Finite Group Databasebeta

← #10690564

Subgroups — #10690564

Order 768.

736 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×C3×#388C211yesmaximal, characteristic
2C2×#4925C211yesmaximal, characteristic, complemented
3C2×#4525C212yesmaximal
4#71531C211yesmaximal, characteristic, complemented
5#67613C212yesmaximal, complemented
6#9080—31nomaximal, Sylow
7C22×C3×C4⋊C4C411yescharacteristic
8C22×C3×C22⋊C4C411yescharacteristic
9C22×C3×C22⋊C4C411yescharacteristic
10C22×C3×C22⋊C4C2211yescharacteristic
11C22×C3×C42C2211yescharacteristic
12C3×#388—21no—
13C3×#388—21no—
14C3×#388—21no—
15C3×#388—21no—
16C2×C3×#95C411yescharacteristic
17C2×C3×#95C2211yescharacteristic
18C2×#830C2212yes—
19C2×#830C2212yes—
20#4925—24no—
21C2×C4⋊(C3⋊C8)—21no—
22C2×C4×C3⋊C8—21no—
23#4525—24no—
24#4524—21no—
25#4524—21no—
26C2×#431—31no—
27C2×#388D311yescharacteristic
28#1817—31no—
29C2×#323—32no—
30#1280—32no—
31C25×C3C2×C411yescharacteristic
32C23×C3×C4C2×C411yescharacteristic
33C23×C3×C4C2×C411yescharacteristic
34C23×C3×C4C2×C411yescharacteristic
35C23×C3×C4C2×C411yescharacteristic
36C23×C3×C4C2×C411yescharacteristic
37C23×C3×C4C2311yescharacteristic
38C2×C3×C4⋊C4—21no—
39C2×C3×C4⋊C4—21no—
40C2×C3×C4⋊C4—42no—
41C2×C3×C22⋊C4—21no—
42C2×C3×C22⋊C4—21no—
43C2×C3×C22⋊C4—22no—
44C2×C3×C22⋊C4—21no—
45C2×C3×C22⋊C4—21no—
46C2×C3×C22⋊C4—21no—
47C2×C3×C22⋊C4—21no—
48C2×C3×C22⋊C4—22no—
49C2×C3×C22⋊C4—22no—
50C2×C3×C22⋊C4—42no—
51C2×C3×C22⋊C4C814yescomplemented
52C2×C3×C42—22no—
53C2×C3×C42—22no—
54C2×C3×C42D411yescharacteristic
55C2×C3×C42D411yescharacteristic
56C2×C3×C42D411yescharacteristic
57C2×C3×C42D411yescharacteristic
58C22×C3⋊C8—22no—
59C22×C3⋊C8—22no—
60C3×#95—21no—
61C3×#95—21no—
62C3×#95—21no—
63C3×#95—21no—
64#830—22no—
65#830—22no—
66#830—22no—
67#830D414yes—
68C4⋊(C3⋊C8)—24no—
69C4×C3⋊C8—24no—
70C22×C4⋊C4Dic311yescharacteristic
71C22×C22⋊C4Dic311yescharacteristic
72C22×C22⋊C4Dic311yescharacteristic
73C22×C22⋊C4C2×D311yescharacteristic
74C22×C42C2×D311yescharacteristic
75#431—64no—
76C2×C4⋊C8—61no—
77C2×C22⋊C8—32no—
78C2×C22⋊C8—32no—
79C2×C4×C8—61no—
80#388—21no—
81#388—21no—
82#388—21no—
83#388—21no—
84C2×#95Dic311yescharacteristic
85C2×#95C2×D311yescharacteristic
86#336—61no—
87#336—61no—
88#323—64no—
89C24×C3—21no—
90C24×C3—21no—
91C24×C3—22no—
92C24×C3—22no—
93C24×C3—41no—
94C24×C3—41no—
95C24×C3—41no—
96C24×C3—41no—
97C24×C3C2×C812yes—
98C24×C3C22×C411yescharacteristic
99C22×C3×C4—22no—
100C22×C3×C4—22no—
101C22×C3×C4—21no—
102C22×C3×C4—22no—
103C22×C3×C4—21no—
104C22×C3×C4—22no—
105C22×C3×C4—21no—
106C22×C3×C4—21no—
107C22×C3×C4—22no—
108C22×C3×C4—22no—
109C22×C3×C4—41no—
110C22×C3×C4—41no—
111C22×C3×C4—41no—
112C22×C3×C4—41no—
113C22×C3×C4—41no—
114C22×C3×C4—41no—
115C22×C3×C4—41no—
116C22×C3×C4—41no—
117C22×C3×C4—41no—
118C22×C3×C4C22⋊C411yescharacteristic
119C22×C3×C4C22⋊C411yescharacteristic
120C22×C3×C4C22⋊C411yescharacteristic
121C22×C3×C4C22⋊C411yescharacteristic
122C22×C3×C4C2×C812yes—
123C22×C3×C4C2×C812yes—
124C22×C3×C4M1612yes—
125C22×C3×C4C2×D412yes—
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4#4112yes—
129C3×C4⋊C4—42no—
130C3×C4⋊C4—42no—
131C3×C22⋊C4—24no—
132C3×C22⋊C4—24no—
133C3×C22⋊C4—42no—
134C3×C22⋊C4—42no—
135C3×C22⋊C4—41no—
136C3×C22⋊C4—42no—
137C3×C22⋊C4—41no—
138C3×C42—21no—
139C3×C42—21no—
140C3×C42—22no—
141C3×C42—22no—
142C3×C42—21no—
143C3×C42—21no—
144C2×C3⋊C8—24no—
145C2×C3⋊C8—42no—
146C2×C3⋊C8—42no—
147C2×C3⋊C8—42no—
148C2×C3⋊C8—42no—
149C2×C3⋊C8—42no—
150C25C2×Dic311yescharacteristic
151C23×C4C2×Dic311yescharacteristic
152C23×C4C2×Dic311yescharacteristic
153C23×C4C2×Dic311yescharacteristic
154C23×C4C2×Dic311yescharacteristic
155C23×C4C2×Dic311yescharacteristic
156C23×C4C22×D311yescharacteristic
157C22×C8—62no—
158C22×C8—62no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—42no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—21no—
164C2×C22⋊C4—21no—
165C2×C22⋊C4—21no—
166C2×C22⋊C4—21no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—21no—
170C2×C22⋊C4—21no—
171C2×C22⋊C4—42no—
172C2×C22⋊C4C3⋊C814yescomplemented
173C2×C42—22no—
174C2×C42—22no—
175C2×C42#6711yescharacteristic
176C2×C42#6711yescharacteristic
177C2×C42#6711yescharacteristic
178C2×C42#6711yescharacteristic
179C4⋊C8—64no—
180C22⋊C8—34no—
181C22⋊C8—62no—
182C22⋊C8—62no—
183C22⋊C8—62no—
184C4×C8—64no—
185#95—21no—
186#95—21no—
187#95—21no—
188#95—21no—
189C23×C3—21no—
190C23×C3—21no—
191C23×C3—21no—
192C23×C3—21no—
193C23×C3—22no—
194C23×C3—24no—
195C23×C3—42no—
196C23×C3—42no—
197C23×C3—42no—
198C23×C3—42no—
199C23×C3—41no—
200C23×C3—41no—
201C23×C3—41no—
202C23×C3—41no—
203C23×C3—41no—
204C23×C3—41no—
205C23×C3—42no—
206C23×C3—82no—
207C23×C3—82no—
208C23×C3—82no—
209C23×C3—82no—
210C23×C3C2×C22⋊C411yescharacteristic
211C23×C3C4×D412yes—
212C23×C3C22×C811yescharacteristic
213C23×C3C2×M1611yescharacteristic
214C23×C3#13112yes—
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—21no—
224C2×C3×C4—21no—
225C2×C3×C4—24no—
226C2×C3×C4—24no—
227C2×C3×C4—21no—
228C2×C3×C4—22no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—42no—
232C2×C3×C4—42no—
233C2×C3×C4—44no—
234C2×C3×C4—42no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4—42no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—82no—
245C2×C3×C4—82no—
246C2×C3×C4C22⋊C812yes—
247C2×C3×C4C22⋊C812yes—
248C2×C3×C4#10012yes—
249C2×C3×C4#12011yescharacteristic
250C2×C3×C4#12112yes—
251C2×C3×C4#12311yescharacteristic
252C3⋊C8—44no—
253C3⋊C8—82no—
254C24—21no—
255C24—21no—
256C24—22no—
257C24—22no—
258C24—41no—
259C24—41no—
260C24—41no—
261C24—41no—
262C24C2×C3⋊C812yes—
263C24C22×Dic311yescharacteristic
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4C2×C3⋊C812yes—
284C22×C4C2×C3⋊C812yes—
285C22×C4#20812yes—
286C22×C4#21711yescharacteristic
287C22×C4#21711yescharacteristic
288C22×C4#21711yescharacteristic
289C22×C4#21711yescharacteristic
290C22×C4D3×D412yes—
291C22×C4#23712yes—
292C22×C4C2×#6711yescharacteristic
293C22×C4C2×#6711yescharacteristic
294C2×C8—64no—
295C2×C8—122no—
296C2×C8—122no—
297C2×C8—122no—
298C2×C8—122no—
299C2×C8—122no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C22⋊C4—24no—
303C22⋊C4—24no—
304C22⋊C4—42no—
305C22⋊C4—42no—
306C22⋊C4—41no—
307C22⋊C4—41no—
308C22⋊C4—42no—
309C42—21no—
310C42—21no—
311C42—21no—
312C42—21no—
313C42—22no—
314C42—22no—
315C22×C3—22no—
316C22×C3—22no—
317C22×C3—21no—
318C22×C3—21no—
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—22no—
322C22×C3—22no—
323C22×C3—42no—
324C22×C3—42no—
325C22×C3—42no—
326C22×C3—84no—
327C22×C3—84no—
328C22×C3—81no—
329C22×C3—81no—
330C22×C3—81no—
331C22×C3—81no—
332C22×C3#38611yescharacteristic
333C22×C3C2×C22⋊C811yescharacteristic
334C22×C3#40811yescharacteristic
335C22×C3#41011yescharacteristic
336C22×C3#41011yescharacteristic
337C22×C3C2×#10011yescharacteristic
338C22×C3#41311yescharacteristic
339C22×C3C8×D412yes—
340C22×C3#43512yes—
341C3×C4—22no—
342C3×C4—22no—
343C3×C4—21no—
344C3×C4—22no—
345C3×C4—21no—
346C3×C4—22no—
347C3×C4—41no—
348C3×C4—42no—
349C3×C4—81no—
350C3×C4—81no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—22no—
356C23—24no—
357C23—42no—
358C23—42no—
359C23—42no—
360C23—42no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—42no—
368C23—82no—
369C23—82no—
370C23—82no—
371C23—82no—
372C23C22×C3⋊C811yescharacteristic
373C23C2×#20811yescharacteristic
374C23D4×Dic312yes—
375C23#94812yes—
376C23C2×#21711yescharacteristic
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—22no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—44no—
396C2×C4—42no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4#83012yes—
409C2×C4#83012yes—
410C2×C4#83312yes—
411C2×C4#93511yescharacteristic
412C2×C4#93711yescharacteristic
413C2×C4#93912yes—
414C8—124no—
415C8—242no—
416C2×C3—22no—
417C2×C3—22no—
418C2×C3—82no—
419C2×C3#145811yescharacteristic
420C2×C3#146011yescharacteristic
421C2×C3#146911yescharacteristic
422C2×C3#147311yescharacteristic
423C2×C3#186911yescharacteristic
424C2×C3#189911yescharacteristic
425C2×C3#190011yescharacteristic
426C22—22no—
427C22—22no—
428C22—22no—
429C22—22no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—42no—
435C22—42no—
436C22—42no—
437C22—84no—
438C22—84no—
439C22—81no—
440C22—81no—
441C22—81no—
442C22—81no—
443C22D4×C3⋊C812yes—
444C22#501112yes—
445C22C2×#83011yescharacteristic
446C22C2×#83311yescharacteristic
447C22#522011yescharacteristic
448C22#523411yescharacteristic
449C22#523711yescharacteristic
450C22#524011yescharacteristic
451C22#524011yescharacteristic
452C4—22no—
453C4—21no—
454C4—22no—
455C4—21no—
456C4—22no—
457C4—22no—
458C4—41no—
459C4—42no—
460C4—81no—
461C4—81no—
462C3#908011yescharacteristic, complemented, Sylow
463C2—22no—
464C2—22no—
465C2—82no—
466C2#7157111yescharacteristic
467C2#7160111yescharacteristic
468C2#7161011yescharacteristic
469C2#7168311yescharacteristic
470C2#7170511yescharacteristic
471C2#7172011yescharacteristic
472C2#7174511yescharacteristic
subgroupG/H
centerC23C2×#217
commutatorC23×C3C22×C8
FrattiniC23×C4C22×D3
FittingC2×C3×#388C2
socleC23×C3C2×C22⋊C4
prime radicalC25×C3C2×C4
cosocleC23×C3×C4C23
hypercenterC2×#388D3
hypocenterC3#9080
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#388

Complements

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

NH ≅ G/Ntype
2. C2×#4925C2semidirect
4. #71531C2semidirect
5. #67613C2semidirect
51. C2×C3×C22⋊C4C8semidirect
172. C2×C22⋊C4C3⋊C8semidirect
462. C3#9080semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3882
C2×#49252
C2×#45252
#715312
#676132
#90803