Finite Group Databasebeta

← #10720452

Subgroups — #10720452

Order 768.

1690 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#126C211yesmaximal, characteristic
2C2×C4×C4⋊Dic3C211yesmaximal, characteristic, complemented
3C2×#4932C211yesmaximal, characteristic, complemented
4#71565C212yesmaximal, complemented
5#71564C212yesmaximal, complemented
6#71564C212yesmaximal, complemented
7#71546C212yesmaximal
8#71545C212yesmaximal
9#71545C212yesmaximal
10#14166—31nomaximal, Sylow
11C2×C3×#126—24no—
12C2×C3×#126—24no—
13C2×C3×#126C418yescomplemented
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C4⋊C4C2211yescharacteristic
17C22×C3×C22⋊C4C2211yescharacteristic
18C22×C3×C22⋊C4C2211yescharacteristic
19C22×C3×C22⋊C4C2211yescharacteristic
20C22×C3×C42C2211yescharacteristic
21C22×#217C2214yes—
22C22×C4⋊Dic3C2212yes—
23C22×C4×Dic3C2212yes—
24C22×C4×Dic3C2212yes—
25C2×C3⋊#95C2212yes—
26C2×C3⋊#95C2212yes—
27C2×C3⋊#95C2214yes—
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2214yes—
30C2×C3⋊#95C2214yes—
31C4×C4⋊Dic3—24no—
32#4932—24no—
33C22×#126D311yescharacteristic
34C2×C4×C4⋊C4—31no—
35C2×#376—31no—
36#1441—32no—
37#1439—32no—
38#1437—32no—
39#1437—32no—
40#1436—32no—
41#1436—32no—
42C25×C3C2311yescharacteristic
43C23×C3×C4C2311yescharacteristic
44C23×C3×C4C2311yescharacteristic
45C23×C3×C4C2311yescharacteristic
46C23×C3×C4C2311yescharacteristic
47C23×C3×C4C2311yescharacteristic
48C23×C3×C4C2311yescharacteristic
49C23×Dic3C2314yes—
50C23×Dic3C2314yes—
51C3×#126—232no—
52C2×C3×C4⋊C4—22no—
53C2×C3×C4⋊C4—22no—
54C2×C3×C4⋊C4—22no—
55C2×C3×C4⋊C4—22no—
56C2×C3×C4⋊C4—22no—
57C2×C3×C4⋊C4—22no—
58C2×C3×C4⋊C4C2×C414yes—
59C2×C3×C4⋊C4C2×C414yes—
60C2×C3×C4⋊C4C2×C414yes—
61C2×C3×C22⋊C4—22no—
62C2×C3×C22⋊C4—22no—
63C2×C3×C22⋊C4—22no—
64C2×C3×C22⋊C4—22no—
65C2×C3×C22⋊C4—22no—
66C2×C3×C22⋊C4—22no—
67C2×C3×C22⋊C4C2×C414yes—
68C2×C3×C22⋊C4C2×C414yes—
69C2×C3×C22⋊C4C2×C414yes—
70C2×C3×C42C2×C414yes—
71C2×C3×C42D414yes—
72C2×C3×C42D414yes—
73C2×#217—28no—
74C2×#217—28no—
75C2×#217—44no—
76C2×C4⋊Dic3—24no—
77C2×C4⋊Dic3—28no—
78C2×C4×Dic3—24no—
79C2×C4×Dic3—28no—
80C2×C4×Dic3—24no—
81C2×C4×Dic3—44no—
82C3⋊#95—24no—
83C3⋊#95—24no—
84C3⋊#95—24no—
85C3⋊#95—28no—
86C3⋊#95—28no—
87C3⋊#95—44no—
88C2×#126—24no—
89C2×#126—24no—
90C2×#126Dic318yescomplemented
91C22×C4⋊C4—32no—
92C22×C4⋊C4C2×D311yescharacteristic
93C22×C4⋊C4C2×D311yescharacteristic
94C22×C4⋊C4C2×D311yescharacteristic
95C22×C22⋊C4—34no—
96C22×C22⋊C4C2×D311yescharacteristic
97C22×C22⋊C4C2×D311yescharacteristic
98C22×C22⋊C4C2×D311yescharacteristic
99C22×C42—32no—
100C22×C42—32no—
101C22×C42C2×D311yescharacteristic
102C4×C4⋊C4—64no—
103#376—64no—
104C2×#95—34no—
105C2×#95—34no—
106C2×#95—32no—
107C2×#95—32no—
108C2×#95—32no—
109C2×#95—34no—
110C24×C3—21no—
111C24×C3—21no—
112C24×C3—24no—
113C24×C3—24no—
114C24×C3—22no—
115C24×C3—22no—
116C24×C3C22×C412yes—
117C24×C3C2411yescharacteristic
118C22×C3×C4—24no—
119C22×C3×C4—24no—
120C22×C3×C4—22no—
121C22×C3×C4—22no—
122C22×C3×C4—21no—
123C22×C3×C4—21no—
124C22×C3×C4—21no—
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—24no—
129C22×C3×C4—24no—
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4C22×C412yes—
133C22×C3×C4C22×C412yes—
134C22×C3×C4C22×C412yes—
135C22×C3×C4C22×C412yes—
136C22×C3×C4C22×C412yes—
137C22×C3×C4C22×C412yes—
138C22×C3×C4C2×D412yes—
139C22×C3×C4C2×D412yes—
140C22×C3×C4C2×D412yes—
141C22×C3×C4C2×D412yes—
142C22×C3×C4C2×D412yes—
143C22×C3×C4C2×D412yes—
144C22×Dic3—24no—
145C22×Dic3—24no—
146C22×Dic3—24no—
147C22×Dic3—24no—
148C22×Dic3—24no—
149C22×Dic3—24no—
150C22×Dic3—24no—
151C22×Dic3—24no—
152C22×Dic3—24no—
153C22×Dic3—24no—
154C22×Dic3—24no—
155C22×Dic3—24no—
156C22×Dic3#4118yes—
157C22×Dic3#4118yes—
158C3×C4⋊C4—28no—
159C3×C4⋊C4—28no—
160C3×C4⋊C4—28no—
161C3×C22⋊C4—28no—
162C3×C22⋊C4—28no—
163C3×C22⋊C4—28no—
164C3×C42C22⋊C4116yescomplemented
165#217—416no—
166C4⋊Dic3—216no—
167C4×Dic3—216no—
168C4×Dic3—48no—
169C25C22×D311yescharacteristic
170C23×C4—34no—
171C23×C4—34no—
172C23×C4C22×D311yescharacteristic
173C23×C4C22×D311yescharacteristic
174C23×C4C22×D311yescharacteristic
175C23×C4C22×D311yescharacteristic
176C23×C4C22×D311yescharacteristic
177C23×C4C22×D311yescharacteristic
178#126—232no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—68no—
186C2×C4⋊C4—64no—
187C2×C4⋊C4C2×Dic314yes—
188C2×C4⋊C4C2×Dic314yes—
189C2×C4⋊C4C2×Dic314yes—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—68no—
197C2×C22⋊C4—68no—
198C2×C22⋊C4—124no—
199C2×C22⋊C4C2×Dic314yes—
200C2×C22⋊C4C2×Dic314yes—
201C2×C22⋊C4C2×Dic314yes—
202C2×C42—68no—
203C2×C42—64no—
204C2×C42—64no—
205C2×C42—124no—
206C2×C42C2×Dic314yes—
207C2×C42#6714yes—
208C2×C42#6714yes—
209#95—68no—
210#95—68no—
211#95—64no—
212#95—64no—
213#95—64no—
214#95—124no—
215C23×C3—22no—
216C23×C3—24no—
217C23×C3—24no—
218C23×C3—24no—
219C23×C3—42no—
220C23×C3—42no—
221C23×C3—42no—
222C23×C3—42no—
223C23×C3—41no—
224C23×C3—41no—
225C23×C3—42no—
226C23×C3—44no—
227C23×C3—44no—
228C23×C3—44no—
229C23×C3—44no—
230C23×C3C23×C411yescharacteristic
231C23×C3C22×D411yescharacteristic
232C23×C3C22×D411yescharacteristic
233C23×C3C2×#4114yes—
234C23×C3C2×#4114yes—
235C23×C3H(5,𝔽2)12yes—
236C23×C32−1+412yes—
237C2×C3×C4—24no—
238C2×C3×C4—24no—
239C2×C3×C4—24no—
240C2×C3×C4—24no—
241C2×C3×C4—24no—
242C2×C3×C4—24no—
243C2×C3×C4—24no—
244C2×C3×C4—22no—
245C2×C3×C4—22no—
246C2×C3×C4—24no—
247C2×C3×C4—22no—
248C2×C3×C4—22no—
249C2×C3×C4—22no—
250C2×C3×C4—24no—
251C2×C3×C4—24no—
252C2×C3×C4—24no—
253C2×C3×C4—42no—
254C2×C3×C4—42no—
255C2×C3×C4—42no—
256C2×C3×C4—42no—
257C2×C3×C4—41no—
258C2×C3×C4—41no—
259C2×C3×C4—42no—
260C2×C3×C4C2×C22⋊C414yes—
261C2×C3×C4C2×C22⋊C414yes—
262C2×C3×C4C2×C22⋊C414yes—
263C2×Dic3—28no—
264C2×Dic3—28no—
265C2×Dic3—28no—
266C2×Dic3—28no—
267C2×Dic3—28no—
268C2×Dic3—28no—
269C2×Dic3—44no—
270C2×Dic3—44no—
271C2×Dic3—44no—
272C2×Dic3—44no—
273C2×Dic3—44no—
274C2×Dic3—44no—
275C2×Dic3—44no—
276C2×Dic3—44no—
277C24—24no—
278C24—24no—
279C24—22no—
280C24—22no—
281C24—21no—
282C24—21no—
283C24C22×Dic312yes—
284C24C23×D311yescharacteristic
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—38no—
300C22×C4—38no—
301C22×C4—64no—
302C22×C4—64no—
303C22×C4—64no—
304C22×C4—64no—
305C22×C4—64no—
306C22×C4—64no—
307C22×C4—64no—
308C22×C4—64no—
309C22×C4—64no—
310C22×C4—64no—
311C22×C4—64no—
312C22×C4—64no—
313C22×C4C22×Dic312yes—
314C22×C4C22×Dic312yes—
315C22×C4C22×Dic312yes—
316C22×C4C22×Dic312yes—
317C22×C4C22×Dic312yes—
318C22×C4C22×Dic312yes—
319C22×C4C2×#6712yes—
320C22×C4C2×#6712yes—
321C22×C4C2×#6712yes—
322C22×C4C2×#6712yes—
323C22×C4C2×#6712yes—
324C22×C4C2×#6712yes—
325C4⋊C4—28no—
326C4⋊C4—28no—
327C4⋊C4—28no—
328C4⋊C4—616no—
329C22⋊C4—28no—
330C22⋊C4—28no—
331C22⋊C4—28no—
332C22⋊C4—1216no—
333C42—616no—
334C42—128no—
335C42#217116yescomplemented
336C22×C3—42no—
337C22×C3—48no—
338C22×C3—42no—
339C22×C3—42no—
340C22×C3—84no—
341C22×C3—82no—
342C22×C3—82no—
343C22×C3C22×C22⋊C411yescharacteristic
344C22×C3C4×#4112yes—
345C22×C3C4×#4112yes—
346C22×C3#52012yes—
347C22×C3#52512yes—
348C22×C3#52512yes—
349C22×C3#53212yes—
350C22×C3#53212yes—
351C22×C3#53511yescharacteristic
352C22×C3#53611yescharacteristic
353C22×C3#53712yes—
354C22×C3#55114yes—
355C22×C3#55214yes—
356C22×C3#55314yes—
357C22×C3#55614yes—
358C3×C4—24no—
359C3×C4—24no—
360C3×C4—24no—
361C3×C4—42no—
362C3×C4—42no—
363C3×C4—42no—
364Dic3—48no—
365Dic3—48no—
366C23—24no—
367C23—24no—
368C23—24no—
369C23—22no—
370C23—44no—
371C23—44no—
372C23—44no—
373C23—44no—
374C23—41no—
375C23—42no—
376C23—42no—
377C23—41no—
378C23—42no—
379C23—42no—
380C23—42no—
381C23D3×#4114yes—
382C23D3×#4114yes—
383C23#100912yes—
384C23#101012yes—
385C23C23×Dic311yescharacteristic
386C23C22×#6711yescharacteristic
387C23C22×#6711yescharacteristic
388C2×C4—22no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—22no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—68no—
412C2×C4—68no—
413C2×C4—68no—
414C2×C4—68no—
415C2×C4—68no—
416C2×C4—68no—
417C2×C4—124no—
418C2×C4—124no—
419C2×C4—124no—
420C2×C4—124no—
421C2×C4—124no—
422C2×C4—124no—
423C2×C4—124no—
424C2×C4—124no—
425C2×C4C2×#21714yes—
426C2×C4C2×#21714yes—
427C2×C4C2×#21714yes—
428C2×C3—82no—
429C2×C3#229711yescharacteristic
430C2×C3#229711yescharacteristic
431C2×C3#231111yescharacteristic
432C2×C3#237314yes—
433C2×C3#260114yes—
434C2×C3#260312yes—
435C2×C3#260412yes—
436C22—42no—
437C22—42no—
438C22—42no—
439C22—48no—
440C22—82no—
441C22—82no—
442C22—84no—
443C22#570214yes—
444C22#570514yes—
445C22#570814yes—
446C22#570914yes—
447C22#582511yescharacteristic
448C22Dic3×#4112yes—
449C22Dic3×#4112yes—
450C22#582712yes—
451C22#582812yes—
452C22#582812yes—
453C22#583011yescharacteristic
454C22#583212yes—
455C22#583212yes—
456C22#583312yes—
457C22C22×#21711yescharacteristic
458C4—24no—
459C4—24no—
460C4—24no—
461C4—42no—
462C4—42no—
463C4—42no—
464C4—128no—
465C4—128no—
466C3#1416611yescharacteristic, complemented, Sylow
467C2—82no—
468C2#8307514yes—
469C2#8308712yes—
470C2#8308812yes—
471C2#8311114yes—
472C2#8408611yescharacteristic
473C2#8408611yescharacteristic
474C2#8408711yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC22×C3×#126C2
socle, cosocleC24×C3C24
prime radicalC25×C3C23
hypercenterC22×#126D3
hypocenterC3#14166
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×#126

Complements

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

NH ≅ G/Ntype
2. C2×C4×C4⋊Dic3C2semidirect
3. C2×#4932C2semidirect
4. #71565C2semidirect
5. #71564C2semidirect
6. #71564C2semidirect
13. C2×C3×#126C4semidirect
90. C2×#126Dic3semidirect
164. C3×C42C22⋊C4semidirect
335. C42#217semidirect
466. C3#14166semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#1262
C2×C4×C4⋊Dic32
C2×#49322
#715652
#715642
#715462
#715452
#141663