Finite Group Databasebeta

← #193821

Subgroups — #193821

Order 512.

1158 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60248C211yesmaximal, characteristic, complemented
2#34972C211yesmaximal, characteristic
3#17480C212yesmaximal
4C2×#1461C211yesmaximal, characteristic, complemented
5#8964C212yesmaximal, complemented
6#8962C212yesmaximal, complemented
7#8960C212yesmaximal, complemented
8#8958C212yesmaximal, complemented
9#7869C212yesmaximal, complemented
10C2×#529C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12#2984—24no—
13#2983—22no—
14#2983C2212yes—
15#2982C2212yes—
16Q8×M16—24no—
17#2920—22no—
18#2920C2212yes—
19C2×#431C2212yes—
20#2604—22no—
21#2565C411yescharacteristic
22#2565C411yescharacteristic
23#2545C411yescharacteristic
24#2545C411yescharacteristic
25#2503C411yescharacteristic
26#2501C411yescharacteristic
27#2436C411yescharacteristic
28#2434C411yescharacteristic
29#2413—22no—
30#2396—22no—
31C4×C4⋊Q8C2211yescharacteristic
32C4×C4⋊Q8C2211yescharacteristic
33C4×#124C2211yescharacteristic
34C4×#124C2211yescharacteristic
35C4×#123—22no—
36C4×#117C2211yescharacteristic
37#1495—24no—
38#1494—24no—
39#1479C2214yes—
40#1478C2214yes—
41#1461—22no—
42#1461C2212yes—
43#1461C2212yes—
44C2×#324—22no—
45#1276C2214yes—
46#1276C2214yes—
47C23×Q8C2×C411yescharacteristic
48C2×C4⋊Q8C2×C411yescharacteristic
49C2×C4⋊Q8C2×C411yescharacteristic
50#529—22no—
51#529—22no—
52#529—22no—
53#529—22no—
54C2×#126—21no—
55C2×#125C2×C411yescharacteristic
56C2×#124C2×C411yescharacteristic
57C2×#124C2×C411yescharacteristic
58C2×#124C2×C411yescharacteristic
59C2×#123—22no—
60C2×#123—21no—
61C2×C22⋊Q8C2×C411yescharacteristic
62C2×#121C2×C411yescharacteristic
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×C4×Q8—22no—
66C2×C4×Q8—22no—
67C2×C4×Q8C2×C411yescharacteristic
68C2×C4×Q8C2×C411yescharacteristic
69C2×C4×Q8D412yes—
70C2×C4×Q8D412yes—
71C2×C4×Q8C2311yescharacteristic
72C2×C4×Q8C2311yescharacteristic
73C2×C4×Q8C2311yescharacteristic
74C2×C4×D4—21no—
75C2×C4×D4C2311yescharacteristic
76C2×#117—21no—
77C2×#117—21no—
78C22×C4⋊C4C2×C411yescharacteristic
79C22×C4⋊C4D412yes—
80C22×C42D412yes—
81C22×C42C2311yescharacteristic
82#446—216no—
83#446—28no—
84C8×Q8—28no—
85#432—24no—
86#432—24no—
87#432—22no—
88#432D414yes—
89#431—24no—
90#431—24no—
91#431—22no—
92#431—22no—
93#431—22no—
94#431—22no—
95#431C2314yes—
96#431C2314yes—
97#423—24no—
98#423D414yes—
99C2×C4⋊C8—22no—
100#407—22no—
101C2×C22⋊C8—22no—
102C4×M16—24no—
103C4×M16—22no—
104C2×C4×C8—22no—
105#400—22no—
106#399—22no—
107#398—22no—
108#398—22no—
109#396—22no—
110#396—22no—
111#395—22no—
112#393—22no—
113#391C2×C411yescharacteristic
114#391C2×C411yescharacteristic
115#391C2×C411yescharacteristic
116#391C2×C411yescharacteristic
117#391C2×C411yescharacteristic
118#391C2×C411yescharacteristic
119#390C2×C411yescharacteristic
120#390C2×C411yescharacteristic
121#389—22no—
122#389—22no—
123#388—22no—
124#388—22no—
125#388—22no—
126#388—22no—
127#387—22no—
128#387—22no—
129#386—22no—
130#385—22no—
131#384C2×C411yescharacteristic
132#384C2×C411yescharacteristic
133#382C2×C411yescharacteristic
134#382C2×C411yescharacteristic
135#381—22no—
136C4×C4⋊C4—22no—
137C4×C4⋊C4—22no—
138C4×C4⋊C4—21no—
139C4×C4⋊C4C2×C411yescharacteristic
140C4×C4⋊C4C2×C411yescharacteristic
141C4×C22⋊C4—21no—
142C4×C22⋊C4—22no—
143C4×C22⋊C4—22no—
144C4×C22⋊C4C2×C411yescharacteristic
145C4×C22⋊C4C2×C411yescharacteristic
146#376—21no—
147C43C2311yescharacteristic
148C43C2311yescharacteristic
149Q8⋊C8—28no—
150D4⋊C8—28no—
151#324—24no—
152#324—24no—
153C22×Q8—24no—
154C22×Q8—22no—
155C22×Q8—22no—
156C22×Q8—42no—
157C22×Q8C22⋊C412yes—
158C22×Q8C22×C411yescharacteristic
159C22×Q8C22×C411yescharacteristic
160C22×Q8C22×C411yescharacteristic
161C22×D4—21no—
162C22×D4C22×C411yescharacteristic
163C23×C4—21no—
164C23×C4C22⋊C412yes—
165C23×C4C22⋊C412yes—
166C23×C4C22×C411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C2×M16—24no—
170C22×C8—42no—
171C4⋊Q8—24no—
172C4⋊Q8—42no—
173#126—42no—
174#126—42no—
175#125—22no—
176#125—22no—
177#124—24no—
178#124—22no—
179#124—22no—
180#124—42no—
181#123—48no—
182#123—42no—
183#123—42no—
184C22⋊Q8—22no—
185C22⋊Q8—22no—
186#121—22no—
187#121—22no—
188C4×Q8—22no—
189C4×Q8—22no—
190C4×Q8—22no—
191C4×Q8—22no—
192C4×Q8—24no—
193C4×Q8—24no—
194C4×Q8—24no—
195C4×Q8—24no—
196C4×Q8—22no—
197C4×Q8—22no—
198C4×Q8—22no—
199C4×Q8—22no—
200C4×Q8—22no—
201C4×Q8—22no—
202C4×D4—22no—
203C4×D4—22no—
204C4×D4—41no—
205C4×D4—41no—
206C4×D4—41no—
207C4×D4—41no—
208#117—44no—
209#117—42no—
210#117—42no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4C22⋊C412yes—
226C2×C4⋊C4C22⋊C412yes—
227C2×C4⋊C4C22⋊C412yes—
228C2×C4⋊C4C22×C411yescharacteristic
229C2×C4⋊C4C22×C411yescharacteristic
230C2×C4⋊C4C22×C411yescharacteristic
231C2×C4⋊C4C22×C411yescharacteristic
232C2×C4⋊C4C2×D412yes—
233C2×C4⋊C4C2×D412yes—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—21no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42C22⋊C412yes—
255C2×C42C22⋊C412yes—
256C2×C42C22×C411yescharacteristic
257C2×C42C22×C411yescharacteristic
258C2×C42C22×C411yescharacteristic
259C2×C42C22×C411yescharacteristic
260C2×C42C22×C411yescharacteristic
261C2×C42C2×D412yes—
262C2×C42C2×D412yes—
263C2×C42C2×D412yes—
264C2×C42C2×D412yes—
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42#4112yes—
268C2×C42#4112yes—
269C2×C42C2411yescharacteristic
270C4⋊C8—28no—
271C4⋊C8—28no—
272C4⋊C8—24no—
273C4⋊C8—24no—
274C4⋊C8—42no—
275C4⋊C8C2×D418yes—
276C22⋊C8—24no—
277C22⋊C8—24no—
278C22⋊C8—24no—
279C22⋊C8—42no—
280#97—28no—
281#97—24no—
282C4×C8—28no—
283C4×C8—24no—
284C4×C8—24no—
285C4×C8—42no—
286#95—21no—
287#95—21no—
288#95—21no—
289#95—21no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95—22no—
298C24—21no—
299C24C2×C22⋊C411yescharacteristic
300C2×Q8—22no—
301C2×Q8—22no—
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—24no—
306C2×Q8—24no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—42no—
311C2×Q8—48no—
312C2×Q8—48no—
313C2×Q8#13114yes—
314C2×D4—22no—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—44no—
318C2×D4—41no—
319C2×D4—41no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—44no—
338C22×C4—44no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4C2×C22⋊C412yes—
342C22×C4C2×C22⋊C411yescharacteristic
343C22×C4C2×C22⋊C412yes—
344C22×C4C2×C22⋊C412yes—
345C22×C4C2×C22⋊C412yes—
346C22×C4C2×C22⋊C411yescharacteristic
347C22×C4C2×C22⋊C411yescharacteristic
348C22×C4C4×D412yes—
349C22×C4C4×D412yes—
350C22×C4#12011yescharacteristic
351C22×C4#12011yescharacteristic
352C22×C4#12112yes—
353C22×C4#12112yes—
354C22×C4#12311yescharacteristic
355C22×C4#12311yescharacteristic
356C22×C4C23×C411yescharacteristic
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D411yescharacteristic
359M16—48no—
360C2×C8—28no—
361C2×C8—44no—
362C2×C8—84no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—44no—
375C4⋊C4—44no—
376C4⋊C4—44no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—44no—
383C22⋊C4—24no—
384C22⋊C4—44no—
385C22⋊C4—44no—
386C22⋊C4—44no—
387C22⋊C4—42no—
388C22⋊C4—41no—
389C22⋊C4—41no—
390C22⋊C4—42no—
391C22⋊C4—42no—
392C22⋊C4—42no—
393C22⋊C4—41no—
394C22⋊C4—41no—
395C22⋊C4—44no—
396C42—22no—
397C42—22no—
398C42—22no—
399C42—22no—
400C42—22no—
401C42—22no—
402C42—22no—
403C42—22no—
404C42—22no—
405C42—22no—
406C42—22no—
407C42—22no—
408C42—42no—
409C42—42no—
410C42—42no—
411C42—42no—
412C42—41no—
413C42—41no—
414C42C4×D412yes—
415C42C4×D412yes—
416C42#12012yes—
417C42#12112yes—
418C42#12112yes—
419C42#12312yes—
420C42C22×D412yes—
421C42C2×#4112yes—
422C23—21no—
423C23—21no—
424C23—22no—
425C23—41no—
426C23—42no—
427C23—42no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23#38611yescharacteristic
432C23#38611yescharacteristic
433C23C22×C22⋊C411yescharacteristic
434Q8—28no—
435Q8—48no—
436Q8—44no—
437Q8—44no—
438D4—44no—
439D4—82no—
440D4—82no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—22no—
444C2×C4—24no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—42no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4#38611yescharacteristic
490C2×C4#38611yescharacteristic
491C2×C4#38612yes—
492C2×C4#38612yes—
493C2×C4#40814yes—
494C2×C4C2×C4×D412yes—
495C2×C4C2×#12011yescharacteristic
496C2×C4C2×#12112yes—
497C2×C4C2×#12311yescharacteristic
498C2×C4C2×#13112yes—
499C2×C4#56812yes—
500C8—48no—
501C8—84no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—42no—
510C22—42no—
511C22—82no—
512C22—82no—
513C22—81no—
514C22—81no—
515C22—81no—
516C22#188412yes—
517C22C2×#38611yescharacteristic
518C22C2×#40811yescharacteristic
519C22#287911yescharacteristic
520C4—22no—
521C4—24no—
522C4—22no—
523C4—22no—
524C4—21no—
525C4—21no—
526C4—21no—
527C4—42no—
528C4—42no—
529C4—42no—
530C4—42no—
531C4—42no—
532C4—81no—
533C4—81no—
534C4—81no—
535C4—81no—
536C4—81no—
537C4#293314yes—
538C4#302012yes—
539C2—21no—
540C2—21no—
541C2—22no—
542C2—41no—
543C2—161no—
544C2#2089711yescharacteristic
545C2C2×#188411yescharacteristic
546C2#2095411yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#386
prime radicalC2×#121C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #60248C2semidirect
4. C2×#1461C2semidirect
5. #8964C2semidirect
6. #8962C2semidirect
7. #8960C2semidirect
8. #8958C2semidirect
9. #7869C2semidirect

Maximal subgroups

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

subgroupindex
#602482
#349722
#174802
C2×#14612
#89642
#89622
#89602
#89582
#78692