Finite Group Databasebeta

← #10849320

Subgroups — #10849320

Order 768.

1076 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5713C211yesmaximal, characteristic
2C2×#5071C211yesmaximal, characteristic
3C2×#4829C211yesmaximal, characteristic, complemented
4#73727C211yesmaximal, characteristic, complemented
5C3×#1982C211yesmaximal, characteristic
6#70473C211yesmaximal, characteristic, complemented
7#70178C212yesmaximal, complemented
8#70178C212yesmaximal, complemented
9#70170C212yesmaximal, complemented
10#70168C212yesmaximal, complemented
11#69475C211yesmaximal, characteristic, complemented
12#25658—31nomaximal, Sylow
13C2×C3×#127C2211yescharacteristic
14C2×#936—21no—
15C2×#936C2212yes—
16C2×#935—22no—
17C2×D4×Dic3—21no—
18C2×D4×Dic3C2212yes—
19#5713—24no—
20#5713—24no—
21C2×#870C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C2×C4×Dic6C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×#102C2212yes—
26C2×C3×#102C2212yes—
27C3×C4×C4⋊C4C2211yescharacteristic
28C2×#832C2212yes—
29C2×#832C2212yes—
30C2×#818C2211yescharacteristic
31C2×#817C2211yescharacteristic
32C2×#816C2211yescharacteristic
33C2×#816C2211yescharacteristic
34#5071—24no—
35#4984C2212yes—
36#4984C2212yes—
37C2×#810C2212yes—
38#4928—22no—
39C2×#803C2211yescharacteristic
40#4829—24no—
41#4661—21no—
42#4656—21no—
43#4550C2212yes—
44#4550C2212yes—
45#4529C2212yes—
46C2×#559—31no—
47C2×#490—31no—
48C2×#479—31no—
49#2584—31no—
50#2042—32no—
51#2042—32no—
52#1985—31no—
53#1982D311yescharacteristic
54#1864—32no—
55#1862—32no—
56#1763—31no—
57C22×C3×D4—21no—
58C22×C3×D4—21no—
59C22×C3×D4C2312yes—
60C23×Dic3—22no—
61C22×Dic6D411yescharacteristic
62C22×Dic6C2311yescharacteristic
63C22×C3×C8C2311yescharacteristic
64C22×C3×C8C2311yescharacteristic
65C3×#127—22no—
66C3×#127D414yes—
67C2×C3×C4⋊C4C2312yes—
68C2×C3×C42—21no—
69C2×C3×C42C2311yescharacteristic
70C2×#217—22no—
71C2×#217—22no—
72C2×#217—22no—
73#936—28no—
74#936—24no—
75#936—24no—
76#935—216no—
77D4×Dic3—28no—
78D4×Dic3—24no—
79D4×Dic3—24no—
80C2×C4⋊Dic3—22no—
81C2×C4⋊Dic3—22no—
82C2×C4⋊Dic3D411yescharacteristic
83C2×C4⋊Dic3C2311yescharacteristic
84C2×C4⋊Dic3C2311yescharacteristic
85C2×C4⋊Dic3C2311yescharacteristic
86C2×Dic3⋊C4—21no—
87C2×Dic3⋊C4—21no—
88C2×Dic3⋊C4—21no—
89C2×Dic3⋊C4D412yes—
90C2×C4×Dic3—21no—
91C2×C4×Dic3C2312yes—
92C22×C3⋊C8C2312yes—
93#870—22no—
94#870—22no—
95C4×Dic6—22no—
96C4×Dic6—22no—
97C4×Dic6—22no—
98C4×Dic6D414yes—
99C3×C4⋊C8—22no—
100C3×#102—24no—
101C3×#102—24no—
102C3×#95—21no—
103#832—24no—
104#832—24no—
105C3⋊#95—22no—
106C3⋊#95—22no—
107#818—22no—
108#817—22no—
109#816—22no—
110#816—22no—
111#810—24no—
112#803—22no—
113#559—64no—
114#559—64no—
115C2×#127C2×D311yescharacteristic
116C2×#125—31no—
117C2×#124—32no—
118C2×#124—61no—
119C2×#123—62no—
120C2×C4×Q8—31no—
121C2×C4×Q8—31no—
122C2×C4×D4—32no—
123C2×C4×D4—61no—
124#490—64no—
125#479—64no—
126C2×#107—31no—
127C2×#106—31no—
128C2×C4⋊C8C2×D311yescharacteristic
129C2×#103—31no—
130C2×#103—31no—
131C2×#103—32no—
132C2×#102—32no—
133C2×#102—32no—
134C2×#102C2×D312yes—
135C2×#102C2×D312yes—
136C2×#97—31no—
137#398—61no—
138#395—61no—
139#389—32no—
140#389—32no—
141#389—62no—
142C4×C4⋊C4C2×D311yescharacteristic
143#340—32no—
144#336—32no—
145#336—32no—
146C24×C3—22no—
147C2×C3×D4—24no—
148C2×C3×D4—24no—
149C2×C3×D4—22no—
150C2×C3×D4—22no—
151C2×C3×D4—22no—
152C2×C3×D4—22no—
153C2×C3×D4—22no—
154C2×C3×D4—44no—
155C2×C3×D4—81no—
156C2×C3×D4—81no—
157C2×C3×D4C2×D414yes—
158C22×C3×C4—22no—
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2411yescharacteristic
162C22×Dic3—21no—
163C22×Dic3—21no—
164C22×Dic3—28no—
165C22×Dic3—21no—
166C22×Dic3—21no—
167C22×Dic3—82no—
168C22×Dic3—82no—
169C22×Dic3C2×D412yes—
170C22×Dic3C2×D412yes—
171C2×Dic6—22no—
172C2×Dic6—22no—
173C2×Dic6—22no—
174C2×Dic6—21no—
175C2×Dic6—42no—
176C2×Dic6C2×D412yes—
177C2×C3×C8—21no—
178C2×C3×C8—21no—
179C2×C3×C8—22no—
180C2×C3×C8—22no—
181C3×C4⋊C4—22no—
182C3×C4⋊C4#4114yes—
183C3×C42—81no—
184C3×C42C2×D412yes—
185C3×C42C2×D412yes—
186#217—28no—
187#217—28no—
188#217—28no—
189C4⋊Dic3—22no—
190C4⋊Dic3—21no—
191C4⋊Dic3—21no—
192C4⋊Dic3—21no—
193C4⋊Dic3—21no—
194C4⋊Dic3—21no—
195C4⋊Dic3—44no—
196C4⋊Dic3—82no—
197C4⋊Dic3C2×D412yes—
198Dic3⋊C4—24no—
199Dic3⋊C4—24no—
200Dic3⋊C4—42no—
201Dic3⋊C4—42no—
202C4×Dic3—24no—
203C4×Dic3—22no—
204C4×Dic3—22no—
205C2×C3⋊C8—22no—
206C2×C3⋊C8—22no—
207C2×C3⋊C8—22no—
208C22×Q8—31no—
209C22×Q8—31no—
210C22×D4—21no—
211C22×D4—21no—
212C22×D4C22×D312yes—
213C23×C4—62no—
214C22×C8—32no—
215C22×C8C22×D311yescharacteristic
216C22×C8C22×D311yescharacteristic
217#127—22no—
218#127D1214yes—
219#125—62no—
220#125—62no—
221#124—64no—
222#124—68no—
223#124—64no—
224#123—616no—
225C4×Q8—34no—
226C4×Q8—62no—
227C4×Q8—62no—
228C4×Q8—62no—
229C4×D4—64no—
230C4×D4—68no—
231C4×D4—64no—
232C2×C4⋊C4—31no—
233C2×C4⋊C4—31no—
234C2×C4⋊C4—32no—
235C2×C4⋊C4—31no—
236C2×C4⋊C4—31no—
237C2×C4⋊C4—61no—
238C2×C4⋊C4—62no—
239C2×C4⋊C4—61no—
240C2×C4⋊C4—61no—
241C2×C4⋊C4—62no—
242C2×C4⋊C4C22×D312yes—
243C2×C22⋊C4—62no—
244C2×C22⋊C4—62no—
245C2×C22⋊C4—62no—
246C2×C42—21no—
247C2×C42—32no—
248C2×C42—61no—
249C2×C42C22×D311yescharacteristic
250#107—62no—
251#106—62no—
252C4⋊C8—22no—
253#103—62no—
254#103—62no—
255#103—64no—
256#102—24no—
257#102—24no—
258#102—64no—
259#102—64no—
260#97—62no—
261#95—21no—
262#95—62no—
263#95—62no—
264C23×C3—24no—
265C23×C3—42no—
266C23×C3—42no—
267C23×C3—42no—
268C23×C3—42no—
269C23×C3—42no—
270C23×C3—42no—
271C23×C3C22×D411yescharacteristic
272C3×D4—24no—
273C3×D4—24no—
274C3×D4—48no—
275C3×D4—42no—
276C3×D4—42no—
277C3×D4—84no—
278C2×C3×C4—22no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—24no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4#12012yes—
285C2×C3×C4#12012yes—
286C2×C3×C4C22×D411yescharacteristic
287C2×C3×C4C22×D411yescharacteristic
288C2×C3×C4C2×#4112yes—
289C2×C3×C42−1+412yes—
290C2×Dic3—22no—
291C2×Dic3—22no—
292C2×Dic3—22no—
293C2×Dic3—22no—
294C2×Dic3—22no—
295C2×Dic3—22no—
296C2×Dic3—22no—
297C2×Dic3—22no—
298C2×Dic3—42no—
299C2×Dic3—41no—
300C2×Dic3—41no—
301C2×Dic3—41no—
302C2×Dic3—41no—
303C2×Dic3—42no—
304C2×Dic3—41no—
305C2×Dic3—41no—
306C2×Dic3—88no—
307Dic6—22no—
308Dic6—22no—
309Dic6—44no—
310Dic6—42no—
311C3×C8—41no—
312C3×C8—41no—
313C3⋊C8—42no—
314C24—22no—
315C2×Q8—32no—
316C2×Q8—62no—
317C2×Q8—61no—
318C2×Q8—62no—
319C2×Q8—62no—
320C2×Q8—122no—
321C2×D4—24no—
322C2×D4—24no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—44no—
329C2×D4—81no—
330C2×D4—81no—
331C2×D4C2×D1214yes—
332C22×C4—22no—
333C22×C4—32no—
334C22×C4—32no—
335C22×C4—61no—
336C22×C4—61no—
337C22×C4—68no—
338C22×C4—61no—
339C22×C4—61no—
340C22×C4—242no—
341C22×C4—242no—
342C22×C4D3×D411yescharacteristic
343C22×C4D3×D411yescharacteristic
344C22×C4C23×D311yescharacteristic
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—22no—
348C2×C8—22no—
349C2×C8—62no—
350C2×C8—62no—
351C2×C8—62no—
352C4⋊C4—22no—
353C4⋊C4—32no—
354C4⋊C4—62no—
355C4⋊C4—64no—
356C4⋊C4—64no—
357C4⋊C4—61no—
358C4⋊C4—61no—
359C4⋊C4—61no—
360C4⋊C4—61no—
361C4⋊C4—61no—
362C4⋊C4—124no—
363C4⋊C4—122no—
364C4⋊C4—122no—
365C4⋊C4—242no—
366C4⋊C4#23714yes—
367C22⋊C4—68no—
368C22⋊C4—68no—
369C22⋊C4—68no—
370C42—64no—
371C42—62no—
372C42—62no—
373C42—81no—
374C42C2×D1212yes—
375C42D3×D412yes—
376C22×C3—44no—
377C22×C3—44no—
378C22×C3—44no—
379C22×C3—82no—
380C22×C3—82no—
381C22×C3—82no—
382C22×C3—82no—
383C22×C3C2×#12011yescharacteristic
384C22×C3#54812yes—
385C22×C3#57512yes—
386C22×C3#57611yescharacteristic
387C22×C3#57711yescharacteristic
388C3×C4—22no—
389C3×C4—22no—
390C3×C4—44no—
391C3×C4#53211yescharacteristic
392C3×C4#53611yescharacteristic
393C3×C4#54712yes—
394Dic3—42no—
395Dic3—42no—
396Dic3—42no—
397Dic3—42no—
398Dic3—81no—
399Dic3—81no—
400C23—24no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23C2×D3×D411yescharacteristic
408Q8—62no—
409Q8—62no—
410Q8—124no—
411Q8—122no—
412D4—24no—
413D4—24no—
414D4—48no—
415D4—42no—
416D4—42no—
417D4—84no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—24no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—62no—
425C2×C4—62no—
426C2×C4—62no—
427C2×C4—62no—
428C2×C4—62no—
429C2×C4—62no—
430C2×C4—62no—
431C2×C4—62no—
432C2×C4—121no—
433C2×C4—121no—
434C2×C4—122no—
435C2×C4—121no—
436C2×C4—121no—
437C2×C4—121no—
438C2×C4—121no—
439C2×C4—122no—
440C2×C4—248no—
441C2×C4#88212yes—
442C2×C4#88212yes—
443C2×C4C22×D1211yescharacteristic
444C2×C4C2×D3×D411yescharacteristic
445C2×C4C2×#23712yes—
446C2×C4#101012yes—
447C8—41no—
448C8—41no—
449C8—122no—
450C2×C3—84no—
451C2×C3#243211yescharacteristic
452C2×C3#300511yescharacteristic
453C2×C3#301211yescharacteristic
454C2×C3#337012yes—
455C2×C3#337112yes—
456C22—44no—
457C22—44no—
458C22—44no—
459C22—82no—
460C22—82no—
461C22—82no—
462C22—82no—
463C22C2×#88211yescharacteristic
464C22#559812yes—
465C22#575711yescharacteristic
466C22#576211yescharacteristic
467C22#577312yes—
468C4—22no—
469C4—22no—
470C4—44no—
471C4—122no—
472C4—122no—
473C4—122no—
474C4—122no—
475C4—241no—
476C4—241no—
477C4#555512yes—
478C4#567511yescharacteristic
479C4#571611yescharacteristic
480C3#2565811yescharacteristic, complemented, Sylow
481C2—84no—
482C2#7644511yescharacteristic
483C2#7874711yescharacteristic
484C2#7892711yescharacteristic
485C2#8032212yes—
486C2#8037312yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1982C2
socleC23×C3C22×D4
prime radicalC2×C3×#127C22
hypercenter#1982D3
hypocenterC3#25658
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1982

Complements

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

NH ≅ G/Ntype
3. C2×#4829C2semidirect
4. #73727C2semidirect
6. #70473C2semidirect
7. #70178C2semidirect
8. #70178C2semidirect
9. #70170C2semidirect
10. #70168C2semidirect
11. #69475C2semidirect
480. C3#25658semidirect

Maximal subgroups

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

subgroupindex
C2×#57132
C2×#50712
C2×#48292
#737272
C3×#19822
#704732
#701782
#701702
#701682
#694752
#256583