Finite Group Databasebeta

← #188426

Subgroups — #188426

Order 512.

1114 conjugacy classes of subgroups.

The subgroup lattice has 535 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34155C211yesmaximal, characteristic
2#21976C211yesmaximal, characteristic, complemented
3#20809C211yesmaximal, characteristic, complemented
4C2×#1496C211yesmaximal, characteristic
5C2×#1480C211yesmaximal, characteristic
6#8569C212yesmaximal, complemented
7#8567C212yesmaximal, complemented
8#8565C212yesmaximal, complemented
9#8563C212yesmaximal, complemented
10#7902C211yesmaximal, characteristic, complemented
11#7891C211yesmaximal, characteristic, complemented
12C2×#532C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#431C2211yescharacteristic
16C22×C4⋊C8C2211yescharacteristic
17#2372—21no—
18#2372—21no—
19#2369C412yes—
20#2366C412yes—
21#2363—22no—
22#2363C412yes—
23#2363C412yes—
24#2297—22no—
25#2293—21no—
26#2293C2212yes—
27C42×D4—21no—
28C42×D4C2212yes—
29C4×#117C2211yescharacteristic
30C4×#117C2211yescharacteristic
31#1940—21no—
32#1940—21no—
33#1850—22no—
34#1843—21no—
35#1842C2212yes—
36#1832—21no—
37#1831C2212yes—
38#1798—22no—
39C4×C4⋊C8C2212yes—
40C4×C4⋊C8C2212yes—
41#1496—24no—
42#1480—24no—
43C2×D4⋊C8C2212yes—
44C2×D4⋊C8C2212yes—
45C2×D4⋊C8C2212yes—
46C2×D4⋊C8C2212yes—
47#1279C2211yescharacteristic
48#1279C2212yes—
49#1279C2211yescharacteristic
50#1279C2211yescharacteristic
51#1279C2212yes—
52#1279C2211yescharacteristic
53C22×#41—21no—
54C23×C8—21no—
55#532—22no—
56#532—22no—
57#532D414yes—
58#532Q814yes—
59C2×C4⋊Q8C2×C411yescharacteristic
60C2×#127C2×C411yescharacteristic
61C2×#124—21no—
62C2×#121—21no—
63C2×#121C2×C412yes—
64C2×C4×D4—22no—
65C2×C4×D4—21no—
66C2×C4×D4C2×C412yes—
67C2×C4×D4C2×C412yes—
68C2×C4×D4C2312yes—
69C2×#117—22no—
70C2×#117C2×C412yes—
71C2×#117D412yes—
72C22×C42—22no—
73C22×C42C2311yescharacteristic
74#432—22no—
75#432—22no—
76#431—22no—
77#431—22no—
78#431—22no—
79#431—22no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8C2311yescharacteristic
90C2×C4⋊C8C2312yes—
91C2×C4⋊C8C2311yescharacteristic
92C2×C22⋊C8—21no—
93C2×C22⋊C8—21no—
94C2×C22⋊C8—21no—
95C2×#97—21no—
96C2×C4×C8C2311yescharacteristic
97C2×C4×C8C2311yescharacteristic
98C2×C4×C8C2312yes—
99#391—22no—
100#391—22no—
101#390—22no—
102#390—22no—
103#390—22no—
104#390C2×C412yes—
105#390C2×C412yes—
106#389—22no—
107#389C2×C412yes—
108#389C2×C412yes—
109#388—22no—
110#388—22no—
111#388—22no—
112#388—22no—
113#386—22no—
114#386—22no—
115#386—22no—
116#386—22no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4C2×C412yes—
119C4×C4⋊C4C2×C412yes—
120C4×C4⋊C4C2×C412yes—
121C4×C4⋊C4C2×C412yes—
122C4×C4⋊C4D411yescharacteristic
123C4×C4⋊C4D411yescharacteristic
124C4×C4⋊C4C2311yescharacteristic
125C4×C4⋊C4C2311yescharacteristic
126C4×C22⋊C4—21no—
127C4×C22⋊C4—21no—
128C4×C22⋊C4—22no—
129C4×C22⋊C4—22no—
130C4×C22⋊C4—22no—
131C4×C22⋊C4—22no—
132C4×C22⋊C4—22no—
133C4×C22⋊C4—22no—
134C4×C22⋊C4C2×C411yescharacteristic
135C4×C22⋊C4C2×C411yescharacteristic
136C4×C22⋊C4D412yes—
137#376—21no—
138#376D411yescharacteristic
139#376D411yescharacteristic
140C43—21no—
141C43C2312yes—
142#336—22no—
143#336—22no—
144#336—22no—
145#336—22no—
146D4⋊C8—24no—
147D4⋊C8—24no—
148D4⋊C8—24no—
149D4⋊C8—24no—
150C2×#41—24no—
151C2×#41—24no—
152C2×#41—81no—
153C2×#41—81no—
154C2×#41—81no—
155C2×#41—81no—
156C22×Q8—21no—
157C22×D4—22no—
158C22×D4—21no—
159C22×D4—21no—
160C22×D4C22×C412yes—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4C22×C411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C22×C8—21no—
168C22×C8—22no—
169C22×C8—21no—
170C22×C8—22no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—42no—
174C22×C8—81no—
175C22×C8—81no—
176C4⋊Q8—22no—
177C4⋊Q8C4⋊C414yes—
178#127—22no—
179#127C4⋊C414yes—
180#124—24no—
181#124—24no—
182#121—24no—
183#121—24no—
184#121—24no—
185#121C4⋊C418yes—
186C4×D4—24no—
187C4×D4—24no—
188C4×D4—24no—
189C4×D4—24no—
190C4×D4—22no—
191C4×D4—22no—
192C4×D4—42no—
193C4×D4—42no—
194C4×D4—48no—
195C4×D4C2×D414yes—
196C4×D4C2×Q814yes—
197#117—24no—
198#117—44no—
199#117—42no—
200#117—82no—
201#117—82no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4C22⋊C412yes—
206C2×C4⋊C4C22⋊C412yes—
207C2×C4⋊C4C22×C412yes—
208C2×C4⋊C4C22×C412yes—
209C2×C4⋊C4C22×C412yes—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4C22⋊C412yes—
220C2×C22⋊C4C22⋊C412yes—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—82no—
228C2×C42—82no—
229C2×C42C22×C412yes—
230C2×C42C22×C412yes—
231C2×C42C22×C411yescharacteristic
232C2×C42C2×D412yes—
233C2×C42C2×D412yes—
234C2×C42C2×D412yes—
235C2×C42C2×D412yes—
236C2×C42C2×D412yes—
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×Q812yes—
240C2×C42#4112yes—
241C2×C42#4112yes—
242C2×C42C2411yescharacteristic
243C4⋊C8—22no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—22no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—44no—
250C4⋊C8—44no—
251C4⋊C8—42no—
252C4⋊C8—42no—
253C4⋊C8—42no—
254C4⋊C8—82no—
255C4⋊C8—82no—
256C22⋊C8—42no—
257C22⋊C8—42no—
258C22⋊C8—42no—
259#97—42no—
260C4×C8—21no—
261C4×C8—21no—
262C4×C8—22no—
263C4×C8—21no—
264C4×C8—21no—
265C4×C8—22no—
266#95—22no—
267#95—22no—
268#95—22no—
269#95—22no—
270#95—22no—
271#95—21no—
272#95—21no—
273#95C22⋊C411yescharacteristic
274#95C22⋊C411yescharacteristic
275#95C22⋊C411yescharacteristic
276#95C22⋊C411yescharacteristic
277#95C22⋊C412yes—
278#95C22⋊C412yes—
279C24—22no—
280C24C2×C22⋊C411yescharacteristic
281#41—84no—
282#41—84no—
283#41—84no—
284#41—84no—
285C2×Q8—24no—
286C2×Q8—81no—
287C2×Q8—81no—
288C2×D4—28no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—24no—
295C2×D4—24no—
296C2×D4—44no—
297C2×D4—81no—
298C2×D4—81no—
299C2×D4—82no—
300C2×D4—82no—
301C2×D4C2×C4⋊C414yes—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—42no—
314C22×C4—44no—
315C22×C4—44no—
316C22×C4—44no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—44no—
322C22×C4—82no—
323C22×C4—82no—
324C22×C4C2×C22⋊C412yes—
325C22×C4C2×C22⋊C412yes—
326C22×C4C2×C22⋊C412yes—
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C2×C22⋊C411yescharacteristic
329C22×C4C2×C22⋊C411yescharacteristic
330C22×C4C2×C22⋊C412yes—
331C22×C4C2×C4⋊C412yes—
332C22×C4#11712yes—
333C22×C4#12112yes—
334C22×C4#12112yes—
335C22×C4C22⋊Q812yes—
336C22×C4C22⋊Q812yes—
337C22×C4C23×C411yescharacteristic
338C22×C4C22×D411yescharacteristic
339C22×C4C22×D411yescharacteristic
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—42no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—42no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—84no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—42no—
364C4⋊C4—82no—
365C4⋊C4—82no—
366C4⋊C4C2×C4⋊C414yes—
367C4⋊C4#13114yes—
368C22⋊C4—24no—
369C22⋊C4—24no—
370C22⋊C4—28no—
371C22⋊C4—24no—
372C22⋊C4—42no—
373C22⋊C4—44no—
374C22⋊C4—44no—
375C22⋊C4—44no—
376C22⋊C4—44no—
377C22⋊C4—44no—
378C22⋊C4—82no—
379C22⋊C4—82no—
380C42—22no—
381C42—22no—
382C42—22no—
383C42—22no—
384C42—22no—
385C42—22no—
386C42—22no—
387C42—22no—
388C42—24no—
389C42—42no—
390C42—44no—
391C42—44no—
392C42—41no—
393C42—41no—
394C42—41no—
395C42—41no—
396C42—82no—
397C42—82no—
398C42C2×C4⋊C412yes—
399C42#11712yes—
400C42#12112yes—
401C42#12112yes—
402C42C22⋊Q812yes—
403C42C22⋊Q812yes—
404C42C22×D411yescharacteristic
405C42C22×Q811yescharacteristic
406C42C2×#4111yescharacteristic
407C42C2×#4111yescharacteristic
408C23—22no—
409C23—21no—
410C23—21no—
411C23—22no—
412C23—24no—
413C23—42no—
414C23—42no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23—42no—
419C23#38012yes—
420C23C22×C22⋊C411yescharacteristic
421Q8—84no—
422D4—24no—
423D4—24no—
424D4—48no—
425D4—42no—
426D4—42no—
427D4—84no—
428D4—88no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—24no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—88no—
468C2×C4—88no—
469C2×C4#38012yes—
470C2×C4#38012yes—
471C2×C4#38012yes—
472C2×C4#42114yes—
473C2×C4C22×C4⋊C411yescharacteristic
474C2×C4C2×#11711yescharacteristic
475C2×C4C2×#12111yescharacteristic
476C2×C4C2×#12111yescharacteristic
477C2×C4C2×C22⋊Q811yescharacteristic
478C2×C4C2×C22⋊Q811yescharacteristic
479C2×C4C2×#13112yes—
480C2×C4#56812yes—
481C2×C4#57611yescharacteristic
482C2×C4#57611yescharacteristic
483C2×C4#57711yescharacteristic
484C2×C4#57711yescharacteristic
485C8—81no—
486C8—81no—
487C8—82no—
488C8—81no—
489C8—82no—
490C8—81no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—42no—
495C22—42no—
496C22—44no—
497C22—44no—
498C22—44no—
499C22—82no—
500C22—82no—
501C22—82no—
502C22—82no—
503C22C2×#38011yescharacteristic
504C22#288012yes—
505C22#289711yescharacteristic
506C22#289711yescharacteristic
507C22C2×#42112yes—
508C4—24no—
509C4—22no—
510C4—22no—
511C4—21no—
512C4—21no—
513C4—22no—
514C4—42no—
515C4—44no—
516C4—42no—
517C4—42no—
518C4—44no—
519C4—82no—
520C4#290612yes—
521C4#292211yescharacteristic
522C4#292311yescharacteristic
523C4#304111yescharacteristic
524C4#304211yescharacteristic
525C4#308111yescharacteristic
526C4#308111yescharacteristic
527C2—42no—
528C2—84no—
529C2#1752112yes—
530C2#1752612yes—
531C2#2063011yescharacteristic
532C2#2065011yescharacteristic
533C2#2065411yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21976C2semidirect
3. #20809C2semidirect
6. #8569C2semidirect
7. #8567C2semidirect
8. #8565C2semidirect
9. #8563C2semidirect
10. #7902C2semidirect
11. #7891C2semidirect

Maximal subgroups

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

subgroupindex
#341552
#219762
#208092
C2×#14962
C2×#14802
#85692
#85672
#85652
#85632
#79022
#78912