Finite Group Databasebeta

← #2044719

Subgroups — #2044719

Order 512.

1978 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17080C211yesmaximal, characteristic, complemented
2#17073C211yesmaximal, characteristic, complemented
3#16387C214yesmaximal, complemented
4#15794C214yesmaximal, complemented
5#15672C214yesmaximal
6#15486C214yesmaximal
7#15393C214yesmaximal
8#14702C212yesmaximal
9#14682C211yesmaximal, characteristic
10#14546C214yesmaximal, complemented
11#14471C212yesmaximal, complemented
12C22×C22⋊Q8C2212yes—
13C23×C42C2211yescharacteristic
14C2×#400C2214yes—
15C2×#398C2214yes—
16C2×#397C2214yes—
17C2×#391C2212yes—
18C2×#391C2212yes—
19C2×#388C2214yes—
20C2×#387C2214yes—
21C2×#382C2212yes—
22C2×#380C2212yes—
23C2×#380C2212yes—
24#1446C2214yes—
25#1446C2214yes—
26#1445C2214yes—
27#1444C2214yes—
28#1444C2214yes—
29#1443C2214yes—
30#1443C2214yes—
31#1443C2214yes—
32#1443C2214yes—
33#1442C2214yes—
34#1442C2214yes—
35#1442C2214yes—
36#1441C2214yes—
37#1441C2214yes—
38#1441C2214yes—
39#1441C2214yes—
40#1440C2214yes—
41#1439C2214yes—
42#1436C2214yes—
43#1436C2214yes—
44#1436C2214yes—
45#1436C2214yes—
46#1436C2214yes—
47#1436C2214yes—
48C4⋊#95C2214yes—
49C4⋊#95C2211yescharacteristic
50C4⋊#95C2211yescharacteristic
51C4⋊#95C2212yes—
52C4⋊#95C2212yes—
53#1434C2214yes—
54C22⋊#95C2214yes—
55C22⋊#95C2214yes—
56C22⋊#95C2214yes—
57C23×Q8C2312yes—
58C24×C4C2312yes—
59C24×C4C2311yescharacteristic
60C2×C22⋊Q8—28no—
61C2×C22⋊Q8—44no—
62C2×C22⋊Q8—44no—
63C22×C4⋊C4C2314yes—
64C22×C4⋊C4C2314yes—
65C22×C4⋊C4C2314yes—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2314yes—
71C22×C4⋊C4C2312yes—
72C22×C22⋊C4C2314yes—
73C22×C22⋊C4C2314yes—
74C22×C22⋊C4C2314yes—
75C22×C42—24no—
76C22×C42—44no—
77C22×C42C2311yescharacteristic
78C22×C42C2312yes—
79C22×C42C2314yes—
80C22×C42C2311yescharacteristic
81#400—48no—
82#398—216no—
83#397—48no—
84#391—28no—
85#391—44no—
86#388—48no—
87#387—48no—
88#382—44no—
89#380—44no—
90#380—44no—
91C2×#95C2314yes—
92C2×#95C2314yes—
93C2×#95C2314yes—
94C2×#95C2314yes—
95C2×#95C2314yes—
96C2×#95C2314yes—
97C2×#95C2314yes—
98C2×#95C2314yes—
99C2×#95C2314yes—
100C2×#95C2314yes—
101C2×#95C2314yes—
102C2×#95C2314yes—
103C2×#95C2314yes—
104C2×#95C2314yes—
105C2×#95C2314yes—
106C2×#95C2314yes—
107C2×#95C2314yes—
108C2×#95C2314yes—
109C2×#95C2314yes—
110C2×#95C2314yes—
111C2×#95C2314yes—
112C2×#95C2314yes—
113C2×#95C2314yes—
114C2×#95C2314yes—
115C2×#95C2314yes—
116C2×#95C2314yes—
117C25C2411yescharacteristic
118C22×Q8—24no—
119C22×Q8—24no—
120C22×Q8—24no—
121C22×Q8—42no—
122C22×Q8—42no—
123C22×Q8—42no—
124C22×Q8—42no—
125C23×C4—24no—
126C23×C4—22no—
127C23×C4—24no—
128C23×C4—22no—
129C23×C4—24no—
130C23×C4—22no—
131C23×C4—24no—
132C23×C4—24no—
133C23×C4—24no—
134C23×C4—42no—
135C23×C4—42no—
136C23×C4—42no—
137C23×C4C2414yes—
138C23×C4C2414yes—
139C23×C4C2414yes—
140C23×C4C2414yes—
141C23×C4C2414yes—
142C23×C4C2414yes—
143C23×C4C2412yes—
144C23×C4C2411yescharacteristic
145C23×C4C2412yes—
146C23×C4C2411yescharacteristic
147C22⋊Q8—432no—
148C2×C4⋊C4—28no—
149C2×C4⋊C4—28no—
150C2×C4⋊C4—28no—
151C2×C4⋊C4—28no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—28no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—44no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—44no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4#4118yes—
175C2×C22⋊C4—28no—
176C2×C22⋊C4—28no—
177C2×C22⋊C4—28no—
178C2×C22⋊C4—28no—
179C2×C22⋊C4—28no—
180C2×C22⋊C4—28no—
181C2×C22⋊C4—44no—
182C2×C22⋊C4—44no—
183C2×C22⋊C4—44no—
184C2×C42—22no—
185C2×C42—24no—
186C2×C42—24no—
187C2×C42—24no—
188C2×C42—416no—
189C2×C42—42no—
190C2×C42—42no—
191C2×C42—44no—
192C2×C42—44no—
193C2×C42—44no—
194C2×C42#4114yes—
195#95—28no—
196#95—28no—
197#95—28no—
198#95—28no—
199#95—28no—
200#95—28no—
201#95—28no—
202#95—28no—
203#95—28no—
204#95—28no—
205#95—28no—
206#95—44no—
207#95—44no—
208#95—44no—
209#95—44no—
210#95—44no—
211#95—44no—
212#95—44no—
213#95—44no—
214#95—44no—
215#95—44no—
216#95—44no—
217#95—44no—
218#95—44no—
219#95—44no—
220#95—44no—
221C24—22no—
222C24—21no—
223C24—21no—
224C24—22no—
225C24—22no—
226C24—22no—
227C24—22no—
228C24H(5,𝔽2)14yes—
229C242−1+412yes—
230C24C2511yescharacteristic
231C2×Q8—48no—
232C2×Q8—48no—
233C2×Q8—48no—
234C2×Q8—84no—
235C2×Q8—84no—
236C2×Q8—84no—
237C2×Q8—84no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—22no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—216no—
246C22×C4—28no—
247C22×C4—24no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—24no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—24no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—48no—
299C22×C4—48no—
300C22×C4—48no—
301C22×C4—48no—
302C22×C4—48no—
303C22×C4—48no—
304C22×C4—44no—
305C22×C4—48no—
306C22×C4—44no—
307C22×C4—48no—
308C22×C4C2×#4118yes—
309C22×C4C2×#4112yes—
310C22×C4C2×#4114yes—
311C22×C4C2×#4114yes—
312C22×C4H(5,𝔽2)12yes—
313C22×C4H(5,𝔽2)14yes—
314C22×C42−1+414yes—
315C22×C42−1+414yes—
316C4⋊C4—48no—
317C4⋊C4—416no—
318C4⋊C4—416no—
319C4⋊C4—84no—
320C4⋊C4—84no—
321C4⋊C4—84no—
322C4⋊C4—84no—
323C4⋊C4—88no—
324C4⋊C4—88no—
325C22⋊C4—416no—
326C22⋊C4—416no—
327C22⋊C4—88no—
328C42—28no—
329C42—44no—
330C42—48no—
331C42—88no—
332C23—24no—
333C23—22no—
334C23—24no—
335C23—24no—
336C23—24no—
337C23—24no—
338C23—24no—
339C23—24no—
340C23—22no—
341C23—24no—
342C23—41no—
343C23—42no—
344C23—42no—
345C23—41no—
346C23—41no—
347C23—42no—
348C23—42no—
349C23—42no—
350C23—41no—
351C23—42no—
352C23#56214yes—
353C23C22×#4111yescharacteristic
354C23C22×#4112yes—
355C23C2×H(5,𝔽2)12yes—
356C23C2×H(5,𝔽2)11yescharacteristic
357C23C2×H(5,𝔽2)12yes—
358C23C2×2−1+412yes—
359C23C2×2−1+412yes—
360C23C2×2−1+411yescharacteristic
361C23#58512yes—
362Q8—816no—
363C2×C4—24no—
364C2×C4—22no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—22no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—28no—
374C2×C4—24no—
375C2×C4—28no—
376C2×C4—24no—
377C2×C4—22no—
378C2×C4—28no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—432no—
382C2×C4—416no—
383C2×C4—41no—
384C2×C4—44no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—44no—
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—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4#54114yes—
430C2×C4#55618yes—
431C22—28no—
432C22—24no—
433C22—44no—
434C22—44no—
435C22—44no—
436C22—42no—
437C22—42no—
438C22—44no—
439C22—44no—
440C22C2×#54111yescharacteristic
441C22#346012yes—
442C22C2×#55612yes—
443C22#348111yescharacteristic
444C22#349012yes—
445C22#349011yescharacteristic
446C22#349211yescharacteristic
447C22#349412yes—
448C22#351512yes—
449C22#352212yes—
450C22#352312yes—
451C22C2×#56211yescharacteristic
452C22#353512yes—
453C22#353612yes—
454C22#353612yes—
455C22#353812yes—
456C22#354412yes—
457C22#355011yescharacteristic
458C22#355311yescharacteristic
459C22#355512yes—
460C22#355812yes—
461C4—28no—
462C4—24no—
463C4—44no—
464C4—48no—
465C4—42no—
466C4—84no—
467C4—84no—
468C4—84no—
469C4—84no—
470C4—84no—
471C2—44no—
472C2#3672811yescharacteristic
473C2#3716812yes—
474C2#3795712yes—
475C2#3876212yes—
476C2#3915712yes—
477C2#4598811yescharacteristic
478C2#4609812yes—
479C2#5319712yes—
480C2#5924811yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #17080C2semidirect
2. #17073C2semidirect
3. #16387C2semidirect
4. #15794C2semidirect
10. #14546C2semidirect
11. #14471C2semidirect

Maximal subgroups

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

subgroupindex
#170802
#170732
#163872
#157942
#156722
#154862
#153932
#147022
#146822
#145462
#144712