Finite Group Databasebeta

← #213655

Subgroups — #213655

Order 512.

1306 conjugacy classes of subgroups.

The subgroup lattice has 505 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic
2#20714C211yesmaximal, characteristic
3#20714C211yesmaximal, characteristic
4#9132C214yesmaximal, complemented
5#9130C214yesmaximal, complemented
6#8724C212yesmaximal, complemented
7#8340C212yesmaximal, complemented
8C22×C4⋊Q8C2211yescharacteristic
9C22×C22⋊Q8—21no—
10C23×C42C2211yescharacteristic
11C2×#431C2212yes—
12C2×#431C2212yes—
13#2857—21no—
14#2857—21no—
15#2857—21no—
16#2857—21no—
17#2857C412yes—
18#2857C412yes—
19#2857C2212yes—
20#2857C2212yes—
21C2×#391—21no—
22C2×#389—21no—
23C2×#389C412yes—
24C2×#384C2211yescharacteristic
25C2×#380—21no—
26C2×#380C412yes—
27#1817—21no—
28#1817—21no—
29#1797C2212yes—
30#1797C2212yes—
31#1797C2212yes—
32#1797C2212yes—
33#1779—21no—
34#1779—21no—
35C4×C22⋊C8—21no—
36C4×C22⋊C8—21no—
37C4×C22⋊C8—21no—
38C4×C22⋊C8—21no—
39C2×#332C2218yescomplemented
40#1296C2218yescomplemented
41C23×Q8—21no—
42C24×C4C2×C411yescharacteristic
43C24×C4D411yescharacteristic
44C24×C4D411yescharacteristic
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8—44no—
48C2×C4⋊Q8C2×C412yes—
49C2×C4⋊Q8C2312yes—
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8—22no—
52C2×C22⋊Q8—22no—
53C2×C22⋊Q8—22no—
54C2×C22⋊Q8—48no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4C2×C412yes—
60C22×C4⋊C4C2×C412yes—
61C22×C22⋊C4—22no—
62C22×C42—44no—
63C22×C42C2×C411yescharacteristic
64C22×C42C2×C411yescharacteristic
65C22×C42C2×C411yescharacteristic
66C22×C42D411yescharacteristic
67C22×C42D411yescharacteristic
68C22×C42D411yescharacteristic
69C22×C42D411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75#431—44no—
76#431—44no—
77C2×C4⋊C8C2314yes—
78C2×C4⋊C8C2314yes—
79C2×C22⋊C8—22no—
80C2×C22⋊C8—22no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—22no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—22no—
85C2×C4×C8—22no—
86C2×C4×C8—22no—
87#391—22no—
88#391—22no—
89#391—22no—
90#391—22no—
91#389—22no—
92#389—22no—
93#389—22no—
94#389—22no—
95#389—22no—
96#389—22no—
97#389C2×C414yes—
98#389C2×C414yes—
99#384—21no—
100#384—21no—
101#384C2×C412yes—
102#384C2312yes—
103#380—22no—
104#380—22no—
105#380—22no—
106#380—22no—
107#380—22no—
108#380—22no—
109#380C2×C414yes—
110#380C2×C414yes—
111C2×#95—22no—
112C2×#95—22no—
113#336—22no—
114#336—22no—
115#336—22no—
116#336—22no—
117#332—48no—
118C25C22⋊C411yescharacteristic
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—44no—
122C22×Q8—44no—
123C22×Q8—44no—
124C23×C4—24no—
125C23×C4—24no—
126C23×C4—24no—
127C23×C4—24no—
128C23×C4—24no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—42no—
132C23×C4—42no—
133C23×C4—42no—
134C23×C4—42no—
135C23×C4C22⋊C411yescharacteristic
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×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C22×C8—24no—
154C22×C8—24no—
155C4⋊Q8—28no—
156C4⋊Q8—44no—
157C4⋊Q8—44no—
158C4⋊Q8—44no—
159C22⋊Q8—48no—
160C22⋊Q8—48no—
161C22⋊Q8—48no—
162C22⋊Q8—48no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—48no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4C22×C414yes—
180C2×C4⋊C4C22×C414yes—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—48no—
184C2×C42—24no—
185C2×C42—24no—
186C2×C42—24no—
187C2×C42—42no—
188C2×C42—42no—
189C2×C42—42no—
190C2×C42—42no—
191C2×C42—42no—
192C2×C42—42no—
193C2×C42—42no—
194C2×C42—42no—
195C2×C42—42no—
196C2×C42C22⋊C411yescharacteristic
197C2×C42C22⋊C411yescharacteristic
198C2×C42C22⋊C411yescharacteristic
199C2×C42C22⋊C411yescharacteristic
200C2×C42C22⋊C411yescharacteristic
201C2×C42C22⋊C411yescharacteristic
202C2×C42C22×C411yescharacteristic
203C2×C42C22×C411yescharacteristic
204C2×C42C22×C411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2411yescharacteristic
212C4⋊C8—28no—
213C4⋊C8—28no—
214C22⋊C8—44no—
215C22⋊C8—44no—
216C22⋊C8—44no—
217C22⋊C8—44no—
218C22⋊C8—44no—
219C22⋊C8—44no—
220C4×C8—44no—
221C4×C8—44no—
222#95—24no—
223#95—24no—
224#95—24no—
225#95—24no—
226C24—22no—
227C24—22no—
228C24—22no—
229C24—22no—
230C24—22no—
231C24—41no—
232C24—41no—
233C24C2×C22⋊C411yescharacteristic
234C24C2×C22⋊C411yescharacteristic
235C24C2×C22⋊C411yescharacteristic
236C2×Q8—44no—
237C2×Q8—44no—
238C2×Q8—44no—
239C2×Q8—44no—
240C2×Q8—44no—
241C2×Q8—44no—
242C2×Q8—816no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
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—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—24no—
281C22×C4—24no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—416no—
295C22×C4—416no—
296C22×C4—416no—
297C22×C4—44no—
298C22×C4—44no—
299C22×C4—44no—
300C22×C4—44no—
301C22×C4—44no—
302C22×C4—44no—
303C22×C4#10114yes—
304C22×C4C2×C22⋊C411yescharacteristic
305C22×C4C2×C22⋊C411yescharacteristic
306C22×C4C2×C22⋊C411yescharacteristic
307C22×C4C2×C22⋊C411yescharacteristic
308C22×C4C2×C22⋊C411yescharacteristic
309C22×C4C2×C22⋊C411yescharacteristic
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4C2×C22⋊C411yescharacteristic
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4#13514yes—
314C22×C4#13514yes—
315C22×C4AGL(1,ℤ/8)14yes—
316C22×C4#13714yes—
317C22×C4C23×C411yescharacteristic
318C22×C4C22×D411yescharacteristic
319C22×C4C22×D411yescharacteristic
320C2×C8—44no—
321C2×C8—44no—
322C2×C8—44no—
323C2×C8—44no—
324C2×C8—44no—
325C2×C8—44no—
326C4⋊C4—28no—
327C4⋊C4—44no—
328C4⋊C4—44no—
329C4⋊C4—48no—
330C4⋊C4—44no—
331C4⋊C4—44no—
332C4⋊C4—44no—
333C4⋊C4—44no—
334C4⋊C4—44no—
335C4⋊C4—44no—
336C4⋊C4—44no—
337C4⋊C4—48no—
338C22⋊C4—48no—
339C22⋊C4—48no—
340C42—22no—
341C42—22no—
342C42—22no—
343C42—22no—
344C42—22no—
345C42—22no—
346C42—22no—
347C42—22no—
348C42—22no—
349C42—41no—
350C42—41no—
351C42—41no—
352C42—41no—
353C42—41no—
354C42—41no—
355C42H(5,𝔽2)14yes—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—28no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—44no—
374C23—44no—
375C23—44no—
376C23—44no—
377C23—44no—
378C23—44no—
379C23C2×#10112yes—
380C23#41612yes—
381C23#41612yes—
382C23#41812yes—
383C23#41912yes—
384C23C22×C22⋊C411yescharacteristic
385Q8—88no—
386Q8—88no—
387C2×C4—28no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—28no—
405C2×C4—28no—
406C2×C4—28no—
407C2×C4—28no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—88no—
434C2×C4—88no—
435C2×C4C2×#10112yes—
436C2×C4C2×#10112yes—
437C2×C4#41312yes—
438C2×C4#41612yes—
439C2×C4#41612yes—
440C2×C4#41812yes—
441C2×C4#41912yes—
442C2×C4#51812yes—
443C2×C4#53412yes—
444C2×C4#53412yes—
445C2×C4C2×#13512yes—
446C2×C4C2×#13512yes—
447C2×C4C2×AGL(1,ℤ/8)12yes—
448C2×C4C2×#13712yes—
449C8—84no—
450C8—84no—
451C22—22no—
452C22—22no—
453C22—22no—
454C22—22no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—416no—
464C22—41no—
465C22—41no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—41no—
471C22—41no—
472C22#154414yes—
473C22#230811yescharacteristic
474C22C22×#10111yescharacteristic
475C22C2×#41311yescharacteristic
476C22C2×#41611yescharacteristic
477C22C2×#41611yescharacteristic
478C22C2×#41811yescharacteristic
479C22C2×#41911yescharacteristic
480C4—22no—
481C4—22no—
482C4—22no—
483C4—22no—
484C4—22no—
485C4—22no—
486C4—22no—
487C4—22no—
488C4—41no—
489C4—41no—
490C4—84no—
491C4—84no—
492C4#318814yes—
493C4#319014yes—
494C2—24no—
495C2—42no—
496C2—42no—
497C2C2×#154411yescharacteristic
498C2#1771711yescharacteristic
499C2#1771811yescharacteristic
500C2#1774211yescharacteristic
501C2#2175311yescharacteristic
502C2#2176111yescharacteristic
503C2#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. #9132C2semidirect
5. #9130C2semidirect
6. #8724C2semidirect
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
#91322
#91302
#87242
#83402