Finite Group Databasebeta

← #365682

Subgroups — #365682

Order 512.

1090 conjugacy classes of subgroups.

The subgroup lattice has 391 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34217C211yesmaximal, characteristic
2C2×#1998C211yesmaximal, characteristic
3#21171C211yesmaximal, characteristic
4#20764C211yesmaximal, characteristic
5#14283C211yesmaximal, characteristic
6#11296C214yesmaximal
7#10899C214yesmaximal
8#10840C211yesmaximal, characteristic
9#10701C211yesmaximal, characteristic
10C22×#125C2211yescharacteristic
11C2×#428C2212yes—
12C2×#423C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14#2370—22no—
15#2370—22no—
16#2345—24no—
17#2330—22no—
18#2316C2212yes—
19C4×#125—21no—
20C4×#125—21no—
21#2285C2211yescharacteristic
22C2×#384C2211yescharacteristic
23#2274C2212yes—
24C4×#117C2211yescharacteristic
25C2×#376C2211yescharacteristic
26#1998C418yescomplemented
27#1948—22no—
28#1947—22no—
29#1836—21no—
30#1836—21no—
31#1834—22no—
32C2×#343C2214yes—
33#1736C2214yes—
34#1736C2212yes—
35C2×#339C2212yes—
36C2×#339C2214yes—
37C2×#339C2212yes—
38#1441—24no—
39C4⋊#95C2214yes—
40C22×M16C2312yes—
41C22×M16C2312yes—
42C2×#125—21no—
43C2×#125—21no—
44C2×#125—22no—
45C2×#125—22no—
46C2×#125—22no—
47C2×#125C2×C414yes—
48C2×#125Q814yes—
49C2×#117C2311yescharacteristic
50C2×#117C2312yes—
51C2×#117C2311yescharacteristic
52C22×C4⋊C4—22no—
53C22×C4⋊C4D414yes—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2312yes—
56C22×C42C2312yes—
57C22×C42C2311yescharacteristic
58#428—28no—
59C2×#107—22no—
60C2×#106—22no—
61#423—22no—
62#423C2×C414yes—
63C2×C4⋊C8—22no—
64C4×M16—22no—
65C4×M16C2×C414yes—
66C2×#97—21no—
67C2×C4×C8—21no—
68#389—24no—
69#389—24no—
70#389—22no—
71#389—22no—
72#389—22no—
73#389—22no—
74#387—24no—
75#387—24no—
76#387—24no—
77#387—24no—
78#384—21no—
79#384—21no—
80#384—22no—
81#384D412yes—
82#382D411yescharacteristic
83#382D411yescharacteristic
84#380—24no—
85#380—22no—
86C4×C4⋊C4—24no—
87C4×C4⋊C4—24no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4—22no—
91C4×C4⋊C4—22no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4D412yes—
94C4×C4⋊C4D412yes—
95C4×C22⋊C4—22no—
96C4×C22⋊C4—24no—
97#376—21no—
98#376—22no—
99#376—22no—
100C2×#95—22no—
101C2×#95—28no—
102C43—21no—
103#343C2×C4116yes—
104#340—24no—
105#340—24no—
106#340—24no—
107#339—24no—
108#339—24no—
109#339—28no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4C2×D414yes—
113C23×C4C2×D412yes—
114C23×C4C2411yescharacteristic
115C2×M16—22no—
116C2×M16—22no—
117C2×M16—24no—
118C2×M16—22no—
119C2×M16—22no—
120C2×M16—24no—
121C2×M16—22no—
122C2×M16—22no—
123C2×M16C22×C414yes—
124C2×M16C22×C414yes—
125C22×C8—22no—
126C22×C8—22no—
127#125—24no—
128#125—24no—
129#125—28no—
130#125—24no—
131#125—24no—
132#125—28no—
133#117—22no—
134#117—24no—
135#117—22no—
136#117—24no—
137#117—22no—
138#117#4114yes—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—28no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—84no—
157C2×C4⋊C4C4⋊C4116yes—
158C2×C4⋊C4C22×C414yes—
159C2×C4⋊C4C2×D412yes—
160C2×C4⋊C4C2×D412yes—
161C2×C4⋊C4C2×D412yes—
162C2×C4⋊C4C2×D411yescharacteristic
163C2×C4⋊C4C2×D411yescharacteristic
164C2×C4⋊C4C2×Q814yes—
165C2×C22⋊C4—21no—
166C2×C22⋊C4—21no—
167C2×C22⋊C4—22no—
168C2×C42—21no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—21no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—24no—
177C2×C42—24no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42C22×C412yes—
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D412yes—
184C2×C42C2×Q812yes—
185C2×C42#4114yes—
186C2×C42#4112yes—
187C2×C42#4112yes—
188#107—44no—
189#106—44no—
190C4⋊C8—28no—
191#97—24no—
192C4×C8—24no—
193#95—24no—
194#95—24no—
195#95—24no—
196#95—24no—
197#95—22no—
198#95—22no—
199#95—22no—
200#95—22no—
201#95—44no—
202#95—88no—
203C24C22×D411yescharacteristic
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—24no—
207C22×C4—24no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—24no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—44no—
215C22×C4—44no—
216C22×C4—44no—
217C22×C4—44no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—48no—
221C22×C4—42no—
222C22×C4—44no—
223C22×C4C2×C4⋊C418yes—
224C22×C4C2×C4⋊C414yes—
225C22×C4#12012yes—
226C22×C4#12012yes—
227C22×C4#12314yes—
228C22×C4C23×C411yescharacteristic
229C22×C4C22×D411yescharacteristic
230C22×C4C22×D411yescharacteristic
231C22×C4C22×Q811yescharacteristic
232C22×C4C2×#4111yescharacteristic
233C22×C4C2×#4111yescharacteristic
234C22×C4C2×#4112yes—
235C22×C4C2×#4112yes—
236C22×C42−1+412yes—
237C22×C42−1+412yes—
238M16—24no—
239M16—28no—
240M16—24no—
241M16—44no—
242M16—42no—
243M16—42no—
244C2×C8—24no—
245C2×C8—24no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—42no—
250C4⋊C4—24no—
251C4⋊C4—24no—
252C4⋊C4—24no—
253C4⋊C4—24no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—216no—
257C4⋊C4—216no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—44no—
265C4⋊C4—82no—
266C4⋊C4—82no—
267C4⋊C4—84no—
268C4⋊C4—84no—
269C4⋊C4—84no—
270C4⋊C4—84no—
271C4⋊C4#12412yes—
272C4⋊C4#12412yes—
273C22⋊C4—24no—
274C22⋊C4—44no—
275C22⋊C4—42no—
276C42—24no—
277C42—24no—
278C42—24no—
279C42—42no—
280C42—42no—
281C42—42no—
282C42—42no—
283C42—42no—
284C42—82no—
285C42—82no—
286C42—84no—
287C42—84no—
288C42C4×D412yes—
289C42C4×D412yes—
290C42C4×D414yes—
291C42C22⋊Q814yes—
292C42C22⋊Q812yes—
293C42C22⋊Q812yes—
294C23—22no—
295C23—22no—
296C23C22×C4⋊C411yescharacteristic
297C23C2×#12011yescharacteristic
298C23C2×#12311yescharacteristic
299C23#54812yes—
300C23#54812yes—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—28no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—42no—
311C2×C4—44no—
312C2×C4—42no—
313C2×C4—44no—
314C2×C4—48no—
315C2×C4—48no—
316C2×C4—44no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—84no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—84no—
330C2×C4—88no—
331C2×C4—82no—
332C2×C4—84no—
333C2×C4#38514yes—
334C2×C4#38514yes—
335C2×C4C2×C4×D411yescharacteristic
336C2×C4C2×C4×D411yescharacteristic
337C2×C4C4×#4112yes—
338C2×C4#51912yes—
339C2×C4C2×C22⋊Q811yescharacteristic
340C2×C4C2×C22⋊Q811yescharacteristic
341C2×C4C2×#12411yescharacteristic
342C2×C4#52912yes—
343C2×C4#53211yescharacteristic
344C2×C4#53611yescharacteristic
345C2×C4#53611yescharacteristic
346C2×C4#54112yes—
347C2×C4#54712yes—
348C2×C4#54812yes—
349C2×C4#55212yes—
350C2×C4#55612yes—
351C2×C4#55712yes—
352C2×C4#55712yes—
353C8—44no—
354C8—44no—
355C22—24no—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—22no—
360C22C2×#38511yescharacteristic
361C22#235112yes—
362C22#242112yes—
363C22#243211yescharacteristic
364C22#243611yescharacteristic
365C22#302112yes—
366C22#302212yes—
367C4—24no—
368C4—24no—
369C4—42no—
370C4—42no—
371C4—48no—
372C4—84no—
373C4—82no—
374C4—82no—
375C4—82no—
376C4—162no—
377C4C4×#12411yescharacteristic
378C4#232711yescharacteristic
379C4#236212yes—
380C4#245312yes—
381C4#256511yescharacteristic
382C4#256811yescharacteristic
383C2—22no—
384C2—22no—
385C2#1397811yescharacteristic
386C2#2085812yes—
387C2#2567012yes—
388C2#2592711yescharacteristic
389C2#2592811yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
26. #1998C4semidirect

Maximal subgroups

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

subgroupindex
#342172
C2×#19982
#211712
#207642
#142832
#112962
#108992
#108402
#107012