Finite Group Databasebeta

← #215131

Subgroups — #215131

Order 512.

1262 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2293C211yesmaximal, characteristic
2#20778C211yesmaximal, characteristic, complemented
3#20778C211yesmaximal, characteristic, complemented
4#9654C212yesmaximal, complemented
5#9653C212yesmaximal
6#9652C214yesmaximal, complemented
7#8300C214yesmaximal, complemented
8C22×C4×D4C411yescharacteristic
9C22×#117C2212yes—
10#2293—21no—
11#2293—21no—
12#2293—21no—
13#2293—21no—
14#2293—21no—
15#2293—21no—
16#2293—21no—
17#2293—21no—
18C2×#390C411yescharacteristic
19C2×#390C2211yescharacteristic
20C2×#389C411yescharacteristic
21C2×#389C2211yescharacteristic
22C2×#388C412yes—
23C2×#388C2212yes—
24C2×C4×C4⋊C4C411yescharacteristic
25C2×C4×C22⋊C4C412yes—
26C2×C43C2211yescharacteristic
27#1839—21no—
28#1839—21no—
29#1831—21no—
30#1831—21no—
31#1817C2212yes—
32#1817C2212yes—
33#1797C2212yes—
34#1797C2212yes—
35#1767—21no—
36#1767—21no—
37#1749C2212yes—
38#1749C2212yes—
39#1745—21no—
40#1745—21no—
41C2×#324C2218yes—
42C2×#323C2218yes—
43C23×D4C2×C411yescharacteristic
44C24×C4C2×C412yes—
45C2×C4×D4—28no—
46C2×C4×D4—21no—
47C2×C4×D4—21no—
48C2×C4×D4—21no—
49C2×C4×D4—21no—
50C2×#117—22no—
51C2×#117—22no—
52C2×#117—22no—
53C2×#117—22no—
54C2×#117—48no—
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C4⋊C4C2×C412yes—
57C22×C4⋊C4C2×C411yescharacteristic
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4C2312yes—
60C22×C22⋊C4C2×C412yes—
61C22×C22⋊C4C2×C412yes—
62C22×C22⋊C4C2×C412yes—
63C22×C22⋊C4C2×C412yes—
64C22×C22⋊C4C2312yes—
65C22×C42C2×C411yescharacteristic
66C22×C42C2×C412yes—
67C22×C42C2×C411yescharacteristic
68C22×C42C2×C411yescharacteristic
69C22×C42C2×C411yescharacteristic
70C22×C42C2311yescharacteristic
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73C2×C4⋊C8—22no—
74C2×C4⋊C8—22no—
75C2×C22⋊C8C2314yes—
76C2×C22⋊C8C2314yes—
77C2×#97—22no—
78C2×#97—22no—
79#390—21no—
80#390—21no—
81#390—21no—
82#390—21no—
83#390—21no—
84#390—21no—
85#390—21no—
86#390—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#388—22no—
96#388—22no—
97#388—22no—
98#388—22no—
99#388—22no—
100#388—22no—
101#388—22no—
102#388—22no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C22⋊C4—22no—
108C4×C22⋊C4—22no—
109C4×C22⋊C4—22no—
110C4×C22⋊C4—22no—
111C2×#95C2×C412yes—
112C2×#95C2×C412yes—
113C2×#95C2×C412yes—
114C43D411yescharacteristic
115C43D411yescharacteristic
116C43D411yescharacteristic
117C43D411yescharacteristic
118C43D411yescharacteristic
119C43D411yescharacteristic
120C43D411yescharacteristic
121C43D411yescharacteristic
122#336—22no—
123#336—22no—
124#336—22no—
125#336—22no—
126#336—22no—
127#336—22no—
128#336—22no—
129#336—22no—
130#324—48no—
131#323—48no—
132C25C22×C412yes—
133C22×D4—24no—
134C22×D4—21no—
135C22×D4—21no—
136C22×D4—42no—
137C22×D4—42no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—24no—
141C23×C4—44no—
142C23×C4—44no—
143C23×C4M1618yes—
144C23×C4C22×C411yescharacteristic
145C23×C4C22×C412yes—
146C23×C4C22×C411yescharacteristic
147C23×C4C22×C411yescharacteristic
148C23×C4C22×C411yescharacteristic
149C23×C4C22×C412yes—
150C23×C4C22×C412yes—
151C23×C4C22×C411yescharacteristic
152C23×C4C22×C411yescharacteristic
153C23×C4C2411yescharacteristic
154C22×C8—24no—
155C22×C8—24no—
156C4×D4—28no—
157C4×D4—28no—
158C4×D4—28no—
159C4×D4—28no—
160#117—48no—
161#117—48no—
162#117—48no—
163#117—48no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—28no—
192C2×C22⋊C4—44no—
193C2×C22⋊C4—44no—
194C2×C22⋊C4—44no—
195C2×C42—24no—
196C2×C42—24no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—42no—
200C2×C42—42no—
201C2×C42—42no—
202C2×C42—42no—
203C2×C42—44no—
204C2×C42—42no—
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×C42C22⋊C411yescharacteristic
216C2×C42C22⋊C411yescharacteristic
217C2×C42C22⋊C411yescharacteristic
218C2×C42C22⋊C411yescharacteristic
219C2×C42C22⋊C411yescharacteristic
220C2×C42C22⋊C411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C4⋊C8—44no—
234C4⋊C8—44no—
235C22⋊C8—24no—
236C22⋊C8—28no—
237C22⋊C8—24no—
238#97—44no—
239#97—44no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—22no—
246C24—22no—
247C24—22no—
248C24—24no—
249C24—24no—
250C24—42no—
251C24—42no—
252C24—42no—
253C24—42no—
254C24C2×M1614yes—
255C24C23×C411yescharacteristic
256C2×D4—24no—
257C2×D4—24no—
258C2×D4—416no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—42no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—28no—
277C22×C4—28no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—416no—
293C22×C4—44no—
294C22×C4—44no—
295C22×C4—44no—
296C22×C4—44no—
297C22×C4—416no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
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×C4C2×C22⋊C411yescharacteristic
311C22×C4C2×C22⋊C411yescharacteristic
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4C2×C22⋊C411yescharacteristic
314C22×C4C2×M1614yes—
315C22×C4C2×M1614yes—
316C22×C4C22×D411yescharacteristic
317C22×C4C22×D411yescharacteristic
318C22×C4H(5,𝔽2)14yes—
319C2×C8—44no—
320C2×C8—44no—
321C2×C8—44no—
322C2×C8—44no—
323C2×C8—44no—
324C2×C8—44no—
325C4⋊C4—24no—
326C4⋊C4—24no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—44no—
330C4⋊C4—44no—
331C4⋊C4—44no—
332C4⋊C4—44no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C22⋊C4—28no—
336C22⋊C4—28no—
337C22⋊C4—44no—
338C22⋊C4—44no—
339C22⋊C4—44no—
340C22⋊C4—44no—
341C22⋊C4—44no—
342C22⋊C4—44no—
343C22⋊C4—44no—
344C22⋊C4—44no—
345C42—22no—
346C42—22no—
347C42—22no—
348C42—22no—
349C42—22no—
350C42—22no—
351C42—22no—
352C42—22no—
353C42—22no—
354C42—22no—
355C42—22no—
356C42—22no—
357C42—22no—
358C42—22no—
359C42—22no—
360C42—22no—
361C42—41no—
362C42—41no—
363C42—41no—
364C42—41no—
365C42—41no—
366C42—41no—
367C42—41no—
368C42—41no—
369C42—41no—
370C42—41no—
371C42—41no—
372C42—41no—
373C42—44no—
374C42—44no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—24no—
380C23—28no—
381C23—44no—
382C23—44no—
383C23—44no—
384C23—44no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—44no—
392C23—84no—
393C23—84no—
394C23—84no—
395C23—84no—
396C23C22×C22⋊C411yescharacteristic
397C23#51812yes—
398C23C22×M1612yes—
399C23#56812yes—
400D4—48no—
401D4—48no—
402C2×C4—28no—
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—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—28no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—24no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—28no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—416no—
444C2×C4—416no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—84no—
457C2×C4—84no—
458C2×C4#40712yes—
459C2×C4#40712yes—
460C2×C4#41012yes—
461C2×C4#41012yes—
462C2×C4#41312yes—
463C2×C4#41312yes—
464C2×C4#53512yes—
465C2×C4#53512yes—
466C8—84no—
467C8—84no—
468C22—22no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—22no—
473C22—22no—
474C22—44no—
475C22—44no—
476C22—44no—
477C22—88no—
478C22—88no—
479C22—82no—
480C22—82no—
481C22—82no—
482C22—82no—
483C22#230911yescharacteristic
484C22C2×#40711yescharacteristic
485C22#287911yescharacteristic
486C22C2×#41011yescharacteristic
487C22C2×#41011yescharacteristic
488C22C2×#41311yescharacteristic
489C22C2×#41311yescharacteristic
490C22#297314yes—
491C4—22no—
492C4—22no—
493C4—22no—
494C4—22no—
495C4—22no—
496C4—22no—
497C4—22no—
498C4—22no—
499C4—44no—
500C4—41no—
501C4—41no—
502C4—41no—
503C4—41no—
504C4—41no—
505C4—41no—
506C4—82no—
507C4—82no—
508C2—24no—
509C2—84no—
510C2#1735711yescharacteristic
511C2#1736111yescharacteristic
512C2#1739511yescharacteristic
513C2#1739911yescharacteristic
514C2#2179011yescharacteristic
515C2#2179411yescharacteristic
516C2#2180011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20778C2semidirect
3. #20778C2semidirect
4. #9654C2semidirect
6. #9652C2semidirect
7. #8300C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22932
#207782
#96542
#96532
#96522
#83002