Finite Group Databasebeta

← #10876243

Subgroups — #10876243

Order 768.

954 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#76428C211yesmaximal, characteristic
2#74224C211yesmaximal, characteristic, complemented
3C3×#2073C211yesmaximal, characteristic, complemented
4#70198C212yesmaximal
5#70156C212yesmaximal, complemented
6#69655C211yesmaximal, characteristic, complemented
7#69557C211yesmaximal, characteristic, complemented
8#68876C212yesmaximal, complemented
9#68791C212yesmaximal, complemented
10#68682C211yesmaximal, characteristic, complemented
11#68665C211yesmaximal, characteristic, complemented
12#27594—31nomaximal, Sylow
13C22×D3×D4C2211yescharacteristic
14C2×#892C2211yescharacteristic
15C2×#872C2211yescharacteristic
16C2×C3×#102C2212yes—
17C3×#390C2211yescharacteristic
18C3×#382C2211yescharacteristic, complemented
19#5221C2211yescharacteristic
20C2×#832C2212yes—
21C2×#821C2212yes—
22C2×(C2×D3)⋊C8C2212yes—
23C2×#817C2212yescomplemented
24#4988C2211yescharacteristic
25#4987C2212yes—
26Dic3×C4⋊C4C2212yescomplemented
27C2×#809C2212yes—
28C2×#807C2212yescomplemented
29#4957—22no—
30#4955—22no—
31#4939C2211yescharacteristic
32#4935C2211yescharacteristic, complemented
33#4670—22no—
34#4655—22no—
35C3×#340C2212yescomplemented
36C3×#336C2211yescharacteristic, complemented
37#4550C2212yescomplemented
38#4529C2212yescomplemented
39#4524C2211yescharacteristic, complemented
40#2314—31no—
41#2309—31no—
42#2094—31no—
43#2093—31no—
44#2073—31no—
45#2073D311yescharacteristic, complemented
46#1926—32no—
47#1925—32no—
48#1854—32no—
49#1818—31no—
50#1786—32no—
51C24×D3—21no—
52C22×C3×D4C2311yescharacteristic
53C22×#67—21no—
54C2×D3×D4—22no—
55C2×D3×D4—22no—
56C2×D3×D4—24no—
57C2×D3×D4—42no—
58C22×D12C2311yescharacteristic
59C22×C4×D3C2311yescharacteristic
60C22×C3×C8C2312yes—
61C2×C3×C4⋊C4C2311yescharacteristic
62C2×C3×C4⋊C4C2312yes—
63C2×C3×C22⋊C4—22no—
64C2×C3×C42C2311yescharacteristic
65C2×#217—22no—
66C2×(C2×D3)⋊C4—21no—
67C2×(C2×D3)⋊C4—22no—
68C2×(C2×D3)⋊C4—22no—
69C2×(C2×D3)⋊C4—22no—
70C2×(C2×D3)⋊C4D412yes—
71C2×C4⋊Dic3—22no—
72C2×C4⋊Dic3C2312yes—
73C2×C4⋊Dic3C2311yescharacteristic
74C2×Dic3⋊C4D412yes—
75C2×C4×Dic3—21no—
76C2×C4×Dic3—22no—
77C2×C4×Dic3C2311yescharacteristic
78C22×C3⋊C8C2312yes—
79#892—42no—
80#872—24no—
81C3×#102—24no—
82C3×#95—22no—
83#832—42no—
84C3⋊#95—22no—
85C3⋊#95—22no—
86#821—42no—
87(C2×D3)⋊C8—24no—
88#817—42no—
89#809—24no—
90#807—42no—
91C23×D4—31no—
92C2×#117—31no—
93C2×#117—31no—
94C2×#107—32no—
95C2×#106—32no—
96C2×#102—32no—
97C2×#102—32no—
98C2×#102—32no—
99C2×#102C2×D312yes—
100C2×C22⋊C8—32no—
101#390—31no—
102#390—31no—
103#390—31no—
104#390—62no—
105#390C2×D311yescharacteristic
106#388—62no—
107#387—62no—
108#384—31no—
109#382C2×D311yescharacteristic, complemented
110#380—32no—
111#379—62no—
112C4×C4⋊C4—32no—
113#340—32no—
114#340C2×D312yescomplemented
115#336—32no—
116#336—31no—
117#336C2×D311yescharacteristic, complemented
118C24×C3—21no—
119C23×D3—21no—
120C23×D3—24no—
121C23×D3—42no—
122C23×D3—42no—
123C23×D3—82no—
124C23×D3—82no—
125C23×D3C2×D411yescharacteristic
126C2×C3×D4—22no—
127C2×C3×D4—21no—
128C2×C3×D4—22no—
129C2×C3×D4#4112yes—
130C22×C3×C4—22no—
131C22×C3×C4—21no—
132C22×C3×C4C2×D412yes—
133C22×C3×C4C2411yescharacteristic
134C2×#67—24no—
135C2×#67—42no—
136C2×#67—42no—
137C22×Dic3—22no—
138C22×Dic3—21no—
139C22×Dic3C2×D412yes—
140C22×Dic3C2×D411yescharacteristic
141D3×D4—48no—
142D3×D4—48no—
143C2×D12—22no—
144C2×D12—22no—
145C2×D12—21no—
146C2×D12#4112yes—
147C2×C4×D3—21no—
148C2×C4×D3—21no—
149C2×C4×D3—22no—
150C2×C4×D3#4114yes—
151C2×C3×C8—22no—
152C2×C3×C8—22no—
153C2×C3×C8—22no—
154C3×C4⋊C4—22no—
155C3×C4⋊C4—22no—
156C3×C4⋊C4—22no—
157C3×C22⋊C4—82no—
158C3×C42—22no—
159#217—82no—
160(C2×D3)⋊C4—24no—
161(C2×D3)⋊C4—42no—
162(C2×D3)⋊C4—82no—
163(C2×D3)⋊C4—82no—
164(C2×D3)⋊C4—82no—
165C4⋊Dic3—22no—
166C4⋊Dic3—22no—
167C4⋊Dic3—22no—
168C4⋊Dic3—82no—
169Dic3⋊C4—24no—
170C4×Dic3—22no—
171C4×Dic3—42no—
172C4×Dic3—82no—
173C2×C3⋊C8—22no—
174C2×C3⋊C8—22no—
175C2×C3⋊C8—22no—
176C25—61no—
177C22×D4—31no—
178C22×D4—64no—
179C22×D4—61no—
180C22×D4—62no—
181C22×D4—62no—
182C22×D4—122no—
183C22×D4C22×D311yescharacteristic
184C23×C4—31no—
185C22×C8—32no—
186C22×C8C22×D312yes—
187#117—64no—
188#117—122no—
189C2×C4⋊C4—32no—
190C2×C4⋊C4—31no—
191C2×C4⋊C4—32no—
192C2×C4⋊C4—62no—
193C2×C4⋊C4C22×D311yescharacteristic
194C2×C4⋊C4C22×D312yes—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—32no—
197C2×C22⋊C4—61no—
198C2×C22⋊C4—62no—
199C2×C22⋊C4—62no—
200C2×C22⋊C4—62no—
201C2×C22⋊C4—62no—
202C2×C42—31no—
203C2×C42—61no—
204C2×C42—62no—
205C2×C42C22×D311yescharacteristic
206#107—122no—
207#106—122no—
208#102—24no—
209#102—64no—
210#102—122no—
211#102—122no—
212C22⋊C8—64no—
213#95—22no—
214#95—62no—
215#95—62no—
216C23×C3—22no—
217C23×C3—41no—
218C23×C3—41no—
219C23×C3—42no—
220C23×C3—42no—
221C23×C3C22×D411yescharacteristic
222C22×D3—21no—
223C22×D3—21no—
224C22×D3—22no—
225C22×D3—22no—
226C22×D3—22no—
227C22×D3—48no—
228C22×D3—41no—
229C22×D3—41no—
230C22×D3—42no—
231C22×D3—42no—
232C22×D3—88no—
233C22×D3—88no—
234C22×D3—84no—
235C22×D3—84no—
236C22×D3#12312yes—
237C3×D4—24no—
238C3×D4—42no—
239C2×C3×C4—22no—
240C2×C3×C4—22no—
241C2×C3×C4—22no—
242C2×C3×C4—42no—
243C2×C3×C4—41no—
244C2×C3×C4—42no—
245C2×C3×C4—42no—
246C2×C3×C4—42no—
247C2×C3×C4C2×#4111yescharacteristic
248C2×C3×C4C2×#4112yes—
249C2×C3×C4C2×#4111yescharacteristic
250C2×C3×C4H(5,𝔽2)12yes—
251#67—48no—
252C2×Dic3—21no—
253C2×Dic3—21no—
254C2×Dic3—22no—
255C2×Dic3—22no—
256C2×Dic3—22no—
257C2×Dic3—41no—
258C2×Dic3—42no—
259C2×Dic3—42no—
260C2×Dic3—42no—
261C2×Dic3—42no—
262C2×Dic3#12312yes—
263D12—24no—
264D12—42no—
265C4×D3—24no—
266C4×D3—24no—
267C3×C8—42no—
268C3⋊C8—42no—
269C24—21no—
270C24—31no—
271C24—64no—
272C24—61no—
273C24—122no—
274C24—122no—
275C24—242no—
276C24—242no—
277C2×D4—22no—
278C2×D4—21no—
279C2×D4—22no—
280C2×D4—32no—
281C2×D4—64no—
282C2×D4—61no—
283C2×D4—62no—
284C2×D4—62no—
285C2×D4—122no—
286C2×D4—128no—
287C2×D4—128no—
288C2×D4—122no—
289C2×D4#23712yes—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—31no—
293C22×C4—34no—
294C22×C4—32no—
295C22×C4—61no—
296C22×C4—61no—
297C22×C4—62no—
298C22×C4—62no—
299C22×C4—61no—
300C22×C4D3×D412yes—
301C22×C4C23×D311yescharacteristic
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—62no—
306C2×C8—62no—
307C2×C8—62no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—64no—
312C4⋊C4—62no—
313C4⋊C4—62no—
314C4⋊C4—62no—
315C4⋊C4—242no—
316C22⋊C4—64no—
317C22⋊C4—82no—
318C22⋊C4—122no—
319C22⋊C4—242no—
320C22⋊C4—242no—
321C22⋊C4—242no—
322C22⋊C4—242no—
323C42—22no—
324C42—62no—
325C42—122no—
326C42—242no—
327C22×C3—44no—
328C22×C3—42no—
329C22×C3—82no—
330C22×C3—81no—
331C22×C3—81no—
332C22×C3C2×#12311yescharacteristic
333C22×C3#52511yescharacteristic
334C22×C3#53511yescharacteristic
335C22×C3#57612yes—
336C22×C3#57712yes—
337C2×D3—24no—
338C2×D3—22no—
339C2×D3—42no—
340C2×D3—41no—
341C2×D3—41no—
342C2×D3—44no—
343C2×D3—42no—
344C2×D3—816no—
345C2×D3—82no—
346C2×D3—81no—
347C2×D3—81no—
348C3×C4—42no—
349C3×C4—82no—
350C3×C4—81no—
351C3×C4#55112yes—
352C3×C4#55112yes—
353Dic3—22no—
354Dic3—42no—
355Dic3—82no—
356Dic3—81no—
357C23—22no—
358C23—32no—
359C23—41no—
360C23—41no—
361C23—42no—
362C23—42no—
363C23—61no—
364C23—62no—
365C23—62no—
366C23—61no—
367C23—62no—
368C23—128no—
369C23—121no—
370C23—121no—
371C23—122no—
372C23—122no—
373C23—248no—
374C23—248no—
375C23—244no—
376C23—244no—
377C23C2×D3×D411yescharacteristic
378D4—24no—
379D4—42no—
380D4—64no—
381D4—128no—
382D4—122no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—32no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—61no—
393C2×C4—61no—
394C2×C4—64no—
395C2×C4—64no—
396C2×C4—62no—
397C2×C4—62no—
398C2×C4—62no—
399C2×C4—121no—
400C2×C4—122no—
401C2×C4—122no—
402C2×C4—122no—
403C2×C4—122no—
404C2×C4C2×#23711yescharacteristic
405C2×C4D3×#4112yes—
406C2×C4D3×#4111yescharacteristic
407C2×C4#100912yes—
408C8—42no—
409C8—122no—
410C2×C3—82no—
411C2×C3#248211yescharacteristic
412C2×C3#309612yes—
413C2×C3#311812yes—
414C2×C3#322011yescharacteristic
415C2×C3#322211yescharacteristic
416D3—42no—
417D3—82no—
418C22—42no—
419C22—44no—
420C22—62no—
421C22—64no—
422C22—81no—
423C22—82no—
424C22—81no—
425C22—121no—
426C22—122no—
427C22—121no—
428C22—122no—
429C22—124no—
430C22—2416no—
431C22—241no—
432C22—242no—
433C22—241no—
434C22#560611yescharacteristic
435C22D3×#12311yescharacteristic
436C22#567411yescharacteristic
437C22#577012yes—
438C22#577412yes—
439C4—42no—
440C4—62no—
441C4—81no—
442C4—82no—
443C4—122no—
444C4—242no—
445C4—241no—
446C4#565812yes—
447C4#570512yes—
448C3#2759411yescharacteristic, complemented, Sylow
449C2—82no—
450C2—122no—
451C2—242no—
452C2#7767611yescharacteristic
453C2#8050912yes—
454C2#8137512yes—
455C2#8164611yescharacteristic
456C2#8171911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2073C2
socleC23×C3C22×D4
prime radicalC22×D3×D4C22
hypercenter#2073D3
hypocenterC3#27594
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2073

Complements

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

NH ≅ G/Ntype
2. #74224C2semidirect
3. C3×#2073C2semidirect
5. #70156C2semidirect
6. #69655C2semidirect
7. #69557C2semidirect
8. #68876C2semidirect
9. #68791C2semidirect
10. #68682C2semidirect
11. #68665C2semidirect
18. C3×#382C22semidirect
23. C2×#817C22semidirect
26. Dic3×C4⋊C4C22semidirect
28. C2×#807C22semidirect
32. #4935C22semidirect
35. C3×#340C22semidirect
36. C3×#336C22semidirect
37. #4550C22semidirect
38. #4529C22semidirect
39. #4524C22semidirect
45. #2073D3semidirect
109. #382C2×D3semidirect
114. #340C2×D3semidirect
117. #336C2×D3semidirect
448. C3#27594semidirect

Maximal subgroups

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

subgroupindex
#764282
#742242
C3×#20732
#701982
#701562
#696552
#695572
#688762
#687912
#686822
#686652
#275943