Finite Group Databasebeta

← #224555

Subgroups — #224555

Order 512.

870 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34945C211yesmaximal, characteristic
2Q8×C4⋊C8C211yesmaximal, characteristic, complemented
3C4×#446C211yesmaximal, characteristic, complemented
4#9675C212yesmaximal
5#9673C211yesmaximal, characteristic, complemented
6#9646C212yesmaximal
7#9637C211yesmaximal, characteristic, complemented
8#8568C212yesmaximal, complemented
9#8563C211yesmaximal, characteristic
10#8180C212yesmaximal, complemented
11C4×D4⋊C8C211yesmaximal, characteristic
12C2×#446—21no—
13C2×C8×Q8—21no—
14#2573—22no—
15#2571C411yescharacteristic
16#2568C411yescharacteristic
17#2547C411yescharacteristic
18#2538C411yescharacteristic
19#2514C412yes—
20#2494—21no—
21#2447C412yes—
22#2421—21no—
23#2401—22no—
24C4×C4⋊Q8C2211yescharacteristic
25C4×#124C2212yes—
26C4×C22⋊Q8—21no—
27#2295C2212yes—
28#2283—21no—
29C42×Q8C2211yescharacteristic
30C42×D4C2211yescharacteristic
31#1939—22no—
32#1939—21no—
33#1842C2211yescharacteristic
34#1842C2212yes—
35#1771—22no—
36#1770—21no—
37C4×C4⋊C8C2211yescharacteristic
38C4×C4⋊C8C2212yes—
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2212yes—
41C4×#97C2212yes—
42C42×C8C2211yescharacteristic
43C2×Q8⋊C8C2212yes—
44C2×Q8⋊C8C2212yes—
45C2×Q8⋊C8C2212yes—
46C2×Q8⋊C8C2212yes—
47C2×D4⋊C8C2211yescharacteristic
48C2×D4⋊C8C2211yescharacteristic
49C2×D4⋊C8C2212yes—
50#1279C2212yes—
51#1279C2211yescharacteristic
52#1279C2211yescharacteristic
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×#124C2×C412yes—
55C2×C22⋊Q8—21no—
56C2×C4×Q8C2×C411yescharacteristic
57C2×C4×Q8D411yescharacteristic
58C2×C4×Q8D411yescharacteristic
59C2×C4×Q8C2312yes—
60C2×C4×D4C2×C411yescharacteristic
61C2×C4×D4C2×C411yescharacteristic
62C2×C4×D4C2311yescharacteristic
63C22×C42—21no—
64#446—28no—
65C8×Q8—44no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—22no—
68C2×C4⋊C8—22no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8C2311yescharacteristic
73C2×C4⋊C8C2312yes—
74C2×C4⋊C8C2311yescharacteristic
75C2×#97—22no—
76C2×#97—22no—
77C2×C4×C8—21no—
78C2×C4×C8—22no—
79C2×C4×C8—21no—
80C2×C4×C8—21no—
81C2×C4×C8C2311yescharacteristic
82C2×C4×C8C2312yes—
83C2×C4×C8C2311yescharacteristic
84#401—22no—
85#400—22no—
86#400—22no—
87#400—22no—
88#397—22no—
89#397—22no—
90#397—22no—
91#396—22no—
92#391—21no—
93#391C2×C412yes—
94#391C2×C412yes—
95#391C2×C412yes—
96#391C2×C412yes—
97#390C2×C412yes—
98#389—21no—
99#389—21no—
100#389—21no—
101#389C2×C412yes—
102#389C2×C412yes—
103#388—22no—
104#388—22no—
105#387—22no—
106#387—22no—
107#387—22no—
108#385—22no—
109#384C2×C411yescharacteristic
110#383—22no—
111#382—21no—
112#382—21no—
113#382C2×C412yes—
114#381—22no—
115#380—21no—
116#380—21no—
117#380—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4C2×C412yes—
125C4×C4⋊C4D412yes—
126C4×C4⋊C4D412yes—
127C4×C4⋊C4D411yescharacteristic
128C4×C4⋊C4D411yescharacteristic
129C4×C4⋊C4C2311yescharacteristic
130C4×C22⋊C4—21no—
131C4×C22⋊C4—21no—
132C4×C22⋊C4—22no—
133C4×C22⋊C4—22no—
134C4×C22⋊C4—21no—
135#376—22no—
136C43C2311yescharacteristic
137C43C2312yes—
138#336—22no—
139#336—21no—
140#336—22no—
141#336—21no—
142Q8⋊C8—24no—
143Q8⋊C8—24no—
144Q8⋊C8—42no—
145Q8⋊C8—42no—
146D4⋊C8—22no—
147D4⋊C8—22no—
148D4⋊C8—42no—
149C22×Q8C22⋊C411yescharacteristic
150C22×Q8C22×C412yes—
151C22×D4C22×C411yescharacteristic
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—22no—
161C4⋊Q8—22no—
162C4⋊Q8—22no—
163#124—24no—
164#124—42no—
165C22⋊Q8—44no—
166C4×Q8—24no—
167C4×Q8—22no—
168C4×Q8—22no—
169C4×Q8—22no—
170C4×Q8—22no—
171C4×Q8—22no—
172C4×Q8—22no—
173C4×Q8—22no—
174C4×Q8—22no—
175C4×D4—22no—
176C4×D4—22no—
177C4×D4—22no—
178C4×D4—21no—
179C4×D4—21no—
180C4×D4—21no—
181C4×D4—21no—
182C4×D4—41no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4C22⋊C411yescharacteristic
191C2×C4⋊C4C22⋊C412yes—
192C2×C4⋊C4C22⋊C411yescharacteristic
193C2×C4⋊C4C22⋊C411yescharacteristic
194C2×C4⋊C4C22⋊C412yes—
195C2×C4⋊C4C22⋊C412yes—
196C2×C4⋊C4C22×C411yescharacteristic
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C4⋊C4C22×C411yescharacteristic
199C2×C4⋊C4C22×C412yes—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—22no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—22no—
210C2×C42—21no—
211C2×C42—42no—
212C2×C42—42no—
213C2×C42C22×C411yescharacteristic
214C2×C42C22×C412yes—
215C2×C42C22×C412yes—
216C2×C42C22×C411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D412yes—
219C2×C42C2×D412yes—
220C2×C42C2×D412yes—
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D412yes—
224C2×C42C2×D411yescharacteristic
225C2×C42C2411yescharacteristic
226C4⋊C8—24no—
227C4⋊C8—28no—
228C4⋊C8—21no—
229C4⋊C8—22no—
230C4⋊C8—22no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—44no—
234C4⋊C8—42no—
235C4⋊C8—81no—
236C4⋊C8—82no—
237C4⋊C8#4112yes—
238#97—28no—
239#97—44no—
240C4×C8—24no—
241C4×C8—21no—
242C4×C8—22no—
243C4×C8—22no—
244C4×C8—21no—
245C4×C8—21no—
246C4×C8—44no—
247C4×C8—42no—
248C4×C8—42no—
249C4×C8#4112yes—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—21no—
255#95—21no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—22no—
263#95—22no—
264#95—21no—
265#95C22⋊C412yes—
266#95C22⋊C412yes—
267#95C22⋊C412yes—
268C24—21no—
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—22no—
277C2×Q8#10414yes—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4#13112yes—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—42no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—81no—
301C22×C4C2×C22⋊C412yes—
302C22×C4C2×C22⋊C412yes—
303C22×C4C2×C22⋊C411yescharacteristic
304C22×C4C2×C22⋊C412yes—
305C22×C4C2×C22⋊C411yescharacteristic
306C22×C4C2×C22⋊C411yescharacteristic
307C22×C4C2×C22⋊C412yes—
308C22×C4C2×C22⋊C411yescharacteristic
309C22×C4C23×C411yescharacteristic
310C22×C4C22×D411yescharacteristic
311C22×C4C22×D411yescharacteristic
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—24no—
317C2×C8—22no—
318C2×C8—41no—
319C2×C8—42no—
320C2×C8—41no—
321C2×C8—42no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—42no—
330C4⋊C4—24no—
331C4⋊C4—22no—
332C4⋊C4—24no—
333C4⋊C4—22no—
334C4⋊C4—24no—
335C4⋊C4—22no—
336C4⋊C4—21no—
337C4⋊C4—22no—
338C4⋊C4—21no—
339C4⋊C4—22no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—44no—
350C4⋊C4—81no—
351C4⋊C4—82no—
352C22⋊C4—42no—
353C22⋊C4—44no—
354C22⋊C4—44no—
355C22⋊C4—41no—
356C22⋊C4—41no—
357C22⋊C4—42no—
358C22⋊C4—44no—
359C22⋊C4—82no—
360C42—21no—
361C42—22no—
362C42—22no—
363C42—22no—
364C42—21no—
365C42—22no—
366C42—22no—
367C42—22no—
368C42—21no—
369C42—22no—
370C42—21no—
371C42—22no—
372C42—22no—
373C42—24no—
374C42—41no—
375C42—42no—
376C42—42no—
377C42—42no—
378C42—41no—
379C42—41no—
380C42—42no—
381C42—42no—
382C42#13112yes—
383C42#13112yes—
384C42AGL(1,ℤ/8)12yes—
385C42#13712yes—
386C42C2×#4111yescharacteristic
387C42C2×#4111yescharacteristic
388C42H(5,𝔽2)11yescharacteristic
389C422−1+411yescharacteristic
390C23—22no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23C22×C22⋊C411yescharacteristic
398Q8—28no—
399Q8—42no—
400Q8—42no—
401Q8—42no—
402Q8—42no—
403D4—22no—
404D4—22no—
405D4—41no—
406D4—41no—
407C2×C4—22no—
408C2×C4—21no—
409C2×C4—22no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—21no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—21no—
422C2×C4—22no—
423C2×C4—21no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—42no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—41no—
442C2×C4—84no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—84no—
448C2×C4#40812yes—
449C2×C4#40812yes—
450C2×C4#41712yes—
451C2×C4#41712yes—
452C2×C4C2×#10414yes—
453C2×C4C2×#10412yes—
454C2×C4C4×#4111yescharacteristic
455C2×C4#52011yescharacteristic
456C2×C4#54811yescharacteristic
457C2×C4#54811yescharacteristic
458C2×C4#55011yescharacteristic
459C2×C4#55011yescharacteristic
460C2×C4C2×#13111yescharacteristic
461C2×C4C2×#13111yescharacteristic
462C2×C4C2×#13111yescharacteristic
463C2×C4#56811yescharacteristic
464C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
465C2×C4C2×#13711yescharacteristic
466C2×C4#57711yescharacteristic
467C2×C4#57811yescharacteristic
468C8—24no—
469C8—42no—
470C8—44no—
471C8—42no—
472C8—81no—
473C8—82no—
474C22—42no—
475C22—42no—
476C22—42no—
477C22—81no—
478C22—81no—
479C22—81no—
480C22—81no—
481C22#234111yescharacteristic
482C22C2×#40811yescharacteristic
483C22#288011yescharacteristic
484C22C2×#41711yescharacteristic
485C22#289811yescharacteristic
486C22C22×#10411yescharacteristic
487C22#290111yescharacteristic
488C4—21no—
489C4—21no—
490C4—24no—
491C4—22no—
492C4—22no—
493C4—42no—
494C4—42no—
495C4—41no—
496C4—42no—
497C4—82no—
498C4—81no—
499C4—82no—
500C4—81no—
501C4—81no—
502C4#148314yes—
503C4#296611yescharacteristic
504C4#296911yescharacteristic
505C4#299311yescharacteristic
506C4#299411yescharacteristic
507C4#336711yescharacteristic
508C4#336911yescharacteristic
509C4#339711yescharacteristic
510C4#339811yescharacteristic
511C2—82no—
512C2C2×#148311yescharacteristic
513C2#1746911yescharacteristic
514C2#1751211yescharacteristic
515C2#1751811yescharacteristic
516C2#2253111yescharacteristic
517C2#2254411yescharacteristic
518C2#2255011yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC22×D4C22×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. Q8×C4⋊C8C2semidirect
3. C4×#446C2semidirect
5. #9673C2semidirect
7. #9637C2semidirect
8. #8568C2semidirect
10. #8180C2semidirect

Maximal subgroups

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

subgroupindex
#349452
Q8×C4⋊C82
C4×#4462
#96752
#96732
#96462
#96372
#85682
#85632
#81802
C4×D4⋊C82