Finite Group Databasebeta

← #195040

Subgroups — #195040

Order 512.

1150 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2921C211yesmaximal, characteristic
2#34634C211yesmaximal, characteristic
3#17486C212yesmaximal, complemented
4C2×#1458C211yesmaximal, characteristic
5#9034C218yesmaximal, complemented
6#8095C212yesmaximal, complemented
7C2×#533C2211yescharacteristic
8C22×#117C2211yescharacteristic
9#2921—24no—
10#2921—22no—
11#2921C2214yes—
12C2×#433—24no—
13C2×#431C2212yes—
14C2×#423C2212yes—
15C2×#407—21no—
16C2×C4×M16—21no—
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—
39#533—22no—
40#533—22no—
41#533—22no—
42#533—22no—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×#125—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8C2×C412yes—
48C2×C4×Q8—21no—
49C2×C4×Q8C2312yes—
50C2×#117—21no—
51C2×#117—21no—
52C2×#117—22no—
53C2×#117—22no—
54C2×#117—22no—
55C2×#117—22no—
56C2×#117C2×C412yes—
57C2×#117D412yes—
58C2×#117D412yes—
59C2×#117C2312yes—
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C4⋊C4D411yescharacteristic
62C22×C22⋊C4C2×C411yescharacteristic
63C22×C22⋊C4D411yescharacteristic
64C22×C42D411yescharacteristic
65C22×C42C2311yescharacteristic
66#433—216no—
67#433—216no—
68#431—22no—
69#431—22no—
70#431—22no—
71#431C2314yes—
72#423—22no—
73#423—22no—
74#423D414yes—
75#423C2314yes—
76C2×C4⋊C8—22no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8D414yes—
79#407—24no—
80#407—24no—
81C2×C22⋊C8—24no—
82C2×C22⋊C8—22no—
83C4×M16—24no—
84C4×M16—24no—
85C2×#97—22no—
86C2×C4×C8—22no—
87C2×C4×C8—22no—
88#400—22no—
89#400C2×C414yes—
90#399—22no—
91#399C2×C414yes—
92#398—24no—
93#398—24no—
94#396—24no—
95#396—24no—
96#395—22no—
97#395C2×C414yes—
98#393—22no—
99#393C2×C414yes—
100#389—22no—
101#389—22no—
102#385—22no—
103#385—22no—
104#384—21no—
105#382C2×C412yes—
106#381—22no—
107#380—22no—
108#380—22no—
109#380—21no—
110#380C2×C412yes—
111#379—22no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—24no—
114C4×C4⋊C4C2312yes—
115C4×C22⋊C4—24no—
116Q8⋊C8—216no—
117#324—24no—
118#323—24no—
119C25C22⋊C411yescharacteristic
120C22×Q8—21no—
121C22×Q8C22×C412yes—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—24no—
127C23×C4—24no—
128C23×C4C22⋊C411yescharacteristic
129C23×C4C22⋊C412yes—
130C23×C4C22⋊C411yescharacteristic
131C23×C4C22⋊C411yescharacteristic
132C23×C4C22×C411yescharacteristic
133C23×C4C2×D412yes—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C2×M16—24no—
137C2×M16—24no—
138C2×M16—44no—
139C2×M16—44no—
140C22×C8—24no—
141C22×C8—42no—
142C4⋊Q8—22no—
143C4⋊Q8—22no—
144C4⋊Q8—22no—
145C4⋊Q8—22no—
146#125—42no—
147#125—42no—
148C22⋊Q8—24no—
149C22⋊Q8—24no—
150C22⋊Q8—42no—
151C22⋊Q8—42no—
152C4×Q8—24no—
153C4×Q8—24no—
154C4×Q8—42no—
155C4×Q8—42no—
156#117—24no—
157#117—24no—
158#117—24no—
159#117—24no—
160#117—24no—
161#117—24no—
162#117—24no—
163#117—24no—
164#117—41no—
165#117—41no—
166#117—41no—
167#117—41no—
168#117—44no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4C22⋊C412yes—
184C2×C4⋊C4C22⋊C412yes—
185C2×C4⋊C4C22×C412yes—
186C2×C4⋊C4C22×C412yes—
187C2×C4⋊C4C2×D412yes—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4C22⋊C412yes—
197C2×C22⋊C4C22⋊C412yes—
198C2×C22⋊C4C22×C412yes—
199C2×C22⋊C4C2×D412yes—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42C22⋊C412yes—
207C2×C42C22×C411yescharacteristic
208C2×C42C2×D412yes—
209C2×C42C2×D412yes—
210C2×C42C2×D412yes—
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42#4112yes—
214C2×C42#4112yes—
215C2×C42C2411yescharacteristic
216C4⋊C8—24no—
217C4⋊C8—24no—
218C4⋊C8—24no—
219C4⋊C8—24no—
220C4⋊C8—42no—
221C4⋊C8C2×D418yes—
222C22⋊C8—216no—
223C22⋊C8—24no—
224C22⋊C8—42no—
225#97—28no—
226C4×C8—28no—
227C4×C8—24no—
228C4×C8—42no—
229#95—24no—
230#95—24no—
231#95—22no—
232#95—24no—
233#95—24no—
234#95C22×C414yes—
235C24—21no—
236C24—21no—
237C24—22no—
238C24—22no—
239C24—81no—
240C24—81no—
241C24C2×C22⋊C412yes—
242C24C2×C22⋊C411yescharacteristic
243C2×Q8—24no—
244C2×Q8—42no—
245C2×Q8—82no—
246C2×Q8—82no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—44no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4—81no—
271C22×C4—81no—
272C22×C4—81no—
273C22×C4—81no—
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C412yes—
276C22×C4C2×C22⋊C412yes—
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4C4×D412yes—
281C22×C4C4×D412yes—
282C22×C4#12011yescharacteristic
283C22×C4#12011yescharacteristic
284C22×C4#12112yes—
285C22×C4#12112yes—
286C22×C4#12311yescharacteristic
287C22×C4#12311yescharacteristic
288C22×C4#13514yes—
289C22×C4#13714yes—
290C22×C4C23×C411yescharacteristic
291C22×C4C22×D411yescharacteristic
292C22×C4C22×D411yescharacteristic
293M16—44no—
294M16—44no—
295M16—48no—
296C2×C8—28no—
297C2×C8—48no—
298C2×C8—44no—
299C2×C8—84no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—44no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—44no—
311C4⋊C4—42no—
312C4⋊C4—44no—
313C4⋊C4—82no—
314C4⋊C4—82no—
315C4⋊C4—84no—
316C4⋊C4—84no—
317C22⋊C4—24no—
318C22⋊C4—24no—
319C22⋊C4—24no—
320C22⋊C4—24no—
321C22⋊C4—42no—
322C22⋊C4—42no—
323C22⋊C4—44no—
324C22⋊C4—84no—
325C22⋊C4—84no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—22no—
330C42—42no—
331C42—42no—
332C42—44no—
333C42C4×D412yes—
334C42C4×D412yes—
335C42#12012yes—
336C42#12112yes—
337C42#12112yes—
338C42#12312yes—
339C42C22×D412yes—
340C42C2×#4112yes—
341C23—22no—
342C23—22no—
343C23—22no—
344C23—41no—
345C23—41no—
346C23—42no—
347C23—42no—
348C23—42no—
349C23—42no—
350C23—42no—
351C23—42no—
352C23—81no—
353C23—81no—
354C23—84no—
355C23—84no—
356C23#38611yescharacteristic
357C23#38611yescharacteristic
358C23#41612yes—
359C23#41912yes—
360C23C22×C22⋊C411yescharacteristic
361Q8—48no—
362Q8—84no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4#38612yes—
396C2×C4#38612yes—
397C2×C4#38611yescharacteristic
398C2×C4#38611yescharacteristic
399C2×C4#41612yes—
400C2×C4#41912yes—
401C2×C4#45214yes—
402C2×C4#45214yes—
403C2×C4#48514yes—
404C2×C4#48514yes—
405C2×C4C2×C4×D412yes—
406C2×C4C2×#12011yescharacteristic
407C2×C4C2×#12112yes—
408C2×C4C2×#12311yescharacteristic
409C2×C4C2×#13512yes—
410C2×C4C2×#13712yes—
411C8—48no—
412C8—84no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—42no—
420C22—42no—
421C22—44no—
422C22—42no—
423C22—42no—
424C22—42no—
425C22—82no—
426C22—84no—
427C22—84no—
428C22#187312yes—
429C22#187312yes—
430C22C2×#38611yescharacteristic
431C22C2×#41611yescharacteristic
432C22C2×#41911yescharacteristic
433C4—22no—
434C4—22no—
435C4—22no—
436C4—22no—
437C4—42no—
438C4—42no—
439C4—42no—
440C4—84no—
441C4—82no—
442C4—82no—
443C4#187312yes—
444C4#187312yes—
445C4C2×#45212yes—
446C4C2×#48512yes—
447C2—21no—
448C2—21no—
449C2—22no—
450C2—22no—
451C2—82no—
452C2C2×#187311yescharacteristic
453C2#2094711yescharacteristic
454C2#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. #9034C2semidirect
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×#29212
#346342
#174862
C2×#14582
#90342
#80952