Finite Group Databasebeta

← #469784

Subgroups — #469784

Order 512.

1004 conjugacy classes of subgroups.

The subgroup lattice has 479 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1987C211yesmaximal, characteristic
2C2×#1920C211yesmaximal, characteristic
3#15143C211yesmaximal, characteristic
4#13921C211yesmaximal, characteristic
5#11384C211yesmaximal, characteristic
6#11357C212yesmaximal
7#11315C212yesmaximal
8#11267C212yesmaximal
9#11257C212yesmaximal
10#11255C211yesmaximal, characteristic
11#11228C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#106C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C2×#395—22no—
16C2×#389C2211yescharacteristic
17C2×#384C2212yes—
18C2×#383—21no—
19C2×C4×C4⋊C4C2211yescharacteristic
20#1987—24no—
21#1920—24no—
22C2×#340C2211yescharacteristic
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2211yescharacteristic
26C2×#340C2211yescharacteristic
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#336C2212yes—
32C2×#336C2211yescharacteristic
33C2×#336C2212yes—
34#1444—21no—
35#1444—21no—
36#1443—22no—
37C4⋊#95C2212yes—
38C4⋊#95C2212yes—
39C4⋊#95C2212yes—
40C4⋊#95C2211yescharacteristic
41#1434—21no—
42C4×#95C2211yescharacteristic
43C23×Q8—21no—
44C23×C8C2312yes—
45C23×C8C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C2×C4⋊Q8—24no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—42no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2312yes—
61C22×C42—21no—
62C22×C42C2312yes—
63C22×C42C2311yescharacteristic
64C2×#106—22no—
65C2×#106—22no—
66C2×#106—22no—
67C2×C4⋊C8—22no—
68C2×C4⋊C8—42no—
69#395—48no—
70#389—22no—
71#389—22no—
72#384—24no—
73#384—42no—
74#383—44no—
75C4×C4⋊C4—22no—
76C4×C4⋊C4D414yes—
77C2×#95—21no—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—21no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—21no—
85C2×#95—22no—
86C2×#95—21no—
87C2×#95D412yes—
88C2×#95D411yescharacteristic
89C2×#95D411yescharacteristic
90#340—22no—
91#340—22no—
92#340—22no—
93#340—24no—
94#340—24no—
95#340—24no—
96#340—24no—
97#340—42no—
98#340—42no—
99#336—24no—
100#336—22no—
101#336—42no—
102C22×Q8—44no—
103C22×Q8—44no—
104C22×Q8—42no—
105C22×Q8—81no—
106C22×Q8—81no—
107C23×C4—22no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—22no—
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D412yes—
115C23×C4C2×D412yes—
116C23×C4C2×D411yescharacteristic
117C23×C4C2411yescharacteristic
118C22×C8—21no—
119C22×C8—22no—
120C22×C8—21no—
121C22×C8—22no—
122C22×C8—21no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—21no—
132C22×C8—21no—
133C22×C8—22no—
134C22×C8#4112yes—
135C4⋊Q8—28no—
136C4⋊Q8—48no—
137C4⋊Q8—44no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—82no—
175C2×C4⋊C4—82no—
176C2×C4⋊C4C2×D412yes—
177C2×C4⋊C4C2×D412yes—
178C2×C4⋊C4#4114yes—
179C2×C4⋊C4#4112yes—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—24no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—42no—
190C2×C42—42no—
191C2×C42C2×D412yes—
192C2×C42#4112yes—
193#106—28no—
194C4⋊C8—44no—
195#95—24no—
196#95—24no—
197#95—22no—
198#95—44no—
199#95—42no—
200#95—44no—
201#95—44no—
202#95—42no—
203#95—44no—
204#95—44no—
205#95—82no—
206#95—81no—
207#95SD1614yes—
208C24C22×D411yescharacteristic
209C2×Q8—48no—
210C2×Q8—84no—
211C2×Q8—84no—
212C2×Q8—88no—
213C2×Q8—88no—
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—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—22no—
236C22×C4—41no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—41no—
255C22×C4—42no—
256C22×C4—41no—
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—41no—
266C22×C4—41no—
267C22×C4—42no—
268C22×C4#12112yes—
269C22×C4#12112yes—
270C22×C4C2×SD1614yes—
271C22×C4C2×SD1612yes—
272C22×C4AGL(1,ℤ/8)14yes—
273C22×C4#13712yes—
274C22×C4C22×D411yescharacteristic
275C22×C4C2×#4112yes—
276C22×C4C2×#4111yescharacteristic
277C22×C4C2×#4111yescharacteristic
278C22×C4C2×#4111yescharacteristic
279C22×C4H(5,𝔽2)12yes—
280C22×C42−1+412yes—
281C22×C42−1+411yescharacteristic
282C22×C42−1+411yescharacteristic
283C22×C42−1+412yes—
284C2×C8—22no—
285C2×C8—24no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
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—22no—
302C4⋊C4—22no—
303C4⋊C4—24no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—24no—
308C4⋊C4—22no—
309C4⋊C4—48no—
310C4⋊C4—42no—
311C4⋊C4—41no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—88no—
325C4⋊C4—84no—
326C4⋊C4—88no—
327C42—22no—
328C42—22no—
329C42—24no—
330C42—42no—
331C42—42no—
332C42—42no—
333C42—42no—
334C42—48no—
335C23C2×#12111yescharacteristic
336C23#53612yes—
337C23#54612yes—
338C23#54711yescharacteristic
339C23#54811yescharacteristic
340C23C22×SD1611yescharacteristic
341C23C2×AGL(1,ℤ/8)12yes—
342C23C2×#13711yescharacteristic
343C23#57611yescharacteristic
344C23#57711yescharacteristic
345C23#57812yes—
346Q8—816no—
347C2×C4—22no—
348C2×C4—24no—
349C2×C4—22no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—22no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—22no—
357C2×C4—24no—
358C2×C4—22no—
359C2×C4—24no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—44no—
371C2×C4—42no—
372C2×C4—44no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—81no—
382C2×C4—82no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—81no—
389C2×C4—82no—
390C2×C4—81no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—81no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—81no—
401C2×C4—82no—
402C2×C4—81no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4#46114yes—
407C2×C4#52511yescharacteristic
408C2×C4#52511yescharacteristic
409C2×C4#53511yescharacteristic
410C2×C4#53611yescharacteristic
411C2×C4#53611yescharacteristic
412C2×C4#53611yescharacteristic
413C2×C4#54112yes—
414C2×C4#54112yes—
415C2×C4#54112yes—
416C2×C4#54212yes—
417C2×C4#54212yes—
418C2×C4#55112yes—
419C2×C4#55212yes—
420C2×C4#55211yescharacteristic
421C2×C4#55612yes—
422C2×C4#55611yescharacteristic
423C2×C4#56312yes—
424C2×C4#56312yes—
425C8—42no—
426C8—81no—
427C8—82no—
428C22#241111yescharacteristic
429C22#241511yescharacteristic
430C22#262111yescharacteristic
431C22#269812yes—
432C22#271612yes—
433C22C2×#46111yescharacteristic
434C22#303811yescharacteristic
435C22#304411yescharacteristic
436C22#304511yescharacteristic
437C22#322712yes—
438C22#323812yes—
439C22#324212yes—
440C22#324312yes—
441C22#329812yes—
442C22#331212yes—
443C22#331812yes—
444C22#332412yes—
445C22#333511yescharacteristic
446C22#334312yes—
447C22#335711yescharacteristic
448C22#336211yescharacteristic
449C22#336712yes—
450C22#336911yescharacteristic
451C4—24no—
452C4—42no—
453C4—44no—
454C4—44no—
455C4—82no—
456C4—82no—
457C4—84no—
458C4—82no—
459C4—161no—
460C4—162no—
461C4#259911yescharacteristic
462C4#263211yescharacteristic
463C4#263411yescharacteristic
464C4#275812yes—
465C4#276212yes—
466C4#282111yescharacteristic
467C2#1666311yescharacteristic
468C2#2448011yescharacteristic
469C2#2449211yescharacteristic
470C2#2468011yescharacteristic
471C2#2469411yescharacteristic
472C2#2986811yescharacteristic
473C2#2987511yescharacteristic
474C2#3168812yes—
475C2#3170212yes—
476C2#3223012yes—
477C2#3224812yes—
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×#19872
C2×#19202
#151432
#139212
#113842
#113572
#113152
#112672
#112572
#112552
#112282