Finite Group Databasebeta

← #121512

Subgroups — #121512

Order 512.

984 conjugacy classes of subgroups.

The subgroup lattice has 569 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21938C211yesmaximal, characteristic
2C2×#1852C211yesmaximal, characteristic
3#13881C211yesmaximal, characteristic
4#10854C211yesmaximal, characteristic
5#10730C211yesmaximal, characteristic
6#8023C211yesmaximal, characteristic
7C2×#1282C211yesmaximal, characteristic
8C23×C4⋊C4C2211yescharacteristic
9C22×#107—21no—
10C22×C22⋊C8C2211yescharacteristic
11C22×C22⋊C8C2211yescharacteristic
12C2×#382—21no—
13C2×#380C2211yescharacteristic
14C2×#380C2211yescharacteristic
15C2×#380C2211yescharacteristic
16C2×C4×C22⋊C4C2211yescharacteristic
17#1853—21no—
18#1853—21no—
19#1853—21no—
20#1853—21no—
21#1852—22no—
22#1852—22no—
23#1807—22no—
24C2×#342—22no—
25C2×#340—22no—
26C2×#340—21no—
27#1442—22no—
28C4⋊#95—21no—
29C22⋊#95—22no—
30#1294—22no—
31#1282C414yescomplemented
32C24×C4—21no—
33C24×C4C2311yescharacteristic
34C23×C8—21no—
35C23×C8D412yes—
36C22×C4⋊C4—21no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4—21no—
41C22×C4⋊C4—21no—
42C22×C4⋊C4—44no—
43C22×C4⋊C4D412yes—
44C22×C4⋊C4D411yescharacteristic
45C22×C4⋊C4D411yescharacteristic
46C22×C4⋊C4D411yescharacteristic
47C22×C4⋊C4D411yescharacteristic
48C22×C22⋊C4—21no—
49C22×C22⋊C4—21no—
50C22×C22⋊C4—21no—
51C22×C22⋊C4D412yes—
52C22×C42—21no—
53C2×#107—24no—
54C2×#107—44no—
55C2×#106—22no—
56C2×#106—22no—
57C2×C22⋊C8—21no—
58C2×C22⋊C8—21no—
59C2×C22⋊C8—21no—
60C2×C22⋊C8—21no—
61C2×C22⋊C8—21no—
62C2×C22⋊C8—21no—
63C2×C22⋊C8—21no—
64C2×C22⋊C8C2×C412yes—
65C2×C22⋊C8C2×C412yes—
66C2×C22⋊C8D412yes—
67C2×C22⋊C8Q812yes—
68C2×C22⋊C8Q812yes—
69#382—44no—
70#380—21no—
71#380—21no—
72#380—21no—
73#380—21no—
74#380—21no—
75#380—21no—
76#380—21no—
77#380—21no—
78#380—21no—
79#380—21no—
80#380—21no—
81#380Q812yes—
82C4×C22⋊C4—21no—
83C4×C22⋊C4—21no—
84C4×C22⋊C4C2×C412yes—
85C4×C22⋊C4Q812yes—
86C2×#95—21no—
87C2×#95—21no—
88C2×#95—24no—
89C2×#95—21no—
90C2×#95—21no—
91C2×#95—24no—
92#342—44no—
93#340—22no—
94#340—22no—
95#340—24no—
96#340—44no—
97C25C2×D411yescharacteristic
98C23×C4—22no—
99C23×C4—22no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—42no—
105C23×C4—42no—
106C23×C4—44no—
107C23×C4—41no—
108C23×C4—41no—
109C23×C4—41no—
110C23×C4—42no—
111C23×C4—42no—
112C23×C4—82no—
113C23×C4C22×C411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2×Q811yescharacteristic
120C23×C4C2×Q811yescharacteristic
121C23×C4C2×Q811yescharacteristic
122C23×C4C2×Q811yescharacteristic
123C23×C4#4111yescharacteristic
124C23×C4#4111yescharacteristic
125C23×C4#4111yescharacteristic
126C23×C4#4111yescharacteristic
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—42no—
133C22×C8—42no—
134C22×C8—42no—
135C22×C8—42no—
136C22×C8C4⋊C414yes—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—44no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—44no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—41no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—42no—
173C2×C22⋊C4—42no—
174C2×C22⋊C4—41no—
175C2×C22⋊C4—41no—
176C2×C22⋊C4—41no—
177C2×C22⋊C4—41no—
178C2×C22⋊C4—41no—
179C2×C22⋊C4—42no—
180C2×C22⋊C4—42no—
181C2×C22⋊C4—42no—
182C2×C22⋊C4—42no—
183C2×C22⋊C4C4⋊C414yes—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—42no—
187C2×C42—42no—
188#107—48no—
189#106—82no—
190#106—82no—
191C22⋊C8—22no—
192C22⋊C8—22no—
193C22⋊C8—22no—
194C22⋊C8—22no—
195C22⋊C8—22no—
196C22⋊C8—22no—
197C22⋊C8—22no—
198C22⋊C8C4⋊C414yes—
199#95—22no—
200#95—22no—
201#95—22no—
202#95—48no—
203#95—41no—
204#95—41no—
205#95—41no—
206#95—41no—
207#95—41no—
208#95—84no—
209C24—41no—
210C24—41no—
211C24—41no—
212C24—41no—
213C24C2×C4⋊C411yescharacteristic
214C24#12011yescharacteristic
215C24#12111yescharacteristic
216C24C22⋊Q811yescharacteristic
217C24#12311yescharacteristic
218C24#12711yescharacteristic
219C24C4⋊Q811yescharacteristic
220C24#13512yes—
221C24#13512yes—
222C24AGL(1,ℤ/8)12yes—
223C24#13712yes—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—44no—
249C22×C4—42no—
250C22×C4—44no—
251C22×C4—44no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—44no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—88no—
272C22×C4—88no—
273C22×C4—82no—
274C22×C4—82no—
275C22×C4—82no—
276C22×C4—81no—
277C22×C4—81no—
278C22×C4—81no—
279C22×C4—81no—
280C22×C4—81no—
281C22×C4—81no—
282C22×C4—81no—
283C22×C4—81no—
284C22×C4—81no—
285C22×C4—81no—
286C22×C4—81no—
287C22×C4—81no—
288C22×C4C2×C4⋊C411yescharacteristic
289C22×C4C2×C4⋊C411yescharacteristic
290C22×C4C4×D411yescharacteristic
291C22×C4C4×D411yescharacteristic
292C22×C4C4×D411yescharacteristic
293C22×C4C4×Q811yescharacteristic
294C22×C4#12011yescharacteristic
295C22×C4#12011yescharacteristic
296C22×C4#12111yescharacteristic
297C22×C4#12111yescharacteristic
298C22×C4C22⋊Q811yescharacteristic
299C22×C4C22⋊Q811yescharacteristic
300C22×C4C22⋊Q811yescharacteristic
301C22×C4C22⋊Q811yescharacteristic
302C22×C4C22⋊Q811yescharacteristic
303C22×C4C22⋊Q811yescharacteristic
304C22×C4C22⋊Q811yescharacteristic
305C22×C4C22⋊Q811yescharacteristic
306C22×C4#12311yescharacteristic
307C22×C4#12311yescharacteristic
308C22×C4#12511yescharacteristic
309C22×C4#12511yescharacteristic
310C22×C4#12511yescharacteristic
311C22×C4C4⋊Q811yescharacteristic
312C22×C4C4⋊Q811yescharacteristic
313C22×C4C4⋊Q811yescharacteristic
314C22×C4C4⋊Q811yescharacteristic
315C22×C4C4⋊Q811yescharacteristic
316C2×C8—24no—
317C2×C8—24no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—44no—
321C2×C8—44no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—44no—
325C2×C8—82no—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—44no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—44no—
334C4⋊C4—44no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—81no—
340C4⋊C4—81no—
341C4⋊C4—82no—
342C22⋊C4—24no—
343C22⋊C4—24no—
344C22⋊C4—42no—
345C22⋊C4—42no—
346C22⋊C4—44no—
347C22⋊C4—82no—
348C22⋊C4—82no—
349C22⋊C4—82no—
350C22⋊C4—81no—
351C22⋊C4—81no—
352C22⋊C4—81no—
353C22⋊C4—81no—
354C42—44no—
355C42—44no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—42no—
373C23—42no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23—42no—
379C23—42no—
380C23—42no—
381C23—42no—
382C23—42no—
383C23—42no—
384C23#38411yescharacteristic
385C23#38511yescharacteristic
386C23#38911yescharacteristic
387C23#39211yescharacteristic
388C23#39511yescharacteristic
389C23#39811yescharacteristic
390C23#39911yescharacteristic
391C23#42712yes—
392C23#42812yes—
393C23#44912yes—
394C23#45212yes—
395C23#46312yes—
396C23#46412yes—
397C23#47812yes—
398C23#47912yes—
399C23#48212yes—
400C23#48512yes—
401C23#49511yescharacteristic
402C23#49511yescharacteristic
403C23#49611yescharacteristic
404C23#49711yescharacteristic
405C23#49912yes—
406C23#50112yes—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—48no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—84no—
447C2×C4—88no—
448C2×C4—84no—
449C2×C4—84no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—84no—
454C2×C4—84no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—161no—
470C2×C4—161no—
471C2×C4—161no—
472C2×C4—161no—
473C2×C4—162no—
474C2×C4—162no—
475C2×C4#38511yescharacteristic
476C2×C4#38511yescharacteristic
477C2×C4#38911yescharacteristic
478C2×C4#38911yescharacteristic
479C2×C4#39511yescharacteristic
480C2×C4#39511yescharacteristic
481C2×C4#39811yescharacteristic
482C2×C4#39811yescharacteristic
483C2×C4B(4,𝔽2)12yes—
484C2×C4#45812yes—
485C8—44no—
486C8—44no—
487C22—21no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—22no—
500C22—22no—
501C22—21no—
502C22—21no—
503C22—22no—
504C22—22no—
505C22—21no—
506C22—21no—
507C22—22no—
508C22—21no—
509C22—22no—
510C22—21no—
511C22—22no—
512C22—21no—
513C22—22no—
514C22—21no—
515C22—42no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22#144411yescharacteristic
524C22#184611yescharacteristic
525C22#184711yescharacteristic
526C22#185412yes—
527C22#194912yes—
528C22#200811yescharacteristic
529C22#200911yescharacteristic
530C22#201211yescharacteristic
531C22#201511yescharacteristic
532C22#202111yescharacteristic
533C22#202211yescharacteristic
534C22#205912yes—
535C22#208512yes—
536C22#209211yescharacteristic
537C22#209711yescharacteristic
538C4—21no—
539C4—21no—
540C4—21no—
541C4—21no—
542C4—21no—
543C4—21no—
544C4—84no—
545C4—84no—
546C4—162no—
547C4—161no—
548C4—161no—
549C4—161no—
550C4—161no—
551C4#186712yes—
552C4#206212yes—
553C2—22no—
554C2—21no—
555C2—22no—
556C2—22no—
557C2—21no—
558C2—21no—
559C2—21no—
560C2—22no—
561C2#994711yescharacteristic
562C2#995311yescharacteristic
563C2#995611yescharacteristic
564C2#995811yescharacteristic
565C2#1132511yescharacteristic
566C2#1133311yescharacteristic
567C2#1134411yescharacteristic
subgroupG/H
center, socleC23#392
commutatorC23×C4C22×C4
Frattini, cosocleC24×C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
31. #1282C4semidirect

Maximal subgroups

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

subgroupindex
#219382
C2×#18522
#138812
#108542
#107302
#80232
C2×#12822