Finite Group Databasebeta

← #222974

Subgroups — #222974

Order 512.

1038 conjugacy classes of subgroups.

The subgroup lattice has 545 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#554C211yesmaximal, characteristic
2#22650C211yesmaximal, characteristic
3#21264C211yesmaximal, characteristic
4#9870C211yesmaximal, characteristic, complemented
5#9864C211yesmaximal, characteristic, complemented
6#9075C212yesmaximal, complemented
7#8942C212yesmaximal, complemented
8#8707C212yesmaximal, complemented
9#8564C211yesmaximal, characteristic, complemented
10C4×Q8⋊C8C211yesmaximal, characteristic, complemented
11#7901C212yesmaximal, complemented
12C2×#554C411yescharacteristic
13C2×#436—21no—
14C2×#431C2212yes—
15C2×#423C2212yes—
16#2370—21no—
17#2369C411yescharacteristic
18#2368—22no—
19#2360C412yes—
20#2355C411yescharacteristic
21#2349—21no—
22D4×C4⋊C4C411yescharacteristic
23#2345C412yes—
24C4×C4⋊Q8C2211yescharacteristic
25C4×#125—21no—
26C4×C22⋊Q8—22no—
27C4×C22⋊Q8C2212yes—
28C42×Q8C2211yescharacteristic
29C42×D4—21no—
30C42×D4C2211yescharacteristic
31C2×C4×C4⋊C4C2212yes—
32#1954—21no—
33#1939—21no—
34#1939—21no—
35#1923—22no—
36#1851—22no—
37#1842C2211yescharacteristic
38#1797C2212yes—
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2211yescharacteristic
41C4×C22⋊C8C2212yes—
42C42×C8C2211yescharacteristic
43C2×Q8⋊C8C2211yescharacteristic
44C2×Q8⋊C8C2211yescharacteristic
45C2×Q8⋊C8C2212yes—
46#1279C2211yescharacteristic
47#1279C2212yes—
48#1279C2212yes—
49#1279C2212yes—
50#1279C2212yes—
51#1279C2212yes—
52#1279C2211yescharacteristic
53C22×M16—22no—
54#554—24no—
55#554—24no—
56C2×C4⋊Q8C2×C411yescharacteristic
57C2×#125—21no—
58C2×C22⋊Q8—22no—
59C2×C22⋊Q8C2×C412yes—
60C2×C4×Q8—21no—
61C2×C4×Q8C2×C411yescharacteristic
62C2×C4×Q8C2×C411yescharacteristic
63C2×C4×Q8C2311yescharacteristic
64C2×C4×D4—21no—
65C2×C4×D4—21no—
66C2×C4×D4C2×C411yescharacteristic
67C2×C4×D4D411yescharacteristic
68C2×C4×D4D411yescharacteristic
69C22×C4⋊C4C2×C412yes—
70C22×C4⋊C4C2×C412yes—
71C22×C42—22no—
72C22×C42C2312yes—
73#436—44no—
74#431—24no—
75#431—42no—
76#423—24no—
77#423—42no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8C2311yescharacteristic
84C2×C4⋊C8C2312yes—
85C2×C4⋊C8C2311yescharacteristic
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C4×C8—21no—
90C2×C4×C8—21no—
91C2×C4×C8—21no—
92C2×C4×C8C2312yes—
93C2×C4×C8C2311yescharacteristic
94C2×C4×C8C2311yescharacteristic
95#391—21no—
96#391—22no—
97#391C2×C412yes—
98#389—21no—
99#389—21no—
100#389—21no—
101#389—22no—
102#389C2×C412yes—
103#389C2×C412yes—
104#387—22no—
105#387—22no—
106#387—22no—
107#387—22no—
108#387—22no—
109#385—22no—
110#385—22no—
111#385—22no—
112#384C2×C411yescharacteristic
113#382—21no—
114#380—22no—
115#380C2×C412yes—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4C2×C411yescharacteristic
126C4×C4⋊C4C2×C412yes—
127C4×C4⋊C4C2×C411yescharacteristic
128C4×C4⋊C4C2×C412yes—
129C4×C4⋊C4C2×C411yescharacteristic
130C4×C4⋊C4D411yescharacteristic
131C4×C4⋊C4D411yescharacteristic
132C4×C4⋊C4C2312yes—
133C4×C4⋊C4C2311yescharacteristic
134C4×C22⋊C4—22no—
135C4×C22⋊C4—22no—
136C4×C22⋊C4—22no—
137C4×C22⋊C4C2×C412yes—
138C4×C22⋊C4D412yes—
139C4×C22⋊C4D412yes—
140C2×#95—22no—
141C43—21no—
142C43C2311yescharacteristic
143#336—22no—
144#336—21no—
145#336—21no—
146#336—22no—
147Q8⋊C8—22no—
148Q8⋊C8—22no—
149Q8⋊C8—42no—
150C22×Q8—21no—
151C22×Q8C22×C411yescharacteristic
152C22×D4C22⋊C411yescharacteristic
153C23×C4—22no—
154C23×C4—22no—
155C23×C4C22×C412yes—
156C23×C4C2×D412yes—
157C23×C4C2×D412yes—
158C2×M16—44no—
159C2×M16—44no—
160C2×M16—82no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—21no—
167C4⋊Q8—22no—
168C4⋊Q8M1614yescomplemented
169#125—28no—
170C22⋊Q8—216no—
171C22⋊Q8—24no—
172C22⋊Q8—24no—
173C4×Q8—22no—
174C4×Q8—22no—
175C4×Q8—21no—
176C4×Q8—21no—
177C4×Q8—21no—
178C4×Q8—21no—
179C4×Q8—41no—
180C4×Q8—81no—
181C4×Q8—81no—
182C4×Q8M1614yescomplemented
183C4×D4—28no—
184C4×D4—22no—
185C4×D4—22no—
186C4×D4—42no—
187C4×D4—44no—
188C4×D4SD1614yes—
189C4×D4Q1614yes—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4C22⋊C411yescharacteristic
206C2×C4⋊C4C22⋊C411yescharacteristic
207C2×C4⋊C4C22⋊C411yescharacteristic
208C2×C4⋊C4C22×C412yes—
209C2×C4⋊C4C22×C411yescharacteristic
210C2×C4⋊C4C22×C411yescharacteristic
211C2×C4⋊C4C22×C412yes—
212C2×C4⋊C4C22×C411yescharacteristic
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4C22⋊C412yes—
216C2×C22⋊C4C22⋊C412yes—
217C2×C22⋊C4C22⋊C412yes—
218C2×C42—21no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—22no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—82no—
232C2×C42—82no—
233C2×C42C22×C411yescharacteristic
234C2×C42C22×C412yes—
235C2×C42C22×C411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D412yes—
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D412yes—
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2411yescharacteristic
243C4⋊C8—21no—
244C4⋊C8—22no—
245C4⋊C8—21no—
246C4⋊C8—22no—
247C4⋊C8—21no—
248C4⋊C8—44no—
249C4⋊C8—44no—
250C4⋊C8—42no—
251C4⋊C8—42no—
252C4⋊C8—81no—
253C4⋊C8#4112yes—
254C22⋊C8—44no—
255C22⋊C8—44no—
256C22⋊C8—44no—
257C4×C8—21no—
258C4×C8—22no—
259C4×C8—21no—
260C4×C8—22no—
261C4×C8—21no—
262C4×C8—42no—
263C4×C8—42no—
264C4×C8—42no—
265C4×C8#4112yes—
266#95—22no—
267#95—22no—
268#95—22no—
269#95—22no—
270#95—22no—
271#95—21no—
272#95—21no—
273#95—22no—
274#95—22no—
275#95—21no—
276#95—22no—
277#95C22⋊C412yes—
278#95C22⋊C412yes—
279#95C22⋊C412yes—
280C24C2×C22⋊C412yes—
281C2×Q8—21no—
282C2×Q8—24no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—21no—
286C2×Q8—21no—
287C2×Q8—81no—
288C2×Q8—81no—
289C2×Q8C2×M1612yes—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4#10314yes—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—28no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—44no—
315C22×C4—44no—
316C22×C4—82no—
317C22×C4—82no—
318C22×C4—82no—
319C22×C4C2×C22⋊C412yes—
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C2×C22⋊C412yes—
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C2×C22⋊C412yes—
325C22×C4C2×C22⋊C411yescharacteristic
326C22×C4C2×SD1614yes—
327C22×C4C2×Q1614yes—
328C22×C4C23×C411yescharacteristic
329C22×C4C22×D411yescharacteristic
330C22×C4C22×D411yescharacteristic
331M16—88no—
332C2×C8—22no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—41no—
350C4⋊C4—28no—
351C4⋊C4—28no—
352C4⋊C4—21no—
353C4⋊C4—22no—
354C4⋊C4—21no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—44no—
364C4⋊C4—42no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—42no—
368C4⋊C4—82no—
369C4⋊C4—81no—
370C4⋊C4—82no—
371C4⋊C4#10314yes—
372C4⋊C4C2×M1612yes—
373C4⋊C4C2×M1614yes—
374C4⋊C4#13112yes—
375C22⋊C4—28no—
376C22⋊C4—24no—
377C22⋊C4—42no—
378C22⋊C4—44no—
379C22⋊C4#10318yes—
380C42—22no—
381C42—22no—
382C42—22no—
383C42—22no—
384C42—22no—
385C42—22no—
386C42—21no—
387C42—21no—
388C42—21no—
389C42—21no—
390C42—24no—
391C42—22no—
392C42—41no—
393C42—42no—
394C42—41no—
395C42—41no—
396C42—41no—
397C42—42no—
398C42—41no—
399C42—42no—
400C42—42no—
401C42—81no—
402C42—82no—
403C42—82no—
404C42C2×M1612yes—
405C42C2×M1612yes—
406C42C2×SD1612yes—
407C42C2×Q1612yes—
408C42C2×#4111yescharacteristic
409C42C2×#4111yescharacteristic
410C42H(5,𝔽2)11yescharacteristic
411C422−1+411yescharacteristic
412C23—22no—
413C23—22no—
414C23—22no—
415C23—22no—
416C23—22no—
417C23—22no—
418C23C2×#10314yes—
419C23C22×C22⋊C411yescharacteristic
420Q8—22no—
421Q8—22no—
422Q8—41no—
423Q8—41no—
424Q8—84no—
425D4—28no—
426C2×C4—22no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—24no—
444C2×C4—21no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—88no—
472C2×C4—88no—
473C2×C4#40712yes—
474C2×C4#40712yes—
475C2×C4C2×#10312yes—
476C2×C4C2×#10314yes—
477C2×C4C2×#10312yes—
478C2×C4#42112yes—
479C2×C4C4×#4111yescharacteristic
480C2×C4#52011yescharacteristic
481C2×C4#54611yescharacteristic
482C2×C4#54611yescharacteristic
483C2×C4#54711yescharacteristic
484C2×C4#54711yescharacteristic
485C2×C4C22×M1611yescharacteristic
486C2×C4C22×M1611yescharacteristic
487C2×C4C2×#13111yescharacteristic
488C2×C4#56811yescharacteristic
489C2×C4C22×SD1611yescharacteristic
490C2×C4C22×Q1611yescharacteristic
491C2×C4#57611yescharacteristic
492C2×C4#57711yescharacteristic
493C8—42no—
494C8—42no—
495C8—42no—
496C8—82no—
497C8—81no—
498C8—82no—
499C22—24no—
500C22—24no—
501C22—24no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22#233911yescharacteristic
507C22C2×#40711yescharacteristic
508C22#288011yescharacteristic
509C22C22×#10311yescharacteristic
510C22#289711yescharacteristic
511C22C2×#42111yescharacteristic
512C22#290111yescharacteristic
513C4—22no—
514C4—22no—
515C4—24no—
516C4—21no—
517C4—21no—
518C4—42no—
519C4—44no—
520C4—41no—
521C4—42no—
522C4—42no—
523C4—42no—
524C4—42no—
525C4—82no—
526C4—81no—
527C4#148714yes—
528C4#296511yescharacteristic
529C4#297111yescharacteristic
530C4#298911yescharacteristic
531C4#299111yescharacteristic
532C4#329511yescharacteristic
533C4#329911yescharacteristic
534C4#333511yescharacteristic
535C4#333811yescharacteristic
536C2—44no—
537C2C2×#148711yescharacteristic
538C2#1745311yescharacteristic
539C2#1750911yescharacteristic
540C2#1752211yescharacteristic
541C2#2247311yescharacteristic
542C2#2248711yescharacteristic
543C2#2250711yescharacteristic
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×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9870C2semidirect
5. #9864C2semidirect
6. #9075C2semidirect
7. #8942C2semidirect
8. #8707C2semidirect
9. #8564C2semidirect
10. C4×Q8⋊C8C2semidirect
11. #7901C2semidirect
168. C4⋊Q8M16semidirect
182. C4×Q8M16semidirect

Maximal subgroups

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

subgroupindex
C4×#5542
#226502
#212642
#98702
#98642
#90752
#89422
#87072
#85642
C4×Q8⋊C82
#79012