Finite Group Databasebeta

← #181537

Subgroups — #181537

Order 512.

1392 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2984C211yesmaximal, characteristic
2#34196C211yesmaximal, characteristic
3C2×#2320C211yesmaximal, characteristic
4#8026C212yesmaximal
5#8025C212yesmaximal
6#8022C211yesmaximal, characteristic
7#8021C211yesmaximal, characteristic
8#8018C212yesmaximal
9#8017C212yesmaximal
10#8014C211yesmaximal, characteristic
11#8013C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13#2984—24no—
14#2984C418yescomplemented
15C2×#446—22no—
16C2×#446—21no—
17C2×C8×Q8—21no—
18C2×#432C2212yes—
19C2×#432C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C2×#431C2212yes—
22#2355C412yescomplemented
23#2355C414yescomplemented
24Q8×C4⋊C4C412yescomplemented
25#2344—22no—
26#2344—21no—
27Q8×C22⋊C4—21no—
28#2320—24no—
29#2320—21no—
30#2320—22no—
31#2320C418yescomplemented
32C2×#391—21no—
33C2×#389—21no—
34C2×#387—22no—
35#2285C2211yescharacteristic
36#2285C2212yes—
37#2283C2211yescharacteristic
38#2283C2212yes—
39C2×#383C2212yes—
40C2×#382C2212yes—
41#2276—21no—
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×#376C2211yescharacteristic
44#1294C2211yescharacteristic
45#1294C2211yescharacteristic
46#1294C2211yescharacteristic
47#1294C2211yescharacteristic
48#1294C2212yes—
49#1294C2212yes—
50#1294C2212yes—
51#1294C2212yes—
52#1294C2212yes—
53#1294C2212yes—
54C23×Q8D411yescharacteristic
55C2×C4×Q8—22no—
56C2×C4×Q8C2×C414yes—
57C2×C4×Q8C2×C411yescharacteristic
58C2×C4×Q8C2×C411yescharacteristic
59C2×C4×Q8C2×C412yes—
60C2×C4×Q8C2×C414yes—
61C2×C4×Q8D411yescharacteristic
62C2×C4×Q8D412yes—
63C2×C4×Q8D414yes—
64C2×C4×Q8D411yescharacteristic
65C2×C4×Q8Q814yes—
66C2×#117—22no—
67C2×#117—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4D412yes—
71C22×C4⋊C4D411yescharacteristic
72C22×C42—21no—
73C22×C42—21no—
74C22×C42C2311yescharacteristic
75C22×C42C2312yes—
76#446—28no—
77#446—216no—
78C8×Q8—28no—
79#432—22no—
80#432—24no—
81#432C2×C418yes—
82#432C2×C414yes—
83#431—22no—
84#431—24no—
85#431C2×C414yes—
86#431C2×C418yes—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—22no—
91C2×C22⋊C8C2311yescharacteristic
92C2×C22⋊C8C2311yescharacteristic
93C2×C22⋊C8C2312yes—
94C2×#97—22no—
95C2×#97—21no—
96C2×C4×C8—21no—
97C2×C4×C8—22no—
98#391—28no—
99#391—22no—
100#391—24no—
101#389—28no—
102#389—24no—
103#389—22no—
104#387—216no—
105#387—24no—
106#387—24no—
107#387—24no—
108#384C2×C412yes—
109#384C2×C414yes—
110#383—24no—
111#383—22no—
112#383—21no—
113#383C2×C418yes—
114#382—24no—
115#382—22no—
116#382—21no—
117#382C2×C412yes—
118#382C2×C414yes—
119#382C2×C418yes—
120#381—22no—
121#381—21no—
122#380—21no—
123#380—22no—
124#380C2311yescharacteristic
125#380C2311yescharacteristic
126#380C2312yes—
127C4×C4⋊C4—24no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—22no—
132C4×C4⋊C4C2×C412yes—
133C4×C4⋊C4C2×C414yes—
134C4×C4⋊C4C2×C412yes—
135C4×C4⋊C4C2×C414yes—
136C4×C4⋊C4C2×C414yes—
137C4×C22⋊C4—21no—
138C4×C22⋊C4—22no—
139#376—22no—
140#376—21no—
141#376—22no—
142#376C2×C414yes—
143C2×#95—21no—
144C2×#95—22no—
145C2×#95—22no—
146C2×#95—22no—
147C2×#95C2311yescharacteristic
148C2×#95C2311yescharacteristic
149C2×#95C2312yes—
150C22×Q8—44no—
151C22×Q8C22⋊C411yescharacteristic
152C22×Q8C22⋊C411yescharacteristic
153C22×Q8C22⋊C414yes—
154C22×Q8C22⋊C412yes—
155C22×Q8C4⋊C414yes—
156C23×C4—21no—
157C23×C4—22no—
158C23×C4—21no—
159C23×C4C2×D412yes—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D412yes—
163C23×C4C2411yescharacteristic
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—22no—
167C4×Q8—48no—
168C4×Q8C4214yes—
169C4×Q8C4214yes—
170C4×Q8C4218yes—
171C4×Q8C22⋊C414yes—
172C4×Q8C22⋊C414yes—
173C4×Q8C22⋊C418yes—
174C4×Q8C4⋊C414yes—
175C4×Q8C4⋊C414yes—
176C4×Q8C4⋊C414yes—
177C4×Q8C4⋊C414yes—
178C4×Q8C4⋊C418yes—
179C4×Q8C4⋊C418yes—
180#117—81no—
181#117—82no—
182#117—82no—
183#117—81no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—82no—
192C2×C4⋊C4—82no—
193C2×C4⋊C4C22⋊C412yes—
194C2×C4⋊C4C22⋊C411yescharacteristic
195C2×C4⋊C4C22⋊C418yes—
196C2×C4⋊C4C22⋊C412yes—
197C2×C4⋊C4C22⋊C412yes—
198C2×C4⋊C4C22⋊C412yes—
199C2×C4⋊C4C22⋊C414yes—
200C2×C4⋊C4C22⋊C412yes—
201C2×C4⋊C4C22⋊C411yescharacteristic
202C2×C4⋊C4C4⋊C414yes—
203C2×C4⋊C4C4⋊C418yes—
204C2×C4⋊C4C22×C412yes—
205C2×C4⋊C4C22×C412yes—
206C2×C4⋊C4C22×C414yes—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—21no—
210C2×C42—24no—
211C2×C42—22no—
212C2×C42—24no—
213C2×C42—22no—
214C2×C42—24no—
215C2×C42—24no—
216C2×C42—44no—
217C2×C42—81no—
218C2×C42—81no—
219C2×C42C22×C412yes—
220C2×C42C22×C414yes—
221C2×C42C22×C412yes—
222C2×C42C22×C411yescharacteristic
223C2×C42C22×C412yes—
224C2×C42C22×C412yes—
225C2×C42C22×C411yescharacteristic
226C2×C42C22×C414yes—
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D412yes—
229C2×C42C2×D412yes—
230C2×C42C2×D412yes—
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D414yes—
233C2×C42C2×Q812yes—
234C2×C42C2×Q814yes—
235C4⋊C8—28no—
236C4⋊C8—24no—
237C4⋊C8—28no—
238C4⋊C8—24no—
239C22⋊C8—21no—
240C22⋊C8—21no—
241C22⋊C8—22no—
242C22⋊C8C22×C412yes—
243C22⋊C8C22×C412yes—
244C22⋊C8C22×C414yes—
245#97—28no—
246#97—24no—
247C4×C8—28no—
248C4×C8—24no—
249#95—24no—
250#95—28no—
251#95—28no—
252#95—28no—
253#95—22no—
254#95—24no—
255#95—22no—
256#95—24no—
257#95—22no—
258#95—24no—
259#95—24no—
260#95—22no—
261#95—21no—
262#95—21no—
263#95—22no—
264#95—21no—
265#95—21no—
266#95—22no—
267#95C22×C412yes—
268#95C22×C412yes—
269#95C22×C414yes—
270C24C22×D411yescharacteristic
271C2×Q8—24no—
272C2×Q8—24no—
273C2×Q8—44no—
274C2×Q8—48no—
275C2×Q8—48no—
276C2×Q8#9514yes—
277C2×Q8#9514yes—
278C2×Q8#9518yes—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—24no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—44no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—44no—
299C22×C4—42no—
300C22×C4—44no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—41no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4C2×C22⋊C412yes—
309C22×C4C2×C22⋊C414yes—
310C22×C4C2×C22⋊C412yes—
311C22×C4C2×C22⋊C411yescharacteristic
312C22×C4C2×C22⋊C412yes—
313C22×C4C2×C22⋊C411yescharacteristic
314C22×C4C2×C22⋊C412yes—
315C22×C4C2×C22⋊C412yes—
316C22×C4C2×C22⋊C411yescharacteristic
317C22×C4C2×C22⋊C412yes—
318C22×C4C2×C22⋊C414yes—
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4C2×C4⋊C414yes—
321C22×C4C2×C4⋊C412yes—
322C22×C4C2×C4⋊C412yes—
323C22×C4C2×C4⋊C414yes—
324C22×C4C23×C411yescharacteristic
325C22×C4C23×C411yescharacteristic
326C22×C4C23×C411yescharacteristic
327C22×C4C22×D411yescharacteristic
328C22×C4C22×D411yescharacteristic
329C22×C4C22×Q811yescharacteristic
330C2×C8—22no—
331C2×C8—24no—
332C2×C8—22no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—44no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—44no—
339C4⋊C4—48no—
340C4⋊C4—48no—
341C4⋊C4—44no—
342C4⋊C4—88no—
343C4⋊C4—82no—
344C4⋊C4—84no—
345C4⋊C4#9518yes—
346C4⋊C4#9518yes—
347C4⋊C4#9518yes—
348C4⋊C4#9518yes—
349C4⋊C4#9518yes—
350C4⋊C4#9514yes—
351C4⋊C4#9514yes—
352C22⋊C4—42no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C22⋊C4—41no—
356C22⋊C4—42no—
357C22⋊C4—41no—
358C42—48no—
359C42—48no—
360C42—44no—
361C42—84no—
362C42—82no—
363C42—84no—
364C42C2×C4212yes—
365C42C2×C4214yes—
366C42C2×C4214yes—
367C42C2×C4212yes—
368C42C2×C22⋊C414yes—
369C42C2×C22⋊C412yes—
370C42C2×C22⋊C414yes—
371C42C2×C22⋊C412yes—
372C42C2×C4⋊C412yes—
373C42C2×C4⋊C412yes—
374C42C2×C4⋊C414yes—
375C42C2×C4⋊C414yes—
376C42C2×C4⋊C414yes—
377C42C2×C4⋊C414yes—
378C42C2×C4⋊C412yes—
379C42C2×C4⋊C412yes—
380C23—22no—
381C23—22no—
382C23C22×C22⋊C411yescharacteristic
383C23C22×C22⋊C411yescharacteristic
384C23C22×C4⋊C411yescharacteristic
385C23#57611yescharacteristic
386C23#57711yescharacteristic
387C23#57711yescharacteristic
388C23#57811yescharacteristic
389Q8—28no—
390Q8—28no—
391Q8—48no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—21no—
396C2×C4—22no—
397C2×C4—24no—
398C2×C4—21no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—24no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—24no—
407C2×C4—22no—
408C2×C4—24no—
409C2×C4—22no—
410C2×C4—24no—
411C2×C4—22no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—44no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—44no—
427C2×C4—42no—
428C2×C4—44no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—42no—
432C2×C4—44no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—82no—
440C2×C4—84no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—84no—
444C2×C4—82no—
445C2×C4C2×#9512yes—
446C2×C4C2×#9514yes—
447C2×C4C2×#9512yes—
448C2×C4C2×#9514yes—
449C2×C4C2×#9512yes—
450C2×C4C2×#9514yes—
451C2×C4C2×#9514yes—
452C2×C4C2×#9512yes—
453C2×C4#42114yes—
454C2×C4C22×C4211yescharacteristic
455C2×C4C22×C22⋊C411yescharacteristic
456C2×C4C22×C4⋊C411yescharacteristic
457C2×C4C22×C4⋊C411yescharacteristic
458C8—42no—
459C8—44no—
460C8—42no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—22no—
466C22—22no—
467C22—22no—
468C22—22no—
469C22C22×#9511yescharacteristic
470C22#288412yes—
471C22#289711yescharacteristic
472C22#289711yescharacteristic
473C22#289811yescharacteristic
474C22#289811yescharacteristic
475C22C2×#42112yes—
476C22#291211yescharacteristic
477C22#291311yescharacteristic
478C4—22no—
479C4—24no—
480C4—24no—
481C4—22no—
482C4—22no—
483C4—24no—
484C4—24no—
485C4—21no—
486C4—21no—
487C4—21no—
488C4—21no—
489C4—82no—
490C4—84no—
491C4—82no—
492C4—82no—
493C4—84no—
494C4—82no—
495C4#173314yes—
496C2—21no—
497C2—21no—
498C2—21no—
499C2—21no—
500C2#1760112yes—
501C2#1760212yes—
502C2C2×#173311yescharacteristic
503C2#2034811yescharacteristic
504C2#2035511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
14. #2984C4semidirect
22. #2355C4semidirect
23. #2355C4semidirect
24. Q8×C4⋊C4C4semidirect
31. #2320C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29842
#341962
C2×#23202
#80262
#80252
#80222
#80212
#80182
#80172
#80142
#80132