Finite Group Databasebeta

← #10824669

Subgroups — #10824669

Order 768.

932 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D24C211yesmaximal, characteristic, complemented
2C2×#4690C211yesmaximal, characteristic, complemented
3#73763C212yesmaximal, complemented
4C3×C4×#106C211yesmaximal, characteristic, complemented
5#70585C212yesmaximal, complemented
6#70581C212yesmaximal, complemented
7#69474C212yesmaximal, complemented
8#68886C211yesmaximal, characteristic
9#68862C211yesmaximal, characteristic, complemented
10#68838C211yesmaximal, characteristic, complemented
11#68830C211yesmaximal, characteristic, complemented
12#23702—31nomaximal, Sylow
13C22×D24C2211yescharacteristic
14C2×#895C2212yes—
15C2×C4×D12C2212yes—
16C2×#872C2211yescharacteristic
17C2×C3×#106C2212yes—
18C2×C3×C4×C8C2211yescharacteristic
19C3×C4×C4⋊C4C2212yes—
20C2×#821C2212yes—
21C2×(C2×D3)⋊C8C2211yescharacteristic
22C2×(C2×D3)⋊C8C2211yescharacteristic
23C2×#818C2211yescharacteristic
24C2×Dic3⋊C8C2211yescharacteristic
25C2×Dic3⋊C8C2211yescharacteristic
26#4988C2212yes—
27#4987C2212yes—
28#4984C2212yes—
29C2×#809C2212yes—
30C2×#809C2212yes—
31#4940—22no—
32#4939—22no—
33C2×#803C2211yescharacteristic
34C4×D24—24no—
35#4690—24no—
36#4668—22no—
37#4665—22no—
38C3×#340C2211yescharacteristic
39C3×#340C2211yescharacteristic
40#4529C2212yes—
41#4529C2212yes—
42C2×C4×D8—31no—
43C2×#433—31no—
44#2324—32no—
45#2086—31no—
46#2075—31no—
47#2065—32no—
48#2054—31no—
49#2034—32no—
50#1999—31no—
51C4×#106D311yescharacteristic, complemented
52#1762—32no—
53C22×D12—22no—
54C22×D12C2312yes—
55C22×C4×D3—22no—
56C22×C4×D3C2311yescharacteristic
57C22×C3×C8C2311yescharacteristic
58C22×C3×C8C2311yescharacteristic
59C2×C3×C4⋊C4C2312yes—
60C2×C3×C4⋊C4C2312yes—
61C2×C3×C42—22no—
62C2×C3×C42C2311yescharacteristic
63C2×(C2×D3)⋊C4—22no—
64C2×(C2×D3)⋊C4—22no—
65C2×(C2×D3)⋊C4—22no—
66C2×(C2×D3)⋊C4—22no—
67C2×(C2×D3)⋊C4—22no—
68C2×(C2×D3)⋊C4D411yescharacteristic
69C2×(C2×D3)⋊C4D411yescharacteristic
70C2×C4⋊Dic3—22no—
71C2×C4⋊Dic3C2312yes—
72C2×Dic3⋊C4D411yescharacteristic
73C2×Dic3⋊C4D411yescharacteristic
74C2×C4×Dic3C2311yescharacteristic
75C22×C3⋊C8C2311yescharacteristic
76C22×C3⋊C8C2311yescharacteristic
77C2×D24—24no—
78C2×D24—22no—
79#895—24no—
80#895—24no—
81C4×D12—24no—
82C4×D12—24no—
83#872—22no—
84#872—22no—
85C3×#106—24no—
86C3×C4×C8D414yescomplemented
87C3×#95—22no—
88C3⋊#95—22no—
89C3⋊#95—22no—
90#821—24no—
91(C2×D3)⋊C8—22no—
92(C2×D3)⋊C8—22no—
93#818—22no—
94Dic3⋊C8—22no—
95Dic3⋊C8—22no—
96#809—24no—
97#809—24no—
98#803—22no—
99C22×D8—31no—
100C2×#121—32no—
101C2×C4×D4—32no—
102C2×#117—31no—
103C4×D8—64no—
104#433—64no—
105C2×#107—31no—
106C2×#106C2×D312yes—
107C2×C4⋊C8—31no—
108C2×C4⋊C8—31no—
109C2×#102—32no—
110C2×#102—32no—
111C2×#102—32no—
112C2×C22⋊C8—31no—
113C2×C22⋊C8—31no—
114C2×#97—31no—
115C2×C4×C8C2×D311yescharacteristic
116#390—32no—
117#390—62no—
118#389—32no—
119#388—62no—
120#386—62no—
121#385—62no—
122#380—32no—
123C4×C4⋊C4C2×D312yes—
124#340—32no—
125#340—32no—
126#340C2×D311yescharacteristic
127#340C2×D311yescharacteristic
128C23×D3—22no—
129C23×D3C2×D411yescharacteristic
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2411yescharacteristic
135C22×Dic3—22no—
136C22×Dic3C2×D411yescharacteristic
137C22×Dic3C2×D411yescharacteristic
138C22×Dic3C2×D411yescharacteristic
139C2×D12—22no—
140C2×D12—22no—
141C2×D12—22no—
142C2×D12—22no—
143C2×D12—22no—
144C2×D12—28no—
145C2×D12—82no—
146C2×D12—82no—
147C2×D12#4114yes—
148C2×C4×D3—21no—
149C2×C4×D3—21no—
150C2×C4×D3—22no—
151C2×C4×D3—22no—
152C2×C4×D3—44no—
153C2×C4×D3—82no—
154C2×C4×D3—82no—
155C2×C3×C8—22no—
156C2×C3×C8—22no—
157C2×C3×C8C2×D412yes—
158C2×C3×C8C2×D412yes—
159C3×C4⋊C4—22no—
160C3×C4⋊C4—22no—
161C3×C4⋊C4—22no—
162C3×C4⋊C4#4114yes—
163C3×C42—82no—
164C3×C42C2×D412yes—
165C3×C42#4112yes—
166(C2×D3)⋊C4—28no—
167(C2×D3)⋊C4—22no—
168(C2×D3)⋊C4—22no—
169(C2×D3)⋊C4—44no—
170(C2×D3)⋊C4—82no—
171(C2×D3)⋊C4—82no—
172(C2×D3)⋊C4—82no—
173C4⋊Dic3—22no—
174C4⋊Dic3—22no—
175C4⋊Dic3—82no—
176Dic3⋊C4—22no—
177Dic3⋊C4—22no—
178C4×Dic3—21no—
179C4×Dic3—21no—
180C2×C3⋊C8—21no—
181C2×C3⋊C8—21no—
182C2×C3⋊C8—22no—
183C2×C3⋊C8—22no—
184D24—28no—
185C22×D4—32no—
186C22×D4—62no—
187C23×C4—31no—
188C23×C4—62no—
189C2×D8—64no—
190C2×D8—62no—
191C22×C8—31no—
192C22×C8—31no—
193C22×C8C22×D311yescharacteristic
194C22×C8C22×D311yescharacteristic
195#121—64no—
196#121—64no—
197C4×D4—64no—
198C4×D4—64no—
199#117—62no—
200#117—62no—
201C2×C4⋊C4—31no—
202C2×C4⋊C4—31no—
203C2×C4⋊C4—32no—
204C2×C4⋊C4—62no—
205C2×C4⋊C4C22×D312yes—
206C2×C4⋊C4C22×D312yes—
207C2×C22⋊C4—31no—
208C2×C22⋊C4—31no—
209C2×C22⋊C4—62no—
210C2×C22⋊C4—62no—
211C2×C22⋊C4—62no—
212C2×C22⋊C4—62no—
213C2×C22⋊C4—62no—
214C2×C42—22no—
215C2×C42—31no—
216C2×C42C22×D311yescharacteristic
217#107—62no—
218#106—24no—
219C4⋊C8—62no—
220C4⋊C8—62no—
221#102—64no—
222#102—64no—
223#102—64no—
224C22⋊C8—62no—
225C22⋊C8—62no—
226#97—62no—
227C4×C8D1214yescomplemented
228#95—22no—
229#95—62no—
230#95—62no—
231C23×C3C22×D411yescharacteristic
232C22×D3—21no—
233C22×D3—24no—
234C22×D3—22no—
235C22×D3—21no—
236C22×D3—21no—
237C22×D3—22no—
238C22×D3—42no—
239C22×D3—42no—
240C22×D3—42no—
241C22×D3—42no—
242C22×D3—42no—
243C22×D3—42no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—22no—
247C2×C3×C4—24no—
248C2×C3×C4—42no—
249C2×C3×C4—42no—
250C2×C3×C4—42no—
251C2×C3×C4—42no—
252C2×C3×C4—42no—
253C2×C3×C4#12112yes—
254C2×C3×C4#12112yes—
255C2×C3×C4C22×D411yescharacteristic
256C2×C3×C4C2×#4112yes—
257C2×C3×C4C2×#4111yescharacteristic
258C2×C3×C4C2×#4112yes—
259C2×Dic3—21no—
260C2×Dic3—21no—
261C2×Dic3—21no—
262C2×Dic3—22no—
263C2×Dic3—21no—
264C2×Dic3—21no—
265C2×Dic3—22no—
266C2×Dic3—42no—
267C2×Dic3—42no—
268C2×Dic3—42no—
269D12—24no—
270D12—24no—
271D12—42no—
272D12—42no—
273D12—88no—
274C4×D3—41no—
275C4×D3—42no—
276C4×D3—41no—
277C4×D3—88no—
278C3×C8—22no—
279C3×C8—22no—
280C3⋊C8—41no—
281C3⋊C8—41no—
282C24—31no—
283C24—62no—
284C2×D4—34no—
285C2×D4—68no—
286C2×D4—62no—
287C2×D4—62no—
288C2×D4—62no—
289C2×D4—62no—
290C2×D4—62no—
291C2×D4—242no—
292C2×D4—242no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—31no—
296C22×C4—31no—
297C22×C4—31no—
298C22×C4—61no—
299C22×C4—62no—
300C22×C4—61no—
301C22×C4—62no—
302C22×C4—62no—
303C22×C4—124no—
304C22×C4—242no—
305C22×C4—242no—
306C22×C4D3×D411yescharacteristic
307C22×C4D3×D411yescharacteristic
308C22×C4C23×D311yescharacteristic
309D8—68no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—61no—
313C2×C8—61no—
314C2×C8—62no—
315C2×C8—62no—
316C2×C8C2×D1212yes—
317C2×C8C2×D1212yes—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—62no—
322C4⋊C4—62no—
323C4⋊C4—62no—
324C4⋊C4—62no—
325C4⋊C4—242no—
326C4⋊C4#23514yes—
327C22⋊C4—68no—
328C22⋊C4—62no—
329C22⋊C4—62no—
330C22⋊C4—124no—
331C22⋊C4—242no—
332C22⋊C4—242no—
333C22⋊C4—242no—
334C42—61no—
335C42—61no—
336C42—82no—
337C42C2×D1212yes—
338C42#23912yes—
339C22×C3C2×#12111yescharacteristic
340C22×C3#52512yes—
341C22×C3#57611yescharacteristic
342C22×C3#57611yescharacteristic
343C22×C3#57712yes—
344C2×D3—42no—
345C2×D3—44no—
346C2×D3—42no—
347C2×D3—44no—
348C2×D3—41no—
349C2×D3—44no—
350C2×D3—41no—
351C2×D3—41no—
352C2×D3—82no—
353C2×D3—82no—
354C2×D3—82no—
355C2×D3—82no—
356C3×C4—22no—
357C3×C4—22no—
358C3×C4—44no—
359C3×C4—82no—
360C3×C4#52511yescharacteristic
361C3×C4#52912yes—
362C3×C4#53211yescharacteristic
363Dic3—41no—
364Dic3—41no—
365Dic3—41no—
366Dic3—82no—
367C23—61no—
368C23—62no—
369C23—61no—
370C23—61no—
371C23—64no—
372C23—62no—
373C23—122no—
374C23—122no—
375C23—122no—
376C23—122no—
377C23—122no—
378C23—122no—
379C23C2×D3×D411yescharacteristic
380D4—64no—
381D4—64no—
382D4—122no—
383D4—122no—
384D4—248no—
385C2×C4—22no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—24no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—61no—
395C2×C4—61no—
396C2×C4—61no—
397C2×C4—62no—
398C2×C4—61no—
399C2×C4—61no—
400C2×C4—62no—
401C2×C4—121no—
402C2×C4—121no—
403C2×C4—122no—
404C2×C4—122no—
405C2×C4—122no—
406C2×C4—122no—
407C2×C4—248no—
408C2×C4#89512yes—
409C2×C4#89512yes—
410C2×C4C22×D1211yescharacteristic
411C2×C4C2×#23512yes—
412C2×C4C2×#23911yescharacteristic
413C2×C4D3×#4112yes—
414C8—22no—
415C8—22no—
416C8—121no—
417C8—121no—
418C2×C3#239211yescharacteristic
419C2×C3#305811yescharacteristic
420C2×C3#306011yescharacteristic
421C2×C3#311812yes—
422C2×C3#311912yes—
423D3—81no—
424D3—84no—
425C22—121no—
426C22—121no—
427C22—124no—
428C22—121no—
429C22—122no—
430C22—124no—
431C22—122no—
432C22—124no—
433C22—242no—
434C22—242no—
435C22—242no—
436C22—242no—
437C22C2×#89511yescharacteristic
438C22#554712yes—
439C22#576212yes—
440C22#576911yescharacteristic
441C22#576911yescharacteristic
442C4—22no—
443C4—22no—
444C4—44no—
445C4—82no—
446C4—121no—
447C4—121no—
448C4—121no—
449C4—242no—
450C4#547711yescharacteristic
451C4#552411yescharacteristic
452C4#558012yes—
453C3#2370211yescharacteristic, complemented, Sylow
454C2—241no—
455C2—244no—
456C2#7377611yescharacteristic
457C2#7912011yescharacteristic
458C2#7918311yescharacteristic
459C2#7969612yes—
460C2#7969712yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×#106C2
socleC23×C3C22×D4
prime radical#68886C2
hypercenterC4×#106D3
hypocenterC3#23702
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#106

Complements

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

NH ≅ G/Ntype
1. C2×C4×D24C2semidirect
2. C2×#4690C2semidirect
3. #73763C2semidirect
4. C3×C4×#106C2semidirect
5. #70585C2semidirect
6. #70581C2semidirect
7. #69474C2semidirect
9. #68862C2semidirect
10. #68838C2semidirect
11. #68830C2semidirect
51. C4×#106D3semidirect
86. C3×C4×C8D4semidirect
227. C4×C8D12semidirect
453. C3#23702semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D242
C2×#46902
#737632
C3×C4×#1062
#705852
#705812
#694742
#688862
#688622
#688382
#688302
#237023