Finite Group Databasebeta

← #10685204

Subgroups — #10685204

Order 768.

1162 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#407C211yesmaximal, characteristic
2C2×#5212C211yesmaximal, characteristic
3C2×C4×#217C211yesmaximal, characteristic
4#67675C214yesmaximal, complemented
5#8019—31nomaximal, Sylow
6C24×C3×C4C2211yescharacteristic
7C22×C3×M16—21no—
8C22×#217—21no—
9C22×#217—21no—
10C22×C4⋊Dic3C412yescomplemented
11C22×C4×Dic3—21no—
12C3×#407—22no—
13C3×#407C414yescomplemented
14C2×C3×C22⋊C8—21no—
15C2×C3×C22⋊C8C2212yes—
16#5212—21no—
17#5212—21no—
18#5212C412yescomplemented
19#5212C2212yes—
20C4×#217—22no—
21C4×#217C414yescomplemented
22C2×C3⋊#95—21no—
23C2×C3⋊#95C2212yes—
24C2×#407D311yescharacteristic
25C2×#380—31no—
26C2×C4×C22⋊C4—31no—
27#1294—34no—
28C25×C3D411yescharacteristic
29C23×C3×C4—44no—
30C23×C3×C4C2×C411yescharacteristic
31C23×C3×C4C2×C412yes—
32C23×C3×C4C2×C411yescharacteristic
33C23×C3×C4C2×C412yes—
34C23×C3×C4D411yescharacteristic
35C23×C3×C4D411yescharacteristic
36C23×C3×C4D411yescharacteristic
37C23×C3×C4D412yes—
38C23×C3×C4Q812yes—
39C23×C3×C4C2311yescharacteristic
40C23×Dic3—22no—
41C23×Dic3—22no—
42C2×C3×M16—24no—
43C2×C3×M16—44no—
44C22×C3×C8—22no—
45C2×#217—28no—
46C2×#217—24no—
47C2×#217—22no—
48C2×#217—22no—
49C2×#217—44no—
50C2×C4⋊Dic3—22no—
51C2×C4⋊Dic3—44no—
52C2×C4⋊Dic3C2×C414yes—
53C2×C4×Dic3—24no—
54C2×C4×Dic3—44no—
55C3×C22⋊C8—24no—
56C3×C22⋊C8—22no—
57C3×C22⋊C8C2×C414yescomplemented
58C3⋊#95—22no—
59C3⋊#95—22no—
60C3⋊#95—22no—
61C3⋊#95C2×C414yescomplemented
62C24×C4C2×D311yescharacteristic
63C22×M16—21no—
64C22×C4⋊C4—32no—
65C22×C22⋊C4—61no—
66C22×C22⋊C4—61no—
67C22×C42—61no—
68#407—22no—
69#407Dic314yescomplemented
70C2×C22⋊C8—21no—
71C2×C22⋊C8C2×D312yes—
72#380—32no—
73#380—32no—
74#380—61no—
75#380—61no—
76C4×C22⋊C4—34no—
77C4×C22⋊C4—62no—
78C2×#95—32no—
79C2×#95—61no—
80C24×C3—24no—
81C24×C3—22no—
82C24×C3—22no—
83C24×C3—42no—
84C24×C3C22⋊C411yescharacteristic
85C24×C3C22⋊C412yes—
86C24×C3C22⋊C411yescharacteristic
87C24×C3C4⋊C412yes—
88C24×C3C2×D411yescharacteristic
89C22×C3×C4—24no—
90C22×C3×C4—24no—
91C22×C3×C4—24no—
92C22×C3×C4—44no—
93C22×C3×C4—44no—
94C22×C3×C4—44no—
95C22×C3×C4—44no—
96C22×C3×C4—42no—
97C22×C3×C4—42no—
98C22×C3×C4—42no—
99C22×C3×C4C4212yes—
100C22×C3×C4C4212yes—
101C22×C3×C4C22⋊C411yescharacteristic
102C22×C3×C4C22⋊C412yes—
103C22×C3×C4C22⋊C411yescharacteristic
104C22×C3×C4C22⋊C412yes—
105C22×C3×C4C22⋊C412yes—
106C22×C3×C4C4⋊C412yes—
107C22×C3×C4C4⋊C412yes—
108C22×C3×C4C4⋊C412yes—
109C22×C3×C4C4⋊C412yes—
110C22×C3×C4C4⋊C412yes—
111C22×C3×C4C22×C411yescharacteristic
112C22×C3×C4C22×C411yescharacteristic
113C22×C3×C4C22×C411yescharacteristic
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×Q811yescharacteristic
117C22×Dic3—28no—
118C22×Dic3—24no—
119C22×Dic3—24no—
120C22×Dic3—44no—
121C22×Dic3—42no—
122C22×Dic3—42no—
123C22×Dic3—44no—
124C22×Dic3—44no—
125C22×Dic3—44no—
126C3×M16—48no—
127C2×C3×C8—24no—
128C2×C3×C8—44no—
129#217—216no—
130#217—44no—
131#217—44no—
132C4⋊Dic3—44no—
133C4⋊Dic3—44no—
134C4×Dic3—48no—
135C25D1211yescharacteristic
136C23×C4—44no—
137C23×C4—62no—
138C23×C4—62no—
139C23×C4Dic612yes—
140C23×C4C4×D311yescharacteristic
141C23×C4C4×D312yes—
142C23×C4C4×D311yescharacteristic
143C23×C4D1211yescharacteristic
144C23×C4C2×Dic312yes—
145C23×C4#6711yescharacteristic
146C23×C4#6711yescharacteristic
147C23×C4#6712yes—
148C23×C4C22×D311yescharacteristic
149C2×M16—24no—
150C2×M16—44no—
151C22×C8—22no—
152C2×C4⋊C4—34no—
153C2×C4⋊C4—62no—
154C2×C4⋊C4—124no—
155C2×C22⋊C4—68no—
156C2×C22⋊C4—64no—
157C2×C22⋊C4—62no—
158C2×C22⋊C4—62no—
159C2×C22⋊C4—124no—
160C2×C42—64no—
161C2×C42—124no—
162C22⋊C8—24no—
163C22⋊C8—22no—
164C22⋊C8C2×Dic314yescomplemented
165#95—34no—
166#95—62no—
167#95—62no—
168#95—62no—
169C23×C3—22no—
170C23×C3—22no—
171C23×C3—21no—
172C23×C3—21no—
173C23×C3—22no—
174C23×C3—24no—
175C23×C3—24no—
176C23×C3—22no—
177C23×C3—24no—
178C23×C3—21no—
179C23×C3—21no—
180C23×C3—22no—
181C23×C3—24no—
182C23×C3—42no—
183C23×C3—42no—
184C23×C3—48no—
185C23×C3—48no—
186C23×C3#9512yes—
187C23×C3#9512yes—
188C23×C3#10414yes—
189C23×C3C2×C22⋊C411yescharacteristic
190C23×C3C2×C22⋊C411yescharacteristic
191C23×C3C2×C4⋊C411yescharacteristic
192C23×C3AGL(1,ℤ/8)12yes—
193C23×C3#13712yes—
194C2×C3×C4—22no—
195C2×C3×C4—22no—
196C2×C3×C4—22no—
197C2×C3×C4—24no—
198C2×C3×C4—24no—
199C2×C3×C4—24no—
200C2×C3×C4—24no—
201C2×C3×C4—24no—
202C2×C3×C4—24no—
203C2×C3×C4—42no—
204C2×C3×C4—42no—
205C2×C3×C4—42no—
206C2×C3×C4—42no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—41no—
211C2×C3×C4#9512yes—
212C2×C3×C4#9512yes—
213C2×C3×C4#9914yes—
214C2×C3×C4C2×C4211yescharacteristic
215C2×C3×C4C2×C22⋊C411yescharacteristic
216C2×C3×C4C2×C4⋊C411yescharacteristic
217C2×C3×C4C2×C4⋊C411yescharacteristic
218C2×Dic3—28no—
219C2×Dic3—44no—
220C2×Dic3—48no—
221C2×Dic3—48no—
222C2×Dic3—44no—
223C2×Dic3—44no—
224C2×Dic3—44no—
225C2×Dic3—44no—
226C2×Dic3—88no—
227C3×C8—44no—
228C24—24no—
229C24—22no—
230C24—22no—
231C24—42no—
232C24C4⋊Dic312yes—
233C24(C2×D3)⋊C411yescharacteristic
234C24(C2×D3)⋊C412yes—
235C24(C2×D3)⋊C411yescharacteristic
236C24C2×D1211yescharacteristic
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—44no—
241C22×C4—44no—
242C22×C4—44no—
243C22×C4—44no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—68no—
248C22×C4—64no—
249C22×C4—64no—
250C22×C4—124no—
251C22×C4—122no—
252C22×C4—122no—
253C22×C4—124no—
254C22×C4—124no—
255C22×C4—124no—
256C22×C4C4×Dic312yes—
257C22×C4C4×Dic312yes—
258C22×C4Dic3⋊C412yes—
259C22×C4Dic3⋊C412yes—
260C22×C4Dic3⋊C412yes—
261C22×C4Dic3⋊C412yes—
262C22×C4C4⋊Dic312yes—
263C22×C4(C2×D3)⋊C412yes—
264C22×C4(C2×D3)⋊C411yescharacteristic
265C22×C4(C2×D3)⋊C411yescharacteristic
266C22×C4#21712yes—
267C22×C4#21712yes—
268C22×C4C2×Dic611yescharacteristic
269C22×C4C2×C4×D311yescharacteristic
270C22×C4C2×C4×D311yescharacteristic
271C22×C4C22×Dic311yescharacteristic
272C22×C4C2×#6711yescharacteristic
273C22×C4C2×#6711yescharacteristic
274M16—48no—
275C2×C8—24no—
276C2×C8—44no—
277C4⋊C4—124no—
278C4⋊C4—124no—
279C22⋊C4—616no—
280C22⋊C4—124no—
281C22⋊C4—124no—
282C42—128no—
283C22×C3—21no—
284C22×C3—21no—
285C22×C3—22no—
286C22×C3—21no—
287C22×C3—22no—
288C22×C3—21no—
289C22×C3—24no—
290C22×C3—24no—
291C22×C3—24no—
292C22×C3—24no—
293C22×C3—22no—
294C22×C3—22no—
295C22×C3—22no—
296C22×C3—48no—
297C22×C3—48no—
298C22×C3—42no—
299C22×C3—42no—
300C22×C3#33914yes—
301C22×C3C2×#9511yescharacteristic
302C22×C3C2×#9912yes—
303C22×C3#41712yes—
304C22×C3#41811yescharacteristic
305C22×C3#41911yescharacteristic
306C22×C3C2×#10412yes—
307C22×C3#42812yes—
308C3×C4—21no—
309C3×C4—22no—
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—21no—
313C3×C4#34214yes—
314Dic3—48no—
315Dic3—84no—
316C23—21no—
317C23—21no—
318C23—22no—
319C23—22no—
320C23—22no—
321C23—21no—
322C23—24no—
323C23—21no—
324C23—24no—
325C23—22no—
326C23—22no—
327C23—24no—
328C23—24no—
329C23—42no—
330C23—42no—
331C23—48no—
332C23—48no—
333C23#82514yes—
334C23C3⋊#9512yes—
335C23C3⋊#9512yes—
336C23#90812yes—
337C23#90912yes—
338C23C2×C4⋊Dic311yescharacteristic
339C23C2×(C2×D3)⋊C411yescharacteristic
340C23C2×(C2×D3)⋊C411yescharacteristic
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—68no—
359C2×C4—124no—
360C2×C4—128no—
361C2×C4—128no—
362C2×C4—124no—
363C2×C4—124no—
364C2×C4—124no—
365C2×C4—124no—
366C2×C4—248no—
367C2×C4#80614yes—
368C2×C4C3⋊#9512yes—
369C2×C4C3⋊#9512yes—
370C2×C4C2×C4×Dic311yescharacteristic
371C2×C4C2×Dic3⋊C411yescharacteristic
372C2×C4C2×Dic3⋊C411yescharacteristic
373C2×C4C2×#21711yescharacteristic
374C8—44no—
375C2×C3—22no—
376C2×C3—24no—
377C2×C3—22no—
378C2×C3—42no—
379C2×C3#150712yes—
380C2×C3#150812yes—
381C2×C3C2×#33911yescharacteristic
382C2×C3#173611yescharacteristic
383C2×C3C2×#34211yescharacteristic
384C22—21no—
385C22—21no—
386C22—22no—
387C22—24no—
388C22—24no—
389C22—24no—
390C22—24no—
391C22—21no—
392C22—22no—
393C22—21no—
394C22—22no—
395C22—22no—
396C22—22no—
397C22—48no—
398C22—48no—
399C22—42no—
400C22—42no—
401C22#455514yes—
402C22C2×#80612yes—
403C22#512011yescharacteristic
404C22#512112yes—
405C22#513111yescharacteristic
406C22#513312yes—
407C22C2×#82512yes—
408C22C2×C3⋊#9511yescharacteristic
409C4—21no—
410C4—22no—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—128no—
415C4—244no—
416C4#452714yes—
417C3#801911yescharacteristic, complemented, Sylow
418C2—22no—
419C2—22no—
420C2—24no—
421C2—42no—
422C2#6997712yes—
423C2#6998012yes—
424C2C2×#452711yescharacteristic
425C2C2×#455511yescharacteristic
426C2#7198511yescharacteristic
subgroupG/H
centerC23C2×(C2×D3)⋊C4
commutatorC2×C3×C4C2×C42
FrattiniC23×C4C22×D3
FittingC2×C3×#407C2
socleC23×C3C2×C22⋊C4
prime radicalC25×C3D4
cosocleC23×C3×C4C23
hypercenterC2×#407D3
hypocenterC3#8019
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#407

Complements

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

NH ≅ G/Ntype
4. #67675C2semidirect
10. C22×C4⋊Dic3C4semidirect
13. C3×#407C4semidirect
18. #5212C4semidirect
21. C4×#217C4semidirect
57. C3×C22⋊C8C2×C4semidirect
61. C3⋊#95C2×C4semidirect
69. #407Dic3semidirect
164. C22⋊C8C2×Dic3semidirect
417. C3#8019semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4072
C2×#52122
C2×C4×#2172
#676752
#80193