Finite Group Databasebeta

← #391827

Subgroups — #391827

Order 512.

1370 conjugacy classes of subgroups.

The subgroup lattice has 525 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic
2#20682C211yesmaximal, characteristic
3#20626C211yesmaximal, characteristic
4#11382C212yesmaximal, complemented
5#11354C212yesmaximal, complemented
6#11204C214yesmaximal, complemented
7#11198C214yesmaximal
8C22×C4⋊Q8C2211yescharacteristic
9C22×C22⋊Q8C412yes—
10C23×C42C2211yescharacteristic
11C2×#423C2212yes—
12C2×#407C2212yes—
13C2×#407C2212yes—
14#2857—21no—
15#2857—21no—
16#2857—21no—
17#2857—21no—
18#2857—21no—
19#2857—21no—
20#2857—21no—
21#2857—21no—
22C2×#391C412yes—
23C2×#389C412yes—
24C2×#389C2212yes—
25C2×#384C2211yescharacteristic
26C2×#380C412yes—
27C2×#380C2212yes—
28#1817—21no—
29#1817—21no—
30#1817—21no—
31#1817—21no—
32#1797—21no—
33#1797—21no—
34#1797—21no—
35#1797—21no—
36C2×#343C2218yes—
37C2×#343C2218yes—
38#1728C2212yes—
39#1728C2212yes—
40#1728C2212yes—
41C23×Q8C2×C412yes—
42C24×C4C2×C411yescharacteristic
43C24×C4C2×C411yescharacteristic
44C24×C4C2311yescharacteristic
45C22×M16C2314yes—
46C22×M16C2314yes—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—44no—
52C2×C22⋊Q8—22no—
53C2×C22⋊Q8—22no—
54C2×C22⋊Q8—22no—
55C2×C22⋊Q8—22no—
56C2×C22⋊Q8—48no—
57C22×C4⋊C4C2×C412yes—
58C22×C4⋊C4C2×C414yes—
59C22×C4⋊C4C2×C412yes—
60C22×C4⋊C4C2×C412yes—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C22⋊C4C2×C414yes—
64C22×C42—44no—
65C22×C42C2×C411yescharacteristic
66C22×C42C2×C411yescharacteristic
67C22×C42D411yescharacteristic
68C22×C42D411yescharacteristic
69C22×C42D411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77#423—44no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—22no—
80#407—44no—
81#407—44no—
82C2×C22⋊C8—22no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86#391—22no—
87#391—22no—
88#391—22no—
89#391—22no—
90#389—22no—
91#389—22no—
92#389—22no—
93#389—22no—
94#389—22no—
95#389—22no—
96#389—22no—
97#389—22no—
98#384—21no—
99#384—21no—
100#384—21no—
101#384—21no—
102#380—22no—
103#380—22no—
104#380—22no—
105#380—22no—
106#380—22no—
107#380—22no—
108#380—22no—
109#380—22no—
110C2×#95C2×C414yes—
111C2×#95C2×C414yes—
112#343—48no—
113#343—48no—
114#336—22no—
115#336—22no—
116#336—22no—
117#336—22no—
118#336—22no—
119#336—22no—
120C25C22×C411yescharacteristic
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—44no—
124C22×Q8—44no—
125C22×Q8—44no—
126C23×C4—24no—
127C23×C4—24no—
128C23×C4—22no—
129C23×C4—24no—
130C23×C4—24no—
131C23×C4—22no—
132C23×C4—42no—
133C23×C4—42no—
134C23×C4—42no—
135C23×C4—42no—
136C23×C4C22⋊C411yescharacteristic
137C23×C4C22⋊C411yescharacteristic
138C23×C4C22⋊C411yescharacteristic
139C23×C4C22⋊C411yescharacteristic
140C23×C4C22⋊C411yescharacteristic
141C23×C4C22⋊C411yescharacteristic
142C23×C4C22⋊C411yescharacteristic
143C23×C4C22⋊C411yescharacteristic
144C23×C4C22×C411yescharacteristic
145C23×C4C22×C411yescharacteristic
146C23×C4C22×C411yescharacteristic
147C23×C4C22×C411yescharacteristic
148C23×C4C22×C411yescharacteristic
149C23×C4C22×C414yes—
150C23×C4C22×C414yes—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2411yescharacteristic
156C2×M16—24no—
157C2×M16—28no—
158C2×M16—28no—
159C2×M16—28no—
160C2×M16—28no—
161C2×M16—28no—
162C2×M16—24no—
163C22×C8—24no—
164C22×C8—24no—
165C4⋊Q8—44no—
166C4⋊Q8—44no—
167C4⋊Q8—44no—
168C4⋊Q8—44no—
169C22⋊Q8—48no—
170C22⋊Q8—48no—
171C22⋊Q8—48no—
172C22⋊Q8—48no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
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—28no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—48no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—48no—
195C2×C42—24no—
196C2×C42—24no—
197C2×C42—42no—
198C2×C42—42no—
199C2×C42—42no—
200C2×C42—42no—
201C2×C42—42no—
202C2×C42—42no—
203C2×C42—42no—
204C2×C42—42no—
205C2×C42—42no—
206C2×C42—42no—
207C2×C42C22⋊C411yescharacteristic
208C2×C42C22⋊C411yescharacteristic
209C2×C42C22⋊C411yescharacteristic
210C2×C42C22⋊C411yescharacteristic
211C2×C42C22⋊C411yescharacteristic
212C2×C42C22⋊C411yescharacteristic
213C2×C42C22⋊C411yescharacteristic
214C2×C42C22⋊C411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C4⋊C8—44no—
224C4⋊C8—44no—
225C22⋊C8—44no—
226C22⋊C8—44no—
227C22⋊C8—44no—
228C22⋊C8—44no—
229#95—24no—
230#95—24no—
231#95—24no—
232#95—24no—
233C24—22no—
234C24—22no—
235C24—22no—
236C24—22no—
237C24—22no—
238C24—41no—
239C24—41no—
240C24C2×C22⋊C411yescharacteristic
241C24C2×C22⋊C411yescharacteristic
242C24C23×C411yescharacteristic
243C2×Q8—44no—
244C2×Q8—44no—
245C2×Q8—44no—
246C2×Q8—44no—
247C2×Q8—44no—
248C2×Q8—44no—
249C2×Q8—816no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—21no—
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—24no—
288C22×C4—24no—
289C22×C4—28no—
290C22×C4—28no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—416no—
306C22×C4—416no—
307C22×C4—48no—
308C22×C4—48no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4—44no—
312C22×C4—44no—
313C22×C4#10014yes—
314C22×C4#10114yes—
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4C2×C22⋊C411yescharacteristic
317C22×C4C2×C22⋊C411yescharacteristic
318C22×C4C2×C22⋊C411yescharacteristic
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C2×C22⋊C411yescharacteristic
325C22×C4C22×D411yescharacteristic
326C22×C4C22×D411yescharacteristic
327C22×C4H(5,𝔽2)14yes—
328C22×C4H(5,𝔽2)12yes—
329C22×C42−1+414yes—
330C22×C42−1+412yes—
331M16—416no—
332M16—48no—
333M16—48no—
334C2×C8—44no—
335C2×C8—44no—
336C2×C8—44no—
337C2×C8—44no—
338C2×C8—44no—
339C2×C8—44no—
340C4⋊C4—44no—
341C4⋊C4—44no—
342C4⋊C4—48no—
343C4⋊C4—44no—
344C4⋊C4—44no—
345C4⋊C4—44no—
346C4⋊C4—44no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—44no—
350C4⋊C4—44no—
351C4⋊C4—48no—
352C4⋊C4—44no—
353C22⋊C4—48no—
354C22⋊C4—48no—
355C42—22no—
356C42—22no—
357C42—22no—
358C42—22no—
359C42—22no—
360C42—22no—
361C42—22no—
362C42—22no—
363C42—41no—
364C42—41no—
365C42—41no—
366C42—41no—
367C42—41no—
368C42—41no—
369C42—41no—
370C42—41no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—28no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—44no—
389C23—44no—
390C23—44no—
391C23—44no—
392C23—44no—
393C23—44no—
394C23C2×#10012yes—
395C23C2×#10112yes—
396C23C22×C22⋊C411yescharacteristic
397C23#51812yes—
398C23#51912yes—
399C23#52012yes—
400Q8—88no—
401Q8—88no—
402C2×C4—28no—
403C2×C4—28no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—88no—
452C2×C4—88no—
453C2×C4C2×#10012yes—
454C2×C4C2×#10012yes—
455C2×C4C2×#10112yes—
456C2×C4C2×#10112yes—
457C2×C4#41312yes—
458C2×C4#41312yes—
459C2×C4#53412yes—
460C2×C4#53412yes—
461C2×C4#53511yescharacteristic
462C2×C4#53612yes—
463C2×C4#53611yescharacteristic
464C2×C4#53612yes—
465C2×C4#53712yes—
466C2×C4#56218yes—
467C2×C4#56318yes—
468C8—84no—
469C8—84no—
470C22—22no—
471C22—22no—
472C22—22no—
473C22—22no—
474C22—22no—
475C22—22no—
476C22—22no—
477C22—22no—
478C22—22no—
479C22—22no—
480C22—22no—
481C22—22no—
482C22—416no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22—41no—
488C22—41no—
489C22—41no—
490C22—41no—
491C22#230811yescharacteristic
492C22#231011yescharacteristic
493C22#231111yescharacteristic
494C22#237814yes—
495C22C22×#10011yescharacteristic
496C22C22×#10111yescharacteristic
497C22C2×#41311yescharacteristic
498C22C2×#41311yescharacteristic
499C4—22no—
500C4—22no—
501C4—22no—
502C4—22no—
503C4—22no—
504C4—22no—
505C4—41no—
506C4—41no—
507C4—41no—
508C4—41no—
509C4—84no—
510C4—84no—
511C4#281414yes—
512C4#281612yes—
513C4#281712yes—
514C2—24no—
515C2—42no—
516C2—42no—
517C2#1466011yescharacteristic
518C2#2175211yescharacteristic
519C2#2175311yescharacteristic
520C2#2182911yescharacteristic
521C2#2183111yescharacteristic
522C2#2187811yescharacteristic
523C2#2187911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11382C2semidirect
5. #11354C2semidirect
6. #11204C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
#206822
#206262
#113822
#113542
#112042
#111982