Finite Group Databasebeta

← #10876308

Subgroups — #10876308

Order 768.

1124 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#380C211yesmaximal, characteristic
2C2×#5212C211yesmaximal, characteristic
3#72126C211yesmaximal, characteristic
4#71555C212yesmaximal, complemented
5#71553C212yesmaximal, complemented
6#69518C214yesmaximal, complemented
7#69504C212yesmaximal, complemented
8#69501C212yesmaximal, complemented
9#27600—31nomaximal, Sylow
10C24×C3×C4C2211yescharacteristic
11C22×C3×C4⋊C4C2211yescharacteristic
12C22×C3×C4⋊C4C2211yescharacteristic
13C22×C3×C22⋊C4—21no—
14C22×#217—21no—
15C22×C4⋊Dic3C2211yescharacteristic
16C22×C4⋊Dic3C2211yescharacteristic
17C3×#380—21no—
18C3×#380—21no—
19C3×#380C2212yes—
20C3×#380C2212yes—
21C2×C3×#95—21no—
22#5212—21no—
23#5212—21no—
24#5212C2212yes—
25#5212C2212yes—
26C2×C3⋊#95—21no—
27C2×#830C2212yes—
28C2×#830C2212yes—
29C2×#830C2212yes—
30C2×#807C2214yescomplemented
31#4529C2214yescomplemented
32#4529C2214yescomplemented
33#4529C2214yescomplemented
34C2×#380—31no—
35C2×#380D311yescharacteristic
36#2075—34no—
37#1852—32no—
38#1852—32no—
39#1823—32no—
40#1823—32no—
41#1779—31no—
42C25×C3D411yescharacteristic
43C23×C3×C4—24no—
44C23×C3×C4—24no—
45C23×C3×C4—21no—
46C23×C3×C4—21no—
47C23×C3×C4—42no—
48C23×C3×C4D411yescharacteristic
49C23×C3×C4D411yescharacteristic
50C23×C3×C4D411yescharacteristic
51C23×C3×C4C2311yescharacteristic
52C23×C3×C4C2311yescharacteristic
53C23×C3×C4C2311yescharacteristic
54C23×Dic3—21no—
55C23×Dic3—21no—
56C2×C3×C4⋊C4—24no—
57C2×C3×C4⋊C4—21no—
58C2×C3×C4⋊C4—21no—
59C2×C3×C4⋊C4—42no—
60C2×C3×C4⋊C4C2312yes—
61C2×C3×C4⋊C4C2312yes—
62C2×C3×C22⋊C4—22no—
63C2×C3×C22⋊C4—22no—
64C2×C3×C22⋊C4—44no—
65C2×#217—22no—
66C2×#217—22no—
67C2×#217—44no—
68C2×C4⋊Dic3—24no—
69C2×C4⋊Dic3—21no—
70C2×C4⋊Dic3—21no—
71C2×C4⋊Dic3—42no—
72C2×C4⋊Dic3C2312yes—
73C2×C4⋊Dic3C2312yes—
74C22×C3⋊C8C2314yes—
75C3×#95—22no—
76C3×#95—22no—
77C3⋊#95—22no—
78C3⋊#95—22no—
79#830—24no—
80#830—24no—
81#830—42no—
82#807—44no—
83C24×C4C2×D311yescharacteristic
84C22×C4⋊C4—31no—
85C22×C4⋊C4—31no—
86C22×C4⋊C4C2×D311yescharacteristic
87C22×C4⋊C4C2×D311yescharacteristic
88C22×C22⋊C4—21no—
89C22×C22⋊C4—61no—
90C2×#107—34no—
91C2×C22⋊C8—32no—
92C2×C22⋊C8—32no—
93C2×C22⋊C8—32no—
94#380—21no—
95#380—21no—
96#380—32no—
97#380—32no—
98#380—61no—
99#380—61no—
100#380C2×D312yes—
101#380C2×D312yes—
102C2×#95—21no—
103C2×#95—61no—
104#340—34no—
105#340—34no—
106#340—34no—
107C24×C3—22no—
108C24×C3—22no—
109C24×C3—22no—
110C24×C3—22no—
111C24×C3—41no—
112C24×C3—41no—
113C24×C3D814yes—
114C24×C3C2×D411yescharacteristic
115C24×C3C2×D411yescharacteristic
116C24×C3C2×D411yescharacteristic
117C22×C3×C4—22no—
118C22×C3×C4—22no—
119C22×C3×C4—22no—
120C22×C3×C4—22no—
121C22×C3×C4—22no—
122C22×C3×C4—22no—
123C22×C3×C4—22no—
124C22×C3×C4—22no—
125C22×C3×C4—22no—
126C22×C3×C4—22no—
127C22×C3×C4—22no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4—41no—
133C22×C3×C4—41no—
134C22×C3×C4—41no—
135C22×C3×C4—416no—
136C22×C3×C4—42no—
137C22×C3×C4—42no—
138C22×C3×C4—42no—
139C22×C3×C4—42no—
140C22×C3×C4—42no—
141C22×C3×C4—42no—
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4#4114yes—
146C22×C3×C4#4114yes—
147C22×C3×C4C2411yescharacteristic
148C22×Dic3—22no—
149C22×Dic3—22no—
150C22×Dic3—42no—
151C22×Dic3—42no—
152C22×Dic3—42no—
153C22×Dic3—42no—
154C22×Dic3—42no—
155C22×Dic3—42no—
156C3×C4⋊C4—24no—
157C3×C4⋊C4—24no—
158C3×C4⋊C4—42no—
159C3×C4⋊C4—42no—
160C3×C22⋊C4—44no—
161C3×C22⋊C4—44no—
162#217—44no—
163#217—44no—
164C4⋊Dic3—24no—
165C4⋊Dic3—24no—
166C4⋊Dic3—42no—
167C4⋊Dic3—42no—
168C2×C3⋊C8—24no—
169C2×C3⋊C8—24no—
170C2×C3⋊C8—24no—
171C25#6711yescharacteristic
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—42no—
177C23×C4—61no—
178C23×C4—61no—
179C23×C4#6711yescharacteristic
180C23×C4#6711yescharacteristic
181C23×C4#6711yescharacteristic
182C23×C4C22×D311yescharacteristic
183C23×C4C22×D311yescharacteristic
184C23×C4C22×D311yescharacteristic
185C22×C8—34no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—32no—
190C2×C4⋊C4—32no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—64no—
193C2×C4⋊C4—61no—
194C2×C4⋊C4—61no—
195C2×C4⋊C4—122no—
196C2×C4⋊C4C22×D312yes—
197C2×C4⋊C4C22×D312yes—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—44no—
201C2×C22⋊C4—62no—
202C2×C22⋊C4—62no—
203C2×C22⋊C4—124no—
204#107—124no—
205C22⋊C8—64no—
206C22⋊C8—64no—
207C22⋊C8—122no—
208#95—22no—
209#95—22no—
210#95—62no—
211#95—62no—
212C23×C3—21no—
213C23×C3—21no—
214C23×C3—21no—
215C23×C3—21no—
216C23×C3—21no—
217C23×C3—21no—
218C23×C3—21no—
219C23×C3—21no—
220C23×C3—21no—
221C23×C3—21no—
222C23×C3—21no—
223C23×C3—21no—
224C23×C3—21no—
225C23×C3—21no—
226C23×C3—28no—
227C23×C3—28no—
228C23×C3—44no—
229C23×C3—44no—
230C23×C3—44no—
231C23×C3—44no—
232C23×C3—44no—
233C23×C3#12312yes—
234C23×C3#12312yes—
235C23×C3C2×D812yes—
236C23×C3C2×D812yes—
237C23×C3C2×D812yes—
238C23×C3#13712yes—
239C23×C3#13712yes—
240C23×C3C22×D411yescharacteristic
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—28no—
246C2×C3×C4—28no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—44no—
251C2×C3×C4—44no—
252C2×C3×C4—44no—
253C2×C3×C4—44no—
254C2×C3×C4—42no—
255C2×C3×C4—42no—
256C2×C3×C4—42no—
257C2×C3×C4—42no—
258C2×C3×C4—42no—
259C2×C3×C4—42no—
260C2×C3×C4—84no—
261C2×C3×C4—84no—
262C2×C3×C4#12312yes—
263C2×C3×C4#12312yes—
264C2×C3×C4C2×#4112yes—
265C2×C3×C4C2×#4112yes—
266C2×C3×C4H(5,𝔽2)12yes—
267C2×Dic3—42no—
268C2×Dic3—42no—
269C2×Dic3—42no—
270C2×Dic3—42no—
271C2×Dic3—42no—
272C2×Dic3—42no—
273C2×Dic3—84no—
274C2×Dic3—84no—
275C3⋊C8—44no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—22no—
280C24—41no—
281C24—41no—
282C24#21314yes—
283C24C2×#6711yescharacteristic
284C24C2×#6711yescharacteristic
285C24C2×#6711yescharacteristic
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—416no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—62no—
312C22×C4—62no—
313C22×C4—122no—
314C22×C4—122no—
315C22×C4—122no—
316C22×C4—122no—
317C22×C4—122no—
318C22×C4—122no—
319C22×C4#23514yes—
320C22×C4#23714yes—
321C22×C4C2×#6711yescharacteristic
322C22×C4C2×#6711yescharacteristic
323C22×C4C2×#6711yescharacteristic
324C22×C4C23×D311yescharacteristic
325C2×C8—64no—
326C2×C8—64no—
327C2×C8—64no—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—64no—
333C4⋊C4—64no—
334C4⋊C4—122no—
335C4⋊C4—122no—
336C22⋊C4—44no—
337C22⋊C4—44no—
338C22⋊C4—124no—
339C22⋊C4—124no—
340C22×C3—22no—
341C22×C3—22no—
342C22×C3—22no—
343C22×C3—22no—
344C22×C3—22no—
345C22×C3—22no—
346C22×C3—22no—
347C22×C3—22no—
348C22×C3—22no—
349C22×C3—22no—
350C22×C3—22no—
351C22×C3—22no—
352C22×C3—22no—
353C22×C3—22no—
354C22×C3—22no—
355C22×C3—416no—
356C22×C3—41no—
357C22×C3—41no—
358C22×C3—41no—
359C22×C3—41no—
360C22×C3#48014yes—
361C22×C3#48014yes—
362C22×C3C2×#12311yescharacteristic
363C22×C3C2×#12311yescharacteristic
364C22×C3#53411yescharacteristic
365C22×C3C22×D811yescharacteristic
366C22×C3C2×#13711yescharacteristic
367C22×C3C2×#13711yescharacteristic
368C22×C3#57511yescharacteristic
369C3×C4—22no—
370C3×C4—22no—
371C3×C4—41no—
372C3×C4—82no—
373C3×C4—82no—
374C3×C4#55114yes—
375Dic3—82no—
376Dic3—82no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—28no—
386C23—21no—
387C23—21no—
388C23—28no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—44no—
394C23—44no—
395C23—44no—
396C23—44no—
397C23—44no—
398C23#92912yes—
399C23C2×#21312yes—
400C23C2×#21312yes—
401C23C2×#21312yes—
402C23#93512yes—
403C23#94212yes—
404C23#95112yes—
405C23C22×#6711yescharacteristic
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—28no—
411C2×C4—28no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—84no—
426C2×C4—84no—
427C2×C4—122no—
428C2×C4—122no—
429C2×C4—122no—
430C2×C4—122no—
431C2×C4—122no—
432C2×C4—122no—
433C2×C4—244no—
434C2×C4—244no—
435C2×C4#92912yes—
436C2×C4#93512yes—
437C2×C4C2×#23512yes—
438C2×C4C2×#23712yes—
439C2×C4#100412yes—
440C8—124no—
441C2×C3—24no—
442C2×C3—24no—
443C2×C3—42no—
444C2×C3#248311yescharacteristic
445C2×C3C2×#48011yescharacteristic
446C2×C3C2×#48011yescharacteristic
447C2×C3#309511yescharacteristic
448C2×C3#309511yescharacteristic
449C2×C3#318611yescharacteristic
450C2×C3#319511yescharacteristic
451C22—22no—
452C22—22no—
453C22—22no—
454C22—22no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—22no—
464C22—22no—
465C22—22no—
466C22—416no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—41no—
471C22#496714yes—
472C22#503314yes—
473C22C2×#92911yescharacteristic
474C22C22×#21311yescharacteristic
475C22C2×#93511yescharacteristic
476C22#580411yescharacteristic
477C22C2×#94211yescharacteristic
478C22#582211yescharacteristic
479C22C2×#95111yescharacteristic
480C4—22no—
481C4—22no—
482C4—41no—
483C4—82no—
484C4—82no—
485C4—242no—
486C4—242no—
487C4#549514yes—
488C3#2760011yescharacteristic, complemented, Sylow
489C2—24no—
490C2—24no—
491C2—42no—
492C2#8211411yescharacteristic
493C2C2×#496711yescharacteristic
494C2#8222011yescharacteristic
495C2C2×#503311yescharacteristic
496C2#8262511yescharacteristic
497C2#8272711yescharacteristic
498C2#8274211yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#380C2
socleC23×C3C22×D4
prime radicalC25×C3D4
hypercenterC2×#380D3
hypocenterC3#27600
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
4. #71555C2semidirect
5. #71553C2semidirect
6. #69518C2semidirect
7. #69504C2semidirect
8. #69501C2semidirect
30. C2×#807C22semidirect
31. #4529C22semidirect
32. #4529C22semidirect
33. #4529C22semidirect
488. C3#27600semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3802
C2×#52122
#721262
#715552
#715532
#695182
#695042
#695012
#276003