Finite Group Databasebeta

← #10822354

Subgroups — #10822354

Order 768.

2018 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#915C212yesmaximal
2C2×D3×C4⋊Q8C211yesmaximal, characteristic, complemented
3C2×#4802C212yesmaximal, complemented
4#76290C211yesmaximal, characteristic
5C3×#2013C211yesmaximal, characteristic, complemented
6#72057C212yesmaximal, complemented
7#71793C211yesmaximal, characteristic, complemented
8#70283C212yesmaximal, complemented
9#69661C211yesmaximal, characteristic, complemented
10#68688C211yesmaximal, characteristic, complemented
11#68660C211yesmaximal, characteristic, complemented
12#23432—31nomaximal, Sylow
13C22×D3×Q8C2212yes—
14C22×#237C2211yescharacteristic
15C22×C3×SD16C2212yes—
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×#944C2212yes—
18C22×#216C2212yes—
19C22×#214C2212yes—
20C2×#915—28no—
21C2×#915—28no—
22C2×#915—28no—
23C22×Dic12C2212yes—
24C22×ΓL(1,𝔽25)C2212yes—
25D3×C4⋊Q8—28no—
26C2×#892C2211yescharacteristic
27C2×D3×C4⋊C4—22no—
28C2×#888—22no—
29C2×C42×D3C2211yescharacteristic
30C2×#869C2211yescharacteristic
31C2×C3×#103C2212yes—
32C3×#390C2211yescharacteristic
33C2×#835C2212yes—
34#5221C2211yescharacteristic
35C2×#816C2212yes—
36#4973C2211yescharacteristic
37C2×#810C2212yes—
38#4956—22no—
39Dic3×C22⋊C4—22no—
40#4929C2211yescharacteristic
41#4802—28no—
42C3×#336C2211yescharacteristic
43#4550C2212yes—
44#4524C2211yescharacteristic
45C22×#137—32no—
46C22×C4⋊Q8—31no—
47C2×#419—32no—
48#2297—31no—
49#2016—32no—
50#2016—31no—
51#2016—31no—
52#2013—31no—
53#2013—32no—
54#2013D311yescharacteristic, complemented
55#1728—31no—
56C22×C3×Q8C2312yes—
57C22×C3×D4C2311yescharacteristic
58C22×#67—21no—
59C23×Dic3—21no—
60C2×D3×Q8—24no—
61C2×D3×Q8—24no—
62C2×D3×Q8—24no—
63C2×D3×Q8—24no—
64C2×D3×Q8—24no—
65C2×D3×Q8D418yes—
66C2×#237—24no—
67C2×#237—24no—
68C2×#237—22no—
69C2×#237D414yes—
70C22×C4×D3—22no—
71C22×C4×D3D412yes—
72C22×C4×D3C2311yescharacteristic
73C22×Dic6—22no—
74C22×Dic6C2311yescharacteristic
75C22×Dic6C2312yes—
76C2×C3×SD16—24no—
77C2×C3×SD16—24no—
78C2×C3×SD16—24no—
79C22×C3×C8C2312yes—
80C3×C4⋊Q8—24no—
81C2×C3×C4⋊C4—22no—
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C22⋊C4—22no—
84C2×C3×C42C2311yescharacteristic
85C2×#217—22no—
86#944—24no—
87#944—24no—
88C2×#216—24no—
89C2×#216—24no—
90C2×#216—24no—
91C2×#214—24no—
92C2×#214—24no—
93C2×#214—24no—
94C2×(C2×D3)⋊C4—21no—
95C2×C4⋊Dic3—22no—
96C2×C4⋊Dic3C2311yescharacteristic
97C2×Dic3⋊C4—24no—
98C2×C4×Dic3—21no—
99C2×C4×Dic3D412yes—
100C2×C4×Dic3C2311yescharacteristic
101C22×C3⋊C8C2312yes—
102#915—264no—
103C2×Dic12—24no—
104C2×Dic12—24no—
105C2×Dic12—24no—
106C2×ΓL(1,𝔽25)—24no—
107C2×ΓL(1,𝔽25)—24no—
108C2×ΓL(1,𝔽25)D418yes—
109#892—24no—
110D3×C4⋊C4—216no—
111#888—216no—
112C42×D3—24no—
113#869—24no—
114C3×#103—24no—
115#835—24no—
116C3⋊#95—22no—
117C3⋊#95—22no—
118#816—24no—
119#810—24no—
120C22×#41—31no—
121C23×Q8—32no—
122C2×#137—68no—
123C2×#137—68no—
124C2×#137—68no—
125C22×Q16—32no—
126C22×Q16—32no—
127C22×SD16—32no—
128C22×SD16C2×D312yes—
129C22×M16—32no—
130C2×C4⋊Q8—32no—
131C2×C4⋊Q8—31no—
132C2×C4⋊Q8—62no—
133C2×C4⋊Q8—68no—
134C2×C4⋊Q8C2×D311yescharacteristic
135C2×#117—31no—
136C22×C4⋊C4—62no—
137C22×C42—31no—
138#419—68no—
139C2×#103—32no—
140C2×#103—32no—
141C2×#103—32no—
142C2×#103C2×D312yes—
143#391—31no—
144#391—31no—
145#390—31no—
146#390C2×D311yescharacteristic
147#386—62no—
148C4×C22⋊C4—62no—
149#336—32no—
150#336—31no—
151#336C2×D311yescharacteristic
152C24×C3—21no—
153C23×D3C2×D411yescharacteristic
154C2×C3×Q8—22no—
155C2×C3×Q8—22no—
156C2×C3×Q8—22no—
157C2×C3×Q8—22no—
158C2×C3×Q8—22no—
159C2×C3×Q8C2×D414yes—
160C2×C3×D4—22no—
161C2×C3×D4—22no—
162C2×C3×D4—21no—
163C2×C3×D4C2×D412yes—
164C22×C3×C4—22no—
165C22×C3×C4—21no—
166C22×C3×C4C2×D412yes—
167C22×C3×C4C2411yescharacteristic
168C2×#67—24no—
169C2×#67—42no—
170C2×#67—42no—
171C22×Dic3—21no—
172C22×Dic3—22no—
173C22×Dic3—24no—
174C22×Dic3—21no—
175C22×Dic3—42no—
176C22×Dic3—42no—
177C22×Dic3C2×D412yes—
178C22×Dic3C2×D411yescharacteristic
179D3×Q8—216no—
180D3×Q8—216no—
181D3×Q8—48no—
182D3×Q8—48no—
183#237—216no—
184#237—48no—
185C2×C4×D3—24no—
186C2×C4×D3—24no—
187C2×C4×D3—24no—
188C2×C4×D3—28no—
189C2×C4×D3—44no—
190C2×C4×D3—44no—
191C2×C4×D3C2×D412yes—
192C2×C4×D3C2×D414yes—
193C2×C4×D3C2×D414yes—
194C2×C4×D3#4112yes—
195C2×Dic6—22no—
196C2×Dic6—22no—
197C2×Dic6—22no—
198C2×Dic6—22no—
199C2×Dic6—22no—
200C2×Dic6—22no—
201C2×Dic6—22no—
202C2×Dic6—21no—
203C2×Dic6—28no—
204C2×Dic6—44no—
205C2×Dic6—44no—
206C2×Dic6C2×D414yes—
207C2×Dic6C2×D412yes—
208C3×SD16—216no—
209C2×C3×C8—22no—
210C2×C3×C8—22no—
211C2×C3×C8C2×D414yes—
212C3×C4⋊C4—28no—
213C3×C4⋊C4—22no—
214C3×C22⋊C4—44no—
215C3×C42—22no—
216#217—44no—
217#216—216no—
218#214—216no—
219(C2×D3)⋊C4—42no—
220C4⋊Dic3—28no—
221C4⋊Dic3—22no—
222Dic3⋊C4—216no—
223C4×Dic3—22no—
224C4×Dic3—24no—
225C4×Dic3—42no—
226C2×C3⋊C8—22no—
227C2×C3⋊C8—22no—
228C2×C3⋊C8C2×D414yes—
229Dic12—216no—
230ΓL(1,𝔽25)—216no—
231C2×#41—34no—
232C2×#41—62no—
233C2×#41—64no—
234C2×#41—64no—
235C22×Q8—38no—
236C22×Q8—32no—
237C22×Q8—31no—
238C22×Q8—62no—
239C22×Q8—64no—
240C22×Q8—64no—
241C22×Q8—64no—
242C22×Q8—64no—
243C22×Q8—64no—
244C22×Q8C22×D312yes—
245C22×D4—61no—
246C22×D4C22×D311yescharacteristic
247C23×C4—31no—
248C23×C4—32no—
249C23×C4—62no—
250C23×C4—61no—
251#137—664no—
252C2×Q16—64no—
253C2×Q16—64no—
254C2×Q16—64no—
255C2×Q16—64no—
256C2×Q16—64no—
257C2×Q16—64no—
258C2×SD16—24no—
259C2×SD16—24no—
260C2×SD16—24no—
261C2×SD16—64no—
262C2×SD16—64no—
263C2×SD16—64no—
264C2×M16—38no—
265C2×M16—64no—
266C2×M16—64no—
267C22×C8—32no—
268C22×C8C22×D312yes—
269C4⋊Q8—24no—
270C4⋊Q8—64no—
271C4⋊Q8—616no—
272C4⋊Q8—64no—
273C4⋊Q8—64no—
274#117—64no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—31no—
277C2×C4⋊C4—616no—
278C2×C4⋊C4—64no—
279C2×C4⋊C4—62no—
280C2×C4⋊C4C22×D311yescharacteristic
281C2×C22⋊C4—22no—
282C2×C22⋊C4—61no—
283C2×C22⋊C4—62no—
284C2×C42—32no—
285C2×C42—31no—
286C2×C42—64no—
287C2×C42—61no—
288C2×C42C22×D311yescharacteristic
289#103—24no—
290#103—64no—
291#103—64no—
292#103—64no—
293#95—62no—
294#95—62no—
295C23×C3—22no—
296C23×C3—41no—
297C23×C3—41no—
298C23×C3—42no—
299C23×C3—42no—
300C23×C3C22×D411yescharacteristic
301C22×D3—22no—
302C22×D3—22no—
303C22×D3#12012yes—
304C22×D3#12112yes—
305C22×D3#12712yes—
306C3×Q8—24no—
307C3×Q8—24no—
308C3×Q8—42no—
309C3×Q8—42no—
310C3×D4—24no—
311C3×D4—42no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—24no—
315C2×C3×C4—42no—
316C2×C3×C4—42no—
317C2×C3×C4—41no—
318C2×C3×C4—42no—
319C2×C3×C4#13714yescomplemented
320C2×C3×C4C22×D411yescharacteristic
321C2×C3×C4C22×D412yes—
322C2×C3×C4C22×D412yes—
323C2×C3×C4C2×#4111yescharacteristic
324#67—48no—
325C2×Dic3—22no—
326C2×Dic3—22no—
327C2×Dic3—24no—
328C2×Dic3—22no—
329C2×Dic3—41no—
330C2×Dic3—42no—
331C2×Dic3—42no—
332C2×Dic3—41no—
333C2×Dic3—48no—
334C2×Dic3—41no—
335C2×Dic3—42no—
336C2×Dic3#12012yes—
337C2×Dic3#12112yes—
338C2×Dic3#12712yes—
339C2×Dic3#13714yescomplemented
340C4×D3—216no—
341C4×D3—416no—
342C4×D3#12018yes—
343C4×D3#12118yes—
344Dic6—24no—
345Dic6—24no—
346Dic6—24no—
347Dic6—42no—
348Dic6—42no—
349Dic6—42no—
350Dic6—416no—
351C3×C8—24no—
352C3⋊C8—24no—
353C24—21no—
354C24—31no—
355#41—616no—
356#41—128no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—22no—
362C2×Q8—34no—
363C2×Q8—32no—
364C2×Q8—68no—
365C2×Q8—616no—
366C2×Q8—616no—
367C2×Q8—62no—
368C2×Q8—61no—
369C2×Q8—62no—
370C2×Q8—62no—
371C2×Q8—62no—
372C2×Q8—62no—
373C2×Q8—62no—
374C2×Q8—62no—
375C2×Q8—124no—
376C2×Q8—124no—
377C2×Q8—128no—
378C2×Q8—128no—
379C2×Q8D3×D414yes—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—21no—
383C2×D4—64no—
384C2×D4—122no—
385C2×D4—122no—
386C2×D4D3×D412yes—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—31no—
390C22×C4—34no—
391C22×C4—34no—
392C22×C4—32no—
393C22×C4—32no—
394C22×C4—32no—
395C22×C4—64no—
396C22×C4—64no—
397C22×C4—68no—
398C22×C4—64no—
399C22×C4—64no—
400C22×C4—61no—
401C22×C4—62no—
402C22×C4—61no—
403C22×C4—122no—
404C22×C4—122no—
405C22×C4—124no—
406C22×C4—124no—
407C22×C4D3×D412yes—
408C22×C4C23×D311yescharacteristic
409Q16—616no—
410Q16—616no—
411SD16—216no—
412SD16—616no—
413M16—616no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—34no—
417C2×C8—62no—
418C2×C8—62no—
419C2×C8D3×D414yes—
420C4⋊C4—28no—
421C4⋊C4—22no—
422C4⋊C4—616no—
423C4⋊C4—68no—
424C4⋊C4—62no—
425C22⋊C4—44no—
426C22⋊C4—124no—
427C22⋊C4—122no—
428C42—22no—
429C42—64no—
430C42—62no—
431C42—122no—
432C22×C3—42no—
433C22×C3—44no—
434C22×C3—81no—
435C22×C3—81no—
436C22×C3—82no—
437C22×C3C2×#12011yescharacteristic
438C22×C3C2×#12111yescharacteristic
439C22×C3C2×#12711yescharacteristic
440C22×C3C2×#13712yes—
441C22×C3C2×#13712yes—
442C2×D3—22no—
443C2×D3—22no—
444C2×D3—24no—
445C2×D3—24no—
446C2×D3#39214yes—
447C3×C4—24no—
448C3×C4—44no—
449C3×C4—81no—
450C3×C4C2×#12012yes—
451C3×C4C2×#12112yes—
452Dic3—24no—
453Dic3—44no—
454Dic3—42no—
455Dic3—81no—
456Dic3#39214yes—
457C23—22no—
458C23—32no—
459C23—32no—
460C23—32no—
461C23—41no—
462C23—42no—
463C23—42no—
464C23—41no—
465C23—62no—
466C23—62no—
467C23C2×D3×D411yescharacteristic
468Q8—24no—
469Q8—24no—
470Q8—42no—
471Q8—42no—
472Q8—64no—
473Q8—64no—
474Q8—64no—
475Q8—1216no—
476Q8—122no—
477Q8—122no—
478Q8—122no—
479D4—24no—
480D4—42no—
481D4—128no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—32no—
486C2×C4—32no—
487C2×C4—32no—
488C2×C4—38no—
489C2×C4—38no—
490C2×C4—34no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—616no—
496C2×C4—62no—
497C2×C4—62no—
498C2×C4—64no—
499C2×C4—62no—
500C2×C4—128no—
501C2×C4—1216no—
502C2×C4—121no—
503C2×C4—122no—
504C2×C4—121no—
505C2×C4—122no—
506C2×C4—121no—
507C2×C4—122no—
508C2×C4#91514yescomplemented
509C2×C4C2×D3×D411yescharacteristic
510C2×C4C2×D3×D412yes—
511C2×C4C2×D3×D412yes—
512C2×C4D3×#4111yescharacteristic
513C8—24no—
514C8—64no—
515C2×C3—82no—
516C2×C3C2×#39211yescharacteristic
517C2×C3#300412yes—
518C2×C3#303912yes—
519C2×C3#315211yescharacteristic
520C2×C3#315211yescharacteristic
521D3—24no—
522C22—34no—
523C22—44no—
524C22—42no—
525C22—64no—
526C22—62no—
527C22—64no—
528C22—62no—
529C22—81no—
530C22—82no—
531C22—81no—
532C22D3×#12011yescharacteristic
533C22D3×#12111yescharacteristic
534C22D3×#12711yescharacteristic
535C22C2×#91512yes—
536C22D3×#13712yes—
537C4—24no—
538C4—34no—
539C4—44no—
540C4—64no—
541C4—81no—
542C4—122no—
543C4—124no—
544C4—241no—
545C4D3×#12012yes—
546C4D3×#12112yes—
547C3#2343211yescharacteristic, complemented, Sylow
548C2—64no—
549C2—82no—
550C2D3×#39211yescharacteristic
551C2#8003912yes—
552C2#8042912yes—
553C2#8176211yescharacteristic
554C2#8187011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2013C2
socleC23×C3C22×D4
prime radicalC22×#237C22
hypercenter#2013D3
hypocenterC3#23432
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2013

Complements

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

NH ≅ G/Ntype
2. C2×D3×C4⋊Q8C2semidirect
3. C2×#4802C2semidirect
5. C3×#2013C2semidirect
6. #72057C2semidirect
7. #71793C2semidirect
8. #70283C2semidirect
9. #69661C2semidirect
10. #68688C2semidirect
11. #68660C2semidirect
54. #2013D3semidirect
319. C2×C3×C4#137semidirect
339. C2×Dic3#137semidirect
508. C2×C4#915semidirect
547. C3#23432semidirect

Maximal subgroups

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

subgroupindex
C22×#9152
C2×D3×C4⋊Q82
C2×#48022
#762902
C3×#20132
#720572
#717932
#702832
#696612
#686882
#686602
#234323