Finite Group Databasebeta

← #401493

Subgroups — #401493

Order 512.

1036 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1949C212yesmaximal
2C2×#1948C211yesmaximal, characteristic
3C2×#1921C211yesmaximal, characteristic
4#15242C211yesmaximal, characteristic
5#13818C211yesmaximal, characteristic
6#11373C211yesmaximal, characteristic
7#11322C211yesmaximal, characteristic
8#11228C211yesmaximal, characteristic
9#11227C212yesmaximal
10#11208C212yesmaximal
11#10908C212yesmaximal
12C22×#107C2211yescharacteristic
13C22×#106C2212yes—
14C22×C4⋊C8C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C2×#389C2211yescharacteristic
17C2×#389C2212yes—
18C2×#384C2211yescharacteristic
19C2×#382C2212yes—
20C2×C4×C4⋊C4C2212yes—
21C2×C4×C4⋊C4C2211yescharacteristic
22#1949—28no—
23#1948—24no—
24#1921—24no—
25C2×#340C2211yescharacteristic
26C2×#340C2211yescharacteristic
27C2×#340C2211yescharacteristic
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#336C2211yescharacteristic
34C2×#336C2212yes—
35#1445—21no—
36#1445—21no—
37#1441—22no—
38#1436—22no—
39C4⋊#95—21no—
40C4⋊#95C2211yescharacteristic
41C4⋊#95C2212yes—
42C4×#95—21no—
43C4×#95C2211yescharacteristic
44C23×C8C2311yescharacteristic
45C23×C8C2311yescharacteristic
46C23×C8C2312yes—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4C2312yes—
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C42—21no—
58C22×C42—22no—
59C22×C42C2311yescharacteristic
60C22×C42C2312yes—
61C2×#107—22no—
62C2×#107—22no—
63C2×#107D414yes—
64C2×#106—24no—
65C2×#106—24no—
66C2×#106—24no—
67C2×C4⋊C8—24no—
68C2×C4⋊C8—24no—
69C2×C4⋊C8—24no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—24no—
72#389—24no—
73#389—22no—
74#389—22no—
75#389Q818yes—
76#384—22no—
77#384—22no—
78#382—24no—
79#382—42no—
80C4×C4⋊C4—22no—
81C4×C4⋊C4—24no—
82C4×C4⋊C4—42no—
83C4×C4⋊C4D414yes—
84C2×#95—22no—
85C2×#95—21no—
86C2×#95—21no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—21no—
91C2×#95—21no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95D411yescharacteristic
95C2×#95D411yescharacteristic
96C2×#95D412yes—
97#340—24no—
98#340—24no—
99#340—22no—
100#340—22no—
101#340—22no—
102#340—24no—
103#340—24no—
104#340—42no—
105#336—22no—
106#336—24no—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—21no—
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D412yes—
115C23×C4C2×D412yes—
116C23×C4C2×D411yescharacteristic
117C23×C4C2411yescharacteristic
118C22×C8—21no—
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—21no—
128C22×C8—21no—
129C22×C8—22no—
130C22×C8—21no—
131C22×C8—22no—
132C22×C8—21no—
133C22×C8C2×D412yes—
134C22×C8#4114yes—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—28no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—82no—
168C2×C4⋊C4—82no—
169C2×C4⋊C4—81no—
170C2×C4⋊C4—81no—
171C2×C4⋊C4C2×D412yes—
172C2×C4⋊C4C2×D412yes—
173C2×C4⋊C4C2×D412yes—
174C2×C4⋊C4C2×D412yes—
175C2×C4⋊C4C2×Q814yes—
176C2×C4⋊C4C2×Q814yes—
177C2×C4⋊C4#4112yes—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—24no—
185C2×C42—44no—
186C2×C42—44no—
187C2×C42—82no—
188C2×C42—82no—
189C2×C42C2×D412yes—
190C2×C42C2×Q814yes—
191C2×C42#4112yes—
192C2×C42#4114yes—
193#107—28no—
194#106—48no—
195C4⋊C8—28no—
196C4⋊C8—216no—
197#95—24no—
198#95—28no—
199#95—22no—
200#95—22no—
201#95—42no—
202#95—44no—
203#95—44no—
204#95—44no—
205#95—42no—
206#95—42no—
207#95—81no—
208#95—82no—
209#95—82no—
210C24C22×D411yescharacteristic
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—24no—
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—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—21no—
234C22×C4—24no—
235C22×C4—22no—
236C22×C4—42no—
237C22×C4—41no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—41no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—41no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—41no—
260C22×C4—42no—
261C22×C4—41no—
262C22×C4—42no—
263C22×C4#12112yes—
264C22×C4#12112yes—
265C22×C4C22⋊Q814yes—
266C22×C4C22⋊Q814yes—
267C22×C4#13512yes—
268C22×C4#13712yes—
269C22×C4C22×D411yescharacteristic
270C22×C4C22×D411yescharacteristic
271C22×C4C22×Q812yes—
272C22×C4C2×#4111yescharacteristic
273C22×C4C2×#4112yes—
274C22×C4C2×#4112yes—
275C22×C4C2×#4111yescharacteristic
276C22×C4H(5,𝔽2)11yescharacteristic
277C22×C42−1+412yes—
278C22×C42−1+411yescharacteristic
279C2×C8—22no—
280C2×C8—24no—
281C2×C8—24no—
282C2×C8—24no—
283C2×C8—24no—
284C2×C8—42no—
285C2×C8—41no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—41no—
290C2×C8—42no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—42no—
295C2×C8—42no—
296C4⋊C4—22no—
297C4⋊C4—24no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—42no—
306C4⋊C4—41no—
307C4⋊C4—42no—
308C4⋊C4—41no—
309C4⋊C4—42no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—88no—
317C4⋊C4—84no—
318C4⋊C4—84no—
319C4⋊C4—84no—
320C4⋊C4#12114yes—
321C4⋊C4C4⋊Q814yes—
322C42—22no—
323C42—24no—
324C42—22no—
325C42—24no—
326C42—24no—
327C42—24no—
328C42—24no—
329C42—84no—
330C42—88no—
331C23C2×#12111yescharacteristic
332C23C2×C22⋊Q812yes—
333C23D4211yescharacteristic
334C23#54712yes—
335C23#54811yescharacteristic
336C23C2×#13511yescharacteristic
337C23C2×#13711yescharacteristic
338C23#57512yes—
339C23#57611yescharacteristic
340C23#57712yes—
341C23#57711yescharacteristic
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—22no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—44no—
358C2×C4—42no—
359C2×C4—44no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—44no—
366C2×C4—42no—
367C2×C4—44no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—42no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—81no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—81no—
397C2×C4—82no—
398C2×C4—81no—
399C2×C4#46414yes—
400C2×C4C2×#12111yescharacteristic
401C2×C4#52511yescharacteristic
402C2×C4#52512yes—
403C2×C4#52912yes—
404C2×C4C2×C4⋊Q811yescharacteristic
405C2×C4#53211yescharacteristic
406C2×C4#53312yes—
407C2×C4#53611yescharacteristic
408C2×C4#53711yescharacteristic
409C2×C4#53912yes—
410C2×C4#54112yes—
411C2×C4#54711yescharacteristic
412C2×C4D4×Q812yes—
413C2×C4#55212yes—
414C2×C4#55311yescharacteristic
415C2×C4#55412yes—
416C2×C4#55712yes—
417C2×C4#55712yes—
418C8—42no—
419C8—44no—
420C8—81no—
421C22#241311yescharacteristic
422C22#241611yescharacteristic
423C22#244912yes—
424C22#245312yes—
425C22#285711yescharacteristic
426C22C2×#46411yescharacteristic
427C22#304011yescharacteristic
428C22#304211yescharacteristic
429C22#304511yescharacteristic
430C22#307612yes—
431C22#307812yes—
432C22#308112yes—
433C22#308412yes—
434C22#326712yes—
435C22#327811yescharacteristic
436C22#328711yescharacteristic
437C22#333712yes—
438C22#334412yes—
439C22#334712yes—
440C22#335112yes—
441C22#336511yescharacteristic
442C22#337511yescharacteristic
443C22#337612yes—
444C4—24no—
445C4—42no—
446C4—44no—
447C4—44no—
448C4—44no—
449C4—44no—
450C4—82no—
451C4—84no—
452C4—161no—
453C4—162no—
454C4#239211yescharacteristic
455C4#246212yes—
456C4#254611yescharacteristic
457C4#258411yescharacteristic
458C4#258512yes—
459C4#266411yescharacteristic
460C2#1481811yescharacteristic
461C2#2457911yescharacteristic
462C2#2459911yescharacteristic
463C2#2473311yescharacteristic
464C2#2474511yescharacteristic
465C2#2634712yes—
466C2#2636012yes—
467C2#2651412yes—
468C2#2652012yes—
469C2#3320211yescharacteristic
470C2#3320511yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ 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×#19492
C2×#19482
C2×#19212
#152422
#138182
#113732
#113222
#112282
#112272
#112082
#109082