C23×C3—21no—
246C23×C3—21no—
247C23×C3—22no—
248C23×C3—22no—
249C23×C3—24no—
250C23×C3—42no—
251C23×C3—42no—
252C23×C3—42no—
253C23×C3—42no—
254C23×C3—42no—
255C23×C3—42no—
256C23×C3C4×D412yes—
257C23×C3C22×D411yescharacteristic
258C3×D4—24no—
259C3×D4—24no—
260C3×D4—42no—
261C3×D4—42no—
262C3×D4—88no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—22no—
266C2×C3×C4—22no—
267C2×C3×C4—22no—
268C2×C3×C4—22no—
269C2×C3×C4—24no—
270C2×C3×C4—42no—
271C2×C3×C4—42no—
272C2×C3×C4C4×D412yes—
273C2×C3×C4C23×C411yescharacteristic
274C2×C3×C4C22×D411yescharacteristic
275C2×C3×C4C2×#4111yescharacteristic
276C2×C3×C4C2×#4111yescharacteristic
277C2×C3×C4C2×#4111yescharacteristic
278C2×C3×C4C2×#4111yescharacteristic
279C2×Dic3—22no—
280C2×Dic3—22no—
281C2×Dic3—22no—
282C2×Dic3—22no—
283C2×Dic3—22no—
284C2×Dic3—42no—
285C2×Dic3—42no—
286C2×Dic3—42no—
287C2×Dic3—42no—
288C2×Dic3—42no—
289C2×Dic3—42no—
290C2×Dic3—88no—
291C2×Dic3#12114yes—
292C3×C8—22no—
293C3×C8C4×D414yes—
294C3⋊C8—22no—
295C3⋊C8—22no—
296C24—22no—
297C24D3×D411yescharacteristic
298C2×D4—28no—
299C2×D4—22no—
300C2×D4—22no—
301C2×D4—22no—
302C2×D4—22no—
303C2×D4—22no—
304C2×D4—82no—
305C2×D4—82no—
306C2×D4C22×Dic314yes—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—32no—
313C22×C4—32no—
314C22×C4—62no—
315C22×C4—62no—
316C22×C4—124no—
317C22×C4—242no—
318C22×C4—242no—
319C22×C4D3×D411yescharacteristic
320C22×C4#23712yes—
321C22×C4C22×Dic312yes—
322C22×C4C23×D311yescharacteristic
323D8—28no—
324M16—68no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—28no—
328C2×C8—32no—
329C2×C8—32no—
330C2×C8—61no—
331C2×C8—61no—
332C2×C8—61no—
333C2×C8—61no—
334C2×C8D3×D412yes—
335C2×C8#23712yes—
336C2×C8C22×Dic312yes—
337C2×C8C22×Dic312yes—
338C4⋊C4—22no—
339C4⋊C4—64no—
340C4⋊C4—62no—
341C4⋊C4—62no—
342C4⋊C4—62no—
343C4⋊C4—62no—
344C4⋊C4—242no—
345C4⋊C4C22×Dic314yes—
346C22⋊C4—28no—
347C22⋊C4—64no—
348C22⋊C4—124no—
349C22⋊C4—242no—
350C22⋊C4—242no—
351C22⋊C4—242no—
352C42—32no—
353C42—32no—
354C42—61no—
355C42—61no—
356C42—242no—
357C22×C3—22no—
358C22×C3—22no—
359C22×C3—22no—
360C22×C3—42no—
361C22×C3—42no—
362C22×C3—44no—
363C22×C3—44no—
364C22×C3—44no—
365C22×C3—82no—
366C22×C3—82no—
367C22×C3—82no—
368C22×C3—82no—
369C22×C3C2×C4×D411yescharacteristic
370C22×C3C2×#12111yescharacteristic
371C22×C3#53211yescharacteristic
372C22×C3#57612yes—
373C22×C3#57712yes—
374C3×C4—22no—
375C3×C4—44no—
376C3×C4C2×C4×D411yescharacteristic
377C3×C4C4×#4111yescharacteristic
378C3×C4#52511yescharacteristic
379C3×C4#52911yescharacteristic
380Dic3—24no—
381Dic3—42no—
382Dic3—82no—
383Dic3—82no—
384C23—22no—
385C23—22no—
386C23—21no—
387C23—21no—
388C23—24no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23D4×Dic312yes—
396C23C2×D3×D411yescharacteristic
397D4—24no—
398D4—24no—
399D4—42no—
400D4—42no—
401D4—88no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—24no—
409C2×C4—34no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—62no—
413C2×C4—62no—
414C2×C4—62no—
415C2×C4—62no—
416C2×C4—62no—
417C2×C4—122no—
418C2×C4—122no—
419C2×C4—122no—
420C2×C4—122no—
421C2×C4—122no—
422C2×C4—122no—
423C2×C4—248no—
424C2×C4D4×Dic312yes—
425C2×C4C2×D3×D411yescharacteristic
426C2×C4C2×#23711yescharacteristic
427C2×C4C2×#23711yescharacteristic
428C2×C4D3×#4111yescharacteristic
429C2×C4D3×#4111yescharacteristic
430C2×C4C23×Dic311yescharacteristic
431C8—22no—
432C8—62no—
433C8—62no—
434C8D4×Dic314yes—
435C2×C3—42no—
436C2×C3—84no—
437C2×C3C4×#12111yescharacteristic
438C2×C3#294811yescharacteristic
439C2×C3#295211yescharacteristic
440C2×C3#306212yes—
441C2×C3#317912yes—
442C22—22no—
443C22—22no—
444C22—22no—
445C22—42no—
446C22—42no—
447C22—44no—
448C22—44no—
449C22—44no—
450C22—82no—
451C22—82no—
452C22—82no—
453C22—82no—
454C22#559611yescharacteristic
455C22D3×#12111yescharacteristic
456C22#575712yes—
457C22#577012yes—
458C22C2×D4×Dic311yescharacteristic
459C4—22no—
460C4—44no—
461C4—64no—
462C4—122no—
463C4—242no—
464C4—242no—
465C4#560011yescharacteristic
466C4#560611yescharacteristic
467C4C2×D4×Dic311yescharacteristic
468C4Dic3×#4111yescharacteristic
469C3#2152911yescharacteristic, complemented, Sylow
470C2—42no—
471C2—84no—
472C2#8070012yes—
473C2#8071812yes—
474C2Dic3×#12111yescharacteristic
475C2#8382311yescharacteristic
476C2#8389911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
Fitting, prime radicalC2×C3×#466C2
socleC23×C3C22×D4
cosocleC22×C3×C4C24
hypercenterC2×#466D3
hypocenterC3#21529
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#466

Complements

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

NH ≅ G/Ntype
2. C2×#5152C2semidirect
3. C2×#5101C2semidirect
4. #82633C2semidirect
5. #72039C2semidirect
6. #71970C2semidirect
7. #71682C2semidirect
8. #70613C2semidirect
9. #70605C2semidirect
10. #70185C2semidirect
11. Dic3×#102C2semidirect
19. C3×#466C4semidirect
63. C22×C3×C8C2×C4semidirect
67. C3×#121C2×C4semidirect
91. C3×#107C2×C4semidirect
92. C3×#102C2×C4semidirect
101. #815D4semidirect
110. #466Dic3semidirect
209. C22×C8C2×Dic3semidirect
213. #121C2×Dic3semidirect
232. #107C2×Dic3semidirect
237. #102C2×Dic3semidirect
469. C3#21529semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4662
C2×#51522
C2×#51012
#826332
#720392
#719702
#716822
#706132
#706052
#701852
Dic3×#1022
#215293