Finite Group Databasebeta

← #413474

Subgroups — #413474

Order 512.

1468 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1970C212yesmaximal
2C2×#1970C211yesmaximal, characteristic
3C2×#1942C212yesmaximal, complemented
4C2×C8×C4⋊C4C211yesmaximal, characteristic, complemented
5#15148C212yesmaximal, complemented
6#11076C212yesmaximal, complemented
7#11066C214yesmaximal, complemented
8#10862C211yesmaximal, characteristic, complemented
9C23×D8C2211yescharacteristic
10C22×#107C2211yescharacteristic
11C22×#107C2212yes—
12C22×C4⋊C8C2211yescharacteristic
13C22×#102C2214yes—
14C22×#102C2212yes—
15C22×C4×C8C2212yes—
16C22×C4×C8C2211yescharacteristic
17C2×#399—22no—
18C2×#390C2212yes—
19C2×#390C2214yes—
20C2×#389C2214yes—
21C2×C4×C4⋊C4C2211yescharacteristic
22#1970—28no—
23#1970—24no—
24#1942—28no—
25C8×C4⋊C4—24no—
26C2×#340C2214yes—
27C2×#340C2212yes—
28C2×#336C2212yes—
29#1439—24no—
30#1439—22no—
31C4⋊#95C2212yes—
32C23×D4C2312yes—
33C22×D8—22no—
34C22×D8—24no—
35C22×D8—24no—
36C22×D8—22no—
37C22×D8Q814yes—
38C23×C8C2311yescharacteristic
39C23×C8C2312yes—
40C23×C8C2311yescharacteristic
41C22×C4⋊C4—24no—
42C22×C4⋊C4D411yescharacteristic
43C22×C4⋊C4D411yescharacteristic
44C22×C4⋊C4C2312yes—
45C22×C4⋊C4C2314yes—
46C22×C22⋊C4—22no—
47C22×C22⋊C4—24no—
48C22×C22⋊C4—24no—
49C22×C22⋊C4—22no—
50C22×C42C2312yes—
51C22×C42C2311yescharacteristic
52C2×#107—24no—
53C2×#107—24no—
54C2×#107—24no—
55C2×#107—22no—
56C2×#107D418yes—
57C2×C4⋊C8—24no—
58C2×C4⋊C8—22no—
59C2×#102—24no—
60C2×#102—28no—
61C2×#102—28no—
62C2×#102—44no—
63C2×#102—44no—
64C2×C4×C8—24no—
65C2×C4×C8—24no—
66C2×C4×C8—24no—
67C2×C4×C8—24no—
68C2×C4×C8D414yes—
69#399—48no—
70#390—28no—
71#390—24no—
72#390—24no—
73#390—44no—
74#389—28no—
75#389—44no—
76C4×C4⋊C4—22no—
77C4×C4⋊C4—22no—
78C2×#95—22no—
79C2×#95—24no—
80C2×#95—22no—
81C2×#95D412yes—
82#340—28no—
83#340—42no—
84#336—24no—
85C25—22no—
86C22×D4—22no—
87C22×D4—24no—
88C22×D4—24no—
89C22×D4—22no—
90C22×D4—22no—
91C22×D4—24no—
92C22×D4—24no—
93C22×D4—22no—
94C22×D4—22no—
95C22×D4C2×Q814yes—
96C23×C4—24no—
97C23×C4—22no—
98C23×C4C2×D411yescharacteristic
99C23×C4C2×D412yes—
100C23×C4C2×D412yes—
101C23×C4C2×D411yescharacteristic
102C23×C4C2411yescharacteristic
103C2×D8—28no—
104C2×D8—216no—
105C2×D8—44no—
106C2×D8—48no—
107C2×D8—44no—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—21no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—21no—
117C22×C8—22no—
118C22×C8C2×D412yes—
119C22×C8C2×D412yes—
120C22×C8C2×D414yes—
121C22×C8C2×Q812yes—
122C22×C8#4114yes—
123C22×C8#4114yes—
124C22×C8#4112yes—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—48no—
139C2×C4⋊C4—48no—
140C2×C4⋊C4—84no—
141C2×C4⋊C4D814yescomplemented
142C2×C4⋊C4C2×D418yes—
143C2×C22⋊C4—48no—
144C2×C22⋊C4—48no—
145C2×C22⋊C4—44no—
146C2×C22⋊C4—44no—
147C2×C22⋊C4—48no—
148C2×C22⋊C4—44no—
149C2×C22⋊C4—48no—
150C2×C22⋊C4—44no—
151C2×C22⋊C4—84no—
152C2×C22⋊C4—84no—
153C2×C22⋊C4—84no—
154C2×C42—22no—
155C2×C42—22no—
156C2×C42—22no—
157C2×C42—22no—
158C2×C42—22no—
159C2×C42—22no—
160C2×C42—22no—
161C2×C42—22no—
162C2×C42C2×D412yes—
163C2×C42#4112yes—
164#107—216no—
165#107—44no—
166C4⋊C8—28no—
167#102—416no—
168#102—48no—
169C4×C8—28no—
170C4×C8—48no—
171#95—24no—
172#95—48no—
173#95—44no—
174#95—82no—
175C24—24no—
176C24—42no—
177C24—44no—
178C24—44no—
179C24—42no—
180C24—42no—
181C24—42no—
182C24—44no—
183C24—42no—
184C24—44no—
185C24—42no—
186C24C22×D411yescharacteristic
187C2×D4—24no—
188C2×D4—28no—
189C2×D4—28no—
190C2×D4—24no—
191C2×D4—42no—
192C2×D4—44no—
193C2×D4—42no—
194C2×D4—44no—
195C2×D4—42no—
196C2×D4—42no—
197C2×D4—42no—
198C2×D4—44no—
199C2×D4—44no—
200C2×D4—44no—
201C2×D4—44no—
202C2×D4—44no—
203C2×D4—42no—
204C2×D4—44no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—21no—
223C22×C4—28no—
224C22×C4—44no—
225C22×C4—44no—
226C22×C4—44no—
227C22×C4—44no—
228C22×C4—44no—
229C22×C4—44no—
230C22×C4—42no—
231C22×C4—44no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—44no—
235C22×C4#12112yes—
236C22×C4#12112yes—
237C22×C4C2×D814yes—
238C22×C4C2×D812yes—
239C22×C4#13514yes—
240C22×C4#13512yes—
241C22×C4C22×D411yescharacteristic
242C22×C4C22×D412yes—
243C22×C4C22×Q811yescharacteristic
244C22×C4C2×#4112yes—
245C22×C4C2×#4111yescharacteristic
246C22×C4C2×#4112yes—
247C22×C4C2×#4111yescharacteristic
248C22×C4H(5,𝔽2)12yes—
249C22×C4H(5,𝔽2)11yescharacteristic
250C22×C42−1+411yescharacteristic
251D8—416no—
252C2×C8—22no—
253C2×C8—24no—
254C2×C8—22no—
255C2×C8—22no—
256C2×C8—24no—
257C2×C8—24no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8C22⋊Q814yes—
265C2×C8C22⋊Q818yes—
266C2×C8#12314yes—
267C2×C8#12318yes—
268C2×C8#12714yes—
269C4⋊C4—28no—
270C4⋊C4—28no—
271C4⋊C4—28no—
272C4⋊C4—28no—
273C4⋊C4—44no—
274C4⋊C4—44no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—44no—
278C4⋊C4—816no—
279C22⋊C4—816no—
280C22⋊C4—816no—
281C22⋊C4—88no—
282C22⋊C4—88no—
283C42—22no—
284C42—22no—
285C42—24no—
286C42—42no—
287C42—42no—
288C42—42no—
289C42—42no—
290C23—44no—
291C23—44no—
292C23—48no—
293C23—48no—
294C23—44no—
295C23—82no—
296C23—82no—
297C23—82no—
298C23—84no—
299C23—84no—
300C23—84no—
301C23—84no—
302C23—84no—
303C23—84no—
304C23—84no—
305C23—84no—
306C23—84no—
307C23—84no—
308C23—82no—
309C23—82no—
310C23—82no—
311C23—82no—
312C23—82no—
313C23C2×#12111yescharacteristic
314C23#53212yes—
315C23D4212yes—
316C23D4×Q811yescharacteristic
317C23#55011yescharacteristic
318C23C22×D811yescharacteristic
319C23C2×#13511yescharacteristic
320C23C2×#13512yes—
321C23#57611yescharacteristic
322C23#57612yes—
323C23#57811yescharacteristic
324D4—44no—
325D4—44no—
326D4—48no—
327D4—82no—
328D4—84no—
329D4—82no—
330C2×C4—22no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—48no—
354C2×C4—48no—
355C2×C4—48no—
356C2×C4—84no—
357C2×C4—84no—
358C2×C4—84no—
359C2×C4—84no—
360C2×C4—84no—
361C2×C4—84no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4#46614yes—
366C2×C4C2×C22⋊Q811yescharacteristic
367C2×C4C2×C22⋊Q812yes—
368C2×C4C2×#12311yescharacteristic
369C2×C4C2×#12312yes—
370C2×C4#52511yescharacteristic
371C2×C4#52912yes—
372C2×C4C2×#12711yescharacteristic
373C2×C4#53511yescharacteristic
374C2×C4#53711yescharacteristic
375C2×C4#54012yes—
376C2×C4#54212yes—
377C2×C4#54612yes—
378C2×C4#54812yes—
379C2×C4#55012yes—
380C2×C4#55412yes—
381C2×C4#55411yescharacteristic
382C2×C4#55611yescharacteristic
383C2×C4#55912yes—
384C8—42no—
385C8—44no—
386C8#39918yescomplemented
387C22—84no—
388C22—84no—
389C22—84no—
390C22—88no—
391C22—88no—
392C22—162no—
393C22—162no—
394C22—162no—
395C22—162no—
396C22—164no—
397C22—164no—
398C22#244512yes—
399C22#245212yes—
400C22#253811yescharacteristic
401C22#254111yescharacteristic
402C22#285411yescharacteristic
403C22C2×#46611yescharacteristic
404C22#305011yescharacteristic
405C22#305811yescharacteristic
406C22#306111yescharacteristic
407C22#316812yes—
408C22#316912yes—
409C22#317512yes—
410C22#317612yes—
411C22D4×D812yes—
412C22#328012yes—
413C22#328512yes—
414C22#328712yes—
415C22Q8×D811yescharacteristic
416C22#338011yescharacteristic
417C22#338012yes—
418C22#339111yescharacteristic
419C22#339311yescharacteristic
420C22#339612yes—
421C4—24no—
422C4—44no—
423C4—44no—
424C4—42no—
425C4—88no—
426C4—162no—
427C4C2×#39911yescharacteristic
428C4#244812yes—
429C4#245412yes—
430C4#254011yescharacteristic
431C4#258011yescharacteristic
432C4#262411yescharacteristic
433C2—164no—
434C2#1514811yescharacteristic
435C2#2625312yes—
436C2#2625412yes—
437C2#2648212yes—
438C2#2648412yes—
439C2#2826311yescharacteristic
440C2#2826711yescharacteristic
441C2#2848411yescharacteristic
442C2#2848511yescharacteristic
443C2#3312911yescharacteristic
444C2#3313811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D8C22
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
3. C2×#1942C2semidirect
4. C2×C8×C4⋊C4C2semidirect
5. #15148C2semidirect
6. #11076C2semidirect
7. #11066C2semidirect
8. #10862C2semidirect
141. C2×C4⋊C4D8semidirect
386. C8#399semidirect

Maximal subgroups

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

subgroupindex
C2×#19702
C2×#19422
C2×C8×C4⋊C42
#151482
#110762
#110662
#108622