Finite Group Databasebeta

← #10824664

Subgroups — #10824664

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×Dic3×D8C211yesmaximal, characteristic, complemented
2#82633C212yesmaximal, complemented
3C2×#4720C211yesmaximal, characteristic, complemented
4C3×#1999C211yesmaximal, characteristic
5#72040C211yesmaximal, characteristic, complemented
6#71680C212yesmaximal, complemented
7#70193C212yesmaximal, complemented
8#70152C212yesmaximal, complemented
9#69938C211yesmaximal, characteristic, complemented
10#68838C211yesmaximal, characteristic, complemented
11#68784C211yesmaximal, characteristic, complemented
12#23702—31nomaximal, Sylow
13C22×C3×D8C2211yescharacteristic
14C2×C3×#121C2212yes—
15C2×D4×Dic3C2212yes—
16C2×#926C2211yescharacteristic
17C2×C3×#102C2212yes—
18C2×C3×C22⋊C8C2211yescharacteristic
19#5221—22no—
20#5221C2212yes—
21#5220—22no—
22C2×#832C2212yes—
23C2×#832C2212yes—
24#5212C2212yes—
25C2×#830C2211yescharacteristic
26Dic3×D8—24no—
27C2×#818C2211yescharacteristic
28C2×#815C2211yescharacteristic
29C2×Dic3⋊C8C2211yescharacteristic
30C2×Dic3⋊C8C2211yescharacteristic
31C2×C8×Dic3C2211yescharacteristic
32#4984C2212yes—
33Dic3×C4⋊C4C2212yes—
34C2×#808C2212yes—
35#4951—22no—
36#4950—22no—
37#4720—24no—
38C3×#340C2211yescharacteristic
39#4551C2212yes—
40#4551C2211yescharacteristic
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#1999D311yescharacteristic
51C4×#106—31no—
52#1762—32no—
53C22×C3×D4—22no—
54C22×C3×D4C2312yes—
55C23×C3×C4C2311yescharacteristic
56C23×Dic3—22no—
57C2×C3×D8—24no—
58C2×C3×D8—22no—
59C22×C3×C8C2311yescharacteristic
60C22×C3×C8C2311yescharacteristic
61C3×#121—24no—
62C3×#121—24no—
63C2×C3×C4⋊C4C2312yes—
64C2×C3×C22⋊C4—22no—
65C2×#217—22no—
66C2×#217—22no—
67C2×#217—22no—
68C2×#217—22no—
69C2×#217D411yescharacteristic
70C2×#217D411yescharacteristic
71D4×Dic3—24no—
72D4×Dic3—24no—
73#926—22no—
74#926—22no—
75C2×C4⋊Dic3—22no—
76C2×C4⋊Dic3D411yescharacteristic
77C2×C4⋊Dic3D411yescharacteristic
78C2×C4⋊Dic3C2312yes—
79C2×C4⋊Dic3C2312yes—
80C2×C4×Dic3—22no—
81C2×C4×Dic3C2311yescharacteristic
82C2×C4×Dic3C2311yescharacteristic
83C22×C3⋊C8C2311yescharacteristic
84C22×C3⋊C8C2311yescharacteristic
85C3×#102—24no—
86C3×C22⋊C8—22no—
87#832—24no—
88#832—24no—
89C3⋊#95—22no—
90C3⋊#95—22no—
91C3⋊#95—22no—
92#830—22no—
93#818—22no—
94#815—22no—
95Dic3⋊C8—22no—
96Dic3⋊C8—22no—
97C8×Dic3D414yescomplemented
98#808—24no—
99C22×D8C2×D311yescharacteristic
100C2×#121C2×D312yes—
101C2×C4×D4—32no—
102C2×#117—31no—
103C4×D8—64no—
104#433—64no—
105C2×#107—31no—
106C2×#106—32no—
107C2×C4⋊C8—31no—
108C2×C4⋊C8—31no—
109C2×#102—32no—
110C2×#102—32no—
111C2×#102C2×D312yes—
112C2×C22⋊C8—31no—
113C2×C22⋊C8C2×D311yescharacteristic
114C2×#97—31no—
115C2×C4×C8—31no—
116#390—32no—
117#390—62no—
118#389—32no—
119#388—62no—
120#386—62no—
121#385—62no—
122#380—32no—
123C4×C4⋊C4—32no—
124#340—31no—
125#340—32no—
126#340—32no—
127#340C2×D311yescharacteristic
128C24×C3—22no—
129C24×C3C2×D411yescharacteristic
130C2×C3×D4—28no—
131C2×C3×D4—22no—
132C2×C3×D4—22no—
133C2×C3×D4—22no—
134C2×C3×D4—22no—
135C2×C3×D4—22no—
136C2×C3×D4—82no—
137C2×C3×D4—82no—
138C2×C3×D4#4114yes—
139C22×C3×C4—22no—
140C22×C3×C4—21no—
141C22×C3×C4—21no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4C2411yescharacteristic
146C22×Dic3—22no—
147C22×Dic3—22no—
148C22×Dic3—44no—
149C22×Dic3—82no—
150C22×Dic3—82no—
151C22×Dic3C2×D411yescharacteristic
152C22×Dic3C2×D411yescharacteristic
153C22×Dic3C2×D411yescharacteristic
154C22×Dic3C2×D411yescharacteristic
155C3×D8—28no—
156C2×C3×C8—22no—
157C2×C3×C8—21no—
158C2×C3×C8—22no—
159C2×C3×C8C2×D412yes—
160C3×C4⋊C4—22no—
161C3×C4⋊C4#4114yes—
162C3×C22⋊C4—28no—
163#217—22no—
164#217—22no—
165#217—44no—
166#217—82no—
167#217—82no—
168#217—82no—
169C4⋊Dic3—22no—
170C4⋊Dic3—22no—
171C4⋊Dic3—22no—
172C4⋊Dic3—22no—
173C4⋊Dic3—22no—
174C4⋊Dic3—22no—
175C4⋊Dic3—82no—
176C4×Dic3—21no—
177C4×Dic3—21no—
178C4×Dic3—82no—
179C4×Dic3C2×D412yes—
180C4×Dic3#4112yes—
181C2×C3⋊C8—22no—
182C2×C3⋊C8—21no—
183C2×C3⋊C8—22no—
184C2×C3⋊C8C2×D412yes—
185C22×D4—22no—
186C22×D4C22×D312yes—
187C23×C4—62no—
188C23×C4C22×D311yescharacteristic
189C2×D8—24no—
190C2×D8—22no—
191C22×C8—31no—
192C22×C8—31no—
193C22×C8C22×D311yescharacteristic
194C22×C8C22×D311yescharacteristic
195#121—24no—
196#121—24no—
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—32no—
205C2×C4⋊C4—62no—
206C2×C4⋊C4C22×D312yes—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—31no—
209C2×C22⋊C4—31no—
210C2×C22⋊C4—62no—
211C2×C22⋊C4—62no—
212C2×C22⋊C4—62no—
213C2×C22⋊C4—62no—
214C2×C42—31no—
215C2×C42—31no—
216C2×C42—62no—
217#107—62no—
218#106—64no—
219C4⋊C8—62no—
220C4⋊C8—62no—
221#102—24no—
222#102—64no—
223#102—64no—
224C22⋊C8—22no—
225C22⋊C8—62no—
226#97—62no—
227C4×C8—34no—
228#95—62no—
229#95—62no—
230#95—62no—
231C23×C3—21no—
232C23×C3—24no—
233C23×C3—22no—
234C23×C3—21no—
235C23×C3—21no—
236C23×C3—22no—
237C23×C3—42no—
238C23×C3—42no—
239C23×C3—42no—
240C23×C3—42no—
241C23×C3—42no—
242C23×C3—42no—
243C23×C3C22×D411yescharacteristic
244C3×D4—24no—
245C3×D4—24no—
246C3×D4—42no—
247C3×D4—42no—
248C3×D4—88no—
249C2×C3×C4—21no—
250C2×C3×C4—22no—
251C2×C3×C4—24no—
252C2×C3×C4—42no—
253C2×C3×C4—42no—
254C2×C3×C4—41no—
255C2×C3×C4—42no—
256C2×C3×C4—41no—
257C2×C3×C4C22×D411yescharacteristic
258C2×C3×C4C2×#4112yes—
259C2×C3×C4C2×#4112yes—
260C2×C3×C4C2×#4111yescharacteristic
261C2×Dic3—21no—
262C2×Dic3—21no—
263C2×Dic3—22no—
264C2×Dic3—21no—
265C2×Dic3—21no—
266C2×Dic3—21no—
267C2×Dic3—21no—
268C2×Dic3—22no—
269C2×Dic3—42no—
270C2×Dic3—42no—
271C2×Dic3—42no—
272C2×Dic3—42no—
273C2×Dic3—42no—
274C2×Dic3—42no—
275C2×Dic3—88no—
276C2×Dic3#12112yes—
277C2×Dic3#12112yes—
278C3×C8—22no—
279C3×C8—41no—
280C3⋊C8—22no—
281C3⋊C8—41no—
282C24—22no—
283C24D3×D411yescharacteristic
284C2×D4—28no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—82no—
291C2×D4—82no—
292C2×D4#23714yes—
293C22×C4—22no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—21no—
298C22×C4—31no—
299C22×C4—31no—
300C22×C4—31no—
301C22×C4—31no—
302C22×C4—62no—
303C22×C4—62no—
304C22×C4—124no—
305C22×C4—242no—
306C22×C4—242no—
307C22×C4D3×D411yescharacteristic
308C22×C4C23×D311yescharacteristic
309D8—28no—
310C2×C8—22no—
311C2×C8—21no—
312C2×C8—22no—
313C2×C8—32no—
314C2×C8—62no—
315C2×C8—61no—
316C2×C8—62no—
317C2×C8D3×D412yes—
318C4⋊C4—22no—
319C4⋊C4—62no—
320C4⋊C4—62no—
321C4⋊C4—62no—
322C4⋊C4—62no—
323C4⋊C4—62no—
324C4⋊C4—62no—
325C4⋊C4—242no—
326C4⋊C4#23714yes—
327C22⋊C4—28no—
328C22⋊C4—62no—
329C22⋊C4—62no—
330C22⋊C4—124no—
331C22⋊C4—242no—
332C22⋊C4—242no—
333C22⋊C4—242no—
334C42—32no—
335C42—32no—
336C42—61no—
337C42—61no—
338C42—242no—
339C22×C3—42no—
340C22×C3—44no—
341C22×C3—42no—
342C22×C3—44no—
343C22×C3—41no—
344C22×C3—44no—
345C22×C3—41no—
346C22×C3—41no—
347C22×C3—82no—
348C22×C3—82no—
349C22×C3—82no—
350C22×C3—82no—
351C22×C3C2×#12111yescharacteristic
352C22×C3#52512yes—
353C22×C3#57611yescharacteristic
354C22×C3#57611yescharacteristic
355C22×C3#57712yes—
356C3×C4—44no—
357C3×C4—41no—
358C3×C4#52511yescharacteristic
359C3×C4#52912yes—
360C3×C4#53211yescharacteristic
361Dic3—22no—
362Dic3—22no—
363Dic3—41no—
364Dic3—41no—
365Dic3—82no—
366Dic3—82no—
367C23—21no—
368C23—21no—
369C23—22no—
370C23—24no—
371C23—22no—
372C23—21no—
373C23—42no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23C2×D3×D411yescharacteristic
380D4—24no—
381D4—24no—
382D4—42no—
383D4—42no—
384D4—88no—
385C2×C4—22no—
386C2×C4—24no—
387C2×C4—21no—
388C2×C4—32no—
389C2×C4—32no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—61no—
396C2×C4—61no—
397C2×C4—62no—
398C2×C4—61no—
399C2×C4—61no—
400C2×C4—62no—
401C2×C4—61no—
402C2×C4—61no—
403C2×C4—122no—
404C2×C4—122no—
405C2×C4—122no—
406C2×C4—122no—
407C2×C4—122no—
408C2×C4—122no—
409C2×C4—248no—
410C2×C4C2×D3×D411yescharacteristic
411C2×C4C2×#23712yes—
412C2×C4C2×#23712yes—
413C2×C4D3×#4111yescharacteristic
414C8—22no—
415C8—41no—
416C8—62no—
417C8—121no—
418C2×C3—81no—
419C2×C3—84no—
420C2×C3#239211yescharacteristic
421C2×C3#305811yescharacteristic
422C2×C3#306011yescharacteristic
423C2×C3#311812yes—
424C2×C3#311912yes—
425C22—41no—
426C22—42no—
427C22—44no—
428C22—41no—
429C22—44no—
430C22—42no—
431C22—44no—
432C22—41no—
433C22—82no—
434C22—82no—
435C22—82no—
436C22—82no—
437C22#558612yes—
438C22D3×#12111yescharacteristic
439C22#575711yescharacteristic
440C22#576911yescharacteristic
441C22#577512yes—
442C4—44no—
443C4—41no—
444C4—62no—
445C4—62no—
446C4—121no—
447C4—121no—
448C4—242no—
449C4—242no—
450C4#558611yescharacteristic
451C4#560012yes—
452C4#571611yescharacteristic
453C3#2370211yescharacteristic, complemented, Sylow
454C2—81no—
455C2—84no—
456C2#7666611yescharacteristic
457C2#7999212yes—
458C2#8007212yes—
459C2#8071611yescharacteristic
460C2#8086611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#1999C2
socleC23×C3C22×D4
hypercenter#1999D3
hypocenterC3#23702
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1999

Complements

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

NH ≅ G/Ntype
1. C2×Dic3×D8C2semidirect
2. #82633C2semidirect
3. C2×#4720C2semidirect
5. #72040C2semidirect
6. #71680C2semidirect
7. #70193C2semidirect
8. #70152C2semidirect
9. #69938C2semidirect
10. #68838C2semidirect
11. #68784C2semidirect
97. C8×Dic3D4semidirect
453. C3#23702semidirect

Maximal subgroups

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

subgroupindex
C2×Dic3×D82
#826332
C2×#47202
C3×#19992
#720402
#716802
#701932
#701522
#699382
#688382
#687842
#237023