Finite Group Databasebeta

← #409801

Subgroups — #409801

Order 512.

1112 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1809C211yesmaximal, characteristic
2C2×#1753C211yesmaximal, characteristic
3#15132C212yesmaximal
4#11379C211yesmaximal, characteristic, complemented
5#11377C211yesmaximal, characteristic, complemented
6#11327C211yesmaximal, characteristic, complemented
7#11257C211yesmaximal, characteristic, complemented
8#11200C212yesmaximal
9#10880C212yesmaximal, complemented
10#10821C212yesmaximal, complemented
11#10722C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×#106C2211yescharacteristic
15C22×C22⋊C8C2211yescharacteristic
16C22×#97C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18C2×#397—22no—
19C2×#389C2212yes—
20C2×#380C2212yes—
21C2×#380C2212yes—
22C2×C4×C22⋊C4C2211yescharacteristic
23#1809—24no—
24#1753—24no—
25C2×#340C2212yes—
26C2×#340C2211yescharacteristic
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2211yescharacteristic
32C2×#340C2212yes—
33C2×#336C2211yescharacteristic
34C2×#336C2211yescharacteristic
35#1443—22no—
36#1442—22no—
37#1440—22no—
38C4⋊#95C2212yes—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41C24×C4C2311yescharacteristic
42C22×M16—24no—
43C22×M16—22no—
44C22×M16—22no—
45C22×M16—42no—
46C22×M16D414yes—
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C22×C4⋊C4—22no—
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2312yes—
56C22×C22⋊C4—22no—
57C22×C22⋊C4—22no—
58C22×C22⋊C4D412yes—
59C22×C42C2311yescharacteristic
60C22×C42C2311yescharacteristic
61C2×#107—22no—
62C2×#107—22no—
63C2×#107—22no—
64C2×#106—22no—
65C2×#106—42no—
66C2×C22⋊C8—22no—
67C2×C22⋊C8—22no—
68C2×C22⋊C8—22no—
69C2×#97—22no—
70C2×#97—24no—
71C2×C4×C8—22no—
72C2×C4×C8—42no—
73#397—48no—
74#389—24no—
75#389—42no—
76#380—24no—
77#380—24no—
78#380—24no—
79#380—42no—
80C4×C22⋊C4—22no—
81C4×C22⋊C4—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95D412yes—
92#340—24no—
93#340—22no—
94#340—24no—
95#340—24no—
96#340—24no—
97#340—22no—
98#340—24no—
99#340—42no—
100#336—22no—
101#336—22no—
102C25C2×D411yescharacteristic
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4C2×D412yes—
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D412yes—
118C23×C4C2×D412yes—
119C23×C4#4112yes—
120C23×C4C2411yescharacteristic
121C2×M16—28no—
122C2×M16—28no—
123C2×M16—44no—
124C2×M16—48no—
125C2×M16—48no—
126C22×C8—22no—
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—21no—
132C22×C8—21no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—21no—
136C22×C8—21no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8C2×D412yes—
140C22×C8C2×D412yes—
141C22×C8#4112yes—
142C22×C8#4112yes—
143C22×C8#4114yes—
144C22×C8#4112yes—
145C22×C8#4112yes—
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—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—82no—
172C2×C4⋊C4#4114yes—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—42no—
176C2×C22⋊C4—44no—
177C2×C22⋊C4—44no—
178C2×C22⋊C4—44no—
179C2×C22⋊C4—44no—
180C2×C22⋊C4—44no—
181C2×C22⋊C4—82no—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190#107—44no—
191#106—44no—
192C22⋊C8—28no—
193#97—44no—
194C4×C8—44no—
195#95—24no—
196#95—44no—
197#95—44no—
198#95—44no—
199#95—44no—
200#95—44no—
201#95—44no—
202#95—44no—
203#95—44no—
204#95—82no—
205C24—21no—
206C24—21no—
207C24—22no—
208C24—22no—
209C24—22no—
210C24—22no—
211C24—22no—
212C24—22no—
213C24#12112yes—
214C24C22×D411yescharacteristic
215C22×C4—22no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
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—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4#12112yes—
279C22×C4AGL(1,ℤ/8)14yes—
280C22×C4C22×D411yescharacteristic
281C22×C4C2×#4112yes—
282C22×C4C2×#4111yescharacteristic
283C22×C4C2×#4111yescharacteristic
284C22×C4C2×#4111yescharacteristic
285C22×C4C2×#4111yescharacteristic
286C22×C4C2×#4112yes—
287C22×C4C2×#4111yescharacteristic
288C22×C4H(5,𝔽2)12yes—
289C22×C42−1+412yes—
290M16—416no—
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—24no—
296C2×C8—24no—
297C2×C8—24no—
298C2×C8—24no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—24no—
302C2×C8—24no—
303C2×C8—41no—
304C2×C8—42no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—42no—
309C2×C8#12314yes—
310C2×C8#12314yes—
311C4⋊C4—24no—
312C4⋊C4—24no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—88no—
328C22⋊C4—48no—
329C22⋊C4—88no—
330C22⋊C4—88no—
331C42—41no—
332C42—41no—
333C42—41no—
334C42—41no—
335C42—42no—
336C42—42no—
337C23—22no—
338C23—22no—
339C23—22no—
340C23—24no—
341C23—24no—
342C23—42no—
343C23—42no—
344C23—42no—
345C23—42no—
346C23—42no—
347C23—41no—
348C23—41no—
349C23—42no—
350C23—41no—
351C23—41no—
352C23—42no—
353C23—42no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23C2×#12111yescharacteristic
359C23#53211yescharacteristic
360C23#53211yescharacteristic
361C23#54612yes—
362C23#54712yes—
363C23C2×AGL(1,ℤ/8)12yes—
364C23#57512yes—
365C23#57712yes—
366C23#57812yes—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—22no—
372C2×C4—24no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—44no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—44no—
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—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4C2×#12311yescharacteristic
421C2×C4C2×#12311yescharacteristic
422C2×C4#52511yescharacteristic
423C2×C4#52911yescharacteristic
424C2×C4#52912yes—
425C2×C4#52912yes—
426C2×C4#52911yescharacteristic
427C2×C4#53211yescharacteristic
428C2×C4#53411yescharacteristic
429C2×C4#53611yescharacteristic
430C2×C4#53912yes—
431C2×C4#54212yes—
432C2×C4#55112yes—
433C2×C4#55112yes—
434C2×C4#55212yes—
435C2×C4#55212yes—
436C2×C4#55312yes—
437C2×C4#55612yes—
438C8—24no—
439C8—24no—
440C8—42no—
441C8—42no—
442C22—44no—
443C22—42no—
444C22—42no—
445C22—44no—
446C22—42no—
447C22—82no—
448C22—82no—
449C22—82no—
450C22—82no—
451C22#248611yescharacteristic
452C22#252112yes—
453C22#254311yescharacteristic
454C22#255312yes—
455C22#285411yescharacteristic
456C22#305512yes—
457C22#306312yes—
458C22#317112yes—
459C22#317311yescharacteristic
460C22#317711yescharacteristic
461C22#317812yes—
462C22#318311yescharacteristic
463C22#318411yescharacteristic
464C22#330712yes—
465C22#330812yes—
466C22#331012yes—
467C22#331312yes—
468C22#335112yes—
469C22#335212yes—
470C22#335312yes—
471C22#335412yes—
472C4—44no—
473C4—42no—
474C4—82no—
475C4—84no—
476C4—162no—
477C4—162no—
478C4—162no—
479C4#239511yescharacteristic
480C4#248311yescharacteristic
481C4#249911yescharacteristic
482C4#253012yes—
483C4#257712yes—
484C4#263811yescharacteristic
485C2—82no—
486C2#1506111yescharacteristic
487C2#2767012yes—
488C2#2809012yes—
489C2#2809212yes—
490C2#2855912yes—
491C2#2886012yes—
492C2#2886112yes—
493C2#3314111yescharacteristic
494C2#3314311yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11379C2semidirect
5. #11377C2semidirect
6. #11327C2semidirect
7. #11257C2semidirect
9. #10880C2semidirect
10. #10821C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18092
C2×#17532
#151322
#113792
#113772
#113272
#112572
#112002
#108802
#108212
#107222