Finite Group Databasebeta

← #10849319

Subgroups — #10849319

Order 768.

1076 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5585C211yesmaximal, characteristic
2C2×C3×#479C211yesmaximal, characteristic, complemented
3#83041C211yesmaximal, characteristic, complemented
4C2×#5067C211yesmaximal, characteristic
5#71631C211yesmaximal, characteristic, complemented
6#70638C212yesmaximal, complemented
7#70584C212yesmaximal, complemented
8#70473C211yesmaximal, characteristic, complemented
9#70457C211yesmaximal, characteristic
10#70279C212yesmaximal, complemented
11#70279C212yesmaximal, complemented
12#25658—31nomaximal, Sylow
13C2×C3×#125C2211yescharacteristic
14C2×C3×C4×Q8C2211yescharacteristic
15C2×#947C2212yes—
16#5585—24no—
17#5585—24no—
18C2×#894—22no—
19C2×#893C2212yes—
20C2×#875—21no—
21C2×#874C2211yescharacteristic
22C2×C4×D12—21no—
23C2×C4×Dic6C2211yescharacteristic
24C3×#479—24no—
25C2×C3×#107C2211yescharacteristic
26C2×C3×#106C2211yescharacteristic
27C2×C3×C4⋊C8C2211yescharacteristic
28C2×C3×#103C2211yescharacteristic
29C2×C3×#103C2211yescharacteristic
30C2×#835C2212yes—
31C2×#821C2212yes—
32C2×#821C2212yes—
33#5067—24no—
34#4981—21no—
35#4979—21no—
36#4976—22no—
37#4972C2212yes—
38#4972C2212yes—
39C2×#809C2212yes—
40C2×#809C2212yes—
41C4×C4⋊Dic3C2211yescharacteristic
42C2×#803C2211yescharacteristic
43#4550C2212yes—
44#4550C2212yes—
45#4529C2212yes—
46C2×#559—31no—
47C2×#490—31no—
48C2×#479D311yescharacteristic, complemented
49#2584—31no—
50#2042—32no—
51#2042—32no—
52#1985—31no—
53#1982—31no—
54#1864—32no—
55#1862—32no—
56#1763—31no—
57C22×C3×Q8C2311yescharacteristic
58C22×D12—21no—
59C22×D12—21no—
60C22×D12C2312yes—
61C22×C4×D3—22no—
62C22×Dic6D411yescharacteristic
63C22×C3×C8C2311yescharacteristic
64C22×C3×C8C2311yescharacteristic
65C3×#125—22no—
66C3×#125—22no—
67C3×C4×Q8—22no—
68C3×C4×Q8D414yes—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4C2311yescharacteristic
73C2×C3×C4⋊C4C2311yescharacteristic
74C2×C3×C4⋊C4C2311yescharacteristic
75C2×C3×C42—21no—
76C2×C3×C42C2311yescharacteristic
77#947—24no—
78#947—24no—
79C2×(C2×D3)⋊C4—22no—
80C2×(C2×D3)⋊C4—22no—
81C2×(C2×D3)⋊C4—22no—
82C2×C4⋊Dic3D411yescharacteristic
83C2×C4⋊Dic3C2312yes—
84C2×Dic3⋊C4—22no—
85C2×Dic3⋊C4—22no—
86C2×Dic3⋊C4D412yes—
87C2×C4×Dic3—21no—
88C2×C4×Dic3C2312yes—
89C22×C3⋊C8C2312yes—
90#894—216no—
91#893—24no—
92#893—24no—
93#875—28no—
94#874—22no—
95#874D414yes—
96C4×D12—28no—
97C4×Dic6—22no—
98C4×Dic6—22no—
99C3×#107—22no—
100C3×#106—22no—
101C3×C4⋊C8—22no—
102C3×#103—22no—
103C3×#103—22no—
104#835—24no—
105C3⋊#95—21no—
106C3⋊#95—22no—
107C3⋊#95—22no—
108#821—24no—
109#821—24no—
110#809—24no—
111#809—24no—
112#803—22no—
113#559—64no—
114#559—64no—
115C2×#127—31no—
116C2×#125C2×D311yescharacteristic
117C2×#124—32no—
118C2×#124—61no—
119C2×#123—62no—
120C2×C4×Q8—31no—
121C2×C4×Q8C2×D311yescharacteristic
122C2×C4×D4—32no—
123C2×C4×D4—61no—
124#490—64no—
125#479—24no—
126C2×#107C2×D311yescharacteristic
127C2×#106C2×D311yescharacteristic
128C2×C4⋊C8C2×D311yescharacteristic
129C2×#103—32no—
130C2×#103C2×D311yescharacteristic
131C2×#103C2×D311yescharacteristic
132C2×#102—32no—
133C2×#102—32no—
134C2×#102—32no—
135C2×#102—32no—
136C2×#97—31no—
137#398—61no—
138#395—61no—
139#389—32no—
140#389—32no—
141#389—62no—
142C4×C4⋊C4—31no—
143#340—32no—
144#336—32no—
145#336—32no—
146C23×D3—22no—
147C2×C3×Q8—21no—
148C2×C3×Q8—22no—
149C2×C3×Q8—22no—
150C2×C3×Q8C2×D412yes—
151C22×C3×C4—21no—
152C22×C3×C4—21no—
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2411yescharacteristic
158C22×Dic3—22no—
159C22×Dic3C2×D412yes—
160C22×Dic3C2×D412yes—
161C2×D12—22no—
162C2×D12—22no—
163C2×D12—22no—
164C2×D12—22no—
165C2×D12—24no—
166C2×D12—22no—
167C2×D12—24no—
168C2×D12—44no—
169C2×D12—81no—
170C2×D12—81no—
171C2×D12C2×D414yes—
172C2×C4×D3—28no—
173C2×C4×D3—82no—
174C2×C4×D3—82no—
175C2×Dic6—22no—
176C2×Dic6—42no—
177C2×C3×C8—21no—
178C2×C3×C8—21no—
179C2×C3×C8—22no—
180C2×C3×C8—22no—
181C3×C4⋊C4—24no—
182C3×C4⋊C4—21no—
183C3×C4⋊C4—21no—
184C3×C4⋊C4—21no—
185C3×C4⋊C4—21no—
186C3×C4⋊C4—21no—
187C3×C4⋊C4—42no—
188C3×C4⋊C4—42no—
189C3×C4⋊C4C2×D412yes—
190C3×C42—24no—
191C3×C42C2×D412yes—
192C3×C42C2×D412yes—
193(C2×D3)⋊C4—28no—
194(C2×D3)⋊C4—28no—
195(C2×D3)⋊C4—28no—
196C4⋊Dic3—22no—
197C4⋊Dic3—22no—
198C4⋊Dic3#4114yes—
199Dic3⋊C4—24no—
200Dic3⋊C4—44no—
201Dic3⋊C4—82no—
202C4×Dic3—22no—
203C4×Dic3—22no—
204C4×Dic3—81no—
205C2×C3⋊C8—22no—
206C2×C3⋊C8—22no—
207C2×C3⋊C8—22no—
208C22×Q8—31no—
209C22×Q8C22×D311yescharacteristic
210C22×D4—32no—
211C22×D4—61no—
212C22×D4—61no—
213C23×C4—62no—
214C22×C8—32no—
215C22×C8C22×D311yescharacteristic
216C22×C8C22×D311yescharacteristic
217#127—34no—
218#127—62no—
219#125—22no—
220#125—22no—
221#124—68no—
222#124—64no—
223#124—64no—
224#123—616no—
225C4×Q8—22no—
226C4×Q8—62no—
227C4×Q8—62no—
228C4×Q8#6714yes—
229C4×D4—68no—
230C4×D4—64no—
231C4×D4—64no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—32no—
236C2×C4⋊C4—31no—
237C2×C4⋊C4—32no—
238C2×C4⋊C4—62no—
239C2×C4⋊C4—62no—
240C2×C4⋊C4C22×D311yescharacteristic
241C2×C4⋊C4C22×D311yescharacteristic
242C2×C4⋊C4C22×D311yescharacteristic
243C2×C22⋊C4—62no—
244C2×C22⋊C4—62no—
245C2×C22⋊C4—62no—
246C2×C42—21no—
247C2×C42—32no—
248C2×C42—61no—
249C2×C42C22×D311yescharacteristic
250#107—22no—
251#106—22no—
252C4⋊C8—22no—
253#103—22no—
254#103—22no—
255#103—64no—
256#102—64no—
257#102—64no—
258#102—64no—
259#102—64no—
260#97—62no—
261#95—62no—
262#95—62no—
263#95—61no—
264C23×C3C22×D411yescharacteristic
265C22×D3—24no—
266C22×D3—42no—
267C22×D3—42no—
268C22×D3—42no—
269C22×D3—42no—
270C22×D3—42no—
271C22×D3—42no—
272C3×Q8—22no—
273C3×Q8—22no—
274C3×Q8—42no—
275C2×C3×C4—22no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—22no—
279C2×C3×C4—22no—
280C2×C3×C4—42no—
281C2×C3×C4—41no—
282C2×C3×C4—41no—
283C2×C3×C4—41no—
284C2×C3×C4—41no—
285C2×C3×C4—42no—
286C2×C3×C4—41no—
287C2×C3×C4—41no—
288C2×C3×C4#12012yes—
289C2×C3×C4#12012yes—
290C2×C3×C4C22×D411yescharacteristic
291C2×C3×C4C22×D411yescharacteristic
292C2×C3×C4C2×#4112yes—
293C2×C3×C42−1+412yes—
294C2×Dic3—22no—
295C2×Dic3—22no—
296C2×Dic3—24no—
297C2×Dic3—22no—
298C2×Dic3—22no—
299C2×Dic3—22no—
300C2×Dic3—22no—
301C2×Dic3—42no—
302C2×Dic3—42no—
303D12—24no—
304D12—24no—
305D12—48no—
306D12—42no—
307D12—42no—
308D12—84no—
309C4×D3—88no—
310Dic6—44no—
311C3×C8—41no—
312C3×C8—41no—
313C3⋊C8—42no—
314C24—62no—
315C2×Q8—21no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—62no—
319C2×Q8—122no—
320C2×Q8C2×#6712yes—
321C2×D4—34no—
322C2×D4—64no—
323C2×D4—64no—
324C2×D4—62no—
325C2×D4—62no—
326C2×D4—62no—
327C2×D4—62no—
328C2×D4—62no—
329C2×D4—124no—
330C2×D4—241no—
331C2×D4—241no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—32no—
337C22×C4—32no—
338C22×C4—68no—
339C22×C4—62no—
340C22×C4—242no—
341C22×C4—242no—
342C22×C4D3×D411yescharacteristic
343C22×C4D3×D411yescharacteristic
344C22×C4C23×D311yescharacteristic
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—62no—
350C2×C8—62no—
351C2×C8—62no—
352C4⋊C4—24no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—34no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—62no—
362C4⋊C4—64no—
363C4⋊C4—62no—
364C4⋊C4—124no—
365C4⋊C4—242no—
366C4⋊C4C2×#6712yes—
367C22⋊C4—68no—
368C22⋊C4—68no—
369C22⋊C4—68no—
370C42—24no—
371C42—62no—
372C42—62no—
373C42—241no—
374C42D3×D412yes—
375C42C2×#6712yes—
376C22×C3C2×#12011yescharacteristic
377C22×C3#54812yes—
378C22×C3#57512yes—
379C22×C3#57611yescharacteristic
380C22×C3#57711yescharacteristic
381C2×D3—44no—
382C2×D3—44no—
383C2×D3—44no—
384C2×D3—82no—
385C2×D3—82no—
386C2×D3—82no—
387C2×D3—82no—
388C3×C4—22no—
389C3×C4—22no—
390C3×C4—42no—
391C3×C4—42no—
392C3×C4—81no—
393C3×C4—81no—
394C3×C4#53211yescharacteristic
395C3×C4#53611yescharacteristic
396C3×C4#54712yes—
397Dic3—44no—
398Dic3—42no—
399Dic3—42no—
400C23—64no—
401C23—122no—
402C23—122no—
403C23—122no—
404C23—122no—
405C23—122no—
406C23—122no—
407C23C2×D3×D411yescharacteristic
408Q8—22no—
409Q8—22no—
410Q8—42no—
411Q8—124no—
412D4—64no—
413D4—64no—
414D4—128no—
415D4—122no—
416D4—122no—
417D4—244no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—64no—
432C2×C4—62no—
433C2×C4—62no—
434C2×C4—62no—
435C2×C4—62no—
436C2×C4—62no—
437C2×C4—62no—
438C2×C4—122no—
439C2×C4—122no—
440C2×C4—248no—
441C2×C4#93712yes—
442C2×C4#93712yes—
443C2×C4C2×D3×D411yescharacteristic
444C2×C4#100712yes—
445C2×C4D3×#4112yes—
446C2×C4C22×#6711yescharacteristic
447C8—41no—
448C8—41no—
449C8—122no—
450C2×C3#243211yescharacteristic
451C2×C3#300511yescharacteristic
452C2×C3#301211yescharacteristic
453C2×C3#337012yes—
454C2×C3#337112yes—
455D3—84no—
456C22—124no—
457C22—124no—
458C22—124no—
459C22—242no—
460C22—242no—
461C22—242no—
462C22—242no—
463C22#563312yes—
464C22#576911yescharacteristic
465C22#577011yescharacteristic
466C22#577712yes—
467C22C2×#93711yescharacteristic
468C4—22no—
469C4—22no—
470C4—42no—
471C4—42no—
472C4—81no—
473C4—81no—
474C4—122no—
475C4—124no—
476C4—122no—
477C4#563112yes—
478C4#581611yescharacteristic
479C4#583211yescharacteristic
480C3#2565811yescharacteristic, complemented, Sylow
481C2—244no—
482C2#8121712yes—
483C2#8121812yes—
484C2#8275911yescharacteristic
485C2#8391211yescharacteristic
486C2#8398011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#479C2
socleC23×C3C22×D4
prime radicalC2×#874C22
hypercenterC2×#479D3
hypocenterC3#25658
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#479

Complements

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

NH ≅ G/Ntype
2. C2×C3×#479C2semidirect
3. #83041C2semidirect
5. #71631C2semidirect
6. #70638C2semidirect
7. #70584C2semidirect
8. #70473C2semidirect
10. #70279C2semidirect
11. #70279C2semidirect
48. C2×#479D3semidirect
480. C3#25658semidirect

Maximal subgroups

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

subgroupindex
C2×#55852
C2×C3×#4792
#830412
C2×#50672
#716312
#706382
#705842
#704732
#704572
#702792
#256583