Finite Group Databasebeta

← #160650

Subgroups — #160650

Order 512.

1048 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#34330C211yesmaximal, characteristic
2#22696C211yesmaximal, characteristic, complemented
3#22008C211yesmaximal, characteristic, complemented
4#7599C212yesmaximal, complemented
5#7597C212yesmaximal, complemented
6#7587C211yesmaximal, characteristic, complemented
7#7586C212yesmaximal, complemented
8#7578C212yesmaximal, complemented
9#7577C211yesmaximal, characteristic, complemented
10#7569C211yesmaximal, characteristic, complemented
11#7562C211yesmaximal, characteristic, complemented
12C2×#550C411yescharacteristic
13C2×C8×D4—21no—
14#2366—21no—
15#2366C411yescharacteristic
16#2364—22no—
17#2363C412yes—
18#2357—21no—
19#2355C411yescharacteristic
20#2354C411yescharacteristic
21#2339C412yes—
22#2333—22no—
23#2328—21no—
24C4×#127C2211yescharacteristic
25C4×#121C2212yes—
26#2297C2212yes—
27#2295C2211yescharacteristic
28#2275—21no—
29C42×D4C2211yescharacteristic
30#1954—21no—
31#1939—21no—
32#1939—21no—
33#1923—22no—
34#1850—22no—
35#1831C2211yescharacteristic
36#1817C2212yes—
37#1797C2212yes—
38#1797C2212yes—
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2211yescharacteristic
41#1744C2211yescharacteristic
42C2×#336C2212yes—
43#1272C2211yescharacteristic
44#1272C2211yescharacteristic
45#1272C2212yes—
46#1271C2211yescharacteristic
47#1271C2211yescharacteristic
48#1271C2212yes—
49#1270C2212yes—
50#1270C2212yes—
51#1270C2212yes—
52#1270C2212yes—
53C22×#41C2×C412yes—
54C23×C8—22no—
55#550—24no—
56#550C818yescomplemented
57C2×#127—21no—
58C2×#127C2×C411yescharacteristic
59C2×#121—22no—
60C2×#121C2×C412yes—
61C2×C4×Q8C2×C411yescharacteristic
62C2×C4×D4—22no—
63C2×C4×D4—21no—
64C2×C4×D4—21no—
65C2×C4×D4—21no—
66C2×C4×D4—21no—
67C2×C4×D4C2×C412yes—
68C2×C4×D4C2×C412yes—
69C2×C4×D4C2×C411yescharacteristic
70C2×C4×D4D411yescharacteristic
71C2×C4×D4D411yescharacteristic
72C2×#117C2×C412yes—
73C22×C42C2312yes—
74C8×D4—44no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—22no—
80C2×C22⋊C8—22no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—22no—
83C2×C4×C8—21no—
84C2×C4×C8—21no—
85C2×C4×C8—21no—
86#391C2×C411yescharacteristic
87#391C2×C412yes—
88#391C2311yescharacteristic
89#390—21no—
90#390—21no—
91#390—22no—
92#390—21no—
93#390—21no—
94#390C2×C412yes—
95#390C2312yes—
96#389C2×C411yescharacteristic
97#389C2311yescharacteristic
98#388—22no—
99#388—22no—
100#388—22no—
101#387—21no—
102#387—21no—
103#387—22no—
104#386—22no—
105#386—22no—
106#386—22no—
107#386—22no—
108#386—22no—
109#386—22no—
110#386—22no—
111#384C2×C411yescharacteristic
112#382—21no—
113#381—22no—
114#380C2×C412yes—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4C2×C411yescharacteristic
119C4×C4⋊C4C2×C412yes—
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4D411yescharacteristic
122C4×C4⋊C4D411yescharacteristic
123C4×C22⋊C4—22no—
124C4×C22⋊C4—22no—
125C4×C22⋊C4—22no—
126C4×C22⋊C4—22no—
127C4×C22⋊C4C2×C412yes—
128C4×C22⋊C4D412yes—
129C4×C22⋊C4D412yes—
130#376—21no—
131C2×#95—22no—
132C43C2311yescharacteristic
133#336—22no—
134#336—21no—
135#336—22no—
136#336—21no—
137#336C2311yescharacteristic
138#336C2311yescharacteristic
139#336C2312yes—
140#336C2312yes—
141#336C2311yescharacteristic
142#336C2311yescharacteristic
143C2×#41—24no—
144C2×#41—82no—
145C2×#41—82no—
146C2×#41C2×C818yescomplemented
147C22×Q8C22×C411yescharacteristic
148C22×D4—22no—
149C22×D4—22no—
150C22×D4—21no—
151C22×D4—21no—
152C22×D4C22⋊C411yescharacteristic
153C22×D4C22×C412yes—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4C22×C412yes—
158C23×C4C2×D412yes—
159C23×C4C2×D412yes—
160C22×C8—21no—
161C22×C8—22no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—22no—
165C22×C8—21no—
166C22×C8—44no—
167C22×C8—44no—
168C22×C8—82no—
169#127—28no—
170#127—22no—
171#127C2×C814yescomplemented
172#121—24no—
173#121—216no—
174#121C2×C818yescomplemented
175C4×Q8—22no—
176C4×Q8C2×C814yescomplemented
177C4×D4—28no—
178C4×D4—22no—
179C4×D4—22no—
180C4×D4—42no—
181C4×D4—44no—
182C4×D4—42no—
183C4×D4—42no—
184C4×D4—81no—
185C4×D4—81no—
186C4×D4—82no—
187C4×D4—82no—
188C4×D4—82no—
189C4×D4—82no—
190C4×D4C2×C814yes—
191C4×D4M1614yes—
192#117—82no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4C22⋊C411yescharacteristic
199C2×C4⋊C4C22⋊C411yescharacteristic
200C2×C4⋊C4C22⋊C411yescharacteristic
201C2×C4⋊C4C22×C411yescharacteristic
202C2×C4⋊C4C22×C411yescharacteristic
203C2×C4⋊C4C22×C412yes—
204C2×C4⋊C4C22×C411yescharacteristic
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4C22⋊C412yes—
215C2×C22⋊C4C22⋊C412yes—
216C2×C22⋊C4C22⋊C412yes—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—82no—
222C2×C42C22×C411yescharacteristic
223C2×C42C22×C412yes—
224C2×C42C22×C411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D412yes—
229C2×C42C2×D412yes—
230C2×C42C2×D411yescharacteristic
231C2×C42C2411yescharacteristic
232C4⋊C8—42no—
233C4⋊C8—42no—
234C4⋊C8—42no—
235C4⋊C8—82no—
236C4⋊C8—82no—
237C22⋊C8—44no—
238C22⋊C8—82no—
239C22⋊C8—82no—
240C4×C8—42no—
241C4×C8—42no—
242C4×C8—81no—
243#95—21no—
244#95—22no—
245#95—22no—
246#95—21no—
247#95—22no—
248#95—21no—
249#95—22no—
250#95—21no—
251#95—21no—
252#95—21no—
253#95—21no—
254#95C22⋊C412yes—
255#95C22⋊C412yes—
256#95C22⋊C412yes—
257C24—22no—
258C24C2×C22⋊C412yes—
259#41—88no—
260#41—88no—
261C2×Q8—21no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8C22×C812yes—
265C2×D4—28no—
266C2×D4—28no—
267C2×D4—22no—
268C2×D4—24no—
269C2×D4—24no—
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—81no—
277C2×D4—81no—
278C2×D4—82no—
279C2×D4—82no—
280C2×D4—82no—
281C2×D4—82no—
282C2×D4C22⋊C814yes—
283C2×D4C22×C814yes—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—28no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—44no—
301C22×C4—44no—
302C22×C4—82no—
303C22×C4—82no—
304C22×C4—82no—
305C22×C4—82no—
306C22×C4—82no—
307C22×C4—82no—
308C22×C4C2×C22⋊C412yes—
309C22×C4C2×C22⋊C412yes—
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4C2×C22⋊C412yes—
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4C2×C22⋊C411yescharacteristic
314C22×C4C2×C22⋊C411yescharacteristic
315C22×C4C22×C814yes—
316C22×C4C2×M1614yes—
317C22×C4C23×C411yescharacteristic
318C22×C4C22×D411yescharacteristic
319C22×C4C22×D411yescharacteristic
320C2×C8—22no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—41no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—88no—
339C4⋊C4—21no—
340C4⋊C4—24no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—41no—
344C4⋊C4—42no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—42no—
349C4⋊C4—81no—
350C4⋊C4—81no—
351C4⋊C4C22⋊C814yes—
352C4⋊C4C22×C812yes—
353C22⋊C4—28no—
354C22⋊C4—24no—
355C22⋊C4—42no—
356C22⋊C4—44no—
357C22⋊C4—42no—
358C22⋊C4—42no—
359C22⋊C4—42no—
360C22⋊C4—42no—
361C22⋊C4—82no—
362C22⋊C4—82no—
363C22⋊C4—82no—
364C22⋊C4—82no—
365C22⋊C4—82no—
366C22⋊C4C22⋊C818yes—
367C42—21no—
368C42—21no—
369C42—24no—
370C42—22no—
371C42—41no—
372C42—42no—
373C42—42no—
374C42—41no—
375C42—42no—
376C42—41no—
377C42—42no—
378C42—82no—
379C42—82no—
380C42C22×C812yes—
381C42C2×M1612yes—
382C42AGL(1,ℤ/8)12yes—
383C42#13712yes—
384C23—22no—
385C23—22no—
386C23—22no—
387C23—22no—
388C23—22no—
389C23—22no—
390C23—24no—
391C23—42no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23C2×C22⋊C814yes—
398C23C22×C22⋊C411yescharacteristic
399Q8—42no—
400Q8—42no—
401D4—28no—
402D4—44no—
403D4—44no—
404D4—48no—
405D4—84no—
406D4—88no—
407D4—88no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—22no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—22no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—21no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—21no—
428C2×C4—22no—
429C2×C4—21no—
430C2×C4—22no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—88no—
451C2×C4—164no—
452C2×C4—164no—
453C2×C4C2×C22⋊C812yes—
454C2×C4C2×C22⋊C814yes—
455C2×C4C2×C22⋊C812yes—
456C2×C4#41712yes—
457C2×C4#41712yes—
458C2×C4#42112yes—
459C2×C4C23×C811yescharacteristic
460C2×C4C22×M1611yescharacteristic
461C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
462C2×C4C2×#13711yescharacteristic
463C2×C4#57611yescharacteristic
464C2×C4#57711yescharacteristic
465C8—42no—
466C8—81no—
467C8—82no—
468C8—81no—
469C8—81no—
470C8—82no—
471C22—24no—
472C22—24no—
473C22—24no—
474C22—42no—
475C22—42no—
476C22—42no—
477C22—42no—
478C22—44no—
479C22—44no—
480C22—44no—
481C22—82no—
482C22—82no—
483C22—82no—
484C22—82no—
485C22C22×C22⋊C811yescharacteristic
486C22#288311yescharacteristic
487C22#288811yescharacteristic
488C22C2×#41711yescharacteristic
489C22#289711yescharacteristic
490C22C2×#42111yescharacteristic
491C22#290111yescharacteristic
492C4—22no—
493C4—22no—
494C4—21no—
495C4—21no—
496C4—21no—
497C4—21no—
498C4—24no—
499C4—21no—
500C4—21no—
501C4—41no—
502C4—42no—
503C4—42no—
504C4—42no—
505C4—42no—
506C4—81no—
507C4—82no—
508C4—81no—
509C4#147714yes—
510C2—44no—
511C2—84no—
512C2#1735011yescharacteristic
513C2C2×#147711yescharacteristic
514C2#1743811yescharacteristic
515C2#1759911yescharacteristic
516C2#1763011yescharacteristic
517C2#1768411yescharacteristic
518C2#1776211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
prime radicalC2×#550C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22696C2semidirect
3. #22008C2semidirect
4. #7599C2semidirect
5. #7597C2semidirect
6. #7587C2semidirect
7. #7586C2semidirect
8. #7578C2semidirect
9. #7577C2semidirect
10. #7569C2semidirect
11. #7562C2semidirect
56. #550C8semidirect
146. C2×#41C2×C8semidirect
171. #127C2×C8semidirect
174. #121C2×C8semidirect
176. C4×Q8C2×C8semidirect

Maximal subgroups

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

subgroupindex
#343302
#226962
#220082
#75992
#75972
#75872
#75862
#75782
#75772
#75692
#75622