Finite Group Databasebeta

← #410708

Subgroups — #410708

Order 512.

936 conjugacy classes of subgroups.

The subgroup lattice has 477 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1834C211yesmaximal, characteristic
2C2×#1771C211yesmaximal, characteristic
3#15196C211yesmaximal, characteristic
4#15152C211yesmaximal, characteristic
5#11371C211yesmaximal, characteristic
6#11357C211yesmaximal, characteristic
7#11315C212yesmaximal
8#11227C212yesmaximal
9#10881C212yesmaximal
10#10821C212yesmaximal
11#10820C211yesmaximal, characteristic
12C22×C4⋊C8C2211yescharacteristic
13C22×#97C2211yescharacteristic
14C22×#97C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16C2×#400—21no—
17C2×#389C2212yes—
18C2×#384C2211yescharacteristic
19C2×#384C2211yescharacteristic
20C2×C4×C4⋊C4C2211yescharacteristic
21#1834—24no—
22#1771—24no—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#336C2211yescharacteristic
32C2×#336C2211yescharacteristic
33#1445—21no—
34#1444—21no—
35#1444—21no—
36#1442—21no—
37#1441—21no—
38#1441—21no—
39#1441—21no—
40C4⋊#95C2212yes—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2212yes—
43C4⋊#95C2212yes—
44C4⋊#95C2211yescharacteristic
45C23×C8C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4D411yescharacteristic
55C22×C4⋊C4D411yescharacteristic
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2312yes—
60C22×C42C2311yescharacteristic
61C22×C42C2311yescharacteristic
62C22×C42C2311yescharacteristic
63C2×C4⋊C8—22no—
64C2×C4⋊C8—22no—
65C2×C4⋊C8—22no—
66C2×#97—22no—
67C2×#97—22no—
68C2×#97—22no—
69C2×#97—22no—
70C2×#97—24no—
71C2×C4×C8—24no—
72C2×C4×C8D414yes—
73#400—44no—
74#389—24no—
75#389—24no—
76#384—22no—
77#384—22no—
78#384—22no—
79#384—22no—
80C4×C4⋊C4—22no—
81C4×C4⋊C4—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—21no—
85C2×#95—21no—
86C2×#95—21no—
87C2×#95—21no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95D411yescharacteristic
94C2×#95D411yescharacteristic
95#340—24no—
96#340—24no—
97#340—24no—
98#340—24no—
99#340—24no—
100#340—24no—
101#340—24no—
102#340—24no—
103#336—22no—
104#336—22no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C23×C4C2411yescharacteristic
116C22×C8—21no—
117C22×C8—21no—
118C22×C8—22no—
119C22×C8—21no—
120C22×C8—21no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—21no—
126C22×C8—22no—
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—21no—
130C22×C8—22no—
131C22×C8C2×D412yes—
132C22×C8C2×D412yes—
133C22×C8#4112yes—
134C22×C8#4112yes—
135C22×C8#4114yes—
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—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—82no—
173C2×C4⋊C4—82no—
174C2×C4⋊C4—82no—
175C2×C4⋊C4—82no—
176C2×C4⋊C4—82no—
177C2×C4⋊C4#4114yes—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—22no—
181C2×C42—21no—
182C2×C42—21no—
183C2×C42—22no—
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42C2×D412yes—
189C2×C42#4112yes—
190C2×C42#4112yes—
191C2×C42#4112yes—
192C4⋊C8—28no—
193#97—28no—
194#97—44no—
195C4×C8—28no—
196#95—22no—
197#95—22no—
198#95—44no—
199#95—44no—
200#95—44no—
201#95—42no—
202#95—42no—
203#95—42no—
204#95—42no—
205#95—82no—
206#95—82no—
207#95—82no—
208#95—82no—
209C24C22×D411yescharacteristic
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—24no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—42no—
236C22×C4—42no—
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—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#12112yes—
263C22×C4#12112yes—
264C22×C4#12412yes—
265C22×C4#12412yes—
266C22×C4AGL(1,ℤ/8)12yes—
267C22×C4#13712yes—
268C22×C4#13712yes—
269C22×C4#13712yes—
270C22×C4C22×D411yescharacteristic
271C22×C4C2×#4112yes—
272C22×C4C2×#4111yescharacteristic
273C22×C4C2×#4111yescharacteristic
274C22×C4C2×#4111yescharacteristic
275C22×C4C2×#4111yescharacteristic
276C22×C4C2×#4112yes—
277C22×C4C2×#4111yescharacteristic
278C22×C42−1+412yes—
279C22×C42−1+412yes—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—24no—
283C2×C8—24no—
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—24no—
289C2×C8—22no—
290C2×C8—24no—
291C2×C8—41no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—42no—
299C2×C8#12314yes—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—44no—
305C4⋊C4—44no—
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—84no—
319C4⋊C4—84no—
320C4⋊C4—84no—
321C4⋊C4—88no—
322C4⋊C4—84no—
323C42—22no—
324C42—22no—
325C42—22no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—24no—
330C42—41no—
331C42—41no—
332C42—42no—
333C23C2×#12111yescharacteristic
334C23C2×#12411yescharacteristic
335C23#53211yescharacteristic
336C23#54712yes—
337C23#54812yes—
338C23C2×AGL(1,ℤ/8)11yescharacteristic
339C23C2×#13711yescharacteristic
340C23C2×#13711yescharacteristic
341C23C2×#13711yescharacteristic
342C23#57612yes—
343C23#57712yes—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—24no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—44no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—82no—
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—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#46814yes—
403C2×C4#47014yes—
404C2×C4C2×#12311yescharacteristic
405C2×C4#52511yescharacteristic
406C2×C4#52511yescharacteristic
407C2×C4#52912yes—
408C2×C4#52912yes—
409C2×C4#52911yescharacteristic
410C2×C4#52911yescharacteristic
411C2×C4#53211yescharacteristic
412C2×C4#53611yescharacteristic
413C2×C4#53611yescharacteristic
414C2×C4#54112yes—
415C2×C4#54112yes—
416C2×C4#55212yes—
417C2×C4#55212yes—
418C2×C4#55212yes—
419C2×C4#55212yes—
420C2×C4#55212yes—
421C2×C4#55612yes—
422C8—24no—
423C8—42no—
424C8—42no—
425C8—42no—
426C22#248611yescharacteristic
427C22#248912yes—
428C22#251912yes—
429C22#254511yescharacteristic
430C22#285011yescharacteristic
431C22C2×#46811yescharacteristic
432C22C2×#47011yescharacteristic
433C22#305711yescharacteristic
434C22#305811yescharacteristic
435C22#313612yes—
436C22#313911yescharacteristic
437C22#314111yescharacteristic
438C22#317912yes—
439C22#318211yescharacteristic
440C22#318511yescharacteristic
441C22#334412yes—
442C22#334512yes—
443C22#334612yes—
444C22#334612yes—
445C22#336612yes—
446C22#336912yes—
447C22#337212yes—
448C22#337512yes—
449C4—24no—
450C4—24no—
451C4—42no—
452C4—42no—
453C4—81no—
454C4—81no—
455C4—84no—
456C4—162no—
457C4—162no—
458C4—162no—
459C4#239511yescharacteristic
460C4#248411yescharacteristic
461C4#252412yes—
462C4#252812yes—
463C4#261111yescharacteristic
464C4#263411yescharacteristic
465C2#1508511yescharacteristic
466C2#2766311yescharacteristic
467C2#2766611yescharacteristic
468C2#2773912yes—
469C2#2774612yes—
470C2#2804612yes—
471C2#2804812yes—
472C2#2864511yescharacteristic
473C2#2864711yescharacteristic
474C2#3304511yescharacteristic
475C2#3304911yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ 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×#18342
C2×#17712
#151962
#151522
#113712
#113572
#113152
#112272
#108812
#108212
#108202