Finite Group Databasebeta

← #10690934

Subgroups — #10690934

Order 768.

1088 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82723C211yesmaximal, characteristic
2C2×#4666C211yesmaximal, characteristic, complemented
3C2×C3×#327C211yesmaximal, characteristic, complemented
4C2×#4537C211yesmaximal, characteristic, complemented
5#69955C211yesmaximal, characteristic, complemented
6#67675C211yesmaximal, characteristic, complemented
7#67612C211yesmaximal, characteristic, complemented
8#9150—31nomaximal, Sylow
9C23×C4×D3C2211yescharacteristic
10C2×C3×#121C2211yescharacteristic
11C2×#938C412yescomplemented
12C2×#937—21no—
13C22×(C2×D3)⋊C4—21no—
14C22×C4×Dic3—21no—
15C2×C3×C22⋊C8C2211yescharacteristic
16C2×C3×#95C2211yescharacteristic
17#5220—21no—
18#5212C2211yescharacteristic
19C2×C3⋊#95C2211yescharacteristic
20C2×#830C2211yescharacteristic
21C2×(C2×D3)⋊C8—22no—
22#4987C412yescomplemented
23#4956—22no—
24#4666—22no—
25#4666—22no—
26C3×#327—22no—
27#4537—22no—
28#2847—31no—
29C2×C4×C22⋊C4—31no—
30#1779—31no—
31C2×#327—31no—
32C2×#327D311yescharacteristic, complemented
33#1294—31no—
34#1294—31no—
35C24×D3D411yescharacteristic
36C22×C3×D4—21no—
37C22×C3×D4C2×C411yescharacteristic
38C23×C3×C4—21no—
39C23×C3×C4C2311yescharacteristic
40C22×#67—22no—
41C23×Dic3—21no—
42C23×Dic3D411yescharacteristic
43C22×C4×D3—21no—
44C22×C4×D3—21no—
45C22×C4×D3—21no—
46C22×C4×D3—21no—
47C22×C4×D3—44no—
48C22×C4×D3C2×C412yes—
49C22×C4×D3D412yes—
50C22×C3×C8—21no—
51C3×#121—21no—
52C3×#121—21no—
53C3×#121—21no—
54C3×#121—21no—
55C2×C3×C4⋊C4C2×C411yescharacteristic
56C2×C3×C22⋊C4—21no—
57C2×#217—21no—
58C2×#217—22no—
59#938—44no—
60#937—42no—
61#937—42no—
62C2×(C2×D3)⋊C4—24no—
63C2×(C2×D3)⋊C4—28no—
64C2×(C2×D3)⋊C4—22no—
65C2×C4⋊Dic3C2×C412yes—
66C2×C4×Dic3—24no—
67C2×C4×Dic3—44no—
68C22×C3⋊C8—21no—
69C3×C22⋊C8—21no—
70C3×C22⋊C8—21no—
71C3×#95—21no—
72C3×#95D412yescomplemented
73C3⋊#95—21no—
74C3⋊#95—21no—
75C3⋊#95—22no—
76C3⋊#95D412yescomplemented
77#830—21no—
78#830—21no—
79(C2×D3)⋊C8—44no—
80C24×C4—31no—
81C2×#121—32no—
82C2×#121C2×D311yescharacteristic
83C2×#120—61no—
84C22×C22⋊C4—61no—
85C22×C42—61no—
86C2×C22⋊C8—31no—
87C2×C22⋊C8—62no—
88C2×C22⋊C8C2×D311yescharacteristic
89#386—62no—
90#386—61no—
91#380—31no—
92#380—32no—
93C4×C22⋊C4—62no—
94C4×C22⋊C4—62no—
95C2×#95—31no—
96C2×#95C2×D311yescharacteristic
97#327—22no—
98#327—62no—
99C24×C3—21no—
100C24×C3C2×D411yescharacteristic
101C23×D3—21no—
102C23×D3—21no—
103C23×D3—24no—
104C23×D3—21no—
105C23×D3—21no—
106C23×D3—21no—
107C23×D3—21no—
108C23×D3—42no—
109C23×D3C22⋊C412yes—
110C2×C3×D4—21no—
111C2×C3×D4—21no—
112C2×C3×D4—42no—
113C2×C3×D4—41no—
114C2×C3×D4—41no—
115C2×C3×D4—41no—
116C2×C3×D4—41no—
117C2×C3×D4—41no—
118C2×C3×D4—41no—
119C22×C3×C4—24no—
120C22×C3×C4—22no—
121C22×C3×C4—21no—
122C22×C3×C4—42no—
123C22×C3×C4—42no—
124C22×C3×C4—41no—
125C22×C3×C4—41no—
126C22×C3×C4C22×C411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4#4111yescharacteristic
131C22×C3×C4#4111yescharacteristic
132C2×#67—44no—
133C2×#67—44no—
134C2×#67—44no—
135C22×Dic3—21no—
136C22×Dic3—21no—
137C22×Dic3—24no—
138C22×Dic3—24no—
139C22×Dic3—22no—
140C22×Dic3—22no—
141C22×Dic3—42no—
142C22×Dic3—41no—
143C22×Dic3—41no—
144C22×Dic3C22⋊C412yes—
145C2×C4×D3—22no—
146C2×C4×D3—22no—
147C2×C4×D3—22no—
148C2×C4×D3—22no—
149C2×C4×D3—24no—
150C2×C4×D3—24no—
151C2×C4×D3—42no—
152C2×C4×D3—42no—
153C2×C4×D3—44no—
154C2×C4×D3—44no—
155C2×C4×D3—44no—
156C2×C4×D3—44no—
157C2×C3×C8—42no—
158C2×C3×C8—41no—
159C3×C4⋊C4—21no—
160C3×C4⋊C4—21no—
161C3×C22⋊C4—41no—
162C3×C22⋊C4—41no—
163#217—41no—
164#217—41no—
165#217—44no—
166(C2×D3)⋊C4—216no—
167(C2×D3)⋊C4—44no—
168C4⋊Dic3—42no—
169C4×Dic3—48no—
170C2×C3⋊C8—42no—
171C2×C3⋊C8—41no—
172C25—31no—
173C22×D4—21no—
174C22×D4—62no—
175C22×D4C4×D311yescharacteristic
176C23×C4—21no—
177C23×C4—31no—
178C23×C4—32no—
179C23×C4—32no—
180C23×C4—61no—
181C23×C4—61no—
182C23×C4—61no—
183C23×C4—61no—
184C23×C4—61no—
185C23×C4—124no—
186C23×C4C22×D311yescharacteristic
187C22×C8—21no—
188C22×C8—61no—
189#121—21no—
190#121—21no—
191#121—21no—
192#121—21no—
193#121—124no—
194#120—122no—
195#120—122no—
196C2×C4⋊C4—32no—
197C2×C4⋊C4C4×D311yescharacteristic
198C2×C22⋊C4—21no—
199C2×C22⋊C4—64no—
200C2×C22⋊C4—68no—
201C2×C22⋊C4—62no—
202C2×C22⋊C4—62no—
203C2×C22⋊C4—61no—
204C2×C42—64no—
205C2×C42—124no—
206C22⋊C8—21no—
207C22⋊C8—21no—
208C22⋊C8—61no—
209C22⋊C8—61no—
210C22⋊C8—124no—
211#95—21no—
212#95—32no—
213#95—62no—
214#95—61no—
215#95—61no—
216#95#6712yescomplemented
217C23×C3—22no—
218C23×C3—22no—
219C23×C3—41no—
220C23×C3—42no—
221C23×C3—41no—
222C23×C3—41no—
223C23×C3—42no—
224C23×C3C2×C22⋊C411yescharacteristic
225C23×C3#12011yescharacteristic
226C23×C3#12311yescharacteristic
227C23×C3#13511yescharacteristic
228C23×C3#13511yescharacteristic
229C23×C3AGL(1,ℤ/8)11yescharacteristic
230C23×C3AGL(1,ℤ/8)11yescharacteristic
231C22×D3—24no—
232C22×D3—24no—
233C22×D3—22no—
234C22×D3—22no—
235C22×D3—22no—
236C22×D3—22no—
237C22×D3—22no—
238C22×D3—22no—
239C22×D3—44no—
240C22×D3—44no—
241C22×D3—44no—
242C22×D3—44no—
243C22×D3—42no—
244C22×D3—42no—
245C22×D3—41no—
246C22×D3—41no—
247C22×D3—41no—
248C22×D3—41no—
249C22×D3#10414yes—
250C3×D4—81no—
251C3×D4—81no—
252C3×D4—81no—
253C3×D4—81no—
254C3×D4—81no—
255C3×D4—81no—
256C2×C3×C4—22no—
257C2×C3×C4—22no—
258C2×C3×C4—24no—
259C2×C3×C4—41no—
260C2×C3×C4—41no—
261C2×C3×C4—41no—
262C2×C3×C4—41no—
263C2×C3×C4—42no—
264C2×C3×C4—44no—
265C2×C3×C4—44no—
266C2×C3×C4—42no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—41no—
270C2×C3×C4C4×D411yescharacteristic
271C2×C3×C4C4×D411yescharacteristic
272C2×C3×C4#12111yescharacteristic
273C2×C3×C4#12111yescharacteristic
274#67—88no—
275C2×Dic3—24no—
276C2×Dic3—24no—
277C2×Dic3—22no—
278C2×Dic3—22no—
279C2×Dic3—24no—
280C2×Dic3—42no—
281C2×Dic3—44no—
282C2×Dic3—44no—
283C2×Dic3—42no—
284C2×Dic3—42no—
285C2×Dic3—42no—
286C2×Dic3—42no—
287C2×Dic3#9914yes—
288C4×D3—44no—
289C4×D3—44no—
290C4×D3—44no—
291C4×D3—44no—
292C3×C8—81no—
293C3⋊C8—81no—
294C24—21no—
295C24—32no—
296C24—64no—
297C24—61no—
298C24—61no—
299C24—61no—
300C24—61no—
301C24—61no—
302C24—61no—
303C24—122no—
304C24D3×D411yescharacteristic
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—42no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—124no—
315C2×D4—124no—
316C2×D4—124no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—32no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—64no—
326C22×C4—64no—
327C22×C4—64no—
328C22×C4—61no—
329C22×C4—61no—
330C22×C4—64no—
331C22×C4—62no—
332C22×C4—62no—
333C22×C4—62no—
334C22×C4—62no—
335C22×C4—62no—
336C22×C4—62no—
337C22×C4—122no—
338C22×C4—122no—
339C22×C4—124no—
340C22×C4—124no—
341C22×C4—124no—
342C22×C4—124no—
343C22×C4—122no—
344C22×C4—121no—
345C22×C4—121no—
346C22×C4C2×C4×D311yescharacteristic
347C22×C4#23511yescharacteristic
348C22×C4D3×D411yescharacteristic
349C22×C4D3×D411yescharacteristic
350C22×C4#23711yescharacteristic
351C22×C4C2×#6711yescharacteristic
352C2×C8—42no—
353C2×C8—41no—
354C2×C8—122no—
355C2×C8—121no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—122no—
359C22⋊C4—41no—
360C22⋊C4—41no—
361C22⋊C4—616no—
362C22⋊C4—124no—
363C22⋊C4—124no—
364C22⋊C4—121no—
365C22⋊C4—121no—
366C42—128no—
367C22×C3—21no—
368C22×C3—21no—
369C22×C3—21no—
370C22×C3—21no—
371C22×C3—22no—
372C22×C3—22no—
373C22×C3—22no—
374C22×C3—22no—
375C22×C3—81no—
376C22×C3—81no—
377C22×C3—81no—
378C22×C3—82no—
379C22×C3—82no—
380C22×C3#38611yescharacteristic
381C22×C3C2×#9912yes—
382C22×C3#41611yescharacteristic
383C22×C3#41811yescharacteristic
384C22×C3C2×#10412yes—
385C22×C3#44911yescharacteristic
386C22×C3#44911yescharacteristic
387C22×C3#48211yescharacteristic
388C22×C3#48211yescharacteristic
389C2×D3—24no—
390C2×D3—24no—
391C2×D3—24no—
392C2×D3—44no—
393C2×D3—44no—
394C2×D3—44no—
395C2×D3—44no—
396C2×D3—42no—
397C2×D3—42no—
398C2×D3—42no—
399C2×D3—42no—
400C3×C4—21no—
401C3×C4—21no—
402C3×C4—21no—
403C3×C4—21no—
404C3×C4—44no—
405C3×C4—81no—
406Dic3—24no—
407Dic3—44no—
408Dic3—82no—
409C23—22no—
410C23—22no—
411C23—34no—
412C23—41no—
413C23—42no—
414C23—41no—
415C23—41no—
416C23—42no—
417C23—64no—
418C23—64no—
419C23—62no—
420C23—62no—
421C23—62no—
422C23—62no—
423C23—62no—
424C23—62no—
425C23—124no—
426C23—124no—
427C23—124no—
428C23—124no—
429C23—122no—
430C23—122no—
431C23—121no—
432C23—121no—
433C23—121no—
434C23—121no—
435C23D3×C22⋊C411yescharacteristic
436C23#88311yescharacteristic
437C23#91111yescharacteristic
438C23#91211yescharacteristic
439C23#91411yescharacteristic
440C23#91611yescharacteristic
441C23#93711yescharacteristic
442D4—81no—
443D4—81no—
444D4—81no—
445D4—81no—
446D4—81no—
447D4—81no—
448D4—248no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—34no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—64no—
465C2×C4—64no—
466C2×C4—62no—
467C2×C4—62no—
468C2×C4—64no—
469C2×C4—124no—
470C2×C4—124no—
471C2×C4—124no—
472C2×C4—124no—
473C2×C4—122no—
474C2×C4—124no—
475C2×C4—124no—
476C2×C4—122no—
477C2×C4—122no—
478C2×C4—122no—
479C2×C4—122no—
480C2×C4#67⋊C411yescharacteristic
481C2×C4#88411yescharacteristic
482C2×C4C4×#6711yescharacteristic
483C2×C4#93811yescharacteristic
484C8—81no—
485C8—241no—
486C2×C3—21no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—161no—
491C2×C3#150112yes—
492C2×C3#150712yes—
493C2×C3#187511yescharacteristic
494C2×C3#189611yescharacteristic
495C2×C3#189811yescharacteristic
496D3—44no—
497C22—21no—
498C22—21no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—22no—
503C22—21no—
504C22—21no—
505C22—64no—
506C22—64no—
507C22—64no—
508C22—81no—
509C22—81no—
510C22—82no—
511C22—82no—
512C22—81no—
513C22—124no—
514C22—124no—
515C22—124no—
516C22—124no—
517C22—122no—
518C22—122no—
519C22—122no—
520C22—122no—
521C22D3×#9912yes—
522C22#477011yescharacteristic
523C22#477211yescharacteristic
524C22#477911yescharacteristic
525C22#478411yescharacteristic
526C22D3×#10412yes—
527C22#495611yescharacteristic
528C22#515811yescharacteristic
529C22#517211yescharacteristic
530C4—21no—
531C4—21no—
532C4—21no—
533C4—21no—
534C4—44no—
535C4—64no—
536C4—81no—
537C4—124no—
538C4—242no—
539C3#915011yescharacteristic, complemented, Sylow
540C2—21no—
541C2—21no—
542C2—21no—
543C2—21no—
544C2—124no—
545C2—161no—
546C2#6830112yes—
547C2#6830312yes—
548C2#7003611yescharacteristic
549C2#7020111yescharacteristic
550C2#7038211yescharacteristic
subgroupG/H
centerC23D3×C22⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#327C2
socleC23×C3C2×C22⋊C4
prime radical#82723C2
cosocleC23×C3×C4C23
hypercenterC2×#327D3
hypocenterC3#9150
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#327

Complements

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

NH ≅ G/Ntype
2. C2×#4666C2semidirect
3. C2×C3×#327C2semidirect
4. C2×#4537C2semidirect
5. #69955C2semidirect
6. #67675C2semidirect
7. #67612C2semidirect
11. C2×#938C4semidirect
22. #4987C4semidirect
32. C2×#327D3semidirect
72. C3×#95D4semidirect
76. C3⋊#95D4semidirect
216. #95#67semidirect
539. C3#9150semidirect

Maximal subgroups

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

subgroupindex
#827232
C2×#46662
C2×C3×#3272
C2×#45372
#699552
#676752
#676122
#91503