Finite Group Databasebeta

← #390863

Subgroups — #390863

Order 512.

980 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
1C2×#2087C211yesmaximal, characteristic
2#14045C212yesmaximal
3#11327C212yesmaximal
4#11231C211yesmaximal, characteristic
5#11219C212yesmaximal
6#10884C212yesmaximal
7#10880C212yesmaximal
8#10834C211yesmaximal, characteristic
9#10806C212yesmaximal
10C22×#107C2211yescharacteristic
11C22×#106C2211yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13C22×C4×C8C2211yescharacteristic
14C2×#389C2212yes—
15C2×#389C2212yes—
16C2×#389C2212yes—
17C2×C4×C4⋊C4C2212yes—
18C2×C4×C4⋊C4C2211yescharacteristic
19#2087C418yescomplemented
20C2×#340C2212yes—
21C2×#340C2211yescharacteristic
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2211yescharacteristic
27C2×#340C2212yes—
28C2×#336C2212yes—
29C2×#336C2212yes—
30#1444—22no—
31#1441—22no—
32#1436—22no—
33C4⋊#95—22no—
34C4⋊#95C2212yes—
35C4⋊#95C2212yes—
36C23×C8C2311yescharacteristic
37C23×C8C2311yescharacteristic
38C23×C8C2312yes—
39C22×C4⋊C4—22no—
40C22×C4⋊C4—22no—
41C22×C4⋊C4—22no—
42C22×C4⋊C4D412yes—
43C22×C4⋊C4C2312yes—
44C22×C4⋊C4C2312yes—
45C22×C4⋊C4C2312yes—
46C22×C4⋊C4C2312yes—
47C22×C42—22no—
48C22×C42C2312yes—
49C22×C42C2311yescharacteristic
50C2×#107—41no—
51C2×#107—41no—
52C2×#107C2×C414yes—
53C2×#106—42no—
54C2×#106—41no—
55C2×C4⋊C8—41no—
56C2×C4⋊C8—41no—
57C2×C4⋊C8C2×C414yes—
58C2×C4×C8—22no—
59C2×C4×C8—41no—
60C2×C4×C8—41no—
61#389—24no—
62#389—24no—
63#389—42no—
64#389—42no—
65#389—42no—
66#389C2×C418yes—
67C4×C4⋊C4—24no—
68C4×C4⋊C4—41no—
69C4×C4⋊C4—41no—
70C4×C4⋊C4—42no—
71C2×#95—22no—
72C2×#95—22no—
73C2×#95—22no—
74C2×#95—22no—
75C2×#95—22no—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95—22no—
79C2×#95D411yescharacteristic
80C2×#95D411yescharacteristic
81#340—24no—
82#340—41no—
83#340—42no—
84#340—42no—
85#340—42no—
86#340C2×C418yes—
87#340C2×C414yes—
88#340Q818yes—
89#336—42no—
90#336—42no—
91C23×C4—22no—
92C23×C4—22no—
93C23×C4—22no—
94C23×C4—22no—
95C23×C4C2×D412yes—
96C23×C4C2×D411yescharacteristic
97C23×C4C2×D412yes—
98C23×C4C2×D411yescharacteristic
99C23×C4C2411yescharacteristic
100C22×C8—22no—
101C22×C8—22no—
102C22×C8—22no—
103C22×C8—22no—
104C22×C8—22no—
105C22×C8—22no—
106C22×C8—22no—
107C22×C8—21no—
108C22×C8—21no—
109C22×C8—21no—
110C22×C8—21no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8C22×C412yes—
115C22×C8C22×C412yes—
116C22×C8C2×Q814yes—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—28no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4—44no—
141C2×C4⋊C4—44no—
142C2×C4⋊C4—44no—
143C2×C4⋊C4—82no—
144C2×C4⋊C4—82no—
145C2×C4⋊C4—82no—
146C2×C4⋊C4—82no—
147C2×C4⋊C4—82no—
148C2×C4⋊C4C22×C414yes—
149C2×C4⋊C4C22×C414yes—
150C2×C4⋊C4C2×Q814yes—
151C2×C4⋊C4C2×Q814yes—
152C2×C4⋊C4#4114yes—
153C2×C42—21no—
154C2×C42—21no—
155C2×C42—21no—
156C2×C42—21no—
157C2×C42—22no—
158C2×C42—22no—
159C2×C42—22no—
160C2×C42—22no—
161C2×C42—22no—
162C2×C42—22no—
163C2×C42—44no—
164C2×C42—44no—
165C2×C42—82no—
166C2×C42C22×C412yes—
167C2×C42#4112yes—
168#107—44no—
169#106—44no—
170C4⋊C8—44no—
171C4×C8—44no—
172#95—28no—
173#95—41no—
174#95—41no—
175#95—44no—
176#95—44no—
177#95—44no—
178#95—44no—
179#95—82no—
180#95—82no—
181#95—82no—
182C24C22×D411yescharacteristic
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—22no—
186C22×C4—24no—
187C22×C4—22no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—24no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—24no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—24no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—24no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4C4×D412yes—
232C22×C4C4×D412yes—
233C22×C4#13514yes—
234C22×C4#13714yes—
235C22×C4C23×C411yescharacteristic
236C22×C4C22×Q812yes—
237C22×C4C2×#4112yes—
238C22×C4C2×#4111yescharacteristic
239C22×C4H(5,𝔽2)11yescharacteristic
240C22×C4H(5,𝔽2)11yescharacteristic
241C22×C4H(5,𝔽2)11yescharacteristic
242C22×C4H(5,𝔽2)12yes—
243C22×C42−1+411yescharacteristic
244C22×C42−1+412yes—
245C2×C8—22no—
246C2×C8—22no—
247C2×C8—24no—
248C2×C8—24no—
249C2×C8—24no—
250C2×C8—24no—
251C2×C8—24no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—41no—
256C2×C8—41no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—41no—
261C2×C8—41no—
262C4⋊C4—24no—
263C4⋊C4—24no—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—24no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—48no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—88no—
279C4⋊C4—88no—
280C4⋊C4—88no—
281C4⋊C4C4×Q818yes—
282C42—22no—
283C42—22no—
284C42—22no—
285C42—22no—
286C42—42no—
287C42—42no—
288C42—42no—
289C42—42no—
290C42—88no—
291C23C2×C4×D411yescharacteristic
292C23#53411yescharacteristic
293C23#53711yescharacteristic
294C23#54612yes—
295C23D4×Q812yes—
296C23C2×#13512yes—
297C23C2×#13712yes—
298C23#57612yes—
299C23#57712yes—
300C2×C4—22no—
301C2×C4—24no—
302C2×C4—22no—
303C2×C4—24no—
304C2×C4—22no—
305C2×C4—28no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—24no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—44no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—42no—
333C2×C4—44no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4#44114yes—
352C2×C4C2×C4×Q812yes—
353C2×C4#51811yescharacteristic
354C2×C4#52011yescharacteristic
355C2×C4#52012yes—
356C2×C4#53811yescharacteristic
357C2×C4#53911yescharacteristic
358C2×C4#54012yes—
359C2×C4#54112yes—
360C2×C4#54412yes—
361C2×C4#54412yes—
362C2×C4#55212yes—
363C2×C4#55412yes—
364C2×C4#55412yes—
365C2×C4#55511yescharacteristic
366C2×C4Q8211yescharacteristic
367C2×C4#56011yescharacteristic
368C2×C4#56212yes—
369C2×C4#56311yescharacteristic
370C8—42no—
371C8—42no—
372C8—44no—
373C22#232511yescharacteristic
374C22#236012yes—
375C22#269911yescharacteristic
376C22#270412yes—
377C22#273411yescharacteristic
378C22C4×#13511yescharacteristic
379C22C2×#44111yescharacteristic
380C22#294612yes—
381C22#319012yes—
382C22#319812yes—
383C22#324812yes—
384C22#325911yescharacteristic
385C22#326311yescharacteristic
386C22#329612yes—
387C22#329712yes—
388C22#332912yes—
389C22#333312yes—
390C22#338012yes—
391C22#338112yes—
392C22#338512yes—
393C22#338712yes—
394C4—24no—
395C4—48no—
396C4—42no—
397C4—44no—
398C4—84no—
399C4—84no—
400C4—82no—
401C4—162no—
402C4#233012yes—
403C4#236911yescharacteristic
404C4#237611yescharacteristic
405C4#276812yes—
406C4#276911yescharacteristic
407C4#279211yescharacteristic
408C2#1458711yescharacteristic
409C2#2218311yescharacteristic
410C2#2219911yescharacteristic
411C2#2296612yes—
412C2#2297112yes—
413C2#3172512yes—
414C2#3187712yes—
415C2#3187812yes—
416C2#3268712yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
19. #2087C4semidirect

Maximal subgroups

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

subgroupindex
C2×#20872
#140452
#113272
#112312
#112192
#108842
#108802
#108342
#108062