Finite Group Databasebeta

← #409391

Subgroups — #409391

Order 512.

1244 conjugacy classes of subgroups.

The subgroup lattice has 491 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1825C212yesmaximal
2#14340C211yesmaximal, characteristic
3#13795C211yesmaximal, characteristic
4#11357C212yesmaximal, complemented
5#11316C212yesmaximal, complemented
6#11315C212yesmaximal, complemented
7#11268C212yesmaximal, complemented
8#10730C212yesmaximal
9#10713C211yesmaximal, characteristic
10C22×#117C2211yescharacteristic
11C22×#117C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C22×C22⋊C8C2212yes—
14C2×#384C2211yescharacteristic
15C2×#382C2211yescharacteristic
16C2×#380C2212yes—
17#1825—28no—
18C2×#340C2212yes—
19C2×#340C2212yes—
20C2×#340C2212yes—
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#336C2212yes—
27C2×#336C2211yescharacteristic
28C2×#336C2211yescharacteristic
29#1441—22no—
30#1439—22no—
31C4⋊#95—22no—
32C4⋊#95C2212yes—
33C4⋊#95C2212yes—
34C4⋊#95C2212yes—
35C22⋊#95—22no—
36C24×C4—21no—
37C24×C4C2311yescharacteristic
38C23×C8C2312yes—
39C23×C8C2312yes—
40C2×#117—22no—
41C2×#117—24no—
42C2×#117—24no—
43C2×#117—24no—
44C2×#117—24no—
45C2×#117—41no—
46C2×#117—41no—
47C2×#117D414yes—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—24no—
50C22×C4⋊C4—24no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—41no—
53C22×C4⋊C4—41no—
54C22×C4⋊C4D412yes—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2312yes—
59C22×C22⋊C4—21no—
60C22×C22⋊C4—22no—
61C22×C22⋊C4D412yes—
62C22×C42—21no—
63C22×C42C2311yescharacteristic
64C22×C42C2311yescharacteristic
65C2×C22⋊C8—24no—
66C2×C22⋊C8—24no—
67C2×C22⋊C8—24no—
68#384—24no—
69#382—41no—
70#382—41no—
71#380—24no—
72#380—24no—
73C2×#95—22no—
74C2×#95—22no—
75C2×#95—22no—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—22no—
81#340—24no—
82#340—24no—
83#340—24no—
84#340—24no—
85#340—24no—
86#340—24no—
87#340—42no—
88#340—42no—
89#336—24no—
90#336—41no—
91#336—41no—
92C25C2×D411yescharacteristic
93C23×C4—22no—
94C23×C4—22no—
95C23×C4—21no—
96C23×C4—21no—
97C23×C4—21no—
98C23×C4—21no—
99C23×C4—22no—
100C23×C4—22no—
101C23×C4—24no—
102C23×C4—22no—
103C23×C4—22no—
104C23×C4—42no—
105C23×C4—42no—
106C23×C4—82no—
107C23×C4—82no—
108C23×C4C2×D412yes—
109C23×C4C2×D412yes—
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D412yes—
112C23×C4#4112yes—
113C23×C4#4114yes—
114C23×C4#4114yes—
115C23×C4C2411yescharacteristic
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130#117—216no—
131#117—48no—
132#117—48no—
133#117—48no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—48no—
158C2×C4⋊C4—48no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—44no—
163C2×C4⋊C4—82no—
164C2×C4⋊C4—82no—
165C2×C4⋊C4C2×D414yes—
166C2×C4⋊C4#4114yes—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—42no—
171C2×C22⋊C4—42no—
172C2×C22⋊C4—42no—
173C2×C22⋊C4—44no—
174C2×C22⋊C4—82no—
175C2×C22⋊C4—82no—
176C2×C42—22no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—24no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—42no—
185C2×C42—42no—
186C2×C42#4114yes—
187C22⋊C8—48no—
188#95—44no—
189#95—44no—
190#95—82no—
191#95—82no—
192#95—82no—
193#95—82no—
194#95—82no—
195#95—82no—
196C24—21no—
197C24—21no—
198C24—21no—
199C24—21no—
200C24—22no—
201C24—22no—
202C24—22no—
203C24—22no—
204C24#12112yes—
205C24#12312yes—
206C24#12412yes—
207C24C22×D411yescharacteristic
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—24no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—24no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—48no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—84no—
269C22×C4—84no—
270C22×C4—88no—
271C22×C4—88no—
272C22×C4#12112yes—
273C22×C4#12314yes—
274C22×C4#12318yes—
275C22×C4#12312yes—
276C22×C4#12412yes—
277C22×C4#12618yes—
278C22×C4C22×D411yescharacteristic
279C22×C4C2×#4112yes—
280C22×C4C2×#4112yes—
281C22×C4C2×#4111yescharacteristic
282C22×C4C2×#4112yes—
283C22×C4C2×#4112yes—
284C22×C4H(5,𝔽2)11yescharacteristic
285C22×C4H(5,𝔽2)11yescharacteristic
286C22×C42−1+411yescharacteristic
287C22×C42−1+411yescharacteristic
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C4⋊C4—28no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—48no—
306C4⋊C4—48no—
307C4⋊C4—44no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—88no—
319C22⋊C4—48no—
320C22⋊C4—48no—
321C22⋊C4—88no—
322C42—24no—
323C42—24no—
324C42—42no—
325C42—42no—
326C42—48no—
327C23—22no—
328C23—24no—
329C23—24no—
330C23—22no—
331C23—22no—
332C23—24no—
333C23—24no—
334C23—22no—
335C23—22no—
336C23—22no—
337C23—24no—
338C23—24no—
339C23—42no—
340C23—42no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—42no—
346C23—42no—
347C23—42no—
348C23—42no—
349C23#39714yes—
350C23C2×#12111yescharacteristic
351C23C2×#12311yescharacteristic
352C23C2×#12311yescharacteristic
353C23#52511yescharacteristic
354C23C2×#12411yescharacteristic
355C23#53411yescharacteristic
356C23#53611yescharacteristic
357C23#57612yes—
358C23#57712yes—
359C23#57712yes—
360C23#57812yes—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—28no—
368C2×C4—28no—
369C2×C4—28no—
370C2×C4—28no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—44no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—816no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4#39714yes—
415C2×C4C2×#12312yes—
416C2×C4C2×#12612yes—
417C2×C4#52912yes—
418C2×C4#53212yes—
419C2×C4#53711yescharacteristic
420C2×C4#53711yescharacteristic
421C2×C4#53911yescharacteristic
422C2×C4#53911yescharacteristic
423C2×C4#55112yes—
424C2×C4#55112yes—
425C2×C4#55112yes—
426C2×C4#55112yes—
427C2×C4#55212yes—
428C2×C4#55212yes—
429C2×C4#55212yes—
430C2×C4#55212yes—
431C8—82no—
432C8—82no—
433C22—28no—
434C22—24no—
435C22—44no—
436C22—44no—
437C22—42no—
438C22—42no—
439C22—42no—
440C22—42no—
441C22—44no—
442C22—42no—
443C22—42no—
444C22C2×#39711yescharacteristic
445C22#248311yescharacteristic
446C22#248411yescharacteristic
447C22#250311yescharacteristic
448C22#256111yescharacteristic
449C22#261811yescharacteristic
450C22#285411yescharacteristic
451C22#304512yes—
452C22#304612yes—
453C22#309612yes—
454C22#309712yes—
455C22#309812yes—
456C22#309912yes—
457C22#312211yescharacteristic
458C22#312311yescharacteristic
459C22#312611yescharacteristic
460C22#312711yescharacteristic
461C22#313612yes—
462C22#313712yes—
463C22#319812yes—
464C22#320112yes—
465C22#324312yes—
466C22#324412yes—
467C4—24no—
468C4—44no—
469C4—84no—
470C4—84no—
471C4—82no—
472C4—162no—
473C4—162no—
474C4#249312yes—
475C4#249312yes—
476C4#249812yes—
477C4#249812yes—
478C2—44no—
479C2#1501911yescharacteristic
480C2#2363412yes—
481C2#2760612yes—
482C2#2762011yescharacteristic
483C2#2762111yescharacteristic
484C2#2797112yes—
485C2#2911511yescharacteristic
486C2#2911711yescharacteristic
487C2#2976912yes—
488C2#3311611yescharacteristic
489C2#3311711yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11357C2semidirect
5. #11316C2semidirect
6. #11315C2semidirect
7. #11268C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18252
#143402
#137952
#113572
#113162
#113152
#112682
#107302
#107132