Finite Group Databasebeta

← #192655

Subgroups — #192655

Order 512.

1020 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
1#34196C211yesmaximal, characteristic
2#21962C211yesmaximal, characteristic
3#20760C211yesmaximal, characteristic
4#20740C211yesmaximal, characteristic
5C22×Q8⋊C8C211yesmaximal, characteristic
6#8896C212yesmaximal, complemented
7#8895C212yesmaximal, complemented
8#8892C212yesmaximal, complemented
9#8891C212yesmaximal, complemented
10#8044C211yesmaximal, characteristic
11#7898C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16#2355—22no—
17#2355C2212yes—
18Q8×C4⋊C4C2212yes—
19#2344—22no—
20#2344—21no—
21Q8×C22⋊C4—21no—
22#2320—22no—
23#2320—21no—
24#2285—21no—
25#2285C2211yescharacteristic
26#2285C2211yescharacteristic
27#2283—21no—
28#2283C2211yescharacteristic
29#2276—21no—
30#1949—22no—
31#1948C2212yes—
32#1947C2212yes—
33#1835—22no—
34#1834C2212yes—
35#1834C2212yes—
36#1823—21no—
37#1808—21no—
38#1807—21no—
39#1803C2212yes—
40#1803C2212yes—
41#1787—21no—
42#1763—21no—
43#1763—21no—
44#1735—22no—
45C2×Q8⋊C8C418yescomplemented
46C2×Q8⋊C8C2211yescharacteristic
47C2×Q8⋊C8C2211yescharacteristic
48C2×Q8⋊C8C2211yescharacteristic
49C2×Q8⋊C8C2211yescharacteristic
50#1296C2212yes—
51#1296C2212yes—
52#1279C2212yes—
53#1279C2212yes—
54C23×Q8D411yescharacteristic
55C23×C8—21no—
56C23×C8—21no—
57C2×C4×Q8—22no—
58C2×C4×Q8—21no—
59C2×C4×Q8—21no—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8C2×C414yes—
63C2×C4×Q8D411yescharacteristic
64C2×C4×Q8D411yescharacteristic
65C2×C4×Q8Q814yes—
66C2×C4×Q8C2311yescharacteristic
67C2×C4×Q8C2311yescharacteristic
68C2×#117—22no—
69C2×#117—21no—
70C2×#117D412yes—
71C22×C4⋊C4—21no—
72C22×C4⋊C4D411yescharacteristic
73C22×C42—21no—
74C22×C42C2311yescharacteristic
75#432—22no—
76#432—22no—
77C2×#107—22no—
78C2×#106—22no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8C2×C414yes—
82C2×C4⋊C8C2312yes—
83C2×C4⋊C8C2311yescharacteristic
84C2×C4⋊C8C2311yescharacteristic
85C2×#103—22no—
86C2×#103—22no—
87C2×C22⋊C8—21no—
88C2×#97—21no—
89C2×C4×C8—21no—
90C2×C4×C8—22no—
91C2×C4×C8C2×C414yes—
92C2×C4×C8C2311yescharacteristic
93C2×C4×C8C2311yescharacteristic
94#391—24no—
95#391—22no—
96#389—22no—
97#389—24no—
98#387—24no—
99#387—24no—
100#387—24no—
101#384—22no—
102#384C2312yes—
103#384C2312yes—
104#383—22no—
105#383—21no—
106#382—22no—
107#382—22no—
108#382—21no—
109#382D412yes—
110#382D412yes—
111#381—22no—
112#381—21no—
113#380—22no—
114#380—21no—
115#380—21no—
116#380—21no—
117#380—21no—
118#380D411yescharacteristic
119#380D411yescharacteristic
120C4×C4⋊C4—22no—
121C4×C4⋊C4—24no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4C2312yes—
130C4×C22⋊C4—22no—
131C4×C22⋊C4—21no—
132#376—22no—
133#376—21no—
134#340—22no—
135#340—22no—
136#340—22no—
137#340—22no—
138#340—24no—
139#340—24no—
140Q8⋊C8C2×C414yes—
141Q8⋊C8C2×C414yes—
142Q8⋊C8C2×C414yes—
143Q8⋊C8C2×C414yes—
144C22×Q8—21no—
145C22×Q8—22no—
146C22×Q8—24no—
147C22×Q8—42no—
148C22×Q8C4⋊C414yes—
149C22×Q8C2×D411yescharacteristic
150C22×Q8C2×D411yescharacteristic
151C23×C4—21no—
152C23×C4—21no—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C22×C8—24no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—24no—
162C22×C8—22no—
163C22×C8—42no—
164C22×C8—42no—
165C22×C8—82no—
166C4×Q8—24no—
167C4×Q8—24no—
168C4×Q8—24no—
169C4×Q8—24no—
170C4×Q8—22no—
171C4×Q8—22no—
172C4×Q8—21no—
173C4×Q8—21no—
174C4×Q8—21no—
175C4×Q8—21no—
176C4×Q8—44no—
177C4×Q8—44no—
178C4×Q8C4⋊C414yes—
179C4×Q8C4⋊C414yes—
180C4×Q8C22×C412yes—
181C4×Q8C22×C412yes—
182C4×Q8C2×Q812yes—
183C4×Q8C2×Q812yes—
184#117—44no—
185#117—44no—
186#117—84no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4C4⋊C414yes—
203C2×C4⋊C4C2×D412yes—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2×D411yescharacteristic
206C2×C4⋊C4C2×D411yescharacteristic
207C2×C4⋊C4C2×D414yes—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—41no—
212C2×C42—24no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—24no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—24no—
219C2×C42—22no—
220C2×C42—24no—
221C2×C42—22no—
222C2×C42—41no—
223C2×C42C22×C412yes—
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×Q812yes—
228C2×C42#4112yes—
229C2×C42#4112yes—
230C2×C42C2411yescharacteristic
231#107—82no—
232#106—82no—
233C4⋊C8—24no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—22no—
237C4⋊C8—22no—
238C4⋊C8C22×C412yes—
239C4⋊C8C22×C412yes—
240#103—44no—
241#103—44no—
242C22⋊C8—42no—
243#97—42no—
244C4×C8—24no—
245C4×C8—21no—
246C4×C8—21no—
247C4×C8C22×C412yes—
248C4×C8C22×C412yes—
249#95—24no—
250#95—24no—
251#95—24no—
252#95—24no—
253#95—21no—
254#95—22no—
255#95—21no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—41no—
261#95—41no—
262C24#12011yescharacteristic
263C2×Q8—21no—
264C2×Q8—21no—
265C2×Q8—22no—
266C2×Q8—24no—
267C2×Q8—28no—
268C2×Q8—22no—
269C2×Q8—22no—
270C2×Q8—44no—
271C2×Q8—44no—
272C2×Q8—48no—
273C2×Q8C2×C4⋊C412yes—
274C2×Q8C2×C4⋊C412yes—
275C2×Q8#13514yes—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—44no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4C2×C4⋊C412yes—
296C22×C4C4×D412yes—
297C22×C4C4×D412yes—
298C22×C4#12011yescharacteristic
299C22×C4#12011yescharacteristic
300C22×C4#12011yescharacteristic
301C22×C4C22⋊Q812yes—
302C22×C4C22⋊Q812yes—
303C22×C4#12312yes—
304C22×C4C22×D411yescharacteristic
305C22×C4C22×D411yescharacteristic
306C22×C4C22×D411yescharacteristic
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—48no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—84no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—44no—
336C4⋊C4—48no—
337C4⋊C4—44no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—84no—
347C4⋊C4C2×C4⋊C412yes—
348C4⋊C4C2×C4⋊C412yes—
349C22⋊C4—42no—
350C22⋊C4—44no—
351C22⋊C4—42no—
352C22⋊C4—82no—
353C42—22no—
354C42—22no—
355C42—22no—
356C42—22no—
357C42—24no—
358C42—24no—
359C42—48no—
360C42—42no—
361C42—42no—
362C42—82no—
363C42—82no—
364C42—84no—
365C42C2×C4⋊C412yes—
366C42C4×D412yes—
367C42C4×D412yes—
368C42C22⋊Q812yes—
369C42C22⋊Q812yes—
370C42#12312yes—
371C42C23×C411yescharacteristic
372C42C22×Q811yescharacteristic
373C42C2×#4111yescharacteristic
374C42C2×#4111yescharacteristic
375C23—22no—
376C23—22no—
377C23#38512yes—
378C23#45412yes—
379C23#45512yes—
380C23C2×#12011yescharacteristic
381Q8—24no—
382Q8—22no—
383Q8—22no—
384Q8—22no—
385Q8—48no—
386Q8—42no—
387Q8#42714yes—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—48no—
408C2×C4—48no—
409C2×C4—42no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—44no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4#38512yes—
427C2×C4#38512yes—
428C2×C4#38512yes—
429C2×C4C22×C4⋊C411yescharacteristic
430C2×C4C2×C4×D411yescharacteristic
431C2×C4C2×C4×D411yescharacteristic
432C2×C4C2×C22⋊Q811yescharacteristic
433C2×C4C2×C22⋊Q811yescharacteristic
434C2×C4C2×#12311yescharacteristic
435C2×C4C2×#13512yes—
436C2×C4#57512yes—
437C2×C4#57712yes—
438C2×C4#57812yes—
439C8—42no—
440C8—42no—
441C8—42no—
442C8—42no—
443C8—82no—
444C22—22no—
445C22—22no—
446C22—24no—
447C22—24no—
448C22#186014yes—
449C22C2×#38511yescharacteristic
450C22#300212yes—
451C22#301312yes—
452C22C2×#45511yescharacteristic
453C22C2×#45411yescharacteristic
454C4—22no—
455C4—24no—
456C4—21no—
457C4—21no—
458C4—22no—
459C4—22no—
460C4—42no—
461C4—42no—
462C4—42no—
463C4—82no—
464C4—84no—
465C4—82no—
466C4—162no—
467C4C2×#42711yescharacteristic
468C4#291111yescharacteristic
469C4#294911yescharacteristic
470C4#295011yescharacteristic
471C4#308212yes—
472C4#309112yes—
473C2—24no—
474C2#1862912yes—
475C2#1863912yes—
476C2#2084311yescharacteristic
477C2C2×#186011yescharacteristic
478C2#2088011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ 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
6. #8896C2semidirect
7. #8895C2semidirect
8. #8892C2semidirect
9. #8891C2semidirect
45. C2×Q8⋊C8C4semidirect

Maximal subgroups

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

subgroupindex
#341962
#219622
#207602
#207402
C22×Q8⋊C82
#88962
#88952
#88922
#88912
#80442
#78982