Finite Group Databasebeta

← #10720453

Subgroups — #10720453

Order 768.

2058 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#935C211yesmaximal, characteristic
2C2×Dic3×C22⋊C4C212yesmaximal, complemented
3C3×#1438C211yesmaximal, characteristic
4#72131C211yesmaximal, characteristic
5#71549C212yesmaximal, complemented
6#71545C214yesmaximal, complemented
7#69406C214yesmaximal, complemented
8#14167—31nomaximal, Sylow
9C23×C3×D4C2211yescharacteristic
10C24×Dic3C2211yescharacteristic
11C22×C3×C22⋊C4C2212yes—
12C22×C3×C22⋊C4C2212yes—
13C22×#217C2212yes—
14C22×#217C2212yes—
15C22×#217C2212yes—
16C22×#217C2211yescharacteristic
17C2×#935—28no—
18C2×#935C418yescomplemented
19C22×Dic3⋊C4C2212yes—
20C22×C4×Dic3C2214yes—
21C2×C3×#95C2212yes—
22C2×C3⋊#95C2214yes—
23C2×C3⋊#95C2214yes—
24C2×C3⋊#95C2214yes—
25C2×C3⋊#95C2212yes—
26Dic3×C22⋊C4—28no—
27C22×#123—31no—
28C2×C4×C22⋊C4—32no—
29#1440—32no—
30#1438—31no—
31#1438D311yescharacteristic
32#1437—34no—
33#1436—34no—
34C25×C3C2311yescharacteristic
35C25×C3C2311yescharacteristic
36C22×C3×D4—24no—
37C22×C3×D4—42no—
38C22×C3×D4—42no—
39C22×C3×D4C2×C414yes—
40C23×C3×C4C2311yescharacteristic
41C23×C3×C4C2314yes—
42C23×Dic3—42no—
43C23×Dic3—42no—
44C23×Dic3C2×C414yes—
45C23×Dic3D418yes—
46C23×Dic3C2312yes—
47C23×Dic3C2314yes—
48C23×Dic3C2312yes—
49C2×C3×C22⋊C4—24no—
50C2×C3×C22⋊C4—28no—
51C2×C3×C22⋊C4—28no—
52C2×C3×C22⋊C4—42no—
53C2×#217—24no—
54C2×#217—24no—
55C2×#217—24no—
56C2×#217—24no—
57C2×#217—28no—
58C2×#217—28no—
59C2×#217—42no—
60C2×#217C2×C414yes—
61C2×#217C2×C418yes—
62#935—232no—
63C2×Dic3⋊C4—24no—
64C2×Dic3⋊C4—24no—
65C2×Dic3⋊C4C2×C418yes—
66C2×C4×Dic3—28no—
67C2×C4×Dic3—44no—
68C2×C4×Dic3—44no—
69C3×#95—42no—
70C3⋊#95—28no—
71C3⋊#95—28no—
72C3⋊#95—28no—
73C3⋊#95—42no—
74C23×D4C2×D311yescharacteristic
75C24×C4—31no—
76C2×#123—38no—
77C2×#123—68no—
78C22×C4⋊C4—32no—
79C22×C22⋊C4—32no—
80C22×C22⋊C4—31no—
81C22×C22⋊C4—32no—
82C22×C22⋊C4—32no—
83C22×C22⋊C4C2×D312yes—
84C22×C22⋊C4C2×D312yes—
85C22×C42—34no—
86C4×C22⋊C4—68no—
87C2×#95—34no—
88C2×#95—34no—
89C2×#95—34no—
90C2×#95—32no—
91C2×#95C2×D312yes—
92C24×C3—24no—
93C24×C3—24no—
94C24×C3—22no—
95C24×C3—22no—
96C24×C3—22no—
97C24×C3—24no—
98C24×C3—24no—
99C24×C3—22no—
100C24×C3—22no—
101C24×C3C22×C412yes—
102C24×C3C22×C412yes—
103C24×C3C2×D414yes—
104C24×C3C2411yescharacteristic
105C2×C3×D4—28no—
106C2×C3×D4—44no—
107C2×C3×D4—44no—
108C2×C3×D4—416no—
109C22×C3×C4—24no—
110C22×C3×C4—22no—
111C22×C3×C4—24no—
112C22×C3×C4—24no—
113C22×C3×C4—24no—
114C22×C3×C4—22no—
115C22×C3×C4—24no—
116C22×C3×C4—22no—
117C22×C3×C4—24no—
118C22×C3×C4C22×C412yes—
119C22×C3×C4#4118yes—
120C22×Dic3—24no—
121C22×Dic3—24no—
122C22×Dic3—24no—
123C22×Dic3—22no—
124C22×Dic3—22no—
125C22×Dic3—24no—
126C22×Dic3—24no—
127C22×Dic3—24no—
128C22×Dic3—24no—
129C22×Dic3—24no—
130C22×Dic3—24no—
131C22×Dic3—24no—
132C22×Dic3—44no—
133C22×Dic3—44no—
134C22×Dic3—416no—
135C22×Dic3C22⋊C4116yescomplemented
136C22×Dic3C22×C414yes—
137C22×Dic3C22×C414yes—
138C22×Dic3C2×D414yes—
139C22×Dic3C2×D414yes—
140C22×Dic3#4118yes—
141C3×C22⋊C4—216no—
142C3×C22⋊C4—48no—
143#217—216no—
144#217—28no—
145#217—216no—
146#217—48no—
147Dic3⋊C4—216no—
148C4×Dic3—416no—
149C25C22×D311yescharacteristic
150C25C22×D311yescharacteristic
151C22×D4—24no—
152C22×D4—42no—
153C22×D4—42no—
154C22×D4C4×D314yes—
155C23×C4—32no—
156C23×C4—34no—
157C23×C4—38no—
158C23×C4—32no—
159C23×C4—34no—
160C23×C4—122no—
161C23×C4—122no—
162C23×C4C22×D311yescharacteristic
163C23×C4C22×D314yes—
164#123—632no—
165C2×C4⋊C4—38no—
166C2×C4⋊C4—64no—
167C2×C4⋊C4—64no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—28no—
170C2×C22⋊C4—28no—
171C2×C22⋊C4—38no—
172C2×C22⋊C4—34no—
173C2×C22⋊C4—42no—
174C2×C22⋊C4—64no—
175C2×C22⋊C4—64no—
176C2×C22⋊C4—68no—
177C2×C22⋊C4—64no—
178C2×C22⋊C4—64no—
179C2×C22⋊C4—68no—
180C2×C22⋊C4—122no—
181C2×C42—68no—
182C2×C42—124no—
183C2×C42—124no—
184#95—42no—
185#95—68no—
186#95—68no—
187#95—68no—
188#95—122no—
189C23×C3—28no—
190C23×C3—22no—
191C23×C3—22no—
192C23×C3—24no—
193C23×C3—24no—
194C23×C3—28no—
195C23×C3—28no—
196C23×C3—22no—
197C23×C3—28no—
198C23×C3—22no—
199C23×C3—22no—
200C23×C3—42no—
201C23×C3—42no—
202C23×C3—44no—
203C23×C3—44no—
204C23×C3—42no—
205C23×C3—42no—
206C23×C3—42no—
207C23×C3—42no—
208C23×C3—44no—
209C23×C3—44no—
210C23×C3—42no—
211C23×C3—42no—
212C23×C3—44no—
213C23×C3—44no—
214C23×C3—42no—
215C23×C3—42no—
216C23×C3C2×C22⋊C414yes—
217C23×C3C23×C411yescharacteristic
218C23×C3C22×D412yes—
219C23×C3C2×#4114yes—
220C23×C3C2×#4114yes—
221C23×C3H(5,𝔽2)12yes—
222C23×C3H(5,𝔽2)12yes—
223C3×D4—416no—
224C2×C3×C4—22no—
225C2×C3×C4—22no—
226C2×C3×C4—22no—
227C2×C3×C4—28no—
228C2×C3×C4—28no—
229C2×C3×C4—28no—
230C2×C3×C4—44no—
231C2×C3×C4—44no—
232C2×C3×C4—44no—
233C2×C3×C4—44no—
234C2×C3×C4—44no—
235C2×Dic3—24no—
236C2×Dic3—24no—
237C2×Dic3—28no—
238C2×Dic3—28no—
239C2×Dic3—24no—
240C2×Dic3—28no—
241C2×Dic3—28no—
242C2×Dic3—28no—
243C2×Dic3—28no—
244C2×Dic3—416no—
245C2×Dic3—42no—
246C2×Dic3—42no—
247C2×Dic3—44no—
248C2×Dic3—44no—
249C2×Dic3—44no—
250C2×Dic3—44no—
251C2×Dic3—44no—
252C2×Dic3C2×C22⋊C418yes—
253C24—24no—
254C24—22no—
255C24—22no—
256C24—24no—
257C24—22no—
258C24—24no—
259C24—22no—
260C24—22no—
261C24—24no—
262C24C2×C4×D312yes—
263C24C2×C4×D312yes—
264C24D3×D414yes—
265C24C23×D311yescharacteristic
266C2×D4—28no—
267C2×D4—44no—
268C2×D4—44no—
269C2×D4—416no—
270C22×C4—24no—
271C22×C4—22no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—34no—
280C22×C4—34no—
281C22×C4—34no—
282C22×C4—316no—
283C22×C4—34no—
284C22×C4—38no—
285C22×C4—64no—
286C22×C4—64no—
287C22×C4—64no—
288C22×C4—62no—
289C22×C4—62no—
290C22×C4—64no—
291C22×C4—64no—
292C22×C4—64no—
293C22×C4—64no—
294C22×C4—64no—
295C22×C4—64no—
296C22×C4—64no—
297C22×C4—124no—
298C22×C4—124no—
299C22×C4—1216no—
300C22×C4C2×C4×D312yes—
301C22×C4#23718yes—
302C4⋊C4—616no—
303C22⋊C4—216no—
304C22⋊C4—48no—
305C22⋊C4—616no—
306C22⋊C4—616no—
307C22⋊C4—68no—
308C22⋊C4—128no—
309C42—1216no—
310C22×C3—24no—
311C22×C3—28no—
312C22×C3—48no—
313C22×C3—48no—
314C22×C3—44no—
315C22×C3—44no—
316C22×C3—42no—
317C22×C3—44no—
318C22×C3—44no—
319C22×C3—44no—
320C22×C3—84no—
321C22×C3—84no—
322C22×C3C22×C22⋊C411yescharacteristic
323C22×C3C4×#4114yes—
324C22×C3#51811yescharacteristic
325C22×C3#51811yescharacteristic
326C22×C3#52512yes—
327C22×C3#52512yes—
328C22×C3#53214yes—
329C22×C3#53414yes—
330C22×C3#55112yes—
331C22×C3#55112yes—
332C22×C3#55314yes—
333C22×C3#55314yes—
334C22×C3#55912yes—
335C22×C3#55912yes—
336C3×C4—42no—
337C3×C4—48no—
338Dic3—28no—
339Dic3—48no—
340Dic3—44no—
341C23—24no—
342C23—22no—
343C23—22no—
344C23—28no—
345C23—24no—
346C23—28no—
347C23—28no—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—28no—
352C23—42no—
353C23—42no—
354C23—42no—
355C23—42no—
356C23—44no—
357C23—44no—
358C23—42no—
359C23—42no—
360C23—42no—
361C23—42no—
362C23—42no—
363C23—42no—
364C23—44no—
365C23—44no—
366C23—44no—
367C23—44no—
368C23D3×C22⋊C414yes—
369C23C22×C4×D311yescharacteristic
370C23C2×D3×D412yes—
371C23C2×#23714yes—
372C23#100412yes—
373C23#100412yes—
374C23D3×#4114yes—
375D4—416no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—28no—
380C2×C4—28no—
381C2×C4—28no—
382C2×C4—38no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—64no—
389C2×C4—64no—
390C2×C4—68no—
391C2×C4—68no—
392C2×C4—68no—
393C2×C4—68no—
394C2×C4—64no—
395C2×C4—68no—
396C2×C4—68no—
397C2×C4—1216no—
398C2×C4—122no—
399C2×C4—124no—
400C2×C4—122no—
401C2×C4—124no—
402C2×C4—124no—
403C2×C4—124no—
404C2×C4—124no—
405C2×C3—44no—
406C2×C3—82no—
407C2×C3#229712yes—
408C2×C3#230811yescharacteristic
409C2×C3#237412yes—
410C2×C3#237412yes—
411C2×C3#260214yes—
412C2×C3#260214yes—
413C22—24no—
414C22—28no—
415C22—44no—
416C22—48no—
417C22—48no—
418C22—44no—
419C22—44no—
420C22—42no—
421C22—44no—
422C22—44no—
423C22—84no—
424C22—84no—
425C22C2×D3×C22⋊C411yescharacteristic
426C22#548611yescharacteristic
427C22#548611yescharacteristic
428C22C4×#23714yes—
429C22#558612yes—
430C22#558712yes—
431C22#558914yes—
432C22#559614yes—
433C22#560114yes—
434C22#561414yes—
435C22#565612yes—
436C22#566412yes—
437C22#567912yes—
438C22#571312yes—
439C4—42no—
440C4—48no—
441C4—68no—
442C4—124no—
443C4—128no—
444C3#1416711yescharacteristic, complemented, Sylow
445C2—44no—
446C2—82no—
447C2#7384911yescharacteristic
448C2#7465812yes—
449C2#7468112yes—
450C2#7630812yes—
451C2#7669414yes—
452C2#7696614yes—
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC3×#1438C2
socle, cosocleC24×C3C24
prime radicalC23×C3×D4C22
hypercenter#1438D3
hypocenterC3#14167
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1438

Complements

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

NH ≅ G/Ntype
2. C2×Dic3×C22⋊C4C2semidirect
5. #71549C2semidirect
6. #71545C2semidirect
7. #69406C2semidirect
18. C2×#935C4semidirect
135. C22×Dic3C22⋊C4semidirect
444. C3#14167semidirect

Maximal subgroups

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

subgroupindex
C22×#9352
C2×Dic3×C22⋊C42
C3×#14382
#721312
#715492
#715452
#694062
#141673