Finite Group Databasebeta

← #239784

Subgroups — #239784

Order 512.

1094 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35279C211yesmaximal, characteristic
2#21959C211yesmaximal, characteristic, complemented
3#21834C211yesmaximal, characteristic, complemented
4#20645C211yesmaximal, characteristic
5C22×#332C211yesmaximal, characteristic, complemented
6C2×#1477C211yesmaximal, characteristic
7#10423C214yesmaximal, complemented
8#10420C214yesmaximal, complemented
9#7893C211yesmaximal, characteristic, complemented
10C22×C4⋊Q8C2211yescharacteristic
11C2×C4×#41C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C2×#416—21no—
16#2857—22no—
17#2628—22no—
18#2568C2212yes—
19#2567—21no—
20#2563—21no—
21#2542C2212yes—
22#2487—22no—
23#2409—22no—
24#2317—21no—
25#2311—21no—
26#2285C2211yescharacteristic
27#2285C2211yescharacteristic
28#1990C2214yes—
29#1987—22no—
30#1948—22no—
31#1948C2214yes—
32#1824—22no—
33#1807—22no—
34#1805—21no—
35#1804—21no—
36#1803C2212yes—
37#1802C2212yes—
38#1735—21no—
39C2×#332—24no—
40C2×#332C2214yes—
41#1477—24no—
42C2×Q8⋊C8C2212yes—
43C2×D4⋊C8C2212yes—
44#1279C2214yes—
45C22×#41D411yescharacteristic
46C23×Q8—21no—
47C23×C8—21no—
48C23×C8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8D412yes—
56C2×C4⋊Q8C2312yes—
57C2×C22⋊Q8—24no—
58C4×#41—22no—
59C4×#41—22no—
60C4×#41—22no—
61C4×#41—22no—
62C2×C4×Q8D411yescharacteristic
63C2×C4×Q8C2311yescharacteristic
64C2×C4×D4—21no—
65C2×C4×D4—21no—
66C2×C4×D4D411yescharacteristic
67C2×C4×D4C2311yescharacteristic
68C2×#117—21no—
69C2×#117—21no—
70C2×#117—21no—
71C2×#117D411yescharacteristic
72C22×C4⋊C4—21no—
73C22×C4⋊C4D412yes—
74C22×C42—21no—
75C22×C42C2311yescharacteristic
76#431—22no—
77#431—22no—
78C2×#106—24no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8D414yes—
86C2×C4⋊C8C2312yes—
87C2×C4⋊C8C2312yes—
88#416—44no—
89C2×#103—22no—
90C2×#102—22no—
91C2×C22⋊C8—21no—
92C2×C4×C8C2312yes—
93#400—24no—
94#399—24no—
95#395—24no—
96#393—24no—
97#391—22no—
98#391—24no—
99#390—22no—
100#389—24no—
101#389—24no—
102#386—24no—
103#385—24no—
104#384—22no—
105#384D412yes—
106#384C2312yes—
107#382—22no—
108#382—21no—
109#382—21no—
110#380—24no—
111#380—24no—
112#380—21no—
113#380—21no—
114#380—21no—
115#380D412yes—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4C2312yes—
120#340—22no—
121#340—22no—
122#340—24no—
123#340—28no—
124#332—28no—
125Q8⋊C8—24no—
126D4⋊C8—24no—
127C2×#41—22no—
128C2×#41—22no—
129C2×#41—22no—
130C2×#41—22no—
131C2×#41—22no—
132C2×#41SD1614yes—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—42no—
136C22×Q8—42no—
137C22×Q8—82no—
138C22×Q8—82no—
139C22×Q8C2×D411yescharacteristic
140C22×D4—21no—
141C22×D4C2×D411yescharacteristic
142C23×C4—21no—
143C23×C4—22no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C22×C8—22no—
150C22×C8—24no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—42no—
154C22×C8—42no—
155C22×C8—42no—
156C22×C8—82no—
157C4⋊Q8—24no—
158C4⋊Q8—24no—
159C4⋊Q8—22no—
160C4⋊Q8—22no—
161C4⋊Q8—22no—
162C4⋊Q8—22no—
163C4⋊Q8—42no—
164C4⋊Q8—42no—
165C4⋊Q8—42no—
166C4⋊Q8—42no—
167C22⋊Q8—84no—
168C22⋊Q8—84no—
169C4×Q8—22no—
170C4×Q8—22no—
171C4×Q8—21no—
172C4×Q8—21no—
173C4×Q8—21no—
174C4×Q8—21no—
175C4×D4—22no—
176C4×D4—22no—
177C4×D4—21no—
178C4×D4—21no—
179C4×D4—21no—
180C4×D4—21no—
181C4×D4—42no—
182C4×D4—42no—
183C4×D4—42no—
184C4×D4—42no—
185#117—22no—
186#117—22no—
187#117—42no—
188#117—42no—
189#117—42no—
190#117—42no—
191#117—42no—
192#117—42no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4C2×D414yes—
209C2×C4⋊C4C2×D411yescharacteristic
210C2×C4⋊C4C2×D411yescharacteristic
211C2×C4⋊C4C2×D414yes—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—41no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—22no—
221C2×C42—21no—
222C2×C42—42no—
223C2×C42—42no—
224C2×C42—41no—
225C2×C42C2×D412yes—
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42#4112yes—
230C2×C42#4112yes—
231C2×C42#4112yes—
232C2×C42C2411yescharacteristic
233#106—84no—
234C4⋊C8—24no—
235C4⋊C8—22no—
236C4⋊C8—22no—
237C4⋊C8—22no—
238C4⋊C8—42no—
239C4⋊C8—42no—
240C4⋊C8C2×D414yes—
241#103—44no—
242#102—44no—
243C22⋊C8—42no—
244C4×C8—22no—
245C4×C8—22no—
246#95—28no—
247#95—28no—
248#95—22no—
249#95—22no—
250#95—41no—
251#95—41no—
252C24—21no—
253C24#12011yescharacteristic
254#41—28no—
255#41—28no—
256#41—44no—
257#41—44no—
258C2×Q8—21no—
259C2×Q8—21no—
260C2×Q8—21no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—44no—
266C2×Q8—44no—
267C2×Q8—44no—
268C2×Q8—84no—
269C2×Q8—88no—
270C2×Q8—84no—
271C2×Q8—84no—
272C2×Q8C2×SD1612yes—
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×SD1612yes—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—24no—
286C22×C4—21no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—21no—
290C22×C4—42no—
291C22×C4—44no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—44no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—84no—
302C22×C4#12011yescharacteristic
303C22×C4#12011yescharacteristic
304C22×C4#12011yescharacteristic
305C22×C4#12112yes—
306C22×C4#12112yes—
307C22×C4#12112yes—
308C22×C4#12312yes—
309C22×C4#12312yes—
310C22×C4#12412yes—
311C22×C4C2×SD1612yes—
312C22×C4#13512yes—
313C22×C4#13712yes—
314C22×C4#13712yes—
315C22×C4C22×D411yescharacteristic
316C22×C4C22×D411yescharacteristic
317C22×C4C22×D411yescharacteristic
318C2×C8—24no—
319C2×C8—44no—
320C2×C8—44no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—84no—
326C2×C8—84no—
327C4⋊C4—24no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—24no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—44no—
336C4⋊C4—44no—
337C4⋊C4—42no—
338C4⋊C4—41no—
339C4⋊C4—42no—
340C4⋊C4—41no—
341C4⋊C4—44no—
342C4⋊C4—44no—
343C4⋊C4—42no—
344C4⋊C4—44no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—88no—
350C22⋊C4—42no—
351C22⋊C4—44no—
352C22⋊C4—41no—
353C22⋊C4—41no—
354C22⋊C4—42no—
355C22⋊C4—88no—
356C22⋊C4—82no—
357C42—41no—
358C42—41no—
359C42—42no—
360C42—42no—
361C42—42no—
362C42—42no—
363C42—82no—
364C42#12112yes—
365C42#12112yes—
366C42#12112yes—
367C42#12312yes—
368C42#12312yes—
369C42#12412yes—
370C42C22×D411yescharacteristic
371C42C2×#4111yescharacteristic
372C42C2×#4111yescharacteristic
373C42C2×#4111yescharacteristic
374C23—22no—
375C23—22no—
376C23—22no—
377C23—41no—
378C23—41no—
379C23—42no—
380C23—42no—
381C23#39412yes—
382C23#44812yes—
383C23#45212yes—
384C23C2×#12011yescharacteristic
385Q8—22no—
386Q8—22no—
387Q8—41no—
388Q8—41no—
389Q8—88no—
390Q8—84no—
391Q8—82no—
392Q8—82no—
393D4—22no—
394D4—22no—
395D4—48no—
396D4—41no—
397D4—41no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—24no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—48no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—48no—
416C2×C4—42no—
417C2×C4—44no—
418C2×C4—42no—
419C2×C4—84no—
420C2×C4—88no—
421C2×C4—82no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4#39412yes—
428C2×C4#39412yes—
429C2×C4#39412yes—
430C2×C4#44812yes—
431C2×C4#45212yes—
432C2×C4#46514yes—
433C2×C4#47014yes—
434C2×C4#48314yes—
435C2×C4#48514yes—
436C2×C4C2×#12111yescharacteristic
437C2×C4C2×#12111yescharacteristic
438C2×C4C2×#12111yescharacteristic
439C2×C4C2×#12311yescharacteristic
440C2×C4C2×#12311yescharacteristic
441C2×C4C2×#12411yescharacteristic
442C2×C4C22×SD1611yescharacteristic
443C2×C4C2×#13511yescharacteristic
444C2×C4C2×#13711yescharacteristic
445C2×C4C2×#13711yescharacteristic
446C2×C4#57612yes—
447C2×C4#57712yes—
448C8—44no—
449C8—82no—
450C8—82no—
451C22—24no—
452C22—22no—
453C22—24no—
454C22—22no—
455C22—42no—
456C22—44no—
457C22—81no—
458C22—82no—
459C22—81no—
460C22#203514yes—
461C22C2×#39411yescharacteristic
462C22C2×#44811yescharacteristic
463C22C2×#45211yescharacteristic
464C22#301212yes—
465C22#302012yes—
466C4—21no—
467C4—22no—
468C4—21no—
469C4—22no—
470C4—42no—
471C4—81no—
472C4—84no—
473C4—81no—
474C4—84no—
475C4—162no—
476C4#203514yes—
477C4#304111yescharacteristic
478C4#304311yescharacteristic
479C4C2×#46511yescharacteristic
480C4C2×#47011yescharacteristic
481C4C2×#48311yescharacteristic
482C4C2×#48511yescharacteristic
483C4#313612yes—
484C2—24no—
485C2—82no—
486C2#1888412yes—
487C2#1888912yes—
488C2C2×#203511yescharacteristic
489C2#2349411yescharacteristic
490C2#2351111yescharacteristic
subgroupG/H
center, socleC23C2×#120
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
2. #21959C2semidirect
3. #21834C2semidirect
5. C22×#332C2semidirect
7. #10423C2semidirect
8. #10420C2semidirect
9. #7893C2semidirect

Maximal subgroups

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

subgroupindex
#352792
#219592
#218342
#206452
C22×#3322
C2×#14772
#104232
#104202
#78932