Finite Group Databasebeta

← #188095

Subgroups — #188095

Order 512.

2410 conjugacy classes of subgroups.

The subgroup lattice has 489 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#552C211yesmaximal, characteristic
2C2×#1821C212yesmaximal, complemented
3#17397C212yesmaximal, complemented
4#17362C212yesmaximal
5#8556C214yesmaximal, complemented
6#8554C214yesmaximal, complemented
7C2×#552—28no—
8C2×#552—24no—
9C2×#552—24no—
10C2×#552—24no—
11C2×#552—24no—
12C22×#126C2212yes—
13C22×#125C411yescharacteristic
14C22×#125C2212yes—
15C22×#123C412yes—
16C22×C22⋊Q8C411yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C22×C4×Q8C411yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C22×#117C411yescharacteristic
21C22×#117C412yes—
22C23×C4⋊C4C2211yescharacteristic
23C2×#407C2214yes—
24#1821—24no—
25#1821C2218yes—
26C2×#343—24no—
27#1472—28no—
28#1465—24no—
29#1463C2214yes—
30#1463C2218yes—
31#1462C2214yes—
32#1459—24no—
33C23×Q8C2×C411yescharacteristic
34C23×D4C2×C411yescharacteristic
35C24×C4C2×C412yes—
36C24×C4C2×C411yescharacteristic
37C24×C4C2311yescharacteristic
38C22×M16—24no—
39#552—232no—
40#552—232no—
41#552—232no—
42#552—232no—
43C2×#126—28no—
44C2×#126—28no—
45C2×#126—28no—
46C2×#125—24no—
47C2×#125—22no—
48C2×#125—22no—
49C2×#125—28no—
50C2×#125—28no—
51C2×#125—28no—
52C2×#125—24no—
53C2×#123—24no—
54C2×#123—24no—
55C2×#123—28no—
56C2×#123—28no—
57C2×C22⋊Q8—24no—
58C2×C22⋊Q8—24no—
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8—22no—
61C2×C22⋊Q8—22no—
62C2×C22⋊Q8—22no—
63C2×C22⋊Q8—22no—
64C2×C22⋊Q8—22no—
65C2×C22⋊Q8D418yes—
66C2×C4×Q8—24no—
67C2×C4×Q8—22no—
68C2×C4×Q8—22no—
69C2×C4×Q8—24no—
70C2×C4×D4—22no—
71C2×C4×D4—24no—
72C2×C4×D4—22no—
73C2×C4×D4D414yes—
74C2×C4×D4D414yes—
75C2×#117—24no—
76C2×#117—28no—
77C2×#117—24no—
78C2×#117—22no—
79C2×#117—22no—
80C2×#117—28no—
81C2×#117—24no—
82C2×#117—24no—
83C22×C4⋊C4—24no—
84C22×C4⋊C4—22no—
85C22×C4⋊C4—22no—
86C22×C4⋊C4C2×C412yes—
87C22×C4⋊C4C2×C412yes—
88C22×C4⋊C4C2×C412yes—
89C22×C4⋊C4C2×C412yes—
90C22×C4⋊C4C2×C411yescharacteristic
91C22×C4⋊C4C2×C411yescharacteristic
92C22×C4⋊C4C2×C411yescharacteristic
93C22×C4⋊C4D414yes—
94C22×C4⋊C4D414yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C22⋊C4C2×C412yes—
99C22×C22⋊C4C2×C412yes—
100C22×C22⋊C4C2×C411yescharacteristic
101C22×C22⋊C4C2×C412yes—
102C22×C22⋊C4C2311yescharacteristic
103C22×C42C2×C412yes—
104C22×C42C2×C412yes—
105C22×C42C2×C411yescharacteristic
106#407—24no—
107#407—24no—
108#407—24no—
109#407C2318yes—
110C2×C22⋊C8—24no—
111C2×C22⋊C8—24no—
112#343—216no—
113#324—28no—
114#324—216no—
115#323—28no—
116C25C22×C411yescharacteristic
117C25C22×C411yescharacteristic
118C22×Q8—21no—
119C22×Q8—22no—
120C22×Q8—21no—
121C22×Q8—44no—
122C22×Q8C22⋊C414yes—
123C22×D4—22no—
124C22×D4—22no—
125C22×D4—44no—
126C22×D4C22⋊C412yes—
127C22×D4C22⋊C412yes—
128C23×C4—24no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—44no—
135C23×C4—44no—
136C23×C4—48no—
137C23×C4C22⋊C418yes—
138C23×C4C22⋊C414yes—
139C23×C4C22×C411yescharacteristic
140C23×C4C22×C412yes—
141C23×C4C22×C412yes—
142C23×C4C22×C411yescharacteristic
143C23×C4C22×C412yes—
144C23×C4C22×C412yes—
145C23×C4C22×C411yescharacteristic
146C23×C4C22×C411yescharacteristic
147C23×C4C2×D412yes—
148C23×C4C2×D414yes—
149C23×C4C2×D412yes—
150C23×C4C2×D412yes—
151C23×C4C2×D412yes—
152C23×C4C2411yescharacteristic
153C2×M16—28no—
154C2×M16—44no—
155C2×M16—416no—
156C22×C8—44no—
157#126—264no—
158#125—264no—
159#125—216no—
160#125—216no—
161#123—232no—
162#123—232no—
163C22⋊Q8—216no—
164C22⋊Q8—216no—
165C22⋊Q8—28no—
166C22⋊Q8—28no—
167C22⋊Q8—28no—
168C22⋊Q8—28no—
169C4×Q8—216no—
170C4×Q8—216no—
171C4×D4—28no—
172C4×D4—28no—
173C4×D4—28no—
174C4×D4—28no—
175#117—216no—
176#117—216no—
177#117—232no—
178#117—232no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4C22⋊C412yes—
209C2×C4⋊C4C22⋊C412yes—
210C2×C4⋊C4C22⋊C412yes—
211C2×C4⋊C4C22⋊C412yes—
212C2×C4⋊C4C22⋊C414yes—
213C2×C4⋊C4C2×D418yes—
214C2×C4⋊C4C2×D414yes—
215C2×C4⋊C4C2×D418yes—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4C22⋊C412yes—
230C2×C22⋊C4C22⋊C412yes—
231C2×C22⋊C4C22⋊C418yes—
232C2×C22⋊C4C2×D414yes—
233C2×C42—24no—
234C2×C42—24no—
235C2×C42—24no—
236C2×C42—24no—
237C2×C42—24no—
238C2×C42—24no—
239C2×C42—22no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42C22⋊C414yes—
243C22⋊C8—28no—
244C22⋊C8—28no—
245C22⋊C8—44no—
246C22⋊C8—44no—
247C24—24no—
248C24—21no—
249C24—21no—
250C24—42no—
251C24—42no—
252C24—42no—
253C24—42no—
254C24—42no—
255C24C2×C22⋊C412yes—
256C24C2×C22⋊C412yes—
257C24C2×C22⋊C412yes—
258C24C2×C22⋊C411yescharacteristic
259C24C2×C22⋊C411yescharacteristic
260C24C23×C411yescharacteristic
261C2×Q8—24no—
262C2×Q8—24no—
263C2×Q8—48no—
264C2×Q8—44no—
265C2×Q8—48no—
266C2×Q8—44no—
267C2×D4—24no—
268C2×D4—24no—
269C2×D4—28no—
270C2×D4—44no—
271C2×D4—48no—
272C2×D4—48no—
273C22×C4—28no—
274C22×C4—24no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—28no—
290C22×C4—28no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—44no—
294C22×C4—48no—
295C22×C4—44no—
296C22×C4—44no—
297C22×C4—42no—
298C22×C4—44no—
299C22×C4—42no—
300C22×C4—44no—
301C22×C4—48no—
302C22×C4—48no—
303C22×C4—48no—
304C22×C4—416no—
305C22×C4—48no—
306C22×C4—44no—
307C22×C4—416no—
308C22×C4—42no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4—42no—
312C22×C4—44no—
313C22×C4C2×C22⋊C414yes—
314C22×C4C2×C22⋊C412yes—
315C22×C4C2×C22⋊C412yes—
316C22×C4C2×C22⋊C411yescharacteristic
317C22×C4C2×C22⋊C412yes—
318C22×C4C2×C22⋊C412yes—
319C22×C4C2×C22⋊C414yes—
320C22×C4C2×C22⋊C412yes—
321C22×C4C2×C22⋊C414yes—
322C22×C4C2×C22⋊C414yes—
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4#12014yes—
325C22×C4#12014yes—
326C22×C4#12014yes—
327C22×C4#12014yes—
328C22×C4C22×D412yes—
329C22×C4C22×D412yes—
330C22×C4C22×D412yes—
331M16—416no—
332M16—88no—
333C2×C8—48no—
334C2×C8—88no—
335C4⋊C4—216no—
336C4⋊C4—216no—
337C4⋊C4—216no—
338C4⋊C4—216no—
339C4⋊C4—216no—
340C4⋊C4—28no—
341C4⋊C4—216no—
342C4⋊C4—24no—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C22⋊C4—216no—
349C22⋊C4—216no—
350C22⋊C4—24no—
351C22⋊C4—24no—
352C22⋊C4—28no—
353C22⋊C4—28no—
354C22⋊C4—28no—
355C42—216no—
356C42—216no—
357C42—24no—
358C42—24no—
359C23—22no—
360C23—22no—
361C23—22no—
362C23—24no—
363C23—22no—
364C23—24no—
365C23—24no—
366C23—24no—
367C23—22no—
368C23—24no—
369C23—41no—
370C23—42no—
371C23—41no—
372C23—41no—
373C23—42no—
374C23—41no—
375C23—416no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—42no—
380C23—42no—
381C23—42no—
382C23—42no—
383C23—42no—
384C23—88no—
385C23#37912yes—
386C23#37912yes—
387C23#41314yes—
388C23C22×C22⋊C411yescharacteristic
389C23C22×C22⋊C411yescharacteristic
390C23C22×C22⋊C411yescharacteristic
391Q8—48no—
392Q8—48no—
393D4—48no—
394D4—48no—
395C2×C4—24no—
396C2×C4—22no—
397C2×C4—28no—
398C2×C4—28no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—28no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—22no—
409C2×C4—24no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—48no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—416no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—42no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—42no—
430C2×C4—416no—
431C2×C4—48no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4#37912yes—
438C2×C4#37912yes—
439C2×C4#41014yes—
440C2×C4C2×#12014yes—
441C8—88no—
442C22—28no—
443C22—21no—
444C22—21no—
445C22—22no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22—22no—
454C22—44no—
455C22—48no—
456C22—48no—
457C22—44no—
458C22—42no—
459C22—42no—
460C22—42no—
461C22—84no—
462C22—84no—
463C22—84no—
464C22#178514yes—
465C22C2×#37911yescharacteristic
466C22C2×#41012yes—
467C22C2×#41312yes—
468C22#288912yes—
469C4—24no—
470C4—22no—
471C4—42no—
472C4—44no—
473C4—42no—
474C4—44no—
475C4—44no—
476C4—44no—
477C4—42no—
478C4#178214yes—
479C2—22no—
480C2—22no—
481C2—22no—
482C2—24no—
483C2—42no—
484C2—82no—
485C2C2×#178211yescharacteristic
486C2C2×#178511yescharacteristic
487C2#2059311yescharacteristic
subgroupG/H
center, socleC22C2×#379
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ 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×#1821C2semidirect
3. #17397C2semidirect
5. #8556C2semidirect
6. #8554C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5522
C2×#18212
#173972
#173622
#85562
#85542