Finite Group Databasebeta

← #10701654

Subgroups — #10701654

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×C4×C22⋊C4C211yesmaximal, characteristic, complemented
2C2×#5213C211yesmaximal, characteristic
3C22×#806C212yesmaximal
4#72127C211yesmaximal, characteristic, complemented
5C2×#4527C212yesmaximal, complemented
6#10960—31nomaximal, Sylow
7C24×C3×C4C2211yescharacteristic
8C23×#67C2211yescharacteristic
9C22×C3×C22⋊C4C2212yes—
10C22×C3×C42—21no—
11C22×#217C2211yescharacteristic
12C22×#217C2212yes—
13C22×(C2×D3)⋊C4C412yescomplemented
14C3×C4×C22⋊C4—24no—
15C2×C3×#95—21no—
16#5213—22no—
17#5213—21no—
18#5213—21no—
19C2×C3⋊#95—22no—
20C2×C3⋊#95C412yescomplemented
21C2×#806—24no—
22C2×#806—24no—
23C2×#806C418yescomplemented
24#4527—24no—
25C22×#99—32no—
26C2×#386—31no—
27C2×C4×C22⋊C4D311yescharacteristic, complemented
28C2×#342—32no—
29C22⋊#95—31no—
30C25×C3C2311yescharacteristic
31C24×D3C2×C411yescharacteristic
32C23×C3×C4—22no—
33C23×C3×C4—22no—
34C23×C3×C4—22no—
35C23×C3×C4—44no—
36C23×C3×C4C2×C412yes—
37C23×C3×C4D412yes—
38C23×C3×C4D411yescharacteristic
39C23×C3×C4D411yescharacteristic
40C22×#67—22no—
41C22×#67—22no—
42C22×#67—21no—
43C22×#67—21no—
44C22×#67—22no—
45C22×#67—22no—
46C22×#67—22no—
47C22×#67C2×C414yes—
48C23×Dic3—22no—
49C23×Dic3C2×C411yescharacteristic
50C23×Dic3C2×C412yes—
51C2×C3×C22⋊C4—22no—
52C2×C3×C22⋊C4—22no—
53C2×C3×C22⋊C4—22no—
54C2×C3×C22⋊C4C2×C414yes—
55C2×C3×C22⋊C4D414yes—
56C2×C3×C22⋊C4D414yes—
57C2×C3×C42—24no—
58C2×C3×C42—44no—
59C2×#217—22no—
60C2×#217—22no—
61C2×#217—22no—
62C2×#217—22no—
63C2×#217—21no—
64C2×#217—21no—
65C2×#217—22no—
66C2×#217—22no—
67C2×#217—22no—
68C2×#217C2×C414yes—
69C2×(C2×D3)⋊C4—22no—
70C2×(C2×D3)⋊C4—22no—
71C2×(C2×D3)⋊C4—44no—
72C3×#95—24no—
73C3⋊#95—22no—
74C3⋊#95—22no—
75C3⋊#95—44no—
76#806—216no—
77C23×D4—31no—
78C24×C4C2×D311yescharacteristic
79C22×C22⋊C4—32no—
80C22×C22⋊C4—31no—
81C22×C22⋊C4—32no—
82C22×C22⋊C4C2×D312yes—
83C22×C42—21no—
84C2×#99—38no—
85C2×#99—64no—
86C2×#99—64no—
87#386—62no—
88#386—61no—
89#386—61no—
90C4×C22⋊C4—24no—
91C2×#95—21no—
92C2×#95—32no—
93C2×#95—62no—
94#342—64no—
95C24×C3—42no—
96C24×C3—42no—
97C24×C3C22×C411yescharacteristic
98C24×C3C22×C412yes—
99C24×C3C2×D411yescharacteristic
100C24×C3C2×D412yes—
101C24×C3C2×D411yescharacteristic
102C24×C3C2×D412yes—
103C24×C3#4112yes—
104C24×C3#4112yes—
105C24×C3#4112yes—
106C23×D3—21no—
107C23×D3—21no—
108C23×D3—22no—
109C23×D3—22no—
110C23×D3—42no—
111C23×D3—42no—
112C23×D3C4212yescomplemented
113C22×C3×C4—21no—
114C22×C3×C4—21no—
115C22×C3×C4—24no—
116C22×C3×C4—24no—
117C22×C3×C4—24no—
118C22×C3×C4—21no—
119C22×C3×C4—24no—
120C22×C3×C4—24no—
121C22×C3×C4—24no—
122C22×C3×C4—21no—
123C22×C3×C4—24no—
124C22×C3×C4—24no—
125C22×C3×C4—42no—
126C22×C3×C4—42no—
127C22×C3×C4—42no—
128C22×C3×C4—48no—
129C22×C3×C4—48no—
130C22×C3×C4—44no—
131C22×C3×C4—44no—
132C22×C3×C4C22⋊C412yes—
133C22×C3×C4C22⋊C412yes—
134C2×#67—24no—
135C2×#67—24no—
136C2×#67—24no—
137C2×#67—24no—
138C2×#67—24no—
139C2×#67—24no—
140C2×#67—41no—
141C2×#67—42no—
142C2×#67—42no—
143C2×#67—42no—
144C2×#67—41no—
145C2×#67—42no—
146C2×#67—42no—
147C2×#67—41no—
148C2×#67—42no—
149C2×#67—41no—
150C22×Dic3—21no—
151C22×Dic3—21no—
152C22×Dic3—22no—
153C22×Dic3—22no—
154C22×Dic3—22no—
155C22×Dic3—24no—
156C22×Dic3—24no—
157C22×Dic3—44no—
158C22×Dic3—44no—
159C22×Dic3—42no—
160C22×Dic3—42no—
161C22×Dic3—42no—
162C22×Dic3—42no—
163C22×Dic3C4212yescomplemented
164C3×C22⋊C4—24no—
165C3×C22⋊C4—24no—
166C3×C22⋊C4—24no—
167C3×C22⋊C4C22⋊C418yescomplemented
168C3×C42—48no—
169#217—24no—
170#217—24no—
171#217—42no—
172#217—42no—
173#217—42no—
174#217—41no—
175#217—41no—
176(C2×D3)⋊C4—44no—
177(C2×D3)⋊C4—44no—
178C25—31no—
179C25C22×D311yescharacteristic
180C22×D4—34no—
181C22×D4—61no—
182C22×D4—62no—
183C22×D4—62no—
184C22×D4—62no—
185C22×D4—62no—
186C22×D4—61no—
187C22×D4—62no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—31no—
192C23×C4—32no—
193C23×C4—44no—
194C23×C4—62no—
195C23×C4C4×D312yes—
196C23×C4D1212yes—
197C23×C4#6711yescharacteristic
198C23×C4#6711yescharacteristic
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—34no—
203C2×C22⋊C4—62no—
204C2×C22⋊C4—62no—
205C2×C22⋊C4—62no—
206C2×C22⋊C4—62no—
207C2×C22⋊C4—62no—
208C2×C22⋊C4—61no—
209C2×C22⋊C4—62no—
210C2×C22⋊C4—62no—
211C2×C22⋊C4—61no—
212C2×C22⋊C4—62no—
213C2×C22⋊C4—62no—
214C2×C22⋊C4—124no—
215C2×C22⋊C4C4×D314yes—
216C2×C22⋊C4D1214yes—
217C2×C22⋊C4#6714yes—
218C2×C42—24no—
219C2×C42—44no—
220#99—616no—
221#95—24no—
222#95—62no—
223#95—62no—
224#95—124no—
225C23×C3—22no—
226C23×C3—22no—
227C23×C3—22no—
228C23×C3—22no—
229C23×C3—22no—
230C23×C3—22no—
231C23×C3—42no—
232C23×C3—42no—
233C23×C3—44no—
234C23×C3—44no—
235C23×C3—42no—
236C23×C3—42no—
237C23×C3—44no—
238C23×C3—44no—
239C23×C3C2×C4211yescharacteristic
240C23×C3C2×C22⋊C412yes—
241C23×C3C2×C22⋊C411yescharacteristic
242C23×C3#11711yescharacteristic
243C23×C3#11712yes—
244C23×C3C4×D412yes—
245C23×C3C4×D412yes—
246C23×C3C4×D412yes—
247C23×C3C4×D412yes—
248C23×C3C4×D412yes—
249C23×C3C4×D412yes—
250C23×C3#12011yescharacteristic
251C23×C3#12112yes—
252C23×C3#12112yes—
253C23×C3#12112yes—
254C23×C3#12311yescharacteristic
255C23×C3#12312yes—
256C23×C3#12411yescharacteristic
257C23×C3#12412yes—
258C23×C3#12612yes—
259C23×C3#12711yescharacteristic
260C22×D3—22no—
261C22×D3—24no—
262C22×D3—44no—
263C22×D3—41no—
264C22×D3—42no—
265C22×D3—41no—
266C22×D3—42no—
267C22×D3—42no—
268C22×D3—44no—
269C22×D3—82no—
270C22×D3—82no—
271C22×D3—84no—
272C2×C3×C4—24no—
273C2×C3×C4—24no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—28no—
277C2×C3×C4—22no—
278C2×C3×C4—22no—
279C2×C3×C4—22no—
280C2×C3×C4—48no—
281C2×C3×C4—48no—
282C2×C3×C4—44no—
283C2×C3×C4—44no—
284C2×C3×C4—48no—
285C2×C3×C4—48no—
286C2×C3×C4—42no—
287C2×C3×C4—42no—
288C2×C3×C4#9914yescomplemented
289#67—44no—
290#67—44no—
291#67—44no—
292#67—44no—
293C2×Dic3—24no—
294C2×Dic3—22no—
295C2×Dic3—41no—
296C2×Dic3—42no—
297C2×Dic3—41no—
298C2×Dic3—48no—
299C2×Dic3—44no—
300C2×Dic3—42no—
301C2×Dic3—42no—
302C2×Dic3—42no—
303C2×Dic3—82no—
304C2×Dic3—82no—
305C2×Dic3—84no—
306C2×Dic3—84no—
307C24—32no—
308C24—42no—
309C24—42no—
310C24—61no—
311C24—61no—
312C24—62no—
313C24—62no—
314C24—122no—
315C24—122no—
316C24C2×C4×D311yescharacteristic
317C24C2×C4×D312yes—
318C24C2×D1211yescharacteristic
319C24C2×D1212yes—
320C24#23512yes—
321C24#23512yes—
322C24#23512yes—
323C24C2×#6711yescharacteristic
324C24C2×#6712yes—
325C2×D4—64no—
326C2×D4—64no—
327C2×D4—64no—
328C2×D4—64no—
329C2×D4—64no—
330C2×D4—64no—
331C2×D4—121no—
332C2×D4—121no—
333C2×D4—122no—
334C2×D4—122no—
335C2×D4—122no—
336C2×D4—121no—
337C2×D4—122no—
338C2×D4—122no—
339C2×D4—122no—
340C2×D4—121no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—21no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—21no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—32no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—48no—
358C22×C4—48no—
359C22×C4—44no—
360C22×C4—44no—
361C22×C4—61no—
362C22×C4—62no—
363C22×C4—61no—
364C22×C4—62no—
365C22×C4—64no—
366C22×C4—62no—
367C22×C4—64no—
368C22×C4—124no—
369C22×C4—124no—
370C22×C4—122no—
371C22×C4—122no—
372C22×C4—122no—
373C22×C4—122no—
374C22×C4(C2×D3)⋊C412yes—
375C22×C4(C2×D3)⋊C412yes—
376C22⋊C4—24no—
377C22⋊C4—24no—
378C22⋊C4—24no—
379C22⋊C4—64no—
380C22⋊C4—64no—
381C22⋊C4—124no—
382C22⋊C4—121no—
383C22⋊C4—122no—
384C22⋊C4—121no—
385C22⋊C4—122no—
386C22⋊C4—122no—
387C22⋊C4—124no—
388C22⋊C4(C2×D3)⋊C418yescomplemented
389C42—48no—
390C22×C3—22no—
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—22no—
394C22×C3—22no—
395C22×C3—21no—
396C22×C3—22no—
397C22×C3—21no—
398C22×C3—22no—
399C22×C3—24no—
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×C3C4×C22⋊C412yes—
415C22×C3C4×C22⋊C411yescharacteristic
416C22×C3#38612yes—
417C22×C3#38611yescharacteristic
418C22×C3#38812yes—
419C22×C3#38812yes—
420C22×C3#38812yes—
421C22×C3#39012yes—
422C22×C3#39011yescharacteristic
423C22×C3C2×#9912yes—
424C22×C3#41012yes—
425C2×D3—42no—
426C2×D3—44no—
427C2×D3—84no—
428C2×D3—81no—
429C2×D3—81no—
430C2×D3—82no—
431C2×D3—84no—
432C3×C4—24no—
433C3×C4—48no—
434C3×C4—42no—
435Dic3—42no—
436Dic3—84no—
437Dic3—82no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—42no—
445C23—42no—
446C23—44no—
447C23—44no—
448C23—44no—
449C23—44no—
450C23—42no—
451C23—42no—
452C23—62no—
453C23—64no—
454C23—124no—
455C23—124no—
456C23—121no—
457C23—122no—
458C23—122no—
459C23—121no—
460C23—122no—
461C23—244no—
462C23—242no—
463C23—242no—
464C23C42×D311yescharacteristic
465C23#87211yescharacteristic
466C23#87212yes—
467C23C4×D1212yes—
468C23C4×D1212yes—
469C23C4×D1212yes—
470C23#87411yescharacteristic
471C23#87511yescharacteristic
472C23#87512yes—
473C23#87612yes—
474C23C2×(C2×D3)⋊C412yes—
475C23C2×(C2×D3)⋊C411yescharacteristic
476C23C4×#6712yes—
477C23C4×#6712yes—
478C23C4×#6712yes—
479C23#92911yescharacteristic
480C23#92912yes—
481C23#93012yes—
482C23#93012yes—
483C23#93012yes—
484C23#95311yescharacteristic
485D4—124no—
486D4—124no—
487D4—124no—
488D4—124no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—28no—
496C2×C4—22no—
497C2×C4—48no—
498C2×C4—48no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—48no—
503C2×C4—48no—
504C2×C4—44no—
505C2×C4—64no—
506C2×C4—62no—
507C2×C4—121no—
508C2×C4—122no—
509C2×C4—121no—
510C2×C4—128no—
511C2×C4—124no—
512C2×C4—122no—
513C2×C4—122no—
514C2×C4—122no—
515C2×C4—244no—
516C2×C4—242no—
517C2×C4—242no—
518C2×C4—244no—
519C2×C4#80614yescomplemented
520C2×C3—22no—
521C2×C3—22no—
522C2×C3—22no—
523C2×C3—22no—
524C2×C3—22no—
525C2×C3—22no—
526C2×C3#143711yescharacteristic
527C2×C3C4×#9912yes—
528C2×C3#189811yescharacteristic
529C2×C3#189911yescharacteristic
530C2×C3#196412yes—
531D3—82no—
532C22—24no—
533C22—24no—
534C22—22no—
535C22—22no—
536C22—24no—
537C22—22no—
538C22—24no—
539C22—22no—
540C22—21no—
541C22—21no—
542C22—22no—
543C22—22no—
544C22—21no—
545C22—22no—
546C22—22no—
547C22—21no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—124no—
557C22—122no—
558C22—244no—
559C22—244no—
560C22—242no—
561C22—241no—
562C22—241no—
563C22C4×(C2×D3)⋊C412yes—
564C22C4×(C2×D3)⋊C411yescharacteristic
565C22#493912yes—
566C22#493911yescharacteristic
567C22#494012yes—
568C22#494012yes—
569C22#494012yes—
570C22C2×#80612yes—
571C22#500612yes—
572C22#521312yes—
573C22#521311yescharacteristic
574C4—24no—
575C4—48no—
576C4—42no—
577C4—122no—
578C4—244no—
579C4—242no—
580C3#1096011yescharacteristic, complemented, Sylow
581C2—22no—
582C2—22no—
583C2—22no—
584C2—22no—
585C2—22no—
586C2—22no—
587C2—242no—
588C2C4×#80612yes—
589C2#6945612yes—
590C2#7153711yescharacteristic
591C2#7154311yescharacteristic
592C2#7159511yescharacteristic
subgroupG/H
centerC23C2×(C2×D3)⋊C4
commutatorC23×C3C2×C42
FrattiniC25C22×D3
FittingC2×C3×C4×C22⋊C4C2
socleC23×C3C2×C22⋊C4
prime radicalC23×#67C22
cosocleC25×C3C23
hypercenterC2×C4×C22⋊C4D3
hypocenterC3#10960
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×C22⋊C4C2semidirect
4. #72127C2semidirect
5. C2×#4527C2semidirect
13. C22×(C2×D3)⋊C4C4semidirect
20. C2×C3⋊#95C4semidirect
23. C2×#806C4semidirect
27. C2×C4×C22⋊C4D3semidirect
112. C23×D3C42semidirect
163. C22×Dic3C42semidirect
167. C3×C22⋊C4C22⋊C4semidirect
288. C2×C3×C4#99semidirect
388. C22⋊C4(C2×D3)⋊C4semidirect
519. C2×C4#806semidirect
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×C4×C22⋊C42
C2×#52132
C22×#8062
#721272
C2×#45272
#109603