Finite Group Databasebeta

← #10690762

Subgroups — #10690762

Order 768.

952 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#407C211yesmaximal, characteristic
2#84031C211yesmaximal, characteristic
3#72127C211yesmaximal, characteristic
4C2×#4467C212yesmaximal, complemented
5#67675C212yesmaximal, complemented
6#9115—31nomaximal, Sylow
7C24×C3×C4C2211yescharacteristic
8C22×C3×M16—21no—
9C22×#217—21no—
10C2×#924C411yescharacteristic, complemented
11C2×#924C411yescharacteristic, complemented
12C2×#924C2211yescharacteristic
13C3×#407—24no—
14C2×C3×C22⋊C8—21no—
15C2×C3×C22⋊C8C2212yes—
16#5247—21no—
17#5212C411yescharacteristic, complemented
18#5212C411yescharacteristic, complemented
19#5212C2211yescharacteristic
20#5210—21no—
21#5210—21no—
22#5210—21no—
23C2×C3⋊#95—21no—
24C2×C3⋊#95C2212yes—
25#4467—24no—
26C2×#407D311yescharacteristic
27#2848—31no—
28C2×#328—32no—
29C22⋊#95—31no—
30#1294—32no—
31C25×C3D411yescharacteristic
32C23×C3×C4—22no—
33C23×C3×C4—22no—
34C23×C3×C4—42no—
35C23×C3×C4—42no—
36C23×C3×C4C2×C411yescharacteristic
37C23×C3×C4C2×C411yescharacteristic
38C23×C3×C4D411yescharacteristic
39C23×C3×C4D411yescharacteristic
40C23×C3×C4D411yescharacteristic
41C23×C3×C4C2311yescharacteristic
42C23×Dic3—22no—
43C22×Dic6C2×C411yescharacteristic
44C2×C3×M16—24no—
45C2×C3×M16—42no—
46C2×C3×M16—42no—
47C22×C3×C8—22no—
48C2×#217—24no—
49C2×#217—21no—
50C2×#217—21no—
51C2×#217—21no—
52C2×#217—21no—
53C2×#217—21no—
54C2×#217—21no—
55C2×#217—42no—
56C2×#217—42no—
57C2×C4⋊Dic3C2×C411yescharacteristic
58C2×C4⋊Dic3C2×C411yescharacteristic
59C2×C4⋊Dic3C2×C411yescharacteristic
60#924—22no—
61#924—22no—
62#924—42no—
63#924—42no—
64C2×Dic3⋊C4—21no—
65C2×Dic3⋊C4—21no—
66C2×Dic3⋊C4—21no—
67C3×C22⋊C8—24no—
68C3×C22⋊C8—22no—
69C3×C22⋊C8D414yes—
70C3⋊#95—21no—
71C3⋊#95—21no—
72C3⋊#95—21no—
73C3⋊#95—22no—
74C3⋊#95—22no—
75C3⋊#95—42no—
76C24×C4C2×D311yescharacteristic
77C22×M16—21no—
78C2×C22⋊Q8—31no—
79C2×C22⋊Q8—31no—
80C2×C22⋊Q8—31no—
81C22×C22⋊C4—61no—
82#407—24no—
83C2×C22⋊C8—21no—
84C2×C22⋊C8C2×D312yes—
85#385—61no—
86#385—61no—
87#385—61no—
88#380—31no—
89#380—31no—
90#380—31no—
91#379—61no—
92C2×#95—32no—
93C2×#95—61no—
94#328—64no—
95C24×C3—22no—
96C24×C3—22no—
97C24×C3—21no—
98C24×C3—21no—
99C24×C3—22no—
100C24×C3—22no—
101C24×C3—41no—
102C24×C3—41no—
103C24×C3C22⋊C411yescharacteristic
104C24×C3C22⋊C411yescharacteristic
105C24×C3C2×D411yescharacteristic
106C22×C3×C4—22no—
107C22×C3×C4—22no—
108C22×C3×C4—21no—
109C22×C3×C4—21no—
110C22×C3×C4—22no—
111C22×C3×C4—22no—
112C22×C3×C4—22no—
113C22×C3×C4—22no—
114C22×C3×C4—22no—
115C22×C3×C4—22no—
116C22×C3×C4—21no—
117C22×C3×C4—21no—
118C22×C3×C4—41no—
119C22×C3×C4—41no—
120C22×C3×C4—41no—
121C22×C3×C4—41no—
122C22×C3×C4—41no—
123C22×C3×C4—41no—
124C22×C3×C4—41no—
125C22×C3×C4—48no—
126C22×C3×C4—48no—
127C22×C3×C4C22⋊C411yescharacteristic
128C22×C3×C4C22⋊C411yescharacteristic
129C22×C3×C4C22×C411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D412yes—
132C22×C3×C4#4112yes—
133C22×Dic3—24no—
134C22×Dic3—21no—
135C22×Dic3—21no—
136C22×Dic3—21no—
137C22×Dic3—21no—
138C22×Dic3—42no—
139C22×Dic3—42no—
140C22×Dic3—42no—
141C22×Dic3—42no—
142C22×Dic3—42no—
143C22×Dic3—42no—
144C2×Dic6—22no—
145C2×Dic6—41no—
146C2×Dic6—41no—
147C3×M16—48no—
148C2×C3×C8—24no—
149C2×C3×C8—42no—
150C2×C3×C8—42no—
151#217—42no—
152#217—42no—
153#217—48no—
154#217—42no—
155#217—42no—
156#217—42no—
157#217—41no—
158#217—41no—
159C4⋊Dic3—22no—
160C4⋊Dic3—41no—
161C4⋊Dic3—41no—
162Dic3⋊C4—42no—
163Dic3⋊C4—42no—
164Dic3⋊C4—41no—
165Dic3⋊C4—41no—
166C25D1211yescharacteristic
167C22×Q8—31no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—42no—
171C23×C4—42no—
172C23×C4—62no—
173C23×C4C4×D311yescharacteristic
174C23×C4C4×D311yescharacteristic
175C23×C4D1211yescharacteristic
176C23×C4#6711yescharacteristic
177C23×C4#6711yescharacteristic
178C23×C4C22×D311yescharacteristic
179C2×M16—24no—
180C2×M16—42no—
181C2×M16—42no—
182C22×C8—22no—
183C22⋊Q8—62no—
184C22⋊Q8—62no—
185C22⋊Q8—122no—
186C22⋊Q8—122no—
187C2×C4⋊C4—31no—
188C2×C4⋊C4—31no—
189C2×C4⋊C4—31no—
190C2×C4⋊C4—61no—
191C2×C4⋊C4—61no—
192C2×C4⋊C4—61no—
193C2×C22⋊C4—64no—
194C2×C22⋊C4—61no—
195C2×C22⋊C4—61no—
196C2×C22⋊C4—61no—
197C2×C22⋊C4—61no—
198C2×C22⋊C4—61no—
199C2×C22⋊C4—61no—
200C2×C22⋊C4—122no—
201C2×C22⋊C4—122no—
202C22⋊C8—24no—
203C22⋊C8—22no—
204C22⋊C8#6714yes—
205#95—61no—
206#95—61no—
207#95—61no—
208#95—62no—
209#95—62no—
210#95—122no—
211C23×C3—21no—
212C23×C3—21no—
213C23×C3—21no—
214C23×C3—21no—
215C23×C3—24no—
216C23×C3—24no—
217C23×C3—21no—
218C23×C3—21no—
219C23×C3—21no—
220C23×C3—21no—
221C23×C3—21no—
222C23×C3—21no—
223C23×C3—21no—
224C23×C3—21no—
225C23×C3—21no—
226C23×C3—21no—
227C23×C3—21no—
228C23×C3—21no—
229C23×C3—44no—
230C23×C3—44no—
231C23×C3—44no—
232C23×C3—44no—
233C23×C3—42no—
234C23×C3—42no—
235C23×C3—42no—
236C23×C3—42no—
237C23×C3#9912yes—
238C23×C3#10412yes—
239C23×C3C2×C22⋊C411yescharacteristic
240C23×C3#12112yes—
241C23×C3AGL(1,ℤ/8)11yescharacteristic
242C23×C3#13711yescharacteristic
243C23×C3#13712yes—
244C2×C3×C4—24no—
245C2×C3×C4—24no—
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—42no—
256C2×C3×C4—42no—
257C2×C3×C4—44no—
258C2×C3×C4—44no—
259C2×C3×C4—44no—
260C2×C3×C4—42no—
261C2×C3×C4—42no—
262C2×C3×C4#9912yes—
263C2×C3×C4C4×D412yes—
264C2×C3×C4#12011yescharacteristic
265C2×C3×C4#12311yescharacteristic
266C2×Dic3—44no—
267C2×Dic3—44no—
268C2×Dic3—44no—
269C2×Dic3—41no—
270C2×Dic3—41no—
271C2×Dic3—41no—
272C2×Dic3—41no—
273C2×Dic3—41no—
274C2×Dic3—41no—
275C2×Dic3—41no—
276C2×Dic3—41no—
277C2×Dic3—41no—
278C2×Dic3—41no—
279C2×Dic3—41no—
280C2×Dic3—41no—
281C2×Dic3—82no—
282C2×Dic3—82no—
283C2×Dic3—82no—
284C2×Dic3—82no—
285Dic6—82no—
286C3×C8—44no—
287C24—22no—
288C24—22no—
289C24—21no—
290C24—21no—
291C24—22no—
292C24—22no—
293C24—41no—
294C24—41no—
295C24(C2×D3)⋊C411yescharacteristic
296C24(C2×D3)⋊C411yescharacteristic
297C24C2×D1211yescharacteristic
298C2×Q8—62no—
299C2×Q8—121no—
300C2×Q8—121no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—48no—
321C22×C4—48no—
322C22×C4—64no—
323C22×C4—61no—
324C22×C4—61no—
325C22×C4—61no—
326C22×C4—61no—
327C22×C4—122no—
328C22×C4—122no—
329C22×C4—122no—
330C22×C4—122no—
331C22×C4—122no—
332C22×C4—122no—
333C22×C4(C2×D3)⋊C411yescharacteristic
334C22×C4(C2×D3)⋊C411yescharacteristic
335C22×C4C2×C4×D311yescharacteristic
336C22×C4#23512yes—
337C22×C4C2×#6711yescharacteristic
338C22×C4C2×#6712yes—
339M16—48no—
340C2×C8—24no—
341C2×C8—42no—
342C2×C8—42no—
343C4⋊C4—62no—
344C4⋊C4—121no—
345C4⋊C4—121no—
346C4⋊C4—122no—
347C4⋊C4—122no—
348C4⋊C4—121no—
349C4⋊C4—121no—
350C22⋊C4—128no—
351C22⋊C4—122no—
352C22⋊C4—122no—
353C22⋊C4—122no—
354C22⋊C4—122no—
355C22⋊C4—121no—
356C22⋊C4—121no—
357C22⋊C4—122no—
358C22×C3—22no—
359C22×C3—22no—
360C22×C3—21no—
361C22×C3—21no—
362C22×C3—22no—
363C22×C3—22no—
364C22×C3—21no—
365C22×C3—21no—
366C22×C3—22no—
367C22×C3—22no—
368C22×C3—22no—
369C22×C3—22no—
370C22×C3—22no—
371C22×C3—22no—
372C22×C3—48no—
373C22×C3—48no—
374C22×C3—41no—
375C22×C3—41no—
376C22×C3—41no—
377C22×C3—41no—
378C22×C3—41no—
379C22×C3—41no—
380C22×C3—41no—
381C22×C3—41no—
382C22×C3#38611yescharacteristic
383C22×C3C2×#9911yescharacteristic
384C22×C3C2×#9911yescharacteristic
385C22×C3#41711yescharacteristic
386C22×C3#41911yescharacteristic
387C22×C3C2×#10411yescharacteristic
388C22×C3#42111yescharacteristic
389C22×C3#46812yes—
390C22×C3#47012yes—
391C3×C4—21no—
392C3×C4—22no—
393C3×C4—22no—
394C3×C4—21no—
395C3×C4—41no—
396Dic3—84no—
397Dic3—81no—
398Dic3—81no—
399Dic3—81no—
400Dic3—81no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—24no—
406C23—24no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—44no—
420C23—44no—
421C23—44no—
422C23—44no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23#80612yes—
428C23#82512yes—
429C23#90811yescharacteristic
430C23#90912yes—
431C23#90911yescharacteristic
432C23C2×(C2×D3)⋊C411yescharacteristic
433C23#93012yes—
434Q8—242no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—124no—
454C2×C4—124no—
455C2×C4—124no—
456C2×C4—121no—
457C2×C4—121no—
458C2×C4—121no—
459C2×C4—121no—
460C2×C4—121no—
461C2×C4—121no—
462C2×C4—121no—
463C2×C4—121no—
464C2×C4—121no—
465C2×C4—121no—
466C2×C4—121no—
467C2×C4—121no—
468C2×C4—242no—
469C2×C4—242no—
470C2×C4—242no—
471C2×C4—242no—
472C2×C4#80612yes—
473C2×C4C4×#6712yes—
474C2×C4#92911yescharacteristic
475C2×C4#95311yescharacteristic
476C8—44no—
477C2×C3—22no—
478C2×C3—22no—
479C2×C3—42no—
480C2×C3—42no—
481C2×C3#150411yescharacteristic
482C2×C3#150811yescharacteristic
483C2×C3#151111yescharacteristic
484C2×C3#151311yescharacteristic
485C2×C3#187111yescharacteristic
486C2×C3#189411yescharacteristic
487C2×C3#189811yescharacteristic
488C22—22no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—48no—
503C22—48no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22C2×#80611yescharacteristic
513C22C2×#80611yescharacteristic
514C22#512011yescharacteristic
515C22#513311yescharacteristic
516C22#513412yes—
517C22#513712yes—
518C22C2×#82511yescharacteristic
519C22#513911yescharacteristic
520C22#521311yescharacteristic
521C4—21no—
522C4—22no—
523C4—21no—
524C4—22no—
525C4—41no—
526C4—244no—
527C4—241no—
528C4—241no—
529C4—241no—
530C4—241no—
531C3#911511yescharacteristic, complemented, Sylow
532C2—22no—
533C2—22no—
534C2—42no—
535C2—42no—
536C2#6997611yescharacteristic
537C2#6997911yescharacteristic
538C2#6998011yescharacteristic
539C2#6998111yescharacteristic
540C2#7154311yescharacteristic
541C2#7198411yescharacteristic
542C2#7198911yescharacteristic
subgroupG/H
centerC23C2×(C2×D3)⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#407C2
socleC23×C3C2×C22⋊C4
prime radicalC25×C3D4
cosocleC23×C3×C4C23
hypercenterC2×#407D3
hypocenterC3#9115
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#407

Complements

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

NH ≅ G/Ntype
4. C2×#4467C2semidirect
5. #67675C2semidirect
10. C2×#924C4semidirect
11. C2×#924C4semidirect
17. #5212C4semidirect
18. #5212C4semidirect
531. C3#9115semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4072
#840312
#721272
C2×#44672
#676752
#91153