Finite Group Databasebeta

← #204471

Subgroups — #204471

Order 512.

958 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3104C211yesmaximal, characteristic
2#17806C211yesmaximal, characteristic
3#17479C211yesmaximal, characteristic
4#17449C211yesmaximal, characteristic
5#17385C211yesmaximal, characteristic
6C2×#1461C211yesmaximal, characteristic
7#17326C211yesmaximal, characteristic
8#9671C214yesmaximal, complemented
9#9669C214yesmaximal, complemented
10C2×#533C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12#3104—22no—
13#3104—22no—
14#3104—21no—
15#3104—21no—
16#3104—22no—
17C2×#484C412yes—
18C2×#477C412yes—
19C2×#462C412yes—
20C2×#451C412yes—
21C2×C4×Q16C2214yes—
22#2920C2211yescharacteristic
23#2920C2211yescharacteristic
24C2×#431C2211yescharacteristic
25C2×#431C2211yescharacteristic
26C2×#431C2211yescharacteristic
27#1574—22no—
28#1573—22no—
29#1564C2214yes—
30#1497—22no—
31#1493—22no—
32#1491—22no—
33#1487—22no—
34#1479C2214yes—
35#1479C2214yes—
36#1472—22no—
37#1470—22no—
38#1461—22no—
39#1461C2212yes—
40#1461C2212yes—
41#1461C2212yes—
42#1461C2212yes—
43C2×#324—22no—
44#1454—22no—
45#1452C2212yes—
46#1451—21no—
47#1450C2212yes—
48C8×M16—21no—
49C23×Q8C2×C411yescharacteristic
50C22×Q16C2×C414yes—
51#533—21no—
52#533—21no—
53#533—21no—
54#533—21no—
55#533—21no—
56#533—21no—
57#533—21no—
58#533—21no—
59C2×C4⋊Q8C2×C411yescharacteristic
60C2×C4⋊Q8C2×C411yescharacteristic
61C2×#125—21no—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8C2×C412yes—
64C2×C4×Q8—21no—
65C2×C4×Q8—21no—
66C2×C4×Q8—21no—
67C2×C4×Q8—21no—
68C2×C4×Q8—21no—
69C2×C4×Q8—21no—
70C2×C4×Q8—21no—
71C2×C4×Q8—21no—
72C2×C4×Q8—21no—
73C2×C4×Q8—21no—
74C2×C4×Q8—21no—
75C2×C4×Q8C2×C412yes—
76C2×C4×Q8C2312yes—
77C2×C4×Q8C2312yes—
78C2×#117—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4C2×C411yescharacteristic
81C22×C42—21no—
82C22×C42C2311yescharacteristic
83#484—22no—
84#484—22no—
85#484—24no—
86#477—24no—
87#477—22no—
88#477—22no—
89#462—24no—
90#462—22no—
91#462—22no—
92#451—22no—
93#451—22no—
94#451—24no—
95C4×Q16—24no—
96C4×Q16—24no—
97C4×Q16—24no—
98C4×Q16—24no—
99#432—21no—
100#432—21no—
101#432—21no—
102#432D412yes—
103#431—21no—
104#431—21no—
105#431—21no—
106#431—21no—
107#431—21no—
108#431—21no—
109#431—21no—
110#431D412yes—
111#431D412yes—
112#431D412yes—
113#431D411yescharacteristic
114#431D411yescharacteristic
115#431D412yes—
116#431C2312yes—
117#431C2312yes—
118#431C2312yes—
119#431C2311yescharacteristic
120#431C2311yescharacteristic
121C2×#107C2×C414yes—
122#423—21no—
123#423D412yes—
124C2×C4⋊C8—21no—
125C2×C4⋊C8—21no—
126C2×C4⋊C8C2×C412yes—
127C2×#103C2×C414yes—
128C2×#103C2×C414yes—
129#407—21no—
130#407—21no—
131C2×C22⋊C8—21no—
132C2×C22⋊C8C2×C412yes—
133C2×C22⋊C8D412yes—
134C4×M16—21no—
135C4×M16—21no—
136C2×C4×C8—21no—
137C2×C4×C8—21no—
138C2×C4×C8C2312yes—
139#335—24no—
140#334—24no—
141Q8⋊C8—28no—
142Q8⋊C8—28no—
143#324—24no—
144#324—24no—
145#324—24no—
146#324—24no—
147#322—22no—
148#322—22no—
149#322—22no—
150C82—22no—
151C22×Q8—24no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8—21no—
155C22×Q8—21no—
156C22×Q8—21no—
157C22×Q8—21no—
158C22×Q8—42no—
159C22×Q8C22×C412yes—
160C22×Q8C22×C412yes—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4C22×C411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4#4111yescharacteristic
166C2×Q16—28no—
167C2×Q16—24no—
168C2×Q16—24no—
169C2×Q16—44no—
170C2×M16—22no—
171C2×M16—41no—
172C22×C8—22no—
173C22×C8—41no—
174C22×C8C22×C414yes—
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8—21no—
182C4⋊Q8—21no—
183#125—41no—
184#125—41no—
185#125—41no—
186#125—41no—
187C22⋊Q8—22no—
188C22⋊Q8—22no—
189C22⋊Q8—22no—
190C22⋊Q8—22no—
191C22⋊Q8—41no—
192C22⋊Q8—41no—
193C22⋊Q8—41no—
194C22⋊Q8—41no—
195C4×Q8—22no—
196C4×Q8—22no—
197C4×Q8—22no—
198C4×Q8—22no—
199C4×Q8—22no—
200C4×Q8—22no—
201C4×Q8—22no—
202C4×Q8—22no—
203C4×Q8—22no—
204C4×Q8—22no—
205C4×Q8—22no—
206C4×Q8—22no—
207C4×Q8—41no—
208C4×Q8—41no—
209C4×Q8—41no—
210C4×Q8—41no—
211C4×Q8—41no—
212C4×Q8—41no—
213C4×Q8—41no—
214C4×Q8—41no—
215C4×Q8—41no—
216C4×Q8—41no—
217C4×Q8—41no—
218C4×Q8—41no—
219#117—41no—
220#117—41no—
221#117—41no—
222#117—41no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4C22×C412yes—
240C2×C4⋊C4C22×C412yes—
241C2×C22⋊C4—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—41no—
246C2×C42—41no—
247C2×C42—41no—
248C2×C42—41no—
249C2×C42C22×C411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42C2×D411yescharacteristic
252C2×C42C2×D411yescharacteristic
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C2×C42#4111yescharacteristic
258C2×C42#4111yescharacteristic
259C2×C42#4111yescharacteristic
260C2×C42C2411yescharacteristic
261#107—24no—
262#107—24no—
263C4⋊C8—22no—
264C4⋊C8—22no—
265C4⋊C8—22no—
266C4⋊C8—22no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—41no—
270C4⋊C8—41no—
271C4⋊C8C22⋊C414yes—
272C4⋊C8C22⋊C412yes—
273C4⋊C8C22⋊C412yes—
274C4⋊C8C2×D414yes—
275#103—24no—
276#103—24no—
277#103—24no—
278#103—24no—
279C22⋊C8—22no—
280C22⋊C8—22no—
281C22⋊C8—22no—
282C22⋊C8—22no—
283C22⋊C8—22no—
284C22⋊C8—41no—
285C22⋊C8—41no—
286C22⋊C8C22⋊C412yes—
287C22⋊C8C22⋊C412yes—
288C22⋊C8C22⋊C414yes—
289C22⋊C8C2×D412yes—
290C22⋊C8C2×D412yes—
291#97—22no—
292#97—41no—
293C4×C8—22no—
294C4×C8—21no—
295C4×C8—22no—
296C4×C8—21no—
297C4×C8—22no—
298C4×C8—22no—
299C4×C8—41no—
300C4×C8C2×D412yes—
301C4×C8C2×D412yes—
302C4×C8C2×D414yes—
303C24C4×D411yescharacteristic
304C2×Q8—24no—
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—24no—
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—24no—
314C2×Q8—44no—
315C2×Q8—44no—
316C2×Q8—44no—
317C2×Q8—44no—
318C2×Q8—42no—
319C2×Q8—41no—
320C2×Q8—41no—
321C2×Q8—41no—
322C2×Q8—41no—
323C2×Q8—82no—
324C2×Q8#13114yes—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—42no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—81no—
339C22×C4—82no—
340C22×C4C2×C22⋊C411yescharacteristic
341C22×C4C2×C22⋊C411yescharacteristic
342C22×C4C4×D411yescharacteristic
343C22×C4C4×D411yescharacteristic
344C22×C4C4×D411yescharacteristic
345C22×C4C4×D411yescharacteristic
346C22×C4C4×D411yescharacteristic
347C22×C4#12111yescharacteristic
348C22×C4#12111yescharacteristic
349C22×C4#12111yescharacteristic
350C22×C4#12111yescharacteristic
351C22×C4#12411yescharacteristic
352C22×C4#12411yescharacteristic
353C22×C4#12711yescharacteristic
354C22×C4#12711yescharacteristic
355C22×C4C23×C411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C2×#4111yescharacteristic
358Q16—48no—
359Q16—48no—
360M16—44no—
361M16—82no—
362C2×C8—24no—
363C2×C8—24no—
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—24no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—82no—
370C2×C8—82no—
371C2×C8C2×C22⋊C414yes—
372C2×C8C2×C22⋊C414yes—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C22⋊C4—42no—
394C22⋊C4—42no—
395C42—42no—
396C42—42no—
397C42—42no—
398C42—42no—
399C42—42no—
400C42—42no—
401C42C2×C22⋊C411yescharacteristic
402C42C2×C22⋊C411yescharacteristic
403C42C4×D411yescharacteristic
404C42C4×D411yescharacteristic
405C42#12111yescharacteristic
406C42#12111yescharacteristic
407C42#12111yescharacteristic
408C42#12111yescharacteristic
409C42#12411yescharacteristic
410C42#12411yescharacteristic
411C42#12711yescharacteristic
412C42#12711yescharacteristic
413C42C22×D411yescharacteristic
414C42C22×D411yescharacteristic
415C42C22×D411yescharacteristic
416C42C2×#4111yescharacteristic
417C23—21no—
418C23—21no—
419C23—22no—
420C23—41no—
421C23#39011yescharacteristic
422C23#39011yescharacteristic
423C23C2×C4×D411yescharacteristic
424Q8—24no—
425Q8—24no—
426Q8—44no—
427Q8—44no—
428Q8—42no—
429Q8—42no—
430Q8—44no—
431Q8—44no—
432Q8—82no—
433Q8—82no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—24no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4#39011yescharacteristic
466C2×C4#39011yescharacteristic
467C2×C4#39011yescharacteristic
468C2×C4#39011yescharacteristic
469C2×C4#39011yescharacteristic
470C2×C4#39011yescharacteristic
471C2×C4#47314yes—
472C2×C4C22×C22⋊C411yescharacteristic
473C2×C4C2×C4×D411yescharacteristic
474C2×C4C2×#12111yescharacteristic
475C2×C4C2×#12111yescharacteristic
476C2×C4C2×#12411yescharacteristic
477C2×C4C2×#12711yescharacteristic
478C2×C4C2×#13112yes—
479C2×C4#56812yes—
480C8—24no—
481C8—24no—
482C8—42no—
483C8—42no—
484C8—44no—
485C8—82no—
486C8—82no—
487C22—21no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—42no—
495C22—42no—
496C22#197912yes—
497C22C2×#39011yescharacteristic
498C22#292911yescharacteristic
499C22#293011yescharacteristic
500C4—21no—
501C4—21no—
502C4—21no—
503C4—21no—
504C4—21no—
505C4—21no—
506C4—21no—
507C4—21no—
508C4—44no—
509C4—82no—
510C4—82no—
511C4—82no—
512C4#197912yes—
513C4#288012yes—
514C4#293312yes—
515C4C2×#47312yes—
516C2—21no—
517C2—21no—
518C2—22no—
519C2—41no—
520C2#2119111yescharacteristic
521C2#2124411yescharacteristic
522C2C2×#197911yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#390
prime radicalC24C4×D4
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
8. #9671C2semidirect
9. #9669C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31042
#178062
#174792
#174492
#173852
C2×#14612
#173262
#96712
#96692