Finite Group Databasebeta

← #10801743

Subgroups — #10801743

Order 768.

1182 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#466C211yesmaximal, characteristic
2C2×#5152C211yesmaximal, characteristic, complemented
3C22×#815C211yesmaximal, characteristic, complemented
4Dic3×#121C212yesmaximal, complemented
5#72039C211yesmaximal, characteristic, complemented
6#71970C211yesmaximal, characteristic, complemented
7#71681C212yesmaximal, complemented
8#70613C211yesmaximal, characteristic, complemented
9#70605C211yesmaximal, characteristic, complemented
10#70187C212yesmaximal, complemented
11#70152C212yesmaximal, complemented
12#21526—31nomaximal, Sylow
13C22×C3×D8C2211yescharacteristic
14C23×C3×C8C2211yescharacteristic
15C2×C3×#121C2212yes—
16C2×D4×Dic3—22no—
17C2×D4×Dic3C2212yes—
18C22×C4×Dic3C2211yescharacteristic
19C23×C3⋊C8C2211yescharacteristic
20C3×#466C418yescomplemented
21C2×C3×#107C2211yescharacteristic
22C2×C3×#102C2212yes—
23#5221C2212yes—
24#5220—22no—
25C2×#832C2212yes—
26C2×#832C2212yes—
27#5212C2212yes—
28#5152—24no—
29C2×#817C2211yescharacteristic
30C2×#817C2211yescharacteristic
31C2×#815—24no—
32C2×#815C2211yescharacteristic
33C2×#815C2211yescharacteristic
34C2×#815C2211yescharacteristic
35C2×#815C2211yescharacteristic
36#4984C2212yes—
37Dic3×C4⋊C4C2212yes—
38#4951—22no—
39Dic3×C22⋊C4—22no—
40#4551C2212yes—
41#4529C2212yes—
42#4529C2212yes—
43C2×#466D311yescharacteristic
44C2×#442—31no—
45C22×#97—31no—
46C4×#121—32no—
47#1972—31no—
48#1943—31no—
49#1936—32no—
50#1893—32no—
51#1807—31no—
52#1776—31no—
53#1762—32no—
54C22×C3×D4—22no—
55C22×C3×D4C2312yes—
56C23×C3×C4C2311yescharacteristic
57C23×Dic3—22no—
58C23×Dic3D412yes—
59C2×C3×D8—22no—
60C2×C3×D8—22no—
61C2×C3×D8C2×C414yescomplemented
62C22×C3×C8—22no—
63C22×C3×C8—22no—
64C22×C3×C8C2×C414yescomplemented
65C22×C3×C8C2311yescharacteristic
66C22×C3×C8C2311yescharacteristic
67C3×#121—24no—
68C3×#121C2×C418yescomplemented
69C2×C3×C4⋊C4C2312yes—
70C2×C3×C22⋊C4—22no—
71C2×#217—22no—
72C2×#217—22no—
73C2×#217—22no—
74C2×#217—22no—
75D4×Dic3—24no—
76D4×Dic3—24no—
77D4×Dic3—48no—
78C2×C4⋊Dic3—22no—
79C2×C4⋊Dic3C2312yes—
80C2×C4⋊Dic3C2312yes—
81C2×C4×Dic3—22no—
82C2×C4×Dic3—22no—
83C2×C4×Dic3—22no—
84C2×C4×Dic3D412yes—
85C2×C4×Dic3C2311yescharacteristic
86C2×C4×Dic3C2311yescharacteristic
87C22×C3⋊C8—22no—
88C22×C3⋊C8—22no—
89C22×C3⋊C8—22no—
90C22×C3⋊C8C2311yescharacteristic
91C22×C3⋊C8C2311yescharacteristic
92C3×#107C2×C414yescomplemented
93C3×#102C2×C418yescomplemented
94#832—24no—
95#832—24no—
96C3⋊#95—22no—
97C3⋊#95—22no—
98C3⋊#95—22no—
99#817—22no—
100#817—22no—
101#815—22no—
102#815—22no—
103#815—22no—
104#815D414yescomplemented
105C22×D8C2×D311yescharacteristic
106C23×C8—31no—
107C23×C8C2×D311yescharacteristic
108C2×#121C2×D312yes—
109C2×C4×D4—32no—
110C2×C4×D4—62no—
111C22×C42—31no—
112#466Dic318yescomplemented
113#442—64no—
114C2×#107C2×D311yescharacteristic
115C2×#106—31no—
116C2×#106—31no—
117C2×#102—32no—
118C2×#102—32no—
119C2×#102C2×D312yes—
120C2×#97—31no—
121C2×#97—31no—
122C2×#97—31no—
123C2×#97—31no—
124C2×#97—64no—
125#390—32no—
126#389—32no—
127#388—62no—
128#386—62no—
129#380—32no—
130C4×C4⋊C4—32no—
131C4×C22⋊C4—62no—
132#340—32no—
133#340—32no—
134#340—32no—
135C24×C3—22no—
136C24×C3C2×D411yescharacteristic
137C2×C3×D4—28no—
138C2×C3×D4—22no—
139C2×C3×D4—22no—
140C2×C3×D4—22no—
141C2×C3×D4—22no—
142C2×C3×D4—22no—
143C2×C3×D4—44no—
144C2×C3×D4—44no—
145C2×C3×D4C22×C414yes—
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—21no—
149C22×C3×C4—21no—
150C22×C3×C4—22no—
151C22×C3×C4C22×C412yes—
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4#4112yes—
154C22×C3×C4C2411yescharacteristic
155C22×Dic3—24no—
156C22×Dic3—24no—
157C22×Dic3—22no—
158C22×Dic3—22no—
159C22×Dic3—42no—
160C22×Dic3—44no—
161C22×Dic3—44no—
162C22×Dic3—44no—
163C22×Dic3C2×D412yes—
164C22×Dic3C2×D412yes—
165C3×D8—28no—
166C2×C3×C8—21no—
167C2×C3×C8—21no—
168C2×C3×C8—28no—
169C2×C3×C8C22×C412yes—
170C2×C3×C8C22×C412yes—
171C2×C3×C8C2×D412yes—
172C2×C3×C8#4112yes—
173C3×C4⋊C4—22no—
174C3×C4⋊C4C22×C414yes—
175C3×C22⋊C4—28no—
176#217—44no—
177#217—44no—
178#217—44no—
179#217—44no—
180C4⋊Dic3—22no—
181C4⋊Dic3—22no—
182C4⋊Dic3—22no—
183C4⋊Dic3—22no—
184C4⋊Dic3—44no—
185C4×Dic3—21no—
186C4×Dic3—21no—
187C4×Dic3—24no—
188C4×Dic3—44no—
189C4×Dic3C2×D412yes—
190C4×Dic3#4112yes—
191C2×C3⋊C8—28no—
192C2×C3⋊C8—21no—
193C2×C3⋊C8—21no—
194C2×C3⋊C8—21no—
195C2×C3⋊C8—21no—
196C2×C3⋊C8C2×D412yes—
197C2×C3⋊C8#4112yes—
198C22×D4—22no—
199C22×D4C22×D312yes—
200C23×C4—32no—
201C23×C4—62no—
202C23×C4C22×D311yescharacteristic
203C2×D8—22no—
204C2×D8—22no—
205C2×D8C2×Dic314yescomplemented
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—31no—
209C22×C8—31no—
210C22×C8—62no—
211C22×C8—62no—
212C22×C8—62no—
213C22×C8C2×Dic314yescomplemented
214C22×C8C22×D311yescharacteristic
215C22×C8C22×D311yescharacteristic
216#121—24no—
217#121C2×Dic318yescomplemented
218C4×D4—64no—
219C4×D4—64no—
220C4×D4—128no—
221C2×C4⋊C4—32no—
222C2×C4⋊C4—32no—
223C2×C4⋊C4—62no—
224C2×C4⋊C4C22×D312yes—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—62no—
227C2×C22⋊C4—62no—
228C2×C22⋊C4—62no—
229C2×C22⋊C4—62no—
230C2×C42—31no—
231C2×C42—32no—
232C2×C42—31no—
233C2×C42—62no—
234C2×C42—62no—
235C2×C42—62no—
236#107C2×Dic314yescomplemented
237#106—62no—
238#106—62no—
239#102—64no—
240#102—64no—
241#102C2×Dic318yescomplemented
242#97—34no—
243#97—62no—
244#97—62no—
245#97—62no—
246#95—62no—
247#95—62no—
248#95—62no—
249C23×C3—21no—
250C23×C3—21no—
251C23×C3—22no—
252C23×C3—24no—
253C23×C3—42no—
254C23×C3—42no—
255C23×C3—42no—
256C23×C3—42no—
257C23×C3—42no—
258C23×C3—42no—
259C23×C3C4×D412yes—
260C23×C3AGL(1,ℤ/8)14yes—
261C23×C3C22×D411yescharacteristic
262C3×D4—24no—
263C3×D4—24no—
264C3×D4—416no—
265C3×D4—42no—
266C3×D4—42no—
267C2×C3×C4—21no—
268C2×C3×C4—21no—
269C2×C3×C4—22no—
270C2×C3×C4—22no—
271C2×C3×C4—22no—
272C2×C3×C4—22no—
273C2×C3×C4—24no—
274C2×C3×C4—42no—
275C2×C3×C4—42no—
276C2×C3×C4C4×D412yes—
277C2×C3×C4C23×C411yescharacteristic
278C2×C3×C4C22×D411yescharacteristic
279C2×C3×C4C2×#4111yescharacteristic
280C2×C3×C4C2×#4111yescharacteristic
281C2×C3×C4C2×#4111yescharacteristic
282C2×C3×C4C2×#4111yescharacteristic
283C2×Dic3—22no—
284C2×Dic3—22no—
285C2×Dic3—22no—
286C2×Dic3—22no—
287C2×Dic3—22no—
288C2×Dic3—48no—
289C2×Dic3—42no—
290C2×Dic3—42no—
291C2×Dic3—42no—
292C2×Dic3—42no—
293C2×Dic3—42no—
294C2×Dic3—42no—
295C2×Dic3—88no—
296C2×Dic3#12114yes—
297C3×C8—22no—
298C3×C8C4×D414yescomplemented
299C3⋊C8—22no—
300C3⋊C8—22no—
301C24—22no—
302C24D3×D411yescharacteristic
303C2×D4—28no—
304C2×D4—22no—
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—22no—
309C2×D4—44no—
310C2×D4—44no—
311C2×D4C22×Dic314yes—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—32no—
318C22×C4—32no—
319C22×C4—64no—
320C22×C4—64no—
321C22×C4—62no—
322C22×C4—62no—
323C22×C4—124no—
324C22×C4—122no—
325C22×C4—124no—
326C22×C4—124no—
327C22×C4D3×D411yescharacteristic
328C22×C4#23712yes—
329C22×C4C22×Dic312yes—
330C22×C4C23×D311yescharacteristic
331D8—28no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—28no—
335C2×C8—32no—
336C2×C8—32no—
337C2×C8—68no—
338C2×C8—61no—
339C2×C8—61no—
340C2×C8—61no—
341C2×C8—61no—
342C2×C8D3×D412yes—
343C2×C8#23712yes—
344C2×C8C22×Dic312yes—
345C2×C8C22×Dic312yes—
346C4⋊C4—22no—
347C4⋊C4—62no—
348C4⋊C4—62no—
349C4⋊C4—62no—
350C4⋊C4—62no—
351C4⋊C4—124no—
352C4⋊C4C22×Dic314yes—
353C22⋊C4—28no—
354C22⋊C4—124no—
355C22⋊C4—124no—
356C22⋊C4—124no—
357C22⋊C4—124no—
358C42—32no—
359C42—32no—
360C42—64no—
361C42—61no—
362C42—61no—
363C42—124no—
364C22×C3—24no—
365C22×C3—22no—
366C22×C3—24no—
367C22×C3—22no—
368C22×C3—44no—
369C22×C3—44no—
370C22×C3—44no—
371C22×C3—82no—
372C22×C3—82no—
373C22×C3—82no—
374C22×C3—82no—
375C22×C3#44214yescomplemented
376C22×C3C2×C4×D411yescharacteristic
377C22×C3C2×#12111yescharacteristic
378C22×C3#53211yescharacteristic
379C22×C3C2×AGL(1,ℤ/8)12yes—
380C22×C3#57512yes—
381C3×C4—22no—
382C3×C4—44no—
383C3×C4C2×C4×D411yescharacteristic
384C3×C4C4×#4111yescharacteristic
385C3×C4#52511yescharacteristic
386C3×C4#52911yescharacteristic
387Dic3—24no—
388Dic3—42no—
389Dic3—82no—
390Dic3—82no—
391C23—22no—
392C23—21no—
393C23—21no—
394C23—24no—
395C23—42no—
396C23—42no—
397C23—42no—
398C23—42no—
399C23—42no—
400C23—42no—
401C23#91114yes—
402C23D4×Dic312yes—
403C23C2×D3×D411yescharacteristic
404D4—24no—
405D4—24no—
406D4—416no—
407D4—42no—
408D4—42no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—24no—
416C2×C4—34no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—62no—
420C2×C4—62no—
421C2×C4—62no—
422C2×C4—62no—
423C2×C4—62no—
424C2×C4—128no—
425C2×C4—122no—
426C2×C4—122no—
427C2×C4—122no—
428C2×C4—122no—
429C2×C4—122no—
430C2×C4—122no—
431C2×C4—248no—
432C2×C4D4×Dic312yes—
433C2×C4C2×D3×D411yescharacteristic
434C2×C4C2×#23711yescharacteristic
435C2×C4C2×#23711yescharacteristic
436C2×C4D3×#4111yescharacteristic
437C2×C4D3×#4111yescharacteristic
438C2×C4C23×Dic311yescharacteristic
439C8—22no—
440C8—62no—
441C8—62no—
442C8D4×Dic314yescomplemented
443C2×C3—24no—
444C2×C3—84no—
445C2×C3C4×#12111yescharacteristic
446C2×C3C2×#44211yescharacteristic
447C2×C3#294311yescharacteristic
448C2×C3#305512yes—
449C2×C3#318012yes—
450C22—24no—
451C22—24no—
452C22—22no—
453C22—22no—
454C22—44no—
455C22—44no—
456C22—44no—
457C22—82no—
458C22—82no—
459C22—82no—
460C22—82no—
461C22#515214yescomplemented
462C22#559611yescharacteristic
463C22D3×#12111yescharacteristic
464C22C2×#91112yes—
465C22#576812yes—
466C22C2×D4×Dic311yescharacteristic
467C4—22no—
468C4—44no—
469C4—64no—
470C4—122no—
471C4—242no—
472C4—242no—
473C4#560011yescharacteristic
474C4#560611yescharacteristic
475C4C2×D4×Dic311yescharacteristic
476C4Dic3×#4111yescharacteristic
477C3#2152611yescharacteristic, complemented, Sylow
478C2—24no—
479C2—84no—
480C2#8070112yes—
481C2#8071512yes—
482C2Dic3×#12111yescharacteristic
483C2C2×#515211yescharacteristic
484C2#8389711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
Fitting, prime radicalC2×C3×#466C2
socleC23×C3C22×D4
cosocleC22×C3×C4C24
hypercenterC2×#466D3
hypocenterC3#21526
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#466

Complements

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

NH ≅ G/Ntype
2. C2×#5152C2semidirect
3. C22×#815C2semidirect
4. Dic3×#121C2semidirect
5. #72039C2semidirect
6. #71970C2semidirect
7. #71681C2semidirect
8. #70613C2semidirect
9. #70605C2semidirect
10. #70187C2semidirect
11. #70152C2semidirect
20. C3×#466C4semidirect
61. C2×C3×D8C2×C4semidirect
64. C22×C3×C8C2×C4semidirect
68. C3×#121C2×C4semidirect
92. C3×#107C2×C4semidirect
93. C3×#102C2×C4semidirect
104. #815D4semidirect
112. #466Dic3semidirect
205. C2×D8C2×Dic3semidirect
213. C22×C8C2×Dic3semidirect
217. #121C2×Dic3semidirect
236. #107C2×Dic3semidirect
241. #102C2×Dic3semidirect
298. C3×C8C4×D4semidirect
375. C22×C3#442semidirect
442. C8D4×Dic3semidirect
461. C22#5152semidirect
477. C3#21526semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4662
C2×#51522
C22×#8152
Dic3×#1212
#720392
#719702
#716812
#706132
#706052
#701872
#701522
#215263