Finite Group Databasebeta

← #391749

Subgroups — #391749

Order 512.

1612 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2854C211yesmaximal, characteristic
2C2×#1797C211yesmaximal, characteristic, complemented
3#14682C211yesmaximal, characteristic, complemented
4#11227C212yesmaximal, complemented
5#11208C212yesmaximal, complemented
6#11200C212yesmaximal, complemented
7#11076C212yesmaximal, complemented
8#11051C212yesmaximal, complemented
9#10969C212yesmaximal
10C22×#123—21no—
11C22×#121C2211yescharacteristic
12C23×C42C2211yescharacteristic
13C22×C4⋊C8C2212yes—
14C22×#102C2214yes—
15C22×C22⋊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—
28#1797—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×C4⋊C8—24no—
70C2×C4⋊C8—44no—
71C2×#102—44no—
72C2×#102—44no—
73C2×#102—44no—
74C2×C22⋊C8—24no—
75C2×C22⋊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—44no—
100#340—44no—
101#340—44no—
102#336—24no—
103C25—21no—
104C25C2×D411yescharacteristic
105C22×D4—24no—
106C22×D4—21no—
107C22×D4—24no—
108C22×D4—22no—
109C22×D4—22no—
110C22×D4—22no—
111C22×D4—22no—
112C22×D4—44no—
113C22×D4—44no—
114C22×D4—44no—
115C22×D4C22×C412yes—
116C23×C4—24no—
117C23×C4—24no—
118C23×C4—24no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—24no—
127C23×C4—42no—
128C23×C4—42no—
129C23×C4—42no—
130C23×C4—42no—
131C23×C4C22⋊C414yes—
132C23×C4C22⋊C414yes—
133C23×C4C22×C412yes—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D412yes—
139C23×C4C2×D411yescharacteristic
140C23×C4C2411yescharacteristic
141C22×C8—24no—
142C22×C8—24no—
143C22×C8—24no—
144C22×C8—24no—
145C22×C8—24no—
146C22×C8—24no—
147C22×C8—24no—
148#123—432no—
149#121—48no—
150#121—48no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4C22×C412yes—
170C2×C4⋊C4C22×C412yes—
171C2×C4⋊C4C22×C412yes—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—24no—
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×C22⋊C4—44no—
185C2×C22⋊C4—44no—
186C2×C22⋊C4—44no—
187C2×C42—24no—
188C2×C42—24no—
189C2×C42—48no—
190C2×C42—48no—
191C2×C42—42no—
192C2×C42—42no—
193C2×C42C22⋊C414yes—
194C2×C42C22⋊C414yes—
195C2×C42C22×C412yes—
196C2×C42C22×C412yes—
197C2×C42C2×D412yes—
198C2×C42C2×D412yes—
199C4⋊C8—48no—
200#102—416no—
201C22⋊C8—48no—
202#95—24no—
203#95—24no—
204#95—24no—
205#95—44no—
206#95—44no—
207#95—44no—
208#95—44no—
209#95—44no—
210C24—22no—
211C24—21no—
212C24—21no—
213C24—22no—
214C24—22no—
215C24—22no—
216C24—22no—
217C24—22no—
218C24—42no—
219C24—41no—
220C24—41no—
221C24—42no—
222C24—42no—
223C24—42no—
224C24—42no—
225C24—42no—
226C24C2×C22⋊C412yes—
227C24#13514yes—
228C24C22×D411yescharacteristic
229C2×D4—24no—
230C2×D4—24no—
231C2×D4—24no—
232C2×D4—48no—
233C2×D4—48no—
234C2×D4—48no—
235C2×D4—42no—
236C2×D4—41no—
237C2×D4—41no—
238C2×D4—42no—
239C2×D4—44no—
240C2×D4—44no—
241C2×D4—44no—
242C2×D4—44no—
243C2×D4—816no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—28no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—28no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—44no—
272C22×C4—44no—
273C22×C4—44no—
274C22×C4—44no—
275C22×C4—416no—
276C22×C4—416no—
277C22×C4—44no—
278C22×C4—44no—
279C22×C4—44no—
280C22×C4—44no—
281C22×C4—42no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—44no—
292C22×C4—44no—
293C22×C4—44no—
294C22×C4—44no—
295C22×C4—44no—
296C22×C4—44no—
297C22×C4—44no—
298C22×C4C2×C22⋊C412yes—
299C22×C4C2×C22⋊C412yes—
300C22×C4C2×C22⋊C412yes—
301C22×C4C2×C22⋊C412yes—
302C22×C4C2×C22⋊C412yes—
303C22×C4AGL(1,ℤ/8)14yes—
304C22×C4C23×C411yescharacteristic
305C22×C4C22×D411yescharacteristic
306C22×C4H(5,𝔽2)14yes—
307C22×C4H(5,𝔽2)12yes—
308C22×C4H(5,𝔽2)12yes—
309C22×C42−1+412yes—
310C22×C42−1+412yes—
311C2×C8—44no—
312C2×C8—44no—
313C2×C8—44no—
314C2×C8—44no—
315C2×C8—44no—
316C2×C8—44no—
317C2×C8—44no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—48no—
322C4⋊C4—42no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—42no—
326C4⋊C4—48no—
327C4⋊C4—44no—
328C4⋊C4—44no—
329C4⋊C4—44no—
330C4⋊C4—44no—
331C22⋊C4—48no—
332C22⋊C4—48no—
333C22⋊C4—48no—
334C22⋊C4—48no—
335C22⋊C4—48no—
336C22⋊C4—48no—
337C22⋊C4—88no—
338C42—24no—
339C42—24no—
340C42—24no—
341C42—24no—
342C42—44no—
343C42—44no—
344C23—22no—
345C23—22no—
346C23—22no—
347C23—22no—
348C23—22no—
349C23—22no—
350C23—28no—
351C23—24no—
352C23—24no—
353C23—28no—
354C23—44no—
355C23—42no—
356C23—42no—
357C23—44no—
358C23—42no—
359C23—42no—
360C23—42no—
361C23—42no—
362C23—42no—
363C23—42no—
364C23—42no—
365C23—42no—
366C23—42no—
367C23—84no—
368C23—84no—
369C23—82no—
370C23—82no—
371C23—82no—
372C23—82no—
373C23—84no—
374C23—84no—
375C23—82no—
376C23—82no—
377C23#41614yes—
378C23C22×C22⋊C411yescharacteristic
379C23#53412yes—
380C23#53511yescharacteristic
381C23#53611yescharacteristic
382C23#53712yes—
383C23C2×#13512yes—
384C23C2×AGL(1,ℤ/8)12yes—
385C23#57512yes—
386C23#57512yes—
387D4—48no—
388D4—816no—
389D4—82no—
390D4—82no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—28no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—416no—
407C2×C4—416no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—48no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—84no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—84no—
429C2×C4—84no—
430C2×C4—84no—
431C2×C4—84no—
432C2×C4—84no—
433C2×C4—84no—
434C2×C4—84no—
435C2×C4—84no—
436C2×C4#41814yes—
437C2×C4#51812yes—
438C2×C4#52012yes—
439C2×C4#52012yes—
440C2×C4#53412yes—
441C2×C4#53511yescharacteristic
442C2×C4#53611yescharacteristic
443C2×C4#53712yes—
444C2×C4#56014yes—
445C2×C4#56214yes—
446C2×C4#56214yes—
447C2×C4#56314yes—
448C8—84no—
449C22—24no—
450C22—24no—
451C22—24no—
452C22—42no—
453C22—42no—
454C22—42no—
455C22—42no—
456C22—48no—
457C22—48no—
458C22—82no—
459C22—82no—
460C22—82no—
461C22—82no—
462C22—82no—
463C22—82no—
464C22—82no—
465C22—164no—
466C22—164no—
467C22#230811yescharacteristic
468C22#231111yescharacteristic
469C22#231111yescharacteristic
470C22#281312yes—
471C22#281412yes—
472C22C2×#41611yescharacteristic
473C22C2×#41811yescharacteristic
474C22#289611yescharacteristic
475C22#289611yescharacteristic
476C22#318814yes—
477C22#319714yes—
478C22#321112yes—
479C22#321412yes—
480C22#323212yes—
481C22#323512yes—
482C22#325014yes—
483C22#325114yes—
484C4—24no—
485C4—42no—
486C4—42no—
487C4—42no—
488C4—42no—
489C4—82no—
490C4—82no—
491C4—82no—
492C4—164no—
493C4#237614yes—
494C4#281312yes—
495C4#281412yes—
496C2—44no—
497C2—162no—
498C2#1465811yescharacteristic
499C2#2176111yescharacteristic
500C2#2176611yescharacteristic
501C2#2186111yescharacteristic
502C2#2186911yescharacteristic
503C2#2188611yescharacteristic
504C2#2189111yescharacteristic
505C2#3272614yes—
506C2#3274614yes—
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×#1797C2semidirect
3. #14682C2semidirect
4. #11227C2semidirect
5. #11208C2semidirect
6. #11200C2semidirect
7. #11076C2semidirect
8. #11051C2semidirect
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×#17972
#146822
#112272
#112082
#112002
#110762
#110512
#109692