Finite Group Databasebeta

← #375651

Subgroups — #375651

Order 512.

1082 conjugacy classes of subgroups.

The subgroup lattice has 497 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2863C211yesmaximal, characteristic
2C4×#423C211yesmaximal, characteristic
3#20364C211yesmaximal, characteristic
4#11204C212yesmaximal
5#11045C212yesmaximal
6#10963C212yesmaximal
7#10963C212yesmaximal
8#10913C212yesmaximal
9#10913C212yesmaximal
10C2×#423C2212yes—
11C22×#101C2214yes—
12C2×C4×M16C2212yes—
13C2×C4×M16C2212yes—
14#2863—21no—
15#2863—21no—
16#2863—21no—
17#2863—21no—
18#2863—21no—
19#2863—21no—
20#2863—21no—
21#2863—21no—
22C2×#391C2211yescharacteristic
23C2×#391C2211yescharacteristic
24C2×#391C2211yescharacteristic
25C2×#389C2211yescharacteristic
26C2×#389C2211yescharacteristic
27C2×#389C2211yescharacteristic
28C2×#387C412yes—
29C2×#387C412yes—
30C2×#387C412yes—
31C2×#383C412yes—
32C2×C43C2211yescharacteristic
33#1767—21no—
34#1767—21no—
35C4×C4⋊C8—21no—
36C4×C4⋊C8—21no—
37#1745—21no—
38#1745—21no—
39#1744—21no—
40#1744—21no—
41C2×#343C2214yes—
42C2×#343C2214yes—
43C2×#343C2214yes—
44#1728C2212yes—
45#1728C2212yes—
46#1728C2212yes—
47C23×Q8C2311yescharacteristic
48C22×M16C2314yes—
49C22×M16C2314yes—
50C22×C4⋊C4C2×C412yes—
51C22×C4⋊C4C2×C412yes—
52C22×C4⋊C4C2×C412yes—
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C42C2×C411yescharacteristic
57C22×C42C2×C411yescharacteristic
58C22×C42C2×C411yescharacteristic
59C22×C42C2×C411yescharacteristic
60C22×C42C2311yescharacteristic
61C22×C42C2311yescharacteristic
62C22×C42C2311yescharacteristic
63#423—28no—
64C2×C4⋊C8—22no—
65C2×C4⋊C8—22no—
66C2×#101—28no—
67C2×#101—28no—
68C2×#101—44no—
69C4×M16—28no—
70C4×M16—44no—
71C2×#97—22no—
72C2×#97—22no—
73C2×C4×C8—22no—
74C2×C4×C8—22no—
75#391—21no—
76#391—21no—
77#391—21no—
78#391—21no—
79#391—21no—
80#391—21no—
81#391—21no—
82#391—21no—
83#391—21no—
84#391—21no—
85#391—21no—
86#391—21no—
87#389—21no—
88#389—21no—
89#389—21no—
90#389—21no—
91#389—21no—
92#389—21no—
93#389—21no—
94#389—21no—
95#389—21no—
96#389—21no—
97#389—21no—
98#389—21no—
99#387—22no—
100#387—22no—
101#387—22no—
102#387—22no—
103#387—22no—
104#387—22no—
105#387—22no—
106#387—22no—
107#387—22no—
108#387—22no—
109#387—22no—
110#387—22no—
111#383—22no—
112#383—22no—
113#383—22no—
114#383—22no—
115C2×#95C2×C412yes—
116C2×#95C2×C412yes—
117C2×#95C2×C412yes—
118C2×#95C2×C412yes—
119C2×#95C2×C412yes—
120C2×#95C2×C412yes—
121C2×#95C2×C412yes—
122C2×#95C2×C412yes—
123C2×#95C2×C412yes—
124C43D411yescharacteristic
125C43D411yescharacteristic
126C43D411yescharacteristic
127C43D411yescharacteristic
128C43D411yescharacteristic
129C43D411yescharacteristic
130C43D411yescharacteristic
131C43D411yescharacteristic
132#343—28no—
133#343—28no—
134#343—44no—
135#336—22no—
136#336—22no—
137#336—22no—
138#336—22no—
139#336—22no—
140#336—22no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—24no—
144C22×Q8—24no—
145C22×Q8—22no—
146C22×Q8—22no—
147C23×C4C22×C412yes—
148C23×C4C22×C411yescharacteristic
149C23×C4C22×C411yescharacteristic
150C23×C4C22×C411yescharacteristic
151C23×C4C22×C411yescharacteristic
152C23×C4C22×C411yescharacteristic
153C23×C4C22×C411yescharacteristic
154C23×C4C22×C412yes—
155C23×C4C22×C412yes—
156C23×C4C22×C412yes—
157C23×C4C2411yescharacteristic
158C2×M16—24no—
159C2×M16—24no—
160C2×M16—24no—
161C2×M16—24no—
162C2×M16—24no—
163C2×M16—24no—
164C2×M16—24no—
165C2×M16—24no—
166C2×M16—24no—
167C2×M16—24no—
168C2×M16#4118yes—
169C2×M16#4118yes—
170C22×C8—24no—
171C22×C8—24no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—44no—
191C2×C42—24no—
192C2×C42—24no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—24no—
196C2×C42—42no—
197C2×C42—42no—
198C2×C42—42no—
199C2×C42C22⋊C411yescharacteristic
200C2×C42C22⋊C411yescharacteristic
201C2×C42C22⋊C411yescharacteristic
202C2×C42C22⋊C411yescharacteristic
203C2×C42C22⋊C411yescharacteristic
204C2×C42C22⋊C411yescharacteristic
205C2×C42C22⋊C411yescharacteristic
206C2×C42C22⋊C411yescharacteristic
207C2×C42C22⋊C411yescharacteristic
208C2×C42C22⋊C411yescharacteristic
209C2×C42C22⋊C411yescharacteristic
210C2×C42C22⋊C411yescharacteristic
211C2×C42C22⋊C411yescharacteristic
212C2×C42C22⋊C411yescharacteristic
213C2×C42C22⋊C411yescharacteristic
214C2×C42C22⋊C411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C4⋊C8—28no—
228C4⋊C8—28no—
229#101—416no—
230#97—28no—
231#97—44no—
232C4×C8—28no—
233C4×C8—44no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—22no—
252C24C23×C411yescharacteristic
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—42no—
260C2×Q8—44no—
261C2×Q8—44no—
262C2×Q8—44no—
263C2×Q8—44no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4C2×C22⋊C411yescharacteristic
299C22×C4C2×C22⋊C411yescharacteristic
300C22×C4C2×C22⋊C411yescharacteristic
301C22×C4C2×C22⋊C411yescharacteristic
302C22×C4C2×C22⋊C411yescharacteristic
303C22×C4C2×C22⋊C411yescharacteristic
304C22×C4C2×C22⋊C411yescharacteristic
305C22×C4C2×C22⋊C411yescharacteristic
306C22×C4C2×C22⋊C411yescharacteristic
307C22×C4C2×C22⋊C411yescharacteristic
308C22×C4C2×C22⋊C411yescharacteristic
309C22×C4C2×C22⋊C411yescharacteristic
310C22×C4C22×D411yescharacteristic
311C22×C4C22×D411yescharacteristic
312C22×C4C2×#4114yes—
313C22×C4C2×#4114yes—
314C22×C4H(5,𝔽2)12yes—
315C22×C42−1+412yes—
316M16—28no—
317M16—28no—
318M16—28no—
319M16—28no—
320M16—44no—
321M16—44no—
322M16—44no—
323M16—44no—
324C2×C8—28no—
325C2×C8—28no—
326C2×C8—44no—
327C2×C8—44no—
328C2×C8—44no—
329C2×C8—44no—
330C4⋊C4—44no—
331C4⋊C4—44no—
332C4⋊C4—44no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—42no—
340C4⋊C4—44no—
341C4⋊C4—44no—
342C4⋊C4—44no—
343C42—22no—
344C42—22no—
345C42—22no—
346C42—22no—
347C42—22no—
348C42—22no—
349C42—22no—
350C42—22no—
351C42—21no—
352C42—21no—
353C42—21no—
354C42—21no—
355C42—22no—
356C42—22no—
357C42—22no—
358C42—22no—
359C42—22no—
360C42—22no—
361C42—22no—
362C42—22no—
363C42—22no—
364C42—22no—
365C42—22no—
366C42—41no—
367C42—41no—
368C42—41no—
369C42—41no—
370C42—41no—
371C42—41no—
372C42#10114yescomplemented
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23C22×C22⋊C411yescharacteristic
378C23C4×#4112yes—
379C23C4×#4112yes—
380C23#52012yes—
381Q8—84no—
382Q8—82no—
383Q8—82no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—48no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—84no—
434C2×C4—84no—
435C2×C4—84no—
436C2×C4C2×#10112yes—
437C2×C4C2×#10112yes—
438C2×C4C2×#10112yes—
439C2×C4#41312yes—
440C2×C4#41312yes—
441C2×C4#41312yes—
442C2×C4#52512yes—
443C2×C4#52512yes—
444C2×C4#53212yes—
445C2×C4#53212yes—
446C2×C4#53511yescharacteristic
447C2×C4#53611yescharacteristic
448C2×C4#53712yes—
449C2×C4#55114yes—
450C2×C4#55214yes—
451C2×C4#55314yes—
452C2×C4#55614yes—
453C8—48no—
454C8—48no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22#229711yescharacteristic
462C22#229711yescharacteristic
463C22#231111yescharacteristic
464C22#237314yes—
465C22C22×#10111yescharacteristic
466C22C2×#41311yescharacteristic
467C22C2×#41311yescharacteristic
468C22C2×#41311yescharacteristic
469C4—22no—
470C4—22no—
471C4—22no—
472C4—22no—
473C4—22no—
474C4—22no—
475C4—41no—
476C4—41no—
477C4—41no—
478C4—41no—
479C4—41no—
480C4—41no—
481C4—82no—
482C4—82no—
483C4—82no—
484C4—82no—
485C4#260114yes—
486C4#260312yes—
487C4#260412yes—
488C2—24no—
489C2#1416611yescharacteristic
490C2#2142511yescharacteristic
491C2#2142511yescharacteristic
492C2#2142611yescharacteristic
493C2#2142611yescharacteristic
494C2#2187911yescharacteristic
495C2#2188011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutator, prime radicalC24C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
372. C42#101semidirect

Maximal subgroups

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

subgroupindex
C2×#28632
C4×#4232
#203642
#112042
#110452
#109632
#109132