Finite Group Databasebeta

← #216186

Subgroups — #216186

Order 512.

1060 conjugacy classes of subgroups.

The subgroup lattice has 505 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2860C211yesmaximal, characteristic
2#20808C211yesmaximal, characteristic, complemented
3#20777C211yesmaximal, characteristic, complemented
4#9879C212yesmaximal, complemented
5#9649C212yesmaximal
6#9080C214yesmaximal, complemented
7#8561C214yesmaximal, complemented
8C22×#126C412yes—
9C2×#431C2212yes—
10#2860—21no—
11#2860—21no—
12#2860—21no—
13#2860—21no—
14#2860—21no—
15#2860—21no—
16#2860—21no—
17#2860—21no—
18C2×#390C411yescharacteristic
19C2×#390C411yescharacteristic
20C2×#390C2211yescharacteristic
21C2×#389C411yescharacteristic
22C2×#389C411yescharacteristic
23C2×#389C2211yescharacteristic
24C2×#388C412yes—
25C2×#388C2212yes—
26C2×#388C2212yes—
27C2×C43C2211yescharacteristic
28#1842—21no—
29#1839—21no—
30#1831—21no—
31#1831—21no—
32#1817C2212yes—
33#1797C2212yes—
34#1797C2212yes—
35#1797C2212yes—
36C4×C4⋊C8—21no—
37C4×C4⋊C8—21no—
38C4×C4⋊C8—21no—
39C4×C22⋊C8C2212yes—
40#1744—21no—
41C2×#323C2218yes—
42#1280C2218yes—
43C23×D4C2×C411yescharacteristic
44C2×#126—22no—
45C2×#126—22no—
46C2×#126—22no—
47C2×#126—22no—
48C2×#126—48no—
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C4⋊C4C2×C412yes—
51C22×C4⋊C4C2×C412yes—
52C22×C4⋊C4C2×C412yes—
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C22⋊C4C2×C412yes—
56C22×C22⋊C4C2×C412yes—
57C22×C22⋊C4C2×C412yes—
58C22×C22⋊C4C2×C412yes—
59C22×C22⋊C4C2×C412yes—
60C22×C22⋊C4C2312yes—
61C22×C42C2×C411yescharacteristic
62C22×C42C2×C411yescharacteristic
63C22×C42C2×C411yescharacteristic
64C22×C42C2×C411yescharacteristic
65C22×C42C2311yescharacteristic
66C22×C42C2311yescharacteristic
67C22×C42C2311yescharacteristic
68#431—44no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8—22no—
73C2×C22⋊C8C2314yes—
74C2×C22⋊C8C2314yes—
75C2×C4×C8—22no—
76C2×C4×C8—22no—
77#390—21no—
78#390—21no—
79#390—21no—
80#390—21no—
81#390—21no—
82#390—21no—
83#390—21no—
84#390—21no—
85#390—21no—
86#390—21no—
87#390—21no—
88#390—21no—
89#389—21no—
90#389—21no—
91#389—21no—
92#389—21no—
93#389—21no—
94#389—21no—
95#389—21no—
96#389—21no—
97#389—21no—
98#389—21no—
99#389—21no—
100#389—21no—
101#388—22no—
102#388—22no—
103#388—22no—
104#388—22no—
105#388—22no—
106#388—22no—
107#388—22no—
108#388—22no—
109#388—22no—
110#388—22no—
111#388—22no—
112#388—22no—
113C2×#95C2×C412yes—
114C2×#95C2×C412yes—
115C2×#95C2312yes—
116C43D411yescharacteristic
117C43D411yescharacteristic
118C43D411yescharacteristic
119C43D411yescharacteristic
120C43D411yescharacteristic
121C43D411yescharacteristic
122C43D411yescharacteristic
123C43D411yescharacteristic
124#336—22no—
125#336—22no—
126#336—22no—
127#336—22no—
128#336—22no—
129#336—22no—
130#323—48no—
131C25C22×C412yes—
132C22×D4—21no—
133C22×D4—21no—
134C22×D4—24no—
135C22×D4—42no—
136C22×D4—42no—
137C23×C4C22×C411yescharacteristic
138C23×C4C22×C411yescharacteristic
139C23×C4C22×C411yescharacteristic
140C23×C4C22×C411yescharacteristic
141C23×C4C22×C411yescharacteristic
142C23×C4C22×C411yescharacteristic
143C23×C4C22×C412yes—
144C23×C4C22×C412yes—
145C23×C4C22×C412yes—
146C23×C4C2411yescharacteristic
147C22×C8—24no—
148C22×C8—24no—
149#126—48no—
150#126—48no—
151#126—48no—
152#126—48no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—44no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—28no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—44no—
187C2×C22⋊C4—44no—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—44no—
190C2×C42—24no—
191C2×C42—24no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—24no—
195C2×C42—42no—
196C2×C42—42no—
197C2×C42—42no—
198C2×C42C22⋊C411yescharacteristic
199C2×C42C22⋊C411yescharacteristic
200C2×C42C22⋊C411yescharacteristic
201C2×C42C22⋊C411yescharacteristic
202C2×C42C22⋊C411yescharacteristic
203C2×C42C22⋊C411yescharacteristic
204C2×C42C22⋊C411yescharacteristic
205C2×C42C22⋊C411yescharacteristic
206C2×C42C22⋊C411yescharacteristic
207C2×C42C22⋊C411yescharacteristic
208C2×C42C22⋊C411yescharacteristic
209C2×C42C22⋊C411yescharacteristic
210C2×C42C22⋊C411yescharacteristic
211C2×C42C22⋊C411yescharacteristic
212C2×C42C22⋊C411yescharacteristic
213C2×C42C22⋊C411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C4⋊C8—44no—
227C4⋊C8—44no—
228C4⋊C8—44no—
229C4⋊C8—44no—
230C22⋊C8—24no—
231C22⋊C8—28no—
232C22⋊C8—24no—
233C4×C8—44no—
234C4×C8—44no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—22no—
241C24—22no—
242C24—24no—
243C24—24no—
244C24—24no—
245C24—42no—
246C24—42no—
247C24—42no—
248C24—42no—
249C24C23×C411yescharacteristic
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4—42no—
256C2×D4—88no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4C2×C22⋊C411yescharacteristic
288C22×C4C2×C22⋊C411yescharacteristic
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4C2×C22⋊C411yescharacteristic
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×C22⋊C411yescharacteristic
294C22×C4C2×C22⋊C411yescharacteristic
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C2×C22⋊C411yescharacteristic
299C22×C4#13114yes—
300C22×C4#13114yes—
301C22×C4C22×D411yescharacteristic
302C22×C4C22×D411yescharacteristic
303C22×C4H(5,𝔽2)12yes—
304C22×C42−1+412yes—
305C2×C8—44no—
306C2×C8—44no—
307C2×C8—44no—
308C2×C8—44no—
309C2×C8—44no—
310C2×C8—44no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—44no—
314C4⋊C4—44no—
315C4⋊C4—44no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—44no—
320C4⋊C4—44no—
321C4⋊C4—44no—
322C4⋊C4—42no—
323C22⋊C4—44no—
324C22⋊C4—44no—
325C22⋊C4—44no—
326C22⋊C4—44no—
327C22⋊C4—42no—
328C22⋊C4—42no—
329C22⋊C4—44no—
330C22⋊C4—44no—
331C22⋊C4—44no—
332C22⋊C4—44no—
333C22⋊C4—44no—
334C22⋊C4—44no—
335C22⋊C4—44no—
336C42—21no—
337C42—21no—
338C42—21no—
339C42—21no—
340C42—22no—
341C42—22no—
342C42—22no—
343C42—22no—
344C42—22no—
345C42—22no—
346C42—22no—
347C42—22no—
348C42—22no—
349C42—22no—
350C42—22no—
351C42—22no—
352C42—22no—
353C42—22no—
354C42—22no—
355C42—22no—
356C42—22no—
357C42—22no—
358C42—22no—
359C42—41no—
360C42—41no—
361C42—41no—
362C42—41no—
363C42—41no—
364C42—41no—
365C42#10014yes—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23—48no—
378C23—84no—
379C23—84no—
380C23—84no—
381C23—84no—
382C23—84no—
383C23—84no—
384C23C22×C22⋊C411yescharacteristic
385C23#52012yes—
386C23C2×#13112yes—
387C23C2×#13112yes—
388D4—84no—
389D4—84no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—28no—
409C2×C4—28no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—84no—
438C2×C4—84no—
439C2×C4—84no—
440C2×C4#40812yes—
441C2×C4#40812yes—
442C2×C4#40812yes—
443C2×C4#40812yes—
444C2×C4#41012yes—
445C2×C4#41012yes—
446C2×C4C2×#10012yes—
447C2×C4C2×#10012yes—
448C2×C4C2×#10012yes—
449C2×C4#41312yes—
450C2×C4#53511yescharacteristic
451C2×C4#53611yescharacteristic
452C2×C4#53712yes—
453C8—84no—
454C8—84no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—82no—
462C22—82no—
463C22—82no—
464C22—82no—
465C22—82no—
466C22—82no—
467C22—82no—
468C22—168no—
469C22#231111yescharacteristic
470C22C2×#40811yescharacteristic
471C22C2×#40811yescharacteristic
472C22C2×#41011yescharacteristic
473C22C2×#41011yescharacteristic
474C22C22×#10011yescharacteristic
475C22C2×#41311yescharacteristic
476C22#299414yes—
477C4—22no—
478C4—22no—
479C4—21no—
480C4—22no—
481C4—21no—
482C4—22no—
483C4—22no—
484C4—22no—
485C4—22no—
486C4—22no—
487C4—41no—
488C4—41no—
489C4—41no—
490C4—41no—
491C4—82no—
492C4—82no—
493C4—82no—
494C4#147314yes—
495C2—24no—
496C2—162no—
497C2#1737611yescharacteristic
498C2C2×#147311yescharacteristic
499C2#1738411yescharacteristic
500C2#1738411yescharacteristic
501C2#2184811yescharacteristic
502C2#2187511yescharacteristic
503C2#2187811yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20808C2semidirect
3. #20777C2semidirect
4. #9879C2semidirect
6. #9080C2semidirect
7. #8561C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28602
#208082
#207772
#98792
#96492
#90802
#85612