Finite Group Databasebeta

← #216324

Subgroups — #216324

Order 512.

858 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2651C211yesmaximal, characteristic
2#21955C211yesmaximal, characteristic, complemented
3#21916C211yesmaximal, characteristic, complemented
4#9881C212yesmaximal, complemented
5#9653C212yesmaximal
6#9084C212yesmaximal, complemented
7#9084C212yesmaximal, complemented
8#8561C212yesmaximal, complemented
9#8561C212yesmaximal, complemented
10C2×#432C2212yes—
11#2651—24no—
12#2651—24no—
13C2×#399C411yescharacteristic
14C2×#398C411yescharacteristic
15C2×#398C411yescharacteristic
16C2×#397C411yescharacteristic
17C2×#397C411yescharacteristic
18C2×#396C411yescharacteristic
19C2×#394C411yescharacteristic
20C2×#394C411yescharacteristic
21C2×#390C2211yescharacteristic
22C2×#388C2211yescharacteristic
23C2×#388C2211yescharacteristic
24C2×#388C2211yescharacteristic
25C2×#388C2211yescharacteristic
26C2×#387C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28#1940—24no—
29#1917—24no—
30#1817C2212yes—
31#1797C2212yes—
32#1797C2212yes—
33#1797C2212yes—
34#1749C2212yes—
35C2×#323C2214yes—
36C2×#323C2214yes—
37#1280C2214yes—
38#1280C2214yes—
39C23×D4C2×C411yescharacteristic
40C22×C4⋊C4C2×C411yescharacteristic
41C22×C4⋊C4C2×C411yescharacteristic
42C22×C4⋊C4C2×C411yescharacteristic
43C22×C4⋊C4C2×C411yescharacteristic
44C22×C4⋊C4C2×C411yescharacteristic
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C4⋊C4C2×C411yescharacteristic
47C22×C4⋊C4C2×C411yescharacteristic
48C22×C4⋊C4C2×C411yescharacteristic
49C22×C22⋊C4C2×C411yescharacteristic
50C22×C22⋊C4C2×C411yescharacteristic
51C22×C22⋊C4C2×C411yescharacteristic
52C22×C22⋊C4C2×C411yescharacteristic
53C22×C22⋊C4C2×C411yescharacteristic
54C22×C22⋊C4C2×C411yescharacteristic
55C22×C22⋊C4C2×C411yescharacteristic
56C22×C22⋊C4C2×C411yescharacteristic
57C22×C22⋊C4C2×C411yescharacteristic
58C22×C22⋊C4C2×C411yescharacteristic
59C22×C22⋊C4C2311yescharacteristic
60C22×C22⋊C4C2311yescharacteristic
61C22×C42C2311yescharacteristic
62C22×C42C2311yescharacteristic
63C22×C42C2311yescharacteristic
64#432—44no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—22no—
68C2×C4⋊C8—22no—
69C2×C22⋊C8C2314yes—
70C2×C22⋊C8C2314yes—
71C2×#97—22no—
72C2×#97—22no—
73#399—24no—
74#398—24no—
75#398—24no—
76#397—24no—
77#397—24no—
78#396—24no—
79#394—24no—
80#394—24no—
81#390—22no—
82#390—22no—
83#388—22no—
84#388—22no—
85#388—22no—
86#388—22no—
87#388—22no—
88#388—22no—
89#388—22no—
90#388—22no—
91#387—22no—
92#387—22no—
93C4×C4⋊C4D414yes—
94C4×C4⋊C4D414yes—
95C2×#95C2×C411yescharacteristic
96C2×#95C2×C411yescharacteristic
97C2×#95C2×C411yescharacteristic
98C2×#95C2×C411yescharacteristic
99C2×#95C2×C411yescharacteristic
100C2×#95C2×C411yescharacteristic
101C2×#95C2×C411yescharacteristic
102C2×#95C2×C411yescharacteristic
103C2×#95C2311yescharacteristic
104C2×#95C2311yescharacteristic
105#336—22no—
106#336—22no—
107#336—22no—
108#336—24no—
109#336—22no—
110#323—44no—
111#323—44no—
112C25C22×C411yescharacteristic
113C25C22×C411yescharacteristic
114C22×D4—22no—
115C22×D4—42no—
116C22×D4—81no—
117C22×D4—81no—
118C23×C4C22×C411yescharacteristic
119C23×C4C22×C411yescharacteristic
120C23×C4C22×C411yescharacteristic
121C23×C4C22×C411yescharacteristic
122C23×C4C22×C411yescharacteristic
123C23×C4C22×C411yescharacteristic
124C23×C4C22×C411yescharacteristic
125C23×C4C22×C411yescharacteristic
126C23×C4C22×C411yescharacteristic
127C23×C4C22×C411yescharacteristic
128C23×C4C22×C411yescharacteristic
129C23×C4C22×C411yescharacteristic
130C23×C4C2411yescharacteristic
131C22×C8—24no—
132C22×C8—24no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4—42no—
142C2×C4⋊C4—42no—
143C2×C4⋊C4—81no—
144C2×C4⋊C4—81no—
145C2×C4⋊C4—81no—
146C2×C4⋊C4—81no—
147C2×C4⋊C4—81no—
148C2×C4⋊C4—81no—
149C2×C4⋊C4C22⋊C414yes—
150C2×C4⋊C4C22⋊C414yes—
151C2×C22⋊C4—22no—
152C2×C22⋊C4—22no—
153C2×C22⋊C4—21no—
154C2×C22⋊C4—21no—
155C2×C22⋊C4—21no—
156C2×C22⋊C4—21no—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—42no—
170C2×C22⋊C4—42no—
171C2×C22⋊C4—81no—
172C2×C22⋊C4—81no—
173C2×C22⋊C4—81no—
174C2×C22⋊C4—81no—
175C2×C22⋊C4—81no—
176C2×C22⋊C4—81no—
177C2×C22⋊C4—81no—
178C2×C22⋊C4—81no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—81no—
182C2×C42—81no—
183C2×C42C2×D412yes—
184C2×C42C2×D412yes—
185C2×C42C2×D412yes—
186C2×C42C2×D412yes—
187C2×C42C2×D412yes—
188C2×C42C2×D412yes—
189C4⋊C8—44no—
190C4⋊C8—82no—
191C4⋊C8—82no—
192C4⋊C8—82no—
193C22⋊C8—24no—
194C22⋊C8—24no—
195C22⋊C8—24no—
196C22⋊C8—24no—
197#97—44no—
198#97—82no—
199#95—22no—
200#95—22no—
201#95—22no—
202#95—22no—
203#95—22no—
204#95—22no—
205#95—21no—
206#95—21no—
207#95—21no—
208#95—21no—
209#95C22⋊C414yes—
210#95C22⋊C414yes—
211C24—21no—
212C24—21no—
213C24—22no—
214C24—22no—
215C24—22no—
216C24—22no—
217C24—22no—
218C24—22no—
219C24—41no—
220C24—41no—
221C24—41no—
222C24—41no—
223C24—41no—
224C24—41no—
225C24—41no—
226C24—41no—
227C24C23×C411yescharacteristic
228C2×D4—44no—
229C2×D4—82no—
230C2×D4—82no—
231C2×D4—88no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4C2×C22⋊C412yes—
277C22×C4C2×C22⋊C412yes—
278C22×C4C2×C22⋊C412yes—
279C22×C4C2×C22⋊C412yes—
280C22×C4C2×C22⋊C412yes—
281C22×C4C2×C22⋊C412yes—
282C22×C4C22×D411yescharacteristic
283C22×C4C22×D411yescharacteristic
284C22×C4H(5,𝔽2)12yes—
285C22×C42−1+412yes—
286C2×C8—44no—
287C2×C8—44no—
288C2×C8—44no—
289C2×C8—44no—
290C2×C8—44no—
291C2×C8—44no—
292C4⋊C4—44no—
293C4⋊C4—44no—
294C4⋊C4—44no—
295C4⋊C4—82no—
296C4⋊C4—82no—
297C4⋊C4—82no—
298C4⋊C4—82no—
299C4⋊C4—82no—
300C4⋊C4—82no—
301C22⋊C4—44no—
302C22⋊C4—44no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—41no—
307C22⋊C4—42no—
308C22⋊C4—42no—
309C22⋊C4—82no—
310C22⋊C4—82no—
311C22⋊C4—82no—
312C22⋊C4—82no—
313C22⋊C4—82no—
314C22⋊C4—82no—
315C22⋊C4—82no—
316C22⋊C4—82no—
317C42—22no—
318C42—22no—
319C42—24no—
320C42—42no—
321C42—42no—
322C42—42no—
323C42—42no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—44no—
343C23—44no—
344C23—82no—
345C23—82no—
346C23—82no—
347C23—82no—
348C23—82no—
349C23—82no—
350C23—82no—
351C23—82no—
352C23—82no—
353C23—82no—
354C23—82no—
355C23—82no—
356C23C22×C22⋊C411yescharacteristic
357C23#52012yes—
358C23#56812yes—
359C23#56812yes—
360D4—88no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
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—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4#53511yescharacteristic
416C2×C4#53611yescharacteristic
417C2×C4#53712yes—
418C8—84no—
419C8—84no—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—22no—
424C22—22no—
425C22—22no—
426C22—81no—
427C22—81no—
428C22—81no—
429C22—81no—
430C22—81no—
431C22—81no—
432C22—81no—
433C22—81no—
434C22—81no—
435C22—81no—
436C22—81no—
437C22—81no—
438C22—81no—
439C22—81no—
440C22—164no—
441C22—164no—
442C22#231111yescharacteristic
443C22#287911yescharacteristic
444C22#287911yescharacteristic
445C22#288311yescharacteristic
446C22#288411yescharacteristic
447C22#288811yescharacteristic
448C22#288811yescharacteristic
449C22#299514yes—
450C4—21no—
451C4—22no—
452C4—21no—
453C4—42no—
454C4—42no—
455C4—42no—
456C4—42no—
457C4—42no—
458C4—42no—
459C4—81no—
460C4—81no—
461C4—81no—
462C4—81no—
463C4—81no—
464C4—81no—
465C2—24no—
466C2—161no—
467C2—161no—
468C2#1740511yescharacteristic
469C2#1740811yescharacteristic
470C2#1740911yescharacteristic
471C2#1740911yescharacteristic
472C2#2185411yescharacteristic
473C2#2187711yescharacteristic
474C2#2188211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21955C2semidirect
3. #21916C2semidirect
4. #9881C2semidirect
6. #9084C2semidirect
7. #9084C2semidirect
8. #8561C2semidirect
9. #8561C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26512
#219552
#219162
#98812
#96532
#90842
#85612