Finite Group Databasebeta

← #10690963

Subgroups — #10690963

Order 768.

1024 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2847C211yesmaximal, characteristic
2C2×#5212C211yesmaximal, characteristic, complemented
3#72126C211yesmaximal, characteristic, complemented
4C2×#4537C212yesmaximal, complemented
5#67624C212yesmaximal, complemented
6#9155—31nomaximal, Sylow
7C24×C3×C4C2211yescharacteristic
8C2×C3×#121C412yescomplemented
9C2×C3×#121C2211yescharacteristic
10C2×C3×#120—21no—
11C22×#217—21no—
12C22×C4⋊Dic3—21no—
13C3×#386—21no—
14C3×#386—22no—
15C3×#380C412yescomplemented
16C3×#380C2211yescharacteristic
17#5212—24no—
18C2×C3⋊#95C2212yes—
19C2×#830—22no—
20C2×#830C2212yes—
21#4537—24no—
22#2847D311yescharacteristic
23C2×#380—31no—
24#1779—31no—
25C2×#327—32no—
26#1294—32no—
27C25×C3D411yescharacteristic
28C22×C3×D4—22no—
29C22×C3×D4—21no—
30C22×C3×D4C2×C411yescharacteristic
31C23×C3×C4—22no—
32C23×C3×C4—22no—
33C23×C3×C4—44no—
34C23×C3×C4C2×C412yes—
35C23×C3×C4D411yescharacteristic
36C23×C3×C4D412yes—
37C23×C3×C4C2311yescharacteristic
38C23×Dic3—22no—
39C3×#121—22no—
40C3×#121—22no—
41C3×#121—44no—
42C3×#120—42no—
43C3×#120—42no—
44C2×C3×C4⋊C4C2×C411yescharacteristic
45C2×C3×C4⋊C4C2×C412yes—
46C2×C3×C22⋊C4—22no—
47C2×C3×C22⋊C4—21no—
48C2×C3×C22⋊C4—22no—
49C2×C3×C22⋊C4—21no—
50C2×#217—24no—
51C2×#217—42no—
52C2×#217—42no—
53C2×C4⋊Dic3—24no—
54C2×C4⋊Dic3—44no—
55C22×C3⋊C8—22no—
56C3×#95—22no—
57C3×#95—21no—
58C3⋊#95—22no—
59C3⋊#95D414yescomplemented
60#830—22no—
61#830—22no—
62#830—44no—
63C24×C4C2×D311yescharacteristic
64C2×#121Dic312yescomplemented
65C2×#121C2×D311yescharacteristic
66C2×#120—21no—
67C22×C4⋊C4—61no—
68C22×C22⋊C4—61no—
69C2×C22⋊C8—32no—
70C2×C22⋊C8—62no—
71#386—21no—
72#386—22no—
73#380—64no—
74#380Dic312yescomplemented
75#380C2×D311yescharacteristic
76C2×#95—32no—
77#327—64no—
78C24×C3—21no—
79C24×C3—21no—
80C24×C3—21no—
81C24×C3—21no—
82C24×C3—21no—
83C24×C3—42no—
84C24×C3—42no—
85C24×C3C22⋊C412yes—
86C24×C3D812yes—
87C24×C3SD1612yes—
88C24×C3C2×D411yescharacteristic
89C2×C3×D4—22no—
90C2×C3×D4—42no—
91C2×C3×D4—44no—
92C2×C3×D4—44no—
93C2×C3×D4—44no—
94C2×C3×D4—41no—
95C2×C3×D4—41no—
96C2×C3×D4—41no—
97C2×C3×D4—41no—
98C2×C3×D4—41no—
99C2×C3×D4—41no—
100C22×C3×C4—21no—
101C22×C3×C4—21no—
102C22×C3×C4—24no—
103C22×C3×C4—24no—
104C22×C3×C4—24no—
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—48no—
113C22×C3×C4—48no—
114C22×C3×C4C22⋊C412yes—
115C22×C3×C4C22×C411yescharacteristic
116C22×C3×C4C2×D412yes—
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4#4112yes—
119C22×Dic3—24no—
120C22×Dic3—44no—
121C22×Dic3—44no—
122C22×Dic3—42no—
123C22×Dic3—42no—
124C3×C4⋊C4—22no—
125C3×C4⋊C4—42no—
126C3×C22⋊C4—44no—
127C3×C22⋊C4—41no—
128C3×C22⋊C4—41no—
129C3×C22⋊C4—44no—
130C3×C22⋊C4—42no—
131#217—48no—
132C4⋊Dic3—48no—
133C2×C3⋊C8—44no—
134C2×C3⋊C8—42no—
135C25#6711yescharacteristic
136C22×D4—22no—
137C22×D4—21no—
138C22×D4C2×Dic311yescharacteristic
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—44no—
142C23×C4—62no—
143C23×C4C2×Dic312yes—
144C23×C4#6711yescharacteristic
145C23×C4#6712yes—
146C23×C4C22×D311yescharacteristic
147C22×C8—62no—
148#121—22no—
149#121—22no—
150#121—44no—
151#120—42no—
152#120—42no—
153C2×C4⋊C4—64no—
154C2×C4⋊C4—124no—
155C2×C4⋊C4C2×Dic311yescharacteristic
156C2×C4⋊C4C2×Dic312yes—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—21no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—21no—
161C2×C22⋊C4—64no—
162C2×C22⋊C4—122no—
163C2×C22⋊C4—122no—
164C22⋊C8—62no—
165C22⋊C8—62no—
166C22⋊C8—124no—
167#95—22no—
168#95—21no—
169#95—34no—
170#95—62no—
171C23×C3—22no—
172C23×C3—22no—
173C23×C3—22no—
174C23×C3—22no—
175C23×C3—22no—
176C23×C3—22no—
177C23×C3—22no—
178C23×C3—22no—
179C23×C3—22no—
180C23×C3—22no—
181C23×C3—22no—
182C23×C3—22no—
183C23×C3—44no—
184C23×C3—44no—
185C23×C3—42no—
186C23×C3—42no—
187C23×C3—42no—
188C23×C3—42no—
189C23×C3—44no—
190C23×C3—44no—
191C23×C3—41no—
192C23×C3—42no—
193C23×C3—41no—
194C23×C3—41no—
195C23×C3—42no—
196C23×C3#10214yes—
197C23×C3C2×C22⋊C411yescharacteristic
198C23×C3#12011yescharacteristic
199C23×C3#12311yescharacteristic
200C23×C3C2×D811yescharacteristic
201C23×C3C2×SD1611yescharacteristic
202C23×C3AGL(1,ℤ/8)11yescharacteristic
203C23×C3#13711yescharacteristic
204C3×D4—82no—
205C3×D4—88no—
206C3×D4—81no—
207C3×D4—81no—
208C3×D4—81no—
209C3×D4—81no—
210C2×C3×C4—24no—
211C2×C3×C4—24no—
212C2×C3×C4—22no—
213C2×C3×C4—22no—
214C2×C3×C4—22no—
215C2×C3×C4—22no—
216C2×C3×C4—42no—
217C2×C3×C4—42no—
218C2×C3×C4—48no—
219C2×C3×C4—48no—
220C2×C3×C4—41no—
221C2×C3×C4—42no—
222C2×C3×C4—41no—
223C2×C3×C4—42no—
224C2×C3×C4—41no—
225C2×C3×C4—42no—
226C2×C3×C4#9914yes—
227C2×C3×C4C4×D412yes—
228C2×C3×C4#12112yes—
229C2×Dic3—44no—
230C2×Dic3—44no—
231C2×Dic3—44no—
232C2×Dic3—84no—
233C2×Dic3—84no—
234C3⋊C8—82no—
235C24—21no—
236C24—21no—
237C24—21no—
238C24—21no—
239C24—21no—
240C24—42no—
241C24—42no—
242C24#21312yes—
243C24#21412yes—
244C24#21712yes—
245C24C2×#6711yescharacteristic
246C2×D4—22no—
247C2×D4—42no—
248C2×D4—44no—
249C2×D4—44no—
250C2×D4—44no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—41no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4—21no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—48no—
270C22×C4—48no—
271C22×C4—64no—
272C22×C4—124no—
273C22×C4—124no—
274C22×C4—122no—
275C22×C4—122no—
276C22×C4#21712yes—
277C22×C4D3×D412yes—
278C22×C4#23712yes—
279C22×C4C22×Dic311yescharacteristic
280C22×C4C2×#6711yescharacteristic
281C2×C8—124no—
282C2×C8—122no—
283C4⋊C4—22no—
284C4⋊C4—42no—
285C4⋊C4—128no—
286C22⋊C4—44no—
287C22⋊C4—41no—
288C22⋊C4—41no—
289C22⋊C4—44no—
290C22⋊C4—42no—
291C22⋊C4—128no—
292C22×C3—21no—
293C22×C3—21no—
294C22×C3—21no—
295C22×C3—21no—
296C22×C3—22no—
297C22×C3—22no—
298C22×C3—22no—
299C22×C3—22no—
300C22×C3—24no—
301C22×C3—24no—
302C22×C3—24no—
303C22×C3—44no—
304C22×C3—44no—
305C22×C3—44no—
306C22×C3—44no—
307C22×C3—42no—
308C22×C3—42no—
309C22×C3—42no—
310C22×C3—42no—
311C22×C3—81no—
312C22×C3—81no—
313C22×C3—81no—
314C22×C3—82no—
315C22×C3—82no—
316C22×C3#38611yescharacteristic
317C22×C3C2×#9912yes—
318C22×C3C2×#10211yescharacteristic
319C22×C3#41711yescharacteristic
320C22×C3#42112yes—
321C22×C3#44711yescharacteristic
322C22×C3#44811yescharacteristic
323C22×C3#48011yescharacteristic
324C22×C3#48111yescharacteristic
325C3×C4—22no—
326C3×C4—24no—
327C3×C4—22no—
328C3×C4—81no—
329C3×C4—82no—
330Dic3—84no—
331C23—22no—
332C23—22no—
333C23—22no—
334C23—22no—
335C23—22no—
336C23—22no—
337C23—22no—
338C23—22no—
339C23—22no—
340C23—22no—
341C23—22no—
342C23—22no—
343C23—44no—
344C23—44no—
345C23—42no—
346C23—42no—
347C23—44no—
348C23—44no—
349C23—42no—
350C23—42no—
351C23—41no—
352C23—42no—
353C23—41no—
354C23—41no—
355C23—42no—
356C23#83214yes—
357C23C2×#21311yescharacteristic
358C23C2×#21411yescharacteristic
359C23#93511yescharacteristic
360C23#93711yescharacteristic
361C23#94911yescharacteristic
362C23#95111yescharacteristic
363C23C2×#21711yescharacteristic
364D4—82no—
365D4—88no—
366D4—81no—
367D4—81no—
368D4—81no—
369D4—81no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—48no—
379C2×C4—48no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—124no—
387C2×C4—124no—
388C2×C4—124no—
389C2×C4—244no—
390C2×C4—244no—
391C2×C4#83414yes—
392C2×C4D4×Dic312yes—
393C2×C4#93912yes—
394C8—242no—
395C2×C3—21no—
396C2×C3—21no—
397C2×C3—21no—
398C2×C3—21no—
399C2×C3—44no—
400C2×C3—161no—
401C2×C3#151312yes—
402C2×C3#151912yes—
403C2×C3#187411yescharacteristic
404C2×C3#189711yescharacteristic
405C2×C3#189811yescharacteristic
406C22—22no—
407C22—22no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—22no—
413C22—22no—
414C22—24no—
415C22—24no—
416C22—24no—
417C22—44no—
418C22—44no—
419C22—44no—
420C22—44no—
421C22—42no—
422C22—42no—
423C22—42no—
424C22—42no—
425C22—82no—
426C22—81no—
427C22—81no—
428C22—82no—
429C22—81no—
430C22#503311yescharacteristic
431C22#503411yescharacteristic
432C22#503611yescharacteristic
433C22#504011yescharacteristic
434C22C2×#83211yescharacteristic
435C22C2×#83412yes—
436C22#522011yescharacteristic
437C22#523211yescharacteristic
438C22#523612yes—
439C4—22no—
440C4—24no—
441C4—22no—
442C4—82no—
443C4—81no—
444C4—244no—
445C3#915511yescharacteristic, complemented, Sylow
446C2—21no—
447C2—21no—
448C2—21no—
449C2—21no—
450C2—44no—
451C2—161no—
452C2#7167011yescharacteristic
453C2#7167512yes—
454C2#7167711yescharacteristic
455C2#7170912yes—
456C2#7171211yescharacteristic
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#9155
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. C2×#5212C2semidirect
3. #72126C2semidirect
4. C2×#4537C2semidirect
5. #67624C2semidirect
8. C2×C3×#121C4semidirect
15. C3×#380C4semidirect
59. C3⋊#95D4semidirect
64. C2×#121Dic3semidirect
74. #380Dic3semidirect
445. C3#9155semidirect

Maximal subgroups

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

subgroupindex
C3×#28472
C2×#52122
#721262
C2×#45372
#676242
#91553