Finite Group Databasebeta

← #215729

Subgroups — #215729

Order 512.

1566 conjugacy classes of subgroups.

The subgroup lattice has 519 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#389C211yesmaximal, characteristic
2#20715C211yesmaximal, characteristic
3#20714C211yesmaximal, characteristic
4#9883C214yesmaximal, complemented
5#9879C214yesmaximal, complemented
6#8698C214yesmaximal
7C23×C4⋊C4C411yescharacteristic
8C23×C4⋊C4C411yescharacteristic
9C23×C4⋊C4C412yes—
10C23×C42C2211yescharacteristic
11C2×#432—21no—
12C2×#431—21no—
13C2×#389—21no—
14C2×#389—21no—
15C2×#389—21no—
16C2×#389—21no—
17C2×#389—21no—
18C2×#389—21no—
19C2×#389—21no—
20C2×#389—21no—
21C2×#389C412yes—
22C2×#389C412yes—
23C2×#389C2212yes—
24C2×#389C2212yes—
25C22×#95C2212yes—
26#1817—21no—
27#1817—21no—
28#1797—21no—
29#1797—21no—
30#1797C2212yes—
31#1797C2212yes—
32#1779C2212yes—
33#1779C2212yes—
34#1749—21no—
35#1749C2212yes—
36C4×C22⋊C8—21no—
37C4×C22⋊C8C2212yes—
38#1280C22116yes—
39C24×C4C2×C411yescharacteristic
40C24×C4C2×C411yescharacteristic
41C24×C4C2×C412yes—
42C24×C4C2×C412yes—
43C24×C4C2311yescharacteristic
44C22×C4⋊C4—21no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—28no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—44no—
58C22×C4⋊C4—48no—
59C22×C4⋊C4C2×C412yes—
60C22×C4⋊C4C2×C412yes—
61C22×C4⋊C4C2×C412yes—
62C22×C4⋊C4C2×C414yes—
63C22×C4⋊C4C2×C414yes—
64C22×C4⋊C4C2×C412yes—
65C22×C42—44no—
66C22×C42C2×C411yescharacteristic
67C22×C42C2×C411yescharacteristic
68C22×C42D411yescharacteristic
69C22×C42D411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78#432—44no—
79#431—44no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—22no—
82C2×C22⋊C8—22no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8C2314yes—
87C2×C22⋊C8C2314yes—
88C2×#97—22no—
89C2×C4×C8—22no—
90#389—22no—
91#389—22no—
92#389—22no—
93#389—22no—
94#389—22no—
95#389—22no—
96#389—22no—
97#389—22no—
98#389—22no—
99#389—22no—
100#389—22no—
101#389—22no—
102#389—22no—
103#389—22no—
104#389—22no—
105#389—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95C2×C414yes—
111C2×#95C2314yes—
112#336—22no—
113#336—22no—
114#336—22no—
115#336—22no—
116C25C22×C411yescharacteristic
117C23×C4—24no—
118C23×C4—24no—
119C23×C4—24no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—24no—
123C23×C4—24no—
124C23×C4—28no—
125C23×C4—24no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—28no—
129C23×C4—42no—
130C23×C4—42no—
131C23×C4—42no—
132C23×C4—44no—
133C23×C4—44no—
134C23×C4—44no—
135C23×C4—44no—
136C23×C4C22⋊C411yescharacteristic
137C23×C4C22⋊C411yescharacteristic
138C23×C4C22⋊C411yescharacteristic
139C23×C4C22⋊C411yescharacteristic
140C23×C4C22⋊C411yescharacteristic
141C23×C4C22⋊C411yescharacteristic
142C23×C4C22⋊C411yescharacteristic
143C23×C4C22⋊C411yescharacteristic
144C23×C4C22×C411yescharacteristic
145C23×C4C22×C411yescharacteristic
146C23×C4C22×C411yescharacteristic
147C23×C4C22×C411yescharacteristic
148C23×C4C22×C414yes—
149C23×C4C22×C411yescharacteristic
150C23×C4C22×C414yes—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2411yescharacteristic
156C22×C8—24no—
157C22×C8—24no—
158C2×C4⋊C4—28no—
159C2×C4⋊C4—28no—
160C2×C4⋊C4—28no—
161C2×C4⋊C4—28no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—28no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—48no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—48no—
181C2×C4⋊C4—48no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—48no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—48no—
187C2×C4⋊C4—48no—
188C2×C4⋊C4M1618yes—
189C2×C42—24no—
190C2×C42—24no—
191C2×C42—42no—
192C2×C42—42no—
193C2×C42—42no—
194C2×C42—42no—
195C2×C42—42no—
196C2×C42—42no—
197C2×C42—42no—
198C2×C42—42no—
199C2×C42—42no—
200C2×C42—42no—
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×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—
219C22⋊C8—28no—
220C22⋊C8—28no—
221C22⋊C8—44no—
222C22⋊C8—44no—
223C22⋊C8—44no—
224C22⋊C8—44no—
225#97—44no—
226C4×C8—44no—
227#95—24no—
228#95—24no—
229#95—24no—
230#95—24no—
231C24—22no—
232C24—22no—
233C24—22no—
234C24—22no—
235C24—22no—
236C24—22no—
237C24C2×C22⋊C411yescharacteristic
238C24C2×C22⋊C411yescharacteristic
239C24#13114yes—
240C24C23×C411yescharacteristic
241C22×C4—232no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—28no—
260C22×C4—28no—
261C22×C4—24no—
262C22×C4—22no—
263C22×C4—24no—
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—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—28no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—416no—
298C22×C4—44no—
299C22×C4—416no—
300C22×C4—432no—
301C22×C4—44no—
302C22×C4—44no—
303C22×C4—44no—
304C22×C4—44no—
305C22×C4—44no—
306C22×C4—44no—
307C22×C4—44no—
308C22×C4—44no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4—44no—
312C22×C4—44no—
313C22×C4—44no—
314C22×C4—816no—
315C22×C4#9914yes—
316C22×C4#10014yes—
317C22×C4#10114yes—
318C22×C4C2×C22⋊C411yescharacteristic
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C2×C22⋊C411yescharacteristic
325C22×C4C2×C22⋊C411yescharacteristic
326C22×C4C2×C22⋊C411yescharacteristic
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C2×M1614yes—
329C22×C4C2×M1618yes—
330C22×C4C22×D411yescharacteristic
331C22×C4C22×D411yescharacteristic
332C22×C42−1+414yes—
333C2×C8—44no—
334C2×C8—44no—
335C2×C8—44no—
336C2×C8—44no—
337C2×C8—44no—
338C2×C8—44no—
339C4⋊C4—28no—
340C4⋊C4—28no—
341C4⋊C4—28no—
342C4⋊C4—28no—
343C4⋊C4—44no—
344C4⋊C4—44no—
345C4⋊C4—48no—
346C4⋊C4—48no—
347C4⋊C4—44no—
348C4⋊C4—44no—
349C4⋊C4—48no—
350C4⋊C4—48no—
351C42—22no—
352C42—22no—
353C42—22no—
354C42—22no—
355C42—22no—
356C42—22no—
357C42—22no—
358C42—22no—
359C42—41no—
360C42—41no—
361C42—41no—
362C42—41no—
363C42—41no—
364C42—41no—
365C42—41no—
366C42—41no—
367C23—28no—
368C23—28no—
369C23—28no—
370C23—28no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—28no—
384C23—28no—
385C23—28no—
386C23#40812yes—
387C23#40812yes—
388C23C2×#9912yes—
389C23#41012yes—
390C23C2×#10012yes—
391C23C2×#10112yes—
392C23C22×C22⋊C411yescharacteristic
393C23#51912yes—
394C23C22×M1612yes—
395C23C2×#13112yes—
396C2×C4—28no—
397C2×C4—28no—
398C2×C4—28no—
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—28no—
414C2×C4—28no—
415C2×C4—28no—
416C2×C4—28no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—28no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—48no—
441C2×C4—416no—
442C2×C4—416no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—48no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—88no—
451C2×C4—88no—
452C2×C4—88no—
453C2×C4#40712yes—
454C2×C4#40712yes—
455C2×C4C2×#9912yes—
456C2×C4C2×#9912yes—
457C2×C4C2×#10012yes—
458C2×C4C2×#10012yes—
459C2×C4C2×#10112yes—
460C2×C4C2×#10112yes—
461C2×C4#53612yes—
462C2×C4#53612yes—
463C8—84no—
464C8—84no—
465C22—232no—
466C22—22no—
467C22—22no—
468C22—22no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—22no—
473C22—22no—
474C22—22no—
475C22—22no—
476C22—22no—
477C22—22no—
478C22—22no—
479C22—22no—
480C22—22no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22#147014yes—
485C22#147114yes—
486C22#231011yescharacteristic
487C22C2×#40711yescharacteristic
488C22C2×#40811yescharacteristic
489C22C22×#9911yescharacteristic
490C22C2×#41011yescharacteristic
491C22C22×#10011yescharacteristic
492C22C22×#10111yescharacteristic
493C22#298414yes—
494C4—22no—
495C4—22no—
496C4—22no—
497C4—22no—
498C4—22no—
499C4—22no—
500C4—22no—
501C4—41no—
502C4—41no—
503C4—48no—
504C4—41no—
505C4—41no—
506C4—84no—
507C4#146314yes—
508C2—24no—
509C2—24no—
510C2—24no—
511C2C2×#146311yescharacteristic
512C2#1736411yescharacteristic
513C2C2×#147011yescharacteristic
514C2C2×#147111yescharacteristic
515C2#2182011yescharacteristic
516C2#2182511yescharacteristic
517C2#2182911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ 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
4. #9883C2semidirect
5. #9879C2semidirect

Maximal subgroups

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

subgroupindex
C22×#3892
#207152
#207142
#98832
#98792
#86982