Finite Group Databasebeta

← #188415

Subgroups — #188415

Order 512.

1062 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34129C211yesmaximal, characteristic
2#21976C211yesmaximal, characteristic, complemented
3#20809C211yesmaximal, characteristic, complemented
4C2×#1488C211yesmaximal, characteristic
5C2×#1477C211yesmaximal, characteristic
6#8569C212yesmaximal, complemented
7#8567C212yesmaximal, complemented
8#8566C212yesmaximal, complemented
9#8564C212yesmaximal, complemented
10#7900C211yesmaximal, characteristic, complemented
11#7893C211yesmaximal, characteristic, complemented
12C2×C4×#41C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#431C2211yescharacteristic
16C22×C4⋊C8C2211yescharacteristic
17#2354—22no—
18#2354C412yes—
19Q8×C4⋊C4C412yes—
20#2346—21no—
21#2346—21no—
22#2346C411yescharacteristic
23#2346C411yescharacteristic
24#2341C411yescharacteristic
25#2341C411yescharacteristic
26#2293—22no—
27#2293C2212yes—
28#2285—21no—
29#2285C2211yescharacteristic
30C42×Q8C2212yes—
31C4×#117—21no—
32C4×#117C2211yescharacteristic
33#1940—21no—
34#1940—21no—
35#1850—22no—
36#1843—21no—
37#1842C2212yes—
38#1832—21no—
39#1831C2212yes—
40#1798—22no—
41C4×C4⋊C8C2212yes—
42C4×C4⋊C8C2212yes—
43#1488—24no—
44#1477—24no—
45C2×Q8⋊C8C2212yes—
46C2×Q8⋊C8C2211yescharacteristic
47C2×Q8⋊C8C2211yescharacteristic
48C2×D4⋊C8C2212yes—
49C2×D4⋊C8C2211yescharacteristic
50C2×D4⋊C8C2211yescharacteristic
51#1279C2212yes—
52#1279C2212yes—
53#1279C2212yes—
54#1279C2212yes—
55C22×#41C2×C411yescharacteristic
56C23×C8—21no—
57C4×#41—22no—
58C4×#41—22no—
59C4×#41D414yes—
60C4×#41Q814yes—
61C2×C4×Q8C2×C412yes—
62C2×C4×Q8C2×C412yes—
63C2×C4×Q8C2×C411yescharacteristic
64C2×C4×Q8C2311yescharacteristic
65C2×C4×D4—21no—
66C2×C4×D4—21no—
67C2×C4×D4C2×C411yescharacteristic
68C2×C4×D4C2311yescharacteristic
69C2×#117—21no—
70C2×#117—22no—
71C2×#117—21no—
72C2×#117—22no—
73C2×#117C2×C411yescharacteristic
74C2×#117D412yes—
75C22×C42—21no—
76C22×C42C2311yescharacteristic
77#432—22no—
78#432—22no—
79#431—22no—
80#431—22no—
81#431—22no—
82#431—22no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2312yes—
94C2×C4⋊C8C2311yescharacteristic
95C2×C22⋊C8—21no—
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×#97—21no—
99C2×C4×C8C2312yes—
100C2×C4×C8C2311yescharacteristic
101C2×C4×C8C2311yescharacteristic
102#390—22no—
103#390—22no—
104#390C2×C412yes—
105#390C2×C412yes—
106#389—22no—
107#389—22no—
108#389C2×C412yes—
109#389C2×C412yes—
110#388—22no—
111#388—22no—
112#388—22no—
113#388—22no—
114#388—22no—
115#388—22no—
116#384—22no—
117#384C2×C412yes—
118#382—21no—
119#382—21no—
120#382—21no—
121#380—21no—
122#380—22no—
123#380—21no—
124#380—21no—
125#380C2×C411yescharacteristic
126#380C2×C411yescharacteristic
127C4×C4⋊C4—22no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—22no—
132C4×C4⋊C4C2×C412yes—
133C4×C4⋊C4C2×C412yes—
134C4×C4⋊C4C2×C412yes—
135C4×C4⋊C4D411yescharacteristic
136C4×C4⋊C4D411yescharacteristic
137C4×C4⋊C4C2312yes—
138C4×C22⋊C4—22no—
139C4×C22⋊C4—21no—
140C4×C22⋊C4—21no—
141C4×C22⋊C4C2×C411yescharacteristic
142C4×C22⋊C4C2×C411yescharacteristic
143C4×C22⋊C4D412yes—
144#376—21no—
145#376—21no—
146#376D411yescharacteristic
147#376D411yescharacteristic
148C43—22no—
149C43C2312yes—
150#336—22no—
151#336—22no—
152#336—22no—
153#336—22no—
154Q8⋊C8—24no—
155Q8⋊C8—22no—
156Q8⋊C8—22no—
157D4⋊C8—24no—
158D4⋊C8—22no—
159D4⋊C8—22no—
160C2×#41—22no—
161C2×#41—24no—
162C2×#41—42no—
163C2×#41C4⋊C414yes—
164C22×Q8C22×C411yescharacteristic
165C22×D4—21no—
166C22×D4C22×C411yescharacteristic
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4C22×C411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C22×C8—21no—
174C22×C8—22no—
175C22×C8—21no—
176C22×C8—22no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—42no—
180C22×C8—81no—
181C22×C8—81no—
182C4×Q8—22no—
183C4×Q8—21no—
184C4×Q8—21no—
185C4×Q8—24no—
186C4×Q8—24no—
187C4×Q8—42no—
188C4×Q8—42no—
189C4×Q8C4⋊C414yes—
190C4×Q8C2×D412yes—
191C4×Q8C2×Q812yes—
192C4×D4—24no—
193C4×D4—24no—
194C4×D4—24no—
195C4×D4—24no—
196C4×D4—22no—
197C4×D4—21no—
198C4×D4—21no—
199C4×D4C4⋊C414yes—
200C4×D4C2×D412yes—
201C4×D4C2×Q812yes—
202#117—22no—
203#117—24no—
204#117—24no—
205#117—44no—
206#117—48no—
207#117—82no—
208#117—82no—
209#117—81no—
210#117—81no—
211#117C4⋊C414yes—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4C22⋊C412yes—
221C2×C4⋊C4C22⋊C412yes—
222C2×C4⋊C4C22×C412yes—
223C2×C4⋊C4C22×C412yes—
224C2×C4⋊C4C22×C411yescharacteristic
225C2×C4⋊C4C22×C411yescharacteristic
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4C22⋊C412yes—
236C2×C22⋊C4C22⋊C412yes—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—24no—
241C2×C42—24no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42C22×C412yes—
248C2×C42C22×C412yes—
249C2×C42C22×C411yescharacteristic
250C2×C42C2×D412yes—
251C2×C42C2×D412yes—
252C2×C42C2×D412yes—
253C2×C42C2×D412yes—
254C2×C42C2×D412yes—
255C2×C42C2×D411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C2×C42C2×Q812yes—
258C2×C42#4112yes—
259C2×C42#4112yes—
260C2×C42C2411yescharacteristic
261C4⋊C8—22no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—22no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—44no—
268C4⋊C8—44no—
269C4⋊C8—42no—
270C4⋊C8—42no—
271C4⋊C8—42no—
272C4⋊C8—82no—
273C4⋊C8—82no—
274C22⋊C8—42no—
275C22⋊C8—42no—
276C22⋊C8—42no—
277#97—42no—
278C4×C8—22no—
279C4×C8—21no—
280C4×C8—21no—
281C4×C8—22no—
282C4×C8—21no—
283C4×C8—21no—
284#95—21no—
285#95—21no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95C22⋊C411yescharacteristic
292#95C22⋊C411yescharacteristic
293#95C22⋊C411yescharacteristic
294#95C22⋊C411yescharacteristic
295#95C22⋊C412yes—
296#95C22⋊C412yes—
297C24—21no—
298C24C2×C22⋊C411yescharacteristic
299#41—28no—
300#41—44no—
301#41—48no—
302C2×Q8—21no—
303C2×Q8—22no—
304C2×Q8C2×C4⋊C412yes—
305C2×Q8#13114yes—
306C2×D4—24no—
307C2×D4—21no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—44no—
311C2×D4C2×C4⋊C412yes—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—42no—
328C22×C4—44no—
329C22×C4—44no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4C2×C22⋊C412yes—
335C22×C4C2×C22⋊C412yes—
336C22×C4C2×C22⋊C412yes—
337C22×C4C2×C22⋊C411yescharacteristic
338C22×C4C2×C22⋊C411yescharacteristic
339C22×C4C2×C22⋊C411yescharacteristic
340C22×C4C2×C22⋊C412yes—
341C22×C4C2×C4⋊C412yes—
342C22×C4#11712yes—
343C22×C4#12112yes—
344C22×C4#12112yes—
345C22×C4C22⋊Q812yes—
346C22×C4C22⋊Q812yes—
347C22×C4C23×C411yescharacteristic
348C22×C4C22×D411yescharacteristic
349C22×C4C22×D411yescharacteristic
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—42no—
360C2×C8—41no—
361C2×C8—42no—
362C2×C8—41no—
363C2×C8—42no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—84no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—24no—
372C4⋊C4—24no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—42no—
378C4⋊C4—44no—
379C4⋊C4—44no—
380C4⋊C4—44no—
381C4⋊C4—82no—
382C4⋊C4—82no—
383C4⋊C4—82no—
384C4⋊C4C2×C4⋊C412yes—
385C4⋊C4C2×C4⋊C412yes—
386C22⋊C4—24no—
387C22⋊C4—22no—
388C22⋊C4—22no—
389C22⋊C4—24no—
390C22⋊C4—24no—
391C22⋊C4—42no—
392C22⋊C4—44no—
393C22⋊C4—44no—
394C22⋊C4—82no—
395C22⋊C4—82no—
396C22⋊C4—81no—
397C22⋊C4—81no—
398C42—22no—
399C42—22no—
400C42—22no—
401C42—22no—
402C42—22no—
403C42—22no—
404C42—22no—
405C42—22no—
406C42—22no—
407C42—22no—
408C42—24no—
409C42—24no—
410C42—24no—
411C42—42no—
412C42—44no—
413C42—44no—
414C42—44no—
415C42—42no—
416C42—42no—
417C42—44no—
418C42C2×C4⋊C412yes—
419C42#11712yes—
420C42#12112yes—
421C42#12112yes—
422C42C22⋊Q812yes—
423C42C22⋊Q812yes—
424C42C22×D411yescharacteristic
425C42C22×Q811yescharacteristic
426C42C2×#4111yescharacteristic
427C42C2×#4111yescharacteristic
428C23—22no—
429C23—21no—
430C23—21no—
431C23—22no—
432C23—22no—
433C23—42no—
434C23—41no—
435C23—41no—
436C23—42no—
437C23#38012yes—
438C23C22×C22⋊C411yescharacteristic
439Q8—22no—
440Q8—24no—
441Q8#42414yes—
442D4—22no—
443D4—22no—
444D4—48no—
445D4—42no—
446C2×C4—22no—
447C2×C4—24no—
448C2×C4—22no—
449C2×C4—21no—
450C2×C4—22no—
451C2×C4—21no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—48no—
476C2×C4—48no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4#38012yes—
491C2×C4#38012yes—
492C2×C4#38012yes—
493C2×C4#42114yes—
494C2×C4C22×C4⋊C411yescharacteristic
495C2×C4C2×#11711yescharacteristic
496C2×C4C2×#12111yescharacteristic
497C2×C4C2×#12111yescharacteristic
498C2×C4C2×C22⋊Q811yescharacteristic
499C2×C4C2×C22⋊Q811yescharacteristic
500C2×C4C2×#13112yes—
501C2×C4#56812yes—
502C2×C4#57611yescharacteristic
503C2×C4#57711yescharacteristic
504C2×C4#57711yescharacteristic
505C2×C4#57811yescharacteristic
506C8—82no—
507C8—81no—
508C8—81no—
509C8—81no—
510C8—82no—
511C8—81no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—42no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—82no—
521C22—82no—
522C22C2×#38011yescharacteristic
523C22#288012yes—
524C22#289711yescharacteristic
525C22#289811yescharacteristic
526C22C2×#42112yes—
527C4—24no—
528C4—22no—
529C4—24no—
530C4—22no—
531C4—21no—
532C4—22no—
533C4—21no—
534C4—42no—
535C4—42no—
536C4—42no—
537C4—42no—
538C4—42no—
539C4—82no—
540C4—82no—
541C4C2×#42411yescharacteristic
542C4#290511yescharacteristic
543C4#292412yes—
544C4#304511yescharacteristic
545C4#304611yescharacteristic
546C4#308311yescharacteristic
547C4#308411yescharacteristic
548C2—42no—
549C2—82no—
550C2#1752412yes—
551C2#1752612yes—
552C2#2062811yescharacteristic
553C2#2065211yescharacteristic
554C2#2065411yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC22×#41C2×C4
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. #8566C2semidirect
9. #8564C2semidirect
10. #7900C2semidirect
11. #7893C2semidirect

Maximal subgroups

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

subgroupindex
#341292
#219762
#208092
C2×#14882
C2×#14772
#85692
#85672
#85662
#85642
#79002
#78932