Finite Group Databasebeta

← #359704

Subgroups — #359704

Order 512.

2166 conjugacy classes of subgroups.

The subgroup lattice has 455 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2921C211yesmaximal, characteristic, complemented
2C2×#2311C211yesmaximal, characteristic
3#20683C211yesmaximal, characteristic, complemented
4C2×#1785C212yesmaximal
5#11045C212yesmaximal, complemented
6#11044C212yesmaximal, complemented
7#10963C212yesmaximal, complemented
8#10962C212yesmaximal, complemented
9#10727C212yesmaximal, complemented
10C23×#41C2211yescharacteristic
11C22×#117C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13#2921—28no—
14C2×#433—24no—
15C2×#423C2212yes—
16C22×#101C2214yes—
17C22×#100C2214yes—
18C2×#407C2212yes—
19C2×#407C2212yes—
20C2×C4×M16C2212yes—
21#2311—24no—
22#2311—24no—
23C2×#391C2211yescharacteristic
24C2×#391C2211yescharacteristic
25C2×#390C2211yescharacteristic
26C2×#390C2211yescharacteristic
27C2×#386C414yes—
28C2×#380C414yes—
29#1818—24no—
30#1785—28no—
31C2×#343C2214yes—
32C2×#343C2214yes—
33#1728C2212yes—
34#1728C2212yes—
35C22×#41—24no—
36C22×#41—24no—
37C22×#41—22no—
38C22×#41—22no—
39C22×#41—24no—
40C22×#41—24no—
41C22×#41—44no—
42C23×Q8C2311yescharacteristic
43C23×D4C2×C412yes—
44C23×D4C2311yescharacteristic
45C24×C4C2×C412yes—
46C24×C4C2×C412yes—
47C24×C4C2311yescharacteristic
48C22×M16C2314yes—
49C22×M16C2314yes—
50C2×#117—24no—
51C2×#117—24no—
52C2×#117D414yes—
53C2×#117D414yes—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—44no—
57C22×C4⋊C4C2×C412yes—
58C22×C4⋊C4C2×C412yes—
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C22⋊C4C2×C412yes—
62C22×C22⋊C4C2×C414yes—
63C22×C22⋊C4C2×C414yes—
64C22×C42C2311yescharacteristic
65C22×C42C2311yescharacteristic
66#433—232no—
67#423—28no—
68C2×C4⋊C8—24no—
69C2×#101—28no—
70C2×#101—28no—
71C2×#101—44no—
72C2×#100—28no—
73C2×#100—28no—
74C2×#100—44no—
75#407—24no—
76#407—24no—
77#407—24no—
78#407D418yes—
79C2×C22⋊C8—24no—
80C2×C22⋊C8—24no—
81C4×M16—28no—
82C2×#97—22no—
83C2×C4×C8—22no—
84#391—22no—
85#391—22no—
86#391—22no—
87#391—22no—
88#390—22no—
89#390—22no—
90#390—22no—
91#390—22no—
92#386—216no—
93#380—216no—
94C2×#95C2×C414yes—
95C2×#95C2×C414yes—
96#343—44no—
97#343—44no—
98#336—24no—
99#336—24no—
100C25C22×C412yes—
101C25C22×C411yescharacteristic
102C2×#41—216no—
103C2×#41—216no—
104C2×#41—416no—
105C2×#41—416no—
106C2×#41—416no—
107C2×#41—416no—
108C2×#41—48no—
109C2×#41—48no—
110C2×#41—44no—
111C2×#41—48no—
112C2×#41—44no—
113C22×Q8—21no—
114C22×Q8—21no—
115C22×Q8—22no—
116C22×Q8—22no—
117C22×Q8—22no—
118C22×Q8—24no—
119C22×Q8#4114yes—
120C22×D4—24no—
121C22×D4—24no—
122C22×D4—28no—
123C22×D4—24no—
124C22×D4—21no—
125C22×D4—21no—
126C22×D4—22no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4—24no—
130C22×D4—44no—
131C22×D4#4114yes—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—24no—
135C23×C4—24no—
136C23×C4—24no—
137C23×C4—24no—
138C23×C4—24no—
139C23×C4—28no—
140C23×C4—44no—
141C23×C4—44no—
142C23×C4—84no—
143C23×C4C22×C412yes—
144C23×C4C22×C411yescharacteristic
145C23×C4C22×C412yes—
146C23×C4C22×C412yes—
147C23×C4C22×C412yes—
148C23×C4C22×C412yes—
149C23×C4C2×D412yes—
150C23×C4C2×D412yes—
151C23×C4C2×D414yes—
152C23×C4#4114yes—
153C23×C4C2411yescharacteristic
154C2×M16—24no—
155C2×M16—24no—
156C2×M16—24no—
157C2×M16—24no—
158C2×M16—24no—
159C2×M16—24no—
160C2×M16—24no—
161C2×M16—24no—
162C2×M16—24no—
163C2×M16—24no—
164C2×M16—24no—
165C2×M16C2×D418yes—
166C22×C8—24no—
167C22×C8—24no—
168#117—28no—
169#117—216no—
170#117—28no—
171C2×C4⋊C4—28no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—48no—
181C2×C4⋊C4—48no—
182C2×C4⋊C4C22⋊C418yes—
183C2×C22⋊C4—28no—
184C2×C22⋊C4—28no—
185C2×C22⋊C4—28no—
186C2×C22⋊C4—84no—
187C2×C22⋊C4—84no—
188C2×C22⋊C4C22⋊C418yes—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42C2×D412yes—
194C2×C42C2×D412yes—
195C2×C42C2×D412yes—
196C2×C42C2×D412yes—
197C4⋊C8—216no—
198#101—416no—
199#100—416no—
200C22⋊C8—216no—
201C22⋊C8—48no—
202#97—28no—
203C4×C8—28no—
204#95—28no—
205#95—28no—
206C24—21no—
207C24—21no—
208C24—22no—
209C24—22no—
210C24—22no—
211C24—24no—
212C24—24no—
213C24—42no—
214C24—42no—
215C24—44no—
216C24—44no—
217C24C2×C22⋊C412yes—
218C24#11714yes—
219C24C4×D414yes—
220C24C23×C411yescharacteristic
221#41—432no—
222#41—464no—
223#41—432no—
224C2×Q8—24no—
225C2×Q8—24no—
226C2×Q8—28no—
227C2×Q8—28no—
228C2×Q8—41no—
229C2×Q8—42no—
230C2×Q8—41no—
231C2×Q8—42no—
232C2×Q8—42no—
233C2×Q8—44no—
234C2×Q8—44no—
235C2×D4—216no—
236C2×D4—24no—
237C2×D4—24no—
238C2×D4—28no—
239C2×D4—28no—
240C2×D4—416no—
241C2×D4—416no—
242C2×D4—48no—
243C2×D4—48no—
244C2×D4—41no—
245C2×D4—42no—
246C2×D4—41no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—44no—
250C2×D4—44no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—28no—
259C22×C4—28no—
260C22×C4—28no—
261C22×C4—28no—
262C22×C4—216no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—416no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—416no—
279C22×C4—48no—
280C22×C4—48no—
281C22×C4—48no—
282C22×C4—44no—
283C22×C4—44no—
284C22×C4—88no—
285C22×C4—816no—
286C22×C4C2×C22⋊C414yes—
287C22×C4C2×C22⋊C414yes—
288C22×C4C2×C22⋊C412yes—
289C22×C4#11714yes—
290C22×C4C4×D414yes—
291C22×C4#12014yes—
292C22×C4#12118yes—
293C22×C4#12314yes—
294C22×C4C22×D412yes—
295C22×C4C22×D411yescharacteristic
296C22×C4C22×D411yescharacteristic
297C22×C4C2×#4112yes—
298C22×C4C2×#4112yes—
299C22×C4C2×#4112yes—
300C22×C4H(5,𝔽2)14yes—
301M16—28no—
302M16—28no—
303M16—44no—
304M16—44no—
305M16—44no—
306M16—44no—
307M16—44no—
308M16—44no—
309C2×C8—28no—
310C2×C8—44no—
311C2×C8—44no—
312C2×C8—44no—
313C2×C8—44no—
314C2×C8—44no—
315C4⋊C4—216no—
316C4⋊C4—48no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C22⋊C4—216no—
322C22⋊C4—88no—
323C22⋊C4—88no—
324C42—22no—
325C42—22no—
326C42—24no—
327C42—24no—
328C42—22no—
329C42—22no—
330C23—22no—
331C23—22no—
332C23—22no—
333C23—24no—
334C23—24no—
335C23—24no—
336C23—24no—
337C23—22no—
338C23—24no—
339C23—28no—
340C23—48no—
341C23—44no—
342C23—44no—
343C23—44no—
344C23—44no—
345C23—48no—
346C23—48no—
347C23—42no—
348C23—44no—
349C23—44no—
350C23—44no—
351C23—42no—
352C23—88no—
353C23—84no—
354C23—84no—
355C23#38614yes—
356C23C22×C22⋊C411yescharacteristic
357C23C2×#11711yescharacteristic
358C23C2×#11711yescharacteristic
359C23C2×C4×D412yes—
360C23#51812yes—
361Q8—44no—
362Q8—48no—
363Q8—44no—
364D4—432no—
365D4—44no—
366D4—48no—
367D4—44no—
368C2×C4—22no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—22no—
372C2×C4—28no—
373C2×C4—216no—
374C2×C4—28no—
375C2×C4—28no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—432no—
384C2×C4—42no—
385C2×C4—44no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—84no—
392C2×C4—84no—
393C2×C4—816no—
394C2×C4—84no—
395C2×C4—84no—
396C2×C4#38614yes—
397C2×C4C2×#12011yescharacteristic
398C2×C4C2×#12112yes—
399C2×C4C2×#12311yescharacteristic
400C2×C4#52512yes—
401C2×C4#52512yes—
402C2×C4#53512yes—
403C2×C4#53512yes—
404C2×C4#54614yes—
405C2×C4#54614yes—
406C2×C4#55114yes—
407C2×C4#55114yes—
408C8—48no—
409C8—84no—
410C22—24no—
411C22—22no—
412C22—22no—
413C22—22no—
414C22—22no—
415C22—28no—
416C22—24no—
417C22—48no—
418C22—48no—
419C22—42no—
420C22—42no—
421C22—44no—
422C22—44no—
423C22—48no—
424C22—88no—
425C22—88no—
426C22—84no—
427C22—82no—
428C22—82no—
429C22#228011yescharacteristic
430C22C2×#38611yescharacteristic
431C22#230911yescharacteristic
432C22#234212yes—
433C22#234212yes—
434C22#288812yes—
435C22#288912yes—
436C4—24no—
437C4—44no—
438C4—44no—
439C4—44no—
440C4—82no—
441C4—82no—
442C4#240412yes—
443C4#241114yes—
444C4#248212yes—
445C2—22no—
446C2—22no—
447C2—44no—
448C2—84no—
449C2#1386811yescharacteristic
450C2#2073112yes—
451C2#2099212yes—
452C2#2180111yescharacteristic
453C2#2180211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×#41C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2921C2semidirect
3. #20683C2semidirect
5. #11045C2semidirect
6. #11044C2semidirect
7. #10963C2semidirect
8. #10962C2semidirect
9. #10727C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29212
C2×#23112
#206832
C2×#17852
#110452
#110442
#109632
#109622
#107272