Finite Group Databasebeta

← #361979

Subgroups — #361979

Order 512.

1522 conjugacy classes of subgroups.

The subgroup lattice has 425 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×Q16C211yesmaximal, characteristic
2C2×#1971C212yesmaximal
3C2×#1846C211yesmaximal, characteristic
4C2×#1805C212yesmaximal
5C2×#1770C211yesmaximal, characteristic
6#13942C212yesmaximal
7#11124C214yesmaximal
8#10863C212yesmaximal
9C23×Q16C2211yescharacteristic
10C22×C4×Q8C2212yes—
11C2×C4×Q16—24no—
12C2×C4×Q16—24no—
13C2×C4×Q16C418yescomplemented
14C22×#107C2211yescharacteristic
15C22×#107C2212yes—
16C22×#106C2212yes—
17C22×C4⋊C8C2211yescharacteristic
18C22×#103C2214yes—
19C22×#103C2212yes—
20C22×#97C2212yes—
21C22×C4×C8C2211yescharacteristic
22C2×#391C2214yes—
23C2×#382C2212yes—
24C2×C4×C4⋊C4C2211yescharacteristic
25#1971—28no—
26#1846—24no—
27#1805—28no—
28#1770—24no—
29C2×#340C2214yes—
30C2×#336C2212yes—
31#1441—24no—
32C4⋊#95C2214yes—
33#1434—24no—
34C23×Q8C2312yes—
35C22×Q16—22no—
36C22×Q16—22no—
37C22×Q16—22no—
38C22×Q16—22no—
39C22×Q16C2×C414yes—
40C22×Q16D414yes—
41C22×Q16Q814yes—
42C23×C8C2312yes—
43C23×C8C2311yescharacteristic
44C23×C8C2311yescharacteristic
45C2×C4×Q8—24no—
46C2×C4×Q8—24no—
47C2×C4×Q8—24no—
48C2×C4×Q8—24no—
49C2×C4×Q8—24no—
50C2×C4×Q8C2×C418yes—
51C22×C4⋊C4—22no—
52C22×C4⋊C4D412yes—
53C22×C4⋊C4C2314yes—
54C22×C4⋊C4C2312yes—
55C22×C42—22no—
56C22×C42C2312yes—
57C22×C42C2311yescharacteristic
58C4×Q16—232no—
59C2×#107—22no—
60C2×#107—22no—
61C2×#107—24no—
62C2×#107—42no—
63C2×#107—42no—
64C2×#107C2×C414yes—
65C2×#106—24no—
66C2×#106—24no—
67C2×#106—24no—
68C2×C4⋊C8—22no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8D414yes—
71C2×#103—24no—
72C2×#103—24no—
73C2×#103—28no—
74C2×#103—28no—
75C2×#103—44no—
76C2×#103C2×C418yes—
77C2×#97—24no—
78C2×#97—24no—
79C2×#97—24no—
80C2×C4×C8C2×C414yes—
81C2×C4×C8D414yes—
82C2×C4×C8D414yes—
83#391—28no—
84#391—44no—
85#382—24no—
86#382—24no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4—22no—
89C2×#95—22no—
90C2×#95—28no—
91C2×#95—28no—
92C2×#95—22no—
93C2×#95D412yes—
94#340—28no—
95#336—24no—
96C22×Q8—22no—
97C22×Q8—22no—
98C22×Q8—22no—
99C22×Q8—22no—
100C22×Q8—22no—
101C22×Q8—22no—
102C22×Q8—22no—
103C22×Q8—22no—
104C22×Q8—22no—
105C22×Q8—22no—
106C22×Q8—22no—
107C22×Q8C22×C414yes—
108C22×Q8C2×D414yes—
109C22×Q8C2×Q814yes—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—24no—
113C23×C4C2×D414yes—
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C23×C4C2411yescharacteristic
117C2×Q16—28no—
118C2×Q16—28no—
119C2×Q16—28no—
120C2×Q16—28no—
121C2×Q16—28no—
122C2×Q16—28no—
123C2×Q16C4⋊C4116yescomplemented
124C22×C8—21no—
125C22×C8—21no—
126C22×C8—21no—
127C22×C8—21no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8C22×C412yes—
135C22×C8C22×C412yes—
136C22×C8C2×D414yes—
137C22×C8C2×D412yes—
138C22×C8C2×D412yes—
139C22×C8C2×D412yes—
140C22×C8C2×D412yes—
141C22×C8C2×D412yes—
142C22×C8C2×Q812yes—
143C22×C8#4112yes—
144C22×C8#4112yes—
145C4×Q8—216no—
146C4×Q8—216no—
147C4×Q8—48no—
148C4×Q8—48no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—28no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—44no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4C22×C414yes—
166C2×C42—28no—
167C2×C42—22no—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—44no—
174C2×C42—44no—
175C2×C42C22×C412yes—
176C2×C42C2×D412yes—
177C2×C42C2×D412yes—
178C2×C42C2×D412yes—
179C2×C42#4112yes—
180C2×C42#4112yes—
181#107—28no—
182#107—48no—
183#106—48no—
184C4⋊C8—28no—
185#103—216no—
186#103—416no—
187#97—48no—
188C4×C8C22⋊C4116yes—
189#95—24no—
190#95—416no—
191#95—416no—
192#95—44no—
193#95—44no—
194C24C22×D411yescharacteristic
195C2×Q8—24no—
196C2×Q8—24no—
197C2×Q8—24no—
198C2×Q8—24no—
199C2×Q8—24no—
200C2×Q8—24no—
201C2×Q8—24no—
202C2×Q8—24no—
203C2×Q8—24no—
204C2×Q8—24no—
205C2×Q8—24no—
206C2×Q8—24no—
207C2×Q8—24no—
208C2×Q8—24no—
209C2×Q8—24no—
210C2×Q8—42no—
211C2×Q8—42no—
212C2×Q8—42no—
213C2×Q8—42no—
214C2×Q8—42no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—42no—
218C2×Q8—42no—
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8—42no—
222C2×Q8C2×C4⋊C418yes—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—28no—
233C22×C4—28no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—48no—
241C22×C4—44no—
242C22×C4—44no—
243C22×C4—44no—
244C22×C4—48no—
245C22×C4—48no—
246C22×C4C4×D412yes—
247C22×C4C4×D412yes—
248C22×C4#12112yes—
249C22×C4#12112yes—
250C22×C4#12712yes—
251C22×C4#12712yes—
252C22×C4C23×C411yescharacteristic
253C22×C4C22×D411yescharacteristic
254C22×C4C22×D411yescharacteristic
255C22×C4C22×D411yescharacteristic
256C22×C4C22×D411yescharacteristic
257C22×C4C22×Q811yescharacteristic
258C22×C4C2×#4111yescharacteristic
259C22×C4C2×#4111yescharacteristic
260C22×C4C2×#4111yescharacteristic
261C22×C4C2×#4111yescharacteristic
262C22×C4H(5,𝔽2)11yescharacteristic
263C22×C4H(5,𝔽2)11yescharacteristic
264C22×C42−1+411yescharacteristic
265C22×C42−1+411yescharacteristic
266Q16—232no—
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—22no—
270C2×C8—22no—
271C2×C8—22no—
272C2×C8—22no—
273C2×C8—24no—
274C2×C8—24no—
275C2×C8—24no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8C2×C22⋊C414yes—
281C2×C8C2×C22⋊C414yes—
282C2×C8C2×C4⋊C414yes—
283C2×C8#11714yes—
284C2×C8#12114yes—
285C2×C8#12114yes—
286C2×C8C22⋊Q814yes—
287C2×C8C22⋊Q814yes—
288C4⋊C4—24no—
289C4⋊C4—24no—
290C4⋊C4—48no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—416no—
294C4⋊C4—44no—
295C4⋊C4—44no—
296C4⋊C4—44no—
297C4⋊C4—44no—
298C42—42no—
299C42—42no—
300C42—42no—
301C42—42no—
302C42—416no—
303C42C2×C22⋊C414yes—
304C42C4×D414yes—
305C42#12114yes—
306C42#12414yes—
307C23C2×C4×D411yescharacteristic
308C23C2×#12111yescharacteristic
309C23C2×#12711yescharacteristic
310C23D4211yescharacteristic
311C23#54711yescharacteristic
312C23D4×Q811yescharacteristic
313C23#55011yescharacteristic
314C23#57514yes—
315C23#57814yes—
316Q8—28no—
317Q8—28no—
318Q8—44no—
319Q8—44no—
320Q8—44no—
321Q8—44no—
322Q8—44no—
323Q8—44no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—28no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—48no—
338C2×C4—48no—
339C2×C4—48no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—84no—
345C2×C4—84no—
346C2×C4—88no—
347C2×C4—84no—
348C2×C4—84no—
349C2×C4—88no—
350C2×C4—84no—
351C2×C4#39014yes—
352C2×C4#39014yes—
353C2×C4C22×C22⋊C411yescharacteristic
354C2×C4C22×C4⋊C411yescharacteristic
355C2×C4C2×#11711yescharacteristic
356C2×C4C2×C4×D411yescharacteristic
357C2×C4#52011yescharacteristic
358C2×C4#52011yescharacteristic
359C2×C4C2×#12111yescharacteristic
360C2×C4C2×#12111yescharacteristic
361C2×C4C2×#12111yescharacteristic
362C2×C4C2×C22⋊Q811yescharacteristic
363C2×C4C2×C22⋊Q811yescharacteristic
364C2×C4C2×#12411yescharacteristic
365C2×C4#53511yescharacteristic
366C2×C4#53611yescharacteristic
367C2×C4#53711yescharacteristic
368C2×C4#53711yescharacteristic
369C2×C4D4211yescharacteristic
370C2×C4#54711yescharacteristic
371C2×C4#54711yescharacteristic
372C2×C4#54711yescharacteristic
373C2×C4D4×Q811yescharacteristic
374C2×C4#55011yescharacteristic
375C2×C4#55411yescharacteristic
376C2×C4#55411yescharacteristic
377C2×C4#55511yescharacteristic
378C2×C4#55511yescharacteristic
379C2×C4#55611yescharacteristic
380C2×C4#55611yescharacteristic
381C8—24no—
382C8—44no—
383C8#38018yes—
384C22C2×#39011yescharacteristic
385C22D4×C4⋊C411yescharacteristic
386C22#235411yescharacteristic
387C22#253711yescharacteristic
388C22#253811yescharacteristic
389C22#254111yescharacteristic
390C22#254211yescharacteristic
391C22#294312yes—
392C22#295012yes—
393C22#305412yes—
394C22#305912yes—
395C22#315012yes—
396C22#315612yes—
397C22#327714yes—
398C22#335214yes—
399C22#339014yes—
400C22#340214yes—
401C4—24no—
402C4—24no—
403C4—48no—
404C4—84no—
405C4—84no—
406C4—164no—
407C4C2×#38011yescharacteristic
408C4#231111yescharacteristic
409C4D4×C4⋊C411yescharacteristic
410C4#235411yescharacteristic
411C4#253711yescharacteristic
412C4#253811yescharacteristic
413C4#254711yescharacteristic
414C4#256311yescharacteristic
415C2#1390611yescharacteristic
416C2#2122811yescharacteristic
417C2#2123211yescharacteristic
418C2#2267512yes—
419C2#2281512yes—
420C2#2823412yes—
421C2#2827112yes—
422C2#2846912yes—
423C2#2853812yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×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
13. C2×C4×Q16C4semidirect
123. C2×Q16C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Q162
C2×#19712
C2×#18462
C2×#18052
C2×#17702
#139422
#111242
#108632