Finite Group Databasebeta

← #10694319

Subgroups — #10694319

Order 768.

742 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2850C211yesmaximal, characteristic
2C2×#4538C212yesmaximal, complemented
3C2×#4537C212yesmaximal
4C4×C3⋊#95C211yesmaximal, characteristic, complemented
5#71530C211yesmaximal, characteristic, complemented
6#9713—31nomaximal, Sylow
7C2×C3×#124C411yescharacteristic, complemented
8C2×C3×C22⋊Q8C2211yescharacteristic
9C2×C3×#121C2211yescharacteristic
10C22×C3×C42C2211yescharacteristic
11C22×C4×Dic3—21no—
12C3×#391C411yescharacteristic, complemented
13C3×#390C411yescharacteristic, complemented
14C3×#388—21no—
15C3×#388—21no—
16C3×#386—21no—
17C3×#386—21no—
18C3×#382C411yescharacteristic, complemented
19C2×C3⋊#95—21no—
20C2×C3⋊#95C2212yes—
21C2×#830C2212yes—
22C2×C4⋊(C3⋊C8)—21no—
23#4538—24no—
24#4537—24no—
25#4524—21no—
26#2850D311yescharacteristic
27#1797—31no—
28C2×#328—32no—
29C2×#327—32no—
30C4×#95—31no—
31C22×C3×Q8C2×C411yescharacteristic
32C22×C3×D4—21no—
33C22×C3×D4C2×C411yescharacteristic
34C23×C3×C4D411yescharacteristic
35C23×C3×C4D411yescharacteristic
36C23×C3×C4C2311yescharacteristic
37C23×Dic3—22no—
38C3×#124—42no—
39C3×C22⋊Q8—22no—
40C3×C22⋊Q8—22no—
41C3×#121—22no—
42C3×#121—22no—
43C2×C3×C4⋊C4—21no—
44C2×C3×C4⋊C4C2×C411yescharacteristic
45C2×C3×C4⋊C4C2×C411yescharacteristic
46C2×C3×C22⋊C4—21no—
47C2×C3×C22⋊C4—21no—
48C2×C3×C22⋊C4—21no—
49C2×C3×C22⋊C4—21no—
50C2×C3×C22⋊C4—21no—
51C2×C3×C22⋊C4—21no—
52C2×C3×C42—42no—
53C2×C3×C42C2×C411yescharacteristic
54C2×C3×C42C2×C411yescharacteristic
55C2×C3×C42D411yescharacteristic
56C2×C3×C42D411yescharacteristic
57C2×C4×Dic3—24no—
58C2×C4×Dic3—42no—
59C2×C4×Dic3—42no—
60C22×C3⋊C8—22no—
61C3×#95—21no—
62C3×#95—21no—
63C3×#95—21no—
64C3×#95—21no—
65C3⋊#95—24no—
66C3⋊#95—22no—
67C3⋊#95D414yes—
68#830—22no—
69#830—22no—
70C4⋊(C3⋊C8)—42no—
71C2×#124Dic311yescharacteristic, complemented
72C2×C22⋊Q8C2×D311yescharacteristic
73C2×#121C2×D311yescharacteristic
74C22×C42—61no—
75C22×C42C2×D311yescharacteristic
76C2×C4⋊C8—61no—
77C2×C22⋊C8—32no—
78#391Dic311yescharacteristic, complemented
79#390Dic311yescharacteristic, complemented
80#388—21no—
81#388—21no—
82#386—21no—
83#386—21no—
84#382Dic311yescharacteristic, complemented
85C2×#95—32no—
86C2×#95—61no—
87#336—61no—
88#328—64no—
89#327—64no—
90C24×C3—21no—
91C24×C3C2×D411yescharacteristic
92C2×C3×Q8—22no—
93C2×C3×Q8—41no—
94C2×C3×Q8—41no—
95C2×C3×D4—22no—
96C2×C3×D4—41no—
97C2×C3×D4—41no—
98C2×C3×D4—42no—
99C2×C3×D4—42no—
100C2×C3×D4—41no—
101C2×C3×D4—41no—
102C22×C3×C4—21no—
103C22×C3×C4—21no—
104C22×C3×C4—22no—
105C22×C3×C4—22no—
106C22×C3×C4—22no—
107C22×C3×C4—22no—
108C22×C3×C4—21no—
109C22×C3×C4—21no—
110C22×C3×C4—21no—
111C22×C3×C4—41no—
112C22×C3×C4—41no—
113C22×C3×C4—41no—
114C22×C3×C4C22⋊C411yescharacteristic
115C22×C3×C4C22⋊C411yescharacteristic
116C22×C3×C4C22⋊C411yescharacteristic
117C22×C3×C4C22⋊C411yescharacteristic
118C22×C3×C4C22×C411yescharacteristic
119C22×C3×C4C2×D412yes—
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4#4112yes—
122C22×Dic3—24no—
123C22×Dic3—24no—
124C22×Dic3—24no—
125C22×Dic3—42no—
126C22×Dic3—42no—
127C22×Dic3—42no—
128C22×Dic3—42no—
129C3×C4⋊C4—22no—
130C3×C4⋊C4—22no—
131C3×C4⋊C4—41no—
132C3×C4⋊C4—41no—
133C3×C22⋊C4—42no—
134C3×C22⋊C4—42no—
135C3×C22⋊C4—42no—
136C3×C22⋊C4—42no—
137C3×C22⋊C4—42no—
138C3×C22⋊C4—41no—
139C3×C22⋊C4—41no—
140C3×C42—22no—
141C3×C42—22no—
142C3×C42—41no—
143C3×C42—41no—
144C4×Dic3—48no—
145C2×C3⋊C8—42no—
146C2×C3⋊C8—42no—
147C2×C3⋊C8—42no—
148C22×Q8C2×Dic311yescharacteristic
149C22×D4—21no—
150C22×D4C2×Dic311yescharacteristic
151C23×C4—62no—
152C23×C4#6711yescharacteristic
153C23×C4#6711yescharacteristic
154C23×C4C22×D311yescharacteristic
155C22×C8—62no—
156#124—42no—
157C22⋊Q8—22no—
158C22⋊Q8—22no—
159#121—22no—
160#121—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4C2×Dic311yescharacteristic
163C2×C4⋊C4C2×Dic311yescharacteristic
164C2×C22⋊C4—21no—
165C2×C22⋊C4—21no—
166C2×C22⋊C4—21no—
167C2×C22⋊C4—21no—
168C2×C22⋊C4—21no—
169C2×C22⋊C4—21no—
170C2×C42—42no—
171C2×C42—64no—
172C2×C42—122no—
173C2×C42—122no—
174C2×C42C2×Dic311yescharacteristic
175C2×C42C2×Dic311yescharacteristic
176C2×C42#6711yescharacteristic
177C2×C42#6711yescharacteristic
178C4⋊C8—122no—
179C22⋊C8—62no—
180C22⋊C8—62no—
181#95—21no—
182#95—21no—
183#95—21no—
184#95—21no—
185#95—34no—
186#95—64no—
187#95—62no—
188C23×C3—21no—
189C23×C3—21no—
190C23×C3—21no—
191C23×C3—21no—
192C23×C3—41no—
193C23×C3—41no—
194C23×C3—41no—
195C23×C3—41no—
196C23×C3—41no—
197C23×C3—41no—
198C23×C3—41no—
199C23×C3C2×C22⋊C411yescharacteristic
200C23×C3#12112yes—
201C23×C3#13512yes—
202C23×C3AGL(1,ℤ/8)11yescharacteristic
203C23×C3#13711yescharacteristic
204C3×Q8—82no—
205C3×D4—82no—
206C3×D4—82no—
207C3×D4—82no—
208C2×C3×C4—24no—
209C2×C3×C4—24no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—21no—
213C2×C3×C4—21no—
214C2×C3×C4—21no—
215C2×C3×C4—21no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
223C2×C3×C4—42no—
224C2×C3×C4—42no—
225C2×C3×C4—44no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4#9912yes—
236C2×C3×C4#10412yes—
237C2×C3×C4#10412yes—
238C2×C3×C4C4×D412yes—
239C2×C3×C4#12411yescharacteristic
240C2×C3×C4#12711yescharacteristic
241C2×Dic3—28no—
242C2×Dic3—44no—
243C2×Dic3—44no—
244C2×Dic3—44no—
245C2×Dic3—44no—
246C2×Dic3—44no—
247C2×Dic3—44no—
248C3⋊C8—82no—
249C24—21no—
250C24C2×#6711yescharacteristic
251C2×Q8—22no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×D4—22no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—42no—
258C2×D4—41no—
259C2×D4—42no—
260C2×D4—41no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—64no—
274C22×C4—64no—
275C22×C4—64no—
276C22×C4—122no—
277C22×C4—122no—
278C22×C4—122no—
279C22×C4—122no—
280C22×C4#21711yescharacteristic
281C22×C4#21711yescharacteristic
282C22×C4#21711yescharacteristic
283C22×C4#21711yescharacteristic
284C22×C4D3×D412yes—
285C22×C4#23712yes—
286C22×C4C22×Dic311yescharacteristic
287C22×C4C2×#6711yescharacteristic
288C2×C8—122no—
289C2×C8—122no—
290C2×C8—122no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C22⋊C4—42no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C22⋊C4—42no—
299C22⋊C4—42no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C42—22no—
303C42—22no—
304C42—41no—
305C42—41no—
306C42—128no—
307C22×C3—22no—
308C22×C3—22no—
309C22×C3—22no—
310C22×C3—22no—
311C22×C3—21no—
312C22×C3—21no—
313C22×C3—21no—
314C22×C3—21no—
315C22×C3—81no—
316C22×C3—81no—
317C22×C3—81no—
318C22×C3—81no—
319C22×C3—81no—
320C22×C3—81no—
321C22×C3—81no—
322C22×C3#39011yescharacteristic
323C22×C3C2×#9911yescharacteristic
324C22×C3#41011yescharacteristic
325C22×C3#41611yescharacteristic
326C22×C3#41711yescharacteristic
327C22×C3C2×#10411yescharacteristic
328C22×C3C2×#10411yescharacteristic
329C22×C3#46312yes—
330C22×C3#46412yes—
331C3×C4—21no—
332C3×C4—22no—
333C3×C4—21no—
334C3×C4—22no—
335C3×C4—22no—
336C3×C4—22no—
337C3×C4—41no—
338C3×C4—81no—
339C3×C4—81no—
340C3×C4—81no—
341Dic3—48no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23#93812yes—
354C23#94911yescharacteristic
355C23#95012yes—
356C23#95111yescharacteristic
357C23C2×#21711yescharacteristic
358Q8—82no—
359D4—82no—
360D4—82no—
361D4—82no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—44no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—68no—
390C2×C4—124no—
391C2×C4—124no—
392C2×C4—124no—
393C2×C4—124no—
394C2×C4—124no—
395C2×C4—124no—
396C2×C4#83412yes—
397C2×C4#83712yes—
398C2×C4#83712yes—
399C2×C4D4×Dic312yes—
400C2×C4#93611yescharacteristic
401C2×C4#94011yescharacteristic
402C8—242no—
403C2×C3—22no—
404C2×C3—22no—
405C2×C3—161no—
406C2×C3#150011yescharacteristic
407C2×C3#150111yescharacteristic
408C2×C3#150411yescharacteristic
409C2×C3#150911yescharacteristic
410C2×C3#196211yescharacteristic
411C2×C3#196411yescharacteristic
412C2×C3#198311yescharacteristic
413C22—22no—
414C22—22no—
415C22—21no—
416C22—21no—
417C22—22no—
418C22—21no—
419C22—21no—
420C22—22no—
421C22—81no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22—81no—
426C22—81no—
427C22—81no—
428C22#505412yes—
429C22#505712yes—
430C22C2×#83411yescharacteristic
431C22#522111yescharacteristic
432C22#523211yescharacteristic
433C22#523311yescharacteristic
434C22C2×#83711yescharacteristic
435C22C2×#83711yescharacteristic
436C22#524011yescharacteristic
437C4—21no—
438C4—22no—
439C4—21no—
440C4—22no—
441C4—22no—
442C4—22no—
443C4—41no—
444C4—81no—
445C4—81no—
446C4—81no—
447C4—128no—
448C3#971311yescharacteristic, complemented, Sylow
449C2—22no—
450C2—22no—
451C2—161no—
452C2#7168611yescharacteristic
453C2#7170711yescharacteristic
454C2#7170811yescharacteristic
455C2#7172111yescharacteristic
456C2#7175411yescharacteristic
457C2#7175511yescharacteristic
458C2#7181511yescharacteristic
subgroupG/H
centerC23C2×#217
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2850C2
socleC23×C3C2×C22⋊C4
prime radicalC2×C3×#121C22
cosocleC23×C3×C4C23
hypercenter#2850D3
hypocenterC3#9713
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2850

Complements

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

NH ≅ G/Ntype
2. C2×#4538C2semidirect
4. C4×C3⋊#95C2semidirect
5. #71530C2semidirect
7. C2×C3×#124C4semidirect
12. C3×#391C4semidirect
13. C3×#390C4semidirect
18. C3×#382C4semidirect
71. C2×#124Dic3semidirect
78. #391Dic3semidirect
79. #390Dic3semidirect
84. #382Dic3semidirect
448. C3#9713semidirect

Maximal subgroups

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

subgroupindex
C3×#28502
C2×#45382
C2×#45372
C4×C3⋊#952
#715302
#97133