Finite Group Databasebeta

← #10944016

Subgroups — #10944016

Order 768.

1436 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×Q8C211yesmaximal, characteristic
2C2×#5228C212yesmaximal
3C2×#5031C212yesmaximal, complemented
4#82076C211yesmaximal, characteristic
5#71632C214yesmaximal, complemented
6#71626C212yesmaximal, complemented
7#71559C212yesmaximal
8#71530C211yesmaximal, characteristic
9#32906—31nomaximal, Sylow
10C23×C3×Q8C2211yescharacteristic
11C2×C3×C4×Q8—28no—
12C2×C3×C4×Q8—21no—
13C2×C3×C4×Q8—21no—
14C2×C3×C4×Q8C2212yes—
15C2×C3×C4×Q8C2212yes—
16C22×C3×C4⋊C4—21no—
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C42—21no—
19C22×C3×C42C2211yescharacteristic
20C22×#216C2214yes—
21C2×#924C2212yes—
22C2×#869C2211yescharacteristic
23#5228—28no—
24C2×#835C2214yes—
25#5212—21no—
26#5212—21no—
27C2×#830C2212yes—
28#5031—28no—
29C2×#810C2214yes—
30#4935C2211yescharacteristic
31#4929C2212yes—
32#4928—21no—
33#4928—21no—
34C2×C4⋊(C3⋊C8)C2212yes—
35#4529C2214yes—
36#4524C2212yes—
37C22×C4×Q8D311yescharacteristic
38C2×#462—32no—
39C2×#451—32no—
40#2857—31no—
41#1878—34no—
42#1872—32no—
43#1803—32no—
44#1797—31no—
45C22×C3×Q8—24no—
46C22×C3×Q8—24no—
47C22×C3×Q8—21no—
48C22×C3×Q8—22no—
49C22×C3×Q8D414yes—
50C22×C3×Q8C2312yes—
51C23×C3×C4—21no—
52C23×C3×C4—21no—
53C23×C3×C4D411yescharacteristic
54C23×C3×C4D411yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C22×Dic6C2312yes—
57C3×C4×Q8—28no—
58C3×C4×Q8—28no—
59C3×C4×Q8—24no—
60C3×C4×Q8—24no—
61C3×C4×Q8—24no—
62C3×C4×Q8D418yes—
63C2×C3×C4⋊C4—28no—
64C2×C3×C4⋊C4—22no—
65C2×C3×C4⋊C4—22no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4—24no—
68C2×C3×C4⋊C4C2312yes—
69C2×C3×C42—28no—
70C2×C3×C42—22no—
71C2×C3×C42—22no—
72C2×C3×C42—24no—
73C2×C3×C42D411yescharacteristic
74C2×C3×C42D411yescharacteristic
75C2×C3×C42C2311yescharacteristic
76C2×C3×C42C2311yescharacteristic
77C2×#217—22no—
78C2×#216—28no—
79C2×#216—28no—
80C2×#216—28no—
81C2×C4⋊Dic3—21no—
82C2×C4⋊Dic3—21no—
83C2×C4⋊Dic3—21no—
84C2×C4⋊Dic3—21no—
85C2×C4⋊Dic3C2312yes—
86#924—24no—
87#924—24no—
88C2×Dic3⋊C4—22no—
89C22×C3⋊C8C2314yes—
90#869—24no—
91#835—28no—
92C3⋊#95—22no—
93C3⋊#95—22no—
94#830—24no—
95#810—28no—
96C4⋊(C3⋊C8)—24no—
97C23×Q8C2×D311yescharacteristic
98C22×Q16—34no—
99C2×C4⋊Q8—31no—
100C2×C22⋊Q8—32no—
101C2×C4×Q8—28no—
102C2×C4×Q8—21no—
103C2×C4×Q8—21no—
104C2×C4×Q8C2×D312yes—
105C2×C4×Q8C2×D312yes—
106C22×C4⋊C4—21no—
107C22×C4⋊C4C2×D311yescharacteristic
108C22×C42—21no—
109C22×C42C2×D311yescharacteristic
110#462—68no—
111#451—68no—
112C2×C4⋊C8—32no—
113C2×#103—34no—
114C2×#103—34no—
115C2×C22⋊C8—32no—
116#391—32no—
117#389—61no—
118#389—61no—
119#384—31no—
120#380—61no—
121#380—61no—
122#340—34no—
123#336—32no—
124C24×C3C2×D411yescharacteristic
125C2×C3×Q8—24no—
126C2×C3×Q8—216no—
127C2×C3×Q8—24no—
128C2×C3×Q8—216no—
129C2×C3×Q8—24no—
130C2×C3×Q8—22no—
131C2×C3×Q8—22no—
132C2×C3×Q8—22no—
133C2×C3×Q8C2×D414yes—
134C2×C3×Q8C2×D414yes—
135C2×C3×Q8#4114yes—
136C22×C3×C4—24no—
137C22×C3×C4—24no—
138C22×C3×C4—24no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—24no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—41no—
149C22×C3×C4—42no—
150C22×C3×C4—42no—
151C22×C3×C4Q1614yescomplemented
152C22×C3×C4C2×D412yes—
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4#4112yes—
159C22×C3×C4C2411yescharacteristic
160C22×Dic3—22no—
161C22×Dic3—22no—
162C2×Dic6—24no—
163C2×Dic6—22no—
164C2×Dic6—22no—
165C2×Dic6—24no—
166C3×C4⋊C4—28no—
167C3×C4⋊C4—24no—
168C3×C4⋊C4—28no—
169C3×C4⋊C4—22no—
170C3×C4⋊C4C2×D414yes—
171C3×C42—28no—
172C3×C42—28no—
173C3×C42—22no—
174C3×C42—22no—
175C3×C42—22no—
176C3×C42C2×D414yes—
177#217—44no—
178#216—416no—
179C4⋊Dic3—22no—
180C4⋊Dic3—22no—
181C4⋊Dic3—42no—
182C4⋊Dic3—42no—
183C4⋊Dic3—42no—
184C4⋊Dic3—42no—
185Dic3⋊C4—44no—
186C2×C3⋊C8—24no—
187C2×C3⋊C8—24no—
188C2×C3⋊C8—24no—
189C22×Q8—24no—
190C22×Q8—24no—
191C22×Q8—22no—
192C22×Q8—21no—
193C22×Q8—32no—
194C22×Q8D1214yes—
195C22×Q8C22×D312yes—
196C23×C4—21no—
197C23×C4—21no—
198C23×C4#6711yescharacteristic
199C23×C4#6711yescharacteristic
200C23×C4C22×D311yescharacteristic
201C2×Q16—68no—
202C2×Q16—68no—
203C2×Q16—68no—
204C22×C8—34no—
205C4⋊Q8—64no—
206C22⋊Q8—64no—
207C22⋊Q8—64no—
208C4×Q8—28no—
209C4×Q8—28no—
210C4×Q8—24no—
211C4×Q8—24no—
212C4×Q8—24no—
213C4×Q8#6718yes—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—32no—
220C2×C4⋊C4—62no—
221C2×C4⋊C4—61no—
222C2×C4⋊C4—61no—
223C2×C4⋊C4—61no—
224C2×C4⋊C4—61no—
225C2×C4⋊C4C22×D312yes—
226C2×C22⋊C4—62no—
227C2×C42—28no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—24no—
231C2×C42#6711yescharacteristic
232C2×C42#6711yescharacteristic
233C2×C42C22×D311yescharacteristic
234C2×C42C22×D311yescharacteristic
235C4⋊C8—64no—
236#103—68no—
237#103—68no—
238C22⋊C8—64no—
239#95—62no—
240#95—62no—
241C23×C3—21no—
242C23×C3—21no—
243C23×C3—21no—
244C23×C3—21no—
245C23×C3#12112yes—
246C23×C3C2×Q1612yes—
247C23×C3#13712yes—
248C23×C3C22×D411yescharacteristic
249C3×Q8—216no—
250C3×Q8—24no—
251C3×Q8—24no—
252C3×Q8—24no—
253C3×Q8#12118yes—
254C2×C3×C4—216no—
255C2×C3×C4—21no—
256C2×C3×C4—24no—
257C2×C3×C4—21no—
258C2×C3×C4—24no—
259C2×C3×C4—24no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—24no—
265C2×C3×C4—24no—
266C2×C3×C4—28no—
267C2×C3×C4—21no—
268C2×C3×C4—24no—
269C2×C3×C4—21no—
270C2×C3×C4—21no—
271C2×C3×C4—44no—
272C2×C3×C4—48no—
273C2×C3×C4—42no—
274C2×C3×C4—42no—
275C2×C3×C4#12012yes—
276C2×C3×C4#12012yes—
277C2×C3×C4#12112yes—
278C2×C3×C4C2×Q1612yes—
279C2×C3×C4C2×Q1612yes—
280C2×C3×C4#13712yes—
281C2×C3×C4C22×D411yescharacteristic
282C2×C3×C4C22×D412yes—
283C2×C3×C4C2×#4112yes—
284C2×C3×C4C2×#4111yescharacteristic
285C2×Dic3—42no—
286C2×Dic3—42no—
287C2×Dic3—42no—
288C2×Dic3—42no—
289C2×Dic3—42no—
290C2×Dic3—42no—
291Dic6—44no—
292Dic6—44no—
293C3⋊C8—44no—
294C24C2×#6711yescharacteristic
295C2×Q8—24no—
296C2×Q8—216no—
297C2×Q8—24no—
298C2×Q8—216no—
299C2×Q8—24no—
300C2×Q8—22no—
301C2×Q8—22no—
302C2×Q8—22no—
303C2×Q8—62no—
304C2×Q8—62no—
305C2×Q8—64no—
306C2×Q8—64no—
307C2×Q8C2×D1214yes—
308C2×Q8#23514yes—
309C2×Q8C2×#6714yes—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—62no—
326C22×C4—62no—
327C22×C4#21614yescomplemented
328C22×C4C2×D1212yes—
329C22×C4#23512yes—
330C22×C4C2×#6711yescharacteristic
331C22×C4C2×#6711yescharacteristic
332C22×C4C2×#6711yescharacteristic
333C22×C4C2×#6711yescharacteristic
334C22×C4C2×#6711yescharacteristic
335C22×C4C23×D311yescharacteristic
336Q16—1216no—
337C2×C8—64no—
338C2×C8—64no—
339C2×C8—64no—
340C4⋊C4—28no—
341C4⋊C4—24no—
342C4⋊C4—28no—
343C4⋊C4—22no—
344C4⋊C4—62no—
345C4⋊C4—62no—
346C4⋊C4—122no—
347C4⋊C4—124no—
348C4⋊C4—122no—
349C4⋊C4—122no—
350C4⋊C4—122no—
351C4⋊C4C2×#6714yes—
352C22⋊C4—124no—
353C42—28no—
354C42—28no—
355C42—22no—
356C42—22no—
357C42—22no—
358C42C2×#6714yes—
359C22×C3—22no—
360C22×C3—22no—
361C22×C3—22no—
362C22×C3—22no—
363C22×C3—22no—
364C22×C3—22no—
365C22×C3#46214yescomplemented
366C22×C3C2×#12011yescharacteristic
367C22×C3C2×#12111yescharacteristic
368C22×C3#52511yescharacteristic
369C22×C3C22×Q1611yescharacteristic
370C22×C3C2×#13711yescharacteristic
371C22×C3C2×#13711yescharacteristic
372C22×C3#57511yescharacteristic
373C3×C4—28no—
374C3×C4—22no—
375C3×C4—22no—
376C3×C4—22no—
377C3×C4—41no—
378C3×C4—44no—
379C3×C4#45114yescomplemented
380C3×C4C2×#12112yes—
381C3×C4#52512yes—
382Dic3—82no—
383Dic3—82no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23#93012yes—
389C23C2×#21612yes—
390C23#95112yes—
391C23C22×#6711yescharacteristic
392Q8—216no—
393Q8—24no—
394Q8—24no—
395Q8—24no—
396Q8—124no—
397Q8—124no—
398Q8#93018yes—
399C2×C4—216no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—24no—
408C2×C4—21no—
409C2×C4—28no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—24no—
416C2×C4—44no—
417C2×C4—48no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—122no—
421C2×C4—122no—
422C2×C4—122no—
423C2×C4—122no—
424C2×C4—122no—
425C2×C4—122no—
426C2×C4#93012yes—
427C2×C4#94212yes—
428C2×C4C2×#21612yes—
429C2×C4C2×#21612yes—
430C2×C4#95312yes—
431C2×C4#95312yes—
432C2×C4C22×D1211yescharacteristic
433C2×C4C2×#23512yes—
434C2×C4C2×#23511yescharacteristic
435C2×C4C22×#6712yes—
436C8—124no—
437C2×C3—24no—
438C2×C3#284711yescharacteristic
439C2×C3C2×#45111yescharacteristic
440C2×C3#300711yescharacteristic
441C2×C3C2×#46211yescharacteristic
442C2×C3#303711yescharacteristic
443C2×C3#310411yescharacteristic
444C2×C3#311311yescharacteristic
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22#503114yescomplemented
452C22C2×#93011yescharacteristic
453C22#579111yescharacteristic
454C22C2×#94211yescharacteristic
455C22C22×#21611yescharacteristic
456C22#582211yescharacteristic
457C22C2×#95111yescharacteristic
458C22C2×#95311yescharacteristic
459C4—28no—
460C4—22no—
461C4—22no—
462C4—22no—
463C4—41no—
464C4—44no—
465C4—242no—
466C4—242no—
467C4#522814yescomplemented
468C4C2×#93012yes—
469C4#579112yes—
470C3#3290611yescharacteristic, complemented, Sylow
471C2—24no—
472C2#8244111yescharacteristic
473C2C2×#503111yescharacteristic
474C2#8244711yescharacteristic
475C2#8244811yescharacteristic
476C2#8403211yescharacteristic
477C2C2×#522811yescharacteristic
478C2#8408411yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC22×C3×C4×Q8C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenterC22×C4×Q8D3
hypocenterC3#32906
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
3. C2×#5031C2semidirect
5. #71632C2semidirect
6. #71626C2semidirect
151. C22×C3×C4Q16semidirect
327. C22×C4#216semidirect
365. C22×C3#462semidirect
379. C3×C4#451semidirect
451. C22#5031semidirect
467. C4#5228semidirect
470. C3#32906semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C4×Q82
C2×#52282
C2×#50312
#820762
#716322
#716262
#715592
#715302
#329063