Finite Group Databasebeta

← #10690311

Subgroups — #10690311

Order 768.

834 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4738C211yesmaximal, characteristic
2#73873C211yesmaximal, characteristic
3#69525C211yesmaximal, characteristic, complemented
4#69424C211yesmaximal, characteristic, complemented
5C2×#4451C211yesmaximal, characteristic
6C3×#1294C211yesmaximal, characteristic, complemented
7#67674C211yesmaximal, characteristic, complemented
8#9039—31nomaximal, Sylow
9C23×Dic6C2211yescharacteristic
10C2×#930C2211yescharacteristic
11C22×#204—21no—
12C2×#67⋊C4—21no—
13C2×C3×C22⋊C8C2211yescharacteristic
14C3×#380C2211yescharacteristic
15C2×C3×#95C2211yescharacteristic
16C2×C3⋊#95—21no—
17C2×C3⋊#95—21no—
18C2×C3⋊#95C2211yescharacteristic
19C2×#830C2211yescharacteristic
20C2×#816—21no—
21#4990—21no—
22#4989—21no—
23C2×#810—21no—
24C2×#810—21no—
25#4958—21no—
26#4738—22no—
27#4738C414yescomplemented
28#4665—21no—
29#4654—21no—
30#4451—22no—
31C2×#448—31no—
32#2325—31no—
33#1787—31no—
34C2×#327—31no—
35#1446—31no—
36#1294D311yescharacteristic, complemented
37#1280—31no—
38C23×C3×C4—21no—
39C23×C3×C4C2311yescharacteristic
40C22×#67—21no—
41C23×Dic3—21no—
42C23×Dic3D411yescharacteristic
43C23×Dic3D411yescharacteristic
44C22×D12D411yescharacteristic
45C22×C4×D3—21no—
46C22×Dic6—21no—
47C22×Dic6—21no—
48C22×Dic6—21no—
49C22×Dic6—21no—
50C22×Dic6—21no—
51C22×Dic6—42no—
52C22×Dic6—42no—
53C22×Dic6C2×C412yescomplemented
54C22×C3×C8—21no—
55C2×C3×C4⋊C4D411yescharacteristic
56C2×C3×C4⋊C4D411yescharacteristic
57C2×C3×C22⋊C4—21no—
58#930—21no—
59#930—21no—
60#930C2×C412yes—
61#930D412yes—
62C2×(C2×D3)⋊C4—21no—
63C2×(C2×D3)⋊C4—21no—
64C2×(C2×D3)⋊C4—21no—
65C2×(C2×D3)⋊C4—21no—
66C2×C4⋊Dic3D411yescharacteristic
67C2×Dic3⋊C4—21no—
68C2×C4×Dic3—21no—
69C22×C3⋊C8—21no—
70C2×#204—24no—
71C2×#204—42no—
72C2×#204—42no—
73#67⋊C4—42no—
74#67⋊C4—42no—
75C3×C22⋊C8—21no—
76C3×C22⋊C8C2×C412yescomplemented
77C3×#95—21no—
78C3×#95—21no—
79C3×#95—21no—
80C3⋊#95—21no—
81C3⋊#95—21no—
82C3⋊#95—21no—
83C3⋊#95—21no—
84C3⋊#95—42no—
85C3⋊#95—42no—
86#830—21no—
87#830—21no—
88#816—24no—
89#810—42no—
90#810—42no—
91C23×Q8—31no—
92C22×SD16—61no—
93C2×#121—31no—
94C2×C4×D4—61no—
95#448—34no—
96#448—62no—
97C2×#103—61no—
98C2×#103—61no—
99C2×#103—61no—
100C2×C22⋊C8—31no—
101C2×C22⋊C8C2×D311yescharacteristic
102#388—61no—
103#387—61no—
104#386—61no—
105#386—61no—
106#385—61no—
107#380C2×D311yescharacteristic
108C2×#95—31no—
109C2×#95—61no—
110C2×#95—61no—
111C2×#95C2×D311yescharacteristic
112#327—62no—
113C24×C3C2×D411yescharacteristic
114C23×D3—21no—
115C22×C3×C4—22no—
116C22×C3×C4—21no—
117C22×C3×C4—21no—
118C22×C3×C4—41no—
119C22×C3×C4—41no—
120C22×C3×C4—41no—
121C22×C3×C4—41no—
122C22×C3×C4—41no—
123C22×C3×C4—41no—
124C22×C3×C4—41no—
125C22×C3×C4—41no—
126C22×C3×C4C22×C411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4#4111yescharacteristic
131C22×C3×C4#4111yescharacteristic
132C2×#67—22no—
133C2×#67—22no—
134C2×#67—41no—
135C2×#67—41no—
136C2×#67—41no—
137C2×#67—41no—
138C22×Dic3—22no—
139C22×Dic3—22no—
140C22×Dic3—22no—
141C22×Dic3—21no—
142C22×Dic3—21no—
143C22×Dic3—21no—
144C22×Dic3—21no—
145C22×Dic3—21no—
146C22×Dic3—21no—
147C22×Dic3—41no—
148C22×Dic3—41no—
149C22×Dic3—41no—
150C22×Dic3—41no—
151C22×Dic3—41no—
152C22×Dic3—41no—
153C22×Dic3—41no—
154C22×Dic3—41no—
155C22×Dic3SD1612yescomplemented
156C2×D12—21no—
157C2×D12—41no—
158C2×D12—41no—
159C2×D12C22⋊C412yes—
160C2×C4×D3—42no—
161C2×C4×D3—81no—
162C2×C4×D3—81no—
163C2×Dic6—22no—
164C2×Dic6—22no—
165C2×Dic6—22no—
166C2×Dic6—22no—
167C2×Dic6—22no—
168C2×Dic6—22no—
169C2×Dic6—44no—
170C2×Dic6—44no—
171C2×Dic6—44no—
172C2×Dic6—44no—
173C2×Dic6—41no—
174C2×Dic6—41no—
175C2×Dic6—41no—
176C2×Dic6—41no—
177C2×Dic6—41no—
178C2×Dic6—41no—
179C2×C3×C8—22no—
180C2×C3×C8—41no—
181C2×C3×C8—41no—
182C3×C4⋊C4—41no—
183C3×C4⋊C4—41no—
184C3×C22⋊C4—41no—
185C3×C22⋊C4—41no—
186(C2×D3)⋊C4—22no—
187(C2×D3)⋊C4—22no—
188(C2×D3)⋊C4—42no—
189(C2×D3)⋊C4—81no—
190(C2×D3)⋊C4—81no—
191C4⋊Dic3—21no—
192C4⋊Dic3C22⋊C412yes—
193Dic3⋊C4—42no—
194C4×Dic3—42no—
195C2×C3⋊C8—41no—
196C2×C3⋊C8—41no—
197C2×C3⋊C8—41no—
198#204—48no—
199C22×Q8—32no—
200C22×Q8—61no—
201C22×Q8—61no—
202C22×Q8—61no—
203C22×Q8—61no—
204C22×Q8—61no—
205C22×Q8—122no—
206C22×Q8—122no—
207C22×D4—31no—
208C22×D4—61no—
209C23×C4—21no—
210C23×C4—31no—
211C23×C4—31no—
212C23×C4—61no—
213C23×C4—61no—
214C23×C4C22×D311yescharacteristic
215C2×SD16—64no—
216C2×SD16—122no—
217C2×SD16—122no—
218C22×C8—21no—
219C22×C8—61no—
220#121—32no—
221#121—32no—
222#121—61no—
223#121—61no—
224C4×D4—122no—
225C4×D4—122no—
226C2×C4⋊C4—31no—
227C2×C4⋊C4—61no—
228C2×C4⋊C4D1211yescharacteristic
229C2×C4⋊C4D1211yescharacteristic
230C2×C22⋊C4—21no—
231C2×C22⋊C4—61no—
232C2×C22⋊C4—61no—
233C2×C22⋊C4—61no—
234C2×C22⋊C4—61no—
235C2×C42—61no—
236#103—64no—
237#103—122no—
238#103—122no—
239C22⋊C8—21no—
240C22⋊C8—61no—
241C22⋊C8—61no—
242C22⋊C8C4×D312yescomplemented
243#95—21no—
244#95—21no—
245#95—21no—
246#95—61no—
247#95—61no—
248#95—61no—
249#95—61no—
250#95—122no—
251#95—122no—
252C23×C3—21no—
253C23×C3—21no—
254C23×C3—21no—
255C23×C3—21no—
256C23×C3C4×D411yescharacteristic
257C23×C3#12011yescharacteristic
258C23×C3#12111yescharacteristic
259C23×C3C2×SD1611yescharacteristic
260C23×C3#13511yescharacteristic
261C23×C3AGL(1,ℤ/8)11yescharacteristic
262C23×C3#13711yescharacteristic
263C22×D3—22no—
264C22×D3—41no—
265C22×D3—41no—
266C22×D3—41no—
267C22×D3—41no—
268C22×D3—41no—
269C22×D3—41no—
270C2×C3×C4—21no—
271C2×C3×C4—21no—
272C2×C3×C4—21no—
273C2×C3×C4—21no—
274C2×C3×C4—41no—
275C2×C3×C4—41no—
276C2×C3×C4—41no—
277C2×C3×C4—41no—
278C2×C3×C4—42no—
279C2×C3×C4—42no—
280C2×C3×C4—42no—
281C2×C3×C4—41no—
282C2×C3×C4—41no—
283C2×C3×C4—41no—
284C2×C3×C4—41no—
285C2×C3×C4—41no—
286C2×C3×C4—41no—
287C2×C3×C4—81no—
288C2×C3×C4—81no—
289C2×C3×C4—81no—
290C2×C3×C4—81no—
291C2×C3×C4C2×C22⋊C411yescharacteristic
292C2×C3×C4C4×D411yescharacteristic
293C2×C3×C4#12111yescharacteristic
294C2×C3×C4#12311yescharacteristic
295#67—42no—
296#67—42no—
297#67—42no—
298#67—42no—
299C2×Dic3—24no—
300C2×Dic3—22no—
301C2×Dic3—22no—
302C2×Dic3—42no—
303C2×Dic3—42no—
304C2×Dic3—42no—
305C2×Dic3—42no—
306C2×Dic3—41no—
307C2×Dic3—41no—
308C2×Dic3—41no—
309C2×Dic3—42no—
310C2×Dic3—42no—
311C2×Dic3—41no—
312C2×Dic3—42no—
313C2×Dic3—42no—
314C2×Dic3—41no—
315C2×Dic3—41no—
316C2×Dic3—41no—
317C2×Dic3—81no—
318C2×Dic3—81no—
319D12—42no—
320D12—81no—
321C4×D3—84no—
322Dic6—24no—
323Dic6—24no—
324Dic6—42no—
325Dic6—42no—
326Dic6—42no—
327Dic6—42no—
328Dic6—42no—
329Dic6—42no—
330C3×C8—42no—
331C3⋊C8—81no—
332C24—61no—
333C24D3×D411yescharacteristic
334C2×Q8—62no—
335C2×Q8—62no—
336C2×Q8—62no—
337C2×Q8—62no—
338C2×Q8—62no—
339C2×Q8—62no—
340C2×Q8—121no—
341C2×Q8—121no—
342C2×Q8—121no—
343C2×Q8—121no—
344C2×Q8—124no—
345C2×Q8—124no—
346C2×Q8—124no—
347C2×Q8—124no—
348C2×Q8—121no—
349C2×Q8—121no—
350C2×D4—32no—
351C2×D4—62no—
352C2×D4—62no—
353C2×D4—61no—
354C2×D4—121no—
355C2×D4—121no—
356C2×D4—121no—
357C2×D4—121no—
358C2×D4—121no—
359C2×D4—121no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—32no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—62no—
373C22×C4—62no—
374C22×C4—61no—
375C22×C4—61no—
376C22×C4—62no—
377C22×C4—61no—
378C22×C4—61no—
379C22×C4—61no—
380C22×C4—61no—
381C22×C4—122no—
382C22×C4—121no—
383C22×C4—121no—
384C22×C4—121no—
385C22×C4—121no—
386C22×C4—121no—
387C22×C4—121no—
388C22×C4—121no—
389C22×C4—121no—
390C22×C4—241no—
391C22×C4—241no—
392C22×C4C2×C4×D311yescharacteristic
393C22×C4C2×D1211yescharacteristic
394C22×C4#23511yescharacteristic
395C22×C4D3×D411yescharacteristic
396C22×C4D3×D411yescharacteristic
397C22×C4#23911yescharacteristic
398SD16—128no—
399C2×C8—22no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—121no—
403C2×C8—121no—
404C2×C8—121no—
405C4⋊C4—32no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—61no—
409C4⋊C4—122no—
410C22⋊C4—41no—
411C22⋊C4—41no—
412C22⋊C4—62no—
413C22⋊C4—62no—
414C22⋊C4—122no—
415C22⋊C4—241no—
416C22⋊C4—241no—
417C42—122no—
418C22×C3—21no—
419C22×C3—21no—
420C22×C3—21no—
421C22×C3—21no—
422C22×C3—21no—
423C22×C3—21no—
424C22×C3—21no—
425C22×C3—21no—
426C22×C3—21no—
427C22×C3—21no—
428C22×C3—21no—
429C22×C3—21no—
430C22×C3#38611yescharacteristic
431C22×C3C4×SD1611yescharacteristic
432C22×C3#44011yescharacteristic
433C22×C3#45011yescharacteristic
434C22×C3#45211yescharacteristic
435C22×C3#45511yescharacteristic
436C22×C3#45611yescharacteristic
437C22×C3B(4,𝔽2)11yescharacteristic
438C22×C3B(4,𝔽2)11yescharacteristic
439C22×C3#46111yescharacteristic
440C22×C3#46311yescharacteristic
441C2×D3—42no—
442C2×D3—42no—
443C2×D3—42no—
444C2×D3—81no—
445C2×D3—81no—
446C2×D3—81no—
447C2×D3—81no—
448C3×C4—21no—
449C3×C4—21no—
450C3×C4—21no—
451C3×C4—21no—
452C3×C4—82no—
453C3×C4—81no—
454C3×C4—81no—
455Dic3—42no—
456Dic3—42no—
457Dic3—44no—
458Dic3—81no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—62no—
464C23—121no—
465C23—121no—
466C23—121no—
467C23—121no—
468C23—121no—
469C23—121no—
470C23#88211yescharacteristic
471C23#88411yescharacteristic
472C23#89311yescharacteristic
473C23#91111yescharacteristic
474C23D3×SD1611yescharacteristic
475C23#91611yescharacteristic
476C23#91811yescharacteristic
477Q8—64no—
478Q8—64no—
479Q8—122no—
480Q8—122no—
481Q8—122no—
482Q8—122no—
483Q8—122no—
484Q8—122no—
485D4—122no—
486D4—122no—
487D4—122no—
488D4—122no—
489D4—122no—
490D4—241no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—64no—
509C2×C4—62no—
510C2×C4—62no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—122no—
516C2×C4—122no—
517C2×C4—122no—
518C2×C4—122no—
519C2×C4—122no—
520C2×C4—122no—
521C2×C4—121no—
522C2×C4—121no—
523C2×C4—121no—
524C2×C4—121no—
525C2×C4—122no—
526C2×C4—122no—
527C2×C4—121no—
528C2×C4—121no—
529C2×C4—121no—
530C2×C4—244no—
531C2×C4—241no—
532C2×C4—241no—
533C2×C4C4×D1211yescharacteristic
534C2×C4D3×C22⋊C411yescharacteristic
535C2×C4#89411yescharacteristic
536C2×C4#89511yescharacteristic
537C8—42no—
538C8—241no—
539C2×C3—21no—
540C2×C3—21no—
541C2×C3—21no—
542C2×C3—21no—
543C2×C3#160111yescharacteristic
544C2×C3#160311yescharacteristic
545C2×C3#160411yescharacteristic
546C2×C3#160611yescharacteristic
547C2×C3#187711yescharacteristic
548C2×C3#188211yescharacteristic
549C2×C3#188911yescharacteristic
550D3—82no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—122no—
564C22—122no—
565C22—122no—
566C22—241no—
567C22—241no—
568C22—241no—
569C22—241no—
570C22#466511yescharacteristic
571C22#474111yescharacteristic
572C22#474111yescharacteristic
573C22#477611yescharacteristic
574C22#478711yescharacteristic
575C22#480811yescharacteristic
576C22#481811yescharacteristic
577C22#482511yescharacteristic
578C22#482611yescharacteristic
579C22#485211yescharacteristic
580C22#488911yescharacteristic
581C4—21no—
582C4—21no—
583C4—21no—
584C4—21no—
585C4—82no—
586C4—81no—
587C4—81no—
588C4—122no—
589C4—124no—
590C4—122no—
591C4—241no—
592C3#903911yescharacteristic, complemented, Sylow
593C2—21no—
594C2—21no—
595C2—21no—
596C2—21no—
597C2—242no—
598C2#6828611yescharacteristic
599C2#6829511yescharacteristic
600C2#6831911yescharacteristic
601C2#6832611yescharacteristic
602C2#6877511yescharacteristic
603C2#6889611yescharacteristic
604C2#6896211yescharacteristic
subgroupG/H
centerC23#882
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1294C2
socleC23×C3#120
prime radicalC2×#930C22
cosocleC23×C3×C4C23
hypercenter#1294D3
hypocenterC3#9039
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
3. #69525C2semidirect
4. #69424C2semidirect
6. C3×#1294C2semidirect
7. #67674C2semidirect
27. #4738C4semidirect
36. #1294D3semidirect
53. C22×Dic6C2×C4semidirect
76. C3×C22⋊C8C2×C4semidirect
155. C22×Dic3SD16semidirect
242. C22⋊C8C4×D3semidirect
592. C3#9039semidirect

Maximal subgroups

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

subgroupindex
C2×#47382
#738732
#695252
#694242
C2×#44512
C3×#12942
#676742
#90393