Finite Group Databasebeta

← #204247

Subgroups — #204247

Order 512.

1368 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2591C211yesmaximal, characteristic
2#21932C211yesmaximal, characteristic, complemented
3#20525C211yesmaximal, characteristic, complemented
4C2×#1471C211yesmaximal, characteristic
5C2×#1471C211yesmaximal, characteristic
6C2×#1463C211yesmaximal, characteristic, complemented
7C22×#324C211yesmaximal, characteristic, complemented
8#9652C214yesmaximal, complemented
9#9651C214yesmaximal, complemented
10C22×#121C2211yescharacteristic
11C23×C4⋊C4C2211yescharacteristic
12C2×#407C2211yescharacteristic
13C2×#407C2211yescharacteristic
14C22×C22⋊C8C2211yescharacteristic
15C22×C22⋊C8C2211yescharacteristic
16#2591—22no—
17#2591—22no—
18#2591—21no—
19#2591—21no—
20#2591—22no—
21C2×#400C412yes—
22C2×#394C412yes—
23C2×#388C2214yes—
24C2×#386C412yes—
25C2×#386C412yes—
26C2×C4×C22⋊C4C2211yescharacteristic
27#1851—22no—
28#1850—22no—
29#1797C2214yes—
30#1753—22no—
31#1752—21no—
32C8×C22⋊C4—21no—
33#1749C2212yes—
34C4×C22⋊C8C2212yes—
35#1471—24no—
36#1471—24no—
37#1463—24no—
38C2×#324—24no—
39C2×#324C2212yes—
40C2×#324C2214yes—
41C2×#324C2212yes—
42C2×#323C2212yes—
43C2×#323C2214yes—
44C2×#323C2212yes—
45C23×D4C2×C412yes—
46C23×D4C2×C411yescharacteristic
47C24×C4C2×C411yescharacteristic
48C24×C4C2×C411yescharacteristic
49C24×C4C2311yescharacteristic
50C22×M16—21no—
51C22×M16—21no—
52C23×C8—21no—
53C23×C8—21no—
54C2×#121—21no—
55C2×#121—21no—
56C2×#121—21no—
57C2×#121—21no—
58C2×#121—22no—
59C2×#121—22no—
60C2×#121—22no—
61C2×#121—22no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4C2×C412yes—
71C22×C4⋊C4C2×C411yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C22⋊C4C2×C414yes—
74C22×C22⋊C4C2×C414yes—
75C22×C22⋊C4C2×C412yes—
76C22×C22⋊C4C2312yes—
77C22×C42C2312yes—
78C2×C4⋊C8—24no—
79#407—22no—
80#407—22no—
81#407—22no—
82#407D414yes—
83C2×C22⋊C8—21no—
84C2×C22⋊C8—21no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—21no—
89C2×C22⋊C8—21no—
90C2×C22⋊C8D414yes—
91C2×C22⋊C8C2312yes—
92C2×C22⋊C8C2312yes—
93C2×C22⋊C8C2311yescharacteristic
94C2×C22⋊C8C2311yescharacteristic
95C2×C22⋊C8C2312yes—
96C2×#97—22no—
97C2×C4×C8—22no—
98#400—24no—
99#400—22no—
100#400—22no—
101#394—24no—
102#394—22no—
103#394—22no—
104#388—24no—
105#388—24no—
106#388—24no—
107#388—24no—
108#386—22no—
109#386—22no—
110#386—24no—
111#386—22no—
112#386—22no—
113#386—24no—
114C4×C22⋊C4D412yes—
115C4×C22⋊C4D412yes—
116C4×C22⋊C4D411yescharacteristic
117C4×C22⋊C4D411yescharacteristic
118C4×C22⋊C4D412yes—
119C2×#95C2×C414yes—
120C2×#95C2×C414yes—
121C2×#95C2×C412yes—
122#336—24no—
123#336—22no—
124#336—22no—
125#324—24no—
126#324—24no—
127#324—28no—
128#323—28no—
129#323—24no—
130#323—24no—
131C25C22×C411yescharacteristic
132C25C22×C412yes—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4—24no—
136C22×D4—24no—
137C22×D4—21no—
138C22×D4—21no—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4—48no—
142C22×D4—82no—
143C23×C4—22no—
144C23×C4—24no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—22no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—42no—
153C23×C4—82no—
154C23×C4—82no—
155C23×C4C22×C412yes—
156C23×C4C22×C412yes—
157C23×C4C22×C412yes—
158C23×C4C22×C414yes—
159C23×C4C22×C411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D412yes—
165C23×C4C2×D412yes—
166C23×C4#4112yes—
167C23×C4#4112yes—
168C23×C4C2411yescharacteristic
169C2×M16—24no—
170C2×M16—42no—
171C2×M16—82no—
172C2×M16—82no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—22no—
176C22×C8—24no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—42no—
180C22×C8—82no—
181C22×C8—82no—
182#121—42no—
183#121—42no—
184#121—42no—
185#121—42no—
186#121—42no—
187#121—42no—
188#121—42no—
189#121—42no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
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—42no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—84no—
226C2×C22⋊C4C22⋊C412yes—
227C2×C22⋊C4C22⋊C412yes—
228C2×C22⋊C4C22⋊C414yes—
229C2×C42—82no—
230C2×C42C2×D412yes—
231C2×C42C2×D412yes—
232C2×C42C2×D414yes—
233C4⋊C8—84no—
234C22⋊C8—24no—
235C22⋊C8—24no—
236C22⋊C8—21no—
237C22⋊C8—21no—
238C22⋊C8—22no—
239C22⋊C8—22no—
240C22⋊C8—22no—
241C22⋊C8—22no—
242C22⋊C8—22no—
243C22⋊C8—42no—
244C22⋊C8—42no—
245C22⋊C8C2×D412yes—
246C22⋊C8C2×D412yes—
247C22⋊C8C2×D414yes—
248#97—82no—
249C4×C8—82no—
250#95—24no—
251#95—24no—
252#95—24no—
253#95—24no—
254#95C22⋊C414yes—
255#95C22⋊C412yes—
256#95C22⋊C412yes—
257C24—22no—
258C24—22no—
259C24—24no—
260C24—22no—
261C24—22no—
262C24—22no—
263C24—24no—
264C24—24no—
265C24—42no—
266C24—44no—
267C24—44no—
268C24C2×C22⋊C411yescharacteristic
269C24C2×C22⋊C411yescharacteristic
270C24C4×D412yes—
271C24C4×D412yes—
272C24C23×C411yescharacteristic
273C2×D4—48no—
274C2×D4—44no—
275C2×D4—44no—
276C2×D4—48no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—82no—
280C2×D4—82no—
281C2×D4—84no—
282C2×D4—84no—
283C2×D4—88no—
284C2×D4—88no—
285C22×C4—28no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—44no—
307C22×C4—44no—
308C22×C4—44no—
309C22×C4—44no—
310C22×C4—48no—
311C22×C4—44no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—44no—
317C22×C4—44no—
318C22×C4—82no—
319C22×C4—82no—
320C22×C4—82no—
321C22×C4—82no—
322C22×C4—82no—
323C22×C4—82no—
324C22×C4—84no—
325C22×C4—84no—
326C22×C4C2×C22⋊C414yes—
327C22×C4C2×C22⋊C414yes—
328C22×C4C2×C22⋊C411yescharacteristic
329C22×C4C2×C22⋊C411yescharacteristic
330C22×C4C4×D412yes—
331C22×C4C4×D412yes—
332C22×C4#12112yes—
333C22×C4#12112yes—
334C22×C4#12112yes—
335C22×C4#12112yes—
336C22×C4#12412yes—
337C22×C4#12412yes—
338C22×C4#12712yes—
339C22×C4#12712yes—
340C22×C4#13114yes—
341C22×C4C22×D411yescharacteristic
342C22×C4C22×D411yescharacteristic
343C22×C4C22×D411yescharacteristic
344C22×C4C22×D411yescharacteristic
345C22×C4C2×#4111yescharacteristic
346C22×C4C2×#4111yescharacteristic
347M16—84no—
348M16—84no—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—24no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—44no—
357C2×C8—44no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—84no—
361C2×C8—84no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—44no—
367C4⋊C4—44no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C22⋊C4—24no—
371C22⋊C4—24no—
372C22⋊C4—44no—
373C22⋊C4—48no—
374C22⋊C4—48no—
375C22⋊C4—42no—
376C22⋊C4—42no—
377C22⋊C4—42no—
378C22⋊C4—42no—
379C22⋊C4—84no—
380C22⋊C4—84no—
381C42—42no—
382C42—42no—
383C42—44no—
384C23—22no—
385C23—22no—
386C23—22no—
387C23—22no—
388C23—22no—
389C23—22no—
390C23—22no—
391C23—22no—
392C23—22no—
393C23—24no—
394C23—24no—
395C23—24no—
396C23—42no—
397C23—42no—
398C23—48no—
399C23—48no—
400C23—44no—
401C23—44no—
402C23—44no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23—44no—
407C23—82no—
408C23—82no—
409C23—82no—
410C23—82no—
411C23—88no—
412C23—88no—
413C23#39012yes—
414C23#39012yes—
415C23#41314yes—
416C23C22×C22⋊C411yescharacteristic
417C23C2×C4×D411yescharacteristic
418C23C2×C4×D411yescharacteristic
419C23C2×#13112yes—
420C23#56812yes—
421D4—84no—
422D4—84no—
423D4—88no—
424D4—88no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—48no—
453C2×C4—48no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—84no—
463C2×C4—84no—
464C2×C4—84no—
465C2×C4—84no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4#39012yes—
471C2×C4#39012yes—
472C2×C4C2×#12111yescharacteristic
473C2×C4C2×#12111yescharacteristic
474C2×C4C2×#12411yescharacteristic
475C2×C4C2×#12711yescharacteristic
476C8—42no—
477C8—42no—
478C8—44no—
479C8—82no—
480C8—82no—
481C22—24no—
482C22—24no—
483C22—28no—
484C22—22no—
485C22—21no—
486C22—21no—
487C22—22no—
488C22—21no—
489C22—21no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—44no—
494C22—44no—
495C22—48no—
496C22—44no—
497C22—88no—
498C22—82no—
499C22—82no—
500C22—82no—
501C22—162no—
502C22—162no—
503C22—164no—
504C22#196714yes—
505C22C2×#39011yescharacteristic
506C22#288012yes—
507C22#288312yes—
508C22C2×#41312yes—
509C22#292911yescharacteristic
510C22#293011yescharacteristic
511C22#293312yes—
512C4—21no—
513C4—21no—
514C4—21no—
515C4—21no—
516C4—22no—
517C4—22no—
518C4—44no—
519C4—44no—
520C4—44no—
521C4—82no—
522C4—82no—
523C2—21no—
524C2—21no—
525C2—22no—
526C2—24no—
527C2—42no—
528C2—162no—
529C2#1742714yes—
530C2#2119111yescharacteristic
531C2#2121311yescharacteristic
532C2C2×#196711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21932C2semidirect
3. #20525C2semidirect
6. C2×#1463C2semidirect
7. C22×#324C2semidirect
8. #9652C2semidirect
9. #9651C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25912
#219322
#205252
C2×#14712
C2×#14632
C22×#3242
#96522
#96512