Finite Group Databasebeta

← #163249

Subgroups — #163249

Order 512.

1191 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36626C211yesmaximal, characteristic
2#20626C212yesmaximal, complemented
3#7715C214yesmaximal, complemented
4#7714C214yesmaximal, complemented
5#7632C212yesmaximal, complemented
6#7631C212yesmaximal, complemented
7C2×#525C411yescharacteristic
8C23×C42C2211yescharacteristic
9C2×#423—22no—
10C2×#407—22no—
11#2855C412yes—
12#2855C2212yes—
13#2854C412yes—
14#2854C2212yes—
15#2853—22no—
16#2849—22no—
17C4×#123—22no—
18C4×#123—22no—
19#2297C411yescharacteristic
20#2297C2211yescharacteristic
21#2296C411yescharacteristic
22#2296C2211yescharacteristic
23#2280C411yescharacteristic
24#1797—22no—
25#1797—22no—
26#1797C2214yes—
27#1797C2214yes—
28#1728C2214yes—
29#1272C2214yescomplemented
30#1271C2218yescomplemented
31#1270C2214yescomplemented
32C22×#41C2×C411yescharacteristic
33C24×C4C2×C411yescharacteristic
34C24×C4D412yes—
35C22×M16—24no—
36#525—44no—
37C2×#123—22no—
38C2×#123—21no—
39C2×#123—22no—
40C2×C22⋊Q8C2×C412yes—
41C2×#121C2×C412yes—
42C2×#120—21no—
43C2×C4×D4—22no—
44C2×#117C2×C411yescharacteristic
45C2×#117C2×C411yescharacteristic
46C2×#117C2×C411yescharacteristic
47C22×C42—44no—
48C22×C42C2×C412yes—
49C22×C42C2×C411yescharacteristic
50C22×C42D412yes—
51C22×C42D412yes—
52C22×C42D412yes—
53C22×C42C2312yes—
54C22×C42C2311yescharacteristic
55#423—48no—
56C2×C4⋊C8—24no—
57C2×C4⋊C8—24no—
58#407—48no—
59C2×C22⋊C8—24no—
60C2×C22⋊C8—24no—
61#391C2×C411yescharacteristic
62#391C2×C412yes—
63#391C2311yescharacteristic
64#390C2×C411yescharacteristic
65#390C2×C412yes—
66#390C2311yescharacteristic
67#389C2×C412yes—
68#389C2×C412yes—
69#389C2×C412yes—
70#389C2312yes—
71#388—22no—
72#388—22no—
73#388—22no—
74#388—22no—
75#387—22no—
76#387—22no—
77#387—22no—
78#387—22no—
79#386—22no—
80#386—22no—
81#386—22no—
82#386—22no—
83#385—22no—
84#385—22no—
85#385—22no—
86#385—22no—
87#381—22no—
88#381—21no—
89#381—22no—
90#380C2×C412yes—
91#380C2×C412yes—
92#379—21no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4—21no—
95C4×C4⋊C4—22no—
96C4×C22⋊C4—22no—
97C4×C22⋊C4—21no—
98C4×C22⋊C4—21no—
99C4×C22⋊C4—22no—
100#336—24no—
101#336C2318yes—
102C25C22⋊C411yescharacteristic
103C2×#41—42no—
104C2×#41—42no—
105C2×#41—42no—
106C22×Q8C22×C411yescharacteristic
107C22×D4—22no—
108C22×D4—21no—
109C22×D4C22×C411yescharacteristic
110C23×C4—24no—
111C23×C4—24no—
112C23×C4—24no—
113C23×C4—21no—
114C23×C4—42no—
115C23×C4—42no—
116C23×C4—44no—
117C23×C4—44no—
118C23×C4C22⋊C412yes—
119C23×C4C22⋊C412yes—
120C23×C4C22⋊C412yes—
121C23×C4C22⋊C412yes—
122C23×C4C22⋊C411yescharacteristic
123C23×C4C22×C411yescharacteristic
124C23×C4C22×C412yes—
125C23×C4C2×D412yes—
126C23×C4C2×D412yes—
127C23×C4C2×D412yes—
128C2×M16—48no—
129C2×M16—48no—
130C2×M16—48no—
131C22×C8—28no—
132#123—48no—
133#123—44no—
134#123—44no—
135#123—44no—
136C22⋊Q8—44no—
137#121—44no—
138#120—44no—
139C4×D4—48no—
140#117—42no—
141#117—42no—
142#117—42no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4C22×C412yes—
149C2×C4⋊C4C22×C412yes—
150C2×C4⋊C4C22×C411yescharacteristic
151C2×C4⋊C4C22×C411yescharacteristic
152C2×C22⋊C4—21no—
153C2×C22⋊C4—21no—
154C2×C22⋊C4—21no—
155C2×C22⋊C4—22no—
156C2×C22⋊C4—22no—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C42—28no—
164C2×C42—21no—
165C2×C42—21no—
166C2×C42—21no—
167C2×C42—44no—
168C2×C42—44no—
169C2×C42—44no—
170C2×C42—44no—
171C2×C42—42no—
172C2×C42—42no—
173C2×C42C22⋊C412yes—
174C2×C42C22⋊C412yes—
175C2×C42C22⋊C411yescharacteristic
176C2×C42C22⋊C411yescharacteristic
177C2×C42C22×C412yes—
178C2×C42C22×C411yescharacteristic
179C2×C42C2×D412yes—
180C2×C42C2×D412yes—
181C2×C42C2×D412yes—
182C2×C42C2411yescharacteristic
183C4⋊C8—48no—
184C4⋊C8—48no—
185C22⋊C8—48no—
186C22⋊C8—48no—
187#95—22no—
188#95—22no—
189#95—22no—
190#95—22no—
191#95—22no—
192#95—22no—
193#95—22no—
194#95—22no—
195#95—22no—
196#95—22no—
197C24—22no—
198C24—24no—
199C24—21no—
200C24—41no—
201C24—41no—
202C24—42no—
203C24C2×C22⋊C411yescharacteristic
204C24C2×C22⋊C412yes—
205#41—88no—
206C2×Q8—22no—
207C2×Q8—41no—
208C2×Q8—41no—
209C2×D4—22no—
210C2×D4—41no—
211C2×D4—41no—
212C2×D4—44no—
213C2×D4—44no—
214C2×D4—42no—
215C2×D4—42no—
216C2×D4—42no—
217C2×D4—44no—
218C22×C4—24no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—416no—
254C22×C4—416no—
255C22×C4—416no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4#9914yes—
260C22×C4C2×C22⋊C412yes—
261C22×C4C2×C22⋊C412yes—
262C22×C4C2×C22⋊C411yescharacteristic
263C22×C4C2×C22⋊C412yes—
264C22×C4C2×C22⋊C412yes—
265C22×C4AGL(1,ℤ/8)14yes—
266C22×C4#13714yes—
267C22×C4C23×C411yescharacteristic
268C22×C4C22×D412yes—
269M16—816no—
270C2×C8—48no—
271C2×C8—48no—
272C2×C8—48no—
273C4⋊C4—24no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—42no—
277C4⋊C4—44no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—44no—
281C4⋊C4—44no—
282C4⋊C4—44no—
283C22⋊C4—44no—
284C22⋊C4—42no—
285C22⋊C4—42no—
286C22⋊C4—44no—
287C22⋊C4—44no—
288C22⋊C4—42no—
289C22⋊C4—44no—
290C22⋊C4—42no—
291C22⋊C4—42no—
292C22⋊C4—44no—
293C22⋊C4—44no—
294C22⋊C4—44no—
295C42—24no—
296C42—24no—
297C42—24no—
298C42—24no—
299C42—41no—
300C42—41no—
301C42—41no—
302C42—42no—
303C42—42no—
304C42—41no—
305C42—42no—
306C42—42no—
307C42—42no—
308C23—21no—
309C23—22no—
310C23—22no—
311C23—21no—
312C23—22no—
313C23—22no—
314C23—22no—
315C23—22no—
316C23—22no—
317C23—22no—
318C23—48no—
319C23—44no—
320C23—44no—
321C23—48no—
322C23—44no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—42no—
327C23—42no—
328C23C2×#9912yes—
329C23#41714yes—
330C23C22×C22⋊C411yescharacteristic
331Q8—82no—
332D4—82no—
333D4—88no—
334D4—84no—
335C2×C4—28no—
336C2×C4—28no—
337C2×C4—28no—
338C2×C4—21no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—48no—
357C2×C4—48no—
358C2×C4—48no—
359C2×C4—48no—
360C2×C4—48no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—84no—
377C2×C4C2×#9914yes—
378C2×C4#41312yes—
379C2×C4#41714yes—
380C2×C4#42114yes—
381C2×C4#56812yes—
382C2×C4C2×AGL(1,ℤ/8)12yes—
383C2×C4C2×#13712yes—
384C8—88no—
385C22—24no—
386C22—24no—
387C22—24no—
388C22—24no—
389C22—22no—
390C22—416no—
391C22—41no—
392C22—42no—
393C22—41no—
394C22—42no—
395C22—42no—
396C22—42no—
397C22—42no—
398C22—81no—
399C22—81no—
400C22—82no—
401C22—82no—
402C22—81no—
403C22#287911yescharacteristic
404C22C22×#9911yescharacteristic
405C22C2×#41311yescharacteristic
406C22C2×#41712yes—
407C22C2×#42112yes—
408C4—22no—
409C4—22no—
410C4—24no—
411C4—24no—
412C4—22no—
413C4—22no—
414C4—24no—
415C4—42no—
416C4—81no—
417C4—81no—
418C4—82no—
419C4—82no—
420C4—81no—
421C2—44no—
422C2—42no—
423C2—161no—
424C2#1739211yescharacteristic
425C2#1749812yes—
426C2#1759212yes—
427C2#1766511yescharacteristic
428C2#1775211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#525C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20626C2semidirect
3. #7715C2semidirect
4. #7714C2semidirect
5. #7632C2semidirect
6. #7631C2semidirect
29. #1272C22semidirect
30. #1271C22semidirect
31. #1270C22semidirect

Maximal subgroups

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

subgroupindex
#366262
#206262
#77152
#77142
#76322
#76312