Finite Group Databasebeta

← #10690974

Subgroups — #10690974

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#82723C211yesmaximal, characteristic
2C2×#4664C211yesmaximal, characteristic, complemented
3C2×C3×#327C211yesmaximal, characteristic, complemented
4C2×#4537C211yesmaximal, characteristic, complemented
5#69955C211yesmaximal, characteristic, complemented
6#67675C211yesmaximal, characteristic, complemented
7#67612C211yesmaximal, characteristic, complemented
8#9158—31nomaximal, Sylow
9C23×C4×D3C2211yescharacteristic
10C2×C3×#121C2211yescharacteristic
11C2×#938C412yescomplemented
12C2×#937—21no—
13C22×(C2×D3)⋊C4—21no—
14C2×C3×C22⋊C8C2211yescharacteristic
15C2×C3×#95C2211yescharacteristic
16#5220—21no—
17#5212C2211yescharacteristic
18C2×C3⋊#95—21no—
19C2×C3⋊#95C2211yescharacteristic
20C2×#830C2211yescharacteristic
21C2×(C2×D3)⋊C8—22no—
22#4987C412yescomplemented
23#4956—22no—
24#4664—22no—
25#4664—22no—
26C3×#327—22no—
27#4537—22no—
28#2847—31no—
29C2×#381—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—22no—
64C2×(C2×D3)⋊C4—44no—
65C2×C4⋊Dic3C2×C412yes—
66C22×C3⋊C8—21no—
67C3×C22⋊C8—21no—
68C3×C22⋊C8—21no—
69C3×#95—21no—
70C3×#95D412yescomplemented
71C3⋊#95—24no—
72C3⋊#95—21no—
73C3⋊#95—21no—
74C3⋊#95—22no—
75C3⋊#95D412yescomplemented
76#830—21no—
77#830—21no—
78(C2×D3)⋊C8—44no—
79C24×C4—31no—
80C2×#121—32no—
81C2×#121C2×D311yescharacteristic
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—31no—
92#380—32no—
93C2×#95—31no—
94C2×#95—61no—
95C2×#95C2×D311yescharacteristic
96#327—22no—
97#327—62no—
98C24×C3—21no—
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—42no—
108C23×D3C22⋊C412yes—
109C2×C3×D4—21no—
110C2×C3×D4—21no—
111C2×C3×D4—42no—
112C2×C3×D4—41no—
113C2×C3×D4—41no—
114C2×C3×D4—41no—
115C2×C3×D4—41no—
116C2×C3×D4—41no—
117C2×C3×D4—41no—
118C22×C3×C4—21no—
119C22×C3×C4—22no—
120C22×C3×C4—42no—
121C22×C3×C4—42no—
122C22×C3×C4—42no—
123C22×C3×C4—41no—
124C22×C3×C4—41no—
125C22×C3×C4C22×C411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4#4111yescharacteristic
130C22×C3×C4#4111yescharacteristic
131C2×#67—44no—
132C2×#67—44no—
133C2×#67—44no—
134C22×Dic3—21no—
135C22×Dic3—21no—
136C22×Dic3—22no—
137C22×Dic3—22no—
138C22×Dic3—42no—
139C22×Dic3—42no—
140C22×Dic3—42no—
141C22×Dic3—41no—
142C22×Dic3—41no—
143C22×Dic3C22⋊C412yes—
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—21no—
159C3×C4⋊C4—21no—
160C3×C22⋊C4—41no—
161C3×C22⋊C4—41no—
162#217—44no—
163#217—41no—
164#217—41no—
165(C2×D3)⋊C4—48no—
166(C2×D3)⋊C4—44no—
167C4⋊Dic3—42no—
168C2×C3⋊C8—42no—
169C2×C3⋊C8—41no—
170C25—31no—
171C22×D4—21no—
172C22×D4—62no—
173C22×D4C4×D311yescharacteristic
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—21no—
188#121—21no—
189#121—21no—
190#121—21no—
191#121—124no—
192#120—122no—
193#120—122no—
194C2×C4⋊C4—32no—
195C2×C4⋊C4C4×D311yescharacteristic
196C2×C22⋊C4—21no—
197C2×C22⋊C4—62no—
198C2×C22⋊C4—64no—
199C2×C22⋊C4—62no—
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—32no—
209#95—62no—
210#95—64no—
211#95—61no—
212#95—61no—
213#95#6712yescomplemented
214C23×C3—22no—
215C23×C3—22no—
216C23×C3—41no—
217C23×C3—42no—
218C23×C3—41no—
219C23×C3—41no—
220C23×C3—42no—
221C23×C3C2×C22⋊C411yescharacteristic
222C23×C3#12011yescharacteristic
223C23×C3#12311yescharacteristic
224C23×C3#13511yescharacteristic
225C23×C3#13511yescharacteristic
226C23×C3AGL(1,ℤ/8)11yescharacteristic
227C23×C3AGL(1,ℤ/8)11yescharacteristic
228C22×D3—24no—
229C22×D3—24no—
230C22×D3—22no—
231C22×D3—22no—
232C22×D3—22no—
233C22×D3—22no—
234C22×D3—22no—
235C22×D3—22no—
236C22×D3—44no—
237C22×D3—44no—
238C22×D3—44no—
239C22×D3—44no—
240C22×D3—42no—
241C22×D3—42no—
242C22×D3—41no—
243C22×D3—41no—
244C22×D3—41no—
245C22×D3—41no—
246C22×D3#9914yes—
247C3×D4—81no—
248C3×D4—81no—
249C3×D4—81no—
250C3×D4—81no—
251C3×D4—81no—
252C3×D4—81no—
253C2×C3×C4—22no—
254C2×C3×C4—22no—
255C2×C3×C4—41no—
256C2×C3×C4—41no—
257C2×C3×C4—41no—
258C2×C3×C4—41no—
259C2×C3×C4—42no—
260C2×C3×C4—42no—
261C2×C3×C4—41no—
262C2×C3×C4—41no—
263C2×C3×C4—41no—
264C2×C3×C4—42no—
265C2×C3×C4—82no—
266C2×C3×C4—82no—
267C2×C3×C4C4×D411yescharacteristic
268C2×C3×C4C4×D411yescharacteristic
269C2×C3×C4#12111yescharacteristic
270C2×C3×C4#12111yescharacteristic
271#67—88no—
272C2×Dic3—22no—
273C2×Dic3—22no—
274C2×Dic3—22no—
275C2×Dic3—42no—
276C2×Dic3—42no—
277C2×Dic3—42no—
278C2×Dic3—42no—
279C2×Dic3—42no—
280C2×Dic3—42no—
281C2×Dic3—42no—
282C2×Dic3—42no—
283C2×Dic3—82no—
284C2×Dic3—82no—
285C4×D3—44no—
286C4×D3—44no—
287C4×D3—44no—
288C4×D3—44no—
289C3×C8—81no—
290C3⋊C8—81no—
291C24—21no—
292C24—32no—
293C24—64no—
294C24—61no—
295C24—61no—
296C24—61no—
297C24—61no—
298C24—61no—
299C24—61no—
300C24—122no—
301C24D3×D411yescharacteristic
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—42no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—124no—
312C2×D4—124no—
313C2×D4—124no—
314C22×C4—21no—
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—62no—
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×C4#23511yescharacteristic
345C22×C4D3×D411yescharacteristic
346C22×C4D3×D411yescharacteristic
347C22×C4#23711yescharacteristic
348C22×C4C2×#6711yescharacteristic
349C2×C8—42no—
350C2×C8—41no—
351C2×C8—122no—
352C2×C8—121no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—122no—
356C22⋊C4—41no—
357C22⋊C4—41no—
358C22⋊C4—128no—
359C22⋊C4—124no—
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—81no—
372C22×C3—81no—
373C22×C3—81no—
374C22×C3—82no—
375C22×C3—82no—
376C22×C3#38611yescharacteristic
377C22×C3C2×#9912yes—
378C22×C3#41611yescharacteristic
379C22×C3#41811yescharacteristic
380C22×C3#42112yes—
381C22×C3#44911yescharacteristic
382C22×C3#44911yescharacteristic
383C22×C3#48211yescharacteristic
384C22×C3#48211yescharacteristic
385C2×D3—24no—
386C2×D3—24no—
387C2×D3—24no—
388C2×D3—44no—
389C2×D3—44no—
390C2×D3—44no—
391C2×D3—44no—
392C2×D3—42no—
393C2×D3—42no—
394C2×D3—42no—
395C2×D3—42no—
396C3×C4—21no—
397C3×C4—21no—
398C3×C4—21no—
399C3×C4—21no—
400C3×C4—82no—
401C3×C4—81no—
402Dic3—42no—
403Dic3—82no—
404Dic3—82no—
405C23—22no—
406C23—22no—
407C23—34no—
408C23—41no—
409C23—42no—
410C23—41no—
411C23—41no—
412C23—42no—
413C23—64no—
414C23—64no—
415C23—62no—
416C23—62no—
417C23—62no—
418C23—62no—
419C23—62no—
420C23—62no—
421C23—124no—
422C23—124no—
423C23—124no—
424C23—124no—
425C23—122no—
426C23—122no—
427C23—121no—
428C23—121no—
429C23—121no—
430C23—121no—
431C23D3×C22⋊C411yescharacteristic
432C23#88311yescharacteristic
433C23#91111yescharacteristic
434C23#91211yescharacteristic
435C23#91411yescharacteristic
436C23#91611yescharacteristic
437C23#93711yescharacteristic
438D4—81no—
439D4—81no—
440D4—81no—
441D4—81no—
442D4—81no—
443D4—81no—
444D4—248no—
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—41no—
455C2×C4—41no—
456C2×C4—41no—
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—122no—
471C2×C4—122no—
472C2×C4—122no—
473C2×C4—122no—
474C2×C4—242no—
475C2×C4—242no—
476C2×C4#67⋊C411yescharacteristic
477C2×C4#88411yescharacteristic
478C2×C4C4×#6711yescharacteristic
479C2×C4#93811yescharacteristic
480C8—81no—
481C8—241no—
482C2×C3—21no—
483C2×C3—21no—
484C2×C3—21no—
485C2×C3—21no—
486C2×C3—161no—
487C2×C3#151012yes—
488C2×C3#151612yes—
489C2×C3#187511yescharacteristic
490C2×C3#189711yescharacteristic
491C2×C3#189811yescharacteristic
492D3—44no—
493C22—21no—
494C22—21no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—21no—
500C22—21no—
501C22—64no—
502C22—64no—
503C22—64no—
504C22—81no—
505C22—81no—
506C22—82no—
507C22—82no—
508C22—81no—
509C22—124no—
510C22—124no—
511C22—124no—
512C22—124no—
513C22—122no—
514C22—122no—
515C22—122no—
516C22—122no—
517C22D3×#9912yes—
518C22#477011yescharacteristic
519C22#477211yescharacteristic
520C22#477911yescharacteristic
521C22#478411yescharacteristic
522C22#482112yes—
523C22#495611yescharacteristic
524C22#515811yescharacteristic
525C22#517211yescharacteristic
526C4—21no—
527C4—21no—
528C4—21no—
529C4—21no—
530C4—82no—
531C4—81no—
532C4—122no—
533C4—242no—
534C4—242no—
535C3#915811yescharacteristic, complemented, Sylow
536C2—21no—
537C2—21no—
538C2—21no—
539C2—21no—
540C2—124no—
541C2—161no—
542C2#6830412yes—
543C2#6830612yes—
544C2#7003611yescharacteristic
545C2#7020111yescharacteristic
546C2#7038311yescharacteristic
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#9158
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×#4664C2semidirect
3. C2×C3×#327C2semidirect
4. C2×#4537C2semidirect
5. #69955C2semidirect
6. #67675C2semidirect
7. #67612C2semidirect
11. C2×#938C4semidirect
22. #4987C4semidirect
32. C2×#327D3semidirect
70. C3×#95D4semidirect
75. C3⋊#95D4semidirect
213. #95#67semidirect
535. C3#9158semidirect

Maximal subgroups

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

subgroupindex
#827232
C2×#46642
C2×C3×#3272
C2×#45372
#699552
#676752
#676122
#91583