Finite Group Databasebeta

← #358057

Subgroups — #358057

Order 512.

1774 conjugacy classes of subgroups.

The subgroup lattice has 429 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#430C211yesmaximal, characteristic
2C2×#2905C211yesmaximal, characteristic
3C22×#344C212yesmaximal
4#20347C211yesmaximal, characteristic
5C2×#1733C212yesmaximal
6#11240C218yesmaximal, complemented
7C23×M16C2211yescharacteristic
8C23×M16C2212yes—
9C23×M16C2211yescharacteristic
10C22×#117C2212yes—
11C22×#117C2211yescharacteristic
12C2×#430—28no—
13C2×#430—22no—
14C2×#430—22no—
15C22×#108C2214yes—
16#2905—24no—
17#2905C418yescomplemented
18C2×#424—22no—
19C2×#424C2214yes—
20C2×#423—22no—
21C2×#344—28no—
22C2×#344C2218yes—
23#1736—24no—
24#1735C2214yes—
25#1733—24no—
26#1733C2218yes—
27C2×#339—24no—
28C24×C4C2311yescharacteristic
29C22×M16—24no—
30C22×M16—24no—
31C22×M16—21no—
32C22×M16—22no—
33C22×M16—24no—
34C22×M16—22no—
35C22×M16—22no—
36C22×M16—22no—
37C22×M16—24no—
38C22×M16—22no—
39C22×M16—22no—
40C22×M16—24no—
41C22×M16—24no—
42C22×M16—24no—
43C22×M16C2×C414yes—
44C22×M16C2×C414yes—
45C22×M16D418yes—
46C22×M16D412yes—
47C22×M16D412yes—
48C22×M16D414yes—
49C22×M16C2312yes—
50C22×M16C2314yes—
51C23×C8—21no—
52C23×C8—22no—
53C23×C8C2312yes—
54C2×#117—21no—
55C2×#117—22no—
56C2×#117—22no—
57C2×#117—21no—
58C2×#117—22no—
59C2×#117—24no—
60C2×#117—24no—
61C2×#117—22no—
62C2×#117—22no—
63C2×#117—24no—
64C2×#117C2×C414yes—
65C2×#117D414yes—
66C2×#117Q818yes—
67C2×#117Q812yes—
68C2×#117C2314yes—
69C2×#117C2312yes—
70C22×C4⋊C4—22no—
71C22×C4⋊C4Q812yes—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—22no—
74C22×C42—21no—
75C22×C42—22no—
76#430—216no—
77#430—216no—
78C2×#108—216no—
79C2×#108—28no—
80#424—216no—
81#424—28no—
82#424C2×C4116yescomplemented
83#423—216no—
84C2×C4⋊C8—28no—
85#344—216no—
86#340—28no—
87#339—216no—
88C25#4111yescharacteristic
89C23×C4—22no—
90C23×C4—42no—
91C23×C4C22×C412yes—
92C23×C4C2×D411yescharacteristic
93C23×C4C2×D411yescharacteristic
94C23×C4C2×D412yes—
95C23×C4C2×D412yes—
96C23×C4C2×D414yes—
97C23×C4C2×Q811yescharacteristic
98C23×C4C2×Q811yescharacteristic
99C23×C4C2×Q814yes—
100C23×C4#4111yescharacteristic
101C23×C4#4112yes—
102C23×C4#4114yes—
103C23×C4#4114yes—
104C23×C4C2411yescharacteristic
105C2×M16—28no—
106C2×M16—28no—
107C2×M16—24no—
108C2×M16—24no—
109C2×M16—28no—
110C2×M16—28no—
111C2×M16—24no—
112C2×M16—24no—
113C2×M16—24no—
114C2×M16—28no—
115C2×M16—22no—
116C2×M16—24no—
117C2×M16—28no—
118C2×M16—28no—
119C2×M16—28no—
120C2×M16—28no—
121C2×M16—28no—
122C2×M16—28no—
123C2×M16—24no—
124C2×M16—24no—
125C2×M16—24no—
126C2×M16—24no—
127C2×M16—24no—
128C2×M16—24no—
129C2×M16—42no—
130C2×M16—42no—
131C2×M16—42no—
132C2×M16—44no—
133C2×M16—44no—
134C2×M16—44no—
135C2×M16—44no—
136C2×M16—42no—
137C2×M16—44no—
138C2×M16—44no—
139C2×M16—44no—
140C2×M16C22⋊C418yes—
141C2×M16C22⋊C418yes—
142C2×M16C22×C414yes—
143C2×M16C2×D418yes—
144C22×C8—24no—
145C22×C8—24no—
146C22×C8—24no—
147C22×C8—28no—
148C22×C8—22no—
149C22×C8—24no—
150C22×C8—42no—
151C22×C8—42no—
152C22×C8—42no—
153C22×C8—42no—
154C22×C8—44no—
155C22×C8C22×C414yes—
156C22×C8C2×D414yes—
157C22×C8C2×D414yes—
158#117—24no—
159#117—24no—
160#117—24no—
161#117—24no—
162#117—24no—
163#117—22no—
164#117—22no—
165#117—28no—
166#117—24no—
167#117—28no—
168#117—24no—
169#117—28no—
170#117—24no—
171#117—44no—
172#117—44no—
173#117—44no—
174#117C4⋊C418yes—
175#117C22×C414yes—
176#117C2×D414yes—
177#117C2×Q818yes—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4C4⋊C418yes—
185C2×C4⋊C4C2×Q814yes—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—28no—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—42no—
193C2×C22⋊C4—44no—
194C2×C42—24no—
195C2×C42—24no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—28no—
199C2×C42—24no—
200C2×C42—44no—
201C2×C42—42no—
202#108—232no—
203C4⋊C8—232no—
204C24—82no—
205C24#11712yes—
206C24#12112yes—
207C24C22⋊Q811yescharacteristic
208C24C22⋊Q811yescharacteristic
209C24C22⋊Q814yes—
210C24#12614yes—
211C24C2×#4111yescharacteristic
212C22×C4—24no—
213C22×C4—28no—
214C22×C4—44no—
215C22×C4—48no—
216C22×C4—84no—
217C22×C4—88no—
218C22×C4C2×C22⋊C412yes—
219C22×C4C2×C22⋊C412yes—
220C22×C4C2×C4⋊C412yes—
221C22×C4C2×C4⋊C412yes—
222C22×C4#11712yes—
223C22×C4C4×D414yes—
224C22×C4C4×D414yes—
225C22×C4C4×Q814yes—
226C22×C4C4×Q814yes—
227C22×C4#12012yes—
228C22×C4#12012yes—
229C22×C4#12114yes—
230C22×C4#12114yes—
231C22×C4#12114yes—
232C22×C4#12114yes—
233C22×C4#12112yes—
234C22×C4#12112yes—
235C22×C4#12112yes—
236C22×C4C22⋊Q812yes—
237C22×C4C22⋊Q812yes—
238C22×C4C22⋊Q814yes—
239C22×C4C22⋊Q814yes—
240C22×C4C22⋊Q814yes—
241C22×C4C22⋊Q814yes—
242C22×C4C22⋊Q811yescharacteristic
243C22×C4C22⋊Q811yescharacteristic
244C22×C4C22⋊Q814yes—
245C22×C4#12312yes—
246C22×C4#12312yes—
247C22×C4#12314yes—
248C22×C4#12314yes—
249C22×C4#12412yes—
250C22×C4#12412yes—
251C22×C4#12514yes—
252C22×C4#12514yes—
253C22×C4#12514yes—
254C22×C4#12514yes—
255C22×C4#12614yes—
256C22×C4C4⋊Q814yes—
257C22×C4C4⋊Q814yes—
258C22×C4C4⋊Q812yes—
259C22×C4C4⋊Q812yes—
260C22×C4C23×C411yescharacteristic
261C22×C4C22×D411yescharacteristic
262C22×C4C22×D411yescharacteristic
263C22×C4C22×D411yescharacteristic
264C22×C4C22×D412yes—
265C22×C4C22×Q811yescharacteristic
266C22×C4C22×Q812yes—
267C22×C4C2×#4112yes—
268C22×C4C2×#4112yes—
269C22×C4C2×#4111yescharacteristic
270M16—216no—
271M16—216no—
272M16—28no—
273M16—28no—
274M16—28no—
275M16—44no—
276M16—44no—
277M16—44no—
278M16—48no—
279M16—48no—
280M16—48no—
281C2×C8—28no—
282C2×C8—24no—
283C2×C8—28no—
284C2×C8—24no—
285C2×C8—24no—
286C2×C8—28no—
287C2×C8—42no—
288C2×C8—44no—
289C2×C8—44no—
290C2×C8—42no—
291C2×C8—48no—
292C2×C8—48no—
293C2×C8—48no—
294C2×C8C2×C22⋊C418yes—
295C4⋊C4—24no—
296C4⋊C4—28no—
297C4⋊C4—28no—
298C4⋊C4—44no—
299C4⋊C4—48no—
300C4⋊C4C2×C4⋊C418yes—
301C22⋊C4—24no—
302C22⋊C4—24no—
303C22⋊C4—28no—
304C22⋊C4—28no—
305C22⋊C4—44no—
306C22⋊C4—48no—
307C42—24no—
308C42—24no—
309C42—28no—
310C42—28no—
311C42—44no—
312C42—48no—
313C23—44no—
314C23—42no—
315C23—44no—
316C23—42no—
317C23—42no—
318C23—84no—
319C23—84no—
320C23#38012yes—
321C23#38012yes—
322C23#38714yes—
323C23#38714yes—
324C23#39512yes—
325C23#39512yes—
326C23#39614yes—
327C23#39614yes—
328C23#40012yes—
329C23#40012yes—
330C23C2×#11711yescharacteristic
331C23C2×#12111yescharacteristic
332C23C2×C22⋊Q811yescharacteristic
333C23C2×C22⋊Q812yes—
334C23C2×#12612yes—
335C2×C4—28no—
336C2×C4—48no—
337C2×C4—48no—
338C2×C4—88no—
339C2×C4—88no—
340C2×C4—816no—
341C2×C4#38012yes—
342C2×C4#38012yes—
343C2×C4#38614yes—
344C2×C4#38614yes—
345C2×C4#38714yes—
346C2×C4#38714yes—
347C2×C4#38914yes—
348C2×C4#38914yes—
349C2×C4#38914yes—
350C2×C4#38914yes—
351C2×C4#39114yes—
352C2×C4#39114yes—
353C2×C4#39414yes—
354C2×C4#39414yes—
355C2×C4#39512yes—
356C2×C4#39512yes—
357C2×C4#39614yes—
358C2×C4#39614yes—
359C2×C4#39814yes—
360C2×C4#39814yes—
361C2×C4#39814yes—
362C2×C4#39814yes—
363C2×C4#39814yes—
364C2×C4#39814yes—
365C2×C4#40012yes—
366C2×C4#40012yes—
367C2×C4C22×C22⋊C411yescharacteristic
368C2×C4C22×C4⋊C411yescharacteristic
369C2×C4C2×C4×D412yes—
370C2×C4C2×C4×Q812yes—
371C2×C4C2×#12011yescharacteristic
372C2×C4C2×#12112yes—
373C2×C4C2×#12112yes—
374C2×C4C2×#12111yescharacteristic
375C2×C4C2×C22⋊Q811yescharacteristic
376C2×C4C2×C22⋊Q812yes—
377C2×C4C2×C22⋊Q812yes—
378C2×C4C2×#12312yes—
379C2×C4C2×#12311yescharacteristic
380C2×C4C2×#12411yescharacteristic
381C2×C4C2×#12512yes—
382C2×C4C2×#12512yes—
383C2×C4C2×C4⋊Q812yes—
384C2×C4C2×C4⋊Q811yescharacteristic
385C8—28no—
386C8—48no—
387C8—44no—
388C22—22no—
389C22—22no—
390C22—24no—
391C22—22no—
392C22—24no—
393C22—44no—
394C22—42no—
395C22—42no—
396C22—44no—
397C22—44no—
398C22—42no—
399C22—42no—
400C22—44no—
401C22—42no—
402C22—42no—
403C22#144114yes—
404C22#144114yes—
405C22C2×#38011yescharacteristic
406C22C2×#38712yes—
407C22C2×#39511yescharacteristic
408C22C2×#39612yes—
409C22C2×#40011yescharacteristic
410C4—48no—
411C4—88no—
412C4#144114yes—
413C4#144114yes—
414C4C2×#38611yescharacteristic
415C4C2×#38912yes—
416C4C2×#39111yescharacteristic
417C4C2×#39411yescharacteristic
418C4C2×#39812yes—
419C4C2×#39811yescharacteristic
420C2—22no—
421C2—21no—
422C2—21no—
423C2—24no—
424C2—24no—
425C2—22no—
426C2C2×#144111yescharacteristic
427C2#2067512yes—
subgroupG/H
centerC2×C4C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#380
prime radicalC25#41
LCSG ▷ C22×C4 ▷ 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
6. #11240C2semidirect
17. #2905C4semidirect
82. #424C2×C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4302
C2×#29052
C22×#3442
#203472
C2×#17332
#112402