Finite Group Databasebeta

← #192707

Subgroups — #192707

Order 512.

988 conjugacy classes of subgroups.

The subgroup lattice has 513 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34214C211yesmaximal, characteristic
2#21894C211yesmaximal, characteristic
3#21362C211yesmaximal, characteristic
4#20764C211yesmaximal, characteristic
5C2×#1485C211yesmaximal, characteristic
6#8896C212yesmaximal, complemented
7#8895C212yesmaximal, complemented
8#8892C212yesmaximal, complemented
9#8891C212yesmaximal, complemented
10#8043C211yesmaximal, characteristic
11#7903C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#423C2211yescharacteristic
15C2×C4×M16C2211yescharacteristic
16#2360—22no—
17#2360—22no—
18#2355—21no—
19#2355C2212yes—
20Q8×C4⋊C4—21no—
21Q8×C4⋊C4C2212yes—
22#2330—22no—
23#2317—21no—
24C4×C4⋊Q8—21no—
25C4×C4⋊Q8—21no—
26#2285C2211yescharacteristic
27C2×#384C2211yescharacteristic
28#2274—21no—
29C4×#117C2211yescharacteristic
30#1988—21no—
31#1981—21no—
32#1948C2211yescharacteristic
33#1948C2211yescharacteristic
34#1947C2211yescharacteristic
35#1947C2211yescharacteristic
36#1894—22no—
37#1854—21no—
38#1854—21no—
39#1836—22no—
40#1834C2212yes—
41#1834C2212yes—
42#1803C2212yes—
43#1803C2212yes—
44C4×#107—21no—
45C4×#106—21no—
46#1736—22no—
47#1485C418yescomplemented
48C2×Q8⋊C8C2212yes—
49C2×Q8⋊C8C2212yes—
50#1296C2212yes—
51#1296C2212yes—
52#1279C2212yes—
53#1279C2212yes—
54C22×M16—21no—
55C22×M16—21no—
56#533—22no—
57#533—22no—
58#533C2×C414yes—
59#533Q814yes—
60C2×C4⋊Q8D411yescharacteristic
61C2×C4⋊Q8D411yescharacteristic
62C2×#125—21no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8D412yes—
65C2×C4×Q8—22no—
66C2×C4×Q8—22no—
67C2×C4×Q8—21no—
68C2×C4×Q8C2312yes—
69C2×#117—21no—
70C2×#117—21no—
71C2×#117D411yescharacteristic
72C2×#117D411yescharacteristic
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4D412yes—
76C22×C42C2311yescharacteristic
77#431—22no—
78#431C2×C414yes—
79C2×#107—21no—
80C2×#107—21no—
81C2×#106—21no—
82C2×#106—21no—
83#423—22no—
84#423—22no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8C2312yes—
87C2×C4⋊C8C2311yescharacteristic
88C2×C4⋊C8C2311yescharacteristic
89C2×#103—22no—
90C2×#103—22no—
91C2×C22⋊C8—21no—
92C4×M16—22no—
93C4×M16C2×C414yes—
94C2×#97—21no—
95C2×C4×C8—21no—
96C2×C4×C8C2312yes—
97#391—22no—
98#391—22no—
99#389—22no—
100#389—22no—
101#389—22no—
102#389—24no—
103#389—22no—
104#389—24no—
105#387—24no—
106#387—24no—
107#385—24no—
108#384—22no—
109#384—21no—
110#384—21no—
111#384D412yes—
112#384C2311yescharacteristic
113#384C2311yescharacteristic
114#384C2312yes—
115#382—22no—
116#382—21no—
117#380—21no—
118#380—21no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4D411yescharacteristic
128C4×C4⋊C4D411yescharacteristic
129C4×C4⋊C4C2311yescharacteristic
130C4×C4⋊C4C2311yescharacteristic
131C4×C22⋊C4—24no—
132C4×C22⋊C4—21no—
133C4×C22⋊C4—21no—
134#376—21no—
135C43—21no—
136#340—22no—
137#340—22no—
138#340—24no—
139#340—24no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144Q8⋊C8C2×C418yescomplemented
145Q8⋊C8C2×C418yescomplemented
146C22×Q8—21no—
147C22×Q8C2×D412yes—
148C23×C4—22no—
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C2×M16—24no—
153C2×M16—42no—
154C2×M16—42no—
155C2×M16—42no—
156C2×M16—81no—
157C2×M16—81no—
158C22×C8—22no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—22no—
162C4⋊Q8—22no—
163C4⋊Q8—22no—
164C4⋊Q8C4⋊C414yes—
165C4⋊Q8C4⋊C414yes—
166#125—24no—
167#125—24no—
168C22⋊Q8—24no—
169C22⋊Q8—24no—
170C22⋊Q8—24no—
171C22⋊Q8C4⋊C418yes—
172C4×Q8—24no—
173C4×Q8—24no—
174C4×Q8—22no—
175C4×Q8—22no—
176C4×Q8—82no—
177C4×Q8—82no—
178C4×Q8—84no—
179C4×Q8C22×C414yes—
180C4×Q8C2×Q814yes—
181#117—24no—
182#117—24no—
183#117—22no—
184#117—22no—
185#117—41no—
186#117—41no—
187#117—44no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—81no—
210C2×C4⋊C4—81no—
211C2×C4⋊C4C2×D411yescharacteristic
212C2×C4⋊C4C2×D411yescharacteristic
213C2×C4⋊C4C2×D412yes—
214C2×C4⋊C4C2×D411yescharacteristic
215C2×C4⋊C4C2×D411yescharacteristic
216C2×C4⋊C4C2×D414yes—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—41no—
221C2×C42—21no—
222C2×C42—22no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—24no—
229C2×C42—24no—
230C2×C42—41no—
231C2×C42C22×C412yes—
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×Q812yes—
236C2×C42#4112yes—
237C2×C42#4112yes—
238C2×C42C2411yescharacteristic
239#107—42no—
240#107—42no—
241#106—42no—
242#106—42no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—22no—
246C4⋊C8—22no—
247C4⋊C8—42no—
248C4⋊C8C22×C412yes—
249C4⋊C8C22×C412yes—
250#103—82no—
251#103—82no—
252C22⋊C8—24no—
253#97—24no—
254C4×C8—24no—
255C4×C8—22no—
256C4×C8C22×C414yes—
257#95—24no—
258#95—24no—
259#95—24no—
260#95—24no—
261#95—21no—
262#95—21no—
263#95—21no—
264#95—21no—
265#95—41no—
266#95—41no—
267C24#12011yescharacteristic
268C2×Q8—22no—
269C2×Q8—24no—
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—22no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8C2×C4⋊C414yes—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—24no—
293C22×C4—21no—
294C22×C4—44no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—82no—
298C22×C4—82no—
299C22×C4C2×C4⋊C412yes—
300C22×C4C4×D412yes—
301C22×C4C4×D412yes—
302C22×C4#12011yescharacteristic
303C22×C4#12011yescharacteristic
304C22×C4#12011yescharacteristic
305C22×C4C22⋊Q812yes—
306C22×C4C22⋊Q812yes—
307C22×C4#12312yes—
308C22×C4C22×D411yescharacteristic
309C22×C4C22×D411yescharacteristic
310C22×C4C22×D411yescharacteristic
311M16—48no—
312M16—84no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—24no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C4⋊C4—24no—
326C4⋊C4—24no—
327C4⋊C4—22no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—28no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—44no—
339C4⋊C4—44no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—44no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—82no—
356C4⋊C4—84no—
357C4⋊C4—84no—
358C4⋊C4C2×C4⋊C414yes—
359C4⋊C4#13712yes—
360C4⋊C4#13712yes—
361C22⋊C4—24no—
362C22⋊C4—28no—
363C22⋊C4—42no—
364C22⋊C4—82no—
365C42—24no—
366C42—24no—
367C42—24no—
368C42—42no—
369C42—41no—
370C42—41no—
371C42—41no—
372C42—41no—
373C42—42no—
374C42—42no—
375C42—84no—
376C42—84no—
377C42—82no—
378C42C2×C4⋊C412yes—
379C42C4×D412yes—
380C42C4×D412yes—
381C42C22⋊Q812yes—
382C42C22⋊Q812yes—
383C42#12312yes—
384C42C23×C411yescharacteristic
385C42C22×Q811yescharacteristic
386C42C2×#4111yescharacteristic
387C42C2×#4111yescharacteristic
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23#38512yes—
395C23C2×#12011yescharacteristic
396Q8—24no—
397Q8—24no—
398Q8—42no—
399Q8—42no—
400Q8—48no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—44no—
432C2×C4—41no—
433C2×C4—44no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—44no—
437C2×C4—41no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—164no—
445C2×C4#38512yes—
446C2×C4#38512yes—
447C2×C4#38512yes—
448C2×C4#45412yes—
449C2×C4#45512yes—
450C2×C4C22×C4⋊C411yescharacteristic
451C2×C4C2×C4×D411yescharacteristic
452C2×C4C2×C4×D411yescharacteristic
453C2×C4C2×C22⋊Q811yescharacteristic
454C2×C4C2×C22⋊Q811yescharacteristic
455C2×C4C2×#12311yescharacteristic
456C2×C4C2×#13711yescharacteristic
457C2×C4C2×#13711yescharacteristic
458C2×C4#57511yescharacteristic
459C2×C4#57511yescharacteristic
460C2×C4#57611yescharacteristic
461C2×C4#57711yescharacteristic
462C2×C4#57711yescharacteristic
463C2×C4#57811yescharacteristic
464C8—44no—
465C8—42no—
466C8—42no—
467C8—82no—
468C22—22no—
469C22—22no—
470C22—22no—
471C22—41no—
472C22—41no—
473C22—41no—
474C22—41no—
475C22C2×#38511yescharacteristic
476C22#300411yescharacteristic
477C22#300511yescharacteristic
478C22#301211yescharacteristic
479C22#301311yescharacteristic
480C22C2×#45511yescharacteristic
481C22C2×#45411yescharacteristic
482C4—22no—
483C4—21no—
484C4—21no—
485C4—22no—
486C4—44no—
487C4—44no—
488C4—42no—
489C4—42no—
490C4—81no—
491C4—84no—
492C4—81no—
493C4—161no—
494C4—161no—
495C4#186014yes—
496C4#291211yescharacteristic
497C4#291311yescharacteristic
498C4C4×#13711yescharacteristic
499C4C4×#13711yescharacteristic
500C4#307811yescharacteristic
501C4#307811yescharacteristic
502C4#309611yescharacteristic
503C4#309711yescharacteristic
504C2—42no—
505C2#1857511yescharacteristic
506C2#1857611yescharacteristic
507C2#1858311yescharacteristic
508C2#1858411yescharacteristic
509C2#2084911yescharacteristic
510C2C2×#186011yescharacteristic
511C2#2087411yescharacteristic
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
47. #1485C4semidirect
144. Q8⋊C8C2×C4semidirect
145. Q8⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#342142
#218942
#213622
#207642
C2×#14852
#88962
#88952
#88922
#88912
#80432
#79032