Finite Group Databasebeta

← #10944012

Subgroups — #10944012

Order 768.

1436 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×Dic6C211yesmaximal, characteristic
2C3×#2857C211yesmaximal, characteristic
3C2×#5090C212yesmaximal, complemented
4C2×#5050C212yesmaximal
5#71692C212yesmaximal
6#71530C211yesmaximal, characteristic
7#69690C214yesmaximal, complemented
8#69588C212yesmaximal, complemented
9#32906—31nomaximal, Sylow
10C23×Dic6C2211yescharacteristic
11C2×C3×C4⋊Q8C2211yescharacteristic
12C2×C3×C22⋊Q8C2212yes—
13C22×C3×C42C2211yescharacteristic
14C22×#216C2214yes—
15C22×C4⋊Dic3C2211yescharacteristic
16C22×Dic3⋊C4—21no—
17C22×C4×Dic3—21no—
18C2×C4×Dic6—21no—
19C2×C4×Dic6—21no—
20C2×C4×Dic6—28no—
21C2×C4×Dic6C2212yes—
22C2×C4×Dic6C2212yes—
23C3×#391C2212yes—
24C3×#389—21no—
25C3×#389—21no—
26C3×#384C2211yescharacteristic
27C3×#380—21no—
28C3×#380—21no—
29C2×#835C2214yes—
30C2×#830C2212yes—
31#5090—28no—
32#5050—28no—
33C2×#810C2214yes—
34C2×C4⋊(C3⋊C8)C2212yes—
35#4529C2214yes—
36#4524C2212yes—
37C22×C4×Q8—31no—
38C2×#462—32no—
39C2×#451—32no—
40#2857D311yescharacteristic
41#1878—34no—
42#1872—32no—
43#1803—32no—
44#1797—31no—
45C22×C3×Q8C2312yes—
46C23×C3×C4D411yescharacteristic
47C23×C3×C4D411yescharacteristic
48C23×C3×C4C2311yescharacteristic
49C23×Dic3—21no—
50C23×Dic3—21no—
51C22×Dic6—21no—
52C22×Dic6—22no—
53C22×Dic6—24no—
54C22×Dic6—24no—
55C22×Dic6D414yes—
56C22×Dic6C2312yes—
57C3×C4⋊Q8—24no—
58C3×C22⋊Q8—24no—
59C3×C22⋊Q8—24no—
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4—22no—
65C2×C3×C4⋊C4C2312yes—
66C2×C3×C22⋊C4—22no—
67C2×C3×C42—24no—
68C2×C3×C42D411yescharacteristic
69C2×C3×C42D411yescharacteristic
70C2×C3×C42C2311yescharacteristic
71C2×C3×C42C2311yescharacteristic
72C2×#216—28no—
73C2×#216—28no—
74C2×#216—28no—
75C2×C4⋊Dic3—21no—
76C2×C4⋊Dic3—24no—
77C2×C4⋊Dic3C2312yes—
78C2×Dic3⋊C4—28no—
79C2×Dic3⋊C4—22no—
80C2×Dic3⋊C4—22no—
81C2×C4×Dic3—28no—
82C2×C4×Dic3—22no—
83C2×C4×Dic3—22no—
84C22×C3⋊C8C2314yes—
85C4×Dic6—28no—
86C4×Dic6—28no—
87C4×Dic6—24no—
88C4×Dic6—24no—
89C4×Dic6—24no—
90C4×Dic6D418yes—
91C3×#95—22no—
92C3×#95—22no—
93#835—28no—
94#830—24no—
95#810—28no—
96C4⋊(C3⋊C8)—24no—
97C23×Q8—31no—
98C22×Q16—34no—
99C2×C4⋊Q8C2×D311yescharacteristic
100C2×C22⋊Q8C2×D312yes—
101C2×C4×Q8—32no—
102C2×C4×Q8—32no—
103C2×C4×Q8—68no—
104C2×C4×Q8—61no—
105C2×C4×Q8—61no—
106C22×C4⋊C4—31no—
107C22×C4⋊C4—61no—
108C22×C42—61no—
109C22×C42C2×D311yescharacteristic
110#462—68no—
111#451—68no—
112C2×C4⋊C8—32no—
113C2×#103—34no—
114C2×#103—34no—
115C2×C22⋊C8—32no—
116#391C2×D312yes—
117#389—21no—
118#389—21no—
119#384C2×D311yescharacteristic
120#380—21no—
121#380—21no—
122#340—34no—
123#336—32no—
124C24×C3C2×D411yescharacteristic
125C2×C3×Q8—24no—
126C2×C3×Q8—22no—
127C2×C3×Q8—24no—
128C2×C3×Q8—22no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4—41no—
138C22×C3×C4Q1614yescomplemented
139C22×C3×C4C2×D412yes—
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4#4112yes—
146C22×C3×C4C2411yescharacteristic
147C22×Dic3—24no—
148C22×Dic3—24no—
149C22×Dic3—24no—
150C22×Dic3—22no—
151C22×Dic3—24no—
152C22×Dic3—22no—
153C22×Dic3—42no—
154C22×Dic3—42no—
155C2×Dic6—24no—
156C2×Dic6—24no—
157C2×Dic6—216no—
158C2×Dic6—24no—
159C2×Dic6—22no—
160C2×Dic6—22no—
161C2×Dic6—22no—
162C2×Dic6—216no—
163C2×Dic6C2×D414yes—
164C2×Dic6C2×D414yes—
165C2×Dic6#4114yes—
166C3×C4⋊C4—22no—
167C3×C4⋊C4—22no—
168C3×C4⋊C4—42no—
169C3×C4⋊C4—42no—
170C3×C4⋊C4—42no—
171C3×C4⋊C4—44no—
172C3×C4⋊C4—42no—
173C3×C22⋊C4—44no—
174C3×C42—22no—
175C3×C42—22no—
176C3×C42—22no—
177C3×C42C2×D414yes—
178#216—416no—
179C4⋊Dic3—24no—
180C4⋊Dic3—22no—
181C4⋊Dic3C2×D414yes—
182Dic3⋊C4—28no—
183Dic3⋊C4—28no—
184C4×Dic3—28no—
185C4×Dic3—28no—
186C2×C3⋊C8—24no—
187C2×C3⋊C8—24no—
188C2×C3⋊C8—24no—
189C22×Q8—34no—
190C22×Q8—32no—
191C22×Q8—64no—
192C22×Q8—64no—
193C22×Q8—62no—
194C22×Q8—61no—
195C22×Q8C22×D312yes—
196C23×C4—61no—
197C23×C4—61no—
198C23×C4#6711yescharacteristic
199C23×C4#6711yescharacteristic
200C23×C4C22×D311yescharacteristic
201C2×Q16—68no—
202C2×Q16—68no—
203C2×Q16—68no—
204C22×C8—34no—
205C4⋊Q8—24no—
206C22⋊Q8—24no—
207C22⋊Q8—24no—
208C4×Q8—38no—
209C4×Q8—64no—
210C4×Q8—64no—
211C4×Q8—68no—
212C4×Q8—68no—
213C4×Q8—64no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—32no—
220C2×C4⋊C4—64no—
221C2×C4⋊C4—68no—
222C2×C4⋊C4—61no—
223C2×C4⋊C4—62no—
224C2×C4⋊C4—62no—
225C2×C4⋊C4C22×D312yes—
226C2×C22⋊C4—22no—
227C2×C42—24no—
228C2×C42—68no—
229C2×C42—62no—
230C2×C42—62no—
231C2×C42#6711yescharacteristic
232C2×C42#6711yescharacteristic
233C2×C42C22×D311yescharacteristic
234C2×C42C22×D311yescharacteristic
235C4⋊C8—64no—
236#103—68no—
237#103—68no—
238C22⋊C8—64no—
239#95—22no—
240#95—22no—
241C23×C3—21no—
242C23×C3—21no—
243C23×C3—21no—
244C23×C3—21no—
245C23×C3#12112yes—
246C23×C3C2×Q1612yes—
247C23×C3#13712yes—
248C23×C3C22×D411yescharacteristic
249C3×Q8—44no—
250C3×Q8—44no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—21no—
256C2×C3×C4—21no—
257C2×C3×C4—24no—
258C2×C3×C4—24no—
259C2×C3×C4—28no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—44no—
264C2×C3×C4—42no—
265C2×C3×C4—42no—
266C2×C3×C4—42no—
267C2×C3×C4—42no—
268C2×C3×C4—42no—
269C2×C3×C4—42no—
270C2×C3×C4#12012yes—
271C2×C3×C4#12012yes—
272C2×C3×C4#12112yes—
273C2×C3×C4C2×Q1612yes—
274C2×C3×C4C2×Q1612yes—
275C2×C3×C4#13712yes—
276C2×C3×C4C22×D411yescharacteristic
277C2×C3×C4C22×D412yes—
278C2×C3×C4C2×#4111yescharacteristic
279C2×C3×C4C2×#4112yes—
280C2×Dic3—216no—
281C2×Dic3—24no—
282C2×Dic3—24no—
283C2×Dic3—24no—
284C2×Dic3—24no—
285C2×Dic3—48no—
286C2×Dic3—42no—
287C2×Dic3—42no—
288Dic6—216no—
289Dic6—24no—
290Dic6—24no—
291Dic6—24no—
292Dic6#12118yes—
293C3⋊C8—44no—
294C24C2×#6711yescharacteristic
295C2×Q8—24no—
296C2×Q8—22no—
297C2×Q8—22no—
298C2×Q8—24no—
299C2×Q8—34no—
300C2×Q8—34no—
301C2×Q8—34no—
302C2×Q8—616no—
303C2×Q8—64no—
304C2×Q8—64no—
305C2×Q8—616no—
306C2×Q8—64no—
307C2×Q8—62no—
308C2×Q8—62no—
309C2×Q8—62no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—41no—
319C22×C4—64no—
320C22×C4—64no—
321C22×C4—64no—
322C22×C4—62no—
323C22×C4—64no—
324C22×C4—62no—
325C22×C4—122no—
326C22×C4—122no—
327C22×C4#21614yescomplemented
328C22×C4D3×D412yes—
329C22×C4#23712yes—
330C22×C4C2×#6711yescharacteristic
331C22×C4C2×#6711yescharacteristic
332C22×C4C2×#6711yescharacteristic
333C22×C4C2×#6711yescharacteristic
334C22×C4C2×#6711yescharacteristic
335C22×C4C23×D311yescharacteristic
336Q16—1216no—
337C2×C8—64no—
338C2×C8—64no—
339C2×C8—64no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—34no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—44no—
347C4⋊C4—42no—
348C4⋊C4—68no—
349C4⋊C4—64no—
350C4⋊C4—68no—
351C4⋊C4—62no—
352C22⋊C4—44no—
353C42—22no—
354C42—22no—
355C42—22no—
356C42—68no—
357C42—68no—
358C42D3×D414yes—
359C22×C3—22no—
360C22×C3—22no—
361C22×C3—22no—
362C22×C3—22no—
363C22×C3—22no—
364C22×C3—22no—
365C22×C3#46214yescomplemented
366C22×C3C2×#12011yescharacteristic
367C22×C3C2×#12111yescharacteristic
368C22×C3#52511yescharacteristic
369C22×C3C22×Q1611yescharacteristic
370C22×C3C2×#13711yescharacteristic
371C22×C3C2×#13711yescharacteristic
372C22×C3#57511yescharacteristic
373C3×C4—22no—
374C3×C4—22no—
375C3×C4—22no—
376C3×C4—41no—
377C3×C4—82no—
378C3×C4—82no—
379C3×C4#45114yescomplemented
380C3×C4C2×#12112yes—
381C3×C4#52512yes—
382Dic3—28no—
383Dic3—44no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23#93812yes—
389C23#94212yes—
390C23C2×#21612yes—
391C23C22×#6711yescharacteristic
392Q8—38no—
393Q8—44no—
394Q8—44no—
395Q8—616no—
396Q8—64no—
397Q8—64no—
398Q8—64no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—28no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—44no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—616no—
419C2×C4—64no—
420C2×C4—64no—
421C2×C4—64no—
422C2×C4—64no—
423C2×C4—128no—
424C2×C4—122no—
425C2×C4—122no—
426C2×C4#93712yes—
427C2×C4#93712yes—
428C2×C4#93812yes—
429C2×C4C2×#21612yes—
430C2×C4C2×#21612yes—
431C2×C4#95112yes—
432C2×C4C2×D3×D411yescharacteristic
433C2×C4C2×D3×D412yes—
434C2×C4C2×#23711yescharacteristic
435C2×C4D3×#4112yes—
436C8—124no—
437C2×C3—24no—
438C2×C3#284711yescharacteristic
439C2×C3C2×#45111yescharacteristic
440C2×C3#300711yescharacteristic
441C2×C3C2×#46211yescharacteristic
442C2×C3#303711yescharacteristic
443C2×C3#310411yescharacteristic
444C2×C3#311311yescharacteristic
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22#509014yescomplemented
452C22C2×#93711yescharacteristic
453C22C2×#93811yescharacteristic
454C22C2×#94211yescharacteristic
455C22C22×#21611yescharacteristic
456C22#582211yescharacteristic
457C22C2×#95111yescharacteristic
458C22#582811yescharacteristic
459C4—22no—
460C4—22no—
461C4—22no—
462C4—41no—
463C4—68no—
464C4—82no—
465C4—82no—
466C4—124no—
467C4#505014yescomplemented
468C4D3×#12112yes—
469C4#560612yes—
470C3#3290611yescharacteristic, complemented, Sylow
471C2—24no—
472C2#8272311yescharacteristic
473C2#8273511yescharacteristic
474C2C2×#505011yescharacteristic
475C2C2×#509011yescharacteristic
476C2#8331411yescharacteristic
477C2#8336811yescharacteristic
478C2#8336911yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2857C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#2857D3
hypocenterC3#32906
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2857

Complements

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

NH ≅ G/Ntype
3. C2×#5090C2semidirect
7. #69690C2semidirect
8. #69588C2semidirect
138. C22×C3×C4Q16semidirect
327. C22×C4#216semidirect
365. C22×C3#462semidirect
379. C3×C4#451semidirect
451. C22#5090semidirect
467. C4#5050semidirect
470. C3#32906semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Dic62
C3×#28572
C2×#50902
C2×#50502
#716922
#715302
#696902
#695882
#329063