Finite Group Databasebeta

← #187765

Subgroups — #187765

Order 512.

2274 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3455C211yesmaximal, characteristic
2C2×#1822C211yesmaximal, characteristic, complemented
3C2×#1820C211yesmaximal, characteristic, complemented
4#17587C214yesmaximal, complemented
5#8521C212yesmaximal, complemented
6#8519C214yesmaximal, complemented
7#8518C212yesmaximal, complemented
8#3455—28no—
9#3455—28no—
10C2×#542C412yescomplemented
11C2×#542C2212yes—
12C2×#541C411yescharacteristic, complemented
13C2×#541C2211yescharacteristic
14C2×#540C411yescharacteristic, complemented
15C2×#540C2211yescharacteristic
16C2×#539—22no—
17C2×#538—22no—
18C2×#533C411yescharacteristic, complemented
19C2×#532C411yescharacteristic, complemented
20C2×#529—22no—
21C2×#529—22no—
22C2×#525C2212yes—
23C2×C4×#41C412yescomplemented
24C22×#117C2211yescharacteristic
25C2×#407C2212yes—
26C2×#381C2212yes—
27#1822—22no—
28#1822C2212yes—
29#1822C2212yes—
30#1820—22no—
31#1820C2212yes—
32#1820C2212yes—
33C2×#343—22no—
34C2×#342—22no—
35#1516—28no—
36#1511C2218yes—
37#1510C2218yes—
38#1504—28no—
39C22×#41C2×C412yes—
40C24×C4C2311yescharacteristic
41C22×M16—22no—
42#542—28no—
43#542—28no—
44#542—28no—
45#542—28no—
46#541—28no—
47#541—28no—
48#540—28no—
49#540—28no—
50#539—216no—
51#539—216no—
52#538—216no—
53#538—216no—
54#533—24no—
55#533—24no—
56#532—24no—
57#532—24no—
58C2×C4⋊Q8C2×C411yescharacteristic, complemented
59C2×C4⋊Q8C2×C412yescomplemented
60C2×#127C2×C411yescharacteristic, complemented
61#529—216no—
62#529—216no—
63#529—216no—
64#529—216no—
65C2×#126—22no—
66C2×#126—22no—
67C2×#126—22no—
68C2×#126—22no—
69C2×#125—22no—
70C2×#125—22no—
71C2×#125C2×C411yescharacteristic, complemented
72C2×#125C2×C411yescharacteristic, complemented
73C2×#124—22no—
74C2×#124—22no—
75C2×#124C2×C412yescomplemented
76#525—24no—
77#525—24no—
78#525D418yes—
79#525D418yes—
80C2×#123—22no—
81C2×#123—22no—
82C2×#123—22no—
83C2×#123—22no—
84C2×#123—22no—
85C2×C22⋊Q8—22no—
86C2×C22⋊Q8—22no—
87C2×C22⋊Q8—22no—
88C2×C22⋊Q8C2×C412yes—
89C2×C22⋊Q8C2312yes—
90C2×#121—22no—
91C2×#121—22no—
92C2×#121—22no—
93C2×#121C2×C412yes—
94C2×#121C2312yes—
95C2×#120—22no—
96C4×#41—24no—
97C4×#41—24no—
98C4×#41—24no—
99C4×#41—24no—
100C2×C4×Q8—22no—
101C2×C4×Q8C2×C412yescomplemented
102C2×C4×Q8C2×C412yescomplemented
103C2×C4×D4—22no—
104C2×C4×D4—22no—
105C2×C4×D4—22no—
106C2×C4×D4—22no—
107C2×C4×D4—22no—
108C2×C4×D4C2×C412yescomplemented
109C2×C4×D4C2×C412yescomplemented
110C2×#117—24no—
111C2×#117—24no—
112C2×#117—22no—
113C2×#117C2×C412yes—
114C2×#117C2×C411yescharacteristic
115C2×#117C2×C411yescharacteristic
116C2×#117D412yes—
117C2×#117D412yes—
118C2×#117C2311yescharacteristic
119C2×#117C2311yescharacteristic
120C22×C4⋊C4D412yes—
121C22×C22⋊C4—22no—
122C22×C22⋊C4D412yes—
123C22×C42—22no—
124C22×C42C2×C411yescharacteristic
125C22×C42C2×C411yescharacteristic
126#407—24no—
127#407—22no—
128#407C2314yes—
129C2×C22⋊C8—24no—
130#381—24no—
131#381—22no—
132#381C2314yes—
133C2×#95—24no—
134#343—24no—
135#343—24no—
136#342—24no—
137#342—24no—
138#328—216no—
139#327—216no—
140C25C2×D411yescharacteristic
141C2×#41—22no—
142C2×#41—22no—
143C2×#41—48no—
144C2×#41C22⋊C414yes—
145C2×#41C22⋊C414yes—
146C22×Q8—22no—
147C22×Q8C22×C412yes—
148C22×D4—22no—
149C22×D4—22no—
150C22×D4—22no—
151C22×D4—22no—
152C22×D4C22×C412yes—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—44no—
157C23×C4—42no—
158C23×C4C22⋊C412yes—
159C23×C4C22⋊C412yes—
160C23×C4C22×C411yescharacteristic
161C23×C4C22×C411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D414yes—
165C23×C4C2×D414yes—
166C23×C4C2×D411yescharacteristic
167C23×C4C2411yescharacteristic
168C2×M16—24no—
169C2×M16—42no—
170C2×M16—48no—
171C22×C8—42no—
172C4⋊Q8—24no—
173C4⋊Q8—24no—
174C4⋊Q8—24no—
175#127—24no—
176#126—216no—
177#126—216no—
178#126—216no—
179#126—216no—
180#125—216no—
181#125—216no—
182#125—24no—
183#125—24no—
184#124—216no—
185#124—216no—
186#124—24no—
187#124—24no—
188#123—216no—
189#123—216no—
190#123—216no—
191#123—216no—
192#123—28no—
193#123—28no—
194C22⋊Q8—216no—
195C22⋊Q8—216no—
196C22⋊Q8—28no—
197C22⋊Q8—28no—
198C22⋊Q8—24no—
199C22⋊Q8—24no—
200C22⋊Q8C22⋊C418yes—
201C22⋊Q8C2×D418yes—
202#121—216no—
203#121—216no—
204#121—28no—
205#121—28no—
206#121—24no—
207#121—24no—
208#121C22⋊C418yes—
209#121C2×D418yes—
210#120—28no—
211#120—28no—
212C4×Q8—28no—
213C4×Q8—28no—
214C4×Q8—24no—
215C4×Q8—24no—
216C4×Q8—24no—
217C4×Q8—24no—
218C4×D4—28no—
219C4×D4—28no—
220C4×D4—28no—
221C4×D4—28no—
222C4×D4—28no—
223C4×D4—28no—
224C4×D4—24no—
225C4×D4—24no—
226C4×D4—28no—
227C4×D4—28no—
228C4×D4—28no—
229C4×D4—28no—
230C4×D4—24no—
231C4×D4—24no—
232#117—28no—
233#117—28no—
234#117—24no—
235#117—24no—
236#117—22no—
237#117—22no—
238#117—24no—
239#117—24no—
240#117—24no—
241#117—24no—
242#117—24no—
243#117—24no—
244#117C22⋊C414yes—
245#117C22⋊C414yes—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4C22⋊C412yes—
257C2×C4⋊C4C22⋊C412yes—
258C2×C4⋊C4C22×C412yes—
259C2×C4⋊C4C22×C412yes—
260C2×C4⋊C4C2×D412yes—
261C2×C4⋊C4C2×D412yes—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—48no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4C22⋊C412yes—
277C2×C22⋊C4C22⋊C412yes—
278C2×C22⋊C4C2×D412yes—
279C2×C22⋊C4C2×D412yes—
280C2×C42—28no—
281C2×C42—28no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42C22⋊C412yes—
290C2×C42C22⋊C412yes—
291C2×C42C22×C411yescharacteristic
292C2×C42C22×C411yescharacteristic
293C2×C42C22×C411yescharacteristic
294C2×C42C22×C411yescharacteristic
295C22⋊C8—28no—
296C22⋊C8—44no—
297#95—28no—
298#95—44no—
299C24—24no—
300C24—22no—
301C24—82no—
302C24C2×C22⋊C411yescharacteristic
303C24C2×C22⋊C411yescharacteristic
304C24#12012yes—
305C24#12012yes—
306C24C22×D411yescharacteristic
307#41—48no—
308#41—48no—
309#41—48no—
310#41—48no—
311C2×Q8—28no—
312C2×Q8—22no—
313C2×Q8—44no—
314C2×Q8—84no—
315C2×Q8C2×C22⋊C414yes—
316C2×D4—24no—
317C2×D4—24no—
318C2×D4—28no—
319C2×D4—28no—
320C2×D4—28no—
321C2×D4—22no—
322C2×D4—44no—
323C2×D4—48no—
324C2×D4—84no—
325C2×D4—84no—
326C2×D4—84no—
327C2×D4C2×C22⋊C414yes—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—28no—
334C22×C4—28no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—48no—
345C22×C4—48no—
346C22×C4—42no—
347C22×C4—48no—
348C22×C4—44no—
349C22×C4—84no—
350C22×C4—84no—
351C22×C4—82no—
352C22×C4—82no—
353C22×C4—84no—
354C22×C4—88no—
355C22×C4C2×C22⋊C412yes—
356C22×C4C2×C22⋊C412yes—
357C22×C4C2×C22⋊C411yescharacteristic
358C22×C4C2×C22⋊C411yescharacteristic
359C22×C4C2×C22⋊C412yes—
360C22×C4C2×C22⋊C412yes—
361C22×C4C2×C22⋊C412yes—
362C22×C4C2×C22⋊C412yes—
363C22×C4C2×C22⋊C414yes—
364C22×C4#12012yes—
365C22×C4#12012yes—
366C22×C4#12014yes—
367C22×C4#12014yes—
368C22×C4C23×C411yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D414yes—
371M16—48no—
372M16—84no—
373C2×C8—44no—
374C2×C8—84no—
375C4⋊C4—28no—
376C4⋊C4—28no—
377C4⋊C4—28no—
378C4⋊C4—28no—
379C4⋊C4—28no—
380C4⋊C4—28no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—28no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—28no—
389C4⋊C4C2×C22⋊C414yes—
390C4⋊C4C2×C22⋊C414yes—
391C22⋊C4—28no—
392C22⋊C4—28no—
393C22⋊C4—28no—
394C22⋊C4—28no—
395C22⋊C4—24no—
396C22⋊C4—24no—
397C22⋊C4—28no—
398C22⋊C4—24no—
399C22⋊C4—24no—
400C22⋊C4—24no—
401C22⋊C4—24no—
402C22⋊C4—28no—
403C22⋊C4—28no—
404C22⋊C4—28no—
405C22⋊C4—48no—
406C22⋊C4—84no—
407C42—28no—
408C42—28no—
409C42—22no—
410C42—22no—
411C42—22no—
412C42—22no—
413C42—22no—
414C42—22no—
415C42—24no—
416C42—24no—
417C42—22no—
418C42—22no—
419C23—24no—
420C23—22no—
421C23—22no—
422C23—24no—
423C23—24no—
424C23—42no—
425C23—44no—
426C23—42no—
427C23—42no—
428C23—44no—
429C23—44no—
430C23—44no—
431C23—44no—
432C23—88no—
433C23#37912yes—
434C23#37911yescharacteristic
435C23#37911yescharacteristic
436C23#41014yes—
437C23C22×C22⋊C411yescharacteristic
438C23C2×#12012yes—
439Q8—44no—
440Q8—44no—
441Q8—88no—
442D4—44no—
443D4—44no—
444D4—48no—
445D4—48no—
446D4—88no—
447D4—88no—
448D4—88no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—22no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—48no—
469C2×C4—48no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—88no—
482C2×C4—88no—
483C2×C4—84no—
484C2×C4—88no—
485C2×C4—168no—
486C2×C4#37912yes—
487C2×C4#37912yes—
488C2×C4C22×C22⋊C412yes—
489C2×C4C2×#12012yes—
490C8—84no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—24no—
497C22—24no—
498C22—44no—
499C22—48no—
500C22—44no—
501C22—44no—
502C22—44no—
503C22—82no—
504C22—82no—
505C22—84no—
506C22—88no—
507C22#178212yes—
508C22#178212yes—
509C22C2×#37911yescharacteristic
510C22C2×#41012yes—
511C4—22no—
512C4—22no—
513C4—22no—
514C4—22no—
515C4—44no—
516C4—44no—
517C4—44no—
518C4—42no—
519C4—42no—
520C4—42no—
521C4—42no—
522C4—164no—
523C2—22no—
524C2—22no—
525C2—22no—
526C2—42no—
527C2—42no—
528C2—84no—
529C2C2×#178211yescharacteristic
530C2#2062312yes—
subgroupG/H
center, socleC22C2×#379
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#3455C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1822C2semidirect
3. C2×#1820C2semidirect
4. #17587C2semidirect
5. #8521C2semidirect
6. #8519C2semidirect
7. #8518C2semidirect
10. C2×#542C4semidirect
12. C2×#541C4semidirect
14. C2×#540C4semidirect
18. C2×#533C4semidirect
19. C2×#532C4semidirect
23. C2×C4×#41C4semidirect
58. C2×C4⋊Q8C2×C4semidirect
59. C2×C4⋊Q8C2×C4semidirect
60. C2×#127C2×C4semidirect
71. C2×#125C2×C4semidirect
72. C2×#125C2×C4semidirect
75. C2×#124C2×C4semidirect
101. C2×C4×Q8C2×C4semidirect
102. C2×C4×Q8C2×C4semidirect
108. C2×C4×D4C2×C4semidirect
109. C2×C4×D4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#34552
C2×#18222
C2×#18202
#175872
#85212
#85192
#85182