Finite Group Databasebeta

← #467246

Subgroups — #467246

Order 512.

1006 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1918C212yesmaximal
2#15151C211yesmaximal, characteristic
3#13910C211yesmaximal, characteristic
4#11227C212yesmaximal
5#11219C212yesmaximal
6#11219C212yesmaximal
7#11207C212yesmaximal
8#11207C212yesmaximal
9#11202C211yesmaximal, characteristic
10C22×#107C2212yes—
11C22×C4⋊C8C2211yescharacteristic
12C2×#398—21no—
13C2×#398—21no—
14C2×#389C2212yes—
15C2×#389C2212yes—
16C2×#389C2212yes—
17C2×#384—21no—
18C2×#384C2211yescharacteristic
19C2×#384C2211yescharacteristic
20C2×C4×C4⋊C4C2211yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22#1918—28no—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#336C2212yes—
31C2×#336C2212yes—
32C2×#336C2211yescharacteristic
33#1445—21no—
34#1445—21no—
35#1444—21no—
36#1444—21no—
37C4⋊#95—21no—
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2212yes—
40C23×C8C2312yes—
41C23×C8C2312yes—
42C22×C4⋊C4—22no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4D411yescharacteristic
50C22×C4⋊C4D411yescharacteristic
51C22×C4⋊C4C2312yes—
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2312yes—
55C22×C42—21no—
56C22×C42C2311yescharacteristic
57C22×C42C2311yescharacteristic
58C22×C42C2311yescharacteristic
59C2×#107—24no—
60C2×#107—24no—
61C2×#107Q818yes—
62C2×C4⋊C8—41no—
63C2×C4⋊C8—42no—
64#398—44no—
65#398—44no—
66#389—24no—
67#389—24no—
68#389—24no—
69#389—24no—
70#389—42no—
71#389—42no—
72#384—41no—
73#384—41no—
74#384—41no—
75#384—41no—
76#384—44no—
77C4×C4⋊C4—22no—
78C4×C4⋊C4—41no—
79C4×C4⋊C4—41no—
80C4×C4⋊C4D414yes—
81C2×#95—21no—
82C2×#95—21no—
83C2×#95—22no—
84C2×#95—21no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95D411yescharacteristic
90C2×#95D411yescharacteristic
91#340—24no—
92#340—24no—
93#340—24no—
94#340—42no—
95#340—42no—
96#340—42no—
97#340—42no—
98#336—24no—
99#336—41no—
100#336—42no—
101C23×C4—22no—
102C23×C4—22no—
103C23×C4—22no—
104C23×C4—22no—
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D411yescharacteristic
108C23×C4C2×D411yescharacteristic
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4C2411yescharacteristic
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—22no—
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×C8C2×Q814yes—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
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—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—44no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—44no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4—44no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—82no—
169C2×C4⋊C4—82no—
170C2×C4⋊C4—82no—
171C2×C4⋊C4D814yes—
172C2×C4⋊C4C2×D414yes—
173C2×C4⋊C4C2×Q814yes—
174C2×C4⋊C4C2×Q814yes—
175C2×C4⋊C4#4114yes—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—24no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—21no—
184C2×C42—22no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—21no—
188C2×C42—22no—
189C2×C42—42no—
190C2×C42—42no—
191C2×C42C2×D412yes—
192C2×C42#4112yes—
193#107—216no—
194C4⋊C8—44no—
195#95—24no—
196#95—41no—
197#95—41no—
198#95—44no—
199#95—44no—
200#95—42no—
201#95—44no—
202#95—44no—
203#95—42no—
204#95—44no—
205C24C22×D411yescharacteristic
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—24no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—24no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—24no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—24no—
232C22×C4—21no—
233C22×C4—21no—
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#12112yes—
260C22×C4#12112yes—
261C22×C4C2×D812yes—
262C22×C4C2×D812yes—
263C22×C4C2×D812yes—
264C22×C4AGL(1,ℤ/8)12yes—
265C22×C4#13712yes—
266C22×C4C22×D411yescharacteristic
267C22×C4C22×Q812yes—
268C22×C4C2×#4112yes—
269C22×C4C2×#4111yescharacteristic
270C22×C4H(5,𝔽2)11yescharacteristic
271C22×C4H(5,𝔽2)11yescharacteristic
272C22×C4H(5,𝔽2)12yes—
273C22×C42−1+411yescharacteristic
274C22×C42−1+411yescharacteristic
275C22×C42−1+412yes—
276C2×C8—24no—
277C2×C8—24no—
278C2×C8—24no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C4⋊C4—24no—
291C4⋊C4—24no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—44no—
296C4⋊C4—44no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—88no—
308C4⋊C4—88no—
309C4⋊C4—84no—
310C4⋊C4—84no—
311C4⋊C4—84no—
312C4⋊C4—84no—
313C4⋊C4—88no—
314C4⋊C4C22⋊Q818yes—
315C42—22no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—42no—
320C42—42no—
321C42—41no—
322C42—41no—
323C42—41no—
324C42—41no—
325C42—48no—
326C23C2×#12111yescharacteristic
327C23#53411yescharacteristic
328C23#53611yescharacteristic
329C23#54612yes—
330C23D4×Q812yes—
331C23C22×D811yescharacteristic
332C23C2×AGL(1,ℤ/8)11yescharacteristic
333C23C2×#13711yescharacteristic
334C23#57511yescharacteristic
335C23#57712yes—
336C23#57812yes—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—28no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—42no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—42no—
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#45914yes—
397C2×C4C2×C22⋊Q812yes—
398C2×C4#53711yescharacteristic
399C2×C4#53712yes—
400C2×C4#53711yescharacteristic
401C2×C4#53911yescharacteristic
402C2×C4#53911yescharacteristic
403C2×C4#54012yes—
404C2×C4#54112yes—
405C2×C4#54412yes—
406C2×C4#54412yes—
407C2×C4#55212yes—
408C2×C4#55412yes—
409C2×C4#55712yes—
410C2×C4Q8211yescharacteristic
411C2×C4#55911yescharacteristic
412C2×C4#56011yescharacteristic
413C2×C4#56312yes—
414C2×C4#56311yescharacteristic
415C8—44no—
416C8—82no—
417C22#241712yes—
418C22#261811yescharacteristic
419C22#270911yescharacteristic
420C22#271312yes—
421C22#273411yescharacteristic
422C22C2×#45911yescharacteristic
423C22#303711yescharacteristic
424C22#304612yes—
425C22#318611yescharacteristic
426C22#319311yescharacteristic
427C22#320011yescharacteristic
428C22#320211yescharacteristic
429C22#322611yescharacteristic
430C22#323411yescharacteristic
431C22#324212yes—
432C22#329412yes—
433C22#331212yes—
434C22#331712yes—
435C22#333012yes—
436C22Q8×D812yes—
437C22#338412yes—
438C22#338712yes—
439C22#339012yes—
440C4—24no—
441C4—42no—
442C4—42no—
443C4—42no—
444C4—42no—
445C4—48no—
446C4—81no—
447C4—84no—
448C4—162no—
449C4—162no—
450C4#258211yescharacteristic
451C4#262912yes—
452C4#279112yes—
453C4#279211yescharacteristic
454C4#279411yescharacteristic
455C4#282311yescharacteristic
456C2#1656311yescharacteristic
457C2#2481112yes—
458C2#2482612yes—
459C2#2975411yescharacteristic
460C2#2976611yescharacteristic
461C2#3202611yescharacteristic
462C2#3203911yescharacteristic
463C2#3213012yes—
464C2#3215912yes—
465C2#3268211yescharacteristic
466C2#3269411yescharacteristic
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×#19182
#151512
#139102
#112272
#112192
#112072
#112022