Finite Group Databasebeta

← #200525

Subgroups — #200525

Order 512.

1226 conjugacy classes of subgroups.

The subgroup lattice has 499 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60248C211yesmaximal, characteristic, complemented
2#34955C211yesmaximal, characteristic
3C2×#1801C211yesmaximal, characteristic, complemented
4#17745C212yesmaximal
5#9406C212yesmaximal, complemented
6#9404C212yesmaximal, complemented
7#9398C212yesmaximal, complemented
8#9396C212yesmaximal, complemented
9#8097C212yesmaximal, complemented
10C2×#532C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12#2984—24no—
13#2983—22no—
14#2983C2212yes—
15#2982C2212yes—
16Q8×M16—24no—
17#2920—22no—
18#2920C2212yes—
19C2×#423C2212yes—
20C22×#101—21no—
21#2592—22no—
22#2591C414yes—
23#2568C2211yescharacteristic
24#2568C2211yescharacteristic
25#2548C2211yescharacteristic
26#2548C2211yescharacteristic
27#2504—21no—
28#2436—21no—
29#2398C414yes—
30C4×#124—21no—
31C4×#124—21no—
32#2297—22no—
33C4×#117C2211yescharacteristic
34#1801—21no—
35#1801—21no—
36#1801C2212yes—
37#1801C2212yes—
38C2×#343—21no—
39#1557—24no—
40#1549C2214yes—
41#1547—24no—
42#1545C2214yes—
43#1303C2214yes—
44#1303C2214yes—
45C22×#41—21no—
46C23×Q8D411yescharacteristic
47C22×M16—22no—
48#532—22no—
49#532—22no—
50#532—22no—
51#532—22no—
52C2×C4⋊Q8C2311yescharacteristic
53C2×#127C2311yescharacteristic
54C2×#124—21no—
55C2×#124—21no—
56C2×#124—21no—
57C2×#121—21no—
58C2×#121C2×C412yes—
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8C2×C412yes—
64C2×C4×Q8D411yescharacteristic
65C2×C4×Q8D411yescharacteristic
66C2×C4×Q8D412yes—
67C2×C4×Q8C2311yescharacteristic
68C2×C4×Q8C2311yescharacteristic
69C2×C4×D4—21no—
70C2×C4×D4C2×C412yes—
71C2×#117—21no—
72C22×C4⋊C4C2×C412yes—
73C22×C4⋊C4D411yescharacteristic
74C22×C42D412yes—
75C22×C42C2311yescharacteristic
76#446—216no—
77#446—28no—
78C8×Q8—28no—
79#432—24no—
80#432—24no—
81#432—22no—
82#432D414yes—
83#431—24no—
84#431—24no—
85#431—22no—
86#431C2314yes—
87#423—24no—
88#423—22no—
89#423—22no—
90#423—22no—
91#423D414yes—
92#423C2314yes—
93C2×C4⋊C8—22no—
94C2×C4⋊C8—22no—
95C2×#101—22no—
96C2×#101—22no—
97C2×#101—44no—
98#407—22no—
99C4×M16—24no—
100C4×M16—22no—
101#400C2×C414yes—
102#400C2×C414yes—
103#397—24no—
104#394C2×C414yes—
105#394C2×C414yes—
106#393—24no—
107#391—21no—
108#391—21no—
109#391—22no—
110#391—22no—
111#390—21no—
112#390—21no—
113#390—22no—
114#390—22no—
115#388—24no—
116#388—24no—
117#386—24no—
118#386—24no—
119#382—21no—
120#382—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4C2311yescharacteristic
123C4×C4⋊C4C2311yescharacteristic
124C4×C22⋊C4—21no—
125C4×C22⋊C4—24no—
126C4×C22⋊C4—24no—
127C4×C22⋊C4C2×C412yes—
128#376C2×C412yes—
129C43—21no—
130#343—22no—
131#343—22no—
132#332—28no—
133#331—28no—
134C2×#41—48no—
135C2×#41—41no—
136C2×#41—41no—
137C2×#41—41no—
138C2×#41—41no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—21no—
144C22×Q8—42no—
145C22×Q8C22⋊C412yes—
146C22×Q8C2×D411yescharacteristic
147C22×Q8C2×D411yescharacteristic
148C22×D4—22no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4C22×C412yes—
152C23×C4—22no—
153C23×C4C22⋊C412yes—
154C23×C4C22⋊C412yes—
155C23×C4C22×C411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C2×M16—24no—
159C2×M16—24no—
160C2×M16—42no—
161C2×M16—44no—
162C2×M16—44no—
163C22×C8—42no—
164C4⋊Q8—22no—
165C4⋊Q8—22no—
166#127—22no—
167#127—22no—
168#124—44no—
169#124—42no—
170#124—42no—
171#124—42no—
172#124—42no—
173#121—24no—
174#121—24no—
175#121—42no—
176#121—42no—
177C4×Q8—24no—
178C4×Q8—24no—
179C4×Q8—24no—
180C4×Q8—22no—
181C4×Q8—22no—
182C4×Q8—22no—
183C4×Q8—22no—
184C4×Q8—24no—
185C4×Q8—22no—
186C4×Q8—22no—
187C4×Q8—22no—
188C4×Q8—22no—
189C4×D4—24no—
190C4×D4—24no—
191C4×D4—42no—
192C4×D4—42no—
193#117—44no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4C22⋊C412yes—
202C2×C4⋊C4C22⋊C412yes—
203C2×C4⋊C4C22⋊C412yes—
204C2×C4⋊C4C22×C412yes—
205C2×C4⋊C4C22×C412yes—
206C2×C4⋊C4C22×C412yes—
207C2×C4⋊C4C2×D411yescharacteristic
208C2×C4⋊C4C2×D411yescharacteristic
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—22no—
224C2×C42C22⋊C412yes—
225C2×C42C22⋊C412yes—
226C2×C42C22×C411yescharacteristic
227C2×C42C2×D412yes—
228C2×C42C2×D412yes—
229C2×C42C2×D412yes—
230C2×C42C2×D412yes—
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42#4112yes—
234C2×C42#4112yes—
235C2×C42C2411yescharacteristic
236C4⋊C8—28no—
237C4⋊C8—28no—
238C4⋊C8—24no—
239C4⋊C8—24no—
240C4⋊C8—24no—
241C4⋊C8—42no—
242C4⋊C8—42no—
243C4⋊C8C2×D418yes—
244#101—44no—
245#101—44no—
246C22⋊C8—24no—
247C22⋊C8—24no—
248#97—28no—
249#97—24no—
250C4×C8—28no—
251C4×C8—24no—
252#95—22no—
253#95—24no—
254#95—24no—
255#95—24no—
256#95—24no—
257#95C22×C412yes—
258#95C22×C412yes—
259C24—22no—
260C24C2×C22⋊C411yescharacteristic
261#41—84no—
262#41—84no—
263#41—84no—
264#41—84no—
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—21no—
268C2×Q8—21no—
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—24no—
272C2×Q8—24no—
273C2×Q8—42no—
274C2×Q8—44no—
275C2×Q8—44no—
276C2×Q8—44no—
277C2×Q8—44no—
278C2×Q8—44no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8AGL(1,ℤ/8)12yes—
282C2×Q8#13712yes—
283C2×D4—24no—
284C2×D4—44no—
285C2×D4—48no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—41no—
289C2×D4—42no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—42no—
294C2×D4—82no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—42no—
308C22×C4—48no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—81no—
312C22×C4—82no—
313C22×C4C2×C22⋊C412yes—
314C22×C4C2×C22⋊C411yescharacteristic
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4C2×C22⋊C412yes—
317C22×C4C2×C22⋊C412yes—
318C22×C4C2×C22⋊C412yes—
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4C4×D412yes—
321C22×C4C4×D412yes—
322C22×C4#12011yescharacteristic
323C22×C4#12011yescharacteristic
324C22×C4#12112yes—
325C22×C4#12112yes—
326C22×C4#12311yescharacteristic
327C22×C4#12311yescharacteristic
328C22×C4C23×C411yescharacteristic
329C22×C4C22×D411yescharacteristic
330C22×C4C22×D411yescharacteristic
331M16—48no—
332M16—44no—
333M16—44no—
334M16—84no—
335C2×C8—28no—
336C2×C8—44no—
337C2×C8—84no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—24no—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C22⋊C4—48no—
350C22⋊C4—44no—
351C22⋊C4—44no—
352C22⋊C4—44no—
353C22⋊C4—48no—
354C22⋊C4—84no—
355C22⋊C4—84no—
356C42—22no—
357C42—22no—
358C42—22no—
359C42—22no—
360C42—22no—
361C42—22no—
362C42—22no—
363C42—22no—
364C42—44no—
365C42—41no—
366C42—41no—
367C42—42no—
368C42—82no—
369C42C4×D412yes—
370C42C4×D412yes—
371C42#12012yes—
372C42#12112yes—
373C42#12112yes—
374C42#12312yes—
375C42C22×D412yes—
376C42C2×#4112yes—
377C23—21no—
378C23—21no—
379C23—22no—
380C23—41no—
381C23—44no—
382C23—44no—
383C23—42no—
384C23—42no—
385C23—42no—
386C23#38611yescharacteristic
387C23#38611yescharacteristic
388C23C22×C22⋊C411yescharacteristic
389Q8—24no—
390Q8—24no—
391Q8—48no—
392Q8—42no—
393Q8—42no—
394Q8—82no—
395Q8—82no—
396D4—84no—
397D4—82no—
398D4—84no—
399D4—84no—
400D4—82no—
401D4—82no—
402D4—82no—
403C2×C4—21no—
404C2×C4—21no—
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—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—84no—
439C2×C4—82no—
440C2×C4—84no—
441C2×C4#38611yescharacteristic
442C2×C4#38612yes—
443C2×C4#38612yes—
444C2×C4#38611yescharacteristic
445C2×C4#41714yes—
446C2×C4#45414yes—
447C2×C4C2×C4×D412yes—
448C2×C4C2×#12011yescharacteristic
449C2×C4C2×#12112yes—
450C2×C4C2×#12311yescharacteristic
451C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
452C2×C4C2×#13711yescharacteristic
453C2×C4#57512yes—
454C8—48no—
455C8—84no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—42no—
464C22—42no—
465C22—84no—
466C22—84no—
467C22—82no—
468C22—82no—
469C22—82no—
470C22#188812yes—
471C22C2×#38611yescharacteristic
472C22C2×#41711yescharacteristic
473C22#289611yescharacteristic
474C4—22no—
475C4—22no—
476C4—21no—
477C4—21no—
478C4—22no—
479C4—24no—
480C4—42no—
481C4—42no—
482C4—42no—
483C4—82no—
484C4—82no—
485C4—82no—
486C4#188812yes—
487C4C2×#45412yes—
488C4#305512yes—
489C4#305612yes—
490C2—21no—
491C2—21no—
492C2—22no—
493C2—41no—
494C2—162no—
495C2C2×#188811yescharacteristic
496C2#2097111yescharacteristic
497C2#2101711yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #60248C2semidirect
3. C2×#1801C2semidirect
5. #9406C2semidirect
6. #9404C2semidirect
7. #9398C2semidirect
8. #9396C2semidirect
9. #8097C2semidirect

Maximal subgroups

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

subgroupindex
#602482
#349552
C2×#18012
#177452
#94062
#94042
#93982
#93962
#80972