Finite Group Databasebeta

← #10876318

Subgroups — #10876318

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×#5212C211yesmaximal, characteristic
2C2×#5212C211yesmaximal, characteristic
3C3×#1779C211yesmaximal, characteristic
4#69940C214yesmaximal, complemented
5#69928C212yesmaximal, complemented
6#69928C212yesmaximal, complemented
7#69925C212yesmaximal, complemented
8#69924C212yesmaximal, complemented
9#27601—31nomaximal, Sylow
10C24×C3×C4C2211yescharacteristic
11C22×#217—21no—
12C22×#217—21no—
13C22×C4⋊Dic3C2211yescharacteristic
14C22×C4⋊Dic3C2211yescharacteristic
15C22×C4⋊Dic3C2211yescharacteristic
16C22×C4⋊Dic3C2211yescharacteristic
17C2×C3×C22⋊C8C2212yes—
18C2×C3×C22⋊C8C2212yes—
19C2×C3×C22⋊C8C2212yes—
20#5212—21no—
21#5212—21no—
22#5212—21no—
23#5212—21no—
24#5212C2212yes—
25#5212C2212yes—
26#5212C2212yes—
27#5212C2212yes—
28C2×C3⋊#95—21no—
29C2×C3⋊#95—21no—
30C2×#817C2214yescomplemented
31#4551C2214yescomplemented
32#4551C2214yescomplemented
33#4551C2214yescomplemented
34C2×#380—31no—
35C2×#380—31no—
36#2077—34no—
37#1853—32no—
38#1853—32no—
39#1824—32no—
40#1823—32no—
41#1779D311yescharacteristic
42C25×C3D411yescharacteristic
43C23×C3×C4—24no—
44C23×C3×C4—24no—
45C23×C3×C4—42no—
46C23×C3×C4D411yescharacteristic
47C23×C3×C4D411yescharacteristic
48C23×C3×C4D411yescharacteristic
49C23×C3×C4C2311yescharacteristic
50C23×C3×C4C2311yescharacteristic
51C23×C3×C4C2311yescharacteristic
52C23×Dic3—21no—
53C23×Dic3—21no—
54C23×Dic3—21no—
55C23×Dic3—21no—
56C22×C3×C8C2314yes—
57C2×#217—22no—
58C2×#217—22no—
59C2×#217—22no—
60C2×#217—22no—
61C2×#217—44no—
62C2×#217—44no—
63C2×C4⋊Dic3—24no—
64C2×C4⋊Dic3—24no—
65C2×C4⋊Dic3—21no—
66C2×C4⋊Dic3—21no—
67C2×C4⋊Dic3—21no—
68C2×C4⋊Dic3—21no—
69C2×C4⋊Dic3—42no—
70C2×C4⋊Dic3—42no—
71C2×C4⋊Dic3C2312yes—
72C2×C4⋊Dic3C2312yes—
73C2×C4⋊Dic3C2312yes—
74C2×C4⋊Dic3C2312yes—
75C3×C22⋊C8—24no—
76C3×C22⋊C8—24no—
77C3×C22⋊C8—42no—
78C3⋊#95—22no—
79C3⋊#95—22no—
80C3⋊#95—22no—
81C3⋊#95—22no—
82#817—44no—
83C24×C4C2×D311yescharacteristic
84C22×C4⋊C4—31no—
85C22×C4⋊C4—31no—
86C22×C4⋊C4—31no—
87C22×C4⋊C4—31no—
88C22×C22⋊C4—61no—
89C22×C22⋊C4—61no—
90C2×#106—34no—
91C2×C22⋊C8C2×D312yes—
92C2×C22⋊C8C2×D312yes—
93C2×C22⋊C8C2×D312yes—
94#380—32no—
95#380—32no—
96#380—32no—
97#380—32no—
98#380—61no—
99#380—61no—
100#380—61no—
101#380—61no—
102C2×#95—61no—
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×C3SD1614yes—
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—41no—
131C22×C3×C4—41no—
132C22×C3×C4—41no—
133C22×C3×C4—416no—
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4#4114yes—
138C22×C3×C4#4114yes—
139C22×C3×C4C2411yescharacteristic
140C22×Dic3—22no—
141C22×Dic3—22no—
142C22×Dic3—22no—
143C22×Dic3—22no—
144C22×Dic3—42no—
145C22×Dic3—42no—
146C22×Dic3—42no—
147C22×Dic3—42no—
148C22×Dic3—42no—
149C22×Dic3—42no—
150C22×Dic3—42no—
151C22×Dic3—42no—
152C22×Dic3—42no—
153C22×Dic3—42no—
154C22×Dic3—42no—
155C22×Dic3—42no—
156C2×C3×C8—24no—
157C2×C3×C8—24no—
158C2×C3×C8—24no—
159#217—44no—
160#217—44no—
161#217—44no—
162#217—44no—
163C4⋊Dic3—24no—
164C4⋊Dic3—24no—
165C4⋊Dic3—24no—
166C4⋊Dic3—24no—
167C4⋊Dic3—42no—
168C4⋊Dic3—42no—
169C4⋊Dic3—42no—
170C4⋊Dic3—42no—
171C25D1211yescharacteristic
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—42no—
175C23×C4—61no—
176C23×C4—61no—
177C23×C4—61no—
178C23×C4—61no—
179C23×C4D1211yescharacteristic
180C23×C4D1211yescharacteristic
181C23×C4D1211yescharacteristic
182C23×C4C22×D311yescharacteristic
183C23×C4C22×D311yescharacteristic
184C23×C4C22×D311yescharacteristic
185C22×C8C22×D314yes—
186C2×C4⋊C4—32no—
187C2×C4⋊C4—32no—
188C2×C4⋊C4—32no—
189C2×C4⋊C4—32no—
190C2×C4⋊C4—64no—
191C2×C4⋊C4—64no—
192C2×C4⋊C4—61no—
193C2×C4⋊C4—61no—
194C2×C4⋊C4—61no—
195C2×C4⋊C4—61no—
196C2×C4⋊C4—122no—
197C2×C4⋊C4—122no—
198C2×C22⋊C4—62no—
199C2×C22⋊C4—62no—
200C2×C22⋊C4—62no—
201C2×C22⋊C4—62no—
202C2×C22⋊C4—124no—
203C2×C22⋊C4—124no—
204#106—124no—
205C22⋊C8—24no—
206C22⋊C8—24no—
207C22⋊C8—42no—
208#95—62no—
209#95—62no—
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×SD1612yes—
236C23×C3C2×SD1612yes—
237C23×C3C2×SD1612yes—
238C23×C3AGL(1,ℤ/8)12yes—
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#12312yes—
255C2×C3×C4#12312yes—
256C2×C3×C4C2×#4112yes—
257C2×C3×C4C2×#4112yes—
258C2×C3×C4H(5,𝔽2)12yes—
259C2×Dic3—42no—
260C2×Dic3—42no—
261C2×Dic3—42no—
262C2×Dic3—42no—
263C2×Dic3—42no—
264C2×Dic3—42no—
265C2×Dic3—42no—
266C2×Dic3—42no—
267C2×Dic3—42no—
268C2×Dic3—42no—
269C2×Dic3—42no—
270C2×Dic3—42no—
271C2×Dic3—84no—
272C2×Dic3—84no—
273C2×Dic3—84no—
274C2×Dic3—84no—
275C3×C8—44no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—22no—
280C24—41no—
281C24—41no—
282C24#20414yes—
283C24C2×D1211yescharacteristic
284C24C2×D1211yescharacteristic
285C24C2×D1211yescharacteristic
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—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—416no—
303C22×C4—62no—
304C22×C4—62no—
305C22×C4—62no—
306C22×C4—62no—
307C22×C4—122no—
308C22×C4—122no—
309C22×C4—122no—
310C22×C4—122no—
311C22×C4—122no—
312C22×C4—122no—
313C22×C4—122no—
314C22×C4—122no—
315C22×C4—122no—
316C22×C4—122no—
317C22×C4—122no—
318C22×C4—122no—
319C22×C4C2×D1211yescharacteristic
320C22×C4C2×D1211yescharacteristic
321C22×C4C2×D1211yescharacteristic
322C22×C4#23714yes—
323C22×C4#23714yes—
324C22×C4C23×D311yescharacteristic
325C2×C8—24no—
326C2×C8—24no—
327C2×C8—24no—
328C4⋊C4—64no—
329C4⋊C4—64no—
330C4⋊C4—64no—
331C4⋊C4—64no—
332C4⋊C4—122no—
333C4⋊C4—122no—
334C4⋊C4—122no—
335C4⋊C4—122no—
336C22⋊C4—124no—
337C22⋊C4—124no—
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#48114yes—
361C22×C3#48314yes—
362C22×C3C2×#12311yescharacteristic
363C22×C3C2×#12311yescharacteristic
364C22×C3#53411yescharacteristic
365C22×C3C22×SD1611yescharacteristic
366C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
367C22×C3C2×#13711yescharacteristic
368C22×C3#57511yescharacteristic
369C3×C4—22no—
370C3×C4—22no—
371C3×C4—41no—
372C3×C4#55114yes—
373Dic3—82no—
374Dic3—82no—
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—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—28no—
393C23—44no—
394C23—44no—
395C23—44no—
396C23—44no—
397C23—44no—
398C23#88612yes—
399C23#88612yes—
400C23C2×#20412yes—
401C23C2×#20412yes—
402C23C2×#20412yes—
403C23#90812yes—
404C23#90912yes—
405C23C22×D1211yescharacteristic
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—122no—
420C2×C4—122no—
421C2×C4—122no—
422C2×C4—122no—
423C2×C4—122no—
424C2×C4—122no—
425C2×C4—122no—
426C2×C4—122no—
427C2×C4—122no—
428C2×C4—122no—
429C2×C4—122no—
430C2×C4—122no—
431C2×C4—244no—
432C2×C4—244no—
433C2×C4—244no—
434C2×C4—244no—
435C2×C4#88612yes—
436C2×C4#88612yes—
437C2×C4C2×#23712yes—
438C2×C4C2×#23712yes—
439C2×C4#100412yes—
440C8—44no—
441C2×C3—24no—
442C2×C3—24no—
443C2×C3—42no—
444C2×C3#248311yescharacteristic
445C2×C3C2×#48311yescharacteristic
446C2×C3C2×#48111yescharacteristic
447C2×C3#309411yescharacteristic
448C2×C3#309511yescharacteristic
449C2×C3#318711yescharacteristic
450C2×C3#319311yescharacteristic
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#472114yes—
472C22#473514yes—
473C22C2×#88611yescharacteristic
474C22C2×#88611yescharacteristic
475C22#549311yescharacteristic
476C22C22×#20411yescharacteristic
477C22C2×#90811yescharacteristic
478C22C2×#90911yescharacteristic
479C22#574811yescharacteristic
480C4—22no—
481C4—22no—
482C4—41no—
483C4—242no—
484C4—242no—
485C4—242no—
486C4—242no—
487C4#558714yes—
488C3#2760111yescharacteristic, complemented, Sylow
489C2—24no—
490C2—24no—
491C2—42no—
492C2#7383311yescharacteristic
493C2C2×#472111yescharacteristic
494C2C2×#473511yescharacteristic
495C2#7861211yescharacteristic
496C2#7862411yescharacteristic
497C2#7862511yescharacteristic
498C2#7864211yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1779C2
socleC23×C3C22×D4
prime radicalC25×C3D4
hypercenter#1779D3
hypocenterC3#27601
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C24×C4 ◁ #1779

Complements

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

NH ≅ G/Ntype
4. #69940C2semidirect
5. #69928C2semidirect
6. #69928C2semidirect
7. #69925C2semidirect
8. #69924C2semidirect
30. C2×#817C22semidirect
31. #4551C22semidirect
32. #4551C22semidirect
33. #4551C22semidirect
488. C3#27601semidirect

Maximal subgroups

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

subgroupindex
C2×#52122
C3×#17792
#699402
#699282
#699252
#699242
#276013