Finite Group Databasebeta

← #10690920

Subgroups — #10690920

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
1#84032C211yesmaximal, characteristic
2C22×C3⋊#95C211yesmaximal, characteristic, complemented
3C3×#1779C211yesmaximal, characteristic, complemented
4C2×#4470C212yesmaximal, complemented
5#67675C212yesmaximal, complemented
6#9147—31nomaximal, Sylow
7C24×C3×C4C2211yescharacteristic
8C24×Dic3—21no—
9C2×#953—21no—
10C2×#930C412yescomplemented
11C2×#930C2211yescharacteristic
12C2×C3×C22⋊C8—22no—
13C2×C3×C22⋊C8C2212yes—
14#5213—21no—
15#5213—22no—
16#5212C412yescomplemented
17#5212C2211yescharacteristic
18C2×C3⋊#95—21no—
19C2×C3⋊#95—24no—
20C2×C3⋊#95C2212yes—
21#4470—24no—
22#2847—31no—
23C22×#95—31no—
24#1779D311yescharacteristic, complemented
25C2×#327—32no—
26#1294—32no—
27C25×C3D411yescharacteristic
28C23×C3×C4—22no—
29C23×C3×C4—22no—
30C23×C3×C4—44no—
31C23×C3×C4C2×C412yes—
32C23×C3×C4D411yescharacteristic
33C23×C3×C4D412yes—
34C23×C3×C4C2311yescharacteristic
35C22×#67—22no—
36C22×#67—21no—
37C23×Dic3—28no—
38C23×Dic3—24no—
39C23×Dic3—22no—
40C23×Dic3—44no—
41C23×Dic3—42no—
42C23×Dic3—42no—
43C22×D12C2×C411yescharacteristic, complemented
44C22×C3×C8—22no—
45#953—42no—
46#953—42no—
47C2×#217—22no—
48C2×#217—21no—
49#930—22no—
50#930—22no—
51#930—44no—
52C2×(C2×D3)⋊C4—22no—
53C2×(C2×D3)⋊C4—21no—
54C2×C4⋊Dic3C2×C411yescharacteristic, complemented
55C2×C4⋊Dic3C2×C412yescomplemented
56C3×C22⋊C8—22no—
57C3×C22⋊C8—22no—
58C3×C22⋊C8—44no—
59C3⋊#95—24no—
60C3⋊#95—22no—
61C3⋊#95—22no—
62C3⋊#95—21no—
63C3⋊#95D414yescomplemented
64C24×C4—61no—
65C24×C4C2×D311yescharacteristic
66C2×#121—31no—
67C2×#121—32no—
68C2×#120—61no—
69C2×C22⋊C8—22no—
70C2×C22⋊C8C2×D312yes—
71#386—61no—
72#386—62no—
73#380—32no—
74#380—31no—
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×C3C22⋊C412yes—
86C24×C3D812yes—
87C24×C3SD1612yes—
88C24×C3C2×D411yescharacteristic
89C23×D3—21no—
90C22×C3×C4—24no—
91C22×C3×C4—24no—
92C22×C3×C4—21no—
93C22×C3×C4—21no—
94C22×C3×C4—24no—
95C22×C3×C4—24no—
96C22×C3×C4—42no—
97C22×C3×C4—42no—
98C22×C3×C4—42no—
99C22×C3×C4—42no—
100C22×C3×C4—48no—
101C22×C3×C4—48no—
102C22×C3×C4C22⋊C412yes—
103C22×C3×C4C22×C411yescharacteristic
104C22×C3×C4C2×D412yes—
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4#4112yes—
107C2×#67—44no—
108C2×#67—44no—
109C2×#67—44no—
110C2×#67—41no—
111C2×#67—41no—
112C2×#67—41no—
113C2×#67—41no—
114C2×#67—41no—
115C2×#67—41no—
116C22×Dic3—216no—
117C22×Dic3—28no—
118C22×Dic3—24no—
119C22×Dic3—21no—
120C22×Dic3—22no—
121C22×Dic3—48no—
122C22×Dic3—416no—
123C22×Dic3—416no—
124C22×Dic3—44no—
125C22×Dic3—48no—
126C22×Dic3—42no—
127C22×Dic3—42no—
128C2×D12—22no—
129C2×D12—42no—
130C2×C3×C8—44no—
131C2×C3×C8—42no—
132#217—44no—
133#217—41no—
134#217—41no—
135(C2×D3)⋊C4—44no—
136(C2×D3)⋊C4—42no—
137C4⋊Dic3—22no—
138C4⋊Dic3—42no—
139C25D1211yescharacteristic
140C22×D4—31no—
141C22×D4—62no—
142C22×D4—61no—
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×C4C4×D312yes—
153C23×C4D1211yescharacteristic
154C23×C4#6712yes—
155C23×C4C22×D311yescharacteristic
156C22×C8—22no—
157#121—62no—
158#121—62no—
159#121—124no—
160#120—122no—
161#120—122no—
162C2×C4⋊C4—31no—
163C2×C4⋊C4—32no—
164C2×C22⋊C4—62no—
165C2×C22⋊C4—61no—
166C2×C22⋊C4—62no—
167C2×C22⋊C4—61no—
168C22⋊C8—22no—
169C22⋊C8—22no—
170C22⋊C8—44no—
171#95—34no—
172#95—64no—
173#95—62no—
174#95—62no—
175#95—61no—
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#9914yes—
195C23×C3#10214yes—
196C23×C3#10414yes—
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
204C22×D3—41no—
205C22×D3—42no—
206C22×D3—41no—
207C22×D3—41no—
208C22×D3—42no—
209C2×C3×C4—22no—
210C2×C3×C4—22no—
211C2×C3×C4—22no—
212C2×C3×C4—22no—
213C2×C3×C4—22no—
214C2×C3×C4—42no—
215C2×C3×C4—42no—
216C2×C3×C4—48no—
217C2×C3×C4—48no—
218C2×C3×C4—42no—
219C2×C3×C4—42no—
220C2×C3×C4C4×D412yes—
221C2×C3×C4#12112yes—
222#67—88no—
223#67—81no—
224#67—81no—
225#67—81no—
226#67—81no—
227C2×Dic3—28no—
228C2×Dic3—432no—
229C2×Dic3—44no—
230C2×Dic3—48no—
231C2×Dic3—48no—
232C2×Dic3—44no—
233C2×Dic3—41no—
234C2×Dic3—42no—
235C2×Dic3—41no—
236C2×Dic3—42no—
237C2×Dic3—41no—
238C2×Dic3—42no—
239D12—82no—
240C3×C8—82no—
241C24—24no—
242C24—24no—
243C24—21no—
244C24—21no—
245C24—21no—
246C24—21no—
247C24—61no—
248C24#20412yes—
249C24D2412yes—
250C24(C2×D3)⋊C412yes—
251C24C2×D1211yescharacteristic
252C2×D4—62no—
253C2×D4—122no—
254C2×D4—124no—
255C2×D4—124no—
256C2×D4—124no—
257C2×D4—121no—
258C2×D4—121no—
259C2×D4—121no—
260C2×D4—121no—
261C2×D4—121no—
262C2×D4—121no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—48no—
274C22×C4—48no—
275C22×C4—616no—
276C22×C4—68no—
277C22×C4—64no—
278C22×C4—61no—
279C22×C4—62no—
280C22×C4—128no—
281C22×C4—1216no—
282C22×C4—124no—
283C22×C4—1216no—
284C22×C4—128no—
285C22×C4—122no—
286C22×C4—122no—
287C22×C4(C2×D3)⋊C412yes—
288C22×C4C2×C4×D311yescharacteristic
289C22×C4D3×D412yes—
290C22×C4#23712yes—
291C22×C4C2×#6711yescharacteristic
292C2×C8—44no—
293C2×C8—42no—
294C4⋊C4—62no—
295C4⋊C4—122no—
296C22⋊C4—124no—
297C22⋊C4—124no—
298C22⋊C4—121no—
299C22⋊C4—121no—
300C22⋊C4—122no—
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#32718yescomplemented
320C22×C3#38611yescharacteristic
321C22×C3C2×#9912yes—
322C22×C3C2×#10211yescharacteristic
323C22×C3#41711yescharacteristic
324C22×C3C2×#10412yes—
325C22×C3#44711yescharacteristic
326C22×C3#44811yescharacteristic
327C22×C3#48011yescharacteristic
328C22×C3#48111yescharacteristic
329C2×D3—81no—
330C2×D3—81no—
331C2×D3—81no—
332C2×D3—82no—
333C2×D3—82no—
334C3×C4—22no—
335C3×C4—22no—
336C3×C4—42no—
337Dic3—48no—
338Dic3—81no—
339Dic3—82no—
340C23—24no—
341C23—24no—
342C23—24no—
343C23—28no—
344C23—24no—
345C23—28no—
346C23—22no—
347C23—22no—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—28no—
353C23—28no—
354C23—22no—
355C23—22no—
356C23—22no—
357C23—22no—
358C23—121no—
359C23—122no—
360C23—122no—
361C23—121no—
362C23—121no—
363C23#80614yes—
364C23#82114yes—
365C23#82514yes—
366C23#88211yescharacteristic
367C23#88611yescharacteristic
368C23C2×#20411yescharacteristic
369C23C2×D2411yescharacteristic
370C23#90811yescharacteristic
371C23#90911yescharacteristic
372C23C2×(C2×D3)⋊C411yescharacteristic
373D4—242no—
374D4—248no—
375D4—241no—
376D4—241no—
377D4—241no—
378D4—241no—
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—68no—
391C2×C4—1232no—
392C2×C4—128no—
393C2×C4—128no—
394C2×C4—124no—
395C2×C4—124no—
396C2×C4—121no—
397C2×C4—122no—
398C2×C4—122no—
399C2×C4—121no—
400C2×C4—121no—
401C2×C4—122no—
402C2×C4#67⋊C412yes—
403C2×C4#93912yes—
404C8—82no—
405C2×C3—21no—
406C2×C3—28no—
407C2×C3—21no—
408C2×C3—21no—
409C2×C3—21no—
410C2×C3C2×#32712yes—
411C2×C3#150412yes—
412C2×C3#187411yescharacteristic
413C2×C3#189611yescharacteristic
414C2×C3#189811yescharacteristic
415D3—161no—
416C22—28no—
417C22—28no—
418C22—28no—
419C22—28no—
420C22—24no—
421C22—22no—
422C22—24no—
423C22—22no—
424C22—21no—
425C22—24no—
426C22—21no—
427C22—22no—
428C22—24no—
429C22—24no—
430C22—22no—
431C22—21no—
432C22—24no—
433C22—21no—
434C22—241no—
435C22—242no—
436C22—241no—
437C22—242no—
438C22—241no—
439C22#447018yescomplemented
440C22#473211yescharacteristic
441C22#473511yescharacteristic
442C22#473611yescharacteristic
443C22#473811yescharacteristic
444C22C2×#80612yes—
445C22#495811yescharacteristic
446C22C2×#82111yescharacteristic
447C22#513311yescharacteristic
448C22C2×#82512yes—
449C4—22no—
450C4—22no—
451C4—42no—
452C4—128no—
453C4—241no—
454C4—242no—
455C3#914711yescharacteristic, complemented, Sylow
456C2—21no—
457C2—28no—
458C2—21no—
459C2—21no—
460C2—21no—
461C2—481no—
462C2#6949611yescharacteristic
463C2#6993211yescharacteristic
464C2C2×#447012yes—
465C2#6996511yescharacteristic
466C2#6997612yes—
subgroupG/H
centerC23C2×(C2×D3)⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1779C2
socleC23×C3C2×C22⋊C4
prime radical#84032C2
cosocleC23×C3×C4C23
hypercenter#1779D3
hypocenterC3#9147
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C24×C4 ◁ #1779

Complements

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

NH ≅ G/Ntype
2. C22×C3⋊#95C2semidirect
3. C3×#1779C2semidirect
4. C2×#4470C2semidirect
5. #67675C2semidirect
10. C2×#930C4semidirect
16. #5212C4semidirect
24. #1779D3semidirect
43. C22×D12C2×C4semidirect
54. C2×C4⋊Dic3C2×C4semidirect
55. C2×C4⋊Dic3C2×C4semidirect
63. C3⋊#95D4semidirect
319. C22×C3#327semidirect
439. C22#4470semidirect
455. C3#9147semidirect

Maximal subgroups

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

subgroupindex
#840322
C22×C3⋊#952
C3×#17792
C2×#44702
#676752
#91473