Finite Group Databasebeta

← #10824666

Subgroups — #10824666

Order 768.

932 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4872C211yesmaximal, characteristic, complemented
2C2×#4720C211yesmaximal, characteristic, complemented
3#73872C212yesmaximal, complemented
4C3×#2075C211yesmaximal, characteristic, complemented
5#70621C211yesmaximal, characteristic, complemented
6#69961C211yesmaximal, characteristic
7#69524C212yesmaximal, complemented
8#68900C212yesmaximal, complemented
9#68862C211yesmaximal, characteristic, complemented
10#68784C211yesmaximal, characteristic, complemented
11#68782C212yesmaximal, complemented
12#23702—31nomaximal, Sylow
13C2×#930C2212yes—
14C2×#926C2211yescharacteristic
15C22×D24C2211yescharacteristic
16C2×#893C2212yes—
17C2×C3×#107C2211yescharacteristic
18C2×C3×C22⋊C8C2211yescharacteristic
19C3×#380C2212yes—
20C2×#830C2211yescharacteristic
21C2×#821C2212yes—
22C2×#815C2211yescharacteristic
23C2×Dic3⋊C8C2211yescharacteristic
24C2×Dic3⋊C8C2211yescharacteristic
25C2×C8×Dic3C2211yescharacteristic
26#4988C2212yes—
27Dic3×C4⋊C4C2212yes—
28#4972C2212yes—
29C2×#809C2212yes—
30C2×#809C2212yes—
31C2×#808C2212yes—
32#4957—22no—
33#4956—22no—
34#4872—24no—
35#4720—24no—
36#4670—22no—
37#4668—22no—
38C3×#340C2212yes—
39C3×#340C2211yescharacteristic
40#4551C2211yescharacteristic
41#4529C2212yes—
42C2×C4×D8—31no—
43C2×#433—31no—
44#2324—32no—
45#2086—31no—
46#2075D311yescharacteristic, complemented
47#2065—32no—
48#2054—31no—
49#2034—32no—
50#1999—31no—
51C4×#106—31no—
52#1762—32no—
53C23×C3×C4C2311yescharacteristic
54C22×#67—22no—
55C22×D12C2312yes—
56C22×C4×D3—22no—
57C22×C3×C8C2311yescharacteristic
58C22×C3×C8C2311yescharacteristic
59C2×C3×C4⋊C4C2312yes—
60C2×C3×C4⋊C4C2312yes—
61C2×C3×C22⋊C4—22no—
62C2×#217D411yescharacteristic
63C2×#217D411yescharacteristic
64#930—24no—
65#930—24no—
66C2×(C2×D3)⋊C4—22no—
67C2×(C2×D3)⋊C4—22no—
68C2×(C2×D3)⋊C4—22no—
69C2×(C2×D3)⋊C4—22no—
70#926—22no—
71#926—22no—
72C2×C4⋊Dic3D411yescharacteristic
73C2×C4⋊Dic3D411yescharacteristic
74C2×C4⋊Dic3C2312yes—
75C2×Dic3⋊C4—22no—
76C2×C4×Dic3—22no—
77C2×C4×Dic3C2311yescharacteristic
78C2×C4×Dic3C2311yescharacteristic
79C22×C3⋊C8C2311yescharacteristic
80C22×C3⋊C8C2311yescharacteristic
81C2×D24—24no—
82C2×D24—22no—
83#893—24no—
84#893—24no—
85C3×#107—22no—
86C3×C22⋊C8—22no—
87C3×#95—22no—
88C3⋊#95—22no—
89C3⋊#95—22no—
90#830—22no—
91#821—24no—
92#815—22no—
93Dic3⋊C8—22no—
94Dic3⋊C8—22no—
95C8×Dic3D414yescomplemented
96#809—24no—
97#809—24no—
98#808—24no—
99C22×D8—31no—
100C2×#121—32no—
101C2×C4×D4—32no—
102C2×#117—31no—
103C4×D8—64no—
104#433—64no—
105C2×#107C2×D311yescharacteristic
106C2×#106—32no—
107C2×C4⋊C8—31no—
108C2×C4⋊C8—31no—
109C2×#102—32no—
110C2×#102—32no—
111C2×#102—32no—
112C2×C22⋊C8—31no—
113C2×C22⋊C8C2×D311yescharacteristic
114C2×#97—31no—
115C2×C4×C8—31no—
116#390—32no—
117#390—62no—
118#389—32no—
119#388—62no—
120#386—62no—
121#385—62no—
122#380C2×D312yes—
123C4×C4⋊C4—32no—
124#340—31no—
125#340—32no—
126#340C2×D311yescharacteristic
127#340C2×D312yes—
128C24×C3C2×D411yescharacteristic
129C23×D3—22no—
130C22×C3×C4—21no—
131C22×C3×C4—21no—
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4—22no—
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2411yescharacteristic
138C2×#67—28no—
139C2×#67—82no—
140C2×#67—82no—
141C22×Dic3—22no—
142C22×Dic3C2×D411yescharacteristic
143C22×Dic3C2×D411yescharacteristic
144C22×Dic3C2×D411yescharacteristic
145C22×Dic3C2×D411yescharacteristic
146C2×D12—22no—
147C2×D12—22no—
148C2×D12—22no—
149C2×D12—22no—
150C2×D12—22no—
151C2×D12#4114yes—
152C2×C4×D3—44no—
153C2×C4×D3—82no—
154C2×C4×D3—82no—
155C2×C3×C8—22no—
156C2×C3×C8—21no—
157C2×C3×C8—22no—
158C2×C3×C8C2×D412yes—
159C3×C4⋊C4—22no—
160C3×C4⋊C4—22no—
161C3×C4⋊C4—22no—
162C3×C4⋊C4—22no—
163C3×C22⋊C4—82no—
164#217—22no—
165#217—22no—
166(C2×D3)⋊C4—28no—
167(C2×D3)⋊C4—44no—
168(C2×D3)⋊C4—82no—
169(C2×D3)⋊C4—82no—
170C4⋊Dic3—22no—
171C4⋊Dic3—22no—
172C4⋊Dic3—22no—
173C4⋊Dic3#4114yes—
174Dic3⋊C4—82no—
175C4×Dic3—21no—
176C4×Dic3—21no—
177C4×Dic3—82no—
178C4×Dic3C2×D412yes—
179C4×Dic3#4112yes—
180C2×C3⋊C8—22no—
181C2×C3⋊C8—21no—
182C2×C3⋊C8—22no—
183C2×C3⋊C8C2×D412yes—
184D24—28no—
185C22×D4—32no—
186C22×D4—62no—
187C23×C4—62no—
188C23×C4C22×D311yescharacteristic
189C2×D8—64no—
190C2×D8—62no—
191C22×C8—31no—
192C22×C8—31no—
193C22×C8C22×D311yescharacteristic
194C22×C8C22×D311yescharacteristic
195#121—64no—
196#121—64no—
197C4×D4—64no—
198C4×D4—64no—
199#117—62no—
200#117—62no—
201C2×C4⋊C4—32no—
202C2×C4⋊C4—31no—
203C2×C4⋊C4—31no—
204C2×C4⋊C4—62no—
205C2×C4⋊C4C22×D312yes—
206C2×C4⋊C4C22×D312yes—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—31no—
209C2×C22⋊C4—31no—
210C2×C22⋊C4—62no—
211C2×C22⋊C4—62no—
212C2×C22⋊C4—62no—
213C2×C22⋊C4—62no—
214C2×C42—31no—
215C2×C42—31no—
216C2×C42—62no—
217#107—22no—
218#106—64no—
219C4⋊C8—62no—
220C4⋊C8—62no—
221#102—64no—
222#102—64no—
223#102—64no—
224C22⋊C8—22no—
225C22⋊C8—62no—
226#97—62no—
227C4×C8—34no—
228#95—22no—
229#95—62no—
230#95—62no—
231C23×C3—21no—
232C23×C3—22no—
233C23×C3—21no—
234C23×C3—21no—
235C23×C3—22no—
236C23×C3C22×D411yescharacteristic
237C22×D3—24no—
238C22×D3—42no—
239C22×D3—42no—
240C22×D3—42no—
241C22×D3—42no—
242C22×D3—42no—
243C22×D3—42no—
244C2×C3×C4—21no—
245C2×C3×C4—22no—
246C2×C3×C4—41no—
247C2×C3×C4—42no—
248C2×C3×C4—41no—
249C2×C3×C4—42no—
250C2×C3×C4—42no—
251C2×C3×C4—42no—
252C2×C3×C4—42no—
253C2×C3×C4—42no—
254C2×C3×C4—42no—
255C2×C3×C4C22×D411yescharacteristic
256C2×C3×C4C2×#4112yes—
257C2×C3×C4C2×#4111yescharacteristic
258C2×C3×C4C2×#4112yes—
259#67—88no—
260C2×Dic3—21no—
261C2×Dic3—21no—
262C2×Dic3—22no—
263C2×Dic3—21no—
264C2×Dic3—21no—
265C2×Dic3—24no—
266C2×Dic3—21no—
267C2×Dic3—21no—
268C2×Dic3—22no—
269C2×Dic3—42no—
270C2×Dic3—42no—
271C2×Dic3#12112yes—
272C2×Dic3#12112yes—
273D12—24no—
274D12—24no—
275D12—42no—
276D12—42no—
277C4×D3—88no—
278C3×C8—22no—
279C3×C8—41no—
280C3⋊C8—22no—
281C3⋊C8—41no—
282C24—62no—
283C24D3×D411yescharacteristic
284C2×D4—34no—
285C2×D4—68no—
286C2×D4—62no—
287C2×D4—62no—
288C2×D4—62no—
289C2×D4—62no—
290C2×D4—62no—
291C2×D4—242no—
292C2×D4—242no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—31no—
300C22×C4—31no—
301C22×C4—31no—
302C22×C4—31no—
303C22×C4—62no—
304C22×C4—124no—
305C22×C4—242no—
306C22×C4—242no—
307C22×C4D3×D411yescharacteristic
308C22×C4C23×D311yescharacteristic
309D8—68no—
310C2×C8—22no—
311C2×C8—21no—
312C2×C8—22no—
313C2×C8—32no—
314C2×C8—62no—
315C2×C8—61no—
316C2×C8—62no—
317C2×C8D3×D412yes—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—34no—
323C4⋊C4—62no—
324C4⋊C4—62no—
325C4⋊C4—62no—
326C4⋊C4—242no—
327C22⋊C4—68no—
328C22⋊C4—62no—
329C22⋊C4—62no—
330C22⋊C4—82no—
331C22⋊C4—124no—
332C22⋊C4—242no—
333C22⋊C4—242no—
334C42—32no—
335C42—32no—
336C42—61no—
337C42—61no—
338C42—242no—
339C22×C3—42no—
340C22×C3—42no—
341C22×C3—41no—
342C22×C3—41no—
343C22×C3—41no—
344C22×C3C2×#12111yescharacteristic
345C22×C3#52512yes—
346C22×C3#57611yescharacteristic
347C22×C3#57611yescharacteristic
348C22×C3#57712yes—
349C2×D3—44no—
350C2×D3—44no—
351C2×D3—44no—
352C2×D3—82no—
353C2×D3—82no—
354C2×D3—82no—
355C2×D3—82no—
356C3×C4—41no—
357C3×C4—82no—
358C3×C4—82no—
359C3×C4#52511yescharacteristic
360C3×C4#52912yes—
361C3×C4#53211yescharacteristic
362Dic3—22no—
363Dic3—22no—
364Dic3—44no—
365Dic3—41no—
366Dic3—41no—
367C23—22no—
368C23—21no—
369C23—21no—
370C23—22no—
371C23—21no—
372C23—64no—
373C23—122no—
374C23—122no—
375C23—122no—
376C23—122no—
377C23—122no—
378C23—122no—
379C23C2×D3×D411yescharacteristic
380D4—64no—
381D4—64no—
382D4—122no—
383D4—122no—
384D4—248no—
385C2×C4—22no—
386C2×C4—21no—
387C2×C4—32no—
388C2×C4—32no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—61no—
399C2×C4—61no—
400C2×C4—62no—
401C2×C4—64no—
402C2×C4—61no—
403C2×C4—61no—
404C2×C4—62no—
405C2×C4—61no—
406C2×C4—61no—
407C2×C4—122no—
408C2×C4—122no—
409C2×C4—248no—
410C2×C4C2×D3×D411yescharacteristic
411C2×C4D3×#4112yes—
412C2×C4D3×#4112yes—
413C2×C4D3×#4111yescharacteristic
414C8—22no—
415C8—41no—
416C8—62no—
417C8—121no—
418C2×C3—81no—
419C2×C3#239211yescharacteristic
420C2×C3#305811yescharacteristic
421C2×C3#306011yescharacteristic
422C2×C3#311812yes—
423C2×C3#311912yes—
424D3—84no—
425C22—41no—
426C22—41no—
427C22—42no—
428C22—42no—
429C22—41no—
430C22—124no—
431C22—124no—
432C22—124no—
433C22—242no—
434C22—242no—
435C22—242no—
436C22—242no—
437C22D3×#12111yescharacteristic
438C22#565612yes—
439C22#575711yescharacteristic
440C22#576911yescharacteristic
441C22#577812yes—
442C4—41no—
443C4—62no—
444C4—62no—
445C4—82no—
446C4—82no—
447C4—121no—
448C4—121no—
449C4—124no—
450C4#565611yescharacteristic
451C4#566811yescharacteristic
452C4#570612yes—
453C3#2370211yescharacteristic, complemented, Sylow
454C2—81no—
455C2—244no—
456C2#7745711yescharacteristic
457C2#8071611yescharacteristic
458C2#8090611yescharacteristic
459C2#8127212yes—
460C2#8149612yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2075C2
socleC23×C3C22×D4
prime radical#69961C2
hypercenter#2075D3
hypocenterC3#23702
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2075

Complements

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

NH ≅ G/Ntype
1. C2×#4872C2semidirect
2. C2×#4720C2semidirect
3. #73872C2semidirect
4. C3×#2075C2semidirect
5. #70621C2semidirect
7. #69524C2semidirect
8. #68900C2semidirect
9. #68862C2semidirect
10. #68784C2semidirect
11. #68782C2semidirect
46. #2075D3semidirect
95. C8×Dic3D4semidirect
453. C3#23702semidirect

Maximal subgroups

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

subgroupindex
C2×#48722
C2×#47202
#738722
C3×#20752
#706212
#699612
#695242
#689002
#688622
#687842
#687822
#237023