Finite Group Databasebeta

← #213651

Subgroups — #213651

Order 512.

1352 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic
2#20714C211yesmaximal, characteristic
3C2×#1797C211yesmaximal, characteristic
4#9131C214yesmaximal, complemented
5#9130C214yesmaximal, complemented
6#8723C212yesmaximal, complemented
7#8340C212yesmaximal, complemented
8C22×C4⋊Q8C2211yescharacteristic
9C22×C22⋊Q8—21no—
10C23×C42C2211yescharacteristic
11C2×#431C2212yes—
12C22×C4⋊C8C2212yes—
13C22×C22⋊C8—21no—
14#2857—21no—
15#2857—21no—
16#2857—21no—
17#2857—21no—
18#2857C412yes—
19#2857C412yes—
20#2857C2212yes—
21#2857C2212yes—
22C2×#391—21no—
23C2×#389—21no—
24C2×#389C412yes—
25C2×#384C2211yescharacteristic
26C2×#380—21no—
27C2×#380C412yes—
28#1817—21no—
29#1797—21no—
30#1797—21no—
31#1797C2212yes—
32#1797C2212yes—
33#1797C2212yes—
34#1797C2212yes—
35#1779—21no—
36C4×C22⋊C8—21no—
37C4×C22⋊C8—21no—
38C2×#336—21no—
39C2×#332C2218yescomplemented
40#1296C2218yescomplemented
41C23×Q8—21no—
42C24×C4C2×C411yescharacteristic
43C24×C4D411yescharacteristic
44C24×C4D411yescharacteristic
45C23×C8—22no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—44no—
49C2×C4⋊Q8C2×C412yes—
50C2×C4⋊Q8C2312yes—
51C2×C22⋊Q8—22no—
52C2×C22⋊Q8—22no—
53C2×C22⋊Q8—22no—
54C2×C22⋊Q8—22no—
55C2×C22⋊Q8—44no—
56C2×C22⋊Q8—44no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4C2×C412yes—
62C22×C4⋊C4C2×C412yes—
63C22×C22⋊C4—22no—
64C22×C42—44no—
65C22×C42C2×C411yescharacteristic
66C22×C42C2×C411yescharacteristic
67C22×C42C2×C411yescharacteristic
68C22×C42D411yescharacteristic
69C22×C42D411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77#431—44no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—44no—
80C2×C4⋊C8C2314yes—
81C2×C4⋊C8C2314yes—
82C2×C22⋊C8—22no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—44no—
88C2×C4×C8—22no—
89#391—22no—
90#391—22no—
91#391—22no—
92#391—22no—
93#389—22no—
94#389—22no—
95#389—22no—
96#389—22no—
97#389—22no—
98#389—22no—
99#389C2×C414yes—
100#389C2×C414yes—
101#384—21no—
102#384—21no—
103#384C2×C412yes—
104#384C2312yes—
105#380—22no—
106#380—22no—
107#380—22no—
108#380—22no—
109#380—22no—
110#380—22no—
111#380C2×C414yes—
112#380C2×C414yes—
113C2×#95—22no—
114C2×#95—22no—
115#336—22no—
116#336—22no—
117#336—22no—
118#336—22no—
119#332—48no—
120C25C22⋊C411yescharacteristic
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—44no—
124C22×Q8—44no—
125C22×Q8—42no—
126C22×Q8—42no—
127C23×C4—24no—
128C23×C4—24no—
129C23×C4—24no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—42no—
133C23×C4—42no—
134C23×C4—42no—
135C23×C4—42no—
136C23×C4—42no—
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×C4D814yes—
147C23×C4SD1614yes—
148C23×C4C22×C411yescharacteristic
149C23×C4C22×C411yescharacteristic
150C23×C4C22×C411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C22×C8—24no—
158C22×C8—24no—
159C22×C8—44no—
160C22×C8—44no—
161C22×C8—44no—
162C4⋊Q8—28no—
163C4⋊Q8—44no—
164C4⋊Q8—44no—
165C4⋊Q8—44no—
166C22⋊Q8—48no—
167C22⋊Q8—48no—
168C22⋊Q8—44no—
169C22⋊Q8—44no—
170C22⋊Q8—44no—
171C22⋊Q8—44no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
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—44no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—48no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4C22×C414yes—
190C2×C4⋊C4C22×C414yes—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—48no—
194C2×C42—24no—
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×C42C22⋊C411yescharacteristic
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×C42C2411yescharacteristic
222C4⋊C8—28no—
223C4⋊C8—28no—
224C4⋊C8—44no—
225C22⋊C8—44no—
226C22⋊C8—44no—
227C22⋊C8—44no—
228C22⋊C8—44no—
229C22⋊C8—44no—
230C4×C8—44no—
231#95—24no—
232#95—24no—
233#95—24no—
234#95—24no—
235C24—22no—
236C24—22no—
237C24—22no—
238C24—22no—
239C24—41no—
240C24—41no—
241C24#10214yes—
242C24C2×C22⋊C411yescharacteristic
243C24C2×C22⋊C411yescharacteristic
244C24C2×C22⋊C411yescharacteristic
245C2×Q8—44no—
246C2×Q8—44no—
247C2×Q8—44no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—44no—
251C2×Q8—44no—
252C2×Q8—88no—
253C2×Q8—88no—
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—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
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—24no—
284C22×C4—24no—
285C22×C4—48no—
286C22×C4—48no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
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—416no—
304C22×C4—416no—
305C22×C4—44no—
306C22×C4—44no—
307C22×C4—44no—
308C22×C4—44no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4#10114yes—
312C22×C4#10214yes—
313C22×C4#10214yes—
314C22×C4#10214yes—
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×D812yes—
325C22×C4C2×D812yes—
326C22×C4C2×D812yes—
327C22×C4C2×SD1612yes—
328C22×C4C2×SD1612yes—
329C22×C4C2×SD1612yes—
330C22×C4#13514yes—
331C22×C4#13714yes—
332C22×C4C23×C411yescharacteristic
333C22×C4C22×D411yescharacteristic
334C22×C4C22×D411yescharacteristic
335C2×C8—44no—
336C2×C8—44no—
337C2×C8—44no—
338C2×C8—44no—
339C2×C8—44no—
340C2×C8—44no—
341C2×C8—88no—
342C4⋊C4—28no—
343C4⋊C4—44no—
344C4⋊C4—44no—
345C4⋊C4—44no—
346C4⋊C4—44no—
347C4⋊C4—44no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—44no—
351C4⋊C4—44no—
352C4⋊C4—48no—
353C4⋊C4—48no—
354C4⋊C4—44no—
355C22⋊C4—48no—
356C22⋊C4—48no—
357C42—22no—
358C42—22no—
359C42—22no—
360C42—22no—
361C42—22no—
362C42—22no—
363C42—22no—
364C42—22no—
365C42—22no—
366C42—41no—
367C42—41no—
368C42—41no—
369C42—41no—
370C42—41no—
371C42—41no—
372C42H(5,𝔽2)14yes—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—28no—
382C23—21no—
383C23—28no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—44no—
390C23—44no—
391C23—44no—
392C23—44no—
393C23—44no—
394C23C2×#10112yes—
395C23#41312yes—
396C23C2×#10212yes—
397C23C2×#10212yes—
398C23C2×#10212yes—
399C23#41612yes—
400C23#41912yes—
401C23C22×C22⋊C411yescharacteristic
402Q8—88no—
403Q8—84no—
404Q8—84no—
405C2×C4—28no—
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—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—28no—
429C2×C4—28no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
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—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—88no—
455C2×C4—88no—
456C2×C4C2×#10112yes—
457C2×C4C2×#10112yes—
458C2×C4C2×#10212yes—
459C2×C4C2×#10212yes—
460C2×C4C2×#10212yes—
461C2×C4#41612yes—
462C2×C4#41912yes—
463C2×C4#51812yes—
464C2×C4#53412yes—
465C2×C4#53412yes—
466C2×C4C22×D811yescharacteristic
467C2×C4C22×SD1611yescharacteristic
468C2×C4C2×#13512yes—
469C2×C4C2×#13712yes—
470C2×C4#57512yes—
471C8—84no—
472C8—84no—
473C22—22no—
474C22—22no—
475C22—22no—
476C22—22no—
477C22—22no—
478C22—22no—
479C22—22no—
480C22—22no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—416no—
489C22—41no—
490C22—41no—
491C22—41no—
492C22—41no—
493C22#154014yes—
494C22#154614yes—
495C22#230811yescharacteristic
496C22C22×#10111yescharacteristic
497C22C2×#41311yescharacteristic
498C22C22×#10211yescharacteristic
499C22C2×#41611yescharacteristic
500C22C2×#41911yescharacteristic
501C22#289611yescharacteristic
502C4—22no—
503C4—22no—
504C4—22no—
505C4—22no—
506C4—22no—
507C4—22no—
508C4—22no—
509C4—22no—
510C4—41no—
511C4—41no—
512C4—84no—
513C4—84no—
514C4#318612yes—
515C4#318712yes—
516C4#319014yes—
517C2—24no—
518C2—24no—
519C2—42no—
520C2C2×#154011yescharacteristic
521C2C2×#154611yescharacteristic
522C2#1771511yescharacteristic
523C2#1774811yescharacteristic
524C2#2175311yescharacteristic
525C2#2175811yescharacteristic
526C2#2176211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9131C2semidirect
5. #9130C2semidirect
6. #8723C2semidirect
7. #8340C2semidirect
39. C2×#332C22semidirect
40. #1296C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
#207142
C2×#17972
#91312
#91302
#87232
#83402