Finite Group Databasebeta

← #470097

Subgroups — #470097

Order 512.

1040 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2076C211yesmaximal, characteristic
2C2×#1818C211yesmaximal, characteristic
3#14264C212yesmaximal
4#11328C212yesmaximal, complemented
5#11316C212yesmaximal, complemented
6#11292C212yesmaximal, complemented
7#11220C212yesmaximal, complemented
8#11199C212yesmaximal
9#11095C211yesmaximal, characteristic
10C22×#117C2211yescharacteristic
11C22×#107C2211yescharacteristic
12C22×#106C2211yescharacteristic
13C22×C22⋊C8C2211yescharacteristic
14C22×C22⋊C8C2211yescharacteristic
15C2×#389C2212yes—
16C2×#389C2212yes—
17C2×#380C2212yes—
18C2×#380C2212yes—
19#2076—24no—
20#1818—24no—
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2211yescharacteristic
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2211yescharacteristic
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#336C2212yes—
30C2×#336C2212yes—
31#1444—22no—
32#1442—22no—
33#1439—22no—
34#1439—22no—
35C4⋊#95C2212yes—
36C4⋊#95C2212yes—
37C24×C4C2311yescharacteristic
38C23×C8C2312yes—
39C23×C8C2311yescharacteristic
40C23×C8C2311yescharacteristic
41C2×#117—22no—
42C2×#117—22no—
43C2×#117—41no—
44C2×#117—41no—
45C2×#117—41no—
46C2×#117—41no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4D411yescharacteristic
50C22×C4⋊C4D411yescharacteristic
51C22×C4⋊C4C2312yes—
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2312yes—
55C22×C22⋊C4—22no—
56C22×C22⋊C4—22no—
57C22×C22⋊C4D412yes—
58C22×C42C2312yes—
59C2×#107—22no—
60C2×#107—41no—
61C2×#107—41no—
62C2×#106—24no—
63C2×#106—41no—
64C2×C22⋊C8—22no—
65C2×C22⋊C8—41no—
66C2×C22⋊C8—41no—
67C2×C22⋊C8—41no—
68C2×C22⋊C8—41no—
69C2×C22⋊C8D414yes—
70#389—42no—
71#389—42no—
72#389—42no—
73#389—42no—
74#380—24no—
75#380—24no—
76#380—24no—
77#380—24no—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86#340—22no—
87#340—24no—
88#340—24no—
89#340—24no—
90#340—24no—
91#340—41no—
92#340—42no—
93#340—42no—
94#336—24no—
95#336—24no—
96C25C2×D411yescharacteristic
97C23×C4—22no—
98C23×C4—22no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—22no—
104C23×C4—22no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4C2×D412yes—
110C23×C4C2×D412yes—
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D412yes—
113C23×C4#4112yes—
114C23×C4C2411yescharacteristic
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—21no—
118C22×C8—21no—
119C22×C8—21no—
120C22×C8—21no—
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—
130C22×C8C2×D412yes—
131C22×C8C2×D412yes—
132#117—44no—
133#117—44no—
134#117—44no—
135#117—44no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
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—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4—82no—
167C2×C4⋊C4—82no—
168C2×C4⋊C4—82no—
169C2×C4⋊C4—82no—
170C2×C4⋊C4#4114yes—
171C2×C4⋊C4#4114yes—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—28no—
174C2×C22⋊C4—42no—
175C2×C22⋊C4—42no—
176C2×C22⋊C4—44no—
177C2×C22⋊C4—82no—
178C2×C22⋊C4—82no—
179C2×C22⋊C4—82no—
180C2×C22⋊C4—82no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187#107—44no—
188#106—44no—
189C22⋊C8—44no—
190C22⋊C8—44no—
191#95—28no—
192#95—44no—
193#95—44no—
194#95—44no—
195#95—82no—
196#95—82no—
197#95—82no—
198#95—82no—
199C24—22no—
200C24—21no—
201C24—21no—
202C24—22no—
203C24—22no—
204C24—22no—
205C24—22no—
206C24—22no—
207C24#12112yes—
208C24C22×D411yescharacteristic
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—24no—
224C22×C4—24no—
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—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—41no—
242C22×C4—42no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—42no—
246C22×C4—41no—
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#12112yes—
274C22×C4C22×D411yescharacteristic
275C22×C4C2×#4111yescharacteristic
276C22×C4C2×#4112yes—
277C22×C4C2×#4112yes—
278C22×C4H(5,𝔽2)11yescharacteristic
279C22×C4H(5,𝔽2)11yescharacteristic
280C22×C4H(5,𝔽2)11yescharacteristic
281C22×C4H(5,𝔽2)11yescharacteristic
282C22×C42−1+412yes—
283C22×C42−1+412yes—
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—24no—
287C2×C8—24no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—44no—
306C4⋊C4—44no—
307C4⋊C4—42no—
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—42no—
319C4⋊C4—88no—
320C4⋊C4—88no—
321C22⋊C4—48no—
322C22⋊C4—88no—
323C22⋊C4—88no—
324C42—42no—
325C42—42no—
326C42—42no—
327C42—42no—
328C23—22no—
329C23—24no—
330C23—22no—
331C23—22no—
332C23—24no—
333C23—41no—
334C23—41no—
335C23—42no—
336C23—42no—
337C23—42no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23—41no—
342C23—42no—
343C23—42no—
344C23—41no—
345C23—42no—
346C23—42no—
347C23—42no—
348C23—42no—
349C23C2×#12111yescharacteristic
350C23#52512yes—
351C23#53411yescharacteristic
352C23#53411yescharacteristic
353C23#53612yes—
354C23#57612yes—
355C23#57712yes—
356C23#57712yes—
357C23#57812yes—
358C2×C4—22no—
359C2×C4—24no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—44no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—44no—
368C2×C4—42no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—44no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
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#53411yescharacteristic
412C2×C4#53511yescharacteristic
413C2×C4#53811yescharacteristic
414C2×C4#53811yescharacteristic
415C2×C4#53912yes—
416C2×C4#53912yes—
417C2×C4#54212yes—
418C2×C4#54212yes—
419C2×C4#54712yes—
420C2×C4#54712yes—
421C2×C4#55111yescharacteristic
422C2×C4#55212yes—
423C2×C4#55212yes—
424C2×C4#55312yes—
425C2×C4#55911yescharacteristic
426C2×C4#56311yescharacteristic
427C2×C4#56312yes—
428C2×C4#56311yescharacteristic
429C8—42no—
430C8—42no—
431C8—82no—
432C22—42no—
433C22—44no—
434C22—44no—
435C22—42no—
436C22—42no—
437C22—82no—
438C22—82no—
439C22—82no—
440C22—82no—
441C22#241312yes—
442C22#260011yescharacteristic
443C22#261711yescharacteristic
444C22#263812yes—
445C22#281711yescharacteristic
446C22#305911yescharacteristic
447C22#306011yescharacteristic
448C22#306212yes—
449C22#311812yes—
450C22#312012yes—
451C22#312412yes—
452C22#312612yes—
453C22#319812yes—
454C22#319911yescharacteristic
455C22#320112yes—
456C22#320112yes—
457C22#320411yescharacteristic
458C22#324112yes—
459C22#324212yes—
460C22#324312yes—
461C22#324412yes—
462C4—42no—
463C4—82no—
464C4—82no—
465C4—84no—
466C4—84no—
467C4—162no—
468C4—162no—
469C4#271711yescharacteristic
470C4#271812yes—
471C4#272411yescharacteristic
472C4#276411yescharacteristic
473C4#276512yes—
474C4#277111yescharacteristic
475C2—82no—
476C2#1667511yescharacteristic
477C2#2461412yes—
478C2#2461712yes—
479C2#2942012yes—
480C2#2973711yescharacteristic
481C2#2974511yescharacteristic
482C2#3017712yes—
483C2#3018212yes—
484C2#3279312yes—
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. #11328C2semidirect
5. #11316C2semidirect
6. #11292C2semidirect
7. #11220C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20762
C2×#18182
#142642
#113282
#113162
#112922
#112202
#111992
#110952