Finite Group Databasebeta

← #10727945

Subgroups — #10727945

Order 768.

2190 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×C4⋊C4C211yesmaximal, characteristic
2C2×#5210C212yesmaximal
3C2×#5212C211yesmaximal, characteristic
4C2×#4981C212yesmaximal, complemented
5#72127C212yesmaximal
6#72127C211yesmaximal, characteristic
7#71566C214yesmaximal, complemented
8#71565C212yesmaximal, complemented
9#14821—31nomaximal, Sylow
10C24×C3×C4C2211yescharacteristic
11C24×C3×C4C2211yescharacteristic
12C24×C3×C4C2211yescharacteristic
13C22×C3×C4⋊C4—24no—
14C22×C3×C4⋊C4—28no—
15C22×C3×C4⋊C4C2212yes—
16C22×C3×C4⋊C4C2212yes—
17C22×#217C2212yes—
18C22×#217C2212yes—
19C22×C4⋊Dic3C2211yescharacteristic
20C22×C4⋊Dic3C2211yescharacteristic
21C22×Dic3⋊C4C2214yes—
22#5212—24no—
23#5210—28no—
24C2×C3⋊#95C2212yes—
25C2×C3⋊#95C2214yes—
26C2×C3⋊#95C2212yes—
27C2×C3⋊#95C2214yes—
28C2×C3⋊#95C2214yes—
29C2×C3⋊#95C2212yes—
30#4981—28no—
31C23×C4⋊C4D311yescharacteristic
32C2×#398—32no—
33C2×#385—32no—
34C2×#380—31no—
35#1441—32no—
36#1441—34no—
37C22⋊#95—31no—
38C22⋊#95—32no—
39C25×C3C2311yescharacteristic
40C23×C3×C4—22no—
41C23×C3×C4—22no—
42C23×C3×C4—24no—
43C23×C3×C4—24no—
44C23×C3×C4—22no—
45C23×C3×C4—24no—
46C23×C3×C4—24no—
47C23×C3×C4—24no—
48C23×C3×C4—24no—
49C23×C3×C4—42no—
50C23×C3×C4—42no—
51C23×C3×C4D414yes—
52C23×C3×C4D414yes—
53C23×C3×C4C2311yescharacteristic
54C23×C3×C4C2312yes—
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2312yes—
57C23×Dic3C2312yes—
58C23×Dic3C2312yes—
59C23×Dic3C2314yes—
60C2×C3×C4⋊C4—232no—
61C2×C3×C4⋊C4—24no—
62C2×C3×C4⋊C4—24no—
63C2×C3×C4⋊C4—24no—
64C2×C3×C4⋊C4—24no—
65C2×C3×C4⋊C4—24no—
66C2×C3×C4⋊C4D418yes—
67C2×#217—24no—
68C2×#217—24no—
69C2×#217—24no—
70C2×#217—24no—
71C2×#217—24no—
72C2×#217Q818yes—
73C2×C4⋊Dic3—22no—
74C2×C4⋊Dic3—22no—
75C2×C4⋊Dic3—24no—
76C2×C4⋊Dic3—42no—
77C2×Dic3⋊C4—28no—
78C2×Dic3⋊C4—28no—
79C2×Dic3⋊C4—28no—
80C3⋊#95—28no—
81C3⋊#95—24no—
82C3⋊#95—24no—
83C3⋊#95—28no—
84C3⋊#95—42no—
85C3⋊#95—44no—
86C24×C4C2×D311yescharacteristic
87C24×C4C2×D311yescharacteristic
88C24×C4C2×D311yescharacteristic
89C22×C4⋊C4—24no—
90C22×C4⋊C4—28no—
91C22×C4⋊C4—31no—
92C22×C4⋊C4—34no—
93C22×C4⋊C4—31no—
94C22×C4⋊C4C2×D312yes—
95C22×C4⋊C4C2×D312yes—
96C22×C22⋊C4—32no—
97C22×C22⋊C4—32no—
98#398—68no—
99#385—68no—
100#380—64no—
101C2×#95—32no—
102C2×#95—34no—
103C2×#95—32no—
104C2×#95—34no—
105C2×#95—32no—
106C2×#95—34no—
107C24×C3—22no—
108C24×C3—22no—
109C24×C3—22no—
110C24×C3—22no—
111C24×C3C2×D412yes—
112C24×C3C2×D412yes—
113C24×C3C2×Q814yes—
114C24×C3#4112yes—
115C24×C3#4114yes—
116C24×C3C2411yescharacteristic
117C22×C3×C4—28no—
118C22×C3×C4—216no—
119C22×C3×C4—28no—
120C22×C3×C4—28no—
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—216no—
128C22×C3×C4—28no—
129C22×C3×C4—28no—
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—24no—
135C22×C3×C4—22no—
136C22×C3×C4—24no—
137C22×C3×C4—28no—
138C22×C3×C4—44no—
139C22×C3×C4—416no—
140C22×C3×C4—416no—
141C22×C3×C4—44no—
142C22×C3×C4—44no—
143C22×C3×C4C2×D414yes—
144C22×C3×C4C2×D412yes—
145C22×C3×C4C2×D414yes—
146C22×C3×C4C2×D414yes—
147C22×C3×C4C2×D412yes—
148C22×C3×C4C2×D414yes—
149C22×C3×C4#4114yes—
150C22×Dic3—24no—
151C22×Dic3—24no—
152C22×Dic3—24no—
153C22×Dic3—24no—
154C22×Dic3—24no—
155C22×Dic3—24no—
156C22×Dic3—22no—
157C22×Dic3—24no—
158C22×Dic3—22no—
159C22×Dic3—22no—
160C22×Dic3—22no—
161C22×Dic3—24no—
162C22×Dic3—22no—
163C22×Dic3—22no—
164C22×Dic3—24no—
165C22×Dic3—24no—
166C22×Dic3C2×Q818yes—
167C3×C4⋊C4—216no—
168C3×C4⋊C4—216no—
169#217—216no—
170#217—48no—
171C4⋊Dic3—44no—
172C4⋊Dic3—44no—
173Dic3⋊C4—416no—
174C25C22×D311yescharacteristic
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—22no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—34no—
185C23×C4—32no—
186C23×C4—32no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4D1214yes—
190C23×C4#6714yes—
191C23×C4C22×D311yescharacteristic
192C23×C4C22×D312yes—
193C23×C4C22×D312yes—
194C23×C4C22×D311yescharacteristic
195C2×C4⋊C4—232no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—68no—
202C2×C4⋊C4—68no—
203C2×C4⋊C4—62no—
204C2×C4⋊C4—62no—
205C2×C4⋊C4—68no—
206C2×C4⋊C4—64no—
207C2×C4⋊C4—122no—
208C2×C4⋊C4#6718yes—
209C2×C22⋊C4—38no—
210C2×C22⋊C4—64no—
211C2×C22⋊C4—64no—
212C2×C22⋊C4—64no—
213C2×C22⋊C4—64no—
214C2×C22⋊C4—64no—
215#95—68no—
216#95—64no—
217#95—64no—
218#95—68no—
219#95—122no—
220#95—124no—
221C23×C3—22no—
222C23×C3—24no—
223C23×C3—22no—
224C23×C3—24no—
225C23×C3—22no—
226C23×C3—22no—
227C23×C3—24no—
228C23×C3—24no—
229C23×C3—28no—
230C23×C3—28no—
231C23×C3—28no—
232C23×C3—28no—
233C23×C3#12114yes—
234C23×C3C22⋊Q818yes—
235C23×C3#12314yes—
236C23×C3#12518yes—
237C23×C3C4⋊Q814yes—
238C23×C3C22×D411yescharacteristic
239C23×C3C22×D411yescharacteristic
240C23×C3C22×D412yes—
241C23×C3C22×Q812yes—
242C23×C3C2×#4111yescharacteristic
243C23×C3C2×#4112yes—
244C23×C3C2×#4112yes—
245C23×C3H(5,𝔽2)12yes—
246C23×C32−1+412yes—
247C2×C3×C4—232no—
248C2×C3×C4—24no—
249C2×C3×C4—24no—
250C2×C3×C4—24no—
251C2×C3×C4—22no—
252C2×C3×C4—24no—
253C2×C3×C4—24no—
254C2×C3×C4—24no—
255C2×C3×C4—22no—
256C2×C3×C4—28no—
257C2×C3×C4—22no—
258C2×C3×C4—24no—
259C2×C3×C4—22no—
260C2×C3×C4—22no—
261C2×C3×C4—28no—
262C2×C3×C4—416no—
263C2×C3×C4—416no—
264C2×C3×C4—42no—
265C2×C3×C4—42no—
266C2×C3×C4—44no—
267C2×C3×C4—44no—
268C2×C3×C4#12014yes—
269C2×C3×C4#12118yes—
270C2×Dic3—28no—
271C2×Dic3—28no—
272C2×Dic3—28no—
273C2×Dic3—44no—
274C2×Dic3—44no—
275C2×Dic3—44no—
276C2×Dic3—44no—
277C2×Dic3—42no—
278C2×Dic3—44no—
279C2×Dic3—42no—
280C2×Dic3—42no—
281C2×Dic3—42no—
282C2×Dic3—44no—
283C2×Dic3—42no—
284C2×Dic3—42no—
285C2×Dic3—44no—
286C2×Dic3—44no—
287C24—22no—
288C24—22no—
289C24—22no—
290C24—22no—
291C24C2×D1212yes—
292C24#23512yes—
293C24#23714yes—
294C24D3×Q814yes—
295C24C2×#6712yes—
296C24C23×D311yescharacteristic
297C22×C4—28no—
298C22×C4—28no—
299C22×C4—216no—
300C22×C4—28no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—216no—
308C22×C4—28no—
309C22×C4—28no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—28no—
318C22×C4—38no—
319C22×C4—44no—
320C22×C4—416no—
321C22×C4—416no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4—64no—
325C22×C4—64no—
326C22×C4—64no—
327C22×C4—64no—
328C22×C4—64no—
329C22×C4—64no—
330C22×C4—62no—
331C22×C4—62no—
332C22×C4—62no—
333C22×C4—62no—
334C22×C4—64no—
335C22×C4—64no—
336C22×C4—64no—
337C22×C4—62no—
338C22×C4—62no—
339C22×C4—64no—
340C22×C4C2×D1214yes—
341C22×C4#23514yes—
342C22×C4C2×#6712yes—
343C22×C4C2×#6714yes—
344C22×C4C2×#6714yes—
345C22×C4C2×#6714yes—
346C22×C4C2×#6712yes—
347C4⋊C4—216no—
348C4⋊C4—216no—
349C4⋊C4—124no—
350C4⋊C4—1216no—
351C4⋊C4—124no—
352C22⋊C4—616no—
353C22⋊C4—128no—
354C22×C3—24no—
355C22×C3—24no—
356C22×C3—24no—
357C22×C3—24no—
358C22×C3—216no—
359C22×C3—24no—
360C22×C3—216no—
361C22×C3—24no—
362C22×C3#39818yescomplemented
363C22×C3C2×#12011yescharacteristic
364C22×C3C2×#12111yescharacteristic
365C22×C3C2×#12112yes—
366C22×C3C2×C22⋊Q812yes—
367C22×C3C2×#12311yescharacteristic
368C22×C3#52511yescharacteristic
369C22×C3#52512yes—
370C22×C3C2×#12512yes—
371C22×C3C2×C4⋊Q811yescharacteristic
372C22×C3#53411yescharacteristic
373C22×C3#53611yescharacteristic
374C22×C3#53712yes—
375C22×C3#53912yes—
376C22×C3#54614yes—
377C22×C3D4×Q814yes—
378C22×C3#55214yes—
379C22×C3#55414yes—
380C3×C4—28no—
381C3×C4—24no—
382C3×C4—42no—
383C3×C4—44no—
384Dic3—48no—
385Dic3—82no—
386Dic3—82no—
387C23—24no—
388C23—28no—
389C23—28no—
390C23—24no—
391C23—24no—
392C23—24no—
393C23—22no—
394C23—22no—
395C23—28no—
396C23—28no—
397C23—22no—
398C23—22no—
399C23#88918yes—
400C23#89718yes—
401C23#93014yes—
402C23#93514yes—
403C23#94414yes—
404C23C22×D1211yescharacteristic
405C23C2×#23511yescharacteristic
406C23C2×#23512yes—
407C23C2×#23712yes—
408C23#100412yes—
409C23C2×D3×Q812yes—
410C23#100712yes—
411C23C22×#6711yescharacteristic
412C23C22×#6712yes—
413C2×C4—232no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—28no—
422C2×C4—24no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—28no—
426C2×C4—22no—
427C2×C4—24no—
428C2×C4—416no—
429C2×C4—416no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—68no—
435C2×C4—68no—
436C2×C4—68no—
437C2×C4—124no—
438C2×C4—124no—
439C2×C4—124no—
440C2×C4—124no—
441C2×C4—124no—
442C2×C4—124no—
443C2×C4—122no—
444C2×C4—122no—
445C2×C4—124no—
446C2×C4—122no—
447C2×C4—124no—
448C2×C4—122no—
449C2×C4—122no—
450C2×C4—122no—
451C2×C4#93018yes—
452C2×C4#95314yes—
453C2×C3—28no—
454C2×C3C2×#39811yescharacteristic
455C2×C3#240511yescharacteristic
456C2×C3#241714yes—
457C2×C3#243411yescharacteristic
458C2×C3#247314yes—
459C2×C3#248411yescharacteristic
460C2×C3#258311yescharacteristic
461C2×C3#261811yescharacteristic
462C2×C3#284711yescharacteristic
463C22—24no—
464C22—24no—
465C22—216no—
466C22—24no—
467C22—24no—
468C22—24no—
469C22—24no—
470C22—216no—
471C22#498118yescomplemented
472C22C2×#88912yes—
473C22C2×#89712yes—
474C22#551012yes—
475C22#551114yes—
476C22#551314yes—
477C22#551512yes—
478C22C2×#93012yes—
479C22C2×#93011yescharacteristic
480C22#579111yescharacteristic
481C22#579112yes—
482C22C2×#93511yescharacteristic
483C22#580411yescharacteristic
484C22#580614yes—
485C22C2×#94411yescharacteristic
486C22Q8×#6714yes—
487C22#581611yescharacteristic
488C22C2×#95311yescharacteristic
489C4—28no—
490C4—24no—
491C4—42no—
492C4—44no—
493C4—128no—
494C4—242no—
495C4—242no—
496C3#1482111yescharacteristic, complemented, Sylow
497C2—28no—
498C2C2×#498111yescharacteristic
499C2#8217414yes—
500C2#8217611yescharacteristic
501C2#8218111yescharacteristic
502C2#8219414yes—
503C2#8219911yescharacteristic
504C2#8403211yescharacteristic
505C2#8404711yescharacteristic
506C2#8406511yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC23×C3×C4⋊C4C2
prime radicalC25×C3C23
hypercenterC23×C4⋊C4D3
hypocenterC3#14821
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C23×C4⋊C4

Complements

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

NH ≅ G/Ntype
4. C2×#4981C2semidirect
7. #71566C2semidirect
8. #71565C2semidirect
362. C22×C3#398semidirect
471. C22#4981semidirect
496. C3#14821semidirect

Maximal subgroups

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

subgroupindex
C23×C3×C4⋊C42
C2×#52102
C2×#52122
C2×#49812
#721272
#715662
#715652
#148213