Finite Group Databasebeta

← #201311

Subgroups — #201311

Order 512.

1429 conjugacy classes of subgroups.

The subgroup lattice has 501 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35317C211yesmaximal, characteristic
2#34371C211yesmaximal, characteristic, complemented
3#21829C211yesmaximal, characteristic, complemented
4#17589C211yesmaximal, characteristic
5#17561C211yesmaximal, characteristic
6#9464C218yesmaximal, complemented
7#7941C211yesmaximal, characteristic, complemented
8#7940C211yesmaximal, characteristic, complemented
9C2×#536—22no—
10C2×#532C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#2920C2211yescharacteristic
13#2920C2211yescharacteristic
14#2868C412yescomplemented
15#2858C412yescomplemented
16#2658—22no—
17#2542—22no—
18#2542C414yescomplemented
19#2501—22no—
20#2501C2214yes—
21#2381—22no—
22#2365—22no—
23#2345—22no—
24Q8×C22⋊C4—22no—
25#2331—22no—
26#2327—22no—
27#2313C2214yes—
28#2283C2211yescharacteristic
29#2283C2211yescharacteristic
30C4×#117C2211yescharacteristic
31#1997—22no—
32#1997—22no—
33#1821C2214yes—
34#1517—22no—
35#1514—22no—
36#1510C2214yes—
37#1510C2214yes—
38#1505—22no—
39#1502—22no—
40#1284C2214yes—
41#1284C2214yes—
42C22×#41—21no—
43C23×Q8D412yes—
44#536—416no—
45#532—21no—
46#532—21no—
47#532—21no—
48#532—21no—
49#532—21no—
50#532—21no—
51#532—21no—
52#532—21no—
53C2×C4⋊Q8—24no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—21no—
61C2×C4⋊Q8—21no—
62C2×C4⋊Q8C2×C411yescharacteristic
63C2×C4⋊Q8C2×C412yes—
64C2×C4⋊Q8C2×C412yes—
65C2×C4⋊Q8D412yes—
66C2×C4⋊Q8D412yes—
67C2×#127C2×C411yescharacteristic
68C2×#124—24no—
69C2×#124—21no—
70C2×#123—28no—
71C2×C22⋊Q8—28no—
72C2×#121—21no—
73C2×#121C2312yes—
74C2×C4×Q8—24no—
75C2×C4×D4—21no—
76C2×C4×D4C2×C412yes—
77C2×#117—21no—
78C22×C4⋊C4D412yes—
79C22×C4⋊C4C2312yes—
80C22×C42C2311yescharacteristic
81#432—21no—
82#432—21no—
83#432—21no—
84#432—21no—
85#431—21no—
86#431—21no—
87#431—21no—
88#431—21no—
89#423—21no—
90#423—21no—
91#407C2312yes—
92#407C2312yes—
93C4×M16—21no—
94C4×M16—21no—
95#399—28no—
96#399—28no—
97#391—24no—
98#391—28no—
99#390—24no—
100#390C2×C418yes—
101#389—24no—
102#387—28no—
103#387—28no—
104#387—28no—
105#384—24no—
106#384C2×C418yes—
107#383—21no—
108#383D412yes—
109#382—21no—
110#382D412yes—
111#381C2312yes—
112#381C2312yes—
113#380—24no—
114#380—21no—
115#380—21no—
116C4×C4⋊C4—24no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4C2×C412yes—
120C4×C4⋊C4D412yes—
121C4×C22⋊C4—24no—
122C4×C22⋊C4—21no—
123C4×C22⋊C4—21no—
124C4×C22⋊C4—21no—
125C4×C22⋊C4C2312yes—
126#376—21no—
127#376—21no—
128#376D412yes—
129C43C2×C412yes—
130#343—28no—
131#327—28no—
132#327—28no—
133C2×#41—48no—
134C2×#41—41no—
135C2×#41—41no—
136C2×#41—41no—
137C2×#41—41no—
138C22×Q8—21no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—48no—
144C22×Q8C22⋊C414yes—
145C22×Q8C22⋊C414yes—
146C22×D4—22no—
147C22×D4—21no—
148C22×D4—21no—
149C22×D4C22×C412yes—
150C23×C4—22no—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D414yes—
153C23×C4C2×D411yescharacteristic
154C23×C4C2411yescharacteristic
155C2×M16—22no—
156C2×M16—22no—
157C4⋊Q8—22no—
158C4⋊Q8—22no—
159C4⋊Q8—22no—
160C4⋊Q8—22no—
161C4⋊Q8—22no—
162C4⋊Q8—22no—
163C4⋊Q8—22no—
164C4⋊Q8—22no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—22no—
170C4⋊Q8—22no—
171C4⋊Q8—22no—
172C4⋊Q8—22no—
173C4⋊Q8—22no—
174C4⋊Q8—22no—
175C4⋊Q8—22no—
176C4⋊Q8—22no—
177C4⋊Q8—416no—
178#127—21no—
179#127—21no—
180#127—21no—
181#127—21no—
182#124—416no—
183#124—41no—
184#124—41no—
185#124—41no—
186#124—41no—
187#123—432no—
188C22⋊Q8—432no—
189#121—22no—
190#121—22no—
191#121—22no—
192#121—22no—
193#121—41no—
194#121—41no—
195#121—41no—
196#121—41no—
197C4×Q8—88no—
198C4×D4—22no—
199C4×D4—22no—
200C4×D4—22no—
201C4×D4—22no—
202C4×D4—41no—
203C4×D4—41no—
204C4×D4—41no—
205C4×D4—41no—
206#117—28no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—216no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4C22⋊C414yes—
222C2×C4⋊C4C22⋊C414yes—
223C2×C4⋊C4C22×C412yes—
224C2×C4⋊C4C22×C414yes—
225C2×C4⋊C4C2×D414yes—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—216no—
231C2×C22⋊C4—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—28no—
237C2×C42C22×C414yes—
238C2×C42C22×C411yescharacteristic
239C2×C42C22×C411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C2×C42#4111yescharacteristic
247C2×C42#4111yescharacteristic
248C2×C42#4111yescharacteristic
249C2×C42#4111yescharacteristic
250C4⋊C8—41no—
251C4⋊C8—41no—
252C4⋊C8—41no—
253C4⋊C8—41no—
254C22⋊C8—22no—
255C22⋊C8—22no—
256C22⋊C8—22no—
257C22⋊C8—22no—
258#97—41no—
259#97—41no—
260C4×C8—41no—
261C4×C8—41no—
262#95—28no—
263#95—28no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—22no—
268#95—22no—
269#95—22no—
270#95—22no—
271#95—22no—
272#95C2×D414yes—
273#95C2×D414yes—
274C24—22no—
275C24C22×D411yescharacteristic
276#41—84no—
277#41—84no—
278#41—84no—
279#41—84no—
280C2×Q8—24no—
281C2×Q8—24no—
282C2×Q8—24no—
283C2×Q8—24no—
284C2×Q8—48no—
285C2×Q8—48no—
286C2×Q8—44no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—48no—
290C2×Q8—84no—
291C2×Q8—84no—
292C2×Q8—84no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—44no—
296C2×D4—44no—
297C2×D4—44no—
298C2×D4—48no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—48no—
315C22×C4—48no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4C2×C22⋊C411yescharacteristic
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C2×C22⋊C418yes—
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4#12011yescharacteristic
324C22×C4#12011yescharacteristic
325C22×C4#12011yescharacteristic
326C22×C4#12011yescharacteristic
327C22×C4#12311yescharacteristic
328C22×C4#12311yescharacteristic
329C22×C4#12311yescharacteristic
330C22×C4#12311yescharacteristic
331C22×C4C23×C411yescharacteristic
332C22×C4C22×D411yescharacteristic
333C22×C4C22×D411yescharacteristic
334C22×C4C22×D411yescharacteristic
335C22×C4C2×#4111yescharacteristic
336C22×C4C2×#4111yescharacteristic
337M16—44no—
338M16—44no—
339C2×C8—42no—
340C2×C8—42no—
341C4⋊C4—28no—
342C4⋊C4—24no—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—24no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—44no—
353C4⋊C4—432no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—44no—
357C4⋊C4—88no—
358C22⋊C4—28no—
359C22⋊C4—432no—
360C22⋊C4—44no—
361C22⋊C4—44no—
362C22⋊C4—42no—
363C22⋊C4—42no—
364C22⋊C4—42no—
365C22⋊C4—42no—
366C42—28no—
367C42—44no—
368C42—44no—
369C42—44no—
370C42—44no—
371C42—88no—
372C42C4×D411yescharacteristic
373C42C4×D411yescharacteristic
374C42C4×D411yescharacteristic
375C42C4×D411yescharacteristic
376C42C4×D411yescharacteristic
377C42C4×D411yescharacteristic
378C42C4×D411yescharacteristic
379C42C4×D411yescharacteristic
380C42#12111yescharacteristic
381C42#12111yescharacteristic
382C42#12111yescharacteristic
383C42#12111yescharacteristic
384C42#12111yescharacteristic
385C42#12111yescharacteristic
386C42#12111yescharacteristic
387C42#12111yescharacteristic
388C23—22no—
389C23—22no—
390C23—41no—
391C23—44no—
392C23—44no—
393C23—44no—
394C23—42no—
395C23C22×C22⋊C411yescharacteristic
396C23C2×#12011yescharacteristic
397C23C2×#12311yescharacteristic
398Q8—48no—
399Q8—48no—
400Q8—84no—
401Q8—84no—
402Q8—82no—
403Q8—82no—
404D4—82no—
405D4—82no—
406D4—82no—
407D4—82no—
408D4—84no—
409D4—84no—
410D4—82no—
411D4—82no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—28no—
423C2×C4—28no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—48no—
434C2×C4—48no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—42no—
443C2×C4—84no—
444C2×C4—84no—
445C2×C4—84no—
446C2×C4#38611yescharacteristic
447C2×C4#38611yescharacteristic
448C2×C4#38611yescharacteristic
449C2×C4#38611yescharacteristic
450C2×C4#38611yescharacteristic
451C2×C4#38611yescharacteristic
452C2×C4#38611yescharacteristic
453C2×C4#38611yescharacteristic
454C2×C4#45314yes—
455C2×C4C2×C4×D411yescharacteristic
456C2×C4C2×C4×D411yescharacteristic
457C2×C4C2×#12111yescharacteristic
458C2×C4C2×#12111yescharacteristic
459C8—82no—
460C8—82no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—44no—
469C22—88no—
470C22—82no—
471C22—82no—
472C22—82no—
473C22C2×#38611yescharacteristic
474C22#288412yes—
475C22C2×#45312yes—
476C4—21no—
477C4—21no—
478C4—21no—
479C4—21no—
480C4—21no—
481C4—21no—
482C4—21no—
483C4—21no—
484C4—21no—
485C4—48no—
486C4—48no—
487C4—84no—
488C4—84no—
489C4—82no—
490C4#188712yes—
491C4#188712yes—
492C4#191212yes—
493C2—22no—
494C2—22no—
495C2—41no—
496C2—162no—
497C2C2×#188711yescharacteristic
498C2#2098611yescharacteristic
499C2C2×#191211yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34371C2semidirect
3. #21829C2semidirect
6. #9464C2semidirect
7. #7941C2semidirect
8. #7940C2semidirect
14. #2868C4semidirect
15. #2858C4semidirect
18. #2542C4semidirect

Maximal subgroups

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

subgroupindex
#353172
#343712
#218292
#175892
#175612
#94642
#79412
#79402