Finite Group Databasebeta

← #360635

Subgroups — #360635

Order 512.

1280 conjugacy classes of subgroups.

The subgroup lattice has 449 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60124C211yesmaximal, characteristic
2#20685C211yesmaximal, characteristic
3#20628C212yesmaximal
4C2×#1740C211yesmaximal, characteristic
5#11345C212yesmaximal, complemented
6#11288C212yesmaximal, complemented
7#10920C214yesmaximal, complemented
8#10738C212yesmaximal
9C22×#117C2212yes—
10C22×#117C2211yescharacteristic
11#2924—24no—
12#2924—24no—
13#2921—22no—
14#2920—22no—
15#2905C2212yes—
16C2×#424C2214yes—
17C2×#407C2211yescharacteristic
18C2×#407C2212yes—
19C22×C22⋊C8C2211yescharacteristic
20#2873C2212yes—
21#1818—22no—
22#1817—22no—
23#1798—24no—
24#1797—24no—
25C2×#344C2214yes—
26C2×#344C2214yes—
27#1740C414yes—
28#1740C2211yescharacteristic
29#1740C2211yescharacteristic
30#1740C2212yes—
31C2×#342C414yes—
32#1729C2212yes—
33#1729C2214yes—
34#1729C2212yes—
35C24×C4C2311yescharacteristic
36C22×M16C2312yes—
37C22×M16C2314yes—
38C23×C8C2312yes—
39C2×#117—22no—
40C2×#117—22no—
41C2×#117—22no—
42C2×#117—22no—
43C2×#117—22no—
44C2×#117—22no—
45C2×#117—28no—
46C2×#117—44no—
47C2×#117C2×C414yes—
48C2×#117C2×C412yes—
49C2×#117D412yes—
50C2×#117Q811yescharacteristic
51C2×#117Q811yescharacteristic
52C2×#117C2312yes—
53C2×#117C2312yes—
54C2×#117C2311yescharacteristic
55C2×#117C2311yescharacteristic
56C22×C4⋊C4—22no—
57C22×C4⋊C4Q811yescharacteristic
58C22×C4⋊C4Q811yescharacteristic
59C22×C22⋊C4C2×C412yes—
60C22×C22⋊C4C2×C414yes—
61C22×C42—22no—
62C22×C42D412yes—
63#433—216no—
64#432—24no—
65#432—24no—
66#431—24no—
67#431—24no—
68#424—28no—
69#424—28no—
70#424—44no—
71#423—24no—
72C2×C4⋊C8—28no—
73#407—22no—
74#407—22no—
75#407—22no—
76#407—22no—
77#407—22no—
78#407—22no—
79C2×C22⋊C8—22no—
80C2×C22⋊C8—22no—
81C2×C22⋊C8—21no—
82C2×C22⋊C8—21no—
83C2×C22⋊C8—21no—
84C2×C22⋊C8—21no—
85C2×C22⋊C8D414yes—
86C2×C22⋊C8D414yes—
87#405—28no—
88C4×M16—24no—
89#344—28no—
90#344—44no—
91#342C2×C414yes—
92#342C2×C418yes—
93#342C2×C414yes—
94#336—28no—
95#336—24no—
96#336—24no—
97C25C22×C411yescharacteristic
98C23×C4—24no—
99C23×C4—22no—
100C23×C4—22no—
101C23×C4—24no—
102C23×C4C4⋊C412yes—
103C23×C4C4⋊C412yes—
104C23×C4C22×C411yescharacteristic
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D412yes—
108C23×C4C2×D412yes—
109C23×C4C2×Q811yescharacteristic
110C23×C4C2×Q811yescharacteristic
111C23×C4#4112yes—
112C23×C4#4112yes—
113C23×C4C2411yescharacteristic
114C2×M16—24no—
115C2×M16—24no—
116C2×M16—24no—
117C2×M16—24no—
118C2×M16—24no—
119C2×M16—24no—
120C2×M16—24no—
121C2×M16—24no—
122C2×M16—22no—
123C2×M16—22no—
124C22×C8—22no—
125C22×C8—24no—
126C22×C8—22no—
127C22×C8—24no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8C2×D418yes—
131#117—24no—
132#117—24no—
133#117—24no—
134#117—22no—
135#117—22no—
136#117—22no—
137#117—22no—
138#117—24no—
139#117—22no—
140#117—22no—
141#117—22no—
142#117—22no—
143#117—24no—
144#117—24no—
145#117—24no—
146#117—28no—
147#117—48no—
148#117—44no—
149#117—48no—
150#117#4118yes—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—48no—
159C2×C4⋊C4C4⋊C412yes—
160C2×C4⋊C4C4⋊C412yes—
161C2×C4⋊C4C2×Q811yescharacteristic
162C2×C4⋊C4C2×Q811yescharacteristic
163C2×C4⋊C4C2×Q811yescharacteristic
164C2×C4⋊C4C2×Q811yescharacteristic
165C2×C22⋊C4—24no—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—28no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—44no—
170C2×C22⋊C4C4⋊C412yes—
171C2×C22⋊C4C4⋊C412yes—
172C2×C22⋊C4C22×C414yes—
173C2×C22⋊C4C22×C412yes—
174C2×C22⋊C4C22×C412yes—
175C2×C22⋊C4C22×C414yes—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—24no—
182C2×C42—22no—
183C2×C42—48no—
184C2×C42C4⋊C412yes—
185C2×C42C4⋊C412yes—
186C2×C42C2×D412yes—
187C2×C42C2×D412yes—
188C4⋊C8—28no—
189C4⋊C8—28no—
190C4⋊C8—416no—
191C22⋊C8—24no—
192C22⋊C8—24no—
193C22⋊C8—24no—
194C22⋊C8—24no—
195C22⋊C8—41no—
196C22⋊C8—41no—
197C22⋊C8—42no—
198C22⋊C8—41no—
199C22⋊C8—41no—
200C22⋊C8—42no—
201#97—28no—
202C4×C8—28no—
203C24—22no—
204C24—21no—
205C24—21no—
206C24—42no—
207C24—81no—
208C24C2×C4⋊C411yescharacteristic
209C24C2×C4⋊C411yescharacteristic
210C24C4×D412yes—
211C24C4×D412yes—
212C24C23×C411yescharacteristic
213C22×C4—22no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—24no—
238C22×C4—48no—
239C22×C4—44no—
240C22×C4—48no—
241C22×C4—82no—
242C22×C4—84no—
243C22×C4C2×C4⋊C411yescharacteristic
244C22×C4C2×C4⋊C411yescharacteristic
245C22×C4C2×C4⋊C412yes—
246C22×C4C2×C4⋊C412yes—
247C22×C4C2×C4⋊C412yes—
248C22×C4C2×C4⋊C412yes—
249C22×C4C4×D412yes—
250C22×C4C4×D412yes—
251C22×C4#12011yescharacteristic
252C22×C4#12011yescharacteristic
253C22×C4#12012yes—
254C22×C4C22⋊Q812yes—
255C22×C4C22⋊Q812yes—
256C22×C4C22⋊Q812yes—
257C22×C4C22⋊Q812yes—
258C22×C4#12311yescharacteristic
259C22×C4#12311yescharacteristic
260C22×C4#12312yes—
261C22×C4C22×D412yes—
262C22×C4C22×D411yescharacteristic
263C22×C4C22×Q811yescharacteristic
264C22×C4C2×#4112yes—
265C22×C4C2×#4114yes—
266C22×C4H(5,𝔽2)12yes—
267C22×C42−1+412yes—
268M16—44no—
269M16—44no—
270M16—44no—
271M16—44no—
272M16—44no—
273M16—44no—
274C2×C8—28no—
275C2×C8—28no—
276C2×C8—28no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—44no—
284C2×C8—44no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—28no—
290C4⋊C4—24no—
291C4⋊C4—24no—
292C4⋊C4—48no—
293C4⋊C4—44no—
294C22⋊C4—24no—
295C22⋊C4—24no—
296C22⋊C4—24no—
297C22⋊C4—24no—
298C22⋊C4—24no—
299C22⋊C4—24no—
300C22⋊C4—24no—
301C22⋊C4—48no—
302C22⋊C4#11718yes—
303C42—22no—
304C42—22no—
305C42—22no—
306C42—22no—
307C42—28no—
308C42—22no—
309C42—22no—
310C42—22no—
311C42—22no—
312C42—48no—
313C42—44no—
314C23—21no—
315C23—22no—
316C23—22no—
317C23—22no—
318C23—24no—
319C23—21no—
320C23—21no—
321C23—22no—
322C23—22no—
323C23—21no—
324C23—41no—
325C23—42no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—42no—
330C23—41no—
331C23—42no—
332C23—84no—
333C23—82no—
334C23—82no—
335C23#38512yes—
336C23#38511yescharacteristic
337C23#38511yescharacteristic
338C23C22×C4⋊C411yescharacteristic
339C23C2×#11712yes—
340C23C2×C4×D412yes—
341C23#51811yescharacteristic
342C23#51911yescharacteristic
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—24no—
353C2×C4—22no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—22no—
359C2×C4—24no—
360C2×C4—22no—
361C2×C4—28no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—816no—
380C2×C4#38511yescharacteristic
381C2×C4#38511yescharacteristic
382C2×C4#38512yes—
383C2×C4C2×#12011yescharacteristic
384C2×C4C2×C22⋊Q812yes—
385C2×C4C2×#12311yescharacteristic
386C2×C4#53212yes—
387C2×C4#53312yes—
388C2×C4#53712yes—
389C2×C4#54412yes—
390C2×C4#54614yes—
391C2×C4#54714yes—
392C2×C4#55114yes—
393C2×C4#55214yes—
394C8—48no—
395C8—82no—
396C8—82no—
397C22—21no—
398C22—21no—
399C22—22no—
400C22—21no—
401C22—22no—
402C22—21no—
403C22—21no—
404C22—22no—
405C22—21no—
406C22—22no—
407C22—21no—
408C22—22no—
409C22—42no—
410C22—41no—
411C22—41no—
412C22—44no—
413C22—42no—
414C22—44no—
415C22—42no—
416C22—82no—
417C22—81no—
418C22—82no—
419C22—81no—
420C22#186812yes—
421C22#228511yescharacteristic
422C22C2×#38511yescharacteristic
423C22#231611yescharacteristic
424C22#234212yes—
425C22#234412yes—
426C4—21no—
427C4—22no—
428C4—21no—
429C4—48no—
430C4—42no—
431C4—42no—
432C4—42no—
433C4—42no—
434C4—84no—
435C4#243012yes—
436C4#243814yes—
437C4#249812yes—
438C2—22no—
439C2—22no—
440C2—21no—
441C2—22no—
442C2—21no—
443C2—41no—
444C2—42no—
445C2#1387811yescharacteristic
446C2C2×#186811yescharacteristic
447C2#2089211yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#385
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #11345C2semidirect
6. #11288C2semidirect
7. #10920C2semidirect

Maximal subgroups

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

subgroupindex
#601242
#206852
#206282
C2×#17402
#113452
#112882
#109202
#107382