Finite Group Databasebeta

← #10690936

Subgroups — #10690936

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#74048C211yesmaximal, characteristic
2C2×#4666C211yesmaximal, characteristic, complemented
3#69955C211yesmaximal, characteristic, complemented
4C2×#4470C211yesmaximal, characteristic, complemented
5C2×#4451C211yesmaximal, characteristic, complemented
6C3×#1294C211yesmaximal, characteristic, complemented
7#67624C211yesmaximal, characteristic, complemented
8#9150—31nomaximal, Sylow
9C23×C4×D3C2211yescharacteristic
10C2×#930C2211yescharacteristic
11C22×(C2×D3)⋊C4—21no—
12C22×C4×Dic3—21no—
13C2×#895C412yescomplemented
14C2×#882—21no—
15C2×C3×C22⋊C8C2211yescharacteristic
16C3×#380C2211yescharacteristic
17C2×C3×#95C2211yescharacteristic
18C2×C3⋊#95C2211yescharacteristic
19C2×#830C2211yescharacteristic
20C2×(C2×D3)⋊C8—22no—
21#4987C412yescomplemented
22#4958—21no—
23#4666—22no—
24#4666—22no—
25#4665—22no—
26#4470—22no—
27#4451—22no—
28#2847—31no—
29C2×C4×C22⋊C4—31no—
30#1779—31no—
31C2×#327—31no—
32C2×#327—31no—
33#1294—31no—
34#1294D311yescharacteristic, complemented
35C24×D3D411yescharacteristic
36C23×C3×C4—21no—
37C23×C3×C4C2311yescharacteristic
38C22×#67—21no—
39C23×Dic3—21no—
40C23×Dic3D411yescharacteristic
41C22×D12—22no—
42C22×D12C2×C411yescharacteristic
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—
51C2×C3×C4⋊C4C2×C412yes—
52C2×C3×C22⋊C4—21no—
53#930—21no—
54#930—21no—
55#930—21no—
56#930—21no—
57C2×(C2×D3)⋊C4—24no—
58C2×(C2×D3)⋊C4—28no—
59C2×(C2×D3)⋊C4—22no—
60C2×(C2×D3)⋊C4—21no—
61C2×(C2×D3)⋊C4—22no—
62C2×C4⋊Dic3C2×C411yescharacteristic
63C2×C4×Dic3—24no—
64C2×C4×Dic3—44no—
65C22×C3⋊C8—21no—
66#895—44no—
67#882—42no—
68#882—42no—
69C3×C22⋊C8—21no—
70C3×C22⋊C8—21no—
71C3×#95—21no—
72C3×#95—21no—
73C3×#95D412yescomplemented
74C3⋊#95—21no—
75C3⋊#95—22no—
76C3⋊#95D412yescomplemented
77#830—21no—
78#830—21no—
79(C2×D3)⋊C8—44no—
80C24×C4—31no—
81C2×#121—31no—
82C2×#121—32no—
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—32no—
92#380C2×D311yescharacteristic
93C4×C22⋊C4—62no—
94C4×C22⋊C4—62no—
95C2×#95—31no—
96C2×#95C2×D311yescharacteristic
97#327—62no—
98#327—62no—
99C24×C3C2×D411yescharacteristic
100C23×D3—21no—
101C23×D3—21no—
102C23×D3—24no—
103C23×D3—21no—
104C23×D3—21no—
105C23×D3—21no—
106C23×D3—21no—
107C23×D3—21no—
108C23×D3—42no—
109C23×D3C22⋊C412yes—
110C22×C3×C4—24no—
111C22×C3×C4—22no—
112C22×C3×C4—22no—
113C22×C3×C4—42no—
114C22×C3×C4—42no—
115C22×C3×C4—41no—
116C22×C3×C4—41no—
117C22×C3×C4C22×C411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4#4111yescharacteristic
122C22×C3×C4#4111yescharacteristic
123C2×#67—41no—
124C2×#67—41no—
125C2×#67—41no—
126C2×#67—41no—
127C2×#67—41no—
128C2×#67—41no—
129C22×Dic3—21no—
130C22×Dic3—21no—
131C22×Dic3—24no—
132C22×Dic3—24no—
133C22×Dic3—22no—
134C22×Dic3—21no—
135C22×Dic3—42no—
136C22×Dic3—41no—
137C22×Dic3—41no—
138C22×Dic3C22⋊C412yes—
139C2×D12—21no—
140C2×D12—21no—
141C2×D12—42no—
142C2×D12—44no—
143C2×D12—44no—
144C2×D12—44no—
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—42no—
160C3×C22⋊C4—41no—
161C3×C22⋊C4—41no—
162(C2×D3)⋊C4—216no—
163(C2×D3)⋊C4—44no—
164(C2×D3)⋊C4—41no—
165(C2×D3)⋊C4—41no—
166(C2×D3)⋊C4—44no—
167C4⋊Dic3—21no—
168C4⋊Dic3—21no—
169C4×Dic3—48no—
170C2×C3⋊C8—42no—
171C2×C3⋊C8—41no—
172C25—31no—
173C22×D4—31no—
174C22×D4—62no—
175C22×D4—61no—
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—61no—
190#121—61no—
191#121—61no—
192#121—61no—
193#121—124no—
194#120—122no—
195#120—122no—
196C2×C4⋊C4—31no—
197C2×C4⋊C4C4×D312yes—
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—21no—
213#95—32no—
214#95—62no—
215#95—61no—
216#95D1212yescomplemented
217C23×C3—22no—
218C23×C3—22no—
219C23×C3C2×C22⋊C411yescharacteristic
220C23×C3#12011yescharacteristic
221C23×C3#12311yescharacteristic
222C23×C3#13511yescharacteristic
223C23×C3#13511yescharacteristic
224C23×C3AGL(1,ℤ/8)11yescharacteristic
225C23×C3AGL(1,ℤ/8)11yescharacteristic
226C22×D3—24no—
227C22×D3—24no—
228C22×D3—22no—
229C22×D3—22no—
230C22×D3—22no—
231C22×D3—22no—
232C22×D3—22no—
233C22×D3—22no—
234C22×D3—44no—
235C22×D3—44no—
236C22×D3—44no—
237C22×D3—44no—
238C22×D3—42no—
239C22×D3—42no—
240C22×D3—41no—
241C22×D3—41no—
242C22×D3—41no—
243C22×D3—41no—
244C22×D3—41no—
245C22×D3—42no—
246C22×D3—41no—
247C22×D3—41no—
248C22×D3—42no—
249C22×D3#10414yes—
250C2×C3×C4—22no—
251C2×C3×C4—22no—
252C2×C3×C4—24no—
253C2×C3×C4—41no—
254C2×C3×C4—41no—
255C2×C3×C4—41no—
256C2×C3×C4—41no—
257C2×C3×C4—42no—
258C2×C3×C4—44no—
259C2×C3×C4—44no—
260C2×C3×C4—42no—
261C2×C3×C4—42no—
262C2×C3×C4—42no—
263C2×C3×C4—42no—
264C2×C3×C4C4×D411yescharacteristic
265C2×C3×C4C4×D411yescharacteristic
266C2×C3×C4#12111yescharacteristic
267C2×C3×C4#12111yescharacteristic
268#67—81no—
269#67—81no—
270#67—81no—
271#67—81no—
272C2×Dic3—24no—
273C2×Dic3—24no—
274C2×Dic3—22no—
275C2×Dic3—22no—
276C2×Dic3—24no—
277C2×Dic3—42no—
278C2×Dic3—44no—
279C2×Dic3—44no—
280C2×Dic3—42no—
281C2×Dic3—41no—
282C2×Dic3—41no—
283C2×Dic3—41no—
284C2×Dic3#9914yes—
285D12—81no—
286D12—81no—
287D12—88no—
288C4×D3—44no—
289C4×D3—44no—
290C4×D3—44no—
291C4×D3—44no—
292C3×C8—81no—
293C3⋊C8—81no—
294C24—32no—
295C24—64no—
296C24—61no—
297C24—61no—
298C24—61no—
299C24—61no—
300C24—61no—
301C24—61no—
302C24—61no—
303C24—122no—
304C24D3×D411yescharacteristic
305C2×D4—61no—
306C2×D4—61no—
307C2×D4—122no—
308C2×D4—124no—
309C2×D4—124no—
310C2×D4—124no—
311C2×D4—121no—
312C2×D4—121no—
313C2×D4—121no—
314C2×D4—121no—
315C2×D4—121no—
316C2×D4—121no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—22no—
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—61no—
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×C4C2×D1211yescharacteristic
348C22×C4#23511yescharacteristic
349C22×C4D3×D411yescharacteristic
350C22×C4D3×D411yescharacteristic
351C22×C4#23911yescharacteristic
352C2×C8—42no—
353C2×C8—41no—
354C2×C8—122no—
355C2×C8—121no—
356C4⋊C4—42no—
357C4⋊C4—61no—
358C4⋊C4—61no—
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#38611yescharacteristic
376C22×C3C2×#9912yes—
377C22×C3#41611yescharacteristic
378C22×C3#41811yescharacteristic
379C22×C3C2×#10412yes—
380C22×C3#44911yescharacteristic
381C22×C3#44911yescharacteristic
382C22×C3#48211yescharacteristic
383C22×C3#48211yescharacteristic
384C2×D3—24no—
385C2×D3—24no—
386C2×D3—24no—
387C2×D3—44no—
388C2×D3—44no—
389C2×D3—44no—
390C2×D3—44no—
391C2×D3—42no—
392C2×D3—42no—
393C2×D3—42no—
394C2×D3—42no—
395C2×D3—81no—
396C2×D3—81no—
397C2×D3—81no—
398C2×D3—82no—
399C2×D3—82no—
400C3×C4—21no—
401C3×C4—21no—
402C3×C4—21no—
403C3×C4—21no—
404C3×C4—44no—
405C3×C4—82no—
406Dic3—24no—
407Dic3—44no—
408Dic3—81no—
409C23—22no—
410C23—22no—
411C23—34no—
412C23—64no—
413C23—64no—
414C23—62no—
415C23—62no—
416C23—62no—
417C23—62no—
418C23—62no—
419C23—62no—
420C23—124no—
421C23—124no—
422C23—124no—
423C23—124no—
424C23—122no—
425C23—122no—
426C23—121no—
427C23—121no—
428C23—121no—
429C23—121no—
430C23—121no—
431C23—122no—
432C23—121no—
433C23—121no—
434C23—122no—
435C23D3×C22⋊C411yescharacteristic
436C23#88211yescharacteristic
437C23#89411yescharacteristic
438C23#91111yescharacteristic
439C23#91411yescharacteristic
440C23#91611yescharacteristic
441C23#91911yescharacteristic
442D4—241no—
443D4—241no—
444D4—248no—
445D4—241no—
446D4—241no—
447D4—241no—
448D4—241no—
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—42no—
462C2×C4—42no—
463C2×C4—42no—
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—121no—
478C2×C4—121no—
479C2×C4—121no—
480C2×C4C4×D1211yescharacteristic
481C2×C4#88411yescharacteristic
482C2×C4#89311yescharacteristic
483C2×C4#89511yescharacteristic
484C8—81no—
485C8—241no—
486C2×C3—21no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3#150112yes—
491C2×C3#150712yes—
492C2×C3#187511yescharacteristic
493C2×C3#189611yescharacteristic
494C2×C3#189811yescharacteristic
495D3—44no—
496D3—161no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—22no—
502C22—22no—
503C22—22no—
504C22—22no—
505C22—64no—
506C22—64no—
507C22—64no—
508C22—124no—
509C22—124no—
510C22—124no—
511C22—124no—
512C22—122no—
513C22—122no—
514C22—122no—
515C22—122no—
516C22—241no—
517C22—241no—
518C22—241no—
519C22—242no—
520C22—242no—
521C22#466511yescharacteristic
522C22D3×#9912yes—
523C22#477011yescharacteristic
524C22#478111yescharacteristic
525C22#480411yescharacteristic
526C22#481011yescharacteristic
527C22D3×#10412yes—
528C22#486611yescharacteristic
529C22#488511yescharacteristic
530C4—21no—
531C4—21no—
532C4—21no—
533C4—21no—
534C4—44no—
535C4—64no—
536C4—82no—
537C4—124no—
538C4—241no—
539C3#915011yescharacteristic, complemented, Sylow
540C2—21no—
541C2—21no—
542C2—21no—
543C2—21no—
544C2—124no—
545C2—481no—
546C2#6830312yes—
547C2#6833312yes—
548C2#6875611yescharacteristic
549C2#6888111yescharacteristic
550C2#6897011yescharacteristic
subgroupG/H
centerC23D3×C22⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1294C2
socleC23×C3C2×C22⋊C4
prime radical#74048C2
cosocleC23×C3×C4C23
hypercenter#1294D3
hypocenterC3#9150
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1294

Complements

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

NH ≅ G/Ntype
2. C2×#4666C2semidirect
3. #69955C2semidirect
4. C2×#4470C2semidirect
5. C2×#4451C2semidirect
6. C3×#1294C2semidirect
7. #67624C2semidirect
13. C2×#895C4semidirect
21. #4987C4semidirect
34. #1294D3semidirect
73. C3×#95D4semidirect
76. C3⋊#95D4semidirect
216. #95D12semidirect
539. C3#9150semidirect

Maximal subgroups

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

subgroupindex
#740482
C2×#46662
#699552
C2×#44702
C2×#44512
C3×#12942
#676242
#91503