Finite Group Databasebeta

← #10877204

Subgroups — #10877204

Order 768.

970 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×#121C211yesmaximal, characteristic
2C2×#4935C211yesmaximal, characteristic, complemented
3#71677C212yesmaximal
4#71604C212yesmaximal, complemented
5#71530C211yesmaximal, characteristic, complemented
6#69656C212yesmaximal, complemented
7#69612C212yesmaximal, complemented
8#69505C212yesmaximal, complemented
9#69467C212yesmaximal, complemented
10#27678—31nomaximal, Sylow
11C2×C3×#121C2211yescharacteristic
12C2×C3×C4×D4—21no—
13C2×C3×C4×D4C2211yescharacteristic
14C22×C3×C42C2211yescharacteristic
15C22×C4⋊Dic3—21no—
16C22×C4⋊Dic3—21no—
17C22×C4⋊Dic3C2212yes—
18C3×#390C2211yescharacteristic
19C3×#389C2211yescharacteristic
20C3×#388—21no—
21C3×#386—21no—
22C3×#380C2211yescharacteristic
23C3×C4×C4⋊C4C2211yescharacteristic
24C3×C4×C22⋊C4—21no—
25C2×#832C2214yes—
26C2×#830C2212yes—
27C2×#808C2214yes—
28#4935—21no—
29#4935—21no—
30#4935C2212yes—
31#4935C2212yes—
32C2×C4⋊(C3⋊C8)C2212yes—
33#4529C2214yes—
34#4529C2214yes—
35#4524C2212yes—
36C4×#121D311yescharacteristic
37C2×#384—31no—
38#2100—32no—
39#2094—32no—
40#1948—32no—
41#1874—32no—
42#1853—32no—
43#1802—32no—
44#1797—31no—
45C22×C3×D4—21no—
46C22×C3×D4C2311yescharacteristic
47C23×C3×C4—21no—
48C23×C3×C4D411yescharacteristic
49C23×C3×C4D411yescharacteristic
50C23×C3×C4C2311yescharacteristic
51C23×Dic3—22no—
52C3×#121—22no—
53C3×#121Q814yes—
54C3×C4×D4—24no—
55C3×C4×D4—44no—
56C2×C3×C4⋊C4—21no—
57C2×C3×C4⋊C4C2311yescharacteristic
58C2×C3×C4⋊C4C2311yescharacteristic
59C2×C3×C4⋊C4C2311yescharacteristic
60C2×C3×C22⋊C4—21no—
61C2×C3×C22⋊C4—21no—
62C2×C3×C22⋊C4—21no—
63C2×C3×C22⋊C4—21no—
64C2×C3×C22⋊C4—21no—
65C2×C3×C42—24no—
66C2×C3×C42—21no—
67C2×C3×C42D411yescharacteristic
68C2×C3×C42D411yescharacteristic
69C2×C3×C42C2311yescharacteristic
70C2×C3×C42C2311yescharacteristic
71C2×C4⋊Dic3—22no—
72C2×C4⋊Dic3—22no—
73C2×C4⋊Dic3—22no—
74C2×C4⋊Dic3—22no—
75C2×C4⋊Dic3—22no—
76C2×C4⋊Dic3—44no—
77C2×C4⋊Dic3—44no—
78C2×C4⋊Dic3—44no—
79C2×C4⋊Dic3C2314yes—
80C22×C3⋊C8C2314yes—
81C3×#95—21no—
82C3×#95—21no—
83C3×#95—21no—
84#832—44no—
85#830—24no—
86#808—44no—
87C4⋊(C3⋊C8)—24no—
88C2×#121C2×D311yescharacteristic
89C2×C4×D4—21no—
90C2×C4×D4C2×D311yescharacteristic
91C22×C4⋊C4—32no—
92C22×C4⋊C4—61no—
93C22×C4⋊C4—61no—
94C22×C42C2×D311yescharacteristic
95C2×#106—34no—
96C2×C4⋊C8—32no—
97C2×#102—34no—
98C2×C22⋊C8—32no—
99#390C2×D311yescharacteristic
100#389C2×D311yescharacteristic
101#388—21no—
102#386—21no—
103#384—32no—
104#384—32no—
105#384—61no—
106#384—61no—
107#380C2×D311yescharacteristic
108C4×C4⋊C4C2×D311yescharacteristic
109C4×C22⋊C4—21no—
110#340—34no—
111#340—34no—
112#336—32no—
113C24×C3—21no—
114C24×C3C2×D411yescharacteristic
115C2×C3×D4—24no—
116C2×C3×D4—22no—
117C2×C3×D4—21no—
118C2×C3×D4—22no—
119C2×C3×D4—42no—
120C2×C3×D4—42no—
121C2×C3×D4C2×Q812yes—
122C22×C3×C4—22no—
123C22×C3×C4—22no—
124C22×C3×C4—22no—
125C22×C3×C4—22no—
126C22×C3×C4—22no—
127C22×C3×C4—22no—
128C22×C3×C4—22no—
129C22×C3×C4—21no—
130C22×C3×C4—21no—
131C22×C3×C4—21no—
132C22×C3×C4—42no—
133C22×C3×C4—42no—
134C22×C3×C4—42no—
135C22×C3×C4SD1614yes—
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×Q812yes—
142C22×C3×C4#4112yes—
143C22×C3×C4C2411yescharacteristic
144C22×Dic3—24no—
145C22×Dic3—44no—
146C22×Dic3—44no—
147C22×Dic3—44no—
148C3×C4⋊C4—21no—
149C3×C4⋊C4—22no—
150C3×C4⋊C4—22no—
151C3×C4⋊C4—42no—
152C3×C4⋊C4C2×Q812yes—
153C3×C22⋊C4—24no—
154C3×C22⋊C4—42no—
155C3×C22⋊C4—42no—
156C3×C22⋊C4—42no—
157C3×C22⋊C4—42no—
158C3×C42—22no—
159C3×C42—22no—
160C3×C42—22no—
161C3×C42—42no—
162C3×C42#4114yes—
163C4⋊Dic3—24no—
164C4⋊Dic3—24no—
165C4⋊Dic3—44no—
166C4⋊Dic3—44no—
167C4⋊Dic3—44no—
168C4⋊Dic3—44no—
169C4⋊Dic3—44no—
170C2×C3⋊C8—24no—
171C2×C3⋊C8—24no—
172C2×C3⋊C8—24no—
173C22×D4—21no—
174C22×D4C22×D311yescharacteristic
175C23×C4—21no—
176C23×C4—62no—
177C23×C4#6711yescharacteristic
178C23×C4#6711yescharacteristic
179C23×C4C22×D311yescharacteristic
180C22×C8—34no—
181#121—22no—
182#121Dic614yes—
183C4×D4—24no—
184C4×D4—44no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—34no—
187C2×C4⋊C4—62no—
188C2×C4⋊C4—62no—
189C2×C4⋊C4—62no—
190C2×C4⋊C4—62no—
191C2×C4⋊C4—62no—
192C2×C4⋊C4—124no—
193C2×C4⋊C4—124no—
194C2×C4⋊C4—124no—
195C2×C4⋊C4C22×D311yescharacteristic
196C2×C4⋊C4C22×D311yescharacteristic
197C2×C4⋊C4C22×D311yescharacteristic
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C42—24no—
204C2×C42—21no—
205C2×C42#6711yescharacteristic
206C2×C42#6711yescharacteristic
207C2×C42C22×D311yescharacteristic
208C2×C42C22×D311yescharacteristic
209#106—124no—
210C4⋊C8—64no—
211#102—124no—
212C22⋊C8—64no—
213#95—21no—
214#95—21no—
215#95—21no—
216C23×C3—21no—
217C23×C3—21no—
218C23×C3—21no—
219C23×C3—21no—
220C23×C3—22no—
221C23×C3—41no—
222C23×C3—41no—
223C23×C3—41no—
224C23×C3—41no—
225C23×C3—41no—
226C23×C3—41no—
227C23×C3C22⋊Q812yes—
228C23×C3C2×SD1612yes—
229C23×C3AGL(1,ℤ/8)12yes—
230C23×C3C22×D411yescharacteristic
231C3×D4—24no—
232C3×D4—48no—
233C3×D4—42no—
234C2×C3×C4—28no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—24no—
244C2×C3×C4—24no—
245C2×C3×C4—28no—
246C2×C3×C4—22no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—41no—
255C2×C3×C4—84no—
256C2×C3×C4C22⋊Q812yes—
257C2×C3×C4#12312yes—
258C2×C3×C4#12312yes—
259C2×C3×C4C2×SD1612yes—
260C2×C3×C4C2×SD1612yes—
261C2×C3×C4AGL(1,ℤ/8)12yes—
262C2×C3×C4#13712yes—
263C2×C3×C4C22×Q811yescharacteristic
264C2×C3×C4C2×#4111yescharacteristic
265C2×C3×C4C2×#4112yes—
266C2×C3×C4H(5,𝔽2)12yes—
267C2×Dic3—44no—
268C2×Dic3—44no—
269C2×Dic3—44no—
270C2×Dic3—88no—
271C3⋊C8—44no—
272C24—21no—
273C24C2×#6711yescharacteristic
274C2×D4—24no—
275C2×D4—21no—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4C2×Dic612yes—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—64no—
295C22×C4—124no—
296C22×C4—124no—
297C22×C4—124no—
298C22×C4#21414yes—
299C22×C4C2×Dic612yes—
300C22×C4#23512yes—
301C22×C4C2×#6711yescharacteristic
302C22×C4C2×#6711yescharacteristic
303C22×C4C2×#6711yescharacteristic
304C22×C4C2×#6711yescharacteristic
305C22×C4C2×#6711yescharacteristic
306C22×C4C23×D311yescharacteristic
307C2×C8—64no—
308C2×C8—64no—
309C2×C8—64no—
310C4⋊C4—21no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—42no—
314C4⋊C4—64no—
315C4⋊C4—64no—
316C4⋊C4—124no—
317C4⋊C4—124no—
318C4⋊C4—124no—
319C4⋊C4—124no—
320C4⋊C4—124no—
321C4⋊C4C2×Dic612yes—
322C22⋊C4—24no—
323C22⋊C4—42no—
324C22⋊C4—42no—
325C22⋊C4—42no—
326C22⋊C4—42no—
327C42—22no—
328C42—22no—
329C42—22no—
330C42—42no—
331C42#23714yes—
332C22×C3—22no—
333C22×C3—22no—
334C22×C3—22no—
335C22×C3—22no—
336C22×C3—22no—
337C22×C3—22no—
338C22×C3—42no—
339C22×C3—42no—
340C22×C3—42no—
341C22×C3—81no—
342C22×C3—81no—
343C22×C3—81no—
344C22×C3—81no—
345C22×C3#47614yes—
346C22×C3C2×C22⋊Q811yescharacteristic
347C22×C3C2×#12311yescharacteristic
348C22×C3#53511yescharacteristic
349C22×C3C22×SD1611yescharacteristic
350C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
351C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
352C22×C3C2×#13711yescharacteristic
353C3×C4—22no—
354C3×C4—22no—
355C3×C4—22no—
356C3×C4—22no—
357C3×C4—42no—
358C3×C4—81no—
359C3×C4—81no—
360C3×C4#48114yes—
361C3×C4#55312yes—
362C3×C4#55412yes—
363Dic3—84no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—22no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23#92412yes—
376C23C2×#21412yes—
377C23#94912yes—
378C23C22×#6711yescharacteristic
379D4—24no—
380D4—48no—
381D4—42no—
382C2×C4—28no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—24no—
392C2×C4—28no—
393C2×C4—24no—
394C2×C4—22no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—84no—
404C2×C4—124no—
405C2×C4—124no—
406C2×C4—124no—
407C2×C4—248no—
408C2×C4#92412yes—
409C2×C4C2×#21412yes—
410C2×C4C2×#21412yes—
411C2×C4#93512yes—
412C2×C4#93512yes—
413C2×C4#94912yes—
414C2×C4#95112yes—
415C2×C4C22×Dic611yescharacteristic
416C2×C4C2×#23511yescharacteristic
417C2×C4C2×#23712yes—
418C2×C4#100912yes—
419C8—124no—
420C2×C3—24no—
421C2×C3—82no—
422C2×C3#248711yescharacteristic
423C2×C3C2×#47611yescharacteristic
424C2×C3#307911yescharacteristic
425C2×C3C2×#48111yescharacteristic
426C2×C3#309211yescharacteristic
427C2×C3#320611yescharacteristic
428C2×C3#321711yescharacteristic
429C22—22no—
430C22—22no—
431C22—22no—
432C22—22no—
433C22—22no—
434C22—22no—
435C22—42no—
436C22—42no—
437C22—42no—
438C22—81no—
439C22—81no—
440C22—81no—
441C22—81no—
442C22#500814yes—
443C22C2×#92411yescharacteristic
444C22C22×#21411yescharacteristic
445C22C2×#93511yescharacteristic
446C22C2×#94911yescharacteristic
447C22C2×#94911yescharacteristic
448C22C2×#95111yescharacteristic
449C22#583011yescharacteristic
450C4—22no—
451C4—22no—
452C4—22no—
453C4—22no—
454C4—42no—
455C4—81no—
456C4—81no—
457C4—244no—
458C4#503414yes—
459C4#554312yes—
460C4#556512yes—
461C3#2767811yescharacteristic, complemented, Sylow
462C2—24no—
463C2—82no—
464C2C2×#500811yescharacteristic
465C2#8237411yescharacteristic
466C2#8250811yescharacteristic
467C2C2×#503411yescharacteristic
468C2#8291411yescharacteristic
469C2#8314611yescharacteristic
470C2#8331711yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×#121C2
socleC23×C3C22×D4
prime radicalC2×C3×#121C22
hypercenterC4×#121D3
hypocenterC3#27678
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×#121

Complements

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

NH ≅ G/Ntype
2. C2×#4935C2semidirect
4. #71604C2semidirect
5. #71530C2semidirect
6. #69656C2semidirect
7. #69612C2semidirect
8. #69505C2semidirect
9. #69467C2semidirect
461. C3#27678semidirect

Maximal subgroups

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

subgroupindex
C3×C4×#1212
C2×#49352
#716772
#716042
#715302
#696562
#696122
#695052
#694672
#276783