Finite Group Databasebeta

← #208014

Subgroups — #208014

Order 512.

968 conjugacy classes of subgroups.

The subgroup lattice has 447 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36635C211yesmaximal, characteristic
2#20778C211yesmaximal, characteristic
3C42×M16C211yesmaximal, characteristic
4#9890C212yesmaximal, complemented
5#9686C212yesmaximal, complemented
6#9122C214yesmaximal
7#8227C214yesmaximal, complemented
8C2×C4×M16—21no—
9C2×C4×M16—21no—
10C2×C4×M16C2212yes—
11#2864C412yes—
12#2863—21no—
13#2863—21no—
14#2862C2211yescharacteristic
15#2859C2211yescharacteristic
16#2857C411yescharacteristic
17#2857C411yescharacteristic
18#2855—21no—
19#2855—21no—
20#2851C2212yes—
21C4×C4⋊Q8C411yescharacteristic
22C4×C4⋊Q8C411yescharacteristic
23C4×#125C2212yes—
24C4×C22⋊Q8—21no—
25C4×C22⋊Q8—21no—
26#2295—21no—
27#2295—21no—
28#2283C412yes—
29C2×C43C2211yescharacteristic
30#1839—21no—
31#1831—21no—
32#1817—21no—
33#1797—21no—
34#1767C2212yes—
35#1749C2212yes—
36#1745C2212yes—
37C4×#97—21no—
38C4×#97C2212yes—
39C4×#97C2212yes—
40C42×C8—21no—
41C2×#330C2218yes—
42#1295C2218yes—
43C22×M16—22no—
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×#125C2312yes—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×C4×Q8—21no—
50C2×C4×Q8—21no—
51C22×C42C2×C411yescharacteristic
52C22×C42C2×C411yescharacteristic
53C22×C42D411yescharacteristic
54C22×C42D411yescharacteristic
55C22×C42D411yescharacteristic
56C22×C42D411yescharacteristic
57C22×C42C2311yescharacteristic
58C2×C4⋊C8—22no—
59C2×C22⋊C8—22no—
60C4×M16—28no—
61C4×M16—28no—
62C4×M16—28no—
63C2×#97—22no—
64C2×#97—22no—
65C2×#97C2314yes—
66C2×#97C2314yes—
67C2×C4×C8—22no—
68C2×C4×C8—22no—
69C2×C4×C8—22no—
70#391—21no—
71#391—21no—
72#391—21no—
73#391—21no—
74#391—21no—
75#391—21no—
76#391—21no—
77#391—21no—
78#389—21no—
79#389—21no—
80#389—21no—
81#389—21no—
82#389—21no—
83#389—21no—
84#389—21no—
85#389—21no—
86#389C2×C412yes—
87#389C2×C412yes—
88#389C2×C412yes—
89#389C2×C412yes—
90#387—22no—
91#387—22no—
92#387—22no—
93#387—22no—
94#387—22no—
95#387—22no—
96#387—22no—
97#387—22no—
98#385—22no—
99#385—22no—
100#384C2×C411yescharacteristic
101#384C2×C411yescharacteristic
102#383—22no—
103#382C2×C412yes—
104#382C2×C412yes—
105#382C2312yes—
106#381—22no—
107#380—21no—
108#380—21no—
109#380C2×C412yes—
110#380C2×C412yes—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4C2×C412yes—
116C4×C4⋊C4C2×C412yes—
117C4×C22⋊C4—22no—
118#376—22no—
119C43C2×C411yescharacteristic
120C43C2×C411yescharacteristic
121C43D411yescharacteristic
122C43D411yescharacteristic
123C43D411yescharacteristic
124C43D411yescharacteristic
125C43C2311yescharacteristic
126C43C2311yescharacteristic
127#336—22no—
128#336—22no—
129#336—22no—
130#336—22no—
131#330—216no—
132C22×Q8—21no—
133C22×Q8—21no—
134C23×C4C22⋊C411yescharacteristic
135C23×C4C22⋊C411yescharacteristic
136C23×C4C22⋊C411yescharacteristic
137C23×C4C22⋊C411yescharacteristic
138C23×C4C22×C411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C2×M16—28no—
142C2×M16—28no—
143C2×M16—28no—
144C22×C8—24no—
145C22×C8—24no—
146C4⋊Q8—42no—
147C4⋊Q8—42no—
148#125—28no—
149C22⋊Q8—44no—
150C22⋊Q8—44no—
151C4×Q8—44no—
152C4×Q8—44no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4C22×C412yes—
166C2×C4⋊C4C22×C412yes—
167C2×C4⋊C4C22×C412yes—
168C2×C4⋊C4C22×C412yes—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C42—24no—
172C2×C42—24no—
173C2×C42—24no—
174C2×C42—24no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—42no—
178C2×C42—42no—
179C2×C42—42no—
180C2×C42C22⋊C411yescharacteristic
181C2×C42C22⋊C411yescharacteristic
182C2×C42C22⋊C411yescharacteristic
183C2×C42C22⋊C411yescharacteristic
184C2×C42C22⋊C411yescharacteristic
185C2×C42C22⋊C411yescharacteristic
186C2×C42C22⋊C411yescharacteristic
187C2×C42C22⋊C411yescharacteristic
188C2×C42C22⋊C411yescharacteristic
189C2×C42C22⋊C411yescharacteristic
190C2×C42C22⋊C411yescharacteristic
191C2×C42C22⋊C411yescharacteristic
192C2×C42C22×C411yescharacteristic
193C2×C42C22×C411yescharacteristic
194C2×C42C22×C411yescharacteristic
195C2×C42C22×C411yescharacteristic
196C2×C42C22×C411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2411yescharacteristic
208C4⋊C8—44no—
209C22⋊C8—44no—
210#97—28no—
211#97—28no—
212#97—24no—
213#97—28no—
214#97#4118yes—
215C4×C8—28no—
216C4×C8—28no—
217C4×C8—28no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232C24C2×C22⋊C411yescharacteristic
233C2×Q8—42no—
234C2×Q8—42no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8—42no—
238C2×Q8—42no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4#10414yes—
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C2×C22⋊C411yescharacteristic
266C22×C4C2×C22⋊C411yescharacteristic
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C2×C22⋊C411yescharacteristic
275C22×C4C23×C411yescharacteristic
276C22×C4C22×D411yescharacteristic
277C22×C4C22×D411yescharacteristic
278M16—232no—
279C2×C8—28no—
280C2×C8—28no—
281C2×C8—28no—
282C2×C8—44no—
283C2×C8—44no—
284C2×C8—44no—
285C4⋊C4—24no—
286C4⋊C4—24no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—44no—
291C4⋊C4—42no—
292C4⋊C4—44no—
293C4⋊C4—44no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—44no—
297C4⋊C4—42no—
298C4⋊C4—44no—
299C4⋊C4—44no—
300C4⋊C4—42no—
301C22⋊C4—44no—
302C22⋊C4—44no—
303C42—22no—
304C42—22no—
305C42—22no—
306C42—22no—
307C42—22no—
308C42—22no—
309C42—22no—
310C42—22no—
311C42—22no—
312C42—22no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—22no—
320C42—22no—
321C42—22no—
322C42—41no—
323C42—41no—
324C42—41no—
325C42—41no—
326C42—41no—
327C42—41no—
328C42—44no—
329C42—44no—
330C42#10114yes—
331C42#10414yes—
332C42C2×#4114yes—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23C2×#10412yes—
340C23C22×C22⋊C411yescharacteristic
341Q8—84no—
342Q8—84no—
343C2×C4—28no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4C2×#10112yes—
391C2×C4C2×#10112yes—
392C2×C4C2×#10112yes—
393C2×C4#41312yes—
394C2×C4C2×#10412yes—
395C2×C4C2×#10412yes—
396C2×C4C2×#10412yes—
397C2×C4C2×#10412yes—
398C2×C4C2×#10412yes—
399C2×C4#42112yes—
400C2×C4#42112yes—
401C2×C4C4×#4112yes—
402C2×C4#52512yes—
403C2×C4#53212yes—
404C8—216no—
405C8—84no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22#229711yescharacteristic
414C22C22×#10111yescharacteristic
415C22C2×#41311yescharacteristic
416C22C22×#10411yescharacteristic
417C22C22×#10411yescharacteristic
418C22C2×#42111yescharacteristic
419C22C2×#42111yescharacteristic
420C4—22no—
421C4—22no—
422C4—22no—
423C4—22no—
424C4—22no—
425C4—22no—
426C4—22no—
427C4—22no—
428C4—22no—
429C4—22no—
430C4—22no—
431C4—41no—
432C4—41no—
433C4—82no—
434C4—82no—
435C4—82no—
436C4—82no—
437C4#152914yes—
438C2—42no—
439C2C2×#152911yescharacteristic
440C2#1764811yescharacteristic
441C2#1764811yescharacteristic
442C2#1765011yescharacteristic
443C2#2138311yescharacteristic
444C2#2141511yescharacteristic
445C2#2142511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9890C2semidirect
5. #9686C2semidirect
7. #8227C2semidirect

Maximal subgroups

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

subgroupindex
#366352
#207782
C42×M162
#98902
#96862
#91222
#82272