Finite Group Databasebeta

← #416985

Subgroups — #416985

Order 512.

1012 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1992C211yesmaximal, characteristic
2C2×#1991C211yesmaximal, characteristic
3C2×#1834C211yesmaximal, characteristic
4C2×#1768C211yesmaximal, characteristic
5#14283C211yesmaximal, characteristic
6#14276C211yesmaximal, characteristic
7#11316C212yesmaximal
8#11315C212yesmaximal
9#11228C212yesmaximal
10#11227C212yesmaximal
11#10843C211yesmaximal, characteristic
12C22×#125C2211yescharacteristic
13C22×C4⋊C8C2212yes—
14C22×#97C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16C2×#389C2212yes—
17C2×#384C2211yescharacteristic
18C2×#384C2211yescharacteristic
19C2×#382C2211yescharacteristic
20C2×#376C2211yescharacteristic
21#1992—24no—
22#1991—24no—
23#1834—24no—
24#1768—24no—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#336C2212yes—
34#1441—22no—
35#1441—22no—
36#1441—22no—
37#1441—22no—
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2212yes—
40C4⋊#95C2211yescharacteristic
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2211yescharacteristic
43C23×C8C2312yes—
44C23×C8C2312yes—
45C2×#125—22no—
46C2×#125—22no—
47C2×#125—22no—
48C2×#125—22no—
49C2×#125—22no—
50C2×#125Q814yes—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C42C2311yescharacteristic
65C22×C42C2311yescharacteristic
66C22×C42C2311yescharacteristic
67C2×C4⋊C8—24no—
68C2×C4⋊C8—24no—
69C2×C4⋊C8—24no—
70C2×#97—22no—
71C2×#97—22no—
72C2×#97—22no—
73C2×C4×C8—22no—
74C2×C4×C8—22no—
75C2×C4×C8D414yes—
76#389—24no—
77#389—24no—
78#384—22no—
79#384—22no—
80#384—22no—
81#384D414yes—
82#382—22no—
83#382—22no—
84#376—22no—
85#376—22no—
86C2×#95—21no—
87C2×#95—21no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95D412yes—
96C2×#95D412yes—
97#340—24no—
98#340—24no—
99#340—24no—
100#340—24no—
101#340—24no—
102#340—24no—
103#340—24no—
104#340—24no—
105#336—24no—
106C23×C4—22no—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4—22no—
112C23×C4C2×D412yes—
113C23×C4C2×D412yes—
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C23×C4C2411yescharacteristic
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×C8C2×D414yes—
130C22×C8#4114yes—
131#125—28no—
132#125—28no—
133#125—44no—
134#125—44no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—82no—
176C2×C4⋊C4—82no—
177C2×C4⋊C4—82no—
178C2×C4⋊C4—82no—
179C2×C4⋊C4—82no—
180C2×C4⋊C4—82no—
181C2×C4⋊C4C2×D414yes—
182C2×C4⋊C4C2×Q812yes—
183C2×C4⋊C4C2×Q812yes—
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—22no—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42C2×D412yes—
192C2×C42C2×D412yes—
193C2×C42C2×Q812yes—
194C2×C42#4114yes—
195C2×C42#4112yes—
196C2×C42#4112yes—
197C2×C42#4112yes—
198C4⋊C8—216no—
199#97—44no—
200C4×C8—28no—
201#95—24no—
202#95—24no—
203#95—42no—
204#95—42no—
205#95—44no—
206#95—82no—
207#95—82no—
208#95—82no—
209#95—82no—
210#95—82no—
211#95—82no—
212C24C22×D411yescharacteristic
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
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—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
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—41no—
249C22×C4—41no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
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#12112yes—
265C22×C4#12112yes—
266C22×C4#12112yes—
267C22×C4#12112yes—
268C22×C4C22⋊Q812yes—
269C22×C4C22⋊Q812yes—
270C22×C4#12414yes—
271C22×C4C22×D411yescharacteristic
272C22×C4C22×D411yescharacteristic
273C22×C4C22×Q811yescharacteristic
274C22×C4C2×#4111yescharacteristic
275C22×C4C2×#4111yescharacteristic
276C22×C4C2×#4111yescharacteristic
277C22×C4C2×#4112yes—
278C22×C4C2×#4112yes—
279C22×C42−1+412yes—
280C22×C42−1+412yes—
281C2×C8—24no—
282C2×C8—24no—
283C2×C8—24no—
284C2×C8—24no—
285C2×C8—24no—
286C2×C8—24no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—48no—
308C4⋊C4—48no—
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—84no—
318C4⋊C4—84no—
319C4⋊C4—84no—
320C4⋊C4—84no—
321C4⋊C4—88no—
322C42—24no—
323C42—24no—
324C42—42no—
325C42—42no—
326C42#12114yes—
327C42C22⋊Q814yes—
328C42C22⋊Q814yes—
329C42#12614yes—
330C23C2×#12111yescharacteristic
331C23C2×#12111yescharacteristic
332C23C2×C22⋊Q811yescharacteristic
333C23C2×#12411yescharacteristic
334C23#53211yescharacteristic
335C23#53611yescharacteristic
336C23#53611yescharacteristic
337C23#57612yes—
338C23#57712yes—
339C23#57712yes—
340C23#57812yes—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
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—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4#39614yes—
398C2×C4#39614yes—
399C2×C4C2×#12111yescharacteristic
400C2×C4C2×C22⋊Q811yescharacteristic
401C2×C4C2×C22⋊Q811yescharacteristic
402C2×C4#52512yes—
403C2×C4C2×#12611yescharacteristic
404C2×C4#52912yes—
405C2×C4#53612yes—
406C2×C4#54112yes—
407C2×C4#54712yes—
408C2×C4#54812yes—
409C2×C4#55212yes—
410C2×C4#55612yes—
411C2×C4#55612yes—
412C2×C4#55612yes—
413C2×C4#55712yes—
414C2×C4#55712yes—
415C8—44no—
416C8—44no—
417C22C2×#39611yescharacteristic
418C22#254511yescharacteristic
419C22#254711yescharacteristic
420C22#256511yescharacteristic
421C22#256811yescharacteristic
422C22#262111yescharacteristic
423C22#285011yescharacteristic
424C22#304111yescharacteristic
425C22#304211yescharacteristic
426C22#304311yescharacteristic
427C22#304411yescharacteristic
428C22#306012yes—
429C22#306112yes—
430C22#308312yes—
431C22#308412yes—
432C22#313612yes—
433C22#313712yes—
434C22#317812yes—
435C22#317912yes—
436C22#324112yes—
437C22#324112yes—
438C22#324212yes—
439C22#324212yes—
440C4—24no—
441C4—24no—
442C4—44no—
443C4—84no—
444C4—82no—
445C4—82no—
446C4—82no—
447C4—162no—
448C4—161no—
449C4—161no—
450C4#241512yes—
451C4#245312yes—
452C4#251412yes—
453C4#258112yes—
454C2#1523711yescharacteristic
455C2#2361011yescharacteristic
456C2#2361111yescharacteristic
457C2#2866611yescharacteristic
458C2#2866711yescharacteristic
459C2#2872811yescharacteristic
460C2#2872911yescharacteristic
461C2#2918812yes—
462C2#2922812yes—
463C2#2988212yes—
464C2#3304712yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#19922
C2×#19912
C2×#18342
C2×#17682
#142832
#142762
#113162
#113152
#112282
#112272
#108432