Finite Group Databasebeta

← #162564

Subgroups — #162564

Order 512.

820 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36635C211yesmaximal, characteristic
2#20777C211yesmaximal, characteristic
3C4×#432C211yesmaximal, characteristic
4#7682C212yesmaximal, complemented
5#7680C212yesmaximal, complemented
6#7661C214yesmaximal
7#7611C214yesmaximal, complemented
8C2×#432—21no—
9#2864—21no—
10#2863—21no—
11#2863C412yes—
12#2862C2211yescharacteristic
13#2859C2211yescharacteristic
14#2857C411yescharacteristic
15#2857C411yescharacteristic
16#2855—21no—
17#2855C2212yes—
18#2851—21no—
19C4×C4⋊Q8C411yescharacteristic
20C4×C4⋊Q8C411yescharacteristic
21C4×#125—21no—
22C4×C22⋊Q8—21no—
23C4×C22⋊Q8C412yes—
24#2295—21no—
25#2295C2212yes—
26#2283—21no—
27C2×C43C2211yescharacteristic
28#1839C2212yes—
29#1831—21no—
30#1817—21no—
31#1797C2212yes—
32#1767C2212yes—
33C4×C4⋊C8—21no—
34C4×C4⋊C8—21no—
35#1749—21no—
36C4×C22⋊C8—21no—
37C4×C22⋊C8C2212yes—
38#1745C2212yes—
39#1744C2212yes—
40C4×#97—21no—
41#1272C2218yes—
42#1271C2218yes—
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×Q8C2×C412yes—
50C22×C42C2×C411yescharacteristic
51C22×C42C2×C411yescharacteristic
52C22×C42D411yescharacteristic
53C22×C42D411yescharacteristic
54C22×C42D411yescharacteristic
55C22×C42D411yescharacteristic
56C22×C42C2311yescharacteristic
57#432—28no—
58C2×C4⋊C8—22no—
59C2×C4⋊C8—22no—
60C2×C22⋊C8—22no—
61C2×C22⋊C8—22no—
62C2×#97—22no—
63C2×#97—22no—
64C2×C4×C8—22no—
65C2×C4×C8—22no—
66#391—21no—
67#391—21no—
68#391—21no—
69#391—21no—
70#391C2×C412yes—
71#391C2×C412yes—
72#391C2×C412yes—
73#391C2312yes—
74#389—21no—
75#389—21no—
76#389—21no—
77#389—21no—
78#389—21no—
79#389—21no—
80#389—21no—
81#389—21no—
82#389C2×C412yes—
83#389C2×C412yes—
84#389C2×C412yes—
85#389C2312yes—
86#387—22no—
87#387—22no—
88#387—22no—
89#387—22no—
90#387—22no—
91#387—22no—
92#387—22no—
93#387—22no—
94#385—22no—
95#385—22no—
96#384C2×C411yescharacteristic
97#384C2×C411yescharacteristic
98#383—22no—
99#382—21no—
100#382—21no—
101#382—21no—
102#381—22no—
103#380—21no—
104#380—21no—
105#380—21no—
106#380C2×C412yes—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4C2×C412yes—
113C4×C22⋊C4—22no—
114#376—22no—
115C43C2×C411yescharacteristic
116C43C2×C411yescharacteristic
117C43D411yescharacteristic
118C43D411yescharacteristic
119C43D411yescharacteristic
120C43D411yescharacteristic
121C43C2311yescharacteristic
122C43C2311yescharacteristic
123#336—22no—
124#336—22no—
125#336—22no—
126#336—22no—
127#336C2314yes—
128#336C2314yes—
129C22×Q8—21no—
130C22×Q8C22×C412yes—
131C23×C4C22⋊C411yescharacteristic
132C23×C4C22⋊C411yescharacteristic
133C23×C4C22⋊C411yescharacteristic
134C23×C4C22⋊C411yescharacteristic
135C23×C4C22×C411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C22×C8—24no—
139C22×C8—24no—
140C4⋊Q8—42no—
141C4⋊Q8—42no—
142#125—44no—
143C22⋊Q8—44no—
144C22⋊Q8—44no—
145C4×Q8—44no—
146C4×Q8—44no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4C22×C412yes—
161C2×C4⋊C4C22×C412yes—
162C2×C4⋊C4C22×C412yes—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—22no—
165C2×C42—24no—
166C2×C42—24no—
167C2×C42—24no—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—42no—
171C2×C42—42no—
172C2×C42—42no—
173C2×C42—42no—
174C2×C42C22⋊C411yescharacteristic
175C2×C42C22⋊C411yescharacteristic
176C2×C42C22⋊C411yescharacteristic
177C2×C42C22⋊C411yescharacteristic
178C2×C42C22⋊C411yescharacteristic
179C2×C42C22⋊C411yescharacteristic
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×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2411yescharacteristic
202C4⋊C8—28no—
203C4⋊C8—44no—
204C22⋊C8—28no—
205C22⋊C8—44no—
206#97—28no—
207#97—44no—
208C4×C8—44no—
209C4×C8—44no—
210#95—22no—
211#95—22no—
212#95—22no—
213#95—22no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224C24C2×C22⋊C411yescharacteristic
225C2×Q8—24no—
226C2×Q8—42no—
227C2×Q8—42no—
228C2×Q8—42no—
229C2×Q8—42no—
230C2×Q8—42no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
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—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4C2×C22⋊C411yescharacteristic
256C22×C4C2×C22⋊C411yescharacteristic
257C22×C4C2×C22⋊C411yescharacteristic
258C22×C4C2×C22⋊C411yescharacteristic
259C22×C4C2×C22⋊C411yescharacteristic
260C22×C4C2×C22⋊C411yescharacteristic
261C22×C4C2×C22⋊C411yescharacteristic
262C22×C4C2×C22⋊C411yescharacteristic
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C2×C22⋊C411yescharacteristic
266C22×C4#13714yes—
267C22×C4C23×C411yescharacteristic
268C22×C4C22×D411yescharacteristic
269C22×C4C22×D411yescharacteristic
270C2×C8—28no—
271C2×C8—44no—
272C2×C8—44no—
273C2×C8—44no—
274C2×C8—44no—
275C2×C8—44no—
276C4⋊C4—24no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—44no—
280C4⋊C4—44no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—44no—
284C4⋊C4—44no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—44no—
289C4⋊C4—42no—
290C4⋊C4—44no—
291C4⋊C4—42no—
292C22⋊C4—44no—
293C22⋊C4—44no—
294C42—22no—
295C42—22no—
296C42—22no—
297C42—22no—
298C42—22no—
299C42—22no—
300C42—22no—
301C42—22no—
302C42—22no—
303C42—22no—
304C42—22no—
305C42—22no—
306C42—22no—
307C42—22no—
308C42—22no—
309C42—22no—
310C42—22no—
311C42—22no—
312C42—41no—
313C42—41no—
314C42—41no—
315C42—41no—
316C42—41no—
317C42—41no—
318C42—41no—
319C42—41no—
320C42—44no—
321C42—44no—
322C42#10414yes—
323C42#13514yes—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23#41912yes—
331C23C22×C22⋊C411yescharacteristic
332Q8—84no—
333Q8—84no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
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—44no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4#41012yes—
384C2×C4#41312yes—
385C2×C4#41612yes—
386C2×C4#41612yes—
387C2×C4#41912yes—
388C2×C4C2×#10412yes—
389C2×C4C2×#10412yes—
390C2×C4C2×#10412yes—
391C2×C4#42112yes—
392C2×C4#56812yes—
393C2×C4C2×#13512yes—
394C2×C4C2×#13712yes—
395C8—48no—
396C8—84no—
397C22—22no—
398C22—22no—
399C22—22no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22#287911yescharacteristic
405C22C2×#41011yescharacteristic
406C22C2×#41311yescharacteristic
407C22C2×#41611yescharacteristic
408C22C2×#41911yescharacteristic
409C22C22×#10411yescharacteristic
410C22C2×#42111yescharacteristic
411C4—22no—
412C4—22no—
413C4—22no—
414C4—22no—
415C4—22no—
416C4—22no—
417C4—22no—
418C4—22no—
419C4—22no—
420C4—41no—
421C4—41no—
422C4—41no—
423C4—41no—
424C4—82no—
425C4—82no—
426C4—82no—
427C4—82no—
428C4#152914yes—
429C2—42no—
430C2#1739511yescharacteristic
431C2#1748911yescharacteristic
432C2#1750111yescharacteristic
433C2#1753311yescharacteristic
434C2#1759611yescharacteristic
435C2C2×#152911yescharacteristic
436C2#1771811yescharacteristic
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 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7682C2semidirect
5. #7680C2semidirect
7. #7611C2semidirect

Maximal subgroups

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

subgroupindex
#366352
#207772
C4×#4322
#76822
#76802
#76612
#76112