Finite Group Databasebeta

← #213534

Subgroups — #213534

Order 512.

1360 conjugacy classes of subgroups.

The subgroup lattice has 475 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#123C211yesmaximal, characteristic
2#20715C212yesmaximal, complemented
3#9086C214yesmaximal, complemented
4#9084C214yesmaximal, complemented
5#8700C212yesmaximal, complemented
6#8301C212yesmaximal
7C22×#123C411yescharacteristic
8C22×C4×D4C2211yescharacteristic
9C23×C42C2211yescharacteristic
10C2×#432C2214yes—
11C4×#123—22no—
12C4×#123—22no—
13C4×#123—22no—
14C4×#123—22no—
15C2×#388C2212yes—
16C2×#387C2212yes—
17C2×#386C411yescharacteristic
18C2×#385C411yescharacteristic
19C2×#381C411yescharacteristic
20C2×C4×C4⋊C4C412yes—
21C2×C4×C22⋊C4C412yes—
22C2×C4×C22⋊C4C2211yescharacteristic
23#1817C2214yes—
24#1797—22no—
25#1797—22no—
26#1779C2214yes—
27#1749—22no—
28#1749—22no—
29C2×#324C2214yes—
30C2×#323C2214yes—
31#1280C2218yes—
32C23×D4C2×C411yescharacteristic
33C24×C4C2×C411yescharacteristic
34C24×C4C2×C411yescharacteristic
35C24×C4C2×C411yescharacteristic
36C24×C4C2311yescharacteristic
37C2×#123—24no—
38C2×#123—21no—
39C2×#123—22no—
40C2×#123—21no—
41C2×#123—24no—
42C2×C4×D4—22no—
43C2×C4×D4—22no—
44C2×C4×D4—44no—
45C22×C4⋊C4C2×C412yes—
46C22×C4⋊C4C2×C412yes—
47C22×C4⋊C4C2311yescharacteristic
48C22×C22⋊C4C2×C412yes—
49C22×C22⋊C4C2×C411yescharacteristic
50C22×C22⋊C4C2×C411yescharacteristic
51C22×C22⋊C4C2×C412yes—
52C22×C22⋊C4C2×C411yescharacteristic
53C22×C22⋊C4C2311yescharacteristic
54C22×C42—44no—
55C22×C42C2×C411yescharacteristic
56C22×C42C2×C412yes—
57C22×C42C2×C412yes—
58C22×C42D412yes—
59C22×C42D412yes—
60C22×C42D412yes—
61C22×C42D412yes—
62C22×C42C2312yes—
63#432—48no—
64C2×C4⋊C8—24no—
65C2×C22⋊C8—24no—
66C2×C22⋊C8—24no—
67C2×C22⋊C8C2318yes—
68C2×#97—24no—
69#388—22no—
70#388—22no—
71#388—22no—
72#388—22no—
73#387—22no—
74#387—22no—
75#387—22no—
76#387—22no—
77#386—21no—
78#386—22no—
79#386—21no—
80#385—21no—
81#385—22no—
82#385—21no—
83#381—21no—
84#381—22no—
85#381—21no—
86C4×C4⋊C4—22no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4—22no—
90C4×C22⋊C4—22no—
91C4×C22⋊C4—22no—
92C4×C22⋊C4—22no—
93C4×C22⋊C4—22no—
94C4×C22⋊C4—22no—
95C4×C22⋊C4—22no—
96C2×#95C2×C412yes—
97C2×#95C2×C412yes—
98C2×#95C2×C412yes—
99C2×#95C2×C412yes—
100C2×#95C2312yes—
101#336—24no—
102#336—24no—
103#324—44no—
104#323—44no—
105C25C22×C411yescharacteristic
106C25C22×C411yescharacteristic
107C22×D4—21no—
108C22×D4—21no—
109C22×D4—22no—
110C22×D4—42no—
111C22×D4—42no—
112C22×D4M1614yes—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—24no—
116C23×C4—22no—
117C23×C4—44no—
118C23×C4—44no—
119C23×C4—42no—
120C23×C4—42no—
121C23×C4—42no—
122C23×C4—42no—
123C23×C4—44no—
124C23×C4C22⋊C411yescharacteristic
125C23×C4C22⋊C411yescharacteristic
126C23×C4C22⋊C412yes—
127C23×C4C22⋊C411yescharacteristic
128C23×C4C22⋊C412yes—
129C23×C4C22⋊C411yescharacteristic
130C23×C4C22×C412yes—
131C23×C4C22×C411yescharacteristic
132C23×C4C22×C411yescharacteristic
133C23×C4C22×C412yes—
134C23×C4C22×C412yes—
135C23×C4C22×C411yescharacteristic
136C23×C4C22×C411yescharacteristic
137C23×C4C22×C412yes—
138C23×C4C2×D412yes—
139C23×C4C2×D412yes—
140C23×C4C2411yescharacteristic
141C22×C8—28no—
142#123—28no—
143#123—216no—
144#123—28no—
145C4×D4—48no—
146C4×D4—48no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—44no—
156C2×C22⋊C4—21no—
157C2×C22⋊C4—21no—
158C2×C22⋊C4—24no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—21no—
164C2×C22⋊C4—21no—
165C2×C22⋊C4—21no—
166C2×C22⋊C4—21no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—42no—
172C2×C22⋊C4—42no—
173C2×C22⋊C4—44no—
174C2×C22⋊C4M1614yes—
175C2×C42—28no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—44no—
180C2×C42—44no—
181C2×C42—44no—
182C2×C42—44no—
183C2×C42—42no—
184C2×C42—44no—
185C2×C42—44no—
186C2×C42C22⋊C412yes—
187C2×C42C22⋊C412yes—
188C2×C42C22⋊C412yes—
189C2×C42C22⋊C412yes—
190C2×C42C2×D412yes—
191C2×C42C2×D412yes—
192C2×C42C2×D412yes—
193C2×C42C2×D412yes—
194C4⋊C8—48no—
195C22⋊C8—28no—
196C22⋊C8—28no—
197C22⋊C8—48no—
198C22⋊C8—48no—
199#97—48no—
200#95—22no—
201#95—22no—
202#95—22no—
203#95—22no—
204#95—22no—
205#95—22no—
206#95—22no—
207#95—22no—
208#95—22no—
209#95—22no—
210C24—22no—
211C24—22no—
212C24—21no—
213C24—21no—
214C24—22no—
215C24—22no—
216C24—42no—
217C24—42no—
218C24—42no—
219C24—42no—
220C24C2×C22⋊C411yescharacteristic
221C24C2×C22⋊C411yescharacteristic
222C24C2×M1612yes—
223C24C2×M1612yes—
224C24C2×M1612yes—
225C24C23×C411yescharacteristic
226C2×D4—24no—
227C2×D4—24no—
228C2×D4—48no—
229C2×D4—42no—
230C2×D4—42no—
231C2×D4—42no—
232C2×D4—42no—
233C2×D4—48no—
234C22×C4—22no—
235C22×C4—24no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—24no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—24no—
249C22×C4—22no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—41no—
264C22×C4—42no—
265C22×C4—41no—
266C22×C4—42no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—42no—
270C22×C4—41no—
271C22×C4—42no—
272C22×C4—41no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—416no—
279C22×C4—44no—
280C22×C4—44no—
281C22×C4—48no—
282C22×C4—48no—
283C22×C4—416no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—44no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—88no—
292C22×C4C2×C22⋊C412yes—
293C22×C4C2×C22⋊C412yes—
294C22×C4C2×C22⋊C412yes—
295C22×C4C2×C22⋊C412yes—
296C22×C4C2×C22⋊C412yes—
297C22×C4C2×M1612yes—
298C22×C4C2×M1614yes—
299C22×C4C22×D412yes—
300C22×C4H(5,𝔽2)14yes—
301C2×C8—48no—
302C2×C8—48no—
303C2×C8—48no—
304C4⋊C4—28no—
305C4⋊C4—28no—
306C4⋊C4—44no—
307C4⋊C4—44no—
308C4⋊C4—44no—
309C22⋊C4—28no—
310C22⋊C4—28no—
311C22⋊C4—24no—
312C22⋊C4—24no—
313C22⋊C4—44no—
314C22⋊C4—42no—
315C22⋊C4—42no—
316C22⋊C4—42no—
317C22⋊C4—42no—
318C22⋊C4—44no—
319C22⋊C4—44no—
320C42—24no—
321C42—24no—
322C42—24no—
323C42—24no—
324C42—42no—
325C42—42no—
326C42—42no—
327C42—42no—
328C42—44no—
329C42—44no—
330C42—44no—
331C23—22no—
332C23—22no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—22no—
338C23—22no—
339C23—22no—
340C23—22no—
341C23—24no—
342C23—24no—
343C23—22no—
344C23—24no—
345C23—44no—
346C23—44no—
347C23—44no—
348C23—44no—
349C23—48no—
350C23—44no—
351C23—44no—
352C23—41no—
353C23—41no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23—82no—
358C23—84no—
359C23—82no—
360C23#40712yes—
361C23#40712yes—
362C23C22×C22⋊C411yescharacteristic
363C23#51812yes—
364C23C22×M1611yescharacteristic
365C23C22×M1611yescharacteristic
366C23#56812yes—
367D4—48no—
368D4—48no—
369C2×C4—28no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—24no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—24no—
388C2×C4—44no—
389C2×C4—48no—
390C2×C4—48no—
391C2×C4—44no—
392C2×C4—48no—
393C2×C4—48no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—48no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—42no—
406C2×C4—44no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—88no—
415C2×C4—84no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4#41014yes—
419C2×C4#41314yes—
420C2×C4#53414yes—
421C8—88no—
422C22—22no—
423C22—22no—
424C22—22no—
425C22—24no—
426C22—22no—
427C22—24no—
428C22—24no—
429C22—22no—
430C22—22no—
431C22—48no—
432C22—48no—
433C22—41no—
434C22—41no—
435C22—42no—
436C22—42no—
437C22—41no—
438C22—41no—
439C22—44no—
440C22—42no—
441C22—84no—
442C22—84no—
443C22—81no—
444C22—82no—
445C22—81no—
446C22#230811yescharacteristic
447C22C2×#40711yescharacteristic
448C22#287911yescharacteristic
449C22C2×#41012yes—
450C22C2×#41312yes—
451C22#297312yes—
452C22#297512yes—
453C4—24no—
454C4—24no—
455C4—24no—
456C4—44no—
457C4—42no—
458C4—42no—
459C4—42no—
460C4—42no—
461C4—81no—
462C4—82no—
463C4—81no—
464C2—22no—
465C2—22no—
466C2—42no—
467C2—42no—
468C2—82no—
469C2#1736312yes—
470C2#1740112yes—
471C2#2174211yescharacteristic
472C2#2174711yescharacteristic
473C2#2175411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20715C2semidirect
3. #9086C2semidirect
4. #9084C2semidirect
5. #8700C2semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1232
#207152
#90862
#90842
#87002
#83012