Finite Group Databasebeta

← #10702880

Subgroups — #10702880

Order 768.

962 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#823C211yesmaximal, characteristic
2C22×#823C211yesmaximal, characteristic
3#82269C211yesmaximal, characteristic
4C2×C3×#344C211yesmaximal, characteristic, complemented
5C2×#4559C211yesmaximal, characteristic, complemented
6C2×#4556C211yesmaximal, characteristic, complemented
7C2×#4533C211yesmaximal, characteristic, complemented
8#11170—31nomaximal, Sylow
9C23×D12C2211yescharacteristic
10C22×C3×M16C2211yescharacteristic
11C22×C3×M16C2211yescharacteristic
12C2×C3×#117C2211yescharacteristic
13C2×#926C2211yescharacteristic
14C22×#208C2211yescharacteristic
15C22×#208C2211yescharacteristic
16C2×#823—22no—
17C2×#823—22no—
18C2×#823—22no—
19C2×#823—22no—
20C2×#823—22no—
21C2×#823—22no—
22#4988—22no—
23#4955C414yes—
24#4939—22no—
25C3×#344—22no—
26#4559—22no—
27#4556—22no—
28#4533—22no—
29C22×#100—31no—
30C22×#100—31no—
31#2309—31no—
32C2×#344—31no—
33C2×#344—31no—
34C2×#344—31no—
35C2×#344D311yescharacteristic, complemented
36C24×D3C2×C412yes—
37C23×C3×C4C2311yescharacteristic
38C22×D12—21no—
39C22×D12—21no—
40C22×D12—21no—
41C22×D12—21no—
42C22×D12—21no—
43C22×D12—21no—
44C22×D12—21no—
45C22×D12—21no—
46C22×D12—21no—
47C22×D12—21no—
48C22×D12—21no—
49C22×D12—21no—
50C22×D12D412yes—
51C22×D12D412yes—
52C2×C3×M16—21no—
53C2×C3×M16—21no—
54C2×C3×M16—21no—
55C2×C3×M16—21no—
56C2×C3×M16—21no—
57C2×C3×M16—21no—
58C2×C3×M16—21no—
59C2×C3×M16—21no—
60C2×C3×M16—21no—
61C2×C3×M16—21no—
62C2×C3×M16D412yes—
63C2×C3×M16D412yes—
64C22×C3×C8—21no—
65C22×C3×C8—21no—
66C3×#117—21no—
67C3×#117—21no—
68C3×#117—21no—
69C3×#117—21no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C22⋊C4C2×C412yes—
72C2×C3×C42—21no—
73C2×#217C2×C412yes—
74C2×(C2×D3)⋊C4—28no—
75#926—21no—
76#926—21no—
77#926—21no—
78#926—21no—
79C2×C4⋊Dic3—21no—
80C2×C4×Dic3—21no—
81C2×#208—21no—
82C2×#208—21no—
83C2×#208—21no—
84C2×#208—21no—
85C2×#208—21no—
86C2×#208—21no—
87C2×#208—21no—
88C2×#208—21no—
89C2×#208—21no—
90C2×#208—21no—
91C2×#208D412yes—
92C2×#208D412yes—
93C22×C3⋊C8—21no—
94C22×C3⋊C8—21no—
95#823—28no—
96#823—28no—
97C23×D4—31no—
98C22×M16—31no—
99C22×M16—31no—
100C22×M16C2×D311yescharacteristic
101C22×M16C2×D311yescharacteristic
102C2×#117—31no—
103C2×#117C2×D311yescharacteristic
104C2×#100—62no—
105C2×#100—62no—
106C2×#100—62no—
107C2×#100—62no—
108C2×#100—62no—
109C2×#100—62no—
110#390—62no—
111#390—62no—
112#379—34no—
113#344—22no—
114#344—62no—
115#344—62no—
116#344—62no—
117C24×C3C22×C411yescharacteristic
118C23×D3—22no—
119C23×D3—22no—
120C23×D3—22no—
121C23×D3—22no—
122C23×D3—22no—
123C23×D3—22no—
124C23×D3—84no—
125C23×D3C22⋊C414yes—
126C22×C3×C4—22no—
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4#4111yescharacteristic
134C22×C3×C4#4111yescharacteristic
135C22×C3×C4#4111yescharacteristic
136C22×C3×C4#4111yescharacteristic
137C22×C3×C4#4111yescharacteristic
138C22×C3×C4#4111yescharacteristic
139C22×C3×C4#4111yescharacteristic
140C22×C3×C4#4111yescharacteristic
141C22×Dic3—22no—
142C2×D12—22no—
143C2×D12—22no—
144C2×D12—22no—
145C2×D12—22no—
146C2×D12—22no—
147C2×D12—22no—
148C2×D12—22no—
149C2×D12—22no—
150C2×D12—22no—
151C2×D12—22no—
152C2×D12—22no—
153C2×D12—22no—
154C2×D12—41no—
155C2×D12—41no—
156C2×D12—41no—
157C2×D12—41no—
158C2×D12—41no—
159C2×D12—41no—
160C2×D12—41no—
161C2×D12—41no—
162C2×D12—41no—
163C2×D12—41no—
164C2×D12—41no—
165C2×D12—41no—
166C2×D12—41no—
167C2×D12—41no—
168C2×D12—41no—
169C2×D12—41no—
170C3×M16—22no—
171C3×M16—22no—
172C3×M16—22no—
173C3×M16—22no—
174C3×M16—41no—
175C3×M16—41no—
176C3×M16—41no—
177C3×M16—41no—
178C2×C3×C8—22no—
179C2×C3×C8—22no—
180C2×C3×C8—41no—
181C2×C3×C8—41no—
182C2×C3×C8—41no—
183C2×C3×C8—41no—
184C3×C4⋊C4—41no—
185C3×C4⋊C4—41no—
186C3×C22⋊C4—22no—
187C3×C22⋊C4—22no—
188C3×C42—41no—
189C3×C42—41no—
190#217—22no—
191#217—22no—
192(C2×D3)⋊C4—88no—
193C4⋊Dic3—41no—
194C4⋊Dic3—41no—
195C4×Dic3—41no—
196C4×Dic3—41no—
197#208—22no—
198#208—22no—
199#208—22no—
200#208—22no—
201#208—41no—
202#208—41no—
203#208—41no—
204#208—41no—
205C2×C3⋊C8—22no—
206C2×C3⋊C8—22no—
207C2×C3⋊C8—41no—
208C2×C3⋊C8—41no—
209C2×C3⋊C8—41no—
210C2×C3⋊C8—41no—
211C25—32no—
212C22×D4—32no—
213C22×D4—32no—
214C22×D4—61no—
215C22×D4—61no—
216C22×D4—61no—
217C22×D4—61no—
218C22×D4—61no—
219C22×D4—61no—
220C22×D4—61no—
221C22×D4—61no—
222C22×D4—61no—
223C22×D4—61no—
224C22×D4—61no—
225C22×D4—61no—
226C23×C4C22×D311yescharacteristic
227C2×M16—21no—
228C2×M16—21no—
229C2×M16—21no—
230C2×M16—21no—
231C2×M16—21no—
232C2×M16—21no—
233C2×M16—21no—
234C2×M16—21no—
235C2×M16—21no—
236C2×M16—21no—
237C2×M16—32no—
238C2×M16—32no—
239C2×M16—61no—
240C2×M16—61no—
241C2×M16—61no—
242C2×M16—61no—
243C2×M16—61no—
244C2×M16—61no—
245C2×M16—61no—
246C2×M16—61no—
247C2×M16—61no—
248C2×M16—61no—
249C2×M16D1212yes—
250C2×M16#6712yes—
251C22×C8—21no—
252C22×C8—21no—
253C22×C8—61no—
254C22×C8—61no—
255#117—21no—
256#117—21no—
257#117—21no—
258#117—21no—
259#117—61no—
260#117—61no—
261#117—61no—
262#117—61no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—61no—
265C2×C22⋊C4—32no—
266C2×C22⋊C4—68no—
267C2×C22⋊C4C4×D312yes—
268C2×C42—21no—
269C2×C42—61no—
270#100—68no—
271#100—68no—
272C23×C3—42no—
273C23×C3C2×C22⋊C411yescharacteristic
274C23×C3#11711yescharacteristic
275C23×C3#11711yescharacteristic
276C23×C3C4×D411yescharacteristic
277C23×C3C4×D411yescharacteristic
278C23×C3C4×D411yescharacteristic
279C23×C3C4×D411yescharacteristic
280C22×D3—24no—
281C22×D3—24no—
282C22×D3—24no—
283C22×D3—42no—
284C22×D3—42no—
285C22×D3—42no—
286C22×D3—42no—
287C22×D3—48no—
288C22×D3—84no—
289C22×D3—84no—
290C22×D3—84no—
291C22×D3—84no—
292C22×D3—84no—
293C22×D3—84no—
294C2×C3×C4—44no—
295C2×C3×C4—42no—
296C2×C3×C4#12011yescharacteristic
297C2×C3×C4#12011yescharacteristic
298C2×C3×C4#12111yescharacteristic
299C2×C3×C4#12111yescharacteristic
300C2×C3×C4#12111yescharacteristic
301C2×C3×C4#12111yescharacteristic
302C2×C3×C4#12111yescharacteristic
303C2×C3×C4#12111yescharacteristic
304C2×C3×C4#12111yescharacteristic
305C2×C3×C4#12111yescharacteristic
306C2×C3×C4#12311yescharacteristic
307C2×C3×C4#12311yescharacteristic
308C2×C3×C4#12311yescharacteristic
309C2×C3×C4#12311yescharacteristic
310C2×C3×C4#12311yescharacteristic
311C2×C3×C4#12311yescharacteristic
312C2×C3×C4#12311yescharacteristic
313C2×C3×C4#12311yescharacteristic
314C2×C3×C4#12311yescharacteristic
315C2×C3×C4#12311yescharacteristic
316C2×C3×C4#12411yescharacteristic
317C2×C3×C4#12411yescharacteristic
318C2×C3×C4#12411yescharacteristic
319C2×C3×C4#12411yescharacteristic
320C2×C3×C4#12611yescharacteristic
321C2×C3×C4#12611yescharacteristic
322C2×C3×C4#12611yescharacteristic
323C2×C3×C4#12611yescharacteristic
324C2×Dic3—44no—
325C2×Dic3—42no—
326D12—42no—
327D12—42no—
328D12—42no—
329D12—42no—
330D12—42no—
331D12—42no—
332D12—42no—
333D12—42no—
334C3×C8—42no—
335C3×C8—42no—
336C3⋊C8—42no—
337C3⋊C8—42no—
338C24—34no—
339C24—62no—
340C24—62no—
341C24—62no—
342C24—62no—
343C24—62no—
344C24—62no—
345C24—244no—
346C24C2×C4×D311yescharacteristic
347C2×D4—62no—
348C2×D4—62no—
349C2×D4—62no—
350C2×D4—62no—
351C2×D4—62no—
352C2×D4—62no—
353C2×D4—62no—
354C2×D4—62no—
355C2×D4—62no—
356C2×D4—62no—
357C2×D4—62no—
358C2×D4—62no—
359C2×D4—121no—
360C2×D4—121no—
361C2×D4—121no—
362C2×D4—121no—
363C2×D4—121no—
364C2×D4—121no—
365C2×D4—121no—
366C2×D4—121no—
367C2×D4—121no—
368C2×D4—121no—
369C2×D4—121no—
370C2×D4—121no—
371C2×D4—121no—
372C2×D4—121no—
373C2×D4—121no—
374C2×D4—121no—
375C22×C4—22no—
376C22×C4—62no—
377C22×C4C2×D1211yescharacteristic
378C22×C4#23511yescharacteristic
379C22×C4#23511yescharacteristic
380C22×C4#23511yescharacteristic
381C22×C4#23511yescharacteristic
382C22×C4D3×D411yescharacteristic
383C22×C4D3×D411yescharacteristic
384C22×C4D3×D411yescharacteristic
385C22×C4D3×D411yescharacteristic
386C22×C4#23711yescharacteristic
387C22×C4#23711yescharacteristic
388C22×C4#23911yescharacteristic
389C22×C4#23911yescharacteristic
390C22×C4C2×#6711yescharacteristic
391M16—22no—
392M16—22no—
393M16—22no—
394M16—22no—
395M16—41no—
396M16—41no—
397M16—41no—
398M16—41no—
399M16—62no—
400M16—62no—
401M16—62no—
402M16—62no—
403M16—121no—
404M16—121no—
405M16—121no—
406M16—121no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—62no—
414C2×C8—62no—
415C2×C8—121no—
416C2×C8—121no—
417C2×C8—121no—
418C2×C8—121no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—121no—
422C4⋊C4—121no—
423C22⋊C4—22no—
424C22⋊C4—22no—
425C22⋊C4—62no—
426C22⋊C4—62no—
427C22⋊C4—248no—
428C42—41no—
429C42—41no—
430C42—121no—
431C42—121no—
432C22×C3—22no—
433C22×C3—22no—
434C22×C3—22no—
435C22×C3—41no—
436C22×C3—41no—
437C22×C3—41no—
438C22×C3—41no—
439C22×C3#38111yescharacteristic
440C22×C3#38611yescharacteristic
441C22×C3#38611yescharacteristic
442C22×C3#38811yescharacteristic
443C22×C3#38811yescharacteristic
444C22×C3#38811yescharacteristic
445C22×C3#38811yescharacteristic
446C2×D3—48no—
447C2×D3—44no—
448C2×D3—84no—
449C2×D3—84no—
450C2×D3—84no—
451C2×D3—84no—
452C2×D3—84no—
453C2×D3—84no—
454C3×C4—82no—
455C3×C4#39411yescharacteristic
456C3×C4#39411yescharacteristic
457C3×C4#39411yescharacteristic
458C3×C4#39411yescharacteristic
459C3×C4#39711yescharacteristic
460C3×C4#39711yescharacteristic
461C3×C4#39711yescharacteristic
462C3×C4#39711yescharacteristic
463Dic3—82no—
464C23—42no—
465C23—64no—
466C23—64no—
467C23—64no—
468C23—128no—
469C23—122no—
470C23—122no—
471C23—122no—
472C23—122no—
473C23—244no—
474C23—244no—
475C23—244no—
476C23—244no—
477C23—244no—
478C23—244no—
479C23#87211yescharacteristic
480C23C4×D1211yescharacteristic
481C23D3×C22⋊C411yescharacteristic
482C23#67⋊C411yescharacteristic
483C23#89211yescharacteristic
484C23#89311yescharacteristic
485C23C4×#6711yescharacteristic
486D4—122no—
487D4—122no—
488D4—122no—
489D4—122no—
490D4—122no—
491D4—122no—
492D4—122no—
493D4—122no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—124no—
497C2×C4—122no—
498C2×C4#87511yescharacteristic
499C2×C4#87611yescharacteristic
500C2×C4#88211yescharacteristic
501C2×C4#88311yescharacteristic
502C2×C4#88311yescharacteristic
503C2×C4#88311yescharacteristic
504C2×C4#88411yescharacteristic
505C2×C4#88411yescharacteristic
506C2×C4#88411yescharacteristic
507C2×C4#88511yescharacteristic
508C2×C4#88511yescharacteristic
509C2×C4#88611yescharacteristic
510C2×C4#89411yescharacteristic
511C2×C4#89411yescharacteristic
512C2×C4#89411yescharacteristic
513C2×C4#89411yescharacteristic
514C2×C4#89411yescharacteristic
515C2×C4#89511yescharacteristic
516C2×C4#89511yescharacteristic
517C2×C4#89811yescharacteristic
518C2×C4#89811yescharacteristic
519C2×C4#89811yescharacteristic
520C2×C4#92911yescharacteristic
521C2×C4#93011yescharacteristic
522C2×C4#93711yescharacteristic
523C2×C4#93811yescharacteristic
524C2×C4#93911yescharacteristic
525C2×C4#94711yescharacteristic
526C8—42no—
527C8—42no—
528C8—122no—
529C8—122no—
530C2×C3—21no—
531C2×C3—21no—
532C2×C3—21no—
533C2×C3—21no—
534C2×C3—21no—
535C2×C3—21no—
536C2×C3#144011yescharacteristic
537C2×C3#182912yes—
538D3—84no—
539C22—22no—
540C22—22no—
541C22—22no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22—124no—
547C22—128no—
548C22—244no—
549C22—244no—
550C22—244no—
551C22—244no—
552C22—244no—
553C22—244no—
554C22#466411yescharacteristic
555C22#466511yescharacteristic
556C22#466911yescharacteristic
557C22#467011yescharacteristic
558C22#494011yescharacteristic
559C22#495611yescharacteristic
560C22#499011yescharacteristic
561C4—82no—
562C4—242no—
563C4#467211yescharacteristic
564C4#467411yescharacteristic
565C4#467511yescharacteristic
566C4#467511yescharacteristic
567C4#495911yescharacteristic
568C4#496111yescharacteristic
569C4#499111yescharacteristic
570C4#499211yescharacteristic
571C3#1117011yescharacteristic, complemented, Sylow
572C2—21no—
573C2—21no—
574C2—21no—
575C2—21no—
576C2—21no—
577C2—21no—
578C2—244no—
579C2#6902612yes—
580C2#6942911yescharacteristic
subgroupG/H
centerC22#4664
commutatorC24×C3C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#344C2
socleC22×C3#381
prime radicalC23×D12C22
cosocleC23×C3×C4C23
hypercenterC2×#344D3
hypocenterC3#11170
LCSG ▷ C24×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#344

Complements

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

NH ≅ G/Ntype
4. C2×C3×#344C2semidirect
5. C2×#4559C2semidirect
6. C2×#4556C2semidirect
7. C2×#4533C2semidirect
35. C2×#344D3semidirect
571. C3#11170semidirect

Maximal subgroups

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

subgroupindex
C22×#8232
#822692
C2×C3×#3442
C2×#45592
C2×#45562
C2×#45332
#111703