Finite Group Databasebeta

← #10849321

Subgroups — #10849321

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×#5713C211yesmaximal, characteristic
2C2×C3×#490C211yesmaximal, characteristic
3#83041C211yesmaximal, characteristic, complemented
4C2×#5063C211yesmaximal, characteristic, complemented
5#71761C211yesmaximal, characteristic
6#70216C212yesmaximal, complemented
7#70216C212yesmaximal, complemented
8#70169C212yesmaximal, complemented
9#70167C212yesmaximal, complemented
10#69838C211yesmaximal, characteristic, complemented
11#69813C211yesmaximal, characteristic, complemented
12#25658—31nomaximal, Sylow
13C2×C3×#127C2211yescharacteristic
14C2×C3×#125C2211yescharacteristic
15C2×#936—21no—
16C2×#936C2212yes—
17C2×#935—22no—
18C2×D4×Dic3—21no—
19C2×D4×Dic3C2212yes—
20#5713—24no—
21#5713—24no—
22C2×C4×Dic6C2211yescharacteristic
23C2×C4×Dic6C2211yescharacteristic
24C3×#490—24no—
25C2×C3×#102C2212yes—
26C2×C3×#102C2212yes—
27C2×C3×#97C2211yescharacteristic
28C2×#832C2212yes—
29C2×#832C2212yes—
30C2×#816C2212yes—
31#5063—24no—
32#4981—21no—
33#4979—21no—
34#4976—22no—
35#4972C2212yes—
36#4972C2212yes—
37C2×#810C2211yescharacteristic
38C2×#810C2211yescharacteristic
39C2×#808C2211yescharacteristic
40C2×#807C2211yescharacteristic
41C4×C4⋊Dic3C2211yescharacteristic
42C2×C4⋊(C3⋊C8)C2211yescharacteristic
43#4551C2212yes—
44#4550C2212yes—
45#4550C2212yes—
46C2×#559—31no—
47C2×#490D311yescharacteristic
48C2×#479—31no—
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×D4—21no—
58C22×C3×D4—21no—
59C22×C3×D4C2312yes—
60C23×Dic3—22no—
61C22×Dic6D411yescharacteristic
62C22×Dic6C2311yescharacteristic
63C22×C3×C8C2312yes—
64C3×#127—22no—
65C3×#127D414yes—
66C3×#125—22no—
67C3×#125—22no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4C2311yescharacteristic
71C2×C3×C4⋊C4C2311yescharacteristic
72C2×C3×C42C2311yescharacteristic
73C2×#217—22no—
74C2×#217—22no—
75C2×#217—22no—
76#936—28no—
77#936—24no—
78#936—24no—
79#935—216no—
80D4×Dic3—28no—
81D4×Dic3—24no—
82D4×Dic3—24no—
83C2×C4⋊Dic3D411yescharacteristic
84C2×C4⋊Dic3C2311yescharacteristic
85C2×C4⋊Dic3C2312yes—
86C2×Dic3⋊C4—21no—
87C2×Dic3⋊C4—22no—
88C2×Dic3⋊C4—22no—
89C2×Dic3⋊C4D412yes—
90C2×C4×Dic3—21no—
91C2×C4×Dic3—21no—
92C2×C4×Dic3C2312yes—
93C22×C3⋊C8C2311yescharacteristic
94C22×C3⋊C8C2311yescharacteristic
95C4×Dic6—22no—
96C4×Dic6—22no—
97C4×Dic6—22no—
98C4×Dic6D414yes—
99C3×#102—24no—
100C3×#102—24no—
101C3×#97—22no—
102#832—24no—
103#832—24no—
104C3⋊#95—21no—
105C3⋊#95—22no—
106C3⋊#95—22no—
107#816—24no—
108#810—22no—
109#810—22no—
110#808—22no—
111#807—22no—
112C4⋊(C3⋊C8)—22no—
113#559—64no—
114#559—64no—
115C2×#127C2×D311yescharacteristic
116C2×#125C2×D311yescharacteristic
117C2×#124—32no—
118C2×#124—61no—
119C2×#123—62no—
120C2×C4×Q8—31no—
121C2×C4×Q8—31no—
122C2×C4×D4—32no—
123C2×C4×D4—61no—
124#490—24no—
125#479—64no—
126C2×#107—31no—
127C2×#106—31no—
128C2×C4⋊C8—31no—
129C2×#103—32no—
130C2×#103—31no—
131C2×#103—31no—
132C2×#102—32no—
133C2×#102—32no—
134C2×#102C2×D312yes—
135C2×#102C2×D312yes—
136C2×#97C2×D311yescharacteristic
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—
146C24×C3—22no—
147C2×C3×D4—24no—
148C2×C3×D4—24no—
149C2×C3×D4—22no—
150C2×C3×D4—22no—
151C2×C3×D4—22no—
152C2×C3×D4—22no—
153C2×C3×D4—22no—
154C2×C3×D4—44no—
155C2×C3×D4—81no—
156C2×C3×D4—81no—
157C2×C3×D4C2×D414yes—
158C22×C3×C4—21no—
159C22×C3×C4—21no—
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2411yescharacteristic
163C22×Dic3—22no—
164C22×Dic3—21no—
165C22×Dic3—21no—
166C22×Dic3—28no—
167C22×Dic3—82no—
168C22×Dic3—82no—
169C22×Dic3C2×D412yes—
170C22×Dic3C2×D412yes—
171C2×Dic6—22no—
172C2×Dic6—22no—
173C2×Dic6—22no—
174C2×Dic6—21no—
175C2×Dic6—42no—
176C2×Dic6C2×D412yes—
177C2×C3×C8—22no—
178C2×C3×C8—22no—
179C2×C3×C8—22no—
180C3×C4⋊C4—21no—
181C3×C4⋊C4—21no—
182C3×C4⋊C4—21no—
183C3×C4⋊C4—21no—
184C3×C4⋊C4—42no—
185C3×C4⋊C4—42no—
186C3×C42C2×D412yes—
187C3×C42C2×D412yes—
188#217—28no—
189#217—28no—
190#217—28no—
191C4⋊Dic3—22no—
192C4⋊Dic3—21no—
193C4⋊Dic3—22no—
194C4⋊Dic3C2×D412yes—
195C4⋊Dic3#4114yes—
196Dic3⋊C4—24no—
197Dic3⋊C4—24no—
198Dic3⋊C4—44no—
199Dic3⋊C4—82no—
200C4×Dic3—24no—
201C4×Dic3—22no—
202C4×Dic3—22no—
203C4×Dic3—81no—
204C2×C3⋊C8—21no—
205C2×C3⋊C8—21no—
206C2×C3⋊C8—22no—
207C2×C3⋊C8—22no—
208C22×Q8—31no—
209C22×Q8—31no—
210C22×D4—21no—
211C22×D4—21no—
212C22×D4C22×D312yes—
213C23×C4—62no—
214C22×C8—31no—
215C22×C8—31no—
216C22×C8C22×D312yes—
217#127—22no—
218#127#6714yes—
219#125—22no—
220#125—22no—
221#124—64no—
222#124—68no—
223#124—64no—
224#123—616no—
225C4×Q8—34no—
226C4×Q8—62no—
227C4×Q8—62no—
228C4×Q8—62no—
229C4×D4—64no—
230C4×D4—68no—
231C4×D4—64no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—31no—
235C2×C4⋊C4—32no—
236C2×C4⋊C4—31no—
237C2×C4⋊C4—32no—
238C2×C4⋊C4—61no—
239C2×C4⋊C4—62no—
240C2×C4⋊C4—62no—
241C2×C4⋊C4C22×D311yescharacteristic
242C2×C4⋊C4C22×D311yescharacteristic
243C2×C22⋊C4—62no—
244C2×C22⋊C4—62no—
245C2×C22⋊C4—62no—
246C2×C42—32no—
247C2×C42—61no—
248C2×C42—61no—
249C2×C42C22×D311yescharacteristic
250#107—62no—
251#106—62no—
252C4⋊C8—62no—
253#103—62no—
254#103—62no—
255#103—64no—
256#102—24no—
257#102—24no—
258#102—64no—
259#102—64no—
260#97—22no—
261#95—62no—
262#95—62no—
263#95—61no—
264C23×C3—24no—
265C23×C3—42no—
266C23×C3—42no—
267C23×C3—42no—
268C23×C3—42no—
269C23×C3—42no—
270C23×C3—42no—
271C23×C3C22×D411yescharacteristic
272C3×D4—24no—
273C3×D4—24no—
274C3×D4—48no—
275C3×D4—42no—
276C3×D4—42no—
277C3×D4—84no—
278C2×C3×C4—22no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—41no—
282C2×C3×C4—41no—
283C2×C3×C4—42no—
284C2×C3×C4—41no—
285C2×C3×C4—41no—
286C2×C3×C4#12012yes—
287C2×C3×C4#12012yes—
288C2×C3×C4C22×D411yescharacteristic
289C2×C3×C4C22×D411yescharacteristic
290C2×C3×C4C2×#4112yes—
291C2×C3×C42−1+412yes—
292C2×Dic3—22no—
293C2×Dic3—22no—
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—
303C2×Dic3—42no—
304C2×Dic3—41no—
305C2×Dic3—41no—
306C2×Dic3—88no—
307Dic6—22no—
308Dic6—22no—
309Dic6—44no—
310Dic6—42no—
311C3×C8—42no—
312C3⋊C8—41no—
313C3⋊C8—41no—
314C24—22no—
315C2×Q8—32no—
316C2×Q8—62no—
317C2×Q8—61no—
318C2×Q8—62no—
319C2×Q8—62no—
320C2×Q8—122no—
321C2×D4—24no—
322C2×D4—24no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—44no—
329C2×D4—81no—
330C2×D4—81no—
331C2×D4C2×#6714yes—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—32no—
335C22×C4—32no—
336C22×C4—62no—
337C22×C4—61no—
338C22×C4—61no—
339C22×C4—68no—
340C22×C4—242no—
341C22×C4—242no—
342C22×C4D3×D411yescharacteristic
343C22×C4D3×D411yescharacteristic
344C22×C4C23×D311yescharacteristic
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—22no—
348C2×C8—61no—
349C2×C8—61no—
350C2×C8—62no—
351C2×C8—62no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—32no—
357C4⋊C4—34no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—62no—
361C4⋊C4—64no—
362C4⋊C4—64no—
363C4⋊C4—62no—
364C4⋊C4—61no—
365C4⋊C4—124no—
366C4⋊C4—242no—
367C22⋊C4—68no—
368C22⋊C4—68no—
369C22⋊C4—68no—
370C42—64no—
371C42—62no—
372C42—62no—
373C42—241no—
374C42D3×D412yes—
375C42C2×#6712yes—
376C22×C3—44no—
377C22×C3—44no—
378C22×C3—44no—
379C22×C3—82no—
380C22×C3—82no—
381C22×C3—82no—
382C22×C3—82no—
383C22×C3C2×#12011yescharacteristic
384C22×C3#54812yes—
385C22×C3#57512yes—
386C22×C3#57611yescharacteristic
387C22×C3#57711yescharacteristic
388C3×C4—22no—
389C3×C4—22no—
390C3×C4—81no—
391C3×C4—81no—
392C3×C4#53211yescharacteristic
393C3×C4#53611yescharacteristic
394C3×C4#54712yes—
395Dic3—44no—
396Dic3—42no—
397Dic3—42no—
398Dic3—42no—
399Dic3—42no—
400C23—24no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23C2×D3×D411yescharacteristic
408Q8—62no—
409Q8—62no—
410Q8—124no—
411Q8—122no—
412D4—24no—
413D4—24no—
414D4—48no—
415D4—42no—
416D4—42no—
417D4—84no—
418C2×C4—22no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—64no—
427C2×C4—62no—
428C2×C4—62no—
429C2×C4—62no—
430C2×C4—62no—
431C2×C4—62no—
432C2×C4—62no—
433C2×C4—62no—
434C2×C4—62no—
435C2×C4—122no—
436C2×C4—122no—
437C2×C4—122no—
438C2×C4—121no—
439C2×C4—121no—
440C2×C4—248no—
441C2×C4#93712yes—
442C2×C4#93712yes—
443C2×C4C2×D3×D411yescharacteristic
444C2×C4D3×#4112yes—
445C2×C4#101012yes—
446C2×C4C22×#6711yescharacteristic
447C8—42no—
448C8—121no—
449C8—121no—
450C2×C3—84no—
451C2×C3#243211yescharacteristic
452C2×C3#300511yescharacteristic
453C2×C3#301211yescharacteristic
454C2×C3#337012yes—
455C2×C3#337112yes—
456C22—44no—
457C22—44no—
458C22—44no—
459C22—82no—
460C22—82no—
461C22—82no—
462C22—82no—
463C22#567712yes—
464C22#577312yes—
465C22#577411yescharacteristic
466C22#577511yescharacteristic
467C22C2×#93711yescharacteristic
468C4—22no—
469C4—22no—
470C4—81no—
471C4—81no—
472C4—122no—
473C4—124no—
474C4—122no—
475C4—122no—
476C4—122no—
477C4#566811yescharacteristic
478C4#567511yescharacteristic
479C4#582412yes—
480C3#2565811yescharacteristic, complemented, Sylow
481C2—84no—
482C2#8171012yes—
483C2#8171112yes—
484C2#8295911yescharacteristic
485C2#8394511yescharacteristic
486C2#8394911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#490C2
socleC23×C3C22×D4
prime radicalC2×C3×#127C22
hypercenterC2×#490D3
hypocenterC3#25658
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#490

Complements

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

NH ≅ G/Ntype
3. #83041C2semidirect
4. C2×#5063C2semidirect
6. #70216C2semidirect
7. #70216C2semidirect
8. #70169C2semidirect
9. #70167C2semidirect
10. #69838C2semidirect
11. #69813C2semidirect
480. C3#25658semidirect

Maximal subgroups

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

subgroupindex
C2×#57132
C2×C3×#4902
#830412
C2×#50632
#717612
#702162
#701692
#701672
#698382
#698132
#256583