Finite Group Databasebeta

← #195038

Subgroups — #195038

Order 512.

1308 conjugacy classes of subgroups.

The subgroup lattice has 487 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2905C211yesmaximal, characteristic
2#34634C211yesmaximal, characteristic
3#17486C212yesmaximal, complemented
4C2×#1458C211yesmaximal, characteristic
5#9033C218yesmaximal, complemented
6#8095C212yesmaximal, complemented
7C23×M16—21no—
8C2×#533C2211yescharacteristic
9C22×#117C2211yescharacteristic
10C2×#431C2212yes—
11#2905—24no—
12#2905—22no—
13#2905C2214yes—
14C2×#424—24no—
15C2×#423—21no—
16C2×#423C2212yes—
17#2629—22no—
18#2628C412yes—
19#2628C412yes—
20#2627—21no—
21#2627C412yes—
22#2626—21no—
23#2626C412yes—
24#2396—22no—
25#2396C2214yes—
26#2296—21no—
27#2285C2211yescharacteristic
28#2280—21no—
29#1493—24no—
30#1487C2218yescomplemented
31#1481—24no—
32#1458—22no—
33#1458C2214yes—
34C2×#324—21no—
35C2×#323—21no—
36#1303C2218yescomplemented
37C24×C4D411yescharacteristic
38C22×M16—22no—
39C22×M16—24no—
40C22×M16—22no—
41C22×M16—24no—
42C22×M16—44no—
43C22×M16—44no—
44C23×C8—22no—
45#533—22no—
46#533—22no—
47#533—22no—
48#533—22no—
49C2×C4⋊Q8C2×C411yescharacteristic
50C2×C4⋊Q8C2×C411yescharacteristic
51C2×#125—21no—
52C2×C22⋊Q8—21no—
53C2×C22⋊Q8C2×C412yes—
54C2×C4×Q8—21no—
55C2×C4×Q8C2312yes—
56C2×#117—21no—
57C2×#117—21no—
58C2×#117—22no—
59C2×#117—22no—
60C2×#117—22no—
61C2×#117—22no—
62C2×#117C2×C412yes—
63C2×#117D412yes—
64C2×#117D412yes—
65C2×#117C2312yes—
66C22×C4⋊C4C2×C411yescharacteristic
67C22×C4⋊C4D411yescharacteristic
68C22×C22⋊C4C2×C411yescharacteristic
69C22×C22⋊C4D411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42C2311yescharacteristic
72#431—22no—
73#431—22no—
74#431—22no—
75#431C2314yes—
76#424—216no—
77#424—216no—
78#423—24no—
79#423—24no—
80#423—22no—
81#423—22no—
82#423D414yes—
83#423C2314yes—
84C2×C4⋊C8—24no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8D414yes—
88C2×C22⋊C8—22no—
89C2×C4×C8—22no—
90#400—22no—
91#400C2×C414yes—
92#399—22no—
93#399C2×C414yes—
94#398—24no—
95#398—24no—
96#396—24no—
97#396—24no—
98#395—22no—
99#395C2×C414yes—
100#393—22no—
101#393C2×C414yes—
102#389—22no—
103#389—22no—
104#385—22no—
105#385—22no—
106#384—21no—
107#382C2×C412yes—
108#381—22no—
109#380—22no—
110#380—22no—
111#380—21no—
112#380C2×C412yes—
113#379—22no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—24no—
116C4×C4⋊C4C2312yes—
117C4×C22⋊C4—24no—
118Q8⋊C8—216no—
119#324—24no—
120#323—24no—
121C25C22⋊C411yescharacteristic
122C22×Q8—21no—
123C22×Q8C22×C412yes—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4C22⋊C411yescharacteristic
131C23×C4C22⋊C412yes—
132C23×C4C22⋊C411yescharacteristic
133C23×C4C22⋊C411yescharacteristic
134C23×C4SD1614yes—
135C23×C4Q1614yes—
136C23×C4C22×C411yescharacteristic
137C23×C4C2×D412yes—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C2×M16—28no—
141C2×M16—24no—
142C2×M16—24no—
143C2×M16—48no—
144C2×M16—416no—
145C2×M16—48no—
146C2×M16—48no—
147C2×M16—44no—
148C2×M16—44no—
149C22×C8—28no—
150C22×C8—24no—
151C22×C8—48no—
152C22×C8—42no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C4⋊Q8—22no—
156C4⋊Q8—22no—
157#125—42no—
158#125—42no—
159C22⋊Q8—24no—
160C22⋊Q8—24no—
161C22⋊Q8—42no—
162C22⋊Q8—42no—
163C4×Q8—24no—
164C4×Q8—24no—
165C4×Q8—42no—
166C4×Q8—42no—
167#117—24no—
168#117—24no—
169#117—24no—
170#117—24no—
171#117—24no—
172#117—24no—
173#117—24no—
174#117—24no—
175#117—41no—
176#117—41no—
177#117—41no—
178#117—41no—
179#117—42no—
180#117—42no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4C22⋊C412yes—
196C2×C4⋊C4C22⋊C412yes—
197C2×C4⋊C4C22×C412yes—
198C2×C4⋊C4C22×C412yes—
199C2×C4⋊C4C2×D412yes—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4C22⋊C412yes—
209C2×C22⋊C4C22⋊C412yes—
210C2×C22⋊C4C22×C412yes—
211C2×C22⋊C4C2×D412yes—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42C22⋊C412yes—
219C2×C42C22×C411yescharacteristic
220C2×C42C2×D412yes—
221C2×C42C2×D412yes—
222C2×C42C2×D412yes—
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42#4112yes—
226C2×C42#4112yes—
227C2×C42C2411yescharacteristic
228C4⋊C8—216no—
229C4⋊C8—24no—
230C4⋊C8—24no—
231C4⋊C8—24no—
232C4⋊C8—24no—
233C4⋊C8—42no—
234C4⋊C8C2×D418yes—
235C22⋊C8—24no—
236C22⋊C8—42no—
237C4×C8—24no—
238C4×C8—42no—
239#95—24no—
240#95—24no—
241#95—22no—
242#95—24no—
243#95—24no—
244#95C22×C414yes—
245C24—21no—
246C24—21no—
247C24—22no—
248C24—81no—
249C24—81no—
250C24#10314yes—
251C24C2×C22⋊C412yes—
252C24C2×C22⋊C411yescharacteristic
253C2×Q8—24no—
254C2×Q8—42no—
255C2×Q8—82no—
256C2×Q8—82no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—81no—
283C22×C4—81no—
284C22×C4—81no—
285C22×C4—81no—
286C22×C4#10314yes—
287C22×C4#10318yes—
288C22×C4C2×C22⋊C412yes—
289C22×C4C2×C22⋊C412yes—
290C22×C4C2×C22⋊C412yes—
291C22×C4C2×C22⋊C411yescharacteristic
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×C22⋊C411yescharacteristic
294C22×C4C4×D412yes—
295C22×C4C4×D412yes—
296C22×C4#12011yescharacteristic
297C22×C4#12011yescharacteristic
298C22×C4#12112yes—
299C22×C4#12112yes—
300C22×C4#12311yescharacteristic
301C22×C4#12311yescharacteristic
302C22×C4C2×SD1614yes—
303C22×C4C2×SD1612yes—
304C22×C4C2×Q1614yes—
305C22×C4C2×Q1612yes—
306C22×C4AGL(1,ℤ/8)12yes—
307C22×C4#13712yes—
308C22×C4C23×C411yescharacteristic
309C22×C4C22×D411yescharacteristic
310C22×C4C22×D411yescharacteristic
311M16—416no—
312M16—44no—
313M16—44no—
314M16—48no—
315C2×C8—28no—
316C2×C8—416no—
317C2×C8—48no—
318C2×C8—44no—
319C2×C8—84no—
320C4⋊C4—24no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—44no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—44no—
331C4⋊C4—42no—
332C4⋊C4—44no—
333C4⋊C4—82no—
334C4⋊C4—82no—
335C4⋊C4—84no—
336C4⋊C4—84no—
337C22⋊C4—24no—
338C22⋊C4—24no—
339C22⋊C4—24no—
340C22⋊C4—24no—
341C22⋊C4—42no—
342C22⋊C4—42no—
343C22⋊C4—44no—
344C22⋊C4—84no—
345C22⋊C4—84no—
346C42—22no—
347C42—22no—
348C42—22no—
349C42—22no—
350C42—42no—
351C42—42no—
352C42—42no—
353C42—42no—
354C42C4×D412yes—
355C42C4×D412yes—
356C42#12012yes—
357C42#12112yes—
358C42#12112yes—
359C42#12312yes—
360C42C22×D412yes—
361C42C2×#4112yes—
362C23—24no—
363C23—22no—
364C23—22no—
365C23—24no—
366C23—41no—
367C23—41no—
368C23—42no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23—81no—
373C23—81no—
374C23—84no—
375C23—84no—
376C23#38611yescharacteristic
377C23#38611yescharacteristic
378C23C2×#10314yes—
379C23C2×#10312yes—
380C23#41712yes—
381C23C22×C22⋊C411yescharacteristic
382Q8—48no—
383Q8—84no—
384C2×C4—22no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—22no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—28no—
391C2×C4—28no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4#38612yes—
418C2×C4#38612yes—
419C2×C4#38611yescharacteristic
420C2×C4#38611yescharacteristic
421C2×C4C2×#10314yes—
422C2×C4C2×#10312yes—
423C2×C4#41712yes—
424C2×C4#45012yes—
425C2×C4#45012yes—
426C2×C4#45112yes—
427C2×C4#45112yes—
428C2×C4#48312yes—
429C2×C4#48312yes—
430C2×C4#48412yes—
431C2×C4#48412yes—
432C2×C4C2×C4×D412yes—
433C2×C4C2×#12011yescharacteristic
434C2×C4C2×#12112yes—
435C2×C4C2×#12311yescharacteristic
436C2×C4C22×SD1611yescharacteristic
437C2×C4C22×Q1611yescharacteristic
438C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
439C2×C4C2×#13711yescharacteristic
440C8—48no—
441C8—84no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—24no—
449C22—42no—
450C22—42no—
451C22—44no—
452C22—48no—
453C22—42no—
454C22—42no—
455C22—82no—
456C22—84no—
457C22#187212yes—
458C22#187212yes—
459C22C2×#38611yescharacteristic
460C22C22×#10311yescharacteristic
461C22C2×#41711yescharacteristic
462C4—24no—
463C4—22no—
464C4—22no—
465C4—22no—
466C4—22no—
467C4—42no—
468C4—42no—
469C4—84no—
470C4—82no—
471C4—82no—
472C4#187212yes—
473C4#187212yes—
474C4C2×#45011yescharacteristic
475C4C2×#45111yescharacteristic
476C4C2×#48311yescharacteristic
477C4C2×#48411yescharacteristic
478C2—21no—
479C2—21no—
480C2—22no—
481C2—22no—
482C2—44no—
483C2C2×#187211yescharacteristic
484C2#2094711yescharacteristic
485C2#2096411yescharacteristic
subgroupG/H
centerC2×C4C2×#120
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. #17486C2semidirect
5. #9033C2semidirect
6. #8095C2semidirect
30. #1487C22semidirect
36. #1303C22semidirect

Maximal subgroups

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

subgroupindex
C2×#29052
#346342
#174862
C2×#14582
#90332
#80952