Finite Group Databasebeta

← #195026

Subgroups — #195026

Order 512.

1219 conjugacy classes of subgroups.

The subgroup lattice has 449 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60125C211yesmaximal, characteristic
2#34952C211yesmaximal, characteristic
3C2×#1487C212yesmaximal, complemented
4#17349C211yesmaximal, characteristic
5#9034C218yesmaximal, complemented
6#8095C212yesmaximal, complemented
7C2×C4×#41C2211yescharacteristic
8C22×C4×Q8C2211yescharacteristic
9#2925—24no—
10#2925—24no—
11#2921C2214yes—
12#2920—22no—
13C2×#431C2212yes—
14C2×#423C2212yes—
15#2879—21no—
16#2873—21no—
17#2582—21no—
18#2581C412yes—
19#2580C412yes—
20#2579—22no—
21#2538—21no—
22#2487C412yes—
23#2409C412yes—
24#2396C2214yes—
25C4×C22⋊Q8—22no—
26#2297—21no—
27#2296—21no—
28#2285C2211yescharacteristic
29#1487—24no—
30#1487C2218yes—
31C2×Q8⋊C8—24no—
32#1461—21no—
33#1460—21no—
34#1459—22no—
35#1458C2214yes—
36#1303C2218yes—
37C22×#41D411yescharacteristic
38C23×Q8C2×C411yescharacteristic
39C22×M16—22no—
40C2×C22⋊Q8—22no—
41C4×#41—22no—
42C4×#41—22no—
43C4×#41—22no—
44C4×#41—22no—
45C2×C4×Q8—21no—
46C2×C4×Q8—21no—
47C2×C4×Q8—22no—
48C2×C4×Q8—22no—
49C2×C4×Q8—22no—
50C2×C4×Q8—22no—
51C2×C4×Q8C2×C412yes—
52C2×C4×Q8C2×C411yescharacteristic
53C2×C4×Q8D411yescharacteristic
54C2×C4×Q8C2312yes—
55C2×C4×D4C2×C412yes—
56C2×C4×D4C2×C411yescharacteristic
57C2×C4×D4D412yes—
58C2×C4×D4D411yescharacteristic
59C2×#117—21no—
60C2×#117D411yescharacteristic
61C2×#117C2312yes—
62C22×C4⋊C4—21no—
63C22×C4⋊C4C2×C411yescharacteristic
64C22×C42—21no—
65C22×C42D412yes—
66C22×C42C2311yescharacteristic
67#433—216no—
68#432—24no—
69#432—24no—
70#431—24no—
71#431—24no—
72#431—22no—
73#431—22no—
74#431—22no—
75#431C2314yes—
76#423—22no—
77#423—22no—
78#423D414yes—
79#423C2314yes—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8D414yes—
83#408—28no—
84#407—24no—
85C2×C22⋊C8—22no—
86#405—28no—
87C4×M16—24no—
88C2×C4×C8—22no—
89#400C2×C414yes—
90#399C2×C414yes—
91#398—24no—
92#396—24no—
93#395C2×C414yes—
94#393C2×C414yes—
95#391—22no—
96#391—22no—
97#390—22no—
98#389—22no—
99#389—22no—
100#389—22no—
101#387—24no—
102#386—22no—
103#386—22no—
104#385—24no—
105#385—22no—
106#385—22no—
107#384—21no—
108#382C2×C412yes—
109#380—22no—
110#380—21no—
111#380C2×C412yes—
112C4×C4⋊C4—24no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4C2312yes—
115C4×C22⋊C4—24no—
116Q8⋊C8—216no—
117#324—24no—
118#323—24no—
119C2×#41—22no—
120C2×#41—22no—
121C2×#41—24no—
122C2×#41—24no—
123C22×Q8—21no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—44no—
127C22×Q8C22⋊C411yescharacteristic
128C22×Q8C22×C412yes—
129C22×D4C22⋊C412yes—
130C22×D4C22⋊C411yescharacteristic
131C23×C4—21no—
132C23×C4—21no—
133C23×C4C22⋊C412yes—
134C23×C4C22⋊C412yes—
135C23×C4C22×C411yescharacteristic
136C23×C4C2×D412yes—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C2×M16—24no—
140C2×M16—24no—
141C2×M16—44no—
142C2×M16—44no—
143C22×C8—24no—
144C22×C8—42no—
145C22⋊Q8—48no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×Q8—22no—
150C4×Q8—24no—
151C4×Q8—24no—
152C4×Q8—24no—
153C4×Q8—24no—
154C4×Q8—41no—
155C4×Q8—41no—
156C4×Q8—42no—
157C4×Q8—42no—
158C4×Q8—41no—
159C4×Q8—41no—
160C4×D4—24no—
161C4×D4—24no—
162C4×D4—24no—
163C4×D4—24no—
164C4×D4—22no—
165C4×D4—22no—
166C4×D4—22no—
167C4×D4—22no—
168#117—22no—
169#117—22no—
170#117—24no—
171#117—24no—
172#117—44no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4C22⋊C411yescharacteristic
186C2×C4⋊C4C22⋊C411yescharacteristic
187C2×C4⋊C4C22×C412yes—
188C2×C4⋊C4C22×C412yes—
189C2×C4⋊C4C2×D412yes—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4C22⋊C412yes—
194C2×C22⋊C4C22×C412yes—
195C2×C22⋊C4C2×D412yes—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—42no—
204C2×C42—42no—
205C2×C42C22⋊C412yes—
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—28no—
223C22⋊C8—28no—
224C22⋊C8—24no—
225C22⋊C8—42no—
226#97—28no—
227C4×C8—28no—
228C4×C8—24no—
229C4×C8—42no—
230#95—24no—
231#95—24no—
232#95—22no—
233#95—24no—
234#95—24no—
235#95C22×C414yes—
236C24C2×C22⋊C412yes—
237C24C2×C22⋊C411yescharacteristic
238#41—44no—
239#41—44no—
240#41—48no—
241C2×Q8—24no—
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8—28no—
246C2×Q8—24no—
247C2×Q8—44no—
248C2×Q8—42no—
249C2×Q8—44no—
250C2×Q8—48no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×D4—24no—
254C2×D4—24no—
255C2×D4—24no—
256C2×D4—22no—
257C2×D4—22no—
258C2×D4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—22no—
270C22×C4—42no—
271C22×C4—44no—
272C22×C4—44no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—81no—
276C22×C4—82no—
277C22×C4—82no—
278C22×C4C2×C22⋊C412yes—
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4C2×C22⋊C412yes—
281C22×C4C2×C22⋊C412yes—
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C4×D412yes—
285C22×C4C4×D412yes—
286C22×C4#12012yes—
287C22×C4#12112yes—
288C22×C4#12112yes—
289C22×C4#12312yes—
290C22×C4#13514yes—
291C22×C4#13714yes—
292C22×C4C23×C411yescharacteristic
293C22×C4C22×D411yescharacteristic
294C22×C4C22×D411yescharacteristic
295M16—44no—
296M16—44no—
297M16—48no—
298C2×C8—28no—
299C2×C8—48no—
300C2×C8—44no—
301C2×C8—84no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—24no—
307C4⋊C4—44no—
308C4⋊C4—48no—
309C4⋊C4—44no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—44no—
315C4⋊C4—82no—
316C4⋊C4—84no—
317C22⋊C4—24no—
318C22⋊C4—24no—
319C22⋊C4—24no—
320C22⋊C4—48no—
321C22⋊C4—44no—
322C22⋊C4—84no—
323C42—22no—
324C42—22no—
325C42—22no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—42no—
330C42—42no—
331C42—44no—
332C42—44no—
333C42C4×D412yes—
334C42C4×D412yes—
335C42#12011yescharacteristic
336C42#12011yescharacteristic
337C42#12112yes—
338C42#12112yes—
339C42#12311yescharacteristic
340C42#12311yescharacteristic
341C42C22×D412yes—
342C42C2×#4112yes—
343C23—22no—
344C23—22no—
345C23—22no—
346C23—41no—
347C23—41no—
348C23—42no—
349C23—42no—
350C23#38612yes—
351C23#41612yes—
352C23#41912yes—
353C23C22×C22⋊C411yescharacteristic
354Q8—28no—
355Q8—44no—
356Q8—44no—
357Q8—44no—
358Q8—44no—
359D4—48no—
360D4—44no—
361C2×C4—22no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—44no—
376C2×C4—42no—
377C2×C4—48no—
378C2×C4—42no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—42no—
386C2×C4—44no—
387C2×C4—42no—
388C2×C4—44no—
389C2×C4—42no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—84no—
393C2×C4#38611yescharacteristic
394C2×C4#38611yescharacteristic
395C2×C4#38611yescharacteristic
396C2×C4#38612yes—
397C2×C4#38611yescharacteristic
398C2×C4#41612yes—
399C2×C4#41912yes—
400C2×C4#45214yes—
401C2×C4#45214yes—
402C2×C4#48518yes—
403C2×C4C2×C4×D412yes—
404C2×C4C2×#12011yescharacteristic
405C2×C4C2×#12112yes—
406C2×C4C2×#12311yescharacteristic
407C2×C4C2×#13512yes—
408C2×C4C2×#13712yes—
409C8—48no—
410C8—84no—
411C22—21no—
412C22—22no—
413C22—21no—
414C22—22no—
415C22—42no—
416C22—42no—
417C22—44no—
418C22—42no—
419C22—42no—
420C22—42no—
421C22—82no—
422C22#187314yes—
423C22C2×#38611yescharacteristic
424C22C2×#41611yescharacteristic
425C22C2×#41911yescharacteristic
426C4—22no—
427C4—21no—
428C4—21no—
429C4—21no—
430C4—22no—
431C4—42no—
432C4—42no—
433C4—44no—
434C4—42no—
435C4—84no—
436C4—82no—
437C4#187314yes—
438C4#189112yes—
439C4C2×#45212yes—
440C4C2×#48512yes—
441C2—21no—
442C2—21no—
443C2—24no—
444C2—82no—
445C2C2×#187311yescharacteristic
446C2#2094711yescharacteristic
447C2C2×#189111yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#386
prime radicalC22×#41D4
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. C2×#1487C2semidirect
5. #9034C2semidirect
6. #8095C2semidirect

Maximal subgroups

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

subgroupindex
#601252
#349522
C2×#14872
#173492
#90342
#80952