Finite Group Databasebeta

← #241356

Subgroups — #241356

Order 512.

966 conjugacy classes of subgroups.

The subgroup lattice has 477 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35167C211yesmaximal, characteristic
2#21959C211yesmaximal, characteristic
3#21170C211yesmaximal, characteristic
4#20816C211yesmaximal, characteristic
5#10582C212yesmaximal, complemented
6#10580C212yesmaximal, complemented
7#10575C212yesmaximal, complemented
8#10574C212yesmaximal, complemented
9#8045C211yesmaximal, characteristic
10#7897C211yesmaximal, characteristic
11C2×#1279C211yesmaximal, characteristic
12C2×#544—21no—
13C2×#428—21no—
14C2×#423C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×C4×C8C2211yescharacteristic
17#2857—22no—
18#2659—22no—
19#2552—21no—
20#2552C2212yes—
21#2551—21no—
22#2551C2212yes—
23#2550—22no—
24#2550—22no—
25#2316—21no—
26#2285C2211yescharacteristic
27#2285C2211yescharacteristic
28#2285C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30#1949—22no—
31#1948—21no—
32#1948—21no—
33#1948C2212yes—
34#1948C2212yes—
35#1948C2212yes—
36#1947C2212yes—
37#1844C2211yescharacteristic
38#1844C2211yescharacteristic
39#1834C2211yescharacteristic
40#1834C2211yescharacteristic
41#1807—22no—
42#1807—22no—
43C4×#106—22no—
44#1736—21no—
45#1296C2212yes—
46#1296C2212yes—
47#1279C2212yes—
48#1279C2212yes—
49#1279C2212yes—
50#1279C2212yes—
51C22×M16—21no—
52C23×C8—21no—
53C23×C8—21no—
54#544—44no—
55#544—44no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×#125—22no—
60C2×#125—22no—
61C2×C22⋊Q8—24no—
62C2×#117—21no—
63C2×#117—21no—
64C2×#117—21no—
65C22×C4⋊C4D411yescharacteristic
66C22×C4⋊C4D411yescharacteristic
67C22×C42—21no—
68C22×C42C2311yescharacteristic
69#428—44no—
70C2×#107—22no—
71C2×#106—22no—
72C2×#106—22no—
73C2×#106—22no—
74C2×#106—22no—
75C2×#106—22no—
76#423—22no—
77#423Q814yescomplemented
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8Q814yescomplemented
82C2×C4⋊C8C2312yes—
83C2×C4⋊C8C2312yes—
84C2×C4×C8—21no—
85C2×C4×C8—22no—
86C2×C4×C8Q814yescomplemented
87C2×C4×C8C2311yescharacteristic
88C2×C4×C8C2311yescharacteristic
89#398—24no—
90#398—24no—
91#395—24no—
92#395—24no—
93#391—24no—
94#389—22no—
95#389—22no—
96#389—24no—
97#389—22no—
98#389—22no—
99#389—22no—
100#389—22no—
101#384—22no—
102#384D411yescharacteristic
103#384D411yescharacteristic
104#384C2311yescharacteristic
105#384C2311yescharacteristic
106#384C2311yescharacteristic
107#384C2311yescharacteristic
108#382—21no—
109#382—21no—
110#382—21no—
111#380—24no—
112#380—21no—
113#380—21no—
114#380—21no—
115#380—21no—
116#380D412yes—
117#380D412yes—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4D412yes—
125C4×C4⋊C4D412yes—
126C4×C4⋊C4C2312yes—
127C4×C4⋊C4C2312yes—
128C2×#95—21no—
129#340—22no—
130#340—22no—
131#340—22no—
132#340—22no—
133#340—22no—
134#340—22no—
135C22×Q8—24no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C2×M16—24no—
142C2×M16—42no—
143C2×M16—42no—
144C22×C8—21no—
145C22×C8—24no—
146C22×C8—21no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—24no—
150C22×C8—42no—
151C22×C8—42no—
152C22×C8—81no—
153C22×C8—81no—
154C4⋊Q8—44no—
155C4⋊Q8—48no—
156C4⋊Q8—44no—
157C4⋊Q8—82no—
158C4⋊Q8—82no—
159#125—44no—
160#125—48no—
161#125—44no—
162C22⋊Q8—416no—
163#117—42no—
164#117—42no—
165#117—42no—
166#117—42no—
167#117—42no—
168#117—42no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4SD1614yes—
193C2×C4⋊C4SD1614yes—
194C2×C4⋊C4C2×D412yes—
195C2×C4⋊C4C2×D412yes—
196C2×C4⋊C4C2×D412yes—
197C2×C4⋊C4C2×D412yes—
198C2×C4⋊C4C2×D412yes—
199C2×C4⋊C4C2×D412yes—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—41no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—41no—
208C2×C42—81no—
209C2×C42—81no—
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×Q812yes—
214C2×C42C2×Q812yes—
215C2×C42C2×Q812yes—
216C2×C42#4112yes—
217C2×C42C2411yescharacteristic
218#107—44no—
219#106—44no—
220#106—44no—
221#106—44no—
222#106—82no—
223#106—82no—
224C4⋊C8—24no—
225C4⋊C8—24no—
226C4⋊C8—22no—
227C4⋊C8—22no—
228C4⋊C8C2×Q814yes—
229C4⋊C8C2×Q814yes—
230C4×C8—24no—
231C4×C8—21no—
232C4×C8—21no—
233C4×C8C2×Q812yes—
234C4×C8C2×Q812yes—
235#95—24no—
236#95—24no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—41no—
244#95—41no—
245C24#12011yescharacteristic
246C2×Q8—48no—
247C2×Q8—84no—
248C2×Q8—84no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—81no—
266C22×C4—81no—
267C22×C4—81no—
268C22×C4—81no—
269C22×C4#12011yescharacteristic
270C22×C4#12011yescharacteristic
271C22×C4#12011yescharacteristic
272C22×C4C22⋊Q812yes—
273C22×C4C22⋊Q812yes—
274C22×C4C22⋊Q812yes—
275C22×C4C4⋊Q812yes—
276C22×C4C4⋊Q812yes—
277C22×C4C4⋊Q812yes—
278C22×C4C2×SD1612yes—
279C22×C4C2×SD1612yes—
280C22×C4AGL(1,ℤ/8)12yes—
281C22×C4#13712yes—
282C22×C4C22×D411yescharacteristic
283C22×C4C22×D411yescharacteristic
284C22×C4C22×D411yescharacteristic
285M16—48no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—24no—
291C2×C8—24no—
292C2×C8—48no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—84no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—48no—
311C4⋊C4—48no—
312C4⋊C4—48no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—81no—
334C4⋊C4—81no—
335C4⋊C4—84no—
336C4⋊C4—84no—
337C4⋊C4C2×SD1614yes—
338C4⋊C4C2×SD1614yes—
339C22⋊C4—44no—
340C22⋊C4—44no—
341C22⋊C4—82no—
342C42—42no—
343C42—42no—
344C42—42no—
345C42—42no—
346C42—44no—
347C42—82no—
348C42—82no—
349C42C22⋊Q812yes—
350C42C22⋊Q812yes—
351C42C22⋊Q812yes—
352C42C4⋊Q812yes—
353C42C4⋊Q812yes—
354C42C4⋊Q812yes—
355C42C22×Q811yescharacteristic
356C42C22×Q811yescharacteristic
357C42C22×Q811yescharacteristic
358C42C2×#4111yescharacteristic
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23#39512yes—
364C23#44812yes—
365C23#45012yes—
366C23C2×#12011yescharacteristic
367Q8—168no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—84no—
399C2×C4—84no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4#39512yes—
407C2×C4#39512yes—
408C2×C4#39512yes—
409C2×C4#44812yes—
410C2×C4#45012yes—
411C2×C4#47514yes—
412C2×C4#47614yes—
413C2×C4#49814yes—
414C2×C4#50114yes—
415C2×C4C2×C22⋊Q811yescharacteristic
416C2×C4C2×C22⋊Q811yescharacteristic
417C2×C4C2×C22⋊Q811yescharacteristic
418C2×C4C2×C4⋊Q811yescharacteristic
419C2×C4C2×C4⋊Q811yescharacteristic
420C2×C4C2×C4⋊Q811yescharacteristic
421C2×C4C22×SD1611yescharacteristic
422C2×C4C22×SD1611yescharacteristic
423C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
424C2×C4C2×#13711yescharacteristic
425C2×C4#57611yescharacteristic
426C2×C4#57711yescharacteristic
427C2×C4#57711yescharacteristic
428C2×C4#57811yescharacteristic
429C8—24no—
430C8—42no—
431C8—44no—
432C8—44no—
433C22—22no—
434C22—22no—
435C22—22no—
436C22—22no—
437C22—22no—
438C22—22no—
439C22#205814yes—
440C22C2×#39511yescharacteristic
441C22C2×#45011yescharacteristic
442C22C2×#44811yescharacteristic
443C22#301211yescharacteristic
444C22#301311yescharacteristic
445C22#301811yescharacteristic
446C22#301811yescharacteristic
447C4—22no—
448C4—21no—
449C4—21no—
450C4—22no—
451C4—44no—
452C4—44no—
453C4—82no—
454C4—82no—
455C4—82no—
456C4—82no—
457C4—161no—
458C4—161no—
459C4#205814yes—
460C4C2×#47611yescharacteristic
461C4C2×#47511yescharacteristic
462C4#308111yescharacteristic
463C4#308211yescharacteristic
464C4C2×#49811yescharacteristic
465C4C2×#50111yescharacteristic
466C4#316311yescharacteristic
467C4#316411yescharacteristic
468C2—24no—
469C2#1866311yescharacteristic
470C2#1866411yescharacteristic
471C2#1867511yescharacteristic
472C2#1867611yescharacteristic
473C2C2×#205811yescharacteristic
474C2#2355711yescharacteristic
475C2#2356011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10582C2semidirect
6. #10580C2semidirect
7. #10575C2semidirect
8. #10574C2semidirect
77. #423Q8semidirect
81. C2×C4⋊C8Q8semidirect
86. C2×C4×C8Q8semidirect

Maximal subgroups

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

subgroupindex
#351672
#219592
#211702
#208162
#105822
#105802
#105752
#105742
#80452
#78972
C2×#12792