Finite Group Databasebeta

← #10824603

Subgroups — #10824603

Order 768.

1062 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4872C211yesmaximal, characteristic, complemented
2C2×#4718C211yesmaximal, characteristic, complemented
3#73865C212yesmaximal, complemented
4C3×#2076C211yesmaximal, characteristic, complemented
5#70622C211yesmaximal, characteristic, complemented
6#69961C211yesmaximal, characteristic
7#69524C212yesmaximal, complemented
8#68900C212yesmaximal, complemented
9#68862C211yesmaximal, characteristic, complemented
10#68784C211yesmaximal, characteristic, complemented
11#68780C212yesmaximal, complemented
12#23697—31nomaximal, Sylow
13C2×#930C2212yes—
14C22×C4×Dic3C2211yescharacteristic
15C22×D24C2211yescharacteristic
16C2×#893C2212yes—
17C2×#67⋊C4—22no—
18C2×C3×#107C2211yescharacteristic
19C2×C3×C22⋊C8C2211yescharacteristic
20C3×#380C2212yes—
21C2×#830C2211yescharacteristic
22C2×#821C2212yes—
23C2×Dic3⋊C8C2211yescharacteristic
24C2×Dic3⋊C8C2211yescharacteristic
25C2×C8×Dic3C2211yescharacteristic
26C2×C8×Dic3C2211yescharacteristic
27#4988C2212yes—
28Dic3×C4⋊C4C2212yes—
29#4972C2212yes—
30C2×#809C2212yes—
31C2×#809C2212yes—
32C2×#808C2212yes—
33#4958—22no—
34#4872—24no—
35#4718—24no—
36#4669—22no—
37#4666—22no—
38C3×#340C2212yes—
39C3×#340C2211yescharacteristic
40#4551C2211yescharacteristic
41#4529C2212yes—
42C2×C4×D8—31no—
43C2×#431—31no—
44C4×#121—32no—
45#2087—31no—
46#2076D311yescharacteristic, complemented
47#2065—32no—
48#2054—31no—
49#2034—32no—
50#1999—31no—
51C4×#106—31no—
52C4×#102—32no—
53C23×C3×C4C2311yescharacteristic
54C22×#67—22no—
55C23×Dic3D411yescharacteristic
56C23×Dic3D411yescharacteristic
57C22×D12C2312yes—
58C22×C4×D3—22no—
59C22×C3×C8C2311yescharacteristic
60C22×C3×C8C2311yescharacteristic
61C2×C3×C4⋊C4C2312yes—
62C2×C3×C4⋊C4C2312yes—
63C2×C3×C22⋊C4—22no—
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—
70C2×C4⋊Dic3C2312yes—
71C2×Dic3⋊C4—22no—
72C2×C4×Dic3—22no—
73C2×C4×Dic3—22no—
74C2×C4×Dic3—22no—
75C2×C4×Dic3D411yescharacteristic
76C2×C4×Dic3D411yescharacteristic
77C2×C4×Dic3C2311yescharacteristic
78C2×C4×Dic3C2311yescharacteristic
79C22×C3⋊C8C2311yescharacteristic
80C22×C3⋊C8C2311yescharacteristic
81C2×D24—24no—
82C2×D24—22no—
83#893—24no—
84#893—24no—
85#67⋊C4—48no—
86C3×#107—22no—
87C3×C22⋊C8—22no—
88C3×#95—22no—
89C3⋊#95—22no—
90C3⋊#95—22no—
91#830—22no—
92#821—24no—
93Dic3⋊C8—22no—
94Dic3⋊C8—22no—
95C8×Dic3—22no—
96C8×Dic3D414yescomplemented
97#809—24no—
98#809—24no—
99#808—24no—
100C22×D8—31no—
101C2×#121—32no—
102C2×C4×D4—32no—
103C2×C4×D4—62no—
104C22×C42—31no—
105C4×D8—64no—
106#431—64no—
107C2×#107C2×D311yescharacteristic
108C2×#106—32no—
109C2×C4⋊C8—31no—
110C2×C4⋊C8—31no—
111C2×#102—32no—
112C2×#102—32no—
113C2×#102—32no—
114C2×C22⋊C8—31no—
115C2×C22⋊C8C2×D311yescharacteristic
116C2×C4×C8—31no—
117C2×C4×C8—31no—
118#390—32no—
119#389—32no—
120#388—62no—
121#386—62no—
122#380C2×D312yes—
123C4×C4⋊C4—32no—
124C4×C22⋊C4—62no—
125#340—31no—
126#340—32no—
127#340C2×D311yescharacteristic
128#340C2×D312yes—
129C24×C3C2×D411yescharacteristic
130C23×D3—22no—
131C22×C3×C4—21no—
132C22×C3×C4—21no—
133C22×C3×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2411yescharacteristic
139C2×#67—28no—
140C2×#67—44no—
141C2×#67—44no—
142C22×Dic3—22no—
143C22×Dic3—22no—
144C22×Dic3—22no—
145C22×Dic3—22no—
146C22×Dic3—22no—
147C22×Dic3—24no—
148C22×Dic3C2×D411yescharacteristic
149C22×Dic3C2×D411yescharacteristic
150C22×Dic3C2×D411yescharacteristic
151C22×Dic3C2×D411yescharacteristic
152C2×D12—22no—
153C2×D12—22no—
154C2×D12—22no—
155C2×D12—22no—
156C2×D12—22no—
157C2×D12#4114yes—
158C2×C4×D3—44no—
159C2×C4×D3—44no—
160C2×C4×D3—44no—
161C2×C3×C8—22no—
162C2×C3×C8—21no—
163C2×C3×C8—22no—
164C2×C3×C8C2×D412yes—
165C3×C4⋊C4—22no—
166C3×C4⋊C4—22no—
167C3×C4⋊C4—22no—
168C3×C4⋊C4—22no—
169C3×C22⋊C4—44no—
170(C2×D3)⋊C4—28no—
171(C2×D3)⋊C4—44no—
172(C2×D3)⋊C4—44no—
173(C2×D3)⋊C4—44no—
174C4⋊Dic3—22no—
175C4⋊Dic3#4114yes—
176Dic3⋊C4—44no—
177C4×Dic3—21no—
178C4×Dic3—21no—
179C4×Dic3—22no—
180C4×Dic3—22no—
181C4×Dic3—44no—
182C4×Dic3C2×D412yes—
183C4×Dic3#4112yes—
184C2×C3⋊C8—22no—
185C2×C3⋊C8—21no—
186C2×C3⋊C8—22no—
187C2×C3⋊C8C2×D412yes—
188D24—28no—
189C22×D4—32no—
190C22×D4—62no—
191C23×C4—31no—
192C23×C4—31no—
193C23×C4—62no—
194C23×C4C22×D311yescharacteristic
195C2×D8—64no—
196C2×D8—62no—
197C22×C8—31no—
198C22×C8—31no—
199C22×C8C22×D311yescharacteristic
200C22×C8C22×D311yescharacteristic
201#121—64no—
202#121—64no—
203C4×D4—64no—
204C4×D4—64no—
205C4×D4—128no—
206C2×C4⋊C4—32no—
207C2×C4⋊C4—62no—
208C2×C4⋊C4C22×D312yes—
209C2×C4⋊C4C22×D312yes—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—62no—
212C2×C22⋊C4—62no—
213C2×C22⋊C4—62no—
214C2×C22⋊C4—62no—
215C2×C42—31no—
216C2×C42—31no—
217C2×C42—31no—
218C2×C42—31no—
219C2×C42—62no—
220C2×C42—62no—
221C2×C42—62no—
222#107—22no—
223#106—64no—
224C4⋊C8—62no—
225C4⋊C8—62no—
226#102—64no—
227#102—64no—
228#102—64no—
229C22⋊C8—22no—
230C22⋊C8—62no—
231C4×C8—34no—
232C4×C8—62no—
233#95—22no—
234#95—62no—
235#95—62no—
236C23×C3—21no—
237C23×C3—22no—
238C23×C3—21no—
239C23×C3—22no—
240C23×C3#13512yes—
241C23×C3C22×D411yescharacteristic
242C22×D3—24no—
243C22×D3—42no—
244C22×D3—42no—
245C22×D3—42no—
246C22×D3—42no—
247C22×D3—42no—
248C22×D3—42no—
249C2×C3×C4—21no—
250C2×C3×C4—22no—
251C2×C3×C4—41no—
252C2×C3×C4—42no—
253C2×C3×C4—41no—
254C2×C3×C4—42no—
255C2×C3×C4—42no—
256C2×C3×C4—42no—
257C2×C3×C4—42no—
258C2×C3×C4—42no—
259C2×C3×C4—42no—
260C2×C3×C4C22×D411yescharacteristic
261C2×C3×C4C2×#4112yes—
262C2×C3×C4C2×#4111yescharacteristic
263C2×C3×C4C2×#4112yes—
264#67—416no—
265C2×Dic3—28no—
266C2×Dic3—21no—
267C2×Dic3—21no—
268C2×Dic3—21no—
269C2×Dic3—21no—
270C2×Dic3—21no—
271C2×Dic3—24no—
272C2×Dic3—22no—
273C2×Dic3—42no—
274C2×Dic3—42no—
275C2×Dic3—44no—
276C2×Dic3#12112yes—
277C2×Dic3#12112yes—
278C2×Dic3#13512yes—
279C2×Dic3AGL(1,ℤ/8)14yes—
280D12—24no—
281D12—24no—
282D12—42no—
283D12—42no—
284C4×D3—88no—
285C3×C8—22no—
286C3×C8—41no—
287C3⋊C8—22no—
288C3⋊C8—41no—
289C24—62no—
290C24D3×D411yescharacteristic
291C2×D4—34no—
292C2×D4—68no—
293C2×D4—62no—
294C2×D4—62no—
295C2×D4—62no—
296C2×D4—62no—
297C2×D4—62no—
298C2×D4—124no—
299C2×D4—124no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—31no—
307C22×C4—31no—
308C22×C4—31no—
309C22×C4—31no—
310C22×C4—62no—
311C22×C4—62no—
312C22×C4—62no—
313C22×C4—64no—
314C22×C4—62no—
315C22×C4—62no—
316C22×C4—124no—
317C22×C4—124no—
318C22×C4—124no—
319C22×C4D3×D411yescharacteristic
320C22×C4C23×D311yescharacteristic
321D8—68no—
322C2×C8—22no—
323C2×C8—21no—
324C2×C8—22no—
325C2×C8—32no—
326C2×C8—62no—
327C2×C8—61no—
328C2×C8—62no—
329C2×C8D3×D412yes—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—34no—
335C4⋊C4—62no—
336C4⋊C4—124no—
337C22⋊C4—44no—
338C22⋊C4—68no—
339C22⋊C4—124no—
340C22⋊C4—124no—
341C22⋊C4—124no—
342C42—32no—
343C42—32no—
344C42—62no—
345C42—62no—
346C42—61no—
347C42—61no—
348C42—124no—
349C22×C3—24no—
350C22×C3—22no—
351C22×C3—42no—
352C22×C3—41no—
353C22×C3—41no—
354C22×C3C2×#12111yescharacteristic
355C22×C3#52512yes—
356C22×C3C2×#13511yescharacteristic
357C22×C3C2×#13511yescharacteristic
358C22×C3C2×AGL(1,ℤ/8)12yes—
359C2×D3—44no—
360C2×D3—44no—
361C2×D3—44no—
362C2×D3—82no—
363C2×D3—82no—
364C2×D3—82no—
365C2×D3—82no—
366C3×C4—41no—
367C3×C4—82no—
368C3×C4—82no—
369C3×C4#52511yescharacteristic
370C3×C4#52912yes—
371C3×C4#53211yescharacteristic
372Dic3—22no—
373Dic3—22no—
374Dic3—44no—
375Dic3—41no—
376Dic3#46914yes—
377C23—22no—
378C23—21no—
379C23—22no—
380C23—21no—
381C23—64no—
382C23—122no—
383C23—122no—
384C23—122no—
385C23—122no—
386C23—122no—
387C23—122no—
388C23#91912yes—
389C23C2×D3×D411yescharacteristic
390D4—64no—
391D4—64no—
392D4—1216no—
393D4—122no—
394D4—122no—
395C2×C4—22no—
396C2×C4—21no—
397C2×C4—34no—
398C2×C4—32no—
399C2×C4—32no—
400C2×C4—32no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—68no—
411C2×C4—61no—
412C2×C4—61no—
413C2×C4—61no—
414C2×C4—61no—
415C2×C4—64no—
416C2×C4—62no—
417C2×C4—61no—
418C2×C4—124no—
419C2×C4—122no—
420C2×C4—122no—
421C2×C4—248no—
422C2×C4C2×D3×D411yescharacteristic
423C2×C4D3×#4112yes—
424C2×C4D3×#4112yes—
425C2×C4D3×#4111yescharacteristic
426C8—22no—
427C8—41no—
428C8—62no—
429C8—121no—
430C2×C3—42no—
431C2×C3#239211yescharacteristic
432C2×C3#305011yescharacteristic
433C2×C3C2×#46911yescharacteristic
434C2×C3#310512yes—
435C2×C3#310612yes—
436D3—84no—
437C22—24no—
438C22—22no—
439C22—41no—
440C22—42no—
441C22—41no—
442C22—124no—
443C22—124no—
444C22—124no—
445C22—242no—
446C22—242no—
447C22—242no—
448C22—242no—
449C22D3×#12111yescharacteristic
450C22#565612yes—
451C22C2×#91911yescharacteristic
452C22D3×#13511yescharacteristic
453C22D3×AGL(1,ℤ/8)12yes—
454C4—34no—
455C4—41no—
456C4—62no—
457C4—62no—
458C4—82no—
459C4—82no—
460C4—124no—
461C4—121no—
462C4#565611yescharacteristic
463C4#566811yescharacteristic
464C4#570612yes—
465C3#2369711yescharacteristic, complemented, Sylow
466C2—42no—
467C2—244no—
468C2#7745711yescharacteristic
469C2#8075811yescharacteristic
470C2D3×#46911yescharacteristic
471C2#8134912yes—
472C2#8145312yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2076C2
socleC23×C3C22×D4
prime radical#69961C2
hypercenter#2076D3
hypocenterC3#23697
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2076

Complements

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

NH ≅ G/Ntype
1. C2×#4872C2semidirect
2. C2×#4718C2semidirect
3. #73865C2semidirect
4. C3×#2076C2semidirect
5. #70622C2semidirect
7. #69524C2semidirect
8. #68900C2semidirect
9. #68862C2semidirect
10. #68784C2semidirect
11. #68780C2semidirect
46. #2076D3semidirect
96. C8×Dic3D4semidirect
465. C3#23697semidirect

Maximal subgroups

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

subgroupindex
C2×#48722
C2×#47182
#738652
C3×#20762
#706222
#699612
#695242
#689002
#688622
#687842
#687802
#236973