Finite Group Databasebeta

← #381268

Subgroups — #381268

Order 512.

1324 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2860C211yesmaximal, characteristic
2C2×C4×C4⋊C8C211yesmaximal, characteristic, complemented
3#14689C211yesmaximal, characteristic, complemented
4#11228C212yesmaximal, complemented
5#11208C212yesmaximal, complemented
6#11076C212yesmaximal
7#11051C212yesmaximal
8#10925C212yesmaximal
9#10880C212yesmaximal, complemented
10C22×#126—21no—
11C22×C4⋊C8C2212yes—
12C22×#102C2214yes—
13C22×C4×C8C2212yes—
14#2860—24no—
15#2860C418yescomplemented
16C2×#390C2211yescharacteristic
17C2×#390C2211yescharacteristic
18C2×#390C2211yescharacteristic
19C2×#389C2211yescharacteristic
20C2×#389C2211yescharacteristic
21C2×#389C2211yescharacteristic
22C2×#388—21no—
23C2×#388—21no—
24C2×#388—21no—
25C2×#382C2212yes—
26C2×C43C2211yescharacteristic
27C4×C4⋊C8—24no—
28C2×#340C2214yes—
29C2×#340C2214yes—
30C2×#340C2214yes—
31C2×#336C2212yes—
32C4⋊#95C2212yes—
33C4⋊#95C2212yes—
34#1434—21no—
35#1434—21no—
36#1434—21no—
37#1434—21no—
38C23×D4C2311yescharacteristic
39C23×C8C2314yes—
40C2×#126—28no—
41C2×#126—48no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4C2314yes—
46C22×C4⋊C4C2311yescharacteristic
47C22×C4⋊C4C2311yescharacteristic
48C22×C4⋊C4C2311yescharacteristic
49C22×C22⋊C4—21no—
50C22×C22⋊C4—21no—
51C22×C22⋊C4—21no—
52C22×C22⋊C4—21no—
53C22×C22⋊C4—21no—
54C22×C22⋊C4—21no—
55C22×C42D411yescharacteristic
56C22×C42D411yescharacteristic
57C22×C42D411yescharacteristic
58C22×C42D411yescharacteristic
59C22×C42C2311yescharacteristic
60C22×C42C2311yescharacteristic
61C22×C42C2311yescharacteristic
62C2×C4⋊C8—28no—
63C2×C4⋊C8—24no—
64C2×#102—28no—
65C2×#102—44no—
66C2×#102—44no—
67C2×C4×C8—24no—
68C2×C4×C8—44no—
69#390—22no—
70#390—22no—
71#390—22no—
72#390C2×C414yes—
73#390C2×C414yes—
74#390C2×C414yes—
75#389—22no—
76#389—22no—
77#389—22no—
78#389C2×C414yes—
79#389C2×C414yes—
80#389C2×C414yes—
81#388—28no—
82#388—28no—
83#388—28no—
84#382—44no—
85C2×#95—21no—
86C2×#95—21no—
87C2×#95—21no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C43C2×C414yes—
95C43D414yes—
96#340—28no—
97#340—44no—
98#340—44no—
99#336—24no—
100C25—21no—
101C22×D4—21no—
102C22×D4—24no—
103C22×D4—22no—
104C22×D4—22no—
105C22×D4—22no—
106C22×D4—22no—
107C22×D4C22×C412yes—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4—24no—
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C23×C4C2411yescharacteristic
119C22×C8—24no—
120C22×C8—24no—
121C22×C8—24no—
122C22×C8—24no—
123C22×C8—24no—
124C22×C8—24no—
125C22×C8#4118yes—
126#126—432no—
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—44no—
147C2×C4⋊C4C22×C412yes—
148C2×C4⋊C4C22×C412yes—
149C2×C4⋊C4C22×C412yes—
150C2×C22⋊C4—24no—
151C2×C22⋊C4—24no—
152C2×C22⋊C4—24no—
153C2×C22⋊C4—24no—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—24no—
156C2×C22⋊C4—44no—
157C2×C22⋊C4—44no—
158C2×C22⋊C4—44no—
159C2×C22⋊C4—44no—
160C2×C22⋊C4—44no—
161C2×C22⋊C4—44no—
162C2×C42—24no—
163C2×C42—24no—
164C2×C42—24no—
165C2×C42—24no—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—42no—
171C2×C42C22⋊C414yes—
172C2×C42C22⋊C414yes—
173C2×C42C22⋊C414yes—
174C2×C42C22⋊C414yes—
175C2×C42C22×C412yes—
176C2×C42C22×C412yes—
177C2×C42C22×C412yes—
178C2×C42C2×D412yes—
179C2×C42C2×D412yes—
180C2×C42C2×D412yes—
181C4⋊C8—216no—
182#102—416no—
183C4×C8—48no—
184#95—24no—
185#95—24no—
186#95—24no—
187#95—44no—
188#95—44no—
189#95—44no—
190#95—44no—
191#95—44no—
192#95—44no—
193C24—22no—
194C24—42no—
195C24—41no—
196C24—41no—
197C24—42no—
198C24—42no—
199C24—42no—
200C24—42no—
201C24—42no—
202C24C22×D411yescharacteristic
203C2×D4—24no—
204C2×D4—24no—
205C2×D4—24no—
206C2×D4—42no—
207C2×D4—41no—
208C2×D4—41no—
209C2×D4—42no—
210C2×D4—44no—
211C2×D4—44no—
212C2×D4—44no—
213C2×D4—44no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—42no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—44no—
245C22×C4—44no—
246C22×C4—44no—
247C22×C4—44no—
248C22×C4—44no—
249C22×C4—44no—
250C22×C4—44no—
251C22×C4C2×C22⋊C412yes—
252C22×C4C2×C22⋊C412yes—
253C22×C4C2×C22⋊C412yes—
254C22×C4C2×C22⋊C412yes—
255C22×C4C2×C22⋊C412yes—
256C22×C4C2×C22⋊C412yes—
257C22×C4#13514yes—
258C22×C4#13514yes—
259C22×C4AGL(1,ℤ/8)14yes—
260C22×C4C23×C411yescharacteristic
261C22×C4C22×D411yescharacteristic
262C22×C4C2×#4114yes—
263C22×C4H(5,𝔽2)12yes—
264C22×C4H(5,𝔽2)12yes—
265C22×C42−1+412yes—
266C22×C42−1+412yes—
267C2×C8—28no—
268C2×C8—28no—
269C2×C8—28no—
270C2×C8—44no—
271C2×C8—44no—
272C2×C8—44no—
273C2×C8—44no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—24no—
277C4⋊C4—48no—
278C4⋊C4—48no—
279C4⋊C4—42no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—42no—
283C4⋊C4—48no—
284C4⋊C4—44no—
285C4⋊C4—44no—
286C4⋊C4—44no—
287C4⋊C4—44no—
288C22⋊C4—48no—
289C22⋊C4—48no—
290C22⋊C4—48no—
291C22⋊C4—48no—
292C22⋊C4—48no—
293C22⋊C4—48no—
294C42—24no—
295C42—24no—
296C42—24no—
297C42—24no—
298C42—22no—
299C42—22no—
300C42—24no—
301C42—28no—
302C42—28no—
303C42—28no—
304C42—44no—
305C23—42no—
306C23—42no—
307C23—42no—
308C23—42no—
309C23—42no—
310C23—42no—
311C23—42no—
312C23—84no—
313C23—84no—
314C23—82no—
315C23—82no—
316C23—82no—
317C23—82no—
318C23—82no—
319C23—82no—
320C23—84no—
321C23—84no—
322C23C22×C22⋊C411yescharacteristic
323C23#53212yes—
324C23#53511yescharacteristic
325C23#53611yescharacteristic
326C23#53712yes—
327C23C2×#13512yes—
328C23C2×#13512yes—
329C23C2×AGL(1,ℤ/8)12yes—
330C23#57512yes—
331D4—48no—
332D4—82no—
333D4—82no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—28no—
340C2×C4—28no—
341C2×C4—28no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—44no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—84no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—84no—
370C2×C4—84no—
371C2×C4—84no—
372C2×C4—84no—
373C2×C4—84no—
374C2×C4—84no—
375C2×C4—84no—
376C2×C4—84no—
377C2×C4#41614yes—
378C2×C4#41614yes—
379C2×C4#41814yes—
380C2×C4C4×#4112yes—
381C2×C4#52012yes—
382C2×C4#52012yes—
383C2×C4#52512yes—
384C2×C4#53511yescharacteristic
385C2×C4#53611yescharacteristic
386C2×C4#53712yes—
387C2×C4#53814yes—
388C2×C4#53914yes—
389C2×C4#54114yes—
390C2×C4#54214yes—
391C8—48no—
392C22—82no—
393C22—82no—
394C22—82no—
395C22—82no—
396C22—82no—
397C22—82no—
398C22—82no—
399C22—164no—
400C22—164no—
401C22#229711yescharacteristic
402C22#231111yescharacteristic
403C22#231111yescharacteristic
404C22#266212yes—
405C22#266412yes—
406C22C2×#41611yescharacteristic
407C22C2×#41611yescharacteristic
408C22C2×#41811yescharacteristic
409C22#289611yescharacteristic
410C22#316814yes—
411C22#317114yes—
412C22#320712yes—
413C22#321212yes—
414C22#322812yes—
415C22#323312yes—
416C22#324714yes—
417C22#325014yes—
418C4—24no—
419C4—24no—
420C4—24no—
421C4—42no—
422C4—42no—
423C4—42no—
424C4—82no—
425C4—82no—
426C4—82no—
427C4—164no—
428C4#233414yes—
429C4#263612yes—
430C4#263712yes—
431C2—162no—
432C2#1429111yescharacteristic
433C2#2140211yescharacteristic
434C2#2140511yescharacteristic
435C2#2186211yescharacteristic
436C2#2186311yescharacteristic
437C2#2188611yescharacteristic
438C2#2188811yescharacteristic
439C2#3034014yes—
440C2#3038114yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
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
2. C2×C4×C4⋊C8C2semidirect
3. #14689C2semidirect
4. #11228C2semidirect
5. #11208C2semidirect
9. #10880C2semidirect
15. #2860C4semidirect

Maximal subgroups

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

subgroupindex
C2×#28602
C2×C4×C4⋊C82
#146892
#112282
#112082
#110762
#110512
#109252
#108802