Finite Group Databasebeta

← #409381

Subgroups — #409381

Order 512.

1186 conjugacy classes of subgroups.

The subgroup lattice has 485 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#11378C212yesmaximal, complemented
5#11316C212yesmaximal, complemented
6#11315C212yesmaximal, complemented
7#11256C212yesmaximal, complemented
8#10722C211yesmaximal, characteristic
9#10721C212yesmaximal
10C23×M16C2211yescharacteristic
11C24×C8C2211yescharacteristic
12C22×#117C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×#106C2211yescharacteristic
15C2×#380C2212yes—
16C2×#380C2212yes—
17#1809—24no—
18C2×#340C2212yes—
19C2×#340C2212yes—
20C2×#340C2212yes—
21C2×#340C2212yes—
22C2×#340C2211yescharacteristic
23C2×#340C2212yes—
24C2×#340C2211yescharacteristic
25C2×#340C2212yes—
26#1729—24no—
27C2×#336C2212yes—
28C2×#336C2212yes—
29#1445—22no—
30#1445—22no—
31#1440—22no—
32#1440—22no—
33C4⋊#95C2212yes—
34C4⋊#95C2212yes—
35C4⋊#95C2212yes—
36C4⋊#95C2212yes—
37C24×C4C2311yescharacteristic
38C22×M16—24no—
39C22×M16—24no—
40C22×M16—41no—
41C22×M16—41no—
42C22×M16D414yes—
43C23×C8—22no—
44C23×C8—24no—
45C23×C8—24no—
46C23×C8—41no—
47C23×C8—41no—
48C23×C8C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2311yescharacteristic
51C2×#117—22no—
52C2×#117—22no—
53C2×#117—24no—
54C2×#117—24no—
55C22×C4⋊C4D412yes—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2312yes—
60C22×C22⋊C4—22no—
61C22×C22⋊C4—22no—
62C22×C22⋊C4D411yescharacteristic
63C22×C22⋊C4D411yescharacteristic
64C22×C42C2312yes—
65C2×#107—22no—
66C2×#107—41no—
67C2×#107—41no—
68C2×#106—22no—
69C2×#106—24no—
70#380—24no—
71#380—24no—
72#380—24no—
73#380—24no—
74C2×#95—22no—
75C2×#95—22no—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—22no—
84#340—24no—
85#340—24no—
86#340—24no—
87#340—24no—
88#340—41no—
89#340—41no—
90#340—42no—
91#340—42no—
92#336—24no—
93#336—24no—
94C25C2×D411yescharacteristic
95C23×C4—22no—
96C23×C4—22no—
97C23×C4—21no—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—22no—
102C23×C4—22no—
103C23×C4—22no—
104C23×C4—22no—
105C23×C4C2×D412yes—
106C23×C4C2×D412yes—
107C23×C4C2×D411yescharacteristic
108C23×C4C2×D412yes—
109C23×C4#4112yes—
110C23×C4#4114yes—
111C23×C4#4114yes—
112C23×C4C2411yescharacteristic
113C2×M16—28no—
114C2×M16—28no—
115C2×M16—44no—
116C2×M16—48no—
117C2×M16—48no—
118C22×C8—22no—
119C22×C8—21no—
120C22×C8—21no—
121C22×C8—28no—
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×C8—21no—
131C22×C8—21no—
132C22×C8—48no—
133C22×C8—48no—
134C22×C8—44no—
135C22×C8—44no—
136C22×C8C2×D412yes—
137C22×C8C2×D412yes—
138C22×C8#4114yes—
139C22×C8#4114yes—
140#117—44no—
141#117—44no—
142#117—48no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—24no—
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—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—44no—
175C2×C22⋊C4—44no—
176C2×C22⋊C4—82no—
177C2×C22⋊C4—82no—
178C2×C22⋊C4—82no—
179C2×C22⋊C4—82no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186#107—44no—
187#106—44no—
188#95—44no—
189#95—44no—
190#95—82no—
191#95—82no—
192#95—82no—
193#95—82no—
194#95—82no—
195#95—82no—
196#95—82no—
197#95—82no—
198C24—22no—
199C24—21no—
200C24—21no—
201C24—22no—
202C24—22no—
203C24—22no—
204C24—22no—
205C24#12112yes—
206C24#12314yes—
207C24C22×D411yescharacteristic
208C22×C4—22no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—22no—
212C22×C4—22no—
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—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—42no—
241C22×C4—42no—
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#12112yes—
269C22×C4#12314yes—
270C22×C4#12314yes—
271C22×C4#12414yes—
272C22×C4#12618yes—
273C22×C4C22×D411yescharacteristic
274C22×C4C2×#4112yes—
275C22×C4C2×#4112yes—
276C22×C4C2×#4112yes—
277C22×C4C2×#4112yes—
278C22×C4C2×#4111yescharacteristic
279C22×C4H(5,𝔽2)11yescharacteristic
280C22×C4H(5,𝔽2)11yescharacteristic
281C22×C42−1+411yescharacteristic
282C22×C42−1+411yescharacteristic
283M16—416no—
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—24no—
287C2×C8—24no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—24no—
291C2×C8—22no—
292C2×C8—24no—
293C2×C8—24no—
294C2×C8—416no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8#12314yes—
302C4⋊C4—48no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
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—
319C22⋊C4—44no—
320C22⋊C4—44no—
321C22⋊C4—88no—
322C22⋊C4—88no—
323C42—42no—
324C42—42no—
325C42—42no—
326C42—42no—
327C23—24no—
328C23—24no—
329C23—24no—
330C23—24no—
331C23—22no—
332C23—22no—
333C23—24no—
334C23—24no—
335C23—24no—
336C23—24no—
337C23—42no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23#39714yes—
348C23C2×#12111yescharacteristic
349C23C2×#12312yes—
350C23#53212yes—
351C23#53711yescharacteristic
352C23#53711yescharacteristic
353C23#57612yes—
354C23#57712yes—
355C23#57712yes—
356C23#57812yes—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—28no—
365C2×C4—28no—
366C2×C4—24no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
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#39714yes—
410C2×C4C2×#12311yescharacteristic
411C2×C4C2×#12311yescharacteristic
412C2×C4#52511yescharacteristic
413C2×C4C2×#12411yescharacteristic
414C2×C4C2×#12612yes—
415C2×C4#52912yes—
416C2×C4#53411yescharacteristic
417C2×C4#53611yescharacteristic
418C2×C4#53911yescharacteristic
419C2×C4#53911yescharacteristic
420C2×C4#55112yes—
421C2×C4#55112yes—
422C2×C4#55112yes—
423C2×C4#55112yes—
424C2×C4#55212yes—
425C2×C4#55212yes—
426C2×C4#55212yes—
427C2×C4#55212yes—
428C8—24no—
429C8—44no—
430C8—42no—
431C22—28no—
432C22—24no—
433C22—42no—
434C22—42no—
435C22—44no—
436C22—44no—
437C22—44no—
438C22—44no—
439C22—44no—
440C22C2×#39711yescharacteristic
441C22#249812yes—
442C22#249812yes—
443C22#261811yescharacteristic
444C22#285411yescharacteristic
445C22#306212yes—
446C22#306312yes—
447C22#309612yes—
448C22#309712yes—
449C22#309812yes—
450C22#309912yes—
451C22#317412yes—
452C22#317512yes—
453C22#317812yes—
454C22#317912yes—
455C22#325712yes—
456C22#325812yes—
457C22#325911yescharacteristic
458C22#326111yescharacteristic
459C22#326311yescharacteristic
460C22#326311yescharacteristic
461C4—24no—
462C4—82no—
463C4—82no—
464C4—162no—
465C4—162no—
466C4—162no—
467C4—162no—
468C4#248311yescharacteristic
469C4#248411yescharacteristic
470C4#249312yes—
471C4#249312yes—
472C4#250311yescharacteristic
473C4#256111yescharacteristic
474C2—44no—
475C2#1501911yescharacteristic
476C2#2364312yes—
477C2#2786712yes—
478C2#2787012yes—
479C2#2787112yes—
480C2#2787412yes—
481C2#2978511yescharacteristic
482C2#2978611yescharacteristic
483C2#3314212yes—
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. #11378C2semidirect
5. #11316C2semidirect
6. #11315C2semidirect
7. #11256C2semidirect

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
#113782
#113162
#113152
#112562
#107222
#107212