Finite Group Databasebeta

← #10797366

Subgroups — #10797366

Order 768.

974 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#389C211yesmaximal, characteristic
2C2×#4964C211yesmaximal, characteristic
3C4×#926C211yesmaximal, characteristic
4#71558C211yesmaximal, characteristic
5#71558C211yesmaximal, characteristic
6#71558C211yesmaximal, characteristic
7#71527C211yesmaximal, characteristic
8#69603C212yesmaximal, complemented
9#69602C212yesmaximal, complemented
10#69601C212yesmaximal, complemented
11#69600C212yesmaximal, complemented
12#21034—31nomaximal, Sylow
13C22×C3×C4⋊C4—21no—
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C42C2211yescharacteristic
17C2×#926C2211yescharacteristic
18C2×#926C2211yescharacteristic
19C22×#208C2211yescharacteristic
20C22×#208C2211yescharacteristic
21C3×#389—21no—
22C3×#389—21no—
23C3×#389C2212yes—
24C3×#389C2212yes—
25C2×C3×#95—21no—
26C4×#217—21no—
27C4×#217—21no—
28#4964C418yescomplemented
29C2×#808C2212yes—
30C2×#807C2212yes—
31C4×C4⋊Dic3C2211yescharacteristic
32C4×C4⋊Dic3C2211yescharacteristic
33C4×C4⋊Dic3C2211yescharacteristic
34C4×C4⋊Dic3C2211yescharacteristic
35#4932—21no—
36C42×Dic3—21no—
37#4529C2212yes—
38#4529C2212yes—
39#4529C2212yes—
40#4529C2212yes—
41#4529C2212yes—
42#4529C2212yes—
43#4524C2212yes—
44#4524C2212yes—
45C2×#428—31no—
46C2×#389D311yescharacteristic
47C4×#117—31no—
48#1921—32no—
49#1920—32no—
50#1919—32no—
51#1918—32no—
52#1736—31no—
53#1736—31no—
54#1736—31no—
55#1728—31no—
56C23×C3×C4—21no—
57C23×C3×C4—21no—
58C23×C3×C4D411yescharacteristic
59C23×C3×C4D411yescharacteristic
60C23×C3×C4C2311yescharacteristic
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4—22no—
63C2×C3×C4⋊C4—22no—
64C2×C3×C4⋊C4—22no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4—22no—
67C2×C3×C4⋊C4—22no—
68C2×C3×C4⋊C4—44no—
69C2×C3×C4⋊C4C2×C414yes—
70C2×C3×C4⋊C4C2312yes—
71C2×C3×C4⋊C4C2312yes—
72C2×C3×C42—22no—
73C2×C3×C42—22no—
74C2×C3×C42D411yescharacteristic
75C2×C3×C42D411yescharacteristic
76C2×C3×C42C2311yescharacteristic
77C2×C3×C42C2311yescharacteristic
78C2×#217—21no—
79C2×#217—21no—
80#926—22no—
81#926—22no—
82#926C2×C414yes—
83#926Q814yes—
84C2×C4⋊Dic3C2311yescharacteristic
85C2×C4⋊Dic3C2311yescharacteristic
86C2×C4⋊Dic3C2311yescharacteristic
87C2×C4⋊Dic3C2311yescharacteristic
88C2×C4×Dic3—21no—
89C2×C4×Dic3—21no—
90C2×C4×Dic3—21no—
91C2×C4×Dic3—21no—
92C2×C4×Dic3—21no—
93C2×C4×Dic3—21no—
94C2×#208—22no—
95C2×#208—22no—
96C2×#208—22no—
97C2×#208—22no—
98C2×#208—22no—
99C2×#208C2×C414yes—
100C22×C3⋊C8C2312yes—
101C22×C3⋊C8C2312yes—
102C3×#95—22no—
103C3×#95—22no—
104C3⋊#95—21no—
105C3⋊#95—21no—
106C3⋊#95—21no—
107C3⋊#95—21no—
108#808C2×C418yescomplemented
109#807C2×C418yescomplemented
110C22×M16—31no—
111C22×M16—31no—
112C2×#117—31no—
113C2×#117—31no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4C2×D311yescharacteristic
116C22×C4⋊C4C2×D311yescharacteristic
117C22×C42C2×D311yescharacteristic
118#428—38no—
119C2×#107—32no—
120C2×#106—32no—
121#389—21no—
122#389—21no—
123#389C2×D312yes—
124#389C2×D312yes—
125C4×C4⋊C4—31no—
126C4×C4⋊C4—31no—
127C4×C4⋊C4—31no—
128C4×C4⋊C4—31no—
129C4×C22⋊C4—61no—
130C4×C22⋊C4—61no—
131#376—61no—
132C2×#95—21no—
133C43—61no—
134#340—32no—
135#340—32no—
136#340—32no—
137#340—32no—
138#340—32no—
139#340—32no—
140#336—32no—
141#336—32no—
142C24×C3C2×D411yescharacteristic
143C22×C3×C4—24no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—41no—
151C22×C3×C4—41no—
152C22×C3×C4—42no—
153C22×C3×C4—42no—
154C22×C3×C4—42no—
155C22×C3×C4—42no—
156C22×C3×C4—42no—
157C22×C3×C4C22×C412yes—
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×Q812yes—
164C22×C3×C4#4112yes—
165C22×C3×C4#4112yes—
166C22×C3×C4#4112yes—
167C22×C3×C4#4112yes—
168C22×C3×C4C2411yescharacteristic
169C22×Dic3—21no—
170C22×Dic3—21no—
171C22×Dic3—21no—
172C22×Dic3—21no—
173C3×C4⋊C4—24no—
174C3×C4⋊C4—22no—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—22no—
177C3×C4⋊C4—44no—
178C3×C4⋊C4—42no—
179C3×C4⋊C4—44no—
180C3×C4⋊C4C22×C414yes—
181C3×C42—22no—
182C3×C42—22no—
183C3×C42—21no—
184C3×C42—21no—
185C3×C42—21no—
186C3×C42—21no—
187#217—24no—
188#217—24no—
189C4⋊Dic3—21no—
190C4⋊Dic3—21no—
191C4⋊Dic3—21no—
192C4⋊Dic3—21no—
193C4⋊Dic3C22×C412yes—
194C4⋊Dic3C22×C412yes—
195C4⋊Dic3C2×Q812yes—
196C4⋊Dic3C2×Q812yes—
197C4×Dic3—24no—
198C4×Dic3—24no—
199C4×Dic3—42no—
200C4×Dic3—42no—
201C4×Dic3—42no—
202C4×Dic3—42no—
203#208—28no—
204#208—44no—
205C2×C3⋊C8—22no—
206C2×C3⋊C8—22no—
207C2×C3⋊C8—22no—
208C2×C3⋊C8—22no—
209C2×C3⋊C8—22no—
210C2×C3⋊C8C22×C414yes—
211C23×C4—21no—
212C23×C4—21no—
213C23×C4#6711yescharacteristic
214C23×C4#6711yescharacteristic
215C23×C4C22×D311yescharacteristic
216C2×M16—34no—
217C2×M16—62no—
218C2×M16—62no—
219C2×M16—62no—
220C2×M16—62no—
221C2×M16—62no—
222C22×C8—32no—
223C22×C8—32no—
224#117—34no—
225#117—34no—
226#117—62no—
227#117—62no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—31no—
236C2×C4⋊C4—31no—
237C2×C4⋊C4—31no—
238C2×C4⋊C4—31no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4C4×D314yes—
241C2×C4⋊C4C22×D312yes—
242C2×C4⋊C4C22×D312yes—
243C2×C22⋊C4—61no—
244C2×C22⋊C4—61no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—61no—
248C2×C42—61no—
249C2×C42—61no—
250C2×C42—61no—
251C2×C42—61no—
252C2×C42—61no—
253C2×C42#6711yescharacteristic
254C2×C42#6711yescharacteristic
255C2×C42C22×D311yescharacteristic
256C2×C42C22×D311yescharacteristic
257#107—38no—
258#106—38no—
259#95—22no—
260#95—22no—
261#95—61no—
262#95—61no—
263#95—61no—
264#95—61no—
265C23×C3—21no—
266C23×C3—21no—
267C23×C3—21no—
268C23×C3—21no—
269C23×C3C4×D412yes—
270C23×C3C22⋊Q812yes—
271C23×C3#12312yes—
272C23×C3C22×D411yescharacteristic
273C2×C3×C4—21no—
274C2×C3×C4—21no—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—24no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—44no—
285C2×C3×C4—44no—
286C2×C3×C4—48no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—42no—
290C2×C3×C4—42no—
291C2×C3×C4—42no—
292C2×C3×C4—84no—
293C2×C3×C4#11714yes—
294C2×C3×C4C4×D412yes—
295C2×C3×C4C4×Q814yes—
296C2×C3×C4C22⋊Q812yes—
297C2×C3×C4#12312yes—
298C2×C3×C4#12514yes—
299C2×C3×C4#12614yes—
300C2×C3×C4AGL(1,ℤ/8)12yes—
301C2×C3×C4#13712yes—
302C2×C3×C4C23×C411yescharacteristic
303C2×C3×C4C22×Q811yescharacteristic
304C2×C3×C4C2×#4111yescharacteristic
305C2×C3×C4C2×#4111yescharacteristic
306C2×C3×C4C2×#4111yescharacteristic
307C2×C3×C4C2×#4111yescharacteristic
308C2×Dic3—22no—
309C2×Dic3—22no—
310C2×Dic3—22no—
311C2×Dic3—22no—
312C2×Dic3—41no—
313C2×Dic3—41no—
314C2×Dic3—41no—
315C2×Dic3—41no—
316C2×Dic3—41no—
317C2×Dic3—41no—
318C2×Dic3—41no—
319C2×Dic3—41no—
320C3⋊C8—24no—
321C3⋊C8—42no—
322C24C2×#6711yescharacteristic
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—61no—
338C22×C4—61no—
339C22×C4—61no—
340C22×C4—61no—
341C22×C4C2×C4×D312yes—
342C22×C4#23512yes—
343C22×C4#23712yes—
344C22×C4#23712yes—
345C22×C4D3×Q812yes—
346C22×C4#23912yes—
347C22×C4C2×#6711yescharacteristic
348C22×C4C2×#6711yescharacteristic
349C22×C4C2×#6711yescharacteristic
350C22×C4C2×#6711yescharacteristic
351C22×C4C2×#6711yescharacteristic
352C22×C4C23×D311yescharacteristic
353M16—68no—
354M16—124no—
355C2×C8—34no—
356C2×C8—62no—
357C2×C8—62no—
358C2×C8—62no—
359C2×C8—62no—
360C2×C8—62no—
361C4⋊C4—24no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—32no—
366C4⋊C4—32no—
367C4⋊C4—32no—
368C4⋊C4—32no—
369C4⋊C4—44no—
370C4⋊C4—44no—
371C4⋊C4—42no—
372C4⋊C4—61no—
373C4⋊C4—61no—
374C4⋊C4—61no—
375C4⋊C4—61no—
376C4⋊C4C2×C4×D314yes—
377C22⋊C4—64no—
378C22⋊C4—64no—
379C42—22no—
380C42—22no—
381C42—21no—
382C42—21no—
383C42—21no—
384C42—21no—
385C42—64no—
386C42—64no—
387C42—122no—
388C42—122no—
389C42—122no—
390C42—122no—
391C22×C3—22no—
392C22×C3—22no—
393C22×C3—22no—
394C22×C3—22no—
395C22×C3—22no—
396C22×C3—22no—
397C22×C3#38714yes—
398C22×C3C2×C4×D411yescharacteristic
399C22×C3C2×C22⋊Q811yescharacteristic
400C22×C3C2×#12311yescharacteristic
401C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
402C22×C3C2×#13711yescharacteristic
403C22×C3#57511yescharacteristic
404C22×C3#57511yescharacteristic
405C3×C4—22no—
406C3×C4—22no—
407C3×C4—41no—
408C3×C4—41no—
409C3×C4—44no—
410C3×C4—82no—
411C3×C4#38714yes—
412C3×C4C2×#11711yescharacteristic
413C3×C4C2×C4×Q811yescharacteristic
414C3×C4C2×#12511yescharacteristic
415C3×C4C2×#12611yescharacteristic
416Dic3—42no—
417Dic3—42no—
418Dic3—42no—
419Dic3—42no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23C4×#6712yes—
425C23#93512yes—
426C23#94612yes—
427C23C22×#6711yescharacteristic
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—24no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—48no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—62no—
448C2×C4—62no—
449C2×C4—62no—
450C2×C4—62no—
451C2×C4—84no—
452C2×C4—121no—
453C2×C4—121no—
454C2×C4—121no—
455C2×C4—121no—
456C2×C4—121no—
457C2×C4—121no—
458C2×C4—121no—
459C2×C4—121no—
460C2×C4#88714yes—
461C2×C4#88914yes—
462C2×C4#89214yes—
463C2×C4#89814yes—
464C2×C4C4×#6712yes—
465C2×C4#93212yes—
466C2×C4#93512yes—
467C2×C4#94212yes—
468C2×C4#94612yes—
469C2×C4C22×C4×D311yescharacteristic
470C2×C4C2×#23511yescharacteristic
471C2×C4C2×#23711yescharacteristic
472C2×C4C2×#23711yescharacteristic
473C2×C4C2×D3×Q811yescharacteristic
474C2×C4C2×#23911yescharacteristic
475C8—64no—
476C8—122no—
477C2×C3—24no—
478C2×C3C2×#38711yescharacteristic
479C2×C3C4×AGL(1,ℤ/8)11yescharacteristic
480C2×C3C4×#13711yescharacteristic
481C2×C3#307911yescharacteristic
482C2×C3#308011yescharacteristic
483C2×C3#309411yescharacteristic
484C2×C3#309511yescharacteristic
485C22—22no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22#497814yes—
492C22C2×C4×#6711yescharacteristic
493C22C2×#93211yescharacteristic
494C22C2×#93511yescharacteristic
495C22C2×#94211yescharacteristic
496C22C2×#94611yescharacteristic
497C22#582211yescharacteristic
498C22#582211yescharacteristic
499C4—22no—
500C4—22no—
501C4—41no—
502C4—44no—
503C4—41no—
504C4—82no—
505C4—122no—
506C4—122no—
507C4—122no—
508C4—122no—
509C4#497814yes—
510C4C2×#88711yescharacteristic
511C4C2×#88911yescharacteristic
512C4C2×#89211yescharacteristic
513C4C2×#89811yescharacteristic
514C3#2103411yescharacteristic, complemented, Sylow
515C2—24no—
516C2C2×#497811yescharacteristic
517C2C4×#93211yescharacteristic
518C2C4×#94211yescharacteristic
519C2#8261611yescharacteristic
520C2#8272711yescharacteristic
521C2#8301711yescharacteristic
522C2#8321211yescharacteristic
subgroupG/H
centerC23C22×#67
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC2×C3×#389C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
cosocleC22×C3×C4C24
hypercenterC2×#389D3
hypocenterC3#21034
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
8. #69603C2semidirect
9. #69602C2semidirect
10. #69601C2semidirect
11. #69600C2semidirect
28. #4964C4semidirect
108. #808C2×C4semidirect
109. #807C2×C4semidirect
514. C3#21034semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3892
C2×#49642
C4×#9262
#715582
#715272
#696032
#696022
#696012
#696002
#210343