Finite Group Databasebeta

← #10945050

Subgroups — #10945050

Order 768.

1216 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×#5812C211yesmaximal, characteristic
2C2×C3×#451C211yesmaximal, characteristic
3#82787C211yesmaximal, characteristic
4C2×#5045C211yesmaximal, characteristic
5#72132C211yesmaximal, characteristic
6#72065C212yesmaximal, complemented
7#72065C212yesmaximal, complemented
8#70237C212yesmaximal, complemented
9#70235C212yesmaximal, complemented
10#69926C211yesmaximal, characteristic
11#69911C211yesmaximal, characteristic
12#33019—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C22×C3×Q16C2212yes—
15C2×C3×C22⋊Q8C2211yescharacteristic
16#5812—24no—
17#5812—24no—
18C2×Q8×Dic3—21no—
19C2×Q8×Dic3—21no—
20C2×Q8×Dic3C2212yes—
21C2×Q8×Dic3C2212yes—
22C2×#926—22no—
23C2×#926C2211yescharacteristic
24C2×#926C2211yescharacteristic
25C3×#451—24no—
26C2×C3×#103C2212yes—
27C2×C3×C22⋊C8C2211yescharacteristic
28#5229C2212yes—
29C2×#835C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×#835C2212yes—
32C2×#835C2212yes—
33#5212C2211yescharacteristic
34C2×#830C2211yescharacteristic
35#5045—24no—
36#4986—21no—
37#4981—21no—
38#4972C2212yes—
39C2×#808C2211yescharacteristic
40C2×#807C2211yescharacteristic
41#4952—22no—
42#4551C2212yes—
43#4551C2212yes—
44#4550C2212yes—
45#4550C2212yes—
46C2×#519—31no—
47C2×#485—31no—
48C2×#451D311yescharacteristic
49#2625—31no—
50#1878—32no—
51#1878—32no—
52#1865—32no—
53#1863—32no—
54#1825—31no—
55#1818—31no—
56#1787—31no—
57C22×C3×Q8—24no—
58C22×C3×Q8—24no—
59C22×C3×Q8—21no—
60C22×C3×Q8—22no—
61C22×C3×Q8D414yes—
62C22×C3×Q8C2311yescharacteristic
63C22×C3×Q8C2312yes—
64C23×C3×C4—21no—
65C23×C3×C4C2311yescharacteristic
66C2×C3×Q16—24no—
67C2×C3×Q16—24no—
68C2×C3×Q16—24no—
69C22×C3×C8C2312yes—
70C3×C22⋊Q8—22no—
71C3×C22⋊Q8—22no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4C2311yescharacteristic
74C2×C3×C22⋊C4—21no—
75C2×#217—21no—
76C2×#217—21no—
77C2×#217D412yes—
78Q8×Dic3—28no—
79Q8×Dic3—28no—
80Q8×Dic3—24no—
81Q8×Dic3—24no—
82Q8×Dic3—24no—
83Q8×Dic3—24no—
84#926—216no—
85#926—22no—
86#926—22no—
87#926—22no—
88#926D414yes—
89C2×C4⋊Dic3—22no—
90C2×C4⋊Dic3—22no—
91C2×C4⋊Dic3D411yescharacteristic
92C2×C4⋊Dic3D411yescharacteristic
93C2×C4⋊Dic3C2312yes—
94C2×C4⋊Dic3C2311yescharacteristic
95C2×C4⋊Dic3C2311yescharacteristic
96C2×Dic3⋊C4—22no—
97C2×C4×Dic3—21no—
98C2×C4×Dic3—22no—
99C2×C4×Dic3—22no—
100C2×C4×Dic3C2312yes—
101C22×C3⋊C8C2311yescharacteristic
102C22×C3⋊C8C2311yescharacteristic
103C3×#103—24no—
104C3×C22⋊C8—22no—
105#835—24no—
106#835—24no—
107#835—22no—
108#835—22no—
109C3⋊#95—21no—
110C3⋊#95—22no—
111C3⋊#95—22no—
112C3⋊#95—22no—
113#830—22no—
114#808—22no—
115#807—22no—
116C23×Q8C2×D311yescharacteristic
117C22×Q16C2×D312yes—
118C2×C22⋊Q8C2×D311yescharacteristic
119#519—64no—
120#519—64no—
121C2×C4×Q8—32no—
122C2×C4×Q8—32no—
123C2×C4×Q8—61no—
124C2×C4×Q8—61no—
125C2×#117—31no—
126C2×#117—31no—
127C2×#117—62no—
128#485—64no—
129#451—24no—
130C2×#107—31no—
131C2×#106—31no—
132C2×#103—31no—
133C2×#103—31no—
134C2×#103—32no—
135C2×#103—32no—
136C2×#103C2×D312yes—
137C2×C22⋊C8—31no—
138C2×C22⋊C8C2×D311yescharacteristic
139#400—61no—
140#398—61no—
141#397—62no—
142#391—32no—
143#389—32no—
144#380—31no—
145#340—32no—
146#340—32no—
147#336—32no—
148#336—32no—
149C24×C3C2×D411yescharacteristic
150C2×C3×Q8—28no—
151C2×C3×Q8—28no—
152C2×C3×Q8—28no—
153C2×C3×Q8—216no—
154C2×C3×Q8—24no—
155C2×C3×Q8—22no—
156C2×C3×Q8—22no—
157C2×C3×Q8—22no—
158C2×C3×Q8—21no—
159C2×C3×Q8—21no—
160C2×C3×Q8—22no—
161C2×C3×Q8—22no—
162C2×C3×Q8C2×D414yes—
163C2×C3×Q8#4114yes—
164C2×C3×Q8#4114yes—
165C22×C3×C4—28no—
166C22×C3×C4—24no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—21no—
171C22×C3×C4—21no—
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D412yes—
174C22×C3×C4C2×D412yes—
175C22×C3×C4C2411yescharacteristic
176C22×Dic3—21no—
177C22×Dic3—21no—
178C22×Dic3—22no—
179C22×Dic3C2×D412yes—
180C22×Dic3C2×D412yes—
181C3×Q16—48no—
182C2×C3×C8—22no—
183C2×C3×C8—22no—
184C2×C3×C8—22no—
185C3×C4⋊C4—21no—
186C3×C4⋊C4—21no—
187C3×C4⋊C4—42no—
188C3×C22⋊C4—42no—
189#217—24no—
190#217—24no—
191#217—81no—
192C4⋊Dic3—28no—
193C4⋊Dic3—28no—
194C4⋊Dic3—22no—
195C4⋊Dic3—22no—
196C4⋊Dic3—21no—
197C4⋊Dic3—21no—
198C4⋊Dic3—22no—
199C4⋊Dic3—22no—
200C4⋊Dic3C2×D412yes—
201C4⋊Dic3C2×D412yes—
202Dic3⋊C4—82no—
203C4×Dic3—28no—
204C4×Dic3—28no—
205C4×Dic3—24no—
206C4×Dic3—22no—
207C4×Dic3—22no—
208C2×C3⋊C8—21no—
209C2×C3⋊C8—21no—
210C2×C3⋊C8—22no—
211C2×C3⋊C8—22no—
212C22×Q8—24no—
213C22×Q8—24no—
214C22×Q8—21no—
215C22×Q8—22no—
216C22×Q8#6714yes—
217C22×Q8C22×D311yescharacteristic
218C22×Q8C22×D312yes—
219C23×C4—21no—
220C23×C4C22×D311yescharacteristic
221C2×Q16—24no—
222C2×Q16—24no—
223C2×Q16—24no—
224C22×C8—31no—
225C22×C8—31no—
226C22×C8C22×D312yes—
227C22⋊Q8—22no—
228C22⋊Q8—22no—
229C4×Q8—64no—
230C4×Q8—64no—
231C4×Q8—68no—
232C4×Q8—68no—
233C4×Q8—64no—
234C4×Q8—64no—
235#117—34no—
236#117—616no—
237#117—62no—
238#117—62no—
239#117—62no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—31no—
242C2×C4⋊C4—31no—
243C2×C4⋊C4—31no—
244C2×C4⋊C4—31no—
245C2×C4⋊C4—32no—
246C2×C4⋊C4—62no—
247C2×C4⋊C4—62no—
248C2×C4⋊C4—62no—
249C2×C4⋊C4C22×D311yescharacteristic
250C2×C22⋊C4—21no—
251C2×C22⋊C4—32no—
252C2×C22⋊C4—61no—
253C2×C22⋊C4—61no—
254C2×C42—32no—
255C2×C42—62no—
256C2×C42—62no—
257C2×C42—61no—
258#107—62no—
259#106—62no—
260#103—24no—
261#103—62no—
262#103—62no—
263#103—64no—
264#103—64no—
265C22⋊C8—22no—
266C22⋊C8—62no—
267#95—62no—
268#95—62no—
269#95—62no—
270#95—61no—
271C23×C3—22no—
272C23×C3—21no—
273C23×C3—22no—
274C23×C3—21no—
275C23×C3#12012yes—
276C23×C3C22×D411yescharacteristic
277C3×Q8—216no—
278C3×Q8—24no—
279C3×Q8—24no—
280C3×Q8—24no—
281C3×Q8—42no—
282C3×Q8—41no—
283C3×Q8—41no—
284C3×Q8#12318yes—
285C2×C3×C4—216no—
286C2×C3×C4—24no—
287C2×C3×C4—24no—
288C2×C3×C4—24no—
289C2×C3×C4—22no—
290C2×C3×C4—24no—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4—41no—
294C2×C3×C4—41no—
295C2×C3×C4—42no—
296C2×C3×C4—41no—
297C2×C3×C4—41no—
298C2×C3×C4#12012yes—
299C2×C3×C4C22×D411yescharacteristic
300C2×C3×C4C22×D411yescharacteristic
301C2×C3×C4C2×#4112yes—
302C2×C3×C4C2×#4112yes—
303C2×Dic3—22no—
304C2×Dic3—22no—
305C2×Dic3—22no—
306C2×Dic3—22no—
307C2×Dic3—22no—
308C2×Dic3—24no—
309C2×Dic3—22no—
310C2×Dic3—41no—
311C2×Dic3—42no—
312C2×Dic3—41no—
313C2×Dic3—42no—
314C2×Dic3—42no—
315C2×Dic3—42no—
316C3×C8—42no—
317C3⋊C8—41no—
318C3⋊C8—41no—
319C24D3×D411yescharacteristic
320C2×Q8—28no—
321C2×Q8—28no—
322C2×Q8—28no—
323C2×Q8—216no—
324C2×Q8—24no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—21no—
329C2×Q8—22no—
330C2×Q8—22no—
331C2×Q8—21no—
332C2×Q8#23714yes—
333C2×Q8#23714yes—
334C2×Q8C2×#6714yes—
335C22×C4—28no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—32no—
343C22×C4—32no—
344C22×C4—61no—
345C22×C4—61no—
346C22×C4—62no—
347C22×C4D3×D411yescharacteristic
348C22×C4D3×D412yes—
349C22×C4C2×#6712yes—
350C22×C4C23×D311yescharacteristic
351Q16—48no—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—61no—
356C2×C8—61no—
357C2×C8—62no—
358C2×C8—62no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—32no—
362C4⋊C4—32no—
363C4⋊C4—42no—
364C4⋊C4—68no—
365C4⋊C4—68no—
366C4⋊C4—62no—
367C4⋊C4—62no—
368C4⋊C4—61no—
369C4⋊C4—61no—
370C4⋊C4—62no—
371C4⋊C4—62no—
372C4⋊C4—242no—
373C22⋊C4—42no—
374C22⋊C4—64no—
375C22⋊C4—64no—
376C22⋊C4—241no—
377C42—68no—
378C42—68no—
379C42—64no—
380C42—62no—
381C42—62no—
382C22×C3—22no—
383C22×C3—22no—
384C22×C3—22no—
385C22×C3—42no—
386C22×C3—42no—
387C22×C3C2×#12011yescharacteristic
388C22×C3#52512yes—
389C22×C3#57711yescharacteristic
390C22×C3#57812yes—
391C22×C3#57811yescharacteristic
392C3×C4—28no—
393C3×C4—22no—
394C3×C4—81no—
395C3×C4—81no—
396C3×C4C2×#12312yes—
397C3×C4#52511yescharacteristic
398C3×C4#52511yescharacteristic
399Dic3—42no—
400Dic3—42no—
401Dic3—42no—
402Dic3—42no—
403Dic3—82no—
404C23—22no—
405C23—21no—
406C23—21no—
407C23—22no—
408C23#93712yes—
409C23C2×D3×D411yescharacteristic
410Q8—216no—
411Q8—24no—
412Q8—24no—
413Q8—24no—
414Q8—42no—
415Q8—41no—
416Q8—41no—
417Q8#93518yes—
418C2×C4—216no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—22no—
423C2×C4—24no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—42no—
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—121no—
440C2×C4—121no—
441C2×C4—122no—
442C2×C4—122no—
443C2×C4—122no—
444C2×C4#93712yes—
445C2×C4C2×D3×D411yescharacteristic
446C2×C4C2×#23712yes—
447C2×C4C2×#23712yes—
448C2×C4C22×#6711yescharacteristic
449C8—42no—
450C8—121no—
451C8—121no—
452C2×C3—42no—
453C2×C3#284911yescharacteristic
454C2×C3#301111yescharacteristic
455C2×C3#301311yescharacteristic
456C2×C3#312112yes—
457C2×C3#312712yes—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—42no—
462C22—42no—
463C22#558612yes—
464C22#576612yes—
465C22#577811yescharacteristic
466C22#577911yescharacteristic
467C22C2×#93711yescharacteristic
468C4—28no—
469C4—22no—
470C4—81no—
471C4—81no—
472C4—122no—
473C4—122no—
474C4—122no—
475C4—122no—
476C4—242no—
477C4#558611yescharacteristic
478C4#558611yescharacteristic
479C4C2×#93512yes—
480C3#3301911yescharacteristic, complemented, Sylow
481C2—42no—
482C2#8017612yes—
483C2#8017712yes—
484C2#8260211yescharacteristic
485C2#8388411yescharacteristic
486C2#8398411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#451C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenterC2×#451D3
hypocenterC3#33019
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#451

Complements

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

NH ≅ G/Ntype
6. #72065C2semidirect
7. #72065C2semidirect
8. #70237C2semidirect
9. #70235C2semidirect
480. C3#33019semidirect

Maximal subgroups

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

subgroupindex
C2×#58122
C2×C3×#4512
#827872
C2×#50452
#721322
#720652
#702372
#702352
#699262
#699112
#330193