Finite Group Databasebeta

← #414894

Subgroups — #414894

Order 512.

1190 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35281C211yesmaximal, characteristic
2C2×#1997C211yesmaximal, characteristic
3#21312C211yesmaximal, characteristic
4C2×#1986C211yesmaximal, characteristic
5C4×#428C211yesmaximal, characteristic
6#14161C211yesmaximal, characteristic
7#11276C214yesmaximal
8#11217C214yesmaximal
9#10852C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12C2×#428C2212yes—
13C2×#423C2211yescharacteristic
14C2×C4×M16C2211yescharacteristic
15#2868—21no—
16#2862—21no—
17#2660—22no—
18#2567—21no—
19#2567—21no—
20#2546—22no—
21#2502—22no—
22#2434—22no—
23#2317C2214yes—
24C4×#117C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27#1997—24no—
28#1989—22no—
29#1988—21no—
30#1987—21no—
31#1986—24no—
32#1776—22no—
33C4×#107—21no—
34C4×#106—21no—
35C2×#343C2212yes—
36C2×#343C2212yes—
37#1736C2212yes—
38#1736C2214yes—
39C2×#339C2212yes—
40C2×#339C2214yes—
41C2×#339C2212yes—
42#1446—22no—
43C4⋊#95—22no—
44C4⋊#95C2214yes—
45C23×Q8D412yes—
46C22×M16C2312yes—
47C22×M16C2311yescharacteristic
48C22×M16C2311yescharacteristic
49#533—22no—
50#533—22no—
51#533—22no—
52#533—22no—
53C2×C4⋊Q8—24no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8D411yescharacteristic
63C2×C4⋊Q8D411yescharacteristic
64C2×#125D412yes—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8—21no—
67C2×C4×Q8—21no—
68C2×C4×Q8—21no—
69C2×#117C2312yes—
70C2×#117C2312yes—
71C22×C4⋊C4—21no—
72C22×C4⋊C4D412yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C42—21no—
76C22×C42C2312yes—
77C22×C42C2311yescharacteristic
78#428—24no—
79#428Q818yescomplemented
80C2×#107—22no—
81C2×#106—22no—
82#423—22no—
83#423—22no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C4×M16—22no—
87C4×M16D414yes—
88C2×#97—21no—
89C2×C4×C8—21no—
90#398—28no—
91#395—24no—
92#395—24no—
93#391—22no—
94#391—22no—
95#391—24no—
96#389—24no—
97#389—24no—
98#385—28no—
99#384—22no—
100#384—22no—
101#384—24no—
102#382—24no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4D411yescharacteristic
110C4×C4⋊C4D411yescharacteristic
111C4×C4⋊C4D412yes—
112C4×C22⋊C4—21no—
113C4×C22⋊C4—21no—
114#376—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—28no—
118C43—21no—
119#343—24no—
120#343—24no—
121#340—24no—
122#340—24no—
123#340—22no—
124#340—22no—
125#339—24no—
126#339—24no—
127#339—28no—
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—24no—
133C22×Q8—24no—
134C22×Q8—84no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D414yes—
139C23×C4C2×D411yescharacteristic
140C23×C4C2411yescharacteristic
141C2×M16—21no—
142C2×M16—22no—
143C2×M16—21no—
144C2×M16—22no—
145C2×M16—22no—
146C2×M16—22no—
147C2×M16—24no—
148C2×M16—22no—
149C2×M16—24no—
150C2×M16C2×D412yes—
151C2×M16C2×D412yes—
152C2×M16C2×Q814yes—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—22no—
156C4⋊Q8—22no—
157C4⋊Q8—22no—
158C4⋊Q8—22no—
159C4⋊Q8—22no—
160C4⋊Q8—42no—
161C4⋊Q8—42no—
162C4⋊Q8—42no—
163C4⋊Q8—42no—
164C4⋊Q8—42no—
165C4⋊Q8—42no—
166C4⋊Q8—42no—
167C4⋊Q8—42no—
168C4⋊Q8—88no—
169#125—24no—
170#125—24no—
171C22⋊Q8—42no—
172C22⋊Q8—42no—
173C22⋊Q8—42no—
174C22⋊Q8—42no—
175C4×Q8—42no—
176C4×Q8—42no—
177C4×Q8—42no—
178C4×Q8—42no—
179#117—22no—
180#117—22no—
181#117—22no—
182#117—22no—
183#117—22no—
184#117—22no—
185#117C2×Q814yes—
186#117#4114yes—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4C2×D412yes—
212C2×C4⋊C4C2×D412yes—
213C2×C4⋊C4C2×D412yes—
214C2×C4⋊C4C2×D412yes—
215C2×C4⋊C4C2×Q814yes—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—24no—
225C2×C42—22no—
226C2×C42—24no—
227C2×C42—44no—
228C2×C42—41no—
229C2×C42—41no—
230C2×C42C2×D412yes—
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42#4112yes—
236C2×C42#4112yes—
237C2×C42#4112yes—
238#107—28no—
239#106—28no—
240C4⋊C8—42no—
241C4⋊C8—42no—
242#97—24no—
243C4×C8—24no—
244#95—28no—
245#95—28no—
246#95—21no—
247#95—21no—
248#95—22no—
249#95—41no—
250#95—41no—
251#95—42no—
252#95—88no—
253C24C22×D411yescharacteristic
254C2×Q8—44no—
255C2×Q8—48no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—44no—
259C2×Q8—44no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—84no—
263C2×Q8—88no—
264C2×Q8—88no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—24no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—22no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—42no—
277C22×C4—48no—
278C22×C4—44no—
279C22×C4—44no—
280C22×C4—44no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—44no—
285C22×C4—42no—
286C22×C4#12011yescharacteristic
287C22×C4#12011yescharacteristic
288C22×C4#12011yescharacteristic
289C22×C4#12011yescharacteristic
290C22×C4#12112yes—
291C22×C4#12112yes—
292C22×C4#12312yes—
293C22×C4#12312yes—
294C22×C4C22×D411yescharacteristic
295C22×C4C22×D411yescharacteristic
296C22×C4C22×D411yescharacteristic
297C22×C4C22×Q812yes—
298C22×C4C2×#4111yescharacteristic
299C22×C4C2×#4111yescharacteristic
300C22×C4C2×#4112yes—
301C22×C4C2×#4111yescharacteristic
302C22×C42−1+412yes—
303C22×C42−1+412yes—
304M16—22no—
305M16—24no—
306M16—22no—
307M16—22no—
308M16—42no—
309M16—42no—
310M16—42no—
311M16—44no—
312M16C4⋊Q814yescomplemented
313C2×C8—22no—
314C2×C8—24no—
315C2×C8—22no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—44no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—24no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—24no—
327C4⋊C4—48no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—44no—
335C4⋊C4—44no—
336C4⋊C4—44no—
337C4⋊C4—44no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—88no—
341C4⋊C4—88no—
342C4⋊C4C22⋊Q814yes—
343C4⋊C4C22⋊Q814yes—
344C22⋊C4—24no—
345C22⋊C4—24no—
346C22⋊C4—42no—
347C22⋊C4—42no—
348C42—24no—
349C42—24no—
350C42—44no—
351C42—42no—
352C42—42no—
353C42—42no—
354C42—42no—
355C42—42no—
356C42—42no—
357C42—44no—
358C42—82no—
359C42—82no—
360C42#12112yes—
361C42#12112yes—
362C42#12112yes—
363C42#12112yes—
364C42#12312yes—
365C42#12312yes—
366C42#12412yes—
367C42#12412yes—
368C23—21no—
369C23—21no—
370C23—22no—
371C23C2×#12011yescharacteristic
372C23C2×#12111yescharacteristic
373C23C2×#12311yescharacteristic
374C23#54812yes—
375C23D4×Q812yes—
376Q8—84no—
377Q8—88no—
378Q8—88no—
379Q8—84no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—44no—
395C2×C4—48no—
396C2×C4—48no—
397C2×C4—48no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—44no—
405C2×C4—48no—
406C2×C4—84no—
407C2×C4—82no—
408C2×C4—84no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—84no—
413C2×C4#39412yes—
414C2×C4#39412yes—
415C2×C4#39412yes—
416C2×C4#39412yes—
417C2×C4#45412yes—
418C2×C4#45612yes—
419C2×C4C2×#12111yescharacteristic
420C2×C4C2×#12111yescharacteristic
421C2×C4C2×C22⋊Q812yes—
422C2×C4C2×#12311yescharacteristic
423C2×C4C2×#12411yescharacteristic
424C2×C4C2×C4⋊Q811yescharacteristic
425C2×C4#53211yescharacteristic
426C2×C4#53312yes—
427C2×C4#53612yes—
428C2×C4#53611yescharacteristic
429C2×C4#53611yescharacteristic
430C2×C4#54112yes—
431C2×C4#54812yes—
432C2×C4D4×Q812yes—
433C2×C4#55612yes—
434C2×C4#55612yes—
435C2×C4#55712yes—
436C2×C4#55712yes—
437C8—24no—
438C8—42no—
439C8—82no—
440C22—22no—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22C2×#39411yescharacteristic
449C22#240912yes—
450C22#244612yes—
451C22#254511yescharacteristic
452C22#254611yescharacteristic
453C22C2×#45411yescharacteristic
454C22C2×#45611yescharacteristic
455C22#301912yes—
456C4—22no—
457C4—22no—
458C4—22no—
459C4—44no—
460C4—44no—
461C4—84no—
462C4—84no—
463C4—82no—
464C4—84no—
465C4—82no—
466C4#203814yes—
467C4#244612yes—
468C4#254511yescharacteristic
469C4#254611yescharacteristic
470C4#256812yes—
471C4#262111yescharacteristic
472C4#285711yescharacteristic
473C2—21no—
474C2—22no—
475C2—21no—
476C2#1518011yescharacteristic
477C2C2×#203811yescharacteristic
478C2#2349311yescharacteristic
479C2#2432112yes—
480C2#2432312yes—
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
79. #428Q8semidirect
312. M16C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
#352812
C2×#19972
#213122
C2×#19862
C4×#4282
#141612
#112762
#112172
#108522