Finite Group Databasebeta

← #476218

Subgroups — #476218

Order 512.

1116 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×#2058C212yesmaximal
2C2×#2058C212yesmaximal
3#15128C211yesmaximal, characteristic
4#15128C211yesmaximal, characteristic
5#11378C212yesmaximal
6#11346C211yesmaximal, characteristic
7#11293C212yesmaximal
8#11292C212yesmaximal
9#11257C212yesmaximal
10C22×C4⋊Q8C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12C22×#106C2212yes—
13C22×#106C2212yes—
14C2×#395—21no—
15C2×#395—21no—
16C2×#389C2212yes—
17C2×#389C2212yes—
18C2×C4×C4⋊C4C2211yescharacteristic
19#2058—28no—
20#2058—28no—
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#336C2212yes—
28C2×#336C2212yes—
29C2×#336C2211yescharacteristic
30C2×#336C2211yescharacteristic
31#1446—21no—
32#1446—21no—
33#1444—21no—
34#1444—21no—
35#1441—21no—
36#1441—21no—
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41C23×Q8—21no—
42C23×Q8—21no—
43C23×C8C2312yes—
44C23×C8C2312yes—
45C2×C4⋊Q8—24no—
46C2×C4⋊Q8—24no—
47C2×C4⋊Q8—41no—
48C2×C4⋊Q8—41no—
49C2×C4⋊Q8—41no—
50C2×C4⋊Q8—41no—
51C2×C4⋊Q8—42no—
52C2×C4⋊Q8—42no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4D411yescharacteristic
58C22×C4⋊C4D411yescharacteristic
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C42C2311yescharacteristic
64C22×C42C2311yescharacteristic
65C22×C42C2311yescharacteristic
66C2×#106—24no—
67C2×#106—24no—
68C2×#106—42no—
69C2×#106—42no—
70C2×#106—42no—
71C2×#106D418yes—
72#395—44no—
73#395—44no—
74#389—24no—
75#389—24no—
76#389—42no—
77#389—42no—
78C4×C4⋊C4—41no—
79C4×C4⋊C4—41no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—21no—
83C2×#95—21no—
84C2×#95—21no—
85C2×#95—21no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95D411yescharacteristic
91C2×#95D411yescharacteristic
92#340—24no—
93#340—24no—
94#340—24no—
95#340—24no—
96#340—42no—
97#340Q818yescomplemented
98#336—24no—
99#336—24no—
100#336—41no—
101#336—41no—
102C22×Q8—44no—
103C22×Q8—44no—
104C22×Q8—42no—
105C22×Q8—42no—
106C22×Q8—42no—
107C22×Q8—42no—
108C22×Q8—42no—
109C22×Q8—42no—
110C22×Q8—82no—
111C22×Q8—82no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2411yescharacteristic
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—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8C2×D414yes—
136C22×C8C2×Q814yes—
137C4⋊Q8—44no—
138C4⋊Q8—44no—
139C4⋊Q8—48no—
140C4⋊Q8—48no—
141C4⋊Q8—44no—
142C4⋊Q8—44no—
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—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—82no—
177C2×C4⋊C4—82no—
178C2×C4⋊C4SD1614yes—
179C2×C4⋊C4C2×D414yes—
180C2×C4⋊C4C2×D414yes—
181C2×C4⋊C4C2×Q814yes—
182C2×C4⋊C4C2×Q814yes—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—22no—
194C2×C42—22no—
195#106—48no—
196#106—48no—
197#95—41no—
198#95—41no—
199#95—44no—
200#95—44no—
201#95—42no—
202#95—44no—
203#95—42no—
204#95—42no—
205#95—44no—
206#95—42no—
207#95—82no—
208#95—82no—
209C24C22×D411yescharacteristic
210C2×Q8—48no—
211C2×Q8—48no—
212C2×Q8—88no—
213C2×Q8—88no—
214C2×Q8—84no—
215C2×Q8—84no—
216C2×Q8—88no—
217C2×Q8—88no—
218C2×Q8—84no—
219C2×Q8—84no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—22no—
250C22×C4—22no—
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—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4C2×SD1612yes—
276C22×C4C2×SD1612yes—
277C22×C4C2×SD1612yes—
278C22×C4AGL(1,ℤ/8)12yes—
279C22×C4#13712yes—
280C22×C4C22×D412yes—
281C22×C4C22×Q812yes—
282C22×C4H(5,𝔽2)11yescharacteristic
283C22×C4H(5,𝔽2)12yes—
284C22×C4H(5,𝔽2)11yescharacteristic
285C22×C42−1+412yes—
286C22×C42−1+412yes—
287C22×C42−1+412yes—
288C2×C8—24no—
289C2×C8—24no—
290C2×C8—24no—
291C2×C8—24no—
292C2×C8—24no—
293C2×C8—24no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—24no—
308C4⋊C4—24no—
309C4⋊C4—24no—
310C4⋊C4—44no—
311C4⋊C4—44no—
312C4⋊C4—48no—
313C4⋊C4—48no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—88no—
323C4⋊C4—88no—
324C42—42no—
325C42—42no—
326C42—41no—
327C42—41no—
328C42—41no—
329C42—41no—
330C42—42no—
331C42—42no—
332C23#53411yescharacteristic
333C23#53611yescharacteristic
334C23#53611yescharacteristic
335C23D4212yes—
336C23D4×Q812yes—
337C23C22×SD1611yescharacteristic
338C23C2×AGL(1,ℤ/8)11yescharacteristic
339C23C2×#13711yescharacteristic
340C23#57511yescharacteristic
341C23#57712yes—
342C23#57712yes—
343Q8—816no—
344Q8—816no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—24no—
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—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
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—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4#53412yes—
406C2×C4#53612yes—
407C2×C4#53612yes—
408C2×C4#54412yes—
409C2×C4#54412yes—
410C2×C4#54412yes—
411C2×C4D4211yescharacteristic
412C2×C4D4×Q812yes—
413C2×C4D4×Q812yes—
414C2×C4Q8211yescharacteristic
415C2×C4H(3,𝔽4)11yescharacteristic
416C2×C4#56311yescharacteristic
417C2×C4#56312yes—
418C2×C4#56312yes—
419C2×C4#56312yes—
420C2×C4#56312yes—
421C8—44no—
422C8—44no—
423C22#267712yes—
424C22#267712yes—
425C22#267911yescharacteristic
426C22#273411yescharacteristic
427C22#281711yescharacteristic
428C22#318711yescharacteristic
429C22#319311yescharacteristic
430C22#320011yescharacteristic
431C22#320011yescharacteristic
432C22#322711yescharacteristic
433C22#322911yescharacteristic
434C22#323511yescharacteristic
435C22#323711yescharacteristic
436C22#324412yes—
437C22#324412yes—
438C22#326512yes—
439C22#326612yes—
440C22#327512yes—
441C22D4×SD1612yes—
442C22Q8×SD1612yes—
443C22#338412yes—
444C22#338712yes—
445C22#338812yes—
446C4—42no—
447C4—42no—
448C4—42no—
449C4—42no—
450C4—42no—
451C4—81no—
452C4—84no—
453C4—84no—
454C4—84no—
455C4—84no—
456C4#267712yes—
457C4#273412yes—
458C4#281712yes—
459C4#284412yes—
460C2#1708511yescharacteristic
461C2#3094912yes—
462C2#3095412yes—
463C2#3096912yes—
464C2#3097412yes—
465C2#3105811yescharacteristic
466C2#3106511yescharacteristic
467C2#3268311yescharacteristic
468C2#3269411yescharacteristic
469C2#3278811yescharacteristic
470C2#3279411yescharacteristic
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

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
97. #340Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#20582
#151282
#113782
#113462
#112932
#112922
#112572