Finite Group Databasebeta

← #415284

Subgroups — #415284

Order 512.

1140 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35281C211yesmaximal, characteristic
2C2×#1997C211yesmaximal, characteristic
3C2×#1995C211yesmaximal, characteristic
4#21323C211yesmaximal, characteristic
5#21172C211yesmaximal, characteristic
6#14282C211yesmaximal, characteristic
7#11276C214yesmaximal
8#11217C214yesmaximal
9#10701C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12C2×#428C2212yes—
13C2×#423C2211yescharacteristic
14C2×C4×M16C2211yescharacteristic
15#2868—21no—
16#2862—21no—
17#2660—22no—
18#2567—22no—
19#2546—22no—
20#2502—22no—
21#2434—22no—
22#2317C2212yes—
23#2317C2212yes—
24C2×#382C2211yescharacteristic
25C4×#117C2211yescharacteristic
26C2×#376C2211yescharacteristic
27#1997—24no—
28#1995—24no—
29#1993—21no—
30#1993—21no—
31#1991—22no—
32#1949—22no—
33#1949—22no—
34C2×#343C2214yes—
35#1736C2212yes—
36#1736C2214yes—
37C2×#339C2214yes—
38C2×#339C2212yes—
39C2×#339C2212yes—
40#1446—22no—
41#1445—22no—
42C4⋊#95C2214yes—
43C23×Q8D412yes—
44C22×M16C2312yes—
45C22×M16C2312yes—
46#533—22no—
47#533—22no—
48#533—22no—
49#533—22no—
50C2×C4⋊Q8—24no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8D411yescharacteristic
60C2×C4⋊Q8D411yescharacteristic
61C2×#125D412yes—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8—21no—
64C2×C4×Q8—21no—
65C2×C4×Q8—21no—
66C2×#117C2312yes—
67C2×#117C2311yescharacteristic
68C2×#117C2311yescharacteristic
69C22×C4⋊C4D412yes—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C42C2312yes—
73C22×C42C2311yescharacteristic
74#428—28no—
75C2×#107—22no—
76C2×#106—22no—
77#423—22no—
78#423D414yes—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C4×M16—22no—
82C4×M16—22no—
83C2×#97—21no—
84C2×C4×C8—21no—
85#398—28no—
86#395—28no—
87#391—24no—
88#391—24no—
89#389—24no—
90#389—24no—
91#385—24no—
92#385—24no—
93#384—24no—
94#384—22no—
95#384—22no—
96#382—22no—
97#382—22no—
98#382—21no—
99#382—21no—
100#382—21no—
101#382—21no—
102C4×C4⋊C4D412yes—
103C4×C4⋊C4D412yes—
104C4×C22⋊C4—21no—
105C4×C22⋊C4—21no—
106#376—21no—
107#376—22no—
108#376—22no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—28no—
114C43—21no—
115#343—28no—
116#340—24no—
117#340—24no—
118#340—24no—
119#339—28no—
120#339—24no—
121#339Q818yescomplemented
122C22×Q8—21no—
123C22×Q8—21no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—24no—
127C22×Q8—24no—
128C22×Q8—84no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4C2×D414yes—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2411yescharacteristic
135C2×M16—22no—
136C2×M16—22no—
137C2×M16—22no—
138C2×M16—22no—
139C2×M16—22no—
140C2×M16—24no—
141C2×M16—24no—
142C2×M16—22no—
143C2×M16C2×D414yes—
144C2×M16C2×Q814yes—
145C22×C8—22no—
146C22×C8—22no—
147C4⋊Q8—22no—
148C4⋊Q8—22no—
149C4⋊Q8—22no—
150C4⋊Q8—22no—
151C4⋊Q8—42no—
152C4⋊Q8—42no—
153C4⋊Q8—42no—
154C4⋊Q8—42no—
155C4⋊Q8—42no—
156C4⋊Q8—42no—
157C4⋊Q8—42no—
158C4⋊Q8—42no—
159C4⋊Q8—88no—
160#125—24no—
161#125—24no—
162C22⋊Q8—42no—
163C22⋊Q8—42no—
164C22⋊Q8—42no—
165C22⋊Q8—42no—
166C4×Q8—42no—
167C4×Q8—42no—
168C4×Q8—42no—
169C4×Q8—42no—
170#117—22no—
171#117—22no—
172#117—22no—
173#117—22no—
174#117—22no—
175#117—22no—
176#117C2×Q814yes—
177#117#4114yes—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4C2×D412yes—
199C2×C4⋊C4C2×D412yes—
200C2×C4⋊C4C2×D412yes—
201C2×C4⋊C4C2×D412yes—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—22no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42C2×D412yes—
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×Q814yes—
221C2×C42#4112yes—
222C2×C42#4112yes—
223C2×C42#4112yes—
224#107—44no—
225#106—44no—
226C4⋊C8—24no—
227C4⋊C8—24no—
228#97—42no—
229C4×C8—42no—
230#95—28no—
231#95—28no—
232#95—22no—
233#95—22no—
234#95—41no—
235#95—41no—
236#95—41no—
237#95—41no—
238#95—42no—
239#95—42no—
240#95—88no—
241C24C22×D411yescharacteristic
242C2×Q8—44no—
243C2×Q8—48no—
244C2×Q8—44no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×Q8—44no—
248C2×Q8—44no—
249C2×Q8—84no—
250C2×Q8—88no—
251C2×Q8—88no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—48no—
263C22×C4—44no—
264C22×C4—44no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4—42no—
270C22×C4—44no—
271C22×C4#12011yescharacteristic
272C22×C4#12011yescharacteristic
273C22×C4#12011yescharacteristic
274C22×C4#12011yescharacteristic
275C22×C4#12112yes—
276C22×C4#12112yes—
277C22×C4#12312yes—
278C22×C4#12312yes—
279C22×C4C22×D411yescharacteristic
280C22×C4C22×D411yescharacteristic
281C22×C4C22×D411yescharacteristic
282C22×C4C22×Q812yes—
283C22×C4C2×#4111yescharacteristic
284C22×C4C2×#4111yescharacteristic
285C22×C4C2×#4111yescharacteristic
286C22×C4C2×#4112yes—
287C22×C42−1+412yes—
288C22×C42−1+412yes—
289M16—24no—
290M16—28no—
291M16—24no—
292M16—42no—
293M16—42no—
294M16—44no—
295C2×C8—24no—
296C2×C8—24no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—44no—
312C4⋊C4—44no—
313C4⋊C4—44no—
314C4⋊C4—44no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—44no—
320C4⋊C4—88no—
321C4⋊C4—88no—
322C4⋊C4#12414yes—
323C4⋊C4C4⋊Q814yes—
324C22⋊C4—24no—
325C22⋊C4—24no—
326C22⋊C4—42no—
327C22⋊C4—42no—
328C42—24no—
329C42—24no—
330C42—24no—
331C42—24no—
332C42—44no—
333C42—44no—
334C42—42no—
335C42—42no—
336C42—42no—
337C42—42no—
338C42#12112yes—
339C42#12112yes—
340C42#12112yes—
341C42#12112yes—
342C42#12312yes—
343C42#12312yes—
344C42#12412yes—
345C42#12412yes—
346C23—22no—
347C23—22no—
348C23C2×#12011yescharacteristic
349C23C2×#12111yescharacteristic
350C23C2×#12311yescharacteristic
351C23#54812yes—
352C23D4×Q812yes—
353Q8—84no—
354Q8—84no—
355Q8—88no—
356Q8—88no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—48no—
372C2×C4—48no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—48no—
379C2×C4—44no—
380C2×C4—84no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—84no—
384C2×C4—88no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4#39412yes—
389C2×C4#39412yes—
390C2×C4#39412yes—
391C2×C4#39412yes—
392C2×C4C2×#12111yescharacteristic
393C2×C4C2×#12111yescharacteristic
394C2×C4C2×#12311yescharacteristic
395C2×C4C2×#12411yescharacteristic
396C2×C4C2×#12411yescharacteristic
397C2×C4C2×C4⋊Q811yescharacteristic
398C2×C4#53312yes—
399C2×C4#53312yes—
400C2×C4#53611yescharacteristic
401C2×C4#53611yescharacteristic
402C2×C4#54112yes—
403C2×C4#54112yes—
404C2×C4#54712yes—
405C2×C4#54812yes—
406C2×C4D4×Q812yes—
407C2×C4D4×Q812yes—
408C2×C4#55612yes—
409C2×C4#55712yes—
410C8—44no—
411C8—44no—
412C22—24no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—24no—
418C22C2×#39411yescharacteristic
419C22#242912yes—
420C22#243411yescharacteristic
421C22#243611yescharacteristic
422C22#244612yes—
423C22#301912yes—
424C22#302012yes—
425C4—22no—
426C4—22no—
427C4—22no—
428C4—22no—
429C4—44no—
430C4—44no—
431C4—84no—
432C4—84no—
433C4—82no—
434C4—82no—
435C4—162no—
436C4#242912yes—
437C4#244612yes—
438C4#254611yescharacteristic
439C4#254711yescharacteristic
440C4#264211yescharacteristic
441C4#286211yescharacteristic
442C2—22no—
443C2—22no—
444C2#1518811yescharacteristic
445C2#2349311yescharacteristic
446C2#2349411yescharacteristic
447C2#2583212yes—
448C2#2592612yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
121. #339Q8semidirect

Maximal subgroups

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

subgroupindex
#352812
C2×#19972
C2×#19952
#213232
#211722
#142822
#112762
#112172
#107012