Finite Group Databasebeta

← #222971

Subgroups — #222971

Order 512.

1032 conjugacy classes of subgroups.

The subgroup lattice has 537 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#556C211yesmaximal, characteristic
2D4×C4⋊C8C211yesmaximal, characteristic
3#21264C211yesmaximal, characteristic
4#9870C211yesmaximal, characteristic, complemented
5#9865C211yesmaximal, characteristic, complemented
6#9073C212yesmaximal, complemented
7#8937C212yesmaximal, complemented
8#8708C212yesmaximal, complemented
9#8564C211yesmaximal, characteristic, complemented
10C4×Q8⋊C8C211yesmaximal, characteristic, complemented
11#7891C212yesmaximal, complemented
12C2×#556C411yescharacteristic
13C2×C8×D4—21no—
14C2×#431C2212yes—
15C22×C4⋊C8C2212yes—
16#2373—22no—
17#2367—21no—
18#2365C411yescharacteristic
19#2362C412yes—
20#2356—21no—
21#2355C411yescharacteristic
22#2354C411yescharacteristic
23#2347C412yes—
24C4×C4⋊Q8C2211yescharacteristic
25C4×#126—22no—
26C4×#124—21no—
27C4×C22⋊Q8C2212yes—
28C42×Q8—21no—
29C42×Q8C2211yescharacteristic
30C42×D4C2211yescharacteristic
31C4×#117C2212yes—
32#1954—21no—
33#1939—21no—
34#1939—21no—
35#1923—22no—
36#1850—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
53C23×C8—22no—
54#556—24no—
55#556C818yescomplemented
56C2×C4⋊Q8C2×C411yescharacteristic
57C2×#126—22no—
58C2×#124—21no—
59C2×C22⋊Q8C2×C412yes—
60C2×C4×Q8—21no—
61C2×C4×Q8—21no—
62C2×C4×Q8—21no—
63C2×C4×Q8C2×C411yescharacteristic
64C2×C4×Q8C2×C411yescharacteristic
65C2×C4×Q8C2311yescharacteristic
66C2×C4×D4C2×C411yescharacteristic
67C2×C4×D4D411yescharacteristic
68C2×C4×D4D411yescharacteristic
69C2×#117C2×C412yes—
70C2×#117C2×C412yes—
71C22×C42C2312yes—
72C8×D4—44no—
73#431—24no—
74#431—42no—
75C2×C4⋊C8—24no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—42no—
82C2×C4⋊C8C2311yescharacteristic
83C2×C4⋊C8C2312yes—
84C2×C4⋊C8C2311yescharacteristic
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C4×C8—21no—
89C2×C4×C8—21no—
90C2×C4×C8—21no—
91C2×C4×C8C2312yes—
92C2×C4×C8C2311yescharacteristic
93C2×C4×C8C2311yescharacteristic
94#391—21no—
95#391—21no—
96#391C2×C412yes—
97#390—21no—
98#389—21no—
99#389C2×C412yes—
100#389C2×C412yes—
101#388—22no—
102#388—22no—
103#388—22no—
104#388—22no—
105#388—22no—
106#387—22no—
107#387—22no—
108#387—22no—
109#387—22no—
110#385—22no—
111#384C2×C411yescharacteristic
112#383—22no—
113#382—21no—
114#380C2×C412yes—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4C2×C411yescharacteristic
125C4×C4⋊C4C2×C412yes—
126C4×C4⋊C4C2×C411yescharacteristic
127C4×C4⋊C4C2×C412yes—
128C4×C4⋊C4C2×C411yescharacteristic
129C4×C4⋊C4D411yescharacteristic
130C4×C4⋊C4D411yescharacteristic
131C4×C4⋊C4C2312yes—
132C4×C4⋊C4C2311yescharacteristic
133C4×C22⋊C4—22no—
134C4×C22⋊C4—22no—
135C4×C22⋊C4—22no—
136C4×C22⋊C4C2×C412yes—
137C4×C22⋊C4D412yes—
138C4×C22⋊C4D412yes—
139#376—22no—
140C43—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×C4C22×C412yes—
154C23×C4C2×D412yes—
155C23×C4C2×D412yes—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—21no—
162C22×C8—44no—
163C22×C8—44no—
164C22×C8—82no—
165C4⋊Q8—22no—
166C4⋊Q8C2×C814yescomplemented
167#126—216no—
168#124—28no—
169C22⋊Q8—24no—
170C22⋊Q8C2×C818yescomplemented
171C4×Q8—28no—
172C4×Q8—22no—
173C4×Q8—22no—
174C4×Q8—21no—
175C4×Q8—21no—
176C4×Q8—21no—
177C4×Q8—21no—
178C4×Q8—41no—
179C4×Q8—44no—
180C4×Q8—42no—
181C4×Q8—42no—
182C4×Q8C2×C814yescomplemented
183C4×D4—22no—
184C4×D4—22no—
185C4×D4—42no—
186C4×D4C2×C814yes—
187C4×D4M1614yes—
188#117—24no—
189#117—24no—
190#117—42no—
191#117C2×C818yescomplemented
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4C22⋊C411yescharacteristic
202C2×C4⋊C4C22⋊C411yescharacteristic
203C2×C4⋊C4C22⋊C411yescharacteristic
204C2×C4⋊C4C22×C412yes—
205C2×C4⋊C4C22×C411yescharacteristic
206C2×C4⋊C4C22×C411yescharacteristic
207C2×C4⋊C4C22×C412yes—
208C2×C4⋊C4C22×C411yescharacteristic
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4C22⋊C412yes—
214C2×C22⋊C4C22⋊C412yes—
215C2×C22⋊C4C22⋊C412yes—
216C2×C42—22no—
217C2×C42—21no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—22no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—22no—
230C2×C42—21no—
231C2×C42C22×C411yescharacteristic
232C2×C42C22×C412yes—
233C2×C42C22×C411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D412yes—
236C2×C42C2×D412yes—
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42C2411yescharacteristic
241C4⋊C8—21no—
242C4⋊C8—22no—
243C4⋊C8—21no—
244C4⋊C8—22no—
245C4⋊C8—21no—
246C4⋊C8—44no—
247C4⋊C8—44no—
248C4⋊C8—42no—
249C4⋊C8—42no—
250C4⋊C8—81no—
251C4⋊C8#4112yes—
252C22⋊C8—44no—
253C22⋊C8—44no—
254C22⋊C8—44no—
255C4×C8—21no—
256C4×C8—22no—
257C4×C8—21no—
258C4×C8—22no—
259C4×C8—21no—
260C4×C8—42no—
261C4×C8—42no—
262C4×C8—42no—
263C4×C8#4112yes—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—22no—
268#95—22no—
269#95—22no—
270#95—21no—
271#95—21no—
272#95—22no—
273#95—21no—
274#95—22no—
275#95C22⋊C412yes—
276#95C22⋊C412yes—
277#95C22⋊C412yes—
278C24C2×C22⋊C412yes—
279C2×Q8—21no—
280C2×Q8—24no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8C22×C812yes—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4C22⋊C814yes—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4C2×C22⋊C412yes—
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4C2×C22⋊C412yes—
314C22×C4C2×C22⋊C412yes—
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4C2×C22⋊C411yescharacteristic
317C22×C4C2×C22⋊C411yescharacteristic
318C22×C4C22×C814yes—
319C22×C4C2×M1614yes—
320C22×C4C23×C411yescharacteristic
321C22×C4C22×D411yescharacteristic
322C22×C4C22×D411yescharacteristic
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—41no—
341C2×C8—88no—
342C4⋊C4—21no—
343C4⋊C4—22no—
344C4⋊C4—21no—
345C4⋊C4—28no—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—42no—
355C4⋊C4—44no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—44no—
359C4⋊C4—42no—
360C4⋊C4—81no—
361C4⋊C4C22⋊C814yes—
362C4⋊C4C22×C812yes—
363C4⋊C4C22×C814yes—
364C4⋊C4#13112yes—
365C22⋊C4—28no—
366C22⋊C4—28no—
367C22⋊C4—24no—
368C22⋊C4—42no—
369C22⋊C4—44no—
370C22⋊C4—82no—
371C22⋊C4C22⋊C818yes—
372C42—22no—
373C42—22no—
374C42—22no—
375C42—22no—
376C42—22no—
377C42—22no—
378C42—21no—
379C42—21no—
380C42—21no—
381C42—21no—
382C42—28no—
383C42—24no—
384C42—22no—
385C42—41no—
386C42—44no—
387C42—42no—
388C42—44no—
389C42—41no—
390C42—41no—
391C42—42no—
392C42—42no—
393C42—41no—
394C42—41no—
395C42—42no—
396C42—81no—
397C42—82no—
398C42C22×C812yes—
399C42C2×M1612yes—
400C42AGL(1,ℤ/8)12yes—
401C42#13712yes—
402C42C2×#4111yescharacteristic
403C42C2×#4111yescharacteristic
404C42H(5,𝔽2)11yescharacteristic
405C422−1+411yescharacteristic
406C23—22no—
407C23—22no—
408C23—22no—
409C23—22no—
410C23—22no—
411C23—22no—
412C23C2×C22⋊C814yes—
413C23C22×C22⋊C411yescharacteristic
414Q8—22no—
415Q8—22no—
416Q8—41no—
417Q8—41no—
418Q8—48no—
419D4—28no—
420C2×C4—22no—
421C2×C4—21no—
422C2×C4—22no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—21no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—21no—
441C2×C4—22no—
442C2×C4—21no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4C2×C22⋊C814yes—
466C2×C4C2×C22⋊C812yes—
467C2×C4C2×C22⋊C812yes—
468C2×C4#41712yes—
469C2×C4#41712yes—
470C2×C4#42112yes—
471C2×C4C4×#4111yescharacteristic
472C2×C4#52011yescharacteristic
473C2×C4#54611yescharacteristic
474C2×C4#54611yescharacteristic
475C2×C4#54711yescharacteristic
476C2×C4#54711yescharacteristic
477C2×C4C23×C811yescharacteristic
478C2×C4C22×M1611yescharacteristic
479C2×C4C2×#13111yescharacteristic
480C2×C4#56811yescharacteristic
481C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
482C2×C4C2×#13711yescharacteristic
483C2×C4#57711yescharacteristic
484C2×C4#57811yescharacteristic
485C8—42no—
486C8—42no—
487C8—42no—
488C8—82no—
489C8—81no—
490C8—82no—
491C22—24no—
492C22—24no—
493C22—24no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22#233911yescharacteristic
499C22C22×C22⋊C811yescharacteristic
500C22#288011yescharacteristic
501C22C2×#41711yescharacteristic
502C22#289811yescharacteristic
503C22C2×#42111yescharacteristic
504C22#290111yescharacteristic
505C4—22no—
506C4—22no—
507C4—24no—
508C4—21no—
509C4—21no—
510C4—44no—
511C4—42no—
512C4—41no—
513C4—42no—
514C4—42no—
515C4—42no—
516C4—42no—
517C4—82no—
518C4—81no—
519C4#147714yes—
520C4C8×#4111yescharacteristic
521C4#297011yescharacteristic
522C4#298811yescharacteristic
523C4#299011yescharacteristic
524C4#332411yescharacteristic
525C4#332811yescharacteristic
526C4#335311yescharacteristic
527C4#335611yescharacteristic
528C2—44no—
529C2C2×#147711yescharacteristic
530C2#1743911yescharacteristic
531C2#1751511yescharacteristic
532C2#1752311yescharacteristic
533C2#2246911yescharacteristic
534C2#2250411yescharacteristic
535C2#2250711yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC22×D4C22⋊C4
LCSG ▷ C2×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. #9865C2semidirect
6. #9073C2semidirect
7. #8937C2semidirect
8. #8708C2semidirect
9. #8564C2semidirect
10. C4×Q8⋊C8C2semidirect
11. #7891C2semidirect
55. #556C8semidirect
166. C4⋊Q8C2×C8semidirect
170. C22⋊Q8C2×C8semidirect
182. C4×Q8C2×C8semidirect
191. #117C2×C8semidirect

Maximal subgroups

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

subgroupindex
C4×#5562
D4×C4⋊C82
#212642
#98702
#98652
#90732
#89372
#87082
#85642
C4×Q8⋊C82
#78912