Finite Group Databasebeta

← #10945032

Subgroups — #10945032

Order 768.

1292 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#520C211yesmaximal, characteristic
2C2×#5239C211yesmaximal, characteristic
3#82263C211yesmaximal, characteristic, complemented
4C2×#4970C211yesmaximal, characteristic, complemented
5#72135C211yesmaximal, characteristic
6#71632C212yesmaximal, complemented
7#71628C212yesmaximal, complemented
8#71612C212yesmaximal, complemented
9#71606C212yesmaximal, complemented
10#71584C211yesmaximal, characteristic, complemented
11#71553C211yesmaximal, characteristic, complemented
12#33017—31nomaximal, Sylow
13C22×C3×#41C2211yescharacteristic
14C3×#520—24no—
15C3×#520—24no—
16C2×C3×C4×Q8C2212yes—
17C2×C3×C4×D4—22no—
18C2×C3×C4×D4C2212yes—
19C2×C3×#117—21no—
20C2×C3×#117C2211yescharacteristic
21C22×C3×C4⋊C4—21no—
22C22×C3×C4⋊C4C2211yescharacteristic
23C22×#216C2211yescharacteristic
24C22×#214C2211yescharacteristic
25C2×#924C2211yescharacteristic
26#5239—24no—
27C2×#835C2211yescharacteristic
28C2×#835C2212yes—
29C2×#832C2211yescharacteristic
30C2×#832C2212yes—
31#5212C2211yescharacteristic
32C2×#830C2211yescharacteristic
33C2×#830C2211yescharacteristic
34#4985—21no—
35#4983—21no—
36#4970—24no—
37C2×#810C2212yes—
38C2×#807C2212yes—
39#4952—22no—
40#4929C2212yes—
41#4928C2212yes—
42#4529C2212yes—
43#4529C2212yes—
44#4524C2212yes—
45#4524C2212yes—
46C2×#520D311yescharacteristic
47C2×#484—31no—
48C2×#452—31no—
49#2625—31no—
50#1878—32no—
51#1877—32no—
52#1863—32no—
53#1862—32no—
54#1823—31no—
55#1818—31no—
56#1788—31no—
57C2×C3×#41—24no—
58C2×C3×#41—24no—
59C2×C3×#41—22no—
60C2×C3×#41D414yes—
61C22×C3×Q8C2311yescharacteristic
62C22×C3×D4—21no—
63C22×C3×D4C2311yescharacteristic
64C23×C3×C4—21no—
65C23×C3×C4—21no—
66C23×C3×C4—22no—
67C23×C3×C4C2311yescharacteristic
68C22×Dic6C2311yescharacteristic
69C3×C4×Q8—24no—
70C3×C4×Q8—24no—
71C3×C4×D4—216no—
72C3×C4×D4—24no—
73C3×C4×D4—24no—
74C3×#117—28no—
75C3×#117—22no—
76C3×#117—22no—
77C2×C3×C4⋊C4—28no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4—22no—
80C2×C3×C4⋊C4—22no—
81C2×C3×C4⋊C4D411yescharacteristic
82C2×C3×C4⋊C4D411yescharacteristic
83C2×C3×C4⋊C4D414yes—
84C2×C3×C4⋊C4C2312yes—
85C2×C3×C22⋊C4—22no—
86C2×C3×C22⋊C4D412yes—
87C2×C3×C42—22no—
88C2×C3×C42C2312yes—
89C2×#217—21no—
90C2×#217—21no—
91C2×#216—22no—
92C2×#216—24no—
93C2×#214—22no—
94C2×#214—24no—
95C2×C4⋊Dic3C2311yescharacteristic
96C2×C4⋊Dic3C2312yes—
97#924—22no—
98#924—22no—
99C2×Dic3⋊C4—21no—
100C2×Dic3⋊C4—22no—
101C22×C3⋊C8C2312yes—
102C22×C3⋊C8C2311yescharacteristic
103C22×C3⋊C8C2311yescharacteristic
104#835—22no—
105#835—24no—
106#832—22no—
107#832—24no—
108C3⋊#95—22no—
109C3⋊#95—22no—
110C3⋊#95—22no—
111C3⋊#95—21no—
112#830—22no—
113#830—22no—
114#810—24no—
115#807—24no—
116C22×#41C2×D311yescharacteristic
117C22×Q16—31no—
118C22×SD16—31no—
119C2×C22⋊Q8—31no—
120#520—24no—
121#520—24no—
122C2×C4×Q8C2×D312yes—
123C2×C4×D4—22no—
124C2×C4×D4C2×D312yes—
125C2×#117—21no—
126C2×#117C2×D311yescharacteristic
127C22×C4⋊C4—21no—
128C22×C4⋊C4C2×D311yescharacteristic
129#484—64no—
130#452—64no—
131C2×#107—32no—
132C2×#103—31no—
133C2×#103—32no—
134C2×#103—32no—
135C2×#102—31no—
136C2×#102—32no—
137C2×C22⋊C8—31no—
138C2×C22⋊C8—31no—
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×C3—21no—
150C24×C3C2×D411yescharacteristic
151C3×#41—216no—
152C3×#41—28no—
153C3×#41—28no—
154C2×C3×Q8—21no—
155C2×C3×Q8—22no—
156C2×C3×Q8C2×D412yes—
157C2×C3×Q8#4114yes—
158C2×C3×D4—28no—
159C2×C3×D4—24no—
160C2×C3×D4—21no—
161C2×C3×D4—22no—
162C2×C3×D4C2×D412yes—
163C2×C3×D4#4114yes—
164C22×C3×C4—28no—
165C22×C3×C4—21no—
166C22×C3×C4—24no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—28no—
171C22×C3×C4—24no—
172C22×C3×C4—82no—
173C22×C3×C4—82no—
174C22×C3×C4—81no—
175C22×C3×C4—81no—
176C22×C3×C4C2×D412yes—
177C22×C3×C4C2×D412yes—
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4C2×D412yes—
180C22×C3×C4C2×D412yes—
181C22×C3×C4C2411yescharacteristic
182C22×Dic3—22no—
183C22×Dic3—21no—
184C22×Dic3—21no—
185C2×Dic6—21no—
186C2×Dic6—21no—
187C2×Dic6—22no—
188C2×Dic6—22no—
189C3×C4⋊C4—28no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—24no—
192C3×C4⋊C4—22no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4C2×D414yes—
195C3×C22⋊C4—28no—
196C3×C22⋊C4—24no—
197C3×C42—28no—
198C3×C42—22no—
199C3×C42—22no—
200#217—42no—
201#217—81no—
202#216—44no—
203#214—44no—
204C4⋊Dic3—21no—
205C4⋊Dic3—21no—
206C4⋊Dic3—22no—
207C4⋊Dic3—22no—
208Dic3⋊C4—42no—
209Dic3⋊C4—82no—
210C2×C3⋊C8—22no—
211C2×C3⋊C8—21no—
212C2×C3⋊C8—21no—
213C2×C3⋊C8—22no—
214C2×C3⋊C8—22no—
215C2×C3⋊C8—22no—
216C2×C3⋊C8—22no—
217C2×#41—24no—
218C2×#41—24no—
219C2×#41—22no—
220C2×#41#6714yes—
221C22×Q8—31no—
222C22×Q8C22×D311yescharacteristic
223C22×D4—21no—
224C22×D4C22×D311yescharacteristic
225C23×C4—21no—
226C23×C4—21no—
227C23×C4—22no—
228C23×C4C22×D311yescharacteristic
229C2×Q16—64no—
230C2×Q16—62no—
231C2×SD16—64no—
232C2×SD16—62no—
233C22×C8—31no—
234C22×C8—31no—
235C22×C8—32no—
236C22⋊Q8—62no—
237C22⋊Q8—62no—
238C4×Q8—24no—
239C4×Q8—24no—
240C4×D4—216no—
241C4×D4—24no—
242C4×D4—24no—
243#117—28no—
244#117—22no—
245#117—22no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—31no—
251C2×C4⋊C4—32no—
252C2×C4⋊C4—61no—
253C2×C4⋊C4—62no—
254C2×C4⋊C4#6711yescharacteristic
255C2×C4⋊C4#6711yescharacteristic
256C2×C4⋊C4#6714yes—
257C2×C4⋊C4C22×D312yes—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—61no—
260C2×C22⋊C4—61no—
261C2×C22⋊C4#6712yes—
262C2×C42—22no—
263C2×C42C22×D312yes—
264#107—64no—
265#103—64no—
266#103—62no—
267#103—64no—
268#102—64no—
269#102—62no—
270C22⋊C8—62no—
271C22⋊C8—62no—
272#95—61no—
273#95—62no—
274#95—62no—
275#95—62no—
276C23×C3—22no—
277C23×C3—22no—
278C23×C3—24no—
279C23×C3—21no—
280C23×C3—21no—
281C23×C3—22no—
282C23×C3—41no—
283C23×C3—41no—
284C23×C3—42no—
285C23×C3#12012yes—
286C23×C3C22×D411yescharacteristic
287C3×Q8—22no—
288C3×Q8—24no—
289C3×Q8#12314yes—
290C3×D4—216no—
291C3×D4—22no—
292C3×D4—24no—
293C3×D4#12314yes—
294C2×C3×C4—216no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—24no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—24no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—24no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—24no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4—84no—
310C2×C3×C4—88no—
311C2×C3×C4#12012yes—
312C2×C3×C4C22×D411yescharacteristic
313C2×C3×C4C22×D411yescharacteristic
314C2×C3×C4C2×#4112yes—
315C2×C3×C4C2×#4112yes—
316C2×Dic3—41no—
317C2×Dic3—41no—
318C2×Dic3—42no—
319C2×Dic3—42no—
320C2×Dic3—41no—
321C2×Dic3—42no—
322C2×Dic3—41no—
323C2×Dic3—42no—
324Dic6—41no—
325Dic6—41no—
326Dic6—42no—
327C3⋊C8—41no—
328C3⋊C8—41no—
329C3⋊C8—42no—
330C24—21no—
331C24C2×#6711yescharacteristic
332#41—216no—
333#41—28no—
334#41—28no—
335C2×Q8—22no—
336C2×Q8—21no—
337C2×Q8—62no—
338C2×Q8—62no—
339C2×Q8—61no—
340C2×Q8—61no—
341C2×Q8#23514yes—
342C2×Q8C2×#6712yes—
343C2×D4—28no—
344C2×D4—24no—
345C2×D4—22no—
346C2×D4—21no—
347C2×D4#23514yes—
348C2×D4C2×#6712yes—
349C22×C4—28no—
350C22×C4—21no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—28no—
356C22×C4—24no—
357C22×C4—62no—
358C22×C4—61no—
359C22×C4—61no—
360C22×C4—82no—
361C22×C4—82no—
362C22×C4—81no—
363C22×C4—81no—
364C22×C4C2×#6712yes—
365C22×C4C2×#6712yes—
366C22×C4C2×#6711yescharacteristic
367C22×C4C2×#6712yes—
368C22×C4C2×#6712yes—
369C22×C4C23×D311yescharacteristic
370Q16—124no—
371SD16—124no—
372C2×C8—62no—
373C2×C8—62no—
374C2×C8—62no—
375C2×C8—62no—
376C2×C8—61no—
377C2×C8—62no—
378C2×C8—61no—
379C4⋊C4—28no—
380C4⋊C4—22no—
381C4⋊C4—24no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—62no—
385C4⋊C4—62no—
386C4⋊C4—61no—
387C4⋊C4—61no—
388C4⋊C4—122no—
389C4⋊C4—242no—
390C4⋊C4C2×#6714yes—
391C22⋊C4—28no—
392C22⋊C4—24no—
393C22⋊C4—122no—
394C22⋊C4—241no—
395C42—28no—
396C42—22no—
397C42—22no—
398C22×C3—22no—
399C22×C3—24no—
400C22×C3—22no—
401C22×C3—22no—
402C22×C3—42no—
403C22×C3—44no—
404C22×C3—42no—
405C22×C3—44no—
406C22×C3—42no—
407C22×C3—42no—
408C22×C3C2×#12011yescharacteristic
409C22×C3#52512yes—
410C22×C3#57611yescharacteristic
411C22×C3#57711yescharacteristic
412C22×C3#57812yes—
413C3×C4—22no—
414C3×C4—24no—
415C3×C4—42no—
416C3×C4—42no—
417C3×C4—44no—
418C3×C4C2×#12311yescharacteristic
419C3×C4C2×#12311yescharacteristic
420C3×C4#52512yes—
421Dic3—81no—
422Dic3—82no—
423Dic3—81no—
424C23—22no—
425C23—24no—
426C23—21no—
427C23—21no—
428C23—22no—
429C23—22no—
430C23—41no—
431C23—41no—
432C23—42no—
433C23#95312yes—
434C23C22×#6711yescharacteristic
435Q8—22no—
436Q8—24no—
437Q8—121no—
438Q8—121no—
439Q8—122no—
440Q8#92914yes—
441D4—216no—
442D4—22no—
443D4—24no—
444D4#92914yes—
445C2×C4—216no—
446C2×C4—22no—
447C2×C4—24no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—24no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—84no—
461C2×C4—88no—
462C2×C4—122no—
463C2×C4—122no—
464C2×C4—122no—
465C2×C4—121no—
466C2×C4—121no—
467C2×C4—122no—
468C2×C4—121no—
469C2×C4—121no—
470C2×C4#95312yes—
471C2×C4C2×#23512yes—
472C2×C4C2×#23512yes—
473C2×C4C22×#6711yescharacteristic
474C2×C4C22×#6711yescharacteristic
475C8—121no—
476C8—121no—
477C8—122no—
478C2×C3—42no—
479C2×C3—44no—
480C2×C3#284911yescharacteristic
481C2×C3#301111yescharacteristic
482C2×C3#301211yescharacteristic
483C2×C3#312012yes—
484C2×C3#312412yes—
485C22—22no—
486C22—24no—
487C22—22no—
488C22—22no—
489C22—42no—
490C22—42no—
491C22—44no—
492C22—42no—
493C22—44no—
494C22—42no—
495C22#579112yes—
496C22#583511yescharacteristic
497C22#583611yescharacteristic
498C22#583812yes—
499C22C2×#95311yescharacteristic
500C4—22no—
501C4—24no—
502C4—42no—
503C4—44no—
504C4—42no—
505C4—241no—
506C4—241no—
507C4—242no—
508C4C2×#92911yescharacteristic
509C4C2×#92911yescharacteristic
510C4#579112yes—
511C3#3301711yescharacteristic, complemented, Sylow
512C2—42no—
513C2—44no—
514C2#8256912yes—
515C2#8257512yes—
516C2#8403311yescharacteristic
517C2#8410211yescharacteristic
518C2#8411711yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#520C2
socleC23×C3C22×D4
prime radicalC22×C3×#41C22
hypercenterC2×#520D3
hypocenterC3#33017
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#520

Complements

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

NH ≅ G/Ntype
3. #82263C2semidirect
4. C2×#4970C2semidirect
6. #71632C2semidirect
7. #71628C2semidirect
8. #71612C2semidirect
9. #71606C2semidirect
10. #71584C2semidirect
11. #71553C2semidirect
511. C3#33017semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5202
C2×#52392
#822632
C2×#49702
#721352
#716322
#716282
#716122
#716062
#715842
#715532
#330173