Finite Group Databasebeta

← #10695427

Subgroups — #10695427

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
1#82130C211yesmaximal, characteristic
2C2×#4949C211yesmaximal, characteristic, complemented
3C2×#4735C211yesmaximal, characteristic, complemented
4C2×C3×#323C211yesmaximal, characteristic, complemented
5C2×#4470C211yesmaximal, characteristic, complemented
6#69925C211yesmaximal, characteristic, complemented
7#67675C211yesmaximal, characteristic, complemented
8#9920—31nomaximal, Sylow
9C22×C3×C22⋊C4C2211yescharacteristic
10C2×#953—21no—
11C22×#217—21no—
12C2×#930C2211yescharacteristic
13C22×C4⋊Dic3C2211yescharacteristic
14C22×Dic3⋊C4—21no—
15C2×C3×C22⋊C8C2211yescharacteristic
16C2×C3×C22⋊C8C2211yescharacteristic
17#5213—21no—
18#5212C2211yescharacteristic
19C2×C3⋊#95C2211yescharacteristic
20C2×#821—21no—
21C2×#817—21no—
22#4961—22no—
23#4960—21no—
24#4953—21no—
25#4949—22no—
26#4949—22no—
27#4735—22no—
28#4735—22no—
29C3×#323—22no—
30#4551—22no—
31#4470—22no—
32C2×#481—31no—
33#2617—31no—
34C2×#399—31no—
35#1824—31no—
36C2×#327—31no—
37C2×#323D311yescharacteristic, complemented
38#1294—31no—
39C25×C3D411yescharacteristic
40C23×C3×C4D411yescharacteristic
41C23×C3×C4C2311yescharacteristic
42C22×#67—21no—
43C22×#67—22no—
44C23×Dic3—21no—
45C23×Dic3—21no—
46C22×D12D411yescharacteristic
47C22×C3×C8—21no—
48C22×C3×C8—21no—
49C2×C3×C22⋊C4—21no—
50C2×C3×C22⋊C4—21no—
51C2×C3×C22⋊C4—21no—
52C2×C3×C22⋊C4—21no—
53C2×C3×C22⋊C4—21no—
54C2×C3×C22⋊C4—21no—
55#953—42no—
56#953—42no—
57C2×#217—24no—
58C2×#217—22no—
59C2×#217—21no—
60C2×#217—44no—
61#930—21no—
62#930—21no—
63#930—21no—
64#930—21no—
65C2×(C2×D3)⋊C4—21no—
66C2×(C2×D3)⋊C4—22no—
67C2×C4⋊Dic3—21no—
68C2×C4⋊Dic3—21no—
69C2×C4⋊Dic3—21no—
70C2×C4⋊Dic3—21no—
71C2×C4⋊Dic3—21no—
72C2×C4⋊Dic3D411yescharacteristic
73C2×C4⋊Dic3D412yes—
74C2×C4⋊Dic3D412yescomplemented
75C2×Dic3⋊C4—24no—
76C2×Dic3⋊C4—44no—
77C3×C22⋊C8—21no—
78C3×C22⋊C8—21no—
79C3×C22⋊C8—21no—
80C3×C22⋊C8D412yescomplemented
81C3⋊#95—22no—
82C3⋊#95—21no—
83C3⋊#95—21no—
84C3⋊#95D412yescomplemented
85#821—24no—
86#817—24no—
87C2×#121—31no—
88C2×#120—61no—
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⋊C8C2×D311yescharacteristic
98C2×C22⋊C8C2×D311yescharacteristic
99#399—62no—
100#399—62no—
101#397—61no—
102#394—62no—
103#392—61no—
104#386—61no—
105#380—31no—
106C2×#95—31no—
107#340—62no—
108#327—62no—
109#323—22no—
110C24×C3—21no—
111C24×C3—21no—
112C24×C3—21no—
113C24×C3—21no—
114C24×C3—21no—
115C24×C3—21no—
116C24×C3—42no—
117C24×C3—42no—
118C24×C3SD1612yes—
119C24×C3C2×D411yescharacteristic
120C23×D3—21no—
121C22×C3×C4—21no—
122C22×C3×C4—21no—
123C22×C3×C4—21no—
124C22×C3×C4—42no—
125C22×C3×C4—42no—
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4#4111yescharacteristic
132C2×#67—44no—
133C2×#67—41no—
134C2×#67—41no—
135C2×#67—41no—
136C2×#67—41no—
137C2×#67—41no—
138C2×#67—41no—
139C2×#67—82no—
140C2×#67—82no—
141C22×Dic3—24no—
142C22×Dic3—22no—
143C22×Dic3—21no—
144C22×Dic3—22no—
145C22×Dic3—22no—
146C22×Dic3—42no—
147C22×Dic3—41no—
148C22×Dic3—41no—
149C22×Dic3—42no—
150C22×Dic3—42no—
151C22×Dic3—41no—
152C22×Dic3—41no—
153C2×D12—21no—
154C2×D12—21no—
155C2×D12—42no—
156C2×C3×C8—22no—
157C2×C3×C8—42no—
158C2×C3×C8—42no—
159C2×C3×C8—41no—
160C3×C22⋊C4—41no—
161C3×C22⋊C4—41no—
162C3×C22⋊C4—41no—
163C3×C22⋊C4—41no—
164#217—48no—
165#217—44no—
166#217—41no—
167#217—41no—
168(C2×D3)⋊C4—41no—
169(C2×D3)⋊C4—41no—
170(C2×D3)⋊C4—82no—
171C4⋊Dic3—21no—
172C4⋊Dic3—22no—
173C4⋊Dic3—22no—
174C4⋊Dic3—21no—
175C4⋊Dic3—41no—
176C4⋊Dic3—41no—
177C4⋊Dic3—42no—
178Dic3⋊C4—48no—
179C25D1211yescharacteristic
180C22×D4—31no—
181C22×D4—61no—
182C22×D4—62no—
183C23×C4—61no—
184C23×C4—61no—
185C23×C4D1211yescharacteristic
186C23×C4C22×D311yescharacteristic
187C22×C8—21no—
188C22×C8—21no—
189#121—61no—
190#121—61no—
191#121—61no—
192#121—61no—
193#120—122no—
194#120—122no—
195C2×C4⋊C4—32no—
196C2×C4⋊C4—31no—
197C2×C4⋊C4—32no—
198C2×C4⋊C4—61no—
199C2×C4⋊C4—61no—
200C2×C4⋊C4—61no—
201C2×C4⋊C4—61no—
202C2×C4⋊C4—64no—
203C2×C4⋊C4—61no—
204C2×C4⋊C4—124no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—64no—
212C2×C22⋊C4—62no—
213C2×C22⋊C4—61no—
214C2×C22⋊C4—61no—
215C2×C22⋊C4—62no—
216C2×C22⋊C4—124no—
217#106—64no—
218#102—64no—
219C22⋊C8—21no—
220C22⋊C8—21no—
221C22⋊C8—21no—
222C22⋊C8#6712yescomplemented
223#95—32no—
224#95—62no—
225#95—61no—
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—82no—
243C23×C3—82no—
244C23×C3—81no—
245C23×C3—81no—
246C23×C3#12011yescharacteristic
247C23×C3#12011yescharacteristic
248C23×C3#12111yescharacteristic
249C23×C3C2×SD1611yescharacteristic
250C23×C3#13511yescharacteristic
251C23×C3AGL(1,ℤ/8)11yescharacteristic
252C23×C3#13711yescharacteristic
253C22×D3—41no—
254C22×D3—41no—
255C22×D3—41no—
256C22×D3—42no—
257C22×D3—42no—
258C2×C3×C4—22no—
259C2×C3×C4—22no—
260C2×C3×C4—41no—
261C2×C3×C4—41no—
262C2×C3×C4—41no—
263C2×C3×C4—41no—
264C2×C3×C4—42no—
265C2×C3×C4—41no—
266C2×C3×C4—42no—
267C2×C3×C4—81no—
268C2×C3×C4#12011yescharacteristic
269C2×C3×C4#12111yescharacteristic
270C2×C3×C4#12111yescharacteristic
271C2×C3×C4#12711yescharacteristic
272#67—88no—
273#67—81no—
274#67—81no—
275#67—81no—
276#67—81no—
277C2×Dic3—24no—
278C2×Dic3—44no—
279C2×Dic3—44no—
280C2×Dic3—42no—
281C2×Dic3—42no—
282C2×Dic3—42no—
283C2×Dic3—42no—
284C2×Dic3—42no—
285C2×Dic3—41no—
286C2×Dic3—42no—
287C2×Dic3—41no—
288C2×Dic3—41no—
289C2×Dic3—42no—
290C2×Dic3—42no—
291C2×Dic3—82no—
292C2×Dic3—82no—
293D12—42no—
294D12—81no—
295C3×C8—42no—
296C3×C8—81no—
297C24—21no—
298C24—21no—
299C24—21no—
300C24—21no—
301C24—21no—
302C24—21no—
303C24—42no—
304C24—42no—
305C24—61no—
306C24#20412yes—
307C24C2×D1211yescharacteristic
308C2×D4—61no—
309C2×D4—61no—
310C2×D4—122no—
311C2×D4—124no—
312C2×D4—121no—
313C2×D4—121no—
314C2×D4—121no—
315C2×D4—121no—
316C2×D4—121no—
317C2×D4—121no—
318C2×D4—242no—
319C2×D4—242no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—64no—
326C22×C4—62no—
327C22×C4—62no—
328C22×C4—61no—
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×C4#23511yescharacteristic
338C22×C4D3×D411yescharacteristic
339C22×C4D3×D411yescharacteristic
340C22×C4D3×D411yescharacteristic
341C22×C4D3×D411yescharacteristic
342C22×C4C2×#6711yescharacteristic
343C2×C8—22no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—41no—
347C4⋊C4—62no—
348C4⋊C4—61no—
349C4⋊C4—62no—
350C4⋊C4—61no—
351C4⋊C4—122no—
352C4⋊C4—128no—
353C4⋊C4—121no—
354C4⋊C4—121no—
355C22⋊C4—41no—
356C22⋊C4—41no—
357C22⋊C4—41no—
358C22⋊C4—41no—
359C22⋊C4—128no—
360C22⋊C4—124no—
361C22⋊C4—121no—
362C22⋊C4—121no—
363C22⋊C4—121no—
364C22⋊C4—121no—
365C22⋊C4—242no—
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#39211yescharacteristic
384C22×C3#44811yescharacteristic
385C22×C3#44911yescharacteristic
386C22×C3#45011yescharacteristic
387C22×C3#45211yescharacteristic
388C22×C3#45312yes—
389C22×C3#45812yes—
390C22×C3#46511yescharacteristic
391C22×C3#46811yescharacteristic
392C2×D3—82no—
393C2×D3—82no—
394C2×D3—81no—
395C2×D3—81no—
396C2×D3—81no—
397C3×C4—21no—
398C3×C4—21no—
399C3×C4—21no—
400C3×C4—21no—
401C3×C4—81no—
402Dic3—44no—
403Dic3—82no—
404Dic3—81no—
405Dic3—82no—
406C23—22no—
407C23—22no—
408C23—22no—
409C23—22no—
410C23—22no—
411C23—44no—
412C23—42no—
413C23—44no—
414C23—42no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—42no—
420C23—42no—
421C23—82no—
422C23—82no—
423C23—81no—
424C23—81no—
425C23—121no—
426C23—121no—
427C23—121no—
428C23—122no—
429C23—122no—
430C23#88211yescharacteristic
431C23#88211yescharacteristic
432C23C2×#20411yescharacteristic
433C23#90411yescharacteristic
434C23#90811yescharacteristic
435C23#90911yescharacteristic
436C23#93011yescharacteristic
437D4—122no—
438D4—241no—
439D4—248no—
440D4—241no—
441D4—241no—
442D4—241no—
443D4—241no—
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—64no—
454C2×C4—81no—
455C2×C4—124no—
456C2×C4—124no—
457C2×C4—122no—
458C2×C4—122no—
459C2×C4—122no—
460C2×C4—122no—
461C2×C4—121no—
462C2×C4—122no—
463C2×C4—122no—
464C2×C4—121no—
465C2×C4—121no—
466C2×C4—122no—
467C2×C4—122no—
468C2×C4—242no—
469C2×C4—242no—
470C2×C4#88411yescharacteristic
471C2×C4#88411yescharacteristic
472C2×C4#93711yescharacteristic
473C2×C4#94011yescharacteristic
474C8—42no—
475C8—81no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—21no—
480C2×C3—82no—
481C2×C3#161312yes—
482C2×C3#161812yes—
483C2×C3#200011yescharacteristic
484C2×C3#200211yescharacteristic
485C2×C3#202211yescharacteristic
486D3—161no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—21no—
495C22—44no—
496C22—44no—
497C22—42no—
498C22—82no—
499C22—82no—
500C22—82no—
501C22—82no—
502C22—81no—
503C22—81no—
504C22—242no—
505C22—242no—
506C22—241no—
507C22—241no—
508C22—241no—
509C22#473111yescharacteristic
510C22#473311yescharacteristic
511C22#473411yescharacteristic
512C22#473811yescharacteristic
513C22#474212yes—
514C22#474712yes—
515C22#496011yescharacteristic
516C22#511411yescharacteristic
517C22#513411yescharacteristic
518C4—21no—
519C4—21no—
520C4—21no—
521C4—21no—
522C4—81no—
523C4—124no—
524C4—242no—
525C4—241no—
526C4—242no—
527C3#992011yescharacteristic, complemented, Sylow
528C2—21no—
529C2—21no—
530C2—21no—
531C2—21no—
532C2—82no—
533C2—481no—
534C2#6811212yes—
535C2#6811312yes—
536C2#6996211yescharacteristic
537C2#7003111yescharacteristic
538C2#7008211yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC2×C3×#323C2
socleC23×C3#120
prime radical#82130C2
hypercenterC2×#323D3
hypocenterC3#9920
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#323

Complements

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

NH ≅ G/Ntype
2. C2×#4949C2semidirect
3. C2×#4735C2semidirect
4. C2×C3×#323C2semidirect
5. C2×#4470C2semidirect
6. #69925C2semidirect
7. #67675C2semidirect
37. C2×#323D3semidirect
74. C2×C4⋊Dic3D4semidirect
80. C3×C22⋊C8D4semidirect
84. C3⋊#95D4semidirect
222. C22⋊C8#67semidirect
527. C3#9920semidirect

Maximal subgroups

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

subgroupindex
#821302
C2×#49492
C2×#47352
C2×C3×#3232
C2×#44702
#699252
#676752
#99203