Finite Group Databasebeta

← #224567

Subgroups — #224567

Order 512.

818 conjugacy classes of subgroups.

The subgroup lattice has 517 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34945C211yesmaximal, characteristic
2#22722C211yesmaximal, characteristic, complemented
3#21339C211yesmaximal, characteristic, complemented
4#9676C212yesmaximal
5#9673C211yesmaximal, characteristic, complemented
6#9640C212yesmaximal
7#9637C211yesmaximal, characteristic, complemented
8#8568C212yesmaximal, complemented
9#8563C211yesmaximal, characteristic
10#8178C212yesmaximal, complemented
11C4×D4⋊C8C211yesmaximal, characteristic
12C2×#446—21no—
13#2573—22no—
14#2571C411yescharacteristic
15#2568C411yescharacteristic
16#2547C411yescharacteristic
17#2538C411yescharacteristic
18#2514C412yes—
19#2494—21no—
20#2447C412yes—
21#2421—21no—
22#2401—22no—
23C4×C4⋊Q8C2211yescharacteristic
24C4×#124C2212yes—
25C4×C22⋊Q8—21no—
26#2295C2212yes—
27#2283—21no—
28C42×Q8C2211yescharacteristic
29C42×D4C2211yescharacteristic
30#1984—21no—
31#1940—22no—
32#1939—21no—
33#1939—21no—
34#1916—22no—
35#1842C2211yescharacteristic
36#1842C2212yes—
37#1767C2212yes—
38C4×C4⋊C8C2211yescharacteristic
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2212yes—
41#1744C2212yes—
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—44no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—22no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8C2311yescharacteristic
72C2×C4⋊C8C2312yes—
73C2×C4⋊C8C2311yescharacteristic
74C2×#97—22no—
75C2×C4×C8—21no—
76C2×C4×C8—21no—
77C2×C4×C8—21no—
78C2×C4×C8C2311yescharacteristic
79C2×C4×C8C2312yes—
80C2×C4×C8C2311yescharacteristic
81#401—22no—
82#400—22no—
83#400—22no—
84#400—22no—
85#397—22no—
86#397—22no—
87#397—22no—
88#396—22no—
89#391—21no—
90#391C2×C412yes—
91#391C2×C412yes—
92#391C2×C412yes—
93#391C2×C412yes—
94#390C2×C412yes—
95#389—21no—
96#389—21no—
97#389—21no—
98#389C2×C412yes—
99#389C2×C412yes—
100#388—22no—
101#388—22no—
102#387—22no—
103#387—22no—
104#387—22no—
105#385—22no—
106#384C2×C411yescharacteristic
107#383—22no—
108#382—21no—
109#382—21no—
110#382C2×C412yes—
111#381—22no—
112#380—21no—
113#380—21no—
114#380—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4C2×C411yescharacteristic
119C4×C4⋊C4C2×C411yescharacteristic
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4C2×C412yes—
122C4×C4⋊C4D412yes—
123C4×C4⋊C4D412yes—
124C4×C4⋊C4D411yescharacteristic
125C4×C4⋊C4D411yescharacteristic
126C4×C4⋊C4C2311yescharacteristic
127C4×C22⋊C4—21no—
128C4×C22⋊C4—21no—
129C4×C22⋊C4—22no—
130C4×C22⋊C4—22no—
131C4×C22⋊C4—21no—
132#376—22no—
133C43C2311yescharacteristic
134C43C2312yes—
135#336—21no—
136#336—22no—
137#336—22no—
138#336—22no—
139#336—22no—
140#336—21no—
141Q8⋊C8—24no—
142Q8⋊C8—24no—
143Q8⋊C8—42no—
144Q8⋊C8—42no—
145D4⋊C8—22no—
146D4⋊C8—22no—
147D4⋊C8—42no—
148C22×Q8C22⋊C411yescharacteristic
149C22×Q8C22×C412yes—
150C22×D4C22×C411yescharacteristic
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8—21no—
159C22×C8—22no—
160C4⋊Q8—22no—
161C4⋊Q8—22no—
162#124—24no—
163#124—42no—
164C22⋊Q8—44no—
165C4×Q8—22no—
166C4×Q8—22no—
167C4×Q8—22no—
168C4×Q8—22no—
169C4×Q8—22no—
170C4×Q8—22no—
171C4×Q8—22no—
172C4×Q8—22no—
173C4×Q8—42no—
174C4×D4—22no—
175C4×D4—22no—
176C4×D4—22no—
177C4×D4—21no—
178C4×D4—21no—
179C4×D4—21no—
180C4×D4—21no—
181C4×D4—41no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4C22⋊C411yescharacteristic
190C2×C4⋊C4C22⋊C412yes—
191C2×C4⋊C4C22⋊C411yescharacteristic
192C2×C4⋊C4C22⋊C411yescharacteristic
193C2×C4⋊C4C22⋊C412yes—
194C2×C4⋊C4C22⋊C412yes—
195C2×C4⋊C4C22×C411yescharacteristic
196C2×C4⋊C4C22×C411yescharacteristic
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C4⋊C4C22×C412yes—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—22no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—22no—
209C2×C42—21no—
210C2×C42—42no—
211C2×C42—42no—
212C2×C42C22×C411yescharacteristic
213C2×C42C22×C412yes—
214C2×C42C22×C412yes—
215C2×C42C22×C411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D412yes—
218C2×C42C2×D412yes—
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D412yes—
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D412yes—
223C2×C42C2×D411yescharacteristic
224C2×C42C2411yescharacteristic
225C4⋊C8—21no—
226C4⋊C8—22no—
227C4⋊C8—22no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—42no—
231C4⋊C8—44no—
232C4⋊C8—44no—
233C4⋊C8—42no—
234C4⋊C8—82no—
235C4⋊C8—81no—
236C4⋊C8#4112yes—
237#97—44no—
238C4×C8—21no—
239C4×C8—22no—
240C4×C8—22no—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8—42no—
244C4×C8—42no—
245C4×C8—42no—
246C4×C8#4112yes—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—21no—
252#95—21no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—21no—
262#95C22⋊C412yes—
263#95C22⋊C412yes—
264#95C22⋊C412yes—
265C24—21no—
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—22no—
274C2×Q8—22no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4#13112yes—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—42no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—81no—
298C22×C4C2×C22⋊C412yes—
299C22×C4C2×C22⋊C411yescharacteristic
300C22×C4C2×C22⋊C412yes—
301C22×C4C2×C22⋊C412yes—
302C22×C4C2×C22⋊C412yes—
303C22×C4C2×C22⋊C411yescharacteristic
304C22×C4C2×C22⋊C411yescharacteristic
305C22×C4C2×C22⋊C411yescharacteristic
306C22×C4C23×C411yescharacteristic
307C22×C4C22×D411yescharacteristic
308C22×C4C22×D411yescharacteristic
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—41no—
322C2×C8—42no—
323C2×C8—41no—
324C2×C8—42no—
325C2×C8—41no—
326C2×C8—42no—
327C4⋊C4—22no—
328C4⋊C4—24no—
329C4⋊C4—22no—
330C4⋊C4—24no—
331C4⋊C4—22no—
332C4⋊C4—21no—
333C4⋊C4—22no—
334C4⋊C4—21no—
335C4⋊C4—22no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—42no—
341C4⋊C4—44no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—81no—
348C4⋊C4—82no—
349C22⋊C4—42no—
350C22⋊C4—44no—
351C22⋊C4—44no—
352C22⋊C4—44no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C22⋊C4—42no—
356C22⋊C4—82no—
357C42—21no—
358C42—22no—
359C42—22no—
360C42—22no—
361C42—21no—
362C42—22no—
363C42—22no—
364C42—22no—
365C42—21no—
366C42—22no—
367C42—21no—
368C42—22no—
369C42—22no—
370C42—41no—
371C42—42no—
372C42—42no—
373C42—42no—
374C42—42no—
375C42—41no—
376C42—41no—
377C42—42no—
378C42—42no—
379C42#13112yes—
380C42#13112yes—
381C42AGL(1,ℤ/8)12yes—
382C42#13712yes—
383C42C2×#4111yescharacteristic
384C42C2×#4111yescharacteristic
385C42H(5,𝔽2)11yescharacteristic
386C422−1+411yescharacteristic
387C23—22no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23C22×C22⋊C411yescharacteristic
395Q8—44no—
396Q8—42no—
397Q8—42no—
398Q8—42no—
399Q8—42no—
400D4—22no—
401D4—22no—
402D4—41no—
403D4—41no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—21no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—21no—
419C2×C4—22no—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—42no—
439C2×C4—41no—
440C2×C4—84no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—84no—
446C2×C4#40812yes—
447C2×C4#40812yes—
448C2×C4#41712yes—
449C2×C4#41712yes—
450C2×C4#42112yes—
451C2×C4C4×#4111yescharacteristic
452C2×C4#52011yescharacteristic
453C2×C4#54811yescharacteristic
454C2×C4#54811yescharacteristic
455C2×C4#55011yescharacteristic
456C2×C4#55011yescharacteristic
457C2×C4C2×#13111yescharacteristic
458C2×C4C2×#13111yescharacteristic
459C2×C4C2×#13111yescharacteristic
460C2×C4#56811yescharacteristic
461C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
462C2×C4C2×#13711yescharacteristic
463C2×C4#57711yescharacteristic
464C2×C4#57811yescharacteristic
465C8—42no—
466C8—42no—
467C8—42no—
468C8—82no—
469C8—81no—
470C8—82no—
471C22—42no—
472C22—42no—
473C22—42no—
474C22—81no—
475C22—81no—
476C22—81no—
477C22—81no—
478C22#234111yescharacteristic
479C22C2×#40811yescharacteristic
480C22#288011yescharacteristic
481C22C2×#41711yescharacteristic
482C22#289811yescharacteristic
483C22C2×#42111yescharacteristic
484C22#290111yescharacteristic
485C4—21no—
486C4—21no—
487C4—22no—
488C4—22no—
489C4—42no—
490C4—42no—
491C4—42no—
492C4—41no—
493C4—42no—
494C4—82no—
495C4—81no—
496C4—82no—
497C4—81no—
498C4—81no—
499C4#149514yes—
500C4#296611yescharacteristic
501C4#296911yescharacteristic
502C4#299311yescharacteristic
503C4#299411yescharacteristic
504C4#336711yescharacteristic
505C4#336911yescharacteristic
506C4#339711yescharacteristic
507C4#339811yescharacteristic
508C2—82no—
509C2C2×#149511yescharacteristic
510C2#1748411yescharacteristic
511C2#1751211yescharacteristic
512C2#1752311yescharacteristic
513C2#2253111yescharacteristic
514C2#2254411yescharacteristic
515C2#2255111yescharacteristic
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. #22722C2semidirect
3. #21339C2semidirect
5. #9673C2semidirect
7. #9637C2semidirect
8. #8568C2semidirect
10. #8178C2semidirect

Maximal subgroups

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

subgroupindex
#349452
#227222
#213392
#96762
#96732
#96402
#96372
#85682
#85632
#81782
C4×D4⋊C82