Finite Group Databasebeta

← #240324

Subgroups — #240324

Order 512.

1146 conjugacy classes of subgroups.

The subgroup lattice has 489 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3039C211yesmaximal, characteristic
2#21847C211yesmaximal, characteristic
3#21833C211yesmaximal, characteristic, complemented
4#20645C211yesmaximal, characteristic
5C2×#1571C211yesmaximal, characteristic, complemented
6C22×#332C211yesmaximal, characteristic, complemented
7#10478C214yesmaximal, complemented
8#10476C214yesmaximal, complemented
9#8045C211yesmaximal, characteristic, complemented
10C22×#137—21no—
11C22×C4⋊Q8C2211yescharacteristic
12C2×C4×#41C2211yescharacteristic
13#3039—24no—
14#3039—24no—
15C2×#462—21no—
16C2×#462C2212yes—
17C2×#461—21no—
18C2×#461C2212yes—
19C2×#431C2211yescharacteristic
20C2×#423C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23C2×#419—21no—
24C2×#416—21no—
25#2285C2211yescharacteristic
26#1990C2214yes—
27#1988—22no—
28#1984C2212yes—
29#1954C2212yes—
30#1869—22no—
31#1823—22no—
32#1805—21no—
33#1804—21no—
34#1803C2212yes—
35#1802C2212yes—
36#1797—22no—
37#1735—21no—
38#1571—24no—
39C2×#335C2214yes—
40C2×#332—24no—
41C2×#332C2214yes—
42#1296C2214yes—
43C22×#41D411yescharacteristic
44C23×Q8—21no—
45C2×#137—28no—
46C2×#137—44no—
47C2×#137—44no—
48C22×Q16—22no—
49C22×SD16—22no—
50C22×M16—21no—
51C23×C8—21no—
52C23×C8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8D414yes—
60C2×C4⋊Q8C2312yes—
61C4×#41—22no—
62C4×#41—22no—
63C4×#41—22no—
64C4×#41—22no—
65C2×C4×Q8D411yescharacteristic
66C2×C4×Q8C2311yescharacteristic
67C2×C4×D4—21no—
68C2×C4×D4—21no—
69C2×C4×D4D411yescharacteristic
70C2×C4×D4C2311yescharacteristic
71C2×#117—21no—
72C2×#117—21no—
73C2×#117D411yescharacteristic
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C42—21no—
77C22×C42C2311yescharacteristic
78#462—28no—
79#462—24no—
80#462—24no—
81#461—28no—
82#461—24no—
83#461—24no—
84#431—22no—
85#431—22no—
86#423—22no—
87#423D414yes—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8D412yes—
96C2×C4⋊C8C2312yes—
97C2×C4⋊C8C2312yes—
98C2×C4⋊C8C2312yes—
99#419—28no—
100#416—44no—
101C2×#103—22no—
102C2×#103—22no—
103C2×#103—22no—
104C2×#102—22no—
105C2×C22⋊C8—24no—
106C2×C22⋊C8D412yes—
107C2×C4×C8C2312yes—
108#384C2312yes—
109#382—21no—
110#380—21no—
111#380—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114#340—28no—
115#340—22no—
116#340—22no—
117#336—24no—
118#335—28no—
119#332—28no—
120C2×#41—22no—
121C2×#41—22no—
122C2×#41—22no—
123C2×#41—22no—
124C2×#41—22no—
125C2×#41D814yes—
126C22×Q8—24no—
127C22×Q8—22no—
128C22×Q8—41no—
129C22×Q8—42no—
130C22×Q8—42no—
131C22×Q8—42no—
132C22×Q8—42no—
133C22×Q8—42no—
134C22×Q8C2×D411yescharacteristic
135C22×D4—21no—
136C22×D4C2×D411yescharacteristic
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—41no—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4#4111yescharacteristic
144#137—432no—
145C2×Q16—28no—
146C2×Q16—44no—
147C2×Q16—44no—
148C2×SD16—28no—
149C2×SD16—44no—
150C2×SD16—44no—
151C2×M16—24no—
152C2×M16—42no—
153C2×M16—42no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—42no—
158C22×C8—42no—
159C22×C8—42no—
160C22×C8—42no—
161C22×C8—82no—
162C22×C8C2×D414yes—
163C4⋊Q8—24no—
164C4⋊Q8—24no—
165C4⋊Q8—24no—
166C4⋊Q8—22no—
167C4⋊Q8—22no—
168C4⋊Q8—22no—
169C4⋊Q8—42no—
170C4⋊Q8—42no—
171C4⋊Q8—42no—
172C4⋊Q8C2×D414yes—
173C4×Q8—22no—
174C4×Q8—22no—
175C4×Q8—21no—
176C4×Q8—21no—
177C4×Q8—21no—
178C4×Q8—21no—
179C4×D4—22no—
180C4×D4—22no—
181C4×D4—21no—
182C4×D4—21no—
183C4×D4—21no—
184C4×D4—21no—
185C4×D4—42no—
186C4×D4—42no—
187C4×D4—42no—
188C4×D4—42no—
189#117—22no—
190#117—22no—
191#117—42no—
192#117—42no—
193#117—42no—
194#117—42no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4C2×D411yescharacteristic
209C2×C4⋊C4C2×D411yescharacteristic
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—41no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—42no—
218C2×C42—42no—
219C2×C42—41no—
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D412yes—
222C2×C42C2×D412yes—
223C2×C42C2×D411yescharacteristic
224C2×C42#4112yes—
225C2×C42#4112yes—
226C2×C42#4111yescharacteristic
227C2×C42C2411yescharacteristic
228C4⋊C8—24no—
229C4⋊C8—24no—
230C4⋊C8—22no—
231C4⋊C8—22no—
232C4⋊C8—22no—
233C4⋊C8—22no—
234C4⋊C8—22no—
235C4⋊C8—42no—
236C4⋊C8—42no—
237C4⋊C8C2×D414yes—
238#103—28no—
239#103—28no—
240#103—44no—
241#102—44no—
242C22⋊C8—24no—
243C22⋊C8—84no—
244C4×C8—22no—
245C4×C8—22no—
246#95—41no—
247#95—41no—
248C24—21no—
249C24#12111yescharacteristic
250#41—28no—
251#41—28no—
252#41—44no—
253#41—44no—
254C2×Q8—21no—
255C2×Q8—21no—
256C2×Q8—24no—
257C2×Q8—21no—
258C2×Q8—21no—
259C2×Q8—21no—
260C2×Q8—42no—
261C2×Q8—44no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—48no—
267C2×Q8—48no—
268C2×Q8—82no—
269C2×Q8—82no—
270C2×Q8—84no—
271C2×Q8—84no—
272C2×Q8C2×D812yes—
273C2×D4—24no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4C2×D812yes—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—24no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—44no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—82no—
300C22×C4—82no—
301C22×C4—82no—
302C22×C4#12012yes—
303C22×C4#12112yes—
304C22×C4#12112yes—
305C22×C4#12111yescharacteristic
306C22×C4#12111yescharacteristic
307C22×C4#12111yescharacteristic
308C22×C4#12312yes—
309C22×C4#12312yes—
310C22×C4#12412yes—
311C22×C4C2×D812yes—
312C22×C4#13512yes—
313C22×C4AGL(1,ℤ/8)12yes—
314C22×C4#13712yes—
315C22×C4C22×D411yescharacteristic
316C22×C4C22×D411yescharacteristic
317C22×C4C2×#4111yescharacteristic
318Q16—416no—
319SD16—416no—
320M16—48no—
321C2×C8—24no—
322C2×C8—24no—
323C2×C8—24no—
324C2×C8—24no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—44no—
332C2×C8—84no—
333C2×C8—84no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—42no—
341C4⋊C4—41no—
342C4⋊C4—42no—
343C4⋊C4—41no—
344C4⋊C4—44no—
345C4⋊C4—44no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—82no—
352C22⋊C4—42no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C22⋊C4—42no—
356C22⋊C4—82no—
357C42—41no—
358C42—41no—
359C42—42no—
360C42—42no—
361C42—82no—
362C42#12012yes—
363C42#12112yes—
364C42#12112yes—
365C42#12312yes—
366C42#12312yes—
367C42#12412yes—
368C42C22×D411yescharacteristic
369C42C22×D411yescharacteristic
370C42C2×#4111yescharacteristic
371C42C2×#4111yescharacteristic
372C23—22no—
373C23—22no—
374C23—22no—
375C23—41no—
376C23—42no—
377C23—41no—
378C23—42no—
379C23#39412yes—
380C23#46612yes—
381C23#46812yes—
382C23C2×#12111yescharacteristic
383Q8—22no—
384Q8—22no—
385Q8—41no—
386Q8—41no—
387Q8—44no—
388Q8—44no—
389Q8—82no—
390Q8—82no—
391Q8—88no—
392D4—22no—
393D4—22no—
394D4—48no—
395D4—41no—
396D4—41no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—24no—
403C2×C4—22no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—48no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—44no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—84no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—88no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4#39412yes—
425C2×C4#39412yes—
426C2×C4#39412yes—
427C2×C4#44714yes—
428C2×C4#45214yes—
429C2×C4#46612yes—
430C2×C4#46812yes—
431C2×C4#48014yes—
432C2×C4#48214yes—
433C2×C4C2×#12011yescharacteristic
434C2×C4C2×#12111yescharacteristic
435C2×C4C2×#12111yescharacteristic
436C2×C4C2×#12311yescharacteristic
437C2×C4C2×#12311yescharacteristic
438C2×C4C2×#12411yescharacteristic
439C2×C4C22×D811yescharacteristic
440C2×C4C2×#13511yescharacteristic
441C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
442C2×C4C2×#13711yescharacteristic
443C2×C4#57712yes—
444C2×C4#57812yes—
445C8—44no—
446C8—44no—
447C8—82no—
448C8—82no—
449C22—24no—
450C22—22no—
451C22—22no—
452C22—24no—
453C22—42no—
454C22—44no—
455C22—81no—
456C22—81no—
457C22—82no—
458C22#203314yes—
459C22C2×#39411yescharacteristic
460C22#304612yes—
461C22C2×#46611yescharacteristic
462C22C2×#46811yescharacteristic
463C22#306912yes—
464C4—22no—
465C4—21no—
466C4—21no—
467C4—22no—
468C4—42no—
469C4—84no—
470C4—81no—
471C4—81no—
472C4—162no—
473C4#203314yes—
474C4C2×#44711yescharacteristic
475C4C2×#45211yescharacteristic
476C4#304311yescharacteristic
477C4#304411yescharacteristic
478C4#306312yes—
479C4C2×#48011yescharacteristic
480C4C2×#48211yescharacteristic
481C2—24no—
482C2—82no—
483C2#1941612yes—
484C2#1945112yes—
485C2C2×#203311yescharacteristic
486C2#2353211yescharacteristic
487C2#2354511yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
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. #21833C2semidirect
5. C2×#1571C2semidirect
6. C22×#332C2semidirect
7. #10478C2semidirect
8. #10476C2semidirect
9. #8045C2semidirect

Maximal subgroups

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

subgroupindex
C2×#30392
#218472
#218332
#206452
C2×#15712
C22×#3322
#104782
#104762
#80452