Finite Group Databasebeta

← #239430

Subgroups — #239430

Order 512.

1454 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2618C211yesmaximal, characteristic
2#21940C211yesmaximal, characteristic, complemented
3#21758C211yesmaximal, characteristic
4C2×#1519C211yesmaximal, characteristic
5C22×#324C211yesmaximal, characteristic, complemented
6#14339C211yesmaximal, characteristic, complemented
7#10397C218yesmaximal, complemented
8#8023C211yesmaximal, characteristic, complemented
9C22×#123—21no—
10C22×#121C2211yescharacteristic
11C23×C4⋊C4C2211yescharacteristic
12C22×C22⋊C8C2211yescharacteristic
13C22×C22⋊C8C2211yescharacteristic
14#2618—22no—
15#2618—22no—
16#2618—22no—
17#2618C2214yes—
18C2×#398—21no—
19C2×#396—22no—
20C2×#394—21no—
21C2×#386—21no—
22C2×#381—21no—
23C2×#380C2211yescharacteristic
24C2×#380C2211yescharacteristic
25C2×#380C2211yescharacteristic
26#1893—22no—
27#1874C2214yes—
28#1852—22no—
29#1852C2214yescomplemented
30#1823—21no—
31#1808—22no—
32#1786—21no—
33#1519—24no—
34C2×#327C2214yes—
35C2×#324—24no—
36C2×#324C2214yescomplemented
37#1441—22no—
38#1441C2214yescomplemented
39C22⋊#95—21no—
40#1294C2214yescomplemented
41C23×D4—21no—
42C23×D4D411yescharacteristic
43C24×C4D411yescharacteristic
44C24×C4D411yescharacteristic
45C24×C4C2311yescharacteristic
46C23×C8—21no—
47C23×C8—21no—
48C2×#123—24no—
49C2×#123—24no—
50C2×#123—44no—
51C2×#123—44no—
52C2×#121—21no—
53C2×#121—21no—
54C2×#121—22no—
55C2×#121—22no—
56C2×#121—22no—
57C2×#121—22no—
58C2×#121D412yes—
59C2×#121C2312yes—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4D412yes—
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4D412yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C22⋊C4—22no—
76C22×C22⋊C4—21no—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—22no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C2×#107—24no—
82C2×#102—24no—
83C2×C22⋊C8—21no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8D414yes—
89C2×C22⋊C8C2312yes—
90C2×C22⋊C8C2312yes—
91#398—24no—
92#398—24no—
93#396—28no—
94#396—28no—
95#394—24no—
96#394—24no—
97#386—24no—
98#386—24no—
99#381—44no—
100#380—21no—
101#380—21no—
102#380—22no—
103#380—22no—
104#380—21no—
105#380—21no—
106#380—21no—
107#380—21no—
108#380D412yes—
109#380C2312yes—
110C2×#95—22no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—24no—
114C2×#95—24no—
115C2×#95C2312yes—
116#340—24no—
117#340—24no—
118#327—28no—
119#324—28no—
120C25—21no—
121C25C2×D411yescharacteristic
122C22×D4—22no—
123C22×D4—22no—
124C22×D4—21no—
125C22×D4—22no—
126C22×D4—22no—
127C22×D4—42no—
128C22×D4—42no—
129C22×D4—42no—
130C22×D4—42no—
131C22×D4—42no—
132C22×D4—42no—
133C22×D4—82no—
134C22×D4C2×D412yes—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—82no—
151C23×C4—82no—
152C23×C4D814yes—
153C23×C4C2×D412yes—
154C23×C4C2×D412yes—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D412yes—
161C23×C4#4112yes—
162C23×C4#4112yes—
163C23×C4#4112yes—
164C23×C4C2411yescharacteristic
165C22×C8—22no—
166C22×C8—24no—
167C22×C8—22no—
168C22×C8—42no—
169C22×C8—82no—
170C22×C8—82no—
171#123—48no—
172#123—416no—
173#123—48no—
174#121—24no—
175#121—24no—
176#121—22no—
177#121—22no—
178#121—22no—
179#121—22no—
180#121—42no—
181#121—42no—
182#121—42no—
183#121—42no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—48no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—82no—
212C2×C4⋊C4—82no—
213C2×C4⋊C4—82no—
214C2×C4⋊C4C2×D412yes—
215C2×C4⋊C4C2×D414yes—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—48no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—41no—
228C2×C22⋊C4—41no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—84no—
235#107—84no—
236#102—84no—
237C22⋊C8—24no—
238C22⋊C8—22no—
239C22⋊C8—22no—
240C22⋊C8—22no—
241C22⋊C8—42no—
242C22⋊C8C2×D414yes—
243#95—28no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—41no—
249#95—41no—
250#95—48no—
251#95—84no—
252C24—22no—
253C24—42no—
254C24—42no—
255C24—44no—
256C24—44no—
257C24—42no—
258C24—42no—
259C24—42no—
260C24#12011yescharacteristic
261C24#12011yescharacteristic
262C24#12112yes—
263C24#12312yes—
264C24C2×D812yes—
265C24#13512yes—
266C24AGL(1,ℤ/8)12yes—
267C24#13712yes—
268C24C22×D411yescharacteristic
269C2×D4—22no—
270C2×D4—22no—
271C2×D4—44no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—44no—
279C2×D4—44no—
280C2×D4—44no—
281C2×D4—42no—
282C2×D4—82no—
283C2×D4—84no—
284C2×D4—84no—
285C2×D4—84no—
286C2×D4—84no—
287C2×D4—84no—
288C2×D4—84no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—28no—
298C22×C4—28no—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—44no—
315C22×C4—44no—
316C22×C4—44no—
317C22×C4—44no—
318C22×C4—44no—
319C22×C4—44no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—44no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—44no—
327C22×C4—44no—
328C22×C4—44no—
329C22×C4—44no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—82no—
337C22×C4—82no—
338C22×C4—82no—
339C22×C4—82no—
340C22×C4—82no—
341C22×C4—84no—
342C22×C4—84no—
343C22×C4#12011yescharacteristic
344C22×C4#12011yescharacteristic
345C22×C4#12112yes—
346C22×C4#12112yes—
347C22×C4#12112yes—
348C22×C4#12112yes—
349C22×C4#12112yes—
350C22×C4#12312yes—
351C22×C4#12312yes—
352C22×C4#12312yes—
353C22×C4#12412yes—
354C22×C4#12412yes—
355C22×C4C2×D814yes—
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D411yescharacteristic
359C22×C4C2×#4111yescharacteristic
360C22×C4C2×#4111yescharacteristic
361C22×C4C2×#4111yescharacteristic
362C2×C8—24no—
363C2×C8—44no—
364C2×C8—44no—
365C2×C8—42no—
366C2×C8—84no—
367C2×C8—84no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—44no—
371C4⋊C4—416no—
372C4⋊C4—44no—
373C4⋊C4—44no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—44no—
379C4⋊C4—44no—
380C4⋊C4—42no—
381C4⋊C4—82no—
382C4⋊C4—84no—
383C22⋊C4—416no—
384C22⋊C4—42no—
385C22⋊C4—42no—
386C22⋊C4—44no—
387C22⋊C4—42no—
388C22⋊C4—42no—
389C22⋊C4—44no—
390C22⋊C4—88no—
391C22⋊C4—82no—
392C22⋊C4—82no—
393C22⋊C4—84no—
394C23—22no—
395C23—22no—
396C23—22no—
397C23—22no—
398C23—22no—
399C23—22no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—44no—
405C23—44no—
406C23—44no—
407C23—44no—
408C23—42no—
409C23—42no—
410C23—42no—
411C23—44no—
412C23—44no—
413C23—84no—
414C23—88no—
415C23—88no—
416C23—84no—
417C23—84no—
418C23#39412yes—
419C23#39412yes—
420C23#44712yes—
421C23#44712yes—
422C23#45212yes—
423C23#45212yes—
424C23#46614yes—
425C23#46814yes—
426C23#48014yes—
427C23#48214yes—
428C23C2×#12011yescharacteristic
429C23C2×#12111yescharacteristic
430C23C2×#12311yescharacteristic
431C23C22×D811yescharacteristic
432C23C2×#13511yescharacteristic
433C23C2×AGL(1,ℤ/8)11yescharacteristic
434C23C2×#13711yescharacteristic
435D4—44no—
436D4—82no—
437D4—84no—
438D4—82no—
439D4—82no—
440D4—82no—
441D4—82no—
442D4—88no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—48no—
469C2×C4—48no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—88no—
480C2×C4—84no—
481C2×C4—84no—
482C2×C4—84no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—84no—
486C2×C4—84no—
487C2×C4—82no—
488C2×C4—84no—
489C2×C4—84no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4#39412yes—
495C2×C4#39412yes—
496C2×C4C2×#12111yescharacteristic
497C2×C4C2×#12111yescharacteristic
498C2×C4C2×#12311yescharacteristic
499C2×C4C2×#12411yescharacteristic
500C8—44no—
501C8—82no—
502C22—22no—
503C22—24no—
504C22—22no—
505C22—24no—
506C22—22no—
507C22—24no—
508C22—22no—
509C22—24no—
510C22—21no—
511C22—22no—
512C22—21no—
513C22—22no—
514C22—21no—
515C22—21no—
516C22—22no—
517C22—22no—
518C22—44no—
519C22—44no—
520C22—44no—
521C22—44no—
522C22—84no—
523C22—84no—
524C22—82no—
525C22—82no—
526C22—82no—
527C22—168no—
528C22#203314yes—
529C22#203314yes—
530C22C2×#39411yescharacteristic
531C22C2×#44711yescharacteristic
532C22C2×#45211yescharacteristic
533C22#302012yes—
534C22#302612yes—
535C22C2×#46611yescharacteristic
536C22C2×#46811yescharacteristic
537C22C2×#48011yescharacteristic
538C22C2×#48211yescharacteristic
539C4—21no—
540C4—21no—
541C4—21no—
542C4—21no—
543C4—22no—
544C4—22no—
545C4—44no—
546C4—84no—
547C4—82no—
548C4—82no—
549C4—162no—
550C4—162no—
551C2—21no—
552C2—24no—
553C2—21no—
554C2—21no—
555C2—21no—
556C2—24no—
557C2—162no—
558C2#1828912yes—
559C2#1829212yes—
560C2C2×#203311yescharacteristic
561C2#2349411yescharacteristic
562C2#2351511yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21940C2semidirect
5. C22×#324C2semidirect
6. #14339C2semidirect
7. #10397C2semidirect
8. #8023C2semidirect
29. #1852C22semidirect
36. C2×#324C22semidirect
38. #1441C22semidirect
40. #1294C22semidirect

Maximal subgroups

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

subgroupindex
C2×#26182
#219402
#217582
C2×#15192
C22×#3242
#143392
#103972
#80232