Finite Group Databasebeta

← #195912

Subgroups — #195912

Order 512.

1232 conjugacy classes of subgroups.

The subgroup lattice has 519 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2922C211yesmaximal, characteristic
2#34979C211yesmaximal, characteristic
3#21849C211yesmaximal, characteristic
4C2×#1493C211yesmaximal, characteristic, complemented
5C2×#1485C211yesmaximal, characteristic, complemented
6#9075C214yesmaximal, complemented
7#9073C214yesmaximal, complemented
8#7903C211yesmaximal, characteristic, complemented
9#7890C211yesmaximal, characteristic, complemented
10C2×#533C2211yescharacteristic
11C2×C4×#41C2211yescharacteristic
12#2922—24no—
13#2922—24no—
14C2×#433—22no—
15C2×#431C2211yescharacteristic
16C2×#431C2212yes—
17C2×#431C2212yes—
18C2×#424—21no—
19C2×#423C2211yescharacteristic
20C2×#408—21no—
21C2×C4×M16C2211yescharacteristic
22C22×C4×C8C2211yescharacteristic
23#2611—22no—
24#2568C411yescharacteristic
25#2547C411yescharacteristic
26#2546C411yescharacteristic
27#2542C411yescharacteristic
28#2499—21no—
29#2432—21no—
30#2415C414yes—
31#2395—22no—
32C4×C4⋊Q8—21no—
33C4×#124—21no—
34C4×C22⋊Q8C2214yes—
35C4×#117C2211yescharacteristic
36#1984—21no—
37#1954—21no—
38#1870—22no—
39#1797C2214yes—
40#1493—24no—
41#1485—24no—
42C2×Q8⋊C8C2214yes—
43C2×Q8⋊C8C2212yes—
44C2×Q8⋊C8C2212yes—
45#1279C2212yes—
46#1279C2214yes—
47#1279C2212yes—
48C22×#41D411yescharacteristic
49C22×M16—21no—
50C22×M16—21no—
51C23×C8—21no—
52#533—22no—
53#533—22no—
54#533—22no—
55#533—22no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8C2×C411yescharacteristic
58C2×C4⋊Q8C2×C411yescharacteristic
59C2×#125—21no—
60C2×#124—21no—
61C2×C22⋊Q8—21no—
62C2×C22⋊Q8C2×C412yes—
63C2×C22⋊Q8C2×C414yes—
64C4×#41—22no—
65C4×#41—22no—
66C4×#41—22no—
67C4×#41—22no—
68C2×C4×Q8—21no—
69C2×C4×Q8C2×C411yescharacteristic
70C2×C4×Q8D411yescharacteristic
71C2×C4×Q8C2312yes—
72C2×C4×D4C2×C412yes—
73C2×C4×D4C2×C411yescharacteristic
74C2×C4×D4D412yes—
75C2×C4×D4D411yescharacteristic
76C2×#117—21no—
77C2×#117C2×C412yes—
78C2×#117D412yes—
79C2×#117D411yescharacteristic
80C22×C42C2312yes—
81C22×C42C2311yescharacteristic
82#433—216no—
83#431—24no—
84#431—24no—
85#431—24no—
86#431—24no—
87#431—22no—
88#431—22no—
89#424—28no—
90#423—22no—
91#423—22no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—22no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8C2312yes—
96C2×C4⋊C8C2312yes—
97#408—28no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8—21no—
101C2×C22⋊C8—24no—
102C4×M16—22no—
103C4×M16D414yes—
104C2×#97—21no—
105C2×C4×C8—21no—
106C2×C4×C8—22no—
107C2×C4×C8—21no—
108C2×C4×C8D414yes—
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2312yes—
112#400—22no—
113#399—22no—
114#398—22no—
115#397—22no—
116#397—24no—
117#394—24no—
118#391—21no—
119#391—21no—
120#391—21no—
121#391C2×C414yes—
122#390—21no—
123#389—24no—
124#389C2×C414yes—
125#388—24no—
126#387—24no—
127#385—24no—
128#384—21no—
129#382—21no—
130#380C2×C414yes—
131C4×C4⋊C4—24no—
132C4×C4⋊C4C2×C411yescharacteristic
133C4×C4⋊C4C2×C411yescharacteristic
134C4×C4⋊C4C2312yes—
135C4×C22⋊C4—21no—
136C4×C22⋊C4—24no—
137C4×C22⋊C4—21no—
138#376—21no—
139C43—21no—
140#336—24no—
141Q8⋊C8—28no—
142Q8⋊C8—24no—
143Q8⋊C8—24no—
144C2×#41—22no—
145C2×#41—22no—
146C2×#41—42no—
147C2×#41—42no—
148C22×Q8—21no—
149C22×Q8C22⋊C411yescharacteristic
150C22×Q8C22×C412yes—
151C22×D4C22⋊C411yescharacteristic
152C22×D4C22⋊C412yes—
153C23×C4C22×C412yes—
154C23×C4C22×C411yescharacteristic
155C23×C4C2×D412yes—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D412yes—
159C2×M16—24no—
160C2×M16—42no—
161C2×M16—42no—
162C2×M16—42no—
163C2×M16—82no—
164C22×C8—24no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—42no—
171C22×C8—42no—
172C4⋊Q8—22no—
173C4⋊Q8—22no—
174C4⋊Q8—22no—
175C4⋊Q8—22no—
176C4⋊Q8—44no—
177#125—42no—
178#125—42no—
179#124—44no—
180C22⋊Q8—24no—
181C22⋊Q8—24no—
182C22⋊Q8—28no—
183C22⋊Q8—44no—
184C22⋊Q8—42no—
185C22⋊Q8—42no—
186C4×Q8—22no—
187C4×Q8—22no—
188C4×Q8—22no—
189C4×Q8—22no—
190C4×Q8—22no—
191C4×Q8—22no—
192C4×Q8—22no—
193C4×Q8—22no—
194C4×Q8—42no—
195C4×Q8—42no—
196C4×D4—24no—
197C4×D4—24no—
198C4×D4—24no—
199C4×D4—24no—
200C4×D4—22no—
201C4×D4—22no—
202C4×D4—22no—
203C4×D4—22no—
204#117—22no—
205#117—22no—
206#117—24no—
207#117—24no—
208#117—24no—
209#117—42no—
210#117—42no—
211#117—42no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4C22⋊C411yescharacteristic
220C2×C4⋊C4C22⋊C412yes—
221C2×C4⋊C4C22⋊C411yescharacteristic
222C2×C4⋊C4C22⋊C412yes—
223C2×C4⋊C4C22×C412yes—
224C2×C4⋊C4C22×C412yes—
225C2×C4⋊C4C22×C412yes—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4C22⋊C412yes—
231C2×C22⋊C4C22⋊C412yes—
232C2×C22⋊C4C22⋊C412yes—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42C22×C412yes—
243C2×C42C22×C411yescharacteristic
244C2×C42C2×D412yes—
245C2×C42C2×D412yes—
246C2×C42C2×D412yes—
247C2×C42C2×D412yes—
248C2×C42C2×D411yescharacteristic
249C2×C42C2×D411yescharacteristic
250C2×C42#4112yes—
251C2×C42#4112yes—
252C2×C42C2411yescharacteristic
253C4⋊C8—28no—
254C4⋊C8—28no—
255C4⋊C8—22no—
256C4⋊C8—22no—
257C4⋊C8—22no—
258C4⋊C8—22no—
259C4⋊C8—42no—
260C22⋊C8—28no—
261C22⋊C8—28no—
262C22⋊C8—42no—
263C22⋊C8—84no—
264#97—24no—
265C4×C8—24no—
266C4×C8—21no—
267C4×C8—21no—
268C4×C8—22no—
269C4×C8—42no—
270C4×C8C2×D412yes—
271C4×C8C2×D412yes—
272C4×C8C2×D414yes—
273#95—21no—
274#95—21no—
275#95—24no—
276#95—24no—
277#95—24no—
278#95—24no—
279#95—22no—
280C24C2×C22⋊C412yes—
281C24C2×C22⋊C411yescharacteristic
282#41—48no—
283#41—48no—
284C2×Q8—22no—
285C2×Q8—22no—
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—22no—
289C2×Q8—24no—
290C2×Q8—24no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×D4—24no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—42no—
299C2×D4—42no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4C2×C22⋊C412yes—
323C22×C4C2×C22⋊C412yes—
324C22×C4C2×C22⋊C412yes—
325C22×C4C2×C22⋊C411yescharacteristic
326C22×C4C2×C22⋊C411yescharacteristic
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C4×D412yes—
329C22×C4C4×D412yes—
330C22×C4#12012yes—
331C22×C4#12112yes—
332C22×C4#12112yes—
333C22×C4#12312yes—
334C22×C4C23×C411yescharacteristic
335C22×C4C22×D411yescharacteristic
336C22×C4C22×D411yescharacteristic
337M16—48no—
338M16—84no—
339C2×C8—22no—
340C2×C8—22no—
341C2×C8—24no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—48no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—44no—
353C2×C8—42no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—48no—
364C4⋊C4—48no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—44no—
370C4⋊C4—42no—
371C4⋊C4#13114yes—
372C22⋊C4—24no—
373C22⋊C4—24no—
374C22⋊C4—24no—
375C22⋊C4—44no—
376C22⋊C4—48no—
377C22⋊C4—48no—
378C22⋊C4—44no—
379C42—22no—
380C42—22no—
381C42—22no—
382C42—22no—
383C42—22no—
384C42—22no—
385C42—44no—
386C42—42no—
387C42—42no—
388C42—42no—
389C42—42no—
390C42—42no—
391C42—42no—
392C42C4×D412yes—
393C42C4×D412yes—
394C42#12012yes—
395C42#12112yes—
396C42#12112yes—
397C42#12312yes—
398C42C22×D411yescharacteristic
399C42C22×D411yescharacteristic
400C42C2×#4111yescharacteristic
401C42C2×#4111yescharacteristic
402C23—22no—
403C23—22no—
404C23—21no—
405C23—21no—
406C23—24no—
407C23—24no—
408C23—22no—
409C23—22no—
410C23—22no—
411C23#38612yes—
412C23C22×C22⋊C411yescharacteristic
413Q8—44no—
414Q8—42no—
415Q8—42no—
416Q8—44no—
417Q8—84no—
418D4—48no—
419D4—44no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—24no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—44no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4#38612yes—
459C2×C4#38612yes—
460C2×C4#38612yes—
461C2×C4#42112yes—
462C2×C4#42112yes—
463C2×C4C2×C4×D411yescharacteristic
464C2×C4C2×C4×D411yescharacteristic
465C2×C4C2×#12011yescharacteristic
466C2×C4C2×#12111yescharacteristic
467C2×C4C2×#12111yescharacteristic
468C2×C4C2×#12311yescharacteristic
469C2×C4C2×#13112yes—
470C2×C4#56812yes—
471C2×C4#57712yes—
472C2×C4#57812yes—
473C8—42no—
474C8—42no—
475C8—44no—
476C8—82no—
477C8—82no—
478C22—22no—
479C22—22no—
480C22—22no—
481C22—42no—
482C22—42no—
483C22—44no—
484C22—44no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22C2×#38611yescharacteristic
489C22#288012yes—
490C22#289812yes—
491C22C2×#42111yescharacteristic
492C22C2×#42111yescharacteristic
493C4—22no—
494C4—22no—
495C4—21no—
496C4—21no—
497C4—44no—
498C4—42no—
499C4—42no—
500C4—42no—
501C4—82no—
502C4—82no—
503C4—82no—
504C4#189714yes—
505C4#293111yescharacteristic
506C4#293211yescharacteristic
507C4#293512yes—
508C4#305711yescharacteristic
509C4#305911yescharacteristic
510C4#306312yes—
511C2—42no—
512C2—82no—
513C2#1752312yes—
514C2#1752512yes—
515C2#2090011yescharacteristic
516C2#2097411yescharacteristic
517C2C2×#189711yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1493C2semidirect
5. C2×#1485C2semidirect
6. #9075C2semidirect
7. #9073C2semidirect
8. #7903C2semidirect
9. #7890C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29222
#349792
#218492
C2×#14932
C2×#14852
#90752
#90732
#79032
#78902