Finite Group Databasebeta

← #194784

Subgroups — #194784

Order 512.

1538 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#400C211yesmaximal, characteristic
2#34178C211yesmaximal, characteristic
3#20700C212yesmaximal
4C2×#1463C211yesmaximal, characteristic
5#9023C214yesmaximal, complemented
6#9022C214yesmaximal, complemented
7#8020C212yesmaximal
8C23×C4⋊C4C2211yescharacteristic
9C23×C4⋊C4C2211yescharacteristic
10C2×#407C2212yes—
11C2×#400—21no—
12C2×#400—21no—
13C2×#400—24no—
14C2×#400C418yescomplemented
15C2×#400C2212yes—
16C2×#400C2212yes—
17#2345—24no—
18#2344—24no—
19#2316C2212yes—
20#2313—21no—
21#2313C2212yes—
22#2280—21no—
23#2280—21no—
24#2280C2212yes—
25#2274—21no—
26C22×#95—21no—
27C22×#95—21no—
28C22×#95C2211yescharacteristic
29#1824C2218yes—
30#1823—24no—
31#1736—24no—
32#1463—24no—
33C2×#324C2212yes—
34C2×#324C2212yes—
35#1294C2218yes—
36C24×C4—21no—
37C24×C4—21no—
38C24×C4D411yescharacteristic
39C24×C4D411yescharacteristic
40C24×C4C2311yescharacteristic
41C22×M16—22no—
42C2×#117—22no—
43C2×#117—22no—
44C2×#117D414yes—
45C22×C4⋊C4—22no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—24no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—24no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4C2×C414yes—
54C22×C4⋊C4C2×C414yes—
55C22×C4⋊C4D411yescharacteristic
56C22×C4⋊C4D411yescharacteristic
57C22×C4⋊C4D414yes—
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61#407—24no—
62#407—24no—
63C2×C22⋊C8—22no—
64C2×C22⋊C8C2314yes—
65#400—28no—
66#400—28no—
67#400—24no—
68#400—24no—
69#400C2×C418yes—
70#400C2×C418yes—
71#389—28no—
72#387—28no—
73#387—28no—
74#381—22no—
75#381—22no—
76#381—22no—
77#380—22no—
78#380—22no—
79#380—22no—
80#380—22no—
81#380D414yes—
82#380C2314yes—
83#379—22no—
84C4×C4⋊C4—28no—
85C4×C22⋊C4—22no—
86C4×C22⋊C4—22no—
87C4×C22⋊C4—22no—
88C4×C22⋊C4—22no—
89C4×C22⋊C4—22no—
90C4×C22⋊C4—22no—
91C2×#95—28no—
92C2×#95—28no—
93C2×#95—21no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95C2×C414yes—
100C2×#95C2312yes—
101#340—216no—
102#324—24no—
103#324—24no—
104C25C2×D411yescharacteristic
105C23×C4—21no—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—22no—
109C23×C4—21no—
110C23×C4—22no—
111C23×C4—24no—
112C23×C4—24no—
113C23×C4—24no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—42no—
117C23×C4—44no—
118C23×C4—44no—
119C23×C4—44no—
120C23×C4—44no—
121C23×C4—48no—
122C23×C4—44no—
123C23×C4—41no—
124C23×C4—41no—
125C23×C4—42no—
126C23×C4C22⋊C414yes—
127C23×C4C22⋊C414yes—
128C23×C4C22×C412yes—
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D412yes—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D412yes—
136C23×C4C2×D412yes—
137C23×C4#4114yes—
138C23×C4C2411yescharacteristic
139C2×M16—48no—
140C2×M16—44no—
141C22×C8—24no—
142#117—44no—
143#117—44no—
144#117—48no—
145#117—84no—
146C2×C4⋊C4—28no—
147C2×C4⋊C4—28no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—28no—
151C2×C4⋊C4—28no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—28no—
159C2×C4⋊C4—28no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4C22⋊C414yes—
171C2×C4⋊C4C22⋊C414yes—
172C2×C4⋊C4C22×C414yes—
173C2×C4⋊C4C22×C412yes—
174C2×C4⋊C4C22×C412yes—
175C2×C4⋊C4C2×D412yes—
176C2×C4⋊C4C2×D412yes—
177C2×C4⋊C4C2×D418yes—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—24no—
181C2×C42—24no—
182C2×C42—24no—
183C2×C42—24no—
184C22⋊C8—24no—
185C22⋊C8—24no—
186C22⋊C8—44no—
187#95—24no—
188#95—24no—
189#95—24no—
190#95—24no—
191#95—28no—
192#95—28no—
193#95—28no—
194#95—28no—
195#95—24no—
196#95—24no—
197#95—22no—
198#95—24no—
199#95C22×C414yes—
200C24—22no—
201C24—22no—
202C24—24no—
203C24—24no—
204C24C2×C22⋊C412yes—
205C24#12011yescharacteristic
206C24#12011yescharacteristic
207C24#12312yes—
208C24C22×D411yescharacteristic
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—22no—
212C22×C4—28no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—24no—
225C22×C4—28no—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4—28no—
229C22×C4—44no—
230C22×C4—44no—
231C22×C4—44no—
232C22×C4—42no—
233C22×C4—44no—
234C22×C4—48no—
235C22×C4—42no—
236C22×C4—48no—
237C22×C4—44no—
238C22×C4—44no—
239C22×C4—44no—
240C22×C4—48no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—48no—
244C22×C4—48no—
245C22×C4—44no—
246C22×C4—44no—
247C22×C4—416no—
248C22×C4—48no—
249C22×C4—48no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—44no—
253C22×C4—44no—
254C22×C4—44no—
255C22×C4—84no—
256C22×C4—82no—
257C22×C4—816no—
258C22×C4—88no—
259C22×C4—84no—
260C22×C4—84no—
261C22×C4C2×C22⋊C412yes—
262C22×C4C2×C22⋊C412yes—
263C22×C4C2×C22⋊C412yes—
264C22×C4C2×C22⋊C412yes—
265C22×C4C2×C22⋊C412yes—
266C22×C4C4×D414yes—
267C22×C4C4×D414yes—
268C22×C4#12012yes—
269C22×C4#12114yes—
270C22×C4#12114yes—
271C22×C4#12312yes—
272C22×C4#13714yes—
273C22×C4C23×C411yescharacteristic
274C22×C4C22×D412yes—
275C22×C4C22×D411yescharacteristic
276C22×C4C2×#4112yes—
277M16—88no—
278C2×C8—48no—
279C2×C8—44no—
280C4⋊C4—24no—
281C4⋊C4—24no—
282C4⋊C4—24no—
283C4⋊C4—24no—
284C4⋊C4—28no—
285C4⋊C4—28no—
286C4⋊C4—28no—
287C4⋊C4—48no—
288C4⋊C4—48no—
289C4⋊C4—44no—
290C4⋊C4—48no—
291C4⋊C4—84no—
292C22⋊C4—44no—
293C22⋊C4—44no—
294C22⋊C4—48no—
295C22⋊C4—48no—
296C42—48no—
297C42—84no—
298C42—84no—
299C23—216no—
300C23—216no—
301C23—24no—
302C23—22no—
303C23—22no—
304C23—24no—
305C23—24no—
306C23—24no—
307C23—24no—
308C23—22no—
309C23—22no—
310C23—22no—
311C23—22no—
312C23—22no—
313C23—22no—
314C23#38612yes—
315C23#38612yes—
316C23#45414yes—
317C23#45814yes—
318C23C22×C22⋊C411yescharacteristic
319C23C2×#12011yescharacteristic
320C23C2×#12311yescharacteristic
321C23C2×#13712yes—
322C23#57512yes—
323C2×C4—24no—
324C2×C4—22no—
325C2×C4—28no—
326C2×C4—28no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—42no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—48no—
344C2×C4—48no—
345C2×C4—42no—
346C2×C4—48no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—48no—
355C2×C4—44no—
356C2×C4—48no—
357C2×C4—48no—
358C2×C4—48no—
359C2×C4—44no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—84no—
363C2×C4—816no—
364C2×C4—88no—
365C2×C4—84no—
366C2×C4—88no—
367C2×C4—88no—
368C2×C4—84no—
369C2×C4#38614yes—
370C2×C4#41914yes—
371C2×C4C2×C4×D412yes—
372C2×C4C2×#12112yes—
373C8—84no—
374C22—28no—
375C22—216no—
376C22—28no—
377C22—22no—
378C22—22no—
379C22—21no—
380C22—21no—
381C22—22no—
382C22—21no—
383C22—21no—
384C22—22no—
385C22—24no—
386C22—24no—
387C22—28no—
388C22—28no—
389C22#188814yes—
390C22#189014yes—
391C22C2×#38611yescharacteristic
392C22C2×#41911yescharacteristic
393C22#289611yescharacteristic
394C22#300712yes—
395C22C2×#45412yes—
396C22C2×#45812yes—
397C22#309512yes—
398C4—24no—
399C4—22no—
400C4—22no—
401C4—24no—
402C4—42no—
403C4—42no—
404C4—42no—
405C4—84no—
406C4—84no—
407C4—88no—
408C2—21no—
409C2—22no—
410C2—21no—
411C2—24no—
412C2—24no—
413C2#1817214yes—
414C2#2091111yescharacteristic
415C2C2×#188811yescharacteristic
416C2C2×#189011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #9023C2semidirect
6. #9022C2semidirect
14. C2×#400C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4002
#341782
#207002
C2×#14632
#90232
#90222
#80202