Finite Group Databasebeta

← #409382

Subgroups — #409382

Order 512.

1186 conjugacy classes of subgroups.

The subgroup lattice has 501 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1809C211yesmaximal, characteristic
2C2×#1729C211yesmaximal, characteristic
3#14343C212yesmaximal
4#11379C211yesmaximal, characteristic, complemented
5#11377C211yesmaximal, characteristic, complemented
6#11316C212yesmaximal, complemented
7#11315C212yesmaximal, complemented
8#11257C211yesmaximal, characteristic, complemented
9#11257C211yesmaximal, characteristic, complemented
10#10722C211yesmaximal, characteristic
11#10721C212yesmaximal
12C23×M16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C22×#106C2211yescharacteristic
17C2×#380C2212yes—
18C2×#380C2212yes—
19#1809—24no—
20C2×#340C2212yes—
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2211yescharacteristic
24C2×#340C2211yescharacteristic
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28#1729—24no—
29C2×#336C2211yescharacteristic
30C2×#336C2211yescharacteristic
31C2×#336C2211yescharacteristic
32C2×#336C2211yescharacteristic
33#1445—22no—
34#1445—22no—
35#1440—22no—
36#1440—22no—
37C4⋊#95C2212yes—
38C4⋊#95C2212yes—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41C24×C4C2311yescharacteristic
42C22×M16—24no—
43C22×M16—22no—
44C22×M16—22no—
45C22×M16—42no—
46C22×M16D414yes—
47C23×C8—22no—
48C23×C8—22no—
49C23×C8—22no—
50C23×C8—24no—
51C23×C8—42no—
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56C2×#117—22no—
57C2×#117—22no—
58C2×#117—24no—
59C2×#117—24no—
60C22×C4⋊C4D412yes—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C22⋊C4—22no—
66C22×C22⋊C4—22no—
67C22×C22⋊C4D412yes—
68C22×C42C2311yescharacteristic
69C22×C42C2311yescharacteristic
70C2×#107—22no—
71C2×#107—22no—
72C2×#107—22no—
73C2×#106—22no—
74C2×#106—42no—
75#380—24no—
76#380—24no—
77#380—24no—
78#380—24no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89#340—24no—
90#340—24no—
91#340—24no—
92#340—22no—
93#340—22no—
94#340—42no—
95#340—42no—
96#340—42no—
97#336—22no—
98#336—22no—
99#336—22no—
100#336—22no—
101C25C2×D411yescharacteristic
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4C2×D412yes—
113C23×C4C2×D412yes—
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D412yes—
116C23×C4#4112yes—
117C23×C4#4114yes—
118C23×C4#4114yes—
119C23×C4C2411yescharacteristic
120C2×M16—28no—
121C2×M16—28no—
122C2×M16—44no—
123C2×M16—48no—
124C2×M16—48no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—28no—
128C22×C8—21no—
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—21no—
136C22×C8—21no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—48no—
141C22×C8—48no—
142C22×C8—44no—
143C22×C8—44no—
144C22×C8C2×D412yes—
145C22×C8C2×D412yes—
146C22×C8#4112yes—
147C22×C8#4112yes—
148C22×C8#4112yes—
149C22×C8#4112yes—
150#117—44no—
151#117—44no—
152#117—48no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
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—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—44no—
184C2×C22⋊C4—44no—
185C2×C22⋊C4—82no—
186C2×C22⋊C4—82no—
187C2×C22⋊C4—82no—
188C2×C22⋊C4—82no—
189C2×C42—21no—
190C2×C42—21no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—21no—
196C2×C42—21no—
197#107—44no—
198#106—44no—
199#95—44no—
200#95—44no—
201#95—82no—
202#95—82no—
203#95—82no—
204#95—82no—
205#95—82no—
206#95—82no—
207#95—82no—
208#95—82no—
209C24—21no—
210C24—21no—
211C24—22no—
212C24—22no—
213C24—22no—
214C24—22no—
215C24—22no—
216C24#12112yes—
217C24#12314yes—
218C24C22×D411yescharacteristic
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
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—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—42no—
279C22×C4#12112yes—
280C22×C4#12314yes—
281C22×C4#12314yes—
282C22×C4#12414yes—
283C22×C4#12618yes—
284C22×C4C22×D411yescharacteristic
285C22×C4C2×#4112yes—
286C22×C4C2×#4111yescharacteristic
287C22×C4C2×#4111yescharacteristic
288C22×C4C2×#4111yescharacteristic
289C22×C4C2×#4111yescharacteristic
290C22×C4C2×#4112yes—
291C22×C4C2×#4111yescharacteristic
292C22×C4H(5,𝔽2)12yes—
293C22×C42−1+412yes—
294M16—416no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—24no—
298C2×C8—24no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—24no—
305C2×C8—24no—
306C2×C8—416no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8#12314yes—
316C4⋊C4—48no—
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—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C22⋊C4—48no—
334C22⋊C4—88no—
335C22⋊C4—88no—
336C42—41no—
337C42—41no—
338C42—41no—
339C42—41no—
340C42—42no—
341C42—42no—
342C23—24no—
343C23—24no—
344C23—24no—
345C23—24no—
346C23—22no—
347C23—22no—
348C23—24no—
349C23—24no—
350C23—24no—
351C23—24no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—42no—
357C23—42no—
358C23—42no—
359C23—42no—
360C23—42no—
361C23—42no—
362C23#39714yes—
363C23C2×#12111yescharacteristic
364C23C2×#12312yes—
365C23#53211yescharacteristic
366C23#53211yescharacteristic
367C23#53712yes—
368C23#57612yes—
369C23#57712yes—
370C23#57712yes—
371C23#57812yes—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—28no—
380C2×C4—28no—
381C2×C4—24no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
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×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4#39714yes—
425C2×C4C2×#12311yescharacteristic
426C2×C4C2×#12311yescharacteristic
427C2×C4#52511yescharacteristic
428C2×C4C2×#12411yescharacteristic
429C2×C4C2×#12612yes—
430C2×C4#52911yescharacteristic
431C2×C4#52911yescharacteristic
432C2×C4#53411yescharacteristic
433C2×C4#53611yescharacteristic
434C2×C4#53912yes—
435C2×C4#55112yes—
436C2×C4#55112yes—
437C2×C4#55112yes—
438C2×C4#55112yes—
439C2×C4#55212yes—
440C2×C4#55212yes—
441C2×C4#55212yes—
442C2×C4#55212yes—
443C8—24no—
444C8—42no—
445C8—42no—
446C8—42no—
447C22—28no—
448C22—24no—
449C22—44no—
450C22—42no—
451C22—42no—
452C22—44no—
453C22—44no—
454C22—44no—
455C22—44no—
456C22C2×#39711yescharacteristic
457C22#249812yes—
458C22#249812yes—
459C22#261811yescharacteristic
460C22#285411yescharacteristic
461C22#306212yes—
462C22#306312yes—
463C22#309612yes—
464C22#309712yes—
465C22#309812yes—
466C22#309912yes—
467C22#317311yescharacteristic
468C22#317311yescharacteristic
469C22#317611yescharacteristic
470C22#317711yescharacteristic
471C22#317812yes—
472C22#317912yes—
473C22#325712yes—
474C22#325812yes—
475C22#326012yes—
476C22#326212yes—
477C4—24no—
478C4—82no—
479C4—82no—
480C4—162no—
481C4—162no—
482C4—162no—
483C4—162no—
484C4#248311yescharacteristic
485C4#248411yescharacteristic
486C4#249312yes—
487C4#249312yes—
488C4#250311yescharacteristic
489C4#256111yescharacteristic
490C2—44no—
491C2#1501911yescharacteristic
492C2#2364312yes—
493C2#2786812yes—
494C2#2786912yes—
495C2#2787212yes—
496C2#2787312yes—
497C2#2978712yes—
498C2#3314011yescharacteristic
499C2#3314111yescharacteristic
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. #11379C2semidirect
5. #11377C2semidirect
6. #11316C2semidirect
7. #11315C2semidirect
8. #11257C2semidirect
9. #11257C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18092
C2×#17292
#143432
#113792
#113772
#113162
#113152
#112572
#107222
#107212