Finite Group Databasebeta

← #192092

Subgroups — #192092

Order 512.

1326 conjugacy classes of subgroups.

The subgroup lattice has 477 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34178C211yesmaximal, characteristic
2#20843C212yesmaximal
3C22×#324C211yesmaximal, characteristic
4#13870C211yesmaximal, characteristic
5#8857C212yesmaximal, complemented
6#8856C212yesmaximal, complemented
7#8855C214yesmaximal, complemented
8#8024C212yesmaximal
9C22×#117C2211yescharacteristic
10C23×C4⋊C4C2211yescharacteristic
11C2×#427—22no—
12C22×C22⋊C8C2212yes—
13#2345—24no—
14#2344—22no—
15#2344—22no—
16#2316C2212yes—
17#2313—21no—
18#2313C2212yes—
19#2280—21no—
20#2280—21no—
21#2280C2212yes—
22C2×#381—21no—
23C2×#381C2211yescharacteristic
24#2274—21no—
25#1854—24no—
26#1853C2214yes—
27#1852C2214yes—
28#1735—22no—
29C2×#324C418yescomplemented
30C2×#324C2212yes—
31C2×#324C2211yescharacteristic
32C2×#324C2211yescharacteristic
33#1441C2212yes—
34#1441C2212yes—
35#1436—21no—
36#1436—21no—
37C22⋊#95—21no—
38#1294C2214yes—
39#1294C2214yes—
40C24×C4D411yescharacteristic
41C24×C4D411yescharacteristic
42C24×C4C2311yescharacteristic
43C23×C8—22no—
44C2×#117—22no—
45C2×#117—21no—
46C2×#117—21no—
47C2×#117—22no—
48C2×#117—22no—
49C2×#117—44no—
50C2×#117D412yes—
51C2×#117D412yes—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—24no—
55C22×C4⋊C4C2×C414yes—
56C22×C4⋊C4D411yescharacteristic
57C22×C4⋊C4D411yescharacteristic
58C22×C4⋊C4Q814yes—
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2312yes—
62C22×C22⋊C4—21no—
63C22×C22⋊C4—21no—
64C22×C42—21no—
65#427—48no—
66C2×#107—24no—
67C2×#106—24no—
68C2×C22⋊C8—22no—
69C2×C22⋊C8—24no—
70C2×C22⋊C8C2×C418yes—
71C2×C22⋊C8C2312yes—
72C2×C22⋊C8C2312yes—
73#389—24no—
74#389—24no—
75#387—24no—
76#387—24no—
77#387—24no—
78#387—24no—
79#381—22no—
80#381—22no—
81#381—22no—
82#381—24no—
83#381—44no—
84#380—21no—
85#380—21no—
86#380—22no—
87#380—22no—
88#380—22no—
89#380D414yes—
90#380C2314yes—
91#379—22no—
92C4×C4⋊C4—24no—
93C4×C4⋊C4—24no—
94C4×C22⋊C4—22no—
95C4×C22⋊C4—22no—
96C4×C22⋊C4—22no—
97C4×C22⋊C4—22no—
98C4×C22⋊C4—22no—
99C4×C22⋊C4—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—21no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95C2312yes—
106#340—24no—
107#340—24no—
108#324C2×C418yes—
109#324C2×C414yes—
110#324C2×C414yes—
111C25C2×D411yescharacteristic
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4—22no—
116C23×C4—21no—
117C23×C4—22no—
118C23×C4—24no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—42no—
122C23×C4—42no—
123C23×C4—42no—
124C23×C4—42no—
125C23×C4—42no—
126C23×C4C4⋊C414yes—
127C23×C4C4⋊C414yes—
128C23×C4C22×C412yes—
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D412yes—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D412yes—
136C23×C4C2×Q812yes—
137C23×C4#4114yes—
138C23×C4C2411yescharacteristic
139C22×C8—28no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—44no—
143C22×C8—44no—
144#117—28no—
145#117—48no—
146#117—44no—
147#117—44no—
148#117—44no—
149#117—44no—
150#117—44no—
151#117—82no—
152#117—82no—
153C2×C4⋊C4—28no—
154C2×C4⋊C4—28no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—28no—
158C2×C4⋊C4—28no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4C4⋊C414yes—
172C2×C4⋊C4C4⋊C414yes—
173C2×C4⋊C4C22×C412yes—
174C2×C4⋊C4C22×C412yes—
175C2×C4⋊C4C2×D414yes—
176C2×C4⋊C4C2×D414yes—
177C2×C4⋊C4C2×Q812yes—
178C2×C4⋊C4C2×Q812yes—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—42no—
185C2×C22⋊C4—41no—
186C2×C22⋊C4—41no—
187C2×C22⋊C4—42no—
188C2×C22⋊C4—42no—
189C2×C22⋊C4—42no—
190C2×C42—24no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—44no—
196C2×C42—42no—
197#107—48no—
198#106—48no—
199C22⋊C8—28no—
200C22⋊C8—22no—
201C22⋊C8—22no—
202C22⋊C8C22×C414yes—
203C22⋊C8C22×C414yes—
204#95—24no—
205#95—24no—
206#95—22no—
207#95—22no—
208#95—24no—
209#95—24no—
210#95—24no—
211#95—24no—
212#95—44no—
213#95—44no—
214#95—44no—
215#95—42no—
216#95—44no—
217C24—22no—
218C24—22no—
219C24—22no—
220C24—22no—
221C24—42no—
222C24C2×C4⋊C412yes—
223C24#12011yescharacteristic
224C24#12011yescharacteristic
225C24#12312yes—
226C24C22×D411yescharacteristic
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—24no—
244C22×C4—28no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—42no—
251C22×C4—44no—
252C22×C4—44no—
253C22×C4—41no—
254C22×C4—42no—
255C22×C4—44no—
256C22×C4—48no—
257C22×C4—41no—
258C22×C4—48no—
259C22×C4—44no—
260C22×C4—44no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—48no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—84no—
282C22×C4—81no—
283C22×C4—81no—
284C22×C4C2×C4⋊C412yes—
285C22×C4C2×C4⋊C412yes—
286C22×C4C2×C4⋊C412yes—
287C22×C4C2×C4⋊C412yes—
288C22×C4C2×C4⋊C412yes—
289C22×C4C4×D414yes—
290C22×C4C4×D414yes—
291C22×C4#12012yes—
292C22×C4C22⋊Q814yes—
293C22×C4C22⋊Q814yes—
294C22×C4#12312yes—
295C22×C4#13514yes—
296C22×C4C23×C411yescharacteristic
297C22×C4C22×D412yes—
298C22×C4C22×Q811yescharacteristic
299C22×C4C2×#4112yes—
300C2×C8—24no—
301C2×C8—24no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—416no—
305C2×C8—42no—
306C2×C8—42no—
307C4⋊C4—24no—
308C4⋊C4—24no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—28no—
312C4⋊C4—28no—
313C4⋊C4—28no—
314C4⋊C4—44no—
315C4⋊C4—44no—
316C4⋊C4—44no—
317C4⋊C4—44no—
318C4⋊C4—44no—
319C4⋊C4—82no—
320C4⋊C4—82no—
321C22⋊C4—48no—
322C22⋊C4—44no—
323C22⋊C4—44no—
324C22⋊C4—44no—
325C22⋊C4—44no—
326C22⋊C4—44no—
327C22⋊C4—44no—
328C22⋊C4—84no—
329C42—48no—
330C42—48no—
331C42—82no—
332C42—82no—
333C42—82no—
334C42—82no—
335C23—22no—
336C23—22no—
337C23—22no—
338C23—21no—
339C23—21no—
340C23—24no—
341C23—24no—
342C23—24no—
343C23—22no—
344C23—22no—
345C23—22no—
346C23—22no—
347C23—21no—
348C23—21no—
349C23—42no—
350C23—42no—
351C23—44no—
352C23—44no—
353C23—44no—
354C23—44no—
355C23#38512yes—
356C23#38512yes—
357C23B(4,𝔽2)12yes—
358C23#45812yes—
359C23C22×C4⋊C411yescharacteristic
360C23C2×#12011yescharacteristic
361C23C2×#12311yescharacteristic
362C23C2×#13512yes—
363C23#57512yes—
364C2×C4—24no—
365C2×C4—28no—
366C2×C4—28no—
367C2×C4—22no—
368C2×C4—24no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—42no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—48no—
387C2×C4—48no—
388C2×C4—42no—
389C2×C4—48no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—84no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4#38514yes—
421C2×C4#42714yescomplemented
422C2×C4C2×C4×D412yes—
423C2×C4C2×C22⋊Q812yes—
424C8—44no—
425C8—44no—
426C22—22no—
427C22—22no—
428C22—24no—
429C22—22no—
430C22—22no—
431C22—24no—
432C22—22no—
433C22—21no—
434C22—21no—
435C22—22no—
436C22—21no—
437C22—21no—
438C22—22no—
439C22—22no—
440C22—22no—
441C22—48no—
442C22—44no—
443C22—44no—
444C22—42no—
445C22—42no—
446C22#186714yes—
447C22C2×#38511yescharacteristic
448C22C2×#42711yescharacteristic
449C22#291111yescharacteristic
450C22#300212yes—
451C22#302012yes—
452C22C2×B(4,𝔽2)11yescharacteristic
453C22C2×#45811yescharacteristic
454C22#309112yes—
455C4—24no—
456C4—22no—
457C4—22no—
458C4—24no—
459C4—42no—
460C4—42no—
461C4—42no—
462C4—84no—
463C4—84no—
464C4—84no—
465C4—162no—
466C2—22no—
467C2—24no—
468C2—21no—
469C2—21no—
470C2—42no—
471C2#1819312yes—
472C2#1820112yes—
473C2#2084311yescharacteristic
474C2#2085711yescharacteristic
475C2C2×#186711yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #8857C2semidirect
6. #8856C2semidirect
7. #8855C2semidirect
29. C2×#324C4semidirect
421. C2×C4#427semidirect

Maximal subgroups

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

subgroupindex
#341782
#208432
C22×#3242
#138702
#88572
#88562
#88552
#80242