Finite Group Databasebeta

← #10814408

Subgroups — #10814408

Order 768.

906 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×D8C211yesmaximal, characteristic
2#82338C211yesmaximal, characteristic, complemented
3#82333C211yesmaximal, characteristic, complemented
4#72037C212yesmaximal
5#71605C211yesmaximal, characteristic, complemented
6#71605C211yesmaximal, characteristic, complemented
7#70609C212yesmaximal, complemented
8#70187C212yesmaximal, complemented
9#70152C212yesmaximal, complemented
10#69763C211yesmaximal, characteristic, complemented
11#69757C211yesmaximal, characteristic, complemented
12#22764—31nomaximal, Sylow
13C22×C3×D8C2211yescharacteristic
14C2×C3×C4×D4C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C2×D4×Dic3C2212yes—
17C3×C4×D8C418yescomplemented
18C2×C3×#107C2211yescharacteristic
19C2×C3×#102C2211yescharacteristic
20C2×C3×#102C2211yescharacteristic
21C2×C3×C4×C8C2211yescharacteristic
22#5221C2212yes—
23C2×#832C2212yes—
24C2×#832C2212yes—
25#5212—21no—
26#5211—21no—
27C4×#217—21no—
28C2×C3⋊#95—21no—
29C2×#815C2212yes—
30#4984C2212yes—
31Dic3×C4⋊C4C2212yes—
32#4951—22no—
33#4950—22no—
34#4936C2211yescharacteristic
35#4936C2211yescharacteristic
36C4×C4⋊Dic3C2211yescharacteristic
37C2×C4⋊(C3⋊C8)C2211yescharacteristic
38#4551C2212yes—
39#4551C2212yes—
40#4550C2212yes—
41#4529C2212yes—
42#4529C2212yes—
43C2×C4×D8D311yescharacteristic
44#2357—31no—
45#2349—31no—
46#1936—32no—
47#1918—32no—
48#1879—32no—
49#1835—31no—
50#1804—31no—
51#1804—31no—
52#1770—31no—
53#1762—32no—
54C22×C3×D4C2311yescharacteristic
55C22×C3×D4C2311yescharacteristic
56C23×C3×C4—21no—
57C23×C3×C4—21no—
58C23×Dic3—22no—
59C2×C3×D8—22no—
60C2×C3×D8—22no—
61C2×C3×D8C2×C414yes—
62C22×C3×C8C2311yescharacteristic
63C22×C3×C8C2311yescharacteristic
64C3×C4×D4—22no—
65C3×C4×D4—22no—
66C3×C4×D4C2×C414yes—
67C3×C4×D4C2×C414yescomplemented
68C2×C3×C4⋊C4C2311yescharacteristic
69C2×C3×C4⋊C4C2311yescharacteristic
70C2×C3×C22⋊C4—21no—
71C2×C3×C22⋊C4—21no—
72C2×C3×C42C2311yescharacteristic
73C2×#217—22no—
74C2×#217—22no—
75C2×#217—22no—
76D4×Dic3—24no—
77D4×Dic3—42no—
78C2×C4⋊Dic3—22no—
79C2×C4⋊Dic3D412yes—
80C2×C4⋊Dic3C2312yes—
81C2×C4⋊Dic3C2312yes—
82C2×C4×Dic3—22no—
83C2×C4×Dic3C2312yes—
84C22×C3⋊C8C2312yes—
85C3×#107C2×C414yescomplemented
86C3×#102C2×C414yes—
87C3×#102C2×C414yescomplemented
88C3×C4×C8C2×C414yescomplemented
89#832—24no—
90#832—42no—
91C3⋊#95—21no—
92C3⋊#95—21no—
93C3⋊#95—21no—
94C3⋊#95—21no—
95C3⋊#95—22no—
96C3⋊#95—22no—
97C3⋊#95D411yescharacteristic
98C3⋊#95D411yescharacteristic
99#815—42no—
100C4⋊(C3⋊C8)—22no—
101C22×D8C2×D311yescharacteristic
102C2×C4×D4—32no—
103C2×C4×D4C2×D311yescharacteristic
104C2×C4×D4C2×D311yescharacteristic
105C4×D8Dic318yescomplemented
106C2×#107C2×D311yescharacteristic
107C2×C4⋊C8—31no—
108C2×#102—32no—
109C2×#102—32no—
110C2×#102C2×D311yescharacteristic
111C2×#102C2×D311yescharacteristic
112C2×#97—32no—
113C2×C4×C8C2×D311yescharacteristic
114#390—32no—
115#389—32no—
116#388—62no—
117#385—62no—
118#382—31no—
119#382—31no—
120#381—61no—
121#380—61no—
122C4×C4⋊C4—31no—
123C4×C4⋊C4—32no—
124C4×C22⋊C4—61no—
125C2×#95—61no—
126#340—32no—
127#340—32no—
128#340—32no—
129#340—32no—
130#336—32no—
131C24×C3—21no—
132C24×C3—21no—
133C2×C3×D4—21no—
134C2×C3×D4—21no—
135C2×C3×D4—21no—
136C2×C3×D4—21no—
137C2×C3×D4—21no—
138C2×C3×D4—21no—
139C2×C3×D4—21no—
140C2×C3×D4—21no—
141C2×C3×D4C22×C412yes—
142C2×C3×D4C22×C412yes—
143C2×C3×D4#4112yes—
144C2×C3×D4#4112yes—
145C22×C3×C4—21no—
146C22×C3×C4—21no—
147C22×C3×C4—24no—
148C22×C3×C4—24no—
149C22×C3×C4—82no—
150C22×C3×C4—82no—
151C22×C3×C4C2×D411yescharacteristic
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2411yescharacteristic
154C22×Dic3—22no—
155C22×Dic3—22no—
156C22×Dic3—44no—
157C22×Dic3—44no—
158C22×Dic3—82no—
159C22×Dic3C2×D412yes—
160C22×Dic3C2×D412yes—
161C3×D8—28no—
162C2×C3×C8—21no—
163C2×C3×C8—21no—
164C2×C3×C8—21no—
165C2×C3×C8—21no—
166C2×C3×C8C22×C412yes—
167C2×C3×C8C22×C412yes—
168C3×C4⋊C4—21no—
169C3×C4⋊C4—21no—
170C3×C4⋊C4C22×C412yes—
171C3×C4⋊C4C22×C412yes—
172C3×C22⋊C4—24no—
173C3×C22⋊C4—24no—
174C3×C42C22×C412yes—
175C3×C42#4112yes—
176#217—44no—
177#217—44no—
178#217—82no—
179C4⋊Dic3—22no—
180C4⋊Dic3—22no—
181C4⋊Dic3—22no—
182C4⋊Dic3—22no—
183C4⋊Dic3—42no—
184C4⋊Dic3—82no—
185C4×Dic3—22no—
186C4×Dic3—22no—
187C4×Dic3—82no—
188C2×C3⋊C8—22no—
189C2×C3⋊C8—22no—
190C2×C3⋊C8—22no—
191C22×D4C22×D311yescharacteristic
192C22×D4C22×D311yescharacteristic
193C23×C4—21no—
194C23×C4—21no—
195C23×C4—62no—
196C2×D8—22no—
197C2×D8—22no—
198C2×D8C2×Dic314yes—
199C22×C8—32no—
200C22×C8C22×D311yescharacteristic
201C22×C8C22×D311yescharacteristic
202C4×D4—22no—
203C4×D4—22no—
204C4×D4—64no—
205C4×D4—122no—
206C4×D4C2×Dic314yes—
207C4×D4C2×Dic314yescomplemented
208C2×C4⋊C4—32no—
209C2×C4⋊C4—32no—
210C2×C4⋊C4—32no—
211C2×C4⋊C4—62no—
212C2×C4⋊C4C22×D311yescharacteristic
213C2×C4⋊C4C22×D311yescharacteristic
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—62no—
217C2×C22⋊C4—62no—
218C2×C22⋊C4—62no—
219C2×C42—32no—
220C2×C42—62no—
221C2×C42C22×D311yescharacteristic
222#107C2×Dic314yescomplemented
223C4⋊C8—62no—
224#102—64no—
225#102—122no—
226#102C2×Dic314yes—
227#102C2×Dic314yescomplemented
228#97—122no—
229C4×C8C2×Dic314yescomplemented
230#95—31no—
231#95—31no—
232#95—62no—
233#95—62no—
234#95—61no—
235#95—61no—
236#95—61no—
237#95—61no—
238C23×C3—22no—
239C23×C3—22no—
240C23×C3—22no—
241C23×C3—22no—
242C23×C3—42no—
243C23×C3—42no—
244C23×C3—41no—
245C23×C3—41no—
246C23×C3—42no—
247C23×C3—42no—
248C23×C3C22×D411yescharacteristic
249C3×D4—22no—
250C3×D4—22no—
251C3×D4—22no—
252C3×D4—22no—
253C3×D4—22no—
254C3×D4—41no—
255C3×D4—41no—
256C3×D4#11714yes—
257C2×C3×C4—22no—
258C2×C3×C4—22no—
259C2×C3×C4—22no—
260C2×C3×C4—22no—
261C2×C3×C4—42no—
262C2×C3×C4—42no—
263C2×C3×C4—84no—
264C2×C3×C4—84no—
265C2×C3×C4C4×D412yes—
266C2×C3×C4C4×D412yes—
267C2×C3×C4C23×C411yescharacteristic
268C2×C3×C4C2×#4111yescharacteristic
269C2×C3×C4C2×#4111yescharacteristic
270C2×C3×C4C2×#4111yescharacteristic
271C2×C3×C4H(5,𝔽2)11yescharacteristic
272C2×C3×C42−1+411yescharacteristic
273C2×Dic3—22no—
274C2×Dic3—22no—
275C2×Dic3—22no—
276C2×Dic3—22no—
277C2×Dic3—22no—
278C2×Dic3—22no—
279C2×Dic3—42no—
280C2×Dic3—42no—
281C2×Dic3—42no—
282C2×Dic3—42no—
283C2×Dic3—42no—
284C2×Dic3—42no—
285C2×Dic3—88no—
286C3×C8—22no—
287C3×C8—22no—
288C3⋊C8—42no—
289C24—21no—
290C24—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4#23512yes—
300C2×D4#23512yes—
301C2×D4C22×Dic312yes—
302C2×D4C22×Dic312yes—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—32no—
308C22×C4—32no—
309C22×C4—62no—
310C22×C4—62no—
311C22×C4—82no—
312C22×C4—82no—
313C22×C4—124no—
314C22×C4—124no—
315C22×C4—242no—
316C22×C4D3×D411yescharacteristic
317C22×C4D3×D411yescharacteristic
318C22×C4C23×D311yescharacteristic
319D8—28no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—62no—
325C2×C8—62no—
326C2×C8—62no—
327C2×C8C22×Dic312yes—
328C2×C8C22×Dic312yes—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—62no—
332C4⋊C4—62no—
333C4⋊C4—62no—
334C4⋊C4—62no—
335C4⋊C4—122no—
336C4⋊C4—242no—
337C4⋊C4C22×Dic312yes—
338C4⋊C4C22×Dic312yes—
339C22⋊C4—24no—
340C22⋊C4—24no—
341C22⋊C4—124no—
342C22⋊C4—124no—
343C22⋊C4—242no—
344C42—62no—
345C42—62no—
346C42—242no—
347C42#23712yes—
348C42C22×Dic312yes—
349C22×C3—24no—
350C22×C3—44no—
351C22×C3—42no—
352C22×C3—42no—
353C22×C3—44no—
354C22×C3—42no—
355C22×C3—42no—
356C22×C3—42no—
357C22×C3—82no—
358C22×C3—82no—
359C22×C3C2×C4×D411yescharacteristic
360C22×C3#54611yescharacteristic
361C22×C3#54811yescharacteristic
362C22×C3#57512yes—
363C22×C3#57612yes—
364C3×C4—22no—
365C3×C4—22no—
366C3×C4—42no—
367C3×C4—42no—
368C3×C4C2×#11711yescharacteristic
369C3×C4#52011yescharacteristic
370C3×C4#55211yescharacteristic
371C3×C4#55311yescharacteristic
372Dic3—42no—
373Dic3—42no—
374Dic3—82no—
375Dic3—82no—
376C23—22no—
377C23—22no—
378C23—22no—
379C23—22no—
380C23—42no—
381C23—42no—
382C23—42no—
383C23—42no—
384C23—41no—
385C23—41no—
386C23C2×D3×D411yescharacteristic
387D4—22no—
388D4—22no—
389D4—22no—
390D4—22no—
391D4—22no—
392D4—41no—
393D4—41no—
394D4#92614yes—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—62no—
402C2×C4—62no—
403C2×C4—62no—
404C2×C4—62no—
405C2×C4—62no—
406C2×C4—62no—
407C2×C4—84no—
408C2×C4—84no—
409C2×C4—122no—
410C2×C4—122no—
411C2×C4—122no—
412C2×C4—122no—
413C2×C4—122no—
414C2×C4—122no—
415C2×C4—248no—
416C2×C4D4×Dic312yes—
417C2×C4D4×Dic312yes—
418C2×C4C2×#23511yescharacteristic
419C2×C4C2×#23511yescharacteristic
420C2×C4C2×#23711yescharacteristic
421C2×C4#100911yescharacteristic
422C2×C4#101011yescharacteristic
423C2×C4C23×Dic311yescharacteristic
424C8—22no—
425C8—22no—
426C8—122no—
427C2×C3—44no—
428C2×C3—82no—
429C2×C3#235211yescharacteristic
430C2×C3#294311yescharacteristic
431C2×C3#294811yescharacteristic
432C2×C3#332212yes—
433C2×C3#337012yes—
434C22—24no—
435C22—42no—
436C22—42no—
437C22—44no—
438C22—44no—
439C22—42no—
440C22—42no—
441C22—42no—
442C22—82no—
443C22—82no—
444C22#555011yescharacteristic
445C22#555211yescharacteristic
446C22#575712yes—
447C22#576812yes—
448C22C2×D4×Dic311yescharacteristic
449C4—22no—
450C4—22no—
451C4—42no—
452C4—42no—
453C4—122no—
454C4—122no—
455C4—242no—
456C4—242no—
457C4#554111yescharacteristic
458C4#556511yescharacteristic
459C4C2×#92611yescharacteristic
460C4#582711yescharacteristic
461C3#2276411yescharacteristic, complemented, Sylow
462C2—44no—
463C2—82no—
464C2#7957212yes—
465C2#7961112yes—
466C2#8233511yescharacteristic
467C2#8382311yescharacteristic
468C2#8389711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC2×C3×C4×D8C2
socleC23×C3C22×D4
prime radicalC22×C3×D8C22
cosocleC22×C3×C4C24
hypercenterC2×C4×D8D3
hypocenterC3#22764
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×D8

Complements

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

NH ≅ G/Ntype
2. #82338C2semidirect
3. #82333C2semidirect
5. #71605C2semidirect
6. #71605C2semidirect
7. #70609C2semidirect
8. #70187C2semidirect
9. #70152C2semidirect
10. #69763C2semidirect
11. #69757C2semidirect
17. C3×C4×D8C4semidirect
67. C3×C4×D4C2×C4semidirect
85. C3×#107C2×C4semidirect
87. C3×#102C2×C4semidirect
88. C3×C4×C8C2×C4semidirect
105. C4×D8Dic3semidirect
207. C4×D4C2×Dic3semidirect
222. #107C2×Dic3semidirect
227. #102C2×Dic3semidirect
229. C4×C8C2×Dic3semidirect
461. C3#22764semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×D82
#823382
#823332
#720372
#716052
#706092
#701872
#701522
#697632
#697572
#227643