Finite Group Databasebeta

← #204594

Subgroups — #204594

Order 512.

1650 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic
2C2×C4×C22⋊C8C211yesmaximal, characteristic
3C2×#1520C214yesmaximal
4#14686C211yesmaximal, characteristic
5#9679C218yesmaximal, complemented
6C22×C4⋊Q8C411yescharacteristic, complemented
7C22×C22⋊Q8C2212yes—
8C23×C42C2211yescharacteristic
9C22×C22⋊C8C2212yes—
10C22×C4×C8—21no—
11#2857—21no—
12#2857—21no—
13#2857—21no—
14#2857—21no—
15#2857C414yescomplemented
16#2857C2214yes—
17C2×#391C412yescomplemented
18C2×#389—22no—
19C2×#384—21no—
20C2×#384C411yescharacteristic, complemented
21C2×#380—22no—
22C2×#380—22no—
23C2×#380C2212yes—
24C4×C22⋊C8—22no—
25C4×C22⋊C8C2214yes—
26C2×#336—21no—
27#1520—216no—
28C2×#328C22116yes—
29C4⋊#95—21no—
30C4⋊#95C2214yes—
31C23×Q8C2×C412yes—
32C24×C4D412yes—
33C24×C4C2311yescharacteristic
34C23×C8—22no—
35C2×C4⋊Q8—21no—
36C2×C4⋊Q8—21no—
37C2×C4⋊Q8—44no—
38C2×C4⋊Q8C2×C414yes—
39C2×C22⋊Q8—22no—
40C2×C22⋊Q8—22no—
41C2×C22⋊Q8—28no—
42C2×C22⋊Q8—28no—
43C2×C22⋊Q8C2×C414yes—
44C2×C22⋊Q8C2314yes—
45C22×C4⋊C4—22no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—24no—
50C22×C4⋊C4C2×C412yes—
51C22×C22⋊C4—22no—
52C22×C22⋊C4—22no—
53C22×C42—44no—
54C22×C42C2×C411yescharacteristic
55C22×C42C2×C412yes—
56C22×C42C2×C411yescharacteristic
57C22×C42D411yescharacteristic
58C22×C42D411yescharacteristic
59C22×C42D411yescharacteristic
60C22×C42D411yescharacteristic
61C22×C42D412yes—
62C22×C42C2312yes—
63C2×C22⋊C8—22no—
64C2×C22⋊C8—24no—
65C2×C22⋊C8D418yes—
66C2×C22⋊C8C2314yes—
67C2×C4×C8—24no—
68C2×C4×C8—44no—
69#391—22no—
70#391—22no—
71#391C2×C418yes—
72#389—24no—
73#389—24no—
74#389—24no—
75#389—24no—
76#384—21no—
77#384—21no—
78#384—44no—
79#384C2×C414yes—
80#380—24no—
81#380—24no—
82#380—24no—
83#380—24no—
84#380—28no—
85#380—48no—
86C2×#95—24no—
87C2×#95—24no—
88C2×#95C2314yes—
89#336—24no—
90#328—232no—
91C25C2×D411yescharacteristic
92C22×Q8—22no—
93C22×Q8—28no—
94C22×Q8—44no—
95C22×Q8—44no—
96C22×Q8C22×C414yes—
97C23×C4—22no—
98C23×C4—22no—
99C23×C4—24no—
100C23×C4—44no—
101C23×C4—44no—
102C23×C4—42no—
103C23×C4—42no—
104C23×C4C22⋊C412yes—
105C23×C4C22⋊C412yes—
106C23×C4C22⋊C412yes—
107C23×C4C22⋊C412yes—
108C23×C4SD1618yes—
109C23×C4Q1618yes—
110C23×C4C22×C411yescharacteristic
111C23×C4C22×C411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D412yes—
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D412yes—
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D414yes—
118C23×C4#4114yes—
119C23×C4C2411yescharacteristic
120C22×C8—28no—
121C22×C8—24no—
122C22×C8—44no—
123C22×C8—44no—
124C4⋊Q8—44no—
125C4⋊Q8—48no—
126C4⋊Q8—44no—
127C22⋊Q8—216no—
128C22⋊Q8—28no—
129C22⋊Q8—28no—
130C22⋊Q8—48no—
131C22⋊Q8—44no—
132C22⋊Q8—44no—
133C2×C4⋊C4—28no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—48no—
142C2×C4⋊C4—48no—
143C2×C4⋊C4—48no—
144C2×C4⋊C4—44no—
145C2×C4⋊C4—48no—
146C2×C4⋊C4—44no—
147C2×C4⋊C4—48no—
148C2×C4⋊C4—44no—
149C2×C4⋊C4—48no—
150C2×C4⋊C4C22×C414yes—
151C2×C22⋊C4—24no—
152C2×C22⋊C4—24no—
153C2×C22⋊C4—44no—
154C2×C22⋊C4—44no—
155C2×C22⋊C4—44no—
156C2×C22⋊C4—48no—
157C2×C42—24no—
158C2×C42—24no—
159C2×C42—24no—
160C2×C42—42no—
161C2×C42—44no—
162C2×C42—44no—
163C2×C42—42no—
164C2×C42—42no—
165C2×C42—44no—
166C2×C42C22⋊C411yescharacteristic
167C2×C42C22⋊C412yes—
168C2×C42C22⋊C411yescharacteristic
169C2×C42C22⋊C411yescharacteristic
170C2×C42C22⋊C412yes—
171C2×C42C22⋊C411yescharacteristic
172C2×C42C22×C412yes—
173C2×C42C22×C412yes—
174C2×C42C2×D412yes—
175C2×C42C2×D412yes—
176C22⋊C8—28no—
177C22⋊C8—24no—
178C22⋊C8C2×D418yes—
179C4×C8—48no—
180#95—28no—
181#95—28no—
182#95—28no—
183#95—48no—
184C24—41no—
185C24—41no—
186C24—42no—
187C24C2×C22⋊C411yescharacteristic
188C24C2×C22⋊C411yescharacteristic
189C24#12412yes—
190C24#12712yes—
191C24C2×SD1614yes—
192C24C2×Q1614yes—
193C24C22×D411yescharacteristic
194C2×Q8—28no—
195C2×Q8—44no—
196C2×Q8—44no—
197C2×Q8—44no—
198C2×Q8—44no—
199C2×Q8—44no—
200C2×Q8—816no—
201C22×C4—24no—
202C22×C4—22no—
203C22×C4—24no—
204C22×C4—22no—
205C22×C4—24no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—416no—
216C22×C4—416no—
217C22×C4—48no—
218C22×C4—48no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—41no—
222C22×C4—42no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—42no—
230C22×C4—41no—
231C22×C4—44no—
232C22×C4—48no—
233C22×C4—44no—
234C22×C4—44no—
235C22×C4—44no—
236C22×C4—44no—
237C22×C4—44no—
238C22×C4—48no—
239C22×C4—44no—
240C22×C4—48no—
241C22×C4#10318yes—
242C22×C4#10318yes—
243C22×C4#10318yes—
244C22×C4#10318yes—
245C22×C4C2×C22⋊C411yescharacteristic
246C22×C4C2×C22⋊C412yes—
247C22×C4C2×C22⋊C412yes—
248C22×C4C2×C22⋊C412yes—
249C22×C4C2×C22⋊C411yescharacteristic
250C22×C4C2×C22⋊C412yes—
251C22×C4C4×D414yes—
252C22×C4C4×D414yes—
253C22×C4#12114yes—
254C22×C4#12114yes—
255C22×C4#12412yes—
256C22×C4#12712yes—
257C22×C4C2×SD1614yes—
258C22×C4C2×SD1614yes—
259C22×C4C2×Q1614yes—
260C22×C4C2×Q1614yes—
261C22×C4C23×C411yescharacteristic
262C22×C4C22×D411yescharacteristic
263C22×C4C22×D412yes—
264C22×C4C2×#4112yes—
265C2×C8—28no—
266C2×C8—44no—
267C2×C8—44no—
268C2×C8—416no—
269C4⋊C4—28no—
270C4⋊C4—48no—
271C4⋊C4—48no—
272C4⋊C4—48no—
273C4⋊C4—44no—
274C4⋊C4—48no—
275C4⋊C4—48no—
276C4⋊C4—48no—
277C4⋊C4—88no—
278C4⋊C4—816no—
279C22⋊C4—44no—
280C22⋊C4—44no—
281C22⋊C4—48no—
282C22⋊C4—88no—
283C42—24no—
284C42—22no—
285C42—22no—
286C42—24no—
287C42—22no—
288C42—24no—
289C42—42no—
290C42—42no—
291C42—42no—
292C42#9914yes—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—22no—
300C23—22no—
301C23—22no—
302C23—42no—
303C23—42no—
304C23—44no—
305C23—44no—
306C23—42no—
307C23—42no—
308C23—44no—
309C23—44no—
310C23#39012yes—
311C23#39012yes—
312C23C2×#10314yes—
313C23C2×#10314yes—
314C23#48718yes—
315C23#49214yes—
316C23#49414yes—
317C23C22×C22⋊C411yescharacteristic
318C23C2×#12411yescharacteristic
319C23C2×#12711yescharacteristic
320C23C22×SD1612yes—
321C23C22×Q1612yes—
322Q8—88no—
323Q8—88no—
324C2×C4—22no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—24no—
337C2×C4—28no—
338C2×C4—28no—
339C2×C4—24no—
340C2×C4—28no—
341C2×C4—216no—
342C2×C4—216no—
343C2×C4—28no—
344C2×C4—48no—
345C2×C4—48no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—48no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—88no—
358C2×C4—88no—
359C2×C4—88no—
360C2×C4—84no—
361C2×C4—84no—
362C2×C4—88no—
363C2×C4—84no—
364C2×C4#39012yes—
365C2×C4#39012yes—
366C2×C4C2×#9912yes—
367C2×C4C2×#9914yes—
368C2×C4#41012yes—
369C2×C4C2×#10314yes—
370C2×C4C2×#10314yes—
371C2×C4C2×#10314yes—
372C2×C4C2×#10314yes—
373C2×C4#42114yes—
374C2×C4C2×C4×D412yes—
375C2×C4C2×#12112yes—
376C8—48no—
377C22—21no—
378C22—22no—
379C22—21no—
380C22—22no—
381C22—22no—
382C22—22no—
383C22—24no—
384C22—24no—
385C22—22no—
386C22—22no—
387C22—24no—
388C22—24no—
389C22—21no—
390C22—22no—
391C22—21no—
392C22—22no—
393C22—48no—
394C22—44no—
395C22—44no—
396C22#195614yes—
397C22#195614yes—
398C22C2×#39011yescharacteristic
399C22C22×#9911yescharacteristic
400C22C2×#41011yescharacteristic
401C22C22×#10312yes—
402C22C2×#42112yes—
403C22C2×#48712yes—
404C22C2×#49211yescharacteristic
405C22C2×#49411yescharacteristic
406C4—24no—
407C4—24no—
408C4—24no—
409C4—22no—
410C4—22no—
411C4—22no—
412C4—84no—
413C4—84no—
414C4—164no—
415C4#152018yescomplemented
416C4#196414yes—
417C2—21no—
418C2—22no—
419C2—21no—
420C2—24no—
421C2—24no—
422C2C2×#152012yes—
423C2#1757512yes—
424C2C2×#195611yescharacteristic
425C2C2×#196411yescharacteristic
426C2#2125211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #9679C2semidirect
6. C22×C4⋊Q8C4semidirect
15. #2857C4semidirect
17. C2×#391C4semidirect
20. C2×#384C4semidirect
415. C4#1520semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C2×C4×C22⋊C82
C2×#15202
#146862
#96792