Finite Group Databasebeta

← #237759

Subgroups — #237759

Order 512.

1282 conjugacy classes of subgroups.

The subgroup lattice has 473 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2659C211yesmaximal, characteristic
2#21759C212yesmaximal
3#14281C211yesmaximal, characteristic
4#10343C214yesmaximal, complemented
5#10341C214yesmaximal, complemented
6#8023C212yesmaximal
7C2×#1280C211yesmaximal, characteristic
8C22×C22⋊Q8—21no—
9C22×C22⋊Q8C2211yescharacteristic
10C22×C22⋊C8C2212yes—
11#2659—24no—
12#2659—22no—
13#2659C2214yes—
14C2×#398—22no—
15C2×#395—22no—
16C2×#380—21no—
17C2×#380C2211yescharacteristic
18C2×#380C2212yes—
19C22×#95C2211yescharacteristic
20#1892—24no—
21#1872—22no—
22#1872—22no—
23#1824C2214yes—
24#1787C2214yes—
25#1446C2212yes—
26#1445C2212yes—
27#1441—22no—
28C22⋊#95—21no—
29C22⋊#95—21no—
30#1294C2218yes—
31#1280C2214yes—
32C23×Q8—21no—
33C23×Q8C2311yescharacteristic
34C24×C4D411yescharacteristic
35C24×C4D411yescharacteristic
36C24×C4C2311yescharacteristic
37C23×C8—22no—
38C2×C22⋊Q8—24no—
39C2×C22⋊Q8—24no—
40C2×C22⋊Q8—22no—
41C2×C22⋊Q8—22no—
42C2×C22⋊Q8—22no—
43C2×C22⋊Q8—24no—
44C2×C22⋊Q8—44no—
45C2×C22⋊Q8—44no—
46C2×C22⋊Q8Q814yescomplemented
47C22×C4⋊C4—22no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4D412yes—
54C22×C4⋊C4D412yes—
55C22×C4⋊C4C2311yescharacteristic
56C22×C22⋊C4—22no—
57C22×C22⋊C4—21no—
58C22×C22⋊C4—21no—
59C2×#103—24no—
60C2×#103—24no—
61C2×C22⋊C8—24no—
62C2×C22⋊C8—24no—
63C2×C22⋊C8—22no—
64C2×C22⋊C8C2314yes—
65#398—28no—
66#398—28no—
67#395—28no—
68#395—28no—
69#380—24no—
70#380—22no—
71#380—22no—
72#380—24no—
73#380—24no—
74#380D414yes—
75#380C2314yes—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95D412yes—
85C2×#95C2312yes—
86C2×#95C2312yes—
87#340—24no—
88#340—24no—
89C25C2×D411yescharacteristic
90C22×Q8—24no—
91C22×Q8—21no—
92C22×Q8—22no—
93C22×Q8—22no—
94C22×Q8—44no—
95C22×Q8—81no—
96C22×Q8—81no—
97C22×Q8—82no—
98C22×Q8—82no—
99C22×Q8C2×Q812yescomplemented
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—22no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—22no—
111C23×C4—81no—
112C23×C4—81no—
113C23×C4—81no—
114C23×C4—81no—
115C23×C4—81no—
116C23×C4—81no—
117C23×C4SD1614yes—
118C23×C4Q1614yes—
119C23×C4C2×D412yes—
120C23×C4C2×D412yes—
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D412yes—
123C23×C4C2×D412yes—
124C23×C4C2×Q812yes—
125C23×C4#4114yes—
126C23×C4#4112yes—
127C23×C4C2411yescharacteristic
128C22×C8—24no—
129C22×C8—44no—
130C22×C8—82no—
131C22×C8—82no—
132C22⋊Q8—28no—
133C22⋊Q8—28no—
134C22⋊Q8—48no—
135C22⋊Q8—416no—
136C22⋊Q8—44no—
137C22⋊Q8—44no—
138C22⋊Q8—48no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—28no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—44no—
150C2×C4⋊C4—48no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—82no—
155C2×C4⋊C4—82no—
156C2×C4⋊C4—82no—
157C2×C4⋊C4—82no—
158C2×C4⋊C4—81no—
159C2×C4⋊C4—81no—
160C2×C4⋊C4C2×D414yes—
161C2×C4⋊C4C2×D414yes—
162C2×C4⋊C4C2×Q812yescomplemented
163C2×C22⋊C4—24no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—41no—
167C2×C22⋊C4—41no—
168C2×C22⋊C4—42no—
169C2×C22⋊C4—44no—
170C2×C22⋊C4—44no—
171C2×C22⋊C4—42no—
172C2×C22⋊C4—42no—
173C2×C22⋊C4—42no—
174#103—84no—
175#103—84no—
176C22⋊C8—24no—
177C22⋊C8—24no—
178C22⋊C8—44no—
179#95—28no—
180#95—24no—
181#95—24no—
182#95—24no—
183#95—24no—
184#95—24no—
185#95—82no—
186#95—82no—
187#95—82no—
188#95—82no—
189C24—41no—
190C24—41no—
191C24—41no—
192C24—41no—
193C24#12011yescharacteristic
194C24#12011yescharacteristic
195C24C22⋊Q812yes—
196C24#12412yes—
197C24C2×SD1612yes—
198C24C2×Q1612yes—
199C24AGL(1,ℤ/8)12yes—
200C24#13712yes—
201C24C22×D411yescharacteristic
202C2×Q8—24no—
203C2×Q8—48no—
204C2×Q8—44no—
205C2×Q8—44no—
206C2×Q8—42no—
207C2×Q8—82no—
208C2×Q8—82no—
209C2×Q8—84no—
210C2×Q8—84no—
211C2×Q8—84no—
212C2×Q8—816no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
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—42no—
248C22×C4—44no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—44no—
254C22×C4—44no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—82no—
263C22×C4—84no—
264C22×C4—82no—
265C22×C4—84no—
266C22×C4—84no—
267C22×C4—81no—
268C22×C4—81no—
269C22×C4—81no—
270C22×C4—81no—
271C22×C4—84no—
272C22×C4—84no—
273C22×C4#12012yes—
274C22×C4C22⋊Q814yes—
275C22×C4C22⋊Q814yes—
276C22×C4C22⋊Q812yes—
277C22×C4#12414yes—
278C22×C4#12414yes—
279C22×C4#12412yes—
280C22×C4C2×SD1614yes—
281C22×C4C2×Q1614yes—
282C22×C4C22×D412yes—
283C22×C4C22×Q811yescharacteristic
284C22×C4C2×#4112yes—
285C22×C4C2×#4111yescharacteristic
286C2×C8—44no—
287C2×C8—44no—
288C2×C8—44no—
289C2×C8—88no—
290C4⋊C4—24no—
291C4⋊C4—44no—
292C4⋊C4—44no—
293C4⋊C4—48no—
294C4⋊C4—416no—
295C4⋊C4—42no—
296C4⋊C4—48no—
297C4⋊C4—82no—
298C4⋊C4—82no—
299C4⋊C4—84no—
300C4⋊C4—84no—
301C4⋊C4—84no—
302C22⋊C4—44no—
303C22⋊C4—44no—
304C22⋊C4—44no—
305C22⋊C4—44no—
306C22⋊C4—48no—
307C22⋊C4—84no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—42no—
321C23—42no—
322C23—42no—
323C23—42no—
324C23—42no—
325C23—42no—
326C23—42no—
327C23—42no—
328C23—42no—
329C23—42no—
330C23—42no—
331C23—42no—
332C23#39312yes—
333C23#39312yes—
334C23#45012yes—
335C23#45012yes—
336C23#45112yes—
337C23#45112yes—
338C23#47514yes—
339C23#47714yes—
340C23#48714yes—
341C23#48914yes—
342C23C2×#12011yescharacteristic
343C23C2×C22⋊Q811yescharacteristic
344C23C2×#12411yescharacteristic
345C23C22×SD1611yescharacteristic
346C23C22×Q1611yescharacteristic
347C23C2×AGL(1,ℤ/8)11yescharacteristic
348C23C2×#13711yescharacteristic
349Q8—48no—
350Q8—84no—
351Q8—816no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—42no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—42no—
363C2×C4—44no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—48no—
368C2×C4—48no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—84no—
388C2×C4—84no—
389C2×C4—88no—
390C2×C4—82no—
391C2×C4—88no—
392C2×C4—82no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4#39314yes—
405C2×C4C2×C22⋊Q812yes—
406C2×C4C2×#12412yes—
407C8—84no—
408C22—22no—
409C22—21no—
410C22—21no—
411C22—22no—
412C22—21no—
413C22—22no—
414C22—21no—
415C22—22no—
416C22—22no—
417C22—21no—
418C22—21no—
419C22—22no—
420C22—21no—
421C22—21no—
422C22—22no—
423C22—22no—
424C22—21no—
425C22—21no—
426C22—22no—
427C22—22no—
428C22—21no—
429C22—21no—
430C22—22no—
431C22—22no—
432C22—44no—
433C22—44no—
434C22—44no—
435C22—44no—
436C22#202414yes—
437C22#202414yes—
438C22C2×#39311yescharacteristic
439C22C2×#45011yescharacteristic
440C22C2×#45111yescharacteristic
441C22#301911yescharacteristic
442C22#302011yescharacteristic
443C22#302611yescharacteristic
444C22#302611yescharacteristic
445C22C2×#47511yescharacteristic
446C22C2×#47711yescharacteristic
447C22C2×#48711yescharacteristic
448C22C2×#48911yescharacteristic
449C4—22no—
450C4—22no—
451C4—24no—
452C4—44no—
453C4—44no—
454C4—84no—
455C4—84no—
456C4—82no—
457C4—82no—
458C4—162no—
459C2—21no—
460C2—21no—
461C2—21no—
462C2—21no—
463C2—24no—
464C2—24no—
465C2#1828511yescharacteristic
466C2#1828711yescharacteristic
467C2#1829311yescharacteristic
468C2#1829511yescharacteristic
469C2C2×#202411yescharacteristic
470C2#2346211yescharacteristic
471C2#2346811yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #10343C2semidirect
5. #10341C2semidirect
46. C2×C22⋊Q8Q8semidirect
99. C22×Q8C2×Q8semidirect
162. C2×C4⋊C4C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26592
#217592
#142812
#103432
#103412
#80232
C2×#12802