Finite Group Databasebeta

← #216295

Subgroups — #216295

Order 512.

974 conjugacy classes of subgroups.

The subgroup lattice has 471 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#125C211yesmaximal, characteristic
2#20810C211yesmaximal, characteristic
3#20778C211yesmaximal, characteristic
4#9883C212yesmaximal
5#9654C212yesmaximal
6#9086C214yesmaximal
7#8560C214yesmaximal
8C22×#125C411yescharacteristic
9C2×#432C2212yes—
10C4×#125—21no—
11C4×#125—21no—
12C4×#125—21no—
13C4×#125—21no—
14C4×#125—21no—
15C4×#125—21no—
16C4×#125—21no—
17C4×#125—21no—
18C2×#389C2211yescharacteristic
19C2×#389C2211yescharacteristic
20C2×#387C412yes—
21C2×#387C2212yes—
22C2×#382C411yescharacteristic
23C2×C4×C4⋊C4C412yes—
24C2×C4×C4⋊C4C411yescharacteristic
25C2×C4×C4⋊C4C411yescharacteristic
26C2×C4×C4⋊C4C2212yes—
27C2×C43C2211yescharacteristic
28#1842—21no—
29#1839—21no—
30#1831—21no—
31#1831—21no—
32#1817C2212yes—
33#1797C2212yes—
34#1797C2212yes—
35#1797C2212yes—
36#1767—21no—
37#1767—21no—
38#1767—21no—
39#1749C2212yes—
40#1745—21no—
41C2×#324C2218yes—
42#1280C2218yes—
43C2×#125—28no—
44C2×#125—21no—
45C2×#125—21no—
46C2×#125—21no—
47C2×#125—21no—
48C22×C4⋊C4C2×C412yes—
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C4⋊C4C2×C411yescharacteristic
51C22×C4⋊C4C2×C412yes—
52C22×C4⋊C4C2×C411yescharacteristic
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C4⋊C4C2×C412yes—
55C22×C4⋊C4C2312yes—
56C22×C42C2×C411yescharacteristic
57C22×C42C2×C411yescharacteristic
58C22×C42C2×C412yes—
59C22×C42C2×C412yes—
60C22×C42C2×C411yescharacteristic
61C22×C42C2×C411yescharacteristic
62C22×C42C2311yescharacteristic
63C22×C42C2311yescharacteristic
64C22×C42C2311yescharacteristic
65#432—44no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—22no—
68C2×C4⋊C8—22no—
69C2×C4⋊C8—22no—
70C2×C22⋊C8C2314yes—
71C2×C22⋊C8C2314yes—
72C2×#97—22no—
73C2×#97—22no—
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#387—22no—
83#387—22no—
84#387—22no—
85#387—22no—
86#387—22no—
87#387—22no—
88#387—22no—
89#387—22no—
90#382—21no—
91#382—21no—
92#382—21no—
93#382—21no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—21no—
99C4×C4⋊C4—21no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C2×#95C2×C412yes—
111C2×#95C2×C412yes—
112C2×#95C2×C412yes—
113C2×#95C2×C412yes—
114C2×#95C2×C412yes—
115C2×#95C2312yes—
116C43D411yescharacteristic
117C43D411yescharacteristic
118C43D411yescharacteristic
119C43D411yescharacteristic
120C43D411yescharacteristic
121C43D411yescharacteristic
122C43D411yescharacteristic
123C43D411yescharacteristic
124#336—22no—
125#336—22no—
126#336—22no—
127#336—22no—
128#336—22no—
129#336—22no—
130#324—48no—
131C23×C4C22×C411yescharacteristic
132C23×C4C22×C411yescharacteristic
133C23×C4C22×C411yescharacteristic
134C23×C4C22×C412yes—
135C23×C4C22×C411yescharacteristic
136C23×C4C22×C411yescharacteristic
137C23×C4C22×C412yes—
138C23×C4C22×C412yes—
139C23×C4C22×C412yes—
140C23×C4C22×C411yescharacteristic
141C23×C4C2411yescharacteristic
142C22×C8—24no—
143C22×C8—24no—
144#125—28no—
145#125—28no—
146#125—28no—
147#125—28no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4M1618yes—
175C2×C42—24no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—24no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—42no—
184C2×C42—42no—
185C2×C42—42no—
186C2×C42—42no—
187C2×C42—44no—
188C2×C42—44no—
189C2×C42C22⋊C411yescharacteristic
190C2×C42C22⋊C411yescharacteristic
191C2×C42C22⋊C411yescharacteristic
192C2×C42C22⋊C411yescharacteristic
193C2×C42C22⋊C411yescharacteristic
194C2×C42C22⋊C411yescharacteristic
195C2×C42C22⋊C411yescharacteristic
196C2×C42C22⋊C411yescharacteristic
197C2×C42C22⋊C411yescharacteristic
198C2×C42C22⋊C411yescharacteristic
199C2×C42C22⋊C411yescharacteristic
200C2×C42C22⋊C411yescharacteristic
201C2×C42C22⋊C411yescharacteristic
202C2×C42C22⋊C411yescharacteristic
203C2×C42C22⋊C411yescharacteristic
204C2×C42C22⋊C411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C4⋊C8—44no—
218C4⋊C8—44no—
219C4⋊C8—44no—
220C4⋊C8—44no—
221C22⋊C8—24no—
222C22⋊C8—28no—
223C22⋊C8—24no—
224#97—44no—
225#97—44no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—22no—
238C24C23×C411yescharacteristic
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—22no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
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×C4C2×C22⋊C411yescharacteristic
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C2×C22⋊C411yescharacteristic
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4C2×M1614yes—
287C22×C4C2×M1614yes—
288C22×C4C2×M1614yes—
289C22×C4C22×D411yescharacteristic
290C22×C4C22×D411yescharacteristic
291C22×C4H(5,𝔽2)12yes—
292C22×C42−1+412yes—
293C2×C8—44no—
294C2×C8—44no—
295C2×C8—44no—
296C2×C8—44no—
297C2×C8—44no—
298C2×C8—44no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—28no—
304C4⋊C4—28no—
305C4⋊C4—28no—
306C4⋊C4—28no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—44no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—44no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—44no—
316C42—21no—
317C42—21no—
318C42—21no—
319C42—21no—
320C42—22no—
321C42—22no—
322C42—22no—
323C42—22no—
324C42—22no—
325C42—22no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—22no—
330C42—22no—
331C42—22no—
332C42—22no—
333C42—22no—
334C42—22no—
335C42—41no—
336C42—41no—
337C42—41no—
338C42—41no—
339C42—41no—
340C42—41no—
341C42—41no—
342C42—41no—
343C42—41no—
344C42—41no—
345C42—44no—
346C42—44no—
347C42—44no—
348C42—44no—
349C42#10114yes—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23C22×C22⋊C411yescharacteristic
355C23#52012yes—
356C23C22×M1612yes—
357C23#56812yes—
358C2×C4—28no—
359C2×C4—28no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—28no—
395C2×C4—44no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—84no—
415C2×C4—84no—
416C2×C4#40712yes—
417C2×C4#40712yes—
418C2×C4#41012yes—
419C2×C4#41012yes—
420C2×C4C2×#10112yes—
421C2×C4C2×#10112yes—
422C2×C4C2×#10112yes—
423C2×C4#41312yes—
424C2×C4#53511yescharacteristic
425C2×C4#53611yescharacteristic
426C2×C4#53712yes—
427C8—84no—
428C8—84no—
429C22—22no—
430C22—22no—
431C22—22no—
432C22—22no—
433C22—22no—
434C22—22no—
435C22#231111yescharacteristic
436C22C2×#40711yescharacteristic
437C22#287911yescharacteristic
438C22C2×#41011yescharacteristic
439C22C2×#41011yescharacteristic
440C22C22×#10111yescharacteristic
441C22C2×#41311yescharacteristic
442C22#298914yes—
443C4—21no—
444C4—22no—
445C4—21no—
446C4—22no—
447C4—22no—
448C4—22no—
449C4—22no—
450C4—22no—
451C4—22no—
452C4—22no—
453C4—22no—
454C4—41no—
455C4—41no—
456C4—44no—
457C4—41no—
458C4—44no—
459C4—41no—
460C4—82no—
461C4—82no—
462C2—24no—
463C2#1736311yescharacteristic
464C2#1736511yescharacteristic
465C2#1739411yescharacteristic
466C2#1739911yescharacteristic
467C2#2185311yescharacteristic
468C2#2187511yescharacteristic
469C2#2187911yescharacteristic
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 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
C2×C4×#1252
#208102
#207782
#98832
#96542
#90862
#85602