Finite Group Databasebeta

← #10701655

Subgroups — #10701655

Order 768.

1526 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#386C211yesmaximal, characteristic
2C22×#834C212yesmaximal
3C2×C4×#217C211yesmaximal, characteristic, complemented
4#72127C211yesmaximal, characteristic, complemented
5C2×#4526C212yesmaximal, complemented
6#10960—31nomaximal, Sylow
7C23×C3×D4C2211yescharacteristic
8C24×C3×C4C2211yescharacteristic
9C22×C3×C22⋊C4C412yescomplemented
10C22×C3×C22⋊C4C2211yescharacteristic
11C22×#217C2212yes—
12C22×#217C2212yes—
13C22×C4×Dic3—21no—
14C3×#386—21no—
15C3×#386—22no—
16C3×#386—21no—
17C2×C3×#95C412yescomplemented
18C2×#834—24no—
19C2×#834—24no—
20C2×#834C418yescomplemented
21C4×#217—24no—
22C2×C3⋊#95—21no—
23C2×C3⋊#95—22no—
24#4526—24no—
25C22×#99—32no—
26C2×#386D311yescharacteristic
27C2×C4×C22⋊C4—31no—
28C2×#342—32no—
29C22⋊#95—31no—
30C25×C3C2×C411yescharacteristic
31C25×C3C2311yescharacteristic
32C22×C3×D4—22no—
33C22×C3×D4—22no—
34C22×C3×D4—21no—
35C22×C3×D4—21no—
36C22×C3×D4—22no—
37C22×C3×D4—22no—
38C22×C3×D4—22no—
39C22×C3×D4C2×C414yes—
40C23×C3×C4—22no—
41C23×C3×C4—22no—
42C23×C3×C4—44no—
43C23×C3×C4C2×C411yescharacteristic
44C23×C3×C4C2×C412yes—
45C23×C3×C4C2×C412yes—
46C23×C3×C4D411yescharacteristic
47C23×C3×C4D412yes—
48C23×C3×C4D411yescharacteristic
49C23×Dic3—22no—
50C23×Dic3—22no—
51C2×C3×C22⋊C4—21no—
52C2×C3×C22⋊C4—21no—
53C2×C3×C22⋊C4—22no—
54C2×C3×C22⋊C4—22no—
55C2×C3×C22⋊C4—22no—
56C2×C3×C22⋊C4—22no—
57C2×C3×C22⋊C4—44no—
58C2×#217—22no—
59C2×#217—22no—
60C2×#217—22no—
61C2×#217—22no—
62C2×#217—22no—
63C2×#217—22no—
64C2×#217—22no—
65C2×#217—22no—
66C2×#217C2×C414yes—
67C2×#217C2×C414yes—
68C2×#217D414yes—
69C2×#217D414yes—
70C2×C4×Dic3—24no—
71C2×C4×Dic3—44no—
72C3×#95—22no—
73C3×#95—22no—
74#834—216no—
75C3⋊#95—24no—
76C3⋊#95—44no—
77C23×D4C2×D311yescharacteristic
78C24×C4C2×D311yescharacteristic
79C22×C22⋊C4—32no—
80C22×C22⋊C4—32no—
81C22×C22⋊C4Dic312yescomplemented
82C22×C22⋊C4C2×D311yescharacteristic
83C22×C42—61no—
84C2×#99—38no—
85C2×#99—64no—
86C2×#99—64no—
87#386—21no—
88#386—22no—
89#386—21no—
90C4×C22⋊C4—64no—
91C2×#95—61no—
92C2×#95—62no—
93C2×#95Dic312yescomplemented
94#342—64no—
95C24×C3—21no—
96C24×C3—21no—
97C24×C3—22no—
98C24×C3—22no—
99C24×C3—42no—
100C24×C3—42no—
101C24×C3—42no—
102C24×C3—42no—
103C24×C3C4212yescomplemented
104C24×C3C22×C412yes—
105C24×C3C22×C411yescharacteristic
106C24×C3C2×D412yes—
107C24×C3C2×D412yes—
108C24×C3C2×D411yescharacteristic
109C24×C3C2×D411yescharacteristic
110C24×C3#4112yes—
111C24×C3#4112yes—
112C24×C3#4112yes—
113C2×C3×D4—24no—
114C2×C3×D4—24no—
115C2×C3×D4—24no—
116C2×C3×D4—24no—
117C2×C3×D4—24no—
118C2×C3×D4—24no—
119C2×C3×D4—41no—
120C2×C3×D4—42no—
121C2×C3×D4—42no—
122C2×C3×D4—42no—
123C2×C3×D4—42no—
124C2×C3×D4—41no—
125C2×C3×D4—42no—
126C2×C3×D4—41no—
127C2×C3×D4—42no—
128C2×C3×D4—41no—
129C22×C3×C4—21no—
130C22×C3×C4—21no—
131C22×C3×C4—24no—
132C22×C3×C4—21no—
133C22×C3×C4—21no—
134C22×C3×C4—24no—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4—21no—
138C22×C3×C4—24no—
139C22×C3×C4—21no—
140C22×C3×C4—24no—
141C22×C3×C4—24no—
142C22×C3×C4—22no—
143C22×C3×C4—24no—
144C22×C3×C4—42no—
145C22×C3×C4—42no—
146C22×C3×C4—42no—
147C22×C3×C4—48no—
148C22×C3×C4—48no—
149C22×C3×C4—42no—
150C22×C3×C4—42no—
151C22×C3×C4C4212yescomplemented
152C22×C3×C4C22⋊C412yes—
153C22×C3×C4C22⋊C412yes—
154C22×Dic3—24no—
155C22×Dic3—24no—
156C22×Dic3—24no—
157C22×Dic3—24no—
158C22×Dic3—44no—
159C22×Dic3—44no—
160C22×Dic3—44no—
161C22×Dic3—44no—
162C22×Dic3—42no—
163C22×Dic3—42no—
164C3×C22⋊C4—44no—
165C3×C22⋊C4—42no—
166C3×C22⋊C4—41no—
167C3×C22⋊C4—41no—
168C3×C22⋊C4—44no—
169#217—24no—
170#217—24no—
171#217—24no—
172#217—24no—
173#217—24no—
174#217—42no—
175#217—42no—
176#217C22⋊C418yescomplemented
177C4×Dic3—48no—
178C25C2×Dic311yescharacteristic
179C25C22×D311yescharacteristic
180C22×D4—22no—
181C22×D4—22no—
182C22×D4—22no—
183C22×D4—22no—
184C22×D4—22no—
185C22×D4—21no—
186C22×D4—21no—
187C22×D4C4×D314yes—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—44no—
191C23×C4—62no—
192C23×C4—62no—
193C23×C4C2×Dic311yescharacteristic
194C23×C4C2×Dic312yes—
195C23×C4C2×Dic312yes—
196C23×C4#6711yescharacteristic
197C23×C4#6712yes—
198C23×C4#6711yescharacteristic
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—34no—
206C2×C22⋊C4—34no—
207C2×C22⋊C4—34no—
208C2×C22⋊C4—34no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—62no—
211C2×C22⋊C4—62no—
212C2×C22⋊C4—62no—
213C2×C22⋊C4—62no—
214C2×C22⋊C4—62no—
215C2×C22⋊C4—62no—
216C2×C22⋊C4—62no—
217C2×C22⋊C4—62no—
218C2×C42—64no—
219C2×C42—124no—
220#99—616no—
221#95—22no—
222#95—22no—
223#95—64no—
224#95—124no—
225C23×C3—22no—
226C23×C3—22no—
227C23×C3—22no—
228C23×C3—22no—
229C23×C3—22no—
230C23×C3—22no—
231C23×C3—22no—
232C23×C3—24no—
233C23×C3—42no—
234C23×C3—42no—
235C23×C3—44no—
236C23×C3—42no—
237C23×C3—44no—
238C23×C3—44no—
239C23×C3—42no—
240C23×C3—44no—
241C23×C3—44no—
242C23×C3—44no—
243C23×C3—41no—
244C23×C3—42no—
245C23×C3—41no—
246C23×C3—42no—
247C23×C3—42no—
248C23×C3—82no—
249C23×C3—82no—
250C23×C3—84no—
251C23×C3C2×C4211yescharacteristic
252C23×C3C2×C22⋊C412yes—
253C23×C3C2×C22⋊C411yescharacteristic
254C23×C3#11712yes—
255C23×C3#11711yescharacteristic
256C23×C3C4×D412yes—
257C23×C3C4×D412yes—
258C23×C3C4×D412yes—
259C23×C3C4×D412yes—
260C23×C3C4×D412yes—
261C23×C3C4×D412yes—
262C23×C3#12011yescharacteristic
263C23×C3#12112yes—
264C23×C3#12112yes—
265C23×C3#12112yes—
266C23×C3#12312yes—
267C23×C3#12311yescharacteristic
268C23×C3#12412yes—
269C23×C3#12411yescharacteristic
270C23×C3#12612yes—
271C23×C3#12711yescharacteristic
272C3×D4—44no—
273C3×D4—44no—
274C3×D4—44no—
275C3×D4—44no—
276C2×C3×C4—24no—
277C2×C3×C4—24no—
278C2×C3×C4—22no—
279C2×C3×C4—22no—
280C2×C3×C4—24no—
281C2×C3×C4—22no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—48no—
286C2×C3×C4—48no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—41no—
290C2×C3×C4—42no—
291C2×C3×C4—41no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—84no—
296C2×C3×C4#9914yescomplemented
297C2×Dic3—28no—
298C2×Dic3—44no—
299C2×Dic3—44no—
300C2×Dic3—48no—
301C2×Dic3—48no—
302C2×Dic3—48no—
303C2×Dic3—44no—
304C2×Dic3—82no—
305C2×Dic3—82no—
306C2×Dic3—84no—
307C24—21no—
308C24—21no—
309C24—22no—
310C24—22no—
311C24—42no—
312C24—42no—
313C24—42no—
314C24—42no—
315C24C4×Dic312yescomplemented
316C24C2×C4×D312yes—
317C24#23512yes—
318C24D3×D412yes—
319C24D3×D412yes—
320C24#23712yes—
321C24#23712yes—
322C24C22×Dic311yescharacteristic
323C24C2×#6711yescharacteristic
324C24C2×#6711yescharacteristic
325C2×D4—24no—
326C2×D4—24no—
327C2×D4—24no—
328C2×D4—24no—
329C2×D4—24no—
330C2×D4—24no—
331C2×D4—42no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4—41no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—24no—
344C22×C4—21no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—24no—
351C22×C4—21no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—48no—
360C22×C4—48no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—64no—
364C22×C4—64no—
365C22×C4—64no—
366C22×C4—64no—
367C22×C4—124no—
368C22×C4—124no—
369C22×C4—124no—
370C22×C4—124no—
371C22×C4—122no—
372C22×C4—122no—
373C22×C4C4×Dic312yescomplemented
374C22×C4#21712yes—
375C22×C4#21712yes—
376C22⋊C4—38no—
377C22⋊C4—44no—
378C22⋊C4—41no—
379C22⋊C4—42no—
380C22⋊C4—41no—
381C22⋊C4—44no—
382C22⋊C4—64no—
383C22⋊C4—64no—
384C22⋊C4—64no—
385C22⋊C4—64no—
386C22⋊C4—64no—
387C22⋊C4—122no—
388C22⋊C4—122no—
389C42—128no—
390C22×C3—21no—
391C22×C3—22no—
392C22×C3—21no—
393C22×C3—22no—
394C22×C3—21no—
395C22×C3—22no—
396C22×C3—22no—
397C22×C3—21no—
398C22×C3—24no—
399C22×C3—22no—
400C22×C3—24no—
401C22×C3—22no—
402C22×C3—24no—
403C22×C3—24no—
404C22×C3—22no—
405C22×C3—22no—
406C22×C3—42no—
407C22×C3—42no—
408C22×C3—42no—
409C22×C3—42no—
410C22×C3—42no—
411C22×C3—42no—
412C22×C3—42no—
413C22×C3—42no—
414C22×C3—42no—
415C22×C3—44no—
416C22×C3—84no—
417C22×C3—84no—
418C22×C3—81no—
419C22×C3—81no—
420C22×C3—82no—
421C22×C3C4×C22⋊C412yes—
422C22×C3C4×C22⋊C411yescharacteristic
423C22×C3#38612yes—
424C22×C3#38611yescharacteristic
425C22×C3#38812yes—
426C22×C3#38812yes—
427C22×C3#38812yes—
428C22×C3#39012yes—
429C22×C3#39011yescharacteristic
430C22×C3C2×#9912yes—
431C22×C3#41012yes—
432C3×C4—24no—
433C3×C4—42no—
434C3×C4—42no—
435C3×C4—82no—
436Dic3—48no—
437Dic3—84no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—24no—
446C23—42no—
447C23—42no—
448C23—44no—
449C23—44no—
450C23—44no—
451C23—42no—
452C23—44no—
453C23—42no—
454C23—44no—
455C23—44no—
456C23—42no—
457C23—42no—
458C23—41no—
459C23—41no—
460C23—42no—
461C23—84no—
462C23—82no—
463C23—82no—
464C23#87712yes—
465C23#87912yes—
466C23D3×C22⋊C412yes—
467C23#67⋊C412yes—
468C23#67⋊C412yes—
469C23#88312yes—
470C23#88412yes—
471C23#88512yes—
472C23C2×C4×Dic311yescharacteristic
473C23#92611yescharacteristic
474C23C4×#6712yes—
475C23C4×#6712yes—
476C23D4×Dic312yes—
477C23D4×Dic312yes—
478C23#93511yescharacteristic
479C23#93611yescharacteristic
480C23#93711yescharacteristic
481C23#93812yes—
482C23#93912yes—
483C23#94011yescharacteristic
484C23C2×#21711yescharacteristic
485D4—44no—
486D4—44no—
487D4—44no—
488D4—44no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—48no—
499C2×C4—48no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—68no—
509C2×C4—84no—
510C2×C4—124no—
511C2×C4—128no—
512C2×C4—124no—
513C2×C4—128no—
514C2×C4—128no—
515C2×C4—124no—
516C2×C4—244no—
517C2×C4—242no—
518C2×C4—242no—
519C2×C4#83414yescomplemented
520C2×C3—22no—
521C2×C3—22no—
522C2×C3—22no—
523C2×C3—22no—
524C2×C3—22no—
525C2×C3—22no—
526C2×C3—82no—
527C2×C3#143711yescharacteristic
528C2×C3C4×#9912yes—
529C2×C3#189811yescharacteristic
530C2×C3#189911yescharacteristic
531C2×C3#196412yes—
532C22—24no—
533C22—24no—
534C22—22no—
535C22—22no—
536C22—21no—
537C22—22no—
538C22—22no—
539C22—21no—
540C22—24no—
541C22—24no—
542C22—22no—
543C22—22no—
544C22—21no—
545C22—21no—
546C22—22no—
547C22—22no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—44no—
558C22—84no—
559C22—84no—
560C22—82no—
561C22—81no—
562C22—81no—
563C22Dic3×C22⋊C412yes—
564C22#494412yes—
565C22#494512yes—
566C22#495112yes—
567C22#495612yes—
568C22#495712yes—
569C22C4×#21711yescharacteristic
570C22C2×#83412yes—
571C22#522011yescharacteristic
572C22#522111yescharacteristic
573C22#524012yes—
574C4—24no—
575C4—42no—
576C4—42no—
577C4—82no—
578C4—128no—
579C4—244no—
580C3#1096011yescharacteristic, complemented, Sylow
581C2—22no—
582C2—22no—
583C2—22no—
584C2—22no—
585C2—22no—
586C2—22no—
587C2—82no—
588C2#7154511yescharacteristic
589C2C4×#83412yes—
590C2#7167011yescharacteristic
591C2#7168612yes—
592C2#7170511yescharacteristic
subgroupG/H
centerC23C2×#217
commutatorC23×C3C2×C42
FrattiniC25C22×D3
FittingC2×C3×#386C2
socleC23×C3C2×C22⋊C4
prime radicalC23×C3×D4C22
cosocleC25×C3C23
hypercenterC2×#386D3
hypocenterC3#10960
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#386

Complements

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

NH ≅ G/Ntype
3. C2×C4×#217C2semidirect
4. #72127C2semidirect
5. C2×#4526C2semidirect
9. C22×C3×C22⋊C4C4semidirect
17. C2×C3×#95C4semidirect
20. C2×#834C4semidirect
81. C22×C22⋊C4Dic3semidirect
93. C2×#95Dic3semidirect
103. C24×C3C42semidirect
151. C22×C3×C4C42semidirect
176. #217C22⋊C4semidirect
296. C2×C3×C4#99semidirect
315. C24C4×Dic3semidirect
373. C22×C4C4×Dic3semidirect
519. C2×C4#834semidirect
580. C3#10960semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3862
C22×#8342
C2×C4×#2172
#721272
C2×#45262
#109603