Finite Group Databasebeta

← #493985

Subgroups — #493985

Order 512.

1053 conjugacy classes of subgroups.

The subgroup lattice has 511 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#536C211yesmaximal, characteristic
2#20727C212yesmaximal, complemented
3#12862C212yesmaximal, complemented
4#12804C212yesmaximal, complemented
5#12803C212yesmaximal, complemented
6#12702C212yesmaximal
7C4×#356C212yesmaximal, complemented
8C4×#355C212yesmaximal, complemented
9C2×#536C2211yescharacteristic
10C2×C4×#41C411yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12#2920—22no—
13#2327C411yescharacteristic
14#2327C2211yescharacteristic
15#2326C411yescharacteristic
16#2326C2211yescharacteristic
17#2323—22no—
18#2322—22no—
19#2317C411yescharacteristic
20#2311C411yescharacteristic
21#2310C2211yescharacteristic
22C4×C4⋊Q8C411yescharacteristic
23C4×C4⋊Q8C2211yescharacteristic
24C4×#124C411yescharacteristic
25C4×#124C2211yescharacteristic
26C4×C22⋊Q8—22no—
27C4×#123—22no—
28C4×#117C411yescharacteristic
29C2×#356C2214yes—
30C2×#355C2214yes—
31#1821—22no—
32#1801—22no—
33#1801C2214yes—
34#1784C2214yes—
35C4×#101—22no—
36C4×#101C2214yes—
37#1394C2214yes—
38#1394C2214yes—
39C22×#41C2×C411yescharacteristic
40C23×Q8C2311yescharacteristic
41#536—24no—
42#536—24no—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×C4⋊Q8C2311yescharacteristic
45C2×#124C2×C411yescharacteristic
46C2×#124C2311yescharacteristic
47C2×#123—22no—
48C2×C22⋊Q8—22no—
49C4×#41—42no—
50C4×#41—42no—
51C2×C4×Q8—42no—
52C2×C4×Q8—42no—
53C2×C4×Q8C2×C411yescharacteristic
54C2×C4×Q8C2×C411yescharacteristic
55C2×C4×Q8C2×C411yescharacteristic
56C2×C4×Q8C2×C411yescharacteristic
57C2×C4×Q8D412yes—
58C2×C4×Q8D412yes—
59C2×C4×Q8C2311yescharacteristic
60C2×C4×Q8C2311yescharacteristic
61C2×C4×D4—21no—
62C2×C4×D4—21no—
63C2×C4×D4C2×C411yescharacteristic
64C2×C4×D4C2×C411yescharacteristic
65C2×#117—21no—
66C2×#117C2×C411yescharacteristic
67C2×#117C2×C411yescharacteristic
68C2×#117C2×C411yescharacteristic
69C22×C4⋊C4C2×C411yescharacteristic
70C22×C4⋊C4D412yes—
71C22×C42—21no—
72C22×C42C2×C411yescharacteristic
73C22×C42D412yes—
74#432—42no—
75#432—42no—
76#431—42no—
77#431—42no—
78#423—24no—
79C2×#101—24no—
80C2×#101C2318yes—
81#407—24no—
82C4×M16—24no—
83#391—22no—
84#391C2×C411yescharacteristic
85#391C2×C411yescharacteristic
86#391C2×C411yescharacteristic
87#391C2×C411yescharacteristic
88#391C2311yescharacteristic
89#391C2311yescharacteristic
90#390C2×C411yescharacteristic
91#390C2×C411yescharacteristic
92#389—22no—
93#389—22no—
94#389—22no—
95#388—22no—
96#388—22no—
97#388—22no—
98#388—22no—
99#387—22no—
100#387—22no—
101#387—22no—
102#387—22no—
103#386—22no—
104#385—22no—
105#385—22no—
106#385—22no—
107#384C2×C411yescharacteristic
108#384C2×C411yescharacteristic
109#382C2×C411yescharacteristic
110#382C2×C411yescharacteristic
111#381—22no—
112#380—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—22no—
117C4×C22⋊C4—21no—
118C4×C22⋊C4—21no—
119C4×C22⋊C4—22no—
120C4×C22⋊C4—22no—
121#376C2×C411yescharacteristic
122#376C2×C411yescharacteristic
123C43C2×C411yescharacteristic
124C43C2×C411yescharacteristic
125#356—28no—
126#355—28no—
127#343—24no—
128#343—24no—
129C2×#41—22no—
130C2×#41—22no—
131C2×#41—42no—
132C2×#41—42no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—42no—
136C22×Q8—42no—
137C22×Q8C22×C411yescharacteristic
138C22×Q8C22×C411yescharacteristic
139C22×Q8C2×D412yes—
140C22×Q8C2411yescharacteristic
141C22×D4—21no—
142C22×D4C22×C411yescharacteristic
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4C22⋊C412yes—
147C23×C4C22⋊C411yescharacteristic
148C23×C4C22⋊C411yescharacteristic
149C23×C4C22×C411yescharacteristic
150C23×C4C2×D412yes—
151C2×M16—28no—
152C4⋊Q8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C4⋊Q8#4114yes—
156#124—22no—
157#124—22no—
158#124—22no—
159#124#4114yes—
160#123—216no—
161C22⋊Q8—216no—
162C4×Q8—41no—
163C4×Q8—42no—
164C4×Q8—41no—
165C4×Q8—42no—
166C4×Q8—42no—
167C4×Q8—42no—
168C4×Q8—42no—
169C4×Q8—41no—
170C4×Q8—42no—
171C4×Q8—41no—
172C4×Q8—42no—
173C4×Q8—41no—
174C4×Q8—41no—
175C4×D4—41no—
176C4×D4—41no—
177C4×D4—42no—
178C4×D4—44no—
179C4×D4—42no—
180C4×D4—42no—
181#117—22no—
182#117—22no—
183#117—44no—
184#117—42no—
185#117—42no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4C22⋊C412yes—
198C2×C4⋊C4C22⋊C412yes—
199C2×C4⋊C4C22⋊C412yes—
200C2×C4⋊C4C22×C411yescharacteristic
201C2×C4⋊C4C22×C411yescharacteristic
202C2×C4⋊C4C22×C411yescharacteristic
203C2×C4⋊C4C22×C411yescharacteristic
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2×D412yes—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—42no—
229C2×C42—42no—
230C2×C42C22⋊C411yescharacteristic
231C2×C42C22⋊C412yes—
232C2×C42C22⋊C412yes—
233C2×C42C22⋊C411yescharacteristic
234C2×C42C22×C411yescharacteristic
235C2×C42C22×C411yescharacteristic
236C2×C42C22×C411yescharacteristic
237C2×C42C22×C411yescharacteristic
238C2×C42C22×C411yescharacteristic
239C2×C42C22×C411yescharacteristic
240C2×C42C2×D412yes—
241C2×C42C2×D412yes—
242C4⋊C8—44no—
243C4⋊C8—44no—
244#101—28no—
245#101—28no—
246#101—48no—
247C22⋊C8—44no—
248C22⋊C8—44no—
249#97—44no—
250C4×C8—44no—
251#95—21no—
252#95—21no—
253#95—21no—
254#95—21no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—22no—
263C24—21no—
264C24C2×C22⋊C411yescharacteristic
265#41—48no—
266#41—48no—
267C2×Q8—21no—
268C2×Q8—21no—
269C2×Q8—21no—
270C2×Q8—24no—
271C2×Q8—24no—
272C2×Q8—41no—
273C2×Q8—42no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—48no—
278C2×Q8—48no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8#11712yes—
282C2×Q8C2×#4112yes—
283C2×Q8C2×#4112yes—
284C2×D4—24no—
285C2×D4—21no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4#11712yes—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—44no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—82no—
318C22×C4—82no—
319C22×C4—81no—
320C22×C4#9912yes—
321C22×C4#9912yes—
322C22×C4C2×C22⋊C412yes—
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C2×C22⋊C411yescharacteristic
325C22×C4C2×C22⋊C411yescharacteristic
326C22×C4C2×C22⋊C412yes—
327C22×C4C2×C22⋊C412yes—
328C22×C4C2×C22⋊C411yescharacteristic
329C22×C4C2×C22⋊C411yescharacteristic
330C22×C4C23×C411yescharacteristic
331C22×C4C22×D412yes—
332M16—48no—
333M16—48no—
334C2×C8—48no—
335C4⋊C4—24no—
336C4⋊C4—28no—
337C4⋊C4—28no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—82no—
351C4⋊C4—82no—
352C22⋊C4—24no—
353C22⋊C4—28no—
354C22⋊C4—28no—
355C22⋊C4—42no—
356C22⋊C4—42no—
357C22⋊C4—42no—
358C22⋊C4—42no—
359C22⋊C4—82no—
360C22⋊C4—82no—
361C42—21no—
362C42—21no—
363C42—22no—
364C42—22no—
365C42—22no—
366C42—21no—
367C42—22no—
368C42—22no—
369C42—22no—
370C42—22no—
371C42—22no—
372C42—21no—
373C42—22no—
374C42—21no—
375C42—22no—
376C42—21no—
377C42—41no—
378C42—41no—
379C42—42no—
380C42—42no—
381C42—42no—
382C42—42no—
383C42—41no—
384C42—41no—
385C42—81no—
386C42—81no—
387C42#11712yes—
388C42#11712yes—
389C23—21no—
390C23—21no—
391C23—22no—
392C23—41no—
393C23—41no—
394C23—42no—
395C23—42no—
396C23—41no—
397C23—41no—
398C23C2×#9912yes—
399C23C22×C22⋊C411yescharacteristic
400Q8—42no—
401Q8—42no—
402Q8—48no—
403Q8—42no—
404Q8—42no—
405Q8—42no—
406Q8—42no—
407D4—42no—
408D4—42no—
409D4—48no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—22no—
415C2×C4—21no—
416C2×C4—22no—
417C2×C4—21no—
418C2×C4—22no—
419C2×C4—24no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—42no—
431C2×C4—41no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—48no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—44no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—84no—
460C2×C4—82no—
461C2×C4C2×#9911yescharacteristic
462C2×C4C2×#9912yes—
463C2×C4C2×#9911yescharacteristic
464C2×C4#41012yes—
465C2×C4C2×#11711yescharacteristic
466C2×C4C2×#11711yescharacteristic
467C2×C4#52514yes—
468C8—88no—
469C22—21no—
470C22—21no—
471C22—22no—
472C22—22no—
473C22—22no—
474C22—42no—
475C22—42no—
476C22—44no—
477C22—42no—
478C22—82no—
479C22—81no—
480C22—81no—
481C22#228011yescharacteristic
482C22C22×#9911yescharacteristic
483C22C2×#41011yescharacteristic
484C4—21no—
485C4—22no—
486C4—24no—
487C4—22no—
488C4—21no—
489C4—21no—
490C4—42no—
491C4—42no—
492C4—42no—
493C4—42no—
494C4—41no—
495C4—41no—
496C4—42no—
497C4—42no—
498C4—81no—
499C4—81no—
500C4—81no—
501C4—81no—
502C4#213312yes—
503C2—21no—
504C2—21no—
505C2—42no—
506C2—82no—
507C2#2072211yescharacteristic
508C2C2×#213311yescharacteristic
509C2#3336911yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×Q8C24
socleC22C22×#99
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20727C2semidirect
3. #12862C2semidirect
4. #12804C2semidirect
5. #12803C2semidirect
7. C4×#356C2semidirect
8. C4×#355C2semidirect

Maximal subgroups

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

subgroupindex
C4×#5362
#207272
#128622
#128042
#128032
#127022
C4×#3562
C4×#3552