Finite Group Databasebeta

← #196776

Subgroups — #196776

Order 512.

1184 conjugacy classes of subgroups.

The subgroup lattice has 543 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2991C211yesmaximal, characteristic
2#35297C211yesmaximal, characteristic
3#21851C211yesmaximal, characteristic
4C2×#1554C211yesmaximal, characteristic, complemented
5C2×#1550C211yesmaximal, characteristic, complemented
6#9137C212yesmaximal, complemented
7#9135C212yesmaximal, complemented
8#9130C212yesmaximal, complemented
9#9128C212yesmaximal, complemented
10#8042C211yesmaximal, characteristic, complemented
11#8039C211yesmaximal, characteristic, complemented
12C2×#533C2211yescharacteristic
13C2×C4×#41C2211yescharacteristic
14#2991—24no—
15#2991—24no—
16C2×C8×Q8—21no—
17C2×#436—21no—
18C2×#436—22no—
19C2×#432C2211yescharacteristic
20C2×#431C2212yes—
21C2×#431C2211yescharacteristic
22C2×#423C2212yes—
23C2×#423C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25#2857C2212yes—
26#2857C2212yes—
27#2625C412yes—
28#2616C412yes—
29#2584—21no—
30#2581—21no—
31#2562—21no—
32#2536—21no—
33#2485—22no—
34#2415—22no—
35#2323C414yes—
36#2285C2211yescharacteristic
37#1985—21no—
38#1955—21no—
39#1869—22no—
40#1797C2212yes—
41#1797C2212yes—
42#1554—24no—
43#1550—24no—
44C2×#333C2212yes—
45C2×#333C2211yescharacteristic
46C2×#333C2211yescharacteristic
47C2×#332C2212yes—
48C2×#332C2211yescharacteristic
49C2×#332C2211yescharacteristic
50#1296C2212yes—
51#1296C2212yes—
52#1296C2212yes—
53#1296C2212yes—
54C22×#41C2×C411yescharacteristic
55C22×M16—22no—
56C22×M16—21no—
57C23×C8—21no—
58#533—22no—
59#533—22no—
60#533—22no—
61#533—22no—
62C2×C4⋊Q8C2311yescharacteristic
63C2×C4⋊Q8C2311yescharacteristic
64C2×#125—21no—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8—22no—
67C2×C22⋊Q8C2×C412yes—
68C2×C22⋊Q8C2×C414yes—
69C4×#41—22no—
70C4×#41—22no—
71C4×#41—22no—
72C4×#41—22no—
73C2×C4×Q8—21no—
74C2×C4×Q8C2×C412yes—
75C2×C4×Q8D411yescharacteristic
76C2×C4×Q8D411yescharacteristic
77C2×C4×D4D411yescharacteristic
78C2×C4×D4D412yes—
79C2×C4×D4D412yes—
80C2×C4×D4D411yescharacteristic
81C2×#117—21no—
82C2×#117—21no—
83C2×#117C2×C412yes—
84C2×#117C2×C411yescharacteristic
85C22×C42C2312yes—
86C22×C42C2311yescharacteristic
87C8×Q8—28no—
88#436—216no—
89#436—28no—
90#432—22no—
91#432—22no—
92#431—22no—
93#431—24no—
94#431—24no—
95#431D414yes—
96#423—22no—
97#423—24no—
98#423—24no—
99#423D414yes—
100C2×C4⋊C8—22no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—22no—
103C2×C4⋊C8—22no—
104C2×C4⋊C8—21no—
105C2×C4⋊C8C2312yes—
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×C4⋊C8C2311yescharacteristic
109C2×C4⋊C8C2312yes—
110C2×C4⋊C8C2311yescharacteristic
111C2×C22⋊C8—22no—
112C2×C22⋊C8—21no—
113C2×C22⋊C8—21no—
114C2×C22⋊C8—22no—
115C2×C22⋊C8—22no—
116C2×#97—21no—
117C2×C4×C8—22no—
118C2×C4×C8—21no—
119#400—22no—
120#398—22no—
121#397—24no—
122#395—22no—
123#394—22no—
124#394—24no—
125#391—22no—
126#391C2×C414yes—
127#389—24no—
128#389—22no—
129#389—22no—
130#389—22no—
131#389C2×C414yes—
132#388—24no—
133#388—24no—
134#387—24no—
135#384C2312yes—
136#382—21no—
137#380—22no—
138#380—22no—
139#380—22no—
140#380—21no—
141#380C2×C414yes—
142#380C2×C412yes—
143C4×C4⋊C4—21no—
144C4×C4⋊C4C2×C412yes—
145#336—22no—
146#336—22no—
147#333—24no—
148#333—22no—
149#333—22no—
150#332—24no—
151#332—22no—
152#332—22no—
153C2×#41—22no—
154C2×#41—22no—
155C2×#41—44no—
156C22×Q8—21no—
157C22×Q8C22⋊C411yescharacteristic
158C22×Q8C22×C412yes—
159C22×D4C22⋊C412yes—
160C22×D4C22⋊C411yescharacteristic
161C23×C4C22×C412yes—
162C23×C4C22×C411yescharacteristic
163C23×C4C2×D412yes—
164C23×C4C2×D412yes—
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C2×M16—28no—
168C2×M16—24no—
169C2×M16—82no—
170C2×M16—82no—
171C2×M16—82no—
172C22×C8—22no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—22no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—42no—
179C22×C8—81no—
180C22×C8—81no—
181C4⋊Q8—21no—
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189#125—42no—
190#125—42no—
191C22⋊Q8—24no—
192C22⋊Q8—24no—
193C22⋊Q8—42no—
194C22⋊Q8—42no—
195C22⋊Q8—48no—
196C22⋊Q8—82no—
197C22⋊Q8—82no—
198C4×Q8—22no—
199C4×Q8—22no—
200C4×Q8—22no—
201C4×Q8—22no—
202C4×Q8—24no—
203C4×Q8—24no—
204C4×Q8—42no—
205C4×Q8—42no—
206C4×D4—24no—
207C4×D4—24no—
208C4×D4—24no—
209C4×D4—24no—
210C4×D4—22no—
211C4×D4—22no—
212C4×D4—22no—
213C4×D4—22no—
214#117—22no—
215#117—22no—
216#117—24no—
217#117—24no—
218#117—42no—
219#117—42no—
220#117—81no—
221#117—81no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4C22⋊C412yes—
232C2×C4⋊C4C22⋊C412yes—
233C2×C4⋊C4C22⋊C411yescharacteristic
234C2×C4⋊C4C22⋊C411yescharacteristic
235C2×C4⋊C4C22×C412yes—
236C2×C4⋊C4C22×C414yes—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4C22⋊C412yes—
242C2×C22⋊C4C22⋊C412yes—
243C2×C22⋊C4C22⋊C412yes—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42C22×C412yes—
251C2×C42C22×C411yescharacteristic
252C2×C42C2×D412yes—
253C2×C42C2×D412yes—
254C2×C42C2×D412yes—
255C2×C42C2×D412yes—
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42#4112yes—
259C2×C42#4112yes—
260C2×C42C2411yescharacteristic
261C4⋊C8—28no—
262C4⋊C8—24no—
263C4⋊C8—22no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—22no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—22no—
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—42no—
273C4⋊C8C2×D412yes—
274C4⋊C8C2×D414yes—
275C4⋊C8C2×D412yes—
276C22⋊C8—28no—
277C22⋊C8—24no—
278C22⋊C8—42no—
279C22⋊C8—82no—
280C22⋊C8—82no—
281#97—42no—
282C4×C8—28no—
283C4×C8—24no—
284#95—22no—
285#95—22no—
286#95—24no—
287#95—24no—
288#95—24no—
289#95—24no—
290C24C2×C22⋊C412yes—
291C24C2×C22⋊C411yescharacteristic
292#41—48no—
293#41—48no—
294C2×Q8—22no—
295C2×Q8—24no—
296C2×Q8—22no—
297C2×Q8—22no—
298C2×Q8—44no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×D4—24no—
306C2×D4—22no—
307C2×D4—24no—
308C2×D4—42no—
309C2×D4—42no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4C2×C22⋊C412yes—
337C22×C4C2×C22⋊C412yes—
338C22×C4C2×C22⋊C411yescharacteristic
339C22×C4C2×C22⋊C411yescharacteristic
340C22×C4C2×C22⋊C411yescharacteristic
341C22×C4C2×C22⋊C412yes—
342C22×C4C4×D412yes—
343C22×C4C4×D412yes—
344C22×C4#12012yes—
345C22×C4#12112yes—
346C22×C4#12112yes—
347C22×C4#12312yes—
348C22×C4C23×C411yescharacteristic
349C22×C4C22×D411yescharacteristic
350C22×C4C22×D411yescharacteristic
351M16—84no—
352M16—88no—
353C2×C8—22no—
354C2×C8—24no—
355C2×C8—22no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—42no—
363C2×C8—41no—
364C2×C8—42no—
365C2×C8—41no—
366C2×C8—42no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—84no—
370C4⋊C4—24no—
371C4⋊C4—24no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—44no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—44no—
388C4⋊C4—44no—
389C4⋊C4—48no—
390C4⋊C4—84no—
391C22⋊C4—24no—
392C22⋊C4—24no—
393C22⋊C4—24no—
394C22⋊C4—48no—
395C22⋊C4—44no—
396C22⋊C4—82no—
397C22⋊C4—84no—
398C42—22no—
399C42—22no—
400C42—22no—
401C42—22no—
402C42—22no—
403C42—22no—
404C42—44no—
405C42—82no—
406C42C4×D412yes—
407C42C4×D412yes—
408C42#12012yes—
409C42#12112yes—
410C42#12112yes—
411C42#12312yes—
412C42C22×D411yescharacteristic
413C42C22×D411yescharacteristic
414C42C2×#4111yescharacteristic
415C42C2×#4111yescharacteristic
416C23—22no—
417C23—21no—
418C23—22no—
419C23—21no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—22no—
424C23—22no—
425C23—22no—
426C23—22no—
427C23#38612yes—
428C23C22×C22⋊C411yescharacteristic
429Q8—28no—
430Q8—82no—
431Q8—82no—
432Q8—81no—
433Q8—81no—
434Q8—81no—
435Q8—81no—
436D4—48no—
437D4—44no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—21no—
442C2×C4—22no—
443C2×C4—21no—
444C2×C4—22no—
445C2×C4—24no—
446C2×C4—22no—
447C2×C4—22no—
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—42no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4#38612yes—
479C2×C4#38612yes—
480C2×C4#38612yes—
481C2×C4#41314yes—
482C2×C4C2×C4×D411yescharacteristic
483C2×C4C2×C4×D411yescharacteristic
484C2×C4C2×#12011yescharacteristic
485C2×C4C2×#12111yescharacteristic
486C2×C4C2×#12111yescharacteristic
487C2×C4C2×#12311yescharacteristic
488C2×C4#57611yescharacteristic
489C2×C4#57712yes—
490C2×C4#57711yescharacteristic
491C2×C4#57711yescharacteristic
492C2×C4#57812yes—
493C2×C4#57811yescharacteristic
494C8—44no—
495C8—42no—
496C8—42no—
497C8—81no—
498C8—81no—
499C8—82no—
500C22—22no—
501C22—22no—
502C22—22no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22—42no—
507C22—42no—
508C22—42no—
509C22—42no—
510C22—42no—
511C22—42no—
512C22C2×#38611yescharacteristic
513C22C2×#41312yes—
514C22#289711yescharacteristic
515C22#289812yes—
516C22#289811yescharacteristic
517C4—22no—
518C4—22no—
519C4—21no—
520C4—21no—
521C4—22no—
522C4—24no—
523C4—42no—
524C4—42no—
525C4—84no—
526C4—82no—
527C4—82no—
528C4#300911yescharacteristic
529C4#301111yescharacteristic
530C4#306011yescharacteristic
531C4#306111yescharacteristic
532C4#306211yescharacteristic
533C4#306311yescharacteristic
534C4#309712yes—
535C2—42no—
536C2—82no—
537C2#1777212yes—
538C2#1777812yes—
539C2#2091311yescharacteristic
540C2#2097611yescharacteristic
541C2#2098911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ 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×#1554C2semidirect
5. C2×#1550C2semidirect
6. #9137C2semidirect
7. #9135C2semidirect
8. #9130C2semidirect
9. #9128C2semidirect
10. #8042C2semidirect
11. #8039C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29912
#352972
#218512
C2×#15542
C2×#15502
#91372
#91352
#91302
#91282
#80422
#80392