Finite Group Databasebeta

← #239662

Subgroups — #239662

Order 512.

1074 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2568C211yesmaximal, characteristic
2#21833C211yesmaximal, characteristic
3#21170C211yesmaximal, characteristic
4#20734C211yesmaximal, characteristic
5C2×#1546C211yesmaximal, characteristic
6C2×#1487C211yesmaximal, characteristic
7#10423C214yesmaximal, complemented
8#10422C214yesmaximal, complemented
9#7892C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#428—21no—
15C2×#423C2211yescharacteristic
16#2568—22no—
17#2568—22no—
18#2568—22no—
19#2568C2214yes—
20C2×#400—22no—
21C2×#391—21no—
22C2×#391—21no—
23C2×#391—21no—
24#2285C2211yescharacteristic
25C2×#382—21no—
26C2×C4×C4⋊C4C2211yescharacteristic
27#1990C2212yes—
28#1990C2212yes—
29#1988—22no—
30#1949—22no—
31#1948C2212yes—
32#1947C2212yes—
33#1824—21no—
34#1823—22no—
35#1803C2212yes—
36#1803C2212yes—
37#1787—21no—
38C4×#103—22no—
39#1736—21no—
40#1546—24no—
41C2×#332C2214yes—
42#1487—24no—
43C2×Q8⋊C8C2214yes—
44#1279C2214yes—
45C23×Q8—21no—
46C23×Q8D411yescharacteristic
47C22×M16—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8D412yes—
55C2×C4⋊Q8C2312yes—
56C2×C4×Q8—21no—
57C2×C4×Q8—21no—
58C2×C4×Q8—22no—
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8D412yes—
63C2×C4×Q8C2312yes—
64C2×#117—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4D412yes—
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4D411yescharacteristic
71C22×C42—21no—
72C22×C42—21no—
73C22×C42C2311yescharacteristic
74#431—22no—
75#431—22no—
76#431—22no—
77#431—22no—
78#428—44no—
79C2×#107—22no—
80C2×#106—22no—
81#423—22no—
82#423D414yes—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8C2312yes—
86C2×C4⋊C8C2312yes—
87C2×#103—22no—
88C2×#103—22no—
89C2×C22⋊C8—21no—
90C2×C22⋊C8—21no—
91C2×C4×C8—21no—
92C2×C4×C8C2312yes—
93#400—28no—
94#400—28no—
95#391—24no—
96#391—24no—
97#391—24no—
98#391—24no—
99#391—24no—
100#391—24no—
101#384C2311yescharacteristic
102#384C2311yescharacteristic
103#382—24no—
104#382—24no—
105#382—21no—
106#380—21no—
107#380—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4D412yes—
113C4×C4⋊C4C2312yes—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95—21no—
119#340—22no—
120#340—22no—
121#340—24no—
122#340—24no—
123#340—22no—
124#340—22no—
125#332—28no—
126Q8⋊C8—28no—
127C22×Q8—24no—
128C22×Q8—21no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—42no—
134C22×Q8—42no—
135C22×Q8—42no—
136C22×Q8—42no—
137C22×Q8—42no—
138C22×Q8—42no—
139C22×Q8—42no—
140C22×Q8C2×D412yes—
141C23×C4—21no—
142C23×C4—22no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C2×M16—24no—
150C2×M16—42no—
151C2×M16—42no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C4⋊Q8—24no—
156C4⋊Q8—24no—
157C4⋊Q8—22no—
158C4⋊Q8—22no—
159C4⋊Q8—22no—
160C4⋊Q8—22no—
161C4⋊Q8—42no—
162C4⋊Q8—42no—
163C4⋊Q8—42no—
164C4⋊Q8—42no—
165C4×Q8—24no—
166C4×Q8—24no—
167C4×Q8—22no—
168C4×Q8—22no—
169C4×Q8—22no—
170C4×Q8—22no—
171C4×Q8—42no—
172C4×Q8—42no—
173C4×Q8—42no—
174C4×Q8—42no—
175#117—42no—
176#117—42no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4C2×D412yes—
204C2×C4⋊C4C2×D414yes—
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4C2×D412yes—
207C2×C22⋊C4—41no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—44no—
213C2×C42—42no—
214C2×C42—42no—
215C2×C42—41no—
216C2×C42C2×D412yes—
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42#4112yes—
221C2×C42#4112yes—
222C2×C42#4112yes—
223C2×C42C2411yescharacteristic
224#107—44no—
225#106—44no—
226C4⋊C8—24no—
227C4⋊C8—22no—
228C4⋊C8—22no—
229C4⋊C8—22no—
230C4⋊C8—42no—
231C4⋊C8C2×D414yes—
232#103—44no—
233#103—44no—
234C22⋊C8—42no—
235C22⋊C8—42no—
236C4×C8—22no—
237C4×C8—22no—
238C4×C8—42no—
239#95—28no—
240#95—28no—
241#95—28no—
242#95—28no—
243#95—22no—
244#95—22no—
245#95—41no—
246#95—41no—
247C24#12011yescharacteristic
248C2×Q8—24no—
249C2×Q8—22no—
250C2×Q8—22no—
251C2×Q8—22no—
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—48no—
255C2×Q8—48no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—44no—
265C2×Q8—44no—
266C2×Q8—44no—
267C2×Q8—84no—
268C2×Q8—84no—
269C2×Q8—84no—
270C2×Q8—84no—
271C2×Q8#13714yes—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—44no—
292C22×C4—42no—
293C22×C4—44no—
294C22×C4—42no—
295C22×C4—44no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4#12011yescharacteristic
299C22×C4#12011yescharacteristic
300C22×C4#12011yescharacteristic
301C22×C4#12112yes—
302C22×C4#12112yes—
303C22×C4#12112yes—
304C22×C4#12312yes—
305C22×C4#12312yes—
306C22×C4#12412yes—
307C22×C4#13512yes—
308C22×C4#13512yes—
309C22×C4AGL(1,ℤ/8)12yes—
310C22×C4#13712yes—
311C22×C4C22×D411yescharacteristic
312C22×C4C22×D411yescharacteristic
313C22×C4C22×D411yescharacteristic
314M16—48no—
315C2×C8—24no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C4⋊C4—22no—
325C4⋊C4—24no—
326C4⋊C4—22no—
327C4⋊C4—24no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—44no—
340C4⋊C4—44no—
341C4⋊C4—44no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C22⋊C4—82no—
351C42—44no—
352C42—42no—
353C42—42no—
354C42—44no—
355C42—44no—
356C42—82no—
357C42#12112yes—
358C42#12112yes—
359C42#12112yes—
360C42#12312yes—
361C42#12312yes—
362C42#12412yes—
363C42C22×D411yescharacteristic
364C42C2×#4111yescharacteristic
365C42C2×#4111yescharacteristic
366C42C2×#4111yescharacteristic
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23#39412yes—
372C23#44912yes—
373C23#45212yes—
374C23C2×#12011yescharacteristic
375Q8—24no—
376Q8—24no—
377Q8—48no—
378Q8—42no—
379Q8—42no—
380Q8—82no—
381Q8—82no—
382Q8—82no—
383Q8—82no—
384Q8—88no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—48no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—44no—
401C2×C4—42no—
402C2×C4—44no—
403C2×C4—42no—
404C2×C4—44no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—84no—
412C2×C4—84no—
413C2×C4—84no—
414C2×C4—88no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4#39412yes—
422C2×C4#39412yes—
423C2×C4#39412yes—
424C2×C4#44912yes—
425C2×C4#45212yes—
426C2×C4#46314yes—
427C2×C4#46414yes—
428C2×C4#48814yes—
429C2×C4#48914yes—
430C2×C4C2×#12111yescharacteristic
431C2×C4C2×#12111yescharacteristic
432C2×C4C2×#12111yescharacteristic
433C2×C4C2×#12311yescharacteristic
434C2×C4C2×#12311yescharacteristic
435C2×C4C2×#12411yescharacteristic
436C2×C4C2×#13511yescharacteristic
437C2×C4C2×#13511yescharacteristic
438C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
439C2×C4C2×#13712yes—
440C2×C4C2×#13711yescharacteristic
441C2×C4#57512yes—
442C8—44no—
443C8—82no—
444C8—82no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22#204214yes—
452C22C2×#39411yescharacteristic
453C22C2×#44911yescharacteristic
454C22C2×#45211yescharacteristic
455C22#300411yescharacteristic
456C22#300511yescharacteristic
457C22#302011yescharacteristic
458C22#302011yescharacteristic
459C4—22no—
460C4—21no—
461C4—21no—
462C4—22no—
463C4—44no—
464C4—82no—
465C4—82no—
466C4—82no—
467C4—84no—
468C4—161no—
469C4—161no—
470C4#204214yes—
471C4C2×#46311yescharacteristic
472C4C2×#46411yescharacteristic
473C4#305612yes—
474C4#309512yes—
475C4C2×#48811yescharacteristic
476C4C2×#48911yescharacteristic
477C2—24no—
478C2#1876011yescharacteristic
479C2#1876111yescharacteristic
480C2#1878011yescharacteristic
481C2#1878111yescharacteristic
482C2#2348111yescharacteristic
483C2#2349411yescharacteristic
484C2C2×#204211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
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
7. #10423C2semidirect
8. #10422C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25682
#218332
#211702
#207342
C2×#15462
C2×#14872
#104232
#104222
#78922