Finite Group Databasebeta

← #381093

Subgroups — #381093

Order 512.

1668 conjugacy classes of subgroups.

The subgroup lattice has 497 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2854C211yesmaximal, characteristic
2C2×C4×C22⋊C8C211yesmaximal, characteristic, complemented
3#14682C211yesmaximal, characteristic, complemented
4#11208C212yesmaximal, complemented
5#11200C212yesmaximal, complemented
6#11051C212yesmaximal, complemented
7#10969C212yesmaximal
8#10925C212yesmaximal, complemented
9#10880C212yesmaximal, complemented
10C22×#123—21no—
11C22×#121C2211yescharacteristic
12C23×C42C2211yescharacteristic
13C22×#102C2214yes—
14C22×C22⋊C8C2212yes—
15C22×C4×C8C2212yes—
16#2854—24no—
17#2854C418yescomplemented
18C2×#390C2211yescharacteristic
19C2×#390C2211yescharacteristic
20C2×#389C2211yescharacteristic
21C2×#389C2211yescharacteristic
22C2×#388—21no—
23C2×#388—21no—
24C2×#386—21no—
25C2×#382C2212yes—
26C2×#380C2211yescharacteristic
27C2×#380C2212yes—
28C4×C22⋊C8—24no—
29C2×#340C2214yes—
30C2×#340C2214yes—
31C2×#340C2214yes—
32C2×#336C2212yes—
33C4⋊#95C2212yes—
34#1434—21no—
35#1434—21no—
36C22⋊#95—21no—
37C22⋊#95—21no—
38C23×D4—21no—
39C23×D4C2311yescharacteristic
40C24×C4D411yescharacteristic
41C24×C4D411yescharacteristic
42C24×C4C2311yescharacteristic
43C23×C8C2314yes—
44C2×#123—28no—
45C2×#123—48no—
46C2×#121—22no—
47C2×#121—44no—
48C2×#121C2×C414yes—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4C2314yes—
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2311yescharacteristic
55C22×C22⋊C4—21no—
56C22×C22⋊C4—21no—
57C22×C22⋊C4—21no—
58C22×C22⋊C4—21no—
59C22×C22⋊C4—21no—
60C22×C22⋊C4—21no—
61C22×C22⋊C4—22no—
62C22×C42—44no—
63C22×C42C2×C414yes—
64C22×C42D411yescharacteristic
65C22×C42D411yescharacteristic
66C22×C42D414yes—
67C22×C42C2311yescharacteristic
68C22×C42C2311yescharacteristic
69C2×#102—28no—
70C2×#102—44no—
71C2×#102—44no—
72C2×C22⋊C8—24no—
73C2×C22⋊C8—28no—
74C2×C4×C8—24no—
75C2×C4×C8—44no—
76#390—22no—
77#390—22no—
78#390C2×C414yes—
79#390C2×C414yes—
80#389—22no—
81#389—22no—
82#389C2×C414yes—
83#389C2×C414yes—
84#388—28no—
85#388—28no—
86#386—28no—
87#382—44no—
88#380—22no—
89#380—44no—
90#380C2×C414yes—
91C2×#95—21no—
92C2×#95—21no—
93C2×#95—21no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99#340—28no—
100#340—44no—
101#340—44no—
102#336—24no—
103C25—21no—
104C25C2×D411yescharacteristic
105C22×D4—24no—
106C22×D4—21no—
107C22×D4—24no—
108C22×D4—24no—
109C22×D4—22no—
110C22×D4—22no—
111C22×D4—44no—
112C22×D4—44no—
113C22×D4—44no—
114C22×D4C22×C412yes—
115C23×C4—24no—
116C23×C4—24no—
117C23×C4—24no—
118C23×C4—24no—
119C23×C4—24no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—24no—
126C23×C4—42no—
127C23×C4—42no—
128C23×C4—42no—
129C23×C4C22⋊C414yes—
130C23×C4C22⋊C414yes—
131C23×C4C22×C412yes—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D412yes—
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2411yescharacteristic
139C22×C8—24no—
140C22×C8—24no—
141C22×C8—24no—
142C22×C8—24no—
143C22×C8—24no—
144C22×C8—24no—
145C22×C8#4118yes—
146#123—432no—
147#121—48no—
148#121—48no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4C22×C412yes—
167C2×C4⋊C4C22×C412yes—
168C2×C4⋊C4C22×C412yes—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—44no—
176C2×C22⋊C4—44no—
177C2×C22⋊C4—44no—
178C2×C22⋊C4—44no—
179C2×C22⋊C4—44no—
180C2×C22⋊C4—44no—
181C2×C22⋊C4—44no—
182C2×C22⋊C4—44no—
183C2×C22⋊C4—44no—
184C2×C42—24no—
185C2×C42—24no—
186C2×C42—48no—
187C2×C42—48no—
188C2×C42—42no—
189C2×C42—42no—
190C2×C42C22⋊C414yes—
191C2×C42C22⋊C414yes—
192C2×C42C22×C412yes—
193C2×C42C22×C412yes—
194C2×C42C2×D412yes—
195C2×C42C2×D412yes—
196#102—416no—
197C22⋊C8—216no—
198C4×C8—48no—
199#95—24no—
200#95—24no—
201#95—24no—
202#95—44no—
203#95—44no—
204#95—44no—
205#95—44no—
206#95—44no—
207C24—22no—
208C24—22no—
209C24—21no—
210C24—21no—
211C24—22no—
212C24—22no—
213C24—22no—
214C24—22no—
215C24—42no—
216C24—42no—
217C24—41no—
218C24—41no—
219C24—42no—
220C24—42no—
221C24—42no—
222C24—42no—
223C24C2×C22⋊C412yes—
224C24#13514yes—
225C24C22×D411yescharacteristic
226C2×D4—24no—
227C2×D4—24no—
228C2×D4—24no—
229C2×D4—48no—
230C2×D4—48no—
231C2×D4—48no—
232C2×D4—42no—
233C2×D4—42no—
234C2×D4—42no—
235C2×D4—44no—
236C2×D4—44no—
237C2×D4—44no—
238C2×D4—44no—
239C2×D4—816no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—28no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—28no—
262C22×C4—28no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—44no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4—416no—
270C22×C4—416no—
271C22×C4—44no—
272C22×C4—44no—
273C22×C4—44no—
274C22×C4—44no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—44no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4C2×C22⋊C412yes—
293C22×C4C2×C22⋊C412yes—
294C22×C4C2×C22⋊C412yes—
295C22×C4C2×C22⋊C412yes—
296C22×C4C2×C22⋊C412yes—
297C22×C4#13514yes—
298C22×C4AGL(1,ℤ/8)14yes—
299C22×C4C23×C411yescharacteristic
300C22×C4C22×D411yescharacteristic
301C22×C4C2×#4114yes—
302C22×C4H(5,𝔽2)14yes—
303C22×C4H(5,𝔽2)12yes—
304C22×C42−1+412yes—
305C2×C8—28no—
306C2×C8—28no—
307C2×C8—28no—
308C2×C8—44no—
309C2×C8—44no—
310C2×C8—44no—
311C2×C8—44no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—24no—
315C4⋊C4—48no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—48no—
320C4⋊C4—44no—
321C4⋊C4—44no—
322C4⋊C4—44no—
323C4⋊C4—44no—
324C22⋊C4—48no—
325C22⋊C4—48no—
326C22⋊C4—48no—
327C22⋊C4—48no—
328C22⋊C4—48no—
329C22⋊C4—48no—
330C22⋊C4—88no—
331C42—24no—
332C42—24no—
333C42—24no—
334C42—24no—
335C42—44no—
336C42—44no—
337C23—22no—
338C23—22no—
339C23—22no—
340C23—22no—
341C23—22no—
342C23—28no—
343C23—28no—
344C23—24no—
345C23—24no—
346C23—22no—
347C23—44no—
348C23—42no—
349C23—42no—
350C23—44no—
351C23—42no—
352C23—42no—
353C23—42no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23—42no—
359C23—42no—
360C23—84no—
361C23—84no—
362C23—82no—
363C23—82no—
364C23—84no—
365C23—84no—
366C23—82no—
367C23—82no—
368C23—82no—
369C23—82no—
370C23#41614yes—
371C23C22×C22⋊C411yescharacteristic
372C23#53212yes—
373C23#53412yes—
374C23#53511yescharacteristic
375C23#53611yescharacteristic
376C23C2×#13512yes—
377C23C2×#13512yes—
378C23C2×AGL(1,ℤ/8)12yes—
379C23#57512yes—
380D4—48no—
381D4—816no—
382D4—84no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—28no—
388C2×C4—28no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—416no—
399C2×C4—416no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—48no—
405C2×C4—48no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—84no—
416C2×C4—84no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—84no—
420C2×C4—84no—
421C2×C4—84no—
422C2×C4—84no—
423C2×C4—84no—
424C2×C4—84no—
425C2×C4—84no—
426C2×C4#41614yes—
427C2×C4#41814yes—
428C2×C4C4×#4112yes—
429C2×C4#51812yes—
430C2×C4#52012yes—
431C2×C4#52512yes—
432C2×C4#53412yes—
433C2×C4#53712yes—
434C2×C4#53814yes—
435C2×C4#53914yes—
436C2×C4#54014yes—
437C2×C4#54214yes—
438C8—48no—
439C22—24no—
440C22—24no—
441C22—24no—
442C22—42no—
443C22—42no—
444C22—48no—
445C22—48no—
446C22—42no—
447C22—42no—
448C22—82no—
449C22—82no—
450C22—82no—
451C22—82no—
452C22—82no—
453C22—82no—
454C22—82no—
455C22—164no—
456C22—164no—
457C22#229711yescharacteristic
458C22#230811yescharacteristic
459C22#231111yescharacteristic
460C22#266112yes—
461C22#266312yes—
462C22C2×#41611yescharacteristic
463C22C2×#41611yescharacteristic
464C22C2×#41811yescharacteristic
465C22#289611yescharacteristic
466C22#316814yes—
467C22#317114yes—
468C22#318814yes—
469C22#319114yes—
470C22#320712yes—
471C22#321112yes—
472C22#322812yes—
473C22#323212yes—
474C4—24no—
475C4—24no—
476C4—42no—
477C4—42no—
478C4—42no—
479C4—82no—
480C4—82no—
481C4—82no—
482C4—164no—
483C4#233514yes—
484C4#263014yes—
485C2—44no—
486C2—162no—
487C2#1428711yescharacteristic
488C2#2140211yescharacteristic
489C2#2140511yescharacteristic
490C2#2176011yescharacteristic
491C2#2176111yescharacteristic
492C2#2186111yescharacteristic
493C2#2186311yescharacteristic
494C2#3032014yes—
495C2#3036014yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊C8C2semidirect
3. #14682C2semidirect
4. #11208C2semidirect
5. #11200C2semidirect
6. #11051C2semidirect
8. #10925C2semidirect
9. #10880C2semidirect
17. #2854C4semidirect

Maximal subgroups

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

subgroupindex
C2×#28542
C2×C4×C22⋊C82
#146822
#112082
#112002
#110512
#109692
#109252
#108802