Finite Group Databasebeta

← #10690976

Subgroups — #10690976

Order 768.

980 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#74048C211yesmaximal, characteristic
2C2×#4664C211yesmaximal, characteristic, complemented
3#69955C211yesmaximal, characteristic, complemented
4C2×#4470C211yesmaximal, characteristic, complemented
5C2×#4451C211yesmaximal, characteristic, complemented
6C3×#1294C211yesmaximal, characteristic, complemented
7#67624C211yesmaximal, characteristic, complemented
8#9158—31nomaximal, Sylow
9C23×C4×D3C2211yescharacteristic
10C2×#930C2211yescharacteristic
11C22×(C2×D3)⋊C4—21no—
12C2×#895C412yescomplemented
13C2×#882—21no—
14C2×C3×C22⋊C8C2211yescharacteristic
15C3×#380C2211yescharacteristic
16C2×C3×#95C2211yescharacteristic
17C2×C3⋊#95—21no—
18C2×C3⋊#95C2211yescharacteristic
19C2×#830C2211yescharacteristic
20C2×(C2×D3)⋊C8—22no—
21#4987C412yescomplemented
22#4958—21no—
23#4665—22no—
24#4664—22no—
25#4664—22no—
26#4470—22no—
27#4451—22no—
28#2847—31no—
29C2×#381—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—22no—
59C2×(C2×D3)⋊C4—21no—
60C2×(C2×D3)⋊C4—22no—
61C2×(C2×D3)⋊C4—44no—
62C2×C4⋊Dic3C2×C411yescharacteristic
63C22×C3⋊C8—21no—
64#895—44no—
65#882—42no—
66#882—42no—
67C3×C22⋊C8—21no—
68C3×C22⋊C8—21no—
69C3×#95—21no—
70C3×#95—21no—
71C3×#95D412yescomplemented
72C3⋊#95—24no—
73C3⋊#95—21no—
74C3⋊#95—22no—
75C3⋊#95D412yescomplemented
76#830—21no—
77#830—21no—
78(C2×D3)⋊C8—44no—
79C24×C4—31no—
80C2×#121—31no—
81C2×#121—32no—
82C2×#120—61no—
83C22×C22⋊C4—61no—
84C2×C22⋊C8—31no—
85C2×C22⋊C8—62no—
86C2×C22⋊C8C2×D311yescharacteristic
87#386—62no—
88#386—61no—
89#381—62no—
90#381—62no—
91#380—32no—
92#380C2×D311yescharacteristic
93C2×#95—31no—
94C2×#95—61no—
95C2×#95C2×D311yescharacteristic
96#327—62no—
97#327—62no—
98C24×C3C2×D411yescharacteristic
99C23×D3—21no—
100C23×D3—21no—
101C23×D3—24no—
102C23×D3—21no—
103C23×D3—21no—
104C23×D3—21no—
105C23×D3—21no—
106C23×D3—21no—
107C23×D3—42no—
108C23×D3C22⋊C412yes—
109C22×C3×C4—22no—
110C22×C3×C4—22no—
111C22×C3×C4—42no—
112C22×C3×C4—42no—
113C22×C3×C4—42no—
114C22×C3×C4—41no—
115C22×C3×C4—41no—
116C22×C3×C4C22×C411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4#4111yescharacteristic
121C22×C3×C4#4111yescharacteristic
122C2×#67—41no—
123C2×#67—41no—
124C2×#67—41no—
125C2×#67—41no—
126C2×#67—41no—
127C2×#67—41no—
128C22×Dic3—21no—
129C22×Dic3—21no—
130C22×Dic3—21no—
131C22×Dic3—22no—
132C22×Dic3—42no—
133C22×Dic3—42no—
134C22×Dic3—42no—
135C22×Dic3—41no—
136C22×Dic3—41no—
137C22×Dic3C22⋊C412yes—
138C2×D12—21no—
139C2×D12—21no—
140C2×D12—42no—
141C2×D12—44no—
142C2×D12—44no—
143C2×D12—44no—
144C2×C4×D3—24no—
145C2×C4×D3—24no—
146C2×C4×D3—22no—
147C2×C4×D3—22no—
148C2×C4×D3—22no—
149C2×C4×D3—22no—
150C2×C4×D3—42no—
151C2×C4×D3—42no—
152C2×C4×D3—44no—
153C2×C4×D3—44no—
154C2×C4×D3—44no—
155C2×C4×D3—44no—
156C2×C3×C8—42no—
157C2×C3×C8—41no—
158C3×C4⋊C4—42no—
159C3×C22⋊C4—41no—
160C3×C22⋊C4—41no—
161(C2×D3)⋊C4—44no—
162(C2×D3)⋊C4—44no—
163(C2×D3)⋊C4—48no—
164(C2×D3)⋊C4—41no—
165(C2×D3)⋊C4—41no—
166C4⋊Dic3—21no—
167C4⋊Dic3—21no—
168C2×C3⋊C8—42no—
169C2×C3⋊C8—41no—
170C25—31no—
171C22×D4—31no—
172C22×D4—62no—
173C22×D4—61no—
174C23×C4—21no—
175C23×C4—31no—
176C23×C4—32no—
177C23×C4—32no—
178C23×C4—61no—
179C23×C4—61no—
180C23×C4—61no—
181C23×C4—61no—
182C23×C4—61no—
183C23×C4—124no—
184C23×C4C22×D311yescharacteristic
185C22×C8—21no—
186C22×C8—61no—
187#121—61no—
188#121—61no—
189#121—61no—
190#121—61no—
191#121—124no—
192#120—122no—
193#120—122no—
194C2×C4⋊C4—31no—
195C2×C4⋊C4C4×D312yes—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—62no—
198C2×C22⋊C4—62no—
199C2×C22⋊C4—64no—
200C2×C22⋊C4—61no—
201C2×C22⋊C4—124no—
202C22⋊C8—21no—
203C22⋊C8—21no—
204C22⋊C8—61no—
205C22⋊C8—61no—
206C22⋊C8—124no—
207#95—21no—
208#95—21no—
209#95—32no—
210#95—62no—
211#95—64no—
212#95—61no—
213#95D1212yescomplemented
214C23×C3—22no—
215C23×C3—22no—
216C23×C3C2×C22⋊C411yescharacteristic
217C23×C3#12011yescharacteristic
218C23×C3#12311yescharacteristic
219C23×C3#13511yescharacteristic
220C23×C3#13511yescharacteristic
221C23×C3AGL(1,ℤ/8)11yescharacteristic
222C23×C3AGL(1,ℤ/8)11yescharacteristic
223C22×D3—24no—
224C22×D3—24no—
225C22×D3—22no—
226C22×D3—22no—
227C22×D3—22no—
228C22×D3—22no—
229C22×D3—22no—
230C22×D3—22no—
231C22×D3—44no—
232C22×D3—44no—
233C22×D3—44no—
234C22×D3—44no—
235C22×D3—42no—
236C22×D3—42no—
237C22×D3—41no—
238C22×D3—41no—
239C22×D3—41no—
240C22×D3—41no—
241C22×D3—41no—
242C22×D3—42no—
243C22×D3—41no—
244C22×D3—41no—
245C22×D3—42no—
246C22×D3#9914yes—
247C2×C3×C4—22no—
248C2×C3×C4—22no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—42no—
254C2×C3×C4—42no—
255C2×C3×C4—42no—
256C2×C3×C4—42no—
257C2×C3×C4—42no—
258C2×C3×C4—42no—
259C2×C3×C4—82no—
260C2×C3×C4—82no—
261C2×C3×C4C4×D411yescharacteristic
262C2×C3×C4C4×D411yescharacteristic
263C2×C3×C4#12111yescharacteristic
264C2×C3×C4#12111yescharacteristic
265#67—81no—
266#67—81no—
267#67—81no—
268#67—81no—
269C2×Dic3—22no—
270C2×Dic3—22no—
271C2×Dic3—22no—
272C2×Dic3—42no—
273C2×Dic3—42no—
274C2×Dic3—42no—
275C2×Dic3—42no—
276C2×Dic3—41no—
277C2×Dic3—41no—
278C2×Dic3—41no—
279C2×Dic3—42no—
280C2×Dic3—82no—
281C2×Dic3—82no—
282D12—81no—
283D12—81no—
284D12—88no—
285C4×D3—44no—
286C4×D3—44no—
287C4×D3—44no—
288C4×D3—44no—
289C3×C8—81no—
290C3⋊C8—81no—
291C24—32no—
292C24—64no—
293C24—61no—
294C24—61no—
295C24—61no—
296C24—61no—
297C24—61no—
298C24—61no—
299C24—61no—
300C24—122no—
301C24D3×D411yescharacteristic
302C2×D4—61no—
303C2×D4—61no—
304C2×D4—122no—
305C2×D4—124no—
306C2×D4—124no—
307C2×D4—124no—
308C2×D4—121no—
309C2×D4—121no—
310C2×D4—121no—
311C2×D4—121no—
312C2×D4—121no—
313C2×D4—121no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—32no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—64no—
323C22×C4—64no—
324C22×C4—61no—
325C22×C4—61no—
326C22×C4—62no—
327C22×C4—62no—
328C22×C4—62no—
329C22×C4—62no—
330C22×C4—61no—
331C22×C4—62no—
332C22×C4—122no—
333C22×C4—122no—
334C22×C4—124no—
335C22×C4—124no—
336C22×C4—124no—
337C22×C4—124no—
338C22×C4—122no—
339C22×C4—122no—
340C22×C4—122no—
341C22×C4—121no—
342C22×C4—121no—
343C22×C4C2×C4×D311yescharacteristic
344C22×C4C2×D1211yescharacteristic
345C22×C4#23511yescharacteristic
346C22×C4D3×D411yescharacteristic
347C22×C4D3×D411yescharacteristic
348C22×C4#23911yescharacteristic
349C2×C8—42no—
350C2×C8—41no—
351C2×C8—122no—
352C2×C8—121no—
353C4⋊C4—42no—
354C4⋊C4—61no—
355C4⋊C4—61no—
356C22⋊C4—41no—
357C22⋊C4—41no—
358C22⋊C4—124no—
359C22⋊C4—128no—
360C22⋊C4—124no—
361C22⋊C4—121no—
362C22⋊C4—121no—
363C22×C3—21no—
364C22×C3—21no—
365C22×C3—21no—
366C22×C3—21no—
367C22×C3—22no—
368C22×C3—22no—
369C22×C3—22no—
370C22×C3—22no—
371C22×C3#38611yescharacteristic
372C22×C3C2×#9912yes—
373C22×C3#41611yescharacteristic
374C22×C3#41811yescharacteristic
375C22×C3#42112yes—
376C22×C3#44911yescharacteristic
377C22×C3#44911yescharacteristic
378C22×C3#48211yescharacteristic
379C22×C3#48211yescharacteristic
380C2×D3—24no—
381C2×D3—24no—
382C2×D3—24no—
383C2×D3—44no—
384C2×D3—44no—
385C2×D3—44no—
386C2×D3—44no—
387C2×D3—42no—
388C2×D3—42no—
389C2×D3—42no—
390C2×D3—42no—
391C2×D3—81no—
392C2×D3—81no—
393C2×D3—81no—
394C2×D3—82no—
395C2×D3—82no—
396C3×C4—21no—
397C3×C4—21no—
398C3×C4—21no—
399C3×C4—21no—
400C3×C4—82no—
401C3×C4—82no—
402Dic3—42no—
403Dic3—82no—
404Dic3—81no—
405C23—22no—
406C23—22no—
407C23—34no—
408C23—64no—
409C23—64no—
410C23—62no—
411C23—62no—
412C23—62no—
413C23—62no—
414C23—62no—
415C23—62no—
416C23—124no—
417C23—124no—
418C23—124no—
419C23—124no—
420C23—122no—
421C23—122no—
422C23—121no—
423C23—121no—
424C23—121no—
425C23—121no—
426C23—121no—
427C23—122no—
428C23—121no—
429C23—121no—
430C23—122no—
431C23D3×C22⋊C411yescharacteristic
432C23#88211yescharacteristic
433C23#89411yescharacteristic
434C23#91111yescharacteristic
435C23#91411yescharacteristic
436C23#91611yescharacteristic
437C23#91911yescharacteristic
438D4—241no—
439D4—241no—
440D4—248no—
441D4—241no—
442D4—241no—
443D4—241no—
444D4—241no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—62no—
458C2×C4—62no—
459C2×C4—62no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—124no—
463C2×C4—124no—
464C2×C4—124no—
465C2×C4—124no—
466C2×C4—122no—
467C2×C4—122no—
468C2×C4—122no—
469C2×C4—122no—
470C2×C4—121no—
471C2×C4—121no—
472C2×C4—121no—
473C2×C4—122no—
474C2×C4—242no—
475C2×C4—242no—
476C2×C4C4×D1211yescharacteristic
477C2×C4#88411yescharacteristic
478C2×C4#89311yescharacteristic
479C2×C4#89511yescharacteristic
480C8—81no—
481C8—241no—
482C2×C3—21no—
483C2×C3—21no—
484C2×C3—21no—
485C2×C3—21no—
486C2×C3#151012yes—
487C2×C3#151612yes—
488C2×C3#187511yescharacteristic
489C2×C3#189711yescharacteristic
490C2×C3#189811yescharacteristic
491D3—44no—
492D3—161no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—64no—
502C22—64no—
503C22—64no—
504C22—124no—
505C22—124no—
506C22—124no—
507C22—124no—
508C22—122no—
509C22—122no—
510C22—122no—
511C22—122no—
512C22—241no—
513C22—241no—
514C22—241no—
515C22—242no—
516C22—242no—
517C22#466511yescharacteristic
518C22D3×#9912yes—
519C22#477011yescharacteristic
520C22#478111yescharacteristic
521C22#480411yescharacteristic
522C22#481011yescharacteristic
523C22#482112yes—
524C22#486611yescharacteristic
525C22#488511yescharacteristic
526C4—21no—
527C4—21no—
528C4—21no—
529C4—21no—
530C4—82no—
531C4—82no—
532C4—122no—
533C4—242no—
534C4—241no—
535C3#915811yescharacteristic, complemented, Sylow
536C2—21no—
537C2—21no—
538C2—21no—
539C2—21no—
540C2—124no—
541C2—481no—
542C2#6830412yes—
543C2#6833812yes—
544C2#6875711yescharacteristic
545C2#6888111yescharacteristic
546C2#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#9158
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×#4664C2semidirect
3. #69955C2semidirect
4. C2×#4470C2semidirect
5. C2×#4451C2semidirect
6. C3×#1294C2semidirect
7. #67624C2semidirect
12. C2×#895C4semidirect
21. #4987C4semidirect
34. #1294D3semidirect
71. C3×#95D4semidirect
75. C3⋊#95D4semidirect
213. #95D12semidirect
535. C3#9158semidirect

Maximal subgroups

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

subgroupindex
#740482
C2×#46642
#699552
C2×#44702
C2×#44512
C3×#12942
#676242
#91583