Finite Group Databasebeta

← #10695423

Subgroups — #10695423

Order 768.

856 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×#2617C211yesmaximal, characteristic
2C2×#5034C211yesmaximal, characteristic, complemented
3C2×#4949C211yesmaximal, characteristic, complemented
4C2×#4537C211yesmaximal, characteristic, complemented
5C2×#4525C211yesmaximal, characteristic, complemented
6#69502C211yesmaximal, characteristic, complemented
7#67624C211yesmaximal, characteristic, complemented
8#9920—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C2×C3×#120—21no—
11C22×C3×C22⋊C4C2211yescharacteristic
12C22×#217—21no—
13C22×C4⋊Dic3C2211yescharacteristic
14C22×Dic3⋊C4—21no—
15C3×#397—21no—
16C3×#394—22no—
17C3×#392—21no—
18C3×#386—21no—
19C3×#380C2211yescharacteristic
20C2×#832—21no—
21C2×C3⋊#95C2211yescharacteristic
22C2×#830C2211yescharacteristic
23C2×#830C2211yescharacteristic
24#5034—22no—
25#5034—22no—
26C2×#808—21no—
27#4949—22no—
28#4949—22no—
29#4537—22no—
30#4529—22no—
31#4525—22no—
32C2×#481—31no—
33#2617D311yescharacteristic
34C2×#399—31no—
35#1824—31no—
36C2×#327—31no—
37C2×#323—31no—
38#1294—31no—
39C25×C3D411yescharacteristic
40C22×C3×D4—21no—
41C22×C3×D4—22no—
42C22×C3×D4D411yescharacteristic
43C23×C3×C4D411yescharacteristic
44C23×C3×C4C2311yescharacteristic
45C23×Dic3—21no—
46C23×Dic3—21no—
47C3×#121—21no—
48C3×#121—21no—
49C3×#121—21no—
50C3×#121—21no—
51C3×#120—42no—
52C3×#120—42no—
53C2×C3×C4⋊C4D411yescharacteristic
54C2×C3×C4⋊C4D412yes—
55C2×C3×C22⋊C4—22no—
56C2×C3×C22⋊C4—21no—
57C2×C3×C22⋊C4—21no—
58C2×C3×C22⋊C4—21no—
59C2×C3×C22⋊C4—21no—
60C2×C3×C22⋊C4—21no—
61C2×C3×C22⋊C4—21no—
62C2×C3×C22⋊C4—21no—
63C2×C3×C22⋊C4—21no—
64C2×C3×C22⋊C4—22no—
65C2×#217—24no—
66C2×#217—44no—
67C2×C4⋊Dic3—21no—
68C2×C4⋊Dic3—21no—
69C2×C4⋊Dic3—21no—
70C2×C4⋊Dic3—21no—
71C2×C4⋊Dic3—21no—
72C2×C4⋊Dic3D412yescomplemented
73C2×Dic3⋊C4—24no—
74C2×Dic3⋊C4—44no—
75C22×C3⋊C8—21no—
76C22×C3⋊C8—21no—
77C3×#95—22no—
78C3×#95—21no—
79#832—24no—
80C3⋊#95—21no—
81C3⋊#95D412yescomplemented
82#830—21no—
83#830—21no—
84#830—21no—
85#830D412yescomplemented
86#808—24no—
87C2×#121C2×D311yescharacteristic
88C2×#120—21no—
89C22×C4⋊C4—31no—
90C22×C4⋊C4—61no—
91C22×C22⋊C4—61no—
92C22×C22⋊C4C2×D311yescharacteristic
93#481—62no—
94#481—62no—
95C2×#106—61no—
96C2×#102—61no—
97C2×C22⋊C8—31no—
98C2×C22⋊C8—31no—
99#399—62no—
100#399—62no—
101#397—21no—
102#394—22no—
103#392—21no—
104#386—21no—
105#380C2×D311yescharacteristic
106C2×#95—31no—
107#340—62no—
108#327—62no—
109#323—62no—
110C24×C3—21no—
111C24×C3—21no—
112C24×C3—21no—
113C24×C3—21no—
114C24×C3—21no—
115C24×C3—21no—
116C24×C3—21no—
117C24×C3—42no—
118C24×C3—42no—
119C24×C3SD1612yes—
120C24×C3C2×D411yescharacteristic
121C2×C3×D4—21no—
122C2×C3×D4—21no—
123C2×C3×D4—42no—
124C2×C3×D4—44no—
125C2×C3×D4—41no—
126C2×C3×D4—41no—
127C2×C3×D4—41no—
128C2×C3×D4—41no—
129C2×C3×D4—41no—
130C2×C3×D4—41no—
131C2×C3×D4—82no—
132C2×C3×D4—82no—
133C22×C3×C4—21no—
134C22×C3×C4—21no—
135C22×C3×C4—22no—
136C22×C3×C4—21no—
137C22×C3×C4—21no—
138C22×C3×C4—42no—
139C22×C3×C4—42no—
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4#4111yescharacteristic
146C22×Dic3—24no—
147C22×Dic3—22no—
148C22×Dic3—22no—
149C22×Dic3—42no—
150C22×Dic3—41no—
151C22×Dic3—41no—
152C22×Dic3—42no—
153C22×Dic3—42no—
154C22×Dic3—41no—
155C22×Dic3—41no—
156C3×C4⋊C4—21no—
157C3×C4⋊C4—21no—
158C3×C4⋊C4—42no—
159C3×C22⋊C4—44no—
160C3×C22⋊C4—41no—
161C3×C22⋊C4—41no—
162C3×C22⋊C4—41no—
163C3×C22⋊C4—41no—
164C3×C22⋊C4—41no—
165C3×C22⋊C4—41no—
166C3×C22⋊C4—41no—
167C3×C22⋊C4—41no—
168C3×C22⋊C4—82no—
169#217—48no—
170C4⋊Dic3—22no—
171C4⋊Dic3—22no—
172C4⋊Dic3—41no—
173C4⋊Dic3—41no—
174Dic3⋊C4—48no—
175C2×C3⋊C8—22no—
176C2×C3⋊C8—42no—
177C2×C3⋊C8—42no—
178C2×C3⋊C8—41no—
179C25#6711yescharacteristic
180C22×D4—21no—
181C22×D4—22no—
182C22×D4#6711yescharacteristic
183C23×C4—61no—
184C23×C4—61no—
185C23×C4#6711yescharacteristic
186C23×C4C22×D311yescharacteristic
187C22×C8—61no—
188C22×C8—61no—
189#121—21no—
190#121—21no—
191#121—21no—
192#121—21no—
193#120—42no—
194#120—42no—
195C2×C4⋊C4—32no—
196C2×C4⋊C4—61no—
197C2×C4⋊C4—61no—
198C2×C4⋊C4—61no—
199C2×C4⋊C4—61no—
200C2×C4⋊C4—64no—
201C2×C4⋊C4—61no—
202C2×C4⋊C4—124no—
203C2×C4⋊C4#6711yescharacteristic
204C2×C4⋊C4#6712yes—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—64no—
216C2×C22⋊C4—124no—
217#106—64no—
218#102—64no—
219C22⋊C8—32no—
220C22⋊C8—61no—
221C22⋊C8—61no—
222C22⋊C8—61no—
223#95—22no—
224#95—21no—
225#95—32no—
226#95—61no—
227C23×C3—22no—
228C23×C3—22no—
229C23×C3—22no—
230C23×C3—22no—
231C23×C3—22no—
232C23×C3—44no—
233C23×C3—42no—
234C23×C3—44no—
235C23×C3—42no—
236C23×C3—41no—
237C23×C3—41no—
238C23×C3—41no—
239C23×C3—42no—
240C23×C3—42no—
241C23×C3—41no—
242C23×C3—41no—
243C23×C3—41no—
244C23×C3—41no—
245C23×C3—42no—
246C23×C3—42no—
247C23×C3—82no—
248C23×C3—82no—
249C23×C3—81no—
250C23×C3—81no—
251C23×C3#12011yescharacteristic
252C23×C3#12011yescharacteristic
253C23×C3#12111yescharacteristic
254C23×C3C2×SD1611yescharacteristic
255C23×C3#13511yescharacteristic
256C23×C3AGL(1,ℤ/8)11yescharacteristic
257C23×C3#13711yescharacteristic
258C3×D4—42no—
259C3×D4—81no—
260C3×D4—88no—
261C3×D4—81no—
262C3×D4—81no—
263C3×D4—81no—
264C3×D4—81no—
265C2×C3×C4—22no—
266C2×C3×C4—22no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—41no—
270C2×C3×C4—41no—
271C2×C3×C4—42no—
272C2×C3×C4—41no—
273C2×C3×C4—41no—
274C2×C3×C4—42no—
275C2×C3×C4—41no—
276C2×C3×C4—41no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4—42no—
280C2×C3×C4—81no—
281C2×C3×C4#12011yescharacteristic
282C2×C3×C4#12111yescharacteristic
283C2×C3×C4#12111yescharacteristic
284C2×C3×C4#12711yescharacteristic
285C2×Dic3—24no—
286C2×Dic3—44no—
287C2×Dic3—44no—
288C2×Dic3—42no—
289C2×Dic3—42no—
290C2×Dic3—42no—
291C2×Dic3—42no—
292C2×Dic3—42no—
293C2×Dic3—82no—
294C2×Dic3—82no—
295C3⋊C8—42no—
296C3⋊C8—81no—
297C24—21no—
298C24—21no—
299C24—21no—
300C24—21no—
301C24—21no—
302C24—21no—
303C24—21no—
304C24—42no—
305C24—42no—
306C24#21412yes—
307C24C2×#6711yescharacteristic
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—42no—
311C2×D4—44no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—82no—
319C2×D4—82no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—64no—
328C22×C4—62no—
329C22×C4—62no—
330C22×C4—122no—
331C22×C4—121no—
332C22×C4—121no—
333C22×C4—122no—
334C22×C4—122no—
335C22×C4—121no—
336C22×C4—121no—
337C22×C4D3×D411yescharacteristic
338C22×C4D3×D411yescharacteristic
339C22×C4D3×D411yescharacteristic
340C22×C4#23711yescharacteristic
341C22×C4C2×#6711yescharacteristic
342C22×C4C2×#6711yescharacteristic
343C2×C8—62no—
344C2×C8—122no—
345C2×C8—122no—
346C2×C8—121no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—42no—
350C4⋊C4—62no—
351C4⋊C4—62no—
352C4⋊C4—128no—
353C4⋊C4—121no—
354C4⋊C4—121no—
355C22⋊C4—44no—
356C22⋊C4—41no—
357C22⋊C4—41no—
358C22⋊C4—41no—
359C22⋊C4—41no—
360C22⋊C4—41no—
361C22⋊C4—41no—
362C22⋊C4—41no—
363C22⋊C4—41no—
364C22⋊C4—82no—
365C22⋊C4—128no—
366C22×C3—22no—
367C22×C3—22no—
368C22×C3—22no—
369C22×C3—22no—
370C22×C3—21no—
371C22×C3—21no—
372C22×C3—21no—
373C22×C3—21no—
374C22×C3—44no—
375C22×C3—44no—
376C22×C3—42no—
377C22×C3—82no—
378C22×C3—82no—
379C22×C3—82no—
380C22×C3—82no—
381C22×C3—81no—
382C22×C3—81no—
383C22×C3—82no—
384C22×C3—82no—
385C22×C3—81no—
386C22×C3—81no—
387C22×C3—81no—
388C22×C3#39211yescharacteristic
389C22×C3#44811yescharacteristic
390C22×C3#44911yescharacteristic
391C22×C3#45011yescharacteristic
392C22×C3#45211yescharacteristic
393C22×C3#45312yes—
394C22×C3#45812yes—
395C22×C3#46511yescharacteristic
396C22×C3#46811yescharacteristic
397C3×C4—21no—
398C3×C4—21no—
399C3×C4—21no—
400C3×C4—21no—
401C3×C4—82no—
402C3×C4—81no—
403C3×C4—81no—
404Dic3—44no—
405Dic3—82no—
406C23—22no—
407C23—22no—
408C23—22no—
409C23—22no—
410C23—22no—
411C23—44no—
412C23—42no—
413C23—42no—
414C23—44no—
415C23—41no—
416C23—41no—
417C23—42no—
418C23—42no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—42no—
425C23—42no—
426C23—82no—
427C23—81no—
428C23—81no—
429C23—82no—
430C23#93211yescharacteristic
431C23C2×#21411yescharacteristic
432C23#93711yescharacteristic
433C23#93911yescharacteristic
434C23#95011yescharacteristic
435C23#95111yescharacteristic
436C23#95311yescharacteristic
437D4—42no—
438D4—81no—
439D4—88no—
440D4—81no—
441D4—81no—
442D4—81no—
443D4—81no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—41no—
452C2×C4—42no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—64no—
460C2×C4—81no—
461C2×C4—124no—
462C2×C4—122no—
463C2×C4—124no—
464C2×C4—122no—
465C2×C4—122no—
466C2×C4—122no—
467C2×C4—122no—
468C2×C4—242no—
469C2×C4—242no—
470C2×C4#93711yescharacteristic
471C2×C4#93811yescharacteristic
472C2×C4#93911yescharacteristic
473C2×C4#94011yescharacteristic
474C8—122no—
475C8—241no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—21no—
480C2×C3—82no—
481C2×C3—161no—
482C2×C3#161312yes—
483C2×C3#161812yes—
484C2×C3#200011yescharacteristic
485C2×C3#200211yescharacteristic
486C2×C3#202211yescharacteristic
487C22—22no—
488C22—22no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—22no—
494C22—22no—
495C22—44no—
496C22—42no—
497C22—44no—
498C22—82no—
499C22—82no—
500C22—82no—
501C22—81no—
502C22—81no—
503C22—82no—
504C22—81no—
505C22—82no—
506C22—81no—
507C22—81no—
508C22—82no—
509C22#503711yescharacteristic
510C22#504011yescharacteristic
511C22#504111yescharacteristic
512C22#504211yescharacteristic
513C22#521811yescharacteristic
514C22#522311yescharacteristic
515C22#523911yescharacteristic
516C22#524112yes—
517C22#524312yes—
518C4—21no—
519C4—21no—
520C4—21no—
521C4—21no—
522C4—81no—
523C4—82no—
524C4—81no—
525C4—124no—
526C4—242no—
527C3#992011yescharacteristic, complemented, Sylow
528C2—21no—
529C2—21no—
530C2—21no—
531C2—21no—
532C2—82no—
533C2—161no—
534C2#7168511yescharacteristic
535C2#7184412yes—
536C2#7184911yescharacteristic
537C2#7185611yescharacteristic
538C2#7188312yes—
subgroupG/H
centerC23#953
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
Fitting, prime radicalC3×#2617C2
socleC23×C3#120
hypercenter#2617D3
hypocenterC3#9920
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2617

Complements

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

NH ≅ G/Ntype
2. C2×#5034C2semidirect
3. C2×#4949C2semidirect
4. C2×#4537C2semidirect
5. C2×#4525C2semidirect
6. #69502C2semidirect
7. #67624C2semidirect
72. C2×C4⋊Dic3D4semidirect
81. C3⋊#95D4semidirect
85. #830D4semidirect
527. C3#9920semidirect

Maximal subgroups

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

subgroupindex
C3×#26172
C2×#50342
C2×#49492
C2×#45372
C2×#45252
#695022
#676242
#99203