Finite Group Databasebeta

← #240326

Subgroups — #240326

Order 512.

1164 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3037C211yesmaximal, characteristic
2#21859C211yesmaximal, characteristic
3#21818C211yesmaximal, characteristic, complemented
4C2×#1803C211yesmaximal, characteristic, complemented
5C2×#1571C211yesmaximal, characteristic, complemented
6C2×#1540C211yesmaximal, characteristic
7#10478C214yesmaximal, complemented
8#10474C214yesmaximal, complemented
9#8043C211yesmaximal, characteristic, complemented
10C22×#137—21no—
11C2×#532C2211yescharacteristic
12C22×C4×Q8C2211yescharacteristic
13#3037—24no—
14#3037—24no—
15C2×#464—22no—
16C2×#462C2212yes—
17C2×#460C2212yes—
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×#423C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C2×#417—21no—
23C22×#103—21no—
24C2×#384C2211yescharacteristic
25#1990C2212yes—
26#1984—22no—
27#1984C2214yes—
28#1957C2212yes—
29#1874—22no—
30#1817—22no—
31#1803—22no—
32#1803C2214yes—
33#1802—22no—
34C2×#340—21no—
35#1571—24no—
36C2×#335C2214yes—
37#1540—24no—
38C2×#332C2212yes—
39C2×#331C2212yes—
40#1296C2214yes—
41C22×#41—21no—
42C23×Q8D411yescharacteristic
43C2×#137—28no—
44C2×#137—44no—
45C2×#137—44no—
46C22×Q16—22no—
47C22×SD16—22no—
48C22×M16—21no—
49C23×C8—21no—
50#532—22no—
51#532—22no—
52#532—22no—
53#532D414yes—
54C2×C4⋊Q8C2311yescharacteristic
55C2×#127C2311yescharacteristic
56C2×#124—21no—
57C2×#121—21no—
58C2×#121—21no—
59C2×C4×Q8—21no—
60C2×C4×Q8—21no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×C4×Q8D412yes—
66C2×C4×Q8C2312yes—
67C2×C4×D4—21no—
68C2×C4×D4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4D411yescharacteristic
74C22×C42—21no—
75C22×C42C2311yescharacteristic
76#464—216no—
77#462—24no—
78#462—24no—
79#460—24no—
80#460—24no—
81#431—22no—
82#431—22no—
83#431—22no—
84#431—22no—
85#423—22no—
86#423D414yes—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8D412yes—
92C2×C4⋊C8C2312yes—
93C2×C4⋊C8C2312yes—
94C2×C4⋊C8C2312yes—
95#417—28no—
96C2×#103—22no—
97C2×#103—24no—
98C2×#103—44no—
99C2×#102—22no—
100C2×#102—24no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8D412yes—
103C2×C4×C8—21no—
104C2×C4×C8C2312yes—
105#384—21no—
106#384—21no—
107#384—21no—
108#384C2312yes—
109#340—24no—
110#340—24no—
111#336—28no—
112#335—28no—
113#332—24no—
114#331—24no—
115C2×#41—24no—
116C2×#41—42no—
117C2×#41—42no—
118C2×#41—42no—
119C2×#41—42no—
120C2×#41—42no—
121C22×Q8—21no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—21no—
128C22×Q8D814yes—
129C22×Q8C2×D412yes—
130C22×D4—21no—
131C22×D4—21no—
132C22×D4—21no—
133C22×D4—41no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—41no—
137C23×C4—41no—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4#4111yescharacteristic
141#137—432no—
142C2×Q16—28no—
143C2×Q16—44no—
144C2×Q16—44no—
145C2×SD16—28no—
146C2×SD16—44no—
147C2×SD16—44no—
148C2×M16—24no—
149C2×M16—42no—
150C2×M16—42no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—42no—
155C22×C8—42no—
156C22×C8—42no—
157C22×C8C2×D414yes—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8C2×D412yes—
162#127—21no—
163#127—21no—
164#127—21no—
165#127C2×D412yes—
166#124—24no—
167#124—42no—
168#121—24no—
169#121—24no—
170#121—42no—
171#121—42no—
172C4×Q8—24no—
173C4×Q8—24no—
174C4×Q8—22no—
175C4×Q8—22no—
176C4×Q8—22no—
177C4×Q8—22no—
178C4×Q8—42no—
179C4×Q8—42no—
180C4×Q8—42no—
181C4×Q8—42no—
182C4×D4—24no—
183C4×D4—24no—
184C4×D4—42no—
185C4×D4—42no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—82no—
204C2×C4⋊C4—82no—
205C2×C4⋊C4C2×D412yes—
206C2×C22⋊C4—41no—
207C2×C22⋊C4—41no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—42no—
211C2×C42—42no—
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D412yes—
214C2×C42C2×D412yes—
215C2×C42C2×D411yescharacteristic
216C2×C42#4112yes—
217C2×C42#4112yes—
218C2×C42#4111yescharacteristic
219C2×C42C2411yescharacteristic
220C4⋊C8—24no—
221C4⋊C8—24no—
222C4⋊C8—22no—
223C4⋊C8—22no—
224C4⋊C8—22no—
225C4⋊C8—22no—
226C4⋊C8—22no—
227C4⋊C8—42no—
228C4⋊C8C2×D414yes—
229#103—28no—
230#103—48no—
231#102—28no—
232#102—84no—
233C22⋊C8—24no—
234C22⋊C8—42no—
235C4×C8—22no—
236C4×C8—22no—
237C4×C8—42no—
238C24—41no—
239C24#12111yescharacteristic
240#41—48no—
241#41—48no—
242#41—84no—
243#41—84no—
244C2×Q8—22no—
245C2×Q8—24no—
246C2×Q8—22no—
247C2×Q8—22no—
248C2×Q8—22no—
249C2×Q8—22no—
250C2×Q8—22no—
251C2×Q8—28no—
252C2×Q8—28no—
253C2×Q8—41no—
254C2×Q8—42no—
255C2×Q8—44no—
256C2×Q8—42no—
257C2×Q8—44no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8C2×D814yes—
263C2×D4—22no—
264C2×D4—22no—
265C2×D4—22no—
266C2×D4—44no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—81no—
277C2×D4—81no—
278C2×D4—82no—
279C2×D4—82no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—44no—
292C22×C4—42no—
293C22×C4—81no—
294C22×C4—81no—
295C22×C4—82no—
296C22×C4—82no—
297C22×C4—82no—
298C22×C4—82no—
299C22×C4—82no—
300C22×C4#12012yes—
301C22×C4#12112yes—
302C22×C4#12112yes—
303C22×C4#12111yescharacteristic
304C22×C4#12111yescharacteristic
305C22×C4#12111yescharacteristic
306C22×C4#12312yes—
307C22×C4#12312yes—
308C22×C4#12412yes—
309C22×C4C2×D812yes—
310C22×C4#13512yes—
311C22×C4AGL(1,ℤ/8)12yes—
312C22×C4#13712yes—
313C22×C4C22×D411yescharacteristic
314C22×C4C22×D411yescharacteristic
315C22×C4C2×#4111yescharacteristic
316Q16—416no—
317SD16—416no—
318M16—48no—
319C2×C8—24no—
320C2×C8—24no—
321C2×C8—24no—
322C2×C8—24no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—84no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—41no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—41no—
340C4⋊C4—42no—
341C4⋊C4—44no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—44no—
345C4⋊C4—82no—
346C4⋊C4—82no—
347C4⋊C4—82no—
348C4⋊C4—82no—
349C22⋊C4—44no—
350C22⋊C4—44no—
351C42—42no—
352C42—42no—
353C42—44no—
354C42#12012yes—
355C42#12112yes—
356C42#12112yes—
357C42#12312yes—
358C42#12312yes—
359C42#12412yes—
360C42C22×D411yescharacteristic
361C42C22×D411yescharacteristic
362C42C2×#4111yescharacteristic
363C42C2×#4111yescharacteristic
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—42no—
369C23—81no—
370C23—81no—
371C23—81no—
372C23—81no—
373C23—81no—
374C23—81no—
375C23#39412yes—
376C23#46612yes—
377C23#46812yes—
378C23C2×#12111yescharacteristic
379Q8—24no—
380Q8—24no—
381Q8—42no—
382Q8—42no—
383Q8—48no—
384Q8—42no—
385Q8—42no—
386Q8—81no—
387Q8—81no—
388D4—42no—
389D4—42no—
390D4—44no—
391D4—82no—
392D4—82no—
393D4—88no—
394D4—81no—
395D4—81no—
396D4—82no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—48no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—84no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—88no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—164no—
423C2×C4#39412yes—
424C2×C4#39412yes—
425C2×C4#39412yes—
426C2×C4#44714yes—
427C2×C4#45214yes—
428C2×C4#46612yes—
429C2×C4#46812yes—
430C2×C4#48014yes—
431C2×C4#48214yes—
432C2×C4C2×#12011yescharacteristic
433C2×C4C2×#12111yescharacteristic
434C2×C4C2×#12111yescharacteristic
435C2×C4C2×#12311yescharacteristic
436C2×C4C2×#12311yescharacteristic
437C2×C4C2×#12411yescharacteristic
438C2×C4C22×D811yescharacteristic
439C2×C4C2×#13511yescharacteristic
440C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
441C2×C4C2×#13711yescharacteristic
442C2×C4#57712yes—
443C2×C4#57812yes—
444C8—44no—
445C8—44no—
446C8—82no—
447C8—82no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22—22no—
454C22—82no—
455C22—82no—
456C22—82no—
457C22—161no—
458C22—161no—
459C22—161no—
460C22—161no—
461C22#203314yes—
462C22C2×#39411yescharacteristic
463C22#304311yescharacteristic
464C22#304411yescharacteristic
465C22C2×#46611yescharacteristic
466C22C2×#46811yescharacteristic
467C22#306811yescharacteristic
468C22#306911yescharacteristic
469C4—21no—
470C4—21no—
471C4—22no—
472C4—22no—
473C4—44no—
474C4—82no—
475C4—82no—
476C4—162no—
477C4#203314yes—
478C4C2×#44711yescharacteristic
479C4C2×#45211yescharacteristic
480C4#304612yes—
481C4#306312yes—
482C4C2×#48011yescharacteristic
483C4C2×#48211yescharacteristic
484C2—24no—
485C2—162no—
486C2#1939811yescharacteristic
487C2#1940411yescharacteristic
488C2#1943311yescharacteristic
489C2#1943911yescharacteristic
490C2C2×#203311yescharacteristic
491C2#2353311yescharacteristic
492C2#2354511yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21818C2semidirect
4. C2×#1803C2semidirect
5. C2×#1571C2semidirect
7. #10478C2semidirect
8. #10474C2semidirect
9. #8043C2semidirect

Maximal subgroups

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

subgroupindex
C2×#30372
#218592
#218182
C2×#18032
C2×#15712
C2×#15402
#104782
#104742
#80432