Finite Group Databasebeta

← #381269

Subgroups — #381269

Order 512.

1222 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2863C211yesmaximal, characteristic
2C2×C4×C4⋊C8C211yesmaximal, characteristic
3#14687C211yesmaximal, characteristic
4#11228C212yesmaximal
5#11207C212yesmaximal
6#11077C212yesmaximal
7#11050C212yesmaximal
8#10926C212yesmaximal
9#10880C212yesmaximal
10C22×C4⋊C8C2212yes—
11C22×#103C2214yes—
12C22×C4×C8C2212yes—
13#2863—24no—
14#2863C418yescomplemented
15C2×#391C2211yescharacteristic
16C2×#391C2211yescharacteristic
17C2×#391C2211yescharacteristic
18C2×#389C2211yescharacteristic
19C2×#389C2211yescharacteristic
20C2×#389C2211yescharacteristic
21C2×#387—21no—
22C2×#387—21no—
23C2×#387—21no—
24C2×#384C2212yes—
25C2×#383—21no—
26C2×#382—21no—
27C2×#382—21no—
28C2×C43C2211yescharacteristic
29C4×C4⋊C8—24no—
30C2×#340C2214yes—
31C2×#340C2214yes—
32C2×#340C2214yes—
33C2×#336C2212yes—
34C4⋊#95—21no—
35C4⋊#95—21no—
36C4⋊#95C2212yes—
37C4⋊#95C2212yes—
38C23×Q8C2311yescharacteristic
39C23×C8C2314yes—
40C22×C4⋊C4—21no—
41C22×C4⋊C4—21no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4—22no—
44C22×C4⋊C4—22no—
45C22×C4⋊C4C2314yes—
46C22×C4⋊C4C2311yescharacteristic
47C22×C4⋊C4C2311yescharacteristic
48C22×C4⋊C4C2311yescharacteristic
49C22×C42D411yescharacteristic
50C22×C42D411yescharacteristic
51C22×C42D411yescharacteristic
52C22×C42D411yescharacteristic
53C22×C42C2311yescharacteristic
54C22×C42C2311yescharacteristic
55C22×C42C2311yescharacteristic
56C2×C4⋊C8—28no—
57C2×C4⋊C8—24no—
58C2×#103—28no—
59C2×#103—44no—
60C2×#103—44no—
61C2×C4×C8—24no—
62C2×C4×C8—44no—
63#391—22no—
64#391—22no—
65#391—22no—
66#391C2×C414yes—
67#391C2×C414yes—
68#391C2×C414yes—
69#389—22no—
70#389—22no—
71#389—22no—
72#389C2×C414yes—
73#389C2×C414yes—
74#389C2×C414yes—
75#387—28no—
76#387—28no—
77#387—28no—
78#384—44no—
79#383—28no—
80#382—44no—
81#382—44no—
82C2×#95—21no—
83C2×#95—21no—
84C2×#95—21no—
85C2×#95—21no—
86C2×#95—21no—
87C2×#95—21no—
88C2×#95—21no—
89C2×#95—21no—
90C2×#95—21no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C43C2×C414yes—
96C43D414yes—
97#340—28no—
98#340—44no—
99#340—44no—
100#336—24no—
101C22×Q8—21no—
102C22×Q8—24no—
103C22×Q8—22no—
104C22×Q8—22no—
105C22×Q8—22no—
106C22×Q8—22no—
107C22×Q8C22×C412yes—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—24no—
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2411yescharacteristic
120C22×C8—24no—
121C22×C8—24no—
122C22×C8—24no—
123C22×C8—24no—
124C22×C8—24no—
125C22×C8—24no—
126C22×C8#4118yes—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—24no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—44no—
145C2×C4⋊C4—44no—
146C2×C4⋊C4—42no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—44no—
149C2×C4⋊C4—44no—
150C2×C4⋊C4—44no—
151C2×C4⋊C4—44no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—44no—
154C2×C4⋊C4C22×C412yes—
155C2×C4⋊C4C22×C412yes—
156C2×C4⋊C4C22×C412yes—
157C2×C42—24no—
158C2×C42—22no—
159C2×C42—22no—
160C2×C42—24no—
161C2×C42—22no—
162C2×C42—22no—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42—42no—
166C2×C42C22⋊C414yes—
167C2×C42C22⋊C414yes—
168C2×C42C22⋊C414yes—
169C2×C42C22⋊C414yes—
170C2×C42SD1614yes—
171C2×C42Q1614yes—
172C2×C42C22×C412yes—
173C2×C42C22×C412yes—
174C2×C42C22×C412yes—
175C2×C42C2×D412yes—
176C2×C42C2×D412yes—
177C2×C42C2×D412yes—
178C4⋊C8—216no—
179#103—416no—
180C4×C8—48no—
181#95—24no—
182#95—24no—
183#95—24no—
184#95—24no—
185#95—24no—
186#95—24no—
187#95—24no—
188#95—24no—
189#95—24no—
190#95—44no—
191#95—44no—
192#95—44no—
193#95—44no—
194C24C22×D411yescharacteristic
195C2×Q8—24no—
196C2×Q8—24no—
197C2×Q8—24no—
198C2×Q8—42no—
199C2×Q8—41no—
200C2×Q8—41no—
201C2×Q8—42no—
202C2×Q8—44no—
203C2×Q8—44no—
204C2×Q8—44no—
205C2×Q8—44no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4—44no—
246C22×C4—44no—
247C22×C4—44no—
248C22×C4—44no—
249C22×C4—44no—
250C22×C4C2×C22⋊C412yes—
251C22×C4C2×C22⋊C412yes—
252C22×C4C2×C22⋊C412yes—
253C22×C4C2×C22⋊C412yes—
254C22×C4C2×C22⋊C412yes—
255C22×C4C2×C22⋊C412yes—
256C22×C4C2×SD1612yes—
257C22×C4C2×SD1612yes—
258C22×C4C2×SD1612yes—
259C22×C4C2×Q1612yes—
260C22×C4C2×Q1612yes—
261C22×C4C2×Q1612yes—
262C22×C4#13514yes—
263C22×C4AGL(1,ℤ/8)12yes—
264C22×C4#13712yes—
265C22×C4C23×C411yescharacteristic
266C22×C4C22×D411yescharacteristic
267C22×C4C2×#4114yes—
268C22×C4H(5,𝔽2)12yes—
269C22×C4H(5,𝔽2)12yes—
270C22×C42−1+412yes—
271C22×C42−1+412yes—
272C2×C8—28no—
273C2×C8—28no—
274C2×C8—28no—
275C2×C8—44no—
276C2×C8—44no—
277C2×C8—44no—
278C2×C8—44no—
279C4⋊C4—24no—
280C4⋊C4—24no—
281C4⋊C4—24no—
282C4⋊C4—48no—
283C4⋊C4—42no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—42no—
287C4⋊C4—48no—
288C4⋊C4—48no—
289C4⋊C4—44no—
290C4⋊C4—44no—
291C4⋊C4—44no—
292C4⋊C4—44no—
293C4⋊C4—88no—
294C4⋊C4—88no—
295C42—24no—
296C42—24no—
297C42—24no—
298C42—24no—
299C42—22no—
300C42—22no—
301C42—24no—
302C42—28no—
303C42—28no—
304C42—44no—
305C42#103116yescomplemented
306C23C22×C22⋊C411yescharacteristic
307C23#53212yes—
308C23#53511yescharacteristic
309C23#53611yescharacteristic
310C23#53712yes—
311C23C22×SD1611yescharacteristic
312C23C22×Q1611yescharacteristic
313C23C2×#13512yes—
314C23C2×AGL(1,ℤ/8)11yescharacteristic
315C23C2×#13711yescharacteristic
316C23#57512yes—
317Q8—48no—
318Q8—82no—
319Q8—82no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—24no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
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—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—84no—
368C2×C4—84no—
369C2×C4—84no—
370C2×C4—84no—
371C2×C4—84no—
372C2×C4—84no—
373C2×C4—84no—
374C2×C4C2×#10314yes—
375C2×C4C2×#10314yes—
376C2×C4C2×#10314yes—
377C2×C4#41614yes—
378C2×C4#41714yes—
379C2×C4C4×#4112yes—
380C2×C4#52012yes—
381C2×C4#52012yes—
382C2×C4#52512yes—
383C2×C4#53511yescharacteristic
384C2×C4#53611yescharacteristic
385C2×C4#53712yes—
386C2×C4#53814yes—
387C2×C4#53914yes—
388C2×C4#54114yes—
389C2×C4#54214yes—
390C8—48no—
391C22#229711yescharacteristic
392C22#231111yescharacteristic
393C22#231111yescharacteristic
394C22#266212yes—
395C22#266412yes—
396C22C22×#10311yescharacteristic
397C22C2×#41611yescharacteristic
398C22C2×#41711yescharacteristic
399C22#289611yescharacteristic
400C22#316712yes—
401C22#317012yes—
402C22#317112yes—
403C22#317212yes—
404C22#320811yescharacteristic
405C22#320911yescharacteristic
406C22#321112yes—
407C22#322911yescharacteristic
408C22#323011yescharacteristic
409C22#323212yes—
410C22#324612yes—
411C22#324712yes—
412C22#324812yes—
413C22#324912yes—
414C4—24no—
415C4—24no—
416C4—24no—
417C4—24no—
418C4—24no—
419C4—42no—
420C4—82no—
421C4—82no—
422C4—82no—
423C4—82no—
424C4—164no—
425C4#233414yes—
426C4#263612yes—
427C4#263712yes—
428C2#1429111yescharacteristic
429C2#2140011yescharacteristic
430C2#2140311yescharacteristic
431C2#2186011yescharacteristic
432C2#2186111yescharacteristic
433C2#2188511yescharacteristic
434C2#2188711yescharacteristic
435C2#3033712yes—
436C2#3033812yes—
437C2#3037712yes—
438C2#3037812yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
14. #2863C4semidirect
305. C42#103semidirect

Maximal subgroups

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

subgroupindex
C2×#28632
C2×C4×C4⋊C82
#146872
#112282
#112072
#110772
#110502
#109262
#108802