Finite Group Databasebeta

← #10727953

Subgroups — #10727953

Order 768.

2046 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×C4⋊C4C211yesmaximal, characteristic
2C2×#5212C211yesmaximal, characteristic
3C2×#4986C212yesmaximal, complemented
4#72127C212yesmaximal
5#72127C212yesmaximal
6#72127C211yesmaximal, characteristic
7#71567C214yesmaximal, complemented
8#71565C212yesmaximal, complemented
9#14822—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
21#5212—24no—
22C2×C3⋊#95C2212yes—
23C2×C3⋊#95C2214yes—
24C2×C3⋊#95C2212yes—
25C2×C3⋊#95C2214yes—
26C2×C3⋊#95C2214yes—
27C2×C3⋊#95C2214yes—
28C2×C3⋊#95C2212yes—
29#4986—28no—
30C23×C4⋊C4D311yescharacteristic
31C2×#400—32no—
32C2×#380—31no—
33#1442—34no—
34#1441—32no—
35C22⋊#95—31no—
36C22⋊#95—32no—
37C22⋊#95—32no—
38C25×C3C2311yescharacteristic
39C23×C3×C4—22no—
40C23×C3×C4—22no—
41C23×C3×C4—24no—
42C23×C3×C4—24no—
43C23×C3×C4—22no—
44C23×C3×C4—24no—
45C23×C3×C4—24no—
46C23×C3×C4—24no—
47C23×C3×C4—24no—
48C23×C3×C4—42no—
49C23×C3×C4—42no—
50C23×C3×C4D414yes—
51C23×C3×C4Q814yes—
52C23×C3×C4C2311yescharacteristic
53C23×C3×C4C2312yes—
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2312yes—
56C23×Dic3C2314yes—
57C23×Dic3C2312yes—
58C23×Dic3C2312yes—
59C2×C3×C4⋊C4—232no—
60C2×C3×C4⋊C4—24no—
61C2×C3×C4⋊C4—24no—
62C2×C3×C4⋊C4—24no—
63C2×C3×C4⋊C4—24no—
64C2×C3×C4⋊C4—24no—
65C2×C3×C4⋊C4D418yes—
66C2×#217—24no—
67C2×#217—24no—
68C2×#217—24no—
69C2×#217—28no—
70C2×#217—24no—
71C2×C4⋊Dic3—22no—
72C2×C4⋊Dic3—22no—
73C2×C4⋊Dic3—24no—
74C2×C4⋊Dic3—42no—
75C3⋊#95—24no—
76C3⋊#95—24no—
77C3⋊#95—28no—
78C3⋊#95—28no—
79C3⋊#95—28no—
80C3⋊#95—42no—
81C3⋊#95—44no—
82C24×C4C2×D311yescharacteristic
83C24×C4C2×D311yescharacteristic
84C24×C4C2×D311yescharacteristic
85C22×C4⋊C4—24no—
86C22×C4⋊C4—28no—
87C22×C4⋊C4—31no—
88C22×C4⋊C4—31no—
89C22×C4⋊C4C2×D312yes—
90C22×C4⋊C4C2×D312yes—
91C22×C22⋊C4—32no—
92C22×C22⋊C4—32no—
93#400—68no—
94#380—64no—
95C2×#95—32no—
96C2×#95—34no—
97C2×#95—34no—
98C2×#95—34no—
99C2×#95—32no—
100C2×#95—34no—
101C2×#95—32no—
102C24×C3—22no—
103C24×C3—22no—
104C24×C3—22no—
105C24×C3—22no—
106C24×C3C2×D412yes—
107C24×C3C2×Q812yes—
108C24×C3#4112yes—
109C24×C3#4114yes—
110C24×C3#4114yes—
111C24×C3C2411yescharacteristic
112C22×C3×C4—28no—
113C22×C3×C4—216no—
114C22×C3×C4—28no—
115C22×C3×C4—28no—
116C22×C3×C4—22no—
117C22×C3×C4—22no—
118C22×C3×C4—22no—
119C22×C3×C4—22no—
120C22×C3×C4—22no—
121C22×C3×C4—22no—
122C22×C3×C4—216no—
123C22×C3×C4—28no—
124C22×C3×C4—28no—
125C22×C3×C4—22no—
126C22×C3×C4—22no—
127C22×C3×C4—22no—
128C22×C3×C4—22no—
129C22×C3×C4—24no—
130C22×C3×C4—22no—
131C22×C3×C4—24no—
132C22×C3×C4—28no—
133C22×C3×C4—44no—
134C22×C3×C4—416no—
135C22×C3×C4—416no—
136C22×C3×C4—44no—
137C22×C3×C4—44no—
138C22×C3×C4C2×D412yes—
139C22×C3×C4C2×D414yes—
140C22×C3×C4C2×D414yes—
141C22×C3×C4C2×D412yes—
142C22×C3×C4C2×D414yes—
143C22×C3×C4C2×Q814yes—
144C22×C3×C4#4114yes—
145C22×Dic3—24no—
146C22×Dic3—24no—
147C22×Dic3—22no—
148C22×Dic3—22no—
149C22×Dic3—24no—
150C22×Dic3—24no—
151C22×Dic3—24no—
152C22×Dic3—24no—
153C22×Dic3—24no—
154C22×Dic3—24no—
155C22×Dic3—24no—
156C22×Dic3—24no—
157C22×Dic3—24no—
158C22×Dic3—24no—
159C22×Dic3—24no—
160C3×C4⋊C4—216no—
161C3×C4⋊C4—216no—
162#217—48no—
163#217—48no—
164C4⋊Dic3—44no—
165C4⋊Dic3—44no—
166C25C22×D311yescharacteristic
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—22no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—32no—
177C23×C4—32no—
178C23×C4—34no—
179C23×C4—42no—
180C23×C4—42no—
181C23×C4Dic614yes—
182C23×C4#6714yes—
183C23×C4C22×D311yescharacteristic
184C23×C4C22×D312yes—
185C23×C4C22×D312yes—
186C23×C4C22×D311yescharacteristic
187C2×C4⋊C4—232no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—62no—
194C2×C4⋊C4—62no—
195C2×C4⋊C4—64no—
196C2×C4⋊C4—122no—
197C2×C4⋊C4#6718yes—
198C2×C22⋊C4—64no—
199C2×C22⋊C4—64no—
200C2×C22⋊C4—64no—
201C2×C22⋊C4—68no—
202C2×C22⋊C4—64no—
203#95—68no—
204#95—64no—
205#95—64no—
206#95—68no—
207#95—68no—
208#95—122no—
209#95—124no—
210C23×C3—22no—
211C23×C3—24no—
212C23×C3—22no—
213C23×C3—24no—
214C23×C3—22no—
215C23×C3—22no—
216C23×C3—24no—
217C23×C3—24no—
218C23×C3—28no—
219C23×C3—28no—
220C23×C3—28no—
221C23×C3—28no—
222C23×C3C22⋊Q814yes—
223C23×C3#12314yes—
224C23×C3#12414yes—
225C23×C3#12518yes—
226C23×C3#12618yes—
227C23×C3C22×D411yescharacteristic
228C23×C3C22×D412yes—
229C23×C3C22×Q811yescharacteristic
230C23×C3C2×#4111yescharacteristic
231C23×C3C2×#4112yes—
232C23×C3C2×#4112yes—
233C23×C3C2×#4112yes—
234C23×C3H(5,𝔽2)12yes—
235C23×C32−1+412yes—
236C2×C3×C4—232no—
237C2×C3×C4—24no—
238C2×C3×C4—24no—
239C2×C3×C4—24no—
240C2×C3×C4—22no—
241C2×C3×C4—24no—
242C2×C3×C4—24no—
243C2×C3×C4—24no—
244C2×C3×C4—22no—
245C2×C3×C4—28no—
246C2×C3×C4—22no—
247C2×C3×C4—24no—
248C2×C3×C4—22no—
249C2×C3×C4—22no—
250C2×C3×C4—28no—
251C2×C3×C4—416no—
252C2×C3×C4—416no—
253C2×C3×C4—42no—
254C2×C3×C4—42no—
255C2×C3×C4—44no—
256C2×C3×C4—44no—
257C2×C3×C4#12014yes—
258C2×C3×C4C22⋊Q818yes—
259C2×Dic3—42no—
260C2×Dic3—42no—
261C2×Dic3—44no—
262C2×Dic3—42no—
263C2×Dic3—42no—
264C2×Dic3—44no—
265C2×Dic3—42no—
266C2×Dic3—42no—
267C2×Dic3—44no—
268C2×Dic3—44no—
269C2×Dic3—48no—
270C2×Dic3—44no—
271C2×Dic3—44no—
272C2×Dic3—48no—
273C2×Dic3—44no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24C2×Dic612yes—
279C24#23512yes—
280C24#23714yes—
281C24#23914yes—
282C24C2×#6712yes—
283C24C23×D311yescharacteristic
284C22×C4—28no—
285C22×C4—28no—
286C22×C4—216no—
287C22×C4—28no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—216no—
295C22×C4—28no—
296C22×C4—28no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—28no—
305C22×C4—44no—
306C22×C4—416no—
307C22×C4—416no—
308C22×C4—44no—
309C22×C4—44no—
310C22×C4—64no—
311C22×C4—64no—
312C22×C4—64no—
313C22×C4—64no—
314C22×C4—64no—
315C22×C4—64no—
316C22×C4—64no—
317C22×C4—64no—
318C22×C4—64no—
319C22×C4—64no—
320C22×C4—64no—
321C22×C4—64no—
322C22×C4—64no—
323C22×C4—62no—
324C22×C4—62no—
325C22×C4C2×Dic614yes—
326C22×C4#23514yes—
327C22×C4C2×#6712yes—
328C22×C4C2×#6714yes—
329C22×C4C2×#6714yes—
330C22×C4C2×#6714yes—
331C22×C4C2×#6712yes—
332C4⋊C4—216no—
333C4⋊C4—216no—
334C4⋊C4—124no—
335C4⋊C4—124no—
336C22⋊C4—128no—
337C22⋊C4—128no—
338C22×C3—24no—
339C22×C3—24no—
340C22×C3—24no—
341C22×C3—24no—
342C22×C3—24no—
343C22×C3—216no—
344C22×C3—216no—
345C22×C3—24no—
346C22×C3#40018yescomplemented
347C22×C3C2×#12011yescharacteristic
348C22×C3C2×C22⋊Q811yescharacteristic
349C22×C3C2×C22⋊Q812yes—
350C22×C3C2×#12311yescharacteristic
351C22×C3#52511yescharacteristic
352C22×C3#52512yes—
353C22×C3C2×#12411yescharacteristic
354C22×C3C2×#12512yes—
355C22×C3C2×#12612yes—
356C22×C3#53411yescharacteristic
357C22×C3#53611yescharacteristic
358C22×C3#53912yes—
359C22×C3#54412yes—
360C22×C3#54614yes—
361C22×C3#54814yes—
362C22×C3#55214yes—
363C22×C3#55314yes—
364C3×C4—28no—
365C3×C4—24no—
366C3×C4—42no—
367C3×C4—44no—
368Dic3—82no—
369Dic3—82no—
370Dic3—84no—
371C23—24no—
372C23—28no—
373C23—28no—
374C23—24no—
375C23—24no—
376C23—24no—
377C23—22no—
378C23—22no—
379C23—28no—
380C23—22no—
381C23—28no—
382C23—22no—
383C23#89018yes—
384C23#89818yes—
385C23#92414yes—
386C23#93514yes—
387C23#94714yes—
388C23C22×Dic611yescharacteristic
389C23C2×#23511yescharacteristic
390C23C2×#23512yes—
391C23C2×#23712yes—
392C23#100412yes—
393C23C2×#23912yes—
394C23#100712yes—
395C23C22×#6711yescharacteristic
396C23C22×#6712yes—
397C2×C4—232no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—28no—
406C2×C4—24no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—28no—
410C2×C4—22no—
411C2×C4—24no—
412C2×C4—416no—
413C2×C4—416no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—124no—
419C2×C4—124no—
420C2×C4—128no—
421C2×C4—124no—
422C2×C4—124no—
423C2×C4—122no—
424C2×C4—124no—
425C2×C4—122no—
426C2×C4—124no—
427C2×C4—122no—
428C2×C4—122no—
429C2×C4—122no—
430C2×C4—128no—
431C2×C4—124no—
432C2×C4—122no—
433C2×C4#92418yes—
434C2×C4#95314yes—
435C2×C3—28no—
436C2×C3C2×#40011yescharacteristic
437C2×C3#240511yescharacteristic
438C2×C3#243611yescharacteristic
439C2×C3#244214yes—
440C2×C3#248114yes—
441C2×C3#248411yescharacteristic
442C2×C3#259111yescharacteristic
443C2×C3#262711yescharacteristic
444C2×C3#284811yescharacteristic
445C22—24no—
446C22—24no—
447C22—24no—
448C22—24no—
449C22—24no—
450C22—24no—
451C22—216no—
452C22—216no—
453C22#498618yescomplemented
454C22C2×#89012yes—
455C22#550012yes—
456C22C2×#89812yes—
457C22#551314yes—
458C22#551414yes—
459C22#551512yes—
460C22C2×#92411yescharacteristic
461C22C2×#92412yes—
462C22#579111yescharacteristic
463C22#579112yes—
464C22C2×#93511yescharacteristic
465C22#580411yescharacteristic
466C22#580614yes—
467C22C2×#94711yescharacteristic
468C22#581611yescharacteristic
469C22#581714yes—
470C22C2×#95311yescharacteristic
471C4—28no—
472C4—24no—
473C4—42no—
474C4—44no—
475C4—242no—
476C4—242no—
477C4—244no—
478C3#1482211yescharacteristic, complemented, Sylow
479C2—28no—
480C2C2×#498611yescharacteristic
481C2#8217214yes—
482C2#8217514yes—
483C2#8217611yescharacteristic
484C2#8218211yescharacteristic
485C2#8218311yescharacteristic
486C2#8403111yescharacteristic
487C2#8404711yescharacteristic
488C2#8406611yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC23×C3×C4⋊C4C2
prime radicalC25×C3C23
hypercenterC23×C4⋊C4D3
hypocenterC3#14822
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C23×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C2×#4986C2semidirect
7. #71567C2semidirect
8. #71565C2semidirect
346. C22×C3#400semidirect
453. C22#4986semidirect
478. C3#14822semidirect

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×#52122
C2×#49862
#721272
#715672
#715652
#148223