Finite Group Databasebeta

← #10876122

Subgroups — #10876122

Order 768.

1128 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#390C211yesmaximal, characteristic
2#82062C211yesmaximal, characteristic, complemented
3#72128C212yesmaximal
4#71555C212yesmaximal, complemented
5#71531C211yesmaximal, characteristic, complemented
6#69655C212yesmaximal, complemented
7#69640C212yesmaximal, complemented
8#69636C214yesmaximal, complemented
9#27582—31nomaximal, Sylow
10C23×C3×D4C2211yescharacteristic
11C22×C3×C4⋊C4C2211yescharacteristic
12C22×C3×C22⋊C4—21no—
13C22×C3×C22⋊C4—21no—
14C22×C3×C42C2211yescharacteristic
15C2×#926—21no—
16C3×#390—21no—
17C3×#390—21no—
18C3×#390C2212yes—
19C3×#390C2212yes—
20C2×#832C2214yes—
21C2×#832C2214yes—
22#5212—21no—
23#5212C2212yes—
24C2×#830C2212yes—
25C2×#807C2214yescomplemented
26#4936—21no—
27#4935C2211yescharacteristic, complemented
28#4935C2211yescharacteristic, complemented
29C4×C4⋊Dic3—21no—
30C4×C4⋊Dic3C2212yescomplemented
31#4529C2214yescomplemented
32#4524C2212yescomplemented
33#4524C2212yescomplemented
34C2×#390D311yescharacteristic
35#2285—31no—
36#2093—32no—
37#2071—32no—
38#1925—34no—
39#1852—32no—
40#1817—31no—
41#1786—32no—
42C25×C3—21no—
43C22×C3×D4—24no—
44C22×C3×D4—21no—
45C22×C3×D4—24no—
46C22×C3×D4—42no—
47C22×C3×D4C2312yes—
48C23×C3×C4—21no—
49C23×C3×C4D411yescharacteristic
50C23×C3×C4D411yescharacteristic
51C23×C3×C4C2311yescharacteristic
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4—42no—
54C2×C3×C4⋊C4C2312yes—
55C2×C3×C22⋊C4—22no—
56C2×C3×C22⋊C4—22no—
57C2×C3×C22⋊C4—22no—
58C2×C3×C22⋊C4—22no—
59C2×C3×C22⋊C4—44no—
60C2×C3×C22⋊C4—44no—
61C2×C3×C42—24no—
62C2×C3×C42D411yescharacteristic
63C2×C3×C42D411yescharacteristic
64C2×C3×C42C2311yescharacteristic
65C2×C3×C42C2311yescharacteristic
66C2×#217—22no—
67#926—44no—
68C2×C4⋊Dic3—21no—
69C2×C4⋊Dic3—21no—
70C2×C4⋊Dic3—21no—
71C2×C4⋊Dic3—21no—
72C2×C4⋊Dic3C2312yes—
73C2×C4⋊Dic3C2312yes—
74C2×C4×Dic3—22no—
75C22×C3⋊C8C2314yes—
76#832—28no—
77#832—44no—
78C3⋊#95—22no—
79C3⋊#95—22no—
80#830—24no—
81#807—44no—
82C23×D4C2×D311yescharacteristic
83C2×#117—61no—
84C22×C4⋊C4C2×D311yescharacteristic
85C22×C22⋊C4—21no—
86C22×C22⋊C4—21no—
87C22×C42C2×D311yescharacteristic
88C2×#107—34no—
89C2×#102—34no—
90C2×#102—34no—
91C2×C22⋊C8—32no—
92#390—21no—
93#390—21no—
94#390C2×D312yes—
95#390C2×D312yes—
96#384—31no—
97#384—31no—
98#382—61no—
99#380—32no—
100#380—61no—
101C4×C4⋊C4—32no—
102C4×C4⋊C4—61no—
103#340—34no—
104#336—32no—
105#336—32no—
106C24×C3—24no—
107C24×C3—22no—
108C24×C3—42no—
109C24×C3—42no—
110C24×C3—42no—
111C24×C3—42no—
112C24×C3—42no—
113C24×C3—42no—
114C24×C3C2×D411yescharacteristic
115C2×C3×D4—24no—
116C2×C3×D4—24no—
117C2×C3×D4—22no—
118C2×C3×D4—24no—
119C2×C3×D4—416no—
120C2×C3×D4—42no—
121C2×C3×D4—42no—
122C2×C3×D4#4114yes—
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—22no—
130C22×C3×C4—41no—
131C22×C3×C4—42no—
132C22×C3×C4—42no—
133C22×C3×C4—42no—
134C22×C3×C4D814yes—
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4#4114yes—
141C22×C3×C4C2411yescharacteristic
142C22×Dic3—22no—
143C22×Dic3—22no—
144C3×C4⋊C4—24no—
145C3×C4⋊C4—42no—
146C3×C22⋊C4—44no—
147C3×C22⋊C4—44no—
148C3×C22⋊C4—44no—
149C3×C22⋊C4—44no—
150C3×C42—22no—
151C3×C42—22no—
152C3×C42—22no—
153C3×C42—22no—
154#217—44no—
155C4⋊Dic3—24no—
156C4⋊Dic3—24no—
157C4⋊Dic3—42no—
158C4⋊Dic3—42no—
159C4⋊Dic3—42no—
160C4⋊Dic3—42no—
161C4×Dic3—44no—
162C2×C3⋊C8—24no—
163C2×C3⋊C8—24no—
164C2×C3⋊C8—24no—
165C25—21no—
166C22×D4—24no—
167C22×D4—24no—
168C22×D4—21no—
169C22×D4—42no—
170C22×D4C22×D312yes—
171C23×C4—21no—
172C23×C4#6711yescharacteristic
173C23×C4#6711yescharacteristic
174C23×C4C22×D311yescharacteristic
175C22×C8—34no—
176#117—124no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—32no—
179C2×C4⋊C4—32no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—61no—
182C2×C4⋊C4—61no—
183C2×C4⋊C4—61no—
184C2×C4⋊C4—61no—
185C2×C4⋊C4C22×D312yes—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—44no—
191C2×C22⋊C4—44no—
192C2×C22⋊C4—62no—
193C2×C42—24no—
194C2×C42—62no—
195C2×C42#6711yescharacteristic
196C2×C42#6711yescharacteristic
197C2×C42C22×D311yescharacteristic
198C2×C42C22×D311yescharacteristic
199#107—124no—
200#102—68no—
201#102—124no—
202C22⋊C8—64no—
203#95—62no—
204#95—62no—
205C23×C3—21no—
206C23×C3—21no—
207C23×C3—21no—
208C23×C3—21no—
209C23×C3—24no—
210C23×C3—48no—
211C23×C3—48no—
212C23×C3—48no—
213C23×C3—42no—
214C23×C3—42no—
215C23×C3—42no—
216C23×C3—42no—
217C23×C3—42no—
218C23×C3—42no—
219C23×C3—84no—
220C23×C3—84no—
221C23×C3—84no—
222C23×C3—84no—
223C23×C3#12312yes—
224C23×C3C2×D812yes—
225C23×C3#13712yes—
226C23×C3C22×D411yescharacteristic
227C3×D4—28no—
228C3×D4—48no—
229C3×D4—44no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—28no—
239C2×C3×C4—28no—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—44no—
244C2×C3×C4—42no—
245C2×C3×C4—42no—
246C2×C3×C4—42no—
247C2×C3×C4—84no—
248C2×C3×C4#12312yes—
249C2×C3×C4C2×D812yes—
250C2×C3×C4C2×D812yes—
251C2×C3×C4AGL(1,ℤ/8)12yes—
252C2×C3×C4C2×#4112yes—
253C2×C3×C4C2×#4112yes—
254C2×C3×C4H(5,𝔽2)12yes—
255C2×Dic3—42no—
256C2×Dic3—42no—
257C2×Dic3—42no—
258C2×Dic3—42no—
259C2×Dic3—42no—
260C2×Dic3—42no—
261C3⋊C8—44no—
262C24—24no—
263C24—22no—
264C24—42no—
265C24—42no—
266C24—42no—
267C24—42no—
268C24—42no—
269C24—42no—
270C24C2×#6711yescharacteristic
271C2×D4—24no—
272C2×D4—22no—
273C2×D4—24no—
274C2×D4—24no—
275C2×D4—416no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4#23514yes—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—62no—
291C22×C4—62no—
292C22×C4#21314yes—
293C22×C4#23714yes—
294C22×C4C2×#6711yescharacteristic
295C22×C4C2×#6711yescharacteristic
296C22×C4C2×#6711yescharacteristic
297C22×C4C2×#6711yescharacteristic
298C22×C4C2×#6711yescharacteristic
299C22×C4C23×D311yescharacteristic
300C2×C8—64no—
301C2×C8—64no—
302C2×C8—64no—
303C4⋊C4—24no—
304C4⋊C4—42no—
305C4⋊C4—64no—
306C4⋊C4—64no—
307C4⋊C4—122no—
308C4⋊C4—122no—
309C4⋊C4—122no—
310C4⋊C4—122no—
311C22⋊C4—44no—
312C22⋊C4—44no—
313C22⋊C4—44no—
314C22⋊C4—44no—
315C22⋊C4—124no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—22no—
320C42—124no—
321C22×C3—22no—
322C22×C3—22no—
323C22×C3—22no—
324C22×C3—22no—
325C22×C3—22no—
326C22×C3—22no—
327C22×C3—44no—
328C22×C3—44no—
329C22×C3—44no—
330C22×C3—816no—
331C22×C3—82no—
332C22×C3—82no—
333C22×C3—82no—
334C22×C3—82no—
335C22×C3#48014yes—
336C22×C3C2×#12311yescharacteristic
337C22×C3#52511yescharacteristic
338C22×C3#53511yescharacteristic
339C22×C3C22×D811yescharacteristic
340C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
341C22×C3C2×#13711yescharacteristic
342C22×C3#57511yescharacteristic
343C3×C4—22no—
344C3×C4—22no—
345C3×C4—22no—
346C3×C4—22no—
347C3×C4—41no—
348C3×C4—82no—
349C3×C4#55114yes—
350Dic3—82no—
351Dic3—82no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—24no—
357C23—48no—
358C23—48no—
359C23—48no—
360C23—42no—
361C23—42no—
362C23—42no—
363C23—42no—
364C23—42no—
365C23—42no—
366C23—84no—
367C23—84no—
368C23—84no—
369C23—84no—
370C23C2×#21312yes—
371C23#93512yes—
372C23#95112yes—
373C23C22×#6711yescharacteristic
374D4—28no—
375D4—48no—
376D4—44no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—28no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—28no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—44no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—84no—
395C2×C4—122no—
396C2×C4—122no—
397C2×C4—122no—
398C2×C4—122no—
399C2×C4—122no—
400C2×C4—122no—
401C2×C4C2×#21312yes—
402C2×C4C2×#21312yes—
403C2×C4#93212yes—
404C2×C4#93512yes—
405C2×C4C2×#23512yes—
406C2×C4C2×#23712yes—
407C2×C4#100412yes—
408C8—124no—
409C2×C3—24no—
410C2×C3—84no—
411C2×C3#248211yescharacteristic
412C2×C3C2×#48011yescharacteristic
413C2×C3#309411yescharacteristic
414C2×C3#310011yescharacteristic
415C2×C3#311011yescharacteristic
416C2×C3#320511yescharacteristic
417C2×C3#321511yescharacteristic
418C22—22no—
419C22—22no—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—22no—
424C22—44no—
425C22—44no—
426C22—44no—
427C22—816no—
428C22—82no—
429C22—82no—
430C22—82no—
431C22—82no—
432C22#503314yes—
433C22#579111yescharacteristic
434C22C22×#21311yescharacteristic
435C22C2×#93211yescharacteristic
436C22C2×#93511yescharacteristic
437C22#580511yescharacteristic
438C22#582211yescharacteristic
439C22C2×#95111yescharacteristic
440C4—22no—
441C4—22no—
442C4—22no—
443C4—22no—
444C4—41no—
445C4—82no—
446C4—242no—
447C4—242no—
448C4#549514yes—
449C3#2758211yescharacteristic, complemented, Sylow
450C2—24no—
451C2—84no—
452C2#8211311yescharacteristic
453C2#8230811yescharacteristic
454C2#8231511yescharacteristic
455C2C2×#503311yescharacteristic
456C2#8261611yescharacteristic
457C2#8294911yescharacteristic
458C2#8316611yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#390C2
socleC23×C3C22×D4
prime radicalC23×C3×D4C22
hypercenterC2×#390D3
hypocenterC3#27582
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
2. #82062C2semidirect
4. #71555C2semidirect
5. #71531C2semidirect
6. #69655C2semidirect
7. #69640C2semidirect
8. #69636C2semidirect
25. C2×#807C22semidirect
27. #4935C22semidirect
28. #4935C22semidirect
30. C4×C4⋊Dic3C22semidirect
31. #4529C22semidirect
32. #4524C22semidirect
33. #4524C22semidirect
449. C3#27582semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3902
#820622
#721282
#715552
#715312
#696552
#696402
#696362
#275823