Finite Group Databasebeta

← #240329

Subgroups — #240329

Order 512.

1114 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3077C211yesmaximal, characteristic
2#21860C211yesmaximal, characteristic, complemented
3#21786C211yesmaximal, characteristic
4C2×#1802C211yesmaximal, characteristic
5C2×#1570C211yesmaximal, characteristic, complemented
6C2×#1542C211yesmaximal, characteristic, complemented
7#10484C214yesmaximal, complemented
8#10480C214yesmaximal, complemented
9#8043C211yesmaximal, characteristic, complemented
10C2×#533C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12#3077—24no—
13#3077—24no—
14C2×#478—22no—
15C2×#476C2212yes—
16C2×#474C2212yes—
17C2×#431C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C2×#428—21no—
20C2×#423C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C2×#418—21no—
23C22×#102—21no—
24C2×#384C2211yescharacteristic
25#1990C2212yescomplemented
26#1956C2212yescomplemented
27#1954—22no—
28#1954C2214yes—
29#1872—22no—
30#1803—22no—
31#1802—22no—
32#1802C2214yes—
33#1779—22no—
34C2×#340—21no—
35#1570—24no—
36C2×#334C2214yescomplemented
37#1542—24no—
38C2×#333C2212yescomplemented
39C2×#332C2212yescomplemented
40#1296C2214yescomplemented
41C23×D4D411yescharacteristic
42C24×C4—21no—
43C22×M16—21no—
44C23×C8—21no—
45#533—22no—
46#533—22no—
47#533—22no—
48#533D414yes—
49C2×C4⋊Q8C2311yescharacteristic
50C2×C4⋊Q8C2311yescharacteristic
51C2×#125—21no—
52C2×C22⋊Q8—21no—
53C2×C22⋊Q8—21no—
54C2×C4×Q8—21no—
55C2×C4×Q8—21no—
56C2×C4×D4—21no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—21no—
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4D412yes—
63C2×C4×D4C2312yes—
64C2×#117—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4D411yescharacteristic
69C22×C22⋊C4—21no—
70C22×C42C2311yescharacteristic
71#478—216no—
72#476—24no—
73#476—24no—
74#474—24no—
75#474—24no—
76#431—22no—
77#431—22no—
78#431—22no—
79#431—22no—
80#428—28no—
81C2×#107—22no—
82C2×#106—22no—
83#423—22no—
84#423D414yes—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8D412yes—
90C2×C4⋊C8C2312yes—
91C2×C4⋊C8C2312yes—
92C2×C4⋊C8C2312yes—
93#418—28no—
94C2×#103—24no—
95C2×#102—22no—
96C2×#102—22no—
97C2×#102—24no—
98C2×#102—44no—
99C2×C22⋊C8—21no—
100C2×C22⋊C8—24no—
101C2×C22⋊C8—24no—
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#334—28no—
112#333—24no—
113#332—24no—
114C25—21no—
115C22×Q8—21no—
116C22×Q8—21no—
117C22×D4—22no—
118C22×D4—21no—
119C22×D4—22no—
120C22×D4—22no—
121C22×D4—22no—
122C22×D4—22no—
123C22×D4SD1614yes—
124C22×D4C2×D412yes—
125C23×C4—21no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—42no—
129C23×C4—42no—
130C23×C4—41no—
131C23×C4—82no—
132C23×C4—82no—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4#4111yescharacteristic
136C2×M16—24no—
137C2×M16—42no—
138C2×M16—42no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—42no—
143C22×C8—42no—
144C22×C8—42no—
145C22×C8C2×D414yes—
146C4⋊Q8—21no—
147C4⋊Q8—21no—
148C4⋊Q8—21no—
149C4⋊Q8—21no—
150C4⋊Q8—21no—
151C4⋊Q8—21no—
152C4⋊Q8C2×D412yes—
153C4⋊Q8C2×D412yes—
154#125—24no—
155#125—42no—
156C22⋊Q8—24no—
157C22⋊Q8—24no—
158C22⋊Q8—42no—
159C22⋊Q8—42no—
160C4×Q8—24no—
161C4×Q8—24no—
162C4×Q8—42no—
163C4×Q8—42no—
164C4×D4—24no—
165C4×D4—24no—
166C4×D4—22no—
167C4×D4—22no—
168C4×D4—22no—
169C4×D4—22no—
170C4×D4—42no—
171C4×D4—42no—
172C4×D4—42no—
173C4×D4—42no—
174#117—24no—
175#117—42no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—82no—
194C2×C4⋊C4—82no—
195C2×C4⋊C4C2×D412yes—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—41no—
199C2×C22⋊C4—42no—
200C2×C22⋊C4—42no—
201C2×C42—41no—
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D412yes—
204C2×C42C2×D412yes—
205C2×C42C2×D411yescharacteristic
206C2×C42#4112yes—
207C2×C42#4112yes—
208C2×C42#4111yescharacteristic
209C2×C42C2411yescharacteristic
210#107—28no—
211#106—28no—
212C4⋊C8—24no—
213C4⋊C8—24no—
214C4⋊C8—22no—
215C4⋊C8—22no—
216C4⋊C8—22no—
217C4⋊C8—22no—
218C4⋊C8—22no—
219C4⋊C8—42no—
220C4⋊C8C2×D414yes—
221#103—84no—
222#102—28no—
223#102—28no—
224#102—48no—
225C22⋊C8—24no—
226C22⋊C8—42no—
227C22⋊C8—84no—
228C22⋊C8—84no—
229C4×C8—22no—
230C4×C8—22no—
231C4×C8—42no—
232C24—22no—
233C24—24no—
234C24—42no—
235C24—42no—
236C24—81no—
237C24—81no—
238C24—81no—
239C24—81no—
240C24#12111yescharacteristic
241C2×Q8—22no—
242C2×Q8—22no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×Q8—81no—
248C2×Q8—81no—
249C2×D4—22no—
250C2×D4—22no—
251C2×D4—22no—
252C2×D4—22no—
253C2×D4—22no—
254C2×D4—28no—
255C2×D4—24no—
256C2×D4—28no—
257C2×D4—42no—
258C2×D4—44no—
259C2×D4—42no—
260C2×D4—44no—
261C2×D4C2×SD1614yes—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—44no—
273C22×C4—44no—
274C22×C4—44no—
275C22×C4—42no—
276C22×C4—82no—
277C22×C4—82no—
278C22×C4—88no—
279C22×C4—88no—
280C22×C4—81no—
281C22×C4—81no—
282C22×C4—82no—
283C22×C4—82no—
284C22×C4—82no—
285C22×C4#12012yes—
286C22×C4#12112yes—
287C22×C4#12112yes—
288C22×C4#12111yescharacteristic
289C22×C4#12111yescharacteristic
290C22×C4#12111yescharacteristic
291C22×C4#12312yes—
292C22×C4#12312yes—
293C22×C4#12412yes—
294C22×C4C2×SD1612yes—
295C22×C4#13512yes—
296C22×C4#13712yes—
297C22×C4#13712yes—
298C22×C4C22×D411yescharacteristic
299C22×C4C22×D411yescharacteristic
300C22×C4C2×#4111yescharacteristic
301M16—48no—
302C2×C8—24no—
303C2×C8—24no—
304C2×C8—24no—
305C2×C8—24no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—84no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—42no—
326C4⋊C4—44no—
327C4⋊C4—44no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—44no—
331C4⋊C4—82no—
332C4⋊C4—82no—
333C4⋊C4—82no—
334C4⋊C4—82no—
335C22⋊C4—44no—
336C22⋊C4—42no—
337C22⋊C4—42no—
338C22⋊C4—44no—
339C42—44no—
340C42#12012yes—
341C42#12112yes—
342C42#12112yes—
343C42#12312yes—
344C42#12312yes—
345C42#12412yes—
346C42C22×D411yescharacteristic
347C42C22×D411yescharacteristic
348C42C2×#4111yescharacteristic
349C42C2×#4111yescharacteristic
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—24no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23—42no—
359C23—48no—
360C23—42no—
361C23—42no—
362C23—84no—
363C23—84no—
364C23—84no—
365C23—84no—
366C23—84no—
367C23—84no—
368C23#39412yes—
369C23#46512yes—
370C23#47012yes—
371C23C2×#12111yescharacteristic
372Q8—44no—
373Q8—82no—
374Q8—82no—
375Q8—82no—
376D4—24no—
377D4—24no—
378D4—42no—
379D4—42no—
380D4—48no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—88no—
397C2×C4—84no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—84no—
403C2×C4—84no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—164no—
408C2×C4#39412yes—
409C2×C4#39412yes—
410C2×C4#39412yes—
411C2×C4#44814yes—
412C2×C4#45214yes—
413C2×C4#46512yes—
414C2×C4#47012yes—
415C2×C4#48314yes—
416C2×C4#48514yes—
417C2×C4C2×#12011yescharacteristic
418C2×C4C2×#12111yescharacteristic
419C2×C4C2×#12111yescharacteristic
420C2×C4C2×#12311yescharacteristic
421C2×C4C2×#12311yescharacteristic
422C2×C4C2×#12411yescharacteristic
423C2×C4C22×SD1611yescharacteristic
424C2×C4C2×#13511yescharacteristic
425C2×C4C2×#13711yescharacteristic
426C2×C4C2×#13711yescharacteristic
427C2×C4#57612yes—
428C2×C4#57712yes—
429C8—44no—
430C8—44no—
431C8—82no—
432C8—82no—
433C22—22no—
434C22—22no—
435C22—22no—
436C22—22no—
437C22—22no—
438C22—22no—
439C22—44no—
440C22—44no—
441C22—44no—
442C22—816no—
443C22—82no—
444C22—82no—
445C22—82no—
446C22—82no—
447C22#203514yes—
448C22C2×#39411yescharacteristic
449C22#304111yescharacteristic
450C22#304211yescharacteristic
451C22C2×#46511yescharacteristic
452C22C2×#47011yescharacteristic
453C22#306811yescharacteristic
454C22#306911yescharacteristic
455C4—21no—
456C4—21no—
457C4—22no—
458C4—22no—
459C4—82no—
460C4—82no—
461C4—82no—
462C4—162no—
463C4#203514yes—
464C4C2×#44811yescharacteristic
465C4C2×#45211yescharacteristic
466C4#304512yes—
467C4#306212yes—
468C4C2×#48311yescharacteristic
469C4C2×#48511yescharacteristic
470C2—24no—
471C2—84no—
472C2#1938411yescharacteristic
473C2#1939011yescharacteristic
474C2#1942311yescharacteristic
475C2#1942911yescharacteristic
476C2C2×#203511yescharacteristic
477C2#2353311yescharacteristic
478C2#2354411yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC23×D4D4
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
2. #21860C2semidirect
5. C2×#1570C2semidirect
6. C2×#1542C2semidirect
7. #10484C2semidirect
8. #10480C2semidirect
9. #8043C2semidirect
25. #1990C22semidirect
26. #1956C22semidirect
36. C2×#334C22semidirect
38. C2×#333C22semidirect
39. C2×#332C22semidirect
40. #1296C22semidirect

Maximal subgroups

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

subgroupindex
C2×#30772
#218602
#217862
C2×#18022
C2×#15702
C2×#15422
#104842
#104802
#80432