Finite Group Databasebeta

← #197416

Subgroups — #197416

Order 512.

1304 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2905C211yesmaximal, characteristic
2#34634C211yesmaximal, characteristic
3#17708C212yesmaximal, complemented
4C2×#1458C211yesmaximal, characteristic
5#9193C214yesmaximal, complemented
6#9191C214yesmaximal, complemented
7#7957C212yesmaximal, complemented
8C23×M16—21no—
9C2×#533C2211yescharacteristic
10C22×#117C2211yescharacteristic
11C2×#431C2212yes—
12#2905—24no—
13#2905—22no—
14#2905C2214yes—
15C2×#424—24no—
16C2×#423—21no—
17C2×#423C2212yes—
18#2629—22no—
19#2628C2212yes—
20#2628C2212yes—
21#2627—21no—
22#2627C412yes—
23#2626—21no—
24#2626C412yes—
25#2396—22no—
26#2396C414yes—
27#2296—21no—
28#2285C2211yescharacteristic
29#2280—21no—
30#1550—24no—
31#1549C2214yescomplemented
32#1548C2214yescomplemented
33#1542—24no—
34#1458—22no—
35#1458C2214yes—
36C2×#324—21no—
37C2×#323—21no—
38#1286C2218yescomplemented
39C24×C4D411yescharacteristic
40C22×M16—22no—
41C22×M16—24no—
42C22×M16—22no—
43C22×M16—24no—
44C22×M16—44no—
45C22×M16—44no—
46C23×C8—22no—
47#533—22no—
48#533—22no—
49#533—22no—
50#533—22no—
51C2×C4⋊Q8C2311yescharacteristic
52C2×C4⋊Q8C2311yescharacteristic
53C2×#125—21no—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8C2×C412yes—
56C2×C4×Q8—21no—
57C2×C4×Q8C2×C412yes—
58C2×#117—21no—
59C2×#117—21no—
60C2×#117—22no—
61C2×#117—22no—
62C2×#117—22no—
63C2×#117—22no—
64C2×#117C2×C412yes—
65C2×#117D412yes—
66C2×#117D412yes—
67C2×#117C2312yes—
68C22×C4⋊C4C2×C411yescharacteristic
69C22×C4⋊C4D411yescharacteristic
70C22×C22⋊C4C2×C411yescharacteristic
71C22×C22⋊C4D411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42C2311yescharacteristic
74#431—22no—
75#431—22no—
76#431—22no—
77#431C2314yes—
78#424—216no—
79#424—216no—
80#423—24no—
81#423—24no—
82#423—22no—
83#423—22no—
84#423D414yes—
85#423C2314yes—
86C2×C4⋊C8—24no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8D414yes—
90C2×C22⋊C8—22no—
91C2×C4×C8—22no—
92#400—22no—
93#400C2×C414yes—
94#399—22no—
95#399C2×C414yes—
96#398—24no—
97#398—24no—
98#396—24no—
99#396—24no—
100#395—22no—
101#395C2×C414yes—
102#393—22no—
103#393C2×C414yes—
104#389—22no—
105#389—22no—
106#385—22no—
107#385—22no—
108#384—21no—
109#382C2312yes—
110#381—22no—
111#380—21no—
112#380—22no—
113#380—22no—
114#380C2×C412yes—
115#379—22no—
116C4×C4⋊C4—24no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4C2×C412yes—
119C4×C22⋊C4—24no—
120#333—28no—
121#332—28no—
122#324—24no—
123#323—24no—
124C25C22⋊C411yescharacteristic
125C22×Q8—21no—
126C22×Q8C22×C412yes—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4C22⋊C411yescharacteristic
134C23×C4C22⋊C412yes—
135C23×C4C22⋊C411yescharacteristic
136C23×C4C22⋊C411yescharacteristic
137C23×C4SD1614yes—
138C23×C4Q1614yes—
139C23×C4C22×C411yescharacteristic
140C23×C4C2×D412yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C2×M16—28no—
144C2×M16—24no—
145C2×M16—24no—
146C2×M16—48no—
147C2×M16—416no—
148C2×M16—48no—
149C2×M16—48no—
150C2×M16—44no—
151C2×M16—44no—
152C22×C8—28no—
153C22×C8—24no—
154C22×C8—48no—
155C22×C8—42no—
156C4⋊Q8—22no—
157C4⋊Q8—22no—
158C4⋊Q8—22no—
159C4⋊Q8—22no—
160#125—42no—
161#125—42no—
162C22⋊Q8—24no—
163C22⋊Q8—24no—
164C22⋊Q8—42no—
165C22⋊Q8—42no—
166C4×Q8—24no—
167C4×Q8—24no—
168C4×Q8—42no—
169C4×Q8—42no—
170#117—24no—
171#117—24no—
172#117—24no—
173#117—24no—
174#117—24no—
175#117—24no—
176#117—24no—
177#117—24no—
178#117—41no—
179#117—41no—
180#117—41no—
181#117—41no—
182#117—42no—
183#117—42no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4C22⋊C412yes—
199C2×C4⋊C4C22⋊C412yes—
200C2×C4⋊C4C22×C412yes—
201C2×C4⋊C4C22×C412yes—
202C2×C4⋊C4C2×D412yes—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4C22⋊C412yes—
212C2×C22⋊C4C22⋊C412yes—
213C2×C22⋊C4C22×C412yes—
214C2×C22⋊C4C2×D412yes—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42C22⋊C412yes—
222C2×C42C22×C411yescharacteristic
223C2×C42C2×D412yes—
224C2×C42C2×D412yes—
225C2×C42C2×D412yes—
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42#4112yes—
229C2×C42#4112yes—
230C2×C42C2411yescharacteristic
231C4⋊C8—216no—
232C4⋊C8—24no—
233C4⋊C8—24no—
234C4⋊C8—24no—
235C4⋊C8—24no—
236C4⋊C8—42no—
237C4⋊C8C2×D418yes—
238C22⋊C8—24no—
239C22⋊C8—42no—
240C4×C8—24no—
241C4×C8—42no—
242#95—24no—
243#95—24no—
244#95—24no—
245#95—24no—
246#95—22no—
247#95C22×C414yes—
248C24—21no—
249C24—21no—
250C24—22no—
251C24—81no—
252C24—81no—
253C24#10314yes—
254C24C2×C22⋊C412yes—
255C24C2×C22⋊C411yescharacteristic
256C2×Q8—24no—
257C2×Q8—42no—
258C2×Q8—82no—
259C2×Q8—82no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—24no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—81no—
286C22×C4—81no—
287C22×C4—81no—
288C22×C4—81no—
289C22×C4#10314yes—
290C22×C4#10318yes—
291C22×C4C2×C22⋊C412yes—
292C22×C4C2×C22⋊C412yes—
293C22×C4C2×C22⋊C412yes—
294C22×C4C2×C22⋊C411yescharacteristic
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C4×D412yes—
298C22×C4C4×D412yes—
299C22×C4#12011yescharacteristic
300C22×C4#12011yescharacteristic
301C22×C4#12112yes—
302C22×C4#12112yes—
303C22×C4#12311yescharacteristic
304C22×C4#12311yescharacteristic
305C22×C4C2×SD1614yes—
306C22×C4C2×SD1612yes—
307C22×C4C2×Q1614yes—
308C22×C4C2×Q1612yes—
309C22×C4AGL(1,ℤ/8)12yes—
310C22×C4#13712yes—
311C22×C4C23×C411yescharacteristic
312C22×C4C22×D411yescharacteristic
313C22×C4C22×D411yescharacteristic
314M16—416no—
315M16—44no—
316M16—44no—
317M16—48no—
318C2×C8—28no—
319C2×C8—416no—
320C2×C8—48no—
321C2×C8—44no—
322C2×C8—84no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—42no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—44no—
336C4⋊C4—44no—
337C4⋊C4—82no—
338C4⋊C4—82no—
339C4⋊C4—84no—
340C4⋊C4—84no—
341C22⋊C4—24no—
342C22⋊C4—24no—
343C22⋊C4—24no—
344C22⋊C4—24no—
345C22⋊C4—42no—
346C22⋊C4—42no—
347C22⋊C4—44no—
348C22⋊C4—84no—
349C22⋊C4—84no—
350C42—22no—
351C42—22no—
352C42—22no—
353C42—22no—
354C42—42no—
355C42—42no—
356C42—44no—
357C42C4×D412yes—
358C42C4×D412yes—
359C42#12012yes—
360C42#12112yes—
361C42#12112yes—
362C42#12312yes—
363C42C22×D412yes—
364C42C2×#4112yes—
365C23—24no—
366C23—22no—
367C23—22no—
368C23—24no—
369C23—41no—
370C23—41no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—42no—
375C23—81no—
376C23—81no—
377C23—84no—
378C23—84no—
379C23#38611yescharacteristic
380C23#38611yescharacteristic
381C23C2×#10314yes—
382C23C2×#10312yes—
383C23#41712yes—
384C23C22×C22⋊C411yescharacteristic
385Q8—84no—
386Q8—84no—
387C2×C4—22no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—22no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—28no—
394C2×C4—28no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4#38612yes—
421C2×C4#38612yes—
422C2×C4#38611yescharacteristic
423C2×C4#38611yescharacteristic
424C2×C4C2×#10314yes—
425C2×C4C2×#10312yes—
426C2×C4#41712yes—
427C2×C4#45012yes—
428C2×C4#45012yes—
429C2×C4#45112yes—
430C2×C4#45112yes—
431C2×C4#48312yes—
432C2×C4#48312yes—
433C2×C4#48412yes—
434C2×C4#48412yes—
435C2×C4C2×C4×D412yes—
436C2×C4C2×#12011yescharacteristic
437C2×C4C2×#12112yes—
438C2×C4C2×#12311yescharacteristic
439C2×C4C22×SD1611yescharacteristic
440C2×C4C22×Q1611yescharacteristic
441C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
442C2×C4C2×#13711yescharacteristic
443C8—48no—
444C8—84no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—24no—
452C22—42no—
453C22—42no—
454C22—44no—
455C22—48no—
456C22—42no—
457C22—42no—
458C22—82no—
459C22—84no—
460C22#187212yes—
461C22#187212yes—
462C22C2×#38611yescharacteristic
463C22C22×#10311yescharacteristic
464C22C2×#41711yescharacteristic
465C4—22no—
466C4—22no—
467C4—22no—
468C4—22no—
469C4—24no—
470C4—42no—
471C4—42no—
472C4—84no—
473C4—82no—
474C4—82no—
475C4#187212yes—
476C4#187212yes—
477C4C2×#45011yescharacteristic
478C4C2×#45111yescharacteristic
479C4C2×#48311yescharacteristic
480C4C2×#48411yescharacteristic
481C2—21no—
482C2—21no—
483C2—22no—
484C2—22no—
485C2—44no—
486C2C2×#187211yescharacteristic
487C2#2100911yescharacteristic
488C2#2101711yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#386
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #17708C2semidirect
5. #9193C2semidirect
6. #9191C2semidirect
7. #7957C2semidirect
31. #1549C22semidirect
32. #1548C22semidirect
38. #1286C22semidirect

Maximal subgroups

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

subgroupindex
C2×#29052
#346342
#177082
C2×#14582
#91932
#91912
#79572