Finite Group Databasebeta

← #391198

Subgroups — #391198

Order 512.

1258 conjugacy classes of subgroups.

The subgroup lattice has 541 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2001C211yesmaximal, characteristic
2#14258C212yesmaximal
3#11316C212yesmaximal, complemented
4#11190C212yesmaximal, complemented
5#11122C212yesmaximal, complemented
6#11095C211yesmaximal, characteristic
7#11024C212yesmaximal, complemented
8#10970C212yesmaximal
9#10713C211yesmaximal, characteristic
10C23×Q16C2211yescharacteristic
11C22×C22⋊Q8C2212yes—
12C22×#117C2211yescharacteristic
13C22×#106C2211yescharacteristic
14C22×#103C2212yes—
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×C22⋊C8C2211yescharacteristic
18C22×C22⋊C8C2211yescharacteristic
19C2×#391C2212yes—
20C2×#391C2212yes—
21C2×#380C2212yes—
22#2001C418yescomplemented
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2211yescharacteristic
27C2×#340C2211yescharacteristic
28C2×#336C2212yes—
29C2×#336C2212yes—
30#1441—22no—
31#1441—22no—
32#1440—22no—
33#1440—22no—
34C4⋊#95C2212yes—
35C4⋊#95C2212yes—
36C23×Q8C2312yes—
37C24×C4C2311yescharacteristic
38C22×Q16—42no—
39C22×Q16—42no—
40C22×Q16—41no—
41C22×Q16—41no—
42C22×Q16C2×C414yes—
43C23×C8C2312yes—
44C23×C8C2311yescharacteristic
45C23×C8C2311yescharacteristic
46C2×C22⋊Q8—24no—
47C2×C22⋊Q8—42no—
48C2×C22⋊Q8—42no—
49C2×C22⋊Q8—42no—
50C2×C22⋊Q8—42no—
51C2×C22⋊Q8C2×C418yes—
52C2×#117—42no—
53C2×#117—42no—
54C2×#117—41no—
55C2×#117—41no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4D412yes—
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2312yes—
61C22×C22⋊C4—22no—
62C22×C22⋊C4—22no—
63C22×C22⋊C4D411yescharacteristic
64C22×C22⋊C4D411yescharacteristic
65C22×C42C2312yes—
66C2×#106—42no—
67C2×#106—41no—
68C2×#103—42no—
69C2×#103—42no—
70C2×#103—42no—
71C2×#103—42no—
72C2×#103—42no—
73C2×#103—42no—
74C2×#103—42no—
75C2×#103—42no—
76C2×#103C2×C418yes—
77C2×C22⋊C8—22no—
78C2×C22⋊C8—41no—
79C2×C22⋊C8—41no—
80C2×C22⋊C8—41no—
81C2×C22⋊C8—41no—
82C2×C22⋊C8C2×C414yes—
83#391—42no—
84#391—42no—
85#391—42no—
86#391—42no—
87#380—42no—
88#380—42no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98#340—42no—
99#340—42no—
100#340—42no—
101#340—41no—
102#340C2×C414yes—
103#336—42no—
104#336—42no—
105C25C2×D411yescharacteristic
106C22×Q8—22no—
107C22×Q8—22no—
108C22×Q8—22no—
109C22×Q8—22no—
110C22×Q8—22no—
111C22×Q8—22no—
112C22×Q8—22no—
113C22×Q8—22no—
114C22×Q8—22no—
115C22×Q8—22no—
116C22×Q8—22no—
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8C22×C414yes—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4C22×C412yes—
133C23×C4C2×D412yes—
134C23×C4C2×D412yes—
135C23×C4C2×D411yescharacteristic
136C23×C4#4112yes—
137C23×C4C2411yescharacteristic
138C2×Q16—44no—
139C2×Q16—48no—
140C2×Q16—48no—
141C2×Q16—44no—
142C2×Q16—44no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—22no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8C22×C412yes—
159C22×C8C22×C412yes—
160C22⋊Q8—48no—
161C22⋊Q8—48no—
162C22⋊Q8—48no—
163C22⋊Q8—48no—
164#117—48no—
165#117—44no—
166#117—44no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—28no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4C22×C414yes—
191C2×C22⋊C4—28no—
192C2×C22⋊C4—41no—
193C2×C22⋊C4—41no—
194C2×C22⋊C4—42no—
195C2×C22⋊C4—42no—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—82no—
200C2×C22⋊C4—82no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207#106—44no—
208#103—48no—
209#103—48no—
210#103—48no—
211C22⋊C8—44no—
212C22⋊C8—44no—
213#95—44no—
214#95—44no—
215#95—44no—
216#95—44no—
217#95—44no—
218#95—82no—
219#95—82no—
220#95—82no—
221#95—82no—
222C24—22no—
223C24—21no—
224C24—22no—
225C24—21no—
226C24—22no—
227C24—22no—
228C24—22no—
229C24—22no—
230C24C4×D412yes—
231C24C22×D411yescharacteristic
232C2×Q8—24no—
233C2×Q8—24no—
234C2×Q8—24no—
235C2×Q8—24no—
236C2×Q8—24no—
237C2×Q8—24no—
238C2×Q8—42no—
239C2×Q8—42no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—42no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—22no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—42no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—41no—
293C22×C4—42no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4C4×D412yes—
325C22×C4C23×C411yescharacteristic
326C22×C4C2×#4111yescharacteristic
327C22×C4H(5,𝔽2)11yescharacteristic
328C22×C4H(5,𝔽2)12yes—
329C22×C4H(5,𝔽2)12yes—
330C22×C4H(5,𝔽2)12yes—
331C22×C4H(5,𝔽2)12yes—
332C22×C4H(5,𝔽2)11yescharacteristic
333C22×C4H(5,𝔽2)11yescharacteristic
334C22×C42−1+411yescharacteristic
335Q16—416no—
336C2×C8—22no—
337C2×C8—22no—
338C2×C8—24no—
339C2×C8—24no—
340C2×C8—41no—
341C2×C8—42no—
342C2×C8—41no—
343C2×C8—42no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C4⋊C4—24no—
354C4⋊C4—24no—
355C4⋊C4—416no—
356C4⋊C4—48no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C22⋊C4—416no—
368C22⋊C4—44no—
369C22⋊C4—44no—
370C22⋊C4—88no—
371C42—42no—
372C42—42no—
373C42—42no—
374C42—42no—
375C23—24no—
376C23—22no—
377C23—24no—
378C23—22no—
379C23—22no—
380C23—42no—
381C23—42no—
382C23—42no—
383C23—42no—
384C23—42no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—42no—
395C23—41no—
396C23C2×C4×D411yescharacteristic
397C23#53412yes—
398C23#53411yescharacteristic
399C23#53512yes—
400C23#53711yescharacteristic
401C23#57712yes—
402C23#57712yes—
403C23#57812yes—
404C23#57812yes—
405Q8—44no—
406Q8—44no—
407Q8—44no—
408Q8—44no—
409Q8—82no—
410Q8—82no—
411Q8—82no—
412Q8—82no—
413C2×C4—24no—
414C2×C4—22no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—44no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4#51812yes—
467C2×C4#51811yescharacteristic
468C2×C4#51812yes—
469C2×C4#52011yescharacteristic
470C2×C4#53812yes—
471C2×C4#53811yescharacteristic
472C2×C4#53812yes—
473C2×C4#53911yescharacteristic
474C2×C4#56011yescharacteristic
475C2×C4#56012yes—
476C2×C4#56012yes—
477C2×C4#56012yes—
478C2×C4#56012yes—
479C2×C4#56011yescharacteristic
480C2×C4#56011yescharacteristic
481C2×C4H(3,𝔽4)11yescharacteristic
482C2×C4#56212yes—
483C2×C4#56212yes—
484C8—42no—
485C8—42no—
486C8—82no—
487C22—42no—
488C22—44no—
489C22—44no—
490C22—42no—
491C22—42no—
492C22—82no—
493C22—82no—
494C22—82no—
495C22—82no—
496C22#232112yes—
497C22#232511yescharacteristic
498C22#281211yescharacteristic
499C22#281312yes—
500C22#281511yescharacteristic
501C22#294712yes—
502C22#295011yescharacteristic
503C22#295211yescharacteristic
504C22#320112yes—
505C22#320112yes—
506C22#320112yes—
507C22#320112yes—
508C22#320212yes—
509C22#320412yes—
510C22#322212yes—
511C22#322312yes—
512C22#322512yes—
513C22#322512yes—
514C22#325812yes—
515C22#326111yescharacteristic
516C22#326311yescharacteristic
517C4—42no—
518C4—84no—
519C4—84no—
520C4—82no—
521C4—82no—
522C4—162no—
523C4—162no—
524C4#237511yescharacteristic
525C4#237712yes—
526C4#238011yescharacteristic
527C4#282411yescharacteristic
528C4#282512yes—
529C4#282611yescharacteristic
530C2—82no—
531C2#1462111yescharacteristic
532C2#2205412yes—
533C2#2205912yes—
534C2#2219211yescharacteristic
535C2#2219811yescharacteristic
536C2#3271412yes—
537C2#3273312yes—
538C2#3273712yes—
539C2#3276712yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11316C2semidirect
4. #11190C2semidirect
5. #11122C2semidirect
7. #11024C2semidirect
22. #2001C4semidirect

Maximal subgroups

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

subgroupindex
C2×#20012
#142582
#113162
#111902
#111222
#110952
#110242
#109702
#107132