Finite Group Databasebeta

← #387122

Subgroups — #387122

Order 512.

1330 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×#2546C211yesmaximal, characteristic
2C2×#1985C211yesmaximal, characteristic
3#13926C211yesmaximal, characteristic
4#11384C212yesmaximal
5#11228C212yesmaximal
6#11207C212yesmaximal
7#10884C212yesmaximal
8#10881C212yesmaximal
9#10807C212yesmaximal
10C22×C4⋊Q8—21no—
11C22×C4⋊Q8C2211yescharacteristic
12C22×C4×Q8C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×#97C2211yescharacteristic
15#2546—24no—
16#2546C418yescomplemented
17C2×#398—21no—
18C2×#398—21no—
19C2×#391—21no—
20C2×#391—21no—
21C2×#389C2212yes—
22C2×#389C2212yes—
23C2×#384C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26#1985—24no—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2212yes—
36C2×#336C2212yes—
37#1441—21no—
38#1441—21no—
39C4⋊#95—21no—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C23×Q8—21no—
43C23×Q8D411yescharacteristic
44C23×C8C2312yes—
45C23×C8C2312yes—
46C2×C4⋊Q8—28no—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8—41no—
49C2×C4⋊Q8—41no—
50C2×C4⋊Q8—41no—
51C2×C4⋊Q8—41no—
52C2×C4⋊Q8—44no—
53C2×C4⋊Q8—44no—
54C2×C4⋊Q8C2×C414yes—
55C2×C4×Q8—41no—
56C2×C4×Q8—41no—
57C2×C4×Q8—41no—
58C2×C4×Q8—41no—
59C2×C4×Q8C2×C414yes—
60C2×C4×Q8D414yes—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2312yes—
73C22×C42—21no—
74C22×C42—21no—
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C2×C4⋊C8—22no—
79C2×C4⋊C8—41no—
80C2×C4⋊C8—41no—
81C2×#97—22no—
82C2×#97—41no—
83C2×#97—41no—
84#398—28no—
85#398—28no—
86#391—28no—
87#391—44no—
88#389—24no—
89#389—24no—
90#389C2×C418yes—
91#389C2×C418yes—
92#384—41no—
93#384—41no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4—41no—
96C4×C4⋊C4—41no—
97C4×C4⋊C4C2×C414yes—
98C2×#95—21no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—21no—
104C2×#95—22no—
105C2×#95—22no—
106#340—24no—
107#340—24no—
108#340—24no—
109#340—42no—
110#340—42no—
111#340—42no—
112#340—42no—
113#340Q818yescomplemented
114#336—24no—
115#336—24no—
116C22×Q8—24no—
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—41no—
122C22×Q8—41no—
123C22×Q8—42no—
124C22×Q8—42no—
125C22×Q8—42no—
126C22×Q8—42no—
127C22×Q8—42no—
128C22×Q8—42no—
129C22×Q8C22⋊C414yes—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2411yescharacteristic
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8C2×Q814yes—
155C4⋊Q8—44no—
156C4⋊Q8—44no—
157C4⋊Q8—44no—
158C4⋊Q8—44no—
159C4⋊Q8—432no—
160C4×Q8—44no—
161C4×Q8—44no—
162C4×Q8—44no—
163C4×Q8—44no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—28no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—82no—
203C2×C4⋊C4C22⋊C414yes—
204C2×C4⋊C4C22⋊C414yes—
205C2×C4⋊C4C22⋊C414yes—
206C2×C4⋊C4C22×C414yes—
207C2×C4⋊C4C22×C414yes—
208C2×C4⋊C4C2×Q814yes—
209C2×C4⋊C4C2×Q814yes—
210C2×C4⋊C4#4114yes—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—24no—
224C2×C42—42no—
225C2×C42—42no—
226C2×C42—42no—
227C2×C42—42no—
228C2×C42—42no—
229C2×C42C22×C412yes—
230C2×C42C22×C412yes—
231C2×C42C22×C412yes—
232C2×C42C2×D412yes—
233C2×C42C2×D412yes—
234C2×C42C2×D412yes—
235C4⋊C8—44no—
236#97—44no—
237#95—24no—
238#95—28no—
239#95—28no—
240#95—42no—
241#95—44no—
242#95—44no—
243#95—82no—
244#95—82no—
245C24C22×D411yescharacteristic
246C2×Q8—28no—
247C2×Q8—28no—
248C2×Q8—44no—
249C2×Q8—44no—
250C2×Q8—44no—
251C2×Q8—44no—
252C2×Q8—44no—
253C2×Q8—48no—
254C2×Q8—48no—
255C2×Q8—48no—
256C2×Q8—84no—
257C2×Q8—84no—
258C2×Q8—84no—
259C2×Q8—84no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—24no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
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×C4C2×C22⋊C412yes—
315C22×C4C2×C22⋊C412yes—
316C22×C4C2×C22⋊C412yes—
317C22×C4C2×C22⋊C412yes—
318C22×C4C2×C22⋊C412yes—
319C22×C4C2×C22⋊C412yes—
320C22×C4AGL(1,ℤ/8)12yes—
321C22×C4AGL(1,ℤ/8)12yes—
322C22×C4AGL(1,ℤ/8)12yes—
323C22×C4#13712yes—
324C22×C4C23×C411yescharacteristic
325C22×C4C22×D411yescharacteristic
326C22×C4C22×Q812yes—
327C22×C4C2×#4112yes—
328C22×C4H(5,𝔽2)11yescharacteristic
329C22×C4H(5,𝔽2)11yescharacteristic
330C22×C4H(5,𝔽2)11yescharacteristic
331C22×C4H(5,𝔽2)11yescharacteristic
332C22×C42−1+412yes—
333C22×C42−1+412yes—
334C2×C8—24no—
335C2×C8—24no—
336C2×C8—24no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C4⋊C4—24no—
349C4⋊C4—24no—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4—24no—
353C4⋊C4—24no—
354C4⋊C4—24no—
355C4⋊C4—44no—
356C4⋊C4—44no—
357C4⋊C4—44no—
358C4⋊C4—48no—
359C4⋊C4—416no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—416no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—84no—
370C4⋊C4—84no—
371C4⋊C4C4×Q818yes—
372C42—22no—
373C42—22no—
374C42—22no—
375C42—22no—
376C42—22no—
377C42—22no—
378C42—22no—
379C42—22no—
380C42—22no—
381C42—22no—
382C42—22no—
383C42—22no—
384C42—48no—
385C42—84no—
386C23C22×C22⋊C411yescharacteristic
387C23#53411yescharacteristic
388C23#53411yescharacteristic
389C23#54812yes—
390C23D4×Q812yes—
391C23C2×AGL(1,ℤ/8)11yescharacteristic
392C23C2×AGL(1,ℤ/8)11yescharacteristic
393C23C2×AGL(1,ℤ/8)11yescharacteristic
394C23C2×#13711yescharacteristic
395C23#57712yes—
396C23#57812yes—
397Q8—416no—
398Q8—816no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—28no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4#41714yes—
462C2×C4#41814yes—
463C2×C4C2×C4×Q812yes—
464C2×C4#51811yescharacteristic
465C2×C4#51811yescharacteristic
466C2×C4#51912yes—
467C2×C4#53411yescharacteristic
468C2×C4#53411yescharacteristic
469C2×C4#54212yes—
470C2×C4#54212yes—
471C2×C4#54412yes—
472C2×C4#54412yes—
473C2×C4#54812yes—
474C2×C4D4×Q812yes—
475C2×C4#55412yes—
476C2×C4#55511yescharacteristic
477C2×C4Q8211yescharacteristic
478C2×C4#56312yes—
479C2×C4#56311yescharacteristic
480C2×C4#56311yescharacteristic
481C8—44no—
482C8—82no—
483C22#230811yescharacteristic
484C22Q8×C22⋊C412yes—
485C22#272912yes—
486C22#273111yescharacteristic
487C22#273411yescharacteristic
488C22C2×#41811yescharacteristic
489C22C2×#41711yescharacteristic
490C22#289812yes—
491C22#319211yescharacteristic
492C22#319211yescharacteristic
493C22#319311yescharacteristic
494C22#319311yescharacteristic
495C22#320011yescharacteristic
496C22#320112yes—
497C22#320211yescharacteristic
498C22#336712yes—
499C22#336812yes—
500C22#337312yes—
501C22#337412yes—
502C22#338412yes—
503C22#338612yes—
504C22#338612yes—
505C22#338712yes—
506C4—24no—
507C4—24no—
508C4—42no—
509C4—42no—
510C4—42no—
511C4—42no—
512C4—48no—
513C4—84no—
514C4—84no—
515C4—162no—
516C4#233112yes—
517C4#236911yescharacteristic
518C4#238111yescharacteristic
519C4#272912yes—
520C4#273111yescharacteristic
521C4#273411yescharacteristic
522C2#1445011yescharacteristic
523C2#2176411yescharacteristic
524C2#2177111yescharacteristic
525C2#2252312yes—
526C2#2252412yes—
527C2#3258912yes—
528C2#3259112yes—
529C2#3264411yescharacteristic
530C2#3264611yescharacteristic
531C2#3269311yescharacteristic
532C2#3269411yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
16. #2546C4semidirect
113. #340Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25462
C2×#19852
#139262
#113842
#112282
#112072
#108842
#108812
#108072