Finite Group Databasebeta

← #358039

Subgroups — #358039

Order 512.

2074 conjugacy classes of subgroups.

The subgroup lattice has 417 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#427C211yesmaximal, characteristic
2C22×#424C211yesmaximal, characteristic
3C2×#1736C212yesmaximal
4C22×#340C211yesmaximal, characteristic
5#13795C212yesmaximal
6#11232C214yesmaximal, complemented
7#11231C214yesmaximal, complemented
8C23×M16C2211yescharacteristic
9C24×C8C2211yescharacteristic
10C22×#117C2212yes—
11C22×#117C2211yescharacteristic
12C23×C4⋊C4C2212yes—
13C2×#427—24no—
14C2×#427—28no—
15C22×#107C2212yes—
16C22×#106C2212yes—
17C2×#424—24no—
18C2×#424—24no—
19C2×#424C418yescomplemented
20C22×C4⋊C8C2212yes—
21C22×C4⋊C8C2212yes—
22#1736—28no—
23C2×#340—24no—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2218yes—
27C4⋊#95—22no—
28C4⋊#95—22no—
29C4⋊#95C2218yes—
30C22⋊#95—24no—
31C24×C4—22no—
32C24×C4C2311yescharacteristic
33C22×M16—22no—
34C22×M16—22no—
35C22×M16—24no—
36C22×M16—24no—
37C22×M16C2×C414yes—
38C23×C8—24no—
39C23×C8—24no—
40C23×C8C2×C414yes—
41C23×C8D414yes—
42C23×C8D414yes—
43C23×C8C2312yes—
44C23×C8C2311yescharacteristic
45C23×C8C2311yescharacteristic
46C2×#117—22no—
47C2×#117—22no—
48C2×#117—22no—
49C2×#117—22no—
50C2×#117—24no—
51C2×#117—24no—
52C2×#117—24no—
53C2×#117—24no—
54C2×#117—24no—
55C2×#117C2×C414yes—
56C2×#117D418yes—
57C2×#117Q814yes—
58C22×C4⋊C4—24no—
59C22×C4⋊C4—24no—
60C22×C4⋊C4—24no—
61C22×C4⋊C4—24no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—24no—
64C22×C4⋊C4D411yescharacteristic
65C22×C4⋊C4D411yescharacteristic
66C22×C4⋊C4Q818yes—
67C22×C4⋊C4C2314yes—
68C22×C4⋊C4C2314yes—
69C22×C22⋊C4—22no—
70C22×C22⋊C4D412yes—
71C22×C42—22no—
72C22×C42C2312yes—
73#427—232no—
74C2×#107—24no—
75C2×#107—28no—
76C2×#106—24no—
77C2×#106—28no—
78#424—232no—
79C2×C4⋊C8—24no—
80C2×C4⋊C8—24no—
81C2×C4⋊C8—24no—
82C2×C4⋊C8—24no—
83C2×C4⋊C8C2×C418yescomplemented
84C2×C4⋊C8C2×C418yescomplemented
85C2×#95—28no—
86C2×#95—28no—
87#340—24no—
88#340—24no—
89#340—216no—
90C25C2×D411yescharacteristic
91C23×C4—24no—
92C23×C4—24no—
93C23×C4—28no—
94C23×C4—44no—
95C23×C4—44no—
96C23×C4—84no—
97C23×C4—84no—
98C23×C4C22×C412yes—
99C23×C4C2×D412yes—
100C23×C4C2×D412yes—
101C23×C4C2×D412yes—
102C23×C4C2×D412yes—
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D414yes—
105C23×C4C2×Q812yes—
106C23×C4C2×Q814yes—
107C23×C4#4112yes—
108C23×C4#4112yes—
109C23×C4#4114yes—
110C23×C4#4114yes—
111C23×C4C2411yescharacteristic
112C2×M16—28no—
113C2×M16—216no—
114C2×M16—44no—
115C2×M16—48no—
116C2×M16—44no—
117C22×C8—216no—
118C22×C8—28no—
119C22×C8—216no—
120C22×C8—28no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—24no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—24no—
129C22×C8C22⋊C4116yes—
130C22×C8C22×C414yes—
131C22×C8C22×C412yes—
132C22×C8C22×C412yes—
133C22×C8C2×D412yes—
134C22×C8C2×D412yes—
135C22×C8C2×D412yes—
136C22×C8C2×D412yes—
137#117—28no—
138#117—28no—
139#117—28no—
140#117—28no—
141#117—216no—
142#117—216no—
143#117—48no—
144#117—48no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—216no—
150C2×C4⋊C4—216no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—28no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—48no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—48no—
166C2×C4⋊C4C4⋊C414yes—
167C2×C4⋊C4C4⋊C414yes—
168C2×C4⋊C4C2×D418yes—
169C2×C4⋊C4C2×Q818yes—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—28no—
173C2×C22⋊C4—44no—
174C2×C22⋊C4—44no—
175C2×C22⋊C4C4⋊C418yes—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—28no—
181C2×C42—44no—
182C2×C42—44no—
183C2×C42C22×C414yes—
184C2×C42C2×Q814yes—
185#107—216no—
186#106—216no—
187C4⋊C8—216no—
188C4⋊C8—216no—
189#95—88no—
190#95—88no—
191C24—22no—
192C24—24no—
193C24—22no—
194C24C2×C4⋊C412yes—
195C24#12112yes—
196C24#12112yes—
197C24#12114yes—
198C24C22⋊Q814yes—
199C24C22×D411yescharacteristic
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—24no—
205C22×C4—28no—
206C22×C4—24no—
207C22×C4—28no—
208C22×C4—28no—
209C22×C4—44no—
210C22×C4—44no—
211C22×C4—44no—
212C22×C4—44no—
213C22×C4—44no—
214C22×C4—48no—
215C22×C4—44no—
216C22×C4—48no—
217C22×C4—44no—
218C22×C4—44no—
219C22×C4—416no—
220C22×C4—816no—
221C22×C4—816no—
222C22×C4—88no—
223C22×C4—88no—
224C22×C4C2×C22⋊C414yes—
225C22×C4C2×C4⋊C414yes—
226C22×C4C2×C4⋊C414yes—
227C22×C4C2×C4⋊C412yes—
228C22×C4#11714yes—
229C22×C4C4×D418yes—
230C22×C4C4×Q818yes—
231C22×C4#12014yes—
232C22×C4#12118yes—
233C22×C4#12112yes—
234C22×C4#12112yes—
235C22×C4#12114yes—
236C22×C4C22⋊Q814yes—
237C22×C4C22⋊Q814yes—
238C22×C4C22⋊Q818yes—
239C22×C4C22⋊Q818yes—
240C22×C4C22⋊Q814yes—
241C22×C4#12314yes—
242C22×C4#12318yes—
243C22×C4#12414yes—
244C22×C4#12518yes—
245C22×C4#12518yes—
246C22×C4#12618yes—
247C22×C4C4⋊Q818yes—
248C22×C4C4⋊Q814yes—
249C22×C4C23×C411yescharacteristic
250C22×C4C22×D411yescharacteristic
251C22×C4C22×D411yescharacteristic
252C22×C4C22×D412yes—
253C22×C4C22×Q812yes—
254C22×C4C22×Q811yescharacteristic
255C22×C4C2×#4111yescharacteristic
256C22×C4C2×#4111yescharacteristic
257C22×C4C2×#4112yes—
258C22×C4C2×#4112yes—
259M16—416no—
260C2×C8—24no—
261C2×C8—24no—
262C2×C8—22no—
263C2×C8—22no—
264C2×C8—24no—
265C2×C8—24no—
266C2×C8—22no—
267C2×C8—22no—
268C2×C8—232no—
269C2×C8—24no—
270C2×C8—28no—
271C2×C8—42no—
272C2×C8—44no—
273C2×C8—42no—
274C2×C8C2×C22⋊C414yes—
275C2×C8C2×C22⋊C414yes—
276C4⋊C4—28no—
277C4⋊C4—28no—
278C4⋊C4—28no—
279C4⋊C4—28no—
280C4⋊C4—28no—
281C4⋊C4—28no—
282C4⋊C4—44no—
283C4⋊C4—44no—
284C4⋊C4—44no—
285C4⋊C4—44no—
286C4⋊C4—416no—
287C22⋊C4—216no—
288C22⋊C4—416no—
289C42—24no—
290C42—24no—
291C42—24no—
292C42—24no—
293C42—416no—
294C23—28no—
295C23—28no—
296C23—24no—
297C23—24no—
298C23—28no—
299C23—28no—
300C23—24no—
301C23—22no—
302C23—22no—
303C23—24no—
304C23—24no—
305C23#38014yes—
306C23#38918yes—
307C23#39414yes—
308C23#39618yes—
309C23#39814yes—
310C23C22×C4⋊C411yescharacteristic
311C23C2×#12112yes—
312C23C2×#12111yescharacteristic
313C23C2×#12111yescharacteristic
314C23C2×C22⋊Q812yes—
315C23#57612yes—
316C23#57714yes—
317C23#57812yes—
318C2×C4—28no—
319C2×C4—28no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—44no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—416no—
328C2×C4—48no—
329C2×C4—48no—
330C2×C4—48no—
331C2×C4—48no—
332C2×C4—832no—
333C2×C4—84no—
334C2×C4—88no—
335C2×C4—84no—
336C2×C4—84no—
337C2×C4—84no—
338C2×C4—84no—
339C2×C4—84no—
340C2×C4#38014yes—
341C2×C4#38618yes—
342C2×C4#387116yes—
343C2×C4#38918yes—
344C2×C4#39118yes—
345C2×C4#39414yes—
346C2×C4#39518yes—
347C2×C4#39618yes—
348C2×C4#398116yes—
349C2×C4#39814yes—
350C2×C4#40018yes—
351C2×C4C22×C22⋊C411yescharacteristic
352C2×C4C2×#11711yescharacteristic
353C2×C4C2×C4×D412yes—
354C2×C4C2×C4×Q812yes—
355C2×C4C2×#12011yescharacteristic
356C2×C4C2×#12112yes—
357C2×C4C2×C22⋊Q811yescharacteristic
358C2×C4C2×C22⋊Q812yes—
359C2×C4C2×C22⋊Q811yescharacteristic
360C2×C4C2×C22⋊Q812yes—
361C2×C4C2×#12312yes—
362C2×C4C2×#12311yescharacteristic
363C2×C4C2×#12411yescharacteristic
364C2×C4C2×#12512yes—
365C2×C4C2×#12512yes—
366C2×C4C2×#12612yes—
367C2×C4C2×C4⋊Q812yes—
368C2×C4C2×C4⋊Q811yescharacteristic
369C8—24no—
370C8—24no—
371C8—44no—
372C22—28no—
373C22—28no—
374C22—28no—
375C22—24no—
376C22—24no—
377C22—216no—
378C22—28no—
379C22#144118yes—
380C22C2×#38011yescharacteristic
381C22C2×#38912yes—
382C22C2×#39411yescharacteristic
383C22C2×#39612yes—
384C22C2×#39811yescharacteristic
385C22#291212yes—
386C22#291312yes—
387C22#304514yes—
388C22#304614yes—
389C22#305712yes—
390C22#305811yescharacteristic
391C22#305911yescharacteristic
392C22#306012yes—
393C22#306112yes—
394C22#308114yes—
395C22#308214yes—
396C4—44no—
397C4—44no—
398C4—88no—
399C4—88no—
400C4#144118yes—
401C4C2×#38611yescharacteristic
402C4C2×#38712yes—
403C4C2×#39111yescharacteristic
404C4C2×#39511yescharacteristic
405C4C2×#39812yes—
406C4C2×#40011yescharacteristic
407C2—28no—
408C2C2×#144111yescharacteristic
409C2#2066211yescharacteristic
410C2#2066311yescharacteristic
411C2#2117514yes—
412C2#2354312yes—
413C2#2361212yes—
414C2#2361312yes—
415C2#2366912yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #11232C2semidirect
7. #11231C2semidirect
19. C2×#424C4semidirect
83. C2×C4⋊C8C2×C4semidirect
84. C2×C4⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4272
C22×#4242
C2×#17362
C22×#3402
#137952
#112322
#112312