Finite Group Databasebeta

← #10691606

Subgroups — #10691606

Order 768.

924 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×D4×C22⋊C4C211yesmaximal, characteristic
2C2×#5218C211yesmaximal, characteristic
3#72128C211yesmaximal, characteristic
4#69510C211yesmaximal, characteristic, complemented
5#69457C211yesmaximal, characteristic, complemented
6#67625C211yesmaximal, characteristic, complemented
7#67617C211yesmaximal, characteristic, complemented
8#9255—31nomaximal, Sylow
9C23×C3×D4C2211yescharacteristic
10C2×C3×C4×D4—21no—
11C22×C3×C22⋊C4—21no—
12C22×#214C412yescomplemented
13C2×#924C2211yescharacteristic
14C3×#390—21no—
15C3×#386—22no—
16C3×#380C2211yescharacteristic
17C3×#379—21no—
18C3×C4×C22⋊C4C2211yescharacteristic
19#5218—22no—
20#5218—21no—
21#5218—21no—
22C2×#832—22no—
23C2×#832C412yescomplemented
24#5212C2211yescharacteristic
25C2×#830C2211yescharacteristic
26C2×#830C2211yescharacteristic
27#4527—21no—
28#4527—21no—
29#4452—22no—
30C2×#450—31no—
31D4×C22⋊C4D311yescharacteristic
32#1996—31no—
33#1786—31no—
34#1520—31no—
35#1289—31no—
36#1282—31no—
37C25×C3—21no—
38C22×C3×D4—21no—
39C22×C3×D4—21no—
40C22×C3×D4—21no—
41C22×C3×D4—44no—
42C22×C3×D4C2×C412yes—
43C22×C3×D4D412yes—
44C22×C3×D4D411yescharacteristic
45C22×C3×D4D411yescharacteristic
46C23×C3×C4—22no—
47C23×C3×C4C2311yescharacteristic
48C22×Dic6C2×C411yescharacteristic, complemented
49C3×C4×D4—44no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C22⋊C4—22no—
52C2×C3×C22⋊C4—22no—
53C2×C3×C22⋊C4—21no—
54C2×C3×C22⋊C4—22no—
55C2×C3×C22⋊C4—42no—
56C2×C3×C22⋊C4—42no—
57C2×C3×C22⋊C4D411yescharacteristic
58C2×C3×C22⋊C4D411yescharacteristic
59C2×C3×C42—21no—
60C2×#217—21no—
61C2×#217—21no—
62C2×#214—22no—
63C2×#214—22no—
64C2×#214—44no—
65C2×C4⋊Dic3—21no—
66C2×C4⋊Dic3C2×C411yescharacteristic, complemented
67#924—21no—
68#924—21no—
69#924—21no—
70#924—21no—
71C2×Dic3⋊C4—21no—
72C22×C3⋊C8—21no—
73C22×C3⋊C8C2×C412yescomplemented
74C3×#95—21no—
75C3×#95D412yes—
76#832—22no—
77#832—22no—
78#832—44no—
79C3⋊#95—21no—
80#830—21no—
81#830—21no—
82#830—21no—
83#830—21no—
84C23×D4C2×D311yescharacteristic
85C22×SD16—32no—
86C2×C22⋊Q8—31no—
87C2×C4×D4—21no—
88C22×C22⋊C4—21no—
89#450—62no—
90#450—61no—
91#450—61no—
92C2×#102—32no—
93C2×#102—62no—
94C2×C22⋊C8—31no—
95C2×C22⋊C8—31no—
96#390—21no—
97#386—22no—
98#380—31no—
99#380C2×D311yescharacteristic
100#379—21no—
101C4×C22⋊C4C2×D311yescharacteristic
102#342—61no—
103#342—61no—
104#328—62no—
105C24×C3—21no—
106C24×C3—21no—
107C24×C3—22no—
108C24×C3—22no—
109C24×C3—44no—
110C24×C3—41no—
111C24×C3—41no—
112C24×C3—41no—
113C24×C3—41no—
114C24×C3—82no—
115C24×C3C2×D411yescharacteristic
116C2×C3×D4—24no—
117C2×C3×D4—24no—
118C2×C3×D4—24no—
119C2×C3×D4—22no—
120C2×C3×D4—22no—
121C2×C3×D4—21no—
122C2×C3×D4—21no—
123C2×C3×D4—42no—
124C2×C3×D4—42no—
125C2×C3×D4—42no—
126C2×C3×D4—44no—
127C2×C3×D4—44no—
128C2×C3×D4—44no—
129C2×C3×D4C22⋊C412yes—
130C2×C3×D4C22⋊C412yes—
131C22×C3×C4—21no—
132C22×C3×C4—21no—
133C22×C3×C4—22no—
134C22×C3×C4—42no—
135C22×C3×C4—44no—
136C22×C3×C4—44no—
137C22×C3×C4—82no—
138C22×C3×C4C22×C411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4#4111yescharacteristic
143C22×C3×C4#4111yescharacteristic
144C22×Dic3—21no—
145C22×Dic3—21no—
146C22×Dic3—41no—
147C2×Dic6—21no—
148C2×Dic6—21no—
149C2×Dic6—42no—
150C3×C4⋊C4—42no—
151C3×C22⋊C4—21no—
152C3×C22⋊C4—21no—
153C3×C22⋊C4—21no—
154C3×C22⋊C4—44no—
155C3×C22⋊C4—44no—
156C3×C22⋊C4—41no—
157C3×C22⋊C4—41no—
158C3×C22⋊C4—82no—
159C3×C22⋊C4SD1612yescomplemented
160C3×C42—42no—
161#217—41no—
162#217—41no—
163#217—41no—
164#217—41no—
165#214—44no—
166#214—44no—
167C4⋊Dic3—21no—
168C4⋊Dic3—21no—
169C4⋊Dic3—42no—
170Dic3⋊C4—41no—
171Dic3⋊C4—41no—
172C2×C3⋊C8—22no—
173C2×C3⋊C8—42no—
174C2×C3⋊C8—42no—
175C2×C3⋊C8—41no—
176C25—21no—
177C22×Q8—31no—
178C22×D4—21no—
179C22×D4—21no—
180C22×D4—21no—
181C22×D4—44no—
182C22×D4C4×D312yes—
183C22×D4D1212yes—
184C22×D4#6711yescharacteristic
185C22×D4#6711yescharacteristic
186C23×C4—22no—
187C23×C4C22×D311yescharacteristic
188C2×SD16—62no—
189C2×SD16—62no—
190C2×SD16—124no—
191C22×C8—32no—
192C22×C8—61no—
193C22⋊Q8—61no—
194C22⋊Q8—61no—
195C22⋊Q8—61no—
196C22⋊Q8—61no—
197C4×D4—44no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—31no—
200C2×C4⋊C4—61no—
201C2×C4⋊C4—61no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—42no—
207C2×C22⋊C4—42no—
208C2×C22⋊C4—61no—
209C2×C22⋊C4—61no—
210C2×C22⋊C4#6711yescharacteristic
211C2×C22⋊C4#6711yescharacteristic
212C2×C42—21no—
213#102—62no—
214#102—62no—
215#102—124no—
216C22⋊C8—61no—
217C22⋊C8—61no—
218C22⋊C8—61no—
219C22⋊C8—61no—
220#95—21no—
221#95—61no—
222#95#6712yes—
223C23×C3—24no—
224C23×C3—22no—
225C23×C3—22no—
226C23×C3—41no—
227C23×C3—41no—
228C23×C3—44no—
229C23×C3—44no—
230C23×C3—42no—
231C23×C3—42no—
232C23×C3—42no—
233C23×C3—41no—
234C23×C3—41no—
235C23×C3—42no—
236C23×C3—42no—
237C23×C3—41no—
238C23×C3—41no—
239C23×C3—42no—
240C23×C3—42no—
241C23×C3—82no—
242C23×C3—82no—
243C23×C3—84no—
244C23×C3—84no—
245C23×C3—84no—
246C23×C3—84no—
247C23×C3C4×D411yescharacteristic
248C23×C3#12011yescharacteristic
249C23×C3#12111yescharacteristic
250C23×C3C2×SD1611yescharacteristic
251C23×C3#13511yescharacteristic
252C23×C3AGL(1,ℤ/8)11yescharacteristic
253C23×C3#13711yescharacteristic
254C3×D4—24no—
255C3×D4—22no—
256C3×D4—22no—
257C3×D4—22no—
258C3×D4—44no—
259C3×D4—44no—
260C3×D4—42no—
261C3×D4#9914yes—
262C2×C3×C4—22no—
263C2×C3×C4—22no—
264C2×C3×C4—22no—
265C2×C3×C4—22no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—41no—
270C2×C3×C4—41no—
271C2×C3×C4—41no—
272C2×C3×C4—42no—
273C2×C3×C4—41no—
274C2×C3×C4—42no—
275C2×C3×C4—42no—
276C2×C3×C4—41no—
277C2×C3×C4—88no—
278C2×C3×C4C2×C22⋊C411yescharacteristic
279C2×C3×C4C4×D411yescharacteristic
280C2×C3×C4#12111yescharacteristic
281C2×C3×C4#12311yescharacteristic
282C2×Dic3—41no—
283C2×Dic3—41no—
284C2×Dic3—81no—
285C2×Dic3—81no—
286C2×Dic3—81no—
287C2×Dic3—81no—
288C2×Dic3—81no—
289Dic6—42no—
290Dic6—81no—
291C3⋊C8—42no—
292C3⋊C8—81no—
293C24—21no—
294C24—21no—
295C24—22no—
296C24—22no—
297C24—44no—
298C24—41no—
299C24—41no—
300C24—41no—
301C24—41no—
302C24—82no—
303C24C2×#6711yescharacteristic
304C2×Q8—61no—
305C2×Q8—61no—
306C2×Q8—122no—
307C2×D4—24no—
308C2×D4—24no—
309C2×D4—24no—
310C2×D4—22no—
311C2×D4—21no—
312C2×D4—21no—
313C2×D4—22no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—42no—
317C2×D4—44no—
318C2×D4—44no—
319C2×D4—44no—
320C2×D4(C2×D3)⋊C412yes—
321C2×D4(C2×D3)⋊C412yes—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—42no—
326C22×C4—44no—
327C22×C4—44no—
328C22×C4—61no—
329C22×C4—61no—
330C22×C4—82no—
331C22×C4—121no—
332C22×C4C2×C4×D311yescharacteristic
333C22×C4C2×D1211yescharacteristic
334C22×C4#23511yescharacteristic
335C22×C4#23511yescharacteristic
336C22×C4C2×#6711yescharacteristic
337C22×C4C2×#6711yescharacteristic
338SD16—124no—
339SD16—124no—
340C2×C8—62no—
341C2×C8—122no—
342C2×C8—122no—
343C2×C8—121no—
344C4⋊C4—42no—
345C4⋊C4—61no—
346C4⋊C4—61no—
347C4⋊C4—122no—
348C4⋊C4—121no—
349C4⋊C4—121no—
350C22⋊C4—21no—
351C22⋊C4—21no—
352C22⋊C4—21no—
353C22⋊C4—44no—
354C22⋊C4—44no—
355C22⋊C4—41no—
356C22⋊C4—41no—
357C22⋊C4—82no—
358C22⋊C4—121no—
359C22⋊C4—121no—
360C22⋊C4—121no—
361C22⋊C4—121no—
362C22⋊C4#21412yescomplemented
363C42—42no—
364C22×C3—21no—
365C22×C3—21no—
366C22×C3—21no—
367C22×C3—21no—
368C22×C3—21no—
369C22×C3—21no—
370C22×C3—24no—
371C22×C3—41no—
372C22×C3—41no—
373C22×C3—41no—
374C22×C3—41no—
375C22×C3—44no—
376C22×C3—44no—
377C22×C3—42no—
378C22×C3—42no—
379C22×C3—42no—
380C22×C3—42no—
381C22×C3—42no—
382C22×C3—84no—
383C22×C3—84no—
384C22×C3—84no—
385C22×C3—84no—
386C22×C3—82no—
387C22×C3—82no—
388C22×C3#38611yescharacteristic
389C22×C3C4×SD1611yescharacteristic
390C22×C3#44011yescharacteristic
391C22×C3#45011yescharacteristic
392C22×C3#45211yescharacteristic
393C22×C3#46111yescharacteristic
394C22×C3#46311yescharacteristic
395C3×C4—21no—
396C3×C4—21no—
397C3×C4—21no—
398C3×C4—42no—
399C3×C4—81no—
400C3×C4—82no—
401C3×C4C2×#9912yes—
402Dic3—81no—
403Dic3—81no—
404Dic3—161no—
405C23—24no—
406C23—22no—
407C23—22no—
408C23—41no—
409C23—41no—
410C23—44no—
411C23—44no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—82no—
424C23—82no—
425C23—84no—
426C23—84no—
427C23—84no—
428C23—84no—
429C23C4×#6711yescharacteristic
430C23#93011yescharacteristic
431C23#93211yescharacteristic
432C23C2×#21411yescharacteristic
433C23#95011yescharacteristic
434C23#95111yescharacteristic
435C23#95311yescharacteristic
436Q8—122no—
437Q8—241no—
438D4—24no—
439D4—22no—
440D4—22no—
441D4—22no—
442D4—44no—
443D4—44no—
444D4—42no—
445D4#80614yes—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—41no—
461C2×C4—88no—
462C2×C4—121no—
463C2×C4—121no—
464C2×C4—241no—
465C2×C4—241no—
466C2×C4—241no—
467C2×C4—241no—
468C2×C4—241no—
469C2×C4C2×(C2×D3)⋊C411yescharacteristic
470C2×C4C4×#6711yescharacteristic
471C2×C4#92911yescharacteristic
472C2×C4#93011yescharacteristic
473C8—122no—
474C8—241no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—21no—
480C2×C3—21no—
481C2×C3—44no—
482C2×C3—82no—
483C2×C3#187711yescharacteristic
484C2×C3#189811yescharacteristic
485C2×C3#190411yescharacteristic
486C22—21no—
487C22—21no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—24no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22—44no—
498C22—44no—
499C22—42no—
500C22—42no—
501C22—42no—
502C22—42no—
503C22—42no—
504C22—84no—
505C22—84no—
506C22—84no—
507C22—84no—
508C22—82no—
509C22—82no—
510C22#501611yescharacteristic
511C22C4×#21411yescharacteristic
512C22#501811yescharacteristic
513C22#501911yescharacteristic
514C22#521311yescharacteristic
515C22#521811yescharacteristic
516C22#523911yescharacteristic
517C4—21no—
518C4—21no—
519C4—21no—
520C4—42no—
521C4—82no—
522C4—81no—
523C4—241no—
524C4—241no—
525C4—481no—
526C4C2×#80612yes—
527C3#925511yescharacteristic, complemented, Sylow
528C2—21no—
529C2—21no—
530C2—21no—
531C2—21no—
532C2—21no—
533C2—21no—
534C2—44no—
535C2—82no—
536C2#7154311yescharacteristic
537C2#7161211yescharacteristic
538C2#7165111yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×D4×C22⋊C4C2
socleC22×C3#386
prime radicalC23×C3×D4C22
cosocleC23×C3×C4C23
hypercenterD4×C22⋊C4D3
hypocenterC3#9255
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ D4×C22⋊C4

Complements

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

NH ≅ G/Ntype
4. #69510C2semidirect
5. #69457C2semidirect
6. #67625C2semidirect
7. #67617C2semidirect
12. C22×#214C4semidirect
23. C2×#832C4semidirect
48. C22×Dic6C2×C4semidirect
66. C2×C4⋊Dic3C2×C4semidirect
73. C22×C3⋊C8C2×C4semidirect
159. C3×C22⋊C4SD16semidirect
362. C22⋊C4#214semidirect
527. C3#9255semidirect

Maximal subgroups

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

subgroupindex
C3×D4×C22⋊C42
C2×#52182
#721282
#695102
#694572
#676252
#676172
#92553