Finite Group Databasebeta

← #10690923

Subgroups — #10690923

Order 768.

1348 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2847C211yesmaximal, characteristic
2C22×C3⋊#95C211yesmaximal, characteristic, complemented
3#72126C211yesmaximal, characteristic, complemented
4C2×#4537C212yesmaximal, complemented
5#67624C212yesmaximal, complemented
6#9147—31nomaximal, Sylow
7C24×C3×C4C2211yescharacteristic
8C24×Dic3—21no—
9C2×C3×#121C412yescomplemented
10C2×C3×#121C2211yescharacteristic
11C2×C3×#120—21no—
12C3×#386—21no—
13C3×#386—22no—
14C3×#380C412yescomplemented
15C3×#380C2211yescharacteristic
16C2×C3⋊#95—21no—
17C2×C3⋊#95—24no—
18C2×C3⋊#95C2212yes—
19C2×#830—22no—
20C2×#830C2212yes—
21#4537—24no—
22#2847D311yescharacteristic
23C22×#95—31no—
24#1779—31no—
25C2×#327—32no—
26#1294—32no—
27C25×C3D411yescharacteristic
28C22×C3×D4—22no—
29C22×C3×D4—21no—
30C22×C3×D4C2×C411yescharacteristic, complemented
31C23×C3×C4—22no—
32C23×C3×C4—22no—
33C23×C3×C4—44no—
34C23×C3×C4C2×C412yes—
35C23×C3×C4D412yes—
36C23×C3×C4D411yescharacteristic
37C23×C3×C4C2311yescharacteristic
38C23×Dic3—28no—
39C23×Dic3—24no—
40C23×Dic3—22no—
41C23×Dic3—44no—
42C23×Dic3—42no—
43C23×Dic3—42no—
44C3×#121—22no—
45C3×#121—22no—
46C3×#121—44no—
47C3×#120—42no—
48C3×#120—42no—
49C2×C3×C4⋊C4C2×C411yescharacteristic, complemented
50C2×C3×C4⋊C4C2×C412yescomplemented
51C2×C3×C22⋊C4—22no—
52C2×C3×C22⋊C4—21no—
53C2×C3×C22⋊C4—22no—
54C2×C3×C22⋊C4—21no—
55C22×C3⋊C8—22no—
56C3×#95—22no—
57C3×#95—21no—
58C3⋊#95—24no—
59C3⋊#95—22no—
60C3⋊#95D414yescomplemented
61#830—22no—
62#830—22no—
63#830—44no—
64C24×C4—61no—
65C24×C4C2×D311yescharacteristic
66C2×#121Dic312yescomplemented
67C2×#121C2×D311yescharacteristic
68C2×#120—21no—
69C2×C22⋊C8—32no—
70C2×C22⋊C8—62no—
71#386—21no—
72#386—22no—
73#380Dic312yescomplemented
74#380C2×D311yescharacteristic
75C2×#95—32no—
76C2×#95—64no—
77C2×#95—61no—
78#327—64no—
79C24×C3—24no—
80C24×C3—24no—
81C24×C3—21no—
82C24×C3—21no—
83C24×C3—21no—
84C24×C3—21no—
85C24×C3—21no—
86C24×C3C22⋊C412yes—
87C24×C3D812yes—
88C24×C3SD1612yes—
89C24×C3C2×D411yescharacteristic
90C2×C3×D4—22no—
91C2×C3×D4—42no—
92C2×C3×D4—44no—
93C2×C3×D4—44no—
94C2×C3×D4—44no—
95C2×C3×D4—41no—
96C2×C3×D4—41no—
97C2×C3×D4—41no—
98C2×C3×D4—41no—
99C2×C3×D4—41no—
100C2×C3×D4—41no—
101C22×C3×C4—24no—
102C22×C3×C4—24no—
103C22×C3×C4—21no—
104C22×C3×C4—21no—
105C22×C3×C4—24no—
106C22×C3×C4—24no—
107C22×C3×C4—21no—
108C22×C3×C4—22no—
109C22×C3×C4—42no—
110C22×C3×C4—42no—
111C22×C3×C4—42no—
112C22×C3×C4—42no—
113C22×C3×C4—48no—
114C22×C3×C4—48no—
115C22×C3×C4C22⋊C412yes—
116C22×C3×C4C22×C411yescharacteristic
117C22×C3×C4C2×D412yes—
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4#4112yes—
120C22×Dic3—216no—
121C22×Dic3—28no—
122C22×Dic3—24no—
123C22×Dic3—48no—
124C22×Dic3—416no—
125C22×Dic3—416no—
126C22×Dic3—44no—
127C22×Dic3—48no—
128C22×Dic3—42no—
129C22×Dic3—42no—
130C3×C4⋊C4—22no—
131C3×C4⋊C4—42no—
132C3×C22⋊C4—44no—
133C3×C22⋊C4—44no—
134C3×C22⋊C4—41no—
135C3×C22⋊C4—41no—
136C3×C22⋊C4—42no—
137C2×C3⋊C8—44no—
138C2×C3⋊C8—42no—
139C25#6711yescharacteristic
140C22×D4—22no—
141C22×D4—21no—
142C22×D4C2×Dic311yescharacteristic, complemented
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—44no—
146C23×C4—64no—
147C23×C4—68no—
148C23×C4—62no—
149C23×C4—124no—
150C23×C4—122no—
151C23×C4—122no—
152C23×C4C2×Dic312yes—
153C23×C4#6712yes—
154C23×C4#6711yescharacteristic
155C23×C4C22×D311yescharacteristic
156C22×C8—62no—
157#121—22no—
158#121—22no—
159#121—44no—
160#120—42no—
161#120—42no—
162C2×C4⋊C4C2×Dic311yescharacteristic, complemented
163C2×C4⋊C4C2×Dic312yescomplemented
164C2×C22⋊C4—22no—
165C2×C22⋊C4—21no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—21no—
168C22⋊C8—62no—
169C22⋊C8—62no—
170C22⋊C8—124no—
171#95—22no—
172#95—21no—
173#95—34no—
174#95—64no—
175#95—62no—
176C23×C3—24no—
177C23×C3—24no—
178C23×C3—28no—
179C23×C3—24no—
180C23×C3—24no—
181C23×C3—28no—
182C23×C3—28no—
183C23×C3—28no—
184C23×C3—22no—
185C23×C3—22no—
186C23×C3—22no—
187C23×C3—22no—
188C23×C3—22no—
189C23×C3—22no—
190C23×C3—22no—
191C23×C3—22no—
192C23×C3—22no—
193C23×C3—22no—
194C23×C3—41no—
195C23×C3—42no—
196C23×C3—41no—
197C23×C3—41no—
198C23×C3—42no—
199C23×C3#9914yes—
200C23×C3#10214yes—
201C23×C3#10414yes—
202C23×C3C2×C22⋊C411yescharacteristic
203C23×C3#12011yescharacteristic
204C23×C3#12311yescharacteristic
205C23×C3C2×D811yescharacteristic
206C23×C3C2×SD1611yescharacteristic
207C23×C3AGL(1,ℤ/8)11yescharacteristic
208C23×C3#13711yescharacteristic
209C3×D4—82no—
210C3×D4—88no—
211C3×D4—81no—
212C3×D4—81no—
213C3×D4—81no—
214C3×D4—81no—
215C2×C3×C4—22no—
216C2×C3×C4—22no—
217C2×C3×C4—22no—
218C2×C3×C4—22no—
219C2×C3×C4—22no—
220C2×C3×C4—42no—
221C2×C3×C4—42no—
222C2×C3×C4—48no—
223C2×C3×C4—48no—
224C2×C3×C4—42no—
225C2×C3×C4—42no—
226C2×C3×C4—41no—
227C2×C3×C4—42no—
228C2×C3×C4—41no—
229C2×C3×C4—42no—
230C2×C3×C4—41no—
231C2×C3×C4—42no—
232C2×C3×C4C4×D412yes—
233C2×C3×C4#12112yes—
234C2×Dic3—28no—
235C2×Dic3—432no—
236C2×Dic3—44no—
237C2×Dic3—48no—
238C2×Dic3—48no—
239C2×Dic3—44no—
240C3⋊C8—82no—
241C24—24no—
242C24—24no—
243C24—21no—
244C24—21no—
245C24—21no—
246C24—21no—
247C24—21no—
248C24#21312yes—
249C24#21412yes—
250C24#21712yes—
251C24C2×#6711yescharacteristic
252C2×D4—22no—
253C2×D4—42no—
254C2×D4—44no—
255C2×D4—44no—
256C2×D4—44no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—41no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—48no—
276C22×C4—48no—
277C22×C4—616no—
278C22×C4—68no—
279C22×C4—64no—
280C22×C4—128no—
281C22×C4—1216no—
282C22×C4—124no—
283C22×C4—1216no—
284C22×C4—128no—
285C22×C4—122no—
286C22×C4—122no—
287C22×C4#21712yes—
288C22×C4D3×D412yes—
289C22×C4#23712yes—
290C22×C4C22×Dic311yescharacteristic
291C22×C4C2×#6711yescharacteristic
292C2×C8—124no—
293C2×C8—122no—
294C4⋊C4—22no—
295C4⋊C4—42no—
296C22⋊C4—44no—
297C22⋊C4—44no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—42no—
301C22×C3—28no—
302C22×C3—28no—
303C22×C3—28no—
304C22×C3—28no—
305C22×C3—24no—
306C22×C3—21no—
307C22×C3—24no—
308C22×C3—21no—
309C22×C3—21no—
310C22×C3—24no—
311C22×C3—21no—
312C22×C3—22no—
313C22×C3—24no—
314C22×C3—22no—
315C22×C3—22no—
316C22×C3—24no—
317C22×C3—24no—
318C22×C3—22no—
319C22×C3—81no—
320C22×C3—81no—
321C22×C3—81no—
322C22×C3—82no—
323C22×C3—82no—
324C22×C3#32718yescomplemented
325C22×C3#38611yescharacteristic
326C22×C3C2×#9912yes—
327C22×C3C2×#10211yescharacteristic
328C22×C3#41711yescharacteristic
329C22×C3C2×#10412yes—
330C22×C3#44711yescharacteristic
331C22×C3#44811yescharacteristic
332C22×C3#48011yescharacteristic
333C22×C3#48111yescharacteristic
334C3×C4—22no—
335C3×C4—22no—
336C3×C4—42no—
337C3×C4—81no—
338C3×C4—82no—
339Dic3—48no—
340C23—24no—
341C23—24no—
342C23—24no—
343C23—28no—
344C23—28no—
345C23—24no—
346C23—22no—
347C23—22no—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—22no—
354C23—22no—
355C23—22no—
356C23—28no—
357C23—28no—
358C23—41no—
359C23—42no—
360C23—41no—
361C23—41no—
362C23—42no—
363C23#83214yes—
364C23#83414yes—
365C23#83714yes—
366C23C2×#21311yescharacteristic
367C23C2×#21411yescharacteristic
368C23#93511yescharacteristic
369C23#93711yescharacteristic
370C23#94911yescharacteristic
371C23#95111yescharacteristic
372C23C2×#21711yescharacteristic
373D4—82no—
374D4—81no—
375D4—81no—
376D4—81no—
377D4—81no—
378D4—88no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—48no—
387C2×C4—48no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—68no—
397C2×C4—1232no—
398C2×C4—128no—
399C2×C4—124no—
400C2×C4—128no—
401C2×C4—124no—
402C2×C4D4×Dic312yes—
403C2×C4#93912yes—
404C8—242no—
405C2×C3—21no—
406C2×C3—28no—
407C2×C3—21no—
408C2×C3—21no—
409C2×C3—21no—
410C2×C3—161no—
411C2×C3C2×#32712yes—
412C2×C3#150412yes—
413C2×C3#187411yescharacteristic
414C2×C3#189611yescharacteristic
415C2×C3#189811yescharacteristic
416C22—28no—
417C22—28no—
418C22—28no—
419C22—28no—
420C22—22no—
421C22—24no—
422C22—24no—
423C22—22no—
424C22—21no—
425C22—24no—
426C22—21no—
427C22—21no—
428C22—24no—
429C22—21no—
430C22—22no—
431C22—24no—
432C22—22no—
433C22—24no—
434C22—82no—
435C22—81no—
436C22—81no—
437C22—82no—
438C22—81no—
439C22#453718yescomplemented
440C22#503311yescharacteristic
441C22#503411yescharacteristic
442C22#503611yescharacteristic
443C22#504011yescharacteristic
444C22C2×#83211yescharacteristic
445C22C2×#83412yes—
446C22#522011yescharacteristic
447C22#523211yescharacteristic
448C22C2×#83712yes—
449C4—22no—
450C4—22no—
451C4—42no—
452C4—82no—
453C4—81no—
454C4—128no—
455C3#914711yescharacteristic, complemented, Sylow
456C2—21no—
457C2—21no—
458C2—21no—
459C2—21no—
460C2—28no—
461C2—161no—
462C2#7167011yescharacteristic
463C2C2×#453712yes—
464C2#7167711yescharacteristic
465C2#7170812yes—
466C2#7171111yescharacteristic
subgroupG/H
centerC23C2×#217
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
Fitting, prime radicalC3×#2847C2
socleC23×C3C2×C22⋊C4
cosocleC23×C3×C4C23
hypercenter#2847D3
hypocenterC3#9147
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C24×C4 ◁ #2847

Complements

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

NH ≅ G/Ntype
2. C22×C3⋊#95C2semidirect
3. #72126C2semidirect
4. C2×#4537C2semidirect
5. #67624C2semidirect
9. C2×C3×#121C4semidirect
14. C3×#380C4semidirect
30. C22×C3×D4C2×C4semidirect
49. C2×C3×C4⋊C4C2×C4semidirect
50. C2×C3×C4⋊C4C2×C4semidirect
60. C3⋊#95D4semidirect
66. C2×#121Dic3semidirect
73. #380Dic3semidirect
142. C22×D4C2×Dic3semidirect
162. C2×C4⋊C4C2×Dic3semidirect
163. C2×C4⋊C4C2×Dic3semidirect
324. C22×C3#327semidirect
439. C22#4537semidirect
455. C3#9147semidirect

Maximal subgroups

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

subgroupindex
C3×#28472
C22×C3⋊#952
#721262
C2×#45372
#676242
#91473